• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 Key to symbols
 Abstract
 Introduction
 Assessment for plasticity theories...
 Soil mechanics theories relevant...
 Hypothesis and research plan
 Confined plasticity
 Unconfined plasticity
 Summary and conclusion
 Future work
 Appendix
 Reference
 Biographical sketch
 Copyright














Title: Plasticity of ceramic particulate systems
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Title: Plasticity of ceramic particulate systems
Series Title: Plasticity of ceramic particulate systems
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Language: English
Creator: Janney, Mark Alan
Publisher: Mark Alan Janney
Publication Date: 1982
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Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    Key to symbols
        Page vi
        Page vii
        Page viii
    Abstract
        Page ix
        Page x
        Page xi
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
    Assessment for plasticity theories for ceramics
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
    Soil mechanics theories relevant to plasticity
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
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        Page 27
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        Page 39
        Page 40
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        Page 43
        Page 44
        Page 45
    Hypothesis and research plan
        Page 46
        Page 47
        Page 48
    Confined plasticity
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
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        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
    Unconfined plasticity
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
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        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
    Summary and conclusion
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
    Future work
        Page 142
        Page 143
        Page 144
    Appendix
        Page 145
        Page 146
        Page 147
    Reference
        Page 148
        Page 149
        Page 150
        Page 151
        Page 152
        Page 153
    Biographical sketch
        Page 154
        Page 155
        Page 156
    Copyright
        Copyright
Full Text










PLASTICITY OF CERAMIC PARTICULATE SYSTEMS


BY

MARK ALAN JANNEY


















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


1982

































To My Family
















ACKNOWLEDGEMENTS


The author would like to thank the members of his

supervisory committee which included Dr. G. Y. Onoda, Jr.,

chairman, Dr. R. T. DeHoff, Dr. L. E. Malvern, and

Dr. E. D. Whitney. Special thanks to his advisor,

Dr. G. Y. Onoda, Jr., for first suggesting the application

of soil mechanics to ceramic processing problems and many

helpful discussions, and to Dr. F. C. Townsend for the

use of his soil test equipment. Finally, the author

wishes to acknowledge the National Science Foundation

grant #CPE-7818077A02 for financial support for this

work.


iii

















TABLE OF CONTENTS

PAGE
ACKNOWLEDGEMENTS .................................. iii

KEY TO SYMBOLS .......... ........................ vi

ABSTRACT ............ ....... ... .... ... ............ ix

CHAPTER

I INTRODUCTION ............................ 1

Background ........... ................. 1
A New Approach ....................... 3
Purpose .......... ..... .... ........... 3
Scope .................... ........... 4

II ASSESSMENT OF PLASTICITY THEORIES
FOR CERAMICS ........................ 6

Introduction ......................... 6
Mechanical Behavior of Wet Clay ...... 6
Theory of Clay Plasticity ........... 10
Weaknesses of the Stretched
Membrane Theory .................... 11
Need for a New Approach to Ceramic
Plasticity ......................... 14
Summary .............................. 15

III SOIL MECHANICS THEORIES RELEVANT
TO PLASTICITY ...................... 17

Introduction .... ................... 17
Description of a Particulate System .. 18
Stress and Strain .................... 19
Principle of Effective Stress ........ 28
Yield Criteria ..... ................ 32

IV HYPOTHESIS AND RESEARCH PLAN .......... 46

V CONFINED PLASTICITY .................. 49

Introduction ....................... 49
Materials and Methods ............... 50











Results and Discussion ............... 60
Application to Ceramic Extrusion ..... 88
Summary .... ........ ................. 99

VI UNCONFINED PLASTICITY .................. 101

Introduction ........................ 101
Materials and Methods ................ 102
Results and Discussion .............. 109
Summary .............................. 133

VII SUMMARY AND CONCLUSION ................ 135

Summary of Experimental Results ...... 136
Application of Soil Plasticity Theory 138
Conclusions .......................... 139

VIII FUTURE WORK ............................ 142

Introduction ........................ 142
Role of Viscous Binders .............. 142
Role of Flocculation ................. 143
New Extrusion Theory ................. 144

APPENDIX ......................................... 145

REFERENCES ........... ..................... ......... 148

BIOGRAPHICAL SKETCH ............................ 154
















KEY TO SYMBOLS


A constant, area

C constant

c cohesion intercept

F* mean adhesion force

FH dimensionless adhesion number

h slope of Mohr-Coulomb line, depth of penetration

K constant

K coefficient of lateral stress

k coefficient of permeability

ko pore shape factor

Lf load at fialure in compression test

1 length

M slope of critical state line, torque

P pressure

Pa applied pressure

Pc consolidation pressure

p mean isotropic stress

q deviator stress

r pore radius

SO surface area/unit volume











T tensile stress

u pore pressure

V volume

v specific volume

v reference specific volume for virgin consolidation
line

vS volume fraction solids

v reference specific volume for swelling line

W weight

w water content

x particle size, ratio of binder weight to solid
particle weight


a extrusion constant, angle of internal friction

y surface tension

e porosity

ES deviatoric strain

eV volumetric strain
P P
des/deV slope of plastic strain-increment vector

0 contact angle

K slope of swelling line

A (1 K/X)

A slope of virgin consolidation line

p density

a normal stress

aeq equilibrium stress (unconfined strength)


vii











oH horizontal stress

OV vertical stress

T shear stress

S- angle of internal friction


viii
















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


PLASTICITY OF CERAMIC PARTICULATE SYSTEMS



By



Mark Alan Janney



May 1982



Chairman: George Y. Onoda, Jr.

Major Department: Materials Science and Engineering

Ceramists have tried for many years to make technical

ceramics such as alumina with plastic properties like

those of wet clay. Some success has been achieved; but,

little understanding of the mechanisms of plasticity in

technical ceramics exists.

The purpose of this dissertation was two-fold. The

first was to study the mechanical behavior of alumina

powder compacts saturated with liquid and compare this

behavior with that for wet clay. The second was to

determine if the soil mechanics approach to particulate









plasticity could be successfully applied to ceramic

particulate systems.

Standard soil mechanics test procedures were

employed to determine pressure-volume relationships,

response tocombined states of isotropic and shear stress,

and unconfined shear strength. In addition, some ex-

periments were conducted in a piston extruder to study

the role of pore fluid pressure in extrusion.

The experimental study showed that the mechanical

behavior of wet alumina was quite similar to that for

wet clay. In consolidation, specific volume was propor-

tional to In (applied stress). The yield envelopes were

generally of the Mohr-Coulomb with cap type. The un-

confined strength, Tf, was influenced by both forming

pressure, PC, and specific volue, v: Tf = 0.125 PC;

and InTfyv. Also, the unconfined strength of remolded

specimens was one-third to one-tenth that of as-formed

specimens.

Critical state soil mechanics was used to analyze

and interpret the data. The trends in unconfined

strength are those predicted by critical state and

quantitative agreement was observed between consolidation

behavior and variation of unconfined strength with

specific volume. Critical state theory was used to

formulate a new phenomenological theory of extrusion.










The theory predicts an extrusion equation which is

identical to established empiricism in ceramics.

Two broad conclusions can be drawn from this study.

First, the mechanical behavior of wet alumina is similar

to that for wet clay. The differences that do exist are

matters of degree, not kind. The nature of plasticity

is the same in both systems. Second, critical state soil

mechanics has been successfully applied to the variation

of unconfined shear strength with forming pressure and

specific volume, and to ceramic extrusion. The success

of the soil mechanics approach for these simple systems

suggests that it may prove useful for more complicated

ceramic systems containing dispersants, binders,

lubricants, etc.















CHAPTER I

INTRODUCTION



Background

Plasticity is a form of behavior whereby a material

deforms permanently without rupture after a stress ex-

ceeding the yield stress is applied, and retains its

shape when the stress is removed.

Plastic behavior is essential to the wet processing

of ceramic articles. Plastic forming operations abound

in the ceramic industry. These include the ancient arts

of hand forming, wheel throwing, and turning, as well as

the more modern techniques of jiggering, extrusion, and

filter pressing. Plasticity also plays an important role

in the strength of green bodies. The body must be strong

enough to be formed and handled without slumping or being

damaged. Also, the strength varies with the processing

conditions such as the forming pressure and the solids

content.

The ideal plastic body in ceramics is wet clay.

This material is easily molded and extruded into shape

and the resulting green body is strong enough to be

handled safely both wet and dried. Electrical insula-

tors, dinnerware, bricks and tile are examples of clay

1










articles which are formed using plastic forming

techniques.

In contrast to clay is a class of ceramic materials

know as "non-plastics" (1). This classification origi-

nally was used to designate the traditional ceramics raw

materials such as flint (silica), feldspar, nepheline

syanite and Cornish stone. They were added to alter the

fired properties of the clay body (fired strength, vitri-

fication, and maturing range); but, they also tended to

reduce the plasticity of the body and decrease the

drying time and shrinkage. In the last 40 years, "non-

plastic" has come to include the technical ceramics such

as oxides, carbides and nitrides.

Most technical ceramics exhibit little plasticity

when mixed with water. They flow poorly under compres-

sive loading which leads to cracking, bleed water out of

the batch material during extrusion (2), and possess low

wet and dry strengths. Fortunately, the plasticity of

these materials can be improved by the addition of high

viscosity binders and other additives such as bentonite

and internal lubricants.

