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UNIVERSITY OF CALIFORNIA Santa Barbara

Finite Element Simulations of Crack Propagation in Laminar Ceramic Composites

A Dissertation submitted in partial satisfaction o f the requirements for the degree Doctor o f Philosophy in Materials Science

by Kais Hbaieb

Committee in charge: Professor Fred F. Lange, Co-Chair Professor Robert M. McMeeking, Co-Chair Professor Glenn E. Beltz Professor Anthony G. Evans Professor Frank W. Zok

September 2002


UMI Number 3064724

UMI’ UMI Microform 3064724 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17. United States Code.

ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor. Ml 48106-1346


The dissertation of Kais Hbaieb is approved.

Glenn E. Beltz A M M

C

D

a

I ('(

*

Anthony G. Evans

Frank W jZok

J

Frederick F. Lange, Co-Chair

P i C P I JZ jg_ ( Robert M. McMeeking, Co-Chair

May 2002


Finite Element Simulations of Crack Bifurcation in laminar ceramic composite

Copyright © 2002 by KaisHbaieb

iii


ACKNOWLEDGEMENTS

In the name o f Allah, the M erciful, the Compassionate. All Praise is due to Allah who guided us to the righteous path, and indeed we wouldn’t have attained this state o f guidance without Him Allah, Praise is due to Allah who forgives all sins and misdeeds except associating others with Him. Praise is due to Allah who gave me success to finish this thesis. May Allah bless our Master Muhammad and his family and Companions and give him peace. I would like to thank my mother and father, Naziha and Rashid Hbaieb for their patience, support and encouragement I would like to acknowledge both my advisors Prof. Lange and Prof. McMeeking for the guidance and support they provided to me in my research. I would also like to thank the rest of my thesis committee members, Prof. Glenn Beltz, Prof. Anthony Evans, Prof. Carlos Levi and Prof. Frank Zok. I also appreciate the useful discussions with my colleagues Masa Rao and Micheal Pontin concerning my research.


VITA OF KAIS HBAIEB May 2002 EDUCATION

Bachelor o f Arts, German Diploma (Dipl.-Ing.) in Mechanical Engineering, Universitaet Karlsruhe, Germany, October 1997. Doctor o f Philosophy in Materials science. University of California, Santa Barbara, September 2002.

PROFESSIONAL EMPLOYMENT

Spring 2000: Teaching Assistant, Department of Materials science, University o f California, Santa Barbara. Spring 1999: Teaching Assistant, Department of Materials science, University o f California, Santa Barbara. 1995-1996: Student Internship, ABB Kraftwerke AG, Mannheim, Germany

PUBLICATIONS

“Threshold Strength Predictions for Laminar Ceramics with Cracks that grow Straight,” K. Hbaieb, R. M. McMeeking, s u b m i t t e d to Mechanics o f Materials.


‘'Ceramics that Exhibit a Threshold Strength,” F. F. Lange, M.P. Rao, K. Hbaieb, and R-M. McMeeking, Proceeding o f the PAC Rim IV Ceramic Armor M aterials by Design Symposium, November 4-8,2001, Maui, Hawaii, Ceramic Transactions of the American Ceramic Society, Columbus, O R

“The Workability of Plastic, Saturated Alumina Powder Compacts,” K. Hbaieb, G. V. Franks, F.F. Lange and R. McMeeking, Proceedings o f the 7* International Conference on Ceramic Processing Science, Inuyama City, Japan, May 15-18,2000 (S. Hirano, G. Messing and N. Claussen, eds.).

“Optimal Threshold Strength o f Laminar Ceramics,” R. M. McMeeking, K. Hbaieb, Zeitschrififur Metallkunde, 90 (12), 1031-1036,1999.

AWARDS

Awarded DAAD Fellowship, 1990.

Scholarship, Undergraduate, Six years in a row, Universitfit Karlsruhe, 1991 - 1997.

Awarded Engineer - In - Training certificate, 2002.

vi


HELDS OF STUDY

Major Field: Fracture mechanics using finite element method.

Studies of mechanical properties, particle packing and (fa lsification of alumina/clay green-body mixtures obtained through processing o f alumina/clay slurry mixtures via colloidal routes with Professor Fred. F. Lange.

Studies of stress intensity factor, energy release rate, threshold strength and T-strcss of laminar ceramic materials for both straight-through and bifurcated cracks using finite element simulations with Prof. Robert M. McMeeking.

vii


ABSTRACT

Finite Element Simulations o f Crack Bifurcation in Ceramic lam inar Composites

by

Kais Hbaieb

Laminar ceramic materials composed of alternate layers o f two different ceramics, in which residual stresses are generated, exhibit a threshold strength. Strength limiting cracks are trapped by the compressive layers and require a minimum (threshold) applied stress to cause them to fail the laminate ceramic.

These cracks are observed to propagate straight through the compressive layers in some laminates. Other laminates, however, undergo crack bifurcation; that is, the crack after penetrating some small distance into the compressive layers branches into two symmetrical cracks. Each branch makes an angle of around 60 degrees with the original crack path. A theoretical analysis is developed to optimize the threshold strength. In this analysis the compressive and tensile layers were assumed to have the same elastic properties. The best result is shown to be associated with the toughest material and the highest residual stress. For each material system, the threshold strength is further optimized by making the layers as thin as possible. Given a laminar

viii


ceramic system with a specific compressive layer thickness, the threshold strength is optimized by selecting a ratio of tensile to compressive layer thickness. Furthermore, finite element calculations were carried out to study the influence o f the elastic modulus mismatch between the alternate tensile and compressive layers. Results were obtained for a variety o f combinations of different ceramics and suggest that threshold strength is further optimized by making the tensile layer material as stiff as possible and the compressive layer material as compliant as possible. Finite element analysis is also carried out to explore the cause of crack bifurcation. In this analysis calculations of energy release rate for both straight crack and cracks bifurcated at 60 degrees are performed. If the crack is considered to propagate through an infinite body, the finite element model predicts bifurcation for only one material combination, whereas in experiments bifurcation is observed in three material combinations. When the effect o f thermal stress induced edge cracks is incorporated to the model, it is shown that the simulation results are in good agreement with the experimental observations. bifurcation is predicted in all three materials combinations.

k


Crack

TABLE OF CONTENTS

1.0

Introduction................................................................................................... 1 1.1 References........................................................................................ 10 12 Figures................................................................................................14

2.0

Experimental Background....................................................................... ..16 2.1

Processing, characterization, and testing o f Alumina/ mullite laminates............................................................................................16

22

2.1.1

Laminate preparation............................................................ 16

2.1.2

laminate fabrication..............................................................17

2.1.3

Specimen preparation........................................................... 19

2.1.4

Mechanical testing............................................................... 20

Establishing the concept of using thin compressive layers to create a threshold strength.............................................................................22

23

2.2.1

Fracture mechanics analysis................................................ 23

222

Experimental study.............................................................. 26

Factors influencing the threshold strength and exploration o f the effect of bifurcation...................................................................................29 2.3.1

Effect

of

the

residual

compressive

stress,

oc 30

2.3.2

Effect of compressive layer thickness, / | ............................. 32

2.3.3

Effect of tensile layer thickness,

....................................... 34

x


2.3.4 Crack paths in bifurcatinglaminates.....................................34 2.4 References.......................................................................................36 2.5 Tables and figures............................................................................38 3.0

Optimization of Threshold Strength Based on Theoretical Analysis -47 3.1

Introduction...................................................................................... 47

32

Threshold strength............................................................................ 49

3.3

Optimal threshold strength................................................................52

3.4

Threshold strength for high toughness laminar ceram ics.................. .57

3.5

Threshold strength for laminar ceramics with layers of equal thickness ..........................................................................................................59

3.6

Optimal threshold strength forlaminarceramics with minimal layer thickness.......................................................................................... 60

4.0

3.7

Discussion.........................................................................................64

3.8

Conclusion........................................................................................ 67

3.9

References.........................................................................................68

