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Fracture mechanics and failure predictions for glass, glass-ceramic, and ceramic systems

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Fracture mechanics and failure predictions for glass, glass-ceramic, and ceramic systems
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Fracture mechanics and failure predictions for glass, glass-ceramic, and ceramic systems
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Palmer, Ronald A.
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Crystallization ( jstor )
Dehydration ( jstor )
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Fracture mechanics ( jstor )
Lithium ( jstor )
Nucleation ( jstor )
Stress corrosion ( jstor )
Stress cycles ( jstor )
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Water tables ( jstor )

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FRACTURE MECHANICS AND FAILURE PREDICTIONS
FOR GLASS, GLASS-CERAMIC, AND CERAMIC SYSTEMS












By
RONALD A. PALMER












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



UNIVERSITY OF FLORIDA


1981








































To EZZen













ACKNOWLEDGEMENTS


The author thanks those who served on his committee,

R. W. Gould, C. S. Hartley, L. E. Malvern, and G. Y. Onoda, Jr.,

for their assistance and academic instruction. Thanks are also due

his head professor, L. L. Hench, who provided the opportunity to

perform this study as well as advice and encouragement throughout.

S. W. Freiman (of NBS) gave invaluable assistancelin sample

preparation and experimental technique.

Before returning to graduate school (and since), the author

received greatly appreciated support from D. C. Greenspan, D. Cronin,

R. V. Caporali, and R. A. Ferguson. Discussions with S. Bernstein,

F. K. Urban, J. W. Sheets, W. J. McCracken, C. L. Beatty and many

others also contributed to this work.

Before finishing this dissertation, the author began working

for Rockwell Hanford Operations in Richland, Washington. The patience

shown by his supervisors, F. M. Jungfleisch, M. J. Kupfer, and

L. P. McRae is greatly appreciated.

Finally, the author wishes to thank his wife, Ellen, for her

love and support during this time.

This work was supported by the Air Force Office of Scientific

Research.









TABLE OF CONTENTS


ACKNOWLEDGEMENTS....................................

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

LIST OF TABLES.........................................

ABSTRACT...............................................

CHAPTER


I INTRODUCTION .............................. .....

II GLASS-CERAMICS... ............................

A. General .....................................
B. Nucleation and Crystallization..............
C. Applications and Advantages.................
D. The Lithia-Silica System....................

III FRACTURE MECHANICS.............................

A. Historical Background...........................
B. Static Fatigue in Glasses and Ceramics......
C. Fracture Mechanics of Glasses and Ceramics..
D. Lifetime Predictions........................
E. Methods Used for this Study.................

IV MATERIALS PREPARATION AND CHARACTERIZATION.....

A. Melting and Casting .........................
B. Annealing, Nucleation, and
Crystallization Schedule .................
C. Final Sample Preparation....................
D. Characterization ...........................

V TEST TECHNIQUES ...............................

A. Biaxial Flexure. ......................... .
B. Double Cantilever Beam Testing.............


Pace

iii

vi

ix

xi



1

6

6
7
12
12

18

18
21
29
33
36

38

38

38
39
40

42

42
44









Page

VI DYNAMIC FATIGUE: RESULTS AND DISCUSSION...... 48

A. Quantitative Microscopy.................... 48
B. Dynamic Fatigue Testing of 33L-Glass....... 48
C. Dynamic Fatigue Testing of 33L-92%......... 68
D. Dynamic Fatigue Testing of 33L-7%.......... 82
E. Dynamic Fatigue Testing of 33L-57%......... 99
F. The Effect of Crystallization on Strength.. 108
G. The Effect of Crystallization on N......... 110
H. The Effect of Crystallization on
Lifetime Predictions..................... 113
I. Summary.. ................................. 114

VII CRACK VELOCITY EXPERIMENTS.................... 115

A. General ................................... 115
B. 33L-Glass ................................ 117
C. 33L-Partially Crystallized...... .......... 117
D. 33L-92% Crystalline........................ 127

VIII DISCUSSION..... .............................. 132

A. Dynamic Fatigue............................ 132
B. Slow Crack Growth........................ .. 142
C. Comparing Dynamic Fatigue and
Slow Crack Growth........................ 143

IX CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK... 147

A. Conclusions................................ 147
B. Suggestions for Future Work................ 148

REFERENCES............................................. 152

BIOGRAPHICAL SKETCH .................................. 156








LIST OF FIGURES


Figure Page
1 Two stage and isothermal heat treatments
for processing a glass-ceramic 9

2 Tammann curves (nucleation or growth rates
vs. temperature) for controlled and uncon-
trolled crystallization 11

3 Phase diagram of the Li2O-Si02 system 14

4 Three modes of failure: I, normal or opening
mode; II, sliding mode; III, tearing mode 20

5 Possible reactions between water and segments
of the glass network 23

6 Hypothetical changes in crack tip geometry
due to stress corrosion 26

7 Universal fatigue curve developed by Mould
and Southwick (1959) 28

8 Equation 19 plotted for 33L-Glass tested in
air 35

9 Lifetime prediction diagram using equation 22
for 33L-Glass 37

10 Specimen configuration for double cantilever
beam constant moment crack growth experiment 41

11 Biaxial flexure test jig 43

12 Constant moment double cantilever beam test
apparatus 45

.13 Representative microstructure of 33L-7%
crystalline 51

14 Representative microstructure of 33L-57%
crystalline 52

15 Representative microstructure of 33L-92%
crystalline 53

16 Dynamic fatigue results for 33L-Glass tested
in air 55

17 Dynamic fatigue results for 33L-Glass tested
in water 57









Figure Page

18 Dynamic fatigue results for 33L-glass tested
after aging one day in water 59

19 Weibull plot for some 33L-92% crystalline
material 66

20 LPD for 33L-glass 67

21 Dynamic fatigue results for 33L-92% crystalline
tested in air 70

22 Dynamic fatigue results for 33L-92% crystalline
tested in water 72

23 Dynamic fatigue results for 33L-92% crystalline
tested after aging one day in water 74

24 Dynamic fatigue results for 33L-92% crystalline
tested after aging one week in water 76

25 LPD for 33L-92% crystalline 83

26 Dynamic fatigue results for 33L-7% crystalline
tested in air 85

27 Dynamic fatigue results for 33L-7% crystalline
tested in water 87

28 Dynamic fatigue results for 33L-7% crystalline
tested after aging one day in water 89

29 Dynamic fatigue results for 33L-7% crystalline
tested after aging one week in water 91

30 LPD for 33L-7% crystalline 98

31 Dynamic fatigue results for 33L-57% crystalline
tested after aging one day in water 101

32 Dynamic fatigue results for 33L-57% crystalline
tested after aging one week in water 103

33 LPD for 33L-57% crystalline 104

34 Theoretical crack velocity vs. stress intensity
factor relationship for brittle materials 116









Figure


V-K relationship for 33L-glass tested in air


V-K relationship for

V-K relationship for
samples

V-K relationship for
samples

Region of slow crack

Region of fast crack

V-K relationship for

Region of slow crack

Region of fast crack


33L-glass tested in water

33L-low crystallinity


33L-high crystallinity


growth in 33L-5% crystalline

growth in 33L-5% crystalline

33L-92% crystalline

growth in 33L-92% crystalline

growth in 33L-92% crystalline


123

125

126

128

130

131


133


Growth of a gel layer at the surface of a glass
containing two depths of flaws

A comparison of the Method of Maximum Likelihood
and the Monte Carlo method for predicting N


viii


Page









LIST OF TABLES


Table Page

1 Commercial Glass-Ceramics 13

2 Results of Quantitative Microscopy for
Determining Per Cent Crystallinity 49

3 Dynamic Fatigue Data for 33L-Glass
Tested in Air 60

4 Dynamic Fatigue Data for 33L-Glass
Tested in Water 61

5 Dynamic Fatigue Data for 33L-Glass
Tested After Aging 1 Day in Water 62

6 Liquid Nitrogen Strength of 33L-Glass 63

7 Dynamic Fatigue Data for 33L-92%
Crystalline Tested in Air 77

8 Dynamic Fatigue Data for 33L-92%
Crystalline Tested in Water 78

9 Dynamic Fatigue Data for 33L-92$
Crystalline Tested After Aging
1 Day in Water 79

10 Dynamic Fatigue Data for 33L-92%
Crystalline Tested After Aging
1 Week in Water 80

11 Liquid Nitrogen Strength of 33L-92%
Crystalline 81

12 Dynamic Fatigue Data for 33L-7%
Crystalline Tested in Air 92

13 Dynamic Fatigue Data for 33L-7%
Crystalline Tested in Water 93

14 Dynamic Fatigue Data for 33L-7%
Crystalline Tested After Aging
1 Day in Water 94

15 Dynamic Fatigue Data for 33L-7%
Crystalline Tested After Aging
1 Week in Water 95









Table Page

16 Liquid Nitrogen Strength of 33L-7%
Crystalline 96

17 Dynamic Fatigue Data for 33L-57%
Crystalline Tested After Aging
1 Day in Water 105

18 Dynamic Fatigue Data for 33L-57%
Crystalline Tested After Aging
1 Week in Water 106

19 Liquid Nitrogen Strength of
33L-57% Crystalline 107

20 Strength of 33L-Glass and Glass-
Ceramics 109

21 Fracture Parameters and (oo/aa) from
Dynamic Fatigue Data 111

22 33L-Glass Fracture Parameters as
Determined by Slow Crack Growth 118

23 33L-Partially Crystalline Fracture
Parameters as Determined by Slow
Crack Growth 121

24 33L-92% Crystalline Fracture Parameters
as Determined by Slow Crack Growth 129

25 Effect of Surface Finish on Aged
Strength of Sodium Disilicate Glass 135













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


FRACTURE MECHANICS AND FAILURE PREDICTIONS
FOR GLASS, GLASS-CERAMIC, A?!D CERAMIC SYSTEMS


By

Ronald A. Palmer

June, 1981

Chairman: Larry L. Hench
Major Department: Materials Science and Engineering


State-of-the-art fracture mechanics techniques are used to determine

the fatigue parameters for the lithium disilicate glass/glass-ceramic

system. Glass-ceramics are ideal materials for examining the effects

of crystallization on fatigue behavior. Two methods (dynamic fatigue

and slow crack growth) were employed and were found to produce different

values for the fatigue parameters. These parameters are used to design

lifetime prediction diagrams and, theoretically, should be independent

of the method of determination.

It was found that aqueous corrosion has a severe effect on the

fatigue parameters, as measured in dynamic fatigue, especially after

aging for up to one week in distilled water. The lithium disilicate

glass, which is known to readily form a gel layer in water, exhibits

the most rapid change in parameters.









The most crystalline composition (92% crystalline) shows the

greatest strength loss due to aging. This is attributed to water

migration along a network of microcracks into the grain boundaries.

At low volume fraction of crystals (7%) a residual stress in

the glassy matrix effects a strength increase and a decrease in the

environmental sensitivity which disappears with aging in water.

No effect of aging in water was found in the slow crack growth

studies. In all cases, the environmental sensitivity was found to

be much higher in these experiments than in dynamic fatigue. The

reasons for the difference are discussed in later chapters.

The limitations of the present fracture mechanics theory are

discussed in the light of the inconsistencies found between these

methods. It is recommended that a combination of techniques be used

when reliable lifetime predictions are required. A matrix of tests,

including static and dynamic fatigue and proof tests, is proposed

as an improved method of characterizing the fatigue behavior of a

material.








CHAPTER I-INTRODUCTION


Since the beginning of the twentieth century, the growth of techno-

logy has put increasingly more stringent demands on materials and

material properties. Advances in the fields of transportation (automo-

biles to aircraft to spacecraft), energy (coal and oil to nuclear and

solar), and others have necessitated improvements in existing materials

and the development of new materials in order to meet the high-performance

requirements of these technologies. The investigation of how to make

materials stronger with lighter weight has been advanced by studying

how materials fail. Macroscopic observation of the growth of cracks

combined with microscopic examination has been the most fruitful method

of learning the mechanisms of material failure. Studies of this type

have led to improvements in both the strength and toughness of materials

through the design and control of specific microstructures which impeded

initiation and/or propagation of cracks.

Recently, concern with the conservation or security of complex sys-

tems has placed additional emphasis on determining how long materials

will survive under-various environmental conditions. Thus,the field of

fracture mechanics has expanded from relatively simple measurement-

taking to making failure or lifetime predictions from those measurements.

In order to make meaningful predictions, parameters such-as-or-iginal

flaw size, critical flaw size, and stress corrosion susceptibility

become important. Also, there is a need to know what happens as a flaw

grows from a relatively insignificant stress concentration to some

critical size where catastrophic failure occurs. This process is termed








subcritical crack growth. Characterization of a mater ial.s_s.ub-r.t-i-t-cal

crack growth behavior and knowledge of the distribution of flaw sizes

permits prediction of the lifetime of a component in a given situation.

In order to predict and subsequently to guarantee the lifetime of

a component, some method of non-destructive evaluation is required.

Proof tests can be designed, in which all components are subjected to

a stress far beyond that expected in service, such that survivors are

assured some minimum service lifetime. By performing a proof test,

the need for large safety factors is reduced and overdesigning can be

avoided.

It is essential that the effect of the service environment on

time to failure be completely and correctly accounted for in fracture

mechanics theories if lifetime predictions are to be realistic. Thus,

a deeper understanding of the physical implications of the stress

corrosion susceptibility parameters is important. The combined effects

of aqueous environment and stress state on the subcritical growth of

flaws of various sizes need to be understood in order to ensure that

aging effects are correctly modeled and/or simulated in the proof

test design.

Therefore, one major objective of this thesis is to examine the

relative importance of aqueous environments, including time-dependent

effects, on the lifetime predictions for a brittle material. In order

to achieve this objective, it is necessary to study a more chemically

reactive material system than has heretofore been examined in lifetime

prediction investigations. By testing a material with known mechanisms

of chemical attack it is hoped that the critical reaction phenomena

for deterioration of mechanical performance can be elucidated.




3



In order for lifetime prediction theories to be of general use,

they must be equally applicable to polycrystalline ceramics, glasses,

and glass-ceramics. The accuracy of predicted lifetimes or their

validity should be independent of the wide range of microstructure

encountered in technical ceramics. Thus,a second major objective of

this thesis is to investigate the effect of microstructure on the

fracture behavior and time to failure of a glass-ceramic possessing

a wide range of crystallinity.

Of particular concern in the thesis is the possible interaction

between microstructural and environmental effects in fracture behavior.

Many technical ceramics are multiphased and the environmental suscepti-

bility of interphase boundaries is the potential "weak link" of their

long term performance. A unique feature of this study is the simulta-

neous examination of both major variables, microstructure and environ-

ment, in the same composition of material.

Glass-ceramics are ideal materials for studying the effects of

microstructure on mechanical properties. By varying the heat treatment,

fully glassy or fully crystalline materials may be fabricated, as well

as partially crystalline materials. The grain size of the crystalline

phase can be varied as well as the volume fraction of crystallinity.

In this thesis, the lithium disilicate system was chosen for

study because the nucleation and crystallization kinetics are well-

documented. The mechanical behavior of this system is also known,

although the environmental effects are not yet understood. Also of

initial importance is the ability to produce a fully crystallized

material having the same composition as the glass. Lithium disilicate

glass and glass-ceramics also fulfill the prime requirement of this

thesis --they react readily with water compared with other materials








which have been examined previously for their fracture behavior.

Studies by Sanders (1973), Ethridge (1977), and Dilmore (1977)

have established the details of aqueous attack of lithium disilicate

glasses. Recently, McCracken (1981) has conducted a parallel corrosion

study on lithium disilicate glass-ceramics of variable fraction of

crystals and determined the mechanisms of corrosion of the two-phased

glass-ceramic materials.

Thus, the lithiumdisilicate glass-ceramic system satisfies criteria

for studying both the effect of microstructure and aging in water on

the strength and fatigue behavior of a material of homogeneous composition.

Results obtained on the synergistic effects of subcritical crack growth

and aqueous corrosion should be important in extending the fracture

mechanics principles utilized in predicting lifetimes. By concurrently

studying the corrosion and fatigue mechanisms of this material system,

it is hoped that new insights will be gained in order to more reliably

predict lifetimes of brittle materials in general.

During this investigation quantitative microscopy was used to

monitor the volume fraction of crystallinity. Strength testing was

done with an Instron machine on discs, using the biaxial flexure test

and fracture mechanics tests were performed using the double cantilever

beam technique. Both optical and scanning electron microscopy were used

for examination of fracture surfaces.

Heat treatments were devised to achieve four microstructures: fully

glassy, 5 percent and 50 percent crystalline, and fully crystalline.

For each, both strength tests and fracture mechanics tests were made

in order to determine the nature of the sub-critical crack growth in




5




the material. Ideally, the two methods should yield identical results.

Where they did not, microstructural analysis of the fracture surfaces

was used to interpret the differences. A statistical analysis of the

strength data is also important in determining the accuracyof these

results and is an important part of this investigation.

Chapters II and III introduce the topics of glass-ceramics and

fracture mechanics, respectively. Chapter IV describes the prepara-

tion and characterization of materials used in this investigation.

Chapter V discusses the mechanical testing methods employed. Chapters

VI and VII discuss the results and the analyses of the fatigue

parameters and lifetime prediction diagrams. Finally, conclusions

and suggestions for future work are presented in Chapters VIII and IX.







CHAPTER II-GLASS CERAMICS


A. General

Glass-ceramics may be described as polycrystalline solids which have

been formed from a glassy melt by carefully controlled heat treatments.

In order for crystals to form, nuclei must be present. The nucleation

may be homogeneous (self-nucleating) or heterogeneous (by addition of

nucleating agents). The crystallization process may be controlled or

uncontrolled. Uncontrolled crystallization is termed devitrification

and is generally though of as a process to avoid. Controlled crystalli-

zation of glasses produces materials known as glass-ceramics. Glass-

ceramics have also been referred to as "vitrocerams," "devitrocerams,"

"sitalls," and "melt-formed ceramics." (See Stewart, 1972.)

Glass makers through the ages have struggled to avoid the destruc-

tive effects of devitrification. However, occasionally attempts were

made to crystallize glass purposely. Reaumur (1739) in the early 1700's

crystallized soda-lime-silica glass bottles by packing them in sand and

gypsum and heating them for a long time. Unfortunately, the crystals

nucleated at the surface, resulting in weak and deformed pieces.

The classical work on nucleation and crystallization was done by

Tammann (1925) in the 1920's. His theoretical and experimental investi-

gations laid the foundation for our present knowledge of these processes.

In the 1940's photosensitive glasses were developed by Stookey (1947)

at the Corning Glass Works. Colloidal dispersants of metals (gold,

copper, or silver) added as nucleants were shown to be activated by

exposure to ultra-violet light. Subsequent heat treatments resulted in







.ne nucleation and growth of crystals. By masking portions of the glass,

intricate and precise patterns could be produced. As the crystalline

phase is less durable than the glassy phase, hydrofluoric acid could

dissolve away the more soluble phase, resulting in a glass part or

pattern made more precisely than any other conventional glass forming

method.

Later, reputedly by accident, Stookey allowed a sample of one of

his glass compositions to be heated too long, resulting in complete

crystallization of the sample. It was found to be remarkably fine

grained and harder and stronger than the parent glass. This led to

the development of PyroceramR and many of the other commercial glass-

ceramics we know today. The various families of glass-ceramic composi-

tions and their applications will be discussed later in the chapter.


B. Nucleation and Crystallization

After the formation of the parent glass, the glass-ceramic process

required a two-step heat treatment. Figure 1 is a schematic of the

treatment. The first step is at a relatively low temperature (about

50-100C above the annealing point) and allows for the formation of

nuclei. In general, more nuclei form with time, so in order to obtain

a fine-grained material, longer times may be utilized. (The mathematics

of nucleation and growth have been well developed and will not be dis-

cussed here. See Bergeron, 1973).

Nucleating agents are generally noble metals or refractory oxides.

Materials which have been used as nucleating agents include Au, Pt, Ag,


R Corning Glass Works, Corning, NY.























Figure 1. Two stage and isothermal heat treatments for
processing a glass-ceramic









(1) Two Stage Heat Treatment


(2) Isothermal Heat Treatment
I


TIME









Cu, TiO2, and P205. It is recognized that there are certain key

characteristics for a nucleating agent to be effective. For nucleus

formation, it is important for the nucleant to be very soluble at

melting and forming temperatures while barely soluble at low temperatures,

to have a low free energy of activation for homogeneous nucleating, and

to be very mobile compared to the major components of the glass at low

temperatures. For the nucleant to promote crystal growth effectively,

there must be a low interfacial energy between it and the glass and

its crystal structure and lattice parameters should be similar to that

of the crystal phase.

Glass-in-glass phase separation has been found to promote glass-

ceramic formation without any nucleating agents. Agents which promote

phase separation, such as P205, are very effective in this manner.

Nucleation may occur either at the phase boundary or within one of the

phases (most likely the one most resembling the crystalline phase.)

The second step in the so-called "ceraming" process is the crystal-

lization treatment. For materials where only one crystalline phase develops,

the crystal growth may be a very simple process. However, if several phases

are developing, all different from the composition of the homogeneous glass,

the composition at the crystal-glass interface is constantly changing and

the growth process becomes exceedingly complex.

Figure 2 shows the Tammann curves (nucleation or growth rate vs.

temperature) for controlled and uncontrolled systems. For a controlled

system, it is easy to find nucleation and growth temperatures (ideally

at the maximum rates) where the other process does not occur. This type

allows the experimenter to have great control over the microstructure



























TEMPERATURE


TEMPERATURE


Tammann curves (nucleation or growth rates vs. temperature)
for controlled and uncontrolled crystallization


Figure 2.










of the final material. In the uncontrolled system, the nucleation and

growth curves overlap considerably, making it very difficult to get

appreciable nucleation without some crystallization and vice versa.

Control over the final microstructure is impossible in this system.

C. Applications And Advantages

Table 1 lists a number of commercial glass-ceramics, their major

crystalline phases, important properties, and applications. The wide

variety of materials and applications are discussed in detail by

Pincus (1972).

The advantages of glass-ceramics over conventionally processed

(slip cast, sintered, etc.) ceramics are listed below:

1. Because the microstructure can be controlled to a certain

extent, properties such as electrical properties, transparency,

and chemical durability can be tailored to specific applications.

2. Glass-forming techniques (blowing, pressing, drawing, and

casting) can be used, providing economical, high-speed,

automatic production.

3. Small dimensional changes during crystallization.

4. Zero porosity.

5. The homogeneity of the melt and the nucleation process leads

to uniform microstructure and properties.

6. Crystalline phases otherwise unattainable can be produced.

D. The Lithia-Silica System

Figure 3 shows the phase diagram for the lithia-silica system. The

composition chosen for this study was the 33.3 mole percent lithia composi-

tion. (This will be referred to as the 33L composition.) Previous work

done on this system has shown that the crystals which form this composition






Table 1

Commercial Glass-Ceramics


Designation

Corning 8603


Corning 9606


Corning 9615


G.E. Re-X


0-I CerVit


Pflaudler
Nucerite



PPG Hercuvit
101


Major Phases

Li20-Si02
Li20-Si02

2MgO-2A1203.5Si02

SiO2
Ti02

B-spodumene solid
solution


Li20-2Si02


B-quartz S.S.


alkali silicates




B-quartz S.S.


Properties

Photochemically
machineable

Transparent to
microwaves
Erosion and
Thermal Shock
resistance

Low T.E., strong,
thermally and
chemically stable,
easy to clean

Sealable to metals
Dielectric

Low T.E.
Polishability

Coating of steel
Chemical durability
Impact resistance
Abrasion resistance

Transparent


Application

Fluidics devices
Printing plate molds


Radomes


Range tops


Housings
Bushings


Telescopic mirrors


Chemical process equipment




High temperature windows
Infrared transparencies





14










L.i,0- Si O,
ico After Kracek, Holmquist.
and Glassor.
16oo a Quartz
\ Cristubalite
1 00__ y Tridymite
I Li;O.2SiO, 11:21
00- o LiO SiO, 11:11

1300


u Ioo I







oo-

,00.

E00- --
0)---*o -- \----\- \ I











Li-O ImoloI1


-igure 3. Phase diagram of the Li20-Si02 system









are solely lithium-disilicate (Li20-2Si02). There has been much work

done on both the binary system and systems with additions of nucleating

agents and aluminum oxide. As presented in Table 1, Li20-2SiO2 is the

major phase in many commercially important glass-ceramic compositions.

For this reason, study of the 33L composition will provide a basis on

which the characterization of the whole family of lithia-silica glass-

ceramics can be done.

The photo-chemically machineable glass-ceramics which first appeared

in the 1940's are lithia-silica based materials. Stookey's (1947, 1950a,

1950b, 1954, 1956) work on these materials is found in the patent literature.

Rindone (1958, 1962)studied the addition of platinum to a Li20.4Si02

glass. He measured (using x-ray diffraction) the amount of Li20-2Si02

crystallized from the glass as a function of time (10 minutes to 32 hours)

and the amount of Pt added (zero to 0.025%). Rindone proposed that lithium-

rich clusters surrounding the Pt form as nuclei for the Li20-2Si02 which

precipitates out. Later, Kinser (1968) showed that a metastable lithium-

metasilicate phase (Li20.Si02) is the precursor to the disilicate crystals

in the 33L glass.

Glasser (1967) studied binary lithium silicates ranging in composi-

tion from 80 to 88 weight percent Si02. He concluded that under certain

conditions a two-stage heat treatment schedule would improve the strength

of the crystallized composition. While accepting the explanation that

the formation of a large number of nuclei results in a fine-grained material,

Glasser also proposed another mechanism. He suggested that the lower

temperature treatment yields a metastable solid solution that is later









exsolved at higher temperatures. This results in a fine-grained

precipitate within the host crystals. Whether this mechanism would

actually occur in complex commercial compositions has yet to be shown.

Freiman (1968) studied the crystallization kinetics of several

lithia-silica compositions including 33L. He showed that an increase

in the nucleation time from three to 24 hours decreased the activation

energy for crystallization and increased the crystallization rate.

He found that the Li20.2SiO2 crystal growth was spherulitic in nature

and that small angle x-ray scattering confirmed the presence of a

metastable phase after nucleation at 4750C.

Freiman (1968) also studied the strength of the glass-ceramics he

produced. He found the strength to be lower than that predicted by

theory due to cracking in the composite because of the stresses set up

between the glass and the crystals during crystallization. He was able

to reduce the cracking by increasing the nucleation time, providing

better bonding between the glassy and crystalline phases.

Kinser(1968) also found the metastable lithium metasilicate phase

present in various lithia-silica compositions. A similar precursor

phase was found in a soda-silica composition.

Nakagawa and Izumitani (1969) studied a: Li20-2.5Si02 glass and a

Li20-Ti02-Si02 glass containing 22.5 weight percent Ti02. In the

binary glass, they found droplets due to liquid-liquid phase separation

occurring independently from the nucleation of Li20-2SiO2 crystals.

The droplets did not act as nuclei for crystallization. In the ternary

glass, they found that similar droplets deposited lithium titanate

crystals which then act as nuclei for crystallization.









Harper, James, and McMillan (1970) studied a lithia-silica glass

(30 percent Li20) with and without P205 as a nucleating agent. Nucleation

due to glass-in-glass phase separation as well as homogeneous nucleation

of lithium disilicate was discussed, but neither mechanism was firmly

established as correct. The reasons for the action of P205 as a nucleant

were discussed (surface energy effects, formation of lithium phosphates,

and more extensive phase separation), but were also left unresolved.

Doremus and Turkalo (1972) studied a lithium silicate glass (-26-27

mole percent lithia) with and without P205 also. They found that the

phosphorous slows the growth rate of the lithium disilicate crystals,

but that both produced similar microstructures. Spherulitic crystals

were also found, as in previous works, but some question was raised as

to whether they are true spherulites or not. No sheaf-like structures

have been shown for Li20.2Si02 crystals, such as can be found in so-called

spherulitic polymers, liquid crystals, and complex minerals.

The choice of 33L as the composition to study in this investigation

was based on the detailed microstructural and kinetic information of the

previous work and the fact that the glass and crystalline phases have

the same composition. A two-stage heat treatment with a long nucleation

time was also selected based upon Freiman's (1968) studies. Optical

and electron microscopy were used to confirm the attainment of materials

described previously.







CHAPTER III -FRACTURE MECHANICS


A. Historical Background

It is generally recognized that Griffith (1921) is the father of

fracture mechanics. He was the first to show a relationship between the

measured strength of a material, its material properties, and the length

of the pre-existing crack responsible for failure. The existence of pre-

existing cracks as precursors to failure had been proved by Inglis (1913)

some years before. Inglis showed that the stress near an elliptical hole

(resembling a crack) would be greater than the applied stress by

o = 2aa(a/p)T (1)

where oa is the applied stress, a is the half-crack length, and p is the

crack tip radius.

Griffith reasoned that the free energy of a cracked body and the

applied forces should not change during crack extension.. That is, the

amount of energy put into breaking the material should be equivalent to

the amount of energy required to create two new surfaces, or

dU dW (2)
da da

where U is the-strain energy due to the crack, W the energy required for

growth, and a the half-crack length.

Griffith calculated the strain energy to be
-2
U f a (3)
E

where E is Young's modulus, and the energy required for growth to be

W = 4ay (4)

where y is the surface energy. Inserting Eqs. (3) and (4) into (2)

gives the Griffith criterion for fracture of an infinite sheet with a







slit crack in plane stress:
= 2yE-y
=f (5)
where of is the stress at the crack tip resulting in failure.
There are three different modes in which a solid may be stressed
(Fig. 4). Mode I is the normal or opening mode; mode II is the sliding
mode; mode III is the tearing mode. The mode I stress field at the
crack tip has been determined by Irwin (1958) to be

aij (27 ) fij() (6)

where r and e define a coordinate system at the crack tip, f.i(e) is a
function accounting for the angular dependence of the stress about the
crack tip, and K is the stress intensity factor. For mode I,

KI =a (a)2 (7)

Failure then will occur at some critical stress intensity factor

KIc = of(wa) (8)

Combining Eqs. (5) and (8)

KIc = (2yE) (9)

We may now define G = 2yas the crack extension force such that
the critical value of G is
K2
G c (10)

for plane stress and for plane strain
K2
G Ic 2c
c (1 2)E

where v is Poisson's ratio. (The crack extension force is also called

the strain energy release rate, havingdimensionsof energy per unit
plate thickness and per unit crack extension which are the same as force



























































Three modes of failure: I, normal or opening
mode; II, sliding mode; III, tearing mode


Figure 4.








per unit crack extension. See Broek (1974).)

Because of the dominant tensile nature of the failure of glasses

and ceramics, only mode I failure under plane stress will be considered

in the following discussions.

B. Static Fatique In Glasses And Ceramics

Among the first to notice the detrimental effects of water on the

strength of glass were Preston and his co-workers (1946). They observed

a decrease in strength under a static load over intervals from minutes

to days for soda-lime and borosilicate glasses and several porcelain

compositions. The loss in strength was observed in tests conducted in

humid air and water, but the effect was more pronounced in water. No

loss in strength was noted when testing in a vacuum or at low temperatures.

Charles (1958) and his co-workers (Charles and Fisher, 1960 and

Hillig and Charles, 1964) introduced the chemical aspects of the delayed

failure problem. Figure 5 shows four reactions between water and segments

of the glass network. Reaction (a) demonstrates the replacement of an

alkali ion (M) with a hydrogen ion at a nonbridging oxygen site. In

his original work, Charles (1958) found that the temperature dependence

of the corrosion rate was identical with the temperature dependence of

alkali ion diffusion, which lends support to this mechanism.

Reaction (b) is less important. It shows aqueous attack on the

covalent bridging oxygen site. As fused silica and crystalline quartz

are relatively insoluble at moderate temperature and neutral pH, this

reaction is much less likely to cause significant damage to the glass

network.

























Figure 5. Possible reactions between water and segments
of the glass network







I I
(a) [-Sli-- [M]] + H20 + [-SiOGH] + M + OH

I I I
(b) E -S*i-O-Si + H 4 -2 2 [-SiOH]

(c) [-Si--O-Si-] + OH [-SiOH] + [-Si-0-]
I I I I

(d) [-Si-O-] + H 0 + [-Si--OH] + OH
I 2 I









Reactions (c) and (d) show a two-step breakdown of the covalent

chain initiated by the OH" ion. As reactions (a) and (d) produce the

same OH ion, the attack of water on a binary alkali silicate glass should

be autocatalytic.

Figure 6 shows the changes in crack tip geometry due to the

corrosion reactions and the presence or absence of stress. From Eq. (1),

we know that as the crack tip sharpens (p-o) the stress at the tip

increases. Figure 6(a) shows the crack tip sharpening under an applied

stress in a corrosive medium (i.e., water), causing weakening or fatigue

in the material. Figure 6(c) shows the crack tip rounding by corrosion

with little or no applied stress, which strengthens the material as it

ages in a corrosive medium. Figure 6(b) demonstrates the concept of a

fatigue limit, where the crack tip undergoes growth and rounding concur-

rently, resulting in no change in the stress at the crack tip.

Mould and Southwick (1959) performed an extensive study on the

strength and static fatigue of soda-lime-silica glass microscope

slides. They developed the concept of a universal fatigue curve.

By plotting a normalized strength (strength/low temperature strength,

a/aN) versus the logarithm of the normalized time to failure (time to

failure/time at which the strength is one-half the low temperature

strength, t/t0.5), it was found that for various abrasion treatments,

all of the data normalized in this manner fell on the same curve, such

that

a log t (12)
N t0.5


Their results are plotted in Figure 7.


























Figure 6. Hypothetical changes in crack tip geometry due to
stress corrosion









(a)


(h) (c)
I I
I
I I I
I I
I
r r
( I
r/ I
(I IL I
c,/


























Figure 7. Universal fatigue curve developed by Mould
and Southwick (1959a)







1.0


0.90


0.8

0.7


0.6


0.5


0.4

0.3

0.2.

0.1


0


-4 -3 -2 -1 0 1 2 3 4 5


log0 (t/t0.5)







Mould and Southwick (1959) also determined the value for the

breaking stress times the square root of the crack depth, a Ya7, which,

according to the Griffith criterion (Eq. 5), should be constant

(assuming constant y). Their value was 280-320 psi-in(0.31-0.35

MPam), which compares well with Griffith's value of 240 psi-in

(0.26 MPam).

Mould and Southwick (1959) also studied the effect of aging in

various media on strength and static fatigue. Water was found to be

as effective as HCL or NaOH in strengthening the glass. Approximately

a 30 percent increase in strength was observed. Very little effect was

observed in the static fatigue behavior, other than the general strength-

ening.

C. Fracture Mechanics of Glasses and Ceramics

Bradt, Hasselman, and Lange (1974) have edited a four volume

series on fracture mechanics of ceramics, which contains papers present-

ing an overview of the subject as well as specific problems. Lange (1974)

introduces the subject and traces the development of fracture mechanics

from the early theories to its present-day use as a tool in materials

development.

Evans (1974a) discusses the techniques used for fracture mechanics

determinations, including the advantages and pitfalls of each method.

The methods outlined (with references to more complete descriptions)

are three-point bend, single edge cracked tension, compact tension,

double cantilever beam (four variations), and double torsion.







Wiederhorn (1974) discusses subcritical crack growth and the

methods of obtaining crack velocity data. In addition to the direct

methods, such as described by Evans (1974a),Wiederhorn outlines the

indirect methods of obtaining the same data by constant load (static

fatigue) and constant strain rate (dynamic fatigue) experiments.

It has been shown (Evans, 1974b) that the crack velocity(V) may

be expressed as a power function of the applied stress intensity

factor(K):

V = AKN (13)

where A and N are constants to be determined experimentally. The

constant N is termed the stress corrosion susceptibility and is used

as a measure of a material's resistance to sub-critical crack growth

in corrosive environments.

By defining the crack velocity as

V dt (14)

and assuming a relationship between the stress, flaw size, and stress

intensity factor

K = aYa (15)

where Y is a constant which depends on crack and loading geometry, a

relationship may be derived giving the time to failure under a constant

load. (Note that Eq. 7 is the same as Eq. -15 with Y = ir.)

