Dominant toughening mechanisms in Barium Aluminosilicate (BAS) glass-ceramics

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Dominant toughening mechanisms in Barium Aluminosilicate (BAS) glass-ceramics
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Thesis (Ph.D.)--University of Florida, 1998.
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Includes bibliographical references (leaves 112-116).
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by Jason Alan Griggs.
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Typescript.
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Vita.

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DOMINANT TOUGHENING MECHANISMS IN BARIUM
ALUMINOSILICATE (BAS) GLASS-CERAMICS














By

JASON ALAN GRIGGS


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














ACKNOWLEDGMENTS


I would like to thank my wife, Amy, for enduring all of the lonely nights. I owe

the timely completion of my graduate studies to her encouragement. Likewise, I would

never have written this dissertation if it were not for my parents teaching me the value of

an education and supporting my interest in the physical sciences.

I am also indebted to my mentors. Dr. Jack Mecholsky filled me with his passion

for the subject of fracture mechanics and taught me the art of failure analysis. Dr. Ken

Anusavice kept me from straying too far from clinical relevance. He coached my writing

and presentation skills and prepared me for survival in the world of academic research.














TABLE OF CONTENTS


page


ACKNOWLEDGMENTS............ .................................................................. ii

LIST O F TA B LE S ......................................................... ............................................. v

LIST OF FIGURES ................ .............................................................vii

ABSTRACT .............................................. ........................ ...................... .....xi

P U R P O S E ......................................................................... .............................................
1.1 Process R ationale ................................................................... ...................... 1
1.2 M material Selection.......................................................... ...... ...................
1.3 R research O objectives ................................................................ ..................... 2
1.3.1 Specific A im ..................................................... ................. 2
1.3.2 Specific A im 2 ................................................................ ..................... 2
1.3.3 Specific A im 3 .................................................................. ................... 3

BACKGROUN D ................................... .................................4
2.1 G lass-C eram ics.................................. ........................................................... 4
2.1.1 Processing of Glass-Ceramics .............................. .........................4
2.1.2 BAS Glass-Ceram ics....................... ..................................................... 7
2.2 Strength of Glass-Matrix Composites..................... ............................... 12
2.2.1 Flaw Size Lim itation ............................................................. .................. 12
2.2.2 Stress Transfer................................................... ............................... 15
2.2.3 Crack-Particle Interaction...................................... ........................................17
2.3 Fracture Toughness of Glass-Matrix Composites.........................................17
2.3.1 C rack B ow ing......................................................................................... 17
2.3.2 Crack Deflection...........................................................19
2.3.3 Crack Bridging.......................................................22
2.3.4 Microcracking..........................................................23
2.3.5 Transformation Toughening.................................................................25

MATERIALS AND METHODS .................................. ..............................................27
3.1 Materials Fabrication ........... ...........................................................27
3.1.1 Glass Composition.............. ......................................................27


iii








3.1.2 Glass M elting and Form ing.................................. ...... ....................... 28
3.1.3 Thermal Crystallization Treatments.............................................28
3.2 Microstructural Analyses .................... ...................................29
3.2.1 Phase Identification..................... ......... ........................29
3.2.2 Crystal Morphology........................ ............................30
3.2.3 Glass Morphology......................... ............................30
3.3 Physical Property Analyses ................................................. ..................... 32
3.3.1 Density and Hardness................................................... ....................32
3.3.2 Elastic Constants.......................... ........ ....... ....................33
3.3.3 Thermal Expansion..................... ........... .....................34
3.3.5 Flexural Strength........................... ....... .................................34
3.3.6 Fracture Toughness.......................... ........ ... ....................36
3.5 Finite Elem ent Analyses ... ............................ .. ................................................40

RESULTS AND DISCUSSION .......................................... ....................... 42
4.1 M icrostructure ........................................................................ .........................42
4.1.1 Phase Identification......................................... ....................................... 42
4.1.2 Crystal Morphology......................... ...... .... ...................43
4.1.3 Glass Morphology......................... ...... ... ......................48
4.2 Physical Properties ........................................ ............................................48
4.2.1 Density and Hardness........................... .... ..... ...................48
4.2.2 Elastic Constants.......................... ................ ...................51
4.2.3 Thermal Expansion.......................... ....... ... ..........................52
4.2.4 Strength and Fracture Toughness.......................... ....................52
4.3 Finite Element Predictions ..........................................................54
4.4 Strengthening and Toughening Mechanisms ................... ......................56
4.4.1 Flaw Size Lim itation ................................................... ......................56
4.4.2 Crack Bowing........................ ........................................................ 57
4.4.3 Wake Process-Zone Mechanisms ...........................................................58
4.4.4 Stress Transfer ............................................................. .. ............... ... 61
4.4.5 Crack D eflection................ ...................................................................... 62
4.5 Process Optim ization.................. .......................................................... 67

C O N C LU SIO N S............................................................................. ............................. 70

APPENDIX A SOURCE CODE FOR FINITE ELEMENT ANALYSES ...................72

APPENDIX B TABULATED DATA .............................................. ......................103

R EFEREN CES ......................................................................................................... 112

BIOGRAPHICAL SKETCH................................................ ......................... ... 117





iv














LIST OF TABLES


Table page

2.1 Properties of BAS glass-ceramics cold pressed at 414 MPa and sintered in air
at various temperatures for 20 h according to Drummond and Bansal (1990)............9

2.2 Properties of ceramics of selected compositions in the BaAl2Si2Og-
BaGa2Ge208 system according to Zaykoski and Talmy (1994)........................... 11

3.1 BAS glass compositions prior to melting and after casting.....................................27

4.1 Results of finite element analyses to predict the maximum stresses in BAS
glass-ceram ics .......................................................................... ......................56

4.2 Comparison of observed strength values with those predicted for flaws of
limited size and flaws of observed size for BAS glass and glass-ceramics................57

4.3 Normalized toughness increase, A4c, observed for BAS glass-ceramics and
toughness increase not accounted for by the contribution of stress transfer,
A .xs ......................... .................................. ......................... 66

4.4 Estimated relative contributions of stress transfer and crack deflection
mechanisms to the observed strengths of BAS glass-ceramics...............................67

B.1 Raw data from BAS glass and glass-ceramic four-point flexural specimens...........104

B.2 Data for calculating the hardness of BAS glass and glass-ceramics.......................105

B.3 Data for calculating the elasticity of BAS glass and glass-ceramics...................... 106

B.4 Dimensions of cross-sections between BAS crystals and stereological fields in
the BAS glass-ceramic produced by crystal growth at 9750C for 0.5 h...............107

B.5 Dimensions of cross-sections between BAS crystals and stereological fields in
the BAS glass-ceramic produced by crystal growth at 9750C for 4 h..................108

B.6 Dimensions of cross-sections between BAS crystals and stereological fields in
the BAS glass-ceramic produced by crystal growth at 9750C for 32 h................109








B.7 Dimensions of cross-sections between BAS crystals and stereological fields in
the BAS glass-ceramic produced by crystal growth at 9750C for 256 h ...............109

B.8 Data for calculation of the crystalline volume fraction and mean free path of
the BAS glass-ceramic produced by crystal growth at 9750C for 0.5 h...............110

B.9 Data for calculation of the crystalline volume fraction and mean free path of
the BAS glass-ceramic produced by crystal growth at 9750C for 4 h ..................110

B.10 Data for calculation of the crystalline volume fraction and mean free path of
the BAS glass-ceramic produced by crystal growth at 9750C for 32 h ...............111

B.11 Data for calculation of the crystalline volume fraction and mean free path of
the BAS glass-ceramic produced by crystal growth at 9750C for 256 h ..............111














LIST OF FIGURES


Figure page

2.1 Diagram of the temperature-time cycle for the controlled crystallization of a
glass-ceram ic body. ................................................................. ........................ 5

2.2 Diagram of the variation of crystal nucleation and growth rates with
tem perature. ................................................ .................................................... 6

2.3 Pressure-temperature phase diagram for barium aluminosilicate according to
Lin and Foster (1968).................... ........... .........................8

2.4 Diagram of the atomic structure ofhexacelsian crystals according to Ito (1956)........8

2.5 Flexural strength of ceramics in the system SrO.A1203o2SiO2-
BaO.A1203.2SiO2 according to Talmy et al. (1992)..............................................11

2.6 The effect of Li20 content on the strength of celsian glass-ceramics according
to Zhou et al. (1997). ........................ ..................... ....................... 12

2.7 Plot of fracture strength versus reciprocal mean free path for glass specimens
containing a dispersion of alumina particles according to Hasselman and
Fulrath (1966). ....................................................................... .............................. 14

2.8 Flexural strength versus alumina content for various diameters of alumina
reinforcements according to Borom (1977).................... .....................16

2.9 Diagram of the crack bowing mechanism proposed by Lange (1970).
Propagating cracks are pinned at adjacent inclusions and bow out between
them until increased local stress intensity enables the crack to break away
from the pinning positions. ..... ......................... ........... ..................................18

2.10 Diagram of the effect of thermal expansion mismatch between the matrix phase
and the particulate phase on the path of a propagating crack.................................20

2.11 Normalized toughness predictions for a crack deflection mechanism for
spherical, rod-shaped, and disk-shaped reinforcing particles with aspect ratios
of 1, 3, and 12 according to Faber and Evans (1983a)........................................... 21








2.12 Diagram of the R-curve behavior of a brittle material subject to wake process-
zone toughening mechanism s. ............................... .......................................... 22

2.13 Diagram of possible toughening mechanisms operating in the wake process
zone: (a) crack bridging, (b) debonding and pullout, and (c) stress-induced
m icrocracking................................. .......................................... ......................23

2.14 Diagram of the effect of particle size on the fracture toughness of a material
susceptible to microcracking. Little toughening is observed until the mean
particle size is close to a critical size, re. Fracture toughness decreases for
larger particles. ................................................................. 25

3.1 Thermal processing conditions for various stages of BAS glass-ceramic
develop ent ............................................................................... .. ......................29

3.2 Diagram of a BAS crystal intersected by a stereological field. Measurements
from the sections of many crystals are used to calculate parameters for the
crystal population ............................................................................. .......................31

3.3 A point counting grid superimposed on a typical stereological field for
determination of crystalline volume fraction and mean free path...........................32

3.4 Diagram of the four-point bending apparatus used for the determination of
flexural strength ........................................................................ ..... ................35

3.5 Diagram of the typical fracture surface features occurring in brittle materials.
The regions are not drawn to scale........................................ ......................38

3.6 Diagram of the location of an intact controlled flaw on a fractured four-point
flexure specimen. The flaws are not drawn to scale ..............................................39

3.7 Finite element model for a pair of BAS crystals in a glass matrix. A zero-
displacement boundary condition is applied at the top and left edges. A
coincident node condition is applied at the bottom and right edges and along
the crystal-glass interfaces. ..................................................... ...................... 40

4.1 X-ray diffraction spectra for BAS glass-ceramics. Hexacelsian (BaAl2Si2Os) is
the only phase present. Peak intensity is a function of processing
temperature. .................................... ....................43

4.2 Atomic force micrograph of the microstructure of a BAS glass-ceramic
produced by a crystal growth treatment at 9750C for 0.5 h.....................................45

4.3 Atomic force micrograph of the microstructure of a BAS glass-ceramic
produced by a crystal growth treatment at 9750C for 4 h......................................45








4.4 Atomic force micrograph of the microstructure of a BAS glass-ceramic
produced by a crystal growth treatment at 9750C for 32 h....................................46

4.5 Atomic force micrograph of the microstructure of a BAS glass-ceramic
produced by a crystal growth treatment at 9750C for 256 h..................................46

4.6 Effect of crystal growth time on the crystalline volume fraction of BAS glass-
ceram ics ........................................ ......................................... .......................47

4.7 Effect of crystal growth time on the morphology of the crystalline phase in
BA S glass-ceram ics. ................................................................ ......................47

4.8 Effect of crystal growth time on the mean free path between crystals in BAS
glass-ceram ics..................................................................... .. ...................48

4.9 Effect of crystal growth time on the apparent density of BAS glass and glass-
ceram ics .................................................... ......................... .....................50

4.10 Effect of crystal growth time on the hardness of BAS glass and glass-ceramics......50

4.11 Effect of crystal growth time on Young's modulus, shear modulus, and bulk
modulus of BAS glass and glass-ceramics.................................... ......................51

4.12 Effect of crystal growth time on the flexural strength of BAS glass and glass-
ceram ics ................................................... ............................................................5 3

4.13 Effect of crystal growth time on the fracture toughness of BAS glass and glass-
ceram ics .................................................... ...........................................................53

4.14 Residual stress field predicted by finite element analysis for a BAS glass-
ceramic produced by a crystal growth time of 0.5 h. The vectors represent the
magnitudes and directions of the principal stresses........................... ........... 55

4.15 Comparison of the observed ratio of BAS glass-ceramic strength to base glass
strength, cr7of/, with that predicted by the theory of crack bowing for the
ratios of mean crystal diameter, d, to mean free path, ......................................58

4.16 Effect of controlled flaw size on the fracture toughness of BAS glass and glass-
ceramics. Controlled flaws were induced using a Vickers diamond under a load
of 4.9 or 9.8 N .................. .......................................................................... 59

4.17 SEM micrograph of a radial crack (from a Vickers indentation induced under a
load of 9.8 N) in a BAS glass-ceramic produced by crystal growth at 975C
for 32 h ....................................................................................... 60