3.10

Figures...............................................................................................69

Threshold Strength Predictions for lam inar Ceramics with Cracks that Grow Straight.....................................................................................72 4.1

Introduction........................................................................................72

42

Overview...........................................................................................73

43

Model description.............................................................................75

xi


5.0

4.4

Simulation results..................................................... ........................77

4.5

Discussion..........................................................................................84

4.6

Conclusion........................................................................................ 92

4.7

References......................................................................................... 92

4.8

Figures............................................................................................... 94

C rack Bifurcation in Lam inar Ceramic Com posites............................I l l 5.1

Introduction.....................................................................................111

5.2

Model description............................................................................ 113

5.3

Energy release rate calculations........................................................115 5.1.1

Simulation of a center crack propagating in an infinite body ............................................................................................ 117

5.1.2

6.0

Simulation of a crack propagating under the edge crack

122

52

Discussion........................................................................................132

5.3

Conclusion...................................................................................... 138

5.4

References....................................................................................... 139

5.5

Appendix........................................................................................ 142

5.6

Tables and figures........................................................................... 144

C onclusion................................................................................................164

Appendix 1. T-stress approach.............................................................................166 A l.l Introduction.....................................................................................166 A1.2 T-stress definition............................................................................167 A1.3 Calculation methods........................................................................ 168

xn


A1.4

J-Integral based m ethod................................................................. 170

A1.5

T-stress calculations........................................................................180

A1.6

References...................................................................................... 183

A1.7

Figures............................................................................................ 186

Appendix 2. Shear stress calculations on the crack faces and the implication for Jintegral evaluation...................................................................................... 192 A2.1

Shear stress calculation..................................................................192

A2.2

Evaluation o f the J-integral with consideration o f the shear tractions

on the crack faces........................................................................................195 A2.3

Figures.......................................................................................... .200

xm


1.0 Introduction

The need of light and high-strength materials was always the concern of design engineers. Despite the diversity of materials available for the different industrial applications, the increasingly technological complexity inquires search for new materials providing the best and optimal characteristics for the meant objective. Although ceramics show very poor toughness and inherent high brittleness they are still very attractive owing to their distinguished stability and strength at high temperatures. However, the challenge of ceramists remains mostly correlated to the poor reliability of such materials and the wide range of strength distributions due to the presence of a variety of dispersed flaws in the material. The strength of ceramics obeys a statistical description (e.g. Weibull) involving a wide distribution of values [1.1], meaning that some components are quite weak and therefore unreliable. The reason for the statistical distribution of strength is the existence of a variety of cracks and crack-like flaws unintentionally introduced during processing or post-processing (such as surface machining) [1.1, 12]. Unlike ductile materials such as metals, ceramics materials lack significant plastic deformation and hence exhibit low resistance to crack propagation. Thus,

1


the strength of brittle ceramics correlates directly with the presence o f flaws and decreases with increasing size of the flaw.

The reliability of the ceramic could be improved by controlling the size of flaws introduced into the material during processing. This can be achieved if a slurry of the designated material is dispersed and then passed through a filter [1.2]. Depending on the fineness of the filter, only heterogeneities with sizes smaller than a critical value can pass through. Thus, threshold strength (and hence a guaranteed reliability) can be determined by the size of the filter, i.e. by defining the largest flaw that can be present in the material.

Many other approaches were under excessive investigations. Toughening of ceramics was alternatively addressed by incorporating second phase particles in the brittle ceramics matrix [13]. These particles would support traction across the crack faces and combined with stress-induced microcracking the toughness is augmented. In another approach, system of laminar composite of Ce-ZrOz and either A120 3 or a mixture of A120 3 and Ce-Zr02 was shown to help enhancing

2


fracture toughness [1.4]. The alumina layers served as barrier layers for extending transformation zones ahead of a crack tip present in the zirconia layers. Just before the transformation zone reaches these layer barriers it extends towards the sides of the crack promoting crack tip shielding and hence toughening [1.4].

The concept of increasing resistance to crack propagation in ceramics materials through introduction of residual compression was first used to mitigate surface flaws introduced during machining [1.5-1.9], The presence of near-surface compressive stress, increasing in magnitude away from the surface, has a stabilizing effect on the propagation of surface cracks. As a consequence, the crack growth resistance is improved. In addition, the surface flaw sensitivity and the strength variability are reduced.

In our approach the latter concept is of great concern. Instead at the surface, Rao. et aL have considered introduction of compressive residual stress in the bulk of the material [1.10, 1.11]. This is possible through development of biaxial residual stresses that can be achieved by applying a mismatch strain

3


between two sets of different alternate layers. The mismatch strain can be induced due to a mismatch of thermal expansion coefficient between the different layers or when one set of layers undergo volume increase due to a chemical reaction or a crystallographic phase transformation.

For the experiments carried out by Rao et al., two sets of alternate layers of different materials with different properties were fused together at high temperature [1.11]. Upon cooling one set of layers has a tensile residual stress while the other has a compressive stress, due to thermal expansion differences. This is schematically illustrated in Fig. 1.1, where compressive layers of thickness t„ having a residual stress are chosen so that

for

each crack length is fixed. The crack is then considered to grow through die material starting with a length 2a just greater than r, so that the crack tip is just inside a compressive layer. The total crack tip stress intensity factor, K, is the sum of the applied load stress intensity factor £ * » and die residual stress intensity factor

. However, for the crack to grow, the total stress intensity

78


factor, K must be equal to the fracture toughness, IC , o f the compressive layer. This allows us to calculate the applied stress intensity factor needed to sustain crack growth as:

=

(4.1)

Defining a parameter S = S (a/ti,ti/t* E il E>) as

S=

tr .

(4.2)

we can compute the applied stress

necessary to sustain crack growth by

combining equation (4.1) and (4.2) to give

tXm rtfi *

------- .----

(6)

Clearly S is given as a function o f a l t i for the case o f u l u = 1, Et/E* = 1.7 by the plot o f K**~/(/£> ranges from 1 to 10. The threshold strength increases with increasing toughness and also increases with increasing elastic modulus ratio E J Ex.

Fig. 4.7 also shows the results for the case where the layer

thicknesses are identical; however, the elastic modulus ratio £>/£■ is now ranging from 1/10 to 1/2. Similarly to Fig. 4.6, the threshold strength increases with increasing toughness. However, the increase o f the threshold strength with elastic modulus ratio EJEx \s only present for £ * /£ .bigger than 1/3. For smaller E J Ex, this trend is only valid for smaller toughnesses. For higher toughnesses and E J Ex smaller than 1/3, the threshold strength increases with decreasing E J Ex. In these latter situations, crack instability takes place in the compressive layer well before the crack tip reaches the interface with the tensile layer. As Ei I Ex decreases in this situation, the starting point o f crack instability moves back to smaller penetration distances within the compressive layer. The case where the tensile layer thickness is one and a half times larger than the

82


compressive layer thickness (Le. ti/t, = 1.5) is presented in Fig. 4.8 for £ 2/ £1 £ 1. The trend for the threshold strength is similar to die results when the thickesses o f the layers are identical (i.e. Fig. 4.6) with hardly any effect o f increasing die thickness ratio to 1.5. For the thickness ratio u l u = 1.5 and elastic modulus ratio E J £ 1ranging froml/10 to 1/2, the results for the threshold strengths are shown in Fig. 4.9. Here the trends are similar to those in Fig. 4.7 except that die differences in magnitudes of the threshold strengths are negligible for high compressive layer toughnesses due to varying the elastic modulus ratio E J £, when it is smaller than 1/3. Fig 4.10 depicts the case where the tensile layer is twice as thick as the compressive layer and E il Ei £ 1.

The trends for the threshold strength are

almost identical to those in Figs. 4.6 and 4.8. Results for elastic modulus ratio E il Ex in the range from 1/10 to 1/2 are shown in Fig 4.11 for t J u = 2 . In this figure the threshold strength results are similar to those in F ig 4.9, except that when the compressive layer toughness is large, there is a noticeable difference in the results for £ i / £ . = 1/3 and £>/£■ = 1 / 4 .