The process of improving the plasticity of non-clay

ceramics has been a matter of trial-and-error since the

first efforts to extrude zirconia light bulb filaments in

the 1890's (3). Little is understood about the mechanism

of plasticity in non-clay bodies and only a few studies










have been published (e.g., 2, 4, 5). Thus, no formalism

exists to guide the ceramic engineer in designing a

plastic non-clay material.



A New Approach

Most previous plasticity studies on non-clay

materials have used extrusion as the plastic test methods

(5). The emphasis has been on the flow behavior of the

material beyond the yield point. Because of this, the

models used have been basically theological ones (5,6).

An alternative approach to the study of ceramic plas-

ticity is to apply soil mechanics which is a branch of

particulate mechanics. Whereas rheology (a branch of

fluid mechanics) emphasizes the mechanisms which control

flow, particulate mechanics (a branch of solid mechanics)

emphasizes the mechanisms which control elastic strain

and plastic deformation prior to flow. Soil mechanics

then is concerned with the effects of combined states of

isotropic and shear stress on behavior, the strains which

result from these combined states of stress, and the

resistance to deformation, or fracture, which is called

strength.



Purpose

The purpose of this dissertation is two-fold. The

first purpose is to study the mechanical behavior of wet










alumina. This includes an evaluation of the usefulness

of soil mechanics test procedures to the alumina system.

The second purpose is to determine whether the soil

mechanics approach to soil plasticity is applicable to

ceramic systems. This approach emphasizes the importance

of combined states of isotropic and shear stress on the

behavior of a particulate system. It is also concerned

with the relationships between stress and strain, the

constitutive equations, which govern the mechanical

response of a soil. This will be based on the results of

the experimental study, and comparison of predictions of

theory with behavior.



Scope

Chapter II is an assessment of the prevailing

theories of plasticity of clay ceramics. It is demon-

strated that while these theories explain some of the

observed phenomena, there are other theories based on

soil mechanics which explain the behavior more complete-

ly. In addition, it is shown that soil mechanics ex-

plains some phenomena which are not even considered by

the ceramists' theories.

Chapter III presents the soil mechanics approach to

soil plasticity as it may apply to ceramic systems.

First, the descriptors of a particulate system are

presented. These include phase relationships and the











stress and strain parameters which are used. Second, the

principle of effective stress, which is one of the

foundations of soil mechanics, is presented. Third, the

concept of a yield criterion and the resulting yield

function are discussed.

Chapter IV is a statement of the general hypotheses

that the experimental work was based on.

Chapter V is an investigation of the plastic be-

havior of wet alumina under confined conditions. Con-

solidation of various systems and their response to

combined states of isotropic and shear stresses is

considered. In addition, a new, phenomenological theory

of extrusion based on critical state soil mechanics (7,

8) is presented.

Chapter VI is a study of the unconfined strength of

wet alumina and how it varies with solids loading and

forming conditions. The observed behavior is compared to

that which is predicted by soil mechanics theories. In

addition, the effect of remolding on the strength and

structure of wet alumina is reported.

Chapters VII and VIII are a summary of the disserta-

tion, general conclusions regarding the work, and ident-

ification of areas of need in research.
















CHAPTER II

ASSESSMENT OF PLASTICITY THEORIES FOR CERAMICS



Introduction

Plasticity, as it relates to ceramics, is a property

associated almost exclusively with clay. This carries as

far as the definition of "non-plastic" raw materials

which are exclusively non-clay. Even when dealing with

the plasticity of non-clay materials, such as alumina,

the effort is for more "clay-like" properties (5).

The purpose of this chapter is three-fold. First,

it reviews the properties of clay related to plasticity.

Second, it assesses the current theories of clay plas-

ticity in ceramics and examines weaknesses in these

theories. Third, it presents a new approach to ceramic

plasticity.



Mechanical Behavior of Wet Clay

To the practicing ceramic engineer, plasticity is a

matter of feel, workability, extrudability, and so on.

While these terms may be valuable in plant work, they are

of little value to scientific study.










Numerous attempts to quantify the plastic behavior

of saturated clay (i.e., clay bodies in which the pore

space is filled with water) have been made and are re-

viewed by Bloor (9). Russell and Hanks (10) showed that

clay and clay based bodies behaved in an elastic-plastic

manner when loaded in unconfined compression (Figure

II-la). The stress at which the stress-strain curves

plateaued they called the equilibrium stress, 'eq; it is

also known as the compressive strength of the body. They

also showed (Figure II-lb) that the compressive strength

of a saturated clay was related to the water content

log Oeq w (II-1)

where aeq= equilibrium stress (strength)
w = water content (dry basis)

Speil (11) and Whittaker (12) have shown that the yield

point (as measured with Norton's torsion test machine

(13)) of a clay mass is inversely proportional to the

particle diameter. The work of Macey (14) has shown that

the consolidation of a large number of saturated clays is

governed by a semi-log relationship between water content,

w, and applied pressure, Pa (Figure II-lc):

w C In Pa (11-2)

Finally, Kingery and Francl (using Norton's shear tester)

(15) showed that workability (product of maximum stress

and maximum extension prior to cracking) was proportional

to the surface tension of the pore water (Figure II-ld).



























0 a02 o0 04 5 06as
E-stress (in/Ni)


SI I I I

04 --


38 40 42 44 46 48 SO 52 54 S
M(% of water) dry basis
b


x

x X

A


A *-* s


.I .x


OL-------i-
32 34 36



Figure II-l.


c
Typical mechanical behavior for clay ceramics. (a) Stress-strain curves
in unconfined compression (10); water contents: 1-0.243; 2-0.268;
3-0.298; 4-0.349. (b) Log shear strength vs. water content (10):
1-Ga. kaolin; 2-china clay; 3-Ky. ball clay; 4-Fla. ball clay; 5-dark
special ball clay. (c) Water content vs. In pressure for various
clays (14).


LOG,P. (C"/,)


sS w

















































0oliI 1 11 I
0 20 40 60 80 100
SURFACE TENSION dynee cm )
d












Figure II-1 (cont). (d) Workability vs. surface tension
for kaolin (15).


I I i i










Theory of Clay Plasticity

The most widely accepted theory of ceramic clay

plasticity is the so-called "stretched membrane theory."

It was first suggested by Macey (14) in 1940 and was

introduced in this country by Norton (16) in 1948.

The stretched membrane theory may be summarized

briefly as follows. First, it is assumed that the clay

particles are charged and repel one another. This causes

the particles to move as far away from each other as the

limited amount of water present will allow. Second, this

repulsive force is balanced by forces of surface tension

between particles at the surface of the clay which tends

to put a compressive stress on the body. It should be

noted that the stretched membrane theory does not

consider the action of any other attractive forces in the

system; especially singled out as unimportant by Norton

(17) are van der Waals forces. Third, there are water

films between the clay particles which are essentially

rigid (18) and whose rigidity is affected by such factors

as the type and concentration of ions in solution.

Norton (16, 17) makes several arguments to support

the stretched membrane theory. His first argument

concerns the effect of particle size on capillary forces.

As the particle size decreases, the size of the pore

channels should also decrease and the resulting capillary

pressure should increase. And, "the masses of finer










particles would have the higher yield points because

greater pressure would be exerted on the particles" (16,

p. 238). He cites the work of Speil (11) and Whittaker

(12) which showed that the yield point (as measured in

torsion) was inversely proportional to the particle size

as confirmation of this point. His second argument con-

cerns the water films between clay particles. It is an

established experimental observation that saturated clays

cease shrinking at a well defined water content, the

shrinkage limit (17). He claims that the only explana-

tion for the shrinkage limit is that "the first water lost

comes from the layers between the particles and not from

the pores" (16, p. 238). Thus, the particles are not in

contact with one another until the shrinkage limit is

reached. His third argument concerns the role of surface

tension. Kingery and Francl (15) showed that lowering the

surface tension of water reduced the workability of the

clay proportionally. Since the capillary force exerted

on the clay is proportional to the surface tension,

Norton claims that this decrease in workability supports

the stretched membrane theory (17).


Weaknesses of the Stretched Membrane Theory

While the stretched membrane theory is in reasonable

agreement with some of the observed behavior of clay,

there are many weak points in the theory. In addition,










there are several examples of clay behavior which are not

explained by this theory.

Water Films

In support of the stretched membrane theory, Norton

cites the following example.

When a child's balloon is filled with dry,
powdered clay, it feels just like dry, powdered
clay. But if the air is slowly evacuated from the
balloon, the atmosphere compresses the clay and it
now feels as if the balloon were filled with wet,
plastic clay with all it workability. (16, p.239)

In this example, there is no need for water films between

particles for plastic behavior to obtain. It may be

inferred from this experiment that water films are not

always necessary to impart plasticity to a body.

Particle Repulsion

In the above example, the particles are not charged

and are obviously touching one another; yet, the material

is plastic. Plasticity was obtained for both clay and

finely ground flint (16). Since the particles are

touching one another and plasticity is still obtained,

overcoming the frictional contacts between particles must

somehow be involved. It may be that it is the isotropic

stress provided by the balloon that is truly important.

Surface Tension

Norton (16, 17), and Kingery and Francl (15) have

claimed that surface tension is the only active attrac-

tive force which needs to be considered in clay plas-

ticity and clay drying. In contrast, Sridharan and










Venkatappa Rao (19, 20) have shown that drying shrinkage

is strongly influenced by the dielectric constant of the

pore fluid and only weakly influenced by surface tension.