4.18 Observed effect of mean crystal diameter on the fracture toughness of BAS
glass and glass-ceramics compared to that predicted by Rice and Freiman
(19 8 1) ................................................................................... ..............................60

4.19 Comparison of the observed relationship between the ratio of composite
strength to base glass strength, of /of, and the ratio of composite elastic
modulus to base glass elastic modulus, E/Eg, with that predicted by the theory
of stress transfer........................................................................ ....................62

4.20 SEM micrograph of a radial crack from a Vickers indentation induced under a
load of 9.8 N in a BAS glass-ceramic produced by crystal growth at 9750C for
0 .5 h ..................... ............ ................................................. .................................. 6 3

4.21 SEM micrograph of a radial crack from a Vickers indentation induced under a
load of 9.8 N in a BAS glass-ceramic produced by crystal growth at 9750C for
4 h .................................................................................... ....................................6 3

4.22 SEM micrograph of a radial crack from a Vickers indentation induced under a
load of 9.8 N in a BAS glass-ceramic produced by crystal growth at 9750C for
32 h............................................... ........................................ 64

4.23 SEM micrograph of a radial crack from a Vickers indentation induced under a
load of 9.8 N in a BAS glass-ceramic produced by crystal growth at 9750C for
256 h............... ........................................................................ 64

4.24 Prediction for the relative increase in fracture toughness associated with crack
deflection by a dispersion of disk-shaped reinforcing particles (aspect ratio =
3) according to Faber and Evans (1983a). .............................................................67














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

DOMINANT TOUGHENING MECHANISMS IN BARIUM
ALUMINOSILICATE (BAS) GLASS-CERAMICS

By

Jason Alan Griggs

August, 1998

Chair: John J. Mecholsky, Jr.
Cochair: Kenneth J. Anusavice
Major Department: Materials Science and Engineering

The purpose of this study was to develop a barium aluminosilicate (BAS) glass-

ceramic with improved strength and fracture toughness by controlling the morphology of

the constituent phases through a series of thermal crystallization treatments. The specific

objectives of this study were as follows: 1) to determine which toughening mechanisms

are active in the BAS system, 2) to provide quantitative estimates of the relative

contributions of those mechanisms, and 3) to identify the processing conditions that

correspond to the glass-ceramic with the highest fracture toughness. The BAS system

was chosen for this study because of its potential applications in CAD-CAM production

of dental prostheses. It was concluded that stress transfer between the glass and crystal

phases and crack deflection are the only major sources of toughening in the BAS system.

Theoretical predictions for toughening associated with stress transfer and crack deflection








were sufficient to account for 100% of the increases observed. The flexural strength and

fracture toughness of BAS glass-ceramics increased with increasing crystal growth time.

The strength and fracture toughness increased from 63 8 MPa and 0.89 + 0.05

MPa-m/2, respectively, for BAS glass to 141 8 MPa and 1.87 + 0.07 MPa.m"2,

respectively, for a glass-ceramic treated at 9750C for 256 h. Fracture toughness also

increased with increasing mean crystal size. The addition of glass network modifiers

resulted in thermal compatibility between the glass and crystal phases, preventing the

weakening effect at large crystal sizes associated with spontaneous microcracking.














CHAPTER 1
PURPOSE

1.1 Process Rationale


The glass-ceramic method offers several advantages over ceramic powder

processes. It is relatively easy to cast glass into complex and thin-walled shapes. Glass-

ceramics can be produced near theoretical density with low processing shrinkage and

without high-temperature drying and sintering operations. Furthermore, glass-ceramics

are well suited for applications requiring specific optical properties because the

translucency of a glass-ceramic can be controlled by varying the crystal size and volume

fraction. A limitation of glass-ceramics currently used for dental applications is that they

tend to have low fracture toughness values (0.7 to 2.0 MPa-m"2). To improve the

structural reliability of these materials, it is necessary to understand how their constituent

phases and microstructures interact to control their mechanical behavior.

1.2 Material Selection


The barium aluminosilicate (BAS) glass system was chosen for this study because

glass compositions containing mica, a crystal phase with morphology and atomic

structure similar to those of BAS crystals, have displayed excellent machinability and are

potentially useful for CAD-CAM systems available in dentistry. Two mica-based glass-

ceramics, Dicor and Dicor MGC, are commercially available as dental prosthetic





2


materials. This research was part of a project to produce a dental prosthetic material with

greater strength, fracture toughness, and translucency than existing materials.

1.3 Research Objectives


The primary objective of this study was to identify the mechanisms responsible

for increasing the fracture toughness of BAS glass-ceramics over that of the base glass and

to estimate the relative contributions of those mechanisms. A secondary objective was to

develop a process for the production of a BAS glass-ceramic with maximum fracture

toughness. The specific aims of this study follow.

1.3.1 Specific Aim 1

Test the hypothesis that sources of toughening in the BAS system include

mechanisms that are active in the frontal process zone, e.g., crack deflection, as well as

mechanisms that are active in the wake process zone, e.g., crack bridging. This was

accomplished by analyzing the dependence of fracture toughness on critical flaw size, i.e.,

resistance-curve behavior, which usually indicates crack-particle interactions within the

wake process zone. Also, a qualitative visual analysis of glass-ceramic fracture surfaces

was used to search for evidence of crack bridging, frictional sliding, crack deflection, and

flaw size limitation.

1.3.2 Specific Aim 2

Test the hypothesis that the dominant toughening mechanism associated with

BAS glass-ceramics increases its contribution with increasing size of the reinforcing

crystals, e.g., crack deflection or crack bridging, and does not decrease its contribution








beyond a critical crystal size, e.g., stress-induced microcracking. This goal was achieved

by the use of stereological analyses to determine the size distribution of the crystal

population in BAS glass-ceramics and by quantitative comparison of experimentally

observed fracture toughness values with those predicted by theory for each toughening

mechanism.

1.3.3 Specific Aim 3

Test the hypothesis that strength and fracture toughness can be optimized by

selecting a microstructure with a maximum crystal size. Comparison of experimental

strength and fracture toughness data revealed the crystal growth time which resulted in

the highest strength and fracture toughness. A review of the results of previous strategies

for processing BAS glass-ceramics suggested a strategy for production of the

microstructure that corresponds to optimal mechanical properties.














CHAPTER 2
BACKGROUND

The following sections describe the methods used in fabrication of glass-ceramic

materials and summarize the previous research that has been published on the synthesis

and mechanical properties of barium aluminosilicate (BAS) glass-ceramics. Also included,

is an in-depth review of several theories about the mechanisms responsible for increasing

the strength and fracture toughness of glass-ceramics and glass-matrix composites in

general above those of the base glass.

2.1 Glass-Ceramics


2.1.1 Processing of Glass-Ceramics

A glass-ceramic is a partly glass, partly crystalline solid prepared by the

controlled crystallization of a glass. In theory, the glass-ceramic method consists of the

following steps as shown in Figure 2.1: 1) melting ceramic powders or glass frits at high

temperature to produce an amorphous liquid; 2) quenching the melt to a high viscosity

before the ions have time to form an ordered crystalline lattice; 3) annealing the glass near

the glass transition temperature to remove thermally induced stresses; 4) heating the glass

to the nucleation temperature to add the ionic mobility necessary for the formation of

crystal nuclei; and 5) heating the glass to the growth temperature to achieve the ionic

diffusion necessary for growth of the nucleated crystals. In practice, the unique








properties of some parent glasses may allow elimination of some processing steps, e.g.,

some phase-separating glasses such as BAS glass nucleate upon casting, allowing

production of a fine microstructure without a crystal nucleation step. Alternatively, it

may be beneficial to add a processing step, e.g., photosensitive glasses produce

heterogeneous nucleation sites when exposed to ultraviolet light.



Meltin


3 Growth
Nucleation
S \Annealin




Time

Figure 2.1. Diagram of the temperature-time cycle for the controlled crystallization of a
glass-ceramic body.


Ideally, at the temperature that corresponds to the maximum rate of crystal

nucleation, very little crystal growth occurs, and negligible crystal nucleation occurs at the

temperature corresponding to the maximum crystal growth rate (Figure 2.2). The

duration of the nucleation and growth treatments can be chosen to produce a crystal

population with whatever spatial distribution and size distribution are desired. The

optical and mechanical properties of the resulting glass-ceramic are a function of this

microstructure. In large-scale production, however, isothermal heat treatments are rarely

used, and the slow rates of heating and cooling that are employed allow less control over

the reaction kinetics.

























Temperature

Figure 2.2. Diagram of the variation of crystal nucleation and growth rates with
temperature.


Because the energy barrier to crystal nucleation is much lower at the external

surfaces than in the bulk of the material, crystallization usually initiates on the external

surfaces. This is followed by the crystals growing into the bulk and producing a

population of large crystals with a wide distribution of sizes. Small crystals of uniform

size are generally desirable for optimal mechanical properties, so nucleating agents such as

P205 are often added to provide heterogeneous nucleation sites and to promote bulk

crystallization.

The durations of nucleation and growth treatments necessary for crystallization

depend on ionic mobility within the glass matrix. Ionic mobility is frequently increased

through the addition of network modifiers, such as fluorides or alkali oxides, which

decrease the number of bridging oxygens in the glass network. This allows the use of

lower temperature furnaces or faster processing schedules.








2.1.2 BAS Glass-Ceramics

BAS glass-ceramics consist of a dispersion of BAS crystals surrounded by a

continuous glass matrix. These glass-ceramics have attracted considerable interest as

electronic and refractory materials because of their low dielectric constants and loss

tangents and their high-temperature stability. BAS exists in three different crystalline

polymorphs (Figure 2.3). The hexagonal form, hexacelsian (BaAl2Si20), is stable from

15900C to 17600C but tends to be the first metastable product of synthesis outside this

temperature range and persists below 17600C. Hexacelsian crystals have a hexagonal

plate morphology with a high aspect ratio. The crystal structure consists of di-

tetrahedral sheets of alternating silica and alumina tetrahedra, which are weakly bonded by

barium, calcium, or strontium ions (Figure 2.4). This weak bonding creates a basal (0001)

cleavage plane similar to that responsible for the excellent machinability of mica-

containing glass-ceramics (Chyung et al., 1972). Below 3000C, hexacelsian crystals

undergo a displasive transformation to the orthorhombic form with a 3% volumetric

shrinkage. This may lead to microcracking and makes hexacelsian-containing ceramics

unsuitable for thermal cycling applications. Previous research has focused mainly on

accelerating the sluggish transformation to the thermodynamically stable monoclinic form,

which is not susceptible to a displasive transformation and has a thermal expansion

coefficient that is compatible with that of the stoichiometric glass phase.














1 atm

o monoclinic U
celsian
0







I I I

0 500 1000 1500 2000
Temperature (OC)

Figure 2.3. Pressure-temperature phase diagram for barium aluminosilicate according to
Lin and Foster (1968).


(Si,AI)04


Figure 2.4. Diagram of the atomic structure of hexacelsian crystals according to Ito (1956).








Drummond and Bansal (1990) investigated BAS glass-ceramics as possible

matrices in high-temperature structural composites. They nucleated glass frits of

stoichiometric composition at either 850, 900, or 9500C for 1 h and then allowed crystal

growth to occur at 1080, 1150, 1220, or 12900C for 4 h. All combinations of thermal

treatments resulted in the precipitation of mostly hexacelsian crystals with minor

amounts of monoclinic celsian and residual glass. The glass-ceramic powders were cold

pressed at 414 MPa and sintered at 1200 to 15000C for 20 h. Three-point flexural

strengths ranged from 62 MPa to 124 MPa with maximum strength corresponding to the

lowest sintering temperature (Table 2.1). Glass-ceramic powders were also hot pressed

at 1200 to 14000C at a pressure of 24 MPa for 20 h. This resulted in a transformation to

mostly monoclinic celsian crystals for all hot pressing temperatures.


Table 2.1. Properties of BAS glass-ceramics cold pressed at 414 MPa and sintered in air
at various temperatures for 20 h according to Drummond and Bansal (1990).

Sintering Temperature Pressing Additive Density Flexural Strength
(C) (wt%) (g/cm3) (MPa)
1200 0 3.06 124
1300 0 3.00 117
1400 0 2.87 62
1500 0 2.91 69
1200 5 3.05 103
1300 5 2.98 83
1400 5 2.86 83
1500 5 2.89 83


Talmy et al. (1992) investigated solid solutions of BAS and strontium

aluminosilicate (SAS). They combined ceramic raw powders by substituting strontia for

baria in concentrations of 0, 2, 5, 10, 25, 50, 75, 90, 95, and 100 mol% and seeding with 1








to 10 wt% of monoclinic celsian crystals. Specimens were cold pressed and sintered at

1050 to 15000C for 5 h. Three-point flexural strengths ranged from 86 to 122 MPa with

maximum strength at 25 mol% SAS (Figure 2.5). Conversion of BAS from the hexagonal

form to the monoclinic form was augmented by both strontia substitution and monoclinic

seeding.

Zaykoski and Talmy (1994) synthesized BAS ceramics by replacing aluminum

with either 0, 50, or 100 mol% gallium and silicon with either 0, 50, or 100 mol%

germanium. Glass frits were sintered at 1250 to 15500C, and the strengths were

determined through three-point flexure. Gallium and germanium substitutions decreased

the sintering temperature necessary for conversion to monoclinic BAS; however, full

substitution produced the paracelsian form of BAS. Flexural strengths ranged from 102

down to 64 MPa with maximum strength corresponding to the unsubstituted BAS

ceramic (Table 2.2). This was attributed to the high volatility of germania that produced a

large amount of surface porosity in the substituted specimens.