For £>/£■£ 1/4 in this high

toughness regime, the differences in the threshold strengths are negligible. Fig 4.12 shows the threshold strength results for the case where h l t x - 2.5 and Ei!E\ >1. Compared with Figs. 4.6, 4.8, and 4.10, F ig 4.12 shows almost identical trends in the threshold strength. Fig 4.13 shows the results also for h it, = 2.5 but for elastic modulus ratios £ j / £ , ranging between 1/10 and 1/2. The results are similar to those in Figs. 4.7, 4.9 and 4.11 except that the

83


monotonic increase of die threshold strength with increasing elastic modulus ratio EilEi is present in all cases for £>/£• as low as 1/6. For £ 2/ £ £ 1/ 7 , the i

differences among the threshold strengths are negligible when die compressive layer toughness is high.

4.5

Discussion

The simulation results show a good agreement with the theoretical model results when the tensile and compressive layers have the same elastic properties. As mentioned above, the theoretical model ceases to be exact for the more realistic case with heterogeneous elastic properties. However, the finite element method can be used to treat the heterogeneous case. We judge the finite element predictions of threshold strength to be reliable because they agree almost exactly with the theoretical model results when we used the finite element method to calculate results for the homogenous case.

Effect of parameter variation

Toughness Kc

84


As shown in Figs. 4.6-4.13 depicting the threshold strength versus toughness, the finite element simulation results indicate that die threshold strength increases with increasing toughness o f the compressive layer. This is expected since a tougher material resists crack propagation and failure more effectively. This feature is true for all cases that we consider.

Residual strain e

The effect of the residual strain, £, on the threshold strength varies for different elastic modulus ratios E il E \. For £>/£. = 1/10, with increasing residual strain, e, the threshold strength decreases to a minimum level and then increases back again. This trend is also observed when

£ 2 / £1

has value up to 1/4.

The minimum level o f the threshold strength moves, however, to lower values of e as E i!E \ is increased to 1/4. In contrast for Ei/Ei > 1 / 4 , the minimum for threshold strength disappears in the range investigated and there is a monotonic increase of threshold strength with increasing e. Such a trend is illustrated in Fig. 4.14, for £ 2 / £ i = 1, 3 and 10 when KdEvfm 71 = 0.00038, /i/r. = 1, so that all parameters except e are held fixed. As shown, the higher the elastic modulus ratio £ 2 / £ i the steeper is the monotonic increase of the threshold strength with increasing residual strain. Similar plots can be constructed for the other values o f the ratio EJE> within the range o f 1/4 to 10 and for other values of toughness

85


and layer thickness ratio tilt, . Subject to detail differences, this trend is reproduced at different toughness levels and for different values o f t i l t , .

Tensile layer modulus Ei

The results also show that the threshold strength depends on the elastic modulus ratio, E J E , .

This is most easily understood in relation to our

presentation of the results if one considers the elastic modulus E, o f the compressive layer to be fixed and the elastic modulus Ei o f the tensile layer to vary, so changing the modulus ratio. In our Figs. 4.6-4.13, the threshold strength and the toughness are normalized by terms containing E,, so that by fixing this modulus we have an unvarying normalization of these terms.

Similarly, we

consider e = AaAT and the layer thicknesses to be fixed while we bring about the change in E i. We can then deduce, from Figs 4.6-4.13, that with everything else held fixed, an increase o f Ei will cause an increase o f the threshold strength. This is understood easily when it is realized that with everything else held fixed an increase o f Ei will have two effects. One effect is apparent from equation (1.1) that the magnitude o f the residual stress in the compressive layer rises thereby reducing the total stress intensity factor at a crack tip in the compressive layer. Therefore, crack growth will require a higher applied stress. The other effect o f increasing Ei with everything else held fixed is that a higher fraction o f

86


the applied load is carried by the tensile layers and a Iowa1 fraction by the compressive layers simply because the tensile layers are stiffer. This means that at a given applied load, increase o f £* with everything else held fixed will cause the applied stress intensity factor to be lowered for cracks that have penetrated well into the compressive layer. This occurs because the reduction of stress near the crack tip in the compressive layer has a greater effect on the stress intensity factor than the increase of stress further away. As a consequence, there is an increase o f the threshold strength with the increase o f modulus ratio £ i / £ i that is monotonic for every case as long as £>/£> is higher than 1/3. When the elastic modulus ratio is reduced below 1/3 we notice sometimes a contrary increase in threshold strength with diminishing £ i / £ i , that is especially obvious for t J t , = 1, when there is high toughness o f the compressive layer. This phenomenon is associated with the situation where the crack propagation becomes unstable while the tip is still well within the compressive layer and therefore the features just discussed do not have a very strong influence.

Compressive layer modulus £■

The effect on the threshold strength o f varying £•, die elastic modulus o f the compressive layer, is also studied. Typical trends o f threshold strength due to

87


changing £. are illustrated in Fig. 4.1S for /.//, = 1 (Additional plots have been constructed for the other thickness ratios; however, no difference from Fig. 4.1S is apparen t) Note that die lines in Fig. 4.1S indicate how the threshold strength varies when all parameters are held fixed except E i. It can be seen that when all other parameters are held fixed, the threshold strength increases slightly with increasing £. for values o f Kt /(-eE’rJxtJ2) below 0.03.

For values of

fC /(-sE'rJitt,Tl) above 0.03, the threshold strength decreases significandy as £■ is increased with all other parameters held fixed. This latter trend would seem to be the conjugate to increasing the modulus £> as in the previous paragraph and is presumably explicable by a similar argument

Tensile layer thickness />

When the elastic modulus ratio is in die range from 1/10 to 1/7, there is a slight increase of the threshold strength as /> decreases, for values o f normalized toughness as low as about 0.1. For values of normalized toughness higher than 0.1, a noticeable decrease o f threshold strength with increasing /> is observed. If ti and other parameters are held fixed when £>/£. is in the range from around 1/6 to 2 and the thickness ti of the tensile layer is varied, there is a slight increase o f the threshold strength with increasing h for low values o f K*/(-fifiWxfi/2) and a more significant increase with decreasing /> for high values of

88


K,/(-£E'nJjB>/2).

However, when Ei/E> is in the range from around 3 to 10,

the threshold strength declines slightly for all values o f K,

as /> is

increased while t, and all other parameters are held fixed. Fig. 4.16 illustrates the latter case, with Ei/E, = 7 showing how the threshold strength varies when t> is adjusted but all other parameters are held fixed. As shown, there is barely noticeable decrease in the threshold strength as ti is increased.

Compressive layer thickness t
while all other parameters are fixed. For example, Fig. 4.17 illustrates the effect on the threshold strength o f varying the compressive layer thickness, t,, with all other parameters held fixed for the case where £ i/ £• = 7 . As the value of K./i-eE'rJnti/l) is made larger, the decrease o f the threshold strength, in Fig. 4.17, with increasing t, is more apparent For £>/£•£ 1/4, however, the threshold strength increases with increasing compressive layer thickness for high values o f £ / ( - f i E W « i / 2 ) , whereas the opposite occurs in a limited fashion for low values.

Relevance ofResults

89


Note that in the experiments [4.1, 4.6] the crack is, in many instances, observed to propagate straight through the compressive layer. However, in other cases a phenomenon called bifurcation takes place.