The stretched membrane theory does not even consider the

role of dielectric constant and the van der Waals forces

which it controls.

Norton (16) also claims that meniscii at the surface

of the clay and the resulting surface tension force are

necessary for plasticity. This is refuted by work in

soil mechanics, such as that by Parry (21). In triaxial

tests on Weald clay plastic behavior was observed even

though all meniscii were eliminated through back pressure

saturation (see Chapter V.)

Particle Size

The inverse relationship between yield point and

particle size for clay is consistent with the capillarity

requirements of the stretched membrane theory. However,

if van der Waals forces are important, one would expect

the same variation in yield point with particle size. In

addition, if the van der Waals view is correct, the yield

point should increase as the dielectric constant of the

pore fluid decreases. Sridharan and Venkatappa Rao (19)

have shown this to be qualitatively true for kaolin with

various organic solvents and water as pore fluids.










Need for a New Approach to Ceramic Plasticity

The foregoing discussion indicates that there is a

need in ceramics for a new way of thinking about plas-

ticity. Ceramists have focused on micromechanisms such

as particle repulsion and water films. A different

approach has been taken in soil mechanics (22). Over the

past 50 years, experimental continuum mechanics has

defined the behavior of clay in broad terms including the

effects of combined states of isotropic and shear stress

and the role of pore fluid pressure in saturated systems.

Empirical laws have been derived based on this broad data

base. In the last 25 years, phenomenological theories

based on the application of plasticity theory to soil

mechanics have been derived (e.g., 7, 23, 24, 25). These

theories have had some success in describing the

mechanical behavior of soil.

Application of Soil Mechanics to Powder Processing

Some soil mechanics test procedures and theories have

been applied to ceramic and metal powder systems. Koerner

(26) has compared triaxial pressing (a soil mechanics

test method which employs both isotropic and shear stresses)

with isopressing for consolidation of iron powders. He

showed that triaxial pressing substantially reduced both

radial and axial density gradients, improved green

density, and doubled the green strength. Strijbos et al.

(27) have shown that experimental stress and strain










distributions in die pressed Fe203 agree fairly well with

calculations based on soil mechanics. Thompson (28) has

derived analytically the conditions that cause endcapping

(elastic cracks) in die pressed bodies. All of these

analyses have been for dry powders or granules.

While some work has been done in dry ceramic

systems, no work has been reported in the ceramics

literature on the application of soil mechanics to liquid

saturated systems. A large part of soil mechanics deals

with soil saturated with water. Several phenomenological

theories have been developed to describe the behavior of

wet clay. The most notable of these is the critical

state theory developed by Roscoe and coworkers at

Cambridge University (7, 8, 23). Critical state theory

has had some success in predicting stress-strain behavior,

yield functions, and variation of strength with solids

content and forming pressure of wet clay. In the next

chapter, the theory of soil mechanics will be presented

as it may apply to saturated ceramic systems.



Summary
The mechanical behavior of wet clay was reviewed and

the ceramists' theory of clay plasticity was presented.

Certain weaknesses in the "stretched membrane" theory of

clay plasticity were pointed out. And, it was proposed

that the soil mechanics approach to the behavior of










particulate systems be applied to the problems of plas-

ticity of non-clay ceramics.

The proponents of the stretched membrane theory

hypothesized that an isotropic stress from capillarity

was necessary to impart plasticity to clay. However,

they did not consider other mechanisms by which an iso-

tropic stress could be imposed on a body. Alternate

sources of isotropic stress include externally applied

stress and van der Waals bonding. It will be shown in

the next chapter that the application of isotropic stress

is necessary to impart plasticity (as understood by the

ceramist) to a particulate body.















CHAPTER III

SOIL MECHANICS THEORIES RELEVANT TO PLASTICITY



Introduction

Soil mechanics is concerned with the stress and

strain behavior of particulate systems. It is a continuum

approach whose objects are 1) to define the locus of

stress states that cause elastic strain, permanent yield-

ing, consolidation, and shear failure; and 2) to develop

constitutive relationships between stress and strain (or

stress-increment and strain-increment). The emphasis has

traditionally been on phenomenological behavior without

considering the micromechanisms of deformation; for

example: what is the effect of normal stress on the

shear stress at failure for sand (22); what is an appro-

priate yield function for wet clay (23)? More recently,

studies of microscopic mechanisms (e.g., 19, 20) and

individual particle-particle interactions (29) have begun

to appear in the literature.

The purpose of this chapter is to introduce the

reader to the soil mechanics approach to soil plasticity

as it may relate to ceramic systems. The material covered

includes 1) definition of descriptors of a particulate

system; 2) definition of appropriate stress and strain










parameters; 3) the principle of effective stress; and 4)

yield criteria.



Description of a Particulate System

A particulate system is an assemblage of interacting

particles together with other materials occupying the

void spaces between the particles. The materials dealt

with in this work are saturated; i.e., the void space is

completely filled with water or other liquid. To describe

the system, then, one is interested in the volume and

weight of the total system and of the solid and liquid

phases.

Two phase relationship descriptors have been used

extensively in this work. The water content, w, is

defined as (22):

w = WL = Wwet Wdry (III-1)
WS Wdry

where WL = weight of liquid

WS = weight of solid

It is an experimental value obtained from weight loss on

drying. The specific volume, v, is defined as (22):

v = VT (III-2)
VS
VT = total volume of system

VS = volume of solid

Note that the specific volume is the inverse of the

volume fraction of solids, vS:









V 1
v = S = (III-3)
S v, v
For a saturated system, v and w are related by

v = 1 + w (PS/PL) (111-4)

where PS = solid density

PL = liquid density

The Atterberg limits (22) are used in soil mechanics

to describe the variation of strength with water content.

The ceramist may think of the Atterberg limits as de-

fining the working range of a clay. Using well estab-

lished test procedures (22) one defines the water contents

of transition between the liquid, plastic, semisolid, and

solid states. These are given by the liquid limit, wL,

the plastic limit, wp, and the shrinkage limit, WS.

Although the Atterberg limits are arbitrarily set, they

are quite useful and have been related to many properties

of soils on an empirical basis (22).



Stress and Strain*

To describe the stress-strain behavior of a system,

one must define a set of parameters which are appropri-

ate. A soil's behavior is determined by both the

isotropic and shear stresses that are imposed upon it.

In turn, a soil may show both volumetric and shear strain

increments when a stress increment is applied. Thus,



* In accordance with standard soil mechanics practice
(22), compressive stresses and strains are taken to be
positive.










isotropic and shear stress, and volumetric and shear

strain, parameters are required.

Stress Parameters

The mean isotropic stress, p, may be defined (7) in

terms the principle stresses, CI, 02, and a3 as
1
p -= (o1 + 02 + a3) (III-5)
In many soils tests, cylindrical specimens are used; for

these conditions of cylindrical symmetry a2 = a3 and
1
p = : (C1 + 203) (III-6)

The shear stress parameter most widely used in soil

mechanics is the deviator stress, q (7):

(III-7)
1 2 2 2 }
q = {(cI 02) + (C2 G3) + (O3 l)
/2

For the case of cylindrical symmetry, q becomes

q = a1 3 (III-8)

This is twice the maximum shear stress,

T max. Both q and T are used in this study.
max max
Strain Parameters

The strain parameters are derived from the principal

strains in a manner analogous to the stress parameters.

The volumetric strain (7) is given.by

(III-9)
ev = el + 2 + 63
which for cylindrical symmetry gives
v = + 2E3 (III-10)

The volumetric strain increment is related to the

specific volume by

dEv = dv (III-11)
v












The deviatoric (shear distortional) strain (30) is given

by
=2 {(- 2 + 232 + -)2 (III-12)
s 3 2(E ( 2-e3) + (63-s1) }


which gives for cylindrical symmetry

2 (III-13)
es = (i 3)

Stress Paths and Soil Tests

The mechanical behavior of a soil depends on both

the applied stress state (whether it is an isotropic

stress, a shear stress, or a combination of both) and

the stress history of the soil. To describe these condi-

tions, a p-q diagram is used, Figure III-1. On a p-q

diagram, the stress paths employed in a number of soils

tests such as isotropic compression, confined compression

(analogous to die pressing), unconfined compression, and

triaxial compression can be described. The stress

history of the soil can also be described.

Isotropic consolidation. The isotropic consolida-

tion test is the simplest soils test. The stress path is

described by a line along the p axis in Figure III-l.

The strain is purely volumetric.

An idealized isotropic consolidation curve (31) is

shown in Figure III-2. Line ABE is the virgin consoli-

dation line which describes permanent volume changes and


























(1)



4-





Isotropic
Stress













Figure III-1. Some common stress paths encountered in
testing soil.
/l
>
<"/O
a
/
/r





















v0




v

v K
Figure -2.























Figure 111-2.


I \


B,D


G


0 In p


Idealized consolidation and swelling curves
for a particulate system: ABE virgin
consolidation line; BC swelling and
recompression line.










line BCD is a swelling line which describes elastic

(recoverable) expansion and recompression. The virgin

consolidation line may be described analytically by

v = v Xln p (III-14)

where vo = specific volume at a reference pressure,

say p = 1 psi or 1 kg/cm2

X = slope of consolidation curve

The swelling curve has a similar form

v = vK Klnp (III-15)

where K = slope of swelling curve

v = current reference specific volume for the

swelling line in question (Figure III-2)

Upon application of pressure, the specimen consolidates

along ABE. If at B the pressure is lowered, the specimen

swells along BC. Reapplication of the load elastically

compresses the specimen up to D, at which point consoli-

dation continues along the virgin consolidation line

toward E. Specimens whose p-v states lie on the virgin

consolidation line are called normally consolidated.