Zhou et al. (1997) prepared BAS glass-ceramics by hot pressing powders that

were derived using the sol-gel method. They investigated stoichiometric BAS with

additions of either 5 wt% monoclinic celsian seeds or 1, 3, or 5 wt% lithia. Specimens

were hot pressed at 1100 to 12000C at a pressure of 7 to 15 MPa for 0.5 h and fractured

in a three-point flexural test. Lithia additions were effective in transforming hexacelsian

crystals to monoclinic celsian crystals, and the addition of monoclinic seeds produced

kinetics similar to those for 1 wt% lithia. The room temperature flexural strengths ranged

from 140 down to 122 MPa with maximum strength corresponding to stoichiometric BAS








(Figure 2.6). Addition of I wt% lithia enhanced the high-temperature strength (148 MPa

at 12000C) but decreased the room-temperature strength.


110


a 100

rs


0
SrO.A1,0O,2SiO


20 40 60 80 100
BaO*Al,0*2SiO2


Figure 2.5. Flexural strength of ceramics in the system SrO.Az203.2SiO2-
BaO.Al203.2SiO2 according to Talmy et al. (1992).




Table 2.2. Properties of ceramics of selected compositions in the BaAl2Si2Og-
BaGa2Ge2O8 system according to Zaykoski and Talmy (1994).


Ga:AI Ge:Si Sintering Temperature Flexural Strength
(C) (MPa)
0:2 0:2 1550 102
0:2 1:1 1500 76
0:2 2:0 1400 72
1:1 0:2 1450 87
1:1 1:1 1325 84
1:1 2:0 1275 76
2:0 0:2 1325 88
2:0 1:1 1250 68
2:0 2:0 1225 67








160
140
a-
S120
-100
80
60
X40
LL 20
0
0 1 2 3 4 5
Li2O (wt%)

Figure 2.6. The effect of LiO2 content on the strength of celsian glass-ceramics according
to Zhou et al. (1997).



2.2 Strength of Glass-Matrix Composites


To explain the improvement of strength in glass-ceramics or glass-matrix

composites in general, researchers have focused on the following three mechanisms:

1) limiting the inherent flaw size in the matrix material by decreasing the mean free path

between second-phase inclusions; 2) increasing the composite elastic modulus by adding

high-modulus second-phase inclusions; and 3) increasing the energy necessary for crack

extension through the interaction of cracks with second-phase inclusions.

2.2.1 Flaw Size Limitation

Griffith (1920) calculated the elastic energy stored in the vicinity of an elliptical

flaw which is oriented with its major axis perpendicular to the direction of applied stress.

Griffith showed that the elastic energy decreased as the flaw extended. He equated the

decrease in stored energy with the energy needed to form two new fracture surfaces








during crack extension, yielding the following equation for the strength of a material, o, in

terms of the flaw size, a:


co = 4E (2.1)
i ta

where y is the surface energy and E is Young's modulus.

Hasselman and Fulrath (1966) hypothesized that a dispersion of hard inclusions

would limit the size of Griffith flaws by decreasing the mean free path between crystals.

They predicted that the mean free path of such a composite would fall within one of two

regions (Figure 2.7). In Region I, the mean free path is larger than the mean flaw size.

Only the largest flaw sizes are eliminated so that the composite strength can be described

by the following expression:

o v (2.2)


where ao is the failure strength of the unreinforced glass, and Vv is the volume fraction of

the dispersed phase.

For higher volume fractions or smaller particle sizes, the mean free path falls

within Region II (Figure 2.7), and the sizes of all flaws are limited by the mean free path.

This results in a greater dependence of composite strength on the volume fraction of the

reinforcing phase, which Hasselman and Fulrath described by the following expression:

3yE Vv 12
o = [ (2.3)
r(I- Vv)








where r is the radius of the reinforcements. The experimental results of Hasselman and

Fulrath, including Regions I and II, are shown in Figure 2.7.

180

150 Region II

R120
20 Region I
|90 ,-,

i 60

30

0
0.00 0.05 0.10 0.15 0.20 0.25 0.30

Mean Free Path ( m-1/2 )

Figure 2.7. Plot of fracture strength versus reciprocal mean free path for glass specimens
containing a dispersion of alumina particles according to Hasselman and Fulrath (1966).


Several subsequent studies on glass-ceramics and other glass-matrix composites

(Bertolotti and Fulrath, 1967; Nivas and Fulrath, 1970; Stett and Fulrath, 1970; Hing and

McMillan, 1973; Sproull and Rindone, 1973) support the theory of flaw size limitation;

however, Borom (1977) re-examined these experimental data and concluded that an

increase in the elastic modulus of the composite structure was more likely to be

responsible for strengthening glass-matrix composites and that the inclusions may have

actually increased the mean flaw size. Mecholsky and Freiman (1980) and Swearengen et

al. (1978) confirmed that some reinforcing particles act as flaws which are larger than

those of the intrinsic population.








2.2.2 Stress Transfer

Frey and Mackenzie (1967) proposed that a glass-matrix composite can be treated

as a constant strain system. If the individual phases in a composite are considered well

bonded at their interfaces, then the strains can be assumed to be equal, i.e.,

d g
E = = E (2.4)

where the superscripts d, g, and c refer to the dispersed phase, the glass matrix, and the

composite structure, respectively. Using Hooke's law, Equation (2.4) can be rewritten as


--- (2.5)
E E E

In other words, the constituent phases share the load in proportion to their elastic moduli

with the more compliant matrix experiencing a lower stress than the rigid inclusions.

Substitution of the rule of mixtures for composite modulus in equation (2.5) gives


F (1-V, + V, (2.6)
"f E I

where the subscript f refers to the conditions at fracture. The strength of the composite

is expected to increase with increasing volume fraction and elastic modulus of the

dispersed phase.

Frey and Mackenzie (1967) tested this theory by adding alumina and zirconia

dispersions to raise the elastic moduli of three glass compositions. They concluded that

the glass-crystal composites acted like constant strain systems in that the strength was

dependent on the elastic modulus. They also noted that the strengthening was due in part





16


to crack interaction with stress fields brought about by thermal expansion mismatch

between the glass and crystalline phases.

Other researchers (Borom et al., 1975; Swearengen et al., 1978; Jessen et al., 1986)

subsequently confirmed the findings of Frey and Mackenzie. However, Jessen et al. and

Swearengen et al. reported a greater increase in strength than could be attributed to stress

transfer alone. Borom (1977) examined the existing literature and found that the data

which had previously supported the flaw size limitation theory of Hasselman and Fulrath

could also be explained in terms of stress transfer (Figure 2.8). He accounted for the

difference between expected and observed strengths by proposing that the dispersed

phase introduced flaws into the matrix in addition to increasing the elastic modulus of the

composite. This explains why strengths lower than that of glass alone are observed for

low volume fraction additions of a dispersed phase.

225


200


175 *15pm
3121 pm
A25pm
$ 150 i X32pm
X42 pm
a 51 pm
125 +60pm


100


75
0 10 20 30 40 50 60
Alumina Content (Vol%)

Figure 2.8. Flexural strength versus alumina content for various diameters of alumina
reinforcements according to Borom (1977).








2.2.3 Crack-Particle Interaction

Another approach to increasing the reliability of glass-matrix composites is to

impart sufficient fracture toughness so that the strength becomes insensitive to the size of

flaws. This approach has the advantage that appreciable postprocessing damage can be

tolerated. Also, fracture toughness provides a more objective measure of structural

reliability than does strength, since fracture toughness is an intrinsic material property

that is not dependent on the flaw distribution or specimen geometry. Numerous

mechanisms have been proposed in which a growing crack interacts with particles of the

reinforcing phase or the local stress fields associated with them to increase the fracture

toughness of composites. These mechanisms are described in the following section.

2.3 Fracture Toughness of Glass-Matrix Composites


2.3.1 Crack Bowing

Lange (1970) proposed that when a crack front encounters inclusions, it is pinned

in place and bows out between them, thereby increasing its overall length (Figure 2.9).

Assuming that a crack front has a line energy per unit length, this bowing would increase

the energy needed for crack propagation. Lange proposed that the increase in crack front

length should be related to the distance between pinning inclusions and the curvature of

the crack front just prior to crack extension. This hypothesis led to the following

expression for fracture energy of the composite material:


Y = Yo + (2.7)








where Yo is the fracture energy of the matrix material, T, is the critical line energy per unit

length of the crack front, and d is the distance between pinning inclusions. This

expression is similar to those derived from the theory of Hasselman and Fulrath

(Equations 2.2 and 2.3) in that both models incorporate the mean free path between

inclusions.














SCrack
h Direction




Figure 2.9. Diagram of the crack bowing mechanism proposed by Lange (1970).
Propagating cracks are pinned at adjacent inclusions and bow out between them until
increased local stress intensity enables the crack to break away from the pinning
positions.


Lange (1971) tested his theory by adding dispersions of alumina to a glass and

measuring the strength and fracture energy. The glass was formulated to minimize the

difference in thermal expansion between the matrix and alumina, thereby ensuring

negligible thermally induced stresses. Lange found that his model fit the experimental data

well if he modified it to include the size of the reinforcing particles.








A basic assumption of Lange's crack bowing model was that the crack front would

be semicircular in shape at the point when crack extension proceeded. Evans (1972) later

showed that this was an extreme case, and he developed a model in which the crack front

geometry was a function of the size and shape of impenetrable inclusions. Green et al.

(1979) subsequently derived a model for the case of weak inclusions and pores, which

predicts strength values higher than those they observed for a glass-nickel system.

2.3.2 Crack Deflection

When there is a difference in thermal expansion between an inclusion and the

surrounding matrix, the two phases contract at different rates upon cooling from high-

temperature processing. This creates a residual thermally induced stress field that is

locally associated with each inclusion. When the inclusion has a higher coefficient of

thermal expansion than the matrix, the resulting stress field in the matrix adjacent to the

inclusion consists of hoop compression and radial tension (Figure 2.10). Binns (1962),

Stett and Fulrath (1970), and Khaund et al. (1977) observed that a propagating crack was

deflected around the compressive stress field, thereby increasing the surface area of the

resulting fracture surface and hence the fracture energy. The contribution to fracture

toughness from an increased surface area mechanism is small; however, it was later

recognized that an important difference exists between crack bowing, which produces a

nonlinear crack front, and crack deflection, which produces a nonplanar crack. A

reduction in stress intensity results when the propagating crack is forced out of its

original plane of advance.





20


Faber and Evans (1983a) were the first to derive a quantitative relationship for

crack deflection contributions to fracture toughness. They predicted that the size of the

reinforcing particles would not be a determining factor, but that particle shape and volume

fraction would be important. For low volume fractions of the reinforcing phase, the crack

alters its path by tilting out of plane. For high volume fractions, the proximity of the

particles sometimes causes the crack to twist with both sides of the crack tilting in

opposite directions. Twist-derived toughening dominates whenever it occurs so that

higher volume fractions yield a greater degree of toughening. Crack deflection toughening

is also sensitive to the aspect ratio of the reinforcing particles since larger aspect ratios

result in higher fracture toughness, especially for plate-shaped inclusions (Figure 2.11).





\-a

matrix > Uparticle- -











amatrix < particle



Figure 2.10. Diagram of the effect of thermal expansion mismatch between the matrix
phase and the particulate phase on the path of a propagating crack.









Faber and Evans confirmed their predictions for spherical and rod-shaped particles

by measuring the fracture toughness of lithium aluminosilicate glass-ceramics and hot-

pressed silicon nitride (1983b). Sakai (1991) has criticized the a priori assumption of

crack deflection made by Faber and Evans without consideration of the strength of the

reinforcing particles or their bonding to the matrix material. Sakai points out that there

have never been experimentally reported toughness increases as high as the theoretical

predictions. More recently, Pezzotti (1993) conducted stereological simulations on

dispersions of plate-shaped reinforcements. Pezzotti refuted the analysis of Faber and

Evans on the basis that the spatial distribution function they employ considers only the

distance of the nearest-neighboring particle in calculation of the mean free path between

reinforcements. Pezzotti predicts that platelets can provide a maximum of 40% increase

in fracture toughness over that of the matrix alone.

4 Rod (AR 12)


Disks (AR = 12)

3-

Rods(AR=3)

P2 Disks (AR = 3)
Sphees (AR = I )




0 0.1 0.2 0.3 0.4
Volume Fracion ofPanrcles, Vf


Figure 2.11. Normalized toughness predictions for a crack deflection mechanism for
spherical, rod-shaped, and disk-shaped reinforcing particles with aspect ratios of 1, 3, and
12 according to Faber and Evans (1983a).








2.3.3 Crack Bridging

As a crack propagates through a composite material, strong reinforcing particles

may be left unfractured in its wake. For large cracks, a growing number of intact

reinforcements bridge the crack providing a closing force. The resistance to fracture

increases with increasing crack size, leading to the classic resistance-curve (R-curve)

behavior which is characteristic of toughening mechanisms that operate in the wake of the

crack-tip process zone (Figure 2.12). Knehans and Steinbrech (1982) demonstrated the

importance of the reinforcements by inducing stable crack propagation in polycrystalline

alumina and then removing the material from the crack wake. Upon reloading, they

observed a decrease in fracture toughness to that corresponding to small cracks.





