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Elastic/viscoplastic models for geomaterials and powder-like materials with applications

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Elastic/viscoplastic models for geomaterials and powder-like materials with applications
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Jin, Jishan, 1962-
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viii, 281 leaves : ill. ; 29 cm.

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Subjects / Keywords:
Compressive stress ( jstor )
Constitutive equations ( jstor )
Experimental data ( jstor )
Hydrostatics ( jstor )
Shear stress ( jstor )
Specimens ( jstor )
Stress relaxation ( jstor )
Stress tests ( jstor )
Triaxial stress ( jstor )
Volumetric strain ( jstor )
Aerospace Engineering, Mechanics and Engineering Science thesis, Ph. D ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics and Engineering Science -- UF ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1996.
Bibliography:
Includes bibliographical references (leaves 270-280).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Jishan Jin.

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ELASTIC/VISCOPLASTIC MODELS
FOR GEOMATERIALS AND POWDER-LIKE MATERIALS
WITH APPLICATIONS











By

JISHAN JIN

















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA



















To My Family,

In Memory of My Father













ACKNOWLEDGEMENTS


I would like to express my deepest gratitude to my supervisory committee chairman,

Graduate Research Professor Nicolae D. Cristescu, for his guidance, tremendous support, and

encouragement during the study. I also sincerely appreciate Professor Cristescu's patience,

kindness, and discussions to his students.

I am deeply indebted to Professor Renwei Mei for helpful discussions on the

numerical analysis aspects during the work and for the critical reading of the early draft of

this dissertation. I am very grateful to Professor Frank C. Townsend for his guidance and

helpful discussions on the experimental aspects. I would like to thank Professors Bhavani V.

Sankar and Edward K. Walsh for their support and for serving on the committee.

Many thanks also go to my wife, Qianhong, for her understanding and full support,

and to my family members for their endless support in my life.

Finally, the financial support from the Engineering Research Center at University of

Florida, Particle Science and Technology, is gratefully acknowledged.














TABLE OF CONTENTS



ACKNOWLEDGEMENT ................ ............................ iii

ABSTRACT ............................................ ............ vii

CHAPTERS

1 INTRODUCTION ............................................... 1
1.1 Background .................................... .......... 1
1.1.1 Rock Salt ......................................... 2
1.1.2 Powder M materials .................................... 4
1.1.3 Models ............................................ 6
1.2 Literature Survey ............................................. 8
1.2.1 Mohr-Coulomb Model and Drucker-Prager Model ............ 8
1.2.2 Cap Model .............. ............................ 10
1.2.3 Cam-Clay M odel ...................................... 13
1.2.4 Lade Model ........................................ 16
1.2.5 Endochronic Model ................................... 17
1.2.6 Models for Compaction (Consolidation) of Powders .......... 18
1.2.7 The Elastic/Viscoplastic Models .......................... 21
1.2.8 Other M odels ........................................ 24
1.2.9 Remark on Recent Development of Plasticity-Based Models ... 25
1.2.10 Finite Element Method ........................... 25
1.3 Objective and Outline of the Present Work ......................... 27

2 THE ELASTIC/VISCOPLASTIC (CRISTESCU) THEORY ................ 31
2.1 The Elastic/Viscoplastic Constitutive Equation ...................... 31
2.2 Determination of the Elastic/Viscoplastic Constitutive Equation ........ 34
2.2.1 Elastic Parameters and Viscosity Coefficient ................ 34
2.2.2 Yield Function .................................. ..... 37
2.2.3 Viscoplastic Potential ................................ 38
2.3 Loading Unloading Conditions ............................ .. 39

3 AN ELASTIC/VISCOPLASTIC MODEL FOR TRANSIENT CREEP OF
ROCK SALT ................... ........... .............. 40








3.1 Determination of the ElasticViscoplastic Constitutive Model .......... 41
3.1.1 Determination of the Elastic Parameters .................... 41
3.1.2 Determination of the Compressibility/Dilatancy Boundary
and the Failure Condition ............................ 43
3.1.3 Determination of the Yield Function ...................... 45
3.1.4 Determination of the Viscoplastic Potential ................. 50
3.2 Comparison with the Data ................................... .. 58
3.3 The Finite Element Analysis .................................... 63
3.3.1 Formulation of the Elastic/iscoplastic Theory .............. 63
3.3.2 Example I: Axial Compression with Confining Pressure ....... 65
3.3.3 Example II. Stress Analysis for A Cylindrical Cavity in Rock
Salt ...................... .. .................... 67
3.4 Discussion and Conclusion ........................... ....... .. 72

4 TRIAXIAL EXPERIMENTAL RESULTS ON ALUMINA POWDERS ....... 74
4.1 Introduction ....................... .................. 75
4.2 Experimental Procedures ............................ ...... 78
4.2.1 Triaxial Equipment Setup ............................ 78
4.2.2 Specimen Preparation ................................ 83
4.2.3 Back Pressure Saturation ........................... 84
4.2.4 Hydrostatic Loading Test ............................ 85
4.2.5 Deviatoric Loading Test .............................. 86
4.3 Experimental Results .. ................... ................ 88
4.3.1 Hydrostatic Tests .......................... ... 88
4.3.2 Deviatoric Tests ...................................... 98
4.4 Discussion and Conclusion ................................ 111

5 THE ELASTIC/ISCOPLASTIC MODEL FOR ALUMINA POWDER A10.. 116
5.1 Determination of the ElasticViscoplastic Model ................... 116
5.1.1 Elastic Parameters .................................... 117
5.1.2 Compressibility/Dilatancy (C/D) Boundary and Failure
Condition ................... .... ............. 119
5.1.3 Yield Function .................................... 121
5.1.4 Viscoplastic Potential ............................... 126
5.2 Comparison with Experimental Data ............................. 135
5.2.1 Triaxial Tests .................. ..................... 135
5.2.2 Hydrostatic Tests ........................... ... 141
5.2.3 Rate Dependent Tests ............................ 144
5.3 Discussion and Conclusion .................................. 147

6 A NEW METHOD TO DETERMINE THE ELASTIC/VISCOPLASTIC
MODEL ............... ................................. 149
6.1 Introduction ............. ................. .............. 149








6.2 Elastic/Viscoplastic Constitutive Model .......................... 151
6.2.1 Elastic/Viscoplastic Constitutive Equations ................ 151
6.2.2 Loading-Unloading Conditions .......................... 153
6.3 Determination of the Elastic/Viscoplastic Model for Alumina Powder
A16-SG ............. .............................. 155
6.3.1 Elastic Parameters .................................. 155
6.3.2 Yield Surface ..................................... 155
6.3.3 Irreversible Strain Rate Orientation Tensor ................ 162
6.3.4 Viscosity Coefficient ............................... 167
6.3.5 Remark ........................................ 168
6.4 Validation of the Model ..................................... 169
6.4.1 Creep Type Formula with Stepwise Stress Variations ........ 169
6.4.2 Conventional Triaxial Tests ............................ 170
6.4.3 Constant Mean Stress Tests ............................ 173
6.4.4 Hydrostatic Creep Tests .............................. 175
6.5 Discussion and Conclusion .................................. 175

7 CONCLUSION AND FUTURE WORK .............................. 179
7.1 Conclusion ................... ........ ..................... 179
7.2 Future W ork ............. .... ....................... ..... 180

APPENDICES

A THE FORMULATION OF ELASTIC/VISCOPLASTICITY FOR THE
FINITE ELEMENT METHOD .......................................... 181

B TRIAXIAL EXPERIMENTAL RESULTS FOR ALUMINA POWDER A10
(DENSE) .................. ............... ............ 191

C TRIAXIAL EXPERIMENTAL RESULTS FOR ALUMINA POWDER A10
(LOOSE) ................... ............... ........ 228

D TRIAXIAL EXPERIMENTAL RESULTS FOR ALUMINA POWDER A16-
SG ......................................... ............ 248

REFERENCES ................................................. 270

BIOGRAPHICAL SKETCH .......... ......................... 281













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ELASTIC/VISCOPLASTIC MODELS
FOR GEOMATERIALS AND POWDER-LIKE MATERIALS
WITH APPLICATIONS

By

JISHAN JIN

December 1996

Chairperson: Graduate Research Professor Nicolae D. Cristescu
Major Department: Aerospace Engineering, Mechanics and Engineering Science

An elastic/viscoplastic model with a nonassociated flow rule for transient creep of

rock salt is formulated based on a set of triaxial tests. A new procedure to determine the yield

surfaces and the potential surfaces is proposed. The asymptotic behavior and the physical

grounds of yield surfaces and of potential surfaces are incorporated into the model so that

these surfaces are smooth and the model better matches the data. This type of model with

both yield surfaces and potential surfaces is implemented for the first time into a finite

element program. It is shown that the prediction of the model matches quite well the

experimental data.

The mechanical behavior of alumina powders is investigated experimentally. Three

series of triaxial tests are performed. It is shown that the mechanical (compaction) behavior

of alumina powders is strongly dependent on the particle size and on the initial density. For








the powder A16-SG with smaller particle size (0.4-lpm), the volumetric strain always

exhibits compressibility, while for the powder A10 with larger particle size (40-200pm), the

volumetric strain exhibits both compressibility and dilatancy.

An elastic/viscoplastic model with a nonassociated flow rule to describe the nonlinear

time-dependent behavior of alumina powder A10 is formulated based on five conventional

triaxial tests. By using such type of tests, a new procedure to determine the yield surfaces and

the potential surfaces is proposed. The model is checked against the data. A good agreement

between the model prediction and the data is obtained.

A new methodology is proposed to determine the elastic/viscoplastic model for

compressible materials. In such models, the irreversible volumetric strain is chosen as a work

hardening parameter. A model to describe the behavior of alumina powder A16-SG is

formulated. The irreversible strain rate orientation tensor is determined. The model is easier

to handle and has fewer parameters than the previous ones. A very good agreement between

the data and the model prediction is obtained.














CHAPTER 1
INTRODUCTION



1.1 Background



The formulation of general constitutive equations of three-dimensional models for

time dependent materials has been studied since the pioneering works by Maxwell in 1868

and by Kelvin in 1875 (see Malvem [1969]). Many materials possess time dependent

properties: for instance, polymers exhibit viscoelastic properties; a rock flows in a long

period of time although it seems strong and hard; a metal creeps at high temperature, etc. A

great number of contributions have been made in this field (see Perzyna [1966]; Cristescu

[1967]; Christensen [1982]; Cristescu & Suliciu [1982]; Phillips [1986]; Cristescu [1989];

Chaboche [1989]; Lubliner [1990]). Recently three-dimensional modeling for rock salt and

powder materials has attracted the attention of many researchers, due to the demand of

various engineering industries (see Hardy & Langer [1984, 1988]; Chenot et al. [1990]; Jinka

& Lewis [1994]; Krishnaswami & Trasorras [1995]; Xu & McMeeking [1995]; Aubertin

[1996]; Tszeng & Wu [1996]). They wish to develop general constitutive equations or

modify the existing ones which, on the one hand, can be used to describe better the behavior

of materials, and, on the other hand, can be incorporated into the finite element method.

Further, the models will be used for optimal designs or/and safety evaluations.










1.1.1 Rock Salt

Rock salt exhibits time dependent properties as revealed by creep, relaxation and rate

dependent tests (Hansen & Carter [1984]; Handin et al. [1986]). It is found under natural

underground conditions. Rock salt consists mainly of halite with little anhydrides and clay,

and has small permeability. Due to the theological behavior and the physical properties of

rock salt, the large underground cavities in mass of rock salt are considered as ideal places

to store radioactive waste or hazardous chemical waste. Many salt cavities were excavated

around the world for this purpose (See Hugout [1988] in France; Eekelen [1988] in

Netherlands; Matalucci & Munson [1988] in USA; Kappei & Gessler [1988] in Germany).

The deformations of those cavities are continuously recorded. The stability and the safety of

those large underground cavities are being evaluated. Since the stability and the safety are

not precisely predicted nowadays, millions of extra dollars are spent on maintaining those

cavities in a workable state. Thus it is highly necessary to formulate a general constitutive

equation in order to predict the mechanical behavior of rock salt and to evaluate the safety

of cavities. Also a general constitutive equation would be beneficial, not only for the

evaluation of the stability and the safety of large underground cavities, but also for other rock

salt mining engineering problems such as the improvement of salt mining design procedures

(Rolnik [1988]).

Most of the research done so far was devoted to the long term behavior of rock salt,

i.e., stationary creep. A variety of models have been proposed (e.g., Langer [1988]; Chan et

al. [1994]; Aubertin et al. [1996]). Most of them can be used to predict the long term

behavior of rock salt by neglecting the deformation in the transient creep phase. However,










occasionally, a great discrepancy is found between the measured creep deformation of

underground cavities or openings and that predicted by the models (Aubertin et al. [1993]).

It is believed that this discrepancy is largely due to the neglect of the transient creep

deformation, because the stress around an opening (or cavity) is redistributed during the

excavation of the opening in the transient creep phase. The redistributed stresses will

influence the stationary creep of rock salt around the opening. Therefore it is crucial to

formulate correctly the constitutive equations for transient creep, if one wishes to describe

the behavior of rock salt in the rock surrounding the excavation during the stationary creep.

Several models for transient creep of rock salt have been formulated (e.g., Langer

[1984]; Munson & Dawson [1984]; Desai & Varadarajan [1987]; Aubertin et al. [1993]). In

most cases one starts from empirical formulae using a viscoelastic or viscoplastic model with

Drucker-Prager type of yield surface or internal state variable concepts. However, these

models either do not give good matching with the experimental data or could hardly be

incorporated into a finite element program. Thus these models should be judged from the

point of view of how well they match the experimental data and how easily they can be

incorporated into a finite element program. In the present thesis, we start from a set of true

triaxial tests on rock salt and follow a procedure to formulate a general three-dimensional

elastic/viscoplastic constitutive equation for the transient creep phase. No empirical formulae

are used since empirical formulae may not be adequate for other cases. Nonassociated flow

rule is explored and the viscoplastic potential is determined based on the experimental data.

The model not only matches better the experimental results, but can also be incorporated

easily into the finite element method. The model will be presented in Chapter 3.










1.1.2 Powder Materials

The mechanical behavior of powder materials has been studied for a long time, since

powder materials are used in many industrial applications such as metal powder metallurgy

(Johnson [1992a, 1992b]), ceramics injection molding (Mutsuddy & Ford [1995]) or food

products (Puri et al. [1995]). Powder materials such as metal powder or ceramics powder,

by some compaction or/and sintering processes, become useful products with very dense and

hard properties. However, in shape forming processes such as uniaxial compressing,

ceramics injection molding or pressure casting, the density distribution in the product is

nonuniform due to the friction between powder and the wall of a die. The nonuniform

density distribution would cause warping or other defects in the product in a later sintering

process. These warpings or defects will damage the products and unqualify the products

(Reed [1988]). Nowadays, most shape forming techniques are the ad hoc ones. Due to the

lack of the correct prediction of density distribution in a mold, many products are not

qualified, and millions of dollars are wasted each year (Johnson [1992b]). Thus it is crucial

to understand the mechanical behavior of powder materials during these processes and it is

urgent for designers to simulate these shape forming processes and to predict the density

distribution in a mold.

In order to predict the density distribution of powder in a mold, and to simulate the

shape forming process, the three-dimensional stress strain relationships, i.e., the constitutive

equations for powders have to be correctly and precisely formulated. Triaxial tests are often

used to formulate constitutive equations (Desai & Siriwardane [1984]; Cristescu [1989]).

Unfortunately only very limited triaxial tests have been reported (Koerner & Quirus [1971];








5

Shima & Mimura [1986]; Brown & Abou-Chedid [1994]; Gurson & Yuan [1995]) for

different applications. Without complete experimental data, it is difficult to formulate

constitutive equations correctly. In order to understand the compaction behavior of powder

materials, to formulate a general three-dimensional model, or/and to examine the existing

models, additional triaxial tests are needed.

Alumina powder (AzO03) has been widely used in ceramic engineering for various


applications due to the specific properties of alumina such as high electrical resistivity, high

erosion resistance, high melting temperature and high abrasiveness etc. For instance, alumina

powder can be used in the manufacture of porcelain, alumina laboratory ware, wear-resistant

parts, sandblast nozzles, medical components, abrasives and refractories etc. Hundreds of

tons of alumina powder and alumina-based articles are produced each year (Richerson

[1982]). However, very few experiments have been performed for alumina powders

regarding to the mechanical (compaction) behavior in three-dimensional stress states. Only

some tests in uniaxial conditions have been reported for ceramics powders (e.g., Stanley-

Wood [1988]; Gethin et al. [1994]; Chen et al. [1994]). Unfortunately, uniaxial tests reveal

mainly one-dimensional relationship between pressure and volume changes. Such tests are

unable to provide the necessary data to formulate three-dimensional constitutive equations.

On the contrary, triaxial tests reveal the deformation characteristics in three dimensions

under both hydrostatic pressure and deviatoric loading conditions. Such tests can furnish the

necessary data for the formulation of the general constitutive equations. In the present thesis,

a series of triaxial tests on alumina powders have been performed. Many interesting results

have been found. The experimental results will be presented in Chapter 4.










1.1.3 Models

There are some geomaterials-type models to describe the three-dimensional behavior

of frictional materials such as the Mohr-Coulomb, Drucker-Prager, Cam-Clay, Cap and Lade

models etc. These models are developed based on mainly the experiments on sand and clay.

In some models, the yield surfaces are a prior assumed and the associated flow rule is used.

In some other models, the non-associated flow rule is used and the irreversible strain

potential is assumed (see the following sections on the review of models). Most models are

time independent ones. It is difficult to use these model to describe the behavior of rock salt

(Langer [1988]). These models have a common disadvantage: they may not predict the

volumetric dilatancy behavior correctly even though some models predict the volumetric

compressibility pretty well (e.g., Desai & Siriwardane [1984] for artificial soil). For example,

the Cap model was modified and many additional constants were introduced in the model

by Chen and Baladi [1985]. The model still does not predict the volumetric dilatancy

precisely although the prediction matches well the other data. The Drucker-Prager model

overpredicts the dilatancy behavior. The Cam-clay model does not predict the dilatancy

behavior until the materials reach the failure stress and become softening. One reason for

these weak aspects is that the assumed yield surface is appropriate for some materials, but

it may not be good for other materials. Another reason is the skepticism of the normality in

the associated flow rule. Many experimental results do show that the behaviors of

geomaterials do not follow the normality conditions (Maier & Hueckel [1979]; Kim & Lade

[1988]; Jin et al. [1991]; Anandarajah et al. [1995]). The non-associated flow rule has been

explored by several researchers (Lade & Kim [1989a, 1989b]; Desai & Hashmi [1989]).








7
However, since in their models the potential surfaces are assumed in a priori form for some

materials, it is difficult to use these models for other type of materials. It is advisable to

examine the models carefully before using them.

There are also several models for the compaction of powders. Usually, the elliptical

yield curves or other type of curve in the (1,, Aj) plane are used, where I, and J' are the

first invariant of stress tensor and the second invariant of deviatoric stress tensor. These

models are mainly determined from the micromechanics analysis for porous metal materials.

The associated flow rule is usually used in these models. Nowadays, no formulation exists

regarding to the non-associated plasticity formulation for powder materials even though

some powder materials do have non-associated flow properties (Brown & Abou-Chedid

[1994]; Gurson & Yaun [1995]). It is important and necessary to explore the non-associated

flow rule. It is urgent and valuable to develop a new methodology to formulate non-

associated constitutive equations based on experimental results.

In section 1.2, we review most models which are often referred in the literature. Both

time dependent models and time independent models are reviewed. The advantages and

disadvantages of these models are highlighted. In Chapter 5 the elastic/viscoplastic model

is used to formulate the constitutive equation for alumina powder. The viscoplastic potential

is determined based on the experimental data presented in Chapter 4. Non-associated flow

rule is used in the model. In Chapter 6, the elastic/viscoplastic model is developed for

compressible materials. The orientation of irreversible strain rate tensor is determined based

on the experimental results on A16-SG alumina powder. A good agreement between the

model prediction and experimental data is obtained.










1.2 Literature Survey



In terms of the yielding or plastic flow behavior, materials can generally be classified

in two types. One type of material is the so-called nonfrictional material. For this kind of

material, such as metals, the yielding or plastic flow behavior is independent on the first

invariant of stress tensor. Several yield surfaces can be used to describe their yielding

behavior, as for instance, the Mises or Tresca yield conditions (see Malvern [1969]). An

associated flow rule is usually assumed for such type of models. The other type of material

is called frictional material. For such type of material, the yielding of plastic flow is

dependent on the first invariant of stress tensor. In other words, if there was friction within

the mass of materials, the frictional resistance will be proportional to the normal force. Sand,

clay, rock, concrete, powder-like materials etc. are falling into this category. The Mises yield

criterion is not any more suitable to describe yield behavior of this type of materials. Other

types of models should be used for such materials. In the following sections, we will review

several models for frictional materials and discuss the advantages and disadvantages of these

models. These models include Mohr-Coulomb model, Drucker-Prager model, Cap model,

Cam-Clay model and Lade model etc. More advanced models are also discussed.



1.2.1 Mohr-Coulomb Model and Drucker-Prager Model

According to the Mohr-Coulomb model, the shear strength (or stress at failure)

increases with increasing normal stress on the failure plane. The failure criteria can be

written as










T=c+otan#( (1.1)



where T is the shear stress on the failure plane, c the cohesion of the material, o the normal

effective stress on the failure plane and (P the angle of internal friction.

However, in the Mohr-Coulomb model, the intermediate stress is not taken into

account for the failure criteria. A generalization to account for the effects of all principal

stress was suggested by Drucker and Prager [1952] by using the invariants of stress tensor.

This generalized criterion can be written as


f= 2-aI,-K (1.2)


where a and K are positive material parameters. I, is the first invariant of stress tensor and


J2 is the second invariant of the deviatoric stress tensor. The associated flow (normality) rule

is assumed in this model. However, if the associated flow rule is assumed, the model usually

over-predicts the dilatant volumetric strain of the material comparing to the volumetric strain

obtained in experiments. Some problems can be solved without using the normality rule. If

the irreversible compressibility is taken into account, a cap is usually added at the open end

of the Drucker-Prager failure surface (see Chen & Baladi [1985]). If the Mohr-Coulomb

failure coincides with the Drucker-Prager failure condition at triaxial compression, the above

coefficients satisfy the following relations

2sin() 6ccos(F
V (3-sin) (3-sin() (1.3)









1.2.2 Cap Model

Based on experimental results, DiMaggio and Sandier [1971] and Sandier et al.

[1976] proposed a cap model. The yield surface for this model consists of two yield segments

(Fig. 1.1). One is a fixed yield surface and the other one is a yield cap surface. The fixed

yield segment can be written as


fA ')=0 (1.4)


There are several possible functions for f,. The expression for f, adopted by DiMaggio and

Sandler [1971] is given by


f, = --ye -'-a=0 (1.5)


and Desai and Siriwardane [1984] use


f,=c -ye -'-I,-a=0 (1.6)


for an artificial soil, where y, p, 0 and a are material parameters. The yield surfaces for yield

caps can be expressed as


f2(i, ,K)=0 (1.7)


where K defines the deformation history. DiMaggio and Sandler [1971] and Sandier et al.

[1976] have used an elliptic cap to represent the yield surface for the cohesionless material

they have considered










f =R 2J2+1-L)2-R2b =0 (1.8)


where Rb=(X-L) (Fig. 1.1), R is the ratio of the major to the minor axis of the ellipse, X the

value of I, at the intersection of the cap with the I -axis, L the value of J, at the center of the

ellipse, and b the value of (J2)n0 when I, =L. Because the two yield surfaces are intersecting

at I,=L, the X-L is related with f, through


X-L=R(ye -'+a+0I) (1.9)


The value of X depends on the plastic volumetric strain ep and is assumed as



X -- Inl--+Z, (1.10)
D WI



where D, Z, and W are material parameters to be determined. Usually the associated flow

rule is assumed for this model.

In the model, the cap surface may shrink or expand dependent on the increase or the

decrease of the irreversible volumetric strain. When the loading stress is on the cap surface

and the stress increment points outside, work-hardening takes place and the irreversible

volumetric strain increases so that the cap surface expands. When the loading stress is on the

failure surface, the volumetric irreversible strain decreases so that the cap surface shrinks.

Thus, this model can describe a softening behavior. However, for some frictional materials,

no softening behavior is present in the region considered. Sander et al. [1976] modified eqn

























L X
z


Fig. 1.1 Yield surfaces for cap model.



(1.10) to prevent this softening behavior. Later, a great lot of progress has been made by

many researchers (see Desai & Siriwardane [1984]; Chen & Baladi [1985]). The model has

been used to describe the behavior of sand, artificial soil, concrete, rock and powder-like

materials etc.

The original cap model is further extended to account for rate effect by means of a

viscoplastic theory proposed by Perzyna [1966]. A lot of research have been done

(Zienkiewicz & Cormeau [1974]; Hughes & Taylor [1978]; Katona [1984]; Simo et al.

[1988]; Hoftstetter et al. [1993]). The model is nowadays extended to account for tensile

stresses and becomes more efficient when incorporated into the finite element method.

However, there are some limitations in this model, for instance, dilatancy is only

obtained when the stress state reaches failure surface. For some materials, this assumption








13
is not correct. Materials such as rock (Brace et al. [1966]; Cristescu [1989]), sand (Kim &

Lade [1988]) and some powders (Jin & Cristescu [1996b]; Jin et al. [1996d]), exhibit

dilatancy far away from the failure surface. Even though the model was modified by Chen

and Baladi [1985], it still would not describe correctly volumetric dilatancy. In other words,

this model could not give the reasonable prediction of volumetric strain at least for some

frictional materials. Another limitation is the assumption in eqn (1.10). It may not describe

correctly the work hardening behavior for some materials.



1.2.3 Cam-Clay Model

The Cam-Clay model was developed by a research group in Cambridge lead by

Roscoe (see Desai & Siriwardane [1984]; Chen & Baladi [1985]). The model is based on the

concept of Critical Void Ratio (or Critical Density). They found that the yielding of loose or

dense soil continues under both drained and undrained conditions until the material reaches

a critical void ratio. After the critical void ratio reaches, the volumetric strain does not

change any more. For normally consolidated soil, the stress state at this critical void ratio

reaches failure, while for over-consolidated soil, the stress state at the critical void ratio

reaches the residual stress after passing through failure and softening. After the critical void

ratio, the void ratio remains constant during subsequent deformations, that is, the material

will reach a state in which the arrangement of the particles is such that no volume change

takes place during shearing. This particular void ratio is called the critical void ratio. This

can be considered as the critical state of the material.








14

This idea can be expressed in the (p, q, e) space, where p is mean stress, q=o0-

o3=(3J2). and e is void ratio as shown in Fig. 1.2. The State Boundary (or Roscoe Surface)

can be used to describe the behavior of normally consolidated clay while the Hvorslev

Surface is used to describe the behavior of over consolidated clay (see Desai & Siriwardane

[1984]).


State boundary
or Roscoe
surface \


Fig.1.2 Critical State Line on the (p,q,e) space.



Fig. 1.3 shows a projection of the critical state line (CSL) on the p-q space together

with projections of typical section of the state boundary surface. The CSL is usually a straight

line passing through the origin. The projections of the state boundary surfaces are represented

by continuous curves and referred to as yield surfaces or yield caps. The yield surfaces can








15
be spherical, elliptical or of other shapes. In this model, the associated flow rule (normality)

is assumed.

If the yield surface is expressed as


-pMlu ( (1.11)

the model is called Cam-Clay model, which was first derived by Roscoe in 1958 (see Desai

& Siriwardane [1984]). If the yield surface is an ellipse

M2p2-M2pop+q2=0 (1.12)


the model is called Modified Cam-Clay model, which was first developed by Roscoe in 1968

(see Desai & Siriwardane [1984]), where M is the slope of the CSL and po is a work-

hardening parameter on the p axis. The above yield surface passes through (po,0) in the (p,q)

space. po can be expressed as a function of irreversible volumetric strain which is obtained

from hydrostatic tests only. It can be shown that the normal to the above yield surface at a

critical point is perpendicular to p axis, i.e., no volume changes at this point.

This model was used by many researchers (see Desai & Siriwardane [1984]; Chen

& Baladi [1985]; Britto & Gunn [1987]; Tripodi et al. [1994]) for normal consolidated clay,

over-consolidated clay, and other materials. This model was incorporated into several

commercial finite element programs (e.g., ABAQUS [1993]). However, this model has some

limitations. For some materials, the critical void ratio does not exist. For some materials, the

critical state line does not coincide with the compressibility/dilatancy boundary. Therefore,

it could not predict volumetric strain correctly.




















yield cap


Fig. 1.3 Yield locus in q-p space, projection of Fig. 1.2 on q-p space.



1.2.4 Lade Model

In this model, Lade and Duncan [1975] first proposed the following function for the

failure surface, based on experimental observation for sand


1-K 113=0 (1.13)


where I, and I, are the first and third invariants of stress tensor, Ki is constant depending on


the density of sand. The expansion of the yield surfaces is defined by function f given by


If
f-- (1.14)
I3








17
whose values vary with loading and can reach the value K, at failure. In the model, the plastic

potential function is assumed as


Q=I-K213 (1.15)


where for a given f, K2 is a constant. The theory allows for a non-associated flow rule.

Subsequently, the theory was modified to include curved surfaces (Lade [1977]).

Later the theory was modified again (Kim & Lade [1988]; Lade & Kim [1989a]; Lade & Kim

[1989b]; Lade [1989]). The new model employs a single, isotropic yield surface. The yield

surfaces, expressed in terms of three invariants of stress tensor, describe the contours of total

plastic work. The non-associated flow rule is used in the model. The potential function is

also assumed as a function of three invariants of stress tensor. The elastic Young's modulus

is expressed as a function of the confining pressure. This model has been used to describe

the behavior of sand, clay and concrete etc. This model was also examined by Reddy and

Saxena [1992] for cemented and uncemented sand. They obtained the model from one

hydrostatic test and three conventional triaxial compression tests, then checked the model

with different loading paths. They concluded that the predicted results match reasonably the

experimental data except for the volumetric strain at the conventional triaxial tests.



1.2.5 Endochronic Model

The endochronic theory is derived from the laws of thermodynamics, i.e., the

conservation of energy and dissipation (Clausius-Duhem) inequality. It is based on the idea

of internal variable development. The theory was first proposed by Valanis ([197 la, 1971b,








18
1975]) by choosing the internal variable to be an intrinsic time scale. It was shown by

Valanis that mechanical properties such as hystersis and hardening in metals can be predicted

accurately by this model. Subsequently, Bazant [1977, 1978] has applied the model for soil

and concrete. Reddy and Saxena [1992] have used this model for cemented sand and

obtained the model with one hydrostatic compression test and one conventional triaxial

compression test. One function in the model was modified. The model was checked against

different loading paths. A good agreement between the tests and the prediction was obtained.

However, the endochronic theory is still in the development stage (see Yeh et al [1994]; Wu

& Ho [1995]) and very few cases of implementation of FEM are available. It was shown by

Sandler [1978] that in the present form, the endochronic model can cause difficulties in

numerical implementation, particularly in accounting for unloading, which may be remedied

by introducing internal barriers. In that case, the theory may exhibit some features similar as

the yield loading conditions from the plasticity theory, which the endochronic theory would

initially like to avoid.



1.2.6 Models for Compaction (Consolidation) of Powders

Based on the micromechanics analysis for porous metal, one kind of models for

porous metal or metal powder was first proposed by Green [1972] and Oyan et al. [1973].

The yield surface ( in this kind model can be expressed as a function of the first invariant

of stress tensor, I,, and the second invariant of deviatoric stress tensor, J2,,


0=A(T))J2+B(Ti)II( -(i)o=0( ,


(1.16)








19

where ri is relative density, A(Tr), B(rl) and 8(rl) are functions of relative density and oy is

yield stress of powder materials at full density. Later, Shima and Oyane [1976], Corapcioglu

and Uz [1978], Doraivelu et al. [1984] and Shima and Mimura [1986] also obtained this kind

of yield surface with different functions of A(T), B(r)) and 8(rl) based on the different

assumptions. In the work of Oyan et al. [1973], for instance, A, B and 6 are expressed as


A= 3; B 1
1+ 2 (1.17)



An associated flow rule is usually assumed in this kind of models.

The yield surface (1.16) is an elliptical surface in the principal stress space with the

major axis coinciding with the principal stress. In (1,J2) space, the yield surfaces are a family

of elliptical curves. Along each elliptical curve shown in Fig. 1.4 the relative density is

constant. The yield surfaces have symmetric properties for the tension and compression cases

and approach the von Mises yield surfaces when the relative density tends towards one, i.e.,

full density. The deformation history was incorporated into the model through the use of

relative density as a state variable. The dependence of deformation history can be generalized

(Brown & Weber [1988]; Krishnaswami & Trasorras [1995]; Tszeng & Wu [1996]) to

include a second state variable representing a scalar measure of the average plastic strain

experienced by the powder particles, thereby allowing the yield stress to change with

inelastic deformation. These models can be improved. However, several parameters are

involved, which become difficult to be determined from experimental data.








20
The model represented by eqn (1.16) was extended to viscoplasticity and thermo-

viscoplasticity (Abouaf et al. [1988]; Chenot et al. [1990]). The finite element methods were

used for the prediction of relative density variations in experimental powder compacts

(Nohara et al. [1988]; Jinka & Lewis [1994]; Krishnaswami & Trasorras [1995]; Oliver et

al. [1996]; Xu & McMeeking [1996]).


Fig.l.4 Typical yield surfaces for the model (1.16).



Another kind of micromechanics model was proposed by Gurson [1977]. For the case

of a spherical cavity within a perfect plastic matrix, the yield function can be expressed as


.2fos(1 1 1 ,
<)=3 +2fcosh I -1-f2=O (1.18)
y li 2a








21
where oy is the yield stress of the matrix and f is the void volume fraction defined by void

volume over total volume of porous metal material. This model is suitable to less porous

metal materials only. Other micromechanics models (Torquato [1991]; Fleck et al. [1992a,

1992b, 1995]; Nemart-Nasser & Hori [1993]) are also available. However, all these

micromechanics models are based on many idealized assumptions. Due to very few

experimental data available, it is difficult to validate the models for the complicated stress

states (Abou-Chedid & Brown [1992]; Brown & Abou-Chedid [1994]; Gurson & Yuan

[1995]).



1.2.7 The ElasticViscoplastic Models

The first viscoplastic model was proposed by Bingham in 1922 (see Malvem [1969]).

He considered the case of a simple shear in the x-direction and supposed that no motion takes

place until the stress T reaches a critical value (see Fig. 1.5), after which the magnitude of rate

deformation D is proportional to the amount by which T exceeds k. It can be shown

schematically by Fig.1.5. The viscoplastic equation for simple shear is assumed as


D 0 for T 2rlD= (1- )T for T>k (1.19)
T


where Tr is a viscosity coefficient.













T T



k

Fig.1.5 Bingham viscoplastic model.



Hohenemser and Prager in 1932 (see Malvem [1969]) assumed incompressibility and

generalized the equation to

f 0 for F<0
2lDq= Fo' for F>0 (1.20)

where oa is stress deviator tensor and F is measurement of overstress given by


k
F=l- ,
S(1.21)



where J2 is the second invariant of deviatoric stress. Subsequently, a lot of research has been

done in the viscoplasticity theory (Perzyna [1966]; Malvem [1969]; see comprehensive

review by Cristescu [1982]). One theory was proposed by Perzyna [1966]. The formulation

can be written as

S 1 1-2v .. F
e= --o +---- j+yo (1.22)
'i 2G ii E Go (1.22)








23

where G, E are elastic shear modulus and Young's modulus respectively, .ijis rate of stress

tensor, 6 is mean stress rate.

F- i-1 (1.23)
k

was suggested by Perzyna [1966]. In this theory, the associated flow rule has been used. F

should satisfy the convex conditions. Drucker's stable material postulate was used to define

F. Due to considerable research works (see Cristescu & Suliciu [1982]; Lubliner [1990]), the

elastic/viscoplastic theory can be reduced to the elasto-plastic theory if time effect is

negligible while the elasto-plastic theory can be generalized to elastic/viscoplastic theory if

time effect is taken into account (see Simo et al. [1988]).

Recently, one kind of elastic/viscoplastic model was proposed by Cristescu [1989,

1991, 1994]. The theory is based on the elastic/viscoplastic theory and the characteristics of

geomaterials (mainly, rock and sand). The basic formulation can be written as

1 1 AI / W(t)\9F(o)
e = +( I )61+k W(t J(o (1.24)
2G 3K 2G \H(o)l ao(



where K, G are elastic moduli, H is yield function, F is viscoplastic potential, W(t) is

irreversible work per unit volume, kT is a viscosity coefficient. This formulation is a

generalized form of the Bingham material. However, Cristescu [1991] gave a procedure to

determine the yield surfaces and potential surfaces based on a set of hydrostatic tests and

conventional triaxial compression tests. There is no a prior assumption regarding the yield

function and viscoplastic potential. This model has been used by many researchers (Florea








24
[1994a,b]; Cristescu et al. [1994]; Dahou et al. [1995]). The procedure to determine the yield

function and the viscoplastic potential was improved by Jin and Cristescu [1996a] and Jin

et al. [1996c] so that the model could better match the experimental results and be easily

incorporated into a finite element method. Based on the conventional triaxial compression

tests on alumina powder (Jin & Cristescu [1996b]; Jin et al. [1996d]) which will be presented

in detail in the present thesis, the improved procedure was used for the modeling of

consolidation of alumina powder A10 (Jin et al. [1996a, 1996b]). Another procedure to

determine the orientation of the irreversible volumetric strain rate was given by Cazacu et

al. [1996] and Cristescu et al. [1996]. They used a discontinuity function to describe the

effect of the compressibility and dilatancy boundary. A methodology to determine the

elastic/viscoplastic model in terms of the irreversible volumetric strain as a working

parameter is developed in this thesis. The model prediction matches well the experimental

data. This methodology is more attractive for the powder materials since the work hardening

parameter volumetricc strain) is directly associated with volumetric reduction, and the

procedure to determine the model is much simpler.



1.2.8 Other Models

There exist also many other models. Green elastic model (existence of a strain energy

function) and Cauchy elastic model (stress is assumed as a function of strain) can be used to

describe nonlinear elastic behavior (see Desai & Siriwardane [1984]). Hypoelasticity (the

increment of stress is expressed as a function of stress and increment of strain) originally

proposed by Truesdell [1955, 1966] could serve as a general nonlinear model (also see Green








25
[1956]). However these models also have some limitations (see comments by Chen & Baladi

[1985]).

There are also many publications regarding to the homogenization approach for

modeling (Christensen [1991]; Nemat-Nasser [1993]; Huang et al. [1994]). However, due

to complicated modeling and assumptions, this approach does not yet mature for practical

problems.



1.2.9 Remark on Recent Development of Plasticity-Based Models

For the plasticity-based models, the isotropic hardening and the kinematic hardening

rules were developed and used. Several theories were developed for work hardening. There

are several models for the kinematic work hardening (see comprehensive review by Desai

& Siriwardane [1984]; Chen & Baladi [1985]; Drucker [1988]). Sometimes, the isotropic

hardening rule and the kinematic hardening rules are incorporated into the above models

presented in the previous sections. However, due to the difficulties to determine the

parameters in the work hardening rule from experiments, the application of these hardening

rules is limited.



1.2.10 Finite Element Method

During the past several decades, the finite element method has rapidly become a very

popular technique for the numerical solution of complex problems in engineering with the

help of large-digit computers (see Bathe [1982]; Zienkiewicz & Taylor [1989]; Hughes

[1989]). Applications range from the stress analysis of structures to the solution of acoustical,








26
neutron physics and fluid dynamics problems. Indeed, the finite element method has been

developed so rapidly that the finite element process is established as a general numerical

method for the solution of partial differential equation systems subject to known boundary

and/or initial conditions.

The success of the finite element method is based largely on the basic finite element

procedures used: (i) the formulation of the problem in variational or weighted residual form

(weak form), (ii) the finite element discretization of this formulation, and (iii) the effective

solution of the resulting finite element equations. These basic steps are the same whichever

problem is considered. These basic steps provide a general framework and a quite natural

approach to engineering analysis in conjunction with the use of the large digit computers.

The basic concepts of finite element method and basic formulation for problems in

the isoparametric finite element representation can be referred in the books by Zienkiewicz

and Taylor [ 1989] and Hughes [1989]. The basic idea and procedures are easy to extend. The

basic formulation can serve for different problems, such as elastic-plasticity and

elastic/viscoplasticity problems.

In Appendix 1, we will give brief review of the discrete formulation of the

elastic/viscoplasticity theory for the finite element method. These formulations have been

implemented into a finite element method program. The program has been used for the

analysis of several problems in the present thesis. The main results will be presented in

Chapter 3.










1.3 Objective and Outline of the Present Work



The accurate constitutive formulation of nonlinear time dependent behavior of

materials is extremely important both for a theoretical analysis and for the finite element

simulation. With the existing models presented in the previous sections, it is difficult to

describe accurately three-dimensional nonlinear time dependent behavior, especially for the

nonlinear volumetric behavior from compressibility to dilatancy. Without an accurate

constitutive equation, there is no way to simulate engineering problems efficiently and

accurately when the nonlinear time dependent materials are involved. Thus, today it is a

challenge to develop a methodology to formulate nonlinear time dependent behavior of

materials accurately.

Cristescu [1989, 1991, 1994], as a pioneer, was trying to formulate nonlinear time

dependent behavior with the elastic/viscoplastic theory based on a set of triaxial tests. In his

formulation, the dilatancy and compressibility of volumetric strain are highly considered. The

yield surfaces and potential surfaces are determined from tests without any a priori

assumption. His formulation was used to describe rock salt and sand. However, his model

should be checked with more experimental data and examined from the point of view of the

application of the finite element method. In the present work, Cristescu's approach is

employed to establish the constitutive equations for rock salt and powder materials. We

found that some steps in his approach of the determination of the elastic/viscoplastic model

should be modified or changed in order to match better experimental data and to be easily

incorporated into the finite element method.








28

The main objective of the present work is to investigate the behavior of powder

materials experimentally by triaxial tests and to formulate three-dimensional constitutive

models for powder materials and for rock salt based on the elastic/viscoplastic theory and the

experimental data. Emphasis is placed on the improvement of the procedure (Cristescu

[1991,1994]) to determine the elastic/viscoplastic models. In the determination of the

models, the asymptotic behavior and the physical requirements of yield surfaces and potential

surfaces are incorporated into these procedures so that the yield surfaces and the viscoplastic

potentials are kept smooth. These models can better match the data and can easily be

incorporated into the finite element program.

The present work is associated with three aspects: (1) triaxial tests, (2) constitutive

modeling, and (3) finite element analysis. First, a constitutive equation for rock salt is

formulated based on the true triaxial experimental data obtained by Hunsche (see Cristescu

& Hunsche [1992]). Besides the properties considered in the Cristescu's approach, we have

considered the asymptotic behavior of yield surfaces and of the potential surfaces in the

procedure to determine the model. A new method to determine the viscoplastic potential

surface is proposed. Non-associated flow rule is used. The model matches very well the data.

Also, the model is incorporated into a finite element program. Several engineering problems

are analyzed with the program. The finite element analysis gives a good agreement with

experimental data. Such type of the elastic/viscoplastic model (both yield surfaces and

potential surfaces) is used for the first time in the finite element programs.

Secondly we have performed a series of triaxial tests on alumina powders using a

triaxial apparatus. The influences of the particle size and of the initial density on the








29

mechanical behavior of alumina powders are investigated experimentally. It is shown that

the volumetric strain behavior is strongly influenced by the particle size of powders. For the

alumina powder A16-SG of smaller particle size (0.4-1 pm), the volumetric strain always

exhibits compressibility in the range of applied stress state, while for the alumina powder

A 0 with larger particle size (40-200pm), the volumetric strain exhibits both compressibility

and dilatancy. The influence of the initial density on the behavior of alumina powder A10

has also been investigated. The elastic parameters of these alumina powders have been

measured according to the loading-reloading process. The set of triaxial tests furnishes the

necessary data for the formulation of three-dimensional constitutive equations.

Thirdly, a three-dimensional elastic/viscoplastic constitutive model for alumina

powder A10 is formulated based on the experimental data (a set of conventional triaxial

tests). The yield surfaces and the potential surfaces are determined. A new procedure to

determine the model is proposed based on such type of conventional triaxial compression

tests. The model is checked carefully against the experimental data with different loading

paths. A good agreement between the data and the model prediction is obtained.

Finally, another methodology to formulate the behavior of the materials which are

compressible only is proposed. The irreversible volumetric strain is taken as a work

hardening parameter instead of the irreversible stress work used in the previous method. The

methodology is used to formulate the behavior of alumina powder A16-SG. The irreversible

strain rate orientation tensor is used and determined in this model. It is found that this model

is much easier to handle and has fewer parameters involved than the previous ones. A

reasonable matching between the experimental data and the model prediction is obtained.








30

The organization of the dissertation is as follows: in the first Chapter, the models

existing in the literature are reviewed. In Chapter 2, the Cristescu's approach is outlined. A

new constitutive equation for transient creep of rock salt is proposed in Chapter 3. In the

fourth Chapter, three sets of triaxial tests on alumina powders (two sets for A10, one set for

A16-SG) have been performed and presented. In Chapter 5, an elastic/viscoplastic

constitutive equation for alumina powder A10 is formulated based on the hydrostatic tests

and the deviatoric tests presented in Chapter 4. In Chapter 6, a new methodology to

formulate the behavior of the materials which are compressible only has been proposed.

Finally, some conclusions and suggestions for the future work are given in Chapter 7.













CHAPTER 2
THE ELASTIC/VISCOPLASTIC (CRISTESCU) THEORY



2.1 The Elastic/iscoplastic Constitutive Equation



Many materials such as rock salt and powder materials can be considered as

homogeneous and isotropic ones. These materials have no preference directions. In many

cases, the deformations and rotations of particles are small. In the framework of the

elastic/viscoplastic theory, it is assumed that (1) the materials considered are homogenous

and isotropic; (2) the deformations and rotations of particles are small.

Based on the above assumptions, the total strain rate tensor e can be obtained by


adding the elastic strain rate tensor c and the irreversible strain rate tensor e :


"E "1
e=e +e (2.1)


For the elastic response of material, let us consider the fact that both longitudinal and

transverse extended body seismic waves can propagate through most materials (Cristescu

[1993a]). The fact suggests that most materials exhibit an elastic "instantaneous response."

The elastic parameters can be obtained from the speeds of longitudinal and transverse

extended body seismic waves. In the framework of the elastic/viscoplastic theory, the elastic








32
parameters can also be obtained by unloading process suggested by Cristescu [1989] for time

dependent materials. Thus, the instantaneous response can be expressed by a rate type elastic

relation

E 1 1
e- +(-. )61 (2.2)
2G 3K 2G "

where 6 is stress rate tensor, 6 mean stress rate, 1 unit tensor and G and K are elastic shear

and bulk moduli respectively.

For the irreversible part, it is assumed that the material follows a viscoplastic type

behavior (Cristescu [1991, 1994]), i.e.,

-k. 1 W(t) OF(o)
H(o)l o (2.3a)


where kT is viscosity coefficient, W(t) is total irreversible stress work per unit volume at time

t, F(o) is the viscoplastic potential and H(o) is the yield function with


H(o(t))=W(t) (2.3b)



the relaxation boundary or the equation of the stabilization boundary (where e =0, 6=0).

The bracket o denotes the positive part of a function, i.e.,


= (A +41)=A =A if A (2.4)
2 0 if A 0. (2.4)

If the yield function coincides with the viscoplastic potential, the flow rule is associated,

otherwise, the non-associated flow rule is assumed. For some materials, the viscoplastic








33
potential may not exist. In that case, the irreversible strain rate tensor is assumed to be

proportional to a tensor N (Cazacu et al. [1996]; Cristescu et al. [1996]):


e' k (l- w ) (2.5)


where N(o) is a tensor governing the orientation of irreversible strain rate tensor.

From eqn (2.1-2.3) and (2.5), the elastic/viscoplastic constitutive equation will be

written as


e 1 (- 1 )61+k 1 W(t) F().6a)
2G 3K 2G H(o) o (aa

or

= +( 1 1 +k I W(t) ) (2.6b)
2G 3K 2G H(o)b

In general, the yield function H and the potential function F (or N) are all dependent on stress

tensor. However, if H and F (or N) are assumed to be dependent on the first stress invariant

1, and on the second invariant J2 of deviatoric stress tensor only, disregarding the influence


of the third invariant of stress tensor, the whole constitutive equation can be determined from

a couple of triaxial tests (Cristescu [1991, 1994]). That is, H and F (or N) are assumed to be

dependent on mean stress

1 1
o=- -(1o+02+0 3) (2.7)
3 3


and octahedral shear stress r











t= -2 (2.8)




or equivalent stress


D=- ^3 (2.9)






2.2 Determination of the Elastic/Viscoplastic Constitutive Equation



In the general constitutive equation (2.6a) or (2.6b), there are three parameters, G, K

and kT, and two functions, H and F (or N) to be determined. Cristescu [1991, 1994] provided

a primary procedure to determine them from a couple of triaxial tests for rock salt and

dry/wet sand. The procedure is shown by scheme in Fig. 2.1. The details in the procedure

may be referred in Chapter 3 and 5.



2.2.1 Elastic Parameters and Viscosity Coefficient

Elastic parameters G and K are involved in all steps of the model determination. Thus

the elastic parameters must first be determined. Generally, there are two methods used to

determine elastic parameters, i.e., dynamic and static methods. In the former case, the seismic

wave velocities propagating through materials are measured, and from them are calculated

the elastic constants. In the latter case, the elastic parameters are determined using an








35
unloading procedure (Cristescu [1989]) for time dependent materials. The main idea of the

unloading procedure is to separate time effect from the pure elastic response during

unloading. For example, in stress-controlled tests, we keep stress constant at a chosen value

and allow the specimen to have a sufficient time to creep until the rates of strains are small

enough so that no significant interference between creep and elastic unloading would take

place during the unloading performed in a comparatively short time period. Elastic

parameters can afterwards be determined from the slope of the first portion of the unloading

curves (usually one-third or one-fourth of total axial stress). This procedure also can be used

for strain-controlled tests (see Chapter 4).

The viscosity coefficient can be determined from creep tests such as hydrostatic creep

tests after the yield surface and the potential surface are determined. If, for instance, a

hydrostatic creep test is performed, we integrate eqn (2.3) and obtain the following formula

for the viscosity coefficient


W (t) 'f H(o) 1
H(o)) l Fo(tf-ti) (2.10)
ao


where ti is the "initial" moment of the creep test, tf is the "final" moment, o(t,)=o(t)=

const., and H is yield function and F is potential surface. If we use a nonassociated

constitutive equation, then kT could be determined simultaneously with F. However, creep

tests can still be used for the adjustment of viscosity coefficient.












Functions & parameters
needed in the model Procedure used


Elastic parameters from both hydrostatic and deviatoric tests
G, K through unloading-reloading




Failure condition from deviatr t
--| from deviatoric tests }



H, from hydrostatic tests
Yield surface H- from deviatoqric ests
H
SAssumption: H=H,+Hd



SHydrostatic potential from hydrostatic tests .


G(o,T) function based on E, in deviatoric tests
Potential surface F
k F
F fkT F = fG(o,t)ds+ g(r)


g(r) based on deviatoric tests



Viscositycoefficient rom creep tests




Fig. 2.1 Schematic of the procedure to determine the elastic/viscoplastic model.









2.2.2 Yield Function

The yield function is defined as the irreversible stress work per unit volume on the

stabilization boundary or relaxation boundary according to the elastic/viscoplasticity theory

(see Cristescu [1967, 1989]; Cristescu & Suliciu [1982]; Lubliner [1990]), i.e.,

H(o(t))=W(t) (2.11)


Thus, the yield function can be obtained by evaluating the irreversible stress work per unit

volume along the relaxation boundary. W(t) can be computed as


W(t) =fo(t ):'(t)dt (2.12)
0



where (:) means double contraction on tensors. W(t) is used as an internal state variable or

as a work-hardening parameter. The constitutive equation (2.6) may be used even if the

material is assumed to have a zero initial yield stress.

The yield surface H is assumed as the sum of two parts, hydrostatic one H and

deviatoric one Hd, according to the loading paths used in the triaxial tests, i.e., H=H,+Hd.

H, is determined from the hydrostatic loading tests, while Hdis determined from deviatoric

tests by the regression of the irreversible stress work along the relaxation boundaries as

indicated in Fig.2.1. For different materials, the yield function might not be the same. For the

details of the determination of yield function, one can refer to Chapter 3 and 5 or refer

Cristescu [1991, 1994] for specific materials.








38

It is noted that most models in the literature (see Chapter 1) end at this step (see Fig.

2.1) and associated flow rule is assumed. However, most friction materials are not satisfied

with the associated flow rule. Non-associated flow rule should be used. The viscoplastic

potential should be explored.



2.2.3 Viscoplastic Potential

The viscoplastic potential F or the tensor N in eqn (2.6) governs the orientation of

irreversible strain rate tensor. It can also be determined from experimental data. Several steps

are required to determine the potential as shown in Fig. 2.1. According to the procedure,

some properties of the material such as a failure condition or/and compressibility/dilatancy

boundary are incorporated into the potential. The potential might not be the same for

different materials. In Chapter 3, 5 and 6, the procedure will be given in details for specific

materials. However, many steps have been changed from the original steps (Cristescu [1991,

1994]). The potentials obtained satisfy additional conditions which are required by physical

observation and by the finite element method.

If the yield function H(o) coincides with the viscoplastic potential, i.e., H(o) = F(o),

the flow rule is associated, while if H(o) F(o), the constitutive equation is called non-

associated. Both the associated and nonassociated flow rule can be used for modeling.

However, in general, the associated flow rule could not give good results, while the

nonassociated flow rule can.










2.3 Loading-Unloading Conditions



Let us assume that at an initial stress state (the so-called "primary" stress) oa=o(t0)


is an equilibrium stress state, i.e.H(o(to))=WP, with WP being the value of W for the


primary stress state. A stress variation from o(t) to o(t) with t>to is called unloading if

H(o(t))
loading, instantaneous elastic response given in eqn (2.2) is assumed. A stress variation

from (t) to o(t) (*o(t)) with t>to is called loading if H(o(t))>W(to) with one of the

three possible subcases defined by the following inequalities satisfied at (t):

dF 8F
:1>0 or >0 compressibility
do ao
OF 1F
-:1=0 or a=0 compress./dilatancy boundary (2.13a,b,c)

-:1<0 or -<0 dilatancy
do ao



The three inequalities correspond to the inequalities to be satisfied by the rate of irreversible

volumetric strain e, involved in eqn (2.3). Also, these inequalities define the region of


compressibility, compressibility/dilatancy boundary, and the region of dilatancy. These

concepts are very important not only in the theory but also in engineering applications

(Cristescu [1993a]).













CHAPTER 3
AN ELASTICNISCOPLASTIC MODEL FOR TRANSIENT CREEP
OF ROCK SALT



In the formulation of transient creep of rock salt, we start from a complete and

accurate set of true triaxial data on Gorleben rock salt obtained by Hunsche (see Cristescu

& Hunsche [1992]). The set of data consists of six true triaxial tests under the conditions of

different constant mean stress, i.e., o=8, 14, 25, 30, 35 and 40 MPa. For the formulation of

hydrostatic constitutive equation, one can refer to Cristescu and Hunsche [1992] and

Cristescu [1993a]. For the formulation of the mechanical behavior of rock salt in transient

creep, we follow Cristescu's approach (Chapter 2 and Cristescu [1989, 1991, 1993a, 1993b,

1994]). In this approach both compressibility and dilatancy properties of rock salt will be

considered, which is an advantage of the model over other models (See Chapter 1). The

concept of compressibility/dilatancy (C/D) boundary is introduced, thus one could well

determine with such a constitutive equation, where around an opening, for instance, the rock

becomes dilatant and where compressible. The procedure in this approach allows the

determination of a nonassociated elastic/viscoplastic constitutive equation able to describe

creep, relaxation, dilatancy and/or compressibility during creep, work-hardening and failure.

In this chapter a new elastic/viscoplastic constitutive equation for transient creep is

proposed. Compared with the early models (Cristescu [1993a, 1994]) the determination of








41
the yield function and of the viscoplastic potential is improved: the yield function has a

singularity at failure and the viscoplastic potential surfaces are requested to satisfy additional

conditions required to connect hydrostatic tests with deviatoric ones. A new procedure to

determine the viscoplastic potential is proposed. The viscoplastic potential is formulated in

simpler analytical expression than that in the previous papers. The present model on the one

side will be in a better agreement with the data and on the other side could be easily

incorporated into a FEM program. An explicit integration on viscoplastic strain components

is used in the constitutive equation. The results obtained by FEM are in good agreement with

the experimental data. The stress distribution around a vertical cylindrical cavern is analyzed.

Some interesting results are found.



3.1 Determination of the Elastic/Viscoplastic Constitutive Model



The procedure to determine the elastic/viscoplastic model has the following steps:

(1) determination of the elastic parameters, (2) determination of the C/D boundary and the

failure condition, (3) determination of the yield function, and (4) determination of the

viscoplastic potential and viscosity coefficient. Each step is directly related to experimental

data. We will discuss in details the above steps in the following sections.



3.1.1 Determination of the Elastic Parameters

The elastic parameters are determined by using the unloading processes in a stress

controlled apparatus mentioned in Chapter 2. For a stress controlled test, stress is kept







42
constant at a chosen level. The specimen is allowed to have a sufficient time to creep until

the rates of strains are small enough so that no significant interference between creep and

unloading phenomena would take place during the unloading performed in a comparatively

short time period as schematically shown in Fig. 3.1. Elastic parameters can afterwards be

determined from the slope of the first portion of the unloading curves (Fig. 3.1). For rock

salt, an average value of bulk modulus K=21.7 GPa and of shear modulus G=l 1.8 GPa have

been measured by Hunsche (see Cristescu & Hunsche [1992]). The values obtained by this

method are very close to those obtained by dynamic method (Cristescu [1989]). The elastic

parameters are calculated with the formula (3.1)


1 1
a, =arctg --- ; a=arctg
1G 9K6G
3G 9K 9K 6G


E2








CX


0 = constant











loading


Transverse strains Axial strain
Fig. 3.1 Procedure to determine the elastic parameters in
unloading processes following short creep periods











3.1.2 Determination of the Compressibilitv/Dilatancv Boundary and the Failure Condition

The compressibility/dilatancy (C/D) boundary is defined by eqn (2.13b). The

boundary has specific meanings in engineering applications (Cristescu [1989, 1993a]). Below

the boundary shown in Fig. 3.2 there is the compressibility region. In the compressibility

region, the rate of irreversible volumetric strain is positive (compression strain is

conventionally taken as positive), which means that the absolute value of irreversible volume

is reduced. The decrease of irreversible volumetric strain may close microcracks and pores

which may be present in geomaterials. In the dilatancy region shown in Fig.3.2, the rate of

irreversible volumetric strain is negative, which means that the absolute value of irreversible

volume is increasing. The increase of irreversible volume is directly related to the opening

of microcracks and of pores, or of any kinds of damage which may be present in

geomaterials. Sometimes in engineering applications, dilatancy is to be avoided. The C/D

boundary can be obtained straightforward from experimental data. In classical triaxial tests

where hydrostatic loading is followed by deviatoric loading under constant confining

pressure, the curve ao-e: is obtained in deviatoric tests. Here o and el are the stress and


volumetric strain during deviatoric loading with respect to the reference configuration at the

end of hydrostatic tests superscriptt R means "relative", i.e., with respect to the mentioned

reference configuration). The C/D boundary is determined as the stress loci where the slopes

(or tangents) of the stress-volumetric strain curves are equal to the elastic ones (i.e., e' =0).


In true triaxial tests where the hydrostatic loading is followed by a deviatoric loading under








44
constant mean stress, the C/D boundary is obtained directly from T eR there where the

slope of this curve is vertical.

For the rock salt from true triaxial tests, Cristescu and Hunsche [1992] have obtained

the C/D boundary as follows


o 02
X(o,r )=--i+ fi + f2- 0 (3.2)
O, j f a

where f,=-0.017, f,=0.9 and o=1 MPa(see Fig. 3.2 where the full line is just this


boundary).

Fig. 3.2 shows by diamonds the locus of maximum values of octahedral shear stresses

obtained in several tests using the data by Hunsche (see Cristescu & Hunsche [1992]); they

correspond to the mean stress values: o = 14, 20, 25, 30, 35 and 40 MPa, respectively. The

following equation can be used to approximate the locus of the maximum stresses


tf(o)=,-Yex2 Y3 (3.3)




where T, (o) is octahedral shear stress at failure, o is mean stress and y1=38.0 (MPa),

Y2=34.9(MPa), and y3=0.04 obtained through curve fitting. The stress states at failure and


the fitting curve are shown in Fig. 3.2.










35

(MPa) 30
Failure condition
25
dilatancy region
20 E <0

15 / =0
C/D boundary
10

5 ev >0
compressibility region

0 10 20 30 40 50 60
a (MPa)


Fig. 3.2 Domains of compressibility, dilatancy, C/D boundary and failure condition



3.1.3 Determination of the Yield Function

The yield function is defined as the stabilization boundary or relaxation boundary

according to the viscoplasticity theory (see Cristescu [1967, 1989]; Cristescu & Suliciu

[1982]; Lubliner [1990]). Due to the available short term experimental data on rock salt, it

is assumed that the stress strain curves are "almost" reaching the relaxation boundary for

transient creep. Thus the yield function can be obtained by evaluating the irreversible stress

work from the experimental data using eqn (2.12). W(t) and H(o,t) can be separated into two

parts according to the two loading paths used in true triaxial tests. In the first stage, the

hydrostatic loading, the stress components are kept equal (o0=o2=03), and increased

according to a certain conventionally chosen time interval t e [0, th). The second stage, the








46
deviatoric loading, takes place in the time interval t e [th, t] with o = constant, and it is only

the octahedral shear stress which is increased. Thus the complete expression of the yield

function is the sum of the yield functions obtained in the two stages (Cristescu [1994]):

H(o,T): =H(o)+Hdo,1) (3.4)


as is the total irreversible stress work:

W(t)=W,(t)+W(t) (3.5)


with the relations

Hh(o)=W,(t); Hd(o,T)= Wdt) (3.6)


where the subscripts h and d indicate hydrostatic and deviatoric stages respectively. Wh and

Wd can be computed from eqn (2.12) using the corresponding integral intervals.

From the calculation of irreversible stress work in hydrostatic tests and curve fitting,

Cristescu and Hunsche [1992] obtained the following expression for H,(a)


hoisin Wo--+(p +hi if as o(
H,(o): = Ga ) (3.7)
h+hI if a >0o


where ho0. 116 MPa, h=-0.103 MPa, o=2.880, (p=-62.60 and 0o=53 MPa.












Wd
(MPa)





0.1

S= 20 (MPa)
0.01



0.001 .I
0 5 10 15 20 25
T (MPa)

Fig. 3.3 Typical variation of irreversible stress work Wd with T for o =20 MPa,
test results points, a straight solid line suggesting a model behavior.



Using the experimental data obtained in the deviatoric loading stage, we calculated

the irreversible stress work using eqn (2.12). Fig. 3.3 shows a typical computed result for the

mean stress 20 MPa. The deviatoric irreversible stress work can be approximated by two

terms, for instance; one is linear in the exponential of T, the other has a singularity at failure.

Thus, the formula for the deviatoric part of the yield function becomes,





Hjo,t)=a(a) exp b(o)- + k(o)-exp 8( -1) (3.8)
Va.8


where










a(o) =a+ac-+..a ; b(o)=P, exp -P,- +
j r 0 [ *)(3.9)
k(o)=Kl a ; T/)=YY2 exp -Yz- .


There are two terms involved in the approximation of Hd(o,r) as shown in eqn (3.8).

First, from the log(Wd) ~ T plot of the experimental data (see Fig. 3.3), it was found that

besides a small region near failure Hd(o,t) can well be approximated by an exponential

function, i.e., Hd(o,t) a exp(bc). However, for a given o, near the maximum octahedral

shear stress t~m it was found that WfT) increases much faster than described by an

exponential function. It looks as if Wd(t) blows up near the maximum octahedral shear

stress. To capture this seemingly singular behavior of H,(o,T) near failure in a simple

manner, a first order algebraic singularity is introduced, HI(o,r) kl (tf-t), in which k and

tf are determined from the fit of the experimental data. Because this function decays too

slowly away from Tf, which would offset the close fit already obtained for the exponential

growth part, an exponential decay factor exp[8(r/Tr 1)] is introduced to reduce the error in

joining these two asymptotes. Thus the second part of Hd(o,t) becomes k/(Tz-)exp[8(t/r-

1)]. The factor 8 is introduced by error and trial method to fit better the curves. In general,

T" obtained from the fit of the data is a little larger than the actual maximum octahedral shear

stress in each of the experiments and the difference between T, and T are less than 1%.

Hence practically T, can be used, in lieu of max, to represent the octahedral shear stress at

failure as shown in Fig. 3.2.










12
(MPa) 40
10 calculated values from curve fitting
++++ calculated value from experiments
8

6 30

20 25
4-
14
2 a=8MPa

0 I
0 5 10 15 20 25 30 35
T (MPa)

Fig. 3.4 Fitting curves of irreversible work. Crosses experimental results,
solid lines prediction of eqn (3.8)



All the coefficients a, b, k and T, depend on o. After analyzing the data for each o,

the dependencies of a, b, k and T, on o are found out as shown in eqn (3.9). The following

constants are obtained:


a,=1.88x10-4(MPa), a2=1.13x10-4(MPa), 3=0.016x10- 4(MPa);
P,=0.566, P,=0.106, P3=0.217; (3.10)
K,=0.00171(MPa), K2=1.356 .


In the process of curve fitting, it was found that the data corresponding to 0=8 MPa

is not consistent with the trend established for o>8 MPa. Since during this test the mean

stress o is not really constant. For this reason, these data are further omitted. The data

corresponding to o=35 MPa were not used for the determination of a, b, k, and ,. Hence the








50
experimental data at o=35 MPa are used as an independent check to test the accuracy of the

Hd(o,r). Fig. 3.4 shows very close matching for o=35 MPa as it is for the data obtained with

other values of o.



3.1.4 Determination of the Viscoplastic Potential



After the determination of the yield function, we can use these results to get the

viscoplastic potential which governs the orientation of viscoplastic strain rate. The potential

has to satisfy several constrains from the elastic/viscoplasticity theory of

compressible/dilatant materials (Cristescu [1993a]). Let us shortly remind them. From eqn

(2.3) and (2.13), we have:


OF E
G(o,t): kT v (1
Oo I VW(t) \ (3.11)
H(o,t)

For the hydrostatic loading, G(o,z) satisfies the following condition


G(o,t)I,= -kT--F =4(o) (3.12)
a IT1=0



where 4(o) is determined from hydrostatic tests, while for general loading paths from eqn

(2.13), G (T,o) is required to satisfy the following conditions:


G(o,t) >0 X(o,t)>0 compressibilityy)
G(o,T) =0 X(o,r)=0 (C/D boundary) (3.13)
G(o,t) <0 X(o,T)<0 dilatancyy)








51
where X(o,t)=0 is C/D boundary. The rate of irreversible volumetric strain will increase in

compressibility domain, decrease in dilatancy domain and be equal to zero at the boundary

between compressibility and dilatancy regions. Also the rate of irreversible volumetric strain

may reach negative infinite values when stress state is approaching the failure condition

denoted as Y/o,r)-=O. The simplest form which has the above properties is

aF X(o,T)(o)
kT a( Y(or) (3.14)
a Y/or,t)



where the function P is related to hydrostatic behavior by

X(o,0) t(o)
Y0o,0) 3.15)



Cristescu [1991, 1994] has employed this form to get the viscoplastic potential. Although

this form satisfies exactly eqn (3.13), it may not match well enough the data within the

compressibility and dilatancy regions and may cause a discontinuity of e' at T-r. In order

to match better the data within the mentioned regions and ensure the continuity of e', we

propose a procedure to determine the irreversible volumetric strain rate directly from

experiments, i.e., to determine the function G(o,t) by fitting the experimental data with the

requirement that G has to satisfy the conditions (3.13).

Let us start from eqn (3.12). G(o,T) may be divided into two parts according to two

physical meanings: compressibility and dilatancy, denoted by G, and G, respectively. Each

part of G(o,T) is considered to be an asymptotic approximation and to be a dominant part in








52
the corresponding region. In order to satisfy the condition G(o,t) = 0 at the C/D boundary,

we take G,(o,t) to be strictly equal to zero and G, to be approximately equal to zero on this

boundary in the sense that G, is dominant in the compressibility region as compared with G,

From eqn (2.6), aF/Ta should be zero along and in the immediate neighborhood of =-O in

order to obtain a correct strain variation along a loading path very close to the hydrostatic

one, i.e., an additional very small deviatoric stress was superposed. This condition is imposed

in order to avoid the presence of a vertex of the surface at the hydrostatic axis and to keep

the continuity of e when passing from hydrostatic to deviatoric loading. Since aG/at is also

related (see the following sections) to aF(o,t)/at, at this stage we take aG/at=0 at T=O. Thus

we take G,/ T-=0 and aG2 / 1 =0 along =-0. Depending on the evaluation of the right-hand

side term of eqn (3.11) using the data in the compressibility region, we will be able to choose

an appropriate form for G, in order to match the data and satisfy the above two conditions.

However, since in the compressibility region the experimental data available have not had

enough accurate digits, the results obtained from the right-hand side of eqn (3.12) show a

large scatter in that region. For this reason we use the simplest form for G,, i.e., a second

order polynomial in t as



Gi(o, )= Q(o) -e(o),, (3.16)



where as before a.= 1 MPa. From the requirement 9G,(o,t)/r-=0 at T-O, follows that a

possible first order term in eqn (3.16) is zero.








53
From hydrostatic tests Cristescu and Hunsche [1992] have obtained for 4(o) involved

in eqn (3.15) and (3.16):


o (-q2 /
()= a if o o00 (3.17)
0 if 0>00



where q,=1.5 x 108 (s l), q2=o, = 53 MPa.

From the above conditions, G,(o,T)lj=0= (o) and from the requirement G,(o,T)=0

along X(o,t)=0, we can determine Q[ and Q2 from eqn (3.17) (note f2 / f, = -q2 / .) as


,(o)=q o-q2 Qq2 () q1
o o)( f2 (3.18)




In the dilatancy region, the function G2 mentioned above should be dominant in that

region, but must have much smaller values than G, in the compressibility region in order to

obtain an asymptotic approximation of the conditions (3.13), Higher order polynomials in

z seem to be good expressions for the approximation of G, in the dilatancy region. Based on

the data available and the approximated evaluation of the right-hand side of eqn (3.11) (see

Cristescu [1991]), we can choose the 5th and 9th order powers to fit the data (Fig. 3.5),


G2(Fo,)=A(O)(i J +B(o)( -J (3.19)
G. a








54
A(o) and B(o) can be determined using the data from true triaxial experiments performed at

several mean stresses. When analyzing the data and trying to choose proper polynomial

functions, we have found that at the beginning of the dilatancy region, by combining G,(o,t)

with a fifth order power function we get a better fitting of the experimental data. Thus, the

A coefficients are determined from several tests performed with various constant values of

a. By combining a ninth order power function with G,(o,T) and the above fifth order

function, we were able to obtain a better fitting of the curves in the dilatancy region at higher

value of T. Thus, the B coefficients are determined. After obtaining the values of A and B

from the tests performed under different mean stresses, we use the least-square method to get

the expression of the functions A(o) and B(o),



A(o)a B(o)=a9 (,) (3.20)


where a5=-5.73x 10-7 (s'), b5= -2.12, a,=-1.63x 108(s-'), b9=-4.77.