An equation defining the time to failure under a constant load may

be derived from Eqs. 13-15. From Eqs. 14-15,

da = 2K dK
02y2

and

Vdt 2K dK
a2y2








Using Eq. 13,

AKNdt = 2K dK
2y2

or
2KI-
dt = dK
2 2
AY 2c

Integrating now from zero to tf (corresponding to Ki, the initial stress

intensity factor, to Kc, the critical stress intensity factor) gives

2 22-N
f AY2(2-N)o2 -c i 2N

where a. is some applied stress. Since N is large and positive and
2-N 2-N
Ki>K This leads, finally, to the equation for the time
1 c 1 c
to failure under a constant applied load,

2-N
2K-N
t = (16)
AY2(N-2)co
a

From Eq. 15,
/ \
Ki a Kc (17)


where ac and Kc are the critical fracture stress and stress intensity

factor in an inert environment respectively. Then we have:
2-N
2(K /ac) -N (18)
f AY2(N-2) a


Kc and ac are determined by testing in liquid nitrogen, so that a

logarithmic plot of t vs. aa gives a straight line with a slope of -N

and an intercept which gives A.








Similarly, an equation relating the fracture strength to the

loading or stressing rate may be derived. (Details of the derivation

are given by Greenspan, 1977.) The equation is


N+1 2(N+1)(K /a )2-N
=- 2 a (19)
AY2(N-2)


where & is the stressing rate, do/dt. Now a logarithmic plot of a vs

a gives a straight line of slope 1+N and an intercept which gives

A. Ritter and co-workers (1971, 1974, 1978, 1979) have used Eqs. 18

and 19 extensively to determine these parameters for various glasses

and ceramics.

Usually, the crack propagation parameters obtained using the

static and dynamic fatigue methods agree with those obtained from

crack velocity experiments. However, several instances have occurred

where the data did not agree (Ritter and Manthuruthil,. 1973). The

materials involved are Pyrex and silica glasses. .Differences in the

chemical environment at the crack tip may account for the discrepancy.

The crack velocity experiments utilize a macroscopic crack to determine

the parameter, while the indirect methods initiate failure at micro-

scopic flaws, so that it is not unusual to imagine different chemistries

at the crack tip. Because the parameters are used for estimating failure

times, there is a practical need to resolve these differences in fracture

behavior.








D. Lifetime Predictions

Using Eqs. 18 and 19 we may construct diagrams to demonstrate

graphically the relationships between time to failure and applied

stress and failure stress and stressing rate. Figure 8 shows the

experimental results for 33L glass tested in air using Eq. 19.

Detailed discussion of this type of diagram is found in Chapter VI.

Rearranging Eq. 19 and taking the logarithm yields

noa = (Nl) [n B + zn (N+1) + (N-2) knoc] + TN?1 zno& (20)

where
2
B 2=
AY2 (N 2)K N-2

and the other terms have been previously defined. This now explicitly

defines the straight line relationship between an6 and an&. In this

case N = 30.5 and kn B = -2.07. (At this point, it is convenient to

express B in a logarithm, and A must be determined after a separate

experiment to find K .)

Because ceramics and glasses exhibit a wide spread in strength

values, lifetime predictions must use the lowest values to assure some

minimum lifetime. In order to increase the confidence in the minimum

lifetime prediction and to allow glasses and ceramics to be designed

to sustain greater loads, a proof test may be performed. In a proof

test, each sample is subjected to a stress greater than that expected

in service. This eliminates the weak ones and assures that every survivor

has some minimum strength or service life. This minimum service life

(tmin) is given by
SB N-2 -N \
t = B P2 aC (21)
min P a



























Figure 8. Equation 19 plotted for 33L-glass tested in
air





c (MPa-s- )


1 6 7.9 16 79 158
---- I-----------1----L1 1____________ L


33L Glass Air


5.00














S4.75














4.50


In c (MPa-s-)


N = 30.5

In B = -2.07


- -----------


r








which is simply Eq. 18 with a the proof test stress, substituted for

the inert strength, a.

A design diagram for lifetime predictions may now be constructed

by rearranging Eq. 21:


tmn = B(ap/a)N 2 (22)
min a p a

Using the crack growth parameters B and N and knowing the desired life-

time at an applied stress, a proof test ratio (a p/a) may be determined

in order to design a suitable proof test. Figure 9 shows the design

diagram for 33L glass tested-in air based on the data shown in Fig. 8.

As an example, if a component is expected to last ten years at

50MPa (horizontal line in diagram), a proof test ratio of 2.81 is needed.

Therefore, samples which had survived a proof test of 140 MPa would be

expected to survive the above conditions.

E. Methods Used For This Study

Both direct and indirect methods are used in determining the crack

growth parameters for the lithium disilicate glass-ceramic system. The

constant moment double cantilever beam method is used to measure the

crack velocity directly. The constant stressing rate method is used for

comparison. Chapters VI and VII will compare the results of the two

methods.









op/. a
1.5 2.0 3.0
-t -- f


4.0 5.0
I f


10 years, 50 MPa


33L-GLASS


TESTED IN AIR


0.5 1.0 1.5


In ( p/oa)


Lifetime prediction diagram using equation 22
for 33L-Glass


34





30






26


C-J


(\J ro
r
o


E
.Fa


22






18


14 -






10


Figure 9.


I I







CHAPTER IV-MATERIALS PREPARATION AND
CHARACTERIZATION


A. Melting and Casting

The parent glass was made by mixing reagent grade Li2CO3 and 5 lm

Min-U-Sil silica sand for one hour on a roller mill in a plastic jar.

Each batch weighed 200-250 g and was melted in a covered platinum crucible

for 24 hours in air in a electric muffle furnace at 13500C.

Casting was done in a graphite mold or graphite forms. Discs

required for the biaxial flexure test were made as follows: Cylinders

25 mm in diameter and about 50 mm long were formed and subsequently cut

into 2 mm thick slices with a diamond wheel. For double cantilever beam

specimens, bricks 20 mm x 25 mm x 80 mm were poured and subsequently cut

into plates with a diamond saw.


B. Annealing, Nucleation, and Crystallization Schedule

The completely glassy specimens were annealed immediately after

casting for four hours at 3500C and allowed to furnace cool. Qualitative

analysis of residual stresses was made using a polariscope with a tint

plate. Samples showing excessive stress were remelted and recast.

Those samples to be crystallized were placed in a tube furnace

directly after casting and held at the nucleation temperature. For all

levels of crystallinity, the nucleation treatment was 24 hours at 4750C.

This treatment was selected based on the work of Freiman (1968). The

size of the samples was also a factor. Because they were so large, a

long nucleation time was required to assure a consistent microstructure

throughout each specimen as well as from specimen to specimen. Two

cylinders or one brick could be treated at a time. To minimize thermal

38







gradients, the specimens were held in steel wire mesh boats and set on

an aluminum block in the center of the furnace.

To crystallize the specimens, the furnace was turned up to 5500C

and left for various periods of time. The furnace reached 5500C from

4750C in 15-20 minutes. Two hours resulted in about 7 percent crystal-

linity; four hours gave about 55 percent; fully crystallized specimens

were obtained by leaving them at 5500C for 24 hours.

After crystallization, the specimens were removed from the tube

furnace and placed in a small furnace at 2000C to prevent thermal shock

and allowed to furnace cool to room temperature.


C. Final Sample Preparation

Discs for biaxial flexure were cut from the cylinders with a high

speed diamond saw. Water was used as a coolant. Because of the rough

finish and reactivity of the materials with water, each disc was polished

dry with SiC paper to a 320 grit finish. The final discs were.about

2.5 mm thick. Discs were kept in dessicator to prevent atmospheric water

attack before testing.

Plates for double cantilever beam (DCB) testing were cut from the

bricks to the approximate dimensions 1 mm x12 mm x 75 mm. These were

cut using a low speed diamond watering saw, with mineral oil or Isocut*

fluid as a coolant. Although surface finish is not important in this

test, care was taken to avoid contact with water. A groove roughly half

the thickness of the specimen was machined down the center. This gives

the propagating crack an easy path to follow. The groove was made using


*Buehler and Co., Evanston, IL.







a milling machine and a diamond tool. Figure 10 shows the shape and

representative dimensions of the DCB specimens.


D. Characterization

Volume fraction crystallinity was evaluated by quantitative micro-

scopy (DeHoff and Rhines, 1968) using a petrographic microscope. A

121 point grid was placed 10 times on each sample measured at 200X to

obtain sufficient data for analysis.

The samples to be evaluated were polished sections, polished using

SiC paper to 600 grit, followed by 6 pm and 1 pm diamond paste. The

microstructure was made visible using an etch in 5 percent HF for one

minute. (See Chapter VI for examples of the microstructures.)
































bi
A j


F--2 h-


SECTION
A-A

0.5mm

1.0mm

6.0mm

18.0mm


Figure 10. Specimen configuration for double cantilever
beam constant moment crack growth experiment


L








CHAPTER V-TEST TECHNIQUES


A. Biaxial Flexure

Figure 11 shows the test jig for the biaxial flexure test. The

sample cup, which is mounted on the load cell of an Instron* testing

machine, has three press-fitted ball bearings which define a support

circle for the disc samples and also allows liquids or gases to be added

for testing in various reactive or non-reactive environments. The

loading pin is mounted on the crosshead of the testing machine and is

centered with respect to the circle defined by the ball bearings.

Detailed stress analysis of this method can be found in the work

of Kirstein and Woolley (1967). More recently, Wachtman, Capps, and

Mandel (1972) evaluated this method in a thirteen-laboratory round-

robin test. For small deflections, the strength, S, in this con-

figuration is

S 3 P(X Y) (23)
d
where

X = (1 + v) ln + -1 2 B\2

Y = (1 + v) [1 + ln ] + (1 ()

A = radius of support circle
B = radius of loaded area
C = radius of specimen
P = load
d = specimen thickness
v = Poisson's ratio.


* Instron Corp., Canton, MA.



























RAM-




3 STEEL BALL-
BEARINGS 120
APART




TO LOAD CELL-


UPPER CROSSHEAD


STEEL PIN


HOLDER


Figure 11. Biaxial flexure test jig








The dimensions used in our testing are A = 9.5 mm, B = 0.8 mm, and

C = 12.7 mm. A Poisson's ratio of 0.24 was used for all materials

studied (Freiman, 1968). These values reduce Eq. 23 to


S = -1.8689 2 (24)
d

Samples were tested in biaxial flexure at five cross head speeds

over three decades (0.2, 0.1, 0.02, 0.01 and 0.002 inches per minute)

in three atmospheres (air, water, and liquid nitrogen). Some samples

were also tested after aging one day or one week in water.

Testing in air was straightforward (22C, 65-70% relative humidity).

Care was taken when testing in water to ensure that the samples tested

at the fastest rate were in water about the same length of time as those

tested at the slowest rate (about two minutes).

In the aging experiments, samples were placed in plastic vials

with sufficient water to achieve a surface area to volume ratio (SA/V)

of 1 cm-1. Discs were then tested in the same water in which they were

aged.

Samples tested in liquid nitrogen were pre-cooled to avoid thermal

shock failure. The test jig was cooled with liquid nitrogen and filled

with the liquid during testing. Because of the absence of static

fatigue at liquid nitrogen temperatures, testing was done only at one

rate of 0.02 inches per minute. The average strength from the liquid

nitrogen testing was used as the ac value in Eqs. 17-20.

B. Double Cantilever Beam Testing

Freiman, Mulville, and Mast (1973) give a detailed analysis of the

constant moment double cantilever beam technique. Figure 12 shows the




































~I


I S.


*1





Ipi


'...-A

IJ
i;






1-V
i
| r
*^ '


.- v- ^
^- *I


Constant moment double cantilever beam test apparatus


Figure 12.








test apparatus for this technique. The test specimen is cemented

using epoxy to the loading arms in a pair of slotted inserts to insure

proper alignment. All the pivot points have suitably low friction

provided by bearings. The load is applied by means of a weight

pan connected through a triangular piece (assuring equal load

distribution) to the loading arms. A constant load provides a

constant moment applied by the arms to the specimen. This yields

a constant stress intensity factor, defined by


K TL (25)



where

T = load

L = moment arm length

I =- b h3 = moment of inertia of the beam

t = web thickness,

as defined in Figure 10.

A starter crack is initiated at the base of the slot in the groove

by tightening a sharp screw against the ungrooved side. The crack will

grow at a constant velocity under a constant load. The range of

velocities measured was from 10-10 to 10-4.m/s. With proper care,

multiple measurements may be made on one sample by changing the load

after taking sufficient readings at one load. Measurements were made

every few hours for very slow velocities and every half minute for

fast velocities. Again, testing was done in air and water at room

temperature (%220C).








Measurement of the crack velocity was made with a traveling

microscope.* The magnification was 32X. The accuracy of the

microscope was 0.0005 mm. Readings were made to the nearest

micrometer.

The above procedure will provide crack velocity data for stress

intensity factors less than Kc. Values for Kc, the critical stress

intensity factor, were determined using this apparatus attached to

an Instron** machine. An initial load was applied to begin the crack

propagation, then the cross head was turned on at a constant speed

of 0.02 inches per minute and the crack then propagated to failure.

The highest value of the load (T) then is substituted into Eq. 25,

and the value for Kc determined.


* Gaertner Scientific Corp., Chicago, IL.

** Instron Corp., Canton, MA.







CHAPTER VI-DYNAMIC FATIGUE: RESULTS AND DISCUSSION


A. Quantitative Microscopy

For the dynamic fatigue specimens, at least ten different discs

representing about half of the cylinders were examined for percent

crystallinity. Table 2 is a compilation of these results.

Figures 13-15 are optical micrographs of representative micro-

structures produced by each treatment. For the remainder of the work,

the nomenclature for the various microstructures will be as follows:

Crystallization Treatment Nomenclature

2 hours 33L- 7% Crystalline

4 hours 33L-57% Crystalline

24 hours 33L-92% Crystalline

For the 33L-92%, the remaining 8 percent is not a glassy phase

but open porosity. The porosity is a result of extensive microcracking

which occurs as the crystalline phase shrinks away from the less dense

glassy phase. Evidence of this microcracking can also be seen in 33L-7%

and 33L-57%. (See Figures 13-14.) There appears to be little if any

residual glassy phase in 33L-92%.

B. Dynamic Fatigue Testing of 33L-Glass

Figures 16-18 are plots of Eq. 20 for 33L-Glass testing in air,

in water, and after aging one day in water. The accompanying tables

(Tables 3-5) show the raw data for each plot. The straight lines in

Figures 16-18 are least squares fits using all data points obtained

in the testing. Table 6 shows the inert strength data from testing

samples as prepared and after aging one day in water.
48









TABLE 2

Results of Quantitative Microscopy for
Determining Per Cent Crystallinity


24 Hour Crystallization Treatment


% Crystalline
92.34
97.14
80.77
87.51
93.68
94.04
93.96
96.87
88.44
95.69
90.11
92.64


Average, x


Std., Dev., s



s/x


4 Hour Crystallization Treatment


Average, x


Std., Dev., s



s/x


Sample
308X
412X
507X
605X
804X
1006X
1102X
1310X
1402X
1907X
2303X
1508X


91.93%


4.65%



5.06%


203Z
402Z
409Z
603Z
704Z
709Z
802Z
907Z
1003Z
1309Z


57.14%


55.08
56.28
52.31
60.37
59.13
56.57
60.29
57.02
63.80
50.58


3.95%



6.91%









Table 2 Continued

2 Hour Crystallization Treatment


% Crystalline
5.08
9.05
9.09
7.93
7.07
6.61
6.41
6.07
5.46
5.54
5.54
6.61
7.27


Average, x



Std. Dev., s


Sample
602W
611W
807W
901W
1101W
111OW
1201W
1209W
1307W
1405W
1608W
1807W
2110W


6.79%



1.27%



18.64%


























or p
o
1)
45
9
r
bBO
*Cn d


.1,

I S


0.


C
"



.

*4 *

-g -


&Ar
4 r *
V~~a 9 .f
*'^' '" ,:


S S
9 B

j *8%


i'(


" cr
6 <


e6


.. i
A3


0
A




A' '6

S5

e~

S -V


Figure 13. Representative microstructure of

33L-7% crystalline


I

$sUI


,i' 8'
r-9 5 i
P


1.


ib ab
%" a


t 8


4]







































I 1Oum ,

Figure 14. Representative microstructure of
33L-57% crystalline











































Figure 15. Representative microstructure of
33L-92% crystalline

























Figure 16. Dynamic fatigue results for 33L-glass tested
in air






(MPa-s 1)


33L Glass Air


5.00












r-

S4.75














4.50


In 0 (MPa-s-l)


N = 30.5

In B = -2.07


-100


I I I r - :'
























Figure 17. Dynamic fatigue results for 33L-glass tested
in water






S(MPa-s1 )


33L Glass H20


N = 11.0


In B = 4.23


-150




-125


U ,
t(>I) -


In o (MPa-s-1)


5.20-





5.00-


4.80-





4.60-


4.40





4.20


)


~I


1 - . .ai-r;
























Figure 18. Dynamic fatigue results for 33L-glass tested
after aging one day in water




. (MPa-s-l)


33L Glass Aged 1 Day


5.50





5.25





5.00





4.75






4.50


In o (MPa-s-1)


N = 37.4

In B =-30.73


-200


S175


150 ,


.125





-100


I -


1_--~-









Table 3


Dynamic Fatigue Data for
Tested in Air


Stressing Rate


158

136
20
15%
32
136
7.4


79

131
13
10%
30
132
11.2


= average strength
= standard deviation
= coefficient of variation
= sample size
= median strength
= Weibull modulus


33L-Glass


(MPa-s1I)


S(MPa)
s(MPa)
s/S
n
S (MPa)
m '


16

125
29
23%
32
123
4.9


7.9

135
37
28%
31
125
4.0


1.6

113
19
17%
32
112
6.7


S
s
s/S
n
S
m
m


I





61



Table 4


Dynamic Fatigue
Tested


Data for 33L-Glass
in Water


Stressing


Rate (MPa-s-1)


158

123
16
13%
32
124
8.0


79

137
30
22%
32
S135
5.1


average strength
standard deviation
coefficient of variation
sample size
median strength
Weibull modulus


16

112
19
17%
26
111
6.2


7.9

100
16
16%
32
101
6.6


S(MPa)
s(MPa)
s/S
n
Sm(MPa)
m



S=
s =
s/ =
n =
S =
m =


1.6

91
22
24%
33
91
4.2









Table 5

Dynamic Fatigue Data for 33L-Glass
Tested After Aging 1 Day in Water


Stressing Rate (MPa-s-1)


158

124
8
7%
9
125
14.5


79

157
49
31%
31
.145
3.6


16

129
37
29%
15
120
3.7


7.9

133
46
35%
31
121
3.3


average strength
standard deviation
coefficient of variation
sample size
median strength
Weibull modulus


S(MPa)
s(MPa)
s/S
n
Sm(MPa)
m



=
s =
s/ =
n =
S =
m =
M


1.6

125
42
33%
31
112
4.0
















Table 6


Liquid Nitrogen Strength of 33L-Glass


As Prepared


S(MPa)

s(MPa)

s/S

n

Sm(MPa)

m


Aged 1 Day


179

3.4

19%

11

178

5.0


211

61

29%

10

224

3.1


= average strength

= standard deviation

= coefficient of variation

= sample size

= median strength

= Weibull modulus


S

s

s/S

n

Sm

m








The decrease in strength when tested in water and subsequent

increase after aging are consistent with previous works on glass

(Mould and Southwick, 1959, and others). However, the change in

stress corrosion susceptibility, N, from 30.5 (in air) to 11.0 (in

water) to 37.4 (after aging) was not anticipated. Because the

mechanism of stress corrosion (aqueous attack at the crack tip) is

the same in air and in water, the N values are expected to remain

constant. This large variation in N suggests that the kinetics of

aqueous corrosion are also involved in stress corrosion.

When testing is performed in air, the concentration of water at

the crack tip is low compared to that when testing in water. Less H

would then be available for ion exchange with Li slowing the reaction

and allowing less stress corrosion (giving a higher N value).

The conditions at the crack tip are altered drastically by aging

in water. The strength increases due to a rounding of the crack tip,

which, from Eq. 1, lowers the amount of stress concentration. During

the aging period, a reaction layer forms on the surface which is

depleted in lithium (due to ion exchange). In order for the flaw to

grow, it must first penetrate the reaction layer before entering the

bulk glass. This reaction layer, even after the flaw has grown through

it, protects the bulk from the aqueous environment. The protection

thus afforded raises the N value to one even higher than that in air.

Weibull Statistics

In the tables showing the raw data, m is a parameter called the

Weibull modulus. This parameter is obtained from (Wiederhorn, 1974)

ln In(l-F)-1 = m In (Si/So) (26)








where F is the failure probability for each strength value S.. The

parameters m and So are the Weibull modulus and scale parameter

respectively. The amount of scatter in the data can be quantified

in terms of m. As m increases the scatter becomes less (the slope of

Eq. 26 increases). Figure 19 is an example of a Weibull plot for

33L-92% crystalline material.

For 33L-Glass, the m value ranges from 3.1 to 14.5, depending

on environment, stressing rate, and aging, but mostly ranging from

about three to eight. This is very low and indicates a large range

of flaw sizes controlling the strength of the glass.

Ideally, the sample size for an adequate Weibull analysis should

be thirty or more. Although this study rarely meets that requirement,

the m values are still reported, but should be taken only as an indica-

tion of the true Weibull modulus of the material.

Lifetime Prediction Diagrams

Figure 20 is the lifetime prediction diagram (LPD) based on Eq. 22

for 33L-Glass. It can be seen that for a component to survive ten years

at 50MPa, different proof test levels are necessary depending on the

environment. In air, a proof test ratio of about 2.81 (140MPa) is

needed; in water, 13.46 (673 MPa); after aging one day, 5.21 (260 MPa).

Obviously, no 33L-Glass component would survive a proof test of

673 MPa, so that a different design level (shorter lifetime or lower







Si (MPa)
290


175


2g5


158 MPa-s '
16 MPa-s-
1.6 MPa-s


.0


33L-92% CRYSTALLINE

TESTED IN WATER


5.2


In S.
*1


5.3


(MPa)


Weibull plot for some 33L-92% crystalline material


0.5





0.0-





-0.5-


-1.0 -


-1.5





-2.0


-0.90





-0.75







.0.50


-0.25











-0.10





Figure 19.






ap/ a


1.5 2.0 2.5


in (ap/aa)


Figure 20. LPD for 33L-Glass


r<)
0.
t~

S26
E

r-



24





22





20


1.0








stress) is mandatory for 33L-Glass in water. However, if the data from

the aged samples are used, a proof test of only 260 MPa is necessary.

There is still a need to lower the design level, but not nearly as much.

The problem is, which data are relevant?

This situation illustrates the dynamics of the problem which have

yet to be discussed, let alone resolved. In order to make reliable

lifetime predictions, the conditions that a component sees must be known

(or assumed). It is also assumed (Wiederhorn 1973) that there is no

change in the flaw during or after proof testing. It is evident from

this work that a corrosive environment will have some effect on the flaw

and its immediately surroundings in the bulk. It would seem, then, that

no reliable predictions can be made using this procedure for materials

in corrosive environments due to the changing character of the flaws.

More discussion will follow after presenting the crack velocity

studies.


C. Dynamic Fatigue Testing of 33L-92%

Figures 21-24 present the dynamic fatigue results for 33L-92% and

Tables 7-10 give the raw data. Tests were conducted in air, in water,

and in water after aging for one day and one week. Table 11 shows the

results of the inert environment testing.

There was the expected decrease in strength when tested in water,

but no increase after aging. After aging one day, there was little

change over testing in water, but after one week, there was substantial

decrease in strength. This decrease can be attributed to the pervasive

attack of the water throughout the continuous microcrack network.

Similar effects have been seen in other porous ceramics (Frakes, Brown,

and Kenner 1974).























Figure 21. Dynamic fatigue results for 33L-92% crystalline
tested in air







S(MPa-s-)

1.6 7 9 1 79 158

5.60

33L 92% Crystalline Air



250
5.50





-. 225
5.40




N = 70.6

5.30 In B =- 5.66 200






5.20

0 1 2 3 4 5


In o (MPa-s-1)
























Figure 22. Dynamic fatigue results for 33L-92% crystalline
tested in water





; (MPa-s-1)
0, (MPa-s )


5.50






5.40







C
5.30






5.20-






5.10


In a (MPa-s-1)
























Figure 23. Dynamic fatigue results for 33L-92% crystalline
tested after aging one day in water






o (MPa-s1)

11.6 7.9 1( 77 158
225
5.40

33L 92% Crystalline.- Aged 1 Day




5.30- 200






5.20-
N = 27.6
175
In B = 0.73 175



5.10





5.00

---5


In o (MPa-s-1)
























Figure 24. Dynamic fatigue results for 33L-92% crystalline
tested after aging one week in water







o (MPa-s- )
\.6 7.9 16 79 15,8

5.25

33L 92% Crystalline Aged I Week

175







5/ 150
c 5.00 *


N = 22.3

In B = 2.37



125




4.75 ",

b 2 5 3 5


In c (MPa-s- )









Table 7


Dynamic Fatigue Data for 33L-92% Crystalline
Tested in Air


Stressing


158

226
9
4%
11
225
24.3


79

228
11
5%
11
227
21.1


Rate (MPa-s-1)


16

226
14
6%
9
226
14.7


7.9

226
18
8%
13
219
11.7


average strength
standard deviation
coefficient of variation
sample size
median strength
Weibull modulus


S(MPa)
s(MPa)
s/5
n
Sm
m




S=
s =
s/S =
n =

Sm =
m=


1.6

209
18
9%
10
205
11.2









Table 8


Dynamic Fatigue Data for 33L-92% Crystalline
Tested in Water


Stressing Rate


158

213
14
7%
8
208
13.8


79

214
10
5%
10
213
20.6


16

199
10
5%
11
196
17.9


(MPa-s- )


7.9

193
6
3%
9
196
29.3


average strength
standard deviation
coefficient of variation
sample size
median strength
Weibull modulus


S(MPa)
s(MPa)
s/S
n
Sm
m






s/ =
n=
S
m =


1.6

181
7
4%
10
180
24.0









Table 9


Dynamic Fatigue Data for 33L-92% Crystalline
Tested After Aging 1 Day in Water


Stressing Rate (MPa-s-1)


158

207
9
5%
6
208
19.9


79

206
6
3%
15
*207
33.7


16

203
10
5%
4
200
15.7


7.9

187
8
5%
15
187
22.7


average strength
standard deviation
coefficient of variation
sample size
median strength
Weibull modulus


S(MPa)
s(MPa)
s/S
n
Sm(MPa)
m



S=
s =
s/S =
n =
S =
m =
M


1.6

179
7
4%
6
181
22.3









Table 10


Dynamic Fatigue Data for 33L-92% Crystalline
Tested After Aging 1 Week in Water


Stressina Rate


158

175
18
10%
14
169
9.8


79

169
15
9%
14
165
11.0


16

160
17
11%
13
156
9.4


(MPa-s-1)


7.9

156
18
11%
14
152
8.9


= average strength
= standard deviation
= coefficient of variation
= sample size
= median strength
= Weibull modulus


S(MPa)
s(MPa)
s/S
n
Sm(MPa).
m


1.6

141
15
11%
14
137
9.6


S
s
s/S
n
Sm
m











Table

Liquid Nitrogen Strength


11

of 33L-92% Crystalline


As Prepared

274

19

7%

9

282

12.7


Aged 1 Day

280

20

7%

15

276

14.2


Aged 1 Week

221

26

12%

10

213

8.1


= average strength

= standard deviation

= coefficient of variation

= sample size

= median strength

= Weibull modulus


S(MPa)

s(MPa)

s/S

n

Sm(MPa)

m


s

s/S

n

Sm

m








The stress corrosion susceptibility again changes with environment.

In air, N = 70.6, but in water it drops to the mid-twenties. Because

the lithium disilicate crystals are much more durable than the glass

(McCracken 1981), the N value is very high compared with the glass.

There is virtually no change in N after aging of this glass-ceramic.

This can be explained by the lack of an extensive glassy phase. The

water attacks the grain boundaries where the glassy phase exists. But

because the glassy phase is so limited in size and extent, no protective'

layer builds up, only dissolution occurs. Therefore, no matter how long

the aging period, the mechanism and the associated kinetics change very

little.

The Weibull moduli for 33L-92% range from 8.9 to 33.7. These are

much higher than those for 33L-Glass, indicating a narrower distribution

of strength controlling flaws.

Figure 25 is the lifetime prediction diagram for 33L-92%. The

lines for the material tested in water are grouped together, again indi-

cating little difference in mechanism or kinetics. The aged one week

line is off-set slightly because of the large decrease in strength

observed. For this material in air, a proof test ratio of 1.62 (81MPa)

is needed to assure a lifetime of ten years at 50 MPa. This is much

lower than that for glass under equivalent conditions. (Note different

scales in Figs. 20 and 25.) For an aqueous environment, the ratio

ranges from 2.91 (145 MPa) to 3.39 (169 MPa).


D. Dynamic Fatigue Testing of 33L-7%

Figures 26-29 are plots of the dynamic fatigue data listed in

Tables 12-15. Table 16 shows the inert strength data for 33L-7%.




83


p/.,5
2.,5


33L 92% Crystalline


70





60 -





50 -





40





30





20


Years, 50 MPa


Aged 1 Week


0.4


0.6


0.8


1.6


In ( pI/a)
pa"


Figure 25. LPD for 33L-92% crystalline


3r0 3.5 4,0 4r5 5,0


Z.0


Air



















Aged 1 Day


m i I m m


1 r _ _
























Figure 26.


Dynamic fatigue results for 33L-7% crystalline
tested in air





o (MPa-s1 )


33L 7% Crystalline Air


N = 116.1

In B = -34.43


In 6 (MPa-s-1)


5.40


5.20 -


a0
?5.00 -

c




4.80






4.60


-200




.175


.150 c
O-
s:
10


125







r100


t -I--


___ Jf
























Figure 27. Dynamic fatigue results for 33L-7% crystalline
tested in water






o (MPa-s-1)


33L 7% Crystalline H20


N = 136.0


In B = -53.93


-200


-175




-150
I/)


-125






-100


0 1 2 3 4 5

In o (MPa-s1)


5.40-


5.20-


5.00-


4.80-






4.60-


. .. ..


----L---------
























Figure 28. Dynamic fatigue results for 33L-7% crystalline
tested after aging one day in water




Full Text
1.0
0.7 -
0.6 -
^ 0.5 *
0.4 -
0.3 -
0.2
0.1 4
UNIVERSAL FATIGUE CURVE
SODA-LIME-SILICA GLASS
0
-4
~T~
-3
-2
-1
T
0
lo9lO (t/t0.5>
r\D
Co
3
T
5
6


TEMPERATURE
(1) Two Stage Heat Treatment
(2) Isothermal Heat Treatment
TIME


Figure 8. Equation 19 plotted for 33L-glass tested in
air


41
T T
Figure 10. Specimen configuration for double cantilever
beam constant moment crack growth experiment


Figure 43. Region of fast crack growth in
33L-92% crystal!ine


| lOOurii |
Figure 15. Representative microstructure of
33L-92% crystal 1ine


Figure 27. Dynamic fatigue results for 33L-7% crystalline
tested in water


Figure 26. Dynamic fatigue results for 33L-7% crystalline
tested in air


135
-Table 25
Effect
of Surface
of Sodium i
120 Grit
Finish on Aged Strength
Disilicate Glass
320 Grit
As Prepared
5 Minutes
1 Hour
As Prepared
5 Minutes
1 Hour
5(MPa)
71
64
73
112
108
171
s(MPa)
11
9
13
24
36
61
s/5
15
14
18
21
34
36
n
10
7
9
10^
8
10
S (MPa)
m
73
63
76
106
114
173
m
5
7
5
5
2
3
5 = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


Figure 32. Dynamic fatigue results for 33L-57% crystalline
tested after aging one wek in water


Figure 23. Dynamic fatigue results for 33L-92% crystalline
tested after aging one day in water


39
gradients, the specimens were held in steel wire mesh boats and set on
an aluminum block in the center of the furnace.
To crystallize the specimens, the furnace was turned up to 550C
and left for various periods of time. The furnace reached 550C from
475C in 15-20 minutes. Two hours resulted in about 7 percent crystal
linity; four hours gave about 55 percent; fully crystallized specimens
were obtained by leaving them at 550C for 24 hours.
After crystallization, the specimens were removed from the tube
furnace and placed in a small furnace at 200C to prevent thermal shock
and allowed to furnace cool to room temperature.
C. Final Sample Preparation
Discs for biaxial flexure were cut from the cylinders with a high
speed diamond saw. Water was used as a coolant. Because of the rough
finish and reactivity of the materials with water, each disc was polished
dry with SiC paper to a 320 grit finish. The final discs were, about
2.5 mm thick. Discs were kept in dessicator to prevent atmospheric water
attack before testing.
Plates for double cantilever beam (DCB) testing were cut from the
bricks to the approximate dimensions 1 mm xl2 mm x 75 mm. These were
cut using a low speed diamond watering saw, with mineral oil or Isocut*
fluid as a coolant. Although surface finish is not important in this
test, care was taken to avoid contact with water. A groove roughly half
the thickness of the specimen was machined down the center. This gives
the propagating crack an easy path to follow. The groove was made using
*Buehler and Co., Evanston, IL.


In S (MPa)
\
o (MPa-s-^)
In (MPa-s~^)
S (MPa)


CHAPTER VIII-DISCUSSION
A. Dynamic Fatigue
It has been shown (in Chapter V) that N, the stress corrosion
susceptibility, for the lithium disilicate glass-ceramic system
changes with aging time in water: Because the fracture mechanics
theory behind the empirical equations describing dynamic fatigue
assumes a constancy of material and environment as well as a constant
character of the flaws, the change in N is indicative of deviation
from these assumptions. The most likely culprit is an alteration
of the character of the flaws during the aging process. There is
ample evidence to suggest that the action of aqueous solutions on
glasses and ceramics can alter the flaws at the surface of a
material.
Sanders and Hench (1973) have shown that a silica-rich surface
layer builds up readily when 33L-Glass is exposed to aqueous solutions.
Other investigators (Greenspan, 1977, Dilmore, 1977 Clark, 1976) have
shown this to be true of many other glasses as well. The formation of
this gel-like surface layer may affect the material to a depth of
1-10 ym. The physical and chemical properties of the material are
almost surely changed within this surface region. In addition, any
flaws within this region would have to be altered in some manner
(probably blunted). This two-fold attack, changing the flaw as well as
the material surrounding it, could account for the changes in N.
Figure 44 shows the growth of a gel layer at the surface of a glass
containing two types of flaws. As the gel grows, the shallower flaws
are the first to be affected. An experiment using a sodium disilicate
132


4
which have been examined previously for their fracture behavior.
Studies by Sanders (1973), Ethridge (1977), and Dilmore (1977)
have established the details of aqueous attack of lithium disilicate
glasses. Recently, McCracken (1981) has conducted a parallel corrosion
study on lithium disilicate glass-ceramics of variable fraction of
crystals and determined the mechanisms of corrosion of the two-phased
glass-ceramic materials.
Thus, the lithiumdisil ica'te glass-ceramic system satisfies criteria
for studying both the effect of microstructure and aging in water on
the strength and fatigue behavior of a material of homogeneous composition.
Results obtained on the synergistic effects of subcritical crack growth
and aqueous corrosion should be important in extending the fracture
mechanics principles utilized in predicting lifetimes. By concurrently
studying the corrosion and fatigue mechanisms of this material system,
it is hoped that new insights will be gained in order to more reliably
predict lifetimes of brittle materials in general.
During this investigation quantitative microscopy was used to
monitor the volume fraction of crystal 1inity. Strength testing was
done with an Instron machine on discs, using the biaxial flexure test
and fracture mechanics tests were performed using the double cantilever
beam technique. Both optical and scanning electron microscopy were used
for examination of fracture surfaces.
Heat treatments were devised to achieve four microstructures: fully
glassy, 5 percent and 50 percent crystalline, and fully crystalline.
For each, both strength tests and fracture mechanics tests were made
in order to determine the nature of the sub-critical crack growth in


(Sr-BdW)
67
o Jo
P a
4.0 5,0 6,0 8,0 10.,0 12,0
32
30
28
C\J fO
D
£ 26
EE
M
24
22 H
20
ln (Vaa)
Figure 20. LPD for 33L-Glass


96
Table 16
Liquid Nitrogen Strength of 33L-7% Crystalline
S(MPa)
s(MPa)
s/S
n
S (MPa)
m
m
As Prepared
233
39
17%
7
220
5.5
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


6.1
Table 4
Dynamic Fatigue Data for 33L-Glass
Tested in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
123
137
112
100
91
s(MPa)
16
30
19
16
22
s/S
13%
22%
17%
16%
24%
n
32
32
26
32
33
Sm(MPa)
124
. 135
111
101
91
m
8.0
5.1
6.2
6.6
4.2
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bul 1 modulus


121
Table 23
33L-Partially Crystallized Fracture Parameters
as Determined by Slow Crack Growth
Experiment
% Crystallinity
Environment
N
mA
41
6
Air
14.0
-212.4
46
6
Water
9.9
-152.0
54
5
Air
12.3
-185.7
63
65
Air
16.3
-248.6
64
66
Air
: 15.1
-230.0
68
62
Water
14.5
-219.2


Figure 16. Dynamic fatigue results for 33L-glass tested
in air


ACKNOWLEDGEMENTS
The author thanks those who served on his committee,
R. W. Gould, C. S. Hartley, L. E. Malvern, and G. Y. Onoda, Jr.,
for their assistance and academic instruction. Thanks are also due
his head professor, L. L. Hench, who provided the opportunity to
perform this study as well as advice and encouragement throughout.
S. W. Freiman (of NBS) gave invaluable assistance^in sample
preparation and experimental technique.
Before returning to graduate school (and since), the author
received greatly appreciated support from D. C. Greenspan, D. Cronin,
R. V. Caporali, and R. A. Ferguson. Discussions with S. Bernstein,
F. K. Urban, J. W. Sheets, W. J. McCracken, C. L. Beatty and many
others also contributed to this work.
Before finishing this dissertation, the author began working
for Rockwell Hanford Operations in Richland, Washington. The patience
shown by his supervisors, F. M. Jungfleisch, M. J. Kupfer, and
L. P. McRae is greatly appreciated.
Finally, the author wishes to thank his wife, Ellen, for her
love and support during this time.
This work was supported by the Air Force Office of Scientific
Research.