Wake Process Zone




Frontal Process Zone

Crack Size

Figure 2.12. Diagram of the R-curve behavior of a brittle material subject to wake process-
zone toughening mechanisms.


Crack bridging is usually associated with ductile reinforcements that can dissipate

energy through plastic deformation instead of fracture; however, brittle reinforcements

with sufficient size and strength can bridge a crack and dissipate energy through

debonding from the matrix and frictional sliding (Figure 2.13). Long particles with high







strength and elastic modulus, as well as, a low interfacial strength are desirable to

maximize the contribution of debonding and pullout mechanisms (Evans and McMeeking,

1986; Budiansky et al., 1988). Although several theoretical models have been developed,

experimental evidence of brittle crack bridging has been limited to fiber-reinforced

composites and coarse-grained alumina (Knehans and Steinbrech, 1982; Swanson et al.,

1987).



(a)





(b)






(c) -'


Figure 2.13. Diagram of possible toughening mechanisms operating in the wake process
zone: (a) crack bridging, (b) debonding and pullout, and (c) stress-induced microcracking.


2.3.4 Microcracking

Binns (1962) and Davidge and Green (1968) found that thermally induced stresses

beyond a critical magnitude can generate a dispersion of cracks on the same size scale as

the microstructure. This spontaneous microcracking was observed to lower the strength

and elastic modulus of glass-matrix composites. When the glass has a lower coefficient of








thermal expansion than the reinforcing phase, microcracks form around the circumference

of the particles as they pull away from the matrix. When the glass has a higher coefficient

of thermal expansion than the reinforcing phase, microcracks propagate radially from the

particles. Coalescence of radial microcracks causes a greater degradation of the mechanical

properties. A similar phenomenon can also develop from thermal anisotropy within the

reinforcing phase (Evans, 1978).

At subcritical magnitudes of thermally induced stress, microcracks can increase

fracture toughness. As the crack tip advances near a pre-stressed inclusion, the stress

field associated with the crack tip is superimposed on the one associated with the

inclusion. If the combined stress surpasses a critical level, then a stress-induced

microcrack is developed. The development of microcracks during stable crack

propagation was detected acoustically by Evans et al. (1974) and explained by Hoagland

et al. (1975), Evans (1976), and Green (1981) who also predicted the shape and size of

the microcracked zone. Hoagland and Embury (1980) have verified the predicted shape of

the microcracked zone using a discrete computer model. Evans and Faber (1981; 1984)

later derived quantitative predictions for microcrack toughening by the following two

mechanisms: 1) dissipation of the stored elastic energy and 2) creation of a low-modulus

zone around the crack tip which shields it from applied loading. They showed that

particle size, interfacial strength, and particle size distribution are the most important

factors affecting microcrack toughening.

The potential for microcrack toughening seems limited because Evans and Faber

predict that a narrow particle size distribution with a mean particle size within 95% of








the critical size for spontaneous microcracking is necessary to achieve a significant effect,

and particle sizes larger than this result in a lower composite fracture toughness than that

for the glass matrix alone (Figure 2.14). Rice and Freiman (1981) also predict a strong

dependence of microcrack toughening on particle size. Because microcracks continue to

shield the crack in the wake process zone, this mechanism is expected to result in R-curve

behavior similar to that produced by crack bridging (Figure 2.12).




microcracked
process zone
0 (stress-induced)

I-I


B general
no discrete microcracking
microcracking (spontaneous)



rc
Mean Crystal Size

Figure 2.14. Diagram of the effect of particle size on the fracture toughness of a material
susceptible to microcracking. Little toughening is observed until the mean particle size is
close to a critical size, re. Fracture toughness decreases for larger particles.


2.3.5 Transformation Toughening

Transformation toughening has great potential as a toughening mechanism. It is

not covered in detail here because it requires volumetric expansion from a metastable

phase, such as partially-stabilized zirconia. The hexacelsian phase is the only metastable





26


phase found in the BAS system, and it undergoes a 3% volumetric shrinkage during the

transformation to orthorhombic celsian.













CHAPTER 3
MATERIALS AND METHODS

3.1 Materials Fabrication


3.1.1 Glass Composition

The BAS glass used in this study had the composition 31.9 wt% SiO2, 20.1 wt%

A1203, 22.0 wt% BaO, 8.43 wt% MgO, 2.97 wt% CaO, 8.23 wt% MgF2, and 6.42 wt%

P205. This is the composition used by Uno et al. (1991) in a study of high-strength mica-

containing glass-ceramics. A 154 kg batch of frit was obtained from Specialty Glass

Company (Oldsmar, FL). Compositional analyses of the glass were performed by Coors

Ceramics Analytical Laboratory (Golden, CO) prior to melting and after casting. A

summary of these analyses is presented in Table 3.1



Table 3.1. BAS glass compositions prior to melting and after casting.

Composition (wt%)
Component As-Received Frit Melt I Melt II
SiO2 31.9 32.1 31.9
A1203 20.1 20.4 20.1
BaO 22.0 21.9 22.0
MgO 8.43 9.07 9.73
MgF2 8.23 7.00 9.73
P205 6.42 6.37 6.43
CaO 2.97 2.99 2.98








3.1.2 Glass Melting and Forming

The glass frit was melted in a covered 150 cm3 ZGS platinum crucible (Johnson

Matthey, Seabrook, NH) in a refractory furnace (Deltech Model DT-31-RS-OS, Deltech

Inc., Denver, CO) heated by electrically resistant molybdenum disilicide heating elements.

The glass was melted at 15000C for 1 h to prevent excessive fluoride volatilization. The

glass melt was cast into a graphite mold (5.1 x 12.7 x 2.1 cm). After casting, the glass

plates were immediately placed in an electric muffle furnace (Model F6020, Thermolyne

Corporation, Dubuque, IA) at 6000C. The glass was annealed at 6000C for 1 h and

furnace cooled to room temperature. Once cooled, the glass plates were sectioned into

flexural test bars (24.0 x 3.5 x 6.0 mm) using a low-speed diamond-wheel saw (Model

650, South Bay Technology Inc., San Clemente, CA).

3.1.3 Thermal Crystallization Treatments

Crystallization of all glass specimens was carried out using one-stage isothermal

treatments. BAS glass is pre-nucleated upon casting, and thus no further nucleation

treatment was used. Crystal growth treatments of 0.5, 4, 32, and 256 h duration were

performed at 9750C to minimize thermocouple deterioration. All thermal treatments were

performed in an electric tube furnace (Model 54577, Lindberg Corporation, Watertown,

WI). After crystal growth, glass-ceramic specimens were annealed at 6000C and furnace

cooled to room temperature. A control group of glass specimens received no crystal

growth treatment but were subjected to the same annealing treatment as the glass-ceramic

specimens. Figure 3.1 illustrates the thermal history of the glass-ceramic specimens.








1600 Melt
1400
1200
1000 Crystal Growth
o 1000
800-
600 nneal Anneal Anneal
400
200


0 1 2 3 4 5 6 7 8 9 10 1112 13 14 15
Time (hours)

Figure 3.1. Thermal processing conditions for various stages of BAS glass-ceramic
development.


3.2 Microstructural Analyses


3.2.1 Phase Identification

X-ray diffraction was employed to identify the crystal phases which precipitated

from BAS glass over a range of temperatures. Glass specimens were heated from room

temperature to 10500C at 5C/min. Beginning at 6750C, specimens were removed from

the furnace and bench cooled at 15-min (750C) intervals. Using a mortar and pestle, each

specimen was ground and sieved to 325 mesh and then mixed with amyl acetate. The

resulting slurry was mounted on a soda-lime glass slide and analyzed in an X-ray

diffractometer (APD3720, Philips Electronic Instruments Inc., Mahwah, NJ) with a Cu

K1, source. Scans were conducted at an intensity of 1000 counts/min over a 20 range of

100 to 1200 at 50/min.








3.2.2 Crystal Morphology

The morphology of the crystalline phase was determined using the stereological methods

of Fullman (1953) for analyzing a distribution of circular plates. Four specimens from

each treatment group were sectioned at 900 and 450 angles to the length-wise direction

using a low-speed diamond-wheel saw (Model 650, South Bay Technology Inc., San

Clemente, CA). The sections were polished to a 1 gm finish using an alumina slurry and

acid etched using 1% aqueous hydrofluoric acid for 10 s to reveal their microstructures.

Two stereological fields were sampled from each section using an atomic force microscope

(Nanoscope III Scanning Probe Microscope, Digital Instruments, Inc., Santa Barbara, CA)

in the tapping mode. Fulhnan derived the following expressions for mean diameter (d),

thickness (t), and aspect ratio (i) of three-dimensional circular plates, respectively:

iGt 37 F
d= t= -,and a 3= (3.2,3.3,3.4)
2E 4E 32E

where E, F, and G are the mean reciprocal length, width, and aspect ratio of the two-

dimensional projections, respectively. Figure 2.2 illustrates the relationship between the

two-dimensional geometry measured and the three-dimensional geometry of interest.

3.2.3 Glass Morphology

Standard stereological techniques were used to determine the crystalline volume

fraction and the mean free path between crystals for each treatment group (Hilliard,

1968). Measurements of the two-dimensional projections of mica plates were made from

the stereological fields used for the determination of crystal volume fraction. A 5 x 5

square grid was placed on each field, and point count measurements were made. The size







of the grid was chosen so that no more than one point fell within a given microstructural

feature. The fraction of points falling within the crystalline phase was taken to equal the

volume fraction of the crystalline phase. Figure 2.3 shows a typical point sampling.

The square grid used for determination of crystalline volume fraction was

overlayed once again on the same stereological fields. A count was taken of the number of

intersections between horizontal grid lines and crystal-glass interfaces per unit of grid line

length. The mean free path between crystals, X, was calculated using the following

expression:

1 -V,
I= l-v (3.1)
0.5N,

where Vv is the crystalline volume fraction, and NL, is the number of lineal intercepts

with crystal-glass interfaces per unit length.










/





Figure 3.2. Diagram of a BAS crystal intersected by a stereological field. Measurements
from the sections of many crystals are used to calculate parameters for the crystal
population.






















Figure 3.3. A point counting grid superimposed on a typical stereological field for
determination of crystalline volume fraction and mean free path.


3.3 Physical Property Analyses


3.3.1 Density and Hardness

The densities of specimens were determined from measurements of mass and

volume. One monolithic specimen from each treatment group was ultrasonically cleaned

in ethanol. The dry weight of each specimen was determined using a precision balance

(Model HL52, Mettler Instruments Corp., Hightstown, NJ). The volume of each

specimen was measured using a helium micropycnometer (Model MPY-1, Quantachrome

Corp., Syosset, NY).

A microhardness tester (Model MO Tukon Microhardness Tester, Wilson

Instruments Inc., Binghamton, NY) with a Vickers diamond was used to measure the

hardness of four specimens from each treatment group. Hardness specimens were

flattened using a metal-bonded diamond abrasive disk. The surface to be indented was

then polished to a I tm finish using an alumina slurry. The specimens were each








indented in four different locations under a 4.9 N load. The dimensions of the indentation

diagonals were measured using an optical microscope with a filar eyepiece. The hardness

values were calculated according to the following equation:


2Psin
H _= (3.7)
a

where P is the indentation load, 0 is the angle between opposite diamond faces (1360),

and a is the mean diagonal length of the indentation.

3.3.2 Elastic Constants

Elastic constants were calculated for eight specimens in each treatment group from

the density and the velocity of sound through the material. The velocity of sound was

measured using an ultrasonic pulse apparatus (Ultima 5100, Nuson Inc., Boalsburg, PA).

Shear and longitudinal waves were generated using 5 MHz piezoelectric transducers

(SC25-5 and WC25-5, Ultran Laboratories, Inc., Boalsburg, PA). The transducers were

coupled to the specimens using honey and glycerin for shear and longitudinal waves,

respectively. The electronic delay in the pulse apparatus was subtracted from the time-

of-flight before calculating the velocity of sound. Poisson's ratios were calculated using

the following expression:




V = (v (3.5)

VL








where vs is the shear velocity, and VL is the longitudinal velocity. Young's modulus was

calculated using the following expression:

2
pv(I +v)(l-2v)
E = (3.6)
1-v

where p is the density.

3.3.3 Thermal Expansion

Square plates were cut to produce a bar, 50 x 5 x 5 mm. One glass specimen from

the control group was ground to a length of 50 0.001 mm for thermal expansion

measurement. The expansion was recorded using a single-pushrod dilatometer (Model

EK2, The Edward Orton Jr. Ceramic Foundation, Columbus, OH). The specimen was

heated from 250C to 6000C at a rate of 30C/min. The coefficient of thermal expansion

was calculated from the cooling curve to eliminate the effects of residual thermally

induced stress.

3.3.5 Flexural Strength

Four-point flexural bars were prepared by flattening all sides of a rectangular bar

(24.0 x 3.5 x 6.0 mm) using a metal-bonded diamond abrasive disk. The side to be placed

in tension was then polished to a 1 [tm finish using an alumina slurry. All specimens

were annealed at 600C for 4 h following polishing to eliminate any compressive surface

stresses. After annealing, the lengthwise edges on the tensile surfaces were beveled to

prevent fracture initiation from the specimen edges. Eight specimens per treatment group

were indented with a Vickers diamond at either of two indentation loads (4.9 or 9.8 N).