In the neighborhood of failure, the dilatancy behavior may become singular.

However, the singularity is very difficult to make precise from the data shown in Fig. 3.5.

Thus, for the expression of G we use both G, describing compressibility and G2 describing

dilatancy. From eqns (3.16) and (3.19), it is easy to show that in the compressibility region

G, is a dominant function as compared to G2. One can use a parameter

p=Ic G2 da/fc Glda (where domain C is the compressibility region) to decide which

term is dominant. The value p=0.11 of this parameter indicates that G, is dominant in the

compressible region. The function G is approximated by











(3.21)


G(o,T)=G,+G2=Qi,(o)+Q2() + A(o) -I '+B(o) -




Then, we integrate eqn (3.11) with respect to a to get

k7F(o,t)=fG(T,o)do+g(T)=F,(o,T) +g(r)


where


F,(o,r):=fG(o,t)do .


(3.22)


(3.23)


0.005
G(o,T)
(s-1) 0

-0.005

-0.01

-0.015
a= 14MPa \
-0.02
20
25 o 30
-0 .025 i I I I
0 5 10 15 20 25 30 35
T (MPa)


Fig. 3.5. Data for various confining pressure shown used to determine the function G(o,t),
symbols experimental data, solid lines prediction of eqn (3.21).







56
In order to obtain g(t), we differentiate eqn (3.22) with respect to T and combine the

irreversible strain components from eqn (2.3)


FW() F 8F aT
S I W) ) (3.24)
SH(o) ~ a o o (3.


Finally we get the formula for g '(s):

1I 1 / 2 *.. ._ .2 *,'i .2 OF
r)=(eI-E2) +(E), (3-e) -" (.5F,
W(t) r2 3 (3.25)
H(o)


In eqn (3.25), all terms are known besides the strain rates involved in the square root

which are obtained from the experimental data (Cristescu [1991]). For the determination of

g'(T) we plot Fig. 3.6 using the data for several constant mean stresses. It was found that g'(r)

is independent on mean stress as predicted by eqn (3.25), which implies that the viscoplastic

potential exists. Thus the procedure to find the potential is logical.

The solid line curve in Fig. 3.6 is an approximation for g'(T), which can be described

by a polynomial function:


g'(C +. (3.26)



where g=0.0013(MPa s '), g,=6.5x106(MPas -), g,=2.8xlO-3(MPa s -).Thusg(r)

can be written as










0.3
g'(C)
(sl) o MPa
cr MPaa
8
0.2 14
20
30
40

0.1





0 5 10 15 20 25 30 35
t (MPa)

Fig. 3.6 Data for various confining pressure shown used to determine g'(t);
symbols test results, solid line prediction of (3.26).




1 aT I3 a 9 9
) g, -- + & i 4 (3.27)




It is noted that in eqn (3.26), we start with the first order power. Since aF/8l should

be equal to zero at =-0, we let g'(t)=0 at T--0 in order to obtain the continuity of the

derivatives of the viscoplastic potential.

Therefore the viscoplastic potential for Gorleben rock salt is completely determined

based on the experimental data and can be written as follows











S 014 2q1 o 2 2 12 2 o
kfl(o,t)=o 4 -- i
4 ,J 3 oI a 2 ,) f2t o,) (3.28)






Using the expression (3.28), we get the shapes of the viscoplastic potential surfaces

while by using eqn (3.4) for H(o,T) we can plot the shapes of the yield surfaces. Fig. 3.7

shows the two sets of curves. It is clear that these two families of curves are quite distinct

both in the compressibility and in the dilatancy regions. Let us note also that the C/D

boundary is quite distinct from the curve aH/Oo=O. Thus, an associated flow rule cannot

predict correctly the C/D boundary. Let us remind that the yield function and the viscoplastic

potential are determined from the data following two distinct procedures. It is found that

FsH. Thus, an associated flow rule is not suitable for this rock salt. Non-associated

constitutive equation should be used with the potential furnished by eqn (3.28).



3.2 Comparison with the Data



One simple way to test a model is to compare the experimental data with the model

prediction. For the model developed in the previous sections, we use three different criteria

to check it. First, this can be done by trying to reproduce with the model the kind of tests

which have been used to formulate the model. Secondly, as an independent check we try to











40

(MPa) 35
awao= 0 ^_ r- 2.0
30 .o05--




5 fail --5 5.02-
25

20

15-


0.1 0.2 CD
5


0 10 20 30 40 50 60
o (MPa)


Fig. 3.7 Shape of yield surfaces (thin solid lines), potential surfaces (thick solid lines),
C/D boundary (dash-dot line) and lWH/o=0 (dashed line).



match a set of data which have not been used to formulate the model. Third, we incorporate

the model into a finite element program and try to describe triaxial tests. In order to

reproduce with the model the data which have been used to formulate the model, let us

consider constitutive equation (2.6a). It will be assumed that stresses are increased by small

successive steps, according to the same law as in the experiments done to establish the

model, and with the same global loading rate as in the experiments (Cristescu [1989, 1991,

1994]). At each very small loading step, the stresses are assumed to increase instantly at time

to, say. Afterwards a creep follows under constant stresses in the time interval t-t,. We can

integrate eqn (2.3) by multiplying it with o to get the following equation for the irreversible

stress work












W(t)=H(o(t))-(H(o)-W(to))exp (to t) (3.29)



The total strains as a result of one stress step can be written as follows



e(t) =E) H 0 1 -exp 8oaF (tot) (3.30)
--o---:F L HO J
H oF



with the initial conditions



e' =0
t=to : e )1 1 1 (3.31)
S -3K 2G 2G



where o(to) is the initial relative stress at each stress step.

The above constitutive equations can be used to reproduce the experimental results.

The viscosity coefficient kT can be determined from creep tests and used as a parameter

which takes into account the speed of the tests as mentioned in the paper by Cristescu [1991,

1994]. The constitutive equations are used to predict the stress-strain curves for the cases

o=14, 0=20, o =40 and 0=35 MPa. The prediction of the model matches well with the

experimental data as shown in Figs. 3.8a-3.8d where circles correspond to the experimental







61
results, and the solid lines to the model prediction. As an independent check, for the

experimental data at 0=35 MPa which were not used to formulate the constitutive equation,

there is also a very close agreement between experimental data and the predicted results (Fig.

3.8d).


20-
S 18
(MPa)
16

14

12

10

8

6

4

2

0
-0.06


-0.04 -0.02


0 0.02 0.04 0.06


Fig. 3.8a Experimental stress-strain curves (circles) compared with
predicted results (solid lines) for a = 14 MPa.










25

(MPa) E, Ev
20


15

S= 20 MPa
10


5


0 I I
-0.04 -0.02 0 0.02 0.04 0.06



Fig. 3.8b Same as in Fig. 3.8a. but for o=20 MPa

35
E2 EF
(MPa)

25

20

15 H= 40 MPa

10

5


-0.1 -0.05 0 0.05 0.1 0.15 0.2


Fig. 3.8c Same as in Fig. 3.8a but for o=40 MPa (note: the data e2 are slightly
different from those for E3)










30
T : E2 Ev 000
(MPa) 25


20

Y= 35 MPa
15

10

5



-0.1 -0.05 0 0.05 0.1 0.15 0.2



Fig. 3.8d Same as in Fig. 3.8c but for o=35 MPa as an independent check



3.3 The Finite Element Analysis



3.3.1 Formulation of the ElasticViscoplasticity Theory

The formulation of elastic/viscoplasticity theory in discrete time form is presented

in the book by Owen and Hinton [1980], with a classic approach by truncated Taylor series

for the rate of viscoplastic strain (Zienkiewicz & Cormeau [1974]; Hughes & Taylor [1978];

Owen & Hinton [1980]; Marques & Owen [1983]; Szabo [1990]; Desai et al. [1995]). There

are many finite element programs with source code available, such as VISCO (Owen &

Hinton [1980]) programs. These codes are usually for education and research purpose, which

are explained in detail. There are also many commercial finite element programs without








64
source code in public, such as ABAQUS [1993]. Several standard open connections in these

commercial programs can be used for users to write their own subroutines, but the open

connections are limited. In order to verify the model developed above, we use VISCO

program (Owen & Hinton [1980]) as our basic program. Several subroutines were

additionally developed for the implementation of the model. A brief description of the finite

element algorithm for elastic/viscoplastic model is given in appendix A.

During the computation, the process of time step marching is stopped at certain time

or when the stresses at all Gauss points satisfy H=W(t) and continues in the next loading

step. The time step At., at step (n+l) was selected subject to the following empirical criteria



A -fn -; Ati lKAta (3.32)



where I, and 1 are second invariants of the viscoplastic strain and strain-rate tensors

respectively; 0 and K are specified constants to control time step. The first criterion selects

a variable time-step size such that the maximum effective viscoplastic strain increment

occurring during next time step is a fraction of the total effective strain accumulated before.

At,,- is evaluated at each Gaussian integration point and the least value is taken for

computation. The second criterion imposes a restriction on the variable step size between

successive intervals calculated by the first criterion to prevent oscillations in the solutions

as steady-state conditions are approaching.

In the following examples, only explicit integration scheme for viscoplastic strains

is employed. For nonassociated flow rule, the stiffness matrix [Kt"] is symmetric and does








65
not vary. The explicit scheme is simpler and easy to implement into the finite element

program. However, the explicit scheme may not be stable when time steps become large. The

restrictions expressed by eqn (3.32) have to be imposed on the scheme so that the explicit

scheme is kept stable. The instability problem can be overcome by implicit schemes.



3.3.2 Example I: Axial Compression with Confining Pressure

In true triaxial tests for rock salt (see Cristescu & Hunsche [1992]), the specimen is

cubic. Stresses in two horizonal directions remain equal during the tests. Thus this kind of

tests will be approximated by an axial symmetrical test where the friction between the

cylindrical specimen and the piston is neglected. Due to the axial symmetry of the cylindrical

specimen, we can use a quarter of the specimen only in order to create a finite element mesh.

6 Serendipity quadratic elements and 29 nodes are used as shown in Fig. 3.9. Each element

has eight nodes. The boundary conditions are also shown in Fig. 3.9.

We will use the same loading procedure as used in the experiments. Hydrostatic

loading is applied first and afterwards the deviatoric loading is performed under constant

mean stress. Let us consider the test for the case o-40 MPa. The total hydrostatic loading is

divided into 10 steps. The computation remains stable for all the steps. The total deviatoric

load is also divided into 10 steps with unequal values. In each loading step, the computation

is carried on until the condition H(o(t))=W(t) in every Gaussian integration point or a

convergence condition is satisfied. As a convergence condition we use here the ratio of the

second invariants of irreversible strain rate evaluated at the initial and the current moment.

The tolerance chosen is 0.1%. In the first eight steps shown in Fig. 3.10, the scheme is stable








66

and convergent. However, when we apply the ninth stress step, the scheme becomes

unstable and blows up. In this computation, Q-0.0009 and = 1.1 were used. The results are

shown in Fig. 3.10. The results obtained with the finite element program give a reasonable

matching with the experimental results although the scheme in the finite element program

blows up near failure. The instability problem in finite element programs can be overcome

by implicit schemes (see Hughes & Taylor [1978]; Simo & Govindjee [1991]).
















"o---


Fig. 3.9 Mesh of a quarter of cylindrical specimen used in the finite element analysis.










3.3.3 Example II: Stress Analysis for A Cylindrical Cavity in Rock Salt

Let us consider now a cylindrical cavity such as a borehole or a shaft with the same

initial or primary stress in all horizonal directions (oa=o h). For simplicity, primary

hydrostatic stress is assumed, i.e., the primary stress in the vertical direction o, is taken to

be the same with that in the horizonal direction. At excavation, the diameter a of the cavity

is assumed to be 1 meter. We make some idealized assumptions: the problem is a plane strain

one with primary stress x =hy =av= 10 MPa in the first example and 30 MPa in the second

example, and that the primary stress is instantaneously released at the cavity surface due to

excavation at time to. Afterwards the rock salt around the cavity creeps according to the

above constitutive equations. We use a quarter profile of the cavity for our computation. The

corresponding boundary conditions are shown in Fig. 3.11. Here the cylindrical coordinate

r is used for the distance from the axis of the cavity to some point inside the rock mass (e.g.,

r=a represents the surface of the wall). Two meshes are used for computation. An area with

7 m in length and 7 m in width is first considered with 253 nodes and 72 elements as shown

in Fig. 3.11. Each element has 8 nodes. In the second variant, we enlarge the area to 20 m in

length and 20 m in width and refine the mesh to a total of 333 nodes and 96 elements. The

computed results obtained with the two meshes do not differ significantly. Figs. 3.12-3.16

show the results obtained with the finer mesh.

In the computation, we choose for the restriction parameters in eqn (3.32): =-0.01

and K=1.3. The computation is stable and convergent. Figs. 12-15 show the variations of the

stresses and octahedral shear stress with the distance from the surface of the wall for the two

cases. A comparison with the elastic solution of the ultimate elastic/viscoplastic one (i.e.,










35

T 30 ev El
(MPa) E2
25

20

15 / 40 (MPa)

10

5


-0.1 -0.05 0 0.05 0.1 0.15
E


Fig. 3.10 Comparison between experimental data (solid line) with FEM results (triangles).



computation was carried out up to stabilization) shows a significant change in stress

distribution. The circumferential stress oe decreases near the wall of the cavity and then

slightly increases at farther distances (Fig. 3.12 and 3.14). The vertical stress oz and the radial

stress o, decrease with respect to the elastic solution. The octahedral shear stress t is smaller

near the surface of the wall, but greater at farther distance (Fig. 3.15 and 3.15). At greater

distances from the surface of the wall, the difference between the two solutions becomes

negligible and the stresses approach asymptotically the primary values. In the case of initial

stress 10 MPa, all regions are compressible. In the case of initial stress 30 MPa, there is a

dilatancy region expanding up to about 0.4 m from the surface of the wall. The other regions

around the opening are compressible. The displacements inside the rock vary in time. Fig.








69
3.16 shows the variation of the displacement at the surface of the wall, starting from the

elastic "instantaneous" response up to stabilization of transient creep. The upper curve

corresponds to the case of initial stress 30 MPa and lower curve to the case of initial stress

10 MPa. A dimensionless quantity kTxt is used for the "time" parameter, where k, is the

viscosity coefficient and t is time. If kT is determined from creep tests (Cristescu 1989), the

time t can also be made precise. Here the intention was to give an example only, obtained

with some simplifying assumptions in order to illustrate the use of the model in a mining

problem. More elaborate engineering problems could also be analyzed.


Fig. 3.11 The finite element mesh used for a vertical cavity.













(MPa)
(MPa)


1 2 3 4 5 6
r/a


Fig. 3.12 Stress distribution with distance from the surface of the cavity wall for the
primary hydrostatic stress 10 MPa (thin lines-instantaneous elastic solution; thick
lines-ultimate elastic/viscoplastic solution obtained when (t-to)xk.=6.6).


8

(MPa) 7

6

5

4

3

2



0
1 3 5 7 9 11 13 15 17


Fig. 3.13 Variation of the octahedral shear stress with distance from surface of the cavity
for the primary hydrostatic stress 10 MPa (thin line instantaneous elastic
solution; thick line ultimate elastic/viscoplastic solution obtained when (t-to)xkT=6.6).








71

60
a
(MPa)
50


40


30


20


10



1 4 7 r/a 10


Fig. 3.14 Same as in Fig. 3.15 but for the primary hydrostatic stress 30 MPa,
(ultimate elastic/viscoplastic solution obtained when (t-to)xkT=5.6).




25

(MPa)
20


15


10


5-


0
1 4 7 10 13 16 a


Fig. 3.15 Same as in Fig. 3.16 but for the primary hydrostatic stress 30 MPa,
(ultimate elastic/viscoplastic solution obtained when (t-to)xkr=5.6).










0.004
u/a
0.0035

0.003

0.0025

0.002

0.0015

0.001

0.0005

0 ....
0 1 2 3 4 5 6 7
kT x (t-to)

Fig. 3.16 Variation of relative radial displacement (u/a) of the surface of the wall with
dimensionless "time" kTx(t-to) (upper curve, for primary stress 30 MPa;
lower curve for primary stress 10 MPa).



3.4 Discussion and Conclusion



In present model transient creep is taken into account only. Since this model is a

phenomenological one based on short term tests, it may underestimate the strain magnitudes.

Actually, stationary creep should also be taken into account when one wants to predict stress

or displacement distributions around some underground openings. Stationary creep may also

contribute to the stress redistribution around an underground opening due to the nonlinear

behavior of rock salt. The finite element program for both transient and stationary creep

should be used in this case. The possible influence of kinematic hardening will also be

considered in the future papers, if enough experimental data would be available. Temperature








73

and humidity are other important factors which influence the behavior of rock salt. We have

not considered them in the present model.

In this formulation, non-associated flow rule was used. It was obtained from

experimental data. Many models adopted nonassociated flow rule (Desai et al. [1987]; Kim

& Lade [19881; Lade & Kim [1989a, 1989b]; Klisinski et al. [1992]) although non-associated

flow rule has been found to cause some problems (Sandier & Rubin [1987]).

Using the triaxial experimental data on Gorleben rock salt, a new nonassociated

elastic/viscoplastic model for transient creep is proposed. This elastic/viscoplastic model can

be used to describe creep, relaxation, dilatancy and/or compressibility and work hardening.

Singularity and asymptotic properties of the yield function and of the viscoplastic potential

which has to satisfy several restrictions are considered. A new procedure to determine the

viscoplastic potential is developed. This procedure allows to better fit data and keep the

viscoplastic potential smooth. The examples given show that the model predicts quite well

the experimental data. Since both the yield function and viscoplastic potential are smooth

functions, the present model can be incorporated quite easily into a finite element program.

An explicit integration scheme with very small time steps for viscoplastic strain components

is used. Several examples show that the model is easy to use in the finite element program.

Instability problems during our computation have not been found except the unstable

computation caused by explicit scheme for viscoplastic strains although non-associated flow

rule has been found to cause some problems (Sandler & Rubin [1987]). Implicit integration

schemes may be better to overcome unstable computation problems.













CHAPTER 4
TRIAXIAL EXPERIMENTAL RESULTS ON ALUMINA POWDERS



In this chapter, the mechanical behavior of alumina powder has been investigated

experimentally under hydrostatic and deviatoric loading conditions in a triaxial apparatus.

Two types of alumina powder, as received A10 and A16-SG, were tested. The specimens

were fully saturated with water. The elastic parameters of alumina powders were measured

based on the loading-reloading processes following a creep or relaxation period. The

experimental results reveal that the mechanical behavior of alumina powder is strongly

dependent on the particle size and on the initial density. The deformation of alumina powder

is time dependent and stress history dependent. The application of deviatoric stress can

produce additional consolidation or dilatancy. The compressibility/dilatancy boundaries and

the failure conditions have been obtained for the tested powders.

Also, the experiments have been performed according to the requirements presented

in the previous chapters to formulate a three dimensional constitutive equation. Thus the

experimental results provide a set of data for the formulation of three-dimensional, either

viscoelastic or viscoplastic, models.

In section 4.2, the basic testing procedure is presented and discussed, which consists

of four sequential steps: (1) specimen preparation, (2) back pressure saturation, (3)

hydrostatic consolidation or hydrostatic loading, and (4) shearing (deviatoric loading). In








75
section 4.3, the experimental results under hydrostatic loading conditions are presented. The

results under deviatoric loading conditions are presented and discussed in section 4.4.



4.1 Introduction



Forming processes are crucial in the ceramics industry today. Among various forming

processes, casting is commonly used for the forming of products. One of the casting

processes is pressure casting. This process has been investigated as a forming technique for

porcelain, complex refractory shapes or hard ferrite magnets, etc. The casting time is usually

controlled by regulation of the external pressure (Reed [1988]). In pressure casting, the mold

serves as a filter. The externally applied pressure is usually increased up to 1.5 MPa. Water

in the cast is squeezed out. This process reduces the water content significantly so that the

drying shrinkage can be reduced. However, pressure casting of slurries or pastes in a mold

creates density gradients due to the cast geometry or the friction between the materials and

the wall of the mold. In other words, the density distribution is not uniform in the cast. This

non-uniform density distribution will cause problems such as warping in the later sintering

process. In order to understand the process and avoid non-uniform density distribution in a

mold, the consolidation behavior, or stress strain relationships, of slurries or pastes must be

investigated.

Slurry or paste of alumina powder fully saturated with water is often used for pressure

casting. Therefore, the consolidation behavior of alumina paste plays an important role in

pressure casting. Whether or not we can get good products from this kind of casting depends








76

on how well the consolidation behavior of powder or powder paste is understood. To

understand the consolidation behavior of alumina paste at different stress states, we need to

know the three dimensional stress strain relationships, i.e., constitutive equations, of these

powders. To formulate constitutive equations, triaxial tests are often used.

Triaxial tests have been successfully used in civil engineering for geomaterials such

as sand, clay, and rock to obtain general three dimensional constitutive equations (Bishop &

Henkel [1962]; Cristescu [1989]). However, powders such as alumina powder or alumina

paste are different from sand or clay. Usually powder possesses more pure constituents and

surface chemistry of particles is also involved (Reed [1988]). Hence the mechanical

properties of powders cannot be obtained directly from those of sand, clay, or rock. Several

triaxial tests for metal powders or dry ceramic powders at high pressure were reported

(Shima & Mimura [1986]; Brown & Abou-Chedid [1994]; Gurson & Yuan [1995]).

Unfortunately, very few triaxial tests have been performed for ceramic paste such as alumina

powder (Stanley-Wood [1988]). Triaxial tests are quite different from uniaxial tests. In a

uniaxial confined test, powder is placed into a rigid cylindrical mold or die and pressure is

applied along axial direction using a piston. Such tests reveal mainly one dimensional

relationship between pressure and volume changes (Gethin et al. [1994]; Chen et al. [1994];

Shapiro [1995]). However, due to the friction between the powder and the wall of a mold,

there exist very complicated stress states in the mold and the final density distribution is

nonuniform. Consequently, uniaxial tests cannot provide good basis for the formulation of

three-dimensional relationship between stress and strain. On the contrary, triaxial tests reveal

the deformability characteristics not only under hydrostatic pressures, but also under










deviatoric loading conditions (e.g., shearing under different constant confining pressures or

under constant mean stress conditions). The stress-strain relationships obtained in deviatoric

tests reveal the three-dimensional consolidation behavior. These tests furnish necessary data

for the formulation of the three dimensional constitutive equations (Cristescu [1994]; Gurson

& Yuan [1995]).

In this chapter, triaxial tests are performed to investigate the volume change behavior

of alumina powder under different stress states. The effects of particle size, initial density,

and time are studied. A series of triaxial tests on three types of alumina powders provided

by ALCOA (Aluminum Company Of American) are performed. The first is as-received A10

alumina powder with an initial volume fraction of 0.36; this type will be labeled as "dense

A10". The second one is also A10 alumina powder, but with a smaller prepared initial

volume fraction of 0.334, labeled as "loose A10". The third one is as-received A16-SG

alumina powder with a prepared initial volume fraction of 0.41. The specifications of these

materials are shown in Table 4.1. The elastic parameters were measured for all three types

of alumina powders based on the method proposed by Cristescu [1989] for time-dependent

materials. The triaxial test results show that the consolidation behavior of alumina powder

is strongly dependent on particle size, initial density, and time. In all cases a higher pressure

results in more consolidation. However, deviatoric stress can lead to either consolidation or

dilatancy. For the stress ranges applied to the A16-SG alumina powder, no dilatancy was

observed, while dilatancy occurred for both A10 alumina powders. The

compressibility/dilatancy boundary and the failure condition have also been measured. These

data can be used to formulate a general visco-elastic or elastic/viscoplastic model to describe








78
the consolidation behavior of these alumina powders. In the present research, a total of thirty-

three triaxial tests have been performed. These tests are listed in Table 4.2 and Table 4.3.

Most of the data are given in Appendices B, C and D.



Table 4.1 Specification of Alumina Powders

type of prepared initial density' Particle relative void ratio'
alumina (g/cm3) size(pm) density"

A10 (dense) 1.42 (two layer+vacuum) 40-200 0.36 1.78

A10 (loose) 1.32 (pluviation+vacuum) 40-200 0.334 1.99

A16-SG 1.62 (tap+vacuum) 0.5-1.0 0.41 1.44

SThe density after saturation.
"Theoretical Density of Alumina=3.95 g/cm3 (see Richerson [1982])
+ void ratio = V/V,,
where Vv is volume of void, Vs is volume of solid material in the specimen.


4.2 Experimental Procedures



4.2.1 Triaxial Equipment Setup

Fig. 4.1 shows the schematic diagram of triaxial equipment used (Bishop & Henkel


[1962]).









Table 4.2 List of triaxial tests for alumina powder (dense and loose A10)


No. date type initial confining "B" loading Number Number failure C/D comments
1995 of test density pressure value rate unloading unloading stress stress
-1996 (g/cm3) (kg/cm') (s") on hydro. on deviat. (kg/cm2) (kg/cm2)
1 2/14-17 CT 1.466 3=(7-4) 5.6e-5 no 1 9.28 5.2-5.6 Dry, try test

2 3/7-8 CT 1.388 3=(6-3) 0.93 5.6e-5 2 1 7.02 5-5.5 not max stress

3 3/13-16 CT 1.424 3=(6-3) 0.95 5.6e-5 1(8steps) 1 8.6 4.5-5

4 4/11-13 CT 1.424 3=(7-4) 0.97 l.le-5 4(7steps) 3 8.99 5-5.2

5 4/17-18 CT 1.424 2=(6-4) 0.96 1.1e-5 4(7steps) 6 6.02 2.75-3 unload too much

6' 4/19-20 CT 1.423 2=(6-4) 0.94 1.le-5 3(4steps) 5 5.99 3-3.2

7' 4/25-26 CT 1.423 1=(5-4) 0.97 1.le-5 1 2 3.11 1-1.2

8 5/1-2 CT 1.421 4=(7-3) 0.97 l.le-5 7(8steps) 5 10.93 8.3-8.5 not max stress

9" 5/3-8 CT 1.42 4=(5.5-1.5) 0.94 1.le-5 3 6 11.38 7.8-8 wrong on hydro

10' 5/10-11 CT 1.428 5=(6.5-1.5) 0.945 l.le-5 5 6 14.41 10.5-10.8

11' 6/27-29 CT 1.415 5=(6.5-1.5) 0.94 .le-5 5 5 13.79 10.6-10.9

12 7/3-5 CT 1.425 5=(6.5-1.5) 0.94 1.le-5 5 0 14.28 10.6-10.9

13' 7/18-19 CT 1.422 3=(4.5-1.5) 0.95 1.le-5 5 3 8.76 5.0-5.2

14' 8/2-3 CT 1.415 4=(5.5-1.5) 0.95 1.le-5 4 0 11.42 7.0-7.2

15" 8/8-9 CT 1.325 4=(5.5-1.5) 0.95 3.4e-5 4 2 9.58 9.3 pluviation test








Table 4.2 Continued

No. date type initial confining "B" loading Number Number failure C/D comments
1995 of test density pressure value rate unloading unloading stress stress
-1996 (g/cm3) (kg/cm2) (s") on hydro on deviat (kg/cm2) (kg/cm')

16" 8/16-17 CT 1.338 5=(6.5-1.5) 0.96 3.4e-5 5 3 12.23 12.1 pluviation test

17" 10/25-26 CT 1.316 3=(4.5-1.5) 0.97 3.4e-5 3 3 7.04 6.85 pluviation test

18" 10/30-11/1 CT 1.325 1=(2.5-1.5) 0.97 3.4e-5 1 1 2.46 2.1 pluviation test

19" 11/7-8 CT 1.313 2=(3.5-1.5) 0.96 3.4e-5 2 2 4.75 4.3-4.4 pluviation test

20 11/14-15 CT 1.412 3.5=(5-1.5) 0.95 l.le-5 4 2 9.64 6-6.1 chamber leak

21 12/12-13 CT 1.44 3.5=(5-1.5) 0.93 l.le-5 N/A N/A too large density

22" 12/19-20 CT 1.421 3.5=(5-1.5) 0.96 l.le-5 4 2 10.17 6-6.1

23' 12/27-28 RL 1.425 4=(5.5-1.5) 0.96 relax (2 steps) (8 steps) 10.8(min) 7.0-7.2 12.6 (max stress)

24 1/3-4 CT 1.411 4=(5.5-1.5) 0.97 5.4e-5 4 0 11.34 7.3-7.5

25' 1/8-9 CT 1.42 4=(5.5-1.5) 0.96 3.3e-4 4 0 11.77 8-8.1 very small dilat.

26' 1/11-12 CP 1.419 4=(5.5-1.5) 0.96 (1 step) 0 5.98 1.6 no effect on fail


Note: CT=conventional triaxial test (Kgnrmfn test); RL=relaxation test; CP=constant mean stress test
Confining pressure = (chamber pressure back pressure)

* The tests (total 11) have been listed in Appendix B
SThe tests (total 5) have been listed in Appendix C










Table 4.3 List of triaxial tests for Alumina powder A16-SG

No. date type initial confining-p "B" loading Number Number failure C/D comments
1996 of test density (cham.-back) value rate unloading unloading stress stress
(g/cm3) (kg/cm2) (s ") on hydro on deviat (kg/cm2) (kg/cm')

1' 1/17-18 CT 1.632 4=(5.5-1.5) 0.96 3.4e-5 4 I 11.88 N/A compress.-only

2* 2/16-18 CT 1.604 2=(3.5-1.5) 0.95 3.4e-5 2 2 5.65 N/A compress.-only

3* 3/6-7 CT 1.635 3=(4.5-1.5) 0.96 3.4e-5 3 3 8.79 N/A compress.-only

4' 3/11-12 CT 1.628 5=(6.5-1.5) 0.95 3.4e-5 5 2 14.83 N/A compress.-only

5' 3/13-14 CT 1.623 1=(2.5-1.5) 0.95 3.4e-5 1 1 2.97 N/A compress.-only

6* 3/26-27 CP 1.634 3=(4.5-1.5) 0.95 3.4e-5 (1 step) 0 4.34 N/A compress.-only

7 4/8-10 CP 1.646 4=(5.5-1.5) 0.98 3.4e-5 (1 step) 0 5.66 N/A compress.-only


Note: CT=conventional triaxial test (KirmAn test); CP=constant mean stress test
Confining pressure = (chamber pressure back pressure)

* the tests (total 6) have been listed in Appendix D















vacuum/backpressure
burette


air pressure dial
(Chamber pressure)

displacement
- measurement


Fig. 4.1. Schematic diagram of triaxial equipment setup.



The equipment consists of primarily a chamber, framework, panel, and base motor.

There are several valves to control the flow of water and/or air. Two inlets/outlets at the

bottom and the top of the specimen are connected to a vacuum or/and a burette on the panel.

The following quantities have been measured: axial force by load cell, axial displacement

of specimen by digital gauge, chamber pressure and backpressure by pressure transducers

respectively as shown in Fig. 4.1. Chamber pressure can provide an all-around pressure on

the top and the lateral surface of specimen. Back pressure acting as pore pressure can provide

the pressure of fluid in the specimen.










4.2.2 Specimen Preparation

The specimen is prepared inside a split mold. The split mold consists of two half

cylindrical shells which are connected with a vacuum pump. First, two clamps are put to hold

the mold. A membrane of about 0.12 mm thickness is put inside of the mold and the ends of

the membrane are tightened on the mold by several O-rings. A porous stone is placed at the

bottom of the mold. The mold is then put on the pedestal which connects all necessary fluid

inputs/outputs. A vacuum is applied to pull the membrane to the mold sides. There are two

methods to prepare the specimen in order to obtain a desired initial density. One is the so-

called layer method. The other one is called pluviation method. In the layer method, the

specimen is prepared in the successive layers. For each layer, a constant density is maintained

by certain height with a given amount of powder material. A small vibration is applied by

taping the side of the mold for densification. The maximum variation in relative density

between specimens can be kept within 0.5%. In the pluviation method, powder is poured into

the mold slowly without tapping the mold until the mold is full with powder. After the mold

is filled with powder, the top porous stone and cap are placed on the specimen. The vacuum

pump is disconnected from the mold and then applied to the top and bottom inlets of the

specimen. The mold is then removed. The specimen is supported by about 100 kPa vacuum

pressure. The average diameter Do of the specimen is obtained by measuring the diameter on

the top, middle, and bottom. The average initial height of the specimen, H0, can also be

obtained by measuring heights at four different points. By knowing the theoretical material

density y, of powder, weight of specimen W (powder is assumed dry), and the volume of the

specimen V, the volume fraction of the powder can be computed:










Y W
relative density=volume fraction (4.1)
Yt Vy, (4)

Afterwards the prepared specimen is put into the chamber.

A cylindrical specimen, about 70 mm in diameter and 145 mm in height, is used for

all tests. It is assumed that the specimens deform uniformly and the end effect is neglected.

Since the particle size is less than 0.2 mm, the membrane penetration effect is also neglected

(Frydman [1973]).



4.2.3 Back Pressure Saturation

The next step is to saturate the specimen. First, water is deaired in the burette for

about 20 minutes. Then water is allowed to flow into the bottom of specimen due to vacuum

pressure existing in the specimen. An alternating sequence of vacuuming the top of specimen

and allowing water in from the bottom is followed until most of the trapped air is removed.

A small confining pressure (03=30 kPa) is maintained during saturation to support the

specimen with a certain back pressure in the specimen. A "B" check is performed to verify

the saturation. Here, "B" is defined by B:=Au/Ao3, in which Ao, is the increment of chamber


pressure and Au is the increment of the pore water pressure in the specimen with undrained

conditions. During the tests, the volume of specimen changes. Saturation is essential as the

volume change is measured via the water volume entering/exiting the sample. Usually the

"B" value should be higher than 90% when the volumetric strain is to be measured under

drained conditions. In order to achieve a high accuracy of the measurement of volumetric

strain, the "B" value is required to be as large as possible. In our tests, municipal tap water








85

is used. In order to get higher B values, the prepared specimen is subjected to a backpressure

of 147 kPa over night. The B valve measured the next day is higher than or at least equal to

93% in all tests. See Table 4.2.

The change of the height of specimen, AH, during saturation is measured by attaching

the dial gage to the piston rod before saturation begins, and noting zero. Subsequently, the

small confining pressure (30 kPa) applied during saturation and the back pressure saturation

will cause volume reduction of the specimen. Since AD (the change of the diameter of

specimen) cannot be measured while AH can, it is assumed that AH / H0= -AD / D (axial


and radial strains are equal). The corrected section area A, after saturation is thus obtained,


Ho*,+2AH
A,=A o =A[1+2e] (4.2)
Ho



where e is axial strain after saturation.



4.2.4 Hydrostatic Loading Test

We begin with an incremental hydrostatic loading test (also called consolidation test

or isostatic test). Hydrostatic stress is computed as the chamber pressure minus the

backpressure. At each loading step, the stress is kept constant for several minutes and the

sample is allowed to shrink by creep until the rate of the volumetric strain is very small.

Usually it takes ten minutes. Afterwards the specimen is unloaded with small stress

increments and the stress is kept constant for 5 minutes. Then the specimen is reloaded to

reach the previous stress level. In this way accurate values of the elastic bulk modulus can








86

be calculated (Cristescu [1989]). From such tests, the volumetric change of specimen due to

both hydrostatic stress and, for a given stress level, time can be obtained. Since the

hydrostatic pressure causes specimen compression, the section area of the specimen is

corrected according to the following formula


H(V -Ac)


where: Vo is initial volume of specimen, AVc is volume change of the specimen during


consolidation at each step, AV,,=3VoAH / I H is the volume change of the specimen during


saturation, and AH is the change in height of the specimen at each loading step plus the

change during saturation.



4.2.5 Deviatoric Loading Test

The second stage of the triaxial test is the deviatoric loading test. There are two ways

to perform deviatoric tests. One is called "conventional triaxial test". In the deviatoric stage

of conventional triaxial tests, the confining pressure (lateral stress) is kept constant and only

the axial stress is increased. The confining pressure is the difference between chamber

pressure and back-pressure. The deviatoric stress is defined as the difference of stress

between the axial stress and the lateral stress. The other is called "constant mean stress test"

or "constant-p test". In the deviatoric stage of constant mean stress tests, the mean stress is

kept constant and only the deviatoric stress is increased.








87

Loading was performed at a constant displacement rate and the following quantities

in all deviatoric tests were measured: axial force, chamber pressure, back pressure, the length

change of specimen, and the volume change of specimen. The axial stress, confining

pressure, axial strain, and lateral strain are computed from the measured quantities. During

the test, the section area of cylindrical specimen changes. The following correction is applied

during tests:


V-AV I -E
A H l-A e ,l (4.4)




where VcH and Acare volume, height, and section area of specimen, respectively after the


hydrostatic test, AV is the volume change, AH is the change of the height during deviatoric

test, Ae, = AV/Vc, and e =AH /H. The deviatoric stress is defined as


P
O d (4.5)


where P is the axial load. Eqn (4.4) is also suitable for area correction for other type of tests

such as constant mean stress test or relaxation test.










4.3 Experimental Results



4.3.1 Hydrostatic Tests

Three series of hydrostatic tests (first stage of a triaxial test) on alumina powders

provided by ALCOA have been performed (see Table 4.2 and Appendices B, C and D). The

results of these hydrostatic tests are presented and discussed below.

The hydrostatic tests were performed with an unloading-reloading process in order

to obtain the elastic parameters. Fig. 4.2a shows a typical relationship between volumetric

strain and time in a hydrostatic test for dense A-10 alumina powder at different stress levels,

while Figs. 4.2b and 4.2c show similar results for loose A10 and A16-SG, respectively. At

each plateau the number shown marks the pressure applied. For each plateau shown, the

successive points correspond to 1.0, 2.0, 5.0, and 10.0 minutes (a total of about 10 minutes)

after each reloading. Afterwards an unloading increment of 20 kPa pressure is performed and

the remaining stress is kept constant for 5 minutes, followed by a reloading with the same

20 kPa, and then the final stress same as the previous one is kept constant for 5 minutes. The

cycle of unloading-reloading curve allows us to calculate the elastic bulk modulus. It was

found that A16-SG exhibits much more volumetric creep deformation than A10's for the

same testing time interval and same pressure, i.e., the A16-SG alumina is much more

compressible than the A 10's. After each loading, the volumetric strain continues to vary in

time (creep) as shown in Figs 4.2a-4.2c. The volumetric strain for A16-SG exceeds 6%,

while the volumetric strain for dense A10 and for loose A10 is only about 0.6 % and 1.1 %

under the same conditions (pressure 490 kPa), although A16-SG is initially more dense than








89
A 10's. This difference in compressibility is mainly due to the difference in particle size of

the powders. The particle size of A10 alumina is in the range of 40-200pm, while the

particle size of A16-SG is in the range of 0.4-lim. The particle shape of both alumina

powders is irregular. However, due to the different particle size, the surface area of particle

in these two powders is also different. The surface chemical properties of particle are

certainly involved, which will affect mechanical properties. This aspect should be

investigated separately.

As already mentioned, the unloading processes are used to obtain the elastic

parameters. The unloading processes also allow us to separate the time effect from elastic

properties. After each loading, the specimen is allowed to creep under constant pressure for

10 minutes. When the rate of volumetric strain becomes very small, the unloading is

performed. During the short time necessary to perform the unloading, the contribution of the

deformation by creep is negligible since then the material has reached nearly a quasi-static

state (no much strain can be obtained with the same stress afterwards as shown in Fig.4.3).

Thus, the influence of the time effect on the unloading process is negligible. The elastic bulk

modulus can be obtained from these unloading processes, as shown in Fig. 4.3, as the slope

of the unloading curve. The bulk modulus obtained by following this procedure is very close

to that obtained by using dynamic method for other materials such as rocks (Cristescu

[1989]). It is expected that for alumina powder the value obtained by the unloading processes

is also close to that obtained by dynamic method.












0.006

Sf 490


392
0.004

294


0.002 = 196 (kPa) dense AI0


98


0 I I
0 0.5 1 1.5 2 2.5

t(h)


Fig. 4.2a Volumetric strain versus time for dense A10 (points-reading data).




0.012

490
0.01

392
0.008


0.006
0.006 a =294(kPa)


0.004 196 loose A10


0.002 98


0 .... i, I. ..1 .liiiI
0 0.5 1 1.5 2 2.5 3
t (h)


Fig. 4.2b Same as 4.2a but for loose A10 (points-reading data).











0.07

0.06
490
0.05
392
0.04 -
294
0.03

0.02 = 196 (kPa)

0.01

0
98

0 0.5 1 1.5 2 2.5 3 3.5
t (h)


Fig. 4.2c Volumetric strain versus time for A16-SG (points-reading data).




It was found that elastic bulk moduli change smoothly with mean stress as shown in

Fig. 4.3 and Fig. 4.4. The elastic bulk moduli may also depend on other material properties

such as the density (Gurson & Yuan [1995]). As a first approximation, it is assumed that the

elastic bulk modulus depends on mean stress only. There are several possible functions

which could be used to fit the data if the behaviors of the elastic moduli at very high pressure

or/and very low pressure are also taken into account (Jin et al. [1996a]). Here, a linear

approximation is used to fit these data. It gives a reasonable fitting of the data in the pressure

interval shown in Fig. 4.4a and 4.4b. The bulk modulus K can be expressed as


K(o)= ko+ kjo (4.6)



where ko and k, are material constants. For comparison, the values of the coefficients k, and








92

k, for different types of alumina powders are listed in Table 4.4. It is seen that the denser


material has larger k,. However, the linear relationship (4.6) is not expected to be valid at

high pressure since K is expected to reach a limiting value when all pores are closed at very

high pressures.







500

(kPa)
400


300


200
dense A10

100


0
0 0.002 0.004 0.006


Fig. 4.3a Hydrostatic stress versus volumetric strain in hydrostatic
loading for dense A10 (points-recorded data).




Full Text

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ELASTICNISCOPLASTIC MODELS FOR GEOMATERIALS AND POWDER-LIKE MATERIALS WITH APPLICATIONS By JISHAN JIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996

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To My Family, In Memory of My Father

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ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my supervisory committee chairman, Graduate Research Professor Nicolae D. Cristescu, for his guidance, tremendous support, and encouragement during the study. I also sincerely appreciate Professor Cristescu's patience, kindness, and discussions to his students. I am deeply indebted to Professor Renwei Mei for helpful discussions on the numerical analysis aspects during the work and for the critical reading of the early draft of this dissertation. I am very grateful to Professor Frank C. Townsend for his guidance and helpful discussions on the experimental aspects. I would like to thank Professors Bhavani V. Sankar and Edward K. Walsh for their support and for serving on the committee. Many thanks also go to my wife, Qianhong, for her understanding and full support, and to my family members for their endless support in my life. Finally, the fmancial support from the Engineering Research Center at University of Florida, Particle Science and Technology, is gratefully acknowledged Ill

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TABLE OF CONTENTS ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . 111 ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vll CHAPTERS 1 IN"TRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . I 1 1. 1 Rock Salt . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Powder Materials . . . . . . . . . . . . . . . . . . . 4 1.1.3 Models . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Mohr-Coulomb Model and Drucker-Prager Model . . . . . . 8 1.2.2 Cap Model . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Cam-Clay Model . . . . . . . . . . . . . . . . . . . 13 1.2.4 Lade Model . . . . . . . . . . . . . . . . . . . . . 16 1.2.5 Endochronic Model . . . . . . . . . . . . . . . . . . 17 1.2.6 Models for Compaction (Consolidation) of Powders . . . . . 18 1.2.7 The ElasticNiscoplastic Models . . . . . . . . . . . . . 21 1.2.8 Other Models . . . . . . . . . . . . . . . . . . . . 24 1.2.9 Remark on Recent Development of Plasticity-Based Models . 25 1.2.10 Finite Element Method . . . . . . . . . . . . . . . . 25 1.3 Objective and Outline of the Present Work . . . . . . . . . . . . 27 2 THE ELASTIC/VISCOPLASTIC (CRISTESCU) THEORY . . . . . . . . 31 2.1 The ElasticNiscoplastic Constitutive Equation . . . . . . . . . . . 31 2.2 Determination of the ElasticNiscoplastic Constitutive Equation . . . . 34 2.2.1 Elastic Parameters and Viscosity Coefficient . . . . . . . . 34 2.2 2 Yield Function . . . . . . . . . . . . . . . . . . . 37 2.2.3 Viscoplastic Potential . . . . . . . . . . . . . . . . . 38 2.3 Loading Unloading Conditions . . . . . . . . . . . . . . . . . 39 3 AN ELASTICNISCOPLASTIC MODEL FOR TRANSIENT CREEP OF ROCK SALT . . . . . . . . . . . . . . . .. . . . . . . . . . . 40 IV

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3 1 Determination of the ElasticNiscoplastic Constitutive Model . . . . . 41 3 1.1 Determination of the Elastic Parameters . . . . . . . . . . 41 3 1 2 Determination of the Compre ss ibility/Dilatancy Boundary and the Failure Condition . . . . . . . . . . . . . . . 43 3.1 3 Detertnination of the Yield Function . . . . . . . . . . . 45 3 .1 4 Deter1nination of the Vi s copla s tic Potential . . . . . . . . 50 3 2 Compari s on with the Data . . . . . . . . . . . . . . . . . . . 58 3 3 The Finite Element Analy s is . . . . . . . . . . . . . . . . . . 63 3 3 1 Formulation of the Ela s ticNi s coplastic Theory . . . . . . . 63 3 3 2 Example I : Axial Compression with Confining Pressure . . . 65 3.3 3 Example II Stre ss Analysis for A Cylindrical Cavity in Ro c k Salt . . . . . . . . . . . . . . . . . . . . . . . 67 3 .4 Dis c u ss ion and Conclusion . . . . . . . . . . . . . . . . . . 72 4 T ~A.-11..
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6.2 ElasticNiscoplastic Constitutive Model . . . . . . . . . . . . . 151 6.2.1 ElasticNiscoplastic Constitutive Equations . . . . . . . . 151 6.2.2 Loading-Unloading Conditions . . . . . . . . . . . . . 153 6.3 Determination of the Elastic/Viscoplastic Model for Alumina Powder Al6-SG . . . . . . . . . . . . . . . . . . . . . . . . 155 6.3.1 Elastic Parameters . . . . . . . . . . . . . . . . . . 155 6. 3 .2 Yield Surf ace . . . . . . . . . . . . . . . . . . . 15 5 6.3.3 Irreversible Strain Rate Orientation Tensor . . . . . . . . 162 6.3.4 Viscosity Coefficient . . . . . . . . . . . . . . . . . 167 6.3.5 Remark . . . . . . . . . . . . . . . . . . . . . . 168 6.4 Validation of the Model . . . . . . . . . . . . . . . . . . . 169 6.4.1 Creep Type For1nula with Stepwise Stress Variations . . . . 169 6.4.2 Conventional Triaxial Tests . . . . . . . . . . . . . . 170 6.4.3 Constant Mean Stress Tests . . . . . . . . . . . . . . 173 6.4.4 Hydrostatic Creep Tests . . . . . . . . . . . . . . . 175 6.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . 175 7 CONCLUSION AND WORK .............................. 179 179 180 7 .1 Conclusion ................................................ 7 .2 Future Work ............................................... APPENDICES A THE FORMULATION OF ELASTICMSCOPLASTICITY FOR THE FINfl'E ELEMENT METH OD . . . . . . . . . . . . . . . . . . 181 B T ,._~ .LLJ EXPERIMENT AL RESULTS FOR ALUMINA POWDER AlO (DENSE) . . . . . . . . . . . . . . . . . . . . . . . . . . 191 C T "'-Lt u.... EXPERIMENTAL RESULTS FOR ALUMINA POWDER AlO (LOOSE) . . . . . . . . . . . . . . . . . . . . . . . . . . 228 D .LLJ EXPERIMENTAL RESULTS FOR ALUMINA POWDER A16248 RE.FERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . 281 VI

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ELASTICNISCOPLASTIC MODELS FOR GEOMATERIALS AND POWDER-LIKE MATERIALS WITH APPLICATIONS By JISHAN JIN December 1996 Chairperson: Graduate Research Professor Nicolae D. Cristescu Major Department: Aerospace Engineering, Mechanics and Engineering Science An elastic/viscoplastic model with a nonassociated flow rule for transient creep of rock salt is f or1nulated based on a set of triaxial tests. A new procedure to determine the yield surfaces and the potential surfaces is proposed. The asymptotic behavior and the physical grounds of yield surf aces and of potential surf aces are incorporated into the model so that these surfaces are smooth and the model better matches the data. This type of model with both yield surfaces and potential surfaces is implemented for the first time into a finite element program. It is shown that the prediction of the model matches quite well the experimental data. The mechanical behavior of alumina powders is investigated experimentally. Three series of triaxial tests are perfortned. It is shown that the mechanical ( compaction) behavior of alumina powders is strongly dependent on the particle size and on the initial density. For Vll

PAGE 8

the powder A16-SG with smaller particle size (0 4~ lm) the volumetric strain alway s exhibits compressibility, while for the powder AlO with larger particle size (40~200m), the volumetric strain exhibits both compressibility and dilatancy. An elastic/viscoplastic model with a nonassociated flow rule to describe the nonlinear time-dependent behavior of alumina powder A 10 is formulated based on five conventional triaxial tests. By using such type of tests, a new procedure to determine the yield surfaces and the potential surfaces is proposed. The model is checked against the data. A good agreement between the model prediction and the data is obtained. A new methodology is proposed to determine the elastic/viscoplastic model for compressible materials. In such models, the irreversible volumetric strain is chosen as a work hardening parameter. A model to describe the behavior of alumina powder A16-SG i s f orn1ulated. The irreversible strain rate orientation tensor is determined. The model is easier to handle and has fewer parameters than the previous ones. A very good agreement between the data and the model prediction is obtained Vlll

PAGE 9

CHAPTER 1 INTRODUCTION 1.1 Background The formulation of general constitutive equations of three-dimensional models for time dependent materials has been studied since the pioneering works by Maxwell in 1868 and by Kelvin in 1875 (see Malvern [1969]). Many materials possess time dependent properties: for instance, polymers exhibit viscoelastic properties; a rock flows in a long period of time although it seems strong and hard; a metal creeps at high temperature, etc. A great number of contributions have been made in this field (see Perzyna [1966]; Cristescu [1967]; Christensen [1982]; Cristescu & Suliciu [1982]; Phillips [1986]; Cristescu [1989]; Chaboche [1989]; Lubliner [1990]). Recently three-dimensional modeling for rock salt and powder materials has attracted the attention of many researchers, due to the demand of various engineering industries (see Hardy & Langer [1984, 1988]; Chenot et al. [1990] ; Jinka & Lewis [1994]; Krishnaswami & Trasorras [1995]; Xu & McMeeking (1995]; Aubertin (1996]; Tszeng & Wu (1996]). They wish to develop general constitutive equations or modify the existing ones which, on the one hand, can be used to describe better the behavior of materials, and on the other hand, can be incorporated into the finite element method. Further, the models will be used for optimal designs or/and safety evaluations. 1

PAGE 10

2 1.1.1 Rock Salt Rock salt exhibits time dependent properties as revealed by creep, relaxation and rate dependent tests (Hansen & Carter [1984]; Handin et al. [1986]). It is found under natural underground conditions. Rock salt consists mainly of halite with little anhydrides and clay, and has small permeability. Due to the rheological behavior and the physical properties of rock salt, the large underground cavities in mass of rock salt are considered as ideal places to store radioactive waste or hazardous chemical waste Many salt cavities were excavated around the world for this purpose (See Hu gout [ 1988] in France; Eekelen [ 1988] in Netherlands; Matalucci & Munson [1988] in USA; Kappei & Gessler [1988] in Germany). The deforrr1ations of those cavities are continuously recorded. The stability and the safety of those large underground cavities are being evaluated. Since the stability and the safety are not precisely predicted nowadays millions of extra dollars are spent on maintaining those cavities in a workable state. Thus it is highly necessary to forn1ulate a general constitutive equation in order to predict the mechanical behavior of rock salt and to evaluate the safety of cavities Also a general constitutive equation would be beneficial, not only for the evaluation of the stability and the s afety of large underground cavities, but also for other rock salt mining engineering problems such as the improvement of salt mining design procedures (Rolnik [1988]). Most of the research done so far was devoted to the long term behavior of rock salt, i.e., stationary creep A variety of models have been proposed (e.g. Langer [1988] ; Chan et al [1994]; Aubertin et al. [1996]). Most of them can be used to predict the long ter1n behavior of rock salt by neglecting the deformation in the transient creep phase However,

PAGE 11

3 occasionally, a great discrepancy is found between the measured creep defor1nation of underground cavities or openings and that predicted by the models (Aubertin et al. [1993]). It is believed that this discrepancy is largely due to the neglect of the transient creep deformation, because the stress around an opening ( or cavity) is redistributed during the excavation of the opening in the transient creep phase The redistributed stresses will influence the stationary creep of rock salt around the opening. Therefore it is crucial to formulate correctly the constitutive equations for transient creep, if one wishes to describe the behavior of rock salt in the rock surrounding the excavation during the stationary creep. Several models for transient creep of rock s alt have been for1nulated (e.g., Langer [1984] ; Munson & Dawson [1984]; Desai & Varadarajan [1987]; Aubertin et al. [1993]). In most cases one s tarts from empirical fo11nulae using a viscoelastic or viscoplastic model with Drucker-Prager type of yield surface or internal state variable concepts. However, these models either do not give good matching with the experimental data or could hardly be incorporated into a finite element program Thus these models should be judged from the point of view of how well they match the experimental data and how easily they can be incorporated into a finite element program In the present thesis, we start from a set of true triaxial tests on rock salt and follow a procedure to forrr1ulate a general three-dimensional elastic/viscoplastic constitutive equation for the transient creep phase. No empirical fo11nulae are used since empirical formulae may not be adequate for other cases. Nonassociated flow rule is explored and the viscoplastic potential is determined based on the experimental data The model not only matches better the experimental results, but can also be incorporated easily into the finite element method. The model will be presented in Chapter 3.

PAGE 12

4 1.1.2 Powder Materials The mechanical behavior of powder materials has been studied for a long time, since powder materials are used in many industrial applications such as metal powder metallurgy (Johnson [1992a, 1992b]), ceramics injection molding (Mutsuddy & Ford [1995]) or food products (Puri et al. [1995]). Powder materials such as metal powder or ceramics powder, by some compaction or/and sintering processes, become useful products with very dense and hard properties. However, in shape for1ning processes such as uniaxial compressing, ceramics injection molding or pressure casting, the density distribution in the product is nonuniform due to the friction between powder and the wall of a die. The nonuniform density distribution would cause warping or other defects in the product in a later sintering process. These warpings or defects will damage the products and unqualify the products (Reed [1988]). Nowadays, most shape forming techniques are the ad hoc ones. Due to the lack of the correct prediction of density distribution in a mold, many products are not qualified, and millions of dollars are wasted each year (Johnson [1992b]). Thus it is crucial to understand the mechanical behavior of powder materials during these processes and it is urgent for designers to simulate these shape for1ning processes and to predict the density distribution in a mold. In order to predict the density distribution of powder in a mold, and to simulate the shape for11ling process, the three-dimensional stress strain relationships, i.e., the constitutive equations for powders have to be correctly and precisely formulated. Triaxial tests are often used to formulate constitutive equations (Desai & Siriwardane [1984]; Cristescu [1989]). Unfortunately only very limited triaxial tests have been reported (Koerner & Quiros [1971];

PAGE 13

5 Sruma & Mimura [1986]; Brown & Abou-Chedid [1994]; Gurson & Yuan [1995]) for different applications. Without complete experimental data, it is difficult to formulate constitutive equations correctly. In order to understand the compaction behavior of powder materials, to f orrnulate a general three-dimensional model, or/and to examine the existing models, additional triaxial tests are needed. Alumina powder (Al 2 0 3 ) has been widely used in ceramic engineering for various applications due to the specific properties of alumina such as high electrical resistivity, high erosion resistance, high melting temperature and high abrasiveness etc. For instance, alumina powder can be used in the manufacture of porcelain, alumina laboratory ware, wear-resistant parts, sandblast nozzles, medical components, abrasives and refractories etc. Hundreds of tons of alumina powder and alumina-based articles are produced each year (Richerson [1982]) However, very few experiments have been performed for alumina powders regarding to the mechanical (compaction) behavior in three-dimensional stress states. Only some tests in uniaxial conditions have been reported for ceramics powders (e.g., Stanley Wood [1988); Gethin et al [1994); Chen et al. [1994]) Unfortunately, uniaxial tests reveal mainly one-dimensional relationship between pressure and volume changes. Such tests are unable to provide the necessary data to formulate three-dimensional constitutive equations On the contrary, triaxial tests reveal the deformation characteristics in three dimensions under both hydrostatic pressure and deviatoric loading conditions Such tests can furnish the necessary data for the for111ulation of the general constitutive equations. In the present thesis, a series of triaxial tests on alumina powders have been performed Many interesting results have been found The experimental results will be presented in Chapter 4.

PAGE 14

6 1.1.3 Models There are some geomaterials-type models to describe the three-dimensional behavior of frictional materials such as the Mohr-Coulomb, Drucker-Prager, Cam-Clay, Cap and Lade models etc. These models are developed based on mainly the experiments on sand and clay. In some models, the yield surfaces are a priori assumed and the associated flow rule is used. In some other models, the non-associated flow rule is used and the irreversible strain potential is assumed (see the following sections on the review of models). Most models are time independent ones. It is difficult to use these model to describe the behavior of rock salt (Langer [ 1988]). These models have a common disadvantage: they may not predict the volumetric dilatancy behavior correctly even though some models predict the volumetric compressibility pretty well (e.g., Desai & Siriwardane [1984] for artificial soil). For example, the Cap model was modified and many additional constants were introduced in the model by Chen and Baladi [1985]. The model still does not predict the volumetric dilatancy precisely although the prediction matches well the other data. The Drucker-Prager model overpredicts the dilatancy behavior The Cam-clay model does not predict the dilatancy behavior until the materials reach the failure stress and become softening. One reason for these weak aspects is that the assumed yield surface is appropriate for some materials, but it may not be good for other materials. Another reason is the skepticism of the nortnality in the associated flow rule Many experimental results do show that the behaviors of geomaterials do not follow the norrr1ality conditions (Maier & Hueckel [ 1979]; Kim & Lade [1988]; Jin et al. [1991]; Anandarajah et al. [1995]). The non-associated flow rule has been explored by several researchers (Lade & Kim [1989a, 1989b]; Desai & Hashmi [1989]).

PAGE 15

7 However, since in their models the potential surfaces are assumed in a priori form for some materials, it is difficult to use these models for other type of materials. It is advisable to examine the models careful] y before using them. There are also several models for the compaction of powders. Usually, the elliptical yield curves or other type of curve in the (/ 1 {Ti) plane are used, where / 1 and 1 2 are the first invariant of stress tensor and the second invariant of deviatoric stress tensor. These models are mainly deterrnined from the micromechanics analysis for porous metal materials. The associated flow rule is usually used in these models Nowadays, no formulation exists regarding to the non-associated plasticity formulation for powder materials even though some powder materials do have non-associated flow properties (Brown & Abou-Chedid [1994]; Gurson & Yaun [1995]). It is important and necessary to explore the non-associated flow rule. It is urgent and valuable to develop a new methodology to for1nulate non associated constitutive equations based on experimental results. In section 1.2, we review most models which are often referred in the literature. Both time dependent models and time independent models are reviewed. The advantages and disadvantages of these models are highlighted. In Chapter 5 the elastic/viscoplastic model is used to formulate the constitutive equation for alumina powder The viscoplastic potential is determined based on the experimental data presented in Chapter 4. Non-associated flow rule is used in the model. In Chapter 6, the elastic/viscoplastic model is developed for compressible materials. The orientation of irreversible strain rate tensor is determined based on the experimental results on A16-SG alumina powder. A good agreement between the model prediction and experimental data is obtained.

PAGE 16

8 1.2 Literature Survey In ter1ns of the yielding or plastic flow behavior, materials can generally be classified in two types. One type of material is the so-called nonfrictionaJ material For this kind of material, such as metals, the yielding or plastic flow behavior is independent on the first invariant of stress tensor. Several yield surfaces can be used to describe their yielding behavior, as for instance, the Mises or Tresca yield conditions (see Malvern [1969]). An associated flow rule is usually assumed for such type of models. The other type of material is called frictional material. For such type of material, the yielding of plastic flow is dependent on the first invariant of stress tensor In other words, if there was friction within the mass of materials, the frictional resistance will be proportional to the nor111al force. Sand, clay, rock, concrete, powder-like materials etc. are falling into this category. The Mises yield criterion is not any more suitable to describe yield behavior of this type of materials. Other types of models should be used for such materials In the following sections, we will review several models for frictional materials and discuss the advantages and disadvantages of these models. These models include Mohr-Coulomb model, Drucker-Prager model, Cap model, Cam -C lay model and Lade model etc. More advanced models are also discussed. 1.2.1 Mohr-Coulomb Model and Drucker-Prager Model According to the Mohr-Coulomb model, the shear strength ( or stress at failure) increases with increa s ing normal stress on the failure plane The failure criteria can be written as

PAGE 17

9 t = c+otancp (1.1) where t is the shear stress on the failure plane, c the cohesion of the material, o the normal effective stress on the failure plane and cp the angle of internal friction. However, in the Mohr-Coulomb model, the intertnediate stress is not taken into account for the failure criteria. A generalization to account for the effects of all principal stress was suggested by Drucker and Prager [1952] by using the invariants of stress tensor This generalized criterion can be written as (1.2) where ex and K are positive material parameters. 1 1 is the first invariant of stress tensor and 1 2 is the second invariant of the deviatoric stress tensor. The associated flow (normality) rule is assumed in this model. However, if the associated flow rule is assumed, the model usually over-predicts the dilatant volumetric strain of the material comparing to the volumetric strain obtained in experiments. Some problems can be solved without using the nonnality rule. If the irreversible compressibility is taken into account, a cap is usually added at the open end of the Drucker-Prager failure surface (see Chen & Baladi [1985]). If the Mohr-Coulomb failure coincides with the Drucker-Prager failure condition at triaxial compression, the above coefficients satisfy the following relations a = 2sin /3(3 sin) K = 6ccos /3(3 -sin) (1.3)

PAGE 18

10 1.2.2 Cap Model Based on experimental re s ults, DiMaggio and Sandler (1971] and Sandler et al (1976] proposed a cap model. The yield surface for this model consists of two yield segments ( Fig 1 1 ) One i s a fixed yield s urface and the other one is a yield cap surface. The fixed yield segment can be written a s ( 1.4 ) There are s everal po s sible function s for f 1 The expression for / 1 adopted by DiMaggio and Sandler (1971] i s given by f, = [Ti y e PI 1 a = 0 ( 1 5 ) and De s ai and Siriwardane [ 1984] u s e f = [Ti ye P / 1 8/ 1 a = O ( 1.6 ) for an artificial s oil where y p 0 and a are material parameters. The yield s urfaces for yield caps can be ex pr es sed a s ( 1.7 ) where K define s the deformation history. DiMaggio and Sandler (1971] and Sandler et al. (1976] have used an elliptic cap to represent the yield surface for the cohesionless material they have con s id e red

PAGE 19

11 (1.8) where Rb=(X-L) (Fig I. I), R is the ratio of the major to the minor axis of the ellipse, X the value of 1 1 at the intersection of the cap with the 1 1 -axis, L the value of 1 2 at the center of the ellipse, and b the value of (Ji) 1 12 when 1 1 =L. Because the two yield surfaces are intersecting at l 1 =L, the X-L is related with f 1 through The value of X depends on the plastic volumetric strain e v P and is assumed as 1 X =-ln D < l W + Z' ( 1.9 ) (1.10) where D, Z, and W are material parameters to be deterrnined. Usually the associated flow rule is assumed for this model. In the model, the cap surface may shrink or expand dependent on the increase or the decrease of the irreversible volumetric strain. When the loading stress is on the cap surface and the stress increment points outside, work-hardening takes place and the irreversible volumetric strain increases so that the cap surface expands. When the loading stress is on the failure surface, the volumetric irreversible strain decreases so that the cap surface shrinks. Thus, this model can describe a softening behavior. However, for some frictional materials no softening behavior is present in the region considered. Sandler et al. [1976] modified eqn

PAGE 20

12 b initial cap .. a=Rb L X Fig.1.1 Yield surf aces for cap model. (1.10) to prevent this softening behavior. Later, a great lot of progress has been made by many researchers (see Desai & Siriwardane (1984]; Chen & Baladi (1985]). The model has been used to describe the behavior of sand, artificial soil, concrete, rock and powder-like materials etc. The original cap model is further extended to account for rate effect by means of a viscoplastic theory proposed by Perzyna (1966]. A lot of research have been done (Zienkiewicz & Cor1 11eau (1974]; Hughes & Taylor (1978]; Katona (1984]; Simo et al. (1988]; Hoftstetter et al. (1993]). The model is nowadays extended to account for tensile stresses and becomes more efficient when incorporated into the finite element method. However, there are some limitations in this model, for instance, dilatancy is only obtained when the stress state reaches failure surface. For some materials, this assumption

PAGE 21

13 is not correct. Materials such as rock (Brace et al. [1966]; Cristescu [1989]), sand (Kim & Lade [1988]) and some powders (Jin & Cristescu [1996b]; Jin et al. [1996d]), exhibit dilatancy far away from the failure surface Even though the model was modified by Chen and Baladi [1985], it still would not describe correctly volumetric dilatancy. In other words, this model could not give the reasonable prediction of volumetric strain at least for some frictional materials. Another limitation is the assumption in eqn ( 1.10). It may not describe correctly the work hardening behavior for some materials. 1.2.3 Cam-Clay Model The Cam-Clay model was developed by a research group in Cambridge lead by Roscoe (see Desai & Siriwardane [1984]; Chen & Baladi [1985]). The model is based on the concept of Critical Void Ratio (or Critical Density). They found that the yielding of loose or dense soil continues under both drained and undrained conditions until the material reaches a critical void ratio. After the critical void ratio reaches, the volumetric strain does not change any more. For normally consolidated soil, the stress state at this critical void ratio reaches failure, while for over-consolidated soil, the stress state at the critical void ratio reaches the residual stress after passing through failure and softening. After the critical void ratio, the void ratio remains constant during subsequent defo1rr1ations, that is, the materia] will reach a state in which the arrangement of the particles is such that no volume change takes place during shearing. This particular void ratio is called the critical void ratio. This can be considered as the critical state of the material.

PAGE 22

14 This idea can be expressed in the (p, q, e) space, where p is mean stress q=o 1 o 3 =(3J i)112 and e is void ratio as shown in Fig. 1.2. The State Boundary (or Roscoe Surface) can be used to describe the behavior of nor1r1ally consolidated clay while the Hvorslev Surface is used to describe the behavior of over con s olidated clay (see Desai & Siriwardane [1984] ) p State boun d ary or R oscoe s u rface no rmal ly consolidate d e _Cri t jca l sta t e li n e H vorslev s urfa ce o v e r co n so lid a ted Fig 1.2 Critical State Line on the (p,q,e) space. q Fig. 1.3 shows a projection of the critical s tate line (CSL) on the p-q space together with projections of typical s ection of the state boundary surface. The CSL is usually a s traight line pas s ing through the origin. The projection s of the state boundary surfaces are repre s ented by continuou s curve s and referred to as yield s urface s or yield caps. The yield surfaces can

PAGE 23

15 be spherical, elliptical or of other shapes. In this model, the associated flow rule (no1rnality) is assumed. If the yield surface is expressed as P o q = pMln p (1.11) the model is called Cam-Clay model, which was first derived by Roscoe in 1958 (see Desai & Siriwardane [1984]). If the yield surface is an ellipse ( 1 12) the model is called Modified Cam-Clay model, which was fust developed by Roscoe in 1968 (see Desai & Siriwardane [1984]), where M is the slope of the CSL and Po is a work hardening parameter on the p axis. The above yield surface passes through (p 0 ,0) in the (p,q) space. p 0 can be expressed as a function of irreversible volumetric strain which is obtained from hydrostatic tests only. It can be shown that the normal to the above yield surface at a critical point is perpendicular top axis, i.e., no volume changes at this point. This model was used by many researchers (see Desai & Siriwardane [1984]; Chen & Baladi [1985]; Britto & Gunn [1987]; Tripodi et al. [1994]) for nor1nal consolidated clay, over-consolidated clay, and other materials. This model was incorporated into several commercial finite element programs (e.g., ABAQUS [1993]). However, this model has some limitations. For some materials, the critical void ratio does not exist. For some materials, the critical state line does not coincide with the compressibility/dilatancy boundary. Therefore, it could not predict volumetric strain correctly.

PAGE 24

16 q yie ld cap P o p Fig 1.3 Yield locus in q-p space, projection of Fig.1.2 on q-p space. 1.2.4 Lade Model In this model, Lade and Duncan [1975] first proposed the following function for the failure surface, based on experimental observation for sand ( 1 13 ) where 1 1 and are the fust and third invariants of stress tensor, K 1 is constant depending on the density of sand. The expansion of the yield s urfaces is defined by function f given by ( 1.14 )

PAGE 25

17 whose values vary with loadjng and can reach the value K 1 at failure. In the model, the plastic potential functjon is assumed as (1.15) where for a given f, K 2 is a constant The theory allows for a non-associated flow rule. Subsequently, the theory was modified to include curved surfaces (Lade [1977)). Later the theory was modified again (Kirn &Lade [1988]; Lade & Kim [1989a]; Lade & Kim [1989b]; Lade [1989]). The new model employs a single, isotropic yield surface. The yield surfaces, expressed in tenns of three invariants of stress tensor, describe the contours of total plastic work. The non-associated flow rule is used in the model. The potential function is also assumed as a function of three invariants of stress tensor. The elastic Young's modulus is expressed as a function of the confining pressure This model has been used to describe the behavior of sand, clay and concrete etc. This model was also examined by Reddy and Saxena [1992] for cemented and uncemented sand. They obtained the model from one hydrostatic test and three conventional triaxial compression tests, then checked the model with different loading paths. They concluded that the predicted results match reasonably the experimental data except for the volumetric strain at the conventional triaxial tests 1.2.5 Endochronic Model The endochronic theory is derived from the laws of ther1nodynamics, i.e., the conservation of energy and dissipation (Clausius-Duhem) inequality. It is based on the idea of internal variable development. The theory was first proposed by Valanis ([ 1971 a, 1971 b

PAGE 26

18 1975]) by choosing the internal variable to be an intrinsic time scale. It was shown by Valanis that mechanical properties such as hystersis and hardening in metals can be predicted accurately by this model. Subsequently, Bazant [1977, 1978] has applied the model for soil and concrete. Reddy and Saxena [ 1992] have used this model for cemented sand and obtained the model with one hydrostatic compression test and one conventional triaxial compression test One function in the model was modified. The model was checked against different loading paths A good agreement between the tests and the prediction was obtained However, the endochronic theory is still in the development stage (see Yeh et al [1994]; Wu & Ho [1995]) and very few cases of implementation of FEM are available. It was shown by Sandler [1978] that in the present form, the endochronic model can cause difficulties in numerical implementation, particularly in accounting for unloading, which may be remedied by introducing internal barriers. In that case, the theory may exhibit some features similar as the yield loading conditions from the plasticity theory, which the endochronic theory would initially like to avoid. 1.2.6 Models for Compaction (Consolidation) of Powders Based on the micromechanics analysis for porous metal, one kind of models for porous metal or metal powder was first proposed by Green [1972] and Oyan et al. [1973]. The yield surface <1> in this kind model can be expressed as a function of the first invariant of stress tensor, 1 1 and the second invariant of deviatoric stress tensor, J 2 ( 1 16)

PAGE 27

19 where Tl is relative density, A(TJ) B(TJ) and O(TJ) are functions of relative density and Oy is yield stress of powder materials at full density. Later, Shima and Oyane [1976), Corapcioglu and Uz [1978), Doraivelu et al. [1984] and Shima and Mimura [1986] also obtained this kind of yield surface with different functions of A(TJ), B(TJ) and O(TJ) based on the different assumptions In the work of Oyan et al. [1973], for instance, A, Band o are expressed as A = 3 B = 1 O = f1 2 2 ' 2 Tl (1.17 ) Tl Tl2 1+ 1 TJ An associated flow rule is usually assumed in this kind of models. The yield surface (1.16) is an elliptical surface in the principal stress space with the major axis coinciding with the principal stress. In (1 1 ,Ji) space, the yield surfaces are a family of elliptical curves. Along each elliptical curve shown in Fig. 1.4 the relative density is constant. The yield surf aces have symmetric properties for the tension and compression cases and approach the von Mises yield surfaces when the relative density tends towards one, i.e., full density The deformation history was incorporated into the model through the use of relative density as a state variable. The dependence of defor111ation history can be generalized (Brown & Weber [1988]; Krishnaswami & Trasorras [1995]; Tszeng & Wu (1996] ) to include a second state variable representing a scalar measure of the average plastic strain experienced by the powder particles thereby allowing the yjeld stress to change with inelastic defo1mation. These models can be improved However, several parameters are involved, which become difficult to be determined from experimental data.