K (xlO-6 Pa-nr2)
In K (Pa-m'2)
Figure 35. V-K relationship for 33L-Glass tested in air


70
60
50'
40
30
20
10
104


92
Table 12
Dyanmic Fatigue Data for 33L-7% Crystalline
Tested in Air
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
166
171
165
157
165
s(MPa)
35
30
33
43
31
s/S
21%
18%
20%
CVJ
19%
n
9
8
10
9
9
S (MPa)
m
153
168
167
139
169
m
4.6
5.4
4.7
3.5
5.0
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


137
estimations obtainable. This also applies, of course, to comparing
dynamic fatigue results with slow crack growth results.
Most investigations (including this one) use the method of least
squares for fitting straight lines to experimental data. Unfortunately,
it may not be the best method. The method of least squares assumes
that each point is independent of any other point and that when using
groups of samples, the groups have equivalent variances. For Weibull
analysis, the data points have been ordered; thus the location of each
point is dependent on adjacent points. In dynamic fatigue testing,
although the groups tested are all from the same population so that
ideally the variances would always be equal, this is rarely the case
in practice. Weighting of the data is necessary to account for these
variations in order to properly employ the method of least squares.
Recently, Baldwin and Palmer (1980) have begun an investigation
using the method of maximum likelihood for fitting straight lines.
This technique first derives a probability density function for the
experimentally observed strength values (S.). Then, a likelihood
function given by the product of the probability density functions
evaluated at each is maximized by iteratively estimating the
parameters (slope and intercept) to be determined. When the likelihood
function is maximized, one obtains the best estimate of the parameters.
Unfortunately, this method was not sufficiently developed for inclusion
in this work.
Rockar and Pletka (1978) have questoned the reliability of dynamic
fatigue data for predicting lifetimes. They had difficulty obtaining
good fits to their straight lines apparently due to large scatter in
the data. Thye also found greater scatter and worse fits at higher N


RATE RATE
11
Figure 2. Tammann curves (nucleation or growth rates vs. temperature)
for controlled and uncontrolled crystallization


136
at the grain boundaries. Dense alumina is also known to be environ
mentally sensitive to water at even modest temperatures (Sinharoy et al.
1978). The precise nature of the growth of the flaws is difficult to
characterize because of the small scale of the phenomenon, but flaw
blunting and the growth of a reaction layer (gel layer) on the surface
of the flaw or ahead of the flaws are not unreasonable scenarios.
In order for proof testing of lifetime prediction methods to be
viable, it is assumed that the initial set of flaws determined by the
first set of tests (the proof test) is the set responsible for subse
quent delayed failure. It is apparent from the experiments described
herein that such a criterion is not likely to be satisfied by 33L,
33N, or by any other material in a highly reactive environment.
Delayed failure is a dynamic effect which is determined not only by
possible chemical reactions but also., by the kinetics of those reactions
There is probably no single set of short term experiments which can
adequately describe this process. This means that long term (NOT
accelerated) tests are needed to better determine the durability and
strength characteristics of brittle materials.
The estimations of the fatigue parameters N and B are determined by
way of statistical analysis of the experimental data. Since the main
objective of this thesis is to study the effects of environment and
microstructure on the fatigue properties of the lithium disilicate
system, it is necessary that the statistical analysis allows us to
differentiate among the obtained values for N and B. For reliable
lifetime predictions, it is also necessary to have the most accurate


98:
In (op/oa)
Figure 30. LPD for 33L-7% crystalline


24:
Reactions (c) and (d) show a two-step breakdown of the covalent
chain initiated by the OH" ion. As reactions (a) and (d) produce the
same OH ion, the attack of water on a binary alkali silicate glass should
be autocatalytic.
Figure 6 shows the changes in crack tip geometry due to the
corrosion reactions and the presence or absence of stress. From Eq. (1),
we know that as the crack tip sharpens (p+o) the stress at the tip
increases. Figure 6(a) shows the crack tip sharpening under an applied
stress in a corrosive medium (i.e., water), causing weakening or fatigue
in the material. Figure 6(c) shows the crack tip rounding by corrosion
with little or no applied stress, which strengthens the material as it
ages in a corrosive medium. Figure 6(b) demonstrates the concept of a
fatigue limit, where the crack tip undergoes growth and rounding concur
rently, resulting in no change in the stress at the crack tip.
Mould and Southwick (1959) performed an extensive study on the
strength and static fatigue of soda-lime-silica glass microscope
slides. They developed the concept of a universal fatigue curve.
By plotting a normalized strength (strength/low temperature strength,
a/c^) versus the logarithm of the normalized time to failure (time to
failure/time at which the strength is one-half the low temperature
strength, t/tg 5), it was found that for various abrasion treatments,
all of the data normalized in this manner fell on the same curve, such
that
a loq
N r0.5 .
Their results are plotted in Figure 7.


150
(as developed by Mecholsky, Freiman, and Rice 1976) can identify
type and location of the flaw as well as discern any subcritical
crack growth which may have occurred during fracture. However
such techniques cannot readily characterize residual stress-fields
or compositional gradients associated with the flaws ab-initio.
As this study has shown, the interactions of the material and
the environment are important in the fatigue process. Many different
types of corrosion experiments have been invented, but the ideal
technique is the one which best models the real-life application
of the material or component. For instance, Frakes et al. (1974) tested
proposed bio-ceramic material (porous alumina) by subcutaneous implan
tation in the backs of rabbits and observed significant reductions in
strength due to the environmental exposure. Unfortunately, modeling
real-life applications is usually not easy and most experiments will
involve some compromise. In the Frakes et al. (1974) experiment
the implants were unloaded, which probably would have made the
environmental attack even more severe. Part of the need for an
improved dynamic fatigue model is to handle situations such as these.
In order to determine a more complete picture of the fatigue
process, the following test scheme is suggested. Lifetime prediction
diagrams should be obtained from both static and dynamic fatigue
experiments, followed by a proof test. (The proof test may be static
or dynamic.) The survivors should be tested to failure statically,
which will provide a failure distribution that can be compared with
those predicted by the earlier static and dynamic fatigue tests.


In S (MPa)
7.9
158
5.40
V6
331 7% Crystalline Air
2£
79
I
5.20
5.00
4.80
-200
-175
- 150
. 125
in
4.60
-100
0
1
2
i i r-
3 4 ,5
(MPa-s-1)
In
S (MPa)


144
to the macroscopic crack. The occurrence of aging effects in
dynamic fatigue but not in slow crack growth confirms this
interpretation.
The results of this study show that lifetime prediction
diagrams drawn using crack growth data would be more conservative
than those drawn using dynamic fatigue data. At this time it is
not known whether they would be more or less accurate. The strength
ening effect of water on the 33L-Glass (indeed, on most glasses)
complicates the decision. It is evident that each material will
need extensive testing of various types (those used here as well as
static fatigue) before an intelligent decision can be made regarding
lifetimes in critical applications.
In their work, Pletka and Wiederhorn (1978) studied a lithium
aluminosilicate glass-ceramic and found very little difference between
the parameters determined by crack growth and those determined by
dynamic fatigue. They attribute this to a more homogeneous micro
structure compared with their magnesium aluminosilicate. This under
scores the need for a variety of tests on whatever material is under
investigation.
Doremus (1980) has made a mathematical analysis of the equations
involved in lifetime predictions. He concludes that:
1) crack propagation data cannot be used to predict
fatigue times reliably;;
2) the "agreement" between measured fatigue strengths
and those calculated from crack propagation data
is not significant;
3) propagation velocities of large (Mem) cracks are probably
related to static fatigue in glass in some way, but the
details of this relationship are uncertain.


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
LIST OF TABLES
Paqe
Commercial Glass-Ceramics
13
Results of Quantitative Microscopy for
Determining Per Cent Crystallinity
49
Dynamic Fatigue Data for 33L-Glass
Tested in Air
60
Dynamic Fatigue Data for 33L-Glass
Tested in Water
61
Dynamic Fatigue Data for 33L-Glass
Tested After Aging 1 Day in Water
62
Liquid Nitrogen Strength of 33L-Glass
63
Dynamic Fatigue Data for 33L-92%
Crystalline Tested in Air
77
Dynamic Fatigue Data for 33L-925&
Crystalline Tested in Water
78
Dynamic Fatigue Data for 33L-92$
Crystalline Tested After Aging
1 Day in Water
79
Dynamic Fatigue Data for 33L-92%
Crystalline Tested After Aging
1 Week in Water
80
Liquid Nitrogen Strength of 33L-92%
Crystalline
81
Dynamic Fatigue Data for 33L-7%
Crystalline Tested in Air
92
Dynamic Fatigue Data for 33L-7%
Crystalline Tested in Water
93
Dynamic Fatigue Data for 33L-7%
Crystalline Tested After Aging
1 Day in Water
94
Dynamic Fatigue Data for 33L-7%
Crystalline Tested After Aging
1 Week in Water
95
IX


82
The stress corrosion susceptibility again changes with environment.
In air, N = 70.6, but in water it drops to the mid-twenties. Because
the lithium disilicate crystals are much more durable than the glass
(McCracken 1981), the N value is very high compared with the glass.
There is virtually no change in N after aging of this glass-ceramic
This can be explained by the lack of an extensive glassy phase. The
water attacks the grain boundaries where the glassy phase exists. But
because the glassy phase is so limited in size and extent, no protective
layer builds up, only dissolution occurs. Therefore, no matter how long
the aging period, the mechanism and the associated kinetics change very
1ittle.
The Wei bull moduli for 33L-92% range from 8.9 to 33.7. These are
much higher than those for 33L-Glass, indicating a narrower distribution
of strength controlling flaws.
Figure 25 is the lifetime prediction diagram for 33L-922L The
lines for the material tested in water are grouped together, again indi
cating little difference in mechanism or kinetics. The aged one week
line is off-set slightly because of the large decrease in strength
observed. For this material in air, a proof test ratio of 1.62 (81MPa)
is needed to assure a lifetime of ten years at 50 MPa. This is much
lower than that for glass under equivalent conditions. (Note different
scales in Figs. 20 and 25.) For an aqueous environment, the ratio
ranges from 2.91 (145 MPa) to 3.39 (169 MPa).
D. Dynamic Fatigue Testing of 331.-7%
Figures 26-29 are plots of the dynamic fatigue data listed in
Tables 12-15. Table 16 shows the inert strength data for 33L-7%.


29
Mould and Southwick (1959) also determined the value for the
breaking stress times the square root of the crack depth, a /a, which,
according to the Griffith criterion (Eq. 5), should be constant
(assuming constant y). Their value was 280-320 psi-in^(0.31-0.35
MPanf*), which compares well with Griffith's value of 240 ps'i-in15
(0.26 MPam**).
Mould and Southwick (1959) also studied the effect of aging in
various media on strength and static fatigue. Water was found to be
as effective as HCL or NaOH in strengthening the glass. Approximately
a 30 percent increase in strength was observed. Very little effect was
observed in the static fatigue behavior, other than the general strength
ening.
C. Fracture Mechanics of Glasses and Ceramics
Bradt, Hasselman, and Lange (1974) have edited a four volume
series on fracture mechanics of ceramics, which contains papers present
ing an overview of the subject as well as specific problems. Lange (1974)
introduces the subject and traces the development of fracture mechanics
from the early theories to its present-day use as a tool in materials
development.
Evans (1974a) discusses the techniques used for fracture mechanics
determinations, including the advantages and pitfalls of each method.
The methods outlined (with references to more complete descriptions)
are three-point bend, single edge cracked tension, compact tension,
double cantilever beam (four variations), and double torsion.


16
exsolved at higher temperatures. This results in a fine-grained
precipitate within the host crystals. Whether this mechanism would
actually occur in complex commercial compositions has yet to be shown.
Freiman (1968) studied the crystallization kinetics of several
1ithia-silica compositions including 33L. He showed that an increase
in the nucleation time from three to 24 hours decreased the activation
energy for crystallization and increased the crystallization rate.
He found that the Li^O^SiC^ crystal growth was spherulitic in nature
and that small angle x-ray scattering confirmed the presence of a
metastable phase after nucleation at 475C.
Freiman (1968) also studied the strength of the glass-ceramics he
produced. He found the strength to be lower than that predicted by
theory due to cracking in the composite because of the stresses set up
between the glass and the crystals during crystallization. He was able
to reduce the cracking by increasing the nucleation time, providing
better bonding between the glassy and crystalline phases.
Kinser(1968) also found the metastable lithium metasilicate phase
present in various lithia-silica compositions. A similar precursor
phase was found in a soda-silica composition.
Nakagawa and Izumitani (1969) studied a Li2*3 2.5Si0^ glass and a
Li2O-TiO2-SO2 glass containing 22.5 weight percent TiC^. In the
binary glass, they found droplets due to liquid-liquid phase separation
occurring independently from the nucleation of L^O^SiC^ crystals.
The droplets did not act as nuclei for crystallization. In the ternary
glass, they found that similar droplets deposited lithium titanate
crystals which then act as nuclei for crystallization.


In S (MPa)
a (MPa-s
S (MPa)


154
Materials Characterization Center (1980). Workshop on Compositional and
Microstructural Analysis of Nuclear Waste Materials, November 11-12,
Seattle, Washington (Proceedings to be published in January 1981).
McCracken, W. J. (1981). Ph.D. Dissertation, University of Florida.
Mecholsky, J. J., Freiman, S. W., and Rice, R. W. (1976). J. Mater. Sci.
11 p. 1310.
Mediratta, M. G., and Petrovic, J. S. (1978). J. Am. Ceram. Soc. 61
[5-6] p. 226.
Miyata, N., and Jinno, H. (1972). J. Mater. Sci. ]_ [9] p. 973.
Mould, R. E. and Southwick, R. D. (1959). J. Am. Ceram. Soc., 4£
p. 542 and 582.
Nadeau, J. S. and Bennett, R. C. (1978). p. 961 in Fracture Mechanics of
Ceramics, Vol. 4, R. C. Bradt, D. P. H. Hasselman, and F. F. Lange,
eds., Plenum, New York.
Nakagawa, K. and Izumitani, T. (1969). Phys. Chm. G1., 1_0 [5] p. 179.
Pincus, A. G. (1972). p. 210 in Advances in Nucleation and Crystallization
in Glass, S. W. Freiman and L. L. Bench, eds., American Ceramic
Society, Columbus, Ohio.
Pletka, B. J., Fuller, E. R., Jr., and Koepke, B. G. (1979). p. 19 in
Fracture Mechanics Applied to Brittle Materials, ASTM STP 678,
S. W. Freiman, ed., American Society for Testing and Materials,
Philadelphia, PA.
Pletka, B. J. and Wiederhorrv, S. M. (1978). p. 745 in Fracture Mechanics
of Ceramics, Vol. 4, R. C. Bradt, D. P. H. Hasselman, and F. F. Lange,
eds., Plenum, New York.
Preston, F. W., Baker, T. C., and Glathart, J. L. (1946). J. Appl.
Phys., T7 p. 162.
Rao, A. S. (1977). Ph.D. Dissertation, University of British Columbia.
Reaumur, M. (1739). Mem. Acad. Sci., p. 370.
Rindone, G. E. (1958). J. Am. Ceram. Soc., 41_ [2] p. 41.
(1962). Ibid, 45 [1] p. 7.
Ritter, J. E. (1974). p. 735 in Fracture Mechanics of Ceramics, Vol. 2,
R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, eds., Plenum,
New York.
Ritter, J. E. (1978). p. 667 in Fracture Mechanics of Ceramics, Vol. 4,
R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, eds., Plenum,
New York.


FRACTURE MECHANICS AND FAILURE PREDICTIONS
FOR GLASS, GLASS-CERAMIC, AND CERAMIC SYSTEMS
By
RONALD A. PALMER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1981


139
50
40 -
30. -
20 -
10 -
0 -
Figure 45.
Method of
Monte Carlo
Method
Maximum
Like!ihood
18.4'
A comparison of the Method of Maximum Likelihood
and the Monte Carlo Method for predicting N


155
Ritter, J. E., Bandyopadhyay, N., and Jakus, K. (1980). To be published
in Bull. Am. Ceram. Soc.
Ritter, J. E., Greenspan, D. C., Palmer, R. A., and Hench, L. L. (1979).
J. Biomed. Mater. Res., p. 251.
Ritter, J. E. and Manthuruthil, J. (1973). G1. Tech., 1^ [2] p. 60.
Ritter, J. E. and Sherburne-, C. L. (1971). J. Am. Ceram. Soc., 54 p. 601.
Rockar, E. M. and Pletka, B. J. (1978). p. 725 in Fracture Mechanics of
Ceramics, Vol. 4, R. C. Bradt, D. P. H. Hasselman, and F. F. Lange,
eds., Plenum, New York.
Sanders, D. M. (1973). Ph.D. Dissertation, University of Florida.
Sanders, D. M. and Hench, L. L. (1973). J. Am. Ceram. Soc., 56_ [7] p. 373.
Sinharoy, S., Levenson, L. L., Ballard, W. V., and Day, D. E. (1978).
Bull. Am. Ceram. Soc. b7 [2] p. 231.
Stewart, D. R. (1972). p. 237 in Introduction to Glass Science, L. D. Pye,
H. J. Stevens, and W. C. LaCourse, eds., Plenum, New York.
Stookey, S. D. (1947). Brit. Patent No. 635,649.
(1950a). U. S. Patent No. 2,515,275.
(1950b). U. S. Patent No. 2,515,941.
(1953). Ind. Eng. Chem., 45 [1] p. 115. .
(1954). U. S. Patent No. 2,684,911.
(1956). Brit. Patent No. 752,243.
Tammann, G. (1925). The States of Aggregation, D. Van Nostrand Co.,
New York.
Theocaris, P. S. and Milios, J. (1980). Eng. Fract. Mech. 1_3 p. 599.
Trantina, G. G. (1978). General Electric Report #78CRD004, January.
Wachtman, J. B., Capps, W., and Mendel, J. (1972). J. Mater. 7_ [2] p. 188.
Wiederhorn, S. M. (1973). J. Am. Ceram. Soc. p. 99.
(1974). p. 613 in Fracture Mechanics of Ceramics,
Vol. 2, R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, eds.,
Plenum, New York.


In S (MPa)
hi
15.8
\
5.25
5.00
1£
(MPa-s-1)
21
33L 92% Crystalline Aged 1 Week
79
I
. 125
4.75 *
~1
0
5
In (MPa-s-1)
4
S (MPa)


7
-ne nucleation and growth of crystals. By masking portions of the glass,
intricate and precise patterns could be produced. As the crystalline
phase is less durable than the glassy phase, hydrofluoric acid could
dissolve away the more soluble phase, resulting in a glass part or
pattern made more precisely than any other conventional glass forming
method.
Later, reputedly by accident, Stookey allowed a sample of one of
his glass compositions to be heated too long, resulting in complete
crystallization of the sample. It was found to be remarkably fine
grained and harder and stronger than the parent glass. This led to
D
the development of Pyroceram and many of the other commercial glass-
ceramics we know today. The various families of glass-ceramic composi
tions and their applications will be discussed later in the chapter.
B. Nucleation and Crystallization
After the formation of the parent glass, the glass-ceramic process
required a two-step heat treatment. Figure lisa schematic of the
treatment. The first step is at a relatively low temperature (about
50-100C above the annealing point) and allows for the formation of
nuclei. In general, more nuclei form with time, so in order to obtain
a fine-grained material, longer times may be utilized. (The mathematics
of nucleation and growth have been well developed and will not be dis
cussed here. See Bergeron, 1973).
Nucleating agents are generally noble metals or refractory oxides.
Materials which have been used as nucleating agents include Au, Pt, Ag,
R Corning Glass Works, Corning, NY.


I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Professor of Materials Science
and Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Professor of Materials Science
and Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
7
R. W. prd'uld
Professor of Materials Science
and Engineering


The most crystalline composition (92% crystalline) shows the
greatest strength loss due to aging. This is attributed to water
migration along a network of microcracks into the grain boundaries.
At low volume fraction of crystals (7%) a residual stress in
the glassy matrix effects a strength increase and a decrease in the
environmental sensitivity which disappears with aging in water.
No effect of aging in water was found in the slow crack growth
studies. In all cases, the environmental sensitivity was found to
be much higher in these experiments than in dynamic fatigue. The
reasons for the difference are discussed in later chapters.
The limitations of the present fracture mechanics theory are
discussed in the light of the inconsistencies found between these
methods. It is recommended that a combination of techniques be used
when reliable lifetime predictions are required. A matrix of tests,
including static and dynamic fatigue and proof tests, is proposed
as an improved method of characterizing the fatigue behavior of a
material.
xn


65
where F is the failure probability for each strength value The
parameters m and SQ are the Wei bull modulus and scale parameter
respectively. The amount of scatter in the data can be quantified
in terms of m. As m increases the scatter becomes less (the slope of
Eq. 26 increases). Figure 19 is an example of a Wei bull plot for
33L-92% crystalline material.
For 33L-Glass, the m value ranges from 3.1 to 14.5, depending
on environment, stressing rate, and aging, but mostly ranging from
about three to eight. This is very low and indicates a large range
of flaw sizes controlling the strength of the glass.
Ideally, the sample size for an adequate Weibull analysis should
be thirty or more. Although this study rarely meets that requirement,
the m values are still reported, but should be taken only as an indica
tion of the true Weibull modulus of the material.
Lifetime Prediction Diagrams
Figure 20 is the lifetime prediction diagram (LPD) based on Eq. 22
for 33L-Glass. It can be seen that for a component to survive ten years
at 50MPa, different proof test levels are necessary depending on the
environment. In air, a proof test ratio of about 2.81 (140MPa) is
needed; in water, 13.46 (673 MPa); after aging one day, 5.21 (260 MPa).
Obviously, no 33L-Glass component would survive a proof test of
673 MPa, so that a different design level (shorter lifetime or lower


LIST OF FIGURES
Figure Page
1 Two stage and isothermal heat treatments
for processing a glass-ceramic 9
2 Tammann curves (nucleation or growth rates -
vs. temperature) for controlled and uncon
trolled crystallization 11
3 Phase diagram of the Li^O-SiO^ system 14
4 Three modes of failure: I, normal or opening
mode; II, sliding mode; III, tearing mode 20
5 Possible reactions between water and segments
of the glass network 23
6 Hypothetical changes in crack tip geometry
due to stress corrosion 26
7 Universal fatigue curve developed by Mould
and Southwick (1959) 28
8 Equation 19 plotted for 33L-Glass tested in
air 35
9 Lifetime prediction diagram using equation 22
for 33L-Glass 37
10 Specimen configuration for double cantilever
beam constant moment crack grov/th experiment 41
11 Biaxial flexure test jig 43
12 Constant moment double cantilever beam test
apparatus 45
.13 Representative microstructure of 33L-7%
crystalline 51
14 Representative microstructure of 33L-57%
crystalline 52
15 Representative microstructure of 33L-92%
crystalline 53
16 Dynamic fatigue results for 33L-Glass tested
in air 55
17 Dynamic fatigue results for 33L-Glass tested
in water 57
vi


CHAPTER V-TEST TECHNIQUES
A. Biaxial Flexure
Figure 11 shows the test jig for the biaxial flexure test. The
sample cup, which is mounted on the load cell of an Instron* testing
machine, has three press-fitted ball bearings which define a support
circle for the disc samples and also allows liquids or gases to be added
for testing in various reactive or non-reactive environments. The
loading pin is mounted on the crosshead of the testing machine and is
centered with respect to the circle defined by the ball bearings.
Detailed stress analysis of this method can be found in the work
of Kirstein and Woolley (1967). More recently, Wachtman, Capps, and
Mande! (1972) evaluated this method in a thirteen-laboratory round-
robin test. For small deflections, the strength, S, in this con
figuration is
(23)
where
A = radius of support circle
B = radius of loaded area
C = radius of specimen
P = load
d = specimen thickness
v = Poisson's ratio.
* Instron Corp., Canton, MA.
42


62
Table 5
Dynamic Fatigue Data for 33L-Glass
Tested After Aging 1 Day in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
5(MPa)
124
157
129
133
125
s(MPa)
8
49
37
46
42
s/S
7%
31 %
29%
35%
33%
n
9
31
15
31
31
Sm(MPa)
125
.145
120
121
112
m
14.5
3.6
3.7
3.3
4.0
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


In S (MPa)
0 1 2 3 4 5
In o (MPa-s-^)
S (MPa)


113
$
For 33L-57% crystalline, although testing was not done in air
or in water without aging, it appears that the same phenomena are
occurring as in 33L-7% crystalline. The interaction may proceed more
slowly (N = 59.2 vs H = 46.3 after aging one day), but the, presence
of considerably more grain boundary area drops the N value to a lower
one after one week (23.6 vs 27.4). Microcracking is also likely to
play a role in the stress corrosion process.
H. The Effect of Crystallization on Lifetime Predictions
Although in two cases there appear to be strength increases after
aging the proof test stress ratio (n/a) indicates that there is
no advantage gained over the long term. (These are also reported in
Table 21.) Except for 33L-Glass (in water versus aged), this ratio
increases with increasing availability of water. This means that to
guarantee a particular lifetime, a component must be tested at a much
higher proof test stress. For example, for 33L-Glass, even though
testing in air and after aging one day result in about' the same strength,
the proof test to guarantee ten years at 50 MPa would be 140 MPa in air
versus 275 MPa after aging.
Recalling Eq. 22
t. a*" = B (a /a )^
min a 'p a'
(22)
as N decreases, the time to failure under a given load after a given
proof test decreases dramatically. Correspondingly, the proof test
ratio for a given lifetime/load condition must increase. The same
is true as B decreases (as n B becomes more negative). This means
that even if a material becomes stronger (in short term testing) after
aging, it will be the fatigue parameters, N and B, which will control
the material's long term behavior.


108
F. The Effect of Crystallization on Strength
Reported in Table 20 are the average inert strength and strength
values for each condition tested at 16 MPa-s-"*. (The reader is also
referred to the previous tables, which are more comprehensive. These
values will serve to show the general trend.) Only 33L-Glass and 33L-7%
give any indication of strengthening after aging. The 33L-92% material
loses strength dramatically after aging one week. Strength decreases,
too, for 33L-57%, but not so precipitously. As previously discussed,
this is due to grain boundary attack.
It is difficult to interpret the effect of crystallization on the
strength of this system because of the microcrack formation at high Vv
and the residual stress at low Vv. Previous investigators (Freiman, 1968,
and Hasselman and Ful rath, 1966) have analyzed their materials' strength
as a function of mean free path between dispersed particles. That is,
the flaw size is limited by the interparticle spacing. However, Freiman
(1968) did not consider the presence of microcracks and Hasselman and
Ful rath (1966) assumed no thermal expansion mismatch between particle
and matrix.
For 33L-7% crystalline, the residual stresses in the glassy phase
induced by the thermal expansion mismatch between the glass and spherulites
is most likely responsible for the strength increase over 33L-Glass.
Freiman (1968) found an increase in Young's modulus with crystallinity,
but not enough to explain the strength increase. (Recall that the failure
strength is proportional to E^.) Miyata and Jinno (1972) re-analyzed
some of the work of Hasselman and Fulrath (1966) and observed a decrease


Dynamic Fatigue Data for 33L-92% Crystalline
Tested in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
213
214
199
193
181
s(MPa)
14
10
10
6
7
s/S
7%
5%
5%
3%
4%
n
8
10
11
9
10
Sm
208
213
196
196
180
m
13.8
. 20.6
17.9
29.3
24.0
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


CHAPTER IV-MATERIALS PREPARATION AND
CHARACTERIZATION
A. Melting and Casting
The parent glass was made by mixing reagent grade Li^CO^ and 5 pm
Min-U-Sil silica sand for one hour on a roller mill in a plastic jar.
Each batch weighed 200-250 g and was melted in a covered platinum crucible
for 24 hours in air in a electric muffle furnace at 1350C.
Casting was done in a graphite mold or graphite forms. Discs
required for the biaxial flexure test were made as follows: Cylinders
25 mm in diameter and about 50 mm long were formed and subsequently cut
into 2 mm thick slices with a diamond wheel. For double cantilever beam
specimens, bricks 20 mm x 25 mm x 80 mm were poured and subsequently cut
into plates with a diamond saw.
B. Annealing, Nucleation, and Crystallization Schedule
The completely glassy specimens were annealed immediately after
casting for four hours at 350C and allowed to furnace cool. Qualitative
analysis of residual stresses was made using a polariscope with a tint
plate. Samples showing excessive stress were remelted and recast.
Those samples to be crystallized were placed in a tube furnace
directly after casting and held at the nucleation temperature. For all
levels of crystallinity, the nucleation treatment was 24 hours at 475C.
This treatment was selected based on the work of Freiman (1968). The
size of the samples was also a factor. Because they were so large, a
long nucleation time was required to assure a consistent microstructure
throughout each specimen as well as from specimen to specimen. Two
cylinders or one brick could be treated at a time. To minimize thermal
38


116
Figure 34. Theoretical crack velocity vs. stress
intensity factor relationship for
brittle materials


64
The decrease in strength when tested in water and subsequent
increase after aging are consistent with previous works on glass
(Mould and Southwick, 1959, and others). However, the change in
stress corrosion susceptibility, N, from 30.5 (in air) to 11.0 (in
water) to 37.4 (after aging) was not anticipated. Because the
mechanism of stress corrosion (aqueous attack at the crack tip) is
the same in air and in water, the N values are expected to remain
constant. This large variation in N suggests that the kinetics of
aqueous corrosion are also involved in stress corrosion.
When testing is performed in air, the concentration of water at
the crack tip is low compared to that when testing in water. Less H+
would then be available for ion exchange with Li+, slowing the reaction
and allowing less stress corrosion (giving a higher N value).
The conditions at the crack tip are altered drastically by aging
in water. The strength increases due to a rounding of the crack tip,
which, from Eq. 1, lowers the amount of stress concentration. During
the aging period, a reaction layer forms on the surface which is
depleted in lithium (due to ion exchange). In order for the flaw to
grow, it must first penetrate the reaction layer before entering the
bulk glass. This reaction layer, even after the flaw has grown through
it, protects the bulk from the aqueous environment. The protection
thus afforded raises the N value to one even higher than that in air.
Wei bull Statistics
In the tables showing the raw data, m is a parameter called the
Wei bul 1 modulus. This parameter is obtained from (Wiederhorn, 1974)
In In(l-F)'1 = m In (S^S )
(26)


(a)
(b)
(c)


In S (MPa)
a. (MPa-s"1)
\
S (MPa)


117
and tested. After surviving this learning experience, the same
information could be obtained with about 25 samples.
B. 33L-Glass
Figures 35-36 show the V-K curves for 33L-Glass as tested in air
and water respectively. Tests on aged samples proved to be no different
from samples tested in water. Table 22 gives the values obtained for In
A and N determined by a least squares to fit the data. The critical
stress intensity factor, Kc, was determined to be about 0.78 MPa-nr2.
The values for N are all about the same (9-13). The curve shifts
slightly to the left in water. For the lowest velocities, measurements
were made over 24-30 hours.
The N values obtained in air and in water are different from those
obtained in dynamic fatigue testing. A proposed explanation and the
ramifications of this result will be discussed in Chapter VIII.
C. 33L-Partially Crystallized
Because of difficulty in crystallizing large bricks, it was not
possible to exactly duplicate the same volume percent crystallinity
for the DCB samples as in the dynamic fatigue samples. Figures 37-38
and Table 23 show the results of these sets of samples. Table 23 also
gives the percent crystallinity for each sample (determined as described
in Chapter VI). Although samples were tested after aging, their results
are plotted with those tested in water because no difference was found.
No Kc values were determined for these samples.
For samples containing low volume percent of crystallinity, the
cracks grew in fits and starts as the crack encountered matrix and


112
Hench (1977) has shown five types of corrosion in glasses,
depending on the composition of both the glass and the corrosive
medium. One type involves the development of a protective silica-
rich layer on the surface. Sanders and Hench (1973) have:sJnown
that 33L undergoes this type of corrosion. Apparently this
reaction layer can slow down the ion exchange process at the crack
tip necessary for stress corrosion, thus resulting in a higher N
value. The partially crystalline materials do not exhibit this
behavior because most of the corrosive attack is along grain
boundaries. This intergranular attack can only be detrimental
to the strength, since the "crack tip" cannot be Grounded, only
extended.
For the crystallized materials, N decreases or stays the same
with aging in water down to the range of values for the glass. This
implies that (eventually) the residual glassy phase is the controlling
material for stress corrosion in the glass-ceramics.
For 33L-92% crystalline, the microcracks and extensive grain
boundary surface area allow: rapid takeover of the process by the
residual glassy phase. The most significant point here is the
loss in strength after aging one week which was discussed earlier.
For 33L-7% crystalline, residual stresses in the glassy phase
obviously affect the kinetics of the aqueous corrosion. (The effect
of this stress on the strength was discussed earlier.) However,
once sufficient aging has allowed penetration of glass/crystal
boundaries, some of the stress is then relieved, at which point the
glass corrosion again becomes the controlling factor.