Each specimen was indented at three positions within the inner span length. Prior to

indentation, a drop of silicone oil was placed on the tensile surface to inhibit moisture-

assisted slow crack growth. All specimens were aged in ambient conditions for 24 h

before fracture to allow stabilization of the controlled flaw size. The specimens were

fractured using a Instron testing machine (Model 1125, Instron Corporation, Canton,

MA) and a four-point bending apparatus (19.9 mm outer span, 6.4 mm inner span).

Figure 3.4 illustrates the geometry of the loading apparatus. The specimens were loaded

monotonically to fracture at a rate of 0.5 mm/min. The flexural strength was calculated

using the following expression:

3Px
o, = (3.8)
wt

where P is the failure load, x is the distance between the inner and outer supports, w is

the specimen width, and t is the specimen thickness.

P/2 P/2




I t




x

Figure 3.4. Diagram of the four-point bending apparatus used for the determination of
flexural strength.








3.3.6 Fracture Toughness

Because fracture toughness was the material property of greatest interest in this

study, three different methods were used to collect fracture toughness data. Each has its

own advantages and disadvantages. Fractographic analysis involves direct examination of

the fracture surface and measurement of the critical flaw from which fracture originated.

Using this method, the indentation can be confirmed to be the site of fracture initiation.

However, this method requires a great deal of experience, and many researchers have not

acquired the skill necessary to make accurate measurements of critical flaw size (Quinn et

al., 1994).

The indentation-strength technique developed by Chantikul et al. (1981) does not

require a measurement of the critical flaw size. Instead, a flaw size is assumed from the

indentation load used to induce controlled flaws. This method eliminates some of the

investigator-related error from data collection, but it is not accurate if environmentally-

assisted slow crack growth occurs before fracture.

The modified indentation technique developed by Cook and Lawn (1983) involves

inducing multiple controlled flaws and measuring the trace of the flaws which remain

intact within the tensile surface after fracture. The boundaries of these intact flaws are

usually easier to discern than those of the critical flaw. Because intact flaws are concealed

within the tensile surface, only their widths and not their depths can be observed. These

data may be misleading if the flaw is semi-elliptical instead of semi-circular in shape

(Chan, 1996).








3.3.6.1 Fractographic analysis

Mecholsky et al. (1978) have established a protocol for determining fracture

toughness via analysis of fracture surface features. The flexural strength specimens from

Section 3.3.5 were used with the flaw size as determined through quantitative

fractography to calculate fracture toughness.

The fractured pieces of each four-point flexural bar were ultrasonically cleaned in

an aqueous detergent, followed by ethanol, and then sputter-coated with a gold-palladium

alloy (Hummer II, Anatech Ltd., Alexandria, VA). The fracture surfaces were examined

with an optical microscope, and the fracture surface features were measured using a filar

eyepiece. The characteristic fracture surface features exhibited by glass-ceramics are

shown in Figure 3.5.

After measuring the major and minor axes of the semi-elliptical flaws, the radii of

the equivalent semi-circular flaws were calculated using the following relationship

(Mecholsky et al., 1977):

c= (ab)' 2 (3.9)

where a and b are the depth and half-width of the flaw, respectively. The fracture

toughness of each specimen was then calculated using the Griffith-Irwin equation:


Kc = Yo,c (3.10)

where of is the strength from Equation (3.7) and Y is a shape parameter that has been

determined by Marshall et al. (1983) to be 1.65 for the residual stress associated with an

indentation.









Critical Flaw Hackle Region


Macroscop'c vvvv vv vv vv
Mr 'gion

Crack Branch-,.. !,, ,, ,,, ,, ,, 1 .r... Region
.\All Vi..1j I.I
\ ''i/I \' i.\ f

ifi \I l I J',' /
*.U;. I ri uL*' < I1i'l /

ri0




Figure 3.5. Diagram of the typical fracture surface features occurring in brittle materials.
The regions are not drawn to scale.



3.3.6.2 Indentation-strength technique


The strength values already determined in Section 3.3.5 were used to determine

fracture toughness using the following equation:



K,c =wrlv H p ",P (3.11)


R
where tv is a geometrical constant that Chantikul et al. (1981) determined to be 0.59 for


a Vickers diamond, E is Young's modulus, H is the hardness, of is the fracture stress from


Equation (3.7), and P is the indentation load (4.9 or 9.8 N).

3.3.6.3 Modified indentation technique


Three controlled flaws were induced at different sites within the constant-stress,

inner-span region of the four-point flexure specimens (Section 3.3.5). Because of slight

inhomogeneities in the loading system, only one of the three flaws grew past critical size;

however, the two remaining intact should have grown to nearly critical dimensions. The








half-width of the surface trace from the largest of the two intact flaws was measured for

the crack size, c,. Figure 3.6 shows the location of the intact flaw on a fracture specimen.

For glass-ceramic materials, the equation suggested by Cook and Lawn (1983) for

calculation of fracture toughness yields values that do not agree with those determined

using the fractographic and indentation-strength techniques (Chan, 1996; Hill, 1998).

Instead, the Griffith-Irwin equation (3.9) with the modified crack size, cm, was used to

calculate fracture toughness. The flaw depth, a, of the intact semi-elliptical flaws was not

visible from the surface trace, so the sizes of equivalent semi-circular flaws were

a
estimated using Equation 3.8 with the mean ellipticity of the critical flaws, = 0.804, as
b

a conversion factor. The resulting equation for calculation of modified-indentation

fracture toughness was as follows:

Kic = YO,(0.804)'/ c2 (3.12)

where of is the strength from Equation (3.7), Y is a shape parameter that has been

determined by Marshall et al. (1983)to be 1.65 for the residual stress associated with an

indentation, and Cm is the half-width of the intact flaw.



Intact Vickers Indentation




Critical Flaw

Figure 3.6. Diagram of the location of an intact controlled flaw on a fractured four-point
flexure specimen. The flaws are not drawn to scale.








3.5 Finite Element Analyses


Finite element stress analyses were used to estimate the magnitude of the

thermally induced stresses that develop adjacent to and between BAS crystals in the glass

upon cooling from processing temperatures to ambient temperature. The ANSYS

computer code (ANSYS 5.3, Swanson Analysis Systems, Inc., Houston, PA) was used to

perform these analyses. Finite element meshes were generated using a two-dimensional,

eight-node quadratic element (PLANE82). Models were developed for pairs of BAS

crystals in a parallel edge-to-edge orientation. This geometry was examined using the

mean crystal size and the mean free path between crystals determined for each treatment

group in Sections 3.2.2 and 3.3.3. Figure 3.7 shows the model geometry and boundary

conditions for the finite element simulations.





















Figure 3.7. Finite element model for a pair of BAS crystals in a glass matrix. A zero-
displacement boundary condition is applied at the top and left edges. A coincident node
condition is applied at the bottom and right edges and along the crystal-glass interfaces.





41


The material properties used to model the glass phase were determined using the

methods described in Sections 3.3.1 and 3.3.3. The following material properties were

used for the glass phase: elastic modulus (E) = 94.1 GPa, Poisson's ratio (v) = 0.284, and

coefficient of thermal expansion (a) = 7.64 ppm/C. The material properties used to

model the crystal phase were calculated using the rule of mixtures and compared to values

in the existing literature (Zhou et al., 1997). The following material properties were used

for the crystal phase: elastic modulus (E) = 110 GPa, Poisson's ratio (v) = 0.257, and

coefficient of thermal expansion (a) = 7.4 ppm/C. Thermally induced stresses were

estimated for cooling from the glass transition temperature, 6000C, to room temperature,

23C. The source codes for these models are listed in Appendix A.













CHAPTER 4
RESULTS AND DISCUSSION

In the following sections, the data collected for microstructural and macroscopic

properties of BAS glass and glass-ceramics are presented primarily in a graphical format

(Sections 4.1 and 4.2). The mean properties + 95% confidence intervals are compared for

glass-ceramics with crystal growth treatments of 0.5, 4, 32, and 256 h in duration and

contrasted with those of the base glass. The data for individual specimens are listed in

Appendix B. The residual thermally induced stress distributions predicted by finite

element analyses are presented (Section 4.3), and the roles of several mechanisms in

strengthening and toughening of BAS glass-ceramics are discussed (Section 4.4). A

quantitative estimate is made of the relative contribution of each mechanism to the

observed strength and fracture toughness, and an optimal process is identified for

producing BAS glass-ceramics from the selected glass composition. Future modifications

for optimizing processing time and mechanical properties are suggested.

4.1 Microstructure


4.1.1 Phase Identification

Figure 4.1 shows the x-ray diffraction spectra for the crystalline phases that

precipitated from the base glass at temperatures ranging from 675 to 10500C at 750C

intervals. Comparison of these data with JC-PDS records revealed that only the









hexacelsian crystal phase was present. Uno et al. (1993) reported that a phlogopite


crystal phase precipitated from base glass of the same composition when subjected to


similar crystallization treatments. The atomic structures and physical properties of


hexacelsian and phlogopite are similar and the crystal morphologies are identical.



(002)
(112)130




(221)


1113)
(OO ) (003) (004)
1050 C






8215, C

750C

675"C


10 20 30 40 50 60 70 80
Diffraction Angle (20)



Figure 4.1. X-ray diffraction spectra for BAS glass-ceramics. Hexacelsian (BaAl2Si2Os) is
the only phase present. Peak intensity is a function of processing temperature.



4.1.2 Crystal Morphology


AFM micrographs of polished and etched BAS glass-ceramic microstructures for


crystal growth treatments of 0.5, 4, 32, and 256 h in duration are shown in Figures 4.2 to


4.5, respectively. The crystals exhibited a hexagonal plate morphology in all specimens.








Figure 4.6 shows that no change in crystalline volume fraction occurred during crystal

growth. Figure 4.7 shows the increase in mean crystal diameter and thickness with

increasing crystal growth time. The crystalline volume fraction was observed to remain

constant at 0.76 0.02 over time. The mean crystal size increased over time, but the

mean crystal aspect ratio, 3.81 0.39, was independent of crystal size.


The crystallization of most glass-ceramic systems involves the transformation of

an increasing amount of glass phase into ceramic phase over time. In BAS glass-ceramics,

this process is completed within 30 min. Thereafter, a coarsening process occurs

whereby the crystals smaller than a critical size dissolve into the matrix, and material from

them is reprecipitated onto the larger crystals. Juma'a and Parker (1981) investigated

calcium fluoride glass-ceramics in which crystal growth occurs through a coarsening

process. They reported that, because the addition of fluoride increased ionic mobility

within the glass, the growth of ceramic crystals was limited by the rate of the

dissolution/precipitation reaction at the crystal surface and not by the diffusion rate of

ions through the glass matrix. Consequently, the mean crystal size increased in direct

proportion to the cube-root of the growth treatment duration, i.e., eight times the

processing time is necessary in order to double the mean crystal size. The growth

kinetics of BAS glass-ceramics fit this model very well (R2 = 0.998).



























4 pm


Figure 4.2. Atomic force micrograph of the microstructure of a BAS glass-ceramic
produced by a crystal growth treatment at 9750C for 0.5 h.


Figure 4.3. Atomic force micrograph of the microstructure of a BAS glass-ceramic
produced by a crystal growth treatment at 9750C for 4 h.



























4. .Am


Figure 4.4. Atomic force micrograph of the microstructure of a BAS glass-ceramic
produced by a crystal growth treatment at 9750C for 32 h.


Figure 4.5. Atomic force micrograph of the microstructure
produced by a crystal growth treatment at 9750C for 256 h.


of a BAS glass-ceramic









1.00




0.70

0.60

0.50







0.1 1 10 100 1000
0.40
0.30 0 : :.. ,-.




0.1 1 10 100 1000

Crystal Growth Time (h)

Figure 4.6. Effect of crystal growth time on the crystalline volume fraction of BAS glass-
ceramics.


2.50


2.00



1.50
N


S1.00
u

0.50


0.00


1 10 100
Crystal Growth Time (h)


Figure 4.7. Effect of crystal growth time on the morphology of the crystalline phase in
BAS glass-ceramics.








4.1.3 Glass Morphology


Figure 4.8 shows the change in the mean free path between crystals over time. The mean

free path was observed to increase over time during crystal growth. Accordingly, as the

microstructure coarsened and the crystals became larger, the glass regions between

crystals also increased in size. This is contrary to the usual case. In most glass-ceramic

systems, the mean free path decreases during crystal growth.



3.00


2.50


2.00


1.50


1.00


0.50


0.00
0.1 1 10 100 1000
Crystal Growth Time (h)

Figure 4.8. Effect of crystal growth time on the mean free path between crystals in BAS
glass-ceramics.



4.2 Physical Properties


4.2.1 Density and Hardness


The mean apparent density values for materials with crystal growth times of 0,

0.5, 4, 32, and 256 h are summarized in Figure 4.9. One-way ANOVA showed that the








apparent density increased with increasing crystal growth time (p = 0.0001). A

theoretical density of 3.941 g/cm3 was calculated for the hexacelsian crystals. According

to the rule of mixtures, an increase from the initial glass density of 3.141 0.004 g/cm3 to

a density of 3.748 g/cm3 for the 0.5 h crystal growth treatment glass-ceramic was

expected. Thereafter, the density was expected to remain constant because the crystalline

volume fraction remained constant for longer crystal growth treatments. Instead, a slow

continuous increase in density was observed. This may have been caused by the

existence of internal porosity that was eliminated during high-temperature processing.