PAGE 28

20 The model represented by eqn ( 1.16) was extended to viscoplasticity and therrno viscoplasticity (Abouaf et al. [1988]; Chenot et al. [1990]). The finite element methods were used for the prediction of relative density variations in experimental powder compacts (Nohara et al [1988]; Jinka & Lewis [1994] ; Krishnaswami & Trasorras [1995] ; Oliver et al. (1996]; Xu & McMeeking [1996] ) 11 =1 =0 5 Fig 1.4 Typical yield surfaces for the model ( 1.16 ). Another kind of microme c hanics model was proposed by Gurson [ 1977]. For the case of a s pherical cavity within a perfect pla s tic matrix the yield function can be expressed as J = 3 2 + 2fcosh a y 1 1 1 2 -1 -f = 0 2 a y (1. 18 )

PAGE 29

21 where o Y is the yield stress of the matrix and f is the void volume fraction defined by void volume over total volume of porous metal material. This model is suitable to less porous metal materials only. Other micromechanics models (Torquato [1991]; Fleck et al. [1992a, 1992b 1995]; Nemart-Nasser & Hori [1993]) are also available. However, all these micromecbanics models are based on many idealized assumptions. Due to very few experimental data available, it is difficult to validate the models for the complicated stress states (Abou-Chedid & Brown [1992]; Brown & Abou-Chedid [1994]; Gurson & Yuan [1995]). 1.2. 7 The ElasticNiscoplastic Models The first viscoplastic model was proposed by Bingham in 1922 (see Malvern [1969]). He considered the case of a simple shear in the x-direction and supposed that no motion takes place until the stress T reaches a critical value (see Fig.1.5), after which the magnitude of rate deformation D is proportional to the amount by which T exceeds k. It can be shown schematically by Fig.1.5. The viscoplastic equation for simple shear is assumed as 0 2 riD = (1 )T T where 11 is a viscosity coefficient. for Tk (1.19)

PAGE 30

22 T T k Fig.1.5 Bingham viscoplastic model. Hohenemser and Prager in 1932 (see Malvern [1969]) assumed incompressibility and generalized the equation to 0 for FO 1J a ij or (1.20) where o~ is stress deviator tensor and F is measurement of overstress given by F = l k fTi ( 1.21) where J 2 is the second invariant of deviatoric stress. Subsequently, a lot of research has been done in the viscoplasticity theory (Perzyna [1966]; Malvern (1969]; see comprehensive review by Cristescu [ 1982]) One theory was proposed by Perzyna [ 1966]. The formulation can be written as 1 1 2 v m(F) aF e .. = -o .. +--ou .. + y<'' >, lj 2G lj E 1J a a .. lj ( 1 22)

PAGE 31

23 where G, E are elastic shear modulus and Young's modulus respectively, 6ijis rate of stress tensor 6 is mean stress rate. (F) is a function of F (1.23) was suggested by Perzyna [1966]. In this theory, the associated flow rule has been used. F should satisfy the convex conditions Drucker's stable material postulate was used to define F. Due to considerable research works (see Cristescu & Suliciu [1982]; Lubliner [1990]), the elastic/viscoplastic theory can be reduced to the elasto-plastic theory if time effect is negligible while the elasto-plastic theory can be generalized to elastic/viscoplastic theory if time effect is taken into account (s ee Simo et al. [1988]). Recently, one kind of elastic/viscoplastic model was proposed by Cristescu [1989, 1991, 1994]. The theory is based on the elastic/viscoplastic theory and the characteristics of geomaterials (mainly, rock and sand). The basic formulation can be written as e = 6 +( 1 I )61 + k 2G 3K 2G l W(t ) H(o) a F(o) a o (1.24) where K, G are elastic moduli, H is yield function, F is viscoplastic potential, W(t) is irreversible work per unit volume, kT is a viscosity coefficient. This formulation is a generalized form of the Bingham material. However, Cristescu [ 1991] gave a procedure to determine the yield surfaces and potential surfaces based on a set of hydrostatic tests and conventional triaxial compression tests. There is no a priori assumption regarding the yield function and viscoplastic potentiaJ. This modeJ has been used by many researchers ( Florea

PAGE 32

24 [1994a,b]; Cristescu et al. (1994]; Dahou et al. [1995]). The procedure to deterntlne the yield function and the viscoplastic potential was improved by Jin and Cristescu [1996a] and Jin et al. [1996c] so that the model could better match the experimental results and be easily incorporated into a finite element method. Based on the conventional triaxial compression te s ts on alumina powder (Jin & Cri s tescu [1996b]; Jin et al. [1996d]) which will be pre s ented in detail in the present thesi s, the improved procedure was used for the modeling of con s olidation of alumina powder AlO ( Jin et al [1996a, 1996b] ) Another procedure to determine the ori e ntation of the irrever s ible volumetric s train rate was given by Cazacu et al (1996] and Cristescu et al [1996]. They used a discontinuity function to de s cribe the effect of the compressibility and dilatancy boundary A methodology to deter111ine the elastic/viscoplasti c model in ter1ns of the irreversible volumetric s train as a working parameter i s developed in thi s the s is The model prediction matches well the experimental data. Thi s methodology is more attractive for the powder materials s ince the work hardening parameter ( volumetric s train ) is directly a ss ociated with volumetric reduction and the procedure to determine the model i s much simpler 1.2.8 Other Models There exi s t al s o many other model s Green elastic model ( exi s tence of a s train e nergy function ) and Cau c hy elastic model ( stre s s is assumed as a function of strain ) can be u s ed to describe nonlinear ela s tic behavior ( see Desai & Siriwardane (1984]). Hypoelasticity ( the in c rement of s tre ss i s expre ss ed as a function of stress and increment of strain ) ori g inally propo s ed by True s dell [1955 1966] could s erve as a general nonlinear model ( al s o s ee Green

PAGE 33

25 [ 1956] ). However the s e models al s o have some limitations ( see comments by Chen & Baladi [ 1985] ). There are also many publications regarding to the homogenization approach for modeling (Chri s tensen [1991]; Nemat-Nasser [1993] ; Huang et al. [1994]). However due to complicated modeling and assumption s, this approach does not yet mature for practical problems. 1 2.9 Remark on Recent Development of Plasticity-Based Models For the plasticity-based models the isotropic hardening and the kinematic hardening rules were developed and used. Several theories were developed for work hardening There are several model s for the kinematic work hardening ( see comprehensive review by Desai & Siriwardane [1984] ; Chen & Baladi [1985]; Drucker [1988] ) Sometimes, the isotropic hardening rule and the kinematic hardening rule s are incorporated into the above model s presented in the previous s ection s. However, due to the difficulties to determine the parameters in the work hardening rule from experiments the application of these hardening rules is limited. 1 2.10 Finite Element Method During th e past several decades, the finite element method has rapidly become a very popular technique for the numeri c al solution of complex problems in engineering with the help of large digit computers ( see Bathe (1982]; Zienkiewicz & Taylor [1989] ; Hughes [1989] ) Applications range from the stre s s analy s is of structures to the solution of acou s tical

PAGE 34

26 neutron physics and fluid dynamics problems. Indeed, the finite element method has been developed so rapidly that the finite element process is established as a general numerical method for the solution of partial differential equation systems subject to known boundary and/or initial conditions. The success of the finite element method is based largely on the basic finite element procedures used: ( i) the forrnulation of the problem in variational or weighted residual form (weak form), (ii) the finite element discretization of this formulation, and (iii) the effective solution of the resulting finite element equations These basic steps are the same whichever problem is considered These basic steps provide a general framework and a quite natural approach to engineering analysis in conjunction with the use of the large digit computers The basic concepts of finite element method and basic forrnulation for problems in the isoparametric fmite element representation can be referred in the books by Zienkiewicz and Taylor [ 1989] and Hughes [ 1989]. The basic idea and procedures are easy to extend. The basic formulation can serve for different problems, such as elastic-plasticity and e l astic/viscoplasticity problems In Appendix 1, we will give brief review of the discrete formulation of the elastic/viscoplasticity theory for the finite element method. These formulations have been implemented into a finite element method program The program has been used for the analysis of several problems in the present thesis. The main results will be presented in Chapter 3.

PAGE 35

27 1.3 Objective and Outline of the Present Work L The accurate constitutive formulation of nonlinear time dependent behavior of materials is extremely important both for a theoretical analysis and for the finite element simulation. With the existing models presented in the previous sections, it is difficult to describe accurately three-dimensional nonlinear time dependent behavior, especially for the nonlinear volumetric behavior from compressibility to dilatancy. Without an accurate constitutive equation, there is no way to simulate engineering problems efficiently and accurately when the nonlinear time dependent materials are involved. Thus, today it is a challenge to develop a methodology to for1nulate nonlinear time dependent behavior of materials accurately. Cristescu [1989, 1991, 1994], as a pioneer, was trying to for1nulate nonlinear time dependent behavior with the elastic/viscoplastic theory based on a set of triaxial tests. In his formulation, the dilatancy and compressibility of volumetric strain are highly considered. The yield surfaces and potential surfaces are determined from tests without any a priori assumption. His formulation was used to describe rock salt and sand However, his model should be checked with more experimental data and examined from the point of view of the application of the finite element method. In the present work, Cristescu's approach is employed to establish the constitutive equations for rock salt and powder materials. We found that some steps in his approach of the determination of the elastic/viscoplastic model should be modified or changed in order to match better experimental data and to be easily incorporated into the finite element method.

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28 The main objective of the present work is to investigate the behavior of powder materials experimentally by triaxial tests and to formulate three-dimensional constitutive models for powder materials and for rock salt based on the elastic/viscoplastic theory and the experimental data. Emphasis is placed on the improvement of the procedure (Cristescu [ 1991, 1994 ]) to determine the elastic/viscoplastic models. In the determination of the models, the asymptotic behavior and the physical requirements of yield surfaces and potential surfaces are incorporated into these procedures so that the yield surf aces and the viscoplastic potentials are kept smooth These models can better match the data and can easily be incorporated into the finite element program. The present work is associated with three aspects: (1) triaxial tests, (2) constitutive modeling, and (3) finite element analysis. First, a constitutive equation for rock salt is formulated based on the true triaxial experimental data obtained by Hunsche (see Cristescu & Hunsche [1992)). Besides the properties considered in the Cristescu's approach, we have considered the asymptotic behavior of yield surf aces and of the potential surf aces in the procedure to detern1ine the model. A new method to deter1nine the viscoplastic potential surface is proposed. Non-associated flow rule is used. The model matches very well the data. Also, the model is incorporated into a finite element program. Several engineering problems are analyzed with the program. The finite element analysis gives a good agreement with experimental data. Such type of the elastic/viscoplastic model (both yield surfaces and potential surfaces) is used for the first time in the finite element programs Secondly we have performed a series of triaxial tests on alumina powders using a triaxial apparatus. The influences of the particle size and of the initial density on the

PAGE 37

29 mechanical behavior of alumina powders are investigated experimentally. It is shown that the volumetric strain behavior is strongly influenced by the particle size of powders. For the alumina powder A16-SG of smaller particle size (0.4~ lm), the volumetric strain always exhibits compressibility in the range of applied stress state, while for the alumina powder A 10 with larger particle size ( 40~ 200m), the volumetric strain exhibits both compressibility and dilatancy. The influence of the initial density on the behavior of alumina powder AlO has also been investigated The elastic parameters of these alumina powders have been measured according to the loading-reloading process. The set of triaxial tests furnishes the necessary data for the for1r1ulation of three-dimensional constitutive equations. Thirdly, a three-dimensional elastic/viscoplastic constitutive model for alumina powder AlO is formulated based on the experimental data (a set of conventional triaxial tests). The yield surfaces and the potential surfaces are deter1r1ined. A new procedure to dete11nine the model is proposed based on such type of conventional triaxial compression tests. The model is checked carefully against the experimental data with different loading paths. A good agreement between the data and the model prediction is obtained. Finally, another methodology to formulate the behavior of the materials which are compressible only is proposed. The irreversible volumetric strain is taken as a work hardening parameter instead of the irreversible stress work used in the previous method. The methodology is used to for111ulate the behavior of alumina powder A16-SG. The irreversible strain rate orientation tensor is used and determined in this model. It is found that this model is much easier to handle and has fewer parameters involved than the previous ones. A reasonable matching between the experimental data and the model prediction is obtained.

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30 The organization of the dissertation is as follows: in the first Chapter, the models exjsting in the literature are reviewed. In Chapter 2, the Cristescu's approach is outlined. A new constitutive equation for transient creep of rock salt is proposed in Chapter 3. In the fourth Chapter, three sets of triaxial tests on alumina powders (two sets for A I 0, one set for A16-SG) have been perfor1ned and presented. In Chapter 5, an elastic/viscoplastic constitutive equation for alumina powder AlO is formulated based on the hydrostatic tests and the deviatoric tests presented in Chapter 4. In Chapter 6, a new methodology to formulate the behavior of the materials which are compressible only has been proposed. Finally some conclusions and suggestions for the future work are given in Chapter 7.

PAGE 39

CHAPTER2 THE ELASTIC/VISCOPLASTIC (CRISTESCU) THEORY 2.1 The Elastic/Viscoplastic Constitutive Equation Many materials such as rock salt and powder materials can be considered as homogeneous and isotropic ones. These materials have no preference directions In many cases, the defor1r1ations and rotations of particles are small. In the framework of the elastic/viscoplastic theory, it is assumed that ( 1) the materials considered are homogenous and isotropic; (2) the deformations and rotations of particles are small Based on the above assumptions, the total strain rate tensor e can be obtained by adding the elastic strain rate tensor eE and the irrever s ible strain rate tensor e 1 : E I e = e + e (2.1) For the elastic re s ponse of material, let u s consider the fact that both longitudinal and transverse extended body seismic waves can propagate through most materials (Cri s tescu [1993a]). The fact suggests that most materials exhibit an elastic 'instantaneous response." The elastic parameter s can be obtained from the speeds of longitudinal and transverse extended body seismic waves In the framework of the elastic/viscoplastic theory the elastic 31

PAGE 40

32 parameters can also be obtained by unloading process suggested by Cristescu (1989] for time dependent materials. Thus, the i,zstantaneous response can be expressed by a rate type elastic relation eE = o + < 1 1 )ol 2G 3K 2G (2 2) where o is stress rate tensor, o mean stress rate, 1 unit tensor and G and K are elastic shear and bulk moduli respectively. For the irreversible part, it is assumed that the material follows a viscoplastic type behavior (Cristescu (1991, 1994]), i.e., J e = k l W(t) H(o) a F(o) a o (2 3a) where kT is viscosity coefficient, W(t) is total irreversible stress work per unit volume at time t, F( o) is the viscoplastic potential and H( o) is the yield function with H(o(t)) = W(t) (2.3b) the relaxation boundary or the equation of the stabilization boundary (where e 1 = 0, o = O) The bracket <> denotes the positive part of a function, i.e A if A > 0 0 ifA ~ O. (2.4) If the yield function coincides with the viscoplastic potential, the flow rule is associated, otherwise, the non-associated flow rule is assumed. For some materials, the viscoplastic

PAGE 41

33 potential may not exist. In that case, the irreversible strain rate tensor is assumed to be proportional to a tensor N (Cazacu et al. [1996]; Cristescu et al. [1996]): I e = k 1 W(t) N ( a) H(a) (2.5) where N( o) is a tensor governing the orientation of irreversible strain rate tensor. From eqn (2.1-2.3) and (2.5), the elastic/viscoplastic constitutive equation will be written as a ( 1 I )'l k e = 2G + 3K 2G O + r or a 1 1 e = + (-)al+k 2G 3K 2G l W(t) H(a) a F(a) aa 1 W(t) N(o) H(a) (2.6a) (2.6b) In general, the yield function H and the potential function F ( or N) are all dependent on stress tensor. However, if H and F ( or N) are assumed to be dependent on the first stress invariant / 1 and on the second invariant 1 2 of deviatoric stress tensor only, disregarding the influence of the third invariant of stress tensor, the whole constitutive equation can be deter1nined from a couple of triaxial tests (Cristescu [ 1991, 1994 ]). That is, H and F ( or N) are assumed to be dependent on mean stress (2 7) and octahedral shear stress 't

PAGE 42

or equivalent stress t = I1 3 2 2.2 Determination of the ElasticNiscoplastic Constitutive Equation 34 (2.8) (2.9) In the general constitutive equation (2.6a) or (2.6b), there are three parameters, G, K and k T, and two functions, Hand F ( or N) to be deter1nined. Cristescu [1991, 1994] provided a primary procedure to determine them from a couple of triax.ial tests for rock salt and dry/wet sand. The procedure is shown by scheme in Fig. 2.1. The details in the procedure may be referred in Chapter 3 and 5 2.2.1 Elastic Parameters and Viscosity Coefficient Elastic parameters G and Kare involved in all steps of the model deter1nination. Thus the elastic parameters must first be determined. Generally, there are two methods used to determine elastic parameters, i.e., dynamic and static methods. In the former case, the seismic wave velocities propagating through materials are measured, and from them are calculated the elastic constant s. In the latter case, the elastic parameters are dete1111ined using an

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35 unloading procedure (Cristescu (1989]) for time dependent materials. The main idea of the unloading procedure is to separate time effect from the pure elastic response during unloading. For example, in stress-controlled tests, we keep stress constant at a chosen value and allow the specimen to have a sufficient time to creep until the rates of strains are small enough so that no significant interference between creep and elastic unloading would talce place during the unloading perforrned in a comparatively short time period. Elastic parameters can afterwards be deterrnined from the slope of the frrst portion of the unloading curves (usually one-third or one-fourth of total axial stress) This procedure also can be used for strain-controlled tests (see Chapter 4). The viscosity coefficient can be determined from creep tests such as hydrostatic creep tests after the yield surface and the potential surface are determined. If, for instance, a hydrostatic creep test is perfor1ned, we integrate eqn (2.3) and obtain the following formula for the viscosity coefficient k =ln T l W(t) H(a) 1 1 H(a) I t, a F a (tJ t;) a a (2.10) where t; is the "initial" moment of the creep test, t 1 is the "final" moment, a(t ;)= a(t 1 ) = const., and H is yield function and F is potential surf ace. If we use a nonassociated constitutive equation, then kT could be determined simultaneously with F. However creep tests can still be used for the adjustment of viscosity coefficient.

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Fun ctio n s & parameters needed in the model Procedure used E l astic parameters G,K + CID& Failure conditio n Yield s urf ace H Potential s u rface F kTF from both hydrostatic and deviatoric tests through unl oa din g-re loadin g fro m deviatoric tests I H v from hydrostatic tests I -( H d from dev iat oric tests I ..____ I Assumption: H=H v +H d )-------H ydrostatic p o t e ntial


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37 2.2.2 Yield Function The yield function is defined as the irreversible stress work per unit volume on the stabilization boundary or relaxation boundary according to the elastic/vi s coplasticity theory (see Cristescu [1967 1989]; Criste sc u & Suliciu [1982] ; Lubliner [1990]), i.e ., H ( o ( t )) = W ( t ) (2. 11 ) Thus, the yield function can be obtained by evaluating the irreversible stress work per unit volume along the relaxation boundary W ( t ) can be computed as t W (t)= f o ( t ): e 1 (t)dt (2. 12 ) 0 where (:) means double contraction on tensors W ( t ) i s used as an internal state variable or as a work-hardening parameter The constitutive equation (2.6) may be used even if the material is assumed to have a zero initial yield s tres s. The yield surface H i s assumed as the sum of two parts, hydrostatic one 1-fv and deviatoric one Hd according to the loading path s u s ed in the triaxial test s, i.e., H = H v + Hd. H v i s detennined from the hydrostatic loading tests while H d is dete11nined from deviatoric test s by the regre ss ion of the irreversible stress work along the relaxation boundaries as indicated in Fig .2.1. For different material s, the yield function might not be the same. For the details of the determination of yield function one can refer to Chapter 3 and 5 or refer Cristescu [ 1991 1994] for s pecific material s.

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38 It is noted that most models in the literature (see Chapter 1) end at this step (see Fig. 2.1) and associated flow rule is assumed. However, most frjction materials are not satisfied with the associated flow rule. Non-associated flow rule should be used. The viscoplastic potential should be explored. 2.2.3 Viscoplastic Potential The viscoplastic potential For the tensor Nin eqn (2.6) governs the orientation of irreversible strain rate tensor. It can also be determined from experimental data. Several steps are required to deter1r1ine the potential as shown in Fig. 2.1. According to the procedure, some properties of the material such as a failure condition or/and compressibility/dilatancy boundary are incorporated into the potential. The potential might not be the same for different materials In Chapter 3, 5 and 6, the procedure will be given in details for specific materials. However, many steps have been changed from the original steps (Cristescu [1991, 1994]). The potentials obtained satisfy additional conditions which are required by physical observation and by the finite element method. If the yield function H( o) coincides with the viscoplastic potential, i.e., H( o) = F( o), the flow rule is associated, while if H( o) F( o), the constitutive equation is called non associated. Both the associated and nonassociated flow rule can be used for modeling. However, in general, the associated flow rule could not give good results, while the nonassociated flow rule can.

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39 2.3 Loading-Unloading Conditions Let us assume that at an initial stress state (the s o-called "primary" stress) oP = o(t 0 ) is an equilibrium stress state, i.e H ( a ( t 0 )) = W P, with W P being the value of W for the primary stress state. A stress variation from a ( t 0 ) to a ( t ) with t> t 0 i s called unloading if H ( a ( t) )< W(t o) and is called neutral loading if H(a ( t )) = W(t 0 ) During unloading or neutral loading, instantaneous elastic response given in eqn (2.2) is assumed. A stre ss variation froma (t 0 ) to o ( t ) (* o ( t 0 )) with t>t 0 i s called loading if H(a ( t) )> W ( t 0 ) with one of the three possible subcases defined by the following inequalities satisfied at o (t): a F:1>0 or a F>O compressibility a a a a a F:1 = 0 or a F = O compress./dilatancy boundary (2 .13a,b,c ) a a a a a F : 1<0 or a F < O dilatancy a a a a The three inequalities correspond to the inequalities to be satisfied by the rate of irreversible volumetric s train e~ involved in eqn (2.3). Al s o, these inequalities define the region of compressibility, compressibility/dilatancy boundary and the region of dilatancy. These concepts are very important not only in the theory but also in engineering applications (Cristescu [ 1993a] ).

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CHAPTER3 AN ELASTICMSCOPLASTIC MODEL FOR TRANSIENT CREEP OF ROCK SALT In the f or1nulation of transient creep of rock salt, we start from a complete and accurate set of true triaxial data on Gorleben rock salt obtained by Hunsche (see Cristescu & Hunsche [1992] ). The set of data consists of six true triaxial tests under the conditions of different constant mean stress, i.e a=8, 14, 25, 30, 35 and 40 MPa. For the formulation of hydrostatic constitutive equation, one can refer to Cristescu and Hunscbe [1992] and Cristescu [1993a] For the formulation of the mechanical behavior of rock salt in transient creep, we follow Cristescu's approach (Chapter 2 and Cristescu [1989 1991, 1993a, 1993b, 1994]) In this approach both compressibility and dilatancy properties of rock salt will be considered, which is an advantage of the model over other models (See Chapter 1 ). The concept of compressibility/dilatancy (CID) boundary is introduced, thus one could well dete11nine with such a constitutive equation, where around an opening, for instance the rock becomes dilatant and where compressible. The procedure in thi s approach allows the determination of a nonassociated elastic/viscoplastic constitutive equation able to describe creep, relaxation, dilatancy and/or compressibility during creep, work-hardening and failure. In this chapter a new elastic/viscoplastic constitutive equation for transient creep is proposed. Compared with the early models (Cristescu [1993a, 1994]) the deter1nination of 40

PAGE 49

41 the yield function and of the viscoplastic potential is improved: the yield function has a singularity at failure and the viscoplastic potential surfaces are requested to satisfy additional conditions required to connect hydrostatic tests with deviatoric ones. A new procedure to determine the viscoplastic potential is proposed The viscoplastic potential is f ortnulated in simpler analytical expression than that in the previous papers. The present model on the one side will be in a better agreement with the data and on the other side could be easily incorporated into a FEM program An explicit integration on viscoplastic strain components is used in the con s titutive equation The results obtained by FEM are in good agreement with the experimental data. The stress distribution around a vertical cylindrical cavern is analyzed. Some interesting results are found. 3 1 Determination of the ElasticNiscoplastic Constitutive Model The procedure to determine the elastic/viscoplastic model has the following steps: ( 1) determination of the elastic parameters, (2) determination of the CID boundary and the failure condition, (3) determination of the yield function, and (4) detertnination of the viscoplastic potential and viscosity coefficient. Each step is directly related to experimental data. We will discuss in details the above steps in the following sections 3 .1.1 Deterrr1ination of the Elastic Parameters The elastic parameters are determined by using the unloading processes in a s tress controlled apparatus mentioned in Chapter 2. For a stress controlled test, stress is kept

PAGE 50

42 constant at a chosen level. The specimen is allowed to have a sufficient time to creep until the rates of strains are small enough so that no significant interference between creep and unloading phenomena would take place during the unloading performed in a comparatively short time period as schematically shown in Fig. 3.1. Elastic parameters can afterwards be detern1ined from the slope of the first portion of the unloading curves (Fig. 3.1). For rock salt, an average value of bulk modulus K=21.7 GPa and of shear modulus G=l 1.8 GPa have been measured by Hunsche (see Cristescu & Hunsche [1992]). The values obtained by this method are very close to those obtained by dynamic method (Cristescu [1989]). The elastic parameters are calculated with the f or1nula (3 .1) 1 a 1 = arctg1 -1 -; -+3G 9K 1 a 2 = arctg 1 1 9K 6G E 2 cr = constant El unloading E 2 (XJ loading Transverse strains Axial strain Fig. 3 .1 Procedure to deter1nine the elastic parameters in unloading processes following short creep periods El (3.1)

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43 3.1.2 Determination of the CompressibilitylDilatancy Boundary and the Failure Condition The compressibility/dilatancy (CID) boundary is defined by eqn (2.13b). The boundary has specific meanings in engineering applications (Cristescu [1989, 1993a]). Below the boundary shown in Fig. 3.2 there is the compressibility region. In the compressibility region, the rate of irreversible volumetric strain is positive (compression strain is conventionally taken as positive), which means that the absolute value of irreversible volume is reduced. The decrease of irreversible volumetric strain may close microcracks and pores which may be present in geomaterials. In the dilatancy region shown in Fig.3.2, the rate of irreversible volumetric strain is negative, which means that the absolute value of irreversible volume is increasing. The increase of irreversible volume is directly related to the opening of microcracks and of pores, or of any kinds of damage which may be present in geomaterials. Sometimes in engineering applications, dilatancy is to be avoided. The CID boundary can be obtained straightforward from experimental data. In classical triaxial tests where hydrostatic loading is followed by deviatoric loading under constant confining h R R. b. d' d . H R d R h d pressure, t e curve a 1 ~ e v 1s o ta1ne 1n ev1ator1c tests. ere a 1 an e v are t e stress an volumetric strain during deviatoric loading with respect to the reference configuration at the end of hydrostatic tests (superscript R means "relative", i.e., with respect to the mentioned reference configuration). The CID boundary is determined as the stress loci where the slopes ( or tangents) of the stress-volumetric strain curves are equal to the elastic ones (i.e., e: =0). In true triaxial tests where the hydrostatic loading is followed by a deviatoric loading under

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44 constant mean stress, the CID boundary is obtained directly from ,; ~ e v R there where the slope of this curve is vertical. For the rock salt from true triaxial tests, Cristescu and Hunsche [1992] have obtained the CID boundary as follows 't 0 X(a,-r ) = -+ f 1 a a 2 a + f2= 0 a ( 3.2) where /~ =0 017, / 2 = 0.9 and a = 1 MPa(see Fig. 3.2 where the full line is just this boundary ) Fig. 3.2 shows by diamonds the locus of maximum values of octahedral shear stresses obtained in several tests using the data by Hunsche ( see Cristescu & Hunsche [1992] ) ; they correspond to the mean stress values: o = 14, 20, 25, 30, 35 and 40 MPa, respectively. The following equation can be used to approximate the locus of the maximum stresses (3 3) where 't r ( o) is octahedral shear stress at failure, o is mean stress and y 1 =38.0 (MPa), Y 2 =34.9(MPa) and y 3 =0.04 obtained through curve fitting. The stress states at failure and the fitting curve are shown in Fig 3.2.

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't (MPa) 30 35 f---~-~I =~---=:=:::=:::::::==7 tv 25 20 15 10 5 failure condition I Ev <0 I Ev >0 dilatancy region I Ev =0 CID bound~ .... ..---compressibility region 0 L....L..&....Ji.....L....L..1....J....&....1.....L....a.....L--l-.&.....L....a.....i....J.--1... ......... _.__.......,__.__Ut...L. ......... 1...,1 0 10 20 30 40 50 60 cr(MPa) 45 Fig. 3.2 Domains of compressibility, dilatancy, CID boundary and failure condition 3.1.3 Determination of the Yield Function The yield function is defined as the stabi lization boundary or relaxation boundary according to the viscoplasticity theory (see Cristescu [1967, 1989]; Cristescu & Suliciu [1982] ; Lubliner [1990]). Due to the available short terrn experimental data on rock salt, it is assumed that the stress strain curves are "almost" reaching the relaxation boundary for transient creep. Thus the yield function can be obtained by evaluating the irreversible stress work from the experimental data using eqn (2.12). W(t) and H( o ;t) can be separated into two parts according to the two loading paths used in true triaxial tests In the first stage, the hydrostatic loading, the s tre ss components are kept equal ( o 1 =o 2 =o 3 ), and increased according to a certain conventionally chosen time interval t E [O, th). The second stage, the

PAGE 54

46 deviatoric loading, takes place in the time interval t E [th, t] with a= constant, and it is only the octahedral shear stress which is increased. Thus the complete expression of the yield function is the sum of the yield functions obtained in the two stages (Cristescu (1994]): (3 4) as is the total irreversible stress work: (3. 5) with the relations (3 .6) where the subscripts hand d indicate hydrostatic and deviatoric stages re s pectively W h and W d can be computed from eqn (2. 12 ) using the corresponding integral intervals. From the calculation of irreversible stress work in hydrostatic tests and curve fitting, Cristescu and Hun sc he (1992] obtained the following expression for Hh ( o ) (3.7) where h 0 =0 l 16 MPa h 1 =0 l 03 MPa, w=2.88 q>= -62.6 and a 0 =53 MPa

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lOr---------------------, 1 0.1 a = 20 (MPa) 0.01 0.001 '---1.-----...---.......... --....__.__ ____ ...____...._.__...__. .......... ___ 0 5 10 15 20 25 't (MPa) Fig. 3.3 Typical variation of irreversible st1ess work Wd with 't for a =20 MPa, test results points, a straight so lid line suggesting a model behavior 47 Using the experimental data obtained in the deviatoric loading stage, we calculated the irreversible stress work using eqn (2.12). Fig. 3.3 s how s a typical computed result for the mean stress 20 MPa. The deviatoric irreversible stress work can be approximated by two terms, for instance~ one is linear in the exponential of 't, the other has a singularity at failure. Thus, the formula for the deviatoric part of the yield function becomes, H d( a,-r) = a( a) exp b( a) 't a* where 8( -r -1) -rJ.a) (3.8)

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2 b(a) = P 1 exp a -y 3a 48 (3.9) There are two terms involved in the approximation of Hd(o;t) as shown in eqn (3.8). First, from the log(Wd) ~ 't' plot of the experimental data (see Fig. 3.3), it was found that besides a small region near failure Hd(o,i-) can well be approximated by an exponential function, i.e., Hd(o,i-) ~ a exp(bi-). However, for a given a, near the maximum octahedral shear stress 't'm a x it was found that Wd(i-) increases much faster than described by an exponential function. It looks as if Wd(i-) blows up near the maximum octahedral shear stress. To capture this seemingly singular behavior of Hd(a,i-) near failure in a simple manner, a first order algebraic singularity is introduced, Hd( o, 't') ~ k I ( i1 -,; ) in which k and 't r are determined from the fit of the experimental data. Because this function decays too slowly away from 'tr, which would offset the close fit already obtained for the exponential growth part, an exponential decay factor exp[8(i/ 't' r -l)] is introduced to reduce the error in joining these two asymptotes. Thus the second part of Hd(o,i-) becomes k / (i-f-i-)exp[8(i/ i1 1)]. The factor 8 is introduced by error and trial method to fit better the curves. In general, 't'f obtained from the fit of the data is a little larger than the actual maximum octahedral shear stress in each of the experiments and the difference between 'tr and 'tmax are less than 1 % Hence practically 't'f can be used, in lieu of 't'max, to represent the octahedral shear stress at failure as shown in Fig. 3.2.

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Wd 12 (MPa ) 10 8 6 4 2 0 0 -c alculated value s from curve fitting 35 ++++ calculated value from experiments 30 20 25 14 cr= 8MPa 40 + 5 10 15 20 25 30 35 t (MPa) Fig 3.4 Fitting curves of irreversible work. Crosses experimental results solid lines prediction of eqn (3 8) 49 All the coefficients a, b k and t r depend on a. After analyzing the data for each o the dependencies of a b, k and t r on o are found out as shown in eqn (3 9). The following constants are obtained: a 1 = l 88x10 4 ( MPa), a 2 = l .13x 10 4 ( MPa), a 3 = 0.0I6xl0 4 ( MPa); p 1 = 0 566, p 2 = 0.106, p 3 = 0 217; K 1 = 0.00171 (MPa ), K 2 = 1 356 (3 10) In the process of curve fitting, it was found that the data corresponding to o=8 MPa is not consistent with the trend established for a>8 MP a. Since during this test the mean stress a is not really constant. For this reason, these data are further omitted The data corresponding to o=35 MPa were not used for the detertnination of a b, k, and t r Hence the

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50 experimental data at a=35 MPa are used as an independent check to test the accuracy of the Hd(a;r). Fig. 3.4 shows very close matching for a=35 MPa as it is for the data obtained with other values of a. 3.1.4 Determination of the Viscoplastic Potential After the determination of the yield function, we can use these results to get the viscoplastic potential which governs the orientation of viscoplastic strain rate. The potential has to satisfy several constrains from the elastic/viscoplasticity theory of compressible/dilatant materials (Cristescu [1993a]). Let us shortly remind them. From eqn (2.3) and (2.13), we have: I aF ev G(a;r): = kr= ---~ a a I W(t) (3.11) H(a;r:) For the hydrostatic loading, G( a;r) satisfies the following condition a F G( a;r:)l't = O = kr -1 = cf>( a) a a -r = O (3.12) where ( a) is deterrnined from hydrostatic tests, while for general loading paths from eqn (2.13), G (-r,a) is required to satisfy the following conditions: G( a -r) >0 G( a, -r) = 0 G( a, -r) <0 X(a,-r)>O X(a,-r) = 0 X(a,-r)
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51 where X(o,t)=O is CID boundary. The rate of irreversible volumetric strain will increase in compressibility domain, decrease in dilatancy domain and be equal to zero at the boundary between compressibility and dilatancy regions. Also the rate of irreversible volumetric strain may reach negative infinite values when stress state is approaching the failure condition denoted as Y r(o,t)=O. The simplest form which has the above properties is k aF =X(o,t)'I'(o) T a a Yja,t) where the function 'I' is related to hydrostatic behavior by X(a,O)'I'(a) = cp(o) ~(a,O) (3.14) (3.15) Cristescu (1991, 1994] has employed this for1n to get the viscoplastic potential. Although this form satisfies exactly eqn (3.13), it may not match well enough the data within the compressibility and dilatancy regions and may cause a discontinuity of e 1 at t=O. In order to match better the data within the mentioned regions and ensure the continuity of e 1 we propose a procedure to determine the irreversible volumetric strain rate directly from experiments, i.e., to determine the function G(o,t) by fitting the experimental data with the requirement that G has to satisfy the conditions (3.13). Let us start from eqn (3.12). G(o,t) may be divided into two parts according to two physical meanings: compressibility and dilatancy, denoted by G 1 and G 2 respectively. Each part of G(o,t) is considered to be an asymptotic approximation and to be a dominant part in

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52 the corresponding region In order to satisfy the condition G( o;t) = 0 at the CID boundary, we take G 1 (o;t) to be strictly equal to zero and G 2 to be approximately equal to zero on this boundary in the sense that G I is dominant in the compressibility region as compared with G -c From eqn (2.6), a F! a r: should be zero along and in the immediate neighborhood of t=O in order to obtain a correct strain variation along a loading path very close to the hydrostatic one, i.e., an additional very small deviatoric stress was superposed. This condition is imposed in order to avoid the presence of a vertex of the surface at the hydrostatic axis and to keep the continuity of e 1 when passing from hydrostatic to deviatoric loading. Since a G/ a r: is also related (see the following sections) to a F(a,r:)/ a r: at this stage we take a G/ a r:==O at t==O. Thus we take a G 1 / a t=O and a G 2 I a r: = 0 along t=O. Depending on the evaluation of the right-hand side term of eqn (3 .11) using the data in the compressibility region, we will be able to choose an appropriate form for G 1 in order to match the data and satisfy the above two conditions. However, since in the compressibility region the experimental data available have not had enough accurate digits, the results obtained from the right-hand side of eqn (3.12) show a large scatter in that region For this reason we use the simplest form for G 1 i.e., a second order polynomial int as 2 (3.16) where as before o = 1 MPa. From the requirement a G,(a,r:)lar:=0 at t=O, follows that a possible first order term in eqn (3.16) is zero.

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53 From hydrostatic tests Cristescu and Hunsche [ 1992] have obtained for ( a) involved in eqn (3.15) and (3 .16): 2 (3.17) where q 1 =1.5 x 10 -s (s1 ), (h=0 0 = 53 MPa. From the above conditions, G 1 (o;t) 1 't = o = (o) and from the requirement G 1 (o,t) = O along X(o,t)=O, we can determine Q 1 and Q 2 from eqn (3.17) (note f 2 / f 1 = -q 2 / a.) as 2 (3.18) In the dilatancy region, the function G 2 mentioned above should be dominant in that region, but must have much smaller values than G 1 in the compressibility region in order to obtain an asymptotic approximation of the conditions (3.13), Higher order polynomials in t seem to be good expressions for the approximation of G 2 in the dilatancy region. Based on the data available and the approximated evaluation of the right-hand side of eqn (3.11) (see Cristescu (1991]), we can choose the 5th and 9th order powers to fit the data (Fig. 3.5), 5 G 2 ( o,t) = A (a) + B(a) a 9 (3.19)

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54 A(a) and B(a) can be detertnined using the data from true triaxial experiments performed at several mean stresses When analyzing the data and trying to choose proper polynomial functions, we have found that at the beginning of the dilatancy region, by combining G 1 (a;t) with a fifth order power function we get a better fitting of the experimental data. Thus, the A coefficients are determined from several tests perforn1ed with various constant values of a. By combining a ninth order power function with G 1 (o;t) and the above fifth order function, we were able to obtain a better fitting of the curves in the dilatancy region at higher value of t. Thus, the B coefficients are determined. After obtaining the values of A and B from the tests perfo1 n1ed under different mean stresses, we use the least-square method to get the expression of the functions A(a) and B(a), (3.20) where as=-5.73x10 7 (s 1 ), b 5 = -2.12, <=-1.63x10 8 (s 1 ), b 9 =-4.77. In the neighborhood of failure, the dilatancy behavior may become singular However, the singularity is very difficult to make precise from the data shown in Fig. 3.5. Thus, for the expression of G we use both G 1 describing compressibility and G 2 describing dilatancy. From eqns (3.16) and (3.19), it is easy to show that in the compressibility region G 1 is a dominant function as compared to G 2 One can use a parameter p = f c G 2 1 da / Jc G 1 I da (where domain C is the compressibility region) to decide which term is dominant. The value p=0.11 of this parameter indicates that G 1 is dominant in the compressible region. The function G is approximated by

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2 5 9 G(a;c) = G 1 +G 2 = Q 1 (a)+Q 2 ( a) + A(a) + B (a) a a a where Then, we integrate eqn (3.11) with respect to a to get k-zF( a;r::) = f G(-r::,a) da +g(1:) = F 1 ( a ; t) +g( -r::) 0.005 ,---------------------, G(cr,-c) (s1 ) 0 t-----w -0.005 -0.01 -0.015 -0.02 0 0 cr= 14MPa 6 0 40 20 25 30 V -0.025 .................................................................................................................................................. __ ..._. 0 5 10 15 20 25 30 35 "t (MPa) 55 (3.21) (3.22) (3.23) Fig. 3.5. Data for various confining pressure show n used to detertnine the function G ( a,-r:: ), symbols experimental data, so lid lines prediction of eqn (3.21).

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56 In order to obtain g(t), we differentiate eqn (3.22) with respect tot and combine the irreversible strain components from eqn (2.3) J e = k l W(t) H(a) B F a a B F B r: ,-+ --a a a a a -r a a Finally we get the formula for g 1 (t): 1 g~t) = ----1 W(t) H(a) (3 24) (3 25) In eqn (3.25), all terms are known besides the strain rates involved in the square root which are obtained from the experimental data ( Cristescu [ 1991 ]) For the deter1nination of g'(t) we plot Fig. 3.6 using the data for several constant mean stresses. It was found that g'('t) is independent on mean stress as predicted by eqn (3.25), which implies that the viscoplastic potential exists. Thus the procedure to find the potential is logical. The solid line curve in Fig. 3.6 is an approximation for g'(t), which can be described by a polynomial function: 't g 2 r: 2 g8 r: 8 + + (3.26) can be written as

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0.3 g'('t) (s ') crMPa 0 8 0.2 14 20 0 30 40 0 1 0 0 5 10 15 20 25 30 35 't (MPa) Fig. 3.6 Data for various confining pre ss ure shown u sed to deterrnine g'(t); symbo l s test re s ult s, so lid line prediction of (3.26). g(-r:) = _!_g 2 I 't 2 1 + -g 3 2 a 9 57 (3.27) It is noted that in eqn (3.26), we start with the first order power Since a Fl a t should be equal to zero at t=O, we let g' ( t)=O at t=O in order to obtain the continuity of the derivatives of the viscoplastic potential. Therefore the viscoplastic potential for Gorleben rock sa lt is completely deterrnined ba sed on the experimental data and can be written as follows

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58 4 3 2 2 2 1 2ql q 2 ql kyF( a, -r) = a a a q2 ql a In a + 14 a 3 a a 2 a a /1 2 a a (3 28) a s b 5 + 1 5 a 9 b 9 + 1 9 a a + g(-r) + + b 5 + 1 a a b 9 +I a a * Using the expression (3.28 ) we get the shapes of the viscoplastic potential surfaces while by using eqn (3.4) for H(a -r) we can plot the shapes of the yield surfaces. Fig 3.7 shows the two sets of curves. It is clear that these two families of curves are quite distinct both in the compressibility and in the dilatancy regions. Let us note also that the CID boundary is quite distinct from the curve a w a a=O Thus, an associated flow rule cannot predict correctly the CID boundary Let us remind that the yield function and the viscoplastic potential are deter1nined from the data following two distinct procedures. It is found that F =t H. Thus, an associated flow rule is not suitable for this rock salt. Non-associated constitutive equation should be used with the potential furnished by eqn (3.28 ) 3.2 Comparison with the Data One simple way to test a model is to compare the experimental data with the model prediction. For the model developed in the previous s ections, we use three different criteria to check it. First, this can be done by trying to reproduce with the model the kind of tests which have been used to fon11ulate the model. Secondly, as an independent check we try to

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59 40 't ( MPa ) 3 5 a H ta cr = o 2 0 30 l. O 5 0 fa il u r e 25 0 4 20 H =l 0 0 2 0 .5 15 , krf=O l 10 .....-........ 0 05 0 1 0 2 C ID 5 0 02 0.005 0 0 1 "' 0 0 10 20 30 40 50 60 cr(MPa ) Fig. 3 7 Shape of yield surfaces ( thin solid lines ) potential surface s ( thick solid lines) CID boundary (dash-dot line ) and a H! a o=O (dashed line ) match a set of data which have not been used to forrnulate the model Third, we incorporate the model into a finite element program and try to describe triaxial te s t s. In order to reproduce with the model the data which have been u s ed to formulate the model, let us consider constitutive equation ( 2 6a ) It will be assumed that stresses are increased by small s ucces s ive steps according to the same law as in the experiments done to e s tabli s h the model, and with the same global loading rate a s in the experiments (Cristescu [ 1989, 1991, 1994]) At each very small loading step the stresses are assumed to increase instantly at time to, say Afterwards a creep follows under constant s tresse s in the time interval t t 0 We can integrate eqn (2 3 ) by multiplying it with o to get the following equation for the irrever s ible stres s work

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k a F W(t) = H(a(t)) -( H(a) W(t 0 ))exp T_: a(t 0 t) H a o The total strains as a result of one stress step can be written as follows 1 _ Wi_(t_ o) a F e( t) = ff (t) + ....:....-.-H---:..._a_a 1 a F ---:a H a a with the initial conditions 1 1 3K 2G 1 exp 1 al+-o' 2G where a (t 0 ) is the initial relative stress at each stress step. 60 (3.29) (3.30) ( 3.31) The above constitutive equations can be used to reproduce the experimental results. The viscosity coefficient kT can be determined from creep tests and used as a parameter which takes into account the speed of the tests as mentioned in the paper by Cristescu [1991 1994]. The constitutive equations are used to predict the stress-strain curves for the cases a= 14, o=20, a =40 and a=35 MP a. The prediction of the model matches well with the experimental data as s hown in Figs. 3.8a-3.8d where circles correspond to the experimental

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61 results, and the solid lines to the model prediction. As an independent check, for the experimental data at a=35 MPa which were not used to formulate the cons titut ive equation, there is also a very c l ose agreement between experimen tal data and the predicted results (Fig 3 8d). 20 ,--------------------(~Pa) 18 16 00000 14 12 10 8 6 4 2 cr= 14 MPa 0 ....... __._...._,.___,___._.....__......_.__.,_._......_.__.,_ __________________ ......._. -0.06 -0.04 -0 .02 0 0.02 0.04 Fig. 3.8a Experime ntal stress-strai n curves (circ l es) compared with predicted results (so lid l ines) for a= 14 MPa. 0.06

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25 -------------------, 't (MPa ) 20 15 10 5 Et cr= 20 MPa 0 "-------___ ....__ __ ___.. ___ __. -0.04 -0.02 0 0.02 0.04 0 06 Fig 3 8b Same as in Fig 3.8a. but for a=20 MPa 35 --------------------, 't (MPa)30 2 5 20 15 10 5 f I + ~ :-~ .... J. ~ l I ) l ; > l IJ (1 II ... '. I , I I , ,, , ,, ,, ,, ,_ , cr= 40 MPa 0 ...__._-'--' __ ..._._-'--'___.__ .__ .................... __.__ .......... ....__., __ ..._._-'--'_..__._........., -0 l -0 05 0 0 05 0 1 0 15 0 2 E Fig. 3.8c Same as in Fig 3.8a but for a=40 MPa (note : the data e 2 are slightly different from those for e 3 ) 62

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't {MPa) 25 20 15 10 5 a= 35 MPa 0 '--'-...a.....&.--'-,.___._...a.....L__,__ _,__,___._...L....I......L.~-'--"--'-...&....l~...i......&--'-.L.....L-' -0.1 -0.05 0 0.05 0 1 0.15 0.2 Fig. 3.8d Same as in Fig. 3.8c but for o=35 MPa as an independent check 3 .3 The Finite Element Analysis 3.3.1 Formulation of the ElasticNiscoplasticity Theory 63 The for 1nulation of elastic/viscoplasticity theory in discrete time forrn is presented in the book by Owen and Hinton [1980], with a classic approach by truncated Taylor series for the rate of viscoplastic strain (Zienkiewicz & Cor111eau [1974]; Hughes & Taylor [1978]; Owen & Hinton [1980]; Marques & Owen [1983]; Szabo [1990]; Desai et al. [1995]). There are many finite element programs with source code available, such as VISCO (Owen & Hinton [ 1980]) programs. These codes are usually for education and research purpose, which are explained in detail. There are also many commercial finite element programs without

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64 source code in public, such as ABAQUS [1993). Several standard open connections in these commercial programs can be used for users to write their own subroutines, but the open connections are limited In order to verify the model developed above, we use VISCO program (Owen & Hinton [ 1980]) as our basic program. Several subroutines were additionally developed for the implementation of the model. A brief description of the finite element algorithm for elastic/viscoplastic model is given in appendix A. During the computation, the process of time step marching is stopped at certain time or when the stresses at all Gauss points satisfy H=W(t) and continues in the next loading step. The time step d1n+i at step (n+ 1) was selected subject to the following empirical criteria I 12 (3.32) where 1 2 and ~ 1 are second invariants of the viscoplastic strain and strain-rate tensors respectively; Q and Kare specified constants to control time step. The first criterion selects a variable time-step size such that the maximum effective viscoplastic strain increment occurring during next time step is a fraction of the total effective strain accumulated before. dt 0 + 1 is evaluated at each Gaussian integration point and the least value is taken for computation. The second criterion imposes a restriction on the variable step size between successive intervals calculated by the first criterion to prevent oscillations in the solutions as steady-state conditions are approaching. In the following examples, only explicit integration scheme for viscoplastic strains is employed. For non associated flow rule, the stiffness matrix [~ 0 ) is symmetric and does

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65 not vary. The explicit scheme is simpler and easy to implement into the finite element program. However, the explicit scheme may not be stable when time steps become large. The restrictions expressed by eqn (3 .32) have to be imposed on the scheme so that the explicit scheme is kept stable The instability problem can be overcome by implicit schemes. 3.3.2 Example I: Axial Compression with Confining Pressure In true triaxial tests for rock salt (see Cristescu & Hunsche [1992]), the specimen is cubic. Stresses in two horizonal directions remain equal during the tests. Thus this kind of tests will be approximated by an axial symmetrical test where the friction between the cylindrical specimen and the piston is neglected. Due to the axial symmetry of the cylindrical specimen, we can use a quarter of the specimen only in order to create a finite element mesh. 6 Serendipity quadratic elements and 29 nodes are used as shown in Fig. 3.9. Each element has eight nodes. The boundary conditions are also shown in Fig. 3.9. We will use the same loading procedure as used in the experiments. Hydrostatic loading is applied first and afterwards the deviatoric loading is performed under constant mean stress. Let us consider the test for the case a=40 MP a. The total hydrostatic loading is divided into 10 steps. The computation remains stable for all the steps. The total deviatoric load is also divided into 10 steps with unequal values. In each loading step, the computation is carried on until the condition H( a(t))=W(t) in every Gaussian integration point or a convergence condition is satisfied. As a convergence condition we use here the ratio of the second invariants of irreversible strain rate evaluated at the initial and the current moment. The tolerance chosen is 0.1 %. In the first eight steps shown in Fig. 3.10, the scheme is stable

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66 and convergent. However when we apply the ninth stress step the scheme becomes unstable and blows up. In this computation, 0=0.0009 and K= 1.1 were used The results are shown in Fig. 3 10 The results obtained with the finite element program give a reasonable matching with the experimental results although the scheme in the finite element program blows up near failure The instability problem in finite element programs can be overcome by implicit scheme s ( see Hughes & Taylor [1978]; Simo & Govindjee [1991] ) / Fig. 3 9 Mesh of a quarter of cylindrical specimen u s ed in the finite element analysi s

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67 3.3.3 Example II: Stress Analysis for A Cylindrical Cavity in Rock Salt Let us consider now a cylindrical cavity such as a borehole or a shaft with the same initial or primary stress in all horizonal directions ( ahx=a hJ For simplicity, primary hydrostatic stress is assumed, i.e the primary stress in the vertical direction a v is taken to be the same with that in the horizonal direction. At excavation, the diameter a of the cavity is assumed to be 1 meter. We make some idealized assumptions: the problem is a plane strain one with primary stress aruc = ah y = a v= 10 MPa in the first example and 30 MPa in the second example, and that the primary stress is instantaneously released at the cavity surface due to excavation at time t 0 Afterwards the rock salt around the cavity creeps according to the above constitutive equations. We use a quarter proftle of the cavity for our computation. The corresponding boundary conditions are shown in Fig. 3.11. Here the cylindrical coordinate r is used for the distance from the axis of the cavity to some point inside the rock mass ( e.g., r=a represents the surface of the wall) Two meshes are used for computation. An area with 7 m in length and 7 m in width is first considered with 253 nodes and 72 elements as shown in Fig. 3 11. Each element has 8 nodes. In the second variant, we enlarge the area to 20 m in length and 20 min width and refine the mesh to a total of 333 nodes and 96 elements. The computed results obtained with the two meshes do not differ significantly. Figs 3 12-3.16 show the results obtained with the finer mesh. In the computation, we choose for the restriction parameters in eqn (3.32): 0=0.01 and K= 1.3. The computation is stable and convergent. Figs. 12-15 show the variations of the stresses and octahedral shear stress with the distance from the surface of the wall for the two cases. A comparison with the elastic solution of the ultimate elastic/viscoplastic one (i e.,

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35 ---------------------, 't 30 (MPa ) 25 20 15 10 5 E v cr=40 (MPa) 0 I..-J.-L-L-.L....L---L........L.....L..-J L...-A1.-,L-'--'--'---'--'-~---'-....&.........__L.....J.___.__. -0.1 -0.05 0 0.05 0.1 0 15 68 Fig. 3.10 Comparison between experimental data ( solid line) with FEM results (triangles) computation was carried out up to stabilization) shows a significant change in stress distribution. The circumferential stress 0 8 decreases near the wall of the cavity and then slightly increases at farther distances (Fig. 3.12 and 3 14). The vertical stress o z and the radial stress or decrease with respect to the elastic solution. The octahedral shear stress 't' is smaller near the surface of the wall, but greater at farther distance (Fig 3 15 and 3.15 ) At greater di s tances from the surface of the wall, the difference between the two s olutions becomes negligible and the stresses approach asymptotically the primary values In the case of initial stress 10 MPa all regions are compressible. In the case of initial stress 30 MP a, there is a dilatancy region expanding up to about 0.4 m from the s urface of the wall. The other regions around the openin g are compressible The displacements inside the rock vary in time Fig.

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69 3 .16 shows the variation of the displacement at the surface of the wall, starting from the elastic "instantaneous" response up to stabilization of transient creep The upper curve corresponds to the case of initial stress 30 MPa and lower curve to the case of initial stress 10 MPa. A dimensionless quantity kTxt is used for the "time" parameter, where kT is the viscosity coefficient and t is time. If kT is determined from creep tests (Cristescu 1989), the time t can also be made precise. Here the intention was to give an example only, obtained with some simplifying assumptions in order to illustrate the use of the model in a mining problem. More elaborate engineering problems could also be analyzed. 7 0 0 7 r/a Fig. 3.11 The finite element mesh used for a vertical cavity.

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cr (MPa) 20 15 10 5 0 L------1---___., ________________ ...__ __ __ 1 2 3 4 5 6 r/a Fig. 3.12 Stress distribution with distance from the surface of the cavity wall for the primary hydro sta tic stress 10 MPa ( thin lines-instantaneous elastic solution; thick lines-ultimate elastic/viscoplastic solution obtained when (t-to)xkT=6 6). 8 't (MPa) 7 6 5 4 3 2 1 1 3 5 7 9 1 1 13 15 17 r/a 70 Fig. 3 .13 Variation of the octahedral shear stress with distance from surface of the cavity for the primary hydrostatic stress 10 MPa ( thin line instantaneous elastic solution; thick line ultimate elastic/viscoplastic solution obtained when (t-to)xkT =6.6).

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60 ,----------------------, O' (MPa) 50 40 30 20 10 0 L--L----1---..,___-"'-_ __._ ___ ___., ___..___. 1 4 7 r/a 10 Fig. 3.14 Same as in Fig. 3.15 but for the primary hydrostatic stress 30 MPa, ( ultim ate e la st ic/vi scop la stic solution obtained when ( tf:o)xkr=5.6). 't (MPa) 20 15 5 1 4 7 10 13 16 / ra Fig 3.15 Same as in Fig. 3.16 but for th e primary hydrostatic st re ss 30 MPa, ( ultim ate elastic/viscoplastic sol ution obtained when (t-to)xkT =5 .6). 71

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u/a 0.0035 0 003 0 0025 0.002 0 0015 0 001 0 .................................... ......._ ..................................... .__ ......... ...__....,_ ___ ......_ _____ 0 1 2 3 4 5 6 7 72 Fig. 3.16 Variation of relative radial displacement (u/a) of the surface of the wall with dimension]ess "time" I<,.x(t-t 0 ) (upper curve, for primary stress 30 MPa; lower curve for primary stress 10 MPa) 3.4 Discussion and Conclusion In present model transient creep is taken into account only. Since this model is a phenomenological one based on s hort ter1n tests, it may underestimate the strain magnitudes Actually sta tionary creep should also be taken into account when one wants to predict stress or displacement distributions around some underground openings. Stationary creep may also contribute to the stress redistribution around an underground opening due to the nonlinear behavior of rock salt. The finite element program for both transient and stationary creep should be used in th i s case The possible influence of kinematic hardening will also be considered in the future papers, if enough experimental data would be available. Temperature

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73 and humidity are other important factors which influence the behavior of rock salt. We have not considered them in the present model. In this formulation, non-associated flow rule was used. It was obtained from experimental data. Many models adopted nonassociated flow rule (Desai et al. [ 1987]; Kim & Lade [ 1988]; Lade & Kim [ 1989a, 1989b]; Klisinski et al. [ 1992]) although non-associated flow rule has been found to cause some problems (Sandler & Rubin [ 1987]). Using the triaxial experimental data on Gorleben rock salt, a new nonassociated elastic/viscoplastic model for transient creep is proposed. This elastic/viscoplastic model can be used to describe creep, relaxation, dilatancy and/or compressibility and work hardening. Singularity and asymptotic properties of the yield function and of the viscoplastic potential which has to satisfy several restrictions are considered. A new procedure to determine the viscoplastic potential is developed. This procedure allows to better fit data and keep the viscoplastic potential smooth. The examples given show that the model predicts quite well the experimental data. Since both the yield function and viscoplastic potential are smooth functions, the present model can be incorporated quite easily into a finite element program. An explicit integration scheme with very small time steps for viscoplastic strain components is used. Several examples show that the model is easy to use in the finite element program. Instability problems during our computation have not been found except the unstable computation caused by explicit scheme for viscoplastic strains although non-associated flow rule has been found to cause some problems (Sandler & Rubin [ 1987]). Implicit integration schemes may be better to overcome unstable computation problems.

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CHAPTER4 llo..i.l L&.J EXPERIMENT AL RESULTS ON ALUMINA POWDERS In this chapter, the mechanical behavior of alumina powder has been investigated experimentally under hydrostatic and deviatoric loading conditions in a triaxial apparatus. Two types of alumina powder, as received AlO and A16-SG, were tested. The specimens were fully saturated with water. The elastic parameters of alumina powders were measured based on the loading-reloading processes following a creep or relaxation period. The experimental results reveal that the mechanical behavior of alumina powder is strongly dependent on the particle size and on the initial density. The deformation of alumina powder is time dependent and stress history dependent. The application of deviatoric stress can produce additional consolidation or dilatancy. The compressibility/dilatancy boundaries and the failure conditions have been obtained for the tested powders. Also, the experiments have been perfor1ned according to the requirements presented in the previous chapters to for1nulate a three dimensional constitutive equation. Thus the experimental results provide a set of data for the for1nulation of three-dimensional, either viscoelastic or viscoplastic, models. In section 4.2, the basic testing procedure is presented and discussed, which consists of four sequential steps: (1) specimen preparation, (2) back pressure saturation, (3) hydrostatic consolidation or hydrostatic loading, and (4) shearing (deviatoric loading). In 74

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75 section 4.3 the experimental results under hydrostatic loading conditions are presented The results under deviatoric loading conditions are presented and discussed in section 4.4. 4 1 Introduction Fortning processes are crucial in the ceramics industry today. Among various forming processes, casting is commonly used for the farming of products. One of the casting processes is pressure casting. This process has been investigated as a farming technique for porcelain, complex refractory shapes or hard ferrite magnets, etc. The casting time is usually controlled by regulation of the external pressure (Reed (1988]). In pressure casting, the mold serves as a filter. The externally applied pressure is usually increased up to 1.5 MPa. Water in the cast is squeezed out. This process reduces the water content significantly so that the drying shrinkage can be reduced. However, pressure casting of slurries or pastes in a mold creates density gradients due to the cast geometry or the friction between the material s and the wall of the mold. In other words, the density distribution is not uniform in the cast. This non-unifor1n density distribution will cause problems such as warping in the later sintering process. In order to understand the process and avoid non-unifor1n density distribution in a mold, the consolidation behavior, or stress strain relationships, of slurries or pastes must be investigated. Slurry or paste of alumina powder fully saturated with water is often used for pressure casting. Therefore the consolidation behavior of alumina paste plays an important role in pressure casting. Whether or not we can get good products from this kind of casting depends

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l 76 on how well the consolidation behavior of powder or powder paste is understood. To understand the consolidation behavior of alumina paste at different stress states, we need to know the three dimensional stress strain relationships, i.e., constitutive equations, of these powders. To forrnulate constitutive equations, triaxial tests are often used. Triaxial tests have been successfully used in civil engineering for geomaterials such as sand, clay, and rock to obtain general three dimensional constitutive equations (Bishop & Henkel [1962]; Cristescu [1989]). However, powders such as alumina powder or alumina paste are different from sand or clay. Usually powder possesses more pure constituents and surface chemistry of particles is also involved (Reed [1988]). Hence the mechanical properties of powders cannot be obtained directly from those of sand, clay, or rock. Several triaxial tests for metal powders or dry ceramic powders at high pressure were reported (Shima & Mimura [1986]; Brown & Abou-Chedid [1994]; Gurson & Yuan (1995]). Unfortunately, very few triaxial tests have been perfor1ned for ceramic paste such as alumina powder (Stanley-Wood [1988]). Triaxial tests are quite different from uniaxial tests. In a uniaxial confined test, powder is placed into a rigid cylindrical mold or die and pressure is applied along axial direction using a piston. Such tests reveal mainly one dimensional relationship between pressure and volume changes (Gethin et al [1994]; Chen et al (1994]; Shapiro [ 1995]). However, due to the friction between the powder and the wall of a mold, there exist very complicated stress states in the mold and the final density distribution is nonuniforrn. Consequently, uniaxial tests cannot provide good basis for the formulation of three-dimensional relationship between stress and strain. On the contrary, triaxial tests reveal the def orrnability characteristics not only under hydrostatic pressures, bt1t also under

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77 deviatoric loading conditions ( e.g., shearing under different constant confining pressures or under constant mean stress conditions). The stress-strain relationships obtained in deviatoric tests reveal the three-dimensional consolidation behavior. These tests furnish necessary data for the formulation of the three dimensional constitutive equations (Cristescu [ 1994]; Gurson & Yuan [1995]). In this chapter, triaxial tests are perfor111ed to investigate the volume change behavior of alumina powder under different stress states. The effects of particle size, initial density, and time are studied. A series of triaxial tests on three types of alumina powders provided by ALCOA (Aluminum Company Of American) are perfor1r1ed. The first is as-received AlO alumina powder with an initial volume fraction of 0.36; this type will be labeled as "dense A 1 O". The second one is also A 10 alumina powder, but with a smaller prepared initial volume fraction of 0.334, labeled as "loose AlO". The third one is as-received A16-SG alumina powder with a prepared initial volume fraction of 0.41. The specifications of these materials are shown in Table 4.1. The elastic parameters were measured for all three types of alumina powders based on the method proposed by Cristescu [ 1989] for time-dependent materials. The triaxial test results show that the consolidation behavior of alumina powder is strongly dependent on particle size, initial density, and time. In all cases a higher pressure results in more consolidation. However, deviatoric stress can lead to either consolidation or dilatancy. For the stress ranges applied to the A16-SG alumina powder, no dilatancy was observed, while dilatancy occurred for both AlO alumina powders. The compressibility/dilatancy boundary and the failure condition have also been measured. These data can be used to for1nulate a general visco-elastic or elastic/viscoplastic model to describe

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78 the consolidation behavior of these alumina powders. In the present research, a total of thirtythree triaxial te sts have been performed. These tests are listed in Table 4.2 and Table 4.3. Most of the data are given in Appendices B, C and D. Table 4.1 Specification of Alumina Powders type of prepared initial density* Particle relative alumina (g/cm 3 ) size(m) density AlO (dense) 1.42 (two layer+vacuum) 40~200 0.36 AIO (loose) 1.32 (pluviation+vacuum) 40~200 0.334 A16-SG 1.62 (tap+vacuum) 0.5~ 1.0 0.41 The density after saturation **Theoretica l Den sity of Alumina=3.95 g/cm 3 (see Richerson [1982]) + void ratio= VvNs, where Vv is volume of void, Vs is volume of solid material in the specimen. 4.2 Experimental Procedures 4.2.1 Triaxial Equipment Setup void ratio+ 1.78 1.99 1.44 Fig. 4.1 shows the schematic diagram of triaxial equipment used (Bishop & Henkel [1962]).