In the latter cases,

immediately after penetrating the compressive layer, the crack divides into two symmetrically located branches propagating at an angle o f around 60° from the original crack path. It was shown [4.1, 4.6] that this phenomenon retards failure and causes an increase in the threshold strength compared to predictions based on the straight propagation o f the crack. Finite element calculations conducted to explore the influence of bifurcation on the threshold strength are presented in chapter S. However, based on the experimental observations, we can assert that the threshold strength will be higher when the cracks bifurcate. Therefore, we believe that the results presented in this chapter can provide guidance on the design o f laminar systems to achieve a high threshold strength. Therefore, we ask the question of what material attributes are desirable in a layered material to produce a high threshold strength. In this regard, the conclusion are little different from those presented in chapter 3 but now with the advantage that we can address the influence of the elastic modulus o f the materials composing the layers. First, the materials should be chosen to possess as high a toughness as can be achieved for ceramics. In addition, the residual strain mismatch between the layers should be as high as feasible. Also, the layers in the system should be made as thin as

90


possible. We also see little advantage in making the compressive layers any thinner or thicker than each other unless the ceramics are very tough, so both types o f layers should be made as thin as possible. The final piece o f the puzzle is drawn from our new results. The tensile layer should be made as stiff as possible (i.e. a high elastic modulus) whereas the compressive layers should be as compliant as feasible (i.e. a low elastic modulus). With this set o f choices and with realistic material values and layer thicknesses, it should be possible to achieve threshold strengths as high as three times the effective residual stress (see Fig. 4.6.) Based on sensible material values, we hazard the prediction that threshold strengths over lGPa are achievable. As an example, we consider the case o f a compressive layer o f vitreous silica sandwiched between layers o f alumina. Vitreous silica has a surface energy o f about 5.2 J/m2 at 25°C [4.7]. Its elastic properties are 73 GPa and 0.165 for the Young’s modulus and Poissson’s ratio, respectively [4.7].

We find an average of the coefficient of thermal

expansion of 0.5*1 O'6 K*‘ for a range o f temperature between 25°C and 1000°C [4.7]. The material properties o f alumina are reported in table 2.1 in chapter 2. Assuming the fracture surface for vitreous silica to be equal to twice the surface energy, the toughness o f the compressive layer is calculated to be Kc = Q.62MPayfm . The thermal (residual) strain is 0.00936. The parameter Kc/(-eE[)^jal /2 is then determined to be equal to 0.13. Using Fig. 4.6, for

91


example, for tensile and compressive layers o f equal thicknesses, we determine a value for the threshold strength o f 1-25(-££,) which is equivalent to 0.88 GPa.

4.6

Conclusion

The finite element method was used to predict threshold strengths o f laminar ceramic materials with straight cracks.

The threshold strength is

demonstrated to exist independently o f the special conditions and combinations o f material parameters employed in the experiments.

The threshold strength

depends on the properties of the layer materials including the elastic stiffness and the thermal expansion coefficient and on the thickness o f the layers. The results suggest that threshold strengths in excess of lGPa are feasible.

4.7

References

[4.1] M. P. Rao, A. J. Sanchez-Herencia, G. E. Beltz, R. M. McMeddng, F. F. Lange, “laminar ceramics that exhibit a threshold strength,” Science, 286, 102 (1999). [4.2] J. R. Rice, “A path independent integral and the approximate analysis o f strain concentration by notches and cracks,” Journal o f Applied Mechanics, 34, 379 (1968).

92


[4.3] T. S. Cook and F. Erdogan, “Stress in bonded materials with a crack perpendicular to the interface.” International Journal o f Engineering Science, 10, 677 (1972). [4.4] ABAQUS Users and Examples Manual. Version 5.8. Hibbitt, Karlsson and Sorensen Inc. Pawtucket, R. L [4.5] B. Moran and C. F. Shih, “A general treatment of crack tip contour integral,” International Journal o f Fracture, 35,295 (1987). [4.6] M. P. Rao and F. F. Lange, “Residual stress induced R-curves in laminar ceramics that exhibit a threshold strength,” Journal o f the American Ceramics Society, 84,2722-24 (2001). [4.7] N. P. Bansal and R. H. Doremus, “Handbook o f glass properties,” Academic Press, 1986.

93


4.8

Figures

Fig. 4.1: Finite element Mesh o f the plane-strain model. The different shades o f gray indicate the ceramic layers. The crack lies along the bottom surface extending from the lower left hand corner. The mesh is refined in the region around the crack tip.

94


Fig. 4.2: Detail o f the finite element mesh showing compressive layers (light shading) and tensile layers (dark shading). The compressive layer on the left contains the crack tip and the crack surface lies on the bottom surface o f the mesh extending from the bottom left comer. The elements in this layer are small for higher accuracy.

95


Fig. 43: Comparison of simulation results with theoretical model results for a homogeneous material. Both tensile and compressive layers have same thickness.

Compressive Layer —^

1—►compressive layer 2-* tensile layer

(simulation)

Siren Intensity Factor

-t

applied applied y 2

(theory )

2

(tkepry) OS

(sim ulation)

•05

AaATE'.J—t-

05

t.5

2

3

15

f

T“

15

4

Crack Length 2atti

96


Fig. 4.4: Simulation results for the case where the elastic modulus in the tensile layer Ei is 1.7 times higher than the elastic modulus in the compressive layer E>. The theoretical model results for homogeneous material is also plotted for comparison.

E i / E i = 1.7, t i / h = 1

1-> compressive layer 2-» tensile layer

C'omprcv>i\e Laycr_^

(th eo ry )

154 Stress Inteasltv Factor

(sim u la tio n )

—t

(sim u la tio n )

■05

0

OS

15

2

25

3

35

4

45

Crack Length 2a lti

97


Fig. 4.5: Plot of the stress needed to sustain crack growth vs. crack length for the case o f ti/t, = \ , £ i/£ i = 1.7 and &/(AaA7£\Vrtf*/2) = 0.123 . The stress is increased monotonically until the crack tip is almost at the interface between the compressive layer and the tensile layer.

Compressive Layer

08

Eil Et - 1.7, tilti = 1 Kc ,-r- TW= 0.123 A a A T E n J x ti/2

1-» compressive layer 2 - * tensile layer

-06 05

1

15

T“ 2

1 r—

-t-

25

3

35

T”

—t

4

45

Crack Length la lti

98


Fig. 4.6: Threshold strength vs. compressive layer toughness for the ratio E J Ei o f the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range from 1 to 10. The thicknesses of the tensile and compressive layers are identical.

t:/t> = I

1-» compressive layer 2-> tensile layer

E i l Ei = 4

Ei/Ei = 2 OS

0

Of

OtS

02

03 9

99


Fig. 4.7: Threshold strength vs. compressive layer toughness for the ratio Eil Ex of the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range 1/10 to 1/2. The thicknesses o f the tensile and compressive layers are identical.

04

1—►compressive layer 2—►tensile layer

0.36

-

Ei! Ex = 1 /2

0.3

■— Ei* E\ * 1 /6

01 5

01 Ezf E\ * 1/10

005

0

006

0.1

015

ft3

02

Ke (-fiT,

/2

100


Fig. 4.8: Threshold strength vs. compressive layer toughness for the ratio Eil E, of the elastic modulus o f the tensile layer to the elastic modulus o f the compressive layer in the range 1 to 10. The thickness o f the tensile layer is 1.5 times the thickness o f the compressive layer.

u

t i l ti —1.5

compressive layer tensile layer

u

u •it

Kc

(-eE \ ) 4 * J 1

101


Fig. 4.9: Threshold strength vs. compressive layer toughness for the ratio Eil Ex o f the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range 1/10 to 1/2. The thickness of the tensile layer is 1.5 times the thickness of the compressive layer.

04 ri/ri = 1.5 1- + compressive layer 2—* tensile layer 09

02 0 T9

E i l £, = 1/8

000 o

01

02

03

04

ao

oo

(~eE\ ) 4 m j l

102


Fig. 4.10: Threshold strength vs. compressive layer toughness for the ratio Ei / Ei o f the elastic modulus o f the tensile layer to the elastic modulus o f the compressive layer in the range 1 to 10. The thickness o f the tensile layer is twice the thickness of the compressive layer.

ti/ti = 2 IS

1-> compressive layer 2-* tensile layer

zs

as

o

at

02

as

as

as

103


Fig. 4.11: Threshold strength vs. compressive layer toughness for the ratio Eil Ex o f the elastic modulus o f the tensile layer to the elastic modulus of the compressive layer in the range 1/10 to 1/2. The thickness o f the tensile layer is twice the thickness o f the compressive layer.