Specimens whose p-v states lie on a swelling line to the

left of the virgin consolidation line are called

overconsolidated. The p-v states to the right of the

virgin consolidation line are inaccessible to the soil.

Confined compression. Confined compression consists

of axially consolidating a soil specimen in a cylindrical

die of fixed diameter, Figure III-3. In this case, the





















Dial gauge

/


Confining ring P


SLoad


Porous
stones


Figure III-3.


The confined compression test is used to
study the consolidation behavior of
particulate systems.


Specimen


---










horizontal reaction stress, aH, is a constant fraction of

the applied vertical stress, a (22) -

aH = Kg (III-16)
H o v
where K = coefficient of lateral stress
o
The stresses for this condition are given by

p 1 (Ov + 2GH) = av (1+ 2 Ko) (III-17)
p) 3


q = v H = v (1-K ) (III-18)



and the stress path is

q/p= 3(1 Ko) (III-19)
1 + 2Ko

For a typical value of K 0 0.5, q/p = 0.75, Figure III-1.

The v-lnp curve for confined compression is a special

case of a v-lnp curve for constant q/p; the curve is

parallel to the virgin consolidation line and displaced

to the left, Figure III-4. Analytically, the permanent

consolidation curve for confined compression is given by

v = v Aino (III-20)
c v
The swelling lines are given by

v = vK Klna (III-21)

Triaxial compression. The triaxial compression test

is the most versatile standard soils test. In the test,

a cylindrical specimen can be subjected to either an

isotropic stress, an axial stress or both. Hence, a wide

range of stress paths can be used to test a specimen.

































2.6 F


o
, 2.4

a.

2.2


10














Figure 11-4.


20 50 100

P[O+20:3 (psi)











Consolidation at constant q/p does not
change the slope of the consolidation
curve. Redrawn from Burland (30).


0 q/p = 0.00
A q/p= 0.27
o q/p = 0. 38


I I I I I I I I I










The stress path for isotropic stress lies along the p

axis. For an axial stress only (unconfined compression

test), p and q are given by

p = axial/3 (1-22)
q = axial
axial
and the stress path is (Figure III-1)

q/p = 3 (III-23)

Very often the triaxial test is run by first applying an

isotropic stress, Po, then applying an additional axial

stress, axial. The stress path is shown in Figure

III-l,

q 0 ,a axial = 0 (III-24)

q = 3(p-po), 0 axial>0



Principle of Effective Stress

When a stress state is applied to a saturated par-

ticulate system, the total stress may be divided between

the particulate skeleton and the pore fluid. Consider

the application of an isotropic stress to a saturated

body from which the water is not allowed to drain (the

system is closed). The volume of the system is essential-

ly constant. Because water is far less compressible than

the soil skeleton, little of the applied stress can be

carried by the particle skeleton and it manifests itself

as pore pressure. Only if drainage of the water is

allowed (the system is open) can the stress in the skele-

ton increase and cause consolidation to occur.










The above relationship may be expressed mathemat-

ically as (22):

T = a' + u (III-25

where 0T = total normal stress

o' = effective normal stress

u = pore fluid pressure

This definition of effective stress applies only to

normal stresses since a fluid cannot support a static

shear stress; hence,

T' = T (III-26


)


)


where T = shear stress

The principle of effective stress states that it is

the effective stress and not the total stress that

controls soil behavior. This has been demonstrated to be

true for compaction and shear failure in both sand and

clay (22).

As an example of the principle of effective stress,

consider a hypothetical, normally consolidated specimen

with a constant total stress, aT' on it and in which the

pore pressure, u, can be varied by an apparatus such as

illustrated in Figure III-5. At the start of the test,

let T = 20 psi and u = 10 psi. Then o' = 10 psi and

the specimen will have a specific volume corresponding to

10 psi on the virgin consolidation line. If the pore

pressure is raised to u = 15 psi, then since o is



















































Figure III-5. Changes in pore pressure at constant total
stress can lead to consolidation and
swelling.










constant at 20 psi, a' must fall to 5 psi. This will

cause the specimen to swell elastically along the K-line.

Conversely, if the pore pressure is lowered to zero

psi, a' increases to 20 psi and the specimen consolidates

along the virgin consolidation line to the new specific

volume corresponding to 20 psi.

For very fine particles such as clay and ceramic

powders, the effective stress concept must be modified to

take into account capillary, electrical double layer, and

van der Waals interparticle forces. The effective stress

equation then becomes (19)

C = O u + A R (III-27)

where

C = modified effective stress

a = externally applied pressure

u = pore water pressure

A = effective interparticle attractive pressure

such as due to van der Waals attraction

R = effective interparticle repulsive pressure

such as due to electrical repulsion

The interparticle forces alter the effective stress*

in a manner analogous to the pore water pressure. Un-

fortunately they cannot be directly measured the way that

the applied stress and pore pressure can. They can be

altered in a systematic manner, though, such as increas-

ing the charge on a particle with surfactants or changes










in pH, or altering the van der Waals forces between

particles by changing the dielectric constant of the pore

fluid. It will be shown in Chapter VI how van der Waals

or capillary bonding can lead to increases in strength

for a particulate system.



Yield Criteria

The concept of a yield criterion is familiar to most

engineers. Consider the uniaxial extension of a tensile

test bar (32). The stress-strain curve is shown in

idealized form in Figure III-6. The curve is character-

ized by an elastic region, an initial yield point (the

elastic limit), and a strain hardening plastic region.

If the specimen is not stressed above the initial yield

point, o all strains will be elastic and no permanent

change in length will be obtained. If however, the

specimen is stressed beyond a to say a1, then unloaded,

a permanent offset will be obtained. The new yield point

is given by a1, and all strains will be elastic for

stresses up to al. The material has hardened, and the

yield point has increased.

The idea of a yield criterion can be expanded to

include biaxial or triaxial stress states (33). The

yield criterion then becomes a curve or a surface, re-

spectively. For example, consider the yielding of a

metal bar in combined tension and torsion, Figure III-7.






















y
0 Initial
Axial Elastic
Stress Limit










Permanent Offset

Axial Strain











Figure III-6. Idealized stress-strain curve for
uniaxial extension of a linear strain
hardening metal.

























-Expanded Elastic
Region Due to
Mo .Strain Hardening




A
Original
Elastic
Region

To T1













Figure III-7. Yield surface for a strain hardening
metal in combined tension and torsion.










The initial yield stress for tension is T and for
o
torsion is M The initial yield stress for combined
0
states of tension and torsion is given by the curve

M -A-T All states of stress within M -A-T -0 are
o o O O
elastic. If the stress is increased beyond A to A', the

material plastically deforms and hardens. A new yield

criterion is then given by M1-A'-T1 and all states of

stress within O-M -A'-TI, are elastic.

Yield Criteria for Soil

The application of the theory of plasticity to soil

mechanics (7, 23, 24, 25, 30, 34-40) has led to the

development of a number of theories to describe the yield

envelopes of soils. The theories vary considerably. Some

focus on compaction and the behavior of normally consol-

idated soil; others are mainly concerned with the shear

failure of overconsolidated soil. However, while the

details of the various theories differ, the main assump-

tions, and the procedures for building the theories, are

similar. Some of the common assumptions and general

conclusions of these theories are outlined below. The

specifics of how the theories are derived may be obtained

from the original papers cited above.

There are five common assumptions. First, the

material is isotropic and stable as defined by Drucker

(41). Second, the soil behaves in an elastic-plastic










manner with respect to volumetric strains; i.e., the

soil behaves in the manner described in Figure III-2. If

after unloading to C, the soil followed the line CG

rather than CD, it would not be considered to have a

yield point with respect to volumetric strains; and it

would not be considered to be elastic-plastic. Third,

the soil possesses a yield function in principle stress

space or q-p space which is a function of the specific

volume of the soil. This function usually takes the form

of the Mohr-Coulomb criterion (22) for overconsolidated

soil, and some form of strain hardening (or softening)

cap (7, 23, 24) for normally consolidated soil, Figure

111-8. These functions will be dealt with in detail

below. Fourth, the normality rule (7, 23, 24) of

plasticity is valid. This states that there is a func-

tional relationship between the plastic strain increment

vector (de deP) and the yield function. Specifically

for the case of the yield function in q-p space (see

Figure III-8) --

d = dq (111-28)
deP dp
V

that is, the plastic strain increment vector is the

outward normal to the yield function (7, 23,24). Fifth,

there exists an ultimate state (7, 23, 24) that is ap-

proached at large shear such that deformation occurs with

no further change in volume or pore pressure.























dEP


Cap
v 1 v2 v3
o P
(a)








dEP





C B \


O A A' A"
p
(b)
Figure III-8. The classic yield surface for soil is the
Mohr-Coulomb with cap model. (a) Yield
surface, vl>v2>v3. (b) Stress paths for
overconsolidated and normally consolidated
specimens in drained loading.