The mean hardness values for materials with crystal growth times of 0, 0.5, 4, 32,

and 256 h are summarized in Figure 4.10. One-way ANOVA showed a significant

difference in hardness between materials with different crystal growth times (p = 0.0001)

with the highest hardness values corresponding to an intermediate (0.5 h) crystal growth

time. The initial increase in hardness corresponds to the initial transformation of the

lower hardness glass phase to the higher hardness crystal phase. For longer treatment

times, the hardness decreases as the fraction of crystalline phase remains constant and the

mean crystal size increases. Chyung et al. (1972) investigated mica-containing glass-

ceramics which evolved microstructures similar to those observed in the present study.

They reported that materials with a large mean crystal size exhibited lower hardness

values because frictional sliding at the glass-mica interfaces allowed crushing to occur.



























1 10 100
Crystal Growth Time (h)


Figure 4.9. Effect of crystal growth time on the apparent density of BAS glass and glass-
ceramics.


;" ":.-.- -o-'." -i..""'"' '.a

P -':, '': "" '
-.J- A'.


hr;


. *.. .. :.. '..9

. 3.., ..... ; .- -- :'. !:.?';
=";t~.l'""--- ^ ^
i *r\ i : ', f
i~fiv?; '***':' -- *' '*^


100


Crystal Growth Time (h)


Figure 4.10. Effect of crystal growth time on the hardness of BAS glass and glass-
ceramics.


3.13 L-
0.1


6.5



S6.0
e.o



5.5



> 5.0








4.2.2 Elastic Constants


Figure 4.11 shows the elastic moduli for BAS glass and glass-ceramics. Although

the crystalline volume fraction remained constant, the Young's modulus, shear modulus,

and bulk modulus increased with increasing crystal growth time. This may have been

caused by an increase in the elastic anisotropy of BAS crystals with increasing crystal

size. Likewise, there is a lack of single-crystal data in the existing literature. The only

elastic moduli available are averaged over all the crystallographic directions because the

microstructures of BAS glass-ceramics are fine. The inaccuracies resulting from averaging

the elasticity of BAS crystals may become more evident at larger mean crystal sizes.



120


100


80




40





0.1 1 10 100 1000
Growth Time (h)

Figure 4.11. Effect of crystal growth time on Young's modulus, shear modulus, and bulk
modulus of BAS glass and glass-ceramics.








4.2.3 Thermal Expansion

Push-rod dilatometry was used to measure the coefficient of thermal expansion for

the base glass, which was 7.64 ppm/C. Zhou et al. (1997) reported that the thermal

expansion of hexacelsian glass-ceramic with 2% residual glass was 7.4 ppm/iC. Although

the weak bonding along the basal (0001) cleavage plane suggests a higher thermal

expansion coefficient along the [0001] direction of BAS crystals, the high mean crystal

aspect ratio indicates that the maximum thermally induced strain occurs in <1000>

directions.

4.2.4 Strength and Fracture Toughness

Figures 4.12 and 4.13 show the strength and fracture toughness, respectively, of

BAS glass and glass-ceramics. Both strength and fracture toughness increase with

increasing crystal growth time over the entire range of treatments investigated.

Because the measurements necessary for fracture toughness calculation via

fractographic analysis require a great deal of skill and experience to collect accurately, two

other methods were used to check the accuracy of the fractographic data. Comparison of

the mean fracture toughness values calculated by the indentation strength technique

(Section 3.3.6.2) and modified-indentation technique (Section 3.3.6.3) confirms the values

calculated using fractographic analysis (Section 3.3.6.1). One-way ANOVA revealed no

significant difference among the three toughness measurement methods (p > 0.05).


















80 ; : ?;

60 -

40

20 ...


0.1 1 10


1000


Crystal Growth Time (h)

Figure 4.12. Effect of crystal growth time on the flexural strength of BAS glass and glass-
ceramics.


-. Ir3r. o lr .:.rSlrergih Ternnir ue
S:;4: A .1.fa '. -3 Ir..n.ac n TEcr.i'.que
U I-. .


Growth Time (h)


Figure 4.13. Effect of crystal growth time on the fracture toughness of BAS glass and
glass-ceramics.


S2.0


1.5


1.0


0.5
o



0.0
0.1








Uno et al. (1991) used glass with the same composition and similar thermal

treatments to develop glass-ceramics with phlogopite as primary crystal phase. They

reported strength and fracture toughness values of 350 MPa and 2.3 MPa.m/2,

respectively, for a crystal growth treatment of 2 h at 10500C. The difference between the

observations of Uno et al. and those of the present study is more likely caused by a

difference in test methods than a difference in crystal phase, because the phlogopite and

BAS crystals have a similar structure and an identical morphology. Uno et al. measured

strength using a three-point flexural test without a surface polish and without the

introduction of a controlled flaw. Because the surface flaw population in their test

specimens was not controlled, it is impossible to make a comparison between the strength

values they reported and those observed in the present study. Uno et al. measured

fracture toughness using a chevron-notched beam test, which can overestimate fracture

toughness. Such an overestimate may be responsible for the large standard deviation they

reported for the treatment group with the highest fracture toughness. Disregarding the

suspect treatment group, the fracture toughness data of Uno et al. are similar to those

observed in the present study and are well described by the predictions of Faber and

Evans (1983a) for toughening caused by crack deflection.

4.3 Finite Element Predictions


Figure 4.14 shows the results of a finite element analysis that predicts the stress

distribution surrounding BAS crystals embedded in a glass matrix. A tensile hoop stress

and a compressive radial stress were predicted to develop in the matrix around each








crystal upon cooling from processing temperatures. Table 4.1 lists the maximum

predicted principal stresses that result from each of the crystal growth treatments

investigated. The stress fields associated with individual crystals overlap in the glass-

ceramic crystallized for 0.5 h; however, the stress fields separated as the mean free path

increased for longer crystal growth times. Actual stress magnitudes may be somewhat

lower than those predicted because the comers of BAS crystals are not likely to be

atomically sharp.


Figure 4.14. Residual stress field predicted by finite element analysis for a BAS glass-
ceramic produced by a crystal growth time of 0.5 h. The vectors represent the
magnitudes and directions of the principal stresses.









Table 4.1. Results of finite element analyses to predict the maximum stresses in BAS
glass-ceramics.


Growth Time 1' Principal Stress* 2nd Principal Stress* 3"d Principal Stresst
0.5 h 14.4 2.74 6.13
4h 10.2 1.92 21.9
32 h 10.5 2.05 21.8
256 h 11.3 2.13 27.5
Stresses are given in MPa.
tStresses are given in 104 MPa.


4.4 Strengthening and Toughening Mechanisms


4.4.1 Flaw Size Limitation

The theory of flaw size limitation (Hasselman and Fulrath, 1966) predicts that the

strengths of glass-matrix composites vary inversely with the square-root of the mean free

path between the reinforcing particles, i.e., composite strength increases with decreasing

mean free path. The opposite trend was observed for BAS glass-ceramics. Summarized

in Table 4.2 are comparisons between the strength values predicted by the flaw size

limitation theory with the experimental values. The observed flaw sizes were larger than

those predicted by the theory of flaw size limitation (Equation 2.3); however the

strengths predicted for the observed flaw sizes can be calculated using the Griffith

equation (2.1). The observed strengths were higher than those predicted by the Griffith

equation. For BAS glass-ceramics and other fine-grained highly-crystalline glass-ceramics,

the theory of flaw size limitation predicts composite strengths which are three orders of

magnitude different from those observed. Flaw size limitation is not responsible for the








strengthening of these materials. Indeed, Mecholsky and Freiman (1980) reported that

the mean flaw size increased instead of being limited for glass-ceramics with increased

fracture toughness.


Table 4.2. Comparison of observed strength values with those predicted for flaws of
limited size and flaws of observed size for BAS glass and glass-ceramics.

Growth Predicted Strength For Predicted Strength For Observed Strength*
Time Limited Flaw Size* Observed Flaw Size*
Oh 63.4 63.4 63.4 + 8.0
0.5 h 2450 71.0 87.0 13.0
4 h 1790 72.4 94.8 7.0
32 h 1300 83.1 118 + 3
256 h 1010 76.0 141 + 12
All strength values are given in MPa.


4.4.2 Crack Bowing

The theory of crack bowing (Lange, 1970; Evans, 1972; Green et al., 1979)

predicts that the strength of a glass-matrix composite will increase as the length of the

subcritical crack front increases. The crack front length is a function of the mean free path

between particles, the surface energy of the matrix, and the strength of the reinforcing

particles. Figure 4.15 shows the relative strength increase predicted by the theory of

crack bowing for the ratios of mean particle diameter to mean free path observed in BAS

glass-ceramics. Glass-matrix composites that toughen through a coarsening reaction, such

as BAS glass-ceramics, exhibit increased strength with increased mean free path. This is

the opposite of the trend predicted when crack bowing is active. Also, the predictions of

Green et al. do not fit the strengths observed for composites reinforced by brittle








particles, such as BAS crystals. A significant degree of bowing is more likely when the

reinforcing particles are ductile, so crack bowing is not responsible for the strengthening

of BAS glass-ceramics.

3.5


rK-

-. ,., ... ... .....'. .- -
3 .0 ;
2.5



2.0
0.0' L" .. .... ** ........-----------.. *...i.. =- s--J .;..
1.0 x :'" --. :


0.5

0 1 2 3 4 5 6 7
2d/X

Figure 4.15. Comparison of the observed ratio of BAS glass-ceramic strength to base
glass strength, oWf/o, with that predicted by the theory of crack bowing for the ratios of
mean crystal diameter, d, to mean free path, X.


4.4.3 Wake Process-Zone Mechanisms

When wake process-zone toughening mechanisms such as crack bridging,

interfacial debonding, frictional pullout, stress-induced microcracking, and phase-

transformation toughening (Sections 2.3.3 to 2.3.5) are active, fracture toughness increases

with increasing flaw size because the wake process zone increases in length as a crack

propagates through a material. Figure 4.16 shows the fracture toughness of BAS glass

and glass-ceramics as a function of flaw size. The flaw sizes ranged from 44 to 98 tm for









2 0 0 ,-] .. .- -. ...
N" 1 8 "^* .' .

4 8 1
1.4





















indentation loads of 4.9 to 9.8 N, and fracture toughness was independent of indentation
0 ".7 +++....






























composite. Figure 4.17 shows a BAS glass-ceramic specimen that was indented to
2 O.6 0.. 6. ,
0.4 44 +4
0.2 L- +; ...;; : '. i
0.0

40 50 60 70 80 90 100
Critical Flaw Radius (.tm)


Figure 4.16. Effect of controlled flaw size on the fracture toughness of BAS glass and
glass-ceramics. Controlled flaws were induced using a Vickers diamond under a load of
4.9 or 9.8 N.


indentation loads of 4.9 to 9.8 N, and fracture toughness was independent of indentation

load (p = 0.50) over this range of flaw sizes. These data indicate that wake process-zone

toughening mechanisms do not make a significant contribution to the fracture toughness of

BAS glass-ceramics for the range of flaw sizes examined.

Crack bridging in particular should be visually evident when it is active in a

composite. Figure 4.17 shows a BAS glass-ceramic specimen that was indented to

produce radial cracks and acid-etched to reveal the microstructure surrounding these

cracks. Visual examination revealed no intact reinforcements lying in the crack wake.





60





















Figure 4.17. SEM micrograph of a radial crack (from a Vickers indentation induced under a
load of 9.8 N) in a BAS glass-ceramic produced by crystal growth at 9750C for 32 h.



2. r:.' .


Z-C










0.0
0 0.4 0.8 1.2 1.6 2
Mean Crystal Diameter (gjm)

Figure 4.18. Observed effect of mean crystal diameter on the fracture toughness of BAS
glass and glass-ceramics compared to that predicted by Rice and Freiman (1981).





61


Microcracking in particular should be strongly dependent on the size of the

reinforcing crystals, within 5% of a critical particle size (Rice and Freiman, 1981).

Figure 4.18 shows the fracture toughness of BAS glass and glass-ceramics as a function of

crystal diameter. Although mean crystal diameters ranged over an order of magnitude, a

precipitous decrease in fracture toughness associated with spontaneous microcracking

was not observed. The results of the finite element analysis presented in Figure 4.14

suggest a residual tensile hoop stress surrounding BAS crystals. Above a critical crystal

diameter, this type of stress distribution results in radially-oriented microcracks which

can be detrimental to glass-ceramic strength. However, the maximum tensile stress

predicted for the largest crystals (1.76 im in diameter) is only 11.3 MPa, much lower

than the strength of the glass matrix (63.4 8.0 MPa) for flaws larger than the mean

crystal diameter. Thus, the critical crystal diameter must be much larger than the

observed mean crystal diameter.

4.4.4 Stress Transfer

The theory of stress transfer (Frey and Mackenzie, 1967) predicts that the ratio

of the strength of a glass-matrix composite to the strength of the base glass will increase in

direct proportion to the ratio of the elastic moduli. Previous studies have verified the

accuracy of this relation (Borom et al., 1975; Borom, 1977; Swearengen et al., 1978;

Jessen et al., 1986). Figure 4.19 shows the predicted and observed relative strengths of

BAS glass-ceramics. BAS glass-ceramics were observed to follow the general trend

predicted by stress transfer; however, stress transfer cannot account for all of the strength








increase that occurred. Jessen et al. reported similar results for glass reinfored with

spherical Fe-Ni-Co alloy particles. Jessen et al. attributed the excess strength increase to

crack deflection caused by residual thermally induced stress around the metal particles.

Because the predicted contributions of stress transfer are well established, 100% of the

strength increase attributed to this mechanism is expected for BAS glass-ceramics.