04

ti/tx = 2

029

1-> compressive layer 2-+ tensile layer

029

£ i/£ i* i/e

02

Ot9

Ot E i /£ i = l/10

0

Ot

02

09

04

09

00

104


Fig. 4.12: Threshold strength vs. compressive layer toughness for the ratio E J Ei of the elastic modulus o f the tensile layer to the elastic modulus o f the compressive layer in the range 1 to 10. The thickness o f the tensile layer is 2.5 times the thickness o f the compressive layer.

u l t i = 2.5

E i l Ei = 10

1-» compressive layer 2 - * tensile layer

E i l Ei = 9 E i l Ei —8

-eE '

£ 2/£ . = 7 £ 2/ Ei —6 E i l Ei = 5

£ 2/£ . = 4 £ 2/£ . = 3 E i l Ei = 2 E i l Ei = 1

105


Fig. 4.13: Threshold strength vs. compressive layer toughness for the ratio EilE, o f the elastic modulus o f the tensile layer to the elastic modulus of the compressive layer in the range 1/10 to 1/2. The thickness o f the tensile layer is 2.5 times the thickness o f the compressive layer.

0.4

ti/ ti = 2.5 1—►com pressive layer 2 -> tensile layer

-e E \ 029

at

o

0

at

u

as

as

u

(~eE\

106


Fig. 4.14: Threshold strength versus residual strain f for the case wherek. iEr.Ja,/2 = 0.00038, /»//. = 1 and £ 2/£ , = 1 ,3 and 10.

0.025 i

h/t, = 1 K. / Er.Jxt. / 2 = 0.0003

0 .0 I S

-

0.01

-

0.005 -

0

0.4

0.6

at

1.2

6(H)

107


Fig. 4.15: Threshold strength versus the ratio £■/£> for various values of the compressive layer toughness with tJ t , = 1.

K c / ( - e £ r i H j g , / 2 =0.152

Ok

K.

-eE '

2 = 0 114

K ,l( - e E 'iU * t* l2 =0.076 ,K . l{ r e E i ^ M x l 2 = 0.038 0.6

,K '/( - s E ,i) J i a J 2 = 0.0076

■

OJ •

o

02

at

1.2

Ei/Ez

108


Fig. 4.16: Threshold strength versus the thickness ratio tilt, for the case where Ez / £i = 7 for various values o f the compressive layer toughness.

0*

15 ■

J t/f-£ £ \W » ./2 =0.1

k

=nn« IJ

2 t i l t I

109


Fig. 4.17: Threshold strength versus the thickness ratio f./fi for the case where E i! E\ = 7 for various values o f the compressive layer thickness.

O*

-e E \

K 'K -eE 'W jai/l = 0 5

K .H - e E 'M x til 2 = 0 J

Kc/{-e£r,)^mii2

=

0.1

=0.05

05

0

tl/t2

no


5.0

Crack Bifurcation in laminar Ceramic Composites

5.1

Introduction

We mentioned in chapter 2 that in certain lam inates, containing compressive layers of large thickness and/or large residual compression, crack bifurcation within these layers takes place [5.1-5.3]. Out of four materials investigated in our simulation three of them exhibit this phenomenon and they are designated: M40, M55 and M40, where the designation M40 corresponds to 40% of mullite, the balance being alumina, etc. In these laminates, edge cracks are produced spontaneously upon cooling after processing at the surfaces of the compressive layers and they propagate along the midplane to arrest in the interior of the material. Note that crack bifurcation during loading does not take place at the surface. Inspection of the fracture surface reveals that the propagating flaw somewhat below the surface upon penetrating the compressive layer branches into two symmetrical kinks. The angle that each kink makes with the pre existing crack path is around 60°. As described in chapter 2, experimental

111


observation suggests that crack bifurcation under load occurs under the same conditions that favors the spontaneous creation of edge cracks during cooling. That is, the bifurcation phenomenon takes places in the M40, M55 and M70 materials as does spontaneous edge cracking. In the M25 material neither crack bifurcation nor edge cracking takes place. Experimental observation also shows that the threshold strength when there is a bifurcated crack is higher than predicted by a theoretical analysis that solely considers straight cracks [5.4]. In this chapter we present a finite element model that compares the energy release rates of bifurcated and straight cracks in an attempt to establish the conditions that cause crack bifurcation. We first consider a crack propagating in an infinite body subjected to the loading applied at the tensile surface of a transverse 4-pt flexural specimen. We also address the boundary value effect and explore the relationship between the formation of edge cracks and crack bifurcation. Moreover, we use a J integral based method [5.5] to calculate T-stresses at the tips of straight cracks with differing lengths. We discuss the validity of criteria based on the maximum energy release rate and the T-stress concept in relation with the stability of the crack in the compressive layer.

112


5.2

Model description

Computations are carried out using the finite element code ABAQUS [5.6]. A two dimensional plane strain model is used throughout with 8-noded quadrilateral plane strain elements. Due to symmetry only one quarter of the whole specimen is modeled. Displacement controlled boundary constraints are applied to the bottom and vertical left line of the modeled mesh so that those nodes remain on those lines. The crack is simulated with nodes free of traction that lie along the bottom surface of the model extending from the lower left comer. Fig. 5.1 shows a portion of the mesh simulating the top right quadrant of the specimen used to calculate the energy release rate in the composite material involving a straight through crack. Appropriate refinement of the mesh is performed near the crack tip to ensure accuracy of the calculations. The portion of the mesh shown includes the lower part of both a compressive layer and half of the central tensile layer. The crack opening due to loading is magnified 30 times for illustration. Fig. 5.2 shows a different mesh containing a crack spanning the tensile layer and penetrating straight through the interface with the

113


compressive layer to eventually bifurcate at an angle of 60° when the crack dp is in the compressive layer. Except for the presence of a kink in the compressive layer, the setup adopted for the bifurcated crack is identical to that for the straight crack. The displacements are magnified 30 times for better illustration of crack branching. Such a mesh is used to determine the energy release rate for a crack bifurcating in the compressive layer.

The model is used to simulate the experiment; therefore, the layers dimensions and properties are the same as in the experiment Similar to what is adopted in the experiment the model contains two compressive layers with a thickness of 55 fun and two and a half tensile layers of thickness 550 ftm with the initial crack located in the half tensile layer, as shown in Fig. 5 3 . The properties of the tensile and compressive layer materials are presented in Table 5.1 with different values for the compressive layers depending on its Mullite composition. In the simulation, we ignore any variation in Poisson’s ratios with variation of the layer materials and set its value equal to 0.24 throughout

114


S3

Energy release rale calculations

As described earlier in this thesis, a pre-crack is produced using an indentor and a load of 5 Kg. The pre-crack size is measured to be around 300 /un. Specimens are subjected to 4-point flexural bending. Ignoring the effect of residual stresses due to the indentation, the applied load causes the crack to propagate unstably through the entire tensile layer. As shown in Fig. 4.5 the stress intensity factor rises very high when the crack tip is in the tensile layer in the vicinity of the interface. Immediately after penetrating the compressive layer, the stress intensity factor decreases again. For this reason we assume that the crack grows straight through the interface and any possibility of bifurcation takes place when the crack tip is in the compressive layer. Fig. 5.4 shows optical micrographs of bifurcated cracks beneath the surface in three different materials. As can be seen, Fig. 5.4 provides evidence that bifurcation takes place after the cracks enters the compressive layer. By inspection of at the third micrograph in Fig. 5.4, we can estimate the site of bifurcation to be located at around 10% into

115


the layer. Therefore, we can argue that the crack that has arrested at this location will either continue to propagate straight or bifurcate depending on the conditions governing crack growth behavior. The model that we suggest to investigate the further growth is based on the illustration in Fig. 5.5. As shown, ahead of the crack there are many microcracks along the crack front at random orientation in the material. A further increase in the applied stress will cause the crack to join with one or more of these microcracks. The microcrack that provides the maximum energy release rate at the new tip will be the one that extends the crack thereafter. Crack deflection then arises if the preferred direction is other than straight ahead. The deflected portion of the crack front is assumed to extend laterally along the crack front and leads to bifurcation rather than just crack deflection where two such deflections grow all the way along. Since we observe the crack to always propagate symmetrically at an angle of around 60 degrees with the major crack orientation, we only calculate the energy release rate of both straight and bifurcated crack at 60 degrees at different crack lengths. We assume that the microcracks are of equal sizes. Thus, we compare the energy release rate of a straight and bifurcated crack with a length equivalent

116


to the length of the major crack plus the length of the microcrack where it is either coaxial with the major crack or oriented at 60 degrees.