General behavior. When a soil is stressed, it may

be observed to dilate (expand), compact, or remain at the

same volume. In general, a pure isotropic stress causes

compaction and a pure shear stress causes dilation. At

some combined state of shear and isotropic stress, no

volume change occurs. Because three distinct types of

response are involved, the yield criterion for a soil

exhibits three different regions, depending on the q/p

ratio, Figure III-8. At high q/p (mostly shear stress),

the Mohr-Coulomb condition governs behavior and the soil

tends to dilate (22). At low q/p (mostly isotropic

stress), the strain hardening cap governs behavior and

the soil tends to compact (23, 24). For q/p r 1, the

perfectly plastic or critical state condition (23, 24)

governs behavior and the volume of the soil remains

constant.

Yield condition for overconsolidated soil. The

shear stress required for simple slip of an overconsol-

idated soil is often (22) assumed to depend on the cohe-

sion of the soil and linearly on the normal stress on the

slip surface. This is commonly known as the Mohr-Coulomb

criterion (22) and may be described analytically as (see

Figure III-8)










qff = Pff tan a+ c (III-29)
where qff = deviator stress at failure on

the failure plane

pff = effective normal isotropic stress

at failure on the failure plane

a = angle of internal friction of

the soil

c = cohesion intercept

Drucker and Prager (42), in the first application of

plasticity theory to soil mechanics, showed that if the

Mohr-Coulomb criterion were assumed to be the yield

function, then "plastic deformation must be accompanied

by an increase in volume if a= o" (42, p. 158). This

can be seen in Figure III-8. The normality rule says

that the plastic strain increment vector (del, de ) is

the outward normal to the yield function. For the Mohr-

Coulomb line, the vector decomposes into a positive shear

component and a negative (i.e., dilative) volumetric

component. Improvements on the Mohr-Coulomb criterion

which allow for a non-linear failure surface and which

can predict stress-strain and pore pressure-strain curves

have been developed by several investigators (25, 37-40).

Yield condition for normally consolidated soil. The

yield condition for normally consolidated soil consists

of a strain-hardening cap (7, 8, 23, 24, 34, 35) on the

Mohr-Coulomb criterion, Figure III-8. The cap has been






40


given a specific name such as the Roscoe surface (8) or

the Rendulic surface (35) depending on the authors.

The specific shape of the strain hardening cap is

not well determined. Some shapes have been suggested for

mathematical convenience; among these are the semicircle

(24), the parabola (23), and the circular arc (25).

Others have been derived from first principles based on

the dissipation of work within a material; these include

the "bullet" (7) and the ellipse (34) (see Appendix I).

Regardless of the cap shape, the general behavior pre-

dicted by the yield envelope is the same.

Consider the drained triaxial testing of a specimen

of soil which has been isotropically consolidated to

point A then elastically rebounded to point 0, Figure

III-8. The behavior of this specimen is defined by the

yield envelope A-B-C and will exhibit only elastic strains

if stressed within this region. If stressed beyond

region A-B-C, the behavior of the specimen will depend on

the particular stress path used. If the stress path

intersects the Mohr-Coulomb line, along OF, the specimen

will exhibit shear failure without further expanding the

yield envelope. In fact, the yield envelope may contract

since the plastic strain-increment vector (the normal to

the Morh-Coulomb line) has a negative volumetric component

(i.e., the specimen dilates). This can lead to strain

softening and a peak in the stress-strain curve for an










overconsolidated specimen, Figure III-9. If the stress

path is along the isotropic stress axis, AA' in Figure

V-8, the specimen will simply consolidate according to

the virgin consolidation line. The plastic strain-

increment vector for this case is a pure volumetric

strain-increment. If the stress path is intermediate

between these two, say along OAD, the specimen tends to

both shear and consolidate. As it does, new yield

envelopes are generated. If the process is stopped at D,

then the current yield envelope is defined by A'DB'C'.

Note also that the plastic strain increment vector

rotates in going from A to D. At A, it was a purely

volumetric increment; at D, it is a combined shear and

volumetric increment; and if shearing is continued, at B'

it will be a pure shear increment. At B", no further

volume change occurs; the specimen is said to be in the

perfectly plastic or critical state condition. The

stress-strain and volume change-strain curves

corresponding to stress paths OF and ADB" are shown in

Figure III-9.

The foregoing discussion was for triaxial tests with

drained conditions. A somewhat different set of results

would be obtained if the tests were conducted under

undrained conditions. Consider a specimen consolidated

to point A, Figure III-10, then tested under undrained

conditions after consolidation (and elastic rebound for

an overconsolidated specimen). For these conditions,





























PI IbJ
II
b
I l
b


S OveIrconsolidated ,c 1
Scm= 120
0.8


0.6 i I
S- Normally consolidated
0.4 = 30b ,n'
/ 30

0.2

0
0 5 10 15 20 25
Axial strain (%)


5


0
5-
j.5" - -' -


5 10 15
Axial strain (%) .


Figure III-9. Axial stress-strain and volume change-
strain curves for normally consolidated
and overconsolidated Weald clay. From
Lambe and Whitman (22)


' 0


,I


20 25






43
















Overconsolidated Normally
Consolidated

TSP B TSP
DE

q II ESP ESP /

I


0 A

p










Figure III-10. Theoretical effective stress paths (ESP)
and total stress paths (TSP) for normally
consolidated and overconsolidated soil.









there will be no change in volume during shear; but,

there will be changes in pore pressure. Thus, there will

be differences between the total stress path and the

effective stress path during testing. For the over-

consolidated specimen, the total stress path is OD. Up

to the Mohr-Coulomb line, the effective stress path will

be slightly to the left of OD because a small positive

pore pressure is generated as the soil skeleton attempts

to elastically compress. Once the Mohr-Coulomb line is

crossed, however, the specimen tries to dilate as in the

drained test. Since the specimen cannot dilate, due to

the undrained loading condition, a negative pore pressure

is generated, Figure III-ll,and the effective stress path

lies to the right of OD as it climbs the Mohr-Coulomb

line, Figure III-10 (8). For the normally consolidated

specimen, the total stress path is AE. In this case,

there is positive pore pressure during the entire loading

sequence since the specimen tries to consolidate from the

beginning, Figure III-11. The effective stress path

always lies to the left of AE. The rate of increase of

the pore pressure decreases as the critical state is

approached and at B, the critical state, the pore

pressure remains constant.































c: 20


S 10

0


S^20 r--,- ^
SL- 10_
Z._
t 10 0

"00 -10_
0 5 10 15 20 0 5 10 15 20
Axial strain (%) Axial strain (%)
(a) (b)


Figure III-11. Undrained triaxial test results for
Weald clay. (a)Normally consolidated,
o = 30 psi. (b)Overconsolidated,
Po = 120 psi, Ptest = 5 psi. From
Lambe and Whitman (22).















CHAPTER IV

HYPOTHESIS AND RESEARCH PLAN



The preceding chapters have surveyed the literature

of plasticity of particulate systems. They have identi-

fied weaknesses in the traditional approaches used by

ceramists and suggested an alternative approach as

employed in soil mechanics. This chapter will outline

the working hypothesis and research plan for this inves-

tigation. Detailed hypotheses will be discussed in the

following two chapters.

It has been shown in Chapter III that the applica-

tion of plasticity theory to soil mechanics can be used

to predict the behavior of clay (7, 8, 23, 24). Quali-

tative agreement has been excellent; quantitative agree-

ment depends on the particular model chosen (7, 34, 35).

The working hypothesis for this dissertation is that

plasticity theory as applied to soil mechanics can be

used to describe the plastic behavior of non-clay ceramic

systems. In particular, it is hypothesized that the

principle of effective stress applies, that a yield

surface of the Mohr-Coulomb with strain hardening cap

type governs behavior, that consolidation behavior can be










described by a virgin consolidation line and swelling

lines, and that an ultimate, perfectly plastic state

exists at large shear deformations as required by theory.

Some caution is in order in applying this hypothesis.

The differences between clay and the so-called "non-

plastic" minerals have been well documented (1). Clay is

platey and quite fine grained; many non-plastics tend to

be equiaxed and somewhat coarser. Clay has a complex

chemistry with large amounts of isomorphous substitution;

non-plastics tend to have simpler chemistries. The

surface chemistry of clay is extremely complex with high

ion exchange capacity, adsorbed species (including water)

and complex charged surfaces; non-plastics tend to be

simpler. Also, it has been suggested by Wilson (43) that

particle morphology is important to the mechanical proper-

ties of wet bodies; the platey minerals he studied were

more plastic than the equiaxed ones.

The work plan for this investigation consisted of

two parts. The first was to determine the mechanical

behavior of the test material (alumina-water). This

included its response to consolidation, unconfined

compression (stress-strain, strength vs. solids content,

etc.), and triaxial compression (shear stress and pore

pressure vs. strain, total and effective stress paths).

The second was to determine if the working hypothesis was

correct based on the available data. If so, material






48



behavior such as strength and how its is affected by

processing, and stress paths during forming could be

predicted. If the hypothesis was not verified, alter-

native explanations would be developed.
















CHAPTER V

CONFINED PLASTICITY



Introduction

The soil plasticity approach to soil mechanics

outlined in Chapter III emphasizes the importance of

stress path and stress history. Such an approach can be

used to predict the mechanical behavior of a soil such as

stress-strain and pore pressure-strain curves in triaxial

tests, variation of shear failure stress with specific

volume and confining pressure, and effective stress

paths.