123
K (xl0~6 Pa-rn^)
In K (Pa-m^)
Figure 38. V~K relationship for some 33L-high crystallinity
samples


109
Table 20
Strength of 33L-Glass and Glass-Ceramics
C<
Inert Strength* Ambient Strength**
Glass
As prepared (Air)
179
125
Water

112
Aged 1 Day
211
129
33L-92% Crystalline
As prepared (Air)
274
226
Water

199
Aged 1 Day
280
203
Aged 1 Week
221
160
33L-7% Crystalline
As prepared (Air)
233
155
Water

141
Aged 1 Day

136
Aged 1 Week

153
33L-57% Crystalline
Aged 1 Day
210
148
Aged 1 Week
216
131
*Average strength (MPa), tested in liquid nitrogen at 16 MPa-s"^
**Average strength (MPa), tested at room temperature at 16 MPa-s"^


Temperature (*C)
14
LijO lmolol.1
"igure 3. Phase diagram of the LigO-SiO^ system


CHAPTER VII-CRACK VELOCITY EXPERIMENTS
A. General
Figure 34 shows the general empirical relationship between the
crack velocity, V, and the stress intensity factor, K. Region I,
where both V and K are relatively small, is the region of prime interest.
Here the empirical equation (Chapter III)
V = A KN (13)
holds, where A and N are the same constants determined in Chapter VI.
In Region I, the crack growth is a function of the reaction rate (of
water or water vapor with the material) at the crack tip.
In Region II, the crack velocity is independent of K. The crack
grows more quickly than in Region I so that the limiting process for
growth is the rate of transport (of water or water vapor) to the crack
tip. Region III shows a very steep dependence of V on K. Catastrophic
failure occurs in this region as K approaches the critical stress
intensity factor, K .
U >
The method of choice for determining the V-K curves for the 33L
system is the double cantilever beam (DCB) constant moment technique
described in Chapter V. This method, first described by Freiman et al.
(1973), is simple, inexpensive, and straightforward. However, as with
most experimental techniques, it is only simple and straightforward
after much trial and error and appropriate tricks have been learned so
that the experiment is conducted properly. For the set of data presented
here, nearly 100 samples (many outright washouts) were machined
115


(S- BdW)
37
1.5 2.0 3.0 4.0 5.0
Figure 9. Lifetime prediction diagram using equation 22
for 33L-Glass


5
the material. Ideally, the two methods should yield identical results.
Where they did not, microstructural analysis of the fracture surfaces
was used to interpret the differences. A statistical analysis of the
strength data is also important in determining the accuracy-of these
results and is an important part of this investigation.
Chapters II and III introduce the topics of glass-ceramics and
fracture mechanics, respectively. Chapter IV describes the prepara
tion and characterization of materials used in this investigation.
Chapter V discusses the mechanical testing methods employed. Chapters
VI and VII discuss the results and the analyses of the fatigue
parameters and lifetime prediction diagrams. Finally, conclusions
and suggestions for future work are presented in Chapters VIII and IX.


99
seen almost immediately. The residual stress in the 33L-7% material
is very likely responsible for the increased resistance to stress
corrosion, as exhibited in Figure 30. Aging apparently eliminated the
beneficial effects of the residual stress by attacking the pre
stressed glass/crystal interface. The consequence is a greatly
increased environmental susceptibility of the aged glass-ceramic
and a large reduction in the predicted lifetime (Fig. 30).
E. Dynamic Fatigue Testing of 33L-57%
Only a limited study vas made on 33L-57%. The results of tests
performed after aging one day. and one week are shown in Figures 31-32
and Tables 17-19.
The inert strengths (Table 19) of both sets are approximately
equal, showing no increase in strength of the one week aged samples
over those aged one day. However, the N value changes dramatically
(59.2 to 23.6). This again indicates a change in the flaws due to the
corrosive action of the water. In this case, any increase in strength
due to aging has been bypassed, and after one week, the full effect of
water attack becomes evident in the sharp increase in the susceptibility
to stress corrosion.
Figure 33 is the LPD for 33L-57% based on tests after aging in water
for one day and one week. Because of the difference in N, there is a
large difference between the two lines. For predictive purposes, the
data from the one week test would be recommended in order to be more
conservative.


crystals alternately during their propagation. Figure 39 is a
photomicrograph of a typical microstructure revealed as the crack
split the sample. The ripple marks show the staccatto nature
of the crack growth.
Nadeau and Bennett (1978) examined crack growth in glass
microscope slides containing grooves filled with glasses of
various thermal expansion coefficients. The ripple marks seen
in Fig. 39 resemble those observed by Nadeau as the crack
encountered a lower expansion inclusion. In the present case,
the thermal expansion coefficients for 33L-Glass and 33L-92%
crystalline were found to be (as measured by diTatometry*) 110 x 10~
-7 -1
and 95 x 10 C respectively, thus confirming Nadeau and Bennett
observations.
Figure 39 shows a region of slow crack growth (-10^ m-s~^).
At these velocities, the crack growth tends to be intergranular.
Figure 40 shows a region of the same sample where -the crack was
-3 -1
propagating catastrophically (~10 m-s ). Here the fracture is
more transgranular in nature, as there is sufficient energy to
drive the crack through the crystals.
The variation of the fracture parameters A and N with per cent
crystallinity will be discussed in Chapter VIII, along with the
comparison with the dynamic fatigue data.
*Harrop Laboratories, Columbus, Ohio


94
Table 14
Dynamic Fatigue Data for 33L-7% Crystalline
Tested After Aging 1 Day in Water
Stressing Rate (MPa-s"1)
158
79
16
7.9
1.6
S(Mpa)
138
164
136
139
134
s(MPa)
30
37
33
27
22
s/S
22%
23%
24%
19%
15%
n
9
9
9
9
6
Sm(MPa) '
134
165
151
145
124
m
4.5
3.8
3.7
4.7
5.1
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Weibull modulus


CHAPTER I-INTRODUCTION
Since the beginning of the twentieth century, the growth of techno
logy has put increasingly more stringent demands on materials and
material properties. Advances in the fields of transportation (automo
biles to aircraft to spacecraft), energy (coal and oil to nuclear and
solar), and others have necessitated improvements in existing materials
and the development of new materials in order to meet the high-performance
requirements of these .technologies. The investigation of how to make
materials stronger with lighter weight has been advanced by studying
how materials fail. Macroscopic observation of the growth of cracks
combined with microscopic examination has been the most fruitful method
of learning the mechanisms of material failure. Studies of this type
have led to improvements in both the strength and toughness of materials
through the design and control of specific microstructures which impeded
initiation and/or propagation of cracks.
Recently, concern with the conservation or security of complex sys
tems has placed additional emphasis on determining how long materials
will survive under-various environmental conditions. Thus,the field of
fracture mechanics has expanded from relatively simple measurement
taking to making failure or lifetime predictions from those measurements.
In order to make meaningful predictions, parameters, such as-or-iginal
flaw size, critica! f1 aw size, and stress corrosion susceptibi1ity
become important. Also, there is a need to know what happens as a flaw
grows from a relatively insignificant stress concentration to some
critical size where catastrophic failure occurs. This process is termed
1


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
FRACTURE MECHANICS AND FAILURE PREDICTIONS
FOR GLASS, GLASS-CERAMIC, AND CERAMIC SYSTEMS
By
Ronald A. Palmer
Dune, 1981
Chairman: Larry L. Hench
Major Department: Materials Science and Engineering
State-of-the-art fracture mechanics techniques are used to determine
the fatigue parameters for the lithium disilicate glass/glass-ceramic
system. Glass-ceramics are ideal materials for examining the effects
of crystallization on fatigue behavior. Two methods (dynamic fatigue
and slow crack growth) were employed and were found to produce different
values for the fatigue parameters. These parameters are used to design
lifetime prediction diagrams and, theoretically, should be independent
of the method of determination.
It was found that aqueous corrosion has a severe effect on the
fatigue parameters, as measured in dynamic fatigue, especially after
aging for up to one week in distilled water. The lithium disilicate
glass, which is known to readily form a gel layer in water, exhibits
the most rapid change in parameters.


81
Table 11
Liquid Nitrogen Strength of 33L-92% Crystalline
As Prepared Aged 1 Day Aged 1 Week
S(MPa)
274
280
221
s(MPa)
19
20
26
s/S
7%
7%
12%
n
9
15
10
Sm(MPa)
282
276
213
m
12.7
14.2
8.1
S =
average strength
s =
standard deviation
s/S =
coefficient of variation
n =
sample size
Sn, -
median strength
m =
Wei bull modulus


In S (MPa)
(MPa-s*1)
S (MPa)


Figure 28. Dynamic fatigue results for 33L-7% crystalline
tested after aging one day in water


105
Table 17
Dynamic Fatigue Data for 33L-57% Crystalline
Tested After Aging 1 Day in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
167
166
148
136
160
s(MPa)
18
13
19
10
11
s/S
11%
8%
13%
8%
7%
n
9
10
9
; 8
9
Sm(MPa)
175
167
146
136
162
m
8.5
12.6
6.9
12.5
14.0
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


15
are solely lithium-disilicate (Li^0-2Si0^) There has been much work
done on both the binary system and systems with additions of nucleating
agents and aluminum oxide. As presented in Table 1, L^O-ZSiC^ is the
major phase in many commercially important glass-ceramic compositions.
For this reason, study of the 33L composition will provide a basis on
which the characterization of the whole family of 1ithia-silica glass-
ceramics can be done.
The photo-chemically machineable glass-ceramics which first appeared
in the 1940's are lithia-silica based materials. Stookey's (1947, 1950a,
1950b, 1954, 1956) work on these materials is found in the patent literature.
Rindon (1958, 1962) studied the addition of platinum to a Li^O^SiOg
glass. He measured (using x-ray diffraction) the amount of LigO*2SiO2
crystallized from the glass as a function of time (10 minutes to 32 hours)
and the amount of Pt added (zero to 0.025%). Rindone proposed that lithium-
rich clusters surrounding the Pt form as nuclei for the LigO^SiOg which
precipitates out. Later, Kinser (1968) showed that a metastable lithium-
metasilicate phase (Li2O SiO2) is the precursor to the disilicate crystals
in the 33L glass. '
Glasser (1967) studied binary lithium silicates ranging in composi
tion from 80 to 88 weight percent SiO^. He concluded that under certain
conditions a two-stage heat treatment schedule would improve the strength
of the crystallized composition. While accepting the explanation that
the formation of a large number of nuclei results in a fine-grained material,
Glasser also proposed another mechanism. He suggested that the lower
temperature treatment yields a metastable solid solution that is later


128
K (xlO^Pa-nr1)
Figure 41. V-K relationship for 33L-92% crystalline
V (m-s


In S (MPa)
0 1 2 3 4 5
In o (MPa-s-1)


In S (MPa)
(MPa-s"1)
In 0 (MPa-s-1)
S (MPa)


80
Table 10
Dynamic Fatigue Data for 33L-92% Crystalline
Tested After Aging 1 Week in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
175
169
160
156
141
s(MPa)
18
15
17
18
15
s/S
10%
9%
11%
11%
11%
n
14
. 14
13
14
14
Sm(MPa).
169
165
156
152
137
m
9.8
11.0
9.4
8.9
9.6
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


7,06
Table 18
Dynamic Fatigue Data for 33L-57% Crystalline
Tested After Aging 1 Week in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
163
159
131
139
140
s(MPa)
9
15
19
14
11
s/S
6%
9%
15%
10%
8%
n
8
10
7
8
6
Sm(MPa)
165
155
126
135
140
m
15.2
10.4
6.2
9.2
11.6
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
S = median strength
m
m = Wei bull modulus


BIOGRAPHICAL SKETCH
The author was born October 21, 1950, in Ft. Knox, Kentucky.
He grew up in various towns in Central Pennsylvania and Western New York,
graduating from Horseheads High School in Horseheads, New York, in 1968.
He earned his B.S. in Glass Science from Alfred University in 1972.
After graduation, he went to work for Metro Container Corp., in Jersey
City, New Jersey, as a Quality Control Engineer. While there, the author
developed an interest in the fracture phenomena in glass and ceramic
materials, and so moved on to graduate school at the University of Florida
in 1976.
In 1975, he married Ellen Goldberg in the Bronx, New York.
Prior to completion of this dissertation, the author began employment
as a Senior Chemist at Rockwell Hanford Operations in Ri.chland, Washington.
156


60
Table 3
Dynamic Fatigue Data for 33L-Glass
Tested in Air
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
136
131
125
135
113
s(MPa)
20
13
29
37
19
s/S
15%
10%
23%
28%
17%
n
32
30
32
31
32
S (MPa)
m
136
132
123
125
112
m '
7.4
11.2
4.9
4.0
6.7
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


r
Figure 40. Region of fast crack growth in
33L- 5% crystal 1 ine


CHAPTER III -FRACTURE MECHANICS
A. Historical Background
It is generally recognized that Griffith (1921) is the father of
fracture mechanics. He was the first to show a relationship between the
measured strength of a material, its material properties, and the length
of the pre-existing crack responsible for failure. The existence of pre
existing cracks as precursors to failure had been proved by Inglis (1913)
some years before. Inglis showed that the stress near an elliptical hole-
(resembling a crack) would be greater than the applied stress by
o = 2aa(a/p)z (1)
where a& is the applied stress, a is the half-crack length, and p is the
crack tip radius.
Griffith reasoned that the free energy of a cracked body and the
applied forces should not change during crack extension. That is, the
amount of energy put into breaking the material should be equivalent to
the amount of energy required to create two new surfaces, or
dU
da
dW
da
(2)
where U is the strain energy due to the crack, W the energy required for
growth, and a the half-crack length.
Griffith calculated the strain energy to be
U =
a
(3)
where E is Young's modulus, and the energy required for growth to be
W = 4ay (4)
where y is the surface energy. Inserting Eqs. (3) and (4) into (2)
gives the Griffith criterion for fracture of an infinite sheet with a
18


I I +
CSi0 [M]] + H20 -> [SiOH] + M + OH
I I I
[SiO-Si] + H20 2 [-si OH]
[Si0Si] + OH' [Si OH ] + [-Si-O']
I I
[-Si-0 ] + H20 [SiOH] + OH


79
Table 9
Dynamic Fatigue Data for 33L-92% Crystalline
Tested After Aging 1 Day in Water
Stressing Rate (MPa-s~b
158
79
16
7.9
1.6
S(MPa)
207
206
203
187
179
s(MPa)
9
6
10
8
7
s/S
5%
3%
5%
5%
4%
n
6
15
4
15
6
Sm(MPa)
208
207
200
187
181
m
19.9
. 33.7
15.7
22.7
22.3
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Weibull modulus


118
Table 22
33L-Glass Fracture Parameters
as Determined by Slow Crack Growth
Experiment
Environment
N
In A
52
Air
9.5
-138.7
55
Air
11.0
-158.1
57
Air
12.6
-179.8
36
Water
9.4
-137.2
52
Water
11.0
-153.6
57
Water
12.7
-176.1
C


129
Table 24
33L-92% Crystalline Fracture Parameters
as Determined by Slow Crack Growth
Experiment
Environment
N
InA
58
Air
10.9
-169.1
59
Air
16.8
-248.9
60
Air
16.8
-255.5
67
Air
13.1
-203.7
61
Water
14.4
-216.2


CHAPTER IX-CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK
A. Conclusions
1. The testing of a material which readily reacts with an
aqueous environment has shown that the present model for describing
fatigue by subcritical crack growth needs to be revised to account
for the dynamics of interaction of the material with its environment.
Because the fatigue parameters (N and B) involved in calculating
lifetimes have been observed to change with time, a single analysis
based on a short-term test is not adequate for reliable predictions.
Two further developments are needed. First, the model must be
extended to account for the changes in the fatigue parameters.
Second, a series of tests needs to be devised to characterize these
changes for a given material.
2. The strength of lithium disilicate materials increases as
the percent of crystallinity increases, but not in a simple straight
forward manner. At low levels of crystallinity, a compressive stress
may be induced in the glassy matrix, resulting in a stronger material.
At high levels of crystallinity, microcracking appears to be an
effective toughening mechanism.
3. For the lithium disilicate glass/glass-ceramic system,
the stress corrosion susceptibility (N) is more dependent on aging
in water than on the percent of crystallinity. The kinetics of
aqueous corrosion of the residual glassy phase and any residual
stresses present play important roles in determining the fatigue
behavior of all materials tested.
147


133
Figure 44. Growth of a gel layer at the surface of
a glass containing two depths of flaws


Figure 21. Dynamic fatigue results for 33L-92% crystalline
tested in air


20
Figure 4.
Three inodes of failure: I, normal or opening
mode; II, sliding mode; III, tearing mode


REFERENCES
Argon, A. S. (1974). p. 154 in Composite Materials, Vol. 5, Fracture
and Fatigue, L. J. Broutman, ed., Academic Press, New York.
Baldwin, J. A. and Palmer, R. A. (1980). To be published.
Bergeron, C. 6. (1972). p. 173 in Introduction to Glass Science,
L. D. Pye, H. 0. Stevens, and W. C. LaCourse, eds., Plenum, New York.
Borom, M. P. (1977). J. Am. Ceramic Soc. 60 [1-2] p. 17.
Borom, M. P., Turkalow, A. M., and Doremus, R. H. (1975). J. Am. Ceram. '.
Soc. 58 [9-10] p. 385.
Bradt, R. C., Hasselman, D. P. H., and Lange., F. F. (1974, 1978).
Fracture Mechanics of Ceramics, Vol. 1-2 (1974), Vol. 3-4 (1978),
Plenum, New York.
Broek, David (1974). p. 16 in Elementary Engineering Fracture Mechanics,
Noordhoff, Leyden.
Brown, S. D. (1978). p. 597 in Vol. 4, Fracture Mechanics of Ceramics,
R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, eds., Plenum,
New York.
Charles, R. J. (1958). J. Appl. Phys. Z9_ p. 1549.
Charles, R. J. and Fisher, J. C. (1960). p. 491 in Conference on Non-
Crystalline Solids, V. D. Frechette, ed., John Wiley and Sons,
New York.
Clark, D. E. (1976). Ph.D. Dissertation, University of Florida.
DeHoff, R. T. and Rhines, R. N. (1968). Quantitative Microscopy,
McGraw-Hill, New York.
Dilmore, M. F. (1977). Ph.D. Dissertation, University of Florida.
Doremus, R. H. (1980). Eng. Fract. Mech., 13^ p. 945.
Doremus, R. H. and Turkalow, A. M. (1972). Phys. Chem. G., 13_ [1] p. 14.
Ethridge, E. C. (1977). Ph.D. Dissertation, University of Florida.
Evans, A. G. (1974a). p. 17 in Fracture Mechanics of Ceramics, Vol. 1,
R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, eds., Plenum,
New York.
Evans, A. G. (1974b). Int. J. Fract., 10^, [2] p. 251.
152


141
For both ZZL-7% and 33L-57% there are instances of the strength
at the lowest stressing rate being higher than at higher stressing rates.
This may be an indication of the competition of the corrosion process
rounding the crack tip and the applied stress extending the crack.
Such a competition indicates that the test has gone beyond the range
of Region I of the V-K relationship. If that is the case, it would
be better to use only the remaining four sets of data to determine N.
The slope would then increase, decreasing the value of N.
The fatigue life of high strength steels has been found to
deviate from the usual log-normal or Wei bul 1 distribution at low
stress levels as well. Ichikawa, Takura, and Tanaka (1980) have
decomposed the distribution of the total life into two parts, one at
crack initiation and one of crack propogation. The departure from
usual statistical distributions at low stress levels is attributed
to the influence of the scatter of the length of the initiation cycle,
which increases as the stress level decreases. The crack tip rounding
process in ceramics is likely to cause similar disruptions.
Finally, it must be remembered that the equations governing the
analysis: of dynamic fatigue experiments are empirical and are derived
using several assumptions (see Chapter III for derivation). These
include:
1) constancy of flaw population,
2) constancy of environment,
3) constancy of chemical properties of the material, and
4) constancy of fatigue parameters N and B.


In S (MPa)
(MPa-s-1)
0 1 2 3 4 5
In (MPa-s-1)
S (MPa)


68
stress) is mandatory for 33L-Glass in water. However, if the data from
the aged samples are used, a proof test of only 260 MPa is necessary.
There is still a need to lower the design level, but not nearly as much.
The problem is, which data are relevant?
This situation illustrates the dynamics of the problem which have
yet to be discussed, let alone resolved. In order to make reliable
lifetime predictions, the conditions that a component sees must be known
(or assumed). It is also assumed (Wiederhorn 1973) that there is no
change in the flaw during or after proof testing. It is evident from
this work that a corrosive environment will have some effect on the flaw
and its immediately surroundings in the bulk. It would seem, then, that
no reliable predictions can be made using this procedure for materials
in corrosive environments due to the changing character of the flaws.
More discussion will follow after presenting the crack velocity
studies.
C. Dynamic Fatigue Testing of 33L-92%
Figures 21-24 present the dynamic fatigue results for 33L-92% and
Tables 7-10 give the raw data. Tests were conducted in air, in water,
and in water after aging for one day and one week. Table 11 shows the
results of the inert environment testing.
There was the expected decrease in strength when tested in water,
but no increase after aging. After aging one day, there was little
change over testing in water, but after one week, there was substantial
decrease in strength. This decrease can be attributed to the pervasive
attack of the water throughout the continuous microcrack network.
Similar effects have been seen in other porous ceramics (Frakes, Brown,
and Kenner 1974).


ni
Table 21
Fracture Parameters and (o /a ) from Dynamic Fatigue Data
P d
N
£nB
(Q/<0*
33L-G1ass
p u
Air
30.5
-2.07
2.8
Water
11.0
4.23
14.1
Aged 1 Day
37.4
-30.73
5.5
33L-92% Crystalline
Air
70.6
-5.66
1.7
Water
25.7
2.10
3.0
Aged 1 Day
27.6
0.73
2.9
Aged 1 Week
22.3
2.37 ;
3.4
33L-7% Crystalline
Air
116.1
-34.43
1.8
Water
136.0
-53.93
1.9
Aged 1 Day
46.3
-14.88
2.6
Aged 1 Week
27.4
-1.13
3.0
33L-57% Crystalline
Aged 1 Day
59.2
-10.16
1.9
Aged 1 Week
23.6
0.06
3.4
*For 10 years at 50 MPa


114
(Note: At this point in time, the parameter B has not
?
been well characterized. In Eq. 22, it must have the units of MPa -s,
but it also has been defined in terms of Kc raised to some power
which depends on N. The theoretical approach employed here cannot
resolve this problem. Other approaches solve this situation, but
complicate the analysis and do not facilitate the construction of
lifetime prediction diagrams described herein.)
For 33L-Glass, the proof test stress ratio increases when
tested in water versus air because of the sharp drop in N. After
aging one day, the increase is due to the change in £n B.
For the crystallized materials, it is always a large change
in N which is responsible for increase ratio. (Almost always, £n B
is increasing.) It may be stated, then, that for the lithium disilicate
glass/glass-ceramic system, the N value controls the predicted lifetime.
I. Summary
For dynamic fatigue testing, it has now been'demonstrated that
knowledge of short-term corrosion effects may be totally inadequate
for predicting very long life-times. The changes in the fatigue
parameters, N and B, with aging time in water are inconsistent with
the basic theories upon which the lifetime predictions are made. The
actions of corrosive media on reactive materials must be characterized
for both the stressed and unstressed states as well as at the surface
and crack tip. Kinetics of reaction and growth of reaction layers
are very important for determining the long-term behavior of ceramics
under load in a corrosive environment.


52
¡
:
!
Figure 14. Representative microstructure of
33L-57% crystal!ine


145
Mu ti-technique testing is the most straightforward way of resolving
his last conclusion. Once that relationship is understood, crack
propagation data may then be useful for lifetime prediction and real
"agreement" may then be possible.
Argon (1974) has stated that "... fracture is not a property of a
material but rather constitutes a behavior." The difference between
fatigue and fracture results found in this study as well as the environ
mental and aging effects demonstrated herein tend to support both Argon
and Doremus' concerns.
In order to use the present lifetime prediction theories, it
must be remembered that the empirical model developed to describe
the fatigue behavior of a ceramic under a static load assumes that
the parameters derived from short term tests remain constant over
the lifetime of the material. Unfortunately, some materials will
react with their environment over time in such a way that the parameters
originally used must change. These changes will then negate the original
analysis. Each material will need to be characterized extensively as to
how the parameters might change if the model is to be valid. For
example, if N increases (as with 33L-Glass after aging in water one day),
the original lifetime prediction will be conservative. If N decreases
(as with 33L-92% crystalline after aging one week), the component's
lifetime may be drastically (and catastrophically) shortened.
Thus, the present model is not designed to handle a dynamic
system. A new model needs to be developed to account for changes
in the parameters which describe the process of subcritical crack


To Ellen


Figure 22. Dynamic fatigue results for 33L-92% crystalline
tested in water


In S (MPa)
(MPa-s-1)
In a (MPa-s-1)
S (MPa)


10
Cu, TiC^, and P20g- It is recognized that there are certain key
characteristics for a nucleating agent to be effective. For nucleus
formation, it is important for the nucleant to be very soluble at
melting and forming temperatures while barely soluble at low temperatures,
to have a low free energy of activation for homogeneous nucleating, and
to be very mobile compared to the major components of the glass at low
temperatures. For the nucleant to promote crystal growth effectively,
there must be a low interfacial energy between it and the glass and
its crystal structure and lattice parameters should be similar to that
of the crystal phase.
Glass-in-glass phase separation has been found to promote glass-
ceramic formation without any nucleating agents. Agents which promote
phase separation, such as P2OJ-, are very effective in this manner.
Nucleation may occur either at the phase boundary or within one of the
phases (most likely the one most resembling the crystalline phase.)
The second step in the so-called "ceraming" process is the crystal
lization treatment. For materials where only one crystalline phase develops,
the crystal growth may be a very simple process. However, if several phases
are developing, all different from the composition of the homogeneous glass,
the composition at the crystal-glass interface is constantly changing and
the growth process becomes exceedingly complex.
Figure 2 shows the Tammann curves (nucleation or growth rate vs.
temperature) for controlled and uncontrolled systems. For a controlled
system, it is easy to find nucleation and growth temperatures (ideally
at the maximum rates) where the other process does not occur. This type
allows the experimenter to have great control over the microstructure


63
Table 6
Liquid Nitrogen Strength of 33L-Glass
As Prepared
Aged 1 Day
S(MPa)
179
211
s(MPa)
34
61
s/S
. 19%
29%
n
11
10
Sm(MPa)
178
224
m
5.0
3.1
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Weibull modulus


97
Again a strength decrease was found when testing in water versus
air, but virtually no change in N was found. After aging one day, both
the strength and N value decreased. Aging for one week allowed the
material to regain most of its strength while dropping N still further.
Apparently, the short time in water has virtually no effect on the
subcritical growth in this microstructure. Aging one day for 33L-7%
appears to be equivalent to testing in water for the uncrystallized
glass. Both result in a drastic loss in strength and increased suscep
tibility to stress corrosion. After aging one week, 33L-7% is remarkably
similar to 33L-Glass after aging one day. The partially crystallized
material is somewhat stronger, but the values for N are similar.
It would be expected that 33L-7% would behave like 33L-Glass because
the glassy phase should dominate environmental sensitivity. However, the
extraordinarily high values of N for 331-7% must be a result of the
kinetics of the water/glass-ceramic reaction. As the crystals have a
lower coefficient of thermal expansion, a compressive stress is induced
in the glass. This residual stress is perhaps inhibiting the stress
corrosion process.
The Weibull moduli are very low (3.3-6.8), as they were for 33L-Glass
Again this indicates a wide range of flaw sizes controlling the strength.
The variability in sample crystallinity is at least partially responsible
for the large scatter.
Figure 30 is the LPD for 33L-7%. The clustering of the air and
water lines apart from the one day and one week aged lines indicates that
some time is needed for the water to have an effect on this system. This
is contrasted with the 33L-92% material, in which the effect of water is


146
growth. The results of this study suggest that the following
innovations should be considered:
1) the kinetics of the chemical reaction at the crack tip
need to be accounted for. These studies would establish
the time dependence of the fatigue parameters and (possibly)
the tip radius and depth of the flaw.
2) the stress dependence of the chemical reaction at the
crack tip must be established. This also affects the
fatigue parameters and flaw characteristics.
3) the chemical changes in the surface layer surrounding
the flaws must be characterized. A change in the
chemistry (e.g., H+ replacing Na+) or state (e.g.,
glass to gel) will certainly affect the growth of
the flaw.
4) the synergism of all three of the above needs to be
characterized.
As more and more materials are studied, more changes will be
identified and their quantitative nature will become clear. In this
manner the necessary components for a complete model will slowly piece
together for a more consistent theory that can handle reactive
materials or the reactive phases of brittle materials. But, for now,
every investigation is an adventure.
0


subcri ti cal crack growth. Characterization, ..of a material 'ss-ubcr-i-t-ical
crack growth behavior and knowledge of the distribution of flaw sizes
permits prediction of the lifetime of a component in a given situation.
In order to predict and subsequently to guarantee the lifetime of
a component, some method of non-destructive evaluation is required.
Proof tests can be designed, in which all components are subjected to
a stress far beyond that expected in service, such that survivors are
assured some minimum service lifetime. By performing a proof test,
the need for large safety factors is reduced and overdesigning can be
avoided.
It is essential that the effect of the service environment on
time to failure be completely and correctly accounted for in fracture
mechanics theories if lifetime predictions are to be realistic. Thus,
a deeper understanding of the physical implications of the stress
corrosion susceptibility parameters is important. The combined effects
of aqueous environment and stress state on the subcritical growth of
flaws of various sizes need to be understood in order to ensure that
aging effects are correctly modeled and/or simulated in the proof
test design.
Therefore, one major objective of this thesis is to examine the
relative importance of aqueous environments, including time-dependent
effects, on the lifetime predictions for a brittle material. In order
to achieve this objective, it is necessary to study a more chemically
reactive material system than has heretofore been examined in lifetime
prediction investigations. By testing a material with known mechanisms
of chemical attack it is hoped that the critical reaction phenomena
for deterioration of mechanical performance can be elucidated.


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Page
VI DYNAMIC FATIGUE: RESULTS AND DISCUSSION 48
A. Quantitative Microscopy 48
B. Dynamic Fatigue Testing of 33L-Glass 48
C. Dynamic Fatigue Testing of 33L-927 A 68
D. Dynamic Fatigue Testing of 33L-7% 82
E. Dynamic Fatigue Testing of 33L-57% 99
F. The Effect of Crystallization on Strength.. 108
G. The Effect of Crystallization on N 110
H. The Effect of Crystallization on
Lifetime Predictions 113
I. Summary 114
VII CRACK VELOCITY EXPERIMENTS 115
A. General 115
B. 33L-G1 ass 117
C. 33L-Partially Crystallized !.. 117
D. 33L-92% Crystalline 127
VIII DISCUSSION 132
A. Dynamic Fatigue 132
B. Slow Crack Growth 142
C. Comparing Dynamic Fatigue and
Slow Crack Growth 143
IX CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK... 147
A. Conclusions 147
B. Suggestions for Future Work 148
REFERENCES 152
BIOGRAPHICAL SKETCH 156
v


127
D. 33L-92% Crystalline
Figure 41 and Table 24 provide the results for the completely
crystallized material. Due to the extensive microcracking, problems
again were encountered in measuring the crack velocity. In addition
to the halting nature of the propagation, the microcrack network
tended to guide the crack along a tortuous, twisted path, rather than
continuous and straight. The K was determined to be 4.19 MPa-m^.
V
Rao (1977) also measured K£ for the lithium disilicate glass and
glass-ceramic using double torsion. His values ranged from 1.015 MPa-nr5
for the glass to 3.53 MPa-m2 for an 85% crystalline material. The range
found in this study is 0.78 MPa-rn2 for 33L-Glass to 4.19 MPa-m52 for
33L-92%. The agreement is reasonable, given the two different techniques.
An increase in Kc with crystallinity can be expected for two reasons.
One, the particles act to pin the crack; and two, the microcracking also
acts as a toughening mechanism. (Faber and Evans, 1980, hav recently
discussed this last point.) Both of these aspects were discussed with
the effects of crystallization on strengthening.
As for the partially crystalline materials, samples tested after
aging in water gave results identical to those tested immediately in
water. Thus, they are plotted in Figure 41 on the same line.
Figures 42-43 show regions of slow and fast crack growth respectively.
Again, slow growth is intergranular and rapid growth is transgranular.
The stress corrosion susceptibility, N, as measured by crack growth
methods differs from that measured by dynamic fatigue, as was found for
the uncrystallized samples. Chapter VIII will discuss this difference.


CHAPTER II-GLASS CERAMICS
A. General
Glass-ceramics may be described as polycrystalline solids which have
been formed from a glassy melt by carefully controlled heat treatments.
In order for crystals to form, nuclei must be present. The nucleation
may be homogeneous (self-nucleating) or heterogeneous (by addition of
nucleating agents). The crystallization process may be controlled or
uncontrolled. Uncontrolled crystallization is termed devitrification
and is generally though of as a process to avoid. Controlled crystalli
zation of glasses produces materials known as glass-ceramics. Glass-
ceramics have also been referred to as "vitrocerams," "devitrocerams,"
"sitalls," and "melt-formed ceramics." (See Stewart, 1972.)
Glass makers through the ages have struggled to avoid the destruc
tive effects of devitrification. However, occasionally attempts were
made to crystallize glass purposely. Reaumur (1739) in the early 1700's
crystallized soda-lime-silica glass bottles by packing them in sand and
gypsum and heating them for a long time. Unfortunately, the crystals
nucleated at the surface, resulting in weak and deformed pieces.
The classical work on nucleation and crystallization was done by
Tammann (1925) in the 1920's. His theoretical and experimental investi
gations laid the foundation for our present knowledge of these processes.
In the 1940's photosensitive glasses were developed by Stookey (1947)
at the Corning Glass Works. Colloidal dispersants of metals (gold,
copper, or silver) added as nucleants were shown to be activated by
exposure to ultra-violet light. Subsequent heat treatments resulted in
6


47
Measurement of the crack velocity was made with a traveling
microscope.* The magnification was 32X. The accuracy of the
microscope was 0.0005 mm. Readings were made to the nearest
micrometer.
The above procedure will provide crack velocity data for stress
intensity factors less than Kc. Values for Kc, the critical stress
intensity factor, were determined using this apparatus attached to
an Instron** machine. An initial load was applied to begin the crack
propagation, then the cross head was turned on at a constant speed
of 0.02 inches per minute and the crack then propagated to failure.
The highest value of the load (T) then is substituted into Eq. 25,
and the value for Kc determined.
* Gaertner Scientific Corp., Chicago, IL.
** Instron Corp., Canton, MA.


149
the "agreement" between the theory and experiment is merely fortuitous.)
The next logical step in developing the theory is to incorporate devices
to accommodate a dynamic system. Brown (1978) has developed a kinetic
theory of crack growth, but it is extremely complex and cumbersome.
What appears to be needed is a complete analysis of the phenomena
occurring at and around the flaw. The flaw, be it controlled damage
(machining damage or indentation) or inherent weakness (pore or
"Griffith microcrack"), must first be known or well characterized.
The environmental changes of the bulk material which contain the flaw
are also critical and need to be incorporated within a dynamic theory.
Because failure is caused by the growth of flaws, characterization
of the type of flaw responsible for the failure is paramount. Although
depth and sharpness are probably the most important parameters, other
characterization features may include shape, residual stresses about
the flaw, and compositional gradients (such as H+) ahead of the flaw.
Because of the complexity of characterizing .flaws,, perhaps the
best method of determining the nature of the flaw is to carefully
and deliberately create it in a controlled manner. Mould and
Southwick (1959) used various grades of emery cloth to create flaws
in glass. Mendiratta and Petrovic (1978) have used the indentation
technique to create known flaws in silicon nitride. For both methods,
the post-damage treatment is very important. Aging in various
environments (water for glass or oxidizing atmospheres at high
temperature for silicon nitride) will have profound effects on the
surface damage but has not been considered in the above studies.
Post-fracture analysis of the specimen is also useful for identifying
the nature of the flaws which cause failure. Fractographic techniques


Figure 31. Dynamic fatigue results for 33L-57% crystalline
tested after aging one day in-water


46
test apparatus for this technique. The test specimen is cemented
using epoxy to the loading arms in a pair of slotted inserts to insure
proper alignment. All the pivot points have suitably low friction
provided by bearings. The load is applied by means of a weight
pan connected through a triangular piece (assuring equal load
distribution) to the loading arms. A constant load provides a
constant moment applied by the arms to the specimen. This yields
a constant stress intensity factor, defined by
K = --T- (25)
x/IT
where
T = load
L = moment arm length
1
I = i~2 b h = moment of inertia of the beam
t = web thickness,
as defined in Figure 10.
A starter crack is initiated at the base of the slot in the groove
by tightening a sharp screw against the ungrooved side. The crack will
grow at a constant velocity under a constant load. The range of
velocities measured was from 10^ to 10^. m/s. With proper care,
multiple measurements may be made on one sample by changing the load
after taking sufficient readings at one load. Measurements were made
every few hours for very slow velocities and every half minute for
fast velocities. Again, testing was done in air and water at room
temperature (^22C).


153
Faber, K. T. and Evans, A. G. (1980). Presented at the Pacific Coast
Section Meeting of the American Ceramic Society, October 29.
Ferber, M. K. and Brown, S. D. (1980). J. Am. Ceram. Soc., 63_, p. 242.
Frakes, T., Brown, S. D., and Kenner, G. H. (1974). Bull. Am. Ceram.
Soc., 53^, p. 183.
Freiman, S. W. (1968). Ph.D. Dissertation, University of Florida.
^Freiman, S. W., Mulville, D. R., and Mast., P. W. (1973). J. Mater. Sci.,
8, p. 1527.
Fuller, E. R., Jr. (1979). p. 3 in Fracture Mechanics Applied to Brittle
Materials, ASTM STP 678, S. W. Freiman, ed., American Society for
Testing and Materials, Philadelphia.
Glasser, F. P. (1967). Phys. Chem. G1., 8 [6] p. 224.
Greenspan, D. C. (1977). Ph.D. Dissertation, University of Florida.
Griffith, A. A. (1921). Phil. Trans. Roy. Soc. (London), 221A p. 163.
Harper, H., James, P. E., and McMillan, P. W. (1970). Disc. Far. Soc.,
50, p. 206.
Hasselman, D. P. H. and Fulrath, R. M. (1966). J. Am. Ceram. Soc., 49
[2] p. 68.
Hench, L. L. (1977). Presentation at the Xlth International Congress on
Glass, Prague, Czechoslovakia.
Hillig, W. B., and Charles, R. J. (1964). p. 682 in High Strength
Materials, V. F. Zackay, ed., John Wiley and Sons, New York.
Ichikawa, M., Takura, T., and Tanaka, S. (1980). Int. J. Fract., Tj5,
p. R251.
Inglis, C. E. (1913). Proc. Inst. Nav. Arch., 55^ p. 219.
Irwin, G. R. (1958). p. 558 in Handbuch der Physik VI, Fliigge Ed.,
Springer Verlag, Berlin.
Kinser, D. W. (1968). Ph.D. Dissertation, University of Florida.
Kirstein, A. F. and Wolley, R. M. (1967). J. Res. Nat. Bur. Std.,
71c [1] p. 1.
Lange, F. F. (1971). J. Am. Ceram. Soc., S4 [12] p. 614.
(1974). p. 1 in Fracture Mechanics of Ceramics, Vol. 1.,
R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, eds.,
Plenum, New York.


151
If the static and dynamic fatigue tests predict different distributions
(because of different N and B parameters), the final failure
distribution will determine the appropriate ("correct") method which
applies.
Theoretically, all N values for static fatigue, dynamic fatigue,
and the final failure experiment should be equal. Depending on the
material and environment, they may not be equal, as shown in this
study. The above methodology coupled with fractographic detailed
analysis of the flaws and the kinetics of the material/environment
interaction should pinpoint the reasons for the differences and
allow appropriate formulations to be devised for a realistic dynamic
model.
0


CHAPTER VI-DYNAMIC FATIGUE: RESULTS AND DISCUSSION
A. Quantitative Microscopy
For the dynamic fatigue specimens, at least ten different discs
representing about half of the cylinders were examined for percent
crystallinity. Table 2 is a compilation of these results.
Figures 13-15 are optical micrographs of representative micro
structures produced by each treatment. For the remainder of the work,
the nomenclature for the various microstructures will be as follows:
Crystallization Treatment Nomenclature
33L- 7% Crystalline
33L-57% Crystalline
33L-92% Crystalline
2 hours
4 hours
24 hours
For the 33L-92%, the remaining 8 percent is not a glassy phase
but open porosity. The porosity is a result of extensive microcracking
which occurs as the crystalline phase shrinks away from the less dense
glassy phase. Evidence of this microcracking can also be seen in 33L-7%
and 33L-57%. (See Figures 13-14.) There appears to be little if any
residual glassy phase in 33L-92%.
B. Dynamic Fatigue Testing of 33L-G1ass
Figures 16-18 are plots of Eq. 20 for 33L-Glass testing in air,
in water, and after aging one day in water. The accompanying tables
(Tables 3-5) show the raw data for each plot. The straight lines in
Figures 16-18 are least squares fits using all data points obtained
in the testing. Table 6 shows the inert strength data from testing
samples as prepared and after aging one day in water.
48


12
of the final material. In the uncontrolled system, the nucleation and
growth curves overlap considerably, making it very difficult to get
appreciable nucleation without some crystallization and vice versa.
Control over the final microstructure is impossible in this system.
C. Applications Arid Advantages
Table 1 lists a number of commercial glass-ceramics, their major
crystalline phases, important properties, and applications. The wide
variety of materials and applications are discussed in detail by
Pincus (1972).
The advantages of glass-ceramics over conventionally processed
(slip cast, sintered, etc.) ceramics are listed below:
1. Because the microstructure can be controlled to a certain
extent, properties such as electrical properties, transparency,
and chemical durability can be tailored to specific applications.
2. Glass-forming techniques (blowing, pressing, drawing, and
casting) can be used, providing economical, high-speed,
automatic production.
3. Small dimensional changes during crystallization.
4. Zero porosity.
5. The homogeneity of the melt and the nucleation process leads
to uniform microstructure and properties.
6. Crystalline phases otherwise unattainable can be produced.
D. The Llthia-SIllca System
Figure 3 shows the phase diagram for the lithia-silica system. The
composition chosen for this study was the 33.3 mole percent lithia composi
tion. (This will be referred to as the 33L composition.) Previous work
done on this system has shown that the crystals which form this composition


19
slit crack in plane stress:
(5)
where is the stress at the crack tip resulting in failure.
There are three different modes in which a solid may be stressed
(Fig. 4). Mode I is the normal or opening mode; mode II is the sliding
mode; mode III is the tearing mode. The mode I stress field at the
crack tip has been determined by Irwin (1958) to be
0.
JL_a.
ij
(2ur)2 ij
f,,(e)
(6)
where r and e define a coordinate system at the crack tip, f .(e) is a
J
function accounting for the angular dependence of the stress about the
crack tip, and K is the stress intensity factor. For mode I,
Kj = o (ira)2 (7)
Failure then will occur at some critical stress intensity factor
Kjc = cf(a)* (8)
Combining Eqs. (5) and (8)
KIc = (2YE)^ (9)
We may now define G = 2yas the crack extension force such that
1C
Ic
the critical value of G is
GIc
for plane stress and for plane strain
,2
K'
Ic
JIc
0 v )E
(10)
(ID
where v is Poisson's ratio. (The crack extension force is also called
the strain energy release rate, having dimensions of energy per unit
plate thickness and per unit crack extension which are the same as force


Figure 17. Dynamic fatigue results for 33L-glass tested
in water


In S CMPa)
(MPa-s-1)
\
S (MPa)


70
60
50
40
30
20
10
83
oT¡ oTi oTi i!o TTi iT i!e
ln (p/aa)
Figure 25. LPD for 33L-92% crystalline


Figure 1. Two stage and isothermal heat treatments for
processing a glass-ceramic


In S (MPa)
(MPa-s'1)
S (MPa)


hnm
45
Figure 12. Constant mor,lent double cantilever beam test apparatus


Figure 5. Possible reactions between water and segments
of the glass network


143
of the double torsion method. The double cantilever beam constant
moment technique is similar to the double torsion method in that it
uses a constant K specimen. However in the DCB method employed
herein, the crack growth is viewed and measured directly, which
allows the microstructural effects to be monitored more closely.
By marking the locations on the sample where the arrest-release
events occur, subsequent microscopic inspection of the crack
surface (as in Figure 39 and others) can lead to a better under
standing and perhaps quantification of the crack/microstructure
interactions.
C. Comparing Dynamic Fatigue and Slow Crack Growth
The N values obtained in the crack growth studies correspond
best to the lowest values obtained in dynamic fatigue. Pletka and
Wiederhorn (1978), contrarily, found larger values for N in crack
growth studies than in dynamic fatigue. Their explanation is that
in a uniaxial stress field the obstacles more readily arrest the
crack, yielding a higher N value in crack velocity experiments
than in a four-point bend test or biaxial flexure.
For the case of the lithium disilicate system studied here,
it is likely that the kinetics of the corrosion reaction at the
crack tip govern the fatigue behavior rather than the macroscopic
stress state. The time required for slow crack growth allows more
of a reaction at the crack tip, resulting in smaller N values for
the test which takes longer to complete. The short fracture time
('v % 2 seconds) in dynamic fatigue is too rapid to allow sufficient
degradation of the material at the microscopic flaw at least relative


Similarly, an equation relating the fracture strength to the
loading or stressing rate may be derived. (Details of the derivation
are given by Greenspan, 1977.) The equation is
N+l
a
2(N+l)(Kc/ac)2_N .
, a
AY(N-2)
09)
where b is the stressing rate, da/dt. Now a logarithmic plot of a vs
b gives a straight line of slope -and an intercept which gives
A. Ritter and co-workers (1971, 1974, 1978, 1979) have used Eqs. 18
and 19 extensively to determine these parameters for various glasses
and ceramics.
Usually, the crack propagation parameters obtained using the
static and dynamic fatigue methods agree with those obtained from
crack velocity experiments. However, several instances have occurred
where the data did not agree (Ritter and Manthuruthil,, 1973). The
materials involved are Pyrex and silica glasses. Differences in the
chemical environment at the crack tip may account for the discrepancy.
The crack velocity experiments utilize a macroscopic crack to determine
the parameter, while the indirect methods initiate failure at micro
scopic flaws, so that it is not unusual to imagine different chemistries
at the crack tip. Because the parameters are used for estimating failure
times, there is a practical need to resolve these differences in fracture
behavior.


33
D. Lifetime Predictions
Using Eqs. 18 and 19 we may construct diagrams to demonstrate
graphically the relationships betv/een time to failure and applied
stress and failure stress and stressing rate. Figure 8 sh&ws the
experimental results for 33L glass tested in air using Eq. 19.
Detailed discussion of this type of diagram is found in Chapter VI.
Rearranging Eq. 19 and taking the logarithm yields
no = Un B + &n (N+l) + (N-2) ncy.] + n (20)
where
AY2(N-2)KcN"2
and the other terms have been previously defined. This now explicitly
defines the straight line relationship between an a and £nc. In this
case N = 30.5 and tn B = -2.07. (At this point, it is convenient to
express B in a logarithm, and A must be determined after a separate
experiment to find Kc-)
Because ceramics and glasses exhibit a wide spread in strength
values, lifetime predictions must use the lowest values to assure some
minimum lifetime. In order to increase the confidence in the minimum
lifetime prediction and to allow glasses and ceramics to be designed
to sustain greater loads, a proof test may be performed. In a proof
test, each sample is subjected to a stress greater than that expected
in service. This eliminates the weak ones and assures that every survivor
has some minimum strength or service life. This minimum service life
(tm. ) is given by
min
min
B a
N-2
a
-N
a
(21)


Table 15
Dynamic Fatigue Data for 33L-7% Crystalline
Tested After Aging 1 Week in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
176
183
153
152
154
s(MPa)
29
30
36
29
21
s/S
16%
17%
24%
19%
14%
n
9
9
8
9
7
Sm(MPa)
175
193
166
159
156
m
5.6
5.1
3.3
4.6
6.8
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF FIGURES vi
LIST OF TABLES ix
ABSTRACT xi
CHAPTER
IINTRODUCTION ; 1
II GLASS-CERAMICS 6
A. General 6
B. Nucleation and Crystallization 7
C. Applications and Advantages 12
D. The Lithia-Silica System 12
III FRACTURE MECHANICS 18
A. Historical Background 18
B. Static Fatigue in Glasses and Ceramics 21
C. Fracture Mechanics of Glasses and Ceramics.. 29
D. Lifetime Predictions 33
E. Methods Used for this Study 36
IV MATERIALS PREPARATION AND CHARACTERIZATION 38
A. Melting and Casting 38
B. Annealing, Nucleation, and
Crystallization Schedule 38
C. Final Sample Preparation 39
D. Characterization 40
V TEST TECHNIQUES 42
A. Biaxial Flexure 42
B. Double Cantilever Beam Testing 44
iv


51
4 >p '
t
G > >
*7 '
i
o
V Jt
*~S ¡$ I *
* l '* \ i
. < Mv V- v*
, V *.; -
* t ',4 ** .
*,? *
c 8
(9
A *
B *,<3
1
&
>
*v
*

o;
* #.4 **
^ .*
t
* <0
' 18 J
K> r ^
6'
. *
_
* i < A :*' >'
- 5 v # '
.A C
*9 v- 9
%A
&
* *

.< *
* -.
, I
o 1
v .
# *
s#
; f>* '
e 0
* *
> "J
| IQOym |
Figure 13. Representative microstructure of
33L-7% crystal 1ine


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31
Using Eq. 13.
AK dt =
2K
a2V2
-dK
or
dt =
2K
1-N
AY 2a2
-dK
Integrating now from zero to t^ (corresponding to K., the initial stress
intensity factor, to K the critical stress intensity factor) gives
AY(2-N)c
i/2-N
2 I c
K
2-N
where o is some applied stress. Since N is large and positive and
a
K. i L I L
to failure under a constant applied load,
/2-N
2K
AY2(N-2)c2
(16)
From Eq. 15,
Ki *
(17)
where a and K are the critical fracture stress and stress intensity
v W
factor in an inert environment respectively. Then we have:
v2-N
2(Kc/oc)'
-N
(18)
AY2(N-2)
Kc and are determined by testing in liquid nitrogen, so that a
logarithmic plot oft^vs. aQ gives a straight line with a slope of -N
and an intercept which gives A.


Figure
M
35
V-K relationship for 33L-glass tested in air
119
36
V-K relationship for 33L-glass tested in water
120
37
V-K relationship for 33L-low crystallinity
samples
122
38
V-K relationship for 33L-high crystallinity
samples
123
39
Region of slow crack growth in 33L-5% crystalline
125
40
Region of fast crack growth in 33L-5% crystalline
126
41
V-K relationship for 33L-92% crystalline
128
42
Region of slow crack growth in 33L-92% crystalline
130
43
Region of fast crack growth in 33L-92% crystalline
131
44
Growth of a gel layer at the surface of a glass
containing two depths of flaws
133
45
A comparison of the Method of Maximum Likelihood
and the Monte Carlo method for predicting N
139
vi i i


17
Harper, James, and McMillan (1970) studied a 1ithia-silica glass
(30 percent Li^0) with and without P205 as a nucleating agent. Nucleation
due to glass-in-glass phase separation as well as homogeneous nucleation
of lithium disilicate was discussed, but neither mechanism was firmly
established as correct. The reasons for the action of P20g as a nucleant
were discussed (surface energy effects, formation of lithium phosphates,
and more extensive phase separation), but were also left unresolved.
Doremus and Turkalo (1972) studied a lithium silicate glass (-26-27
mole percent lithia) with and without P20,- also. They found that the
phosphorous slows the growth rate of the lithium disilicate crystals,
but that both produced similar microstructures. Spherulitic crystals
were also found, as in previous works, but some question was raised as
to whether they are true spherulites or not. No sheaf-like structures
have been shown for Li20-2Si02 crystals, such as can be found in so-called
spherulitic polymers, liquid crystals, and complex minerals.
The choice of 33L as the composition to study in this investigation
was based on the detailed microstructural and kinetic information of the
previous work and the fact that the glass and crystalline phases have
the same composition. A two-stage heat treatment with a long nucleation
time was also selected based upon Freiman's (1968) studies. Optical
and electron microscopy were used to confirm the attainment of materials
described previously.


142
Violations such as in the above paragraph or changing the nature of
the flaws (as discussed earlier) will destroy the validity of the
analysis. Therefore, it is evident from this study that great care
must be taken in designing, analyzing, and applying dynamic fatigue
experiments. The kinetics of the corrosion reaction and the empirical
nature of the equations may very well restrict the application of this
method for making lifetime predictions.
B. Slow Crack Growth
In the slow crack growth experiments, the value of N, the stress
corrosion susceptibility, was found to be unaffected by aging in water
and only slightly increased by crystallization. 'These observations are
consistent with previous investigations.
Crack front interaction with the microstructure made determination
of the V-K relationship difficult for all materials except 33L-Glass.
Pletka and Wiederhorn (1978) also noticed this effect in a magnesium
aluminosilicate glass-ceramic. The crack arrest at the particle results
in the stopping of the crack followed by a release with sufficient energy
built up to allow the crack velocity to increase by an inordinate amount.
Theocaris and Milios (1980) have studied cracks propagating through a
bi-material interface and have quantified the increase in velocity as
a function of relative ductilities of the two materials. However,
application of their findings to materials with a brittle matrix and
brittle inclusions is not straightforward.
Ferber and Brown (1980), using the double torsion method, conclude
that because of crack/microstructure interactions, reliable V-K data
could not be obtained on their alumina samples. Studies by Fuller
(1979) and Pletka, Fuller and Koepke (1979) also cast doubt on use


125
Figure 39. Region of slow crack growth in
33L- 5% crystalline


21
per unit crack extension. See Broek (1974).)
Because of the dominant tensile nature of the failure of glasses
and ceramics, only mode I failure under plane stress will be considered
in the following discussions.
B. Static Fatigue Iri Glasses And Ceramics
Among the first to notice the detrimental effects of water on the
strength of glass were Preston and his co-workers (1946). They observed
a decrease in strength under a static load over intervals from minutes
to days for soda-lime and borosilicate glasses and several porcelain
compositions. The loss in strength was observed in tests conducted in
humid air and water, but the effect was more pronounced in water. No
loss in strength was noted when testing in a vacuum or at low temperatures
Charles (1958) and his co-workers (Charles and Fisher, 1960 and
Hillig and Charles, 1964) introduced the chemical aspects of the delayed
failure problem. Figure 5 shows four reactions between water and segments
of the glass network. Reaction (a) demonstrates the replacement of an
alkali ion (M) with a hydrogen ion at a nonbridging oxygen site. In
his original work, Charles (1958) found that the temperature dependence
of the corrosion rate was identical with the temperature dependence of
alkali ion diffusion, which lends support to this mechanism.
Reaction (b) is less important. It shows aqueous attack on the
covalent bridging oxygen site. As fused silica and crystalline quartz
are relatively insoluble at moderate temperature and neutral pH, this
reaction is much less likely to cause significant damage to the glass
network.


Figure 6. Hypothetical changes in crack tip geometry due to
stress corrosion


140
slopes of the lines would be normal. However, since the slope is
equal to 1/(N+1), the distribution of N cannot possibly be normal
(except at very large sample sizes).
As to finding large scatter with large N, large N with large
scatter has been found here as well. Because the slopes of the lines
are so small 0.010 0.100), a small change in the slope can result
in a large change in N. (Recall that the slope is equal to 1/(N+1).)
Thus, greater variability at high N values is to be expected from the
nature of its determination. For example, the same percentage of
variability at large slopes (0.100 0.050) compared with smaller slopes
(0.0100.005) yields a range of 7-20 versus 67-200.
There is also a question as to which statistic of the strength
test should be used for determining the straight line. Ritter et al.
used the median; others use the mean; this study prefers to use all
data points. Because the mean is easily affected by outliers (either
abnormally weak or strong), the median is clearly a better choice
between the two. Using all data points, however, results in utilizing
every shred of information contained in the experiment. After all,
if the samples have been prepared in the same manner as the components
they simulate, the strength statistics of the components will duplicate
those of the samples. The method of maximum likelihood maximizes the
amount of information obtained by using all data points.
Rockar and Pletka (1978) used a different method. They ordered
their data (ten samples per stressing rate), then performed ten linear
regressions, taking as their final line the average of ten. This method
should yield similar results to those obtained here by using all data
points to generate one line.


Figure Page
18 Dynamic fatigue results for 33L-glass tested
after aging one day in water 59
19 Wei bul 1 plot for some 33L-92% crystalline
material 66
20 LPD for 33L-glass 67
21 Dynamic fatigue results for 33L-92% crystalline
tested in air 70
22 Dynamic fatigue results for 33L-92% crystalline
tested in water 72
23 Dynamic fatigue results for 33L-92% crystalline
tested after aging one day in water 74
24 Dynamic fatigue results for 33L-92% crystalline
tested after aging one week in water 76
25 LPD for 33L-92% crystalline 83
26 Dynamic fatigue results for 33L-7% crystalline
tested in air 85
27 Dynamic fatigue results for 33L-7% crystalline
tested in water 87
28 Dynamic fatigue results for 33L-7% crystalline
tested after aging one day in water 89
29 Dynamic fatigue results for 33L-7% crystalline
tested after aging one week in water 91
30 LPD for 33L-7% crystalline 98
31 Dynamic fatigue results for 33L-57% crystalline
tested after aging one day in water 101
32 Dynamic fatigue results for 33L-57% crystalline
tested after aging one week in water 103
33 LPD for 33L-57% crystalline 104
34 Theoretical crack velocity vs. stress intensity
factor relationship for brittle materials 116


FRACTURE MECHANICS AND FAILURE PREDICTIONS
FOR GLASS, GLASS-CERAMIC, AND CERAMIC SYSTEMS
By
RONALD A. PALMER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1981

To Ellen

ACKNOWLEDGEMENTS
The author thanks those who served on his committee,
R. W. Gould, C. S. Hartley, L. E. Malvern, and G. Y. Onoda, Jr.,
for their assistance and academic instruction. Thanks are also due
his head professor, L. L. Hench, who provided the opportunity to
perform this study as well as advice and encouragement throughout.
S. W. Freiman (of NBS) gave invaluable assistance^in sample
preparation and experimental technique.
Before returning to graduate school (and since), the author
received greatly appreciated support from D. C. Greenspan, D. Cronin,
R. V. Caporali, and R. A. Ferguson. Discussions with S. Bernstein,
F. K. Urban, J. W. Sheets, W. J. McCracken, C. L. Beatty and many
others also contributed to this work.
Before finishing this dissertation, the author began working
for Rockwell Hanford Operations in Richland, Washington. The patience
shown by his supervisors, F. M. Jungfleisch, M. J. Kupfer, and
L. P. McRae is greatly appreciated.
Finally, the author wishes to thank his wife, Ellen, for her
love and support during this time.
This work was supported by the Air Force Office of Scientific
Research.

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF FIGURES vi
LIST OF TABLES ix
ABSTRACT xi
CHAPTER
IINTRODUCTION ; 1
II GLASS-CERAMICS 6
A. General 6
B. Nucleation and Crystallization 7
C. Applications and Advantages 12
D. The Lithia-Silica System 12
III FRACTURE MECHANICS 18
A. Historical Background 18
B. Static Fatigue in Glasses and Ceramics 21
C. Fracture Mechanics of Glasses and Ceramics.. 29
D. Lifetime Predictions 33
E. Methods Used for this Study 36
IV MATERIALS PREPARATION AND CHARACTERIZATION 38
A. Melting and Casting 38
B. Annealing, Nucleation, and
Crystallization Schedule 38
C. Final Sample Preparation 39
D. Characterization 40
V TEST TECHNIQUES 42
A. Biaxial Flexure 42
B. Double Cantilever Beam Testing 44
iv

Page
VI DYNAMIC FATIGUE: RESULTS AND DISCUSSION 48
A. Quantitative Microscopy 48
B. Dynamic Fatigue Testing of 33L-Glass 48
C. Dynamic Fatigue Testing of 33L-927 A 68
D. Dynamic Fatigue Testing of 33L-7% 82
E. Dynamic Fatigue Testing of 33L-57% 99
F. The Effect of Crystallization on Strength.. 108
G. The Effect of Crystallization on N 110
H. The Effect of Crystallization on
Lifetime Predictions 113
I. Summary 114
VII CRACK VELOCITY EXPERIMENTS 115
A. General 115
B. 33L-G1 ass 117
C. 33L-Partially Crystallized !.. 117
D. 33L-92% Crystalline 127
VIII DISCUSSION 132
A. Dynamic Fatigue 132
B. Slow Crack Growth 142
C. Comparing Dynamic Fatigue and
Slow Crack Growth 143
IX CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK... 147
A. Conclusions 147
B. Suggestions for Future Work 148
REFERENCES 152
BIOGRAPHICAL SKETCH 156
v

LIST OF FIGURES
Figure Page
1 Two stage and isothermal heat treatments
for processing a glass-ceramic 9
2 Tammann curves (nucleation or growth rates -
vs. temperature) for controlled and uncon
trolled crystallization 11
3 Phase diagram of the Li^O-SiO^ system 14
4 Three modes of failure: I, normal or opening
mode; II, sliding mode; III, tearing mode 20
5 Possible reactions between water and segments
of the glass network 23
6 Hypothetical changes in crack tip geometry
due to stress corrosion 26
7 Universal fatigue curve developed by Mould
and Southwick (1959) 28
8 Equation 19 plotted for 33L-Glass tested in
air 35
9 Lifetime prediction diagram using equation 22
for 33L-Glass 37
10 Specimen configuration for double cantilever
beam constant moment crack grov/th experiment 41
11 Biaxial flexure test jig 43
12 Constant moment double cantilever beam test
apparatus 45
.13 Representative microstructure of 33L-7%
crystalline 51
14 Representative microstructure of 33L-57%
crystalline 52
15 Representative microstructure of 33L-92%
crystalline 53
16 Dynamic fatigue results for 33L-Glass tested
in air 55
17 Dynamic fatigue results for 33L-Glass tested
in water 57
vi

Figure Page
18 Dynamic fatigue results for 33L-glass tested
after aging one day in water 59
19 Wei bul 1 plot for some 33L-92% crystalline
material 66
20 LPD for 33L-glass 67
21 Dynamic fatigue results for 33L-92% crystalline
tested in air 70
22 Dynamic fatigue results for 33L-92% crystalline
tested in water 72
23 Dynamic fatigue results for 33L-92% crystalline
tested after aging one day in water 74
24 Dynamic fatigue results for 33L-92% crystalline
tested after aging one week in water 76
25 LPD for 33L-92% crystalline 83
26 Dynamic fatigue results for 33L-7% crystalline
tested in air 85
27 Dynamic fatigue results for 33L-7% crystalline
tested in water 87
28 Dynamic fatigue results for 33L-7% crystalline
tested after aging one day in water 89
29 Dynamic fatigue results for 33L-7% crystalline
tested after aging one week in water 91
30 LPD for 33L-7% crystalline 98
31 Dynamic fatigue results for 33L-57% crystalline
tested after aging one day in water 101
32 Dynamic fatigue results for 33L-57% crystalline
tested after aging one week in water 103
33 LPD for 33L-57% crystalline 104
34 Theoretical crack velocity vs. stress intensity
factor relationship for brittle materials 116

Figure
M
35
V-K relationship for 33L-glass tested in air
119
36
V-K relationship for 33L-glass tested in water
120
37
V-K relationship for 33L-low crystallinity
samples
122
38
V-K relationship for 33L-high crystallinity
samples
123
39
Region of slow crack growth in 33L-5% crystalline
125
40
Region of fast crack growth in 33L-5% crystalline
126
41
V-K relationship for 33L-92% crystalline
128
42
Region of slow crack growth in 33L-92% crystalline
130
43
Region of fast crack growth in 33L-92% crystalline
131
44
Growth of a gel layer at the surface of a glass
containing two depths of flaws
133
45
A comparison of the Method of Maximum Likelihood
and the Monte Carlo method for predicting N
139
vi i i

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
LIST OF TABLES
Paqe
Commercial Glass-Ceramics
13
Results of Quantitative Microscopy for
Determining Per Cent Crystallinity
49
Dynamic Fatigue Data for 33L-Glass
Tested in Air
60
Dynamic Fatigue Data for 33L-Glass
Tested in Water
61
Dynamic Fatigue Data for 33L-Glass
Tested After Aging 1 Day in Water
62
Liquid Nitrogen Strength of 33L-Glass
63
Dynamic Fatigue Data for 33L-92%
Crystalline Tested in Air
77
Dynamic Fatigue Data for 33L-925&
Crystalline Tested in Water
78
Dynamic Fatigue Data for 33L-92$
Crystalline Tested After Aging
1 Day in Water
79
Dynamic Fatigue Data for 33L-92%
Crystalline Tested After Aging
1 Week in Water
80
Liquid Nitrogen Strength of 33L-92%
Crystalline
81
Dynamic Fatigue Data for 33L-7%
Crystalline Tested in Air
92
Dynamic Fatigue Data for 33L-7%
Crystalline Tested in Water
93
Dynamic Fatigue Data for 33L-7%
Crystalline Tested After Aging
1 Day in Water
94
Dynamic Fatigue Data for 33L-7%
Crystalline Tested After Aging
1 Week in Water
95
IX

Table
Page
16
Liquid Nitrogen Strength of 33L-7%
Crystalline
96
17
Dynamic Fatigue Data for 33L-57% .
Crystalline Tested After Aging
1 Day in Water
105
18
Dynamic Fatigue Data for 33L-57%
Crystalline Tested After Aging
1 Week in Water
106
19
Liquid Nitrogen Strength of
33L-57% Crystalline
107
20
Strength of 33L-Glass and Glass-
Ceramics
109
21
Fracture Parameters and (aD/aa) from
Dynamic Fatigue Data
111
22
33L-Glass Fracture Parameters as
Determined by Slow Crack Growth
118
23
33L-Partially Crystalline Fracture
Parameters as Determined by Slow
Crack Growth
121
24
33L-92% Crystalline Fracture Parameters
as Determined by Slow Crack Growth
129
25
Effect of Surface Finish on Aged
Strength of Sodium Disilicate Glass
135
X

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
FRACTURE MECHANICS AND FAILURE PREDICTIONS
FOR GLASS, GLASS-CERAMIC, AND CERAMIC SYSTEMS
By
Ronald A. Palmer
Dune, 1981
Chairman: Larry L. Hench
Major Department: Materials Science and Engineering
State-of-the-art fracture mechanics techniques are used to determine
the fatigue parameters for the lithium disilicate glass/glass-ceramic
system. Glass-ceramics are ideal materials for examining the effects
of crystallization on fatigue behavior. Two methods (dynamic fatigue
and slow crack growth) were employed and were found to produce different
values for the fatigue parameters. These parameters are used to design
lifetime prediction diagrams and, theoretically, should be independent
of the method of determination.
It was found that aqueous corrosion has a severe effect on the
fatigue parameters, as measured in dynamic fatigue, especially after
aging for up to one week in distilled water. The lithium disilicate
glass, which is known to readily form a gel layer in water, exhibits
the most rapid change in parameters.

The most crystalline composition (92% crystalline) shows the
greatest strength loss due to aging. This is attributed to water
migration along a network of microcracks into the grain boundaries.
At low volume fraction of crystals (7%) a residual stress in
the glassy matrix effects a strength increase and a decrease in the
environmental sensitivity which disappears with aging in water.
No effect of aging in water was found in the slow crack growth
studies. In all cases, the environmental sensitivity was found to
be much higher in these experiments than in dynamic fatigue. The
reasons for the difference are discussed in later chapters.
The limitations of the present fracture mechanics theory are
discussed in the light of the inconsistencies found between these
methods. It is recommended that a combination of techniques be used
when reliable lifetime predictions are required. A matrix of tests,
including static and dynamic fatigue and proof tests, is proposed
as an improved method of characterizing the fatigue behavior of a
material.
xn

CHAPTER I-INTRODUCTION
Since the beginning of the twentieth century, the growth of techno
logy has put increasingly more stringent demands on materials and
material properties. Advances in the fields of transportation (automo
biles to aircraft to spacecraft), energy (coal and oil to nuclear and
solar), and others have necessitated improvements in existing materials
and the development of new materials in order to meet the high-performance
requirements of these .technologies. The investigation of how to make
materials stronger with lighter weight has been advanced by studying
how materials fail. Macroscopic observation of the growth of cracks
combined with microscopic examination has been the most fruitful method
of learning the mechanisms of material failure. Studies of this type
have led to improvements in both the strength and toughness of materials
through the design and control of specific microstructures which impeded
initiation and/or propagation of cracks.
Recently, concern with the conservation or security of complex sys
tems has placed additional emphasis on determining how long materials
will survive under-various environmental conditions. Thus,the field of
fracture mechanics has expanded from relatively simple measurement
taking to making failure or lifetime predictions from those measurements.
In order to make meaningful predictions, parameters, such as-or-iginal
flaw size, critica! f1 aw size, and stress corrosion susceptibi1ity
become important. Also, there is a need to know what happens as a flaw
grows from a relatively insignificant stress concentration to some
critical size where catastrophic failure occurs. This process is termed
1

subcri ti cal crack growth. Characterization, ..of a material 'ss-ubcr-i-t-ical
crack growth behavior and knowledge of the distribution of flaw sizes
permits prediction of the lifetime of a component in a given situation.
In order to predict and subsequently to guarantee the lifetime of
a component, some method of non-destructive evaluation is required.
Proof tests can be designed, in which all components are subjected to
a stress far beyond that expected in service, such that survivors are
assured some minimum service lifetime. By performing a proof test,
the need for large safety factors is reduced and overdesigning can be
avoided.
It is essential that the effect of the service environment on
time to failure be completely and correctly accounted for in fracture
mechanics theories if lifetime predictions are to be realistic. Thus,
a deeper understanding of the physical implications of the stress
corrosion susceptibility parameters is important. The combined effects
of aqueous environment and stress state on the subcritical growth of
flaws of various sizes need to be understood in order to ensure that
aging effects are correctly modeled and/or simulated in the proof
test design.
Therefore, one major objective of this thesis is to examine the
relative importance of aqueous environments, including time-dependent
effects, on the lifetime predictions for a brittle material. In order
to achieve this objective, it is necessary to study a more chemically
reactive material system than has heretofore been examined in lifetime
prediction investigations. By testing a material with known mechanisms
of chemical attack it is hoped that the critical reaction phenomena
for deterioration of mechanical performance can be elucidated.

3
In order for lifetime prediction theories to be of general use,
they must be equally applicable to polycrystalline ceramics, glasses,
and glass-ceramics. The accuracy of predicted lifetimes or their
validity should be independent of the wide range of microstructure
>
encountered in technical ceramics. Thus,a second major objective of
this thesis is to investigate the effect of microstructure on the
fracture behavior and time to failure of a glass-ceramic possessing
a wide range of crystal!inity.
Of particular concern in the thesis is the possible interaction
between microstructural and environmental effects in fracture behavior.
Many technical ceramics are multiphased and the environmental suscepti
bility of interphase boundaries is the potential "weak link" of their
long term performance. A unique feature of this study is the simulta
neous examination of both major variables, microstructure and environ
ment, in the same composition of material.
Glass-ceramics are ideal materials for studying the effects of
microstructure on mechanical properties. By varying the heat treatment,
fully glassy or fully crystalline materials may be fabricated, as well
as partially crystalline materials. The grain size of the crystalline
phase can be varied as well as the volume fraction of crystal!inity.
In this thesis, the lithium disilicate system was chosen for
study because the nucleation and crystallization kinetics are well-
documented. The mechanical behavior of this system is also known,
although the environmental effects are not yet understood. Also of
initial importance is the ability to produce a fully crystallized
material having the same composition as the glass. Lithium disilicate
glass and glass-ceramics also fulfill the prime requirement of this
thesis they react readily with water compared with other materials

4
which have been examined previously for their fracture behavior.
Studies by Sanders (1973), Ethridge (1977), and Dilmore (1977)
have established the details of aqueous attack of lithium disilicate
glasses. Recently, McCracken (1981) has conducted a parallel corrosion
study on lithium disilicate glass-ceramics of variable fraction of
crystals and determined the mechanisms of corrosion of the two-phased
glass-ceramic materials.
Thus, the lithiumdisil ica'te glass-ceramic system satisfies criteria
for studying both the effect of microstructure and aging in water on
the strength and fatigue behavior of a material of homogeneous composition.
Results obtained on the synergistic effects of subcritical crack growth
and aqueous corrosion should be important in extending the fracture
mechanics principles utilized in predicting lifetimes. By concurrently
studying the corrosion and fatigue mechanisms of this material system,
it is hoped that new insights will be gained in order to more reliably
predict lifetimes of brittle materials in general.
During this investigation quantitative microscopy was used to
monitor the volume fraction of crystal 1inity. Strength testing was
done with an Instron machine on discs, using the biaxial flexure test
and fracture mechanics tests were performed using the double cantilever
beam technique. Both optical and scanning electron microscopy were used
for examination of fracture surfaces.
Heat treatments were devised to achieve four microstructures: fully
glassy, 5 percent and 50 percent crystalline, and fully crystalline.
For each, both strength tests and fracture mechanics tests were made
in order to determine the nature of the sub-critical crack growth in

5
the material. Ideally, the two methods should yield identical results.
Where they did not, microstructural analysis of the fracture surfaces
was used to interpret the differences. A statistical analysis of the
strength data is also important in determining the accuracy-of these
results and is an important part of this investigation.
Chapters II and III introduce the topics of glass-ceramics and
fracture mechanics, respectively. Chapter IV describes the prepara
tion and characterization of materials used in this investigation.
Chapter V discusses the mechanical testing methods employed. Chapters
VI and VII discuss the results and the analyses of the fatigue
parameters and lifetime prediction diagrams. Finally, conclusions
and suggestions for future work are presented in Chapters VIII and IX.

CHAPTER II-GLASS CERAMICS
A. General
Glass-ceramics may be described as polycrystalline solids which have
been formed from a glassy melt by carefully controlled heat treatments.
In order for crystals to form, nuclei must be present. The nucleation
may be homogeneous (self-nucleating) or heterogeneous (by addition of
nucleating agents). The crystallization process may be controlled or
uncontrolled. Uncontrolled crystallization is termed devitrification
and is generally though of as a process to avoid. Controlled crystalli
zation of glasses produces materials known as glass-ceramics. Glass-
ceramics have also been referred to as "vitrocerams," "devitrocerams,"
"sitalls," and "melt-formed ceramics." (See Stewart, 1972.)
Glass makers through the ages have struggled to avoid the destruc
tive effects of devitrification. However, occasionally attempts were
made to crystallize glass purposely. Reaumur (1739) in the early 1700's
crystallized soda-lime-silica glass bottles by packing them in sand and
gypsum and heating them for a long time. Unfortunately, the crystals
nucleated at the surface, resulting in weak and deformed pieces.
The classical work on nucleation and crystallization was done by
Tammann (1925) in the 1920's. His theoretical and experimental investi
gations laid the foundation for our present knowledge of these processes.
In the 1940's photosensitive glasses were developed by Stookey (1947)
at the Corning Glass Works. Colloidal dispersants of metals (gold,
copper, or silver) added as nucleants were shown to be activated by
exposure to ultra-violet light. Subsequent heat treatments resulted in
6

7
-ne nucleation and growth of crystals. By masking portions of the glass,
intricate and precise patterns could be produced. As the crystalline
phase is less durable than the glassy phase, hydrofluoric acid could
dissolve away the more soluble phase, resulting in a glass part or
pattern made more precisely than any other conventional glass forming
method.
Later, reputedly by accident, Stookey allowed a sample of one of
his glass compositions to be heated too long, resulting in complete
crystallization of the sample. It was found to be remarkably fine
grained and harder and stronger than the parent glass. This led to
D
the development of Pyroceram and many of the other commercial glass-
ceramics we know today. The various families of glass-ceramic composi
tions and their applications will be discussed later in the chapter.
B. Nucleation and Crystallization
After the formation of the parent glass, the glass-ceramic process
required a two-step heat treatment. Figure lisa schematic of the
treatment. The first step is at a relatively low temperature (about
50-100C above the annealing point) and allows for the formation of
nuclei. In general, more nuclei form with time, so in order to obtain
a fine-grained material, longer times may be utilized. (The mathematics
of nucleation and growth have been well developed and will not be dis
cussed here. See Bergeron, 1973).
Nucleating agents are generally noble metals or refractory oxides.
Materials which have been used as nucleating agents include Au, Pt, Ag,
R Corning Glass Works, Corning, NY.

Figure 1. Two stage and isothermal heat treatments for
processing a glass-ceramic

TEMPERATURE
(1) Two Stage Heat Treatment
(2) Isothermal Heat Treatment
TIME

10
Cu, TiC^, and P20g- It is recognized that there are certain key
characteristics for a nucleating agent to be effective. For nucleus
formation, it is important for the nucleant to be very soluble at
melting and forming temperatures while barely soluble at low temperatures,
to have a low free energy of activation for homogeneous nucleating, and
to be very mobile compared to the major components of the glass at low
temperatures. For the nucleant to promote crystal growth effectively,
there must be a low interfacial energy between it and the glass and
its crystal structure and lattice parameters should be similar to that
of the crystal phase.
Glass-in-glass phase separation has been found to promote glass-
ceramic formation without any nucleating agents. Agents which promote
phase separation, such as P2OJ-, are very effective in this manner.
Nucleation may occur either at the phase boundary or within one of the
phases (most likely the one most resembling the crystalline phase.)
The second step in the so-called "ceraming" process is the crystal
lization treatment. For materials where only one crystalline phase develops,
the crystal growth may be a very simple process. However, if several phases
are developing, all different from the composition of the homogeneous glass,
the composition at the crystal-glass interface is constantly changing and
the growth process becomes exceedingly complex.
Figure 2 shows the Tammann curves (nucleation or growth rate vs.
temperature) for controlled and uncontrolled systems. For a controlled
system, it is easy to find nucleation and growth temperatures (ideally
at the maximum rates) where the other process does not occur. This type
allows the experimenter to have great control over the microstructure

RATE RATE
11
Figure 2. Tammann curves (nucleation or growth rates vs. temperature)
for controlled and uncontrolled crystallization

12
of the final material. In the uncontrolled system, the nucleation and
growth curves overlap considerably, making it very difficult to get
appreciable nucleation without some crystallization and vice versa.
Control over the final microstructure is impossible in this system.
C. Applications Arid Advantages
Table 1 lists a number of commercial glass-ceramics, their major
crystalline phases, important properties, and applications. The wide
variety of materials and applications are discussed in detail by
Pincus (1972).
The advantages of glass-ceramics over conventionally processed
(slip cast, sintered, etc.) ceramics are listed below:
1. Because the microstructure can be controlled to a certain
extent, properties such as electrical properties, transparency,
and chemical durability can be tailored to specific applications.
2. Glass-forming techniques (blowing, pressing, drawing, and
casting) can be used, providing economical, high-speed,
automatic production.
3. Small dimensional changes during crystallization.
4. Zero porosity.
5. The homogeneity of the melt and the nucleation process leads
to uniform microstructure and properties.
6. Crystalline phases otherwise unattainable can be produced.
D. The Llthia-SIllca System
Figure 3 shows the phase diagram for the lithia-silica system. The
composition chosen for this study was the 33.3 mole percent lithia composi
tion. (This will be referred to as the 33L composition.) Previous work
done on this system has shown that the crystals which form this composition

13-
Table 1
Commercial Glass-Ceramics
Designation
Major Phases
Properties
Application
Corning 8603
Li00* Si0o
Photochemically
Fluidics devices
Li ^O-Si02
machineable
Printing plate molds
Corning 9606
2Mg0-2Al203-5Si02
Si 02
Ti 02
Transparent to
microwaves
Erosion and
Thermal Shock
resistance
Radomes
Corning 9615
B-spodumene solid
solution
Low T.E., strong,
thermally and
chemically stable,
easy to clean
Range tops
G.E. Re-X
Li20-2Si02
Sealable to metals
Dielectric
Housings
Bushings
0-1 CerVit
B-quartz S.S.
Low T.E.
Polishability
Telescopic mirrors
Pflaudler
. alkali silicates
Coating of steel
Chemical process equipment
Nucerite
Chemical durability
Impact resistance
Abrasion resistance
PPG Hercuvit
101
B-quartz S.S.
Transparent
High temperature windows
Infrared transparencies

Temperature (*C)
14
LijO lmolol.1
"igure 3. Phase diagram of the LigO-SiO^ system

15
are solely lithium-disilicate (Li^0-2Si0^) There has been much work
done on both the binary system and systems with additions of nucleating
agents and aluminum oxide. As presented in Table 1, L^O-ZSiC^ is the
major phase in many commercially important glass-ceramic compositions.
For this reason, study of the 33L composition will provide a basis on
which the characterization of the whole family of 1ithia-silica glass-
ceramics can be done.
The photo-chemically machineable glass-ceramics which first appeared
in the 1940's are lithia-silica based materials. Stookey's (1947, 1950a,
1950b, 1954, 1956) work on these materials is found in the patent literature.
Rindon (1958, 1962) studied the addition of platinum to a Li^O^SiOg
glass. He measured (using x-ray diffraction) the amount of LigO*2SiO2
crystallized from the glass as a function of time (10 minutes to 32 hours)
and the amount of Pt added (zero to 0.025%). Rindone proposed that lithium-
rich clusters surrounding the Pt form as nuclei for the LigO^SiOg which
precipitates out. Later, Kinser (1968) showed that a metastable lithium-
metasilicate phase (Li2O SiO2) is the precursor to the disilicate crystals
in the 33L glass. '
Glasser (1967) studied binary lithium silicates ranging in composi
tion from 80 to 88 weight percent SiO^. He concluded that under certain
conditions a two-stage heat treatment schedule would improve the strength
of the crystallized composition. While accepting the explanation that
the formation of a large number of nuclei results in a fine-grained material,
Glasser also proposed another mechanism. He suggested that the lower
temperature treatment yields a metastable solid solution that is later

16
exsolved at higher temperatures. This results in a fine-grained
precipitate within the host crystals. Whether this mechanism would
actually occur in complex commercial compositions has yet to be shown.
Freiman (1968) studied the crystallization kinetics of several
1ithia-silica compositions including 33L. He showed that an increase
in the nucleation time from three to 24 hours decreased the activation
energy for crystallization and increased the crystallization rate.
He found that the Li^O^SiC^ crystal growth was spherulitic in nature
and that small angle x-ray scattering confirmed the presence of a
metastable phase after nucleation at 475C.
Freiman (1968) also studied the strength of the glass-ceramics he
produced. He found the strength to be lower than that predicted by
theory due to cracking in the composite because of the stresses set up
between the glass and the crystals during crystallization. He was able
to reduce the cracking by increasing the nucleation time, providing
better bonding between the glassy and crystalline phases.
Kinser(1968) also found the metastable lithium metasilicate phase
present in various lithia-silica compositions. A similar precursor
phase was found in a soda-silica composition.
Nakagawa and Izumitani (1969) studied a Li2*3 2.5Si0^ glass and a
Li2O-TiO2-SO2 glass containing 22.5 weight percent TiC^. In the
binary glass, they found droplets due to liquid-liquid phase separation
occurring independently from the nucleation of L^O^SiC^ crystals.
The droplets did not act as nuclei for crystallization. In the ternary
glass, they found that similar droplets deposited lithium titanate
crystals which then act as nuclei for crystallization.

17
Harper, James, and McMillan (1970) studied a 1ithia-silica glass
(30 percent Li^0) with and without P205 as a nucleating agent. Nucleation
due to glass-in-glass phase separation as well as homogeneous nucleation
of lithium disilicate was discussed, but neither mechanism was firmly
established as correct. The reasons for the action of P20g as a nucleant
were discussed (surface energy effects, formation of lithium phosphates,
and more extensive phase separation), but were also left unresolved.
Doremus and Turkalo (1972) studied a lithium silicate glass (-26-27
mole percent lithia) with and without P20,- also. They found that the
phosphorous slows the growth rate of the lithium disilicate crystals,
but that both produced similar microstructures. Spherulitic crystals
were also found, as in previous works, but some question was raised as
to whether they are true spherulites or not. No sheaf-like structures
have been shown for Li20-2Si02 crystals, such as can be found in so-called
spherulitic polymers, liquid crystals, and complex minerals.
The choice of 33L as the composition to study in this investigation
was based on the detailed microstructural and kinetic information of the
previous work and the fact that the glass and crystalline phases have
the same composition. A two-stage heat treatment with a long nucleation
time was also selected based upon Freiman's (1968) studies. Optical
and electron microscopy were used to confirm the attainment of materials
described previously.

CHAPTER III -FRACTURE MECHANICS
A. Historical Background
It is generally recognized that Griffith (1921) is the father of
fracture mechanics. He was the first to show a relationship between the
measured strength of a material, its material properties, and the length
of the pre-existing crack responsible for failure. The existence of pre
existing cracks as precursors to failure had been proved by Inglis (1913)
some years before. Inglis showed that the stress near an elliptical hole-
(resembling a crack) would be greater than the applied stress by
o = 2aa(a/p)z (1)
where a& is the applied stress, a is the half-crack length, and p is the
crack tip radius.
Griffith reasoned that the free energy of a cracked body and the
applied forces should not change during crack extension. That is, the
amount of energy put into breaking the material should be equivalent to
the amount of energy required to create two new surfaces, or
dU
da
dW
da
(2)
where U is the strain energy due to the crack, W the energy required for
growth, and a the half-crack length.
Griffith calculated the strain energy to be
U =
a
(3)
where E is Young's modulus, and the energy required for growth to be
W = 4ay (4)
where y is the surface energy. Inserting Eqs. (3) and (4) into (2)
gives the Griffith criterion for fracture of an infinite sheet with a
18

19
slit crack in plane stress:
(5)
where is the stress at the crack tip resulting in failure.
There are three different modes in which a solid may be stressed
(Fig. 4). Mode I is the normal or opening mode; mode II is the sliding
mode; mode III is the tearing mode. The mode I stress field at the
crack tip has been determined by Irwin (1958) to be
0.
JL_a.
ij
(2ur)2 ij
f,,(e)
(6)
where r and e define a coordinate system at the crack tip, f .(e) is a
J
function accounting for the angular dependence of the stress about the
crack tip, and K is the stress intensity factor. For mode I,
Kj = o (ira)2 (7)
Failure then will occur at some critical stress intensity factor
Kjc = cf(a)* (8)
Combining Eqs. (5) and (8)
KIc = (2YE)^ (9)
We may now define G = 2yas the crack extension force such that
1C
Ic
the critical value of G is
GIc
for plane stress and for plane strain
,2
K'
Ic
JIc
0 v )E
(10)
(ID
where v is Poisson's ratio. (The crack extension force is also called
the strain energy release rate, having dimensions of energy per unit
plate thickness and per unit crack extension which are the same as force

20
Figure 4.
Three inodes of failure: I, normal or opening
mode; II, sliding mode; III, tearing mode

21
per unit crack extension. See Broek (1974).)
Because of the dominant tensile nature of the failure of glasses
and ceramics, only mode I failure under plane stress will be considered
in the following discussions.
B. Static Fatigue Iri Glasses And Ceramics
Among the first to notice the detrimental effects of water on the
strength of glass were Preston and his co-workers (1946). They observed
a decrease in strength under a static load over intervals from minutes
to days for soda-lime and borosilicate glasses and several porcelain
compositions. The loss in strength was observed in tests conducted in
humid air and water, but the effect was more pronounced in water. No
loss in strength was noted when testing in a vacuum or at low temperatures
Charles (1958) and his co-workers (Charles and Fisher, 1960 and
Hillig and Charles, 1964) introduced the chemical aspects of the delayed
failure problem. Figure 5 shows four reactions between water and segments
of the glass network. Reaction (a) demonstrates the replacement of an
alkali ion (M) with a hydrogen ion at a nonbridging oxygen site. In
his original work, Charles (1958) found that the temperature dependence
of the corrosion rate was identical with the temperature dependence of
alkali ion diffusion, which lends support to this mechanism.
Reaction (b) is less important. It shows aqueous attack on the
covalent bridging oxygen site. As fused silica and crystalline quartz
are relatively insoluble at moderate temperature and neutral pH, this
reaction is much less likely to cause significant damage to the glass
network.

Figure 5. Possible reactions between water and segments
of the glass network

I I +
CSi0 [M]] + H20 -> [SiOH] + M + OH
I I I
[SiO-Si] + H20 2 [-si OH]
[Si0Si] + OH' [Si OH ] + [-Si-O']
I I
[-Si-0 ] + H20 [SiOH] + OH

24:
Reactions (c) and (d) show a two-step breakdown of the covalent
chain initiated by the OH" ion. As reactions (a) and (d) produce the
same OH ion, the attack of water on a binary alkali silicate glass should
be autocatalytic.
Figure 6 shows the changes in crack tip geometry due to the
corrosion reactions and the presence or absence of stress. From Eq. (1),
we know that as the crack tip sharpens (p+o) the stress at the tip
increases. Figure 6(a) shows the crack tip sharpening under an applied
stress in a corrosive medium (i.e., water), causing weakening or fatigue
in the material. Figure 6(c) shows the crack tip rounding by corrosion
with little or no applied stress, which strengthens the material as it
ages in a corrosive medium. Figure 6(b) demonstrates the concept of a
fatigue limit, where the crack tip undergoes growth and rounding concur
rently, resulting in no change in the stress at the crack tip.
Mould and Southwick (1959) performed an extensive study on the
strength and static fatigue of soda-lime-silica glass microscope
slides. They developed the concept of a universal fatigue curve.
By plotting a normalized strength (strength/low temperature strength,
a/c^) versus the logarithm of the normalized time to failure (time to
failure/time at which the strength is one-half the low temperature
strength, t/tg 5), it was found that for various abrasion treatments,
all of the data normalized in this manner fell on the same curve, such
that
a loq
N r0.5 .
Their results are plotted in Figure 7.

Figure 6. Hypothetical changes in crack tip geometry due to
stress corrosion

(a)
(b)
(c)

Figure 7. Universal fatigue curve developed by Mould
and Southwick (1959a)

1.0
0.7 -
0.6 -
^ 0.5 *
0.4 -
0.3 -
0.2
0.1 4
UNIVERSAL FATIGUE CURVE
SODA-LIME-SILICA GLASS
0
-4
~T~
-3
-2
-1
T
0
lo9lO (t/t0.5>
r\D
Co
3
T
5
6

29
Mould and Southwick (1959) also determined the value for the
breaking stress times the square root of the crack depth, a /a, which,
according to the Griffith criterion (Eq. 5), should be constant
(assuming constant y). Their value was 280-320 psi-in^(0.31-0.35
MPanf*), which compares well with Griffith's value of 240 ps'i-in15
(0.26 MPam**).
Mould and Southwick (1959) also studied the effect of aging in
various media on strength and static fatigue. Water was found to be
as effective as HCL or NaOH in strengthening the glass. Approximately
a 30 percent increase in strength was observed. Very little effect was
observed in the static fatigue behavior, other than the general strength
ening.
C. Fracture Mechanics of Glasses and Ceramics
Bradt, Hasselman, and Lange (1974) have edited a four volume
series on fracture mechanics of ceramics, which contains papers present
ing an overview of the subject as well as specific problems. Lange (1974)
introduces the subject and traces the development of fracture mechanics
from the early theories to its present-day use as a tool in materials
development.
Evans (1974a) discusses the techniques used for fracture mechanics
determinations, including the advantages and pitfalls of each method.
The methods outlined (with references to more complete descriptions)
are three-point bend, single edge cracked tension, compact tension,
double cantilever beam (four variations), and double torsion.

30
Wiederhorn (1974) discusses subcritical crack growth and the
methods of obtaining crack velocity data. In addition to the direct
methods, such as described by Evans (1974a), Wiederhorn outlines the
indirect methods of obtaining the same data by constant load (static
fatigue) and constant strain rate (dynamic fatigue) experiments.
It has been shown (Evans, 1974b) that the crack velocity(v) may
be expressed as a power function of the applied stress intensity
factor(K):
V = AKN (13)
where A and N are constants to be determined experimentally. The
constant N is termed the stress corrosion susceptibility and is used
as a measure of a material's resistance to sub-critical crack growth
in corrosive environments.
By defining the crack velocity as
da
V =
dt
(14)
and assuming a relationship between the stress, flaw size, and stress
intensity factor
K = aYa'
(15)
where Y is a constant which depends on crack and loading geometry, a
relationship may be derived giving the time to failure under a constant
load. (Note that Eq. 7 is the same as Eq. 15 with Y = n2.)
An equation defining the time to failure under a constant load may
be derived from Eqs. 13-15. From Eqs. 14-15,
and
da
2K
a2y2
dK
2K
o2Y2
dK
Vdt

31
Using Eq. 13.
AK dt =
2K
a2V2
-dK
or
dt =
2K
1-N
AY 2a2
-dK
Integrating now from zero to t^ (corresponding to K., the initial stress
intensity factor, to K the critical stress intensity factor) gives
AY(2-N)c
i/2-N
2 I c
K
2-N
where o is some applied stress. Since N is large and positive and
a
K. i L I L
to failure under a constant applied load,
/2-N
2K
AY2(N-2)c2
(16)
From Eq. 15,
Ki *
(17)
where a and K are the critical fracture stress and stress intensity
v W
factor in an inert environment respectively. Then we have:
v2-N
2(Kc/oc)'
-N
(18)
AY2(N-2)
Kc and are determined by testing in liquid nitrogen, so that a
logarithmic plot oft^vs. aQ gives a straight line with a slope of -N
and an intercept which gives A.

Similarly, an equation relating the fracture strength to the
loading or stressing rate may be derived. (Details of the derivation
are given by Greenspan, 1977.) The equation is
N+l
a
2(N+l)(Kc/ac)2_N .
, a
AY(N-2)
09)
where b is the stressing rate, da/dt. Now a logarithmic plot of a vs
b gives a straight line of slope -and an intercept which gives
A. Ritter and co-workers (1971, 1974, 1978, 1979) have used Eqs. 18
and 19 extensively to determine these parameters for various glasses
and ceramics.
Usually, the crack propagation parameters obtained using the
static and dynamic fatigue methods agree with those obtained from
crack velocity experiments. However, several instances have occurred
where the data did not agree (Ritter and Manthuruthil,, 1973). The
materials involved are Pyrex and silica glasses. Differences in the
chemical environment at the crack tip may account for the discrepancy.
The crack velocity experiments utilize a macroscopic crack to determine
the parameter, while the indirect methods initiate failure at micro
scopic flaws, so that it is not unusual to imagine different chemistries
at the crack tip. Because the parameters are used for estimating failure
times, there is a practical need to resolve these differences in fracture
behavior.

33
D. Lifetime Predictions
Using Eqs. 18 and 19 we may construct diagrams to demonstrate
graphically the relationships betv/een time to failure and applied
stress and failure stress and stressing rate. Figure 8 sh&ws the
experimental results for 33L glass tested in air using Eq. 19.
Detailed discussion of this type of diagram is found in Chapter VI.
Rearranging Eq. 19 and taking the logarithm yields
no = Un B + &n (N+l) + (N-2) ncy.] + n (20)
where
AY2(N-2)KcN"2
and the other terms have been previously defined. This now explicitly
defines the straight line relationship between an a and £nc. In this
case N = 30.5 and tn B = -2.07. (At this point, it is convenient to
express B in a logarithm, and A must be determined after a separate
experiment to find Kc-)
Because ceramics and glasses exhibit a wide spread in strength
values, lifetime predictions must use the lowest values to assure some
minimum lifetime. In order to increase the confidence in the minimum
lifetime prediction and to allow glasses and ceramics to be designed
to sustain greater loads, a proof test may be performed. In a proof
test, each sample is subjected to a stress greater than that expected
in service. This eliminates the weak ones and assures that every survivor
has some minimum strength or service life. This minimum service life
(tm. ) is given by
min
min
B a
N-2
a
-N
a
(21)

Figure 8. Equation 19 plotted for 33L-glass tested in
air

In S (MPa)
0 1 2 3 4 5
In o (MPa-s-^)
S (MPa)

36
which is simply Eq. 18 with a the proof test stress, substituted for
the inert strength, a .
A design diagram for lifetime predictions may now be constructed
by rearranging Eq. 21:
tmin a = BN'2 <22>
Using the crack growth parameters B and N and knowing the desired life
time at an applied stress, a proof test ratio (a /a ) may be determined
P 3
in order to design a suitable proof test. Figure 9 shows the design
diagram for 33L glass tested-in air based on the data shown in Fig. 8.
As an example, if a component is expected to last ten years at
50MPa (horizontal line in diagram), a proof test ratio of 2.81 is needed.
Therefore, samples which had survived a proof test of 140 MPa would be
expected to survive the above conditions.
E. Methods Used For This Study
Both direct and indirect methods are used in determining the crack
growth parameters for the lithium disilicate glass-ceramic system. The
constant moment double cantilever beam method is used to measure the
crack velocity directly. The constant stressing rate method is used for
comparison. Chapters VI and VII will compare the results of the two
methods.

(S- BdW)
37
1.5 2.0 3.0 4.0 5.0
Figure 9. Lifetime prediction diagram using equation 22
for 33L-Glass

CHAPTER IV-MATERIALS PREPARATION AND
CHARACTERIZATION
A. Melting and Casting
The parent glass was made by mixing reagent grade Li^CO^ and 5 pm
Min-U-Sil silica sand for one hour on a roller mill in a plastic jar.
Each batch weighed 200-250 g and was melted in a covered platinum crucible
for 24 hours in air in a electric muffle furnace at 1350C.
Casting was done in a graphite mold or graphite forms. Discs
required for the biaxial flexure test were made as follows: Cylinders
25 mm in diameter and about 50 mm long were formed and subsequently cut
into 2 mm thick slices with a diamond wheel. For double cantilever beam
specimens, bricks 20 mm x 25 mm x 80 mm were poured and subsequently cut
into plates with a diamond saw.
B. Annealing, Nucleation, and Crystallization Schedule
The completely glassy specimens were annealed immediately after
casting for four hours at 350C and allowed to furnace cool. Qualitative
analysis of residual stresses was made using a polariscope with a tint
plate. Samples showing excessive stress were remelted and recast.
Those samples to be crystallized were placed in a tube furnace
directly after casting and held at the nucleation temperature. For all
levels of crystallinity, the nucleation treatment was 24 hours at 475C.
This treatment was selected based on the work of Freiman (1968). The
size of the samples was also a factor. Because they were so large, a
long nucleation time was required to assure a consistent microstructure
throughout each specimen as well as from specimen to specimen. Two
cylinders or one brick could be treated at a time. To minimize thermal
38

39
gradients, the specimens were held in steel wire mesh boats and set on
an aluminum block in the center of the furnace.
To crystallize the specimens, the furnace was turned up to 550C
and left for various periods of time. The furnace reached 550C from
475C in 15-20 minutes. Two hours resulted in about 7 percent crystal
linity; four hours gave about 55 percent; fully crystallized specimens
were obtained by leaving them at 550C for 24 hours.
After crystallization, the specimens were removed from the tube
furnace and placed in a small furnace at 200C to prevent thermal shock
and allowed to furnace cool to room temperature.
C. Final Sample Preparation
Discs for biaxial flexure were cut from the cylinders with a high
speed diamond saw. Water was used as a coolant. Because of the rough
finish and reactivity of the materials with water, each disc was polished
dry with SiC paper to a 320 grit finish. The final discs were, about
2.5 mm thick. Discs were kept in dessicator to prevent atmospheric water
attack before testing.
Plates for double cantilever beam (DCB) testing were cut from the
bricks to the approximate dimensions 1 mm xl2 mm x 75 mm. These were
cut using a low speed diamond watering saw, with mineral oil or Isocut*
fluid as a coolant. Although surface finish is not important in this
test, care was taken to avoid contact with water. A groove roughly half
the thickness of the specimen was machined down the center. This gives
the propagating crack an easy path to follow. The groove was made using
*Buehler and Co., Evanston, IL.

40
a milling machine and a diamond tool. Figure 10 shows the shape and
representative dimensions of the DCB specimens.
D. Characterization
Volume fraction crystallinity was evaluated by quantitative micro
scopy (DeHoff and Rhines, 1968) using a petrographic microscope. A
121 point grid was placed 10 times on each sample measured at 200X to
obtain sufficient data for analysis.
The samples to be evaluated were polished sections, polished using
SiC paper to 600 grit, followed by 6 ym and 1 ym diamond paste. The
microstructure was made visible using an etch in 5 percent HF for one
minute. (See Chapter VI for examples of the microstructures.)

41
T T
Figure 10. Specimen configuration for double cantilever
beam constant moment crack growth experiment

CHAPTER V-TEST TECHNIQUES
A. Biaxial Flexure
Figure 11 shows the test jig for the biaxial flexure test. The
sample cup, which is mounted on the load cell of an Instron* testing
machine, has three press-fitted ball bearings which define a support
circle for the disc samples and also allows liquids or gases to be added
for testing in various reactive or non-reactive environments. The
loading pin is mounted on the crosshead of the testing machine and is
centered with respect to the circle defined by the ball bearings.
Detailed stress analysis of this method can be found in the work
of Kirstein and Woolley (1967). More recently, Wachtman, Capps, and
Mande! (1972) evaluated this method in a thirteen-laboratory round-
robin test. For small deflections, the strength, S, in this con
figuration is
(23)
where
A = radius of support circle
B = radius of loaded area
C = radius of specimen
P = load
d = specimen thickness
v = Poisson's ratio.
* Instron Corp., Canton, MA.
42

43
Figure 11. Biaxial flexure test jig

44
The dimensions used in our testing are A = 9.5 mm, B = 0.8 mm, and
C = 12.7 mm. A Poisson's ratio of 0.24 was used for all materials
studied (Freiman, 1968). These values reduce Eq. 23 to
S = -1.8689 % (24)
d
Samples were tested in biaxial flexure at five cross head speeds
over three decades (0.2, 0.1, 0.02, 0.01 and 0.002 inches per minute)
in three atmospheres (air, water, and liquid nitrogen). Some samples
were also tested after aging one day or one week in water.
Testing in air was straightforward (22C, 65-70% relative humidity).
Care was taken when testing in water to ensure that the samples tested
at the fastest rate were in water about the same length of time as those
tested at the slowest rate (about two minutes).
In the aging experiments, samples were placed in plastic vials
with sufficient water to achieve a surface area to volume ratio (SA/V)
of 1 cm" ^. Discs were then tested in the same water in which they were
aged.
Samples tested in liquid nitrogen were pre-cooled to avoid thermal
shock failure. The test jig was cooled with liquid nitrogen and filled
with the liquid during testing. Because of the absence of static
fatigue at liquid nitrogen temperatures, testing was done only at one
rate of 0.02 inches per minute. The average strength from the liquid
nitrogen testing was used as the aQ value in Eqs. 17-20.
B. Double Cantilever Beam Testing
Freiman, Mulville, and Mast (1973) give a detailed analysis of the
constant moment double cantilever beam technique. Figure 12 shows the

hnm
45
Figure 12. Constant mor,lent double cantilever beam test apparatus

46
test apparatus for this technique. The test specimen is cemented
using epoxy to the loading arms in a pair of slotted inserts to insure
proper alignment. All the pivot points have suitably low friction
provided by bearings. The load is applied by means of a weight
pan connected through a triangular piece (assuring equal load
distribution) to the loading arms. A constant load provides a
constant moment applied by the arms to the specimen. This yields
a constant stress intensity factor, defined by
K = --T- (25)
x/IT
where
T = load
L = moment arm length
1
I = i~2 b h = moment of inertia of the beam
t = web thickness,
as defined in Figure 10.
A starter crack is initiated at the base of the slot in the groove
by tightening a sharp screw against the ungrooved side. The crack will
grow at a constant velocity under a constant load. The range of
velocities measured was from 10^ to 10^. m/s. With proper care,
multiple measurements may be made on one sample by changing the load
after taking sufficient readings at one load. Measurements were made
every few hours for very slow velocities and every half minute for
fast velocities. Again, testing was done in air and water at room
temperature (^22C).

47
Measurement of the crack velocity was made with a traveling
microscope.* The magnification was 32X. The accuracy of the
microscope was 0.0005 mm. Readings were made to the nearest
micrometer.
The above procedure will provide crack velocity data for stress
intensity factors less than Kc. Values for Kc, the critical stress
intensity factor, were determined using this apparatus attached to
an Instron** machine. An initial load was applied to begin the crack
propagation, then the cross head was turned on at a constant speed
of 0.02 inches per minute and the crack then propagated to failure.
The highest value of the load (T) then is substituted into Eq. 25,
and the value for Kc determined.
* Gaertner Scientific Corp., Chicago, IL.
** Instron Corp., Canton, MA.

CHAPTER VI-DYNAMIC FATIGUE: RESULTS AND DISCUSSION
A. Quantitative Microscopy
For the dynamic fatigue specimens, at least ten different discs
representing about half of the cylinders were examined for percent
crystallinity. Table 2 is a compilation of these results.
Figures 13-15 are optical micrographs of representative micro
structures produced by each treatment. For the remainder of the work,
the nomenclature for the various microstructures will be as follows:
Crystallization Treatment Nomenclature
33L- 7% Crystalline
33L-57% Crystalline
33L-92% Crystalline
2 hours
4 hours
24 hours
For the 33L-92%, the remaining 8 percent is not a glassy phase
but open porosity. The porosity is a result of extensive microcracking
which occurs as the crystalline phase shrinks away from the less dense
glassy phase. Evidence of this microcracking can also be seen in 33L-7%
and 33L-57%. (See Figures 13-14.) There appears to be little if any
residual glassy phase in 33L-92%.
B. Dynamic Fatigue Testing of 33L-G1ass
Figures 16-18 are plots of Eq. 20 for 33L-Glass testing in air,
in water, and after aging one day in water. The accompanying tables
(Tables 3-5) show the raw data for each plot. The straight lines in
Figures 16-18 are least squares fits using all data points obtained
in the testing. Table 6 shows the inert strength data from testing
samples as prepared and after aging one day in water.
48

49
TABLE 2
Results of Quantitative Microscopy for
Determining Per Cent Crystallinity
24 Hour Crystallization Treatment
Sample
% Crystalline
308X
92.34
412X
97.14
507X
80.77
Average, x
91.93%
605X
87.51
804X
93.68
1006X
94.04
Std., Dev., s
4.65%
1102X
93.96
131 OX
96.87
1402X
88.44
s/x
5.06%
1907X
95.69
2303X
90.11
1508X
92.64
4 Hour Crystallization Treatment
203Z
55.08
402Z
' 56.28
Average, x
57.14%
409Z
52.31
603Z
60.37
704Z
59.13
Std., Dev'., s
3.95%
709Z
56.57
802Z
60.29
907Z
57.02
s/x
6.91%
1003Z
63.80
1309Z
50.58

50
Table 2 Continued
2 Hour Crystallization Treatment
Sample
% Crystalline
602W
5.08
611W
9.05
807W
9.09
Average, x
901W
7.93
1101W
7.07
mow
6.61
Std. Dev., s
1201W
6.41
1209W
6.07
1307W
5.46
s/x
1405W
5.54
1608W
5.54
1807W
6.61
211OW
7.27
6.79%
1.27%
18.64%

51
4 >p '
t
G > >
*7 '
i
o
V Jt
*~S ¡$ I *
* l '* \ i
. < Mv V- v*
, V *.; -
* t ',4 ** .
*,? *
c 8
(9
A *
B *,<3
1
&
>
*v
*

o;
* #.4 **
^ .*
t
* <0
' 18 J
K> r ^
6'
. *
_
* i < A :*' >'
- 5 v # '
.A C
*9 v- 9
%A
&
* *

.< *
* -.
, I
o 1
v .
# *
s#
; f>* '
e 0
* *
> "J
| IQOym |
Figure 13. Representative microstructure of
33L-7% crystal 1ine

52
¡
:
!
Figure 14. Representative microstructure of
33L-57% crystal!ine

| lOOurii |
Figure 15. Representative microstructure of
33L-92% crystal 1ine

Figure 16. Dynamic fatigue results for 33L-glass tested
in air

In S (MPa)
0 1 2 3 4 5
In o (MPa-s-1)

Figure 17. Dynamic fatigue results for 33L-glass tested
in water

In S (MPa)
(MPa-s"1)
In 0 (MPa-s-1)
S (MPa)

Figure 18. Dynamic fatigue results for 33L-glass tested
after aging one day in water

In S (MPa)
a. (MPa-s"1)
\
S (MPa)

60
Table 3
Dynamic Fatigue Data for 33L-Glass
Tested in Air
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
136
131
125
135
113
s(MPa)
20
13
29
37
19
s/S
15%
10%
23%
28%
17%
n
32
30
32
31
32
S (MPa)
m
136
132
123
125
112
m '
7.4
11.2
4.9
4.0
6.7
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

6.1
Table 4
Dynamic Fatigue Data for 33L-Glass
Tested in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
123
137
112
100
91
s(MPa)
16
30
19
16
22
s/S
13%
22%
17%
16%
24%
n
32
32
26
32
33
Sm(MPa)
124
. 135
111
101
91
m
8.0
5.1
6.2
6.6
4.2
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bul 1 modulus

62
Table 5
Dynamic Fatigue Data for 33L-Glass
Tested After Aging 1 Day in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
5(MPa)
124
157
129
133
125
s(MPa)
8
49
37
46
42
s/S
7%
31 %
29%
35%
33%
n
9
31
15
31
31
Sm(MPa)
125
.145
120
121
112
m
14.5
3.6
3.7
3.3
4.0
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

63
Table 6
Liquid Nitrogen Strength of 33L-Glass
As Prepared
Aged 1 Day
S(MPa)
179
211
s(MPa)
34
61
s/S
. 19%
29%
n
11
10
Sm(MPa)
178
224
m
5.0
3.1
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Weibull modulus

64
The decrease in strength when tested in water and subsequent
increase after aging are consistent with previous works on glass
(Mould and Southwick, 1959, and others). However, the change in
stress corrosion susceptibility, N, from 30.5 (in air) to 11.0 (in
water) to 37.4 (after aging) was not anticipated. Because the
mechanism of stress corrosion (aqueous attack at the crack tip) is
the same in air and in water, the N values are expected to remain
constant. This large variation in N suggests that the kinetics of
aqueous corrosion are also involved in stress corrosion.
When testing is performed in air, the concentration of water at
the crack tip is low compared to that when testing in water. Less H+
would then be available for ion exchange with Li+, slowing the reaction
and allowing less stress corrosion (giving a higher N value).
The conditions at the crack tip are altered drastically by aging
in water. The strength increases due to a rounding of the crack tip,
which, from Eq. 1, lowers the amount of stress concentration. During
the aging period, a reaction layer forms on the surface which is
depleted in lithium (due to ion exchange). In order for the flaw to
grow, it must first penetrate the reaction layer before entering the
bulk glass. This reaction layer, even after the flaw has grown through
it, protects the bulk from the aqueous environment. The protection
thus afforded raises the N value to one even higher than that in air.
Wei bull Statistics
In the tables showing the raw data, m is a parameter called the
Wei bul 1 modulus. This parameter is obtained from (Wiederhorn, 1974)
In In(l-F)'1 = m In (S^S )
(26)

65
where F is the failure probability for each strength value The
parameters m and SQ are the Wei bull modulus and scale parameter
respectively. The amount of scatter in the data can be quantified
in terms of m. As m increases the scatter becomes less (the slope of
Eq. 26 increases). Figure 19 is an example of a Wei bull plot for
33L-92% crystalline material.
For 33L-Glass, the m value ranges from 3.1 to 14.5, depending
on environment, stressing rate, and aging, but mostly ranging from
about three to eight. This is very low and indicates a large range
of flaw sizes controlling the strength of the glass.
Ideally, the sample size for an adequate Weibull analysis should
be thirty or more. Although this study rarely meets that requirement,
the m values are still reported, but should be taken only as an indica
tion of the true Weibull modulus of the material.
Lifetime Prediction Diagrams
Figure 20 is the lifetime prediction diagram (LPD) based on Eq. 22
for 33L-Glass. It can be seen that for a component to survive ten years
at 50MPa, different proof test levels are necessary depending on the
environment. In air, a proof test ratio of about 2.81 (140MPa) is
needed; in water, 13.46 (673 MPa); after aging one day, 5.21 (260 MPa).
Obviously, no 33L-Glass component would survive a proof test of
673 MPa, so that a different design level (shorter lifetime or lower

66
175
0.5 *
O 158 MPa-s'1
+ 16 MPa-s"1
O 1.6 MPa-s"1
0.0-
-0.5-
1.0 -
-1.5
-2.0
Si (MPa)
2p0
+
+
-0.90
33L-92% CRYSTALLINE
TESTED IN WATER
i
5.2
5.3
In S. (MPa)
5.4
-0.75
0.50
-0.25
*0.10
Figure 19. Weibull plot for some 33L-92% crystalline material

(Sr-BdW)
67
o Jo
P a
4.0 5,0 6,0 8,0 10.,0 12,0
32
30
28
C\J fO
D
£ 26
EE
M
24
22 H
20
ln (Vaa)
Figure 20. LPD for 33L-Glass

68
stress) is mandatory for 33L-Glass in water. However, if the data from
the aged samples are used, a proof test of only 260 MPa is necessary.
There is still a need to lower the design level, but not nearly as much.
The problem is, which data are relevant?
This situation illustrates the dynamics of the problem which have
yet to be discussed, let alone resolved. In order to make reliable
lifetime predictions, the conditions that a component sees must be known
(or assumed). It is also assumed (Wiederhorn 1973) that there is no
change in the flaw during or after proof testing. It is evident from
this work that a corrosive environment will have some effect on the flaw
and its immediately surroundings in the bulk. It would seem, then, that
no reliable predictions can be made using this procedure for materials
in corrosive environments due to the changing character of the flaws.
More discussion will follow after presenting the crack velocity
studies.
C. Dynamic Fatigue Testing of 33L-92%
Figures 21-24 present the dynamic fatigue results for 33L-92% and
Tables 7-10 give the raw data. Tests were conducted in air, in water,
and in water after aging for one day and one week. Table 11 shows the
results of the inert environment testing.
There was the expected decrease in strength when tested in water,
but no increase after aging. After aging one day, there was little
change over testing in water, but after one week, there was substantial
decrease in strength. This decrease can be attributed to the pervasive
attack of the water throughout the continuous microcrack network.
Similar effects have been seen in other porous ceramics (Frakes, Brown,
and Kenner 1974).

Figure 21. Dynamic fatigue results for 33L-92% crystalline
tested in air

In S (MPa)
(MPa-s-1)
0 1 2 3 4 5
In (MPa-s-1)
S (MPa)

Figure 22. Dynamic fatigue results for 33L-92% crystalline
tested in water

In S (MPa)
5.50-
ht
(MPa-s-1)
Id ]t
d is.
33L 92% Crystalline J^O
In (MPa-s"^)
S (MPa)

Figure 23. Dynamic fatigue results for 33L-92% crystalline
tested after aging one day in water

In S (MPa)
a (MPa-s
S (MPa)

Figure 24. Dynamic fatigue results for 33L-92% crystalline
tested after aging one week in water

In S (MPa)
hi
15.8
\
5.25
5.00
1£
(MPa-s-1)
21
33L 92% Crystalline Aged 1 Week
79
I
. 125
4.75 *
~1
0
5
In (MPa-s-1)
4
S (MPa)

77
Table 7
Dynamic Fatigue Data for 33L-92% Crystalline
Tested in Air
Stressing Rate (MPa-s"^)
158
79
16
7.9
1.6
S(MPa)
226
228
226
226
209
s(MPa)
9
11
14
18
18
s/S
4%
5%
6%
8%
9%
n
n
n
9
13
10
sm
m
225
227
226
219
205
m '
24.3
21.1
14.7
11.7
11.2
S = average strength
. s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

Dynamic Fatigue Data for 33L-92% Crystalline
Tested in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
213
214
199
193
181
s(MPa)
14
10
10
6
7
s/S
7%
5%
5%
3%
4%
n
8
10
11
9
10
Sm
208
213
196
196
180
m
13.8
. 20.6
17.9
29.3
24.0
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

79
Table 9
Dynamic Fatigue Data for 33L-92% Crystalline
Tested After Aging 1 Day in Water
Stressing Rate (MPa-s~b
158
79
16
7.9
1.6
S(MPa)
207
206
203
187
179
s(MPa)
9
6
10
8
7
s/S
5%
3%
5%
5%
4%
n
6
15
4
15
6
Sm(MPa)
208
207
200
187
181
m
19.9
. 33.7
15.7
22.7
22.3
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Weibull modulus

80
Table 10
Dynamic Fatigue Data for 33L-92% Crystalline
Tested After Aging 1 Week in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
175
169
160
156
141
s(MPa)
18
15
17
18
15
s/S
10%
9%
11%
11%
11%
n
14
. 14
13
14
14
Sm(MPa).
169
165
156
152
137
m
9.8
11.0
9.4
8.9
9.6
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

81
Table 11
Liquid Nitrogen Strength of 33L-92% Crystalline
As Prepared Aged 1 Day Aged 1 Week
S(MPa)
274
280
221
s(MPa)
19
20
26
s/S
7%
7%
12%
n
9
15
10
Sm(MPa)
282
276
213
m
12.7
14.2
8.1
S =
average strength
s =
standard deviation
s/S =
coefficient of variation
n =
sample size
Sn, -
median strength
m =
Wei bull modulus

82
The stress corrosion susceptibility again changes with environment.
In air, N = 70.6, but in water it drops to the mid-twenties. Because
the lithium disilicate crystals are much more durable than the glass
(McCracken 1981), the N value is very high compared with the glass.
There is virtually no change in N after aging of this glass-ceramic
This can be explained by the lack of an extensive glassy phase. The
water attacks the grain boundaries where the glassy phase exists. But
because the glassy phase is so limited in size and extent, no protective
layer builds up, only dissolution occurs. Therefore, no matter how long
the aging period, the mechanism and the associated kinetics change very
1ittle.
The Wei bull moduli for 33L-92% range from 8.9 to 33.7. These are
much higher than those for 33L-Glass, indicating a narrower distribution
of strength controlling flaws.
Figure 25 is the lifetime prediction diagram for 33L-922L The
lines for the material tested in water are grouped together, again indi
cating little difference in mechanism or kinetics. The aged one week
line is off-set slightly because of the large decrease in strength
observed. For this material in air, a proof test ratio of 1.62 (81MPa)
is needed to assure a lifetime of ten years at 50 MPa. This is much
lower than that for glass under equivalent conditions. (Note different
scales in Figs. 20 and 25.) For an aqueous environment, the ratio
ranges from 2.91 (145 MPa) to 3.39 (169 MPa).
D. Dynamic Fatigue Testing of 331.-7%
Figures 26-29 are plots of the dynamic fatigue data listed in
Tables 12-15. Table 16 shows the inert strength data for 33L-7%.

70
60
50
40
30
20
10
83
oT¡ oTi oTi i!o TTi iT i!e
ln (p/aa)
Figure 25. LPD for 33L-92% crystalline

Figure 26. Dynamic fatigue results for 33L-7% crystalline
tested in air

In S (MPa)
7.9
158
5.40
V6
331 7% Crystalline Air
2£
79
I
5.20
5.00
4.80
-200
-175
- 150
. 125
in
4.60
-100
0
1
2
i i r-
3 4 ,5
(MPa-s-1)
In
S (MPa)

Figure 27. Dynamic fatigue results for 33L-7% crystalline
tested in water

In S CMPa)
(MPa-s-1)
\
S (MPa)

Figure 28. Dynamic fatigue results for 33L-7% crystalline
tested after aging one day in water

In S (MPa)
\
o (MPa-s-^)
In (MPa-s~^)
S (MPa)

Figure 29. Dynamic fatigue results for 33L-7% crystalline
tested after aging one week in water

In S (MPa)
(MPa-s*1)
S (MPa)

92
Table 12
Dyanmic Fatigue Data for 33L-7% Crystalline
Tested in Air
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
166
171
165
157
165
s(MPa)
35
30
33
43
31
s/S
21%
18%
20%
CVJ
19%
n
9
8
10
9
9
S (MPa)
m
153
168
167
139
169
m
4.6
5.4
4.7
3.5
5.0
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

93
Table 13
Dynamic Fatigue Data for 33L-7% Crystalline
Tested in Water
Stressing Rate (MPa-s~~*)
158
79
16
7.9
1.6
5(MPa)
159
148
156
131
155
s(MPa)
32
24
24
21
22
s/S
20%
17%
15%
16%
14%
n
9
8
8
9
10
sm(MPa)
152
150
160
124
162
m
4.6
5.3
5.8
6.1
6.4
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

94
Table 14
Dynamic Fatigue Data for 33L-7% Crystalline
Tested After Aging 1 Day in Water
Stressing Rate (MPa-s"1)
158
79
16
7.9
1.6
S(Mpa)
138
164
136
139
134
s(MPa)
30
37
33
27
22
s/S
22%
23%
24%
19%
15%
n
9
9
9
9
6
Sm(MPa) '
134
165
151
145
124
m
4.5
3.8
3.7
4.7
5.1
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Weibull modulus

Table 15
Dynamic Fatigue Data for 33L-7% Crystalline
Tested After Aging 1 Week in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
176
183
153
152
154
s(MPa)
29
30
36
29
21
s/S
16%
17%
24%
19%
14%
n
9
9
8
9
7
Sm(MPa)
175
193
166
159
156
m
5.6
5.1
3.3
4.6
6.8
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

96
Table 16
Liquid Nitrogen Strength of 33L-7% Crystalline
S(MPa)
s(MPa)
s/S
n
S (MPa)
m
m
As Prepared
233
39
17%
7
220
5.5
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

97
Again a strength decrease was found when testing in water versus
air, but virtually no change in N was found. After aging one day, both
the strength and N value decreased. Aging for one week allowed the
material to regain most of its strength while dropping N still further.
Apparently, the short time in water has virtually no effect on the
subcritical growth in this microstructure. Aging one day for 33L-7%
appears to be equivalent to testing in water for the uncrystallized
glass. Both result in a drastic loss in strength and increased suscep
tibility to stress corrosion. After aging one week, 33L-7% is remarkably
similar to 33L-Glass after aging one day. The partially crystallized
material is somewhat stronger, but the values for N are similar.
It would be expected that 33L-7% would behave like 33L-Glass because
the glassy phase should dominate environmental sensitivity. However, the
extraordinarily high values of N for 331-7% must be a result of the
kinetics of the water/glass-ceramic reaction. As the crystals have a
lower coefficient of thermal expansion, a compressive stress is induced
in the glass. This residual stress is perhaps inhibiting the stress
corrosion process.
The Weibull moduli are very low (3.3-6.8), as they were for 33L-Glass
Again this indicates a wide range of flaw sizes controlling the strength.
The variability in sample crystallinity is at least partially responsible
for the large scatter.
Figure 30 is the LPD for 33L-7%. The clustering of the air and
water lines apart from the one day and one week aged lines indicates that
some time is needed for the water to have an effect on this system. This
is contrasted with the 33L-92% material, in which the effect of water is

98:
In (op/oa)
Figure 30. LPD for 33L-7% crystalline

99
seen almost immediately. The residual stress in the 33L-7% material
is very likely responsible for the increased resistance to stress
corrosion, as exhibited in Figure 30. Aging apparently eliminated the
beneficial effects of the residual stress by attacking the pre
stressed glass/crystal interface. The consequence is a greatly
increased environmental susceptibility of the aged glass-ceramic
and a large reduction in the predicted lifetime (Fig. 30).
E. Dynamic Fatigue Testing of 33L-57%
Only a limited study vas made on 33L-57%. The results of tests
performed after aging one day. and one week are shown in Figures 31-32
and Tables 17-19.
The inert strengths (Table 19) of both sets are approximately
equal, showing no increase in strength of the one week aged samples
over those aged one day. However, the N value changes dramatically
(59.2 to 23.6). This again indicates a change in the flaws due to the
corrosive action of the water. In this case, any increase in strength
due to aging has been bypassed, and after one week, the full effect of
water attack becomes evident in the sharp increase in the susceptibility
to stress corrosion.
Figure 33 is the LPD for 33L-57% based on tests after aging in water
for one day and one week. Because of the difference in N, there is a
large difference between the two lines. For predictive purposes, the
data from the one week test would be recommended in order to be more
conservative.

Figure 31. Dynamic fatigue results for 33L-57% crystalline
tested after aging one day in-water

In S (MPa)
(MPa-s'1)
S (MPa)

Figure 32. Dynamic fatigue results for 33L-57% crystalline
tested after aging one wek in water

In S (MPa)
(MPa-s-1)
In a (MPa-s-1)
S (MPa)

70
60
50'
40
30
20
10
104

105
Table 17
Dynamic Fatigue Data for 33L-57% Crystalline
Tested After Aging 1 Day in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
167
166
148
136
160
s(MPa)
18
13
19
10
11
s/S
11%
8%
13%
8%
7%
n
9
10
9
; 8
9
Sm(MPa)
175
167
146
136
162
m
8.5
12.6
6.9
12.5
14.0
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

7,06
Table 18
Dynamic Fatigue Data for 33L-57% Crystalline
Tested After Aging 1 Week in Water
Stressing Rate (MPa-s~^)
158
79
16
7.9
1.6
S(MPa)
163
159
131
139
140
s(MPa)
9
15
19
14
11
s/S
6%
9%
15%
10%
8%
n
8
10
7
8
6
Sm(MPa)
165
155
126
135
140
m
15.2
10.4
6.2
9.2
11.6
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
S = median strength
m
m = Wei bull modulus

T07
Table 19
Liquid Nitrogen Strength of 33L-57% Crystalline
Aged 1 Day
Aged 1 Week
S(MPa)
210
216
s(MPa)
23
30
s/S
11%
14%
n
10
10
Sm(MPa)
206
223
m
8.7
6.3
S = average
strength
s = standard
deviation
.. *
s/S = coefficeint of variation
n = sample size
Sm = median strength
m = Weibull modulus

108
F. The Effect of Crystallization on Strength
Reported in Table 20 are the average inert strength and strength
values for each condition tested at 16 MPa-s-"*. (The reader is also
referred to the previous tables, which are more comprehensive. These
values will serve to show the general trend.) Only 33L-Glass and 33L-7%
give any indication of strengthening after aging. The 33L-92% material
loses strength dramatically after aging one week. Strength decreases,
too, for 33L-57%, but not so precipitously. As previously discussed,
this is due to grain boundary attack.
It is difficult to interpret the effect of crystallization on the
strength of this system because of the microcrack formation at high Vv
and the residual stress at low Vv. Previous investigators (Freiman, 1968,
and Hasselman and Ful rath, 1966) have analyzed their materials' strength
as a function of mean free path between dispersed particles. That is,
the flaw size is limited by the interparticle spacing. However, Freiman
(1968) did not consider the presence of microcracks and Hasselman and
Ful rath (1966) assumed no thermal expansion mismatch between particle
and matrix.
For 33L-7% crystalline, the residual stresses in the glassy phase
induced by the thermal expansion mismatch between the glass and spherulites
is most likely responsible for the strength increase over 33L-Glass.
Freiman (1968) found an increase in Young's modulus with crystallinity,
but not enough to explain the strength increase. (Recall that the failure
strength is proportional to E^.) Miyata and Jinno (1972) re-analyzed
some of the work of Hasselman and Fulrath (1966) and observed a decrease

109
Table 20
Strength of 33L-Glass and Glass-Ceramics
C<
Inert Strength* Ambient Strength**
Glass
As prepared (Air)
179
125
Water

112
Aged 1 Day
211
129
33L-92% Crystalline
As prepared (Air)
274
226
Water

199
Aged 1 Day
280
203
Aged 1 Week
221
160
33L-7% Crystalline
As prepared (Air)
233
155
Water

141
Aged 1 Day

136
Aged 1 Week

153
33L-57% Crystalline
Aged 1 Day
210
148
Aged 1 Week
216
131
*Average strength (MPa), tested in liquid nitrogen at 16 MPa-s"^
**Average strength (MPa), tested at room temperature at 16 MPa-s"^

in strength at low Vv (< 20%) with no connection to the mean free
path. Rao (1977), studying lithium disilicate containing 2-5 .ym
spherulites, attributed the strength increase to an increase in fracture
surface energy due to pinning by the crystals. ~
The decrease then increase in strength which occurs on exposure
to water in 33L-7% is analogous to that which occurs in 33L-Glass.
The residual stress merely postpones the effect.
For 33L-57% and 33L-92% crystalline, microcracks are prevalent
throughout the material. It has been shown that microcracking can
be effective in toughening glass-ceramics (Faber and Evans, 1980, for
lithium aluminoslicates). The microcracks provide multiple branches
(which acting by themselves will reduce the energy of a propagating
crack) leading to the spherulites which pin the crack.
As discussed previously, the weakening on aging is due to the
migration of water into the grain boundaries (via the raicrocracks),
thus dissolving the residual glassy phase and destroying the cohesiveness
of the network.
G. The Effect of Crystallization on N
The N values for all test conditions are shown in Table 21. Except
for 33L-Glass, the N value decreases or stays the same as the availability
of water increases. This underscores the importance of the kinetics of
the reaction at the crack tip.
While the glass shows an initial decrease in N from testing in air
to testing in water, the subsequent increase after aging one day is
indicative of another mechanism acting during aqueous corrosion.

ni
Table 21
Fracture Parameters and (o /a ) from Dynamic Fatigue Data
P d
N
£nB
(Q/<0*
33L-G1ass
p u
Air
30.5
-2.07
2.8
Water
11.0
4.23
14.1
Aged 1 Day
37.4
-30.73
5.5
33L-92% Crystalline
Air
70.6
-5.66
1.7
Water
25.7
2.10
3.0
Aged 1 Day
27.6
0.73
2.9
Aged 1 Week
22.3
2.37 ;
3.4
33L-7% Crystalline
Air
116.1
-34.43
1.8
Water
136.0
-53.93
1.9
Aged 1 Day
46.3
-14.88
2.6
Aged 1 Week
27.4
-1.13
3.0
33L-57% Crystalline
Aged 1 Day
59.2
-10.16
1.9
Aged 1 Week
23.6
0.06
3.4
*For 10 years at 50 MPa

112
Hench (1977) has shown five types of corrosion in glasses,
depending on the composition of both the glass and the corrosive
medium. One type involves the development of a protective silica-
rich layer on the surface. Sanders and Hench (1973) have:sJnown
that 33L undergoes this type of corrosion. Apparently this
reaction layer can slow down the ion exchange process at the crack
tip necessary for stress corrosion, thus resulting in a higher N
value. The partially crystalline materials do not exhibit this
behavior because most of the corrosive attack is along grain
boundaries. This intergranular attack can only be detrimental
to the strength, since the "crack tip" cannot be Grounded, only
extended.
For the crystallized materials, N decreases or stays the same
with aging in water down to the range of values for the glass. This
implies that (eventually) the residual glassy phase is the controlling
material for stress corrosion in the glass-ceramics.
For 33L-92% crystalline, the microcracks and extensive grain
boundary surface area allow: rapid takeover of the process by the
residual glassy phase. The most significant point here is the
loss in strength after aging one week which was discussed earlier.
For 33L-7% crystalline, residual stresses in the glassy phase
obviously affect the kinetics of the aqueous corrosion. (The effect
of this stress on the strength was discussed earlier.) However,
once sufficient aging has allowed penetration of glass/crystal
boundaries, some of the stress is then relieved, at which point the
glass corrosion again becomes the controlling factor.

113
$
For 33L-57% crystalline, although testing was not done in air
or in water without aging, it appears that the same phenomena are
occurring as in 33L-7% crystalline. The interaction may proceed more
slowly (N = 59.2 vs H = 46.3 after aging one day), but the, presence
of considerably more grain boundary area drops the N value to a lower
one after one week (23.6 vs 27.4). Microcracking is also likely to
play a role in the stress corrosion process.
H. The Effect of Crystallization on Lifetime Predictions
Although in two cases there appear to be strength increases after
aging the proof test stress ratio (n/a) indicates that there is
no advantage gained over the long term. (These are also reported in
Table 21.) Except for 33L-Glass (in water versus aged), this ratio
increases with increasing availability of water. This means that to
guarantee a particular lifetime, a component must be tested at a much
higher proof test stress. For example, for 33L-Glass, even though
testing in air and after aging one day result in about' the same strength,
the proof test to guarantee ten years at 50 MPa would be 140 MPa in air
versus 275 MPa after aging.
Recalling Eq. 22
t. a*" = B (a /a )^
min a 'p a'
(22)
as N decreases, the time to failure under a given load after a given
proof test decreases dramatically. Correspondingly, the proof test
ratio for a given lifetime/load condition must increase. The same
is true as B decreases (as n B becomes more negative). This means
that even if a material becomes stronger (in short term testing) after
aging, it will be the fatigue parameters, N and B, which will control
the material's long term behavior.

114
(Note: At this point in time, the parameter B has not
?
been well characterized. In Eq. 22, it must have the units of MPa -s,
but it also has been defined in terms of Kc raised to some power
which depends on N. The theoretical approach employed here cannot
resolve this problem. Other approaches solve this situation, but
complicate the analysis and do not facilitate the construction of
lifetime prediction diagrams described herein.)
For 33L-Glass, the proof test stress ratio increases when
tested in water versus air because of the sharp drop in N. After
aging one day, the increase is due to the change in £n B.
For the crystallized materials, it is always a large change
in N which is responsible for increase ratio. (Almost always, £n B
is increasing.) It may be stated, then, that for the lithium disilicate
glass/glass-ceramic system, the N value controls the predicted lifetime.
I. Summary
For dynamic fatigue testing, it has now been'demonstrated that
knowledge of short-term corrosion effects may be totally inadequate
for predicting very long life-times. The changes in the fatigue
parameters, N and B, with aging time in water are inconsistent with
the basic theories upon which the lifetime predictions are made. The
actions of corrosive media on reactive materials must be characterized
for both the stressed and unstressed states as well as at the surface
and crack tip. Kinetics of reaction and growth of reaction layers
are very important for determining the long-term behavior of ceramics
under load in a corrosive environment.

CHAPTER VII-CRACK VELOCITY EXPERIMENTS
A. General
Figure 34 shows the general empirical relationship between the
crack velocity, V, and the stress intensity factor, K. Region I,
where both V and K are relatively small, is the region of prime interest.
Here the empirical equation (Chapter III)
V = A KN (13)
holds, where A and N are the same constants determined in Chapter VI.
In Region I, the crack growth is a function of the reaction rate (of
water or water vapor with the material) at the crack tip.
In Region II, the crack velocity is independent of K. The crack
grows more quickly than in Region I so that the limiting process for
growth is the rate of transport (of water or water vapor) to the crack
tip. Region III shows a very steep dependence of V on K. Catastrophic
failure occurs in this region as K approaches the critical stress
intensity factor, K .
U >
The method of choice for determining the V-K curves for the 33L
system is the double cantilever beam (DCB) constant moment technique
described in Chapter V. This method, first described by Freiman et al.
(1973), is simple, inexpensive, and straightforward. However, as with
most experimental techniques, it is only simple and straightforward
after much trial and error and appropriate tricks have been learned so
that the experiment is conducted properly. For the set of data presented
here, nearly 100 samples (many outright washouts) were machined
115

116
Figure 34. Theoretical crack velocity vs. stress
intensity factor relationship for
brittle materials

117
and tested. After surviving this learning experience, the same
information could be obtained with about 25 samples.
B. 33L-Glass
Figures 35-36 show the V-K curves for 33L-Glass as tested in air
and water respectively. Tests on aged samples proved to be no different
from samples tested in water. Table 22 gives the values obtained for In
A and N determined by a least squares to fit the data. The critical
stress intensity factor, Kc, was determined to be about 0.78 MPa-nr2.
The values for N are all about the same (9-13). The curve shifts
slightly to the left in water. For the lowest velocities, measurements
were made over 24-30 hours.
The N values obtained in air and in water are different from those
obtained in dynamic fatigue testing. A proposed explanation and the
ramifications of this result will be discussed in Chapter VIII.
C. 33L-Partially Crystallized
Because of difficulty in crystallizing large bricks, it was not
possible to exactly duplicate the same volume percent crystallinity
for the DCB samples as in the dynamic fatigue samples. Figures 37-38
and Table 23 show the results of these sets of samples. Table 23 also
gives the percent crystallinity for each sample (determined as described
in Chapter VI). Although samples were tested after aging, their results
are plotted with those tested in water because no difference was found.
No Kc values were determined for these samples.
For samples containing low volume percent of crystallinity, the
cracks grew in fits and starts as the crack encountered matrix and

118
Table 22
33L-Glass Fracture Parameters
as Determined by Slow Crack Growth
Experiment
Environment
N
In A
52
Air
9.5
-138.7
55
Air
11.0
-158.1
57
Air
12.6
-179.8
36
Water
9.4
-137.2
52
Water
11.0
-153.6
57
Water
12.7
-176.1
C

K (xlO-6 Pa-nr2)
In K (Pa-m'2)
Figure 35. V-K relationship for 33L-Glass tested in air

120
K (x10"6 Pa-m^)
0.25 0.50 0.75
In K (Pa-n£)
Figure 36. V-K relationship for 33L-Glass tested in water

121
Table 23
33L-Partially Crystallized Fracture Parameters
as Determined by Slow Crack Growth
Experiment
% Crystallinity
Environment
N
mA
41
6
Air
14.0
-212.4
46
6
Water
9.9
-152.0
54
5
Air
12.3
-185.7
63
65
Air
16.3
-248.6
64
66
Air
: 15.1
-230.0
68
62
Water
14.5
-219.2

122
K (xl0~6 Pa-m^)
In K (Pa-m^)
Figure 37. V-K relationship for some 33L-low crystallinity
samples

123
K (xl0~6 Pa-rn^)
In K (Pa-m^)
Figure 38. V~K relationship for some 33L-high crystallinity
samples

crystals alternately during their propagation. Figure 39 is a
photomicrograph of a typical microstructure revealed as the crack
split the sample. The ripple marks show the staccatto nature
of the crack growth.
Nadeau and Bennett (1978) examined crack growth in glass
microscope slides containing grooves filled with glasses of
various thermal expansion coefficients. The ripple marks seen
in Fig. 39 resemble those observed by Nadeau as the crack
encountered a lower expansion inclusion. In the present case,
the thermal expansion coefficients for 33L-Glass and 33L-92%
crystalline were found to be (as measured by diTatometry*) 110 x 10~
-7 -1
and 95 x 10 C respectively, thus confirming Nadeau and Bennett
observations.
Figure 39 shows a region of slow crack growth (-10^ m-s~^).
At these velocities, the crack growth tends to be intergranular.
Figure 40 shows a region of the same sample where -the crack was
-3 -1
propagating catastrophically (~10 m-s ). Here the fracture is
more transgranular in nature, as there is sufficient energy to
drive the crack through the crystals.
The variation of the fracture parameters A and N with per cent
crystallinity will be discussed in Chapter VIII, along with the
comparison with the dynamic fatigue data.
*Harrop Laboratories, Columbus, Ohio

125
Figure 39. Region of slow crack growth in
33L- 5% crystalline

r
Figure 40. Region of fast crack growth in
33L- 5% crystal 1 ine

127
D. 33L-92% Crystalline
Figure 41 and Table 24 provide the results for the completely
crystallized material. Due to the extensive microcracking, problems
again were encountered in measuring the crack velocity. In addition
to the halting nature of the propagation, the microcrack network
tended to guide the crack along a tortuous, twisted path, rather than
continuous and straight. The K was determined to be 4.19 MPa-m^.
V
Rao (1977) also measured K£ for the lithium disilicate glass and
glass-ceramic using double torsion. His values ranged from 1.015 MPa-nr5
for the glass to 3.53 MPa-m2 for an 85% crystalline material. The range
found in this study is 0.78 MPa-rn2 for 33L-Glass to 4.19 MPa-m52 for
33L-92%. The agreement is reasonable, given the two different techniques.
An increase in Kc with crystallinity can be expected for two reasons.
One, the particles act to pin the crack; and two, the microcracking also
acts as a toughening mechanism. (Faber and Evans, 1980, hav recently
discussed this last point.) Both of these aspects were discussed with
the effects of crystallization on strengthening.
As for the partially crystalline materials, samples tested after
aging in water gave results identical to those tested immediately in
water. Thus, they are plotted in Figure 41 on the same line.
Figures 42-43 show regions of slow and fast crack growth respectively.
Again, slow growth is intergranular and rapid growth is transgranular.
The stress corrosion susceptibility, N, as measured by crack growth
methods differs from that measured by dynamic fatigue, as was found for
the uncrystallized samples. Chapter VIII will discuss this difference.

128
K (xlO^Pa-nr1)
Figure 41. V-K relationship for 33L-92% crystalline
V (m-s

129
Table 24
33L-92% Crystalline Fracture Parameters
as Determined by Slow Crack Growth
Experiment
Environment
N
InA
58
Air
10.9
-169.1
59
Air
16.8
-248.9
60
Air
16.8
-255.5
67
Air
13.1
-203.7
61
Water
14.4
-216.2

130
Figure 42. Region of slow crack growth in
33L-92% crystalline

Figure 43. Region of fast crack growth in
33L-92% crystal!ine

CHAPTER VIII-DISCUSSION
A. Dynamic Fatigue
It has been shown (in Chapter V) that N, the stress corrosion
susceptibility, for the lithium disilicate glass-ceramic system
changes with aging time in water: Because the fracture mechanics
theory behind the empirical equations describing dynamic fatigue
assumes a constancy of material and environment as well as a constant
character of the flaws, the change in N is indicative of deviation
from these assumptions. The most likely culprit is an alteration
of the character of the flaws during the aging process. There is
ample evidence to suggest that the action of aqueous solutions on
glasses and ceramics can alter the flaws at the surface of a
material.
Sanders and Hench (1973) have shown that a silica-rich surface
layer builds up readily when 33L-Glass is exposed to aqueous solutions.
Other investigators (Greenspan, 1977, Dilmore, 1977 Clark, 1976) have
shown this to be true of many other glasses as well. The formation of
this gel-like surface layer may affect the material to a depth of
1-10 ym. The physical and chemical properties of the material are
almost surely changed within this surface region. In addition, any
flaws within this region would have to be altered in some manner
(probably blunted). This two-fold attack, changing the flaw as well as
the material surrounding it, could account for the changes in N.
Figure 44 shows the growth of a gel layer at the surface of a glass
containing two types of flaws. As the gel grows, the shallower flaws
are the first to be affected. An experiment using a sodium disilicate
132

133
Figure 44. Growth of a gel layer at the surface of
a glass containing two depths of flaws

134
glass (33N) was performed to demonstrate this effect. (This composition
was chosen because of its rapid reactivity with water (Ethridge, 1977).)
Two sets of 33N discs (prepared as in the 33L dynamic fatigue experi
ments) were tested for strength after receiving different polishing
treatments and after aging in water. One set was polished to a 120
grit finish, the other to a 320 grit. Both sets were tested in air
(no aging), after five minutes in water, and after one hour in water.
The results are shown in Table 25.
The strength of discs containing the deeper flaws (M25 urn for
120 grit) was virtually unaffected by the aging treatment. However,
after one hour there was a 50 percent increase in the strength for the
set of discs containing shallower flaws (M5 urn for 320 grit).
This clearly supports our hypothesis that aging affects the
distribution of flaw sizes and thereby the strength.
Additionally, fracture mechanics can be used to predict one set
of flaw sizes from the other by comparing failure strength. Thus,
a320
a120
using S.|2q = 71 MPa, S^q = 112 MPa, and a^g = 125 ym from Table 25
one obtains ag2Q = 50 ym, which correlates well with the estimated
45 ym.
For crystalline or partially crystalline materials, the (usually)
glassy intergranular phase present is susceptible to the same corrosive
attack. Extremely durable and refractory materials such as alumina
(Ferber and Brown, 1980) and silicon nitride (Trantina, 1978) at high
temperatures are now known to undergo stress corrosion due to attack

135
-Table 25
Effect
of Surface
of Sodium i
120 Grit
Finish on Aged Strength
Disilicate Glass
320 Grit
As Prepared
5 Minutes
1 Hour
As Prepared
5 Minutes
1 Hour
5(MPa)
71
64
73
112
108
171
s(MPa)
11
9
13
24
36
61
s/5
15
14
18
21
34
36
n
10
7
9
10^
8
10
S (MPa)
m
73
63
76
106
114
173
m
5
7
5
5
2
3
5 = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus

136
at the grain boundaries. Dense alumina is also known to be environ
mentally sensitive to water at even modest temperatures (Sinharoy et al.
1978). The precise nature of the growth of the flaws is difficult to
characterize because of the small scale of the phenomenon, but flaw
blunting and the growth of a reaction layer (gel layer) on the surface
of the flaw or ahead of the flaws are not unreasonable scenarios.
In order for proof testing of lifetime prediction methods to be
viable, it is assumed that the initial set of flaws determined by the
first set of tests (the proof test) is the set responsible for subse
quent delayed failure. It is apparent from the experiments described
herein that such a criterion is not likely to be satisfied by 33L,
33N, or by any other material in a highly reactive environment.
Delayed failure is a dynamic effect which is determined not only by
possible chemical reactions but also., by the kinetics of those reactions
There is probably no single set of short term experiments which can
adequately describe this process. This means that long term (NOT
accelerated) tests are needed to better determine the durability and
strength characteristics of brittle materials.
The estimations of the fatigue parameters N and B are determined by
way of statistical analysis of the experimental data. Since the main
objective of this thesis is to study the effects of environment and
microstructure on the fatigue properties of the lithium disilicate
system, it is necessary that the statistical analysis allows us to
differentiate among the obtained values for N and B. For reliable
lifetime predictions, it is also necessary to have the most accurate

137
estimations obtainable. This also applies, of course, to comparing
dynamic fatigue results with slow crack growth results.
Most investigations (including this one) use the method of least
squares for fitting straight lines to experimental data. Unfortunately,
it may not be the best method. The method of least squares assumes
that each point is independent of any other point and that when using
groups of samples, the groups have equivalent variances. For Weibull
analysis, the data points have been ordered; thus the location of each
point is dependent on adjacent points. In dynamic fatigue testing,
although the groups tested are all from the same population so that
ideally the variances would always be equal, this is rarely the case
in practice. Weighting of the data is necessary to account for these
variations in order to properly employ the method of least squares.
Recently, Baldwin and Palmer (1980) have begun an investigation
using the method of maximum likelihood for fitting straight lines.
This technique first derives a probability density function for the
experimentally observed strength values (S.). Then, a likelihood
function given by the product of the probability density functions
evaluated at each is maximized by iteratively estimating the
parameters (slope and intercept) to be determined. When the likelihood
function is maximized, one obtains the best estimate of the parameters.
Unfortunately, this method was not sufficiently developed for inclusion
in this work.
Rockar and Pletka (1978) have questoned the reliability of dynamic
fatigue data for predicting lifetimes. They had difficulty obtaining
good fits to their straight lines apparently due to large scatter in
the data. Thye also found greater scatter and worse fits at higher N

138
values. Ritter et al. (1980) have proposed that small sample sizes
(less than 100) may be responsible for this unreliability. Using a
Monte Carlo technique to generate artificial data sets from a real
set of data, they demonstrate the variability of N with sample size,
the Wei bull modulus, and N itself. The variability increases as
sample size and Weibull modulus decrease and as N increases.
Baldwin and Palmer (1980) employed the method of maximum
likelihood as a comparison to the method of Ritter et al. in order
to evaluate the importance of statistical techniques in interpreting
fatigue data. The Baldwin and Palmer (1980) method utilizes the same
basic equations as Ritter et al. and also assumes a Weibull distribution.
A preliminary comparison of the results for predicting by the method
of maximum likelihood and by the Monte Carlo method of Ritter et al.
is shown in Figure 45. The initial estimates of the fracture and
Weibull parameters are N = 18.4, B = 0.18, m = 8.2, and SQ = 138.0.
The case shown is based on 100 simulations with five samples at each
of two stressing rates (0.01 and 53 MPa-s ). The initial estimates
were taken from Ritter et al. (1980). This shows that while the two
methods usually predict values close to the original estimate, the
Monte Carlo technique is more likely to produce a grossly inaccurate
value of N.
Ritter et al. used the method of least squares to fit their
straight lines (including Weibull analysis). The method of maximum
likelihood is probably a better technique.
Also, Ritter et al. assume that the distribution of N values
obtained from the Monte Carlo simulations is normal. Assuming total
randomness, it is reasonable to expect that the distribution of the

139
50
40 -
30. -
20 -
10 -
0 -
Figure 45.
Method of
Monte Carlo
Method
Maximum
Like!ihood
18.4'
A comparison of the Method of Maximum Likelihood
and the Monte Carlo Method for predicting N

140
slopes of the lines would be normal. However, since the slope is
equal to 1/(N+1), the distribution of N cannot possibly be normal
(except at very large sample sizes).
As to finding large scatter with large N, large N with large
scatter has been found here as well. Because the slopes of the lines
are so small 0.010 0.100), a small change in the slope can result
in a large change in N. (Recall that the slope is equal to 1/(N+1).)
Thus, greater variability at high N values is to be expected from the
nature of its determination. For example, the same percentage of
variability at large slopes (0.100 0.050) compared with smaller slopes
(0.0100.005) yields a range of 7-20 versus 67-200.
There is also a question as to which statistic of the strength
test should be used for determining the straight line. Ritter et al.
used the median; others use the mean; this study prefers to use all
data points. Because the mean is easily affected by outliers (either
abnormally weak or strong), the median is clearly a better choice
between the two. Using all data points, however, results in utilizing
every shred of information contained in the experiment. After all,
if the samples have been prepared in the same manner as the components
they simulate, the strength statistics of the components will duplicate
those of the samples. The method of maximum likelihood maximizes the
amount of information obtained by using all data points.
Rockar and Pletka (1978) used a different method. They ordered
their data (ten samples per stressing rate), then performed ten linear
regressions, taking as their final line the average of ten. This method
should yield similar results to those obtained here by using all data
points to generate one line.

141
For both ZZL-7% and 33L-57% there are instances of the strength
at the lowest stressing rate being higher than at higher stressing rates.
This may be an indication of the competition of the corrosion process
rounding the crack tip and the applied stress extending the crack.
Such a competition indicates that the test has gone beyond the range
of Region I of the V-K relationship. If that is the case, it would
be better to use only the remaining four sets of data to determine N.
The slope would then increase, decreasing the value of N.
The fatigue life of high strength steels has been found to
deviate from the usual log-normal or Wei bul 1 distribution at low
stress levels as well. Ichikawa, Takura, and Tanaka (1980) have
decomposed the distribution of the total life into two parts, one at
crack initiation and one of crack propogation. The departure from
usual statistical distributions at low stress levels is attributed
to the influence of the scatter of the length of the initiation cycle,
which increases as the stress level decreases. The crack tip rounding
process in ceramics is likely to cause similar disruptions.
Finally, it must be remembered that the equations governing the
analysis: of dynamic fatigue experiments are empirical and are derived
using several assumptions (see Chapter III for derivation). These
include:
1) constancy of flaw population,
2) constancy of environment,
3) constancy of chemical properties of the material, and
4) constancy of fatigue parameters N and B.

142
Violations such as in the above paragraph or changing the nature of
the flaws (as discussed earlier) will destroy the validity of the
analysis. Therefore, it is evident from this study that great care
must be taken in designing, analyzing, and applying dynamic fatigue
experiments. The kinetics of the corrosion reaction and the empirical
nature of the equations may very well restrict the application of this
method for making lifetime predictions.
B. Slow Crack Growth
In the slow crack growth experiments, the value of N, the stress
corrosion susceptibility, was found to be unaffected by aging in water
and only slightly increased by crystallization. 'These observations are
consistent with previous investigations.
Crack front interaction with the microstructure made determination
of the V-K relationship difficult for all materials except 33L-Glass.
Pletka and Wiederhorn (1978) also noticed this effect in a magnesium
aluminosilicate glass-ceramic. The crack arrest at the particle results
in the stopping of the crack followed by a release with sufficient energy
built up to allow the crack velocity to increase by an inordinate amount.
Theocaris and Milios (1980) have studied cracks propagating through a
bi-material interface and have quantified the increase in velocity as
a function of relative ductilities of the two materials. However,
application of their findings to materials with a brittle matrix and
brittle inclusions is not straightforward.
Ferber and Brown (1980), using the double torsion method, conclude
that because of crack/microstructure interactions, reliable V-K data
could not be obtained on their alumina samples. Studies by Fuller
(1979) and Pletka, Fuller and Koepke (1979) also cast doubt on use

143
of the double torsion method. The double cantilever beam constant
moment technique is similar to the double torsion method in that it
uses a constant K specimen. However in the DCB method employed
herein, the crack growth is viewed and measured directly, which
allows the microstructural effects to be monitored more closely.
By marking the locations on the sample where the arrest-release
events occur, subsequent microscopic inspection of the crack
surface (as in Figure 39 and others) can lead to a better under
standing and perhaps quantification of the crack/microstructure
interactions.
C. Comparing Dynamic Fatigue and Slow Crack Growth
The N values obtained in the crack growth studies correspond
best to the lowest values obtained in dynamic fatigue. Pletka and
Wiederhorn (1978), contrarily, found larger values for N in crack
growth studies than in dynamic fatigue. Their explanation is that
in a uniaxial stress field the obstacles more readily arrest the
crack, yielding a higher N value in crack velocity experiments
than in a four-point bend test or biaxial flexure.
For the case of the lithium disilicate system studied here,
it is likely that the kinetics of the corrosion reaction at the
crack tip govern the fatigue behavior rather than the macroscopic
stress state. The time required for slow crack growth allows more
of a reaction at the crack tip, resulting in smaller N values for
the test which takes longer to complete. The short fracture time
('v % 2 seconds) in dynamic fatigue is too rapid to allow sufficient
degradation of the material at the microscopic flaw at least relative

144
to the macroscopic crack. The occurrence of aging effects in
dynamic fatigue but not in slow crack growth confirms this
interpretation.
The results of this study show that lifetime prediction
diagrams drawn using crack growth data would be more conservative
than those drawn using dynamic fatigue data. At this time it is
not known whether they would be more or less accurate. The strength
ening effect of water on the 33L-Glass (indeed, on most glasses)
complicates the decision. It is evident that each material will
need extensive testing of various types (those used here as well as
static fatigue) before an intelligent decision can be made regarding
lifetimes in critical applications.
In their work, Pletka and Wiederhorn (1978) studied a lithium
aluminosilicate glass-ceramic and found very little difference between
the parameters determined by crack growth and those determined by
dynamic fatigue. They attribute this to a more homogeneous micro
structure compared with their magnesium aluminosilicate. This under
scores the need for a variety of tests on whatever material is under
investigation.
Doremus (1980) has made a mathematical analysis of the equations
involved in lifetime predictions. He concludes that:
1) crack propagation data cannot be used to predict
fatigue times reliably;;
2) the "agreement" between measured fatigue strengths
and those calculated from crack propagation data
is not significant;
3) propagation velocities of large (Mem) cracks are probably
related to static fatigue in glass in some way, but the
details of this relationship are uncertain.

145
Mu ti-technique testing is the most straightforward way of resolving
his last conclusion. Once that relationship is understood, crack
propagation data may then be useful for lifetime prediction and real
"agreement" may then be possible.
Argon (1974) has stated that "... fracture is not a property of a
material but rather constitutes a behavior." The difference between
fatigue and fracture results found in this study as well as the environ
mental and aging effects demonstrated herein tend to support both Argon
and Doremus' concerns.
In order to use the present lifetime prediction theories, it
must be remembered that the empirical model developed to describe
the fatigue behavior of a ceramic under a static load assumes that
the parameters derived from short term tests remain constant over
the lifetime of the material. Unfortunately, some materials will
react with their environment over time in such a way that the parameters
originally used must change. These changes will then negate the original
analysis. Each material will need to be characterized extensively as to
how the parameters might change if the model is to be valid. For
example, if N increases (as with 33L-Glass after aging in water one day),
the original lifetime prediction will be conservative. If N decreases
(as with 33L-92% crystalline after aging one week), the component's
lifetime may be drastically (and catastrophically) shortened.
Thus, the present model is not designed to handle a dynamic
system. A new model needs to be developed to account for changes
in the parameters which describe the process of subcritical crack

146
growth. The results of this study suggest that the following
innovations should be considered:
1) the kinetics of the chemical reaction at the crack tip
need to be accounted for. These studies would establish
the time dependence of the fatigue parameters and (possibly)
the tip radius and depth of the flaw.
2) the stress dependence of the chemical reaction at the
crack tip must be established. This also affects the
fatigue parameters and flaw characteristics.
3) the chemical changes in the surface layer surrounding
the flaws must be characterized. A change in the
chemistry (e.g., H+ replacing Na+) or state (e.g.,
glass to gel) will certainly affect the growth of
the flaw.
4) the synergism of all three of the above needs to be
characterized.
As more and more materials are studied, more changes will be
identified and their quantitative nature will become clear. In this
manner the necessary components for a complete model will slowly piece
together for a more consistent theory that can handle reactive
materials or the reactive phases of brittle materials. But, for now,
every investigation is an adventure.
0

CHAPTER IX-CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK
A. Conclusions
1. The testing of a material which readily reacts with an
aqueous environment has shown that the present model for describing
fatigue by subcritical crack growth needs to be revised to account
for the dynamics of interaction of the material with its environment.
Because the fatigue parameters (N and B) involved in calculating
lifetimes have been observed to change with time, a single analysis
based on a short-term test is not adequate for reliable predictions.
Two further developments are needed. First, the model must be
extended to account for the changes in the fatigue parameters.
Second, a series of tests needs to be devised to characterize these
changes for a given material.
2. The strength of lithium disilicate materials increases as
the percent of crystallinity increases, but not in a simple straight
forward manner. At low levels of crystallinity, a compressive stress
may be induced in the glassy matrix, resulting in a stronger material.
At high levels of crystallinity, microcracking appears to be an
effective toughening mechanism.
3. For the lithium disilicate glass/glass-ceramic system,
the stress corrosion susceptibility (N) is more dependent on aging
in water than on the percent of crystallinity. The kinetics of
aqueous corrosion of the residual glassy phase and any residual
stresses present play important roles in determining the fatigue
behavior of all materials tested.
147

148
4. For slow crack growth experiments, the interactions of the
crack front with the microstructure greatly influence the determina
tion of the V-K relationship. A method such as the constant moment
double cantilever beam technique, wherein the crack growth is visually
monitored, is ideal for monitoring this interaction. Marking the
locations of features of interest, combined with subsequent fracto-
graphic analysis, provides a powerful tool in studying crack/
microstructure interactions.
5. The N values obtained in crack growth studies more closely
correspond to the lowest values obtained in dynamic fatigue. This
is likely due to the kinetics of aqueous corrosion governing the
growth of macroscopic cracks. This finding is in contrast with less
reactive systems which have yielded higher values of N for crack
growth studies, which were attributed to the macroscopic stress
state controlling the fracture.
6. In dynamic fatigue, the reliability of the~N value obtained
is a function of the magnitude of N itself. This is an inherent
feature of the calculations used in its determination.
7. It is suggested that the method of maximum likelihood may be
an improved technique for treating the statistical nature of the
problem. Any improvement in the estimation of the fatigue parameters
will lead to more reliable lifetime predictions.
B. Suggestions For Future Work
For materials exposed to mechanical stresses in environments where
no change in the flaws or surrounding material occurs, the state of the
art empirical theory appears to be adequate. (Doremus (1980) believes

149
the "agreement" between the theory and experiment is merely fortuitous.)
The next logical step in developing the theory is to incorporate devices
to accommodate a dynamic system. Brown (1978) has developed a kinetic
theory of crack growth, but it is extremely complex and cumbersome.
What appears to be needed is a complete analysis of the phenomena
occurring at and around the flaw. The flaw, be it controlled damage
(machining damage or indentation) or inherent weakness (pore or
"Griffith microcrack"), must first be known or well characterized.
The environmental changes of the bulk material which contain the flaw
are also critical and need to be incorporated within a dynamic theory.
Because failure is caused by the growth of flaws, characterization
of the type of flaw responsible for the failure is paramount. Although
depth and sharpness are probably the most important parameters, other
characterization features may include shape, residual stresses about
the flaw, and compositional gradients (such as H+) ahead of the flaw.
Because of the complexity of characterizing .flaws,, perhaps the
best method of determining the nature of the flaw is to carefully
and deliberately create it in a controlled manner. Mould and
Southwick (1959) used various grades of emery cloth to create flaws
in glass. Mendiratta and Petrovic (1978) have used the indentation
technique to create known flaws in silicon nitride. For both methods,
the post-damage treatment is very important. Aging in various
environments (water for glass or oxidizing atmospheres at high
temperature for silicon nitride) will have profound effects on the
surface damage but has not been considered in the above studies.
Post-fracture analysis of the specimen is also useful for identifying
the nature of the flaws which cause failure. Fractographic techniques

150
(as developed by Mecholsky, Freiman, and Rice 1976) can identify
type and location of the flaw as well as discern any subcritical
crack growth which may have occurred during fracture. However
such techniques cannot readily characterize residual stress-fields
or compositional gradients associated with the flaws ab-initio.
As this study has shown, the interactions of the material and
the environment are important in the fatigue process. Many different
types of corrosion experiments have been invented, but the ideal
technique is the one which best models the real-life application
of the material or component. For instance, Frakes et al. (1974) tested
proposed bio-ceramic material (porous alumina) by subcutaneous implan
tation in the backs of rabbits and observed significant reductions in
strength due to the environmental exposure. Unfortunately, modeling
real-life applications is usually not easy and most experiments will
involve some compromise. In the Frakes et al. (1974) experiment
the implants were unloaded, which probably would have made the
environmental attack even more severe. Part of the need for an
improved dynamic fatigue model is to handle situations such as these.
In order to determine a more complete picture of the fatigue
process, the following test scheme is suggested. Lifetime prediction
diagrams should be obtained from both static and dynamic fatigue
experiments, followed by a proof test. (The proof test may be static
or dynamic.) The survivors should be tested to failure statically,
which will provide a failure distribution that can be compared with
those predicted by the earlier static and dynamic fatigue tests.

151
If the static and dynamic fatigue tests predict different distributions
(because of different N and B parameters), the final failure
distribution will determine the appropriate ("correct") method which
applies.
Theoretically, all N values for static fatigue, dynamic fatigue,
and the final failure experiment should be equal. Depending on the
material and environment, they may not be equal, as shown in this
study. The above methodology coupled with fractographic detailed
analysis of the flaws and the kinetics of the material/environment
interaction should pinpoint the reasons for the differences and
allow appropriate formulations to be devised for a realistic dynamic
model.
0

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8, p. 1527.
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154
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155
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Plenum, New York.

BIOGRAPHICAL SKETCH
The author was born October 21, 1950, in Ft. Knox, Kentucky.
He grew up in various towns in Central Pennsylvania and Western New York,
graduating from Horseheads High School in Horseheads, New York, in 1968.
He earned his B.S. in Glass Science from Alfred University in 1972.
After graduation, he went to work for Metro Container Corp., in Jersey
City, New Jersey, as a Quality Control Engineer. While there, the author
developed an interest in the fracture phenomena in glass and ceramic
materials, and so moved on to graduate school at the University of Florida
in 1976.
In 1975, he married Ellen Goldberg in the Bronx, New York.
Prior to completion of this dissertation, the author began employment
as a Senior Chemist at Rockwell Hanford Operations in Ri.chland, Washington.
156

I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Professor of Materials Science
and Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Professor of Materials Science
and Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
7
R. W. prd'uld
Professor of Materials Science
and Engineering

I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
L. E. Malvern
Professor of Engineering Sciences
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as
partial fulfillment of the requirements for the degee of Doctor of
Philosophy.
June, 1981
Dean, College of Engineering
Dean for Graduate Studies and
Research

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122
K (xl0~6 Pa-m^)
In K (Pa-m^)
Figure 37. V-K relationship for some 33L-low crystallinity
samples


40
a milling machine and a diamond tool. Figure 10 shows the shape and
representative dimensions of the DCB specimens.
D. Characterization
Volume fraction crystallinity was evaluated by quantitative micro
scopy (DeHoff and Rhines, 1968) using a petrographic microscope. A
121 point grid was placed 10 times on each sample measured at 200X to
obtain sufficient data for analysis.
The samples to be evaluated were polished sections, polished using
SiC paper to 600 grit, followed by 6 ym and 1 ym diamond paste. The
microstructure was made visible using an etch in 5 percent HF for one
minute. (See Chapter VI for examples of the microstructures.)


120
K (x10"6 Pa-m^)
0.25 0.50 0.75
In K (Pa-n£)
Figure 36. V-K relationship for 33L-Glass tested in water


77
Table 7
Dynamic Fatigue Data for 33L-92% Crystalline
Tested in Air
Stressing Rate (MPa-s"^)
158
79
16
7.9
1.6
S(MPa)
226
228
226
226
209
s(MPa)
9
11
14
18
18
s/S
4%
5%
6%
8%
9%
n
n
n
9
13
10
sm
m
225
227
226
219
205
m '
24.3
21.1
14.7
11.7
11.2
S = average strength
. s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


93
Table 13
Dynamic Fatigue Data for 33L-7% Crystalline
Tested in Water
Stressing Rate (MPa-s~~*)
158
79
16
7.9
1.6
5(MPa)
159
148
156
131
155
s(MPa)
32
24
24
21
22
s/S
20%
17%
15%
16%
14%
n
9
8
8
9
10
sm(MPa)
152
150
160
124
162
m
4.6
5.3
5.8
6.1
6.4
S = average strength
s = standard deviation
s/S = coefficient of variation
n = sample size
Sm = median strength
m = Wei bull modulus


Figure 29. Dynamic fatigue results for 33L-7% crystalline
tested after aging one week in water


148
4. For slow crack growth experiments, the interactions of the
crack front with the microstructure greatly influence the determina
tion of the V-K relationship. A method such as the constant moment
double cantilever beam technique, wherein the crack growth is visually
monitored, is ideal for monitoring this interaction. Marking the
locations of features of interest, combined with subsequent fracto-
graphic analysis, provides a powerful tool in studying crack/
microstructure interactions.
5. The N values obtained in crack growth studies more closely
correspond to the lowest values obtained in dynamic fatigue. This
is likely due to the kinetics of aqueous corrosion governing the
growth of macroscopic cracks. This finding is in contrast with less
reactive systems which have yielded higher values of N for crack
growth studies, which were attributed to the macroscopic stress
state controlling the fracture.
6. In dynamic fatigue, the reliability of the~N value obtained
is a function of the magnitude of N itself. This is an inherent
feature of the calculations used in its determination.
7. It is suggested that the method of maximum likelihood may be
an improved technique for treating the statistical nature of the
problem. Any improvement in the estimation of the fatigue parameters
will lead to more reliable lifetime predictions.
B. Suggestions For Future Work
For materials exposed to mechanical stresses in environments where
no change in the flaws or surrounding material occurs, the state of the
art empirical theory appears to be adequate. (Doremus (1980) believes


Figure 18. Dynamic fatigue results for 33L-glass tested
after aging one day in water


138
values. Ritter et al. (1980) have proposed that small sample sizes
(less than 100) may be responsible for this unreliability. Using a
Monte Carlo technique to generate artificial data sets from a real
set of data, they demonstrate the variability of N with sample size,
the Wei bull modulus, and N itself. The variability increases as
sample size and Weibull modulus decrease and as N increases.
Baldwin and Palmer (1980) employed the method of maximum
likelihood as a comparison to the method of Ritter et al. in order
to evaluate the importance of statistical techniques in interpreting
fatigue data. The Baldwin and Palmer (1980) method utilizes the same
basic equations as Ritter et al. and also assumes a Weibull distribution.
A preliminary comparison of the results for predicting by the method
of maximum likelihood and by the Monte Carlo method of Ritter et al.
is shown in Figure 45. The initial estimates of the fracture and
Weibull parameters are N = 18.4, B = 0.18, m = 8.2, and SQ = 138.0.
The case shown is based on 100 simulations with five samples at each
of two stressing rates (0.01 and 53 MPa-s ). The initial estimates
were taken from Ritter et al. (1980). This shows that while the two
methods usually predict values close to the original estimate, the
Monte Carlo technique is more likely to produce a grossly inaccurate
value of N.
Ritter et al. used the method of least squares to fit their
straight lines (including Weibull analysis). The method of maximum
likelihood is probably a better technique.
Also, Ritter et al. assume that the distribution of N values
obtained from the Monte Carlo simulations is normal. Assuming total
randomness, it is reasonable to expect that the distribution of the


Figure 24. Dynamic fatigue results for 33L-92% crystalline
tested after aging one week in water


Table
Page
16
Liquid Nitrogen Strength of 33L-7%
Crystalline
96
17
Dynamic Fatigue Data for 33L-57% .
Crystalline Tested After Aging
1 Day in Water
105
18
Dynamic Fatigue Data for 33L-57%
Crystalline Tested After Aging
1 Week in Water
106
19
Liquid Nitrogen Strength of
33L-57% Crystalline
107
20
Strength of 33L-Glass and Glass-
Ceramics
109
21
Fracture Parameters and (aD/aa) from
Dynamic Fatigue Data
111
22
33L-Glass Fracture Parameters as
Determined by Slow Crack Growth
118
23
33L-Partially Crystalline Fracture
Parameters as Determined by Slow
Crack Growth
121
24
33L-92% Crystalline Fracture Parameters
as Determined by Slow Crack Growth
129
25
Effect of Surface Finish on Aged
Strength of Sodium Disilicate Glass
135
X


in strength at low Vv (< 20%) with no connection to the mean free
path. Rao (1977), studying lithium disilicate containing 2-5 .ym
spherulites, attributed the strength increase to an increase in fracture
surface energy due to pinning by the crystals. ~
The decrease then increase in strength which occurs on exposure
to water in 33L-7% is analogous to that which occurs in 33L-Glass.
The residual stress merely postpones the effect.
For 33L-57% and 33L-92% crystalline, microcracks are prevalent
throughout the material. It has been shown that microcracking can
be effective in toughening glass-ceramics (Faber and Evans, 1980, for
lithium aluminoslicates). The microcracks provide multiple branches
(which acting by themselves will reduce the energy of a propagating
crack) leading to the spherulites which pin the crack.
As discussed previously, the weakening on aging is due to the
migration of water into the grain boundaries (via the raicrocracks),
thus dissolving the residual glassy phase and destroying the cohesiveness
of the network.
G. The Effect of Crystallization on N
The N values for all test conditions are shown in Table 21. Except
for 33L-Glass, the N value decreases or stays the same as the availability
of water increases. This underscores the importance of the kinetics of
the reaction at the crack tip.
While the glass shows an initial decrease in N from testing in air
to testing in water, the subsequent increase after aging one day is
indicative of another mechanism acting during aqueous corrosion.


50
Table 2 Continued
2 Hour Crystallization Treatment
Sample
% Crystalline
602W
5.08
611W
9.05
807W
9.09
Average, x
901W
7.93
1101W
7.07
mow
6.61
Std. Dev., s
1201W
6.41
1209W
6.07
1307W
5.46
s/x
1405W
5.54
1608W
5.54
1807W
6.61
211OW
7.27
6.79%
1.27%
18.64%


66
175
0.5 *
O 158 MPa-s'1
+ 16 MPa-s"1
O 1.6 MPa-s"1
0.0-
-0.5-
1.0 -
-1.5
-2.0
Si (MPa)
2p0
+
+
-0.90
33L-92% CRYSTALLINE
TESTED IN WATER
i
5.2
5.3
In S. (MPa)
5.4
-0.75
0.50
-0.25
*0.10
Figure 19. Weibull plot for some 33L-92% crystalline material


49
TABLE 2
Results of Quantitative Microscopy for
Determining Per Cent Crystallinity
24 Hour Crystallization Treatment
Sample
% Crystalline
308X
92.34
412X
97.14
507X
80.77
Average, x
91.93%
605X
87.51
804X
93.68
1006X
94.04
Std., Dev., s
4.65%
1102X
93.96
131 OX
96.87
1402X
88.44
s/x
5.06%
1907X
95.69
2303X
90.11
1508X
92.64
4 Hour Crystallization Treatment
203Z
55.08
402Z
' 56.28
Average, x
57.14%
409Z
52.31
603Z
60.37
704Z
59.13
Std., Dev'., s
3.95%
709Z
56.57
802Z
60.29
907Z
57.02
s/x
6.91%
1003Z
63.80
1309Z
50.58


3
In order for lifetime prediction theories to be of general use,
they must be equally applicable to polycrystalline ceramics, glasses,
and glass-ceramics. The accuracy of predicted lifetimes or their
validity should be independent of the wide range of microstructure
>
encountered in technical ceramics. Thus,a second major objective of
this thesis is to investigate the effect of microstructure on the
fracture behavior and time to failure of a glass-ceramic possessing
a wide range of crystal!inity.
Of particular concern in the thesis is the possible interaction
between microstructural and environmental effects in fracture behavior.
Many technical ceramics are multiphased and the environmental suscepti
bility of interphase boundaries is the potential "weak link" of their
long term performance. A unique feature of this study is the simulta
neous examination of both major variables, microstructure and environ
ment, in the same composition of material.
Glass-ceramics are ideal materials for studying the effects of
microstructure on mechanical properties. By varying the heat treatment,
fully glassy or fully crystalline materials may be fabricated, as well
as partially crystalline materials. The grain size of the crystalline
phase can be varied as well as the volume fraction of crystal!inity.
In this thesis, the lithium disilicate system was chosen for
study because the nucleation and crystallization kinetics are well-
documented. The mechanical behavior of this system is also known,
although the environmental effects are not yet understood. Also of
initial importance is the ability to produce a fully crystallized
material having the same composition as the glass. Lithium disilicate
glass and glass-ceramics also fulfill the prime requirement of this
thesis they react readily with water compared with other materials


13-
Table 1
Commercial Glass-Ceramics
Designation
Major Phases
Properties
Application
Corning 8603
Li00* Si0o
Photochemically
Fluidics devices
Li ^O-Si02
machineable
Printing plate molds
Corning 9606
2Mg0-2Al203-5Si02
Si 02
Ti 02
Transparent to
microwaves
Erosion and
Thermal Shock
resistance
Radomes
Corning 9615
B-spodumene solid
solution
Low T.E., strong,
thermally and
chemically stable,
easy to clean
Range tops
G.E. Re-X
Li20-2Si02
Sealable to metals
Dielectric
Housings
Bushings
0-1 CerVit
B-quartz S.S.
Low T.E.
Polishability
Telescopic mirrors
Pflaudler
. alkali silicates
Coating of steel
Chemical process equipment
Nucerite
Chemical durability
Impact resistance
Abrasion resistance
PPG Hercuvit
101
B-quartz S.S.
Transparent
High temperature windows
Infrared transparencies


T07
Table 19
Liquid Nitrogen Strength of 33L-57% Crystalline
Aged 1 Day
Aged 1 Week
S(MPa)
210
216
s(MPa)
23
30
s/S
11%
14%
n
10
10
Sm(MPa)
206
223
m
8.7
6.3
S = average
strength
s = standard
deviation
.. *
s/S = coefficeint of variation
n = sample size
Sm = median strength
m = Weibull modulus


134
glass (33N) was performed to demonstrate this effect. (This composition
was chosen because of its rapid reactivity with water (Ethridge, 1977).)
Two sets of 33N discs (prepared as in the 33L dynamic fatigue experi
ments) were tested for strength after receiving different polishing
treatments and after aging in water. One set was polished to a 120
grit finish, the other to a 320 grit. Both sets were tested in air
(no aging), after five minutes in water, and after one hour in water.
The results are shown in Table 25.
The strength of discs containing the deeper flaws (M25 urn for
120 grit) was virtually unaffected by the aging treatment. However,
after one hour there was a 50 percent increase in the strength for the
set of discs containing shallower flaws (M5 urn for 320 grit).
This clearly supports our hypothesis that aging affects the
distribution of flaw sizes and thereby the strength.
Additionally, fracture mechanics can be used to predict one set
of flaw sizes from the other by comparing failure strength. Thus,
a320
a120
using S.|2q = 71 MPa, S^q = 112 MPa, and a^g = 125 ym from Table 25
one obtains ag2Q = 50 ym, which correlates well with the estimated
45 ym.
For crystalline or partially crystalline materials, the (usually)
glassy intergranular phase present is susceptible to the same corrosive
attack. Extremely durable and refractory materials such as alumina
(Ferber and Brown, 1980) and silicon nitride (Trantina, 1978) at high
temperatures are now known to undergo stress corrosion due to attack


44
The dimensions used in our testing are A = 9.5 mm, B = 0.8 mm, and
C = 12.7 mm. A Poisson's ratio of 0.24 was used for all materials
studied (Freiman, 1968). These values reduce Eq. 23 to
S = -1.8689 % (24)
d
Samples were tested in biaxial flexure at five cross head speeds
over three decades (0.2, 0.1, 0.02, 0.01 and 0.002 inches per minute)
in three atmospheres (air, water, and liquid nitrogen). Some samples
were also tested after aging one day or one week in water.
Testing in air was straightforward (22C, 65-70% relative humidity).
Care was taken when testing in water to ensure that the samples tested
at the fastest rate were in water about the same length of time as those
tested at the slowest rate (about two minutes).
In the aging experiments, samples were placed in plastic vials
with sufficient water to achieve a surface area to volume ratio (SA/V)
of 1 cm" ^. Discs were then tested in the same water in which they were
aged.
Samples tested in liquid nitrogen were pre-cooled to avoid thermal
shock failure. The test jig was cooled with liquid nitrogen and filled
with the liquid during testing. Because of the absence of static
fatigue at liquid nitrogen temperatures, testing was done only at one
rate of 0.02 inches per minute. The average strength from the liquid
nitrogen testing was used as the aQ value in Eqs. 17-20.
B. Double Cantilever Beam Testing
Freiman, Mulville, and Mast (1973) give a detailed analysis of the
constant moment double cantilever beam technique. Figure 12 shows the


Figure 7. Universal fatigue curve developed by Mould
and Southwick (1959a)


43
Figure 11. Biaxial flexure test jig


130
Figure 42. Region of slow crack growth in
33L-92% crystalline


I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
L. E. Malvern
Professor of Engineering Sciences
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as
partial fulfillment of the requirements for the degee of Doctor of
Philosophy.
June, 1981
Dean, College of Engineering
Dean for Graduate Studies and
Research


36
which is simply Eq. 18 with a the proof test stress, substituted for
the inert strength, a .
A design diagram for lifetime predictions may now be constructed
by rearranging Eq. 21:
tmin a = BN'2 <22>
Using the crack growth parameters B and N and knowing the desired life
time at an applied stress, a proof test ratio (a /a ) may be determined
P 3
in order to design a suitable proof test. Figure 9 shows the design
diagram for 33L glass tested-in air based on the data shown in Fig. 8.
As an example, if a component is expected to last ten years at
50MPa (horizontal line in diagram), a proof test ratio of 2.81 is needed.
Therefore, samples which had survived a proof test of 140 MPa would be
expected to survive the above conditions.
E. Methods Used For This Study
Both direct and indirect methods are used in determining the crack
growth parameters for the lithium disilicate glass-ceramic system. The
constant moment double cantilever beam method is used to measure the
crack velocity directly. The constant stressing rate method is used for
comparison. Chapters VI and VII will compare the results of the two
methods.


In S (MPa)
5.50-
ht
(MPa-s-1)
Id ]t
d is.
33L 92% Crystalline J^O
In (MPa-s"^)
S (MPa)


30
Wiederhorn (1974) discusses subcritical crack growth and the
methods of obtaining crack velocity data. In addition to the direct
methods, such as described by Evans (1974a), Wiederhorn outlines the
indirect methods of obtaining the same data by constant load (static
fatigue) and constant strain rate (dynamic fatigue) experiments.
It has been shown (Evans, 1974b) that the crack velocity(v) may
be expressed as a power function of the applied stress intensity
factor(K):
V = AKN (13)
where A and N are constants to be determined experimentally. The
constant N is termed the stress corrosion susceptibility and is used
as a measure of a material's resistance to sub-critical crack growth
in corrosive environments.
By defining the crack velocity as
da
V =
dt
(14)
and assuming a relationship between the stress, flaw size, and stress
intensity factor
K = aYa'
(15)
where Y is a constant which depends on crack and loading geometry, a
relationship may be derived giving the time to failure under a constant
load. (Note that Eq. 7 is the same as Eq. 15 with Y = n2.)
An equation defining the time to failure under a constant load may
be derived from Eqs. 13-15. From Eqs. 14-15,
and
da
2K
a2y2
dK
2K
o2Y2
dK
Vdt