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Table 4.2 List of triaxial t ests for alumina pow d er (dense and loose AlO) No. date type initial confining "B" loading Number Number failure CID comments 1995 of t est density pressure value r ate u nloading u nl oading stress stress -1996 (g/cm 3 ) (kg/cm 2 ) (s-1) on hydro. on deviat. (kg/cm 2 ) (kg/cm 2 ) 1 2/14-17 CT 1.466 3=(7-4) 5.6e-5 no 1 9.28 5.2 -5. 6 Dry, try test 2 3/7-8 CT 1 .38 8 3=(6-3) 0 93 5.6e-5 2 1 7.02 5-5.5 not max stress 3 3/13-16 CT 1.424 3=(6-3) 0.95 5.6e-5 1(8steps) 1 8.6 4.5-5 4 4/11 13 CT 1.424 3=(7-4) 0.97 l.le-5 4(7steps) 3 8.99 5-5.2 5 4/17-18 CT 1.424 2=(6-4) 0 96 l le-5 4 ( 7steps) 6 6.02 2.75-3 unload too mu c h 6. 4/19-20 CT 1.423 2=(6-4) 0.94 l .le-5 3(4steps) 5 5.99 3-3.2 7~ 4/25-26 CT 1.423 1 = ( 5-4 ) 0.97 1.le-5 1 2 3.11 1-1.2 8 5/1-2 CT 1.421 4=(7-3 ) 0.97 l. le-5 7 (8s teps) 5 10.93 8.3-8.5 not max stress 9 5/3-8 CT 1.42 4=(5.5-1.5 ) 0.94 l.le-5 3 6 11 .38 7 8-8 wro n g on hydro 10 5/1011 CT 1.428 5= ( 6.5-1.5) 0.945 l. le-5 5 6 14.4 1 10 5-10.8 11 6/27-29 CT 1.415 5=(6.5-1.5) 0.94 l.le-5 5 5 13.79 10.6-10.9 12 7/3-5 CT 1.425 5=(6.5-1.5 ) 0.94 l. le-5 5 0 14.28 10.6-10.9 13 7/18-19 CT 1.422 3=( 4.51 .5) 0.95 l. l e-5 5 3 8.76 5 0-5.2 14 8/2-3 CT 1.415 4=(5 51 .5) 0.95 l.le-5 4 0 11.42 7 0-7 .2 15 .. 8/8-9 CT 1.325 4=(5 5-1 5) 0.95 3.4e-5 4 2 9.58 9.3 p l uviation test

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No. 16 .. 17 .. 20 21 22 23" 24 25 26* Table 4.2 Contin u ed date type initial confining "B" l oading Number Number 1995 of test density pressure value rate unloading unloading -1996 (g/cm 3 ) ( kg/cm 2 ) (s-1) on hydro on deviat 8/16-17 CT 1 338 5=(6.5-1.5) 0.96 3.4e-5 5 3 10/25-26 CT 1.316 3=(4.5-1.5) 0.97 3 4e-5 3 3 10/30-1111 CT 1.325 1=(2.5-1.5) 0 97 3.4e-5 1 1 lln-8 CT 1 .313 2=(3.5-1.5) 0.96 3.4e-5 2 2 11114-15 CT 1.412 3.5=(5-1.5) 0 95 l. le-5 4 2 12112-13 CT 1.44 3.5=(5-1.5) 0.93 1.le-5 NIA NIA 12119 20 CT 1.421 3.5=(5-1.5) 0.96 l .le-5 4 2 12127-28 RL 1.425 4=(5.5 1.5) 0.96 relax (2 steps) (8 steps) 113-4 CT 1.411 4=(5.5-1.5) 0 97 5 4e-5 4 0 118-9 CT 1.42 4=(5.5-1 5) 0.96 3.3e-4 4 0 1111-12 CP 1.4 1 9 4=(5.5-1 5) 0.96 (1 step) 0 Note: CT=conventional triaxial test (Karman test); RL=relaxation test ; CP=constant mean stress test Confining pressure = ( chamber pressure back pressure) The tests (total 11) have been listed in Appendix B The tests ( total 5) have been listed in Appendix C failure CID stress stress (kglcm 2 ) (kg/cm 2 ) 12.23 12.1 7.04 6.85 2.46 2.1 4.75 4.3-4.4 9.64 6-6.1 1 0 17 6-6. 1 10.8 ( min) 7.0-7.2 11 34 7.3 7.5 11.77 8-8 1 5 98 1.6 comments p l uviation test pluviation test pluviation t est pluviation test chamber l eak too large density 12 6 (max stress) very smal l dilat. no effect on fail 00 0

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No. 1 + 2+ 3+ 4+ 5 6+ 7 Table 4.3 List of triaxial tests for A lu mi n a powder A 16-SG date type initial co nfming-p "B" loading N u mber 1996 of test density ( cham. -back ) value rate unloading (g/c m 3 ) (kg/c m 2 ) (s') on hydro 1117-18 CT 1 .632 4=(5.5-1.5) 0.96 3.4e-5 4 2116-18 CT 1.604 2=(3.5-1.5) 0.95 3.4e-5 2 316-7 CT 1 .63 5 3=( 4 .51 .5) 0.96 3.4e-5 3 3111-12 CT 1 .628 5= (6. 5-1.5 ) 0.95 3.4e-5 5 311314 CT 1 .623 1 =(2.5-1.5) 0.95 3.4e-5 l 3126-27 CP 1.634 3=( 4.5-1.5) 0.95 3.4e-5 ( I s tep) 418-10 CP 1 .646 4=(5.5-1.5) 0 .98 3.4e-5 (1 s tep ) Note : CT= conve ntio n al triaxia l t est ( Karman test ); CP=constant mean stress test Confining pressure= (c hamber pressure ba ck pre ss ure ) + the tests ( total 6) have bee n listed in Appendix D Number failure CID comments unloading stress s tres s on deviat (kg/cm 2 ) (kg/cm 2 ) 1 11.88 NIA compress.-only 2 5.65 N I A compress. -only 3 8.79 NIA com pres s. -o n l y 2 14 .83 NIA compress. -only 1 2.97 NIA compress.-only 0 4.34 NIA compress. -o nl y 0 5 .66 NIA compress. o nly

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essure vac uum/ backpr burette axial force dial \ \ I > '/ \'. , "T ' alumina powder specunen .. .. I [ base motor ] kpressure e vacuum/bac I v 1....water ,_ .. ~) ( air pressure dial Chamber pressure) displacement measurement _.,..... c hamh er 11 orous s ton e merobra ne r ate of displacemen1 loading direction Fig. 4.1. Schematic diagram of triaxial equipment setup. 82 The equipment consists of primarily a chamber, framework, panel, and base motor. There are several valves to control the flow of water a nd/or air. Two inlets/outlets at the bottom and the top of the specimen are connected to a vacuum or/and a burett e on the panel. The following quantities have been measured: axial force by load cell, axial displacement of specimen by digital ga uge chamber pre ss ure and backpressure by pressure tran sduce r s re spect iv ely as shown in Fig. 4.1. Chamber pressure can provide an all-around pressure on the top and the lateral surface of specimen. Back pressure acting as pore pressure can provide the pressure of fluid in the specimen

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83 4.2.2 Specimen Preparation The specimen is prepared inside a split mold. The split mold consists of two half cylindrical shells which are connected with a vacuum pump. First, two clamps are put to hold the mold. A membrane of about 0.12 mm thlckness is put inside of the mold and the ends of the membrane are tightened on the mold by several 0-rings. A porous stone is placed at the bottom of the mold. The mold is then put on the pedestal which connects all necessary fluid inputs/outputs. A vacuum is applied to pull the membrane to the mold sides. There are two methods to prepare the specimen in order to obtain a desired initial density One is the so called layer method The other one is called pluviation method. In the layer method, the specimen is prepared in the successive layers. For each layer, a constant density is maintained by certain height with a given amount of powder material. A small vibration is applied by taping the side of the mold for densification. The maximum variation in relative density between specimens can be kept within 0.5%. In the pluviation method, powder is poured into the mold slowly without tapping the mold until the mold is full with powder. After the mold is filled with powder, the top porous stone and cap are placed on the specimen. The vacuum pump is disconnected from the mold and then applied to the top and bottom inlets of the specimen. The mold is then removed. The specimen is supported by about 100 kPa vacuum pressure The average diameter D 0 of the specimen is obtained by measuring the diameter on the top middle, and bottom. The average initial height of the specimen, H 0 can also be obtained by measuring heights at four different points. By knowing the theoretical material density Yt of powder, weight of specimen W (powder is assumed dry), and the volume of the specimen V, the volume fraction of the powder can be computed:

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84 relative density = volume fraction = Y = W Yt Vy, (4.1) Afterwards the prepared specimen is put into the chamber. A cylindrical specimen, about 70 mm in diameter and 145 mm in height, is used for all tests. It is assumed that the specimens deform uniformly and the end effect is neglected Since the particle size is less than 0.2 mm, the membrane penetration effect is also neglected (Frydman [1973]). 4.2.3 Back Pressure Saturation The next step is to saturate the specimen. First, water is deaired in the burette for about 20 minutes. Then water is allowed to flow into the bottom of specimen due to vacuum pressure existing in the specimen. An alternating sequence of vacuuming the top of specimen and allowing water in from the bottom is followed until most of the trapped air is removed. A small confining pressure ( a 3 =30 kPa) is maintained during saturation to support the specimen with a certain back pressure in the specimen. A "B" check is performed to verify the saturation. Here "B" is defined by B : =Llu/Lla 3 in which Lla 3 is the increment of chamber pressure and Llu is the increment of the pore water pressure in the specimen with undrained conditions. During the tests, the volume of specimen changes Saturation is essential as the volume change is measured via the water volume entering/exiting the sample. Usually the "B" value should be higher than 90% when the volumetric strain is to be measured under drained conditions In order to achieve a high accuracy of the measurement of volumetric strain, the "B" value is required to be as large as possible In our tests, municipal tap water

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85 is used. In order to get higher B values, the prepared specimen is subjected to a backpressure of 147 k.Pa over night. The B valve measured the next day is higher than or at least equal to 93o/o in all tests. See Table 4.2. The change of the height of specimen, ~H, during saturation is measured by attaching the dial gage to the piston rod before saturation begins, and noting zero. Subsequently, the small confining pressure (30 kPa) applied during saturation and the back pressure saturation will cause volume reduction of the specimen. Since ~D (the change of the diameter of specimen) cannot be measured while ~H can, it is assumed that ~HI H 0 = t:JJ I D 0 (axial and radial strains are equal). The corrected section area A s after saturation is thus obtained, (4.2) where e is axial strain after saturation. 4.2.4 Hydrostatic Loading Test We begin with an incremental hydrostatic loading test ( also called consolidation test or isostatic test) Hydrostatic stress is computed as the chamber pressure minus the backpressure. At each loading step, the stress is kept constant for several minutes and the sample is allowed to shrink by creep until the rate of the volumetric strain is very small. Usually it talces ten minutes. Afterwards the specimen is unloaded with small stress increments and the stress is kept constant for 5 minutes. Then the specimen is reloaded to reach the previous stress level. In this way accurate values of the elastic bulk modulus can

PAGE 94

86 be calculated (Cristescu [1989]). From such tests, the volumetric change of specimen due to both hydrostatic stress and, for a given stress level, time can be obtained. Since the hydrostatic pressure causes specimen compression, the section area of the specimen is corrected according to the foil owing formula ( v tiv tiV) A = 0 s at c C H !:JI 0 (4.3) where: VO is initial volume of specimen, ti V e is volume change of the specimen during consolidation at each step, ti V s at = 3 V 0 !iH 9 I H 0 is the volume change of the specimen during saturation, and !iH is the change in height of the specimen at each loading step plus the change during saturation. 4.2.5 Deviatoric Loading Test The second stage of the triaxial test is the deviatoric loading test. There are two ways to perlorm deviatoric tests One is called "conventional triaxial test". In the deviatoric stage of conventional triaxial tests, the confining pressure (lateral stress) is kept constant and only the axial stress is increased. The confining pressure is the difference between chamber pressure and back-pressure. The deviatoric stress is defined as the difference of stress between the axial stress and the lateral stress. The other is called "constant mean stress test" or ' constant-p test". In the deviatoric stage of constant mean stress tests, the mean stress is kept constant and only the deviatoric stress is increased.

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87 Loading was performed at a constant displacement rate and the following quantities in all deviatoric tests were measured: axial force chamber pressure back pres s ure the length change of specimen, and the volume change of specimen. The axial stress, confining pressure axial strain, and lateral strain are computed from the measured quantities During the test the section area of cylindrical specimen changes. The following correction is applied during te s ts: V ~ V 1 e A I= C = A V c H ~ c 1 e c l ( 4.4 ) where V c H c and A c are volume height, and section area of specimen, respectively after the hydrostatic test, V i s the volume change, ~H is the change of the height during deviatoric test, ~e v = V N c and e 1 = ~ / H The deviatoric stress is defined as ( 4.5 ) where Pi s the axial load. Eqn ( 4 4 ) is also suitable for area correction for other type of tests such as constant mean stress te s t or relaxation te s t

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88 4.3 Experimental Results 4.3.1 Hydrostatic Tests Three series of hydrostatic tests (first stage of a triaxial test) on alumina powders provided by ALCOA have been perforrr1ed (see Table 4 2 and Appendices B, C and D) The results of these hydrostatic tests are presented and discussed below The hydrostatic tests were perforrned with an unloading-reloading process in order to obtain the elastic parameters. Fig. 4.2a shows a typical relationship between volumetric strain and time in a hydrostatic test for dense A-10 alumina powder at different stress levels, while Figs 4.2b and 4.2c show similar results for loose AlO and A16-SG, respectively At each plateau the number shown marks the pressure applied. For each plateau shown, the successive points correspond to 1.0, 2.0, 5 0, and 10.0 minutes (a total of about 10 minutes) after each reloading. Afterwards an unloading increment of 20 kPa pressure is perf onned and the remaining stres s is kept constant for 5 minutes, followed by a reloading with the same 20 kPa, and then the final stress same as the previous one is kept constant for 5 minutes. The cycle of unloading-reloading curve allows us to calculate the elastic bulk modulus. It was found that A16 SG exhibits much more volumetric creep defor1nation than AlO's for the same testing time interval and same pressure, i e the Al6-SG alumina is much more compre s sible than the A 1 O's. After each loading, the volumetric strain continues to vary in time (creep) as shown in Figs 4.2a-4 2c. The volumetric strain for A16-SG exceeds 6 % while the volumetric s train for dense AlO and for loose AlO is only about 0.6 % and 1.1 % under the same conditions (pressure 490 kPa), although Al 6-SG is initially more dense than

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89 A 1 O's. This difference in compressibility is mainly due to the difference in particle size of the powders. The particle size of AlO alumina is in the range of 40~200m, while the particle size of A16-SG is in the range of 0.4~ lm. The particle shape of both alumina powders is irregular. However, due to the different particle size, the surface area of particle in these two powders is also different. The surface chemical properties of particle are certainly involved, which will affect mechanical properties. This aspect should be investigated separately. As already mentioned, the unloading processes are used to obtain the elastic parameters. The unloading processes also allow us to separate the time effect from elastic properties. After each loading, the specimen is allowed to creep under constant pressure for IO minutes. When the rate of volumetric strain becomes very small, the unloading is performed. During the short time necessary to perfor1n the unloading, the contribution of the deformation by creep is negligible since then the material has reached nearly a quasi-static state (no much strain can be obtained with the same stress afterwards as shown in Fig.4.3). Thus, the influence of the time effect on the unloading process is negligible. The elastic bulk modulus can be obtained from these unloading processes, as shown in Fig. 4 3, as the slope of the unloading curve. The bulk modulus obtained by following this procedure is very close to that obtained by using dynamic method for other materials such as rocks (Cristescu [ 1989]) It is expected that for alumina powder the value obtained by the unloading processes is also close to that obtained by dynamic method.

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0.006 490 392 0.004 ,...,.,., ... 294 0.002 r"\.J ... a= 196 (kPa ) den se AlO ,.."'\..J .. 98 0 .__ ___ ..___ ________ '-___ .___ __ __. 0 0.5 1 1.5 2 2.5 t (h) Fig. 4.2a Volumetric strain versus time for den se AlO (poi nts-reading data). 0.012 490 0.01 392 0.008 0.006 cr =29 4(kPa ) 0 004 196 loose AlO 0 ....,a....1-.a.....&....L-.L....L-'-.L....L-'-.L....L....L....L...L.....L...'--'-...L-'--'-...L-'--'-...L-L...L.-L...J 0 0.5 1 1 .5 2 2.5 t (h) Fig. 4 2b Same as 4.2a but for loo se Al O ( point s -reading data ). 3 90

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91 0.07 Ev 0.06 490 0.05 392 0.04 294 0.03 0.02 cr = 196 ( kP a) 0.01 98 0 0 0.5 1 1.5 2 2.5 3 3.5 t ( h ) Fig. 4 2c Volumetric strain versus time for A16-SG (points-reading data). It was found that elastic bulk moduli change smoothly with mean stress as shown in Fig. 4.3 and Fig. 4.4. The elastic bulk moduli may also depend on other material properties such as the density ( Gurson & Yuan [1995]). As a first approximation it is assumed that the elastic bulk modulus depends on mean stress only. There are several possible functions which could be used to fit the data if the behaviors of the elastic moduli at very high pressure or/and very low pressure are also taken into account (Jin et al. [1996a]). Here, a linear approximation is used to fit these data. It gives a reasonable fitting of the data in the pressure interval s hown in Fig. 4.4a and 4.4b. The bulk modulus K can be expressed a s ( 4.6) where ko and k 1 are material constants. For comparison, the values of the coefficients k 0 and

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92 k 1 for different types of alumina powders are listed in Table 4.4 It is seen that the denser material has larger k 1 However the linear relationship ( 4.6) is not expected to be valid at high pressure since K is expected to reach a limiting value when all pores are closed at very high pressures O' ( kPa ) 500 r-----------------;:::;;:..-, 400 300 200 dense AlO 100 0 .__ __ __.__ __ ____._ ___ ...__ __ __,_ ___ .___-----J 0 0 002 0.004 0.006 Fig 4 .3 a Hydrostatic stress versus volumetric strain in hydrostatic loading for dense AlO (points-recorded data)

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O' (kPa) 500 r--------------.:=.:::::.;:.:::::.-----, 400 300 200 A16-SG 100 0 0.01 0 02 0 03 0.04 0 05 0.06 0.07 Ev Fig. 4.3b Hydrostatic stress versus volumetric strain in hydrostatic loading for A16-SG (points-recorded data). K (kPa ) 3x10 5 2x10 5 5 lxlO Ox 1 o 0 "---"---.11.-_....._ __., __ ...L..._ __.,_ __ _,__ __. 0 100 200 300 400 cr (kPa) Fig. 4.4a Variation of elastic bulk modulus with hydrostatic stress for dense alumina A-10 (points-data from different tests). 93

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94 lxl0 6 --------------------, K (kPa ) 8 x10 5 6x l0 5 4xl0 5 2x 1 0 5 O xl0 L--____., __,_ __.__~ _,___...._ __ ___. __ ___,_ __ 0 10 0 2 00 3 00 400 500 cr (kPa ) Fig. 4.4b Same as Fig. 4.4a but for A16-SG. Table 4 4 Elastic bulk modulus and Poisson ratio v values Tvoe of alumina ko (kPa) k, V loose AlO 68675 426 0.25~0.29 dense AlO 47525 834 0.3~0.35 A16-SG 34200 1640 0.39~0.40 For the alumina powder of same particle size, the initial density has an effect on the hydrostatic behavior as shown in Fig 4.5 for the AlO's For the same pressure level the loose AlO alumina deforms more than the dense AlO. Also, during the same period of time, the loose AlO exhibits more creep deformation than the dense AlO which is intuitively obvious. Fig. 4.6 also shows that the relative density of loose AlO is always smaller than that of dense

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95 AIO at all stress levels. fu the pressure interval considered, the relative density of loo s e AlO cannot catch up that of the dense A 10. cr (kPa ) 4 0 0 1 .--~.------------~,::::;.:=:i..==::.7 A-10 ( d e n se) 30 0 A-16 SG A-10 ( l oose) 0 L...----J..-...L---...Ji..,___.i..,_.._____. _..__-__,, _.._ __ ____. 0 0. 01 0 .0 2 0.0 3 0 04 0 05 0. 06 E v Fig. 4.5 Stress-strain relationship for three types of alumina powder s The particle size has a large effect on the hydrostatic behavior The alumina with smaller particle size A16-SG ( 0.4~ lm ) is easily compressed even though the initial density of A16-SG is larger than that of A-10 ( particle s ize 40~200m ) For the same period of time, A16-SG exhibits much more deforn1ation by creep than Al O's under the same pres s ure as shown in Fig. 4.6. For A16-SG there is more than 2 % volume reduction However for the same time period and the same pre ss ure as for A16-SG, the increase in relative density for dense AlO is about 0.15 % only and that for loose AlO is about 0 3 % only. More volume reduction can be obtained if the pre s sure is applied during a longer time period.

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cr ( kP a) 400 ...-------~----------~ = =----, A 16 SG 300 200 A 10 ( loo se) A 10 (de n se) 100 0 '---'---------------...__-____ _._____...._ ____ _, 0.33 0 .3 5 0 .37 0.39 0 41 0 43 0.45 relative den s ity Fig. 4.6 Variation of relative density (volume fraction) with pressure and time. 96 Now let us consider for each loading step the points at the end of creep period shown in Fig. 4 3 These points represent the quasi-stable states reached at each loading step and thus, with some approximation, can be considered to belong to the relaxation boundary. Fig. 4 7a-b shows these points (diamonds) for various hydrostatic tests for dense AlO and A16SG alumina powder respectively. The relationship between stress and volumetric strain can be approximated as a linear one. Since during the process of saturation, a 30 k.Pa hydrostatic pressure was applied before tests, the primary strain should also be taken into account with the boundary condition e~ I o = O = 0. Thus, the relation between stress and volumetric strain can be written as

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97 (4.7) where a = l.26x10 5 k:Pa t and 0.000138 k:Pa 1 for dense AlO and A16-SG alumina powder respectively as shown in Fig. 4.7a-b. This linear relationship is valid only for the pressure range considered. It is not expected that eqn (4.7) be valid at high pressure since the volume of powder cannot be reduced to zero. 0.006 --------------------0.005 0.004 0.003 0.002 dense AlO (a) 0.001 0 L--~'----'----'----'----'----L---L---L---L-...---1 0 100 200 300 400 500 cr (kPa) Fig. 4. 7 a Relationship between volumetric strain versus stress in quasi-static states from the hydrostatic tests for dense AIO; diamonds-test results, solid line-linear fitting curve.

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0 07 -----------------------, E v 0 06 0 05 0 04 0 0 3 0 02 0 01 0 0 01 0 4.3.2 Deviatoric Tests 100 200 Al6-SG ( b ) 300 400 Fig. 4 7b Same as Fig. 4.7a but for A16-SG. 500 cr ( kPa ) 98 Figs. 4.8a and 4.8b show typical stress strain curves obtained under constant confining pressure 196 kPa and 392 kPa respectively in deviatoric tests on dense AlO alumina powder. Compressive stress and strain are taken a s positive here. These tests were perfor1ned at a constant rate of axial displacement, of 0.1 mm/min ( about 1.1x10 5 1/s mean strain rate) up to failure. Failure is def med as maximum stress reached in stress-strain curves. We use the unloading processes to determine the elastic Young's modulus E (Cristescu [1989]). The procedure is as follows. At a certain stress level, the loading is stopped by keeping the axial displacement constant Both the axial displacement and confining pressure are kept constant for about 15 minutes. During this period the axial stress relaxes under the constant axial strain. Fig. 4.9 shows a typical axial stress variation in time for a given axial

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99 strain level. Towards the end of relaxation period, the rate of stress change becomes very small, as shown in Fig. 4.10. The axial stress almost reaches a constant value. At this moment the specimen is partially unloaded. Usually it takes about twenty seconds to unload. The amount of unloading stress is chosen to be 60~ 120 kPa. After the unloading we reload and stop the reloading just before the stress has reached the previous stress state level. The slopes of the unloading and of the reloading curves are quite close to straight lines as shown in Fig. 4.8. They are parallel to each other and almost coincide. Thus, the hysteresis loop does not exist in the unloading-reloading cycle. With this procedure the influence of time effects and irreversible deforrnation by creep become negligible during the unloading/reloading processes Thus the unloading process can be considered to be pure elastic unloading. The Young's modulus is obtained from the slope of the axial stress axial strain. Since both axial stress and strain can be calculated accurately, accurate values of the Young's modulus are thus obtained. Other parameters can also be calculated, such as the bulk modulus or the Poisson ratio. However it should be mentioned that the unloading-reloading cycle causes a complicated change of s tress states at the end of the specimen due to the end effect exerted by the piston on the specimen. No correct volumetric strain can be recorded during the unloading processes. Since the volumetric strain measured is inaccurate during the unloading reloading processes, it is not used in the analysis. The tests described above have been performed for dense AlO, loose AlO and A16SG alumina. The typical stress strain curves are shown in Fig. 4.8 for dense A 10, in Fig. 4.11 for loose AlO and in Fig. 4 12 for A16-SG. These experimental data are li s ted in Appendices B, C and D. It was found that the elastic Young's modulus depends mainly on

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100 mean stress. It may also depend on other quantities such as the current density and loading path. Since the slopes of axial stress axial strain curves in the unloading process depend on the mean stress, the magnitude of unloading stress affects the value of Young s modulus so that the values of the Young's modulus show a little scatter As a first approximation the Young's modulus can be considered as a function depending on the mean stress alone From the dete11nined values of Kand E, the Poisson ratio in the range of 0.3~0.35 is obtained for dense AlO, or in the range of 0 25~0.29 for loose AlO, and in the range of 0.39~0.4 for Al6SG as listed in Table 4.4. For numerical computation it is better to use Kand the Poisson ratio rather than Kand E. Fig. 4.13 shows a typical stress volumetric strain curve obtained in a deviatoric test at confming pressure a 3 =490 kPa for dense AlO alumina. At low stress levels the unloading curves are on the right-hand side of the curve. After a certain stress level, the unloading curves are on the left-hand side of the curve. Compressibility takes place when the unloading curves are on the right-hand side of the stress-volumetric strain curve. Dilatancy takes place when the unloading curves are on the left-hand side of the curve. A stress state separating the volumetric compressibility from the volumetric dilatancy exists for most materials and is well determined for each confining pressure. All those stress states as obtained for various confining pressures define the compressibility/dilatancy (CID) boundary in a ( a;t')-plane. This stress state is sometimes difficult to pinpoint on the stress-volumetric strain curves as can be seen from Fig. 4.13 Instead of a single point, a small smooth transition zone between compressibility and dilatancy is observed in most of the cases From the deviatoric stress volumetric strain curves obtained for various confining pressure as shown in Fig. 4 14 for

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600 ---------------------, 400 300 200 cr 3 = 196 (kPa) 100 0 l..-..aL--ll--.l"---'----I.--L---&.----1.--i.---'-_..___.___,____._ __ __,____,__ ____ ...., -0.04 -0.02 0 0.02 0.04 0.06 Fig. 4.8a Stress strain curves in deviatoric loading test with unloading processes (points-recorded data) for o 3 = 196 kPa. cr 1cr 3 1,200 -----------------------, (kPa) Ev 1 ,000 800 600 400 200 0 ____ ......._ ____ ...._ ___________ _..._____,..____. -0.04 -0.02 0 0.02 0.04 0.06 0.08 E Fig. 4.8b Same with Fig. 4.8a but for o 3 =392 kPa. 101

PAGE 110

480 460 440 420 '-----'__ .__ _.._ ___,ji.,__ _._ __. ____ ____. 0 4 8 12 16 t ( min ) Fig. 4.9 A typical relaxation curve (points-da ta ) for den se AlO at fixed strain. ~(cr 1cr 3) ~t ( kPa/ s) 1.2 ""'T"""-------------------.. 1 0.8 0.6 0.4 0.2 o -L-~.:==:,:=::::, =::;:, ==~===;:::=,=,;:::::=::::;,;=::::::;,== 0 2 4 6 8 10 t (min) Fig. 4.10 Variation of the rate of relaxation stress (points-data) 102

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PAGE 112

1,600 ---------------------, t) 1,200 800 cr 3 = 490 (kPa) 400 0 ...._......____._ ___ __,_....__ ___ __._ ___ ......_ __ ..__. -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Fig. 4.12 Stress strain curves in deviatoric loading te s t with unloading pro cesses ( points-recorded data ) for A16-SG for o 3 =490 kPa. 1 600 .---------------------, 1,200 800 o 3 = 490 (kPa) 400 0 ....._ _____ _.___ _____ __._ _____ ___. -0.0025 0 0.0025 0.005 Fig. 4.13 A typical stress volumetric strain curve with unloading process for den se AlO at confining pressure o 3 =490 kPa. 104

PAGE 113

1 600 ,-----------------------. 1 200 800 400 0 -0. 02 + +++++ -1 +++++ + +++ dense AlO 490 + ,t \ + t+ I +++++++41-1-t 3 9 2 ""1 4 fl+ + H llll l l l l ++t+t~ 294 +++++ ~ t ++ t llitill14 11 + + + ++f+t + + +++ t+t-+++ I+ I++++ + + 196 ~ , 1 J + + ++ + + + + + + + + + + ++-H-1-1-4-i.u_.. i : ++ ++ ... ++ + 4, cr,=98 (kPa) ) : : :: + + -0.0 1 0 0.01 105 Fig. 4.14 Deviatoric stress versus volumetric strain (hydrostatic strain being incorporated ) relationship for various confining pressure for dense AlO. 1,400 1 ,20 0 1,000 800 600 400 200 0 0 490 -+') ++ + loose Al O +1-+ + #+ ++4, + + + + .. ++f 392 ++ + + +++ +++ ++ + ++ ++++ .. ++ J 294 + + +++ ,..,. t+ ++ .. ++ t-+ t++ +++ ~ + + + tT ":-:} + + + + + ++++++ ,.i-+ cr 3 = 196 (k:Pa) + + + ',tft" + + ++ +++r ++ + ++++ + + +++ + +++ +++ + +-f'll+i~ ++ ++ + ++ + ~ _.,,. +-it-ic~ t t# 98 0.005 0.01 0.015 0.02 0.025 0.03 Fig. 4.15 Volumetric strain versus deviatoric stress for various confining pres s ure for loose AlO alumina powder.

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106 1 600 ----------------------, 1 200 A16-SG 800 400 0.01 0.02 0.03 0.04 0.05 0.06 0.07 f.v Fig. 4.16 Same as Fig. 4.15 but for Al6-SG. dense AlO, we can determine these transition points (shown in Fig. 4.14 by a dark point symbol). All these points constitute the CID boundary. The CID boundary is a very useful concept, not only in the modeling of the behavior of particulate materials with the elastic/viscoplastic theory (Cristescu [1989]), but also in engineering applications. In the design of a powder forming process in a mold, dilatancy has to be avoided since dilatancy can break the binder in the materials and create pores. Fig. 4.14 shows deviatoric stress volumetric strain curves for dense AlO for various confining pressures 98, 196, 294, 392, and 490 kPa. At low levels of deviatoric stress the volumetric strain increases (volume reduction). At high levels of deviatoric stresses, the volumetric strain decreases (volume expansion). The CID boundary is distinct from the failure condition. Thus if we want to avoid volumetric expansion in a compaction process,

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107 the deviatoric stress in the process should not be larger than some value, i.e. the stress states always remain in the compressibility regions rather than in the dilatancy regions. Tests for dense A 10 have been performed with the loading strain rate 1 1x10 5 1/s. At this loading speed, data can be easily recorded Also at this loading rate, the whole test lasts several hours. Fig. 4.15 shows deviatoric stress volumetric strains for loose AlO alumina powder for several confining pressures shown and the loading strain rate is 3.3x10 5 1/s, which is three times faster than that used for dense AlO. Dilatancy is always exhibited just before failure. Fig. 4 16 shows deviatoric stress-volumetric strain curves for various confining pressures for A16-SG obtained with the loading strain rate 3.3x10 5 1/s. It is seen that additional volume reduction can be obtained when a deviatoric stress is superposed over a previously applied confining pressure. Thus, as a general conclusion we can state that the volume change of a powder is not only produced by mean stress, but also by a pure deviatoric stress, superposed over the mean stress. Fig. 4.17 a shows a deviatoric stress volumetric strain curve for den s e A 10 under constant mean stress condition. It is seen that the pure deviatoric stress superposed over the mean stress produces mainly dilatancy rather than compressibility Fig. 4.17b shows a deviatoric stress volumetric strain curve for A 16-SG under constant mean stress. It is shown that an additional volume reduction results from the pure deviatoric stress. No dilatancy is created by the pure deviatoric stress at the applied pressure interval. From Fig s 4.14 and 4.15 it is seen that the volumetric behavior of dense AlO is distinct from that of loose AIO. Loose AlO exhibits more compressibility before dilatancy

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108 than dense AIO. The difference in behavior can be seen clearly from Fig. 4.18 where the failure and the CID boundaries for dense AlO and loose AlO are distinct. The failure and CID boundary of loose AlO lie between the failure line and CID boundary of dense Al 0 The tests performed with three types of alumina powder show a clear size effect No dilatancy is exhibited in Fig 4.16 for A16-SG in the range of stresses applied while the Al O 's always exhibit some dilatancy before failure. Also the A16-SG powder i s much ea s ier to compress than th e Al O's. At the s ame pres s ure level the volumetric strain of A16-SG is much larger than the AlO's. From the data (symbols) obtained at various confining pressures and for different types of alumina powders shown in Fig. 4.18 it follows that for the pressure range considered the CID boundaries and the failure conditions can be approximated by straight lines of the form ( 4 8 ) where ( o 1 o 3 ) is the deviatoric stres s o the mean stress a and b are curve fitting con s tants. The numerical value s for those parameters are given in Table 4.5 Several intere s ting characteristics are noted. The alumina powders are almost cohesionless materials This property follows also from the property of the deformed specimen, which i s ea s y to break with a very small applied force. No large cohesion force exist s even in the deformed specimen at least in the range of pressure applied. The deforn1ed s pecimen of A16-SG can be handled after the test, with some care without breaking while the deformed s pecimen of A I O s breaks when the membrane i s removed. It is difficult to

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700 ---------------------, 600 500 A l O 400 300 a=402 (k P a) 200 (a) 1 00 0 '--'--'--'--"--..__..__..__i.......li.......li.......l~~--------------------------O. O l 0.005 0 0.005 0.01 Fi g 4.1 7 a D e viatoric s tre ss volumetric strain curve under con s tant mean s tre s s loadin g test f or den se A 10 o=402 k.Pa 500 ---------------------, 400 A16-SG 300 200 cr=30 4 (kPa) 100 0 .._ __._ __ ...._ __._ __ ...._ __._ __ ...._ __._ ___. 0 0 01 0.02 0.03 0.04 Fi g 4.1 7 b Deviatoric s t ress volumetri c s train curve under con s tant m e an s tre ss loading te s t for Al 6 SG o=304 k.Pa. 10 9

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1 600 ------------------1 200 1 000 800 600 400 200 CID & Failure ( loose AIO ) Failure ( den s e AIO ) Failure (A16SG) CID ( dense A I 0 ) 0 L....-----li.....-------1.-----'-........... -_.___...._ __ __,_----1,, __,j 0 200 400 600 800 1 000 o ( k:Pa ) Fig. 4.18 Failure and CID boundary (points-test) for dense, loose AlO and A16-SG. T bl 4 5 C ff" t f CID b d a e oe 1c1en so oun .ar an d f at ure su rf ace CID boundary Failure condition 1 oe of alumina a (kPa) b a (kPa) b AlO (loose) -31 2 1.36 0.0 1.35 AlO (dense) -86.4 1.32 13.7 1.45 A16-SG 0.0 1.5 110 measure cohesion force by this method. The values of a in eqn (4.8) is obtained by extrapolation from curve fitting. It is not real cohesion force in the sense that cohesion stress equals to the shear stress in a material under zero mean (norn1al stress) stress. The slope b involved in the failure condition increases with the density of the materials. This property depends not only on the density but also on the particle size and surface force of particles. It should be also mentioned that in the neighborhood of o=O the CID boundaries are not

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111 linear in a. The shape of this boundary for much higher values of a, is unknown due to the limitation of the testing apparatus Figs 4.19-4.21 show complete stress-strain relationships for dense A 10, loose Al 0 and A 16-SG alumina, respectively, for several confining pressures. With the se data and hydrostatic data we can for1nulate a three dimensional general model for the behavior of these alumina powders. Since these materials are time dependent (see Figs 4.2 and 4.3), visco-elastic or viscoplastic models should be used. Two elastic/viscoplastic model s have been formulated (Jin et al [1996a]; Jin et al. [1996b]; Cazacu et al. [1996] ) for dense Al 0. The modeling for the alumina powders will be given in Chapters 5 and 6. An experiment to reveal the strain rate influence of the mechanical behavior of alumina powders has also been perfor111ed. Fig. 4.21 shows the stress strain curves obtained with two different axial strain rates (3 3x10 4 s 1 l.lxlo 5 s 1 ) for dense AlO. The influence of the strain rate on the mechanical response is obvious. The stress-strain curves at the higher strain rate are above those obtained at the slower strain rate. Also the deviatoric stress volumetric strain curve shows that deviatoric stress at a lower strain rate creates more compressibility and more dilatancy than at a higher strain rate. 4 4 Discussion and Conclusion In order to reveal the possible existing scatterring between the tests under the same conditions, several deviatoric tests under the same conditions have been perfo1med. Fig. 4.22 shows the comparison between two tests on dense A 10 perfor1ned under the same conditions

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1,600 0' 1 -0' 3 den se AlO 490 .1 (kPa) .v 1 ,200 392 294 800 196 400 o -----------------------------0.05 0 0.05 0. 1 E Fig. 4.19 Stress-strain c urv es for dense AlO at different confining pressure 98, 1 96, 29 4 392 and 490 (kPa). 1 400 0'1-0':3 loose AIO 490 (kPa) 1 ,200 E 3 .1 1,000 Ev 392 800 294 600 196 400 200 0 .................. _...._ ........... .................. _...._ ........... .................. _...._ ........... __._-.... ......... ,.__, ........ -0. l -0.05 0 0.05 0.1 0. 1 5 0.2 E F i g. 4.20 Same as Fig. 4.19 but for loose AIO. 112

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0'1-0' 3 ( kPa) cr,-0"3 (kPa) 1 600 490 Al6-SG 1,200 .3 tJ 392 294 800 196 400 cr 3 = 98 (kPa ) 0 L....-........____.__.._____.____._ _.____._____._....__ ___ L....-........____.____,, -0. 1 -0.05 0 0 .05 0.1 0.15 0.2 0.25 e Fig. 4.20 Sarne as F i g 4 .1 9 b u t for A16S G. 1,200 .-------------------1 ,0 00 800 600 400 200 3.3xlo-4 dense AlO 0':3 = 392 (kPa ) o -----------------------------0.05 0 0.05 0 1 Fig. 4.2 1 Comparison of s tr ess-strai n c u rves for d ense A l O a t differe n t l oading ra t e 113

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114 (initial density, confining pressure, and strain rate etc.). One test, with an unloading process, is shown by solid lines with dots while the other test is shown by solid lines. These two tests give almost the same results (the two sets of lines are superposed). Several other tests under the same loading history, same confining pressure and same strain rate show almost identical results if the initial densities of the specimen are close enough. We have found that if the initial densities of the specimens are distinct, the corresponding stress-strain curves will also be quite different This is why the initial density of the specimen must be controlled as carefully as possible. Fig. 4.22 also shows the stress-strain curves for loose A 10 (full line), which has a different initial density from dense Al 0. The difference between the stress-strain curves for dense AlO and for loose AlO is quite significant. Since the volumetric strain is measured by water drained out from the specimen, it was speculated that the permeability of the specimen may influence the measurement of volumetric strain. However, the speed of water drained out from the specimen during loading is much slower than the one during saturation due to the small loading rate used. Thus, the influence of the pertneability on the volumetric strain measurement can be considered to be negligible. Triaxial hydrostatic and deviatoric tests have shown that the mechanical behavior of alumina powder is loading history dependent and time dependent. Initial density influences the mechanical behavior. Under the same pressure applied, the particle size has a significant effect on the consolidation characteristics of alumina powders Alumina powder A 16-SG with small particle size (0.4~ 1 m) exhibits much more consolidation than alumina powder AlO with large particle size (40~200m). The amount of volume reduction of the A16-SG

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115 is much larger than that of A 1 O's. The volumetric behavior during deviatoric tests is quite different for the two a lumina powders. A 16-SG always exhibits com pr essibil it y, while the A 1 O's always become dilatant before failure. Increasing deviatori c stress produces compressibi li ty first and dilatancy afterwards for the AlO's. Additional vo lu metric compressibility can be obtained for Al 6-SG by superposing a deviatoric stress over a hydrostatic pressure. 1,200 0 1 -~ (k:Pa ) E v dense AlO 1 000 3 800 loose AlO 600 400 ~= 3 92 (k:Pa ) 200 0 l.-,l,__.&..-..L..-...L.......L--'--....&-......._---'---'---'---I.---L--l---l~L..-.1....-....1 0.06 0 03 0 0 03 0.06 0 09 0 12 Fig 4.22 Comparison between tests under the same conditions for dense AlO at confining pressure 392 kPa a nd compar i son of st r ess strain curves between loose AlO (lower initial density) and dense Al O (hig her initial density)

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CHAPTERS THE ELASTICNISCOPLASTIC MODEL FOR ALUMINA POWDER AlO In this chapter, we are dealing with the modeling of powder materials rather than rock salt. In chapter 3, the constitutive equation is formulated based on the true triaxial stress controlled tests on rock salt. Now we have a set of data for conventional triaxial straincontrolled tests. In this type of tests, the confining pressure is kept constant throughout tests rather than the mean stress is kept constant in the true triaxial tests. The question is: can the procedure developed in chapter 3 be used for such type of data? How to determine the model based on the conventional triaxial experimental results presented in chapter 4? In the following sections, we will explore the procedures to determine the model based on the conventional triaxial experimental data for alumina powder AlO (dense). The procedure to determine the model will be presented in detail. Some new procedures are developed for this type of tests in the following sections. 5.1 Determination of the ElasticNiscoplastic Model The determination of the elastic/viscoplastic constitutive equation is to determine all the parameters ( elastic parameters and viscosity coefficient) and all functions (yield surface and viscoplastic potential) involved in the model shown in eqn (2.6). The deterrnination of 116

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117 the model is also divided into four steps: (1) the determination of the elastic parameters, (2) the determination of the compressibility/dilatancy boundary and the failure condition, (3) the dete11nination of the yield surface, and (4) the determination of the potential surface plus the viscosity coefficient. Fig. 2.1 shows a schematic of the procedure. Once these functions and parameters are determined, one can put them into the general elastic/viscoplastic form (2.6). Thus, the constitutive equation is determined. 5 .1.1 Elastic Parameters Elastic parameters are determined by the unloading/reloading procedure presented in Chapters 2 and 4. The experimental results show that the elastic parameters are dependent mainly on confining pressure (shown in Fig. 4.4). Since there are not yet enough data to for,nulate the dependence of elastic parameters on other factors such as the current density, it is assumed that the elastic parameters are dependent on the mean stress only. The elastic bulk modulus K and the Young modulus E are deterrnined with the unloading procedure in the hydrostatic tests and deviatoric tests respectively, as described in Chapter 4. Figs. 5. la-b show the elastic moduli obtained from several tests with the symbols denoting the experimental results. It can be approximately expressed by the following equations which are represented by solid lines in Fig.5.1 a and 5.2b (5.1) where

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K 0 = 1.373x10 7 (kPa); k 1 = l.369x10 7 (kPa); k 2 = 0.618xl0 4 (kPa 1 ) E 0 = 2.745x10 6 (kPa); e 1 = 2.680x10 6 (kPa); e 2 = 0.327x10 3 (kPa 1 ) 118 (5.2) The function s expressing the dependence of the elastic moduli on the mean stress are different from the ones presented in Chapter 4. Here we have taken into account that the values of elastic moduli at very high mean stress tend towards a constant value, i.e K( 00 )=constant. Because at very large mean stress, all pores are closed and thus the elastic bulk modulus is independent on the mean stress. However, due to Jack of data at high pressure, K( 00 ) was largely detennined by trial and error to fit the data The same procedure was used for E( 00 ) as well. K (kPa ) 3x l d 2x ld lxl d 0 100 2 00 3 00 400 cr(kPa ) Fig. 5. la Variation of K with a, square-tests solid-predicted by eqn (5 la )

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E (kPa) 6xlci 4xlci 2xlci 0 200 400 600 800 cr (kPa) Fig. 5.lb Variation ofE with a, square-tests, solid-predicted by eqn (5.lb). 5 1.2 CompressibilitylDilatancy (CID) Boundary and Failure Condition 119 The compressibility/dilatancy (CID) boundary is deterrnined from the experimental data. The variations of volumetric strains in the triaxial tests perfor1ned under several confining pressures are shown in Fig. 4 14. The CID boundary is deterrnined by the stress state where the slope of deviatoric stress volumetric strain curve is the same as the slope of the elastic unloading curve. This boundary as shown in Fig. 5.2 by symbols can be approximated by a straight line: X( a, -r) = -r d 0 -d 1 a = 0 ( 5.3) where

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d 0 = -40 .73(kPa ), d 1 = 0.622 120 (5.4) However, from Figs. 4.13 and 4.14, it is seen that this boundary cannot be exactly pinpointed sometimes. Indeed it is a transition zone as shown by dash-lines in Fig. 5.2. Below the CID boundary, there is a compressible region. In this region, the irreversible volumetric strain increases, thus the volume of materials reduces. Above the CID boundary, there is a dilatant region In this region, the irreversible volumetric strain decreases so that the volume of materials expands. More pores and microcracks are created by dilatancy. In the powder compaction process, the dilatancy has to be avoided since it can break binders in the materials and create pores. The failure condition is determined from the deviatoric stress and strain curve there where in each test the octahedral shear stress reaches its maximum value. Fig. 5 .2 shows, by triangles for each confining pressure, the maximum stress state reached. We use the least square method to fit the data. It results in a linear relationship between octahedral shear stress and mean stress: (5.5) where y 0 = 6.445(kPa), y 1 = 0.684 (5.6)

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't (kPa) 600 400 200 failure ----. Dilatant regio _______ ..... transition zone , CID boundary Compressible region 0 L..-----1...____., __._ _..._ ____ _,__ _______ 0 200 400 600 800 1 000 a(kPa) Fig. 5.2 Failure and CID boundary (points-test) fitted by a linear curve. 5.1.3 Yield Function 121 First, consider the yield function for the hydrostatic part as shown in Fig. 2.1. The yield function is equal to the irreversible stress work on the relaxation boundaries based on the elastic/viscoplastic theory as shown in eqns (2.3) and (2.11). It is assumed that in each loading step followed by creep showed in Fig. 4.2 the final stress strain state has reached quasi-static state, i.e., it is quite close to the relaxation boundary The curve connecting these final stress strain s tates reached at the end of creep in each loading step, constitutes the relaxation boundary. Thus, the irreversible stress work per unit volume can be evaluated as T W V= f aedt ( 5.7) 0

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122 along the relaxation boundary. However, only several loading steps are performed in each hydrostatic test, thus only a few points (same number as loading steps) can be obtained from each test to deterrnine the relaxation boundary. In order to obtain more accurate results for the integration (5.7), spline curve fitting technique was used for the approximation of the relaxation boundary based on the available points Then the integration was performed by using the trapezoidal rule with 100 equally-spaced discretization for the entire region. The irreversible stress works in several tests are shown in Fig. 5.3. These data can be approximated by a polynomial 3 (5.8) where a. = 1 kPa and al = 0.27x 10 5 kPa; a2 = 0.6x 1 o s kPa (5.9) According to the theory, the yield function for hydrostatic loading has been determined as 3 (5.10) Hv( o) is expressed in kPa.

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1 5 .------------------1 0 5 0 1............. 0 100 200 t t t .. .. i'. 300 400 cr(kPa) 500 Fig. 5 3 Variation of the irreversible work at hydrostatic loading, points-tests with interpolation, solid line-curve fitting by eqn (5.8). 123 Next we consider its deviatoric part. The deviatoric stress-strain curves are assumed to reach quasi-static state, i.e., to be quite close to the relaxation boundary, since the loading strain rate is very small (about 10 -s s 1 ). We can calculate the irreversible stress work along these curves. The formula to calculate the irreversible stress work is 1 I 2o + 6G 9K 1 1 3G + 9K 01 I 1 (5.11) + 3G 9K

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124 where a 0 is confining pressure, a 1 is deviatoric stress, e 1 is axial strain and e 3 is circumferential strain. The irreversible works obtained in the deviatoric loading in different tests, are shown by crosses in Fig. 5 4. The data can be approximated by the following expression 0 1 't
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120 (kPa ) 80 40 0r98(kPa) OL....w......... 0 400 294 196 800 + .. 49Q. + + + 392 + + + + + + + + t + .. -1+ + + I 1,200 1 600 Fig 5 4 Irreversible stress work for different confining pressures, s ymbols-test result solid line-prediction by eqn (5.12). 125 (5 15) Based on the assumption in the theory that the yield surface is the irreversible stress work along relaxation boundaries, the yield surface is written as (5 16) This expression i s important for compressible materials. In the previous work (Cristescu [1991]) the yield surface was assumed for conventional tria xial tests as

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126 (5.17) The assumption may not be valid if the material exhibits strong compressible behavior. 5.1 4 Viscoplastic Potential First consider the potential for hydrostatic loading. From the constitutive equation, the irreversible volumetric strain for a hydrostatic test can be written based on eqn (2.3) as or rewritten as I e = k V 1 W v( t) aF H v( o) a o I k a F = e v Tao 1 W v (t) Hv(o) (5.18) (5.19) Each term on the right-hand side of the above equation can be evaluated from the experimental data. It is assumed that at each loading step, the strain or stress is instantaneously applied at time l with an instantaneous elastic response followed by creep or relaxation from time l tot. We approximate eqn (5.19) by I I k a F = -+--e_ v _(t)_ _e v _(t+o) __ Tao 1 H.(o(t o)) (t-t) Hv(o(t)) 0 (5.20) It is also assumed that kr<3F/ a o is independent on time but is dependent on stress only. Here k T is considered to be a constant during each loading step (a total of 100 loading steps have

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127 been generated by interpolation with spline curve fitting technique on the relaxation boundary in each test). Figure 5.5 shows the calculated results (dots) of the right-hand side of eqn (5.20). The following fortnula is used to approximate these data 2 : = ( a) (5.21) where (5.22) The viscoplastic potential for hydrostatic loading is shown by the solid lines in Fig.5 5. 5x10 5 ----------------ki,aF/acr 4x10 5 3x10 5 2x10 5 lx10 5 0 100 200 300 400 cr(kPa) 500 Fig. 5.5 Variation of kTaFtaa with a in hydrostatic loading points-computed from tests, solid line-predicted by eqn (5.21).

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128 Now let us consider the yield function in the deviatoric part. First consider the irreversible volumetric strain change during deviatoric loading. From the constitutive equation (2.3), the irreversible volumetric strain can be written as I e = k V l W(t) a F H(a) a a (5.23) or as I a F e v G ( o;r:): = kr = ---a a l W(t) (5 24) H(a) From the experimental data, we can estimate the terms on the right-hand side of eqn (5 24 ) The same approximation used in eqn (5.20) is used again. However the function G has to satisfy the following conditions due to the physical requirements as presented in eqn ( 2.13) G( o;r:) >0 G(O ,' t) = 0 G(o ,t) <0 X(a;r:)>0 X(a ; r:) = 0 X(a;r:)<0 where X=O is the CID boundary. (compressibility) ( CID boundary) ( dilatancy) ( 5.25a) (5.25b) (5.25c) The symbols in Fig. 5.6 show the variation of G(o;r:) with t for different confining pressures In order to better fit the data we separate G(o,t) into two parts denoted by G 1 and G 2 Each one bas an asymptotic behavior in the corresponding regions. G 1 is used for both the compressibility and the dilatancy regions. G 2 is employed for the dilatancy region which is very close to failure. In the compressibility region, we have

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129 G( o, t) ,-r = o = 4>( o) (5.25d) for hydrostatic loading. Also from eqn (5.25b) we have G(o,t)=O at X(o,t)=O. In order to ensure the continuity of e 1 as explained in Chapter 3 in eqn (3.16), it is assumed a G(o,t)/ a t=O at t=O. Since G 1 is responsible for both the compressibility and dilatancy regions, it follow s that G 1 (o,t) should satisfy these conditions as well. We use a third order polynomial to approximate the computed results in the compressibility and the dilatancy regions as shown in Fig. 5.6, 3 (5.26) Note that the first order term vanishes due to the condition a G 1 / a t=O at t=O. Using the restriction posed by eqn (5.25b), i.e., G 1 =0 at t = d 0 + d 1 a on the CID boundary, we can determine f 2 ( o) as (5.27) where f 3 ( a) is to be determined from experimental data . According to the experimental results in the compressibility and dilatancy regions shown in Fig.5.6, f 3 ( a) can be determined as (5.28)

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130 where (5 .29 ) Note that the data are obtained in the conventional tests, thus, This condition was incorporated into the curve fittings. For dilatancy regions near failure, there exists a singularity. In order to approximate the volumetric strain behavior near the failure point, we use a singular function to describe it (5.30) where a 4 = -6x10 1 0 s l is determined from the data. G 2 is very smal] in both the compressibility and the dilatancy regions and has a singularity at failure which can capture the volumetric strain rate behavior near the failure condition. Thus G(o,r:) can be expressed as the sum of G 1 and G 2 as (5 31) where f 2 (o) and f 3 (o) are given by eqns (5 27-28). Fig. 5.6 shows the comparison of G(o,r:) between the experimental results and the results predicted by eqn (5.31) for the tests with different confining pressures.

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0.0002 ------------x ----, G(cr 't) (kPa s ~ 0 -0.0002 -0.0004 0 0"]=98(kPa) 100 200 0 196 294 300 400 X X 490 X 392 500 600 700 't (kPa) Fig. 5.6 Variation of G with iunder different confining pressures symbols-experimental results; so lid lines-predicted by eqn (5.3 1 ). 131 Now we integrate eqn (5 .24 ) with respect to a to get the viscoplastic potential k,j?( a, i) = f G( a,i-)da +g(i-) = F 1 ( o, i-) +g(i-) (5 32) where (5.33) g(i-) is a function of ito be determined. In order to obtain function g(i-), consider the irreversible part of the deformation rate from the constitutive equation (2 .3 )

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I E = k where 1 W(t) ,<_ a _F __ a _a + a F_ a -r H(a) a a a a a -r a a 132 (5.34) (5.35) Differentiating eqn (5.32) with respect to 1' and substituting them into (5.34), then considering eqn ( 5.35) we finally obtain the formula for g'(-r), 1 '2 1 1 a F1 g (-r) = ~.....;..v_ ~-.-( e e ) 1 W(t) t 3 a -r (5.36) H(a) All terrns except the rate of irreversible strains on the right-hand side of eqn (5.36) are known from the previous eqns (5.16) and (5.33). The rates of irreversible strains were evaluated from the experimental data. The same approximation used in eqn (5.20) to eqn (5.19) is also used here. Thus we can obtain g'(-r) from the data available as shown in Fig 5.7 with symbols. Also, we can use a curve to fit the data shown in Fig. 5.7 (5.37) where g 1 = 0.79xlo 5 s 1 It is shown in Fig. 5.7 that the data are represented well except for several points near failure where a singularity is present. The data in Fig. 5 .7 show that g'(-r)

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133 i s independent on o as predicted by ( 5.32). It implies that the viscoplastic potential exists. Thus the procedure to find the potential is logical. g' (' t ) 0.005 0 004 V 0 003 V V V 'ii 0 002 0 0 001 V + .. ~ .,, , . t, k 0 V a + a /+ a ,ti -1+ + a /-l+ co +++ + .... ... 111 ..._'!, .. ... ... + conf-p=98 (kPa) 196 294 392 490 approximated 0 ._ v0 __.._...__~---IL..--......___.__.._ __ --1._..__........____,1,,_...___-J 0 100 200 300 400 500 600 700 't ( k.Pa) Fig 5 7 Variation of g'(t) with tat different confining pressure, symbo ls-test results, solid line-approximated line (5.37). Hence the viscoplastic potential surface for dense Al O alumina powder is completely determined based on the experimental data and can be written as

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134 ( d o a 6+ d1 ) a 5 + 2 exp a 0 I 2 a 6 (5.38) asdt + exp a 6 a 6 0 1 't' o /2 o 0 2 o o as o exp a 6 a 6 o fi o I t t 3 In( t+yo +y1 o) + a4 o Y1 o :t 2 b, 0 2 b 2 0 3 gl + -1+ -1+ -12 2 o 3 o 2 o Using this expression, we can get the shapes of potential surfaces. Using eqn (5.16), we can draw the contours of the yield function in o-t plane The two families of curves are distinct a s shown in Fig 5 8 The CID boundary is a little bit different from the curve with a w a o=O. IT the associated flow rule i s u sed, the formula can approximately predict the CID boundary. However, the yield surface and potential surface are distinct both in the lower values of (o,t) where i s the stress range applied in the tests, and in higher values of (o,t) where the values of the yield surface s and the potential surfaces are obtained by extrapolation Thus a non associated flow rule should be used with the potential given by eqn (5.38).

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1,400 -----------------------, 't (kPa) 1,200 1,000 800 600 400 200 CID boundary ALUMINA POWDER dense AlO Dilatancy .. / Failure --. '-. 68 6 0.05 kF=0.02 3 92 h'""'::="o:. ~c)H/acr =0 / Compressible 0.2 H=l9 6 ( kPa) 8 0 1 9. 49 0.3 39.2 0.4 \ 0 L--..__.,___........_........_......._......_ ____ ..t,,,&,.,_,,_ _..... __.____.1...1... ___ .......__. 0 500 1,000 1,500 2,000 cr(kPa ) Fig. 5.8 Yield surface, potential surface, domains of compressibility dilatancy, failure criterion, CID boundary and line awao=O. 5.2 Comparison with Experimental Data 5.2.1 Triaxial Tests 135 In order to reproduce with the model the data which have been used to forn1ulate the model, let us consider the ''creep formula" (Cristescu [1989]). In this type of forrr1ula, it is assumed that stresses are increased instantaneously at time t 0 Afterwards a creep follows under constant stresses from t 0 to t. We can integrate eqn (2.3) by multiplying it with o to get the following equation for the irreversible stress work

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k a F W(t) = H(o(t ))-( H(o) W(t 0 ) )exp r :o(t 0 t) H a a The total strains can be written as follows 1 W(t o) a F H a o kr a F e(t) = eE(t) + ----1 exp---:o(t 0 -t) 1 a F H a a -:o H a o with the initial conditions 1 I 3K 2G where o (to) =o(t ) =constant. 1 ol+-o. 2G 136 (5.39) (5. 40) (5.41) For the general loading condition, it is assumed that stresses are increased by small successive steps. At each very small loading steps, the stresses are assumed to increase instantaneously at time toAfterwards a creep follows under constant stresses from time to tot. In order to apply the above creep for1nulae (5.39-41) to the general loading conditions, let us consider the above equations in incremental forrns. For each small stress increment the corresponding variation of the strain tensor is obtained from

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137 1 W(to) a F E H a o kr a F ~e(t) = ~e (t) +__,__ __ ...___ 1 -exp ---:a (t 0 t) 1 a F H a o --:o (5 42) H a o All functions in eqn (5.42) are computed at time t, and~ is the variation of the function in the time interval t-t 0 Also both strains and stresses in the variation have "relative" meaning, i.e., they are taken with respect to the state existing at the end of the previous loading process. to itself is the beginning of a reloading process. Thus these fortnulae can be used to describe creep te s ts or any loading histories which can be approximated by successive stepwise stress variations. We can use the above creep for1r1ulae ( 5.42) to reproduce the experimental data. The predicted results for the conventional triaxial tests are shown in Figs. 5 9-5.15 As an independent check, consider a test with confining pressure o 3 =343 kPa. This test was not used in formulating the constitutive equation. In the loading plane ( o;t) the deviatoric loading path for this confining pressure is in the middle of two loading paths with which the tests have been used for the forn1ulation. Fig.5 14 shows the comparison between the test and predicted results.

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400 -------------------300 200 100 I t+ + I I I + I + 03= 98 ( kPa ) 0 L....----'--.L.---'---"-'-----L..... -L....-___.__.L.-__.__...,____. -0 02 -0 01 0 0.01 0 02 0 03 0 04 Fig. 5 .9 Comparison of stress strain curves between experimental data ( crosses ) 138 and predicted results (solid lines) by eqn (5.42) under confining pressure o 3 =98 kPa. 600 ------------------1 iillt + + ~ 4+tt-i + + t+ -fi++ 0'1-
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0' 1-
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1 600 -------------------, 1 200 800 400 -f t + + + + + cr 3 = 490 (kPa) 0 L--____.__...__--'-_ .......... ____., __ __..___.__.,___.__...______. -0 06 0.0 3 0 0.03 0 06 0 09 0 12 e Fig. 5.13 Same as Fig.5.9 but under a 3 =490 kPa. 1 100 ....--------------------, 0' 1 -
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141 Next, let us consider the tests with different loading paths, for instance, constant mean pressure (constant-p) tests. In this type of tests, the mean stress is kept constant. When the axial force increases, the confining pressure decreases in order to keep the mean stress constant during the tests. Fig. 5.15 shows the comparison between the test and predicted results for constant mean stress tests. Good agreement is observed. 700 cr1-
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142 static states (relaxation boundaries ) between the experimental data and predicted results as shown in Fig.5.16. Secondly let us consider hydrostatic creep tests. Although we obtain both viscoplastic potential and vi s cosity coefficient simultaneously, the viscosity coefficient generally need s to be adjusted to fit better creep curve s The coefficient, k c is introduced a s an adju s tment parameter in the following equations. It can be determined by creep test s. For the hydrostatic creep tests the formula in eqn ( 5 42 ) becomes (H W ( t ) ) k ~e ( t) = ~e ;( t )+ v v O 1 exp c (a ) a(t 0 t ) v a H V ( 5.43) H v and ( a ) are provided by eqn ( 5 10 ) and (5.21 ) respectively i.e., 2 3 H ( a ) = a V 1 a a a ( 5.10 ) and 2 a a ( a ) : = b 1 + b 2 a a ( 5.21 ) is the adjustment c oefficient for the viscosity coefficient.

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500 -----------------0' (kPa) 400 3 00 200 Te 100 .. 0 L.....:;..---'-----'-----'----------___. __ __, 0 0.002 0 004 0 006 E v Fig. 5.16 Comparison of quasi -s tatic states between hydrostatic te sts (symbo ls ) and predicted results (soli d line ). Ev 0.006 ..-------------::::--------.---, 0 .005 0 004 0.003 0.002 0 0 .5 1 1 5 2 2.5 t (hour) Fig. 5 .17 Compar ison between a hydrostatic creep test and predicted re sults, dots experimental data, so lid line predicted re s ults k c = 1. 143

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144 From the data shown by dots in Fig. 15.7, we can adjust the viscosity coefficient. It is obtained from curve fitting. Through trial and error, k c = ls i was obtained. Thus, for this type of tests, the viscosity coefficient does not need to be adjusted. If we use k c = 0.5 s l, the following Fig. 5.18 is obtained. The matching in Fig. 5.18 is not as good as that in Fig. 5.17. From these results, it is shown that the viscosity coefficient does not need to be modified in this case. It is obtained together with the viscoplastic potential 0.006 . 0.005 0.004 0.003 0.002 0 .00 1 0 ._--'----'----'----'----L--..L---L--..1...--..1...-----1 0 0.5 1 1.5 2 2.5 time ( h) Fig. 5.18 Comparison between a hydrostatic creep test and predicted results dots experimental data, solid line predicted results, k c =0.5. 5.2.3 Rate Dependent Tests For the prediction of high strain rate loading tests, it is difficult to apply directly strain rate and strain in the model to obtain stress as performed in the tests. Here, we use another

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145 way to reproduce the data with the model. From the deviatoric stress axial strain curves in a high strain rate loading test as shown in Fig. 4.21, the variation of stress with time i s first calculated out. Then we apply stress and time in the constitutive equation to compute strains rather than to use strains and strain rate to compute stress. In this way, it is easier to use the constitutive equation For the test with strain loading rate 3.3x10 4 s t, the variation of the stress with time is computed from the deviatoric stress -axial strain curve as shown in Fig. 5 .19. The comparison of stress-strain curves between different loading rates is shown in Fig.5.20. From curve fitting by trial and error, the adjusted coefficient is obtained as ~=5.5 for this case. It is different from that used in the hydrostatic creep tests. It might be because the viscosity coefficient is not a constant for all loading paths. The viscosity coefficient might depend on other factors such as irreversible work. Also the creep formula (5.42 ) may not be accurate if large stress increments are used. For instance, if we use 100 incremental steps for stress and time the adjusted coefficient should be taken k c =50 in order to reproduce the experimental results. This topic should be studied separately Another drawback of reproducing the h i gher strain rate loading tests is that the maximum stress could not be reproduced. This is because in the formulation, the failure surface for lower strain rate loading tests was incorporated into the formulation If the stresses in higher strain rate loading tests are larger than the failure surface, the formulation is not valid any more The formulation is only suitable for the stress bounded by the failure sutface deter1nined by low strain rate loading tests.

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146 1 000 800 600 400 200 0 L----'------'"-----L------'---......__---.L. __ ..__ ___. 0 50 100 150 200 time (s) Fig. 5.19 Variation of stress with time for the test with loading strain rate 3.3xl0 4 s1 1,200 -.
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147 5.3 Discussion and Conclusion According to the procedure presented in this chapter, the yield surface and potential surface involved in the model are determined from hydrostatic tests and deviatoric tests under constant confining pressures without any a priori assumption. It is impossible to give general f orrns for the yield surf aces and for the potential surf aces. Both the yield surf aces and viscoplastic potential surfaces may not be the same for different materials. Actually, the forms of the potential and yield surfaces are generally distinct for distinct materials. In the present chapter, five conventional triaxial compression tests with different confining pressures ( a 3 =98 kPa, 196, 294, 392, and 490) are used to forrnulate the model. The number of tests could be reduced to three, for instance a 3 =98 kPa, 294, and 490. However, if there are only few test data, an accurate fonnulation cannot be obtained. According to my experience, five complete triaxial tests (both hydrostatic and deviatoric tests) should be enough to f ortnulate the model in the stress range considered. If more accurate formulation is needed, additional tests should be required. For the checking of the model, the tests with different loading paths from those used to for1nulate the model were used. The predicted results match well with the experimental data. Although there are a number of parameters involved in the model, only five tests were used for their determination. Most parameters are obtained from curve fitting. If simpler expressions for the functions are chosen, fewer parameters are necessary. The viscosity coefficient may be not a constant for all loading paths. It might depend on time or irreversible work, etc. At this stage, it is difficult to make precise how the

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148 viscosity parameter may depend on some other parameters. However, as it was shown, a constant viscosity coefficient gives good agreement with the data for creep tests. Because the yield surface and the viscoplastic potential are determined directly from experimental data without any a priori assumption, this method can easily be extended to other kind of materials, such as metal powders, rocks, or other frictional materials. One has to apply only the procedure. Compared to other models, this model is more appropriate to describe nonlinear time dependent properties. The elastic/viscoplastic model was formulated based on the five conventional triaxial tests. The procedure to determine the model is presented. In the detertnination of the potential function, the physical and asymptotic behaviors were considered. Several curve fitting technique s were used for the deterrnination of the model. The model predicts well not only the experimental data which have been used for its formulation, but also matches well some other experimental data which have not been used for its f or1nulation. Since both the yield surface and the potential surf ace are smooth, the model can easily be incorporated into a finite element program.

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CHAPTER6 A NEW METHOD TO DETERMINE THE ELASTICNISCOPLASTIC MODEL From Chapters 3 and 5, one may think that the procedure to determine the model is quite long: there are several functions to be determined by curve fitting and many parameters are involved in the procedure. One can ask a question: can the procedure be simplified in some way so that the procedure would be easier to handle and fewer functions and parameters need to be determined? The answer is "yes", at least for some materials such as alumina powder A16-SG which exhibits compressibility only. In the present chapter, we will develop a new method to determine the constitutive equations. In the above chapters, the irreversible stress work is taken as a work hardening parameter. In the following sections, we will present the elastic/viscoplastic theory with the irreversible volumetric strain as a work hardening parameter and the procedures to determine the elastic/viscoplastic constitutive equations. It is shown that the present forn1ulation is much simpler than the previous ones. 6.1 Introduction In the theory developed for geomaterials (Cristescu [1991, 1994]) described in the above chapters, the irreversible stress work is taken as an internal variable or work hardening 149

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150 parameter since these materials are compressible at small values of the octahedral shearing stress but dilatant at higher values. A model with the equivalent irreversible strain as a work hardening parameter has been developed for rocks (Cristescu [1989]). However, for some loose particulate system which exhibits compressibility only, it is more meaningful and convenient to use the irreversible volumetric strain as a work hardening parameter than to use the irreversible stress work. For instance, in powder metallurgy or ceramics powder compaction, powder materials ( 40~60% volume fraction) are pressed to reduce their volume and increase their densities. The irreversible volumetric strain can be considered to some extent as the density reduction of powder material. The irreversible volumetric strain as a work hardening parameter can give directly some accurate and meaningful insight to the volume reduction of powders. Also if the irreversible volumetric strain is taken as a work hardening parameter rather than the irreversible stress work, the procedure to determine the elastic/viscoplastic model could be much simpler. In other models (see Chapter 1) the irreversible volumetric strain is also taken as a work hardening parameter. In the following sections, we first present the framework of the elastic/viscoplastic theory with the irreversible volumetric strain as a work hardening parameter. Then the procedure of the deter111ination of the model is developed. Finally, the model is checked against the data.

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151 6 2 ElasticNiscoplastic Constitutive Model 6 2 1 ElasticNiscoplastic Constitutive Equations In this model, the materials considered are assumed to be homogeneous and isotropic. The deformation and particle rotations are assumed small. Thus, the elastic response and irreversible response are assumed to be additive, E I e = e + e (6 1) where eE and e 1 are the total, elastic, and irreversible strain rate tensors, respectively. The elastic response is expressed in the form of rate as E
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152 (6 3b) where N( a) is a tensor governing the orientation of the irreversible strain rate tensor (see Cazacu et al. [1996]) Based on the elastic/viscoplastic theory ( see Cristescu [1989]; Lubliner [ 1990] ), in the pre s ent chapter, the relaxation boundary is defined as H( a(t) ) = e~ ( t ) Thus the yield surfaces (defined by setting H=const. ) are a family of surfaces with constant volumetric strain. The bracket <> in eqn ( 6.3 ) denotes the positive part of a function i.e. if A> 0 if A ~ o ( 6.4 ) From eqn ( 6.1 ) -(6.3 ), the elastic/viscoplastic constitutive equation will be written as or a ( 1 1 )'l k e = 2G + 3K 2G O + r e~ ( t ) 1 H(a) a F(a) a a I e = a + ( I l ) al + k 1 e v (t ) N ( a) 2G 3K 2G r H(a) ( 6.Sa ) (6 Sb ) In general the yield function H and the potential function F ( or N) are all dependent on the stress tensor. However if H and F ( or N) are assumed to depend on the fust stre s s invariant, 1 1 and on the second invariant of deviatoric stress ten s or, J 2 only the whole constitutive equation can be deter1nined from a couple of triaxial tests (both hydrostatic and deviatoric tests) That is, Hand F ( or N) are assumed to depend on the mean stress

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and on the octahedral shear stress t or equivalent stress o t = 'l:..1 3 2 153 (6.6) (6.7a) (6.7b) In the determination of the elastic/viscoplastic constitutive equation, it is assumed that all constitutive functions involved in the model depend on a and o only, disregarding the influence of the third invariant of stress tensor. 6.2.2 Loading-Unloading Conditions Let us assume that at the initial stress state (the so-called "primary" stress) aP = a(t 0 ) is an equilibrium stress state, i.e., H(o(t 0 ))=e:, with e 1 P being the value of e~ for the primary IP stress state. e v is determined by the previous loading history to which the body was subjected. Here we assume that e~ is known A stress variation from a(t 0 ) to o(t) with t >

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154 t 0 is called unloading if H(o(t)) to is called loading if H( a(t))>e~(t 0 ) with one of the two possible subcases defined by the following inequalities satisfied by a (t): a F:1 >0 or e~ >0 a a a F:1 = 0 or e~ = 0 a a compressibility compressibility/dilatancy boundary. (6.8a,b) The above two relations define the concepts of Compressibility ( 6.8a ) and Compressibility/Dilatancy (CID) Boundary (6.8b). In this formulation, we consider the materials which exhibit compres s ibility only. For compressible/dilatant materials, one can use the formulation developed in the previous chapters The present formulation is suitable up to the compressibility/dilatancy boundary According to the experimental results presented in Chapter 4, alumina powder A16-SG exhibits compressibility only, thus this material can be described by the theory developed in this chapter. For loose AIO alumina powder, the compressibility/dilatancy boundary is very close to the failure surface. Thus the behavior of this material can be described either by the theory presented in the previous chapters or, for certain applications, by the theory described in the present chapter.

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155 6.3 Determination of the ElasticNiscoplastic Model for Alumina Powder Al 6-SG The determination of the elastic/viscoplastic model is to determine all functions and parameters in eqn (6.5). The procedure is separated into three steps: (1) the determination of the elastic parameters, (2) the determination of the yield surface, and (3) the determination of the potential surf ace and the viscosity coefficient. 6.3.1 Elastic Parameters The determination of the elastic parameters is similar to the procedure described in the previous chapters. The elastic bulk modulus was determined from the experimental results given in Chapter 4 (Table 4.2) in the pressure range considered O a < 500 kPa. It is dependent on mean stress only and can be expressed as (6.9) where ko=34200 kPa; k 1 =1640 kPa. The Poisson ratio v = 0.40 was also obtained from experimental results (see Chapter 4). For higher pressure, eqn (6.9) may not be linear, as explained in Chapter 4. 6.3.2 Yield Surface For some powder materials, compressibility is exhibited up to failure without any dilatancy or the CID boundary practically coincides with the failure surface. A16-SG alumina powder is such a material (see Figs. 4.16 and 4.20), which has no CID boundary. The yield

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156 surfaces can be determined based on the experimental data on such type of materials. From the experimental results obtained it is evident that the elastic strains are much smaller than the irreversible strain (about 1 % ). Since A16-SG alumina powder is a highly compressible material, in the deterrnination of the yield surface we can neglect the contribution of the elastic strains as compared with the irreversible components. The irreversible strain will be considered to be total strain. Alternatively one could also consider the elastic strains in the determination of the yield surf ace, if necessary. The yield function is defined to be equal to the irreversible volumetric strain on the relaxation boundaries, i.e. H( a(t) a(t)) = e~(t). It can be determined following a wellestablished test procedure, using hydrostatic tests and deviatoric tests as presented in Chapter 4. The total irreversible volumetric strain consists of two parts: the hydrostatic part and the deviatoric part, i.e (6.10) where the superscripts h and d correspond to the hydrostatic part and the deviatoric part respectively The yield function consists also of two parts: the hydrostatic and the deviatoric parts, i e (6.11) It is assumed that the following relations exist on the relaxation boundaries (6.12)

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157 Note that a 3 = a-a/3 is the confining pressure involved in hydrostatic tests. Thus, the total yield surf ace can be written as H = Hh(a-!!...)+Hd(a,a) 3 (6.13) First let us consider the irreversible volumetric strain in hydrostatic tests. Based on the experimental results presented in Chapter 4, the volumetric strains on the quasi-static relaxation boundary for several hydrostatic tests obtained with several pressures are shown in Fig. 6.1 (Fig.4.7) The relationship between the volumetric strain and the hydrostatic stress on the relaxation boundary can be expressed as (6.14) Where a=0.000138 kPa 1 is obtained by curve fitting. Note that a "primary'' volumetric strain exists at zero stress state since the volumetric strain started to be recorded at the hydrostatic stress 30 kPa (see Chapter 4). On the physical reasons, we expect the volumetric strain to be zero when stress is zero. Hence if the primary volumetric strain is taken account into eqn (6.14), we have a 0 =0. Next let us consider the deviatoric part of the yield function in triaxial tests. As stated previously, the slow deviatoric tests (low strain rate tests) are assumed to be quite close to the quasi-static relaxation boundaries. The yield surfaces can be obtained from these tests since the surfaces are assumed to be equal to the irreversible volumetric strain on the quasi static relaxation boundaries Thus, we intend to get the relationship between the volumetric

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E v 0.06 0 .05 0.04 0.0 3 0 02 0 01 0 -0.01 "---.L--.L--.L--.L--.L--.L--------'"'------' 0 100 200 300 400 500 cr (kPa) 158 Fig. 6.1 Volumetric strain along quasi-static relaxation boundaries versus hydrostatic stress, points-experimental data, solid line-linear fitting curve (6.14). strain and the deviatoric stress along these boundaries. The curves relating the vo1umetric strain to the deviatoric stress for several deviatoric tests as shown in Fig. 4.16 are redrawn in Fig.6.2. For each curve, the relationship between volumetric strain and deviatoric stress can be approximated by a linear function: (6.15) The least-square method is used for the curve fitting and the coefficients are given in Table 6.1.