Several requirements must be satisfied before the

soil plasticity approach can be applied to a particulate

system. The particulate system must behave in an elastic-

plastic manner with respect to volumetric strains, as

outlined in Chapter III, Figure III-2. The yield envelope

for the material should be of the Mohr-Coulomb-with-strain-

hardening-cap type, Figure III-8. The effective stress

paths in undrained loading for normally consolidated and

overconsolidated specimens should map out the strain

hardening cap and Mohr-Coulomb surfaces respectively

Figure III-8. The deviator stress-strain and pore










pressure-strain curves should be similar to those in

Figure III-11

The purpose of this chapter is two-fold. First, it

will be determined whether the behavior of alumina-water

can be described in terms of the soil plasticity approach.

This will include determination of its consolidation and

triaxial compression behavior. Second, it will test the

applicability of the critical state theory of soil plas-

ticity (7, 8) to ceramic extrusion. A theory of extru-

sion based on critical state theory is developed and its

predictions compared with experimental results and litera-

ture findings.



Materials and Methods

Materials

The alumina-water system was chosen as the model

particulate system for study because of its importance as

a ceramic material. Two alumina powders were studied:

RC152DBM* and RCHPDBM* (Table V-I).

The systems studied were RCHP/water, RC152/oleic

acid /water and RC152/Y-alumina**/water. The relative

degree of flocculation in these systems is measured by



* Reynolds Metals Company, Bauxite, AR.
Eastman Chemicals, Rochester, NY.

** SF-85, Linde Div., Union Carbide Corp., Indianapolis,
IN.











Table V-I. Characteristics of Alumina Powders.


Material

Particle Size

Distribution


Phases


Grinding

Additives


RC152DBM

1.5 pm

Narrow
Log normal

100% a


Fully ground

None


RCHPDBM

0.5 pm

Narrow
Log normal

95% a
5% Y

Fully ground

0.05 w/o MgO
(as MgCO3)










their sediment volume fraction, Figure V-1. For a given

powder, e.g. RC152, the lower the sediment volume

fraction, the higher the degree of flocculation (44).

RC152/water was only slightly flocculated. As

mixed, the material was a thin slurry at 45 v/o solids; a

rapid transition from slurry to hard, dilatant solid

occurred at 56 to 57 v/o solids. The addition of

1.1 v/o oleic acid as an immiscible liquid flocculant

(45) caused the character of RC152/water to change. At

35 v/o solids, it was a thin paste and above 45 v/o it

had enough strength to hold its shape. It was somewhat

plastic as determined from hand moldability. Similar

results could be achieved through the addition of v10

w/o Y-alumina to RC152.

RCHP/water was also highly flocculated. This was

thought to be due to the finer particle size and the high

(5 w/o) Y-alumina content which could act as a van der

Waals flocculant because of its very fine (0.01 pm) size.

These systems were chosen for study because 1) they

were somewhat plastic which was the phenomenon of interest;

2) they contained no organic binders or surface active

agents which might complicate the analysis; and

3) they had a low viscosity pore phase which allowed con-

solidation to occur in a reasonable amount of time.


















RC152/water


RC152/1.42 v/o oleic
acid/water

RC152/ -alumina/water



RCHP/water


RCHP/milled in water


RCHP/isobutanol


RCHP/milled in
isobutanol


Figure V-l.


i -:.r l















-\-- -L




0 5 10 15 20 25 30 35

v/o solids in sediment


The higher the volume percent solids in the
sediments of dilute (7 v/o solids) slurries
of alumina, the lower their degree of
flocculation.










Slurry Preparation

The powders were mixed with water in a deairing

paddle mixer* at 35 to 45 v/o solids depending on the

particular slurry. The slurries were allowed to age '24

hour prior to use. Material for consolidation tests was

used directly. Samples for triaxial testing were con-

solidated in a filter press with diameter 1.4 inches and

final height 3.0 inches.

There was some concern that the variation in volume

fraction solids from top to bottom might be large for the

1.4 by 3.0 inch specimens. Several specimens were made

and sectioned, top to bottom, to determine this variation.

A typical example is given in Table V-II. The mean for

this specimen was 0.497 with standard deviation of 0.008.

This was considered acceptable in comparison to estab-

lished practice (46).

Experimental Procedures

Consolidation test procedure. Consolidation experi-

ments (i.e., one dimensional confined compression) were

conducted in a fixed-ring consolidometer,** Figure III-3.

Tests were stress controlled. An increment of stress was

made and consolidation allowed to proceed. The equilib-

rium displacement was recorded from the dial gauge. The




* Whip-Mix Corp., Louisville, KY.

** Anteus Corporation, Mt. Vernon, NY.












Table V-II. Variation of volume fraction solids in a
filter pressed triaxial test specimen.




Distance from top
of specimen VS
(inches)

0.3 0.505

0.9 0.492

1.5 0.498

2.1 0.494


2.7


0.505










time required for full consolidation (i.e., no more

change in volume) varied from 15 minutes for highly

flocculated RC152/oleic acid/water to 60 minutes for

RCHP/water. Logarithmic pressure increments were used

throughout as per standard practice (31). To reduce

friction and stress variation in the specimen, a small

L/D ('1:3) and lubricated side walls (10% lecithin solu-

tion in hexane) were used.

Triaxial compression procedure. The triaxial com-

pression test is the standard method for studying the

mechanical behavior of soils (22). It is a most versa-

tile test. The isotropic and shear stress components can

be varied independently which allows for testing with a

variety of stress paths. Because of this, both normally

consolidated and overconsolidated soil can be studied.

The triaxial apparatus is shown in overview in

Figure V-2a. It consists of a pressure cell for applying

an isotropic stress and a vertical push rod for applying

an axial stress. Figure V-2b shows a detail of the

specimen mounted between the pedestal and the top cap.

Porous stones were used top and bottom to allow drainage

during consolidation and measurement of pore pressure

during undrained shear testing. The specimen is isolated

from the pressure chamber by a thin rubber membrane.

After specimen mounting, the test procedure is as

follows. The specimen is consolidated, with the drain









































Connection to pressure
supply, force fit, or -
screwed and sweated


Axial load


1. Sample enclosed
by membrane cap
and pedestal

2. Sealed sample
placed within
pressure vessel,
and confining
pressure applied

3. Drainage from
sample controlled
by valve

4. Axial load applied
by plunger
protruding into
pressure vessel
until failure occurs


- Drainage or pore
pressure connection


Figure V-2. Overview (a) and detail (b) of triaxial compression apparatus used
to test soils. From Lambe and Whitman (22).










valve open, to some isotropic pressure, po. When con-

solidation is complete (and the pore pressure has dis-

sipated), the drain valve is closed and the specimen is

sheared at constant rate. Both the axial stress and the

pore water pressure are recorded. The specimen is

strained to 17% axial strain or failure, whichever comes

first. For overconsolidated soil tests, the specimen is

allowed to swell after consolidation (drain valve open)

to some pressure lower than po. When swelling is complete,

the procedure continues as above. All tests were run

with a back pressure (47, 48) of 15 psi to ensure satura-

tion and to prevent cavitation when negative pore

pressures were encountered. Detailed operating instruc-

tions as given in Laboratory Soils Testing (47) were

followed. Theoretical details are covered by Bishop and

Henkel (48).

Extrusion. Extrusion experiments were conducted in

a piston extruder with a moving die Figure V-3. This is

called inverted extrusion and was used because the extru-

sion force is constant along the length of travel (49).

The extruder was mounted in an Instron* mechanical testing

machine. Tests were run at several crosshead speeds and

the load on the die was measured. A pressure transducer**




* Instron, Inc., Canton, MA.

** Sensotec, Inc., Columbus, OH.









Axial
Stress


Extruder
Barrel


-Hollow push rod






Extrudate




"O" rings




SSaturated cotton
plug


Load to
recorder


Figure V-3. Schematic of inverted extrusion setup used
in extrusion experiments.










was mounted in the extruder barrel to measure the pore

pressure within the extrusion batch.



Results and Discussion

Consolidation and Springback

Consolidation tests were run on the following

systems, Figures V-4 to V-6:

1) RCHP/water

2) RC152/oleic acid/water; oleic acid concen-

tration = 0.0, 0.11, 0.5, and 1.14 v/o based

on volume of alumina

3) Dry RC152

Figures V-4 and V-5 show both the permanent consolidation

and elastic swelling for RCHP/water and RC152/0.5 v/o

oleic acid/water. Over the range of vertical stress used

in these experiments, RCHP/water and RC152/0.5 v/o oleic

acid/water exhibit linear v-lnO consolidation behavior;
v
they also exhibit linear v-lno elastic swelling and
v
recompression with little hysteresis in elastic response.

This type of response is similar to that often observed

for ceramic clay and engineering soils (14, 7, 22) (cf

Figure III-2). It is also of the form commonly assumed

in constructing soil plasticity theories (7, 34, 35).

Figure V-6 summarizes the consolidation behavior for

RC152/oleic acid/water as a function of oleic acid concen-

tration. For clarity, the elastic response has not been


































































2 5 10 20 50 100 200


r
0











C
m


-4-
0
r-


z
cc


t')




U,
on


VERTICAL STRESS, OV (PSI)














Fibure V-4. One dimensional consolidation of RCHP/water
followed classic v-lnc, behavior; X = 0.090,
K= 0.005.










































2 5 10 20 50 100 200


VERTICAL STRESS (PSI)


Figure V-5.


One dimensional consolidation of RC152/0.5
v/o oleic acid/water; X = 0.078, K = 0.005.





































-I

t.,
! 2.0
U
a-


1.9




1.8




1.7




1.6


2 5 10 20

VERTICAL STRESS (PSI)


Figure V-6.