In this section we compare the energy release rate of a crack propagating straight through the compressive layer with that for a crack branching symmetrically in the compressive layer. The J-integral [5.7], whose magnitude corresponds to the energy release rate value for a linear elastic material, is calculated from the finite element model.

5.3.1

Simulation o f a center crack propagating in an infinite body

In this section we present the calculations of the energy release rate for a center crack propagating through a very large body. That is, the crack size is very small compared with the width of the specimen and the latter is smaller than the specimen length. This allows us to simulate effectively a fracture analysis in an infinite body. In addition, the simulation is reduced to a two-dimensional plane strain problem. A remote tensile load is applied to the model. The magnitude of

117


this load is chosen to be the value of the bending stress on the top surface of the specimen. Using ABAQUS [5.6], thermal stresses are added to the remote tensile load. These thermal stresses correspond to the residual stresses induced by cooling the specimen by 1200 °C from the stress-free temperature. These residual stresses are present in the interior of the material. At the free surface, however, these stresses vanish and a different stress distribution exists. Note that our simulation of an infinite body overlooks this complication. A further assumption is to disregard any effect of the edge crack on the stress distribution around the crack that propagates through the compressive layer perpendicular to the edge crack.

Calculations for both the straight and bifurcated cracks in an elastic homogeneous material with tensile and compressive layers with the same properties are first carried out to check the accuracy and reliability of the simulations. For the straight crack, the simulation results are found to agree well with exact theoretical results for a laminar composite material with layers having the same mechanical properties but with different thermal expansion coefficients.

118


For a crack with symmetric branches in the compressive layer the finite element results are compared with numerical results for the a homogenous material calculated using a dislocation model [5.8].

Fig. 5.6 shows a reasonable

agreement between our simulations and Vitek’s numerical results [5.9] for branches at 15°, 30°, 45°, and 60° to the direction of original propagation. A different finite element calculation for a crack having one single kink at its tip at an angle of 30° is carried out in order to compare our results with those obtained by Rice and Cotterell [5.10]. A different finite element model is generated for this purpose. Since the kink employed in this model is relatively big, an extrapolation of the results back to a vanishing crack length of the kinked crack is necessary. The extrapolated result, not shown here, is in a very good agreement with the result given by Rice and Cotterell [5.10].

After ensuring good accuracy of our simulation results, we now employ our model for the heterogeneous case where the tensile and compressive layers composing the composite material have different properties. Note that in our model, as well as observed in the experiment, the crack propagates straight to

119


depth 10% through the compressive layer. If the crack is to deflect from its original path, the energy release rate along the new path has to be higher than that of the straight crack of equal size. As described above and as shown in Fig. 5.5 the length, b, corresponding to the size of a microcrack ahead of the major crack is added to the original length, a, to produce either a straight or a branched crack. We only consider a symmetrically branched crack on either sides of the major crack. We first determine the applied stress for which the longer straight crack has a stress intensity factor equal to the toughness of the material. The fracture toughness of the material is taken to be 2MPa-Jm for all materials. The same stress is then also applied to the bifurcated crack.

Fig. 5.7 shows the energy release rate results for the materials M25, M40, M55 and M70 versus the crack length, which is the length of the original crack plus the length of the joined microcrack. The crack length is normalized by half the thickness of the tensile layer, t i l l . The energy release rate is normalized by the toughness of the material, Gc. The applied stress is determined such that the energy release rate of the straight crack is equal to the toughness of the material,

120


Gc. This is indicated in Fig. 5.7 by the value of one for the normalized energy release rate of the straight crack. Because this value is normalized, the energy release rate for a straight crack in the four different materials is represented by a single horizontal line. The stress required to sustain crack growth is plotted in Fig. 5.8 for the M70 material. Similar plots may be constructed for the three other materials. The same applied stress is applied to the model simulating the bifurcated crack in the M70 material at the same net crack length a +• b. Comparison of the results corresponding to both straight and bifurcated cracks, as depicted in Fig. 5.7, shows that bifurcation is only possible for the M70 material within the range o f microcrack lengths considered. Bifurcation, however, is observed in practice for all materials except for the M25 materials. Therefore, the model does not predict this experimental observation although it gives the right trend in the sense that the energy release rate for branched cracks in M70 is higher than in M55 which is in turn exceeds that for M40 and so on. As a consequence, we need to account for additional features in the model to better approximate the conditions present in the experiment.

121


5.3.2

Simulation o f a crack propagating under the edge crack

Except for the case where the material is M70, the calculations simulating a crack in a very large body (infinite body) predict that the crack propagates straight through the compressive layer. Given the fact that this result is contrary to the experimental observation, a more complicated calculation involving consideration of edge crack effects and the stress distribution induced by the presence of the edge crack is carried out and presented in this section. Note that this incorporation of the effect of the edge crack in our model concerns only the compressive layer materials, M40, MSS, and M70 where the edge crack is produced. Due to the absence of edge cracking, the simulation of the crack configuration for the M2S material is considered complete by the model in the infinite body, as described in the previous section.

As mentioned in chapter 2 and in the introduction of this chapter, an edge crack in the compressive layer emanating from the surface and propagating deep into the compressive layer is produced due to the triaxial stress state near the

122


surface of the compressive layer [5.11]. Fig. 5.9 illustrates a compressive layer with an edge crack along the mid plane extending from the surface to the interior of the layer. Because the layer is not constrained at the surface, an out-of plane tensile stress,

is present and obeys the following equation:

°yy (*)|y=0 = “ [ 0 '

20] ° c

(5.1)

where oc is the compressive residual stress in the layer and the angle 0 is defined by:

=i

(5.2)

The tensile stress, o„, has a maximum value on the surface at the center of the layer, and diminishes to zero in the interior of the layer away from the surface, where the stress is compressive and biaxial. Because the strain energy associated with the tensile stress is highly localized at the surface, one can show that the

123


condition for crack extension is given by the strain energy release rate function, G, as follows:

(53) E

where t is the thickness and E is the elastic modulus of the layer. Equation (S3) shows that G not only depends on the magnitude of the compressive stress and the elastic properties of the compressive layer, but also, on the layer thickness. For a given compressive stress, a critical layer thickness exists (tj, above which, a crack will extend along the centerline to a depth that is proportional to the layer thickness [5.11]. Although edge cracking is produced upon cooling the specimen from a high temperature and before any mechanical load is applied, it turns out that it is of importance in our study as described below in the next paragraph.

As mentioned earlier, 4-point flexure loading is applied to the specimen parallel to the layers. The pre-crack in the tensile layer is observed to rapidly propagate straight until reaching the compressive layer. No further propagation is

124


observed until the applied load is increased. Once a critical load level is reached, the crack is observed to propagate stably and straight at the surface until it meets the edge crack already existing in the compressive layer. It is only when a large increase in load is employed that the crack propagates further. However, after the specimen is ground down to a certain distance into the compressive layer, edge cracking is once again produced and two branching cracks emanating from a location close to the interface with the tensile layer are observed. It is believed that the crack bifurcates, forming these two branches - well inside the interior of the layer and away from the surface - adjacent to the edge crack. Because this phenomenon takes place adjacent to the edge crack, a connection between the stress distribution in this region due to the presence of the edge crack and the phenomenon of crack propagation is assumed to exist. The stress distribution argument is only valid adjacent to the edge crack tip since the stresses at the surface down to about the length of the edge crack are diluted by the relaxation of the strain energy caused by the formation of the edge crack.