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1 ,600 ----------------------, 1 ,200 Al 6 -SG 800 400 + 0 t:....I......L....IL-.L..L..J.-t-.L....L...L..L-.L...L..J.....L-.L..L~L.L..J...J~J....L_,f,_.L..L..J..JL...&,..:.L.;1;,14,..L...J 0 0 .0 1 0.02 0.03 0.04 0.05 0.06 0.07 Fig.6.2 Relationship (ex perimental data) between volumetric strain and deviatoric stress for various confining pressure shown. 2.4x l0 4 ..---.----------------------, -4 2.0x lO -4 1.6x 10 -4 l.2x10 8.0x 1 o 5 -5 4.0xlO 50 100 150 200 250 300 350 400 450 500 0' 3 (kPa) Fig.6.3 Data regression for the relation between p and o 3 point s from Table 6.1 so lid line corresponding to curve fitting using eqn (6 .16 ). 159

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160 Table 6.1 Coefficients obtained by curve fitting in eqn (6.15). o, (kPa) 98 196 294 392 490 B ( 0 1 ) (kPa 1 ) 2.29xl0 4 l. l 7xl0 4 7.9x10 5 5.28x105 4.8x105 By using the data shown by squares in Fig. 6.3, further data regression results in the fallowing relation for p (6. 16 ) where y=0.0229. Thus eqn (6.15) becomes (6.17) The total volumetric s train in both hydrostatic and deviatoric tests for general loading paths can now be expressed as: a h d ( l_) e = e + e = a a -a +y-. v v v 3 l a -a 3 (6. 18 ) Based on the elastic/viscoplastic theory and eqns (6.10-14), the yield function can be expressed as

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H(cr,O) = o:(cr-o)+y 0 1 a -a 3 161 (6.19) The yield surfaces are contours with the same constant density. Thus H has some other dimension than that in the previous chapters. The contours of the yield function in a different stress-mean stress (a, a) plane are shown in Fig. 6.4. As it is seen from Fig 6.4 if the associated flow rule is used, i.e., the yield surfaces act as potential surfaces, the model will predict dilatancy at the high values of 't where the nortnal of the surfaces points to the negative direction of a. This result contradicts the experimental results which reveal that the material exhibits compressibility only. Thus, the model with the associated flow rule cannot predict the behavior correctly. A non-associated flow rule has to be developed for this material It is noted that in both hydrostatic and deviatoric tests, the linear fitting of volumetric strain may not be suitable for higher confining pressures. This tendency is suggested by the deviatoric test performed under the highest confining pressure used ( a 3 = 490 kPa ) and shown in Fig. 6 2. Also, for the hydrostatic tests, a possible nonlinear relationship is suggested by the tests shown in Fig. 6.1.

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162 800 ----------------------, 700 600 0.09 500 0.08 400 0.07 300 200 H =0.06 0 05 100 0 04 0 02 0 03 0 0 100 20 0 300 400 500 cr ( kPa ) Fig.6.4 The contours of yield surface for alumina powder A15-SG. 6.3.3 Irreversible Strain Rate Orientation Tensor The constitutive equation (6.5b) is employed in the formulation of alumina powder A16-SG. The orientation tensor N(a,o) must be determined. Here, we assume that the orientation tensor is dependent on a and o only as done for the viscoplastic potential. We rewrite the constitutive equation for the irreversible strain rate part: I [ e v( t) e = k 1 --r H(a,o) I ev(t) = k 1 -r H(a,o) 1 3a 1 _,n l+tn 3 't' I '1' 2 2o (6.20) where 1 is unit tensor a' deviatoric stress tensor, and q> 1 q> 2 are functions to be determined.

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163 If a q> 1 = a q> 2 there exists the viscoplastic potential, otherwise the vi s coplastic potential does a o a a not exist. q> 1 can be obtained by taking the "trace" in eqn (6.20) l e v q> 1 (a o) = -.,,__ __ ____, e ~( t) k 1 r H(o o) ( 6.21) This equation will be evaluated according to the so-called creep pattern at each loading increment. In the creep pattern, it is assumed that the increment stress is loaded elastically and instantaneously at time to, followed by a creep from t 0 to t 1 until the creep defor1nation reaches the relaxation boundary If it is denoted that the irreversible volumetric strain on the relaxation boundary at to is e~(t 0 ) while at t 1 the corresponding irreversible volumetric strain on the boundary i s e ~( t 1 ) = e~ ( t 0 )+ de ~, equation (6 21 ) becomes ( 6.22) However, e~(t ) changes in the interval (t 0 t 1 ) In order to simplify the evaluation of eqn ( 6.22 ) in the loading s tage an averaging procedure i s used in the proce s s of evaluating eqn ( 6.22 ). The value of e ~( t ) at t 0 is taken for the approximation of e ~ (t) in eqn ( 6.22 ) (Euler forward scheme ) Since His equal to the irreversible volumetric s train on the relaxation

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164 boundary, eqn (6.22) is simplified for a small stress increment from t 0 to t 1 on the relaxation boundary as dH HdH H

(t 1 t 0 ) k.Jt 1 -t 0 ) H 1 o (6.23) At each loading increment in a constant strain rate loading test, the quantity kT (t 1 -fo)=C is assumed to be a constant throughout the whole averaging intervals This assumption is reasonable. For some materials like sand, the strain rate does not influence too much the stress strain curves during loading if the strain rates used are small (Katona [1984]). This means that t 1 t 0 changes while H does not. kT might be dependent on history as well. Therefore we combine kT with t 1 -t 0 and as s ume it to be a constant. On the other hand


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e~(t) 1-H 165 (6.26) Further we consider the same averaging precess as the above one (see eqn (6.21-24)): I I 2 I I 2 de 1 de 3 -(de 1 de 3 ) 3 I I dev de v 3 ( 6.27) 2 = H-dH k/_t 1 t 0 ) k (tl -to) 1 H The same assumption is also made concerning k/_t 1 t 0 )= Cin all the average steps. Also, q> 2 will be evaluated according to the creep pattern for each increment loading. Sincee ~= l/2(e~ e ~), we have H -c (6.28) Since k T is a con s tant to be determined from creep tests, combining 1/C with in eqn (6.20), we can define a new constant k. Further we still use the letter kT as the viscosity coefficient instead of k. Thus we choose for cp 1 and q> 2 the simplest possible for1n without changing the results

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166 (6.29) In the expression of cp 1 and


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dc:i: dc 1 8 7 6 5 4 3 2 1 0 181 a! ti::. cr 3 =98 (kPa) 1111 IS:. 196 A + A ll!'l + em ll!'l 181 Qll 200 400 0 600 0 0 -il 0 D 0 00 D 294 0 392 l I I I I 490 I 0 ++ 0 CD 800 1 000 1 200 1 ,4 00 1 ,6 00 cr1 cr 3 (kPa) Fig. 6.5 Data for the determination of the ratio de v I de 1 symbols-experimental data solid line-predicted by eqn (6.30). 6.3 .4 Vi sc osity Coefficient 167 kT can be determined from creep tests Here we consider hydrostatic creep tests. We integrate the above equation (6.20) for hydrostatic loading condition and obtain the following equation (6. 32) where t 0 i s the "initial" moment of the creep test, t is the "final" moment, H( a ) is yield function, and a i = a =const. e:(t 0 ) is initial value of irreversible volumetric strain. Arranging the above equation gives

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or /( ) = TJ1 kl,t to>) + J(t ) kl, t to) e v t 11.\ e e v O e k = l 1n T t t 0 Therefore the viscosity coefficient can be determined from eqn (6.34). 168 (6.33) (6.34) For alumina A16-SG, the coefficient can be obtained by fitting the creep strain curves. kT ~ 0.01 (1/s) was obtained. Thus the constitutive equation (6.31) is completely determined. 6 .3 .5 Remark As already discussed in Chapter 3 and 5, from the physical point of view, the constitutive equation along hydrostatic axis should be (6.35) in order to avoid any artificial error along hydrostatic loading paths. We can solve this problem by smoothing the discontinuity between eqns (6 31) and (6.35) in order to apply this constitutive equation to FEM. Let us introduce an exponential decay function as

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169 I de1 1 ,.,,. n,.. a2 = -e .1 + w (1 e .1u)+w --13 I 2 2' ~ v a ~ ( 6.36) 3 where r can be any large number, for instance, r=IO.O. With this procedure the constitutive equation can be incorporated into the finite element method. 6.4 Validation of the Model 6.4 1 Creep Type Formula with Stepwise Stress Variations Let us u s e the elastic/viscoplastic model to reproduce the data which have been used for the formulation of the model. The type of creep formula introduced in Chapter 3.3. is used for the prediction. It will be assumed in eqn ( 6.5) that stresses are increased by s mall s ucces s ive steps according to the same law as in the experiments done to establish the model At each very small loading step performed at time t 0 the s tresses are assumed to increase instantaneou s ly. Afterwards in the time interval t to a creep talces place under constant stress. After each small increment stress in the time interval t-to, the corresponding irreversible volumetric strain is obtained by integrating eqn (6.20) ( 6.37) where e ~( t 0 ) is the initial value of e ~ at t=to As a result of one stress s tep the increment strain tensor can be written as follow s

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with the initial conditions N( o,o) (t exp (k,l_t 0 t))) 1 ~ol + -~o, 2G 170 (6.38) (6.39) where o ( to)=o(t)=constant at each stress step ~o. These formulae can be used to describe the creep tests or the tests with any other loading histories which can be approximated by successive stepwi s e stress variations. For the A16-SG alumina powder eqn ( 6.38) is written a s ~e(t) = ~r:E(t) + (H e ~( t 0 ) ) 1 -1+ 3 W o 2 2 1 3a 1 w ---1 2 3 l 20 a -a 2 (t exp (k,l_t 0 t))) We can use the above equation to predict the experimental data. 6 4 2. Conventional Triaxial Tests ( 6 40) We use the above equation s to predict the data. Figs. 6 7 a-e show the comparison between the data and the predicted results at different confining pressures

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171 300 <>1-0J Ev (kPa) \ I 0 00 3 0 0 00 8 0 250 0 a a a 0 0 0 a a 0 0 0 a 0 0 a 0 a a a 0 0 0 0 0 200 0 0 0 0 0 0 150 100 CJ.3=98 (kPa) 50 0 0 L....&..........._......._,i.....,a_.........,...&-.1, ............ ......_...L..-1.,_,___._,__.&....l..-'--L--'........ ...L..-l--'---'.....I -0.1 -0.05 0 0.05 0.1 0.15 0.2 Fig.7a Comparison of experiment results (circles) with predicted results (solid line) at confining pressure 98 (kPa). 600 -----------------500 400 300 200 0:3=196 (kPa) 100 0 ,____, _,__ __ ""'---''"--'------------'--.,___--L __.____. -0.04 0 0.04 0.08 0.12 0.16 Fig.6.7b Same as Fig.6.2a but at confining pressure 196 (kPa).

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0'1-03 (kPa) 900 800 700 600 500 400 300 200 100 3 00000000 Ev E1 0 L.....J..-L............ -L-.L.L..L.--'-.M....1.--'.......... ___._ ........... __.__ .......... _...__.....__. __ ....__. -0.05 0 0 05 0.1 0.15 0.2 Fig.6.7c Same as Fig.6.2a but at confining pressure 294 (kPa). 1 200 --------------------. 00 00000 00 1,000 800 600 400 200 ' ' 0 0 0 0 0 0 0 0 0 0 03=392 (kPa) 0 ................. _.__.__.__.L....L.. .................... __.__ ............................. i....... .................... __.,____,.,___,._.1...-L...1 -0.05 0 0 05 0 1 0 15 0 2 0 25 Fig.6 7d Same as Fig.6.2a but at confining pressure 392 (kPa). 172

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1 600 ,-------------------, 0'1-C>J (kPa) 1 400 1 200 1,000 800 600 400 200 0 0 O'j=490 (kPa) 0 L.....L,....L....l,__._.__,_ .........._ ....,__........._...__.__.__._...._ .......... ...L....I,........_.__,_ .......... _.__......_ ....... -0 05 0 0.05 0.1 0.15 0.2 0.25 Fig.6.7e Same as Fig 6.2a but at confining pressure 490 (kPa). 6.4.3 Constant Mean Stress Tests 173 This model can also be checked in the constant mean stress tests. These tests were not used for the for1nulation of the model. The loading paths in these tests are different from that in conventional triaxial tests For constant mean stress tests, the comparisons between the tests and the results predicted by the model are shown in Fig. 6.8a-b. The creep for1nula (6.40) is also used The results show that the matching is reasonable especially for the volumetric strain.

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500 --------------------. 0 0 o O E1 400 0 0 300 0 200 0 0 o=304 (kPa) 0 100 0 0 0 0 L...L......_._..__ .......................................... _._,_~ ................ ...,L...1....&....,;........_ ......... ..__..__._ ............ -0 1 -0 05 0 0 05 0.1 0.15 0.2 0.25 Fig.6.8a Comparison between experimental data (circle) with predicted results (solid line) at constant mean stress= 304 kPa. 700 0'1-0' 3 ( kPa ) 600 3 0 0 0 0 0 500 0 0 0 0 0 400 0 0 0 t y 0 0 0 0 0 0 300 0 0 0 0 0 cr=397 (kPa) 200 100 0 L....l......a....&--L....L...L...&....IL....1.....&..-L-A-&....1,,...u...&--L....L...L...&....IL--L....L.....1.....&....L...L...&....11.....1 -0.1 -0 05 0 0 05 0.1 0 15 0.2 Fig 6.8b Same as Fig 6.8b but at constant mean stress = 397 (kPa). 174

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175 6.4 4 Hydrostatic Creep Tests We can use formula (6.40) to predict the hydrostatic creep tests The formula (6.40) for hydrostatic creep tests becomes (6.41) with the initial conditions: (6.42) where o(t)=o(t:o)=constant in each stress increment Ila. Fig. 6.9 shows for hydrostatic creep tests the comparison between the experimental data and the predicted results. The matching is reasonable. For the model we used the value 0.01(1/s) for the viscosity coefficient Furthermore we can use eqn (6.41) to predict the quasi-static relaxation boundary in hydrostatic creep tests. The duration of creep taken in the model prediction is to be the same as in the tests. The predicted results are shown in Fig. 6.10. 6 5 Discussion and Conclusion In the model formulation, we have considered the compressible materials. For the materials exhibiting dilatancy, one can use the formulation presented in chapter 4 and 5. For dilatant materials, the volumetric strain and stress state are not related by a one-to-one mapping. This is why for dilatancy materials, one should choose irreversible stress work as

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tv 0.0 7 0.0 6 0 0 5 0 04 0 .03 0 0 2 0 01 0 0 .._._-...-.....---................ --............ ...__----........ ...__--0 0 5 1 1 5 2 2.5 3 time ( h ) 3. 5 Fig. 6 9a Compari s on for hydrostatic creep tests between data ( circles ) and predicted results ( solid line) 0.06 0.05 0 00 00 0.0 4 0.03 0.02 0.0 1 0 ....................... ...___ ......... ._........,___._........., ____ ~__,__ ......... __ ......... __.__ ......... __.__........., 0 10 2 0 30 4 0 50 60 cr (k Pa ) Fig 6 9b Same a s in Fig 6 9a but for another hydrostatic creep test 176

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Ev 0.06 0.05 0.04 0.03 0.02 0.01 0 L..L...1....L..L..L...1..1,...L..L..L...1..1,...L..L..L...l..l,...L..L..L...1..1,...L..L. ............................................................. ...&..I. ......... ............ 0 50 100 150 200 250 300 350 400 450 500 cr (kPa ) Fig. 6.10 Comparison between s everal quasi-static relaxation boundaries obtained in hydrostatic tests and predicted results by eqn (6.41) 177 work-harding parameter. However, for some compressible materials, one can use the irreversible volumetric strain as a work hardening parameter This parameter is straightforward and meaningful in the compaction techniques This model can be incorporated into the finite element method. However since the stiffness matrix for this model is very s tiff implicit schemes should be used when this model is implemented into a finite element program. In the pre se nt chapter a new methodology to fortnulate the elastic/viscoplastic model is given. In this model, the irreversible volumetric strain is taken as a work hardening parameter The model matches well the data. It was shown that the procedure to determine the model is much simpler than the one presented in Chapter 3 and 5 In the model for A 16

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178 SG, only nine constants are used. Among them four constants are used to describe elastic behavior; two constants to describe yield surfaces; two constants to describe irreversible strain rate orientation tensor; and one is the viscosity coefficient. This model can easily be used for other kind of materials.

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CHAPTER 7 CONCLUSION AND FUTURE WORK 7 .1 Conclusion In this dissertation, elastic/viscoplastic constitutive models are discussed and determined for different materials (rock and powder materials). Some new and improved steps in the procedure to dete11 nine the elastic/viscoplastic model are proposed. The complete constitutive equations are determined using a set of triaxial tests (both hydrostatic and deviatoric tests), which includes: elastic parameters, viscosity coefficient, yield surfaces and viscoplastic potential. Normally, five triaxial tests are recommended for the model determination. The elastic parameters are determined from an unloading-reloading process Yield surfaces are expressed as the contours of the irreversible stress work or the irreversible volumetric strain on the relaxation boundaries. A viscoplastic potential and viscosity coefficient are determined simultaneously from tests. The viscosity coefficient could also be adjusted from creep tests. The elastic/viscoplastic models, initial] y proposed by Cristescu [ 1991, 1994], and improved in the present dissertation, can describe nonlinear time dependent behavior of different materials. The properties of irreversible volumetric compressibility and dilatancy ru:e incorporated in the models. The creep, relaxation and other time dependent phenomena 179

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180 can be described by the models The models describe the compressibility/dilatancy boundary correctly and match the data reasonable well. The models are easily incorporated into the finite element programs due to the fact that the yield surfaces and potential surface are smooth. The model involving the irreversible stress work as a work hardening parameter, has been incorporated into finite element programs. Several practical problems are analyzed. This type of models is promising. I am convinced that they will be used in the future. 7 2 Future Work There are several aspects which should be studied further. What concerns the theoretical aspect first, the models should be fo1mulated in the framework of finite def or1nation and finite strain theory in order to study large defor1nation and large strains. How to determine the model should be emphasized. Secondly, in the models presented, the tension tests are not used in the deter1nination process. How to incorporate tension tests into the models is another challenge. Additional practical problems will be analyzed with the finite element program using these models. The implicit scheme will be incorporated into the finite element programs. This type of models should be f or1nulated for other kind of materials. Metal powders, other ceramics powders, other rocks, etc. can be investigated by using such type of models.

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APPENDIX A DISCRETE FORMUL_ATION OF ELASTICNISCOPLASTICITY FOR FINil'E ELEMENT METHOD In this appendix, we will consider the finite element formulation for the elastic/viscoplasticiy problems. First we formulate the elastic/viscoplasticity theory in discrete time for1n depending on the classic approach with truncated Taylor series in viscoplastic strain (Zienkiewicz & Cor111eau [1974]; Hughes & Taylor [1978]; Owen & Hinton [1980]; Marques & Owen [1983]; Szabo [1990]) in details Afterwards other methods are reviewed. In the elastic/viscoplastic problems, the stresses and strains are time dependent. Time t is considered as an independent variable. t is discretized in increment step form. Let us consider a typical time step ~t m = tm + t -tm The loading in this interval changes from Rm to ~+ 1 i.e., ~Rm = Rm + t R m At the beginning of the time step t,n, corresponding to load Rni' the displacements am, strains em, stresses am and viscoplastic strains e~, are already computed. We intend to calculate every quantity at the end of this time step ( at time n+i) From the previous chapters we have the constitutive equation for the rate of viscoplastic stra ins a F e = k <>= k T a a l W(t) aF = k H(t) aa 1 W(t) N H(t) (A.1) 181

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182 where = l W(t ) and N = a F. The viscoplastic strains can be calculated by implicit scheme H(t ) a a from equation (A l) ( A.2 ) where ( A.3) where e~, and e~ + I are the rates of viscoplastic strains at time 1m and tm + l respectively If 0 = 0, we have the explicit scheme (Euler forward scheme) for the integration of viscoplastic strains in which the increments of viscoplastic strain is fully determined by the quantities at time On the other hand, if 0 = 1, we have the fully implicit scheme (Euler backward scheme) for the integration, with the strain increment being determined from the strain rate corresponding to the end of time interval. The case 0 = 1/2 results in the so-called ''implicit trapezoidal'' scheme. I In order to define em + t in equation (A.3), we can use a limited Taylor series expansion (Zienkiewicz & Cormeau [1974]; Hughes & Taylor [1978]; Owen & Hinton [1980]; Marques & Owen [1983]) and write I I e = e + G maa +P '"lle 1 m + l m r11 ( A.4 ) where

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I ,n ae I ,n a e a a a w a w ,n a e 1 183 (A 5) and /ie~ 1 and /iom are the change of irreversible strain and the change of stress respectively occurring in the time interval lit"' = t,n + 1 t ni. Thus the equation (A.2) can be rewritten as I I I lie = e lit +C "'lio + Q m1ie m ,n m m ,n ( A.6) where C '"= 01it Gm. Q m = 01it P rn m m (A.7) It is noted that the eigenvalues in the matrix G deter1nine the limiting time step length lin (Cormeau [ 1975] ). The matrix G depends on the stress level and no difficulty arises in its evaluation in principle if F and H are given where G = k a N' + d~ bN, a a dH b =a H a a' ( A.8) ( A.9) Therefore, if an associative flow law (H = F) is used, G is a symmetric matrix; if a non associated flow law is employed, H F, G is a non-symmetric matrix. It is assumed that the strains are small. The total increment strains at time can be divided into elastic and viscoplastic parts

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184 ( A.10 ) The elastic part is obtained from the Hooke's law ( A I I ) where Dis an elastic constant matrix Introducing equation ( A.6 ) into equation ( A.11 ), we have ( A 12 ) where ( A.13 ) Now con s idering the equilibrium equation at time + i we have 'P = L ceT J B Ta rtt + ldV R m + l = 0 ( A.14 ) e or ( A 15 ) e e with the help of geometrical relation lle = Blla e = B c Ila e ( A.16 ) where 6-a e i s the in c rement di s placements on the node s in each element Ila i s the increment

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185 displacements on the nodes for the entire elements Bis displacement-strain matrix, and c e is transfor1nation matrix which is used to assemble each element into a global stiffness matrix ( see Yin [1987] ; Hughes [1989]) We have 'P = K Ila /:J?. = 0 m m m ( A 17 ) where K m is the tangent stiffness matrix llRm is ter1ned as incremental pseudo-loads, f( m = L Ce T J B T jj ni BdV Ce e ( A.18 ) e e + L ce T J B Tjj n p mfle~0/lt m dV. From the equation ( A 17 ) we can get the displacement increment, fl~. Then substituting fl~ back into eqn s ( A.16 ) and ( A 12 ) we obtain the stress increment /lo rn, and thu s a = a + Ila n1 + l m tn' a m + l = a, n + lla, n ( A 19 ) Equation ( A.10 ) allows us to get the incremental viscoplastic strain and vi s coplastic strain I I I e, n + l = e, 11 + /le m ( A 20 ) The above procedure i s called tangential s tiffness method Now let us con s ider the case 0 =0 in the above equations From the equations ( A 2 ) and ( A 13 ), we get

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and the equations (A.17) and (A.18) give respectively 'P = K!::..a, 11 /lR = O K = Lc : fBTDBdVc e e !::..R = R,n + I L ceT f BT a,ndv+ L ceT f B rve;nt::..t,ndV e e 186 (A.21) ( A.22) K is a constant stiffness matrix dependent on the elastic constants. The third term in the left hand side of last equation is evaluated at time This method is the well-known Euler forward method (Owen & Hinton [1980]). The Euler forward method has some advantages: the stiffness K is not needed to form at each time step and to be inverted; The solving of equation (A.22) is just back substitution procedure, where no iteration is needed. Thus, a lot of computational CPU time can be saved. However the accuracy of the Euler forward method is often not good and computational in s tability can occur if the time step t::..n is too large. The instability problem was analyzed by Corrr1eau [ 1975]. When 0 0.5, the tangent method is unconditional stable. This implies that the time marching scheme is numerical stable, but does not guarantee the accuracy of the solution at any stage. For 0 < 0.5, the integration process is only conditionally stable and the numerical integration can proceed only for the value of !::..n less than some critical value, (A.23)

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187 where at c rit can be detern1ined in some way. Now we will discuss several methods to detennine the critical time step length. Schemes can be employed in which the time step length can be a constant or vary for each time interval In the variable scheme, the magnitude of the time step is controlled by a f factor Q which limit s the maximum effective viscoplastic strain increment ae, n, as a fraction of total effective strain e m so that / ae = ,n 1 2~ I } 2 ( e .. ) ( e .. ) at < Cle = Q 3 lJ 111 11 m m n1 ( A.24 ) For isoparametric elements all strains are evaluated at the Gaussian integration points. Therefore an must be computed to s atisfy the above equation ( A 24 ) at each such point and the least value is taken for analysis. The minimum an involved in equation ( A 24) is then taken over all integrating point s in the material. The value of the time increment parameter Q mu s t be specified by the user For explicit time marching s cheme s accurate results have been obtained ( Zienkiewicz & Cormeau [1974]; Corrneau [1975] ) for the range 0 01 < Q < 0 15 For implicit scheme s values of Q up to 10 have been found to give stable solution though the a c curacy deteriorates In the computation in Chapter 3 '2=0.0009 and 0.01 were used. Satisfied re s ults were obtained Another u s eful limit can be impo s ed while using the variable time stepping s cheme The change in the time step length between any two interval s is limited according to ( A 25 ) where K is a s peci f i e d con s tant. Experience sugge s ts a value of K = 1 5 i s suitable, although

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188 there are no well established criteria for its estimation. In the computation in Chapter 3, K= 1.1 and 1.3 were used. The essential steps based on the Euler one step scheme in the solution process can be summarized as follows. The solution of the problem must begin from knowing the initial conditions at time t = 0, which are, of course, the solution of the static elastic solution. At this stage, a 0 e 0 a 0 and / 0 are known and e~ = O. The time marching scheme described above can then be used to advance the solution by one time step at a time. The solution sequence adopted is as follows: (1) For time step A1ro = t m+J t m, the following quantities (A.26) are known. Calculate (b) K'n = L, Ce JB Tjj m BdV C e e (A.27) e e + c 1 JB Tf)p ,nAe 1 0At dV L.,; e mm e (2) Compute the displacement increment A~ Aa, n = (K ,n) 1 AR (A.28)

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189 and stress increment (A.29) (3) Compute the total di s placements and total stresses at Gaussian points and other quantities a, n + I = a ,n + lla ,n, I I I a,, 1 + 1 = am + lla, n, e,n + t = e, n + /le 11 (A 30) (4) Continue to the next step from (1) to (4) The steady state can be reached if the viscoplastic strain rate e~ + t is acceptably close to zero at each Gaussian integrating point throughout the domain (i.e., within a specified tolerance). If so, steady state condition is deemed to have been achieved and the solution is either terminated or the next loading increment is applied. It can be shown that the static solution is the solution obtained with time-independent elastic-plastic solution (Zienkiewicz & Cormeau (1974] ; Simo & Govindjee [1991]). For elastic/viscoplasticity problems, there are two other new kinds of methods that have been developed, except the one we have discussed in the previous section. One group of methods is reviewed by Simo and Govindjee (1991]. They s tudied and discus se d a generalized midpoint rule with the assumption of convexity of the yield surface. The mid point version of this class of algorithms (0 = 1/2) is second order accurate in the standard sense defined by local truncated error. This algorithm is B-stable and includes the classical Euler backward method or return mapping algorithm (Simo & Taylor (1985]; Simo et al. [ 1988]) as a particular case. For the application to multiple yield surfaces, see Simo et al

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190 (1988]. Auricchio and Taylor (1994] also generalized the return mapping algorithm for computing viscoplastic strain with the concept of limit function. It turns out a third order polynomial equation for plastic multiplier or rate factor dl. The other method was proposed by Nemat-Nasser and Li (1992]. They reversed the procedures of returning mapping algorithm (i.e., the elastic solution as predictor and the plastic solution as corrector) and adapted the plastic solution as predictor and the elastic solution as corrector because the plastic strains are usually much larger than the elastic strains. They claimed that this kind scheme can give very good results no matter what case is used, small strains or finite strains

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APPENDENIX B T "'-i.4~EXPERIMENTALDATA FOR ALUMINA POWDE R AlO (DENSE) NOTE: ( 1 ) Total eleven test s (see table 4.2) are given in thi s appendix. (2) Area of burette section = 0.71 c m 2 (3) The unit of s tress is kg/cm 2 (4) 1 kg/cm 2 = 98.04 kPa. (5) all specimens were prepared with a vacuum pump. TEST 1 No: 6 1 1UAXI._.1\l_ COMPRESSION TEST ON ALUMINA POWDER Date: Aoril 19-20, 1995 Two l ayers oacked Sample: A l umina oowder AlO (dense Soecimen Weight : 844.9 (g) Diameter: I 2.820+2.825+2.824 /3 -2* 0.012( i n) = 2.799 I in) = 7 10946 (c m) Height: (8.9+8.91+8.9 1 5+8.91)/4-3.021 in) = 5.88875 (in) = 14.9574 (c m Vol u me: (7.10946)"2*14.95743*3.1416 I 4 593.774 (cm"3 densitv: 844.9 / 593.7739 = 1 42293 (kg/cm 3 B "Check= 0 94 backo r ess u re =4 0(kl!/cm"2 Hvdrostatic Loading time tlme-c oress u re burette time-cc confin-o strain-v K-value 10:15 10.25 4.3 24.01 0 0.3 0 10: 1 5 10.25 4.6 23.7 0 0.6 0.00037 10: 1 6 J 0.2667 4.6 23.69 0.0 1 667 0 6 0 00038 10: 19 1 0.3167 4.6 23.68 0.06667 0.6 0 0004 10:24 10 4 4.6 23.67 0.15 0.6 0.00041 10:25 10.4167 5 23.28 0.16667 I 0 00087 10:26 1 0.4333 5 23.27 0.18333 1 0.00089 191

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192 10 :29 1 0.4833 5 23.26 0 .23333 1 0 0009 10 :34 10 5667 5 23.26 0.31667 1 0 0009 10 :35 10.5833 4.8 23.38 0.33333 0.8 0.00075 1391 88 10 :3 6 10.6 4.8 23.38 0 .3 5 0.8 0.00075 10 :39 10 .6 5 4 .8 23.38 0.4 0.8 0.00075 10 :40 10 .6667 5 23.26 0.41667 1 0.0009 1 0:4 1 1 0.6833 5 23.26 0.43333 1 0.0009 10 :44 10 .7333 5 23 .26 0.48333 1 0.0009 10 :45 10.75 5.5 22.88 0.5 1 .5 0.00135 10 :46 L0 7667 5.5 22.87 0 51667 1 .5 0.00137 10 :49 10 .8167 5.5 22.86 0.56667 1 .5 0.00138 10 :54 1 0.9 5 5 22.86 0 .65 1 .5 0.00138 10 : 55 10 9167 5.3 22.96 0.66667 1 .3 0.00126 1670 25 10 :56 10 9333 5 .3 22.96 0.68333 1 .3 0.00126 10 :59 10.9833 5 .3 22.96 0.73333 1 .3 0.00126 11:00 11 5.5 22.86 0.75 1.5 0.00138 11 :0 1 11 .0 1 67 5.5 22.86 0.76667 1.5 0.00138 11:04 11.0667 5.5 22.86 0.81667 1.5 0.00138 11:05 11 .0833 6 22.5 0.83333 2 0.00181 11:06 11.1 6 22.49 0.85 2 0.00182 11:09 11 .15 6 22.47 0.9 2 0.00184 11 : 14 11 2333 6 22.45 0 .98 333 2 0.00187 11:15 11.25 5.8 22.52 1 1 .8 0.00178 2386.07 11: 16 11 .2667 5.8 22.52 1.01667 1 .8 0.00178 11 : 19 11 .3 1 67 5.8 22.52 1.06667 1.8 0.00178 11:20 11 .3333 6 22.45 1.08333 2 0 00187 11:21 11.35 6 22.45 1 .1 2 0.00187 11 :24 11.4 6 22 44 1 .15 2 0.00188 11 :30 11.5 6 22.44 1 .2 5 2 0.00188 1 2:3 0 1 2.5 6 22.4 2.25 2 0.00193 D evia tori c l oad ing confi ning pressure= 6-4=2 (kg/cm"2) rate=O. l (mm/min) diso or ess ur e burette stra in -I stra1 n-v s train -3 area-c stress E-va lu e 0 11.35 23.99 0 0 0 39.6976 0.28591 0. 1 33.6 23.91 0.00067 0.0001 -0.00 03 39.7203 0.84591 0.2 69 23.75 0 00134 0.00029 -0 .0005 39.7393 1.73632 0 .3 86 23.65 0 00201 0 00041 -0.0008 39.7612 2.1629 1 0.4 97 23.49 0.00267 0.0006 -0.00 1 39.7802 2.4384 0.432 100 23.49 0.00289 0 0006 -0.0 011 39 .7 887 2.51327 0.434 89.2 23.55 0.0029 0.00053 -0.00 1 2 39.792 1 2.24165 0 .398 60 23.59 0.00266 0.00048 -0.00 11 39.7844 1 .508 13 3047.6665 0.418 81.2 23.53 0.00279 0.00055 -0.00 11 39.7869 2.04087

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193 0.5 106.6 23.51 0.00334 0.00057 -0.0014 39.8078 2.67786 0.6 115 23.49 0.00401 0 0006 -0.0017 39.8336 2.88701 0.7 123 23.47 0 00468 0.00062 -0.002 39.8594 3.08585 0.8 130 .8 23.46 0.00535 0.00063 -0.0024 39.8857 3.27937 0.9 137.5 23.48 0.00602 0.00061 -0.0027 39.9135 3.44495 1 144 23.5 0.00669 0 00059 -0.003 39.9413 3 60529 1.098 150 23.53 0 00734 0.00055 -0.0034 39.9691 3.7529 1 104 135 23.53 0.00738 0.00055 -0.0034 39.9707 3 37747 1.064 100.6 23.54 0.0071 1 0.00054 -0.0033 39.9605 2.51749 3215.7784 1 108 140 23.51 0.0074 1 0 00057 -0.0034 39.9709 3.50255 1.2 156 23.55 0.00802 0.00053 -0.0037 39.9976 3.90024 1.3 161.3 23.55 0.00869 0.00053 -0.004 1 40.0245 4.03003 1.4 166 23.64 0.00936 0.00042 -0.0045 40.0559 4.14421 1.5 170 5 23.79 0.01003 0.00024 -0.0049 40.0901 4.25292 1.6 174 .8 23.92 0.0107 0.00008 -0.0053 40.1235 4.35655 1 .7 178 6 24.01 0.01137 -2e-05 -0.0057 40.1549 4.44777 1 .8 182 24.14 0.01203 -0.0002 -0.0061 40.1884 4.52868 1.9 185 .6 24.28 0.0127 -0.0003 -0.0065 40.2223 4.61436 2 189 24.41 0.01337 -0.0005 -0.0069 40.2558 4.69497 2 1 192 24.56 0.01404 -0.0007 -0.0074 40.2904 4.76541 2.2 194 6 24.7 0.01471 -0.0009 -0.0078 40.3244 4 82586 2.3 197 .2 24.86 0.01538 -0.001 -0.0082 40.3596 4.88608 2.408 200 25.01 0 0161 -0.0012 -0.0087 40 3964 4.95093 2.41 182 25 0.01611 -0.0012 -0.0087 40.3965 4 50534 2.374 150 5 24.99 0 01587 -0.0012 -0.0085 40.3861 3 72653 3235.8527 2.42 190 24.99 0.01618 -0.0012 -0.0087 40.3987 4.70312 2.5 202.9 25.11 0 01671 -0.0013 -0.009 40.4265 5.01898 2.6 204.7 25.28 0.01738 -0.0015 -0.0095 40.4623 5.05904 2.7 206 7 25.47 0.01805 -0.0018 0.0099 40.499 5.10383 2.8 208.7 25.62 0.01872 -0.002 -0 0103 40.5339 5.14878 2.9 210.5 25.8 0 01939 -0.0022 -0.0108 40 5702 5.18854 3 212.4 25.96 0.02006 -0.0024 -0.0112 40 6057 5.2308 3.2 216.1 26.35 0 02139 -0.0028 -0.0 1 2 1 40.6801 5.31218 3.4 219.2 26.74 0.02273 -0.0033 -0.013 40.7547 5.37852 3.6 222.1 27.13 0 02407 -0.0038 -0.0139 40.8295 5.43969 3.8 224.8 27.47 0.0254 1 -0.0042 0 0148 40.9021 5.49604 4 227 .3 27.85 0.02674 -0.0046 -0.0157 40.9769 5.54703 4 .2 230 28.22 0.02808 -0.0051 -0.0166 41.0514 5.60274 4.4 232.8 28.6 1 0 02942 -0.0055 -0.0 1 75 41 127 5.66051 4.6 235 28.98 0.03075 -0.006 -0.0184 41.2019 5.70362 4.8 236.7 29.36 0 03209 -0.0064 -0.0193 41.2775 5 73436 5 238 29.79 0.03343 -0.0069 -0.0202 41.3557 5 .75 494 5.004 219 29.78 0 03345 -0.0069 -0.0202 41.3564 5.29543

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194 4.96 170 29.65 0.03316 -0.0068 -0.02 41.3374 4. 1 125 4021.2917 5.022 230 29.67 0.03358 -0.0068 -0.0202 41 3 5 6 1 5.56 1 45 5.2 240.6 30 0.03477 -0.0072 -0.021 41.4234 5.80832 5.4 241.7 30.49 0.036 1 -0.0078 -0.02 1 9 41.505 5.8234 5.6 243 30.9 1 0.03744 -0.0083 -0.0229 4 1 .5834 5.84368 5.8 244.2 31 35 0.03878 -0.0088 -0.0238 4 1 .663 5.86132 6 245.3 31 75 0.0401 1 -0.0093 0.0247 41.7408 5.87674 6.5 247.8 32.75 0.04346 -0.0105 -0.027 4 1 .9364 5.90895 7 250 33.8 0.0468 -0.0117 -0.0293 42.1358 5.933 1 9 7.5 251.64 34.79 0.0501 4 -0.0 1 29 -0.0315 42.3337 5.9442 8 253.2 35.8 0.053 4 9 0 01 41 -0.0338 42.5339 5.9 5 29 8.004 221.34 35.9 0.0535 1 -0 0 1 43 -0.0339 42. 5 401 5 20309 7.958 180 35.8 0.0532 -0.0141 -0.0337 42.52 1 3 4 233 1 7 3 1 53.78 5 9 7.996 220.1 35.82 0.05346 -0.0142 -0.0338 42.5337 5.17472 8.1 255.4 36.05 0.05415 -0 0144 -0.0343 42 .5 765 5.9986 1 8.2 255.2 36.3 0.05482 -0.0147 -0.0348 42.6192 5.98791 8.3 254.7 36.5 0.05549 -0.0 1 5 -0.0352 42.6595 5 97054 8.5 254.4 36.9 0.05683 -0.0155 -0.036 1 42 740 1 5.95226 9 254.9 37.85 0.060 1 7 -0.0166 -0.0384 42.9402 5.93617 c h anl!e rate to 2.0 (mm/mi n ) 9 5 255 39.2 10 253 39.8 1 1 250 41.4 1 2 247 42.8 15 239 45.8 17 236 47.3 TEST2 No: 7 T R IAXIAL COMPRESSI O N TEST ON ALUMINA PO WD E R D ate: Ao ril 25 26, 1995 Two 1 aver me th o d Sa mol e: Alu mina powder Sp ecimen W e i e: h t : 845 {gl Di ameter: (2.825+2.825+2.823)/3 -2* 0.012(in) = 2.80033 (i n ) = 7.11285 (cm) H eil! h t: (8.92+8.9+8.9 I +8 89 /4-3.02 (i n )= 5 885 ( i n) = 1 4.9479 (cm)

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195 Volume: (7.112847 "2*14.9479*3.1415926 / 4 593.96 (cm"3) density : 845 / 593.9597 = 1.42266 11 B II Check= 0.97 hydrostatic l oading time-c camber-o burette time-cc confi n -p K-value time stram-v 11: 15 1 1 .25 4.3 26.5 0 0.3 0 11:15 11.25 5 25.88 0 1 0.00074 11 :16 11.2667 5 25.88 0.01667 1 0.00074 11: 19 11.3167 5 25.87 0.06667 1 0.00075 11:24 11.4 5 25.86 0.15 l 0.00077 11:25 11.4167 4.8 25 99 0. 1 6667 0.8 0.00061 1285.21 11:26 11.4333 4.8 25.98 0.18333 0.8 0.00062 11:29 11 4833 4.8 25.98 0.23333 0.8 0.00062 11 :30 11 5 4.8 25.98 0.25 0 8 0.00062 11 :31 I 1.5167 5 25.85 0.26667 1 0.00078 11 :32 11.5333 5 25.85 0.28333 1 0.00078 11 :35 11 5833 5 25.85 0.33333 1 0.00078 11 :40 11.6667 5 25.84 0 41667 1 0.00079 12:30 12 5 5 25.84 1.25 1 0 00079 13:30 13.5 5 25.84 2.25 1 0.00079 deviato r ic loadi n g co n fini n g oressure = 5 (chamber) 4 (backoressure) =1 1 kg/cm"2) displacement rate =0.1 (mm/min) d i sol. force burette strain-I stra 1 n -v strain-3 area-c stress E-va l ue 0 19.56 25.75 0 0 0 39.7353 0.49226 0.1 36 25.6 0.00067 0 00018 -0.0002 39.7548 0.90555 0.2 43.9 25.6 0.00134 0 000 1 8 -0 0006 39.78 1 4 1.10353 0.3 50.7 25.59 0 00201 0.00019 -0.0009 39 8076 1.27363 0.4 57 25.62 0 00268 0 00016 -0.0013 39.8357 1.43088 0.5 62 4 25.69 0.00334 0.00007 -0 00 1 6 39.8658 1.56525 0.6 68 25.78 0 00401 4e-05 -0.002 39.8969 1 70439 0.7 72.8 25.88 0.00468 -0.0002 -0.0024 39.9285 1.82326 0 .7 48 75 25 92 0.005 -0.0002 -0.0026 39.9433 1.87766 0.748 68.45 25.93 0.005 -0.0002 -0.0026 39.9438 1.71366 0.716 50 25.92 0.00479 -0.0002 -0.0025 39.9347 1 .25204 2156.3047 0.744 70 25.9 0 00498 -0 0002 -0.0026 39.94 1 3 1 .75257 0.8 77.3 25.99 0.00535 -0.0003 -0.0028 39.9606 1.9344 0.9 81.3 26.15 0.00602 -0 0005 -0.0032 39.9952 2.03275

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196 1 84.8 26.3 0.00669 -0.0007 -0.0 037 40.0293 2.11845 1.1 87.9 26.5 0.00736 -0.0009 0.0041 40.0658 2.19389 1.2 90.8 26.7 0 00803 -0.001 l 0.0046 40 1025 2.2642 1 .3 93.5 26.95 0.0087 -0 0014 -0.0051 40.1415 2.32926 1.4 95 .7 27.15 0 00937 -0.0017 -0.0055 40.1782 2.38189 1 5 98 27.36 0 01003 0.0019 0 006 40.2155 2.43687 1.596 100 27.54 0 .0 1068 -0.0021 -0 0064 40.2502 2.48446 1.596 91.25 27.54 0.01068 -0.0021 -0 0064 40.2502 2 .2 6707 1.554 64 27.5 0.0104 0 0021 -0.0062 40.2369 1.59058 2407.6343 1.606 95 27.49 0.01074 -0.0021 -0.0064 40.2505 2.36022 1.7 102 .6 27.75 0.01137 -0.0024 0 0069 40.2887 2.54662 1.8 104 .2 27.98 0.01204 0 0027 -0.0074 40.327 2 58388 1 .9 105 .7 28.2 0.01271 -0.0029 -0.0078 40.3649 2.61861 2 107 28 48 0 01338 -0.0033 -0.0083 40.4058 2.64813 2.1 108 4 28.71 0.01405 -0.0035 0.0088 40.4443 2.68023 2.2 109.6 28.97 0.01472 -0.0039 0.0093 40.4843 2.70722 2.3 110 .6 29.2 0.01539 -0.0041 0.0098 40.5229 2.72932 2.4 111 .88 29.46 0.01606 0.0044 0.0102 40.5631 2.75817 2.5 112 .7 29.74 0 01672 -0 0048 -0.0108 40.6042 2.77557 2.7 114 .7 1 30.02 0 01806 -0.0051 -0 0116 40.6731 2 82029 3 117 .0 5 30.95 0.02007 0.0062 -0.0131 40.8015 2.86876 3.2 118.61 31.48 0 02141 0 0069 0 0141 40.8831 2.9012 3.4 119 .84 31.95 0.02275 0.0074 -0.0151 40.9619 2.92564 3.6 121.01 32.48 0 .024 08 0.0081 0 .0 161 41.0439 2.9483 3.8 122 11 33.02 0.02542 -0.0087 -0.0171 41 1266 2.96912 4 12 3. 18 33.47 0 .026 76 -0.0092 -0.018 41.2052 2.98943 4.5 125 .2 5 34 7 0.0301 -0.0107 -0.0204 41.4076 3.02481 5 126.9 35.96 0 03345 -0.0122 0.0228 41.6 1 29 3.04954 5.5 128 .32 37.1 0.03679 -0.0136 -0.0252 4 1 .8137 3.06885 6 129 .53 38.25 0.04014 -0.015 -0.0276 42.0164 3.08284 7 132 40.3 0 04683 0.0174 -0 0321 42.4 1 36 3.11221 TEST 3 No : 9 TRIAXIAL COMPRESSION TEST ON ALUMINA POWDER Date: Mav 3-8, 1995 Two laver method Samole : Alumina powder Specimen Weight : 860.3 (g) Diameter : (2.82 0+2 .823 +2.825)/3 -0 025967(in) = 2.7967 (in) = 7.10362 (cm)

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197 H ei2 h t: (9.03+9.05+9.06+9.0 1 /4-3.02 ( in )= 6.0 1 75 ( i n) = 15 28 4 5 cm) Vo lu me : (7.103617)"2* 1 5.284 4 5*3.1416 / 4 605.759 (cm"3) de n s i ty: 860.3 / 605.7589 = 1 .4202 b ackp r ess ur e = 1 .5 (kg/cm"2'i B C h eck = 0.94 Hy dr ostatic l oa din 2 time C h am b -o b ur ette time-c co n fi n -o stra 1 n-v K -va lu e 10:20 1.8 23.6 0 0 3 0 10:21 2.5 22.93 0.0 1 667 1 0.00079 1 0:22 2.5 22.92 0.03333 1 0.0008 10:25 2.5 22.9 1 0.08333 1 0.00081 1 0:30 2.5 22.9 0.16667 1 0 00082 10:31 2.3 23 02 0.18333 0 8 0.00068 14 1 9 97 1 0:32 2.3 23.015 0.2 0.8 0.00069 1 0:36 2.5 22.89 0.26667 1 0.00083 10:37 2.5 22.89 0 28333 1 0.00083 1 0:40 2.5 22.89 0.33333 1 0 00083 10:4 1 3.5 22.06 0.35 2 0.00181 10:42 3.5 22.03 0.36667 2 0.00 1 84 10:4 5 3.5 21.995 0. 41 667 2 0.00 1 88 1 0:50 3.5 2 1 .995 0 5 2 0.00 1 88 1 0: 51 3.3 22 07 0.5 1 667 1.8 0.00 1 8 227 1 .95 10:52 3 3 22 07 0.53333 1.8 0.0018 1 0:5 5 3.3 22.07 0.58333 1.8 0.0018 10:56 3.5 21.99 0.6 2 0.00189 1 0:57 3.5 21.99 0.61667 2 0.00189 1 1 : 0 0 3.5 21 98 0.66667 2 0.0019 1 1 :0 1 4.5 2 1 1 9 0.68333 3 0.00283 1 1 :02 4.5 21.06 0.7 3 0.00298 11 : 1 0 4.5 2 1 .0 1 0.83333 3 0.00304 1 1: 11 4.3 2 1 .09 0.85 2.8 0.00295 2129.95 11:12 4.3 2 1 .09 0.86667 2.8 0.0 0 295 11 : 1 5 4.3 2 1. 09 0.91667 2.8 0.00295 1 1 : 16 4.5 21.005 0.93333 3 0.00305 1 1 : 1 7 4.5 2 1 0.95 3 0.00305 11:20 4.5 21 1 3 0.00305

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198 11 :21 5 5 20.06 1.01667 4 0.00416 11 :22 5.5 20.08 1.03333 4 0.00413 stop Deviatoric l oadini! co n finini!-nress ur e= 5.5(chamber-J) 1.5 ( ba ck-o) = 4 (k ucm"2) rate=0 1 mm/min displace force burette strain-I strrun-v strain-3 area-c stress E-va lu e 0 20 20.24 0 0 0 39.6324 0.50464 0.1 31 6 20.18 0.00065 0 00007 -0.0003 39.6555 0 79686 0.2 55.6 20 05 0.00131 0 00022 0.0005 39.6754 1.40137 0.3 80 19.86 0.00196 0.00045 -0.0008 39.6926 2.01549 0.402 100 19.6 8 0.00263 0 00066 -0.001 39.7108 2 .518 21 0.406 86 .2 19 .66 0 00266 0 00068 -0.001 39.7 109 2.17069 0.378 60 19.69 0.00247 0.00065 -0.0009 39.705 1.51115 3600.278 0.396 80 19.66 0.00259 0.00068 -0.00 1 39.7083 2.01469 0.5 118.4 19 48 0.00327 0.00089 -0.00 12 39.727 2.98034 0.6 1 34.5 19.28 0.00393 0.00113 -0 .00 14 39.7437 3.38418 0.7 150 19 .09 0.00458 0.00135 -0.0016 39.761 3.77254 0.8 164 18 .93 0.00523 0.00154 -0.0018 39 7796 4.12271 0.9 177 1 8.8 0.00589 0 00169 -0.0021 39 .799 7 4.44727 1 189 18 .67 0.00654 0 00184 -0 .002 3 39.8199 4.74638 1.082 200 1 8.53 0.00708 0.00201 -0.0025 39.8348 5 02073 1 .092 178 5 18 51 0.00714 0 00203 -0 .00 26 39.8365 4.48082 1 .058 140 18.56 0 .0 0692 0.00197 -0.0025 39.8299 3.51495 4341 .9 959 1.088 180 18.49 0.00712 0.00205 -0.0025 39.8345 4.51869 1.2 214 18 4 0.00785 0.00216 -0.0028 39.8597 5.36883 1.3 225.2 1 8.29 0.00851 0.00229 -0.003 1 39.8809 5.64682 1 .4 235.2 18.23 0.00916 0.00236 0.0034 39.9044 5.89409 1 5 245 18.16 0 00981 0.00244 -0 0037 39.9275 6.13613 1 .558 250 18.12 0.01019 0.00249 -0.0039 39.9409 6 .2 5925 1 .568 224.2 18 .06 0.0 1 026 0.00256 -0.0039 39.9407 5.61332 1.536 180 1 8.1 0.01005 0.00251 -0.0038 39.9341 4.50742 5282.2078 1 .558 220 18 .04 0.01019 0.00258 -0.0038 39.9371 5.50866 1.6 250 1 8.0 1 0 01047 0.00262 -0.0039 39.9 46 8 6.25832 1.7 264 17 .99 0.01112 0.00264 -0.0042 39.9723 6.60458 1 .8 272 .6 17.97 0 01178 0.00266 -0.0046 39.9978 6 81537 1.9 280.3 17 94 0.01243 0.0027 -0.0049 40.0229 7.00349 2 288 1 7.88 0 01309 0 00277 0.0052 40.0466 7. 19162 2.1 295 17.86 0.01374 0 00279 -0.0055 40.0722 7.3617 1 2 .1 76 300 17 .84 0.01424 0.00282 -0.00 5 7 40.0915 7.48288 2.186 271 6 17 .79 0.0143 0.00288 0.0057 40.0918 6.77445 2.166 240 17.83 0.01417 0.00283 -0 0057 40 0884 5.98678 6019.6158 2.182 270 17.78 0.01428 0.00289 -0 .0057 40.0903 6.7348

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199 2.3 309 17.78 0.01505 0.00289 -0.0061 40.1217 7.70157 2.4 315 17 .8 0.0157 0.00286 -0.0064 40.1493 7.84572 2.5 321 17.84 0.01636 0.00282 -0.0068 40.1779 7.98947 2.6 327 17.88 0.01701 0.00277 -0.0071 40.2065 8.13301 2.7 332 17.89 0.01767 0.00276 -0.0075 40.2338 8.25177 2 8 337 17 .9 1 0.01832 0.00273 -0 0078 40.2615 8.37027 2.9 342.2 17.97 0.01897 0.00266 -0 0082 40 .29 12 8.49316 3 347 17.985 0.01963 0.00265 -0.0085 40.3 1 88 8.6064 3.066 350 18 0.02006 0.00263 -0.0087 40.3373 8.67683 3.074 317.1 17 97 0.02011 0.00266 -0.0087 40.338 7.86106 3.046 266 17.96 0.01993 0.00268 -0.0086 40.33 6 59558 6907.9365 3.074 320 17 .88 0.02011 0.00277 -0. 0087 40.3338 7.9338 3.1 344 17 .89 0.02028 0.00276 -0.0088 40.3413 8.52725 3.2 357.7 17 99 0.02094 0.00264 -0.009 1 40.373 8.85989 3.3 361.5 18.06 0.02159 0.00256 -0.0095 40.4033 8.94729 3.4 365.5 18 .1 0.02224 0.00251 -0.0099 40.4322 9.03982 3.5 368.8 18 16 0.0229 0.00244 -0.0102 40 .4622 9.11469 3.6 373 18 .23 0.02355 0.00236 -0.0 106 40.4926 9.21156 3.7 376.5 18.29 0.02421 0.00229 -0. 011 40.5226 9.29111 3.8 380 18.36 0.02486 0.00221 -0. 0113 40.553 1 9.37042 3.9 383.3 18.43 0.02552 0.00212 -0 .01 17 40.5837 9.44468 4 386.6 18 49 0.02617 0.00205 -0.0121 40 .6 138 9.51892 4.2 392.9 18.66 0.02748 0.00185 -0.0128 40 .6766 9.65911 4 4 398.9 1 8.82 0 02879 0.00167 -0 .01 36 40.7391 9.79158 4.446 400 18.85 0.02909 0.00163 -0.0 137 40.7531 9.81519 4 452 364.l 18 .78 0.02913 0.00171 -0.0137 40.7514 8.93465 4 .422 320 18.77 0.02893 0.00173 -0 .0136 40.7427 7.85416 5504.9007 4.466 380 1 8.76 0.02922 0.00174 -0.0 137 40.7543 9.32416 4 .6 406 18.89 0.0301 0.00 1 58 -0.0 143 40.7974 9.95161 4.8 410.9 19.09 0.0314 0.00135 -0.015 40.8621 10.0558 5 415 8 19 .28 0.03271 0.00113 -0.0158 40 .9265 10.1597 5.5 426.7 19 .79 0 03598 0.00053 -0 .0 1 77 41 .09 10.3845 6 4 36.6 20.3 1 0.03926 -8e-05 -0.0197 41 .2 551 10.5829 6.5 445 5 20.85 0.04253 -0.0007 -0.0216 41.4223 10 .7 551 7 453.2 2 1.46 0.0458 -0.00 14 -0.0236 41.594 10.8958 7.5 460.1 2 1 .96 0.04907 -0.002 -0.0255 41 .76 16 11 .0 173 8 466 3 22.68 0.05234 -0.0029 -0.0276 41.9411 11.118 8.5 47 2.1 23.32 0 05561 -0.0036 -0.0296 42.1179 11.209 9 477 2 23.95 0.05888 -0.0044 -0.0316 42.2954 11 .2825 9.5 481 .45 24.59 0.06215 -0.0051 -0.0336 42.4747 11 .335 10 485 .5 25 25 0.06543 -0.0059 -0.0357 42.6562 11 3817

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200 TEST4 No : 10 Triaxial compression test on alumina powder May 1995 10-1 1, Two laver method Sample : Alumina oowder Specimen Weil!ht : 865.9 ( ~) Diameter: (2.82 4+2 .827+2.8281/3 -0.025967 in)= 2.8003663 . 7.1129305 (cm) m = He ight: (9.03+9.02+9.04+9.02 /4-3.02 ( in)= 6.0075 ( in ) = 15.25905 (cm) Volume: I 7.11293 1"2*15.25905*3 141925 / 4 606.40167 (cm"3) densitv: 865.9 I 606.4017 = 1 .42793 14 B Check = 0.945 co n finin,g oressure = 6.5 chambe r -o) 1.5 back-o) = 5 .0 (kulcm"2 ) Hydr ostatic loadinl! time camber-o burette time -c confin-o strai nv K -v alue 10:00 1 .8 29.84 0 0.3 0 10:01 2.5 29.11 0.0166667 I 0.0008559 1 0:02 2.5 29. 11 0.0333333 1 0 0008559 10 :05 2.5 29.11 0.0833333 1 0 0008559 10:10 2.5 29.1 1 0.1666667 1 0.0008 55 9 10:11 2.3 29.22 0.1833333 0 8 0.0007269 1550 .70 1 2 10:12 2.3 29.22 0.2 0 .8 0.0007269 10:15 2.3 29.22 0.25 0.8 0.0007269 10:16 2.5 29. 105 0 .266 6667 l 0.0008618 10 : 1 7 2.5 29. 1 05 0 .2833333 1 0.0008618 10 :20 2.5 29. 105 0.3333333 1 0.0008618 10 :2 1 3.5 28.41 0.35 2 0.0016767 10:22 3.5 28.39 0.3666667 2 0.001700 1 10:25 3 .5 28.35 0.4166667 2 0 001747 10 :30 3.5 28.35 0.5 2 0.001747 l 0:31 3.2 28.44 0.5166667 1 .7 0.0016415 2842 .9 522 10 :32 3.2 28.44 0.5333333 1.7 0 0016415 10:35 3.2 28.44 0.5833333 1 .7 0.0016415 10:36 3.5 28.26 0.6 2 0.0018525 10:37 3.5 28.26 0.6166667 2 0 0018525

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201 10:40 3.5 28.26 0.6666667 2 0.0018525 10:41 4.5 27.53 0.6833333 3 0.0027085 1 0:42 4.5 27.5 0.7 3 0.0027436 10:45 4.5 27.45 0.75 3 0.0028023 1 0:50 4.5 27.43 0.8333333 3 0.0028257 10 :51 4.2 27.525 0.85 2.7 0.0027143 2693.3231 10:52 4.2 27.525 0.8666667 2.7 0.0027143 10:55 4.2 27.52 0.9166667 2.7 0.0027202 10:56 4.5 27.41 0.9333333 3 0.0028492 10:57 4.5 27.41 0.95 3 0.0028492 11:00 4.5 27.41 1 3 0.0028492 11:01 5.5 26.71 1 .0 166667 4 0.0036699 11:02 5.5 26.66 1.0333333 4 0.0037285 11 :05 5.5 26.6 1 .0833333 4 0.0037989 11 :10 5.5 26.53 1 1 666667 4 0.0038809 11 : 11 5.2 26.63 1 1 833333 3.7 0.0037637 2558.657 11 : 12 5.2 26 62 1 .2 3.7 0 .0 037754 11: 15 5.2 26.6 1 5 1.25 3.7 0.00378 1 3 11: 16 5.5 26.51 1 .2666667 4 0.0039044 11 : 17 5 .5 26.51 1.2833333 4 0.0039044 11:20 5.5 26.5 1 .3333333 4 0.0039161 11 :21 6.5 25.85 1.35 5 0.0046782 11 :22 6.5 25.78 1.3666667 5 0.0047603 11 :25 6.5 25.7 1 .4166667 5 0.0048541 11:30 6.5 25.64 1.5 5 0.0049245 11 :3 1 6.2 25.705 1.5166667 4.7 0.0048482 3936.3953 11 :32 6.2 25.7 1 .5333333 4.7 0.0048541 11:35 6.2 25.7 1. 5833333 4.7 0.0048541 11:36 6.5 25.6 1 .6 5 0.00497 1 4 11:37 6.5 25.59 1 .6 166667 5 0.004983 1 11 :40 6.5 25.59 1 .6666667 5 0.004983 1 12:50 6.5 25.48 2.8333333 5 0.0051121 Deviatoric l oading confming-p =6.5 (c hamb er-o)1 .5 (back-o >=5 (kg/cmA2) rate=O.l mm/min diso l force burette strai n -1 strain-v stra in-3 area-c s tress E-value 0 26.4 25.31 0 0 0 39.740462 0.6643103 0.1 50 25.2 0.0006553 0.000 1 29 -0.000263 39.76 1 394 1 .25750 12 0.2 74 25.0 1 0.0013107 0.0003517 -0.000479 39.778621 1.8602957 0.3 95 24.8 0.001966 0.000598 -0. 000684 39.794937 2.3872383 0.328 100 24.75 0.0021495 0.0006566 -0.000746 39.799921 2.5125678 0.33 84 24.68 0.002 1 627 0.0007387 -0. 000712 39.797175 2.1107026 0.298 58.6 24.72 0.0019529 0.00069 1 8 -0.00063 1 39.79068 1 .4727067 3042.2536

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202 0.326 80.6 24.69 0.0021364 0.0007269 -0 .000705 39.796596 2.0252988 0.4 116.8 24.53 0.0026214 0.0009145 -0.000853 39.808472 2.9340488 0.5 138.1 24.26 0.0032767 0.0012311 -0.001023 39.822024 3.4679302 0.6 158.8 24.04 0.0039321 0.0014891 -0.001222 39.837933 3.9861506 0.7 178 23.79 0.0045874 0.0017822 -0.001403 39.852458 4.4664748 0.8 195 23.58 0.0052428 0.0020284 -0.001607 39.868877 4.8910332 0.822 200 23.53 0.005387 0.002087 -0.00 165 39.872313 5.0160119 0.834 175.1 23.49 0.0054656 0.0021339 -0.00 1666 39.873592 4.3913776 0.802 140 23.51 0.0052559 0.0021105 -0. 001573 39.866123 3.51 17536 4194.4459 0.834 180 23.48 0.0054656 0.0021457 -0 .00 166 39.873124 4.514319 0.9 212 23.35 0.0058981 0.0022981 -0.00 18 39.884379 5.3153642 1 230 23.18 0.0065535 0.0024974 -0.002028 39.902716 5.7640186 1.1 245 23.01 0.0072088 0.0026967 -0 .002256 39.921078 6.1371089 1 .2 259 22.86 0.0078642 0.0028726 -0.002496 39.940402 6.4846617 1. 3 273 22.7 0.0085195 0.0030602 -0 .00273 39.959283 6.8319544 1.4 285 22.56 0.0091749 0.0032243 -0.002975 39.979129 7.1287196 1.5 298 22.46 0.0098302 0.0033416 -0.003244 40.000884 7.4498354 1.518 300 22.45 0.0099482 0.0033533 -0.003297 40.005179 7.4990291 1.528 267.5 22.35 0.0100137 0.0034706 -0.003272 40.003121 6.6869783 1 .506 230 22.38 0.0098696 0.0034354 -0.0032 17 39.998707 5.7501858 6497.5289 1.528 271 22.35 0.0100137 0.0034706 -0.003272 40.003121 6.7744715 1.6 308 22.28 0.0104856 0.0035526 -0.003466 40.0189 7.6963635 1 .7 321 22.19 0.0111409 0.0036582 -0.00374 1 40.041181 8.0167466 1.8 331 22.11 0.0117963 0.003752 -0.004022 40.063963 8.2617888 1 .9 340 22.04 0.0124516 0.003834 -0.004309 40.087247 8.4815004 2 350 21.96 0.013107 0.0039278 -0.00459 40.11009 8.7259839 2.1 359 21.89 0.0137623 0.0040099 -0.004876 40.133436 8.9451599 2.2 366.7 21.85 0.0144177 0.0040568 -0.00518 40.158231 9.1313784 2.3 375.1 21.8 0.015073 0.0041154 -0.005479 40.182586 9.3348896 2.4 382.6 21.75 0.0157284 0.0041741 -0.005777 40.206973 9.5157623 2.5 390 21.71 0.0163837 0.004221 -0.006081 40.231867 9.6938082 2.6 397 2 1 .67 0.0170391 0.0042679 -0.006386 40.256794 9.8616895 2.644 400 21.66 0.0173274 0.0042796 -0 .006524 40.268132 9.9334133 2.654 359.2 21.59 0.017393 0.0043617 -0.006516 40.267499 8.9203455 2.626 309.7 21.595 0.0172095 0.0043558 -0.006427 40.260217 7.6924572 6691.5744 2.652 360.7 21.47 0.0173798 0.0045024 -0 .006439 40.261271 8.9589819 2.7 400 21.46 0.0176944 0.0045141 -0.00 659 40.27369 9.9320425 2.8 411.5 21.52 0.0183498 0.0044437 -0.0069 53 40.303424 10 .2 10051 2.9 417.8 21.54 0.0190051 0.0044203 -0.007292 40.331299 10.3592 3 423.7 21.54 0.0196605 0.0044203 -0.00762 40.35826 10.49847 3.1 429 21.54 0.0203158 0.0044203 -0 .007948 40.385257 10.622688 3.2 434.7 21.55 0.0209712 0.0044086 -0.008281 40.412767 10.756502 3.3 440 21.55 0.0216265 0.0044086 -0.008609 40.439836 10.880361

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203 3.4 445 2 1 .56 0.0222819 0.0043968 0.008943 40.467419 10 996501 3 498 450 2 1 56 0.0229241 0.0043968 -0.009264 40.494019 11 11 2752 3.506 405 .5 21 49 0.0229765 0.0044789 -0.009249 40 .492853 10 .014 11 3 3.468 349.2 2 1 .49 0.0227275 0.0044789 -0.009 1 24 40 .482535 8.6259421 5574.2543 3.502 406.2 21.49 0.0229503 0.0044789 -0.009236 40.491767 10 .031669 3.6 455 2 1 .49 0 0235926 0.0044789 -0.009557 40.518401 11.229466 3.7 461 2 1 .5 1 0.0242479 0.0044555 0.009896 40.546569 11 .369643 3.8 465.5 2 1 53 0.0249033 0.004432 -0.0 10 236 40.574776 11 .472645 3.9 469.7 2 1. 57 0.0255586 0.0043851 -0.0 10587 40.603977 11 .567832 4 473 .9 2 1.6 0.026214 0.0043499 0.010932 40.632738 11.663009 4.2 480 2 1 .65 0.0275246 0.0042913 -0.011617 40.689899 11.79654 4 4 489.6 2 1 .73 0.0288353 0.0041975 -0.0123 19 40.748653 12.015121 4.6 496.8 21.8 1 0.030146 0.0041037 -0.0 13021 40.807565 12.174213 4.8 504 21.9 1 0.0314567 0 0039865 -0.013735 40.8676 12.332508 5 510 .5 22 0.0327674 0.0038809 -0.014443 40.927315 12.473332 5.2 516 .8 22.1 0.0340781 0 0037637 -0.015157 40.987675 1 2.608668 5.4 522.9 22.21 0.0353888 0 0036347 -0.015877 41.048682 1 2.738533 5.6 528.8 22.33 0.0366995 0 003494 -0.0 1660 3 41.110338 12 .862944 5.8 534.2 22.48 0.0380102 0.0033181 -0 017346 41.173616 1 2.974328 6 539.2 22.61 0.0393209 0.0031657 -0.0 18 078 41.236096 1 3.075922 6.3 546.5 22.84 0.041287 0.0028961 -0.019195 41 .33 18 38 1 3.222252 6.448 550 22.94 0.0422569 0 0027788 -0.0 19 739 41 378561 13 291907 6.462 498.79 22.84 0.0423486 0.0028961 -0.019726 41.377659 12.054573 6.426 459.3 22.84 0.0421127 0.0028961 -0.019608 41.367468 11.102928 4033 .6 668 6.462 501 2 22.8 1 0.0423486 0.0029312 -0.019709 41 .3 762 12 11 32 44 6 5 546 22.86 0.0425977 0.0028726 -0.019863 41.389396 1 3. 191 785 6.6 555 22.97 0.043253 0.0027436 -0.020255 41.423104 13.39832 6.8 559 23. 13 0.04 456 37 0.002556 0.021004 41.487732 1 3.473863 7 563 23.3 0.0458744 0.0023567 0.021759 41.553026 1 3. 54 89 5 3 7.5 572 23 .71 0.0491512 0.001876 -0.023638 41.716315 1 3.7 1166 2 8 581 24. 19 0.0524279 0.0013132 0.025557 41 884175 13 .871588 8.5 5 88.2 24 .68 0.0557046 0.0007387 -0.027483 42.053693 13.986881 9 595 25. 1 8 0 0589814 0 0001524 -0.029414 42.224887 14 .0912 1 6 9.5 601.2 2 5.68 0.0622581 -0 .000434 0.031346 42 .3 97278 14 1 801 5 6 1 0 606.1 26.18 0 0655349 -0.00102 0.033277 42.570877 14.237433 1 0.5 611 26.7 0.0688 11 6 -0.00 1 63 0.035221 42 746699 14.293501 11 616.1 27.22 0.0720884 -0.002239 -0.037164 42 .923 763 14 .353355 11.5 620.3 27.73 0 0753651 -0.002837 -0.039101 43.101578 14.391584 12 623.7 28.25 0.0786419 -0.003447 -0. 04 1044 43 281163 1 4.4 1 0426 1 2.5 626.7 28.8 0.0819186 -0.004092 -0.043005 43.463553 14.418978 1 3 629.2 29.33 0.0851953 -0.004713 -0.044954 43.646231 14 .4 15907 1 3.5 631 .3 29.93 0.0884721 -0.005 41 7 -0.046944 43.8338 14 402128 14 633.3 30 .42 0.0917488 -0.00599 1 -0.0 48 87 44 01708 14 .387597

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204 1 4.5 634.55 30.95 0.0950256 -0.006613 -0.050819 44.203746 14.355118 15 635.6 31.28 0.0983023 -0.007 -0.052651 44.381434 14.321304 15.5 635.78 32.03 0.1015791 -0.007879 -0.054729 44.58220 1 14.260848 TESTS No: 11 TRIAXIAL COMPRESSION TEST ON ALUMINA POWDER Date: J u ne 27-29, 1 995 Two layer method Sam p le: Al u mina powder Specimen Weight : 865.9 I {J Diameter: (2.824+2.822+2.826)/3 -0.0258(in) = 2 .7 982 (in) = 7.10743 (cm) Height: (9.08+9.10+9.09 + 9. 1 0)/4-3 02 i'in) = 6.0725 (in = 15 .4 242 1cm) Volume: (7.107428 "2*15.42415*3.1415926 / 4 611 95 (cm"3 ) densitv : 865.9 I 611.9502 = 1.41498 B Check = 0.94 confi n ing oressure = 6.51 c h ambe r p)-l 5 1 back o)=5.0 nlcm"2 > Hydrostatic l oading tune camber-o burette tinec co nfin-o strain-v K-value 10 :3 0 1 .8 37.52 0 0.3 0 10:31 2.5 36.82 0.01667 1 0 00081 10:32 2.5 36.81 0.03333 1 0.00082 10 :3 5 2.5 36.81 0 08333 1 0 00082 10:40 2.5 36.81 0 1 6667 1 0 00082 10:41 2.3 36.92 0 18333 0.8 0.0007 1 564 89 10:42 2.3 36.92 0 .2 0.8 0.0007 10:45 2.3 36.92 0 .2 5 0.8 0.0007 10:46 2.5 36.81 0 .2 6667 1 0 00082 10:47 2 .5 36.81 0 .28333 1 0.00082 10:50 2.5 36.81 0.3 3333 1 0.00082 10 : 51 3.5 36.03 0.35 2 0.00173 1 0:52 3.5 36.01 0.36667 2 0.00175 10 :5 5 3.5 3 6.005 0.41667 2 0.00176 11 : 00 3.5 36 0.5 2 0.00 1 77 11 : 01 3.2 36.105 0.51667 1.7 0.00164 2459.11

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205 11:02 3 .2 36.105 0.53333 1.7 0.00164 11:05 3.2 36.1 0.58333 1.7 0.00165 11:06 3.5 36 0.6 2 0 00177 JI :07 3.5 36 0.61667 2 0.00177 11 :10 3.5 36 0 .66667 2 0.00177 11: 11 4 .5 35.23 0.68333 3 0.00266 11: 12 4 .5 35.19 0.7 3 0.0027 1 11: 15 4.5 35.12 0.75 3 0.00279 11 :20 4.5 35.11 0.83333 3 0.0028 11 :2 1 4.2 35.205 0.85 2.7 0.00269 2717 97 11 :22 4.2 35.205 0.86667 2.7 0.00269 11 :25 4.2 35.205 0.91667 2.7 0.00269 11 :26 4.5 35.1 1 0.93333 3 0.0028 11 :27 4 .5 35. 11 0.95 3 0.0028 11:30 4.5 35.11 1 3 0.0028 11 :31 5 5 34.35 1 .0 16 67 4 0 00368 J 1:32 5.5 34.25 1.03333 4 0.0038 11 :35 5.5 34. 1 8 1.08333 4 0.00388 11 :40 5.5 34. 15 1.16667 4 0.00392 11 :4 1 5.2 34.205 1 1 8333 3.7 0 00385 4694.67 11 :42 5.2 34.205 1.2 3.7 0.00385 11:45 5.2 34.205 1 .25 3.7 0.00385 11:46 5.5 34. 1 5 1.26667 4 0.00392 11 :47 5.5 34. 14 1.28333 4 0.00393 11 :50 5.5 34. 1 4 1 .33333 4 0.00393 11 :51 6.5 33.48 1 .35 5 0.00469 11:52 6.5 33.31 1.36667 5 0 00489 l 1 :55 6.5 33.26 1.4 1 667 5 0.00495 1 2:00 6.5 33.21 1.5 5 0.00501 1 2:0 1 6.2 33.29 1 .5 1 667 4.7 0.0049 1 3227.59 12:02 6.2 33.29 1 53333 4.7 0.00491 12:05 6.2 33.29 1 .58333 4.7 0.00491 1 2:06 6.5 33.2 1.6 5 0.00502 1 2:07 6.5 33.2 1 .61667 5 0.00502 12: 10 6.5 33.19 1 .66667 5 0 00503 13:30 6.5 33.07 3 5 0.00517 Deviatoric l oadinQ co nfinin -n =6.5 1 c h amber-o )1 .5 i'back-n )=5 'kf?/c m"2 ) r ate=0. 1 mm/min disol force burette strain-1 stra tn v stra in3 area-c stress E-value 0 24 32.9 1 0 0 0 39.6752 0.60491 0.1 52 32.78 0.00065 0.00015 -0.0002 39.6949 1 .3 0999 0.2 74 32.58 0 .00 13 0.00038 -0.0005 39.7115 1 .86344

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206 0.3 97 32.31 0.00195 0.0007 -0.0006 39.7248 2.4418 0.318 101 32.28 0.00206 0.00073 -0.0007 39.728 2.54228 0.326 84.3 32.21 0.00211 0.00081 -0.0007 39.7269 2.12199 0.308 60.5 32.24 0.002 0.00078 -0.0006 39.7236 1.52302 5132.52 0.318 81.2 32.22 0.00206 0.0008 -0.0006 39.7253 2.04404 0.34 100 32.22 0.0022 0.0008 -0.0007 39.731 2.51693 0.4 121 31.98 0.00259 0.00108 -0.0008 39.7354 3 04515 0.5 140 3 1.78 0.00324 0.00131 -0.00 1 39.7 52 3.52184 0.6 158 31.51 0.00389 0.00163 -0.0011 39.7653 3.97331 0.7 174 3 1.28 0.00454 0.00189 -0.0013 39.7806 4.37399 0.8 189 31.11 0.00519 0.00209 -0.00 15 39 7986 4.74891 0.868 200 30.95 0.00563 0.00228 -0.0017 39.8089 5.02401 0.876 174 .3 30.89 0.00568 0.00235 -0.00 17 39.8082 4.3785 0.848 140 30.91 0.0055 0.00232 -0.0016 39.8018 3.51743 4743.3252 0.878 180 30.87 0.00569 0.00237 -0.0017 39.8077 4.52173 0.9 19 7 30.8 0.00584 0.00245 -0.0017 39.8102 4 94848 1 220 30.65 0.00648 0.00263 -0.0019 39.8292 5.52358 1.1 232 30.48 0.00713 0.00282 -0.0022 39.8473 5.82222 1.2 245 30.29 0.00778 0.00304 -0.0024 39.8646 6.14581 1.3 257 30. 11 0.00843 0.00325 -0.0026 39.8822 6.44397 1 .4 269 30.04 0.00908 0.00333 -0.0029 39.9051 6.741 1.5 279 29.91 0.00973 0.00349 -0.003 1 39.9252 6.98807 1 .6 289 29.78 0.01037 0.00364 -0.0034 39.9453 7.2349 1 .7 14 300 29.64 0.01111 0.0038 -0.0037 39.9686 7.50589 1.728 267 29.58 0.0112 0.00387 -0.0037 39.9695 6.6801 1.706 230 29.6 0.01106 0 00385 -0.0036 39.9646 5.75509 6485.2305 1 .728 270 29.58 0.0112 0.00387 -0 .0037 39.9695 6.75516 1.8 307 29.49 0.01167 0.00397 -0.0038 39.9841 7.67804 1 .9 317.6 29.39 0.01232 0.00409 -0.004 1 40.0057 7.93886 2 326 29.3 0.01297 0.00419 -0.0044 40 .0278 8.14434 2.1 333.8 29.21 0.01362 0.0043 -0.0047 40 0499 8.3346 2.2 341 29. 17 0.01426 0.00435 -0.005 40.0744 8 50918 2.3 348 29. l 0.01491 0.00443 -0.0052 40.0975 8.67885 2.4 355 29.02 0.01556 0.00452 -0.0055 40.1201 8.84843 2.5 362 28.98 0.01621 0.00457 -0.0058 40.1447 9.01738 2.6 369 28.94 0.01686 0.00461 -0.006 1 40.1693 9.18612 2.7 375 28.89 0.01751 0.00467 -0.0064 40.1935 9.32988 2.8 381 28.81 0.01815 0.00476 -0.0067 40.2162 9.47378 2.9 387 28.78 0.0188 0.0048 -0.007 40 .24 14 9.61696 3 392 28.76 0.01945 0.00482 -0.0073 40 2671 9.735 3.1 397.5 28.72 0.0201 0 00487 -0.0076 40.2918 9.86552 3. 144 400 28.71 0.02038 0 00488 -0.0078 40.3031 9.9248 3. 158 360 28.64 0.02047 0.00496 -0.0078 40 .3035 8.93222

PAGE 215

207 3.122 310 28.67 0.02024 0.00493 -0.0077 4 0.2953 7.6932 5308.5757 3.154 360 28.61 0.02045 0.005 -0.0077 4 0.3011 8.93277 3.2 399 28.6 0.02075 0.00501 -0.0079 40.3129 9.89759 3.3 409 28.6 0.0214 0.00501 -0.0082 40.3396 10.1389 3.4 414 28.59 0.02204 0.00502 -0.0085 40.36 5 8 10.2562 3.5 418 28.58 0.02269 0.00503 -0.0088 40.3921 10 3485 3.6 422 28.58 0.02334 0.00503 -0.0092 40.4 1 9 10 4406 3.7 427 28.57 0.02399 0.00504 -0.0095 40.4453 10. 5 575 3 8 431 28.56 0.02464 0.00505 -0.0098 40. 4 7 1 7 10.6494 3.9 4 35 28.55 0.02529 0.00507 -0.0 1 0 1 40.4982 10.7412 4 439.8 28.54 0.02593 0.00508 -0.0 1 0 4 40.52 4 7 10.8526 4.2 443 28.55 0.02723 0.00507 -0.0 111 40. 5 792 1 0.9169 4.3 447.2 28.56 0.02788 0.00 5 05 -0.01 1 4 40.6067 11.0 1 3 4.278 450 28. 5 6 0.02774 0 00505 -0.0113 40.6007 1 1.0835 4.292 407.5 28.5 0.02783 0.00512 -0.0114 40.60 1 7 1 0.0365 4.252 350 28.49 0.02757 0.00514 0.0 11 2 4 0.5904 8.62273 5451.6537 4.288 400 28.47 0 0278 0.00516 -0.0 1 13 40.5992 9.8524 1 4.4 455 28.5 1 0.02853 0.00511 -0.0 11 7 40.6314 11. 1 982 4.6 462 28.57 0.02982 0.0050 4 -0.0 1 2 4 40.6886 11 .3545 4.8 468.5 28.59 0.03 11 2 0.00502 0.0 1 3 1 40.74 4 11 .4986 5 475.5 28.66 0.03242 0 .00494 -0.0 1 37 40.80 1 9 11.6 5 39 5.2 48 1 .5 28.7 0.03371 0.00489 -0.0 1 44 4 0.8586 1 1.7845 5.4 487.4 28.73 0.0350 1 0.00486 -0.0151 40.9 1 49 1 1.912 5 5.6 492.8 28.81 0.0363 1 0.00476 -0.0 1 58 4 0.9738 1 2.0272 5.8 498.3 28.9 0.0376 0.00 4 66 0.0 1 65 4 1 .0333 12. 1 438 6 503.4 28.99 0.0389 0.00455 0 .01 72 4 1. 093 1 2.2503 6.2 508.2 29.02 0.0402 0.00452 -0.0 1 78 4 1 1 5 12.35 6.4 5 1 2.8 29.1 l 0.04 1 49 0.00442 -0.0 1 85 4 1. 2099 1 2.4436 6.6 5 1 7.3 29.2 0.04279 0.0043 1 -0.0192 41.270 1 1 2.5345 6.8 52 1 .4 29.3 0.04409 0.004 1 9 -0.0199 4 1.3309 1 2.6 1 53 7 525.6 29.4 4 0.04538 0.00403 -0.0207 4 1 .3938 1 2.697 5 7.5 535.4 29.75 0.04863 0.00367 -0.022 5 4 1 .5499 12.8857 8 544.l 30.03 0.05187 0.00335 0.0243 4 1 .7 0 5 5 13 0462 8.5 552.1 30.3 0.05511 0.00303 -0.026 4 1 .8618 13. 1 886 9 559. l 30.7 0.05835 0.00257 -0.0279 42.0255 13.3038 9.5 565.4 31.08 0.06159 0.002 1 3 -0.0297 42. 1 893 1 3.40 1 5 1 0 57 1 .7 3 1 .46 0.06483 0.00168 -0.0316 42.3543 1 3.498 1 0.5 577.4 3 1 .8 1 0.06808 0.00128 -0.0334 42.519 1 3.5798 1 1 582.5 32.2 0.07132 0.00082 -0.0352 42.6867 1 3.6459 11 .5 586.9 32.6 1 0.07 4 56 0.00035 -0 037 1 42.8567 1 3.6945 1 2 59 1 .1 33.03 0.0778 -0 0001 -0.039 43.0283 1 3.7375 1 2.55 594.9 33.32 0.08 1 37 -0.0005 -0.0409 4 3.2099 1 3.7677 1 3 5 98 33.89 0.08428 -0.0011 -0.0 4 27 4 3.3763 1 3.7863

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208 13 5 600.64 34.31 0.08753 -0.00 1 6 -0.0446 43.5516 1 3.79 15 14 603.2 34.71 0.09077 -0.0021 -0.0464 43.7271 1 3.7946 14.5 605.6 35.18 0.09401 -0.0026 -0.0483 43.9075 13.7926 TEST6 No: 13 TRIAXIAL COMPRESSION TEST ON ALUMINA POWDER Dat e : Julv 18-19 1995 Two l ayer method Samole: Alumin a oowder AlO Soecimen W eig ht: 865.29 (g) Diameter : (2 805+2.8 l 8+2.825)/3 0.0258 in = 2.7902 ( in ) = 7 .0 87 11 (cm) H eigh t : i'9.09+9.08+9. l 0+9.09 ,/4-3.02 ( in)= 6.07 (i n ) = 15 .4 1 78 (c m) V o lum e: (7.087 10 8)"2* 15 .4178*3. 1415 926 / 4 608.206 (cm"3) densitv: 865.29 I 608.20 5 6 = 1 .42269 (l!!cm 3 ) B Check= 0 .9 4 co nfining oressure = 4.5 ( c hamb er-o) 1 .5 ( back-o) = 3.0 (kl!!cm"2) Hydr os t atic l oa din g ttme c hamb er-o burette time-c confi n -o strain-v K -va ul e 10:50 1.8 10 0 0 3 0 10:51 2.5 9.2 0.01667 1 0.00094 10 : 52 2.5 9. 1 9 0 03333 l 0 00095 1 0:55 2.5 9.19 0.08333 1 0.00095 11 :00 2.5 9.18 0.16667 1 0.00096 11 :0 1 2.3 9.3 0 1 8333 0 .8 0 00082 1424 .86 11 :02 2.3 9.3 0 .2 0 8 0.00082 11 :05 2.3 9.3 0.25 0.8 0.00082 11:06 2.5 9.19 0.26667 1 0.00095 11 : 07 2.5 9 1 9 0.28333 1 0.00095 11 : 10 2.5 9. 19 0.33333 I 0.00095 11 : 11 3 8.7 0.35 1 5 0 .0 0152 11: 12 3 8.69 0 .3 6667 1 5 0.00153 11 : 1 5 3 8.68 0 .4 1667 1 5 0.00154

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209 11:20 3 8.65 0.5 1 .5 0.00158 11 :2 1 2.8 8.76 0.51667 1 .3 0 .0 0145 1554.39 11:22 2.8 8.76 0.53333 1.3 0 00145 11:25 2.8 8.76 0.58333 1 .3 0.00145 11 :26 3 8.64 0 6 1.5 0.00159 11: 27 3 8.64 0.61667 1.5 0.00159 11:30 3 8.62 0.66667 1.5 0.00161 11:31 3.5 8.2 0.68333 2 0.00211 11:32 3.5 8 15 0.7 2 0 00216 11 : 35 3.5 8.13 0.75 2 0.00219 11:40 3.5 8.1 0.83333 2 0 00222 11 :41 3.3 8.2 0 85 1.8 0 00211 1709.83 11 :42 3.3 8.2 0.86667 1.8 0 .0 0211 11 :45 3.3 8.2 0.91667 1.8 0 00211 11:46 3.5 8.1 0.93333 2 0.00222 11:47 3.5 8.1 0.95 2 0.00222 11:50 3.5 8.09 1 2 0.00223 11:51 4 7 .7 1 .0 1 667 2.5 0.00269 11 : 52 4 7 65 1.03333 2.5 0.00275 11:55 4 7.59 1 08333 2.5 0.00282 12:00 4 7.58 1.16667 2 .5 0.00283 12:01 3 8 7 .65 1.18 333 2 3 0.00275 2442 .61 12:02 3.8 7.65 1.2 2.3 0.00275 1 2:05 3.8 7.65 1.25 2 3 0.00275 1 2:06 4 7.58 1.26667 2.5 0.00283 12:07 4 7.58 1 .28333 2.5 0.00283 12:10 4 7.57 1 .33333 2 .5 0.00284 1 2:11 4 .5 7.2 1 35 3 0 00328 12:12 4 5 7 1 1 .3666 7 3 0.00339 12:15 4 .5 7.06 1 41667 3 0.00344 12:20 4.5 7.03 1.5 3 0.00347 12:21 4.3 7. 1 1 .5 1667 2 8 0.00339 2442.61 12:22 4.3 7.1 1 .53333 2.8 0.00339 12:25 4.3 7.1 1.58333 2.8 0.00339 1 2:26 4.5 7.03 1.6 3 0.00347 1 2:27 4.5 7.02 1.61667 3 0.00349 12:30 4.5 7.02 1.6 6667 3 0.00349 13:40 4 .5 6 .99 2.83333 3 0.00352 Deviatoric loading co nfinin l!1 > =4.5 (chamber-p)-1.5 (back-p)=3 kg/cm"2) rate=O.l mm/min disol. force burette strain-I strru n -v strain-3 area-c stress E-value 0 49 6.45 0 0 0 39.4483 1 .242 13

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210 0.1 73.9 6.2 0.00065 0.00029 -0.0 002 39.4623 1.87267 0.2 91 .6 6 0 0013 0.00053 -0.0004 39.4787 2.32024 0.3 107 5 8 0.00195 0.00076 -0.0006 39.4951 2.70919 0.4 121 5 65 0.00259 0.00094 -0.0008 39.5139 3.06221 0.5 134 5.5 0 00324 0.00111 -0.0011 39.5327 3.3896 0 .6 146 5.38 0 00389 0.00125 -0.0013 39.5529 3.69126 0.628 150 5.35 0.00407 0.00129 -0.0014 39.5587 3.79184 0.636 131 4 5.32 0.00413 0.00132 -0.0014 39.5593 3.32159 0.618 100 5.35 0.00401 0.00129 -0.0014 39 5561 2.52805 6796 993 0.632 1 30 5.31 0.0041 0.00133 -0.0014 39.5579 3.28633 0 7 157 5 .2 4 0.00454 0.00141 -0.0016 39.5721 3.96744 0.8 169 5.2 0.00519 0.00146 -0.0019 39.5961 4.2681 0.9 179 5.12 0.00584 0.00155 -0.0021 39.6182 4.51812 1 188 5.08 0.00649 0.0016 -0.0024 39.6422 4.74242 1.1 196.8 5 05 0.00713 0.00164 -0.0027 39.6667 4.96134 1.138 200 5.05 0.00738 0.00164 0.0029 39.6766 5.04076 1.146 180 5.04 0 00743 0.00165 0 0029 39.6782 4 5365 1.1 3 150 5 06 0.00733 0.00162 0.0029 39.675 3.78072 7282.7558 1. 1 44 180 5 01 0.00742 0.00168 -0.0029 39.6763 4.53672 1 .2 205 5 0 00778 0 00 1 7 0.003 39.6903 5.16499 1.3 213.4 5.02 0 00843 0.00167 0.0034 39.7172 5 .3 7298 1.4 220.5 5.04 0 00908 0.00165 -0.0037 39.7441 5.54799 1.5 227.3 5.08 0 00973 0 0016 -0.0041 39.772 5.71507 1.6 233.9 5 1 0 01038 0 00158 0 0044 39.799 5 87703 1.7 239.9 5 18 0.01103 0.00148 -0.0048 39.8289 6.02327 1.8 245 5 5.21 0 01167 0.00145 -0 005 1 39.8564 6.15961 1 .9 250 2 5.27 0.01232 0.00138 0.0055 39.8854 6.27297 2 255.7 5.33 0.01297 0.00131 -0.0058 39.9144 6.40621 2.1 260.1 5.4 0.01362 0.00123 -0.0062 39.9439 6.51 I 63 2.2 264.9 5.51 0.01427 0.0011 0 0066 39.9753 6.62658 2.3 269.2 5.59 0.01492 0 00101 -0.007 40.0054 6.72909 2.4 273 5.69 0.01557 0.00089 -0.0073 40 0365 6.81879 2.5 277 5.8 0 01622 0 00076 -0 0077 40.068 6.91325 2.6 280.7 5.92 0.01686 0.00062 0 0081 40. 1 001 6.99999 2.7 284.3 6.03 0.0175 1 0 00049 -0.0085 40.1317 7.08418 2.8 287.7 6.13 0.01816 0.00037 -0 0089 40.1629 7 16333 2.9 290.6 6 26 0.01881 0.00022 -0 0093 40.1956 7.22965 3 293.5 6.39 0 01946 0 00007 0 0097 40.2283 7.29586 3.1 296.4 6.52 0 02011 -8e-05 -0.0101 40 261 7.36196 3.2 298.9 6.64 0.02076 -0. 0002 -0.0105 40.2933 7 .418 I 3.242 300 6.7 0.02103 -0.0003 -0.0107 40.3074 7.44281 3.252 273 6.66 0 .021 09 0.0002 -0.0107 40.3082 6.77282 3.23 240 6.65 0.02095 0.0002 -0 0106 40.3018 5.95507 5730.9036

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211 3.254 280 6.62 0.02111 -0.0002 -0.0107 40.3068 6.94672 3.3 302 6.69 0.0214 -0.0003 -0.0108 40.3224 7.48963 3.4 306 6.83 0.02205 -0.0004 -0.0112 40.3557 7.58256 3.5 308.2 6.98 0.0227 -0.0006 -0.0117 40.3896 7 63068 3.6 310.2 7.13 0.02335 -0.0008 -0.0121 40.4235 7.67375 3.7 312.7 7.29 0.024 -0.001 -0.0125 40.4579 7.72902 3.8 314.8 7.45 0.02465 -0.0012 -0.0129 40.4924 7.7743 3.9 316.9 7.57 0.0253 -0.0013 -0.0133 40.525 7.81986 4 318.9 7.73 0.02594 -0.0015 -0.0137 40.5596 7.86251 4.2 322 8 0.02724 -0 .0018 -0.0145 40.6265 7 92587 4.4 326.2 8.3 0.02854 -0.0022 -0.0154 40 .695 8.01573 4.6 329.3 8.63 0.02984 -0.0025 -0.0162 40.7651 8.078 4.8 332.5 8.95 0.03113 -0.0029 -0.017 40.8349 8.14255 5 335.4 9.26 0.03243 -0.0033 -0.0179 40.9044 8.19961 5.2 338.4 9.58 0.03373 -0.0037 -0.0 187 40.9746 8.25878 5.4 340.9 9.89 0.03502 -0.004 -0.0195 41.0445 8.30563 5.6 343.4 10.18 0.03632 -0.0044 -0.0203 41.1136 8.35247 5.8 345.8 10.51 0.03762 -0.00 47 -0.0212 41.1848 8.3963 6 347.9 10.85 0.03892 -0 0051 -0.022 41.2567 8.43256 6.2 350 11.2 0.04021 -0.0056 -0.0229 41.3293 8.46857 6.4 352 11.53 0.04151 -0 0059 -0.0237 41.4011 8.50219 6.6 353.9 11.85 0.04281 -0.0063 -0.0246 41.4726 8.53334 6.8 355.1 12 .2 0.0441 -0 .0067 -0.0254 41.5458 8.54719 7 357 12.52 0.0454 -0.0071 -0.0262 41.6177 8.57808 7.2 358.6 12 .89 0.0467 -0.0075 -0 .0271 41.6923 8.60112 7.4 360.1 13.2 0.048 -0.0079 -0.0279 41.7641 8.62224 7.6 361.6 13 .58 0.04929 -0.0083 -0.0288 41.8395 8.64255 7.8 362.4 13.9 0.05059 -0.0087 -0.0297 41.9122 8.64664 8 364 14.27 0.05189 -0.0091 -0.0305 41.9876 8.66924 8.2 365.9 14.65 0.05319 -0.0096 -0.0314 42 0636 8.69874 8.4 367.3 14.99 0.054 48 -0.0 1 -0.0322 42.1379 8.71662 8.6 369.1 15.3 0.05578 -0 .0103 -0.0331 42.2109 8.74419 8.8 369.9 15.67 0.05708 -0.0108 -0.0339 42.2871 8.74735 9 370.9 16 0.05837 -0.0 112 -0.0348 42.3615 8.75559 9.2 371.7 16 .34 0.05967 -0.0116 -0.0356 42.4366 8.75895 9.4 372.5 16 .65 0.06097 -0.0119 -0.0364 42.5105 8.76255 9.6 373.2 17 0.06227 -0 .0123 -0.0373 42.5865 8.76335 9.8 373.9 17. 36 0.06356 -0.0128 -0.0382 42.6632 8.764 10 374.6 17 .69 0.06486 -0.013 1 -0.039 42.7386 8.7649 10.5 376.2 18.51 0.0681 -0.0141 -0.04 11 42.928 8.76352 11 377.5 19 .36 0.07135 -0.0151 -0.0432 43.1201 8.75462

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212 TEST7 No : 14 TRIAXIAL COMPRESSION TEST ON ALUMINA POWDER Date : August 2-3, 1995 Two l ayer method Sample : Alumina powder AlO Specimen Weight : 866.4 (g) Diam e t er: (2.819+2.823 +2 .823)/3 -0. 0245(in) = 2.79717 (i n ) = 7.1048 cm) Height : (9. 10+ 9.08+9 .1 3 +9.09 )/4-3.02 ( in )= 6.08 (in I = 15.4432 i'cm) Volume: (7.104803 "2* 15.44 32*3.1415926 / 4 612 253 cm"3) density: 866.4 I 612.2535 = 1.4151 (g/c m 3 ) B Check= 0 94 confini ng or essure = 5.5(chamber-p)-l.5 (back-p) = 4.0 1 ki! /cm"2> Hvdro static Loading time c ham -p burette time-c co nfin -p strai nv K-vaule 11: 10 1 .8 10.52 0 0.3 0 11 : 11 2.5 9.85 0.01667 1 0 00078 11: 12 2.5 9.83 0.03333 1 0.0008 11 : 15 2.5 9.82 0.08333 1 0.00081 11:20 2.5 9.81 0. 1 6667 1 0.00082 11 :2 1 2.3 9.96 0.18333 0.8 0.00065 1148.15 11:22 2.3 9.96 0.2 0.8 0 00065 11 :25 2.3 9 96 0.25 0 .8 0.00065 11 :26 2.5 9.81 0.26667 1 0.00082 11:27 2.5 9.8 0.28333 1 0.00084 11:30 2.5 9.8 0 .33333 1 0.00084 11:31 3.5 8.96 0.35 2 0.00181 11 :32 3.5 8.89 0.36667 2 0.00189 11 :35 3.5 8.86 0.41667 2 0.00193 11:40 3.5 8.85 0.5 2 0.00194 11 :41 3 3 8.95 0.51667 1.8 0.00182 17 22.23 11:42 3.3 8.95 0.53333 1 .8 0.00182 11:45 3.3 8.95 0.58333 1.8 0.00182 11:46 3.5 8.84 0.6 2 0.00195 11 :47 3.5 8.83 0.61667 2 0.00196 11 :50 3.5 8.83 0.66667 2 0.00196

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213 11 :51 4.5 8 0 68333 3 0.00293 11 : 52 4.5 7.9 0 7 3 0.00304 11:55 4 .5 7.85 0 75 3 0.0031 12 : 00 4.5 7.83 0.83333 3 0.00312 12:01 4.3 7.91 0 85 2 8 0.00303 2152.79 12 : 02 4.3 7 .9 1 0.86667 2 .8 0.00303 12 : 05 4.3 7.91 0.91667 2.8 0.00303 12 : 06 4 5 7 .81 0.93333 3 0 00315 12:07 4 5 7.81 0.95 3 0.00315 12 : 10 4 5 7.8 1 3 0.00316 12: 1 1 5 5 7 1.01667 4 0.00409 12:12 5.5 7 1.03333 4 0.00409 12:15 5.5 6.8 1.08333 4 0.00432 12:20 5 5 6.77 1 16667 4 0.00435 12:21 5.3 6.82 1.18333 3.8 0.0043 3444.46 12:22 5.3 6.82 1.2 3.8 0.0043 12:25 5.3 6.82 1.25 3.8 0.0043 12:26 5 .5 6.76 1.26667 4 0.00437 12 : 27 5.5 6.75 1.28333 4 0 00438 12:30 5 5 6.74 1.33333 4 0.00439 14:00 5 5 6.6 2.83333 4 0.00455 Deviatoric loading co n fining-p =5.5 (chamber-p)-1.5 (back-o =4 kg/cmA2) diso l. force burette strain-I strain-v strain-3 area-c stress E-vaJue 0 35 6.28 0 0 0 39.6455 0.88282 0.1 72 7 6 0.00065 0.00033 -0.0002 39 6583 1.83316 0 2 99 5 5.75 0.0013 0.00062 -0.0003 39.6725 2.50804 0.3 121 5.5 0.00194 0.00091 -0.0005 39.6867 3.04888 0 4 140 5.3 0 00259 0 00114 -0.0007 39 7032 3.52616 0.5 158 5.1 0.00324 0.00137 0 0009 39.7198 3.97787 0.6 173.5 4.95 0.00389 0.00154 -0.0012 39 7387 4 36602 0.7 188 4.81 0.00453 0.00171 -0.0014 39.758 4 7286 0.8 201 4.68 0.00518 0.00186 -0.0017 39.7779 5.05306 0.9 214 4.53 0.00583 0 00203 -0.0019 39.7969 5 3773 I 1 226 4.4 0.00648 0.00218 -0.0021 39.8168 5.676 1.1 236 .7 4.33 0.00712 0.00226 -0.0024 39.8395 5 94134 1.2 247.7 4.25 0.00777 0.00236 -0.0027 39.8618 6.21397 1 .3 257 4.18 0 00842 0.00244 0 003 39.8846 6.44359 1 4 266 4 12 0.00907 0 00251 -0.0033 39.9078 6.66536 1.5 275 4 06 0.00971 0.00258 -0.0036 39.9312 6.88685 1.6 283.4 4 02 0.01036 0.00262 -0.0039 39.9554 7.09291 1.7 290.7 4 0.01101 0.00265 -0.0042 39.9806 7.27102

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214 1.8 298 5 3.98 0.01166 0.00267 -0.0045 40.0059 7.4614 1.9 305 3.99 0.0123 0.00266 -0 0048 40.0326 7.61879 2 312 3.98 0.01295 0.00267 -0.0051 40.0584 7.78863 2.1 318 3.98 0.0136 0 00267 -0.0055 40 0847 7.9332 2.2 323 3.99 0.01425 0.00266 -0.0058 40.1115 8.05255 2.3 329.5 4 0.01489 0.00265 -0.0061 40.1383 8.20911 2.4 334.5 4 0.01554 0.00265 -0.0064 40.1647 8 3282 2.5 339 4.02 0.01619 0.00262 -0.0068 40.1921 8.43449 2.6 344.8 4.04 0.01684 0.0026 -0.0071 40.2195 8.57295 2.7 349 7 4.08 0.01748 0.00255 -0 0075 40.2479 8.68865 2.8 353.9 4.1 0.01813 0.00253 -0.0078 40.2754 8.78701 2.9 358 4.13 0.01878 0.0025 -0.0081 40 3034 8.88263 3 362 4.15 0.01943 0.00247 -0.0085 40.3309 8.97575 3.1 366 4.23 0.02007 0.00238 -0.0088 40.3613 9.06809 3.2 369.6 4 29 0.02072 0.00231 -0.0092 40.3908 9.15059 3.3 373.1 4.32 0.02137 0.00228 -0.0095 40.419 9.23081 3.4 376.6 4.36 0.02202 0.00223 -0.0099 40.4476 9.31081 3.6 383 4.5 0.02331 0.00207 -0.0106 40.5078 9.45496 3.7 387 4.55 0.02396 0 00201 -0.011 40.5371 9.54682 3.8 390 4.65 0 02461 0.00189 -0.0114 40.5687 9.61332 3.9 393 4.72 0.02525 0.00181 -0.0117 40.599 9.68005 4 396 4.79 0.0259 0.00173 -0.0121 40.6293 9.74667 4.1 398.6 4.86 0.02655 0.00165 -0.0124 40.6596 9.80334 4.2 401 4.94 0.0272 0.00156 -0.0128 40.6904 9.85489 4.3 403 5.01 0.02784 0.00147 -0.0132 40.7209 9.89665 4.4 406.4 5.1 0.02849 0.00137 -0.0136 40.7523 9.97245 4.5 409 5.18 0.02914 0.00128 -0.0139 40.7832 10.0286 4.6 411 5.25 0.02979 0.0012 -0.0143 40.8138 10.0701 4.7 413.8 5.33 0 03043 0.0011 -0.0147 40.8448 10.131 4.8 415 5.41 0.03108 0.00101 -0.015 40 8759 10.1527 4 9 418.08 5.5 0.03173 0.00091 -0.0154 40.9076 10.2201 5 420.5 5.61 0.03238 0.00078 -0.0158 40.9402 10.2711 5.2 424.6 5.78 0.03367 0.00058 -0.0165 41.0031 10.3553 5.4 428.6 5 99 0.03497 0.00034 -0.0173 41 0682 10.4363 5.6 432.4 6.19 0.03626 0 0001 -0 0181 4 1 .1329 10.5123 5.8 436.4 6.38 0.03756 -0.0001 -0.0188 41.1974 10.5929 6 439.5 6.6 0.03885 -0.0004 -0.0196 41.2634 10.6511 6.2 442.8 6.8 0.04015 -0.0006 -0.0204 41 3287 10.7141 6.4 445.9 7.01 0.04144 -0.0008 -0.0211 41.3946 10.7719 6.6 448.7 7.28 0.04274 -0.0012 -0.0219 41.4636 10.8215 6.8 451.6 7.51 0.04403 -0.0014 -0.0227 41.5308 10.8738 7 455 7.71 0.04533 -0.0017 -0.0235 41 5968 10.9383 7.2 458 2 7.91 0.04662 -0.0019 -0.0243 41.663 10.9978

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215 7.4 460.6 8 .1 8 0.04792 -0.0022 -0.0251 41.7327 11.0369 7 .6 462 .7 8.4 0.04921 -0.0025 -0.0258 41.8002 11.0693 7 .8 464.6 8.68 0.05051 -0.0028 -0.0266 41.8708 11.096 8 467 .3 8.88 0.0518 -0.003 -0.0274 41.9377 11.1427 8.5 472 9.52 0.05504 -0.0038 -0.0294 42.1126 11.2081 9 476.9 10 1 0.05828 -0.0044 -0.0314 42.2857 11 .278 9.5 480.8 10.75 0.06152 -0.0052 -0.033 4 42.4635 11.3227 10 484.9 11.35 0.06475 -0.0059 -0.0353 42.64 11.3719 1 0.5 487.9 11.98 0.06799 -0.0066 -0.0373 42.8193 11.3944 11 490.6 1 2.6 0.07123 -0.0073 -0.0393 42.9993 11.4095 11 .5 493.2 13 25 0.07447 -0. 0081 -0.04 1 3 43.182 11.4214 12 495.1 13 .86 0.0777 -0.0088 -0.0433 43.3641 11.4173 12.5 496.7 14 48 0.08094 -0.0095 -0.0452 43.5479 11 .4058 13 498 15 .15 0.08418 -0.0103 -0.0472 43.7355 11.3866 13.5 499 13 15 .76 0.08742 -0.0 11 -0.0 492 43.9215 11.3642 14 498 .5 16 .38 0.09065 -0.0 11 7 0 0512 44.1092 11.3015 TESTS No : 22 TRIAXIAL COMPRESSION TEST ON ALUMINA POWDER D ate : Dec. 19 -20, 1995 Sample: two l ayer method So ecime n Weii!ht : 868.2 (g I Diameter : (2.81+2.822+2.825 1 /3 -0.02(in) = 2 799 ( in ) = 7. 10946 cm Height: (23.03+22.96+22.96+23.09)/4-3.0*2.54 15 .39 (cm) Volume: (7. 1094 6)"2* 15 .39*3. 14159 26 / 4 610.944 i'c m" 3) density : 868.2 / 610.9445 = 1.42108 (g/c m 3 ) B Check= 0.96 confi ninl! oressure = 5.0 (c hamb e r -o)-1.5 ( ba ck-p ) = 3.5 (k g/cm"2) tune c hamb er-o burette t1m e-c conf rn -o strai n -v K -va ule 11:05 1.8 33.18 0 0.3 0 11 :06 2.5 32.3 0.01667 1 0 00102 11 :07 2.5 32.3 0.03333 1 0.00102 11: 10 2.5 32.3 0.08333 1 0.00102 11:15 2.5 32.3 0.16667 1 0 00102 11: 16 2.3 32.42 0.18333 0 8 0.00088 1432.12 11: 17 2.3 32.42 0.2 0.8 0.00088

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2 1 6 11 :20 2.3 32.42 0 25 0.8 0.00088 1 1 :21 2.5 32.3 0.26667 1 0.00102 11: 22 2.5 32.3 0.28333 1 0.00 1 02 11: 25 2.5 32.3 0.33333 1 0.001 0 2 11 :26 3.5 3 1 .4 0.3 5 2 0.00207 1 1:27 3.5 31.3 0.36667 2 0. 0 02 1 9 1 1:30 3.5 3 1 .29 0.41667 2 0.0022 11 :3 5 3.5 3 1 .29 0.5 2 0.0022 11:36 3.3 31.39 0.5 1 667 1 .8 0.00208 17 1 8 55 11 :37 3.3 31.39 0.53333 1.8 0.00208 11 :40 3.3 31.38 0.58333 1.8 0.00209 11 :4 1 3.5 3 1 .28 0 6 2 0.0022 1 11 :42 3.5 31.28 0.61667 2 0 0022 1 11 :4 5 3.5 31.28 0.66667 2 0 0022 1 11 :46 4.5 30.4 0.68333 3 0 00324 1 1 :47 4.5 30.3 0.7 3 0.00335 11:50 4 5 30 27 0.75 3 0.00339 1 1 :55 4.5 30.25 0.83333 3 0.00341 11 :56 4.3 30.33 0.8 5 2.8 0.00332 2 1 48.19 11:57 4.3 30.33 0.86667 2.8 0.00332 12:00 4.3 30 32 0.9 1 667 2.8 0.00333 12:01 4.5 30.2 5 0.93333 3 0 0034 1 1 2:02 4.5 30.25 0.95 3 0.0034 1 1 2 : 05 4.5 30.25 1 3 0.0034 1 1 2:06 5 30 1.01667 3. 5 0.0037 1 2: 0 7 5 29.9 1 .03333 3.5 0.00382 12: 1 0 5 29 8 5 1 .08333 3.5 0.00388 12: 1 5 5 29.83 1 .16667 3.5 0.0039 1 2: 1 6 4.8 29.9 1 1.18333 3.3 0.00381 2 1 48. 1 9 12: 1 7 4.8 29.91 1.2 3.3 0.0 0 38 1 12:20 4.8 29 91 1.25 3.3 0.00381 1 2 : 2 1 5 29.83 1 .26667 3.5 0.0039 1 2 : 22 5 29.83 1 28333 3.5 0.0039 12:25 5 29.83 1.33333 3. 5 0.0039 1 4: 05 5 29.78 3 3.5 0 00396 rate=O.l mm/min diso l fo r ce bu.rette s tr ain-I s tr a rn -v strain-3 area-c stress E-val u e 0 20 29.43 0 0 0 39.6975 0.5038 1 0. 1 40.9 29 3 1 0.00065 0 000 1 4 -0.0003 39.7178 1.02977 0.2 68 29.1 0.0013 0.00038 -0. 00 05 39 7339 1.71139 0.3 90 28.89 0.00195 0.00063 -0.0007 39.75 2 26415 0.35 1 00 28.8 0.00227 0 00073 -0.0008 39.7588 2.5 1 5 1 7 0.356 87.3 28.78 0.00231 0.00076 -0.0008 39.7594 2.1957

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2 1 7 0.342 70 28.8 0.00222 0.00073 -0.0007 39.7567 1.76071 4781.8629 0.356 90 28.78 0.00231 0.00076 -0 0008 39.7594 2.26361 0.4 1 08.7 28 6 1 0.0026 0 00095 -0.0008 39.763 2.7337 0.5 1 26.8 28.5 0 00325 0.00 1 08 -0.0011 39.7838 3.18723 0.6 142 28.32 0.0039 0.00129 -0.0013 39.80 1 4 3 5677 1 0 7 156.1 28.22 0 0045 5 0.00141 -0.00 1 6 39.8227 3 9 1 987 0.8 1 69.4 28 12 0.0052 0.00 1 52 0 00 1 8 39.8 44 1 4.25157 0.9 181.8 28.02 0.00585 0.00 1 64 -0.002 1 39 8655 4.56034 1 1 93.4 27 92 0.0065 0.00 1 76 -0.0024 39.8869 4.8 4 871 1.0 5 4 200 27.89 0.0068 5 0.00179 0.002 5 39.8996 5.01258 1 .062 180 27.87 0.0069 0.00182 -0.0025 39.9008 4.51119 1.034 150 27.92 0.00672 0.00 1 76 -0.0025 39.8958 3.7598 4129.9929 1 .066 190 27.82 0.00693 0.00 1 87 0 .0025 39.8995 4.76197 1.1 203.7 27.8 0.00715 0.00 1 9 -0. 0 026 39 9074 5.1043 1 1.2 216.6 27.78 0.0078 0.00192 0.0029 39.9326 5 42413 1.3 226.6 27.77 0.00845 0.00 1 93 -0.0033 39.9583 5.67091 1 .4 235.6 27 76 0 0091 0.00 1 9 4 -0.0036 39.98 4 1 5.89235 1 5 244.5 27 74 0.00975 0.00197 -0.0039 40 0 0 94 6.11107 1 .6 252 9 27.72 0.0104 0.00 1 99 -0.0042 40.0347 6.31702 1.7 260.8 27.72 0.0 1 10 5 0.00199 -0.00 45 40.061 6.51007 1 .8 268 4 27.76 0.0117 0.00 1 9 4 -0.00 4 9 40.0892 6.69507 1.9 275.6 27.8 0.0 1 235 0. 0 019 -0.0052 40. 11 75 6.86982 2 282.3 27.82 0 013 0.00187 -0.0056 40.1448 7.03204 2 1 288.6 27.87 0.0136 5 0.00182 -0.005 9 40. 1 736 7.18382 2.2 294 4 27 92 0.0 1 429 0.00 1 76 -0.0063 40.2024 7.32294 2.3 300 28 0.0 1 49 4 0.00 1 66 -0.0066 4 0.2327 7. 4 5662 2 4 305.2 28.05 0.01559 0.00161 -0.007 4 0.2616 7 .5 8042 2.5 3 1 0.5 28. 1 0.0 1 624 0. 0 0 1 55 -0 0073 40.2905 7.70652 2.6 3 1 5.1 28. 1 8 0.0 1 689 0.00145 -0.0077 40.3209 7.8148 2.7 3 1 9.7 28.27 0 0 1 754 0.00 1 35 -0.0081 40.35 1 8 7 9228 1 2.8 323.8 28.35 0.01819 0.00 1 26 -0.0085 40.3823 8.0 1 836 2.9 328 28.46 0.0 1 884 0.00113 -0 0089 40 4 14 2 8.11595 3 331.8 28.55 0.01949 0.00 1 02 -0.0092 40.44 5 2 8.20368 3 1 335 8 28.65 0.02014 0.0009 1 -0.0096 40.4768 8.29611 3.2 339 28.76 0.02079 0.00078 -0.01 40.5088 8.36854 3.3 342.7 28.88 0.02144 0.0006 4 -0.0104 40.5414 8.45309 3.5 348.9 29.08 0.02274 0.0004 1 -0.0112 40.60 4 8 8.59259 3.6 353.3 29.2 0.02339 0.00027 -0.01 1 6 4 0.6375 8.69395 3.7 357 29 3 0.02404 0.00015 -0. 0 1 1 9 40.6692 8.77813 3.8 360 29. 4 0.02469 0.00003 -0.0 1 23 4 0.70 1 1 8.8 4 497 3.9 363.2 29.5 0.02534 -8e-05 -0 0 1 27 40.733 8.9 1 661 4 365.3 29.61 0.02599 -0.0002 -0.0 1 31 40 7653 8.9610 4 4. 1 367.7 29.7 1 0.02664 -0.0003 -0.0 1 35 40.7973 9.0128 5

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2 1 8 4 2 370.1 29.89 0.02729 -0.000 5 -0.0139 40 8331 9.06372 4.3 372.3 30 0.02794 -0.0007 -0.0143 40.8656 9.11035 4.4 375 30. 1 0.02859 -0.0008 -0.0 1 47 4 0.8977 9.16922 4.5 377 30.2 0.02924 -0.0009 0.0 1 51 40.9299 9.21088 4.6 379.1 30.3 0 02989 -0.001 -0.0 1 5 5 40.962 9.25 4 9 1 4.8 383.6 30.55 0.03119 -0.00 1 3 -0.0 1 62 4 1 .0289 9.3495 1 4.9 385.3 30.8 0.03 1 84 -0.00 1 6 -0.0167 4 1. 0684 9.38192 5 387.5 30 9 1 0.032 4 9 -0.00 1 7 -0.0 1 7 1 41. 1 0 1 2 9.42795 5. 1 389.1 3 1. 04 0.033 14 -0.00 1 9 -0 0 1 75 4 1 1 35 9.45909 5.2 391.1 31.2 0.03379 -0.002 1 -0.0 1 79 4 1 1 703 9.49956 5.3 392.8 31.35 0.0344 4 -0.0022 -0.0 1 83 41.2052 9.53277 5.4 394.6 31.5 0.03509 -0.0024 -0.0 1 87 41 2402 9.5683 4 5.5 396.3 31.64 0.03574 -0.0026 -0.0 1 92 41 .2747 9 601 5 4 5.6 398 3 1 .78 0.03639 -0.0027 -0.0196 41 .3092 9 63466 5.7 399.4 31.9 0.03704 -0. 0 029 -0.02 4 1 .3 4 28 9 66068 5.8 401.2 32.05 0.03769 -0.003 -0.0204 4 1 .3779 9.69599 5.9 402.5 32.2 0.0383 4 -0.0032 -0.0208 4 1 .413 1 9.719 1 5 6 403.9 32 36 0.03899 -0.0034 -0.02 1 2 4 1 .4488 9.744 5 5 6.2 406.8 32.6 0.04029 0.0037 -0.022 4 1 5 165 9.798 5 2 6.4 409 3 32.95 0.04159 -0.004 1 -0.0228 41. 5 896 9.84139 6.6 411.8 33.25 0.04288 -0.0044 -0.0237 41.6606 9.88464 6.8 414.3 33.55 0.04418 -0.0048 -0.02 4 5 4 1 .7317 9.9277 7 4 1 6.6 33.9 0.04548 -0.0052 -0.0253 41.8055 9.9652 7.2 418.6 34.2 0.04678 -0.0056 -0.0262 4 1 .877 9.99593 7.4 420.5 34.5 0.04808 -0.0059 -0.027 41 9488 10.02 4 1 7.6 422.4 34.8 0.04938 -0.0062 -0.0278 4 2.0207 1 0.0522 7.8 424 35.2 0.05068 -0.0067 -0.0287 42.0977 10 0718 8 425.6 35.3 0 05198 -0.0068 -0.0294 42.1603 10.0948 8.5 429.4 36.3 0.05523 -0.008 -0.03 1 6 42.3541 10.1383 9 432.2 37.1 0.05848 -0.0089 -0.0337 42.5395 10 16 9.5 434.7 37.9 0.06173 -0 0099 -0.0358 42.7262 10.1741 1 0 436.5 38.7 0.06498 -0.0108 -0.0379 42.9 1 42 10.17 1 5 10.5 438 39.5 0.06823 -0.0117 -0.04 43. 1 035 10.16 1 6 11 440 40.2 0.07147 -0.0 1 25 0.0 4 2 43.2892 10.1642 TEST9 No: 23 RELAXATION TEST ON ALUMINA P O WD E R D ate: Dec. 27-28, 1995 two l ave r method Sa m p l e: Al umin a powder AlO S p ec im en W e i g ht : 868.2 (g)

PAGE 227

2 1 9 Diam e t er: (2 795+2.8+2 823)/3 -0. 0 2( i n) = 2.786 I in ) = 7.07644 (cm) H e i g h t : (23.02+23.19+23. 1 3+23.08)/4-3.0*2 5 4 15. 4 85 (c m, V o l ume: (7.07644>"2* 1 5. 4 85*3.1415926 I 4 = 609.0 1 9 cm"3) d e n s i ty : 868.2 / 609.0 1 9 = 1. 4 2 55 7 ( g/c m 3 ) B C h ec k= 0 .96 co n fi n ing pr ess ur e= 5 5 c hamb e r-:> ) -l .5(bac k p ) = 4 .0 (k~c m "2) h y dr os t a t ic l oad in g test ti me c h am-p bure tt e s tr a tn -v stra m -v 0 2 15 0 0.00242 0.25 2 1 3.0 1 0.0 0 232 0.00455 0.5 2 1 3 0.00233 0. 0 0467 0 75 2 1 2.9 8 0.00236 0.00469 1 2 1 2.98 0.00236 0.00474 1.5 2 1 2.98 0.00236 0.00 4 79 3 2 12 97 0.00237 0.0048 1 6 2 1 2.96 0 .00238 0.00488 10 2 1 2.95 0.00239 0.00489 15 2 1 2.94 0.0024 0.0049 20 2 1 2.93 0.00242 0.00493 20 4 1 2.93 0.00242 20.25 4 11 1 0.00455 20.5 4 11 0.00467 20.75 4 1 0.98 0 00469 21 4 1 0.9 4 0. 00 474 2 1 .5 4 1 0.9 0.00479 23 4 1 0.88 0.0048 1 26 4 1 0 .82 0.00488 30 4 1 0.81 0.00489 35 4 1 0.8 0.0049 40 4 10.78 0.00493 d evia t oric l oad in g for relax ati o n tes t bur e t te rea din g: 0.00323 0.00646 0.0 0 969 0.01292 0 0 1 937 0 03229 0.05 1 66 0.077 4 9 strai n 0.5 I 1 .5 2 3 5 8 1 2 diso l 9.5 9.25 8.65 8.39 8.44 8.14 1 1 .53 15.89

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220 9.3 8.71 8.47 8 .5 8 24 11.7 16.05 22.2 9 3 8.71 8.46 8.5 8 22 11.65 1 6 22.19 9.28 8.71 8.45 8. 5 8.2 11.6 15 98 22.15 9.26 8.68 8.4 8.47 8. 1 7 11.55 15.95 22.1 9.25 8.66 8.4 8.45 8.15 11.54 1 5.9 22.07 9.25 8.65 8.39 8.4 4 8.14 11 .53 1 5 89 22.05 vol um etric strain 0 0.00029 0.00099 0.0013 0.00124 0.00159 -0.0024 -0.0075 0.00023 0.00092 0.0012 0.00117 0.001 4 7 -0.0026 -0.0076 -0.0148 0.0 0 023 0.00092 0.0012 1 0.00117 0.00 1 49 -0.0025 0 .0076 -0.0 1 48 0.00026 0.00092 0.00123 0.00117 0.00152 -0.0025 -0.0076 -0.0148 0.00028 0.00096 0.00 1 28 0 0012 0.00155 -0.0024 -0.0075 -0.0147 0.00029 0.00098 0.00128 0.00123 0.00 1 58 -0.0024 -0.0075 -0.0147 0.00029 0.00099 0 0013 0.00124 0.00 1 59 -0.0024 -0.0075 -0.0147 time b u rette stra.i n -v 0 9.5 0 0.25 9 .3 0.00023 0.5 9.3 0.00023 1 9.28 0.00026 3 9.26 0.00028 5 9 .2 5 0.00029 10 9.25 0.00029 10 9.25 0.00029 1 0.25 8.71 0.00092 1 0.5 8.71 0.00092 1 1 8.71 0.00092 13 8.68 0.00096 1 5 8.66 0 00098 20 8.65 0.00099 20 8.65 0.00099 20.25 8.47 0.00 1 2 20.5 8.46 0.00 1 21 21 8.45 0.00123 23 8.4 0.00 1 28 25 8.4 0.00128 30 8.39 0.00 1 3 30 8 .5 0.00117 30.25 8.5 0.00117 30.5 8.5 0.001 1 7

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221 31 8.5 0.00117 33 8 .47 0.0012 3 5 8.45 0.00123 40 8.44 0.00124 40 8.25 0.00146 40.25 8 .2 4 0.00147 40.5 8 .22 0.00149 41 8 .2 0.00152 43 8 17 0.00155 45 8 15 0.00 1 58 50 8.14 0.00159 50 11.7 -0 0026 50.25 11.7 -0.0026 50.5 11.65 -0.0025 51 11.6 -0.0025 53 11.55 -0.0024 55 11.51 -0.0023 60 11.53 -0.0024 60 16 .0 5 -0.0076 60.25 16 .0 5 -0.0076 60.5 16 0 0076 61 15.98 -0.0076 63 15 .95 -0 0075 65 15 9 -0.0075 70 15 .89 -0.0075 70 22.2 -0.0148 70 .25 22 .2 -0.0148 70.5 22.19 -0.0148 71 22.15 -0.0148 73 22.1 -0.0147 75 22.07 -0.0147 80 22.05 -0.0147 force (kQ:) 0.00323 0.00646 0.00969 0 .0 1292 0.01937 0.03229 0.05166 0.07749 -strain 0.5 1 1 5 2 3 5 8 12 disol 160 218 272 316 367 425 474 494 147.94 202.31 258.2 297.16 348 397 439.1 473 140.22 201 252.2 290.06 338.22 389.02 434.21 454 3 136.18 200 14 247.1 285.46 332.41 382.42 426.75 444.1 132.95 194 96 241.06 278.9 324.76 374 416.7 434.6 131.3 192.72 238.51 275 .2 4 321.41 370 412.76 430.57 128.94 190 .3 1 235.13 271.63 317.31 365.75 407.6 425.57 stress:

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222 4.06823 5. 54296 6.9 15 99 8.03 4 7 5 9.33 1 5 1 0.8 0 62 1 2.052 1 12 .5 6 0 7 3. 76 1 59 5.14402 6 .5 651 7.55572 8.8484 10.0943 1 1.1 6 4 7 1 2.0267 3. 5 6529 5.1 1 071 6.4 1 2 55 7.37519 8.59973 9.89 1 39 1 1 0 40 4 11.55 1 2 3.46257 5.0888 5 6.28287 7.25823 8.4 5 2 9.723 5 8 1 0.8 5 07 11.2919 3.38044 4 .95714 6. 1 293 7.09 1 43 8.25749 9.509 4 8 1 0.59 5 2 1 1.0 5 03 3 33849 4. 90018 6.06 44 6 6 99837 8.17231 9.40778 1 0.495 1 0.9479 3.27848 4 .8389 5.97852 6 90658 8.06806 9.29972 1 0.3638 10 8207 axia l f o r ce time force(kg) stress (kg/c m 2 0 160 4 068 1 8 0.25 1 47.94 3.76 1 54 0.5 1 40.22 3.56 5 25 1 136.18 3.462 5 3 3 132.95 3.38041 5 1 31.3 3.33845 10 128.94 3.27845 10 218 5. 54 29 1 0.25 202.31 5. 1 4396 1 0.5 201 5.1 1 065 11 200.14 5.08879 13 194.96 4 .95708 15 192.72 4.90013 20 1 90.31 4 83885 20 272 6.9 1 59 1 20.25 258.2 6. 5 6 5 03 20.5 2 5 2.2 6.4 1 247 21 247.1 6.2828 23 24 1 .06 6. 1 2923 2 5 238.51 6.06439 30 235.13 5 .97845 30 316 8.03466 30.25 297. 1 6 7.55 5 63 30 5 290.06 7.3751 1 31 285.46 7 2 5 8 1 5 33 278.9 7 .0 9 1 35 35 275.24 6.99829 40 271.63 6.9065 40 367 9.33 1 39 40.25 348 8.8483 40.5 338.22 8.59963 41 332.41 8.45 1 9 43 32 4 .76 8 25739

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223 45 32 1 .41 8. 1 7222 50 3 1 7.31 8.06797 50 425 1 0.8 0 6 1 50.25 397 1 0 0942 50.5 389.02 9.89 1 28 51 382.42 9.72347 53 374 9.50938 55 370 9. 4 0767 60 365.75 9.2996 1 60 474 1 2. 0 52 60.25 439.1 1 1. 1 646 60.5 43 4 .21 1 1 0403 61 426.75 10.8 5 06 63 4 1 6.7 1 0. 5 951 65 412.76 1 0. 4 949 70 407.6 1 0.3637 70 494 1 2.5605 70.25 473 1 2 .0 266 70.5 4 5 4.3 1 1.5 51 1 7 1 444 1 11 .29 1 7 73 434.6 1 1 .0 5 02 75 4 30.57 1 0.9 4 77 80 425.57 1 0.8206 TE S T 10 N o: 25 T RIA XIAL CO MP RESS ION TEST O N ALUMINA PO WD E R D ate : Jan. 8-9, 1 996 Samole: two l aver me th o d S o ec im e n W e i l! ht : 866.2 (g) D iame t er: (2.804+2.816 + 2.827)/3 -0.02(i n ) = 2.79567 ( in ) = 7.10099 (cm) H e i l! ht : (22.93+23. 1 3+23 08+22.91)/4-3. *2.54 (i n )= 15.3925 (cm) Vo lum e: (7.101 A2*15.3925*3.1415926 I 4 609. 5 9 (cmA3) d e n s i tv : 866 2 / 609.59=1.421 (g/cm 3 ) rate=3 mm / min rate=3/ 1 53 .9/60= 0.0003249 B C h eck = 0.96 confm in g press u re= 5.5(c h ambe r o )1 .5 (back-p) = 4 0 (kg/cmA21

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224 time camber-p burette t ime-c co nfin-o stra1n-v K-vaule 12:00 1.8 25.5 0 0.3 0 12 : 01 2.5 24.7 0.01667 1 0 00093 12:02 2.5 24.68 0.03333 1 0.00096 12:05 2.5 24.67 0.08333 1 0.00097 12:10 2.5 24.67 0.16667 1 0.00097 12:11 2.3 24.8 0.18333 0 .8 0.00082 1319.03 12: 12 2.3 24.8 0.2 0.8 0.00082 12:15 2.3 24.8 0.25 0.8 0.00082 12:16 2.5 24.67 0.26667 1 0 00097 12: 1 7 2.5 24.67 0.28333 1 0.00097 12:20 2 5 24.67 0 .33333 1 0.00097 1 2 : 21 3.5 23.79 0.35 2 0.00199 1 2:22 3.5 23.74 0.36667 2 0.00205 12:25 3.5 23.73 0.4 l667 2 0 00206 12:30 3.5 23.72 0 5 2 0.00208 12:31 3.3 23.82 0.51667 1.8 0.00196 1714.74 12:32 3.3 23.82 0 .533 33 1.8 0.00196 12:35 3.3 23.81 0.58333 1.8 0.00197 12:36 3.5 23.7 1 0.6 2 0.00209 12:37 3.5 23.71 0 61667 2 0.00209 1 2:40 3.5 23.71 0 66667 2 0.00209 1 2:4 1 4.5 22 97 0.68333 3 0 00295 12:42 4 .5 22.89 0.7 3 0.00304 12:45 4.5 22.85 0.75 3 0 00309 12:50 4 5 22 .8 0 .83333 3 0.00315 12:51 4.3 22.9 0.85 2.8 0.00303 1714 74 12:52 4.3 22 9 0.86667 2.8 0.00303 12:55 4 3 22.89 0.91667 2.8 0.00304 12:56 4 .5 22.79 0.93333 3 0.00316 1 2:57 4 5 22 79 0.95 3 0.00316 13 : 00 4.5 22.79 1 3 0.00316 13:01 5.5 22.0 5 1.01667 4 0.00402 13:02 5.5 21.95 1.03333 4 0.00414 13:05 5.5 21 .9 1 .08333 4 0 0042 1 3: 10 5.5 2 1 .85 1 .16667 4 0 00426 13:11 5.3 21 .9 2 1 18333 3.8 0.00418 2449.63 13:12 5.3 2 1.9 2 1.2 3.8 0 00418 13:15 5 .3 2 1 .91 1 .2 5 3 8 0.00419 13: 16 5.5 21.84 1.26667 4 0 00427 13 : 17 5.5 2 1 .84 1.28333 4 0.00427 13:20 5.5 21.83 1 .33333 4 0.00428

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225 15:00 5.5 21.75 3 4 0.00437 rate=3 mm/min disol. force burette stra in -1 stra m -v stra in3 area-c stress 0 41.32 20.13 0 0 0 39.6031 1.04335 1 232 19.75 0.0065 0.00044 -0.003 39.8 444 5.82266 2 318 19. 6 0.01299 0.00062 -0.0062 40.0996 7.93025 3 375 20.3 0.01949 -0.0002 -0 .0098 40.3983 9.28258 4 416 21.4 0.02599 -0.0015 -0.0137 40.7199 10.2161 5 444 22.7 0.03248 -0.003 -0.0 1 77 41.0554 10.8147 6 467 24.1 0.03898 -0.0046 -0.02 1 8 41.4002 11.2801 7 481 25.75 0.04548 -0.0066 -0.026 41.7618 11.5177 8 493 27.4 0.05197 -0.0085 -0.0302 42.1284 11.7023 9 500 28.9 0.05847 -0.0102 -0.0343 42.4927 11.7667 10 503 30.5 0.06497 -0.0121 -0.0385 42.867 11 .734 11 501 32. 1 0.07146 -0.0 14 -0.0 4 27 43.2465 11.5848 12 499 33.2 0.07796 -0.0152 -0.0 466 43.6063 11 4433 13 492 34.5 0.08446 -0.0168 -0.0 506 43.9813 11.1866 14 482 35.5 0.09095 -0.0179 -0.0544 44 3465 10.869 15 464 36.1 0 .0 9745 -0.0 1 86 0.058 44.6964 10.3812 16 445 36.2 0.10395 -0.0187 -0.0613 45.0256 9.88326 TEST 11 No: 26 TRIAXIAL CONSTANT MEAN STRESS TEST ON ALUMINA POWDER Date: Jan. 11-12, 1996 two l ayer method Sample: Alumina powder Specimen Weight : 866.2 (g) Diameter: (2.808+2.814+2.825)/3 -0.02(in) = 2.7957 (in) = 7.10 1 (cm) H e i ght: (23.09+22.96+23.07+23)/4 3*2.54 (i n ) 15.41 (cm) Volume: (7 10 0993 "2* 15 .41 *3. 14159 26 / 4 610.28 (cm"3> densitv : 866 .2 / 610.2822= 1 .4193 ( /cm 3 )

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226 B Check= 0.96 co n f min sr o ressure = 5 5 (c h a m be r o 1.5 (back -o ) = 4 .0 (1. ~!c m "2) t ime c h amber-o bu rette time-c confi n -o strain-v 10:20 1.8 13.8 0 0.3 0 10:21 :1 5 5.5 10 5 0.0208 4 0.0038 1 0:2 1 5.5 1 0.1 0.0 1 67 4 0.0043 1 0 : 22 5.5 9.92 0.0333 4 0.0045 10:23 5 5 9.88 0.05 4 0 0046 1 0:25 5.5 9.84 0.0833 4 0.0046 I0:27 5.5 9.8 0.1167 4 0.0047 10:30 5.5 9.78 0. 1 667 4 0.0047 10:35 5.5 9.75 0.25 4 0.0047 10: 4 0 5.5 9.72 0.3333 4 0.0048 1 0:50 5.5 9.7 0 5 4 0.0048 11:00 5.5 9.68 0.6667 4 0.0048 1 3:30 5.5 9 65 3.1667 4 0.0048 re du ce confi n i n g p r ess u re to keep constant mea n stress force co nfin -o burette Dis-1 Di s-2 strai n -I s tr a in3 s tr ru n -v c-area stress-I str-3 6.64 4 9.33 0 0 0.004 0 0.002 39.447 0 4 1 8.52 3.9 9 33 0 0 -0 004 0 0.002 39.4 4 7 0.3012 3.9 35.4 3.9 9.27 0.608 0.608 0 7e-05 3e-05 39.6 0.7263 3.9 42 28 3.8 9.27 0.624 0.624 0.0001 7e-05 -2e-05 39.604 0.8999 3.8 54.16 3.7 9.25 0.662 0.66 0.0003 9e-05 -le-04 39.613 1.1996 3.7 66.04 3.6 9.2 0.698 0.698 0 0006 0 0002 -2e-04 39 62 1 .4992 3.6 77.92 3.5 9. 1 5 0.742 0.742 0.0009 0.0002 -3e-04 39.629 1 .7987 3.5 89.8 3.4 9 1 0.798 0.798 0.0012 0.0003 -5e-04 39 6 4 1 2.0978 3.4 10 1 .68 3.3 9.07 0.856 0.854 0.0016 0.0003 -6e-04 39.654 2.3967 3.3 1 13.59 3.2 9.03 0.922 0.922 0.002 0.0003 -8e-04 39.67 2 696 3 2 1 25.44 3.1 9 0.994 0.99 0 0 025 0.000 4 -0.001 39.687 2.9935 3 1 1 37.32 3 8 98 1.102 1.1 02 0.0032 0.0004 -0.001 39.714 3.2905 3 1 49.2 2.9 9 l.192 1 1 94 0.0038 0 .0004 -0.002 39.738 3.5875 2.9 161.08 2.8 9.05 1 .336 1 .334 0.0047 0.0003 -0.002 39.778 3.8826 2.8 1 72.96 2 7 9.18 1 .458 1 4 58 0.0055 0.0002 -0.003 39.816 4 1 773 2.7 184.8 2.6 9.31 1 .62 1.6 1 8 0 0066 2e-05 -0.003 39.863 4.4693 2.6 196.7 2.5 9.6 1 .8 1 8 1.818 0.0079 -3e-04 -0.004 39 929 4.76 2.5 208 6 2.4 10 2 038 2.034 0.0093 -8e-04 -0.005 40.004 5.0484 2.4 220 5 2.3 1 0.65 2.374 2.374 0.0115 -0.002 -0.006 40.124 5.33 2.3 232.36 2.2 11 .7 2.794 2.792 0.0142 -0.003 -0 008 40.283 5.6033 2.2 244 2 2.1 1 3.9 3.568 3.568 0.0192 -0.005 -0.012 40.593 5.8522 2.1

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227 256 12 2 21.4 6 274 6.278 0.0368 -0.014 -0.025 41.693 5.9837 2 mean-s ta u ta u = ( a 1 0 3 ) (kPa) 4 0 0 4.0004 0 142 13.919 4.1421 0.3424 33.565 4.1 0 4242 4 1.59 4.0999 0 5655 55.442 4.0997 0.7067 69.29 4.0996 0.8479 83.129 4.0993 0.9889 96.954 4.0989 1.1298 110.77 4.0987 1.2709 124.6 4.0978 1 .4 1 11 138.35 4.0968 1.5512 152.08 4.0958 1.69 1 1 1 65.8 4.0942 1.8303 179.44 4 .0924 1.9692 193.06 4 0898 2. 1 068 206.55 4.0867 2.2439 219.99 4.0828 2.3799 233.32 4.0767 2.5126 246.34 4.0678 2.6414 258.97 4.0507 2.7587 270.47 3 9946 2.8207 276.55

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APPENDIXC T_.L,,.__Ll L~ ~EXPERIMENTAL DATA FOR ALUMINA POWDER Al O (LOOSE) NOTE: (1) Total five tests corresponding to confining pressure s 4, 5, 3, 1, 2 (kg/cm 2 ) are given in this appendix (see tab l e 4.2). (2) Area of burette sec tion= 0.71 cm 2 (3) The unit of stress is kg/cm 2 (4) 1 kg/cm 2 = 98 .0 4 kPa. (5) all specimens were prepared with a vacuum pump TEST 1 No: 15 Triaxial comoression test on alumina oowder AlO-loose Date: Au2:ust 8-9, 1995 Pluviation oacked Sample: Alumina oowder Soecimen W eie:ht : 757.1 (g) Diameter : (2.775+2.770+2.8 >/3 -0.0245(in) = 2.75717 I in) = 7.0032 (c m) Height: (8.88+8.89+8.83+8.84)/4-3.02 I in)= 5.84 (in) = 14 .8336 (cm> Volume : ( 7.003203 )"2*1 4.8336*3.1415926 / 4 571.386 (cm"3 > Densitv: 757 1/571.353 = 1.3251 (e/cm 3 ) bulk den s it = 757.1 / 608.7 =1.2438 (g/cm 3 ) (wi thout hittin2: and vacuum in" for spec imen) "B "Check= 0.94 confini n g pressure= 5 5 (chamber-o)-1.5 (back-o) = 4.0 ( ke/cm"2 Hvdro s tatic loadinl! time camber-o burette time-c co n fin-o strain-v K-vaule 11:30 1.8 10.95 0 0.3 0 11:31 2.5 10 .15 0.01667 1 0.001 11:32 2.5 10.14 0.03333 1 0 00101 11 :3 5 2.5 10.11 0.08333 1 0 .00 105 228

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229 1 1 :40 2.5 1 0. 1 0.16667 1 0.00 1 06 1 1 :41 2.3 10 25 0.18333 0 8 0.00087 1071 .5 2 11: 4 2 2.3 1 0.25 0.2 0.8 0.00087 11:45 2.3 10.25 0 25 0.8 0.00087 11: 4 6 2.5 10.1 0.26667 1 0.001 0 6 11;47 2.5 1 0.1 0.28333 1 0.00 1 06 1 1 : 5 0 2.5 1 0.1 0.33333 1 0.00 1 06 11 :51 3 5 8.76 0.35 2 0.00273 1 1 : 5 2 3.5 8 57 0.36667 2 0.00296 11 :55 3.5 8.4 0.41667 2 0.003 1 7 12:00 3.5 8.39 0.5 2 0.00319 12:01 3.3 8.49 0 51667 1 .8 0.00306 1 607.28 12 : 02 3.3 8.49 0 53333 1.8 0.003 0 6 12:05 3.3 8.49 0 58333 1.8 0.00306 1 2:06 3.5 8.39 0.6 2 0 003 1 9 1 2:07 3.5 8 38 0.61667 2 0 0032 1 2:10 3.5 8.37 0.66667 2 0.00321 12:11 4.5 6.5 0.68333 3 0.00 5 54 12: 1 2 4.5 6.1 0.7 3 0.00604 12:15 4.5 5.9 0.75 3 0.00628 1 2:20 4.5 5.8 0.83333 3 0.0064 1 1 2:21 4.3 5.9 0.85 2 8 0.00628 16 0 7.28 12 : 22 4.3 5.9 0.86667 2.8 0.0 0 628 12 : 25 4.3 5.9 0 91667 2.8 0.00628 1 2:26 4.5 5.8 0.93333 3 0.00641 12:27 4.5 5.8 0.95 3 0.0064 1 12:30 4.5 5.76 1 3 0 00646 12:31 5.5 4.3 1 .01667 4 0.00827 1 2:32 5.5 3.7 1.03333 4 0.00902 12:35 5.5 3.5 1.08333 4 0.00927 12:40 5.5 3.39 1 1 6667 4 0.0094 1 12:41 5.3 3.45 1.18333 3.8 0.00933 2678.79 1 2 : 42 5.3 3.45 1.2 3.8 0.00933 1 2:45 5.3 3. 4 2 1.25 3.8 0.00937 1 2:46 5.5 3.35 1 .26667 4 0 00946 12:47 5.5 3.34 1.28333 4 0 009 4 7 12:50 5.5 3.28 1 .33333 4 0.00954 13:30 5.5 3.07 2 4 0.0098 1 Deviatoric l oading confininl!or ess ur e =5.5 (c h ambero )1 5 ( b ack-o)=4 (k2/cm"2) diso l force bu r e tt e strain-I strain-v strain-3 area-c stress E-value 0 20 10.05 0 0 0 38. 51 97 0.51921 r ate=0 1

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230 0. 1 39 9.77 0 00067 0.00035 -0.0002 38 .5 323 1.0 1 2 1 4 (mm/mi n ) 0.2 50 9.5 0.0013 5 0.00068 -0. 0 003 38.54 5 3 1.297 1 7 0.3 58 9 9.2 0.00202 0.00106 -0.000 5 38.557 1 52761 0.4 68.8 8.86 0.0027 0.001 4 8 -0 0006 38.5667 1.78392 0.5 76.2 8.54 0.00337 0 00 1 88 -0.0 0 07 38. 5 774 1.97525 0.6 83.5 8.2 0.0040 4 0.0023 -0 00 0 9 38.5871 2.16393 0.7 90 4 7.85 0.00472 0. 0 0274 -0.00 1 38 .5 96 4 2.34219 0.8 96.8 7.54 0.00539 0.003 1 2 -0.00 11 38.6076 2.50728 0.846 100 7.37 0 0057 0 00333 -0.0 0 12 38.6 115 2.5899 0 .848 81 7.28 0.00572 0.00345 -0.00 1 1 38 6077 2 09803 0 816 60.1 7.29 0.00 55 0.00343 -0.00 1 38.5998 1.557 2507 9204 0.858 90.8 7.2 0 00578 0.00355 -0.00 11 38.6064 2.35194 0.9 102 7.08 0.00607 0.0037 0.00 1 2 38.6 11 7 2.64169 1 1 09.5 6.84 0.00674 0.00399 -0.00 1 4 38.6263 2.83486 1 l 1 1 5.2 6.55 0.00742 0 00436 -0.00 1 5 38.6385 2 98148 1. 2 120.1 6.22 0.00809 0.00477 -0.0 0 17 38.6488 3. 1 0747 1.3 125.3 5.93 0 00876 0.005 1 3 0.00 1 8 38.66 1 1 3 24099 1.4 1 30.5 5 .6 5 0.00944 0.00548 -0 0 02 38.6738 3.37437 1.5 1 35.4 5.35 0.0 1 01 1 0.00585 -0 0 0 2 1 38.6857 3.5000 1 1.6 140 5.07 0.0 1 079 0 0062 -0.0023 38.6985 3.6 1 772 1 .7 1 44.5 4.8 0.0 1 146 0.006 5 3 -0. 0 02 5 38.71 1 8 3.73272 1 .8 1 48.9 4.53 0.0 1 2 1 3 0. 00 687 -0.0026 38.7251 3.84505 1 9 153 4 .27 0.0128 1 0.00719 0.0028 38.7389 3 94952 2 157.2 4.02 0.01348 0.0075 -0.003 38.7532 4.056 4 4 2.1 161.2 3 7 0.0 1 416 0.0079 -0.003 1 38.76 4 2 4.15848 2.2 165 2 3.52 0.0 1 483 0.008 1 3 -0.0034 38.78 1 9 4.25972 2.3 168.8 3.28 0 .0 1 5 51 0.00842 -0.0035 38.7968 4.35087 2.4 1 73.2 3.02 0.016 1 8 0.00875 -0.0037 38.8107 4.46268 2.5 1 76.7 2.81 0.0 1 685 0.0090 1 -0.0039 38.8271 4.55095 2.6 1 80.4 2.56 0.0 1 753 0.00932 -0.0041 38 84 1 5 4.64451 2.7 183.9 2.36 0.0182 0.00957 -0.0043 38.8584 4.732 5 6 2 8 187.5 2. 1 5 0.01888 0.00983 0.0045 38.8749 4.823 1 7 2.9 190.6 1. 95 0.0195 5 0.01008 -0.0047 38.89 1 8 4 90077 3 194 1. 7 0.02022 0.0 1 039 -0.0049 38.9064 4.98633 3.1 1 97 2 1.5 0.0209 0.0 1 064 -0.00 51 38 923 4 5 06637 3.188 200 1 .35 0.02149 0.01083 0.0053 38.9396 5. 1 3616 3. 1 98 175.5 1 .2 0.02156 0.0 11 01 -0.0053 38.9349 4.50752 3. 1 98 1 75.4 1 2 0.02156 0.01101 -0.00 5 3 38.93 4 9 4.5049 5 3. 1 64 1 40 1 1.98 0.02133 0.0 11 04 -0 00 5 1 38.9248 3 59667 3962.648 3.2 1 80 1 1.9 0.02157 0.01114 -0.0052 38.9306 4.62361 3.3 204.6 11. 78 0.02225 0.01 1 29 -0.0055 38.95 1 5 5 25268 3.4 207.9 1 1 6 0.02292 0.01 1 51 -0.0057 38.9696 5.33493 3.5 210 9 11 .4 0.0236 0.01176 -0.0059 38.9867 5.40954

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231 3.6 213.5 11.2 0.02427 0.01201 -0.0061 39 0038 5.47383 3.7 216.2 1 1.01 0.02494 0.01224 -0.0063 39.02 1 4 5.54055 3.8 219.1 10.82 0.02562 0.0 1 248 -0.0066 39.039 1 5 61233 4 224.7 1 0.48 0.02697 0.0129 -0.007 39.0764 5.75027 4.1 227 1 0.3 0.02764 0 0 1 3 1 3 -0.0073 39.09 4 6 5 80642 4.2 229.6 1 0 .1 4 0.0283 1 0.0 1 333 -0 0075 39.1139 5.87004 4.3 232.2 10 0.02899 0.0135 -0.0077 39. 1 34 1 5.93344 4 4 234.5 9.83 0.02966 0.0 1 371 -0.008 39.1529 5.98934 4.5 237 9.65 0.03034 0.0 1 394 -0.0082 39.17 1 2 6 05036 4.6 240 9 9. 4 5 0.0310 1 0.0 1 419 -0.0084 39.1886 6.1472 4.7 243.7 9.29 0.03168 0.0 1 438 0.0087 39.2079 6.215 5 8 4.8 246.4 9.15 0.03236 0.01 4 56 -0.0089 39.2283 6.28118 4.9 249 8.98 0.03303 0.01477 -0.009 1 39.2 4 72 6.34439 5 251.2 8.82 0.03371 0.0 1 497 -0. 0 094 39 2667 6.39728 5.2 255.4 8.52 0.03506 0.01534 -0.0099 39.3067 6.49763 5 4 259.7 8.25 0.0364 0.01568 -0.0 1 04 39.3482 6 60004 5.6 264. l 7 98 0.03775 0.0 1 60 1 -0.0 1 09 39.3899 6 70476 5.8 268.5 7.72 0.0391 0.0 1 63 4 -0.0 1 14 39 4322 6.809 1 5 6 272 8 7.48 0 04045 0 0 1 664 -0.0119 39.4756 6.91059 6.2 276.8 7.21 0.04 1 8 0.01697 -0.0124 39.5177 7.00446 6.4 280.9 6.95 0.04315 0.0173 -0.0 1 29 39.5603 7. 1 0055 6.6 284.9 6.73 0.04449 0 0 1 757 -0.0135 39.6051 7. 1 9351 6.8 288.7 6.5 0.04584 0.0 1 786 -0.0 1 4 39.6495 7.28 1 3 7 292.3 6.28 0.04719 0.01813 -0.014 5 39.6946 7.36373 7.2 295.9 6.06 0 04854 0.0184 0.0 1 5 1 39.7397 7.44595 7.4 299.5 5.88 0.04989 0.01863 -0.0 1 56 39.787 7.52758 7.6 303 5.7 0.0512 4 0.0 1 885 -0.0 1 62 39 8345 7.60647 7.8 306 5.5 0.05258 0.0191 -0.0 1 67 39.88 1 1 7.6728 1 8 309.9 5.3 0.05393 0.0 1 935 -0.0 1 73 39.9278 7.76152 c h a n ge to 8.2 316 5.2 0.05528 0.0 1 947 -0.0179 39.9797 7.90402 r a t e=0.3 8.4 318.6 5.08 0.05663 0.0 1 962 -0.0 1 8 5 40. 0 307 7.95889 mm/min 8.6 321 4 .96 0.05798 0.01977 -0.0 1 9 1 4 0.08 1 9 8.0086 8.8 323 4.8 0.05932 0.01997 -0.0197 40. 1 3 1 2 8.0486 9 326 4.67 0.06067 0.020 1 3 -0.0203 40. 1 822 8 11305 9.5 333 2 4.35 0.06404 0.02053 -0.02 1 8 40.3 1 05 8.26584 10 339.7 4 1 0.0674 1 0.02084 -0.0233 4 0.4434 8 3994 1 0.5 3 4 5.8 3.89 0.07079 0.02 11 -0.0248 40.5792 8.5216 1 1 351 3.68 0.07416 0.02 1 37 -0.0264 40 7161 8.62067 11.5 356.9 3.49 0.07753 0.0216 -0.028 4 0.8 5 5 8.73577 12 361.4 3.37 0.0809 0.02175 -0.0296 40.9986 8.8 1 494 1 2.5 368 1 3 .1 8 0.08427 0.02 1 99 -0 03 1 1 41.1395 8.9476 13 373.7 3.04 0.08764 0.02216 -0.0327 41.2842 9.05189 13.5 378.8 2.9 1 0.09101 0.02232 -0. 0 343 4 1 .4304 9 .1 4304

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232 14 383.2 2 88 0.09438 0 02236 -0.036 4 1 .583 9.2153 1 4.5 387.5 2.81 0.09775 0.02245 -0.0377 4 1 .7347 9.28485 1 5 39 1 .4 2.79 0. 1 0112 0.02247 -0.0393 4 1. 890 1 9.3435 15.5 395 2.79 0 1 0449 0.02247 -0.04 1 42.0478 9 39408 16 398.7 2.79 0.10786 0.02247 -0.0 4 27 42.2066 9. 4 4638 1 6.5 402.3 2.8 0.11123 0.02246 -0.04 44 4 2.3672 9.495 5 4 17 4 05 6 2.8 0.1146 0.02246 0.0 4 6 1 42.5285 9 537 1 2 1 7.5 408.2 2 85 0.1 1 798 0 0224 -0 0 478 42.6938 9.56111 1 8 4 1 0.9 2.9 0.12135 0.02234 -0.049 5 4 2.86 0 3 9.58696 18.5 412.6 2.9 0. 1 2472 0.02234 -0.0512 43.0254 9.5897 TE S T2 No: 16 Tri axial com ore ss i on test o n alumin a powde r AlO l oose D a t e: A U l! U St 16 17, 1995 olu v i atio n packed Sam o le: Al u mina powde r A 10 Spec im e n W e i g h t : 750.33 i' g) Di ame t er: 1 2.765+2.759+2.792)/3 -0.0245(i n ) = 2.7475 ( in ) = 6.97865 (cm) He i l! h t: (8 83+8.76+8.83+8.75)/4-3.02 (i n = 5 7725 (in) = 14 6622 (cm) Vo lum e: (6.97865)"2* 14 .662 1 5*3. 1 4 1 5926 / 4 560.829 (cm"3) dens i tv 750.33/ 560.8289 = 1 .33789 g/cm 3 ) B C h eck = 0 .96 co n fi nin g o r ess ur e = 6.5 ; c h amber-o) 1. 5 ( b ack-o) = 5.0 ;kl!/cm"2 tim e ca mb er p b ur e tt e time-c confi n -n stra in -v K -va ul e 11:20 1.8 19.51 0 0.3 0 1 1 :2 1 2.5 1 8.35 0.0 1 667 1 0.00147 1 1:22 2.5 1 8.3 0.03333 1 0.00153 11 :2 5 2.5 18.28 0.08333 1 0.0 0 156 1 1:30 2.5 1 8.27 0.16667 1 0.001 5 7 11 :3 1 2.3 1 8.4 1 0 18333 0.8 0.00139 1 1 26.84 11 :32 2.3 1 8.4 1 0.2 0.8 0 00139 11 :35 2.3 1 8.4 1 0.25 0 8 0.00139 11 :36 2.5 1 8.27 0.26667 1 0.00157 11 :37 2.5 1 8 27 0.28333 1 0 00157 11:40 2.5 1 8.27 0.33333 1 0.00157

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233 11 :4 1 3.5 17 1 0.35 2 0.00306 11:42 3.5 16 98 0.36667 2 0.00321 11 : 45 3.5 16 9 0.41667 2 0.00331 11 : 50 3.5 16.88 0.5 2 0.00333 11 :51 3.3 17 0.51667 1 .8 0.003 18 1314.65 11 :52 3 3 16.99 0 53333 1.8 0 00319 11:55 3.3 16.99 0.58333 1 .8 0.003 19 11 : 56 3.5 16.87 0.6 2 0.00335 11:57 3.5 16. 87 0.61667 2 0.00335 12:00 3.5 16.85 0.66667 2 0.00337 1 2:01 4.5 15 .5 0.68333 3 0.00508 12:02 4 .5 1 5.2 0.7 3 0.00546 12:05 4 5 15 0.75 3 0.00572 12:10 4 .5 14 95 0.83333 3 0.00578 12: 11 4.3 15.01 0.85 2.8 0.0057 2629.3 1 2: 12 4.3 15.01 0.86667 2.8 0.0057 1 2: 15 4.3 15 0.91667 2.8 0.00572 1 2:16 4.5 14. 94 0.93333 3 0.00579 12: 17 4.5 14.93 0.95 3 0.00581 1 2:20 4.5 14 .9 1 3 0.00584 12:21 5.5 13 .6 1.01667 4 0.00749 12:22 5.5 1 3.2 1 .03333 4 0.008 12:25 5.5 12.95 1.08333 4 0.00832 1 2:30 5.5 12.85 1.16667 4 0 00844 12 :31 5.2 1 2.94 1 18333 3.7 0.00833 2629.3 12:32 5.2 1 2 .9 4 1 .2 3.7 0.00833 12:35 5.2 1 2.93 1.25 3.7 0.00834 1 2:36 5.5 12.81 1.26667 4 0.00849 12:37 5.5 12.8 1 .28333 4 0.00851 12:40 5.5 1 2.79 1. 33333 4 0.00852 1 2:4 1 6.5 11.6 1.35 5 0.01003 1 2:42 6.5 11.2 1.36667 5 0.01054 12:45 6.5 10.9 1 .4 1 667 5 0.01092 12:50 6.5 10 .76 1.5 5 0.01109 12:51 6.2 10. 85 1.51667 4.7 0 01098 2629.3 12:52 6.2 10 .85 1.53333 4.7 0 01098 12 : 55 6.2 10.82 1.58333 4.7 0.01102 12:56 6.5 10.69 1 6 5 0.01118 12:57 6.5 10.68 1 .61667 5 0.01119 13:00 6.5 10 .65 1 .66667 5 0.01123 14 :00 6.5 10 4 8 2.66667 5 0.01145 Deviatoric Ioadin ~ rate=0.3 mm/min co nfinin oressure= 5 ( ki! /c m 2 ) disol. force burette stra in -1 stram-v s train -3 area-c stress E-val u e

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234 0 18.31 25.58 0 0 0 38.2501 0.47869 0.1 40.7 25.4 0.00068 0.00023 -0.0002 38.2675 1 .06357 0 2 61.4 25.02 0.00136 0.00071 -0.0003 38.2752 1.60417 0.3 74.8 24.65 0.00205 0.00118 -0.0004 38.2833 1 .95385 0.4 85.4 24.25 0.00273 0.00169 -0.0005 38.2901 2.23034 0.5 95 23.85 0.00341 0.00219 -0.0006 38.2968 2.48062 0.6 104 23.5 0.00409 0.00264 -0.0007 38.306 2 71498 0.7 112 23.1 0.00477 0.00314 -0.0008 38.3128 2.92331 0.8 120 .6 22.65 0.00546 0.00371 -0.0009 38.3171 3.14742 1 134 22.1 0.00682 0.0044 1 -0.00 1 2 38.3429 3.49478 1 1 141 21.75 0.0075 0.00486 -0.0013 38.3521 3.67646 1.226 150 21.35 0.00836 0 00536 -0.00 1 5 38.3658 3.90973 1 .236 11 8.7 21.18 0.00843 0.00558 -0.00 14 38.3601 3.09436 1.198 80 21.21 0.00817 0.00554 -0.0013 38.3516 2.08596 3890.8522 1.262 144.6 21.05 0.00861 0.00574 -0.0014 38.3606 3.76949 1.3 153 20.96 0.00887 0.00586 -0.001 5 38.3662 3.98788 1 .4 162 20.7 0.00955 0.00619 -0.0017 38.3799 4.22096 1.5 167 20.3 0.01023 0.00669 -0.00 18 38.3868 4 35046 1 .6 173 20 0.01091 0.00707 -0.0019 38.3985 4.50538 1 .7 178 .3 19.88 0.0 11 59 0.00723 -0.0022 38.4192 4.64091 1 .8 184 19.5 0.0 1 228 0.00771 -0.0023 38.427 4.7883 1 .9 189.2 19.25 0.01296 0.00802 -0.0025 38.44 1 3 4.92179 2 194 18 .95 0.01364 0.00841 -0.0026 38.4531 5.0451 2.1 198 1 8.7 0.01432 0.00872 -0.0028 38.4674 5.14721 2.2 203 18.45 0.015 0.00904 -0.003 38.4818 5.27523 2.3 208 18 .2 0.01569 0.00936 -0.0032 38.4961 5.40314 2.4 213 17.9 0.01637 0.00974 -0.0033 38.508 5.53132 2.5 217 17.65 0.01705 0.01005 -0 0035 38.5224 5.63309 2.6 221 17 .4 0.01773 0.01037 -0.0037 38.5368 5.73478 2.7 226 17 .2 0.01841 0.01062 -0.0039 38.5537 5.86195 2.8 229 I 6.9 0.0191 0.011 -0.004 38.5657 5.93792 2.9 233 16 .75 0 01978 0.01119 -0.0043 38.5851 6.0386 3 237.4 16 5 0.02046 0.01151 -0.0045 38.5996 6.15033 3.1 241.6 16 .25 0.02114 0.01183 -0.0047 38.6141 6.25678 3.2 245.7 16.05 0.02182 0.0 1 208 -0.0049 38 6311 6.36016 3.3 249 15.85 0.02251 0.01234 -0.00 51 38.6481 6.44274 3.4 252.9 15.6 0.02319 0.01265 -0.00 53 38.6627 6.54119 3.5 256.6 15.4 0.02387 0.01291 -0.005 5 38.6798 6.63396 3.6 260 15.2 0.02455 0.01316 -0 .0057 38.6969 6.71889 3.7 263.7 15 0.02524 0.01341 -0.0059 38.714 6.81149 3.8 266.8 14.8 0.02592 0.01367 -0.0061 38.73 1 2 6.88851 3.9 270 14.5 0.0266 0.01405 -0.0063 38.7434 6.96894 4 273.9 1 4.38 0.02728 0.0142 -0.0065 38.7645 7.06574

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235 4.1 277 14.2 0.02796 0.01443 -0.0068 38.7828 7.14235 4.2 280 14 0.02865 0.01468 -0.007 38.8 7.21649 4.3 283.1 13.8 0.02933 0.01493 -0 0072 38.8173 7.29314 4.4 286.6 13.6 0.03001 0.01519 -0.0074 38.8346 7.38002 4.5 290 13.4 0.03069 0.01544 -0.0076 38.8519 7 46424 4.6 293 13.25 0.03137 0.01563 -0.0079 38.87 1 7 7.53761 4.7 296 13.1 0 03206 0.01582 -0 .008 1 38.8916 7.6109 4.8 298 12.9 0.03274 0.01608 -0.0083 38.909 7.65889 4.846 300 12 .85 0.03305 0.01614 -0.0085 38.9191 7.70829 4.862 258.5 12 .7 0.03316 0.01633 -0.0084 38 916 6.64251 4 .8 18 196 12.69 0.03286 0.01634 -0.0083 38.9034 5.03812 5346.337 4 .876 282 12.6 0.03326 0.01646 -0.0084 38.9148 7.2466 5 306.8 12.45 0.0341 0.01665 -0.0087 38.9414 7.87851 5.2 312 12.15 0.03547 0.01703 -0.0092 38.9814 8.00383 5.4 318 11.81 0.03683 0.01746 -0.0097 39.0194 8.14978 5.6 323.2 11.52 0.03819 0.01782 -0.0102 39.0602 8.27442 5.8 328.6 11.25 0.03956 0.01817 -0.0107 39.102 8.40366 6 333.9 10.95 0.04092 0.01855 -0.0112 39.1424 8.53038 6.2 338.8 10 .65 0.04229 0.01893 -0.0117 39.183 8.64661 6.4 344 10 .38 0.04365 0.01927 -0.0122 39.2252 8.76987 6.6 348.8 10 .12 0.04501 0.0196 -0.0127 39.268 8.88254 6.8 353 4 9.88 0.04638 0.0199 -0.0132 39.312 8 98962 7 358 9.6 0.04774 0.02026 -0.0 137 39.354 9.0969 7.2 362 7 9.4 0.04911 0.02051 -0.01 43 39 .4 003 9.20551 7.4 367 1 9.16 0.05047 0.02082 -0 0148 39.4446 9.30671 7.6 37 1 .5 8.95 0.05183 0.02108 -0.0154 39.4907 9.40729 7.8 375.5 8.7 0.0532 0.0214 -0.0159 39.5347 9.49798 8 379.5 8.5 0.05456 0.02165 -0.0165 39.58 1 5 9.58781 8.2 383.7 8.3 0.05593 0.02191 -0.017 39.6284 9.68244 8.4 387.6 8.1 0.05729 0.02216 -0.0176 39.6755 9.76925 8.6 391.5 7.9 0.05865 0.02241 -0.0 181 39.7227 9.85583 8.8 395.2 7.71 0.06002 0.02265 -0.0187 39.7705 9.93701 9 398.5 7.55 0.06138 0.02286 -0.0193 39.8201 10 0075 9.08 400 7.5 0.06193 0.02292 -0.0195 39.8406 10 04 9.094 346.8 7.32 0.06202 0.02315 -0.0194 39.8354 8.70583 9.062 300 7.32 0.06181 0.02315 -0.0193 39.8261 7.53275 5374.9732 9.1 362 7.28 0 06206 0.0232 -0.0194 39.8351 9.08747 9.4 407.6 7.1 0.06411 0.02343 -0.0203 39.9128 10.2123 9.6 410.3 7 0.06547 0.02356 -0.021 39.9659 I 0.2663 9.8 413.8 6.8 0.06684 0.02381 -0.02 15 40.0139 10 .3414 10 417.1 6.7 0.0682 0.02394 -0.022 1 40.0673 10.41 10.2 420.6 6.55 0.06957 0.02413 -0.0227 40.1 1 82 10 .484 10.4 424.1 6.38 0.07093 0.02434 -0.0233 40.1682 10.5581

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236 10.6 427.6 6.25 0.07229 0.02451 -0.0239 40 2205 10.6314 10.8 430 .6 6. 1 0.07366 0 .02 47 -0.0245 40.2719 10.6923 11 433 .7 5.98 0 07502 0.02485 -0.0251 40.325 10.7551 11.5 441.2 5.7 0.07843 0.0252 0.0266 40.4595 10 9047 12 448.3 5.4 0.08184 0.02558 0.0281 40.5939 11 0435 12.5 455 .2 5 .2 0.08525 0.02584 -0.0297 40.7346 11 1748 13 461 .8 5 0.08866 0.02609 -0.0313 40.8764 11.2975 13.5 467 4.82 0.09207 0.02632 -0.0329 41.0203 11 3846 14 473 .6 4 .6 5 0.09548 0.02653 -0.0345 41.1658 1 1 .5047 14.5 479.2 4 5 0 09889 0.02672 -0.0361 41.3136 11.5991 15 484 .2 4.38 0.1023 0 02688 -0.0377 41.464 11.6776 15.5 489.2 4.28 0.10571 0 027 -0.0394 41.6167 11.7549 16 493.7 4.2 0 109 1 2 0 0271 0 041 41.77 1 7 11.819 16.5 497.8 4.15 0 1 1 253 0.02717 -0.0427 41.9294 11.8723 17 503.6 4.05 0.11594 0.0273 -0.0443 42.0857 11.9661 17.5 507.6 4.02 0 11935 0.02733 -0.046 42.247 12.0151 18 510 .4 4.02 0.12277 0.02733 -0.0477 42.4112 12 0345 18.5 514 4.02 0.12618 0.02733 -0.0494 42.5768 12 0723 19 518.6 4 0.12959 0.02736 -0.051 1 42.7424 12.1331 19.5 521.6 4 0.133 0.02736 -0.0528 42.9106 12.1555 20 524 .7 4.02 0.13641 0.02733 -0.0545 43.0811 12 1793 20.5 527.5 4.08 0.13982 0.02726 -0.0563 43.2553 12 195 21 528.8 4 12 0 14323 0.02721 0.058 43.4297 12.176 21.5 532.7 4 15 0.14664 0 02717 -0.0597 43 605 12.2165 22 535 .2 4.19 0.15005 0.02712 -0.0615 43.7822 12.224 1 22.5 537 .8 4 .25 0.15346 0.02704 -0.0632 43.962 12.2333 23 539 .7 4.3 0 15687 0.02698 -0.0649 44.1427 12.2263 23.5 541 .7 4.34 0.16028 0.02693 -0.0667 44.3243 12.2213 24 543.6 4.39 0 16369 0.02686 0.0684 44.5079 12.2136 24.5 545 .2 4.47 0. 1 671 0.02676 -0.0702 44 .69 48 12.1983 25 547 4.5 0. 1 7051 0.02672 -0.0 719 44 .88 03 12.188 25.5 548.6 4.6 0 17392 0.0266 -0.0737 45.0714 12.1718 26.1 550 4.68 0.17801 0 0265 -0 0758 45.3005 12 1411 26.6 551 4 .75 0 18142 0 02641 -0.0775 45.4934 12.1116 27.1 552 .2 4 8 0.18483 0 02634 -0.0792 45.6867 12.0867 25.6 553 4.9 0.1746 0.02622 -0.0742 45 1263 12.2545 TEST3 No: 17 1'.llJ.A_~ COMPRESSION TEST ON ALUMINA POWDER Date: Oct. 25-26, 1995

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237 pluviation packed Sample: Alumina powder AlO Specimen Weie:ht : 734.74 (g) Diameter: '2.73 4+2 .7 46+2.8)/3 -0.02 in ) = 2.74 (i n ) = 6.9596 cm) H e i g ht : (8.84+8.84+8.81 +8.8 /4-3.02 :i n )= 5.8025 ,. ) I 1n = 14.7384 (cm) Volume: (6.9596)"2* 1 4.73835*3.1415926 / 4 560.67 (cm"3 densitv: 737.74/ 560.67 = 1.31582 (_g/cm 3 B Check= 0.97 co nfming pr essure= 4 .5 (c hamb er-p)l .5(backp ) = 3.0 (kg/cm"2) t ime cambe r-p burette ttme-c co n fin-o stram-v K -va uJ e 10 :10 1.8 23.25 0 0.3 0 10: 11 2.5 22.3 0.01667 1 0.00 1 2 1 0: 12 2.5 22.28 0.03333 1 0.00123 10 : 15 2.5 22 .2 8 0.08333 1 0.00123 10:20 2.5 22.28 0.16667 1 0.00123 10:21 2.3 22.39 0.18333 0.8 0.00 10 9 14 33.76 10:22 2.3 22.39 0.2 0.8 0.00109 10:25 2.3 22.39 0.25 0.8 0 00109 10 :26 2.5 22.25 0.26667 1 0.00127 10 : 27 2.5 22 .2 5 0.28333 I 0 .0 0127 10 :3 0 2.5 22.25 0.33333 1 0.00127 10:31 3.5 20.6 0.35 2 0 00336 10:32 3.5 20.4 0.36667 2 0.00361 10:35 3.5 20.2 0 41667 2 0 00387 1 0:40 3.5 20. 1 2 0.5 2 0.00397 10 :4 1 3.3 20.23 0 51667 1 .8 0.00383 1433 .7 6 10 :42 3.3 20.23 0.53333 1 .8 0.00383 10:45 3.3 20.23 0.58333 1 .8 0.00383 1 0:46 3.5 20.11 0.6 2 0.00398 10 :47 3.5 20. 11 0.61667 2 0 00398 10:50 3.5 20. 11 0 66667 2 0.00398 10 :51 4.5 18 0.68333 3 0.00666 10 :52 4.5 1 7.4 0.7 3 0 00742 10 :55 4.5 1 7. 1 0 .7 5 3 0.0078 11 :00 4.5 17 0 .83333 3 0.00793 1 I :0 1 4. 3 17 l 0.85 2.8 0 0078 1577 1 3 11 :02 4.3 17.1 0.86667 2.8 0 0078 11 :05 4.3 17. l 0.91667 2.8 0.0078

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238 11:06 4.5 17 0.93333 3 0.00793 11 : 07 4.5 16 98 0.95 3 0.00795 11 : 10 4.5 16 .97 1 3 0.00796 rate=0.3 mm/min co nfining lo r essure= 3(kg/cm 3 ) di s ol. ( mm ) force (kg) bur e tt e s train-I s trrunv s train3 area-c s tre ss E-value 0 20 16 .2 0 0 0 38.0416 0.52574 0. 1 26.6 16 0.00068 0.00025 0.0002 38.0577 0.69894 0.2 33 15 .8 0.00136 0 00051 0.0004 38.0739 0.86673 0.3 39.5 15 .6 0.00204 0.00076 -0 0006 38.0902 1.03701 0.4 46.2 15 .3 0.00271 0.00114 -0 0008 38.10 16 1.21255 0.5 5 2.4 14 .9 5 0.00339 0 00159 0 0009 38. 1106 1.37495 0.6 58 14 .62 0.00407 0.002 -0.00 1 38.120 5 1.52149 0 7 62 6 14 .25 0.00475 0.00247 -0.00 11 38. 1 286 1.64181 0.8 67 1 3.9 5 0 00543 0.00285 -0.0013 38.1401 1.75668 0.9 71 .6 1 3.6 0.00611 0 0033 -0.00 1 4 38.1491 1.87685 1 75 7 1 3.25 0.00679 0.00374 -0.0015 38 .1 582 1 .98385 1 1 79 12 .95 0.00746 0 00412 -0 0017 38.1697 2.06971 1 .2 83 12.65 0.00814 0 0045 -0.00 1 8 38 .1 812 2. 1 7385 1 .3 87.7 1 2.35 0 00882 0 00488 0.002 38.1927 2.29625 1 4 91 12. l 0.0095 0.0052 -0.0021 38.2067 2.38178 1 .5 94.4 11 .8 0 01018 0 00558 -0.0023 38.2183 2.47002 1 .6 97 11.56 0 01086 0.00588 -0.0025 38 2328 2.53709 1 .672 100 11 .35 0.01134 0 00615 -0.0026 38.2414 2.61496 1 .676 80.8 11 .29 0.01137 0.00623 -0 .0026 38.2396 2. 113 1 .652 60 11.2 8 0.01121 0.00624 -0.0025 38.2328 1 .56933 3338.6104 1 .66 90 11 .2 0 01126 0.00634 0 0025 38.231 2.3 5411 1 .8 104 .6 10.98 0 01221 0.00662 -0.0028 38.257 2.734 14 1 .9 107 .7 10 .7 0 01289 0.00697 -0.003 38.2696 2.8142 4 2 1 1 0.6 10 .48 0.01357 0.00725 -0.0032 38.2852 2.88885 2.1 11 3 10 .22 0 .0 1425 0.00758 -0.0033 38.2988 2.95048 2.2 11 6 10 0 01493 0.00786 -0.0035 38.3 14 4 3.02758 2.3 118 .6 9.75 0.01561 0 00818 -0 0037 38.3286 3 0943 2.4 1 21 9.52 0.01628 0.00847 -0.0039 38.3 4 37 3.15567 2.5 124 9 .3 0.01696 0.00875 -0.00 41 38.3594 3.23259 2.6 126 4 9.06 0.01764 0.00905 -0 0043 38.3741 3.29389 2.7 1 29 8.85 0.01832 0 00932 -0.0045 38.3903 3.36022 2.8 131 .6 8.66 0 019 0 00956 0.0047 38.4075 3. 4 264 1 2.9 133 .7 8.4 1 0.01968 0 00988 -0.0049 38.42 18 3.4798 3 1 36.2 8.2 0.02036 0 01015 -0.005 1 38.438 3.54336 3.1 1 38.2 8 0.02 10 3 0.0104 0.0053 38. 454 8 3 59 383 3.2 14 0 7.8 0.02171 0.01065 -0 0055 38.47 16 3.6390 4

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239 3.3 143 7.6 0.02239 0.0 1 091 -0.0057 38.4885 3 7154 3.4 145 7.4 0.02307 0.011 1 6 -0 .0 06 38. 5 0 5 3 3.7657 1 3.5 147.2 7.2 0.02375 0.01141 -0.0062 38.5222 3.82117 3.6 1 49.7 7 0.02443 0.0 11 67 -0.0064 38.539 1 3.88437 3.7 1 5 1 .1 6.8 0.025 1 0.0 1 192 0 0 0 66 38.556 3.91897 3.8 153.5 6.6 0.02578 0 01217 -0.0068 38.573 3.979 4 7 3.9 155.7 6.45 0.02646 0.01236 -0.007 38. 5 92 4 4.03447 4 157.7 6.28 0.027 1 4 0.0 1 258 -0.0073 38.6 1 09 4.08434 4.1 159.5 6. 1 0 02782 0.01281 -0.0075 38.6289 4.12903 4.2 161.6 5.98 0.028 5 0.01296 -0 0078 38.65 4.181 1 2 4.3 163.3 5 .78 0.02918 0.0 1 321 -0.008 38.667 4.2232 4 4.4 165 5.65 0.02985 0.0 1 338 -0.0 0 82 38 6876 4.26493 4.5 167 5 .48 0.03053 0.0 1 359 -0.008 5 38.7062 4.31455 4.6 168.8 5.3 0.0312 1 0.01382 0.0087 38.7244 4.3590 1 4.7 170 4 5. 1 8 0.03189 0.01397 -0.009 38.74 5 5 4.39793 4.8 1 72.4 5 0.03257 0.0 1 42 -0.0092 38.7637 4.44746 4.9 174 4.86 0.03325 0 0 1 438 -0.009 4 38.7839 4.48639 5 176 4.7 1 0.03393 0.01457 -0 0097 38 8037 4.53565 5. 1 177.6 4.55 0.0346 0.01477 -0.009 9 38.823 4 57461 5.2 179.3 4.36 0.03528 0.01501 -0.0 1 01 38.8408 4.61628 5.3 1 80.8 4.28 0.03596 0.015 1 2 -0.0 1 04 38.8641 4 652 11 5.4 1 82.4 4 13 0.03664 0.0 1 53 1 -0.0 1 07 38.884 4 .69088 5.5 184 4 0.03732 0.0 1 547 -0.0109 38.9049 4.72949 5.6 1 85.7 3.88 0.038 0.01562 0.0 11 2 38.9263 4.77056 5.7 187.2 3.75 0.03867 0 0 1 579 -0.011 4 38.9472 4.8065 5.8 188.9 3.6 0 03935 0.01598 -0.0 11 7 38 9672 4.84767 5.9 190.4 3.48 0.04003 0.01613 -0.012 38.9887 4.88346 6 1 91.9 3.38 0.04071 0.0 1 626 -0.0 1 22 39.0 11 3 4.91909 6. 1 193.3 3 25 0.04139 0 0 1 6 4 2 -0 0 1 25 39.0323 4.9523 6.2 194.8 3. 1 0.04207 0.0166 1 -0.0127 39.0 5 24 4.988 1 7 6.3 196 2.97 0.04275 0.0 1 678 -0.0 1 3 39.0736 5.0 1 618 6.4 197.6 2 88 0.0 4 342 0.0 1 689 -0.0 1 33 39.0967 5.05413 6.5 200 2.76 0.04 41 0 0 1 704 -0.0 1 35 39. 11 84 5.11268 6.574 166.8 1 8.52 0.0446 0.01704 -0.0 1 38 39.139 4.26173 6.53 130 1 8.5 4 0.04431 0.01702 -0.0136 39. 1 278 3.3224 5 3 1 46 2582 6.574 170 1 8.5 0.0446 0.01707 -0.0 1 38 39. 1 38 4.3436 1 6.7 203 8 1 8.48 0.04546 0.01709 -0.0142 39 .1 72 5 20269 6.8 205.6 18.4 1 0.046 1 4 0.01718 -0.0145 39.1963 5.24539 6.9 206 6 1 8.35 0.0 4 682 0.01726 -0.0 14 8 39.2212 5.26756 7 207 6 1 8.26 0.0475 0.0 1 737 -0.015 1 39.24 4 6 5.2899 7.1 208.9 1 8. 1 8 0.04817 0.0 1 747 -0.0 15 3 39.2685 5.31978 7.2 211. l 18.1 0.04885 0.01758 -0 0 1 56 39.2925 5.37253 7.3 211.3 1 8 0.04953 0.0 1 77 0.0159 39.3 15 4 5.37448

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240 7.4 212.4 17.9 0.05021 0.01783 -0.0162 39.3384 5.3993 7.5 213 1 7.8 0.05089 0.01796 -0.0165 39.3615 5.41138 7.6 217.2 17.7 0.05157 0.01808 -0.0167 39.3846 5.51485 7.7 216 17.65 0.05224 0.01815 -0.017 39.4102 5.48081 7.8 217.2 17.55 0.05292 0.01827 -0.0173 39.4333 5.50803 7.9 2 1 8.4 17.45 0.0536 0.0184 -0.0176 39 4565 5.53521 8 219.3 17.38 0.05428 0.01849 -0.0179 39 4813 5.55453 8.1 220.8 17.28 0.05496 0.01862 -0.0182 39.5045 5.58924 8.2 221.7 17.18 0.05564 0.01874 -0.0184 39.5278 5.60871 8.3 223.1 17.06 0.05632 0.0189 -0.0187 39.5501 5.64095 8.4 224 17 0.05699 0.01897 -0.019 39.5754 5.66007 8.5 225.2 16 92 0.05767 0.01907 -0.0193 39.5998 5.68689 8 6 226.2 16.84 0.05835 0.01917 -0.0196 39.6243 5.70862 8.7 227.2 16.8 0.05903 0.01922 -0.0199 39.6508 5.73002 8.8 228.4 16.72 0 05971 0.01933 -0.0202 39.6753 5.75673 8.9 229.6 16.66 0.06039 0.0194 -0.0205 39.7009 5.78325 9 230.4 16 6 0.06107 0 01948 -0 0208 39 7265 5.79966 9.1 231.6 16.52 0.06174 0.01958 -0.0211 39.7511 5.82625 9.2 232.7 1 6.48 0.06242 0.01963 -0.0214 39.7778 5.85 9.3 233.5 16.38 0.0631 0.01976 -0.0217 39.8015 5 86662 9.4 234.5 16.3 0.06378 0.01986 -0.022 39.8262 5.88808 9.5 235 6 16.25 0.06446 0.01992 -0.0223 39.8525 5.9118 9.6 236.6 16.2 0.06514 0.01999 -0.0226 39.8788 5.93297 9.7 237.5 16.15 0.06581 0.02005 -0.0229 39 9052 5.9516 9.8 238.5 16.08 0.06649 0.02014 -0.0232 39.9306 5.97286 9.9 239.4 16.02 0.06717 0.02021 -0.0235 39.9566 5.99151 10 240.5 16.06 0.06785 0.02016 -0.0238 39.9877 6 01435 1 0.2 242.3 1 5.85 0.06921 0.02043 -0.0244 40.0351 6.05219 10.4 244 15.75 0.07056 0.02056 -0.025 40.0884 6.08655 10.6 245.9 15.66 0.07192 0.02067 -0.0256 40.1423 6.1257 10.8 247.5 15.53 0.07328 0 02084 -0.0262 40.1943 6.15758 1 1 249.4 15.45 0.07464 0.02094 -0.0268 40.2491 6.19641 11.2 251 15.38 0 07599 0.02103 -0.0275 40.3046 6.22758 11.4 252.7 15.28 0.07735 0.02115 -0.0281 40.3586 6.26137 11.6 254.5 15.18 0.07871 0.02 1 28 -0.0287 40.4128 6.29751 11.8 255.9 15.12 0.08006 0.02136 -0.0294 40.4693 6.32331 12 257.5 15.05 0.08142 0.02144 -0.03 40.5254 6.35404 12.2 259 15 0.08278 0 02151 -0.0306 40.5827 6.38203 12.4 260.6 14.94 0.08413 0.02158 -0.0313 40.6397 6.41245 12.6 262.1 14 88 0.08549 0.02166 -0.0319 40.6968 6 44031 12.8 263.7 14 8 0.08685 0.02176 -0.0325 40.7531 6.47068 13 265.1 14.75 0.08821 0.02182 -0.0332 40.8111 6.49578 13.2 266.4 14.7 0.08956 0.02189 -0 0338 40.8693 6.51835

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241 13.4 267.7 14.65 0.09092 0.02195 -0 .0 345 40.9276 6.54082 1 3.6 269. 1 14.6 0.09228 0.02201 -0.035 1 40.9861 6 56563 1 3.8 270.2 14.57 0 09363 0.02205 -0.0 358 41.0459 6.58287 14 271.8 14.55 0 09499 0.02208 -0.0365 41.1064 6.61211 14.5 275.2 14.47 0.09838 0.02218 -0.0381 41.2568 6.67042 1 5 278.5 14 .38 0.10178 0.02229 -0.0397 41.4078 6.72579 15 5 281.4 14.35 0.10517 0 02233 -0.0414 41.5631 6.77042 16 284 14.32 0.10856 0.02237 -0.0431 41.7197 6.80734 16.5 286.6 14 .3 0 11195 0.0224 -0.0448 41.878 6.84369 17 288.8 14 .3 0.11535 0.0224 -0.0 465 42.0386 6.86988 17.5 291.2 14 .32 0.11874 0 02237 -0.0482 42.2015 6.90023 18 293 4 14 .35 0.12213 0 02233 0.0499 42.3662 6.92532 1 8.442 260.3 14 .2 0 12513 0.02252 -0.0513 42 5032 6.12424 18 416 220 14 .18 0.12495 0.02255 -0.0512 42.4935 5. 1 7726 5368 .08 03 18 .8 296.8 14.35 0 1 2756 0.02233 -0.0526 42.6298 6.96226 19 297.2 14.4 0.12892 0.02227 0.0533 42.699 6.96035 1 9.5 299.6 14.45 0.13231 0.0222 -0.0551 42.8687 6 .9887 8 20 301.6 14.5 0.1357 0.02214 -0.0568 43.0398 7.00747 20.5 303.2 14 .58 0.13909 0.02204 -0.0585 43 .2139 7.01626 21 304.5 14 .65 0 14249 0.02195 -0.0603 43 3888 7 .01794 21.5 306.4 14 75 0.14588 0.02182 -0.062 43.5668 7.03288 22 307.8 14. 85 0.14927 0.0217 -0 .06 38 43.7462 7.03604 22.5 309.3 14.95 0.15266 0.02157 -0.0655 43.927 7.04123 23 3 10 .2 15 0 15606 0.02151 -0.0673 44 1064 7.03299 23.5 311.6 15 1 0.15945 0.02138 -0.069 44.2902 7.03542 TEST4 No: 18 11~TAA (AL COMPRESSION TEST ON ALUMINA POWDER Date : Nov. 3-4, 1995 oluviati on pa cked Sample: Alumina oowder AlO Specimen W ei2: ht : 747 .41 ( 1!) Diameter: (2 .778 +2.770+2.8 16 )/3 0.04(in) = 2.748 (in) = 6 97992 (cm) H eig ht : (8.85 +8 .89+8.76+8 .8 )/ 4-3.02 ( in) = 5.805 ( in ) = 14 .7 447 (cm) Volume: (6.97992)"2* 1 4.7447*3.141 5 926 / 4 564 192 (cm"3) den s ity: 747.41/ 564.19 17 = 1 .3 2474 (1!/Cm 3 )

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242 B Check = 0.97 confming pressure= 2 5 (chamber-o)-1.5 i'back-o) = 1.0 (kl!/cm"2) time camber-o burette time-c confin-p strain-v K-vau l e 1:15 1.8 25.48 0 0 3 0 1 :16 2.5 24.65 0.01667 1 0.00105 1: 17 2.5 24.64 0.03333 1 0.00106 1:20 2.5 24.62 0.08333 1 0.00108 1 :25 2.5 24.6 0.16667 1 0.00111 1:26 2.3 24.73 0.18333 0.8 0.00095 1220.8 1:27 2.3 24.73 0 2 0.8 0.00095 1:30 2.3 24.73 0.25 0.8 0.00095 1:31 2.5 24.6 0.26667 1 0.00111 1:32 2.5 24.59 0.28333 1 0 001 1 2 1:3 5 2.5 24.58 0.33333 1 0.00 1 13 2:00 2.5 24 54 0.75 1 0.00118 rate=0.3 mm/min, confming pressure= 1 (kg/cm 2 ) diso l .(mm) force (kg) burette strain-I stram-v strain-3 area-c stress E-value 0 20 24 32 0 0 0 38.264 0.52268 0.1 25.7 24.2 0.00068 0 00015 -0.0003 38.2842 0.67129 0.2 28.8 24.1 0.00136 0.00028 -0.0005 38.3054 0.75185 0.3 30 8 24.01 0.00203 0.00039 -0.0008 38.3271 0.80361 0 4 33.2 23.88 0.00271 0.00055 -0.0011 38.3468 0 86578 0.5 35 23.73 0.00339 0.00074 -0.0013 38.3657 0.91227 0.6 36.7 23 6 0.00407 0.00091 -0.00 1 6 38.3855 0.95609 0.7 38.2 23.5 0.00475 0 00103 -0.0019 38.4068 0.99461 0.8 39.6 23 4 0.00543 0.00116 -0 002 1 38 4282 1.03049 0.9 41 23 3 0.0061 0.00129 -0.0024 38.4495 1.06633 1 42 23.18 0.00678 0.00144 -0.0027 38.47 1.09176 1.1 43.3 23.06 0.00746 0.00159 -0.0029 38.4904 1.12496 1.2 44.6 22.96 0.00814 0.00171 -0.0032 38.5119 1.15808 1.3 45.7 22.86 0.00882 0.00184 -0.0035 38.5334 1.18599 1.4 46.8 22.76 0.00949 0.00197 -0 0038 38.5549 1.21385 ] .5 47.8 22.66 0.01017 0.00209 -0.004 38 5764 1.2391 1 6 48.7 22.56 0.01085 0.00222 -0.0043 38.598 1.26172 1 7 49.6 22.46 0.01153 0.00234 -0.0046 38.6196 1.28432 1.8 51 22.36 0.01221 0.00247 -0.0049 38.6412 1 31983 1.9 51.8 22 26 0.01289 0.0026 -0.0051 38.6629 1.33979 2 52 7 22.16 0.01356 0.00272 -0.0054 38.6846 1 3623 2.1 53.4 22.06 0.01424 0.00285 -0.0057 38.7063 1.37962 2.2 54.8 21.96 0.01492 0.00297 -0 006 38.7281 1.41499 2.3 55.5 2 1 .82 0.0156 0.00315 -0.0062 38.7479 1 43234 2.5 57.2 21.74 0.01696 0.00325 -0.0069 38.7974 1.47432

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243 2.854 60 21.42 0.01936 0.00365 -0 0079 38.8767 1.54334 2.856 52 21.39 0.01937 0.00369 -0.0078 38.8758 1 33759 2.834 40 21.39 0.01922 0.00369 -0.0078 38.8698 1 02908 2067.7371 2.862 56 21 35 0.01941 0.00374 -0.0078 38.8754 1 4405 3 61 21.3 0.02035 0.00381 0.0083 38.9101 1.56772 3.1 61.9 21 22 0.02102 0.00391 -0.0086 38.9331 1.58991 3.2 62.4 2 1. 16 0.0217 0.00398 -0.0089 38 9571 1.60176 3.3 63.1 21.09 0.02238 0.00407 -0.0092 38.9807 1.61875 3.4 63.6 21 03 0.02306 0.00415 -0.0095 39.0048 1.63057 3.5 64.2 20.98 0.02374 0.00421 -0.0098 39 0294 1.64491 3.6 64.9 20.93 0.02442 0 00427 -0.0101 39.0541 1.6618 3.7 65.6 20.86 0.02509 0.00436 -0.0104 39.0778 1.6787 3.8 66 20.8 0.02577 0.00444 -0.0107 39.102 1.68789 3.9 66.7 20.74 0.02645 0.00451 -0.011 39.1263 1.70474 4 67.4 20.68 0.02713 0.00459 -0.0113 39.1506 1.72156 4.1 67.9 20.63 0.02781 0.00465 -0.0116 39.1754 1.73323 4.2 68.4 20.58 0.02848 0.00471 -0.0119 39.2003 1.74488 4.3 68.9 20.51 0.02916 0.0048 -0.0122 39.2242 1.75657 4.4 69.5 20.48 0.02984 0.00484 -0.0125 39.2501 1.77069 4.5 70.1 20.45 0.03052 0.00488 -0.0128 39.2761 1.7848 4.6 70.8 20.39 0.0312 0.00495 -0.0131 39.3006 1.8015 4.7 71.3 20.35 0.03188 0.005 -0.0134 39.3262 1.81304 4.8 71.8 20.3 0.03255 0.00507 -0.0137 39.3512 1.82459 4.9 72.4 20.27 0.03323 0.0051 -0.0141 39.3773 1.83862 5 72.8 20.22 0.03391 0.00517 -0.0144 39.4025 1 8476 5.2 73.8 20.12 0.03527 0.00529 -0.015 39 4529 1.87059 5.4 74.8 20.02 0.03662 0.00542 -0.0156 39.5034 1.89351 5.6 75.5 19.97 0.03798 0.00548 -0.0162 39.5566 1.90866 5.8 76.5 19.92 0.03934 0.00554 -0.0169 39.61 1 93133 6 77.3 1 9.88 0.04069 0.0056 -0.0175 39.664 1.94887 6.2 78.2 19.83 0.04205 0.00566 -0.0182 39.7176 1.9689 6.4 79 19.78 0.04341 0.00572 -0.0188 39.7714 1.98635 6.6 79.7 19.73 0.04476 0.00578 -0.0195 39.8254 2.00124 6.8 80.5 19.68 0 04612 0.00585 -0.0201 39.8795 2.01858 7 81.3 19.64 0.04747 0.0059 -0.0208 39.9342 2 03585 7.2 82.2 19.6 0.04883 0.00595 -0.0214 39.9891 2.05556 7.4 82.8 19.58 0.05019 0.00597 -0.0221 40.0452 2.06766 7 6 83.4 19.57 0.05154 0.00599 -0.0228 40 102 2.0797 7.8 84.3 19.56 0.0529 0.006 -0.0235 40.1589 2.09916 8 84.9 19.55 0.05426 0.00601 -0.0241 40.216 2.1111 8.5 86.6 19.54 0.05765 0.00602 -0 0258 40.3602 2.14568 9 88.2 19.54 0.06104 0.00602 -0.0275 40.506 2.17746 9.5 89.6 19.57 0.06443 0.00599 -0.0292 40.6543 2.20395

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2 4 4 10 91 19 62 0.06782 0.00592 0.0309 40.8048 2.23013 10.5 92 19.68 0.0712 1 0.00585 -0.0327 40.9569 2 24626 11 93 3 1 9.78 0.0746 0.00572 -0.03 44 41 1 1 22 2 2694 1 1.5 94 5 19 88 0.07799 0 0056 -0 0362 4 1 .2686 2.28987 12 95 71 19 98 0 08139 0.00547 0.038 4 1 4 262 2.3 1 037 1 2.5 96.56 20 1 0.08478 0.00532 -0.0397 4 1 5861 2.32 1 93 13 97.63 20.25 0.08817 0 00513 0.04 15 4 1 .7 4 86 2.33852 13.5 98 59 20.39 0.09156 0.00 4 95 -0.0433 4 1 .9 11 9 2.35231 14 99 52 20 53 0.09495 0.00478 -0.0 4 5 1 42 0764 2.36522 14.5 100.38 20 7 0.09834 0.00456 -0.0469 42 2 4 37 2.3762 1 15 101.13 20.86 0.10173 0.00436 -0.0487 42.4 11 8 2.38 44 8 1 5.5 1 02.03 21.03 0.10512 0.004 1 5 -0 050 5 42.5817 2 396 1 1 6 1 02.9 2 1 .2 0. 1 0851 0.00393 -0.0523 42.7528 2.40686 1 6.5 103.56 2 1 .38 0.1119 0 00371 -0.054 1 42 9259 2. 41 2 5 3 1 7 1 04.11 21 .5 6 0.1153 0 003 4 8 -0 0 55 9 43. 1 002 2 4 1 553 17.5 105.28 21.73 0. 11 869 0. 0 0326 -0.0 5 77 43 2754 2. 4 3279 18 1 06.07 21.92 0.12208 0.00302 -0.0 5 9 5 43 4 5 29 2. 4 4 1 03 18. 5 107 2 22 1 0. 1 2547 0.0028 -0.0613 4 3.63 1 4 2.45695 19 1 07.6 22.26 0 12886 0 0026 -0.0631 4 3.810 1 2.45606 19 5 108 22.5 0.13225 0.00229 -0.065 43.99 4 6 2.45485 20 1 08 65 22.68 0.13564 0.00207 -0.0668 4 4 .1772 2.459 41 TE S TS No: 1 9 TRIAXIAL CO MP RESS I ON TEST ON ALUMINA P O WD E R D ate: N ov 7-8, 1995 ol uvia t io n oacked Samo l e: Alumin a oowde r A l O Soeci m en W ei~ h t: 730.63 (q) Di a m e t er: (2.74+2.752+2.775)/3 -0.02(in) = 2.73 5 67 ,. ) 1 lil = 6.948 5 9 (cm) H e i ~ h t: (8 80+8.78+8.82+8.78)/4-3.02 ( in )= 5.775 (i n ) = 14 668 5 (cm > V o lu me: (6.948593)1\2*14.6685*3.1415926 / 4 556.249 (c m 3 ) de n s i tv: 730 63 / 556.2491 = 1.31349 l (g/cm 3 B '' Check= 0 96 confini n g o ress u re = 3.5 (chamber-o)1 .5 (back-ol = 2.0 (l .. lcmA2) t ime ca m ber-o bure t te ttme-c con.fi n -o s tr ain-v K -va u le

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245 10:30 1 .8 3 1 .85 0 0.3 0 10:31 2.5 30 98 0.01667 1 0 00111 10:32 2.5 30.94 0.03333 1 0.00116 10:35 2.5 30 92 0.08333 1 0.00119 10:40 2.5 30 91 0.16667 1 0 0012 10:41 2.3 31.06 0.18333 0.8 0 00101 1043.13 10:42 2.3 31.06 0.2 0.8 0 00101 10:45 2.3 31 06 0.25 0.8 0.00101 10:46 2.5 30.91 0.26667 1 0.0012 10:47 2.5 30.91 0 .2833 3 1 0.0012 10:50 2.5 30 91 0 .33333 1 0.0012 10:51 3.5 29.45 0.35 2 0 00307 10:52 3.5 29.25 0.36667 2 0.00332 10 :55 3 5 29 .1 0.41667 2 0.00352 11:00 3.5 29.07 0.5 2 0.00355 11 :0 1 3.3 29.18 0 51667 1.8 0.00341 14 22 45 11:0 2 3.3 29.18 0.53333 1.8 0.00341 11:05 3.3 29.18 0.58333 1.8 0.00341 11 :06 3.5 29.07 0.6 2 0 00355 11 :07 3.5 29.07 0.61667 2 0.00355 11 : 10 3.5 29.06 0.66667 2 0 00357 13:20 3.5 28.9 2.83333 2 0.00377 rate=0.3 mm/min co nfinin !?: pressure= 2(ki!/Cm 2 ) disol. force burette strain-I s train-v strain-3 area-c stress E-value 0 20 28.23 0 0 0 37.92 1 3 0.52741 0 1 27.9 28 0.00068 0.00029 -0.0002 37.936 0 73545 0.2 33.2 27 77 0.00136 0 00059 0.0004 37.9508 0.87482 0.3 37.9 27.37 0.00205 0.0011 -0 0005 37.9573 0 99849 0.4 41 .6 27. l 0.00273 0.00144 -0.0006 37.9701 1.0956 0.5 44 .9 26.8 5 0.0034 1 0.00176 -0.0008 37.9839 1 1 8208 0.6 48.1 26.6 0.00409 0.00208 -0.001 37.9977 1 .2 6586 0.7 51.4 26.3 0.00477 0 00247 -0 .00 1 2 38.0092 1.35231 0 8 54 .4 26.02 0.00545 0 00282 -0.00 1 3 38.0216 1 43077 0 .9 56.6 25.8 0.00614 0.00311 0.0015 38.0369 1.48803 1 59 .2 25 .55 0 00682 0.00343 -0.00 17 38.0508 1.55581 1 1 61.8 25 3 0 .007 5 0.00375 -0.0019 38 0648 1.62355 1 .2 64 .2 2 5 .05 0.00818 0.00406 -0.0021 38.0787 1.68598 1.3 66 .9 2 4 85 0.00886 0 00432 -0.0023 38.0951 1.75613 1.4 68.5 24.55 0.00954 0 0047 -0.0024 38.1067 1 .79759 1.5 70 .9 24 3 0.01023 0.00502 -0.0026 38.1207 1 85988 1.6 72.8 24.1 0 .0 1091 0 00528 0.0028 38 .1 371 1 .9 089 1 .7 74.9 23.9 0 .0 1159 0.00553 -0.003 38.1536 1.96312

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246 1.8 76.9 23.7 0.01227 0.00579 -0 .00 32 38. 1 701 2.01466 1.9 78.7 23 5 0 01295 0.00605 -0.0035 38.1867 2.06093 2 80.5 23.3 0.01363 0.0063 -0 .00 37 38.2033 2.10715 2.1 82.4 23. 1 0.01432 0.00656 -0.0039 38 .2198 2.15595 2.2 84.2 22.88 0.015 0 00684 -0 .0041 38.2355 2.20214 2.3 86.2 22.69 0.01568 0 00708 -0.0043 38.2526 2.25344 2.4 87 22.5 0 01636 0.00732 0.0045 38.2697 2.27334 2.5 89.3 22.3 0.01704 0.00758 -0.0047 38.286 4 2.33242 2.6 91 22.05 0.01773 0.0079 -0.0049 38 3007 2.37594 2.7 92 21.95 0 01841 0.00803 -0.0052 38.3223 2.40069 2.8 94.5 21.76 0 01909 0.00827 0.0054 38.3396 2 46482 0 95 21.6 0.01909 0.00847 0.0053 38.3317 2.47837 0 80.8 21.55 0.01909 0 00854 -0.00 53 38.3292 2.10805 -0 .0 38 60 21.5 0.01883 0.0086 -0.005 1 38.3166 1 5659 2092.782 -0.008 80 21.5 0.01903 0.0086 -0.0052 38.3246 2.08743 0.2 99 5 21.2 0.02045 0.00899 -0.0057 38.36 5 2 2.59349 0.4 1 02.1 20.9 0.02182 0.00937 -0.0062 38.4038 2.65859 0.6 105 20.6 0.02318 0 00975 -0.0067 38.4 42 5 2.73 1 35 0.8 1 07.5 20.3 0.02454 0.01014 -0.0072 38 4814 2.79356 l 110 20 .05 0.02591 0.01046 -0.0077 38.5228 2.85545 1 .2 112 .8 19 77 0.02727 0.01081 -0.0082 38.5628 2.925 1 1.4 115.2 1 9.5 0.02863 0.01116 -0.0087 38.6035 2 .9 84 1 9 1.6 117 .6 19 .25 0.03 0.01148 -0.0093 38.64 53 3.04306 1 .8 119.7 19 0.03136 0.0118 -0 .0098 38.6872 3.09405 2 122 18.75 0.03272 0.01212 -0.0103 38 .7 292 3.15008 2.2 124 4 1 8. 5 0.03409 0.01244 -0.0108 38.7713 3.20856 2.4 12 6.6 18.27 0.03545 0.01273 -0.01 14 38.8 145 3.26 16 6 2.6 128.6 18.08 0.03681 0.01297 -0.0 119 38.8 599 3.30932 2.8 130 .8 17.85 0.03818 0.01327 -0.0125 38.9034 3.36217 3 132 .9 17.68 0.03954 0.01349 -0.013 38.9501 3.41206 3.2 1 34.6 17 .49 0.0409 0.01373 0.0136 38.9958 3.45165 3.4 136.7 17 .29 0.04227 0.01398 -0.014 1 39.0 41 2 3. 50143 3.6 138 5 17.09 0 .0 4363 0.01424 -0.0147 39.0867 3.5434 3.8 140 5 16.9 0.04499 0.01448 -0.0153 39.1329 3.59033 4 142.1 16 .75 0.04636 0.01467 -0.01 58 39.1812 3.62674 4.2 144 16.6 0.04772 0.01487 -0.0164 39.2297 3.67069 4.4 145 .5 16.5 0.04908 0.01499 0.017 39.2809 3 70409 4.6 147 .2 16.35 0.05045 0.01519 0.0176 39.3296 3.74273 4.8 1 48.7 16.2 0.05181 0.01538 -0.0 182 39 3785 3.77617 4.956 150 1 6. 1 0.05288 0.0155 -0.0187 39.4176 3.8054 1 4.96 131.9 16 05 0.0529 0.01557 -0.0 187 39.4162 3 34634 4 .922 101 15.97 0.05264 0.01567 -0 0185 39.40 1 3 2.56337 3022.3875 4.968 1 40 1 5.9 0 05296 0.01576 0.0186 39.4108 3.55233

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2 4 7 5.2 152.7 15.87 0.05454 0.0158 -0.0194 39.475 1 3.86826 5 4 154.2 1 5.8 0.0559 0.01589 0.02 39. 5 286 3 90098 5.6 155.3 15.7 0.05727 0.01602 -0.0206 39.5806 3.92364 5.8 156.5 1 5.6 0.05863 0.01614 -0.02 1 2 39.6328 3.9 4 875 6 158.1 1 5.5 1 0.05999 0.0 1 626 -0.0219 39.68 5 6 3.98381 6.2 159.5 15.42 0.06136 0.0 1 637 -0.0225 39.7386 4.01373 6.4 1 60.6 15.35 0.06272 0.0 1 646 -0.023 1 39.7928 4.03591 6 6 162 1 5.28 0.06408 0.0 1 655 -0.0238 39.8 4 7 1 4 06554 6 8 1 63.2 1 5.2 0.0654 5 0.0 1 666 -0.02 4 4 39 90 1 1 4.0901 1 7 1 64.6 1 5.13 0.0668 1 0.0 1 674 0 .025 39.9558 4.1 1 955 7. 5 1 67.7 1 5 0.07022 0.0 1 691 -0.0267 4 0 .0955 4 1 825 1 8 170.3 1 4.88 0.07363 0.01706 -0.0283 40.2368 4.23245 8.5 173 1 4.8 0.07704 0.0 1 7 1 7 -0.0299 40 38 1 2 4 .28418 9.5 178 1 4.62 0.08385 0.0174 -0.0332 40.6721 4.37646 10 1 81.4 1 4.6 0.08726 0.0 1 742 -0.0349 4 0 .823 4.443 5 8 10.5 183.5 14 58 0.09067 0 0 1 745 -0.0366 40 9749 4.47835 11 185.8 14.58 0.09408 0.0 1 745 -0.0383 4 1.1 29 1 4.5 1 748 11 .5 187.5 1 4.6 0.09749 0.01742 -0.04 4 1 .28 5 5 4.54 1 55 12 189 4 14 .65 0 1009 0.01736 -0.0 4 18 41.4 4 47 4 56994 12.5 191.2 14.7 0. 1 043 1 0.01729 -0.0 4 35 4 1 .605 1 4.595 5 9 13 192.9 14.77 0.1077 1 0.0172 -0.0 4 53 41.7679 4.6 1 838 13.5 194.6 1 4.85 0.11112 0.0 1 71 -0.047 41.9324 4.6408 14 196. l 14 98 0.11453 0.0 1 694 -0.0 4 88 42.101 4.65785 1 4.5 197.8 1 5.06 0.11794 0.0 1 683 -0.0 5 06 42.268 4.67966 1 5 1 99.2 1 5 .15 0.12135 0.0 1 672 -0.0 5 23 42 437 4.69402 15.5 200 1 5.28 0. 1 2476 0.016 5 5 -0.0541 42.6095 4.69379 16 201.7 15.4 0.12817 0.0164 -0.0559 42.7827 4.71452 16.5 203 15.52 0.13157 0.0 1 625 -0.0577 42.9573 4.72562 17 204 1 5.68 0.13498 0.0 1 604 -0.0595 43 .1 356 4.72927 1 7.5 205 1 5.8 0. 1 3839 0.01589 -0 06 1 3 43.3 1 3 4 .73299 1 8 205.98 15.96 0. 1 418 0.0 1 568 -0.063 1 43.4941 4.73 5 82 1 8.5 207.3 16.1 0.14521 0.0 1 55 -0.0649 43.6754 4.74637 1 9 208.3 1 6.2 0 14862 0.01538 -0.0666 43.856 4.7 4 964 19.5 208.8 16.38 0 15203 0.01515 -0.0684 44.0426 4.74087 20 209.1 16 54 0.15544 0.0 1 494 -0.0702 44.2295 4 .7 276 1

PAGE 256

APPENDIXD T ................ EXPERThIBNTALDATA FOR ALUMINA POWDER A16-SG NOTE: (1) Total six tests corresponding t o constant confining pressures 4, 2, 3, 5, 1 (kg/c m 2 ) and to one constant mean stress 3 (kg/cm 2 ) are given in this appendix (see table 4.2). (2) Area of burette's section= 0.73 cm 2 (3) The unit of stress is kg/cm 2 ( 4) 1 kg/cm 2 = 98.04 kPa (5) all specimens were prepared with a vacuum pump TEST 1 fo r A16-SG No: 1 1&A_J1.__L~ COMPRESSION TEST ON ALUMINA POWDER A16-SG Jan 16-17, 1996 oluviation packed Samnle : Alumina powder Al 6-SG Specimen W eil!ht : 750.76 (g) Diameter : (2.6 50+2 .58+2.67 5)/3 -0.02 = 2.615 (in) = 6.6421 I Cm) H eight: 21.78+21.76+21.78+21.73)/4-3*2.54 (in)= 14 1425 (cm) Volume : (6 6421)"2*14.1425*3.1415926 / 4 490.035 (cm"3 > saturated axia l strain c han2:e=0.288/14 .06= 0.02048 volume strain=0.02048*3= 0.06144 so initial vo lum e for saturated soecimen is 490.03*(1 -0. 06144)= 459.923 initial height=l4.1425*(1-0.020484 >= 13 .8 528 B Check= 0.96 den s ity: 750.76 / 459.9226= l .632362(1!/cm 3 ) vo lum e fraction= 0.4132557 co nfining pressure= 5.5 (c hamber-o) 1.5 (backp ) = 4 .0 1cocm"2) time ca mber-p burette time-(h) confin-p strain-v K -v aule 248

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2 4 9 11:00 1.8 10.05 0 0.3 0 11:01 2.5 8.85 0.01667 1 0.0019 11 :02 2.5 8 0.03333 1 0.00325 1 1:05 2.5 6.9 0.08333 1 0.005 11: 1 0 2.5 6.4 0.16667 1 0.00579 1 1: 11 2.3 6.5 1 0.18333 0.8 0.00562 114 5 .5 1 11 :12 2.3 6 5 0.2 0.8 0.00563 1 1 : 15 2.3 6 48 0.25 0.8 0 00567 11: 16 2.5 6.3 0.26667 1 0.00595 11: 17 2.5 6.2 0.28333 1 0.006 1 1 11 : 20 2.5 6 0.33333 1 0.00643 11:20 2.5 1 8 0.33333 1 0.00643 11 : 2 1 3.5 1 4 0.35 2 0.01278 11 :22 3.5 1 2 0.36667 2 0.0 1 595 1 1:25 3.5 8.8 0 41667 2 0.02 1 03 11 :30 3.5 7.7 0. 5 2 0 02278 1 1:31 3.3 7 8 0.51667 1 .8 0.02262 1260.06 11:32 3.3 7.7 5 0.53333 1. 8 0.0227 11:35 3.3 7.6 0.58333 1 .8 0.02294 11:36 3.5 7. 5 0.6 2 0.02309 11 :37 3.5 7.4 0.61667 2 0.02325 11 :40 3.5 7.1 0.66667 2 0.02373 1 1:40 3.5 46.8 0.66667 2 0.02373 1 1 :41 4.5 42 0.68333 3 0.03135 11:42 4.5 40 0.7 3 0.03452 11: 4 5 4.5 38 0.75 3 0.0377 11 :50 4.5 37 0.83333 3 0.03928 11 : 5 1 4.3 37.08 0.8 5 2.8 0.03916 15 75.08 1 1 :52 4.3 37 0.86667 2.8 0.03928 1 1 :55 4.3 36.8 0.91667 2.8 0.0396 11:56 4.5 36.72 0.93333 3 0.03973 1 1 :57 4.5 36.6 0.95 3 0.03992 12:00 4.5 36.2 1 3 0.04055 12:0 1 5.5 34.2 1.01667 4 0.0 4 373 1 2:02 5.5 32. 1 1 .03333 4 0.04706 12:05 5.5 30.2 1 .08333 4 0.05008 1 2: 1 0 5.5 29.39 1.16667 4 0.05136 1 2: 11 5.3 29.4 1 .18333 3.8 0.05135 12600 6 12:15 5.3 29.35 1 .2 5 3.8 0.05143 12:16 5.3 29.2 1 .26667 3 8 0.05 1 66 12:17 5.5 29.12 1 .28333 4 0.05179 1 2:20 5.5 28.8 1.33333 4 0.0523 13:40 5.5 26.3 2.66667 4 0.05627