Increasing the concentration of oleic acid
increased the degree of flocculation and the
slope of the consolidation curves. Dry RC152
had much higher v and X values than RC152/water.
A typical engineering clay is shown for compar-
ison.










included in Figure V-6. As the amount of oleic acid is

increased, the degree of flocculation increases as

reflected in the low pressure specific volumes. The

slopes of the consolidation curves also increase as the

amount of oleic acid is increased. In addition, the

consolidation curve for the highest oleic acid concen-

tration, 1.14 v/o (the most flocculated system) is not

linear but rather concave up. This behavior is similar

to that observed for silts and clays (22); as v

increases, A also increases and the consolidation curve

tends to become non-linear.

Increasing the oleic acid concentration strengthens

the alumina-oleic acid-water system. The consolidation

curves in Figure V-6 are translated to higher vertical

stresses as the oleic acid concentration is increased

from 0.0 to 1.14 v/o. These data have been replotted in

Figure V-7 as vertical stress vs. oleic acid concentra-

tion at constant specific volume. The vertical stress

required to produce a given specific volume increases

quite rapidly as the oleic acid concentration is in-

creased in this range. For example, at 0.11 v/o oleic

acid, a vertical stress of only 6.4 psi is required to

give a specific volume of 2.0; however, at 1.0 v/o oleic

acid, a vertical stress of 37 psi is required, a six-fold

increase.
























200 I-


100



50



20



10


0.0


v 2.0


V 1. 8


0.2


0.4


0.6


0.8


1.0


1.2


oleic acid concentration









Figure V-7. Increasing the amount of oleic acid in the
slurry raises the equilibrium vertical stress
needed to ahcieve a given specific volume, v.


III


- I--~ ---L~-i---n --~---u-~-*~R--~ylr~--~__- --r










The increase in strength with increase in oleic acid

concentration is thought to be due to the increase in

capillary bonding provided by the oleic acid. The situa-

tion of oleic acid as an immiscible liquid flocculant in

water is analogous to dry powder which has a small amount

of water bonding it together at capillary necks. Rumpf

(50) and Rumpf and Schubert (51) have analyzed the related

problem of tensile strength for single particle size

spheres. His equation for variation of tensile strength

with volume fraction and ratio of liquid to solid volume

is
1 F*
ot- x- (v-l)

where at = tensile strength

E = porosity

F* = mean adhesion force

x = particle size

The factor F* is further related to the interfacial

tension, y, the particle size, x, and the dimensionless

adhesion number, FH, by

F* = FH x Y (V-2)

F is a complicated function which-depends on a number of

geometric parameters; it is derived by Schubert et al.

(52). For the purposes at hand, FH can be determined from

Figure V-8 (51). It has been determined (51) that the mean

a/x for real powder compacts is "0.05 which will be used

here. Table V-III shows typical values of at as a function

of oleic acid concentration and specific volume, v. The







67



















--- 1 --- ----/iv --- i-'

3.0 -Y-x -
v Cost. F + F

S2.0



Y. x
1.5

1.0

0.5 \\ \\10-


0 0.05 0.10 0.15 0.20
















Figure V-8. Dimensionless adhesion force F*/Cy'x)
of a liquid.bridge between two spheres
as a function of distance ration a/x.
From Rumpf and Schubert (51).











Table V-III. Variation of strength with oleic acid
concentration based on Rumpf's analysis
(51, 52).


v/o oleic
acid


0.1%

0.5%

1.0%


0.75

1.6


1.9


at(v=2.0)
(dyne/cm2)


at(v=1.9)
(dyne/cm2)


.4 4- _


3.75 x 105

8.0 x 105

9.5 x 105


4.1 x 105

8.8 x 10s

10.5 x 105


at (v=1.8)
(dyne/cm2)


4.7 x 10s

10.0 x 105

11.9 x 105










increase in strength with increasing oleic acid concen-

tration is of the same order as that observed for consoli-

dation. However, the increase with decreasing specific

volume lags far behind the increase observed for consolida-

tion. The Rumpf analysis predicts an increase in strength

of 25% when v is decreased from 2.0 to 1.8. The observed

increase in strength was about an order of magnitude.

The variance may be due to the difference between tensile

and compressive loading of the material.

Another interesting feature of the consolidation

curves in Figure V-6 is the difference in compaction

behavior between wet and dry RC152. Dry RC152 is highly

flocculated and highly compressible. Wet RC152 is only

slightly flocculated and not very compressible. These

changes in flocculation and compressibility are similar

to changes observed for wet and dry kaolin (19, 20).

Sridharan and Venkatappa Rao have shown that lowering the

dielectric constant of the pore fluid increased the low

pressure specific volume and increased the compress-

ibility of kaolin (20). They interpreted their results

in terms of increasing van der Waals bonding at low

dielectric constant (e). That alumina followed a similar

trend is shown in Figure V-9. RCHP milled in isobutanol

(e=17) had a higher v and higher compressibility than

RCHP milled in water (e=80). Also, decreasing increased

the strength of the specimen at constant specific volume.



















5.0








4.0





o*r


u 3.0








2.0
1 2 5 10 20 50 100

Vertical stress (psi)






Figure V-9. RCHP milled in isobutanol had a much higher
v and a higher average X than RCHP milled
in water.










While the consolidation behavior of alumina/water is

similar to that for engineering soils, there are some

differences, especially in X and K values and time of

consolidation. Figure V-8 shows a comparison of the data

for alumina and for a typical clay. The clay is more

compressible (higher X) and has a higher v value than

any of the alumina materials tested in water. Clay X

values range from 0.2 to 0.5 for kaolinitic clays to as

high as 1.0 to 2.0 for montmorillonitic clays; their K

values range from 0.05 to 0.15 for kaolinitic clay to

0.2 to 0.5 for montmorillonitic clays. The ranges for

alumina are X = 0.05 to 0.11 and K' 0.005. Times of

consolidation were much shorter for RC152 (410 min.) and

RCHP (r1 hour) as compared with clay which for laboratory

tests can range from hours to days depending on the

particular clay (31). This difference in consolidation

time is probably related to the differences in permeabil-

ity of clay and alumina. The coefficient of permeability

of RC152 and RCHP can be estimated from the Kozeny-Carmen

equation (53) --

k 980 2 ....... 3 (V-3)
k = 980 ( ()) V3
ko n So2 (1-E)2

where k = coefficient of permeability (cm/sec)
.Re
(-T) = hydraulic tortuosity (%2.0)

k = pore shape factor ("4 (54))

n = viscosity (poise)

S = surface area/unit volume (cm2/cm3)
o










For 50 v/o solids, the permeability of RC152 is 2.7 x

10-7 cm/sec and of RCHP is 3 x 10-8 cm/sec. These are

about one to two orders of magnitude higher than the

corresponding value for clay. The variation of consolid-

ation times is of the same order as the variation of

permeabilities.

The change in compressibility of a powder with

change in pore fluid could be of practical use in the

formation of ceramic pellets. By filter pressing with a

high dielectric constant fluid rather than die pressing

in air, a more uniform, higher density pellet can be

obtained. The specimen will be of higher density because

the compaction curve lies at low specific volume; the

density will be more uniform because the range of density

available for a given difference in stress will be smaller.

Consider RC152, Figure V-6, as an example. Die pressing

to 224 psi produced a relative density (1/v) of 52.1%;

filter pressing in water to 224 psi produced a relative

density of 58.8%, an increase of 6.7%. If there were a

stress distribution in the compact of 50 to 200 psi (an

exaggerated assumption used to make a point), there would

be a density variation of 4.2% in the dry pressed pellet

(from 47.9 to 52.1%) but only 2.6% in the filter pressed

pellet (from 56.6 to 59.2%). Such an increase in green

density can increase the fired density of a pellet.

Prochazka et al. (55) have shown that increasing the










green density of SiC from 55% to 59.3% increased the

fired density from 98.1% to 99.2%. If one is working

with an optical material such an increase could be quite

important. Also, a smaller difference in green density

will lead to a smaller chance of cracking due to differen-

tial shrinkage during firing.

Yield Criterion For Alumina/Water

The application of any of the current theories of

soil plasticity (7, 23, 24, 25, 34-40) to a system

requires that its yield envelop generally be of the

Mohr-Coulomb with strain hardening cap type (Chapter III,

Figure III-8). To test the applicability of this hy-

pothesis to the alumina/water system, consolidated,

undrained (closed system) triaxial compression tests were

conducted on RCHP/water and RC152/1.14 v/o oleic acid/

water. The results of these tests were analyzed in terms

of stress-strain, pore pressure-strain, and total and

effective stress paths.

Effect of stress history on behavior. The yield

envelop approach states that the stress history of a body

should affect its q-E, u-E and p'-q behavior. Figures

V-10, V-ll, and V-12 show the q-C and u-e curves for

RC152 and RCHP in undrained, triaxial compression. There

is considerable difference in response between normally

consolidated and overconsolidated specimens. The normally

consolidated specimens showed large, positive pore
















































1.0 2.0 3.0 4.0 5.0 6.0


1.0 2.0 3.0 4.0 5.0 6.0


AXIAL STRAIN ()



Figure V-10. Stress-strain and pore pressure-strain curves

for triaxial tests on RC152/1.14% oleic acid.