125


To account for the stresses adjacent to the edge crack, a separate finite element calculation is carried out. A new model is generated with the same geometrical configuration and material properties as adopted in the experiment, so that the specimen upon cooling and before being subjected to the mechanical loading is realistically simulated. A crack perpendicular to the surface and extending along the mid plane of the compressive layer is formed as shown in Fig. S. 10. Fig. S. 10 is magnified to illustrate the shape and orientation of the crack. The length of the edge crack is determined by calculating the energy release rate of the edge crack. We assert that the edge crack continues to grow until the energy release rate reaches the toughness of the layer material, G,- The edge crack size is the critical length corresponding to this value Ge for the energy release rate. The calculations of the edge crack sizes yields the following values: 17/un for M40, 29/un for M55 and 46/un for the M70 material. The residual stresses in the layers can be calculated either analytically or using a separate simple finite element model where a negative temperature difference of -1200 °C is applied to the model. However, these thermal stresses are true for an infinite body. At the surface, there are no tractions. To impose this condition in the

126


model with the edge crack, stresses with opposite sign are applied normal to the surfaces of the layers. The finite element calculation is then carried out and the stresses adjacent to the edge crack are extracted from the model. The stresses we need are those in the through thickness direction relative to Fig. S.10 along a horizontal line passing through the crack tip. Across this region are six elements. Therefore, six values corresponding to these six elements are recorded and plotted versus distance from the interface. A plot of these stresses is depicted in Fig. 5.11 for the M70 material. A polynomial curve is fitted to the calculation points and the equivalent equation is resolved. This equation is then used to calculate the stress values elements in the different finite element models shown in Fig. 5.1 and 5.2 according to their distance from the edge crack plane. In the same manner the values of the stresses in the tensile layer due to the boundary condition at the surface are extracted and averaged across the whole tensile layer. From this method, the tractions that are determined have to be applied to the surfaces of the cracks in Fig. 5.1 and 5.2 to represent in an approximate way the effect of the free surface and the edge crack. Relative to Fig. 5.1 and 5.2, the free surface is parallel to the plane represented by these finite element models

127


and is a distance above this plane equal to the length of the edge crack. The edge crack penetrates down from the free surface and its front lies along a vertical line in the middle of the compressive layer in Fig. S.l and 5.2.

Calculations are performed for the two different models simulating the configuration of the straight and bifurcated crack with the stresses due to the thermal strain mismatch, the applied stress load, the effect of the free surface and the edge crack taken into account by superposition. Adding these stresses to these models is perfectly permissible since we are dealing with linear elasticity that allows superposition of stress fields in arbitrary ways. However, a problem arises at this stage since we are adding local stresses distributed across a local region of the model to remote stresses and residual stresses applied to the whole body. To overcome this difficulty we need to consider a further superposition where we translate the remote applied stresses and the residual stresses to local tractions applied solely on the crack surfaces. That is, our model configuration is equivalent to a superposition of two stress field solutions. The first stress Held is composed of a specimen subjected to applied stresses and thermal residual

128


stresses applied to an infinite specimen without a crack and a central crack subjected to exactly the opposite of these stresses as illustrated in Fig. 5.12. The stress field with tractions applied to the crack provides the energy release rate for the crack. The superposition of tractions on the crack surface includes the effect of the applied load and the thermal residual stresses in an infinite body treated in this way, plus the effect of the free surface and the edge crack obtained in the way previously described. Although, the residual compression is constant throughout the whole compressive layer, the bending stress is maximum at the surface and diminishes the deeper in the compressive layer the considered location. The bending applied stress used also depends on length of the crack shown in Fig. 5.1 and 5.2, since for the crack to stably propagate across the compressive layer, an increase in the bending stress is required. In addition, the bending stress is represented by two components, each applied in separate layers. Due to the strain constraint of the composite body, a relationship between the stress components exists and can be giving by: (5-4)

129


where ctL-e-j and (TlbemiiMg are the bending stress components and

and Ey are

the elastic modulii in the tensile and compressive layers, respectively. The bending stress components added to the residual stress added to the stresses due to the formation of the edge crack constitute the complete stress distribution for the model simulating the propagation of the straight crack. For the bifurcated crack a transformation of axis is employed. The resultant traction on the faces of the bifurcated cracks can be split into normal and shear components and therefore a mixed mode of fracture is considered.

Fig. 5.13 shows the energy release rate calculations for the material system alumina/M40 (alumina-mullite mixture with 40% mullite fraction), for both the straight and a bifurcated crack at angle 60°. That is, the crack that has propagated straight through the interface and arrested at a small penetration distance in the compressive layer will join with a finite microcrack oriented either on the same path or 60° off the original propagation path. Similar to the model of the crack in the infinite body, the applied stress is determined to be the stress required for the energy release rate of the straight crack (that has now

130


joined with a branch oriented along its path) to be equal to the toughness of the material,

where the length a + b for the bifurcated crack and the straight crack

are the same. The energy release rate is normalized by G,_ The results for the straight crack therefore should equal 1 in Fig. S.13. However, approximations in the approach made it difficult to find the applied stress precisely to make G = G,Therefore, the results for the straight crack does not have G/Gc exactly equal to 1. Two finite element models are used. A mesh with coarse elements is generated to calculate the energy release rate at the tip of big branches.

A

different mesh with very small and fine elements is used to calculate the energy release rate at the tip of small branches. The value of the energy release rate at an infinitesimally small kink can then be extrapolated from these results. This value is always smaller that for a straight crack with equivalent length a + b. This well known result is due to the fact that the energy release rate for a nominally Mode I crack is always maximum for a straight extension of the crack [5.10]. Fig. 5.13 shows that the energy release rate for the bifurcated crack increases with length of the crack branch and reaches the toughness of the material at a reasonable Irink length. For longer branch lengths the energy

131


release rate for the bifurcated crack surpasses that of the straight crack. This implies that provided a flaw in the material of such a size exists in the material, bifurcation takes place. This result is in accordance with the experimental observation of crack bifurcation in the M40 material. Fig. 5.14 depicts the results of the energy release rate of the straight and bifurcated cracks in the MSS material. These results are similar to those presented in Fig. S. 13 except that now the bifurcated crack exceeds the toughness value of the material at a smaller kink length. For the M70 material, the energy release rate for an infinitesimal kink is smaller than that of a straight crack with same length but increases steeply for longer branch lengths, as shown in Fig. S.IS. For this material the energy release rate of the bifurcated crack with quite small kinks reaches the toughness of the material.

S.4

Discussion

The phenomenon of bifurcation is closely associated with the path of maximal energy release rate. It was confirmed that the crack chooses the

132


direction that releases maximum energy. This is deduced when investigating the results that account for the effect of the edge crack. Given the fact that the driving force for crack growth is the release of energy and that a physical system always tends to lower its total energy, the results obtained are within our expectation. It is only adjacent to the edge crack that the bifurcated crack reaches an energy release rate value that is higher than that of a straight crack. That is, quite similarly to the experimental observation, the model predicts that the crack propagates straight at the surface. In the interior of the material, the crack either bifurcates or propagates straight depending on the existence of the edge crack. The edge crack induces stresses that are higher in the propagation direction of the investigated crack than those in the through thickness direction and perpendicular to i t The latter stresses provide an additional component to the applied bending stress. However, due to higher magnitude of the stresses in the crack propagation direction it seems that these stresses have a greater effect on the original major crack and cause it to bifurcate.

133


Although the concept that the crack is to choose the direction that releases maximum of energy is sufficient to relate fracture mechanics to the experimental observation of bifurcation, it is relevant to calculate the T-stress ahead of the crack tip in the compressive layer. Rice and Cotterell [S. 10] suggested that when a small kink is produced, due to a perturbation, at the tip of a straight crack whose loaction is characterized by a pure mode I condition (i.e. Kg = 0), the possibility of the original straight crack to deviating from its original path depends on the uniform non singular stress term (T stress) acting parallel to the crack, in the Irwin-Williams expansion of the stress function in the crack tip field. That is, the stability of a crack on its path, under nominally mode I loading, depends on the sign of T stress. If the T-stress is positive, the crack deviates from the straight path whereas with a negative T-stress, the crack remains straight The calculation of the T stress parallel to the straight crack in the compressive layer would thus seem to be relevant with respect to the stability or instability of the straight crack and its tendancy or otherwise to form a bifurcated crack.

134


Williams [S.12] demonstrated that the near crack tip stresses in an isotropic homogeneous elastic material are expressed as an infinite power series. The leading term in this expansion equation exhibits a 1/Vr singularity, the second term is uniform and independent of r, the third term is proportional to Vr, and so on [5.13], Classical fracture mechanics uses only the leading singular term of the Williams expansion equation, offering thereby a single-parameter description of the near crack tip stresses and fracture criterion[S.13]. This is generally a reasonable assumption because the third and higher terms in Williams' expression, which have positive exponents on r, vanish quickly as the crack tip is approached [5.13]. However, the second term, known as the T-stress, is finite at the crack tip and does contribute to the near crack tip stresses - at non vanishing small distances from the crack tip - and thus affect the behavior of the crack. For instance, accounting for this contribution alters significantly the plastic zone shape and the stresses deep inside the plastic zone [5.14-5.15].

The method used to calculate the T-stress was first introduced by Kfouri [5.5], where three finite element solutions are considered (See Appendix for

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more details.) First, a point load is applied on the crack tip in the crack propagation direction and is resisted by an external traction applied on the boundary. The corresponding J-integral is calculated. Next, a solely external load is considered and the J-integral is calculated. Finally, the preceding finite element solutions are superposed and the corresponding J-integral is calculated. When the traction resisting the point force is of the same magnitude, Kfouri showed that the T-stress is calculated from the following equation [5.5]:

K F ,f,t)= m *J{fM TflE

(5.5)

where 7(F) is the value of J-integral when the specimen is subjected to an external Load F; alone; J(f, t) is the value of the J-integral of a specimen subjected to a point force/in the crack propagation direction and resisted by an external traction t, and 7 (F ,/,r) is the J-integral when the preceding stress fields (corresponding to F, / and /) are superposed [5.5].

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This same method is applied to our problem, however, the external load is here the sum of all the stresses applied on the crack faces including residual, bending stress components as well as the stress components induced by the presence of the edge crack in the compressive layer.

For simplicity, t was simulated by distributed line tractions on the external lateral side of the model, alone. The total line traction is applied opposite to the crack propagation direction and o f magnitude —, where L is the L length of the simulated specimen. Note that in our modeled quadrant, the point force is applied only to the top half section of the specimen, thus, is only -j.

Before proceeding with our composite laminar system, this method is verified for a homogenous elastic material with a known theoretical solution. The example taken for this purpose is a body of linear elastic material subjected to uni-axial external applied load Ow. The T-stress for this system is equal to —OU.. Our model gives results that agreed within 6% with the exact theoretical results.

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Fig. S. 16 shows the T-stress versus the crack length for a crack propagating straight through a composite laminar ceramic containing alumina tensile layers and compressive layers of alumina/mullite mixture materials at different mullite fractions. M25, M40, M55 and M70. The calculated results show that the T-stress is positive for all cases when the crack tip is in the compressive layer. According to Rice and Cotterell [S. 10] this implies that the crack is unstable on its original path and shall deviate if any perturbation causes a generation of a small kink at its dp. This indicates that bifurcation shall take place for every case including the case where the crack is propagating through the compressive layer material M25. However, it is well established through experimental observation that the crack always propagates straight through the compressive layer material M25. Therefore, the T-stress concept cannot be used to successfully explain the bifurcation phenomenon.

5.5

Conclusion

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The phenomenon of bifurcation is observed to take place in laminar ceramic composites. Finite element calculations are carried out to explore the cause of bifurcation and predict its path. The calculations suggest that the crack only initiates bifurcation adjacent to the edge crack, which exists only for M40, MSS and M70 materials. This result is in a good agreement with the experimental observation. The T-stress concept for crack deflection is not applicable because our calculations show that the conditions for its validity cannot be m et

S.6

References

[5.1]

A. J. Sanchez-Herencia; L. James; F. F. Lange, “Bifurcation in Alumina

Plates Produced by a Phase Transformation in Central, Alumina/Zirconia Thin Layers,” Journal o f the European Ceramic Society, 20 [9] 1297-1300 (2000). [5.2]

A. J. Sanchez-Herencia; C. Pascual; J. He; F. F. Lange, “Zr02/Zr02

Layered Composites for Crack Bifurcation, Journal o f the American Ceramic Society, 82 [6] 1512-1518 (1999).

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[5.3]

M. Oechsner, C. Hillman; F. F. Lange, “Crack Bifurcation in Laminar

Ceramic Composites,” Journal o f the American Ceramics Society, 79 [7] 18341838 (1996). [5.4]

M. P. Rao, “Lamina Ceramics that Exhibit a Threshold Strength,” Ph.D.

Dissertation, University of California, Santa Barbara (2001). [5.5]

A. P. Kfouri, “Some Evaluations of the Elastic T-term using Eshelby’s

Method,” International Journal o f Fracture, 30,301-315 (1986). [5.6]

ABAQUS users and Examples Manual, Version 5.8. Hibbit, Karlsson and

Sorensen Inc. Pawtucket, R. I. [5.7] J. R. Rice, “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks.” Journal o f Applied Mechanics, 35,379-386 (1968). [5.8]

K. Hbaieb, R. M. McMeeking, “Threshold Strength Predictions for

Laminar Ceramics with Cracks that Grow Straight,” submitted to Mechanics o f M aterials (2002). [5.9]

V. Vitek,, “Plane Strain Stress Intensity Factor for branched Cracks.”

International Journal o f Fracture, 13,481-501 (1977).

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[5.10] B. Cotterell and J. R. Rice, “Slightly curved or kinked cracks,” International Journal o f Fracture, 16,155-169 (1980). [5.11] S. Ho, C. Hillman, F. F. Lange and Z. Suo, “Surface Cracking in Layers Under Biaxial, Residual Compressive Stress,” Journal o f the American Ceramics Society, 78,2353-2359 (1995). [5.12] M. L. Williams, “On the Stress Distribution at the Base of Stationary Crack,” Journal o f Applied Mechanics, 24,109-114 (1957). [5.13] T. L. Anderson, Fracture Mechanics. Fundamentals and Applications, 2nd. Ed., Boca Raton, CRC Press, 1995. [5.14] B. A. Bilby, G. E. Cardew, M. R. Goldthorpe and 1. C. Howard, “A Finite Element Investigation of the Effects of Specimen Geometry on the Fields of Stress and Strain at the Tips of Stationary Cracks,” Size Effect in Fracture, Institution of Mechanical Engineers, London, 37-46 (1986). [5.15] C. Betegon, and J. W. Hancock, “Two Parameter Characterization of Elastic-Plastic Crack Tip Fields,” Journal o f Applied Mechanics, 58, 1991,104110(1991).

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Appendix

5.7

In order to calculate the T-stress in a isotropic linear elastic material, Kfouri [16] has proposed a J-integral based method. First he considered a semi finite crack in an infinite body subjected to plane strain conditions. A point force is applied on the crack tip parallel to the crack. External tractions are applied on the boundary (see Fig. 5.17). The solutions for the corresponding stresses in a polar coordinate system are giving by:

Orr = -/C O S 0/(7ir), (Tea = Or* = 0

(1)

When the body is finite the external tractions applied to the boundary and resisting the point force/can assume any distribution provided that their total value, r, is equivalent to -/ That is, the tractions t are giving by:

t» =