2.4

2.2 Oz-er ed








1.0
1.4





1.0 1.1 1.2 1.3 1.4 1.5
E/Eg

Figure 4.19. Comparison of the observed relationship between the ratio of composite
strength to base glass strength, o~ lo/, and the ratio of composite elastic modulus to base
glass elastic modulus, E/Eg, with that predicted by the theory of stress transfer.



4.4.5 Crack Deflection

For reinforcing particles with a lower coefficient of thermal expansion than the

matrix, the theory of crack deflection predicts that a crack will be deflected towards the

particles, decreasing the stress intensity at the crack tip. This is the condition that exists

in BAS glass-ceramics. Figures 4.20 to 4.23 show micrographs of BAS glass-ceramics





























Figure 4.20. SEM micrograph of a radial crack from a Vickers indentation induced under a
load of 9.8 N in a BAS glass-ceramic produced by crystal growth at 9750C for 0.5 h.


Figure 4.21. SEM micrograph of a radial crack from a Vickers indentation induced under a
load of 9.8 N in a BAS glass-ceramic produced by crystal growth at 9750C for 4 h.





























Figure 4.22. SEM micrograph of a radial crack from a Vickers indentation induced under a
load of 9.8 N in a BAS glass-ceramic produced by crystal growth at 9750C for 32 h.


Figure 4.23. SEM micrograph of a radial crack from a Vickers indentation induced under a
load of 9.8 N in a BAS glass-ceramic produced by crystal growth at 9750C for 256 h.








that were indented with a Vickers diamond under 9.8 N loads to produce radial cracks and

subsequently acid-etched to reveal the microstructure surrounding these cracks. Crack

deflection was visually confirmed in the glass-ceramic with the smallest mean crystal

diameter (Figure 4.20). In the glass-ceramic with the largest mean crystal diameter (Figure

4.23), the crack front was observed to follow the interface, at some points turning

perpendicular to the initial direction of propagation.

After subtracting the contribution of stress transfer, the remaining strength

increase that must be accounted for by other mechanisms can be calculated. The

toughness, Qc, can be expressed in terms of the fracture toughness, Kic, elastic modulus,

E, failure stress, of, and critical flaw size, c, as follows:


v (Ya.rEc)2
tc= KC- E (4.1)
E E

The extra increase in normalized toughness can be expressed as

_GC,,xp -aC,
A9. C, (4.2)


where Qc,,. is the experimentally observed toughness, C,, is the theoretical toughness


caused by stress transfer, and ( is the toughness of the base glass. Substitution of

Equation 4.1 for the toughness values in Equation 4.2 yields the following equation for

the extra increase in normalized toughness:


A 0E0g2 (4.3)

Eof








where E is the elastic modulus of the glass-ceramic, E' is the elastic modulus of the base

glass, of,, is the experimentally observed failure stress, a,, is the theoretical failure


stress associated with stress transfer, and o, is the failure stress of the base glass.

Table 4.3 shows the normalized toughness increase observed for BAS glass-

ceramics and the extra increase not accounted for by the contribution of stress transfer.

Faber and Evans (1983) predicted a normalized toughness increase of 0.62 without crack

front twisting and an increase of 1.43 with maximum twisting for disk-shaped plates with

an aspect ratio of 3.0 (Figure 4.24). After subtracting the stress transfer contribution, the

experimental data for BAS glass-ceramics show an increase in toughness within the range

predicted by Faber and Evans. Table 4.4 shows the estimated relative contributions of

stress transfer and crack deflection mechanisms to the observed strength increases. The

relative contribution from crack deflection is larger for BAS glass-ceramics with a larger

mean crystal size. This may be caused by an increase in the proportion of crack twisting

versus crack tilting during crack deflection.


Table 4.3. Normalized toughness increase, A;c, observed for BAS glass-ceramics and
toughness increase not accounted for by the contribution of stress transfer, A s.


Growth Time AQe Aqxs
0.5 h 0.51 0.00
4 h 0.78 0.02
32 h 1.13 0.05
256 h 3.58 1.13








2.6 "
T.II uric. T -._1i .- .% ., ..,, ,
2.4 IlI P.r* u *""

2 2.4 -- T, r Exira'... ai. .-I" +.+, -...m+-
2.2 ,T, ..', tj _


2.


1.20 .



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Volume Fraction of Disks

Figure 4.24. Prediction for the relative increase in fracture toughness associated with crack
deflection by a dispersion of disk-shaped reinforcing particles (aspect ratio = 3) according
to Faber and Evans (1983a).



Table 4.4. Estimated relative contributions of stress transfer and crack deflection
mechanisms to the observed strengths of BAS glass-ceramics.


Growth Time Stress Transfer Crack Deflection
0.5 h 100% 0%
4 h 97% 3%
32 h 96% 4%
256 h 68% 32%


4.5 Process Optimization


The highest strength (141 + 12 MPa) and fracture toughness (1.87 0.07

MPa-m12) were observed for the BAS glass-ceramic with the longest crystal growth time

(256 h). This glass-ceramic also had the largest mean crystal diameter (1.76 0.25 gm)








and the longest mean free path between crystals (2.44 0.51 im). The highest possible

fracture toughness resulting from stress sharing and crack deflection mechanisms for this

material is estimated to be 2.18 MPa-m"2. Fracture toughness was observed to increase

with increasing mean crystal size over the entire range of glass-ceramics examined. This

trend is expected to continue for larger crystal sizes without a decrease in fracture

toughness caused by spontaneous microcracking. Finite element stress analyses predicted

the maximum interfacial stress (11.3 MPa) to be much lower than the fracture stress of

the base glass for a mean flaw radius of 76.6 im (63.4 8.0 MPa). Thus, only a 10.8%

increase in interfacial stress was predicted for a 238% increase in crystal size. If this

interfacial stress-size relationship continues linearly for larger crystals, then 60 lim would

be the critical crystal diameter for the onset of spontaneous microcracking. This

represents a 1150% increase in crystal size. Therefore, crystal growth treatments for

BAS glass-ceramics should be carried out for as long as economically feasible.

The crystal growth temperature used in the present study was limited to 9750C in

an effort to extend the lifetime of the chromel-alumel thennocouples in the tube furnace.

An increase in crystal growth temperature is desirable for large-volume production of

BAS glass-ceramics with large crystals, because the Arrhenius dependence of crystal

growth rate on temperature causes an exponential increase in the crystal growth at higher

temperatures. Pilot studies using differential thermal analysis estimated the energy barrier

to crystallization for this glass composition to be 40 kJ/mol (Griggs and Anusavice,








1997). This indicates that the maximum crystal growth rate at higher treatment

temperatures would be less than 3.14 times the crystal growth observed at 975C.

Substitution of strontia for baria in the glass composition may decrease the

processing temperature necessary to achieve rapid crystallization. Bansal et al. (1991)

have explored the eutectic solubility that exists between barium aluminosilicate and

strontium aluminosilicate. They found that partial substitution of baria with strontia

produced the highest strength ceramics (96 to 103 MPa) at low processing temperatures

(9000C). Care must be taken; however, to preserve the thermal compatibility between

the glass phase (a = 7.64 ppm/C) and the hexacelsian crystal phase (a = 7.4 ppm/C).

Bansal et al. reported that strontia substitution can change the primary crystal phase to

monoclinic celsian (a = 2.1 ppm/C), which may result in spontaneous microcracking and


a decrease in fracture toughness.














CHAPTER 5
CONCLUSIONS

The objectives of this study were to 1) determine which toughening mechanisms

are active in the BAS glass-ceramic system, 2) estimate the relative contribution of each

active mechanism to the increase in glass-ceramic strength and toughness over that of the

base glass, and 3) identify a process for producing a glass-ceramic with the highest

strength and fracture toughness.

Based on the results of this study, the following conclusions can be made:

1. Only toughening mechanisms that operate in the frontal process zone significantly

increase the strength and fracture toughness of glass-ceramics containing BAS crystals

with a mean diameter less than 1.76 gm. The lack of resistance-curve behavior

excludes from consideration any mechanisms that operate solely within the wake

process zone. Strengthening mechanisms that increase strength as the mean free path

between reinforcements decreases are not active in BAS glass-ceramics.

2. Stress transfer is the primary mechanism responsible for an increase in BAS glass-

ceramic strength. It can account for 68% of the observed strengthening, for glass-

ceramics with coarse microstructures, and 100% of the observed strengthening, for

glass-ceramics with fine microstructures. Crack deflection is the secondary





71


mechanism responsible for the remainder of the observed strengthening in glass-

ceramics with the coarsest microstructures.

3. The strength and fracture toughness of BAS glass-ceramics increased with increasing

duration of crystal growth treatment over the entire range of durations investigated.

To achieve the highest possible strength and fracture toughness, the glass should be

processed at temperatures in excess of 9750C for as long as economically feasible.














APPENDIX A
SOURCE CODE FOR FINITE ELEMENT ANALYSES

A.1 Model of Glass-Ceramic Subjected to Crystal Growth for 0.5 h


/BATCH
/input,menust,tmp ,,,,,,,,,,,,,,,,, 1
/PREP7
K, ,,,,
K, ,0.125e-3,,,
K, ,0.16e-3,,,
K, ,0.16e-3,0.1083e-3,,
K, ,0.0625e-3,0.1083e-3,,
K, ,0.0e-3,0.1083e-3,,
K, ,0.0e-3,0.1689e-3,,
K, ,0.0975e-3,0.1689e-3,,
K, ,0.16e-3,0.1689e-3,,
K, ,0.16e-3,0.2771e-3,,
K, ,0.035e-3,0.2771e-3,,
K,,0.0e-3,0.2771e-3,,
LSTR, 11, 10
LSTR, 10, 9
LSTR, 9, 8
LSTR, 8, 11
LSTR, 6, 5
LSTR, 5, 2
LSTR, 2, 1
LSTR, 1, 6
FLST,2,4,4
FITEM,2,1
FITEM,2,4
FITEM,2,2
FITEM,2,3
AL,P51X
FLST,2,4,4
FITEM,2,5
FITEM,2,6








FITEM,2,8
FITEM,2,7
AL,P51X
LSLA
KSLL
NSLA,,
ESLN,,1
CM,cryst-1,LINE
CM,cryst-k,KP
CM,cryst-a,AREA
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FLST,3,6,3,ORDE,6
FITEM,3,2
FITEM,3,5
FITEM,3,-6
FITEM,3,8
FITEM,3,-9
FITEM,3,11
KGEN,2,P51X .,, ,. 0
CMSEL,U,CRYST-A
CMSEL,U,CRYST-K
CMSEL,U,CRYST-L
LSTR, 12, 18
LSTR, 18, 16
LSTR, 16, 17
LSTR, 17, 4
LSTR, 4, 3
LSTR, 3, 13
LSTR, 13, 14
LSTR, 14, 15
LSTR, 15, 7
LSTR, 7, 12
LSTR, 7, 16
LSTR, 16, 14
LSTR, 14, 4
FLST,2,4,4
FITEM,2,18
FITEM,2,9








FITEM,2,10
FITEM,2,19
AL,P51X
FLST,2,4,4
FITEM,2,19
FITEM,2,20
FITEM,2,17
FITEM,2,16
AL,P51X
FLST,2,4,4
FITEM,2,20
FITEM,2,11
FITEM,2,21
FITEM,2,12
AL,P51X
FLST,2,4,4
FITEM,2,21
FITEM,2,15
FITEM,2,13
FITEM,2,14
AL,P51X
CM,matrix-a,AREA
CM,matrix-1,LINE
CM,matrix-k,KP
VSEL,ALL
ASELALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
ET,1,PLANE82
UIMP, ,EX, ,,94000,
UIMP, ,DENS, ,,,
UIMP,1,ALPX, ,7.6e-6,
UIMP,1,REFT, ,
UIMP,1,NUXY,, 0.28,
UIMP,1,PRXY,,,,
UIMP,1,GXY, ,
UIMP,1,DAMP, ,,,
UIMP,2,EX,,,110000,
UIMP,2,DENS, ,,,
UIMP,2,ALPX, ,7.4e-6,
UIMP,2,REFT, ,,,








UIMP,2,NUXY, ,0.26,
UIMP,2,PRXY, ,,,
UIMP,2,GXY, ,,,
UIMP,2,DAMP, ,,,
FLST,2,21,4,ORDE,2
FITEM,2,1
FITEM,2,-21
LESIZE,P51X,,,12,1,
FLST,5,4,5,ORDE,2
FITEM,5,3
FITEM,5,-6
CM,_Y,AREA
ASEL,,,,P51X
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
ESHAPE,2,0
AMESH,_Y1
ESHAPE,0,0
CMDEL, Y
CMDEL, YI
CMDEL,_Y2
CM,matrix-e,ELEM
CM,matrix-n,NODE
CMGRP,matrix,MATRIX-A,MATRIX-E,MATRIX-K,MATRIX-L,MATRIX-N
CMSEL,U,MATRIX
TYPE,1,
MAT,2,
REAL, ,
ESYS,0,
FLST,5,2,5,ORDE,2
FITEM,5,1
FITEM,5,-2
CM,_Y,AREA
ASEL, ,P51X
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
ESHAPE,2,0
AMESH, YI
ESHAPE,0,0
CMDEL,_Y
CMDEL, Y1