0

U)


C) .-4
N



U)
W
in
w
I -l

C/)
M D 0
< I-
w
UJ


0.0


N


a-


\00

0.0

































0 1.0 2.0 3.0 4.0 5.0 6.0 7.0


0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
AXIAL STRAIN (')


Figure V-ll.


Stress-strain and pore pressure-strain
behavior for two triaxial tests on RCHP
alumina-water compacts.






























0.4 0.8 1.2
Axial strain, %


0.4 0.8
Axial strain,


Figure V-12.


Deviator stress-strain and pore pressure-
strain for RCHP/water filter pressed at
72 psi then tested in triaxial compression
with OR = 0 psi.


0


0.



-1.


-2.0










pressures during shear; the overconsolidated specimens

showed a small, positive pore pressure at small strains

(due to elastic compression), then a negative pore

pressure at larger strains. All of the normally consol-

idated specimens showed peaks in the q-e curve at "1.5%

strain, then q dropped to a lower, ultimate value which

remained constant with further increase in strain. The

q-E curves for the overconsolidated specimens showed a

rapid increase at small strains (up to "0.25%) then a

transition to a much slower, although steady, climb at

higher strains. The transition region for the over-

consolidated q-E curve corresponded with the change in

pore pressure from positive to negative. This behavior

is quite similar to that observed for normally consol-

idated and overconsolidated clay in undrained tests,

Figure III-11. The main difference is that the normally

consolidated clay did not exhibit a peak in its q-E

curve.

Role of effective stress. The effective stress

paths and total stress paths for RC152 and RCHP tested in

undrained triaxial compression are shown in Figure V-13,

V-14, and V-15. The effective stress paths for over-

consolidated behavior, Figures V-13, and V-15, are classic

examples of Mohr-Coulomb failure for undrained conditions

(22). For small elastic strains, the effective stress

path falls slightly to the left of the total stress path























































































a m c
(iSd) SS3U1S HOiVIAha


o
0
0




0




0








C/)




-
C-
0 "--




oo
U,



0

n-
0
F-
o
o v,


*H







r14
0

0


4'4

m I




HE

m0


P4J

ta
n





4-4(
4J 0
.cnC
Ci)




r-I




4)U




44 -U
l rd
-<

aC

nl~









































0 20 40 60 80 100
EFFECTIVE HYDROSTATIC STRESS (PSI)













Figure V-14. Effective stress paths for RCHP/water
tested in triaxial compression.









































0 2 4 6


8 10


Isotropic stress (psi)


Figure V-15. Effective and total stress paths for
RCHP/water. Compact formed at 75 psi
in filter press and tested in triaxial
compression with aR = 0 psi.










indicating a small, positive pore pressure due to elastic

compression. As the strain is increased and the specimen

starts to plasticlaly deform, the effective stress path

starts to climb the Mohr-Coulomb surface and negative

pore pressures start to be generated. The pore pressure

continues to decrease and the effective stress path

eventually lies to the right of the total stress path.

The effective stress paths for normally consolidated

specimens trace out a cap for the Mohr-Coulomb line.

There is an immediate, positive deviation between the

effective and total stress paths which continues to grow

until the ultimate condition is reached. The projection

of the effective stress path for overconsolidated RC152

(Figure V-13) shows that, at very large strains, it would

intersect with the ultimate condition for normally consol-

idated RC152. This is one of the requirements of soil

plasticity theory (e.g., 7, 24, 25).

That it is the effective stresses that are important

in describing the behavior of alumina/water and not the

total stresses can be seen from Figure V-16. Shown here

are the undrained triaxial compression test results for

RC152/1.14 v/o oleic acid/water specimens that were

normally consolidated to 87 psi, then allowed to swell to

lower confining pressures of 75, 40, 20, 10, and 0 psi.

In terms of the total stresses, there is really no pattern

for either peak or residual failure. In terms of the



































20 40 60 80 100
Isotropic stress, psi


Figure V-16.


Effective and total stress paths for RC152/1.14 v/o oleic
acid/water consolidated to 87 psi then allowed to swell prior
to triaxial undrained testing. Total stress paths shown
as - -










effective stresses, however, a definite pattern for the

ultimate stress states is observed. The ultimate condi-

tions of all of the specimens lie on an extension of the

effective stress path for the specimen swelled to zero

confining pressure, i.e., all of the ultimate conditions

lie on the Mohr-Coulomb failure line. Looking only at

the total stress paths no insight is obtained as to why

there is a peak failure stress followed by a lower

residual failure stress. By examining the effective

stress paths, it is appreciated that q (peak) and q

(residual) occur at different values of effective stress

p'; hence, the conditions at peak and residual are quite

different. The reasons for the peak in deviator stress

and other deviations from ideal behavior will be dis-

cussed in Chapter VI in regard to remolding and sensi-

tivity.

Geometric similitude. A requirement of general soil

plasticity theory as outlined in Chapter III is that the

normally consolidated effective stress paths should be

geometrically similar. To test this, the effective

stress paths for normally consolidated RC152/oleic acid/

water and RCHP/water (Figures V-13 and V-14) were replot-

ted in normalized variables, p'/p and q/p where po is

the isotropic consolidation stress. These results are

shown in Figure V-17 and V-18. The normallized curves

for both RC152 and RCHP tend to fall one on top of the
























0.8



0.6



0.4



0.2


0.0


0 0.2 0.4 0.6 0.8


1.0 1.2


Normalized isotropic stress










Figure V-17. Normalized effective stress paths for
RC152/1.14 v/o oleic acid/water.








































0.2 0.4 0.6 0.8 1.0


NORMALIZED EFFECTIVE STRESS


Figure V-18.


Normalized effective stress paths for
RCHP/water.


0.6



0.4



0.2



0.0
0.0










other indicating that they are geometrically similar.

Effect of flocculation on Roscoe surface. The floc-

culation of RC152 with either oleic acid or y-alumina

markedly changed its plastic response. RC152/water was a

slurry up to 156 v/o solids and a hard, dilatant solid

above %57 v/o solids. This is typical behavior for a

"non-plastic" material. RC152/water could not be molded

by hand since tension cracks formed easily (indicating

low cohesion). In contrast, RC152/oleic acid/water and

RC152/Y-alumina/water were pastes at 35 v/o solids and

their consistency increased slowly as the solids loading

increased. (They had a long working range in contrast to

the very short working range of RC152/water.) They could

be easily hand molded since tension cracks formed only on

sharp edges (indicating high cohesion).

Consolidated undrained tests were run for RC152/

water and RC152/y-alumina/water to determine if the hand

molding behavior was reflected in the nature of the

Mohr-Coulomb cap surface. The results of these tests are

shown in Figure V-19. It is observed that the effective

stress paths for highly flocculated RC152/Y-alumina/water

and RC152/oleic-acid/ alumina (Figure V-13) are similar

while the effective stress path for RC152/water is quite

different. The most striking difference is in the

position of the peak shear stress. For the highly floc-

culated systems, peak is at p/p 0.65 with q(peak)h



















120

O RC152/10%
100 -alumina/water
O RC152/water

S80
o) /

6 60


m 40
O I
0 0 /
> 20


0
0 20 40 60 80 100 120 140 160

Isotropic stress (psi)







Figure V-19. Flocculating RC152 with Y-alumina drastically
changed the shape of its normally consolidated
effective stress path. Total stress paths
shown as - -.










0.5 po while for RC152/water, peak is at p/po 0.95 with

q(peak) n 0.87 po. Thus, RC152/water is relatively harder

to shear and the effective stresses are relatively higher

than for the highly flocculated systems. One may interpret

this in terms of the differences in structure of the

systems. The open structure of the flocculated systems

is relatively easier to shear than the closer-packed

structure of RC152/water. During shear, more pore

pressure is mobilized in the flocculated specimens causing

the effective stress to fall rapidly so the ultimate

resistance to shear is not so high. Also, as was noted

earlier, the cohesion of the highly flocculated systems

was higher than for RC152/water. This was due to

capillary bonds in the oleic acid flocculated specimen

and to van der Waals bonding in the y-alumina flocculated

specimen.



Application to Ceramic Extrusion

Given that alumina-water has been shown to follow a

Mohr-Coulomb with cap type yield criterion, one may ask

what insights into ceramic processing will this give us.

In this section it will be shown that critical state

theory (7, 8) predicts a functional form for extrusion

which is identical to established empiricism in ceramics.

In addition, the role of pore pressure in extrusion,

which has not been treated previously in ceramics, will

be presented.










Extrusion Theory

The derivation starts with the assumption of a

Mohr-Coulomb with cap yield function, Figure III-8.

Also, it is assumed that the material is fully saturated

with water. The stress path followed is that for confined

compression as that is the condition in a piston extruder.

The stress state is given by the K condition: the

applied pressure is v and the reaction stress against

the barrel walls is aH. Thus

P = V + H = (1 + 2K) (V-4a)



q = H = v (1 K) (V-4b)


q/p = 3(1 K) (V-4c)
1 = 2Ko


For a typical K value of 0.5, q/p = 0.75. This puts the

loading path on the cap side of the yield surface in

Figure III-8. Undrained loading conditions are also

assumed. This requires a low permeability for the material

in question, i.e., no binder bleed. A fine pore size (as

in clay) or a high binder viscosity (as in alumina and

other technical ceramics) will produce essentially un-

drained conditions. Figure V-20 gives the binder vis-

cosity required to prevent bleeding as a function of

particle size and solids fraction. The calculation was

based on the Kozeny-Carmen equation (Equation V-3). A




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