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2 5 0 disp l acement rate= 0.3 mm/min, confining oresuure=4 (kg cm 2 ) diso l .(mm) force (kg) burette strain-I stra i n-v strain-3 area-c stress E value 0 17.39 25.5 0 0 0 31.9315 0.5446 0.1 35.5 25.11 0.00072 0.000619 -5e-05 31.934 1.11167 0 2 41 9 24.7 0.00144 0.00127 -9e-05 3 1 .9355 1.31202 0.3 46.7 24.3 0.00217 0.001905 -0.0001 3 1. 9375 1.46223 0 4 50.8 23.85 0.00289 0.002619 -0.000 1 31.9368 1.59064 0.5 54 .3 23.45 0.00361 0 003254 -0.0002 31 9389 1.70012 0.6 57.7 23 0.00433 0.003968 -0 0002 31.9382 1.80662 0.7 60.7 22.6 0.00505 0.004603 -0.0002 31.9402 1.90043 0.8 63 6 22.28 0.00578 0.005111 -0.0003 31.9466 1.99082 0.9 66.5 21.75 0 0065 0.005952 -0.0003 31.9416 2.08193 1 69.3 21.35 0.00722 0.006587 -0.0003 3 1 .9436 2.16945 1.1 72.3 21 0 00794 0.007143 -0.0004 31.9483 2 26303 1.2 74.7 20.55 0.00866 0.007857 -0.0004 31.9477 2.3382 1.3 77.5 20.28 0.00938 0.008285 -0.0005 31.9568 2.42515 1.4 80.l 1 9 8 0.01011 0.009047 -0.0005 3 1 .9545 2 50669 1.5 82.7 19 43 0.01083 0.009634 -0.0006 3 1 .9581 2.58776 1 .6 85.2 1 9 05 0.01155 0.0 1 0238 -0.0007 31 9613 2.66573 1.7 87.8 18.68 0.01227 0.0 1 0825 -0.0007 31.965 2.74676 1.8 90.2 18 3 0.01299 0.011428 -0.0008 31.9681 2.82156 1.9 92.7 17.95 0.01372 0.011984 -0.0009 31.9729 2.89933 2 95.2 1 7.6 0.01444 0.012539 -0.0009 31.9777 2.97707 2.1 97.5 17.2 0.01516 0.0 1 3174 -0.001 31.9798 3.0488 2.202 100 16 85 0.0159 0.0 1 3729 -0.0011 3 1 .9851 3. 1 2646 2.218 68.5 1 6.6 1 0.01601 0.014 1 1 -0.00 1 31.9758 2.14225 2.208 50 16.6 0.01594 0.014126 -0.0009 3 1 .9729 1.56383 8012.7231 2.268 90 16.48 0.0 1 637 0.014317 -0.00 1 31 9807 2.8 1 42 2886.8654 2.3 96 16.37 0.0 1 66 0.014491 -0.0011 31.9823 3.00166 811 50842 2 4 102.6 16.05 0.01733 0.014999 -0.0012 31.9888 3 20738 2.5 106 15.7 0.01805 0.0 1 5555 -0.0012 31.9936 3.31316 2.6 108.4 15.4 0 01877 0.0 1 6031 -0.0014 32.0012 3 38738 2.7 1 1 0 8 1 5.1 0.0 1 949 0.016507 -0.0015 32.0087 3.46155 2.8 113.1 14 75 0.0202 1 0.017063 -0.0016 32.0136 3.53287 2.9 115.5 14.4 0.02093 0.017618 -0.0017 32.0184 3.6073 3 117.7 1 4.1 0.02166 0.018094 -0.0018 32.0261 3.67513 3.1 120.1 13.75 0.02238 0.01865 -0.0019 32.0309 3.7495 3.2 122.4 13.47 0.023 1 0.019094 -0.002 32.0397 3 82026 3.3 124 7 13.15 0.02382 0.019602 -0.002 1 32.0462 3 89126 3.4 127 12.83 0 02454 0.02011 -0.0022 32.0528 3.96222 3.5 129.5 1 2.53 0 02527 0.020586 -0.0023 32.0604 4.03925 3.6 131.7 1 2.2 0 02599 0.02 11 1 -0.0024 32.0665 4.1071

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251 3.7 133 .8 11.92 0.02671 0 021554 -0.0026 32.0752 4 17144 3.8 136 11.6 0 02743 0.022062 -0.0027 32.0818 4.23916 3.9 138 .5 11.3 0 02815 0.022539 -0.0028 32.0896 4.31605 4 140 .9 11 0.02888 0.023015 -0.0029 32 0973 4.38978 4 1 142 .9 10.7 0.0296 0.023491 -0.0031 32.105 4.45102 4.2 145.3 10.4 0.03032 0.023967 -0.0032 32.1128 4.52468 4 3 147.3 10.15 0.03104 0.024364 -0 0033 32.1233 4.58546 4.4 149.8 9.85 0.03176 0.02484 -0.0035 32.1311 4.66215 4.5 151 .8 9 55 0.03248 0.025316 0.0036 32.1389 4.72326 4.6 154.3 9.3 0.03321 0.025713 -0 0037 32.1494 4 79946 4.7 156 .3 9 0.03393 0.026189 -0 0039 32.1572 4.86049 4.8 158 .6 8 75 0.03465 0.026586 -0.004 32.1678 4.93039 4.9 160.8 8.45 0.03537 0.027062 -0.0042 32.1757 4.99756 5 163.3 8.2 0.03609 0.027459 -0.0043 32.1863 5.07358 5 2 167.8 7.65 0 03754 0.028332 -0.0046 32.2048 5.2104 5.4 172 1 7.1 0.03898 0.029205 -0.0049 32.2234 5 34083 5.6 176 .5 6.55 0.04043 0.030078 -0 0052 32.2421 5.47421 5.8 180.7 6.05 0.04187 0.030871 -0.0055 32.2636 5.60074 6 185 5.55 0.04331 0.031665 -0.0058 32.2851 5.73019 6.2 189.2 5.05 0 04476 0.032459 -0.0061 32.3068 5.85635 6.4 193 .6 4 6 0.0462 0.033173 -0 0065 32.3313 5.988 6.6 197.6 4.12 0.04764 0.033935 -0.0069 32.3542 6.1074 6.8 201.6 3.7 0.04909 0.034601 -0.0072 32.3806 6.22595 7 205.8 3.22 0.05053 0.035363 -0.0076 32.4036 6.35114 7.038 206.7 33.2 0.05081 0.035363 -0.0077 32.4132 6.37703 7.5 223 .2 32.1 0.05414 0 037109 0.0085 32.4673 6.87462 8 232 31 1 0.05775 0.038697 -0.0095 32.5371 7.13032 8.5 242.8 30.1 0.06136 0.040284 -0.0105 32.6075 7.44615 9 252.4 29.25 0.06497 0.041633 -0.0117 32 687 7.72172 9.5 261 28.4 0.06858 0.042982 0.0128 32.7672 7.96528 10 270.8 27.65 0.07219 0.044172 -0.014 32.8539 8.24256 10.5 279.7 26.85 0.0758 0.045442 -0.0152 32.9383 8.49165 11 288.8 26.1 0.07941 0.046633 -0.0164 33 0263 8.74456 11.5 297 .3 25.42 0.08302 0.047712 -0.0177 33.1191 8.9767 12 305.5 24.8 0.08663 0.048696 -0.019 33.2162 9.19733 12.5 313.7 24.15 0.09023 0.049728 -0 0203 33.3123 9.41695 13 321.2 23.4 0.09384 0.050918 -0.0215 33.4032 9.61583 13.5 328 23.15 0 09745 0.051315 -0.0231 33 5248 9.78382 14 336 22 6 0.10106 0.052188 -0.0244 33.6293 9 99129 14 5 343 22.1 0.10467 0.052982 -0.0258 33.7377 10 1667 15 349.7 21.7 0.10828 0.053616 -0.0273 33.8531 10.3299 15 5 356.2 21.25 0.11189 0.054331 -0.0288 33.9663 10.4869 16 362.2 20.85 0 1155 0.054966 -0 0303 34.0836 10.6268

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252 16.5 367.8 20.5 0.1191 1 0.055521 -0.0318 34 2049 10.7528 17 373.4 20.18 0.12272 0.056029 -0.0333 34.3291 10.8771 17.5 378.7 19.81 0 12633 0.056616 -0.0349 34.4512 10 9924 18 383.9 19.52 0.12994 0.057077 -0.0364 34.5793 11.102 18.5 388.8 19.25 0.13355 0.057505 -0.038 34.7097 11.2015 19 393.7 19 0.13716 0 057902 -0.0396 34.8426 11.2994 20 402.1 18.5 0.14438 0 058696 -0 0428 35.1116 11.452 21 409.8 18.1 0.15159 0.05933 -0.0461 35.3917 11 579 22 417 17.7 0.15881 0 059965 -0.0494 35.6767 11.6883 23 422 .6 17.38 0 16603 0.060473 -0.0528 35 972 11.748 0 422 .6 17 36 0 16603 0 060505 -0.0528 35.9707 11.7485 1 428.5 17 0.17325 0.061076 -0.0561 36.2686 11 8146 2 433.5 16.75 0.18047 0.061473 -0 0595 36.5791 11.851 3 438.5 16 5 0.18769 0.06187 -0.0629 36.8952 11 885 4 441 16 28 0.19491 0.062219 -0.0663 37.2192 11 8487 5 443.2 16.05 0.20213 0.062584 -0.0698 37.5485 11.8034 6 442.7 15.83 0.20934 0.062933 0.0732 37.8846 11.6855 7 444 .7 15.6 0.21656 0.063298 -0.0766 38.2264 11 6333 8 448 15.4 0.22378 0.063616 0.0801 38.5768 11 6132 TEST2 for A l 6-SG No: 2 Triaxial comoression test on alumina powder Al6 SG Date : Feb. 16-17 1996 oluviation packed Sample: Alumina oowder SG-16 Specimen Wei!!ht : 748.07 ( !! ) Diameter: (2 625+2.568+2.71 )/3 -0.02(in) = 2.61433 (in) = 6 64041 (cm) Height: (21.92+22.04+21.97+21.94)/4-3*2.54 (in)= 14.3475 (cm) Vo l ume : (6.640407)"2*14.3475*3.1415926 I 4 496.885 (cm"3) saturated axial strain c hange=0.288/14.06= 0.02048 volume strain=0.02048*3= 0.0614 the height=l4.3475*(1-0.020484)= 14 0536 (cm) so initial volume for sa turated soecimen is 496.885*(1 0.06144) =466.356 density : 748.07/466.36= 1.60406 ( !!/cm 3 ) volume fraction= 0.4061

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2 5 3 B Check = 0.95 confi ni ng oressure = 3.5 i'cam b er-1J) 1.5 (backo ) = 2.0 (kl!/cm"2) time c h amb-p b ur et t e t 1 rne-c co n fin-p strai nv K -va ul e 2:20 1.8 46.3 0 0.3 0 2:2 1 2.5 44.5 0.01667 1 0.00282 2:22 2.5 43.4 0.03333 1 0.0045 4 2:25 2.5 42.5 0.08333 1 0.0059 5 2:30 2.5 42.05 0.16667 1 0.00665 2:31 2.3 42.1139 0. 1 8333 0.8 0.00655 1 999.4 2:32 2.3 42. 1 139 0.2 0.8 0.00655 2:35 2.3 42. 11 39 0.25 0.8 0.0065 5 2:36 2.5 42. 0 159 0.26667 1 0.0067 1 2:37 2.5 42 0.28333 1 0.00673 2: 4 0 2.5 4 1 .7 0.33333 1 0.0072 2:4 1 3.5 38 0.35 2 0.01299 2: 4 2 3.5 33.5 0.36667 2 0.02004 2:45 3.5 31.2 0.4 1 667 2 0.02364 2:50 3.5 30.3 0.5 2 0.02 5 0 5 2:5 1 3.3 30.3341 0.51667 1.8 0.02499 3748.87 2:52 3.3 30.3298 0.53333 1 .8 0.025 2:55 3.3 30.2787 0.58333 1 .8 0.02508 2:56 3.5 30.2 0.6 2 0.0252 2:57 3.5 30. 1 0.61667 2 0.02536 3:00 3.5 29.8 0 .66667 2 0.02583 3: 1 0 3.5 29.2 0.83333 2 0.02677 r ate= 0.3 mm/ mi n Confi nin g o r ess ur e = 3.5 1 c h a mb ero )l .5(backp, =2. 0 (kg/cm"21 di s ol .( mm ) f o r ce (kg) b ur e tt e s tr a in -I s tr a in -v s tr ai n -3 area-c s tr ess E va lu e 0 1 5 28.1 0 0 0 32.5869 0.4603 1 0. 1 21.9 27.7 0.0007 1 0.00063 -4e-05 32.5893 0 672 0.2 24.5 27.3 0.001 4 2 0.00 1 25 -9e-05 32.59 1 8 0.75 1 72 0.3 26.8 26.85 0.002 1 3 0.00 1 96 -9e05 32.59 1 6 0.8223 0.4 28.4 26.45 0.00285 0 .00258 -0.000 1 32. 5 94 0.87 1 33 0. 5 30 26 0.00356 0.00329 -0.000 1 32.5938 0.92042 0.6 31.7 25.55 0.00427 0.00399 -0.0001 32. 5 936 0.97258 0.7 33 25. 1 5 0.00498 0.00462 -0.0 0 02 32. 5 96 1 1.01239 0.8 35 2 4 .75 0.00569 0.00524 -0.0002 32. 5 985 1.07367 0.9 36 24.35 0.0064 0.00587 0.0003 32.60 1 1 1 0426 I 37.6 24 0.00712 0.00642 -0.0003 32.6061 1 1 5316 1 1 39 23.55 0.00783 0.007 1 2 -0.000 4 32.60 5 9 1 1 961 1 .2 40.4 23.15 0.0085 4 0.00775 -0.000 4 32.6084 1.23894 1 .3 41.7 22.8 0.00925 0.0083 -0.000 5 32.6 1 35 1.27861 1 .4 43.2 22.45 0.00996 0.00884 -0.0006 32.6 1 86 1 .3244

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254 1.5 44.4 22.05 0.01067 0.00947 -0.0006 32.6211 1.36108 1.6 45.7 21.65 0.01138 0.0101 -0.0006 32.6236 1 .40083 1.7 46.9 21.3 0.0121 0.01064 -0.0007 32.6287 1.43738 1.8 48.5 20.9 0.0128 1 0.01127 -0.0008 32.6312 1.48631 1.9 49.7 20.55 0.01352 0.01182 -0.0009 32.6364 1 .52284 2 51 20.2 0.01423 0.01237 -0.0009 32.6415 1.56243 2 2 53.5 19.5 0.01565 0.01346 -0.00 11 32.6519 1.6385 2.4 55.8 1 8.8 0.01708 0.01456 -0.0013 32.6623 1 .7 0839 2.6 58 3 1 8. l 0.0185 0.01565 -0.00 14 32.6727 1.78437 2.8 60.8 1 7.4 0.01992 0.0 1 675 -0.00 1 6 32 6831 1.86029 3 63.1 16.75 0.02135 0.01777 -0.0018 32.6963 1.92988 3.2 65.4 16 1 0.02277 0.01878 -0. 002 32.7094 1.99942 3. 4 67.7 15.45 0.02419 0.0198 -0.0022 32.7227 2.0689 3.6 70.1 14.8 0.02562 0.02082 -0.0024 32.7359 2.14138 3.8 72.4 14.2 0.02704 0.02176 -0.0026 32.7519 2.21056 4 74.5 13.62 0.02846 0.02267 -0.0029 32.7691 2.27349 4.2 76.7 13 0.02989 0.02364 -0.0031 32.7841 2.33955 4.4 79 12.5 0.03131 0.02442 -0.0034 32.8056 2.40812 4.6 81.2 11 .9 0.03273 0.02536 -0.0037 32.8218 2 47396 4.8 83.2 11.4 0.03415 0.02614 -0.004 32.8435 2 53323 5 85.4 1 0.85 0.03558 0.027 -0.0043 32.8625 2.59871 5.2 87.4 10 .32 0.037 0.02783 -0.00 46 32.8827 2.65794 5.4 89.5 9.8 0.03842 0.02865 -0.0049 32.9034 2.72008 5.6 91.6 9.3 0.03985 0.02943 -0.0052 32.9254 2.78205 5.8 93.7 8.8 0.04127 0.03021 -0.0055 32.9473 2.84393 6 95.7 8.3 0.04269 0.03099 0.0059 32.9694 2.90269 6.2 97 9 7.8 0.04412 0.03 1 78 -0.0062 32.9915 2.96743 6.416 100 7.3 0 04565 0.03256 -0.0 065 33.0177 3.02868 6.416 100 30.3 0.04565 0.03256 -0.0065 33.0177 3.02868 6.428 74 30.l 0.04574 0.03287 -0.006 4 33.0097 2.24177 6.398 50 30.0489 0.04553 0.03295 -0.0063 32.9994 1.51518 3403.7 6.418 70 30.1 0.04567 0.03287 -0.0064 33.0072 2.12075 6.6 101 29.75 0.04696 0.03342 -0.0068 33.0332 3 05753 6.8 104 29.35 0.04839 0.03405 -0.0072 33.061 3.1457 7 1 06.4 28.9 0.0498 1 0.03475 -0.0075 33.0862 3.21584 7.2 10 8.3 28.45 0.05 1 23 0.03545 -0.0079 33.1 11 4 3.27078 7.4 110.5 28.05 0.05266 0.03608 -0.0083 33.1395 3.33439 7.6 112.4 27.62 0.05408 0.03675 -0.0087 33. 166 3.38901 7.8 114.4 27.2 0.0555 0.03741 -0.009 33.1931 3 4465 8 116 .4 26.75 0.05692 0.03812 -0.0094 33.2187 3.50405 8.2 118 .6 26.35 0.05835 0.03874 -0.0098 33.2471 3.56723 8. 4 120 .5 25.95 0.05977 0.03937 -0.0 102 33.2756 3.62128 8.6 122.2 25.56 0.06119 0.03998 -0.0106 33.3047 3.66915

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255 8.8 124.3 25.2 0.06262 0.04054 -0.0 11 33.3356 3.72875 9 126 24.8 0.06404 0.04117 -0.0114 33.3644 3.77648 9.5 130.8 23.95 0.0676 0.0425 -0.0126 33.4451 3.91089 10 135 .4 23.05 0.07116 0.04391 -0.0136 33.5236 4 03894 10 5 140 22.38 0 .074 71 0.04496 -0.0149 33.6158 4.16471 11 144 21.55 0.07827 0.04626 -0.016 33.6996 4.27305 11.5 148.2 20.8 0.08183 0.04743 -0.0172 33.7886 4.38609 12 152.1 20.1 0.08539 0 04852 0.0184 33.8812 4 48922 12.5 156 .3 19.5 0.08895 0.04946 -0.0 197 33.9803 4.59973 13 160 .2 18.85 0.0925 0.05048 -0 021 34.0772 4.70109 13 5 164 18.25 0.09606 0.05142 -0.0223 34.1779 4.79843 14 167.7 17 .6 5 0 09962 0 05236 0.0236 34 .2 793 4.89216 14.5 171 3 1 7.1 0 10318 0 05322 0 025 34.3845 4.9819 15 1 74 7 1 6.65 0.10673 0.05393 -0.0264 34.4964 5.0643 16 181 1 15.75 0.11385 0.05533 -0.0293 34.7229 5.21557 17 187 .3 15 0.12097 0.05651 0.0322 34.9621 5.35723 18 1 92.5 14.25 0.12808 0.05768 -0 0352 35.2052 5.46794 19 1 97.6 13.6 0.1352 0 0587 -0.0382 35.4585 5.57272 19.408 200 13.32 0.1381 0 05914 0.0395 35.5621 5.62396 19.416 164 13 2 0.13816 0.05933 0.0394 35.5572 4.61229 19.388 1 3 0 13.2 0.13796 0.05933 0 .03 93 35 5489 3.65694 4795.1 19.438 180 13 .2 0.13831 0.05933 -0.0395 35.5637 5.06134 20 213 13 0.1423 1 0.05964 0.0413 35.719 5.96321 21 271.8 12.4 0 14943 0.06058 -0.0444 35.984 7.55336 relaxation recorded at 100 ke: axial force level : time(m) force(kg stress stress rate(l/s) rate(l/s) 0 1 00 3.02942 297.004 0 0202 1 98003 0 .25 90 2.72647 267.303 0.00404 0.39601 0.5 88 2.66589 261.363 0.00343 0.3366 1 84.6 2.56289 251.265 0.00121 0.1188 3 79.8 2.41747 237.009 0.00071 0.0693 5 77 2.33265 228.693 0.0004 0.0396 8 74.6 2.25994 221.565 kg/cm 2 kPa re l axation r ecorded at 200 ke: axial force level: time(m ) force strss(ke:) strss(kpa) rate 1/s, rate(l/s) 0 200 5.75988 564.699 0 02707 2 65409 0 25 1 85.9 5.35381 524.888 0.00749 0 73411 0.5 1 82 5 .24 149 513.876 0.00374 0 .3 6705 1 178. l 5.12918 502.864 0.00168 0.1647 3 171.1 4.92758 483.1 0 00079 0.07765 5 167.8 4.83254 473.782 0.0005 1 0.0502 8 164 .6 4.74038 464.747 1 1 kg/cmA2 (kPa)

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256 TEST3 for Al6-SG No: 3 Triaxial compression test on alumina powder Al6-SG Date : March 67, 1996 oluviation oacked Samole: Alumina oowder A16-SG Specimen W eieht : 745 ( i) Diameter: (2.555+2.621 +2.717)/3 -0.02( in) = 2.6 11 (i n ) = 6.63194 cm 1 H e i e ht : (2 1 .69+2 1 .62+2 1 .68+21.72)/4-3*2.54 (i n )= 14.0575 , cm Volume: (6.63 1 94)A2* 14 0575*3.1415926 / 4 485.601 (cmA3) sa tur ated axial strain cha ng e=0.288/ 1 4 06= 0.02048 volume strain = 0.06144 so initi a l vo lum e for sa turated soeci m e n is 485.6*(1-0.06144)= 455.765 i nitial h eight=l 4.0575* ( 1-0.020484)= 1 3.7695 (cm1 densitv: 745 I 455.76 = 1.63463 (g/c m 3 vo lum e fraction= 0.4 1 38228 B C h eck= 0.96 co nfinine oressure = 4.5(chamber-o) 1 .5 (back-o)=3.0 (ke/cmA2) t1me c hamb -o burette timec co nfin-p strru n -v K -va ul e 1:25 1 .8 45 0 0.3 0 1:26 2.5 43.2 I 1 0.00288 1 :27 2.5 42 2 1 0 .00 481 1 :30 2.5 41 .2 5 1 0 00609 1 :35:45 2.5 40.7 10 .75 1 0.00689 1 :36 2.3 40.7639 11 0.8 0.00678 195 3.98 1 :37 2.3 40.7639 12 0.8 0.00678 1 :40:45 2.3 40.7639 15 .75 0.8 0.00678 1 : 41 2.5 40 .6 617 1 6 1 0.00695 1:4 2 2.5 40.65 17 1 0.00697 1:45:45 2.5 40 5 20.75 1 0.0072 1 1 :46 3.5 36 2 1 2 0.01442 1 :47 3.5 32.2 22 2 0.0205 1:50 3.5 30.5 25 2 0.02322 1 : 55:45 3.5 29.45 30.7 5 2 0.02491 1 :56 3.3 29.4841 31 1.8 0.02485 3663.7

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257 l :57 3.3 29.4798 32 1 .8 0.02486 2:00:45 3.3 29.4287 35.75 1.8 0.02494 2:01 3.5 29 3 36 2 0.02515 2:02 3.5 29.2 37 2 0.02531 2:05:45 3.5 28 85 40.75 2 0.02587 2:06 4 .5 25.5 41 3 0.03123 2:07 4.5 22 6 42 3 0.03588 2:10 4 5 20.5 45 3 0.03924 2: 15:45 4 5 19.96 50 .7 5 3 0.04011 2: 1 6 4.3 19.9792 51 2.8 0.04008 6513 25 2:17 4.3 19.96 52 2.8 0.04011 2:20:45 4.3 19.8492 55.75 2.8 0.04028 2:21 4.5 19 .9 4 56 3 0.04014 2:22 4.5 19.94 57 3 0.04014 2:25 4 5 19 92 60 3 0.04017 2:30 4.5 19.9 65 3 0 0402 rate=0.3 (mm/min), co nfining ore ss ure= 3 (kg/cm 2 ) di s o l. force burette strain-I strain-v s train-3 area-c stress E value 0 8 37.2 0 0 0 32.1992 0.24845 0 1 9.2 37 1 0.00073 0.00016 -0.0003 32.2176 0.28556 0.2 12 .3 37 0.00145 0 00032 0.0006 32.2359 0.38156 0.3 23.2 36.8 0.00218 0 00064 -0. 0008 32.2489 0.7194 0 4 28.5 36.3 0.0029 0.00144 -0.0007 32.2458 0.88384 0.5 31 .4 35.9 0 00363 0 00208 -0.0008 32 2481 0.9737 0.6 34 4 35.6 0.00436 0 00256 -0.0009 32.2557 1.06648 0 .7 37 35 0.00508 0.00352 -0.0008 32.2472 1 14739 0.8 39.2 34.55 0.0058 1 0 00424 0.0008 32.2467 1.21563 0.9 41.5 34.12 0.00654 0.00493 -0.0008 32.2474 1.28693 1 43.8 33 7 0.00726 0 00561 -0.0008 32.2485 1.3582 1.1 45.7 33.25 0.00799 0.00633 -0.0008 32.2481 1.41714 1.2 47 .9 32.9 0.00871 0.00689 -0.0009 32.2531 1.48513 1.3 50 1 32.4 0.00944 0 00769 -0.0009 32.2499 1.55349 1.4 52 32. 1 0.01017 0 00817 -0 00 1 32.2576 1.61202 1 5 54 .2 31.65 0.01089 0.00889 -0.00 1 32.2572 1.68025 1.6 56.1 31.25 0.01162 0 00953 -0.001 32.2594 1.73903 1 .7 58 .2 30.82 0 01235 0.0 1 022 -0.0011 32.2601 1.80409 1.8 60 30 5 0.01307 0.01073 -0.0012 32.2667 1.8595 1 9 62.4 30 1 0.0138 0.01137 -0.0012 32.269 1.93374 2 64 29.75 0 .0 1452 0 01193 -0.0013 32.274 1.98302 2.2 67.7 29.05 0.01598 0.01305 0 0015 32.2841 2.09701 2.4 71.25 28.25 0.01743 0.01434 -0 0015 32.2887 2.20665 2.6 75 1 27.55 0.01888 0 01546 -0.0017 32.2988 2.32516 2.8 79 26.9 0.02033 0 0 1 65 0.0019 32.3117 2.44493

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258 3 82.4 26.2 0.02179 0.01762 -0.0021 32.3219 2.54935 3.2 86 25.55 0.02324 0.01866 -0.0023 32.3349 2 65967 3.4 89.4 24.9 0.02469 0.0197 -0.0025 32.3479 2.76371 3.6 93.2 24.2 0.02614 0.02082 -0.0027 32.3581 2.88026 3.8 96.6 23.6 0.0276 0.02178 -0.0029 32.374 2.98388 4.001 100 22.9 0.02906 0.0229 -0.0031 32.3846 3.08789 4.006 67 22.7 0 02909 0.02322 -0 0029 32 3747 2.06952 3.97 48 22.7 0.02883 0.02322 -0.0028 32.3659 1.48304 2243.2577 4 68.7 22.7 0 02905 0 02322 -0.0029 32.3733 2.12212 4.2 102.7 22.7 0.0305 0.02322 -0.0036 32.4224 3.16756 4.4 106.6 22.2 0.03195 0.02403 -0.004 32.444 3.28566 4.6 110 .2 2 1 .75 0.03341 0.02475 -0.0043 32.4684 3.39407 4.8 113.8 21.05 0.03486 0.02587 -0.0045 32.479 3.50381 5 117.4 20.55 0.03631 0.02667 -0.0048 32.5007 3.61223 5.2 120 5 20 0.03776 0.02755 -0.0051 32.5197 3.70544 5 4 123 .7 1 9.55 0.03922 0.02827 0.0055 32.5444 3.80096 5.6 127 18 .9 0.04067 0.02931 -0.0057 32.558 3.90074 5.8 130 18.35 0.04212 0.03019 -0.006 32.5772 3.99053 6 133.6 17 .8 5 0.04357 0.03099 -0.0063 32.5992 4.09826 6.2 137 17.3 0.04503 0.03187 -0.0066 32.6185 4.20007 6 .4 140 16 .8 0 04648 0 03267 0.0069 32.6407 4.28912 6.6 144 16.3 0.04793 0 03348 -0.0072 32.663 4.40866 6.8 146 6 1 5.8 0.04938 0 03428 -0.0076 32.6853 4.48519 7 149 .7 15.3 0.05084 0.03508 -0.0079 32.7077 4.5769 7 .2 153 1 4 .85 0 .0 5229 0.0358 -0.0082 32.7331 4.67417 7.4 156 l 1 4.35 0.05374 0.0366 -0.0086 32.7556 4.7656 7.6 159 1 1 3.9 0.05519 0.03732 -0.0089 32.7811 4.85341 7.8 162 13.5 0.05664 0.03796 -0.0093 32.8095 4.9376 8 165 13.05 0.0581 0.03868 -0.0097 32.8351 5.02511 8.2 168 .9 12.6 0.05955 0.0394 -0.0101 32.8608 5.13986 8.4 171 4 12.2 0.061 0.04004 0 0105 32.8895 5 2114 8.6 174 .3 11.8 0.06245 0.04068 -0.0109 32.9 1 82 5.29494 8.8 177 1 11.4 0.0639 1 0.04132 -0.0113 32 947 5.37529 9 1 79 .9 l I 0 .0 6536 0.04196 -0.01 1 7 32.976 5 .4 5549 9.5 187 .3 1 0.1 0.06899 0.04341 -0.0 1 28 33.0545 5.66641 10 1 94.5 9.2 0.07262 0.04485 -0.0 1 39 33. 1 336 5.87018 1 0.404 200 8.5 0 07556 0 04597 -0.0148 33. 1 995 6.02418 10.4 1 4 157 4 8.35 0 .07563 0.04621 -0.0147 33. 1 935 4.7419 10.378 120 8.35 0.07537 0.0462 1 -0.0146 33 1839 3.61621 4305.7654 10.444 180 8.35 0.07585 0.04621 -0.0148 33.20 1 4 5 42146 1 0.444 180 30.8934 0 07585 0 04621 -0.0148 33 20 1 4 5.42146 11 208.7 29.7978 0.07988 0.04796 0 .0 16 33.2852 6 27005 11.5 215.5 29.25 0.08351 0.04884 -0.0173 33.3869 6.45463

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259 12 222 28.65 0.08715 0.0498 -0.0187 33.4863 6.62958 12.5 228.l 28 0.09078 0.05084 -0.02 33.5836 6.79201 1 3 234.2 27.35 0 09441 0.05188 -0.02 13 33.68 16 6.95335 13.5 240 26.72 0 09804 0 05289 -0.0226 33.78 17 7.10445 14 246 26. 15 0.10167 0.05381 0.0239 33.886 1 7 .2596 1 14.5 251.8 25.6 0.1053 0.05469 -0.0253 33.9927 7 4074 8 15 257 25.l 0.10893 0.05549 -0.0267 34.1031 7.53597 15 .5 262 24.65 0 11256 0.05621 -0.0282 34.2175 7 65691 16 266.8 24.2 0. 11 6 1 9 0.05693 -0.0296 34.3328 7.77099 1 6.5 27 1 1 23.8 0.11983 0.05757 0.03 l l 34.4522 7.86888 1 7 275 .9 23.4 0.12346 0.05821 0.0326 34.5725 7.98032 1 7.5 280.2 23.05 0.12709 0.05877 -0.0342 34.697 8.07 56 2 1 8 284.5 22 7 0.13072 0.05933 -0.0357 34.8225 8. 1 7 1 8.5 288.2 22.4 0.13435 0.05981 -0.0373 34.9522 8.24554 19 292. 1 22.1 0 1 3798 0 06029 -0.0388 35.083 1 8.32596 19.5 295.5 21.85 0.1416 1 0.06069 -0.0405 35.2181 8.39057 20 298.4 21.6 0.14524 0.06109 -0.0421 3 5 3544 8.44026 20.218 300 2 1 .5 0.14683 0.06125 -0.0428 3 5 4147 8.47106 20.232 248 21.3 0.14693 0.06157 -0.0427 35.4064 7.00439 0 247.8 2 1 .3 0.14693 0.06157 0.0427 35 4064 6.99874 -0.03 200 21.3 0. 14 67 1 0.06 1 57 -0.0426 3 5 .3972 5.65017 6189.9551 0.038 280 2 1 .3 0. 1 472 0.06157 -0.0428 35.4 18 7 90558 1 305.9 2 1 0.15419 0.06206 -0.046 1 35.6962 8.56955 1 .5 308.8 20.85 0.15782 0.0623 -0.04 78 35.843 8.61536 2 3 11 5 20.65 0.16145 0.06262 -0.0494 35.9878 8.6557 2.5 314 20.5 0.16508 0.06286 -0 0511 36.1372 8.689 1 3 3 1 6.4 20.35 0.1687 1 0.0631 -0.0528 36.2879 8.7 1 9 15 3.5 319 20.2 0.17235 0.06334 -0 0545 36.4 4 8 75412 4 320.7 20.05 0.17598 0 06358 -0.0562 36.5934 8 76387 4.5 322.5 19 .9 0 1 796 1 0.06382 -0.0579 36.7482 8.77594 5 324.5 1 9.75 0.18324 0.06406 -0.0596 36.9044 8.79299 TEST4 fo r Al6-SG No: 4 T r iaxial com or ession test o n alumina powd er Al 6-SG D ate: Mar c h 11 -12, 1996 oluviation oacked Samnl e: Alumina powder Al6 -SG Specimen Weight : 750.61 lg) Diameter : (2.60+2.584+2.68 l l/3 -0.02(in) = 2.60 1 67 ( in ) = 6 60823 (cm)

PAGE 268

260 Height: (21.99+22+2 1 .9+2 l .9)/4-3*2.54 i'i n 1 = 14.3275 ( cm Vol u me: (6.608)"2*14.3275*3. 1 4 1 5926 I 4 49 1 36 1 (cm 3 ) density: 750.61 / 491.36= 1 .52762( cm 3 saturated axial strain change=0.288/ 1 4.06= 0.02048 vo l ume strai n =0.02048*3= 0.06144 so init i al vo lum e for sa tu rated soec i me n is 491.36*(1-0 06144 I= 461. 1 71 i ni tial h eig h t=l4.3275*i'l-0.020484)= 14.034 densitv: 750.6 I 461.17= 1.6276(1 /cm 3 ) vol u me fraction = 0.4120506 11 B II C h eck= 0.95 confini n g oress u re=6.5 (c h amber1 J 1 1.5 (backo ) = 5.0 (kg/cm"2) time c h amb-p burette time-c confi n -o Kva ul e stra 1 n-v 1 0:30:45 1 .8 47.7 0 0.3 0 1 0:3 1 2.5 46 0.00417 1 0.00269 1 0:32 2.5 45 0.02083 1 0.00427 1 0:35 2.5 43.9 0.07083 l 0.00602 1 0:40:45 2.5 43.4 0.16667 1 0.00681 1 0:41 2.3 43. 4 895 0.17083 0.8 0.00666 1412.25 1 0:42 2.3 43.4895 0.1875 0.8 0.00666 1 0:45:45 2.3 43. 4 937 0 .2 5 0.8 0.00666 10:46 2.5 43.4 0.25417 1 0.0068 1 1 0:47 2.5 43.4 0.27083 1 0.00681 10:50:45 2.5 43.3 0.33333 1 0.00696 1 0:51 3.5 39.7 0.3375 2 0 01266 10:52 3.5 37.3 0.354 1 7 2 0.0 1 646 10:55 3.5 35.95 0.40417 2 0.0 1 86 11 :00:45 3.5 35. 1 0.5 2 0 01994 11 :01 3.3 35. 14 26 0.50417 1.8 0 0 1 988 2965.72 11:02 3.3 35.1383 0.52083 1.8 0.0 1 988 11 :05:45 3.3 35. 1 043 0 58333 1.8 0.01994 11 :06 3.5 35. 1 0.5875 2 0.01994 1 1 :07 3.5 34.95 0.60417 2 0.02018 11: 1 0:45 3.5 34.7 0.66667 2 0.02058 1 1 : 1 1 4.5 3 1 .5 0.67083 3 0.02564 11 : 1 2 4 5 28.2 0.6875 3 0 03087 11 : 1 5 4.5 26 5 0.7375 3 0.03356 11 :20:45 4 .5 25.5 0.83333 3 0.035 1 4

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261 11 :21 4.3 25.5213 0.8375 2.8 0.03511 5931.45 11:22 4.3 2 5 50 85 0.854 1 7 2.8 0.03513 11 :2 5 4.3 25.4659 0.90417 2.8 0.0352 11:26 4 .5 25 .3 7 0.92083 3 0.03535 11 :27 4 5 25.3 0.937 5 3 0.03546 11 :30:45 4 5 24.9 1 3 0.03609 11 :3 1 5.5 22.5 1.00417 4 0.03989 11 :32 5.5 20.4 1.02083 4 0.04321 11 :35 5.5 1 8.8 1.07083 4 0.04575 11 :40:45 5.5 1 7.8 1.16667 4 0.04733 11 :4 1 5.3 17 .82 1 3 1.17083 3.8 0.0473 5931.45 11 :42 5 .3 17 .7957 1 1875 3.8 0.04734 11 : 45:45 5 3 1 7.6722 1.25 3.8 0.04753 11 :46 5.5 1 7.7 1 .2 5417 4 0.04749 11 :47 5.5 1 7.6 1.27083 4 0.04765 11 :50:45 5.5 1 7 .3 1.33333 4 0.04812 11 : 51 6.5 1 6.2 1.3375 5 0.04986 11 :52 6.5 14 .2 1 .3 5417 5 0.05303 11 :5 5 6.5 1 2.6 1.40417 5 0.05556 12 :00:45 6.5 11 .62 1.5 5 0.05711 1 2:0 1 6.3 11.6349 1.50417 4.8 0.05709 8 4 73.5 1 2:02 6.3 11 .603 1.52083 4.8 0.057 14 12:05:45 6.3 11 .5007 1.58333 4 .8 0.0573 12 : 06 6 5 11 .3807 1.5875 5 0.05749 12:07 6.5 11 .2807 1 .60417 5 0.05765 12:10 6.5 11.0807 1 .654 17 5 0.05797 13:40 6.5 9.1807 1 3. 15417 5 0.06097 rate=0 .3 mm/min strai n rat e= 0.3/60/140.34 = 0.0000356 ( 1 /s) co nfinin j?; pr ess ur e= 5 (kg/cm 2 ) displ. force burette strain-I strai n -v strai n3 area-c s tre ss E value 0 10 35.5 -0.0043 -0.0005 0.0019 31.3761 0.31871 0.1 12 .8 35 4 -0.00 36 -0.0003 0.00163 31.3936 0.40773 0.2 13.5 35.35 -0 .00 29 -0.0002 0 00131 3 1 .4138 0.42975 0.3 14.8 35.3 -0.002 1 -0 0002 0.00099 31.434 0.47083 0.4 15 .2 35.3 -0.0014 -0.0002 0.00063 31.4569 0.4832 0.5 15 .8 35.25 -0.0007 -8e-05 0.00032 31 4772 0 50195 0.6 18 35.2 0 0 0 31.4974 0.57147 0.7 35 35 0.00071 0 00032 -0.0002 3 1 .5098 1.11077 0 .8 48 34.55 0 00143 0.00103 -0.0002 3 1 .5089 1.52338 0.9 54.8 34.1 0.00214 0.00174 -0.0002 3 1 .5079 1.73924 1 59.6 33 65 0 .0 0286 0.00245 -0.0002 3 1 507 1.89164 1 1 63.8 33.25 0.00357 0.00309 -0.0002 31 5087 2.02483

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262 1.2 67.9 32.9 0.00428 0.00364 -0.0003 31.5132 2.15466 1.3 71.8 32.4 0.005 0.00443 -0.0003 31.5096 2.27867 ] .4 74.9 32.05 0.00571 0.00499 -0.0004 3 J 5 J 4 2.37672 1 5 78.4 31.6 0.00643 0.0057 -0.0004 31.513 2.48786 1.6 81.8 31.2 0.00714 0.00633 -0.0004 31.5148 2.59561 1.7 85.2 30.8 0.00786 0.00696 -0.0004 31.5165 2.70334 1.8 89 30.4 0.00857 0 0076 -0 0005 31.5183 2.82376 1.9 91.7 30 0.00928 0.00823 -0.0005 3 1 .52 2.90926 2 95 29.65 0.01 0.00879 -0.0006 31.5245 3.01353 2.1 98 29.25 0.01071 0.00942 -0.0006 31.5262 3.10852 2.152 100 29 0 01108 0.00981 -0.0006 31.5249 3.1721 2.3 101.8 28 3 0.01214 0.01092 -0.0006 31.52 1 7 3.22952 2.4 106 .8 27.95 0.01285 0.01148 -0.0007 31.5261 3.38766 2.5 110 .3 27.6 0.01357 0.0 1 203 -0.0008 31.5306 3.49819 2.6 113 27 3 0.01428 0.01251 -0.0009 31.5378 3.58301 2.7 116 .3 26.95 0.015 0.01306 -0.001 31.5422 3.68712 2.8 119 .3 26.55 0.01571 0.01369 -0.001 31.544 3.78202 2.9 122 .6 26.2 0.01642 0.01425 -0.0011 31.5485 3.88608 3 125 .7 25 9 0.01714 0.01472 0.0012 31.5557 3.98343 3.1 127.8 25.55 0.01785 0.01528 -0.0013 31 5602 4.0494 3.2 131.2 25.2 0.01857 0.01583 -0.0014 31.5647 4 15654 3.3 133 .8 24.85 0.01928 0 .0 1638 -0.0014 31 5692 4.23831 3.4 137 24.5 0.02 0.01694 -0. 0015 31.5737 4.33905 3.5 1 39.9 24.2 0.0207 1 0.01741 0.0016 31 581 4.42988 3.6 142 .7 23.85 0 02142 0 .0 1797 -0.0017 31.5855 4.51789 3.7 145.3 23.55 0.02214 0.01844 -0.0018 31.5928 4.59915 3.8 1 48 .2 23.2 0.02285 0.019 -0.0019 31.5973 4.69027 3.9 151 2 22.9 0.02357 0 01947 -0.002 31.6046 4.78411 4 153 .6 22.6 0.02428 0.01994 -0.0022 31.61 1 9 4.85893 4.2 159.6 21.95 0.0257 1 0.02097 -0.0024 31.6238 5.04684 4.4 1 65 21.35 0.02714 0.02192 -0.0026 31.6384 5.21518 4.6 170.6 20.75 0.02856 0.02287 -0.0028 31.6531 5.38967 4.8 176 20 15 0.02999 0.02382 0.0031 31.6679 5 55768 5 181 .2 19.6 0.03142 0.02469 -0.0034 31.6854 5.71872 5.2 186 .8 19 0 .0328 5 0.02564 -0.0036 31.7002 5.8927 5.4 191 .9 18.45 0 03428 0.02651 -0.0039 31.7179 6.05022 5.6 197 .3 17 95 0.03571 0.02731 -0.0042 31.7383 6.21647 5.728 200 17.5 0.03662 0 02802 -0.0043 31.7442 6.30036 5.732 144 .4 1 7.35 0.03665 0 02826 -0.0042 31.7369 4.54991 5.694 96 1 7.3372 0.03638 0.02828 -0.0041 31.7271 3.02581 5616.4692 5.746 160 1 7.2179 0.03675 0.02846 -0.0041 31.733 5.04207 5.8 187 1 7.0679 0.03713 0.0287 -0.0042 31.7377 5.89205 6 207 1 6.75 0.03856 0.02921 -0. 0047 31.7683 6.51593

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263 6.2 214.3 16 .2 5 0.03999 0.03 -0.005 31.7889 6.74135 6.4 219.6 15.75 0 04142 0.03079 -0.0053 31.8096 6.90358 6.6 224.2 15.25 0.04285 0.03158 -0.0056 31.8303 7.0436 6.8 230 14.8 0.04427 0.03229 -0.006 31.8539 7.22047 7 235.2 14.25 0.0457 0.03316 -0.0063 31.872 7.37953 7.2 240.l 13.8 0.04713 0.03387 -0.0066 31.8957 7.52767 7.4 244.6 13.3 0.04856 0.03467 -0 0069 31.9167 7.66371 7.6 249.7 12.85 0.04999 0.03538 -0.0073 31.9405 7.81766 7.8 254.7 12.4 0.05142 0.03609 -0.0077 31 9644 7.96823 8 259.8 12 0.05284 0.03672 -0.0081 31.99 1 3 8. 1 2097 8.2 264 8 11 55 0.05427 0.03744 -0.0084 32.0 1 53 8.27104 8.4 269 4 11.1 0.0557 0.03815 -0.0088 32.0395 8.40837 8.6 274.5 10.65 0.05713 0.03886 -0.0091 32.0637 8.56108 8.8 279 10.25 0.05856 0.03949 -0.0095 32.0908 8.69407 9 284 9.9 0 05999 0.04005 -0.01 32.1209 8.84 I 6 9.2 289 9.5 0.06141 0.04068 -0.0104 32.1482 8.98962 9.4 293.5 9.1 0.06284 0.04131 0.0108 32.1756 9.12183 9.6 298 8.7 0 06427 0.04195 -0.0112 32 .2 03 9.25378 9.69 300 8.5 0.06491 0.04226 -0.0113 32 2 1 43 9.31264 9.706 235.4 8.25 0.06503 0.04266 -0.0112 32.2041 7.30963 9.706 235.4 28.2 0.06503 0.04266 -0.0112 32.2041 7.30963 9.682 198 28.1957 0.06486 0 04267 -0 0111 32.1978 6 14948 6769.1529 9.724 260 28.1191 0.06516 0 04279 -0.0112 32.204 8 07353 9.8 295 28 1 0 0657 0.04282 -0.0114 32.2221 9.15522 10 308 27.8 0.06713 0.04329 -0.0119 32.2554 9.54879 I 0.5 319.3 27 0.0707 0.04456 -0.0131 32.3363 9.87435 11 330.5 26.1 0.07427 0.04598 -0.0141 32 412 1 10 1 968 11.5 341 25.35 0.07784 0.04717 -0.0153 32.4972 10.4932 12 351.5 24.6 0.08141 0.04836 -0.0165 32.5829 10.7879 12.5 361.8 23.85 0 08498 0.04955 -0.0177 32.6694 11.0746 13 371.7 23.2 0.08855 0.05057 -0.019 32.7623 11.3453 13.5 381.5 22.52 0.09212 0.05165 -0.0202 32.8543 11 6119 14 391.1 21.9 0.09569 0.05263 -0.0215 32.9505 11.8693 14.5 400.1 21.35 0.09926 0.0535 -0.0229 33.05 1 6 12.1053 15 408.2 20.8 0.10283 0.05437 -0.0242 33.1536 12.3124 15.5 416.2 20.35 0.1064 0.05509 -0.0257 33 2623 12.5126 16 424 19.9 0.10997 0.0558 -0.0271 33.372 12.7053 16.5 431.8 19 45 0.1 1 354 0.05651 -0.0285 33 4825 12.8963 17 439.6 19 0.11711 0.05722 -0.0299 33 594 13.0857 17.5 446 6 18.65 0.12068 0 05778 -0.0315 33.7124 13.2473 18 455.6 18.25 0.12426 0 05841 -0.0329 33.8288 13.4678 18.5 463 17.65 0.12783 0 05936 -0.0342 33.934 13.6441 19 470 17 0.1314 0.06039 -0.0355 34.037 13.8085

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264 19.5 477 1 6.5 0.13497 0.06118 -0. 0369 34.150 1 1 3.9678 0 477 1 6.5 0.13497 0.06118 -0.0369 34.1501 1 3.9678 0 5 484.5 16 0.13854 0.06197 -0.0383 34.2641 14 1402 1 491.4 1 5.95 0.14211 0 06205 -0. 04 34.407 1 14 282 1 5 496 .4 1 5.8 0.14568 0.06229 0.0417 34.545 14 3697 2 50 0.8 15.65 0.14925 0.06253 -0.0434 3 4 .6842 14 .4389 2.5 505.2 1 5.5 0.15282 0.06276 -0. 045 3 4 .8245 14 507 3 509 1 15 .3 0 156 39 0.06308 -0.0467 34.9629 14 5612 3.5 51 2.7 15.15 0 15996 0 06332 0 0483 35.1057 14 6045 4 516 .6 15 0 16 353 0 06355 -0 05 35 2497 14 6555 4.5 5 20.5 14 .86 0.1671 0.06378 -0.05 17 35.3956 14.7052 5 523 .8 14.75 0.17067 0.06395 -0.0 5 34 35 5448 14 .7363 5.5 526 .8 14.6 0.17424 0.06419 -0 055 3 5 .6926 14 .7593 6 529 .9 14 5 0 1 778 1 0.06435 0.0567 3 5 .845 1 14 7831 6.5 532 .6 14.35 0.18138 0.06458 -0.0 584 35.9956 14 .7962 7 535.5 1 4 .2 0.18496 0.06482 -0.060 1 36.1476 14.8143 7.5 538 .2 14.08 0.18853 0.06501 0.0618 36.3028 14 .8253 8 539 .6 13.97 0.1921 0.06519 -0.0635 36.4601 14.799 7 8.5 542 .6 1 3.86 0 19 567 0.06536 -0 .065 2 36.6189 14.8175 9 545 7 1 3 .76 0.19924 0 06552 -0.0669 36.7797 14 .837 10 548 8 13.56 0.20638 0.06583 -0.0703 37.1059 14 7901 11 550 .7 13.37 0.2 1 352 0.06613 -0.0737 37.4389 14 7093 1 2 551 4 1 3.2 0.22066 0.0664 -0.0771 37 7795 14 595 2 13 548 13 0 .22 78 0.06672 -0.0805 38.1245 14 .374 14 546 12.7 0.23494 0.0672 -0.0839 38.4691 14 .1932 15 548 .6 1 2.5 0.24208 0.06751 -0.0873 38.8274 14.1292 relaxati o n recorded at 100 kg axial force . tun e i min, fo r ce (kg) 0 100 0.25 84 0.5 80 1 76 3 67.4 5 63 4 8 59 TESTS f o r Al6 -SG No: 5 Triax.ial co mor ession test o n al umin a oowder Al6 SG Dat e: Mar c h 13-14, 1996

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265 pluviation packed Samole : Alumina oowder A16-SG Soecimen Weight : 750.19 (g) Diameter: (2.65+2.613+2.675)/3 -0.02( in ) = 2.626 (in) = 6.67004 (cm) H eig ht : (2 1 .72 +21.7+21.73+21.71/4-3*2.54 1 in) = 14 .092 5 (cm Volume : (6.6 7)"2*14.0925*3.1415926 I 4 492.413 (cm"3) d e n s i tv: 750.19 I 492 4131= 1.5235 g/cm 3 saturated ax ial s train c h a ng e=0.288/14.06= 0.02048 volume strai n=0 .02048*3= 0.06144 so initial vo lum e for saturated soecimen is 492.413*(1-0.06144 462.15915 initial h eig nt = 14.0925* 1 -0.0 20484 )= 13.8038 sat ur ated densitv= 750. 19 /462.16= 1 .62323 (g/cm 3 ) volume fraction= 0.4109433 B C h eck= 0.95 confi nin oressure = 2.5 (chamber-o)-1.5 (back-01=1.0 0c?/cm"2) time chamb-o burette time-c confi n o strai nv K-vaule 10:50 : 45 1.8 46.8 0 0.3 0 10:51 2.5 45.5 0.00417 1 0.00205 10:5 2 2.5 45.2 0.02083 1 0.00253 10:55 2.5 44 .9 0 .0 7083 1 0.003 1 l :00:45 2.5 44.7 0.16667 1 0.00332 11 :01 2.3 44.85 1 2 0.17083 0 .8 0.00308 837.208 11 :02 2.3 44.864 0 1875 0.8 0.00306 11 :05:45 2.3 44.8725 0.25 0.8 0 .003 04 11:06 2.5 44.7 0.25417 1 0 00332 11 :07 2.5 44.6 0.27083 1 0.00347 11: 1 0 2.5 44.5 0.32083 1 0.00363 1 3:10 2.5 43 .6 2.32083 I 0.00505 rate= 0.3 mm/min Confining ore ss ure = 2.51 .5 = 1 .0 (kg/cm 2 ) disol. force (kg) burette s train-I strai n-v strain-3 area-c s tress E value 0 8 43.5 0 0 0 33.3676 0.23975 0. 1 14 43.1 0.00072 0.00063 -5e05 33.3706 0.41953 0.2 16.6 42.75 0.00145 0.001 1 8 -0.0001 33.3763 0.49736 0.3 18.4 42.4 0.002 1 7 0.00174 -0.0002 33 382 0.55 119

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266 0.4 19.8 42.l 0.0029 0.0022l -0.0003 33.3904 0.59299 0.5 2 1.2 41 7 0.00362 0.00284 -0.0004 33.3934 0.63486 0.6 22.3 41 4 0.00435 0.00332 -0.0005 33.4018 0.66763 0.7 23.2 41.1 0.00507 0 00379 -0.0006 33.4102 0.6944 0.8 24.3 40.7 0.0058 0.00442 -0.0007 33.4133 0.72726 0.9 25.5 40 3 0.00652 0.00505 -0.0007 33.4 1 63 0.7631 1 26.2 40 0.00724 0.00553 -0.0009 33.4248 0 78385 1 1 27 39.6 0.00797 0.00616 -0.0009 33.4279 0.80771 1.2 28 39.3 0.00869 0.00663 -0.00 1 33.4363 0.83741 1.3 28.8 39 0 00942 0.00711 -0.0012 33.4447 0.86 11 2 1.4 29.7 38.6 0.01014 0 00774 -0.00 1 2 33.4479 0 88795 1.5 30.4 38.3 0.01087 0.00821 -0.0013 33.4563 0.90865 1.6 31.3 38 0.01159 0.00869 -0.0015 33.4648 0.93531 1.7 32 37.7 0.01232 0.00916 -0.0016 33.4733 0.95599 1.8 32.7 37.3 0.01304 0.00979 -0.0016 33.4764 0.97681 1.9 33.4 37 0.0 1 376 0.0 1 027 -0.0017 33 485 0.99746 2 34.1 36.7 0.01449 0 01074 -0.0019 33.4935 1.018 1 1 2.2 35.7 36.08 0.0 1 594 0.0 1 172 -0.002 1 33.5095 1.06537 2.4 37 35.4 0.01739 0.01279 -0.0023 33.5224 1.10374 2.6 38.3 34.8 0 0 1 884 0.01374 -0 0025 33.5395 1 14 1 94 2.8 39.6 34. 1 0.02028 0.0 1 485 -0.0027 33.5514 1.18028 3 40.7 33.6 0.02173 0.0 1 564 -0 003 33.5741 1.21224 3.2 41.9 33 0.02318 0.01659 -0.0033 33.5914 1.24734 3.4 43 .2 32.45 0.02463 0.01745 -0.0036 33.6115 1.28527 3.6 44.3 31.9 0.02608 0.01832 -0.0039 33.6317 1.31721 3.8 45.6 31.35 0.02753 0.01919 -0.0042 33.6519 1 .35505 4 46.8 30.78 0.02898 0.02009 -0.0044 33.6711 1.38991 4 2 47.8 30.25 0 03043 0.02093 -0.0047 33.6926 1.4187 1 4.4 49 29.8 0.03188 0.02164 -0.0051 33.7185 1.45321 4.6 50 .2 29.25 0.03332 0.02251 -0.0054 33.739 1 48789 4.8 51 .3 28.8 0 03477 0 02322 -0.0058 33.765 1.51932 5 52 3 28.25 0.03622 0.02409 -0.0061 33.7856 1.548 5.2 53 4 27.8 0.03767 0.0248 0 0064 33.81 1 8 1 57933 5 4 54.5 27 .3 5 0.03912 0 02551 -0.0068 33.8381 1 61061 5.6 55 6 26.9 0.04057 0.02622 -0 0072 33.8644 1 64 1 84 5.8 56 .6 26.45 0 04202 0.02693 -0.0075 33.8908 1.67007 6 57 5 26 0.04347 0.02764 -0.0079 33.9173 1.6953 6.2 58 .6 25.6 0.04492 0.02827 -0.0083 33.9467 1.72624 6 4 59 6 25 15 0.04636 0.02898 -0.0087 33.9734 1 75432 6.6 60 .7 24 75 0 04781 0.02962 0.0091 34.0029 1 78514 6.8 61.7 24.3 0.04926 0.03033 -0.0095 34.0297 1.8 1 312 7 62 .7 23 9 0.05071 0.03096 -0.0099 34.0594 1.8409 7.5 65 .2 22 9 0.05433 0 03254 -0.0109 34. 1 3 41 1.91011

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267 8 67.6 21.9 0.05796 0.03412 -0.0119 34.2093 1 .97607 8.5 70.1 2 1 0 06158 0 03554 -0.013 34.2908 2.04428 9 72.4 20. 15 0.0652 0.03688 -0 0142 34.3758 2.10614 9.5 74.6 19 35 0.06882 0.038 15 -0.0 15 3 34.4642 2. 1645 7 1 0 76.9 18 .6 5 0 .0 7244 0.03925 -0 .0 166 34.559 2.22518 10.5 79.2 17.85 0 07607 0.04052 0.0178 34.6489 2.28579 11 81.1 17.15 0 07969 0.04162 -0.0 19 34.7452 2.334 1 3 11 .5 83.2 16.5 0 08331 0 04265 -0 0203 34 .8 452 2.3877 12 85 15.85 0 .08693 0.04367 -0. 0216 34.946 2 43233 1 2.5 87 15.25 0.09055 0.04462 -0.023 35.0505 2.482 14 1 3 88.8 1 4.7 0.09418 0.04549 0.0243 35.1587 2.52 56 9 1 3.5 90.7 14.1 0 .097 8 0 04644 0.0257 35.2649 2.57196 14 92.3 1 3.65 0 1014 2 0.047 15 -0.027 1 35.3808 2 60876 14 .5 94.1 1 3. 15 0 1 0504 0.0479 4 -0.0286 35.4947 2.65 11 15 95.7 1 2.7 0.10867 0.04865 -0.03 3 5 .6 1 24 2.68726 15 5 97.2 1 2.3 0.11229 0 04928 -0 .0 3 15 35.7341 2.72009 16 98.7 11 .95 0.1159 1 0.04983 -0.033 35.8598 2.7 52 38 17 101 .2 10 .9 0.12315 0.05149 0.0358 36.0932 2.80385 1 7.5 102.7 10 .5 0.12678 0 05212 -0.0373 36.2 1 9 2.83553 1 8 1 03.7 10 .3 0.1304 0.05244 -0 039 36.3579 2.8522 1 8.5 104.9 10 .05 0 .134 02 0.05284 -0.0 406 36.495 2.87437 19 105 .9 9 .83 0 13764 0.053 1 8 -0.0422 36.6351 2.89067 1 9.5 1 07 9 .6 0.14127 0 05355 -0.0 4 39 36.7757 2.90953 20 10 7.9 9 .3 8 0.14489 0.05389 -0.0455 36.9 1 82 2.92268 20.5 108 .8 9. 1 8 0.1485 1 0.05421 -0.0 4 72 37 .063 1 2.93553 21 1 09 8 9 0.15213 0 05449 -0 0488 37.2 1 05 2.95078 21.5 11 0.6 8.82 0.15575 0.05478 0 0505 37.3592 2.96045 22 111. 2 8.65 0.15938 0 05505 -0.0522 37.5098 2 .9 6456 22.5 111 .7 8.48 0 163 0.05532 -0.0538 37.6616 2.96588 23 11 2.2 8.35 0.16662 0.05552 -0.0555 37.8174 2.96689 23.5 11 2.5 8.2 0.17024 0 05576 -0.0572 37.9733 2.96261 24 112.7 8.08 0.17387 0.05595 -0.059 38. 1324 2.95549 24 .5 11 2.8 7.98 0.17749 0.05611 -0.0607 38.2942 2.94561 25 11 2.7 7.85 0.1811 1 0.05631 -0.0624 38.45 55 2 93066 25 .1 38 111 .2 7.85 0.18211 0.05631 -0.0629 38.5026 2.8881 1 25.146 88.13 7.6 0.18217 0 05671 -0.0627 38.4892 2.28973 25. 10 2 60 7.38 0.18185 0.05705 -0.0624 38. 4599 1.56007 2289.13 4 9 25.14 90 7.3 0.18212 0.05718 -0. 0625 38.4677 2.33963 2831.81 1 2 25.042 40 6.65 0.18141 0.05821 -0.06 16 38.3922 1.04188 18 27.9428 24 .82 8 20 4.85 0.17986 0.06 10 5 0.0594 38.2032 0.52352 334.36277 24.598 10 3.3 0.1782 0.0635 0.0573 38.0256 0.26298 156.36477 24.404 5 2.2 0.17679 0.06523 -0.0558 37.8897 0.13196 93.224695 24.226 2.4 1.4 0. 1 755 0.0665 -0. 0545 37.7789 0.06353 53 07051

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268 TEST6 for A16-SG No: 6 constant mean stress test on alumina Jowder A l 6-SG Date : March 26-27, 1996 pluviation packed Samo l e: Alumina oowder A16 SG Specimen Weight 750 1g Diameter (2.586+2.61 +2.665 1/3 -0.02(in) = 2.6003 (in) = 6.6048 (cm) Height : (21.8+21.79+2 l.98+21.92)/4-3*2.5 14.253 (cm) 4= Volume: (6.605)"2*14.2525*3.1415926/4 = 488.34 (cm"3) densitv: 750/ 488.3448= 1.5358 (kg/cm 3 saturated axia l s tra i n =0.288/14.06= 0.0205 volume strain=0.02048*3= 0.0614 so initial volume of soecimen is 488 .3 448 *(1-0.06144 >= 458.34 initial heijlht= 14.25 25*( 1-0.020484) =13.961 Area=(6.605*(1 0.020484))"2/4*3. l 415926= 32.873 B Check = 0 95 co nfining pressure=4.5(chamber-o )1.5 (back-o) = 3.0 (kg/cm"2) time cham-o burette time-c confin-p strain-v K-vaule 10:30:45 1 .8 44.5 0 0.3 0 10:31 4.5 32 0.25 3 0.0199 10:31:15 4.5 27.5 0.5 3 0 0271 10:31:30 4.5 26.2 0.75 3 0.0291 10:31 :45 4.5 25.5 1 3 0.0303 10:32 4.5 25 1.25 3 0 .03 11 10:32:30 4 .5 24.3 1.75 3 0.0322 10:33 4 5 23.9 2.25 3 0.0328 10 :3 4 4 5 23.3 3.25 3 0.0338 1 0:35 4 5 22.9 4.25 3 0.0344 10:37 4 .5 22.4 6.25 3 0.0352 1 0:40 4 .5 21.9 9.25 3 0.036 10:45 4 .5 21.3 14.25 3 0.037 10 : 50 4.5 21 19.25 3 0.0374

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269 10:55 4.5 20.7 24.25 3 0.0379 1 3:40 4.5 1 7.8 1 89.25 3 0.0425 force1 kg1 co nfing burette di so l -1 disol-2 area-c stres m ea n -s volume-s strain-I strai n -2 10 3 37 0 0 32.832 0.3046 3.101 5 0 0 0 20 2.9 36. 1 0 0 32.785 0.61 3.1033 0 00 1 4 0 0.0007167 30 2.8 35.5 0.248 0.248 32.8 12 0.9143 3. 1048 0.0024 0.0018 0.0003063 40 2.7 35 0.394 0.388 32.82 1 .2 18 8 3. 1063 0 0032 0.0028 0.0001923 50 2.6 3 4 0 .7 1 8 0.716 32.844 1 .5223 3.1074 0.0048 0.0051 0.000179 60 2.5 32. 4 1 1 94 1.192 32.873 1 .8252 3. 1084 0.0073 0.0085 -0.00061 70 2.4 30.3 1 .878 1.878 32.925 2. 126 3. 1087 0 0 1 07 0.0135 -0.001391 80 2.3 28.3 2.6 16 2.6 14 32.996 2.4246 3. 1082 0.0 1 39 0.0187 -0.002438 90 2.2 26. 1 3. 478 3.472 33.086 2.7202 3.1067 0.0 1 74 0.0249 -0.003766 100 2.1 24 4.456 4.452 33.2 12 3.0109 3. 1036 0.0207 0.0319 -0.00 56 110 2 2 1 .8 5.654 5.562 33.378 3.2955 3.098 5 0.0242 0.0402 -0.007981 120 1 .9 19 .7 6.982 6.978 33.608 3.5706 3.0902 0.0276 0.05 -0.011223 130 1 8 17.5 8.666 8.664 33.9 18 3.8328 3.0776 0.03 I 1 0.0621 -0.015506 140 1 .7 15 .6 1 0.772 10 768 34.365 4.074 3.058 0.034 1 0.0771 -0.02 153 3 150 1 .6 14 1 1 3.5 1 8 13.516 3 5 .026 4 .2825 3.027 5 0.0365 0 0968 -0.030177 155 1 5 1 3.7 1 7.2 17.2 36 056 4 .2988 2.9329 0.0371 0.1232 -0.04305 160 1 5 1 3.3 19.8 19.804 36.8 15 4 .3 46 2.9487 0.0377 0.1418 -0.0520 51 15 8.7 1 4 1 3.2 30.2 30.2 40 308 3.9372 2.7124 0.0379 0.2163 -0.089213 166. 2 1 .4 1 2. 1 38 2 1 76.8 1 4 10.9 48.45 co n s tant 18 3.8 1 4 1 0.1 55.45 mean 58 1 .4 7.2 stress 15 1 4 4.9 0 1 .4 3 0 1 .4 2.8 0 0.5 3.9

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REFERENCES ABAQUS [1993], Version 5.3, Theory Manual, Hibbitt, Karlsson & Sorensen Inc., Pawtucket, Rhode Island. Abou-Chedid, G. and Brown, S. B., (1992], ''On the mechanical behavior of metal powder compaction'', in Compaction and Other Consolidation Processes, Advances in Powder Metallurgy & Particulate Materials-1992, Vol.2, pp.1-9. Abouaf, M., Chenot, J. L Rai sson G. and Bauduin, P., (1988], ''Finite element simulation of hot isostatic pressing of metal powders'', Int. J. Numer. Meth. Engng., Vol.25, pp.191-212. Aubertin, M., (editor), [1996], ''The Proceedings of 4th Conference on The Mechanical Behavior of Salt'', Montreal, Canada, June 17-18, 1996. Aubertin, M. and Gill, D. E. and Ladanyi, B., [1993], ''Modeling the transient inelastic flow of rocksalt'', the Seventh Symposium on Salt, Elsevier Science Publications, Amsterdam, Vol. I, pp. 93-104. Aubertin, M., Sgaoula, J ., Servant, S., Gill, D. E., Julien, M. and Ladanyi, B., [ 1996], ''A recent version of a constitutive model for rock salt'', In the ''Proceedings of 4th Conference on The Mechanical Behavior of Salt'', Montreal, Canada, June 1718, 1996, Preprint, pp.129-134. Auricchio, F. and Taylor, R. L., [1994], ''A generalized visco-plasticity model and its algorithmic implementation'', Comp. Struct., Vol. 53, pp.637-647. Bathe, K. J., [1982], ''Finite Element Procedures in Engineering Analysis'', Prentice-Hall, Englewood Cliffs, New Jersey. Bazant, Z. P., [1978], ''Endochronic inelasticity and incremental plasticity'', Int. J. Solids Structures, Vol.14, pp 691-714. Bazant, Z. P. and Bhat, P., [1977], ''Prediction of hysteresis of reinforced concrete beams'', J. Struct. Eng. Div., ASCE, Vol.31, pp.153-167. 270

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280 Truesdell, C., [1966], ''Continu um Mechanics I, The Mechanical Foundations of Elasticity and Fluid Dynamics'', Gordon and Breach Science Publishers, Inc., New York. Tszeng, T. C. and Wu, W. T., [1996], ''A study of the coefficients in yield functions modeling metal powder deforrnation'', Acta Materials. Vol.44, pp.3543-3552. Valanis, K. C., [1971a], ''A theory of viscoplasticity without a yield surface, Part I. General theory'', Arch. Mech ., Vol 23, pp.517-533. Valanis, K. C., [1971b], ''A theory of viscoplasticity without a yield surface, Part II. Application to mechanical behavior of metals'', Arch. Mech., Vol. 23, pp.535-551 Valanis, K. C., [ 1975], ''On the foundations of the endocbronic theory of viscoplasticity'', Archives of Mechanics, Vol.27, pp 857-868. Wu, H. C. and Ho, C. C., [1995], ''An investigation of transient creep by means of endocbronic viscoplasticty and experiment'', J. Eng. Materials and Technology, Vol.117, pp.260-268. Xu, J. and McMeeking, R. M., [ 1995], ''Finite element simulation of powder consolidation in the formation of fiber reinforced composite materials'', Int. J Mech. Sci., Vol.37, pp.883-897. Yeh, W.-C., Cheng, J.-Y. and Her, R.-S., [1994], ''Analysis of plastic behavior to cyclically uniaxial tests using an endochronic approach'', J. Eng. Materials and Technology, Vol .116, pp.62-67. Yin, Youquan, [ 1987], ''Introduction of Non-linear Finite Element Method in Solid Mechanics'', Peking University Publication (in Chinese), Beijing, China. Zienkiewicz, 0. C. and Cor111eau, I. C., (1974], ''Viscoplasticityplasticity and creep in elastic solid A unified numerical solution approach'', Int. J. Numer. Meth. in Eng. Vol. 8, pp. 821-845. Zienkiewicz, 0 C. and Taylor, R. L. (1989], ''The Finite Element Method'', Fourth Edition, Vol.I, 2, McGraw-Hill Book Company, Maidenhead, England.

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BIOGRAPlllCAL SKETCH Jishan Jin was born on January 23, 1962, in Shandong Province, People's Republic of China He received his B. S. degree in engineering mechanics from Zhejiang University in China in July 1982. Afterwards he worked as an instructor in Qingdao Ocean University for two years. In July 1987, he received a Master of Science in mechanics from Peking University. Then he joined in the fustitute of Geophysics, Chinese Academy of Sciences, where he was hired as a research associate fellow. In August 1992, he was admitted to the graduate program at the University of Florida. He received his Ph.D from the University of Florida in December 1996. 281

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I certify that I have read this study and that in my opinion it confo1n1s to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Nicolae D Cristescu, Chairtnan Graduate Research Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it confor1ns to acceptable standards of scholarly presentation and is fully adequate, in scope and quality as a dissertation for the degree of Doctor of Philosophy Frank C Townsend Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standar d s of sc hol ar ly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philo so phy. Renwei Mei Associate Profe sso r of Aerospace Engineering, Mechanics and Engineering Science I certify that I have r ead this study and that in my opinion it confor1ns to acceptable sta ndard s of scho larly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Bhavani V Sankar Profe ssor of Aerospace Engineering Mechani cs and Engineering Science

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I certify that I have read this study and that in my opinion it confo1111s to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Edward K. Walsh Professor of Aerospace Engineering, Mechanics and Engineering Science This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1996 > L/}_ Winfred M. Phillips D e an, College of Engineering Karen A. Holbrook Dean, Graduate School

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