CMDEL,_Y2
CM,cryst-n,NODE
CM,cryst-e,ELEM
CMGRP,crystal,CRYST-A,CRYST-E,CRYST-K,CRYST-L,CRYST-N
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FLST,4,50,1,ORDE,5
FITEM,4,2
FITEM,4,26
FITEM,4,-49
FITEM,4,1850
FITEM,4,-1874
CP,1,UY,P51X
FLST,4,74,1,ORDE,8
FITEM,4,938
FITEM,4,962
FITEM,4,-985
FITEM,4,1394
FITEM,4,-1417
FITEM,4,1851
FITEM,4,1875
FITEM,4,-1898
CP,2,UX,P51X
FLST,5,4,1,ORDE,4
FITEM,5,26
FITEM,5,962
FITEM,5,1850
FITEM,5,1875
NSEL,U,,,P51X
CPINTF,ALL,1 e-7,
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FLST,5,2,1,ORDE,2
FITEM,5,26
FITEM,5,1850








NSEL,S,,,P51X
FLST,5,2,1,ORDE,2
FITEM,5,26
FITEM,5,1850
NSEL,S,,,P51X
CPINTF,UX,le-07,
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FLST,5,2,1,ORDE,2
FITEM,5,962
FITEM,5,1875
NSEL,S,, ,P51X
CPINTF,UY,le-07,
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FINISH
/SOLU
TUNIF,577,
TREF,577,
FLST,2,74,1,ORDE,9
FITEM,2,1
FITEM,2,-25
FITEM,2,506
FITEM,2,530
FITEM,2,-552
FITEM,2,2331
FITEM,2,2380
FITEM,2,2404
FITEM,2,-2426
D,P51X, ,0,,, ,UX
FLST,2,50,1,ORDE,6
FITEM,2,1394
FITEM,2,1418
FITEM,2,-1441
FITEM,2,2356








FITEM,2,2380
FITEM,2,-2403
D,P51X,,0 ,,,,UY
FLST,2,2811,1,ORDE,2
FITEM,2,1
FITEM,2,-2811
BF,P51X,TEMP,0,
SOLVE
FINISH
POST1
SET,LAST
CMSEL,S,MATRIX
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FINISH

A.2 Model of Glass-Ceramic Subjected to Crystal Growth for 4 h


/BATCH
/input,menust,tmp ,,,,,,,,,,,,,,,,,
/PREP7
K, ,,,,
K, ,0.26e-3,,,
K, ,0.765e-3,,,
K, ,0.765e-3,0.2252e-3,,
K, ,0.13e-3,0.2252e-3,,
K, ,0.0e-3,0.2252e-3,,
K, ,0.0e-3,1.le-3,,
K,,0.635e-3,1.le-3,,
K,,0.765e-3,1.le-3,,
K, ,0.0e-3,1.325e-3,,
K, ,0.505e-3,1.325e-3,,
K, ,0.765e-3,1.325e-3,,
LSTR, 11, 12
LSTR, 12, 9
LSTR, 9, 8
LSTR, 8, 11
LSTR, 6, 5








LSTR, 5, 2
LSTR, 2, 1
LSTR, 1, 6
FLST,2,4,4
FITEM,2,1
FITEM,2,4
FITEM,2,2
FITEM,2,3
AL,P51X
FLST,2,4,4
FITEM,2,5
FITEM,2,8
FITEM,2,7
FITEM,2,6
AL,P51X
LSLA
KSLL
NSLA,,l
ESLN,,
CM,cryst-a,AREA
CM,cryst-1,LINE
CM,cryst-k,KP
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FLST,3,6,3,ORDE,6
FITEM,3,2
FITEM,3,5
FITEM,3,-6
FITEM,3,8
FITEM,3,-9
FITEM,3,11
KGEN,2,P51X,,,,,,,0
CMSEL,U,CRYST-A
CMSEL,U,CRYST-K
CMSEL,U,CRYST-L
LSTR, 10, 18
LSTR, 18, 16
LSTR, 16, 7
LSTR, 7, 10








LSTR, 16, 17
LSTR, 17, 4
LSTR, 16, 14
LSTR, 14, 4
LSTR, 7, 15
LSTR, 15, 14
LSTR, 14, 13
LSTR, 13, 3
LSTR, 3, 4
FLST,2,4,4
FITEM,2,9
FITEM,2,11
FITEM,2,10
FITEM,2,12
AL,P51X
FLST,2,4,4
FITEM,2,11
FITEM,2,18
FITEM,2,15
FITEM,2,17
AL,P51X
FLST,2,4,4
FITEM,2,15
FITEM,2,13
FITEM,2,14
FITEM,2,16
AL,P51X
FLST,2,4,4
FITEM,2,16
FITEM,2,20
FITEM,2,19
FITEM,2,21
AL,P51X
CM,matrix-k,KP
CM,matrix-1,LINE
CM,matrix-a,AREA
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
ET,1,PLANE82








KEYOPT,1,3,0
KEYOPT,1,5,0
KEYOPT, 1,6,0
UIMP,1,EX, ,94000,
UIMP, ,DENS, ,
UIMP,1,ALPX, ,7.6e-6,
UIMP,1,REFT, ,
UIMP,1,NUXY, ,0.28,
UIMP, ,PRXY, ,
UIMP, ,GXY, ,,,
UIMP,1,DAMP, ,,,
UIMP,2,EX, ,110000,
UIMP,2,DENS ,,,.
UIMP,2,ALPX, ,,7.4e-3,
UIMP,2,REFT, ,
UIMP,2,NUXY, ,0.26,
UIMP,2,PRXY, ,,,
UIMP,2,GXY, ,,,
UIMP,2,DAMP, ,,,
ESHAPE,2,0
LESIZE,ALL ,,,10,1,1
FLST,2,3,4,ORDE,3
FITEM,2,14
FITEM,2,-15
FITEM,2,17
LESIZE,P51X,,,16,1,
FLST,2,3,4,ORDE,3
FITEM,2,14
FITEM,2,-15
FITEM,2,17
LESIZE,P51 X,,,20,1,
FLST,2,2,4,ORDE,2
FITEM,2,16
FITEM,2,20
LESIZE,P51X,,,10,2,
FLST,2,2,4,ORDE,2
FITEM,2,16
FITEM,2,20
LESIZE,P51X,,,10,3,
FLST,2,2,4,ORDE,2
FITEM,2,9
FITEM,2,11
LESIZE,P51X,,,10,3,








FLST,2,1,4,ORDE,
FITEM,2,9
LESIZE,P51 X, ,,10,0.33333,
FLST,2,3,4,ORDE,3
FITEM,2,11
FITEM,2,16
FITEM,2,20
LESIZE,P51X,,,10,4,
FLST,2,1,4,ORDE,1
FITEM,2,9
LESIZE,P51X,,,10,0.25,
TYPE,1,
MAT,1,
REAL,1,
ESYS,O,
FLST,5,4,5,ORDE,2
FITEM,5,3
FITEM,5,-6
CM,_Y,AREA
ASEL ,, ,P51X
CM,_YI,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
AMESH, Y1
CMDEL, Y
CMDEL,_Y1
CMDEL, Y2
CM,matrix-e,ELEM
CM,matrix-n,NODE
CMGRP,matrix,MATRIX-A,MATRIX-E,MATRIX-K,MATRIX-L,MATRIX-N
TYPE,1,
MAT,2,
REAL,1,
ESYS,0,
FLST,5,2,5,ORDE,2
FITEM,5,1
FITEM,5,-2
CM,_Y,AREA
ASEL ,,,,P51X
CM,_Yl,AREA
CHKMSH,'AREA'
CMSEL,S, Y
AMESH, Y1








CMDEL, Y
CMDEL,_Y1
CMDEL,_Y2
CMSEL,U,MATRIX
CM,cryst-n,NODE
CM,cryst-e,ELEM
CMGRP,crystal,CRYST-A,CRYST-E,CRYST-K,CRYST-L,CRYST-N
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FLST,5,4,1,ORDE,4
FITEM,5,2
FITEM,5,1002
FITEM,5,1922
FITEM,5,1943
NSEL,U,,,P51X
CPINTF,ALL, 1e-6,
FLST,4,40,1,ORDE,5
FITEM,4,1
FITEM,4,3
FITEM,4,-21
FITEM,4,1923
FITEM,4,-1942
CP,I,UY,P51X
FLST,4,80,1,ORDE,8
FITEM,4,982
FITEM,4,1003
FITEM,4,-1041
FITEM,4,1602
FITEM,4,-1621
FITEM,4,1923
FITEM,4,1944
FITEM,4,-1962
CP,2,UX,P51X
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL








FLST,5,2,1,ORDE,2
FITEM,5,2
FITEM,5,1922
NSEL,S,,,P51X
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
CPDELE,1,2,,ANY
FLST,4,42,1,ORDE,4
FITEM,4,1
FITEM,4,-21
FITEM,4,1922
FITEM,4,-1942
CP,1,UY,P51X
FLST,4,82,1,ORDE,8
FITEM,4,982
FITEM,4,1002
FITEM,4,-1041
FITEM,4,1602
FITEM,4,-1621
FITEM,4,1923
FITEM,4,1943
FITEM,4,-1962
CP,2,UX,P51X
FLST,5,2,1,ORDE,2
FITEM,5,2
FITEM,5,1922
NSEL,S,,,P51X
CPINTF,UX,Ie-06,
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FLST,5,2,1,ORDE,2
FITEM,5,1002
FITEM,5,1943
NSEL,S,,,P51X
CPINTF,UY,le-06,








VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FINISH
/SOLU
FLST,2,82,1,ORDE,10
FITEM,2,1
FITEM,2,42
FITEM,2,62
FITEM,2,-80
FITEM,2,342
FITEM,2,-381
FITEM,2,2263
FITEM,2,2304
FITEM,2,2324
FITEM,2,-2342
D,P51X,,0,,,,UX
FLST,2,42,1,ORDE,6
FITEM,2,1602
FITEM,2,1622
FITEM,2,-1641
FITEM,2,2284
FITEM,2,2304
FITEM,2,-2323
D,P51X,,0,,,,UY
TUNIF,577,
TREF,0,
FLST,2,2603,1,ORDE,2
FITEM,2,1
FITEM,2,-2603
BF,P51X,TEMP,0,
SOLVE
FINISH
POSTI
SET,LAST
FINISH
/SOLU
FINISH
/PREP7
VSEL,ALL








ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FINISH
/SOLU
TUNIF,577,
TREF,577,
FLST,2,2603,1 ,ORDE,2
FITEM,2,1
FITEM,2,-2603
BF,P51X,TEMP,0,
SOLVE
FINISH
POST1
SET,LAST
CMSEL,S,MATRIX
FINISH
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
/PREP7
FINISH
POST1
SET,LAST
FINISH
/SOLU
ANTYPE,0
TUNIF,577,
TREF,577,
FLST,2,2603,1,ORDE,2
FITEM,2,1
FITEM,2,-2603
BF,P51X,TEMP,0,
SOLVE
FINISH
POSTI
SET,LAST
FINISH








/PREP7
FINISH
POSTI
CMSEL,S,MATRIX
/PREP7
UIMP,2,EX ,, 110000,
UIMP,2,DENS, ,,,
UIMP,2,ALPX, ,7.4e-6,
UIMP,2,REFT, ,,,
UIMP,2,NUXY, ,0.26,
UIMP,2,PRXY, ,,,
UIMP,2,GXY, ,,,
UIMP,2,DAMP, ,,,
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FINISH

A.3 Model of Glass-Ceramic Subjected to Crystal Growth for 32 h


/BATCH
/input,menust,tmp ,,,,,,,,,,,,,,,
/PREP7
K, ,,,
K, ,0.48e-3,,,
K, ,1.43e-3,,,
K, ,1.43e-3,0.4157e-3,,
K, ,0.24e-3,0.4157e-3,,
K, ,0.0e-3,0.4157e-3,,
K, ,0.0e-3,2.0611e-3,,
K, ,0.1.16e-3,2.0611e-3,,
K,8,1.16e-3,2.0611e-3,,
K,8,1.43e-3,2.0611e-3,,
K,8,1.43e-3,2.477e-3,,
K,8,1.16e-3,2.0611e-3,,
K, ,1.43e-3,2.0611e-3,,
K, ,1.43e-3,2.477e-3,,
K, ,0.95e-3,2.477e-3,,
K, ,0.e-3,2.477e-3,,








LSTR, 11, 10
LSTR, 10, 9
LSTR, 9, 8
LSTR, 8, 11
LSTR, 6, 5
LSTR, 5, 2
LSTR, 2, 1
LSTR, 1, 6
FLST,2,4,4
FITEM,2,1
FITEM,2,4
FITEM,2,2
FITEM,2,3
AL,P51X
FLST,2,4,4
FITEM,2,5
FITEM,2,6
FITEM,2,8
FITEM,2,7
AL,P51X
LSLA
KSLL
NSLA,,1
ESLN,,
CM,cryst-a,AREA
CM,cryst-1,LINE
CM,cryst-k,KP
VSEL,ALL
ASEL,ALL
LSEL,ALL
KSEL,ALL
ESEL,ALL
NSEL,ALL
FLST,3,6,3,ORDE,6
FITEM,3,2
FITEM,3,5
FITEM,3,-6
FITEM,3,8
FITEM,3,-9
FITEM,3,11
KGEN,2,P51X,,,,,,,0
CMSEL,U,CRYST-A
CMSEL,U,CRYST-K




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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E1BH1BAUN_EBQJZ1 INGEST_TIME 2013-01-23T15:55:07Z PACKAGE AA00012952_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES