crack propogation in bimaterial multilayered periodically microcracking composite media

Crack propogation in bimaterial multilayered periodicallymicrocracking composite media

Sandeep Muju 1

Applied Mechanics, The Ohio State University, Columbus, OH 43210, USA

Abstract

The macroscopically anisotropic homogenization of a multilayered medium implicitly assumes that the spatial wavelength of

material inhomogeneity is smaller than the macroscopic quantity of interest and hence, is a reasonable approximation of the bulkbehavior. However, close to the crack tip, gradients in ®eld quantities are strongly in¯uenced by the local heterogeneity, which theisotropic or anisotropic homogenization fails to capture. In the present work it is shown that, to the ®rst order, the e�ect of moduli

inhomogeneity, residual stresses and inelastic strains on crack tip stress intensity factor are superposable in a multilayered inho-mogeneous medium with smooth interfaces. This method provides an e�cient means to study thermoelastic crack problems incomplex heterogeneous media, alleviating the numerical or analytical di�culties associated with the traditional methods. Theresults show that the material inhomogeneity plays a signi®cant role in e�ecting the crack tip driving force. This method is used

further in an R-curve analysis, [1], of a crack propagating through a similar medium with periodic microcracking in one of the twoconstituent materials. # 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

A mesomechanical model is developed to study thein¯uence of periodic inhomogeneities on the crack-tipdriving force for cracks propagating through multi-layered systems. Within the context of enhancedmechanical integrity of material systems [2], in¯uence offunctionally graded interfaces is incorporated.Asymptotic studies of cracks perpendicular and par-

allel to a bimaterial interface and in a layer between twomaterials have been very widely researched over the pasttwo decades [3±5]. Recently with the advent of thecompositional grading in traditional composites, nano-composites and other multi-material structural systemshas spurred interest in the fracture mechanics of suchinhomogeneous materials [2,6±9]. The problems con-sidered typically assume a functional form for thematerial property variation, usually in one directiononly, and calculate the driving force at the crack tip fora crack at some orientation to the direction of variation.The interest in FGM's is due to their superior perfor-mance as interfacial layers in general and in particular

in high temperature applications [2]. Discontinuities inmaterial properties between joined material constituentsact as stress concentrators, especially due to thermalexpansion mismatches under thermal processing or ser-vice loads. FGM as an interfacial layer smoothes thediscontinuity and reduces the stress concentrations.Based on the eigenfunction expansion approach due

to Williams [6,10] it can be shown that the Beltrami-Mitchel equation for inhomogeneous media is the sumof biharmonic homogeneous ODE, corresponding tothe case of a crack in homogeneous media, and otherhigher order terms which are functions of the inhomo-geneous medium properties. Further, it is seen that themost singular part is the homogeneous biharmonicODE and hence reveals that for the smoothly inhomo-geneous medium the order of the stress singularity atthe crack tip remains square-root. However, the stressintensity factor will depend on the particular functionalform of the smooth inhomogeneity.The present work studies the case of bimaterial mul-

tilayered media with functionally graded interfaces.Therefore, the order of the singularity will remain assquare-root. In order to study the e�ect of layeredinhomogeneity on the Stress Intensity Factor(s), theBueckner-Rice weight function approach for homo-geneous media [11±13] is incorporated here to study the

0266-3538/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

PI I : S0266-3538(00 )00016-6

Composites Science and Technology 60 (2000) 2213±2221

www.elsevier.com/locate/compscitech

1 Now at Allied Signal Aerospace, M/S 301-227, 111 S. 34th Street,

Phoenix, AZ 85034, USA.

multilayered media problem. Based on three dimen-sional weight functions, Rice [13] and Gao [14,15] havestudied the e�ect of inelastic transformations in ahomogeneous medium.Further, Gao [15] included the e�ect of moduli inho-

mogeneity through a moduli purterbation approach.In this work the moduli perturbation approach is

further extended to the case of multilayered media,especially in a functionally gradient material sense. Alsothe e�ect of a periodically varying residual stress ®eldarising from a coe�cient of thermal expansion mis-match between the constituent layers is included. It isshown that to the ®rst order, the e�ect of moduli inho-mogeneity, residual stresses and inelastic strains oncrack-tip stress intensity factor are superposable. Thismethod allows one to study thermoelastic crack pro-blems in complex heterogeneous media, alleviating thenumerical or analytical di�culties associated with thetraditional methods.

2. Crack-tip stress intensity factor

The interaction e�ect of an internal stress source on amacrocrack, especially for 3D crack problems can bestudied with relative ease using the weight functionstechnique. The internal stress source may be understoodin the sense of the Eshelby e�ective inclusion transfor-mation [11±16]. As graphically shown in Fig. 1, Eshel-by's e�ective inclusion transformation states that aninclusion in a matrix material can be represented interms of an equivalent set of body forces in the matrixmaterial along the surface of the inclusion boundary,where the body forces are obtained by satisfying equili-brium and compatibility along the interface. The theoryas advanced by Bueckner [12], Rice [11,13] and Gao[14,15] has been quite successfully used for varioushomogeneous media crack problems, [17,18].

The three dimensional weight function vectors aredenoted by hm � h� x; z0� �, � � I; II; III� �. The compo-nent h�i x; z0� �, i � 1; 2; 3� � is equal to the Mode � StressIntensity factor K�

ÿ �at a point z0 on the 3D crack edge

due to a unit point load at x in the i direction. There-fore, for a body bounded by a surface @, the stressintensity factors may be written as,

K��z0� ��

h��x; z0��f�x�d��@

h��x; z0��t�x�dA �1�

where the surface tractions t(x) have been treated interms of a Dirac layer (in®nitesimally thin layer with themathematical properties similar to a dirac delta func-tion) of body forces along the surface, and f(x) is thebody force vector.Weight functions for many 2D and 3D homogeneous

media crack con®gurations with both unbounded and®nite geometries have been obtained by various analy-tical and numerical methods. For a known force ®eldand the corresponding weight functions, Eq. (1) is aconvenient avenue to determine the stress intensity fac-tors. By treating the inhomogeneous medium as a per-turbation from a homogeneous one [15], a very similar3D weight function equation can be obtained for a het-erogeneous medium. Further, inelastic strains arisingfrom thermal property mismatch and/or transformationstrains can be conveniently introduced. Though theweight function function theory is applicable for all thethree fracture modes, in this work only Mode-I pro-blems are considered.

2.1. Moduli perturbation

Consider the spatially variable elastic sti�ness tensorC~ ijkl�x� of the inhomogeneous medium be a perturba-tion from an arbitrary homogeneous media with sti�-ness Cijkl, i.e.

C~ ijkl�x� � Cijkl ��Cijkl�x� �2�

Henceforth all quantities with the tilde `�' re¯ect theinhomogeneity. As the equilibrium equations are mate-rial independent, the equilibrium and stress-tractionrelations are,

�~ ij;j � fi � 0 in �3a�

�~ ijnj � ti on @ �3b�

The strain-displacement relation for the elasticallyheterogeneous medium may be written as

"~ij � 1

2u~ i;j � u~j;iÿ � �4�Fig. 1. Graphical representation of Eshelby's e�ective inclusion

method.

2214 S. Muju /Composites Science and Technology 60 (2000) 2213±2221

Consider now that an inelastic strain ®eld also existsin addition to the inhomogeneity, such that the elasticstrains may be written as

"~eij � "~ij ÿ "�ij �5�

where, "�ij is the inelastic transformation strain ®eld.Further consider a pre-existing residual stress ®eld aris-ing from, e.g. mismatch in the coe�cient of thermalexpansion coe�cients and elevated temperature manu-facturing processes. Therefore, discounting any interac-tion between the residual stress ®eld and transformationstrains, the elastic constitutive relation for the hetero-geneous medium may be written as

�~ ij � C~ ijkl "~ij ÿ "�ij� �

� �Rij �x� �6�

where �Rij �x�, is the spatially varying residual stress ®eld.Substituting Eqs. (4) and (6) into Eq. (3) gives,

C~ ijmnu~m;n

� �;j�fi � ��Rij �;j ÿ C~ ijmn"

�mn

� �;j� 0 �7a�

njC~ ijmnu~m;n ÿ njC~ ijmn"�mn � �Rij nj � ti �7b�

Upon further simpli®cation Eqs (7) may be written inthe form,

In :

Cijmnu~m;nj�

fi � �Rij � �Cijmnu~m;nÿ �

;jÿ C~ ijmn"

�mn

� �;j

� �� 0

�8a�

On @ :

Cijmnu~m;nnj �ti ÿ�Cijmnu~m;nnj � C~ ijmn"

�mnnj ÿ �Rij nj

n o �8b�

Recognizing the terms in curly brackets as an e�ectivebody force f eff and e�ective tractions t eff , respectively,the form of equilibrium and stress-traction Eq. (8) issimilar to the ones for homogeneous media. Thus in theEshelby e�ective inclusion transformation sense, theheterogeneous media problem is similar to the homo-geneous medium problem with the body forces andtractions over the transformation region appropriatelymodi®ed,

In :

f effi � fi � �Rij � �Cijmnu~m;n

ÿ �;jÿ C~ ijmn"

�mn

� �;j

�9a�

On @ :

t effi � ti ÿ�Cijmnu~m;nnj � C~ ijmn"

�mnnj ÿ �Rij nj

�9b�

As is apparent from Eq. (9), apart from the knownresidual stress and transformation strains, the force ®eldfor the heterogeneous body depends on the actual dis-placement ®eld ù~' weighted by the inhomogeneity.Since u~ is an unknown ®eld quantity the problembecomes non-linear. However, in Eq. (9), u~ may beexpanded in Taylor series about the homogeneousmedium solution ù', and considering only the ®rst orderterm, ù' linearises the system, allowing for a closedform solution.Therefore, the modi®ed Bueckner-Rice weight func-

tion theory for heterogeneous media may be written inthe form

K~ � ��

h�j�x�f effj �x�dv�

�@

h�j�x�t effj �x�ds �10�

Thus, provided the weight functions for homo-geneous, cracked bodies (®nite or in®nite) are known,the near tip e�ects of residual stresses, transformationstrains and heterogeneity can be studied simply byappropriately modifying the e�ective body forces andtractions.Incorporating Eqs. (9) into Eq. (10) and application

of the divergence theorem to the resulting form, leads tothe stress intensity factor Ktip

� for the heterogeneousmedia crack problem with transformation strains andresidual stresses,

Ktip� � Ko

� � K�C� � K"

�

� � K�R

� �11a�

where, the e�ect of Inhomogeneity is represented as,

K�C� � ÿ

�

h�j;i�Cijmnum;ndv �11b�

the e�ect of transformation strain ®eld is represented as,

K"�

� ��

h�j;iC~ ijmn"�mndv �11c�

the e�ect of residual stress ®eld is represented as,

K�R

� � ÿ�

h�j;i�Rij dv �11d�

and

Ko� � Mode � Stress intensity factor in homogeneous

medium, without any transformation strains and resi-dual stresses.

S. Muju /Composites Science and Technology 60 (2000) 2213±2221 2215

Thus provided the integral(s) in Eq. 11 are integrable,i.e. the integrand singularity is weaker than �ÿ3, where �is the distance from the crack tip, these equations pro-vide the ®rst order e�ect of an inelastic strain ®eld,residual stress ®eld and moduli inhomogeneity on thecrack driving force.The rather signi®cant feature as evident in Eq. (11a) is

that to the ®rst order the stress intensity factor at thecrack tip is superposition of the stress intensity factorsindividually due to the moduli inhomogeneity, trans-formation strain ®eld, residual stress ®eld and thehomogeneous media solution.

2.2. Weight functions

Bueckner [12] obtained the full set of three dimen-sional weight functions h�j j � 1; 2; 3� � for semi-in®nitecracks in homogeneous media, for all the three modes,� � I; II; III. These weight functions for semi-in®nite3D cracks may be expressed in terms of a Papkovitch-Neuber potential [15]

G�x; y; zÿ z0� � ÿ 1

4�1ÿ ��3=21

�log

q� �qÿ ��

�12�

where

q ��2�

pcos ��=2�

� ��x2 � y2

p; � � tanÿ1�y=x�

� ��x� i�zÿ z0�

pFor Mode I, the weight functions for 3-D problems

may be obtained as [16,18]

hIx � ÿ�1ÿ 2��G;x ÿ yG;xy �13a�

hIy � 2�1ÿ ��G;y ÿ yG;yy �13b�

hIz � ÿ�1ÿ 2��G;z ÿ yG;zy �13c�

For the 2-D case the weight functions, Eq. (13) maybe reduced to,

h�Ix � 1

4�1ÿ �� 2��p

ÿ2�1ÿ 2�� cos��=2� � sin�� sin�3�=2�� 14a�

h�Iy � 1

4�1ÿ �� 2��p

4�1ÿ 2�� cos��=2� ÿ sin�� cos�3�=2�� 14b�

h�Iz � 0 �14c�

In the integrals (11c) and (11d) the only singularitycomes from the weight functions, and is equal to �ÿ5=2.In the ®rst integral, the integrand singularity is proble-matic, i.e. equal to �ÿ3, leading to a logarithmic singu-larity. Since the integrand singularity must be weakerthan �ÿ3, which necessitates that the moduli perturba-tion, �Cijmn � ��, where � > 0, i.e. �Cijmn ! 0 in thelimit of � ! 0. Hence in case of the moduli inhomo-geneity integral, the moduli perturbation is implicitlyconsidered to be about the crack-tip value of the mod-uli. Which in e�ect implies that the modi®cation to thee�ective body force ®eld and tractions from the inho-mogeneity approach zero at the crack tip. The other twointegrals are well de®ned everywhere.Consider an isotropic elastic inhomogeneous medium

with the reference homogeneous sti�ness tensor (modulivalue at the crack tip),

Cijmn � 2�1

2�im�jn � �in�jmÿ ��

1ÿ 2��ij�mn

� ��15�

As is apparent from Eq. (11) the quantities of interestare the quantities derived from the weight functions, i.e.h�j;i�Cijkl, h�j;iCijkl and h�j;i, which may be expressed inthe form

2�U�mn � Cijmnh�j;i �16a�

2��U�mn � �Cijmnh�j;i �16b�

where,

U�mn � �h�m;n � h�n;m�=2� ��=�1ÿ 2��mnh�k;k �17a�

�U�mn � ��h�m;n � h�n;m�=2� ��~=�1ÿ 2�~��mnh�k;k��~ =��

�17b�

3. Crack in an inhomogeneous medium

The particular problem of interest is the Mode I crackin an inhomogeneous medium, with a residual stress


®eld. Using the 2D weight functions, Eq. (11) may beexpressed as,

K�CI � ÿ

��ÿ�

�10

�UIij�� ij�d�d� �18a�

where �� ij � 2�"ij

K"�

I � 2�

�Axy

UIij"�ijdxdy� 2�

�Axy

�UIij"�ijdxdy �18b�

The integrand in Eq. (18a) satis®es the integrabilitycondition as r ! 0, i.e.,

lim�!0

�Cijkl � 0 �19�

In the far ®eld limit the integrand is proportional to1=r, leading to a logarithmic singularity for � ! 1,unless C1ijkl � Cijkl. However, the general solution forEq. (18a), [15,18], results into an expression of the form,Fig. 2,

KtipI =K

1I �

��=�o

p 1

ÿ��ÿ�

�R��0

�~ ��; �� ÿ ��

II��

d�d�

ÿ �o ÿ ��

��ÿ�

JI��R0�� sin�� R�� cos��

R�� d�

!�20�

where

II � 1

64�1ÿ ��11 cos�� 8 cos�2�� ÿ 3 cos�3��

ÿ 16��cos�� cos�2��

JI � 1

32�1ÿ �� 5� 4 cos�� ÿ cos�2�� ÿ 8��1� cos��

and R�� de®nes the radial boundary of the inhomoge-neous region surrounding the crack tip.The ®rst part of Eq. (18b) for dilatational inelastic

strain agrees with the results of McMeeking et al. [17]and Rice [13],

K"�

I ��1� ��

3�1ÿ �� 2�p

�Axy

�ÿ3=2 cos�3�=2��T�x; y�dxdy �21�

The second integral in Eq. (18b) incorporates theinteraction of the inelastic strain ®eld and the moduliinhomogeneity. Eqs. (11d) and (20), incorporating the

e�ect of residual stresses and moduli inhomogeneity,respectively, are of direct relevance to the present studyof layered media, and, hence are pursued further.Results of Eq. (20), incorporating the e�ect of an

arbitrary size R�� heterogeneous region surroundingthe crack tip, when applied to the problem of a Mode Icrack tip in a circular inclusion are found to be in verygood agreement with the exact result in the modulirange, 0:54�o=�41:5, [15,19,20].Further, Eq. (20) reveals an interesting result. For a

radially inhomogeneous circular inclusion, i.e. R�� Rand �~ ��; �� ~ ��, apart from the moduli values at thetip (�~ �� 0� � �) and the value at the boundary(�~ �� R� � �o), the near tip stress intensity factor isinvariant with respect to the exact moduli variationbetween r � 0 and r � R, as long as it is continuous.This result is consistent with the continuum damageanalyses [21], which were arrived at by enforcing thecomplementary energy to be of order two in stress andthe J integral to be path independent.

4. Crack in a bimaterial multilayered media

For the case of a crack in a bimaterial multilayeredmedia the moduli variation may be represented either interms of a square wave or a modi®ed square wavewithout jump discontinuities, Fig. 3. In cases where the

Fig. 2. Crack in an inhomogenous medium.

Fig. 3. Pictorial representation of bimaterial multilayered media

moduli variation.


material interfaces may be atomistically sharp a squarewave representation is more appropriate, though due todi�usional transport and the manufacturing processes,sharp interfaces are typically not realizable in physicalenvironments. In the case of functionally graded mate-rials, the interfaces are designed as thin zones of con-tinuous moduli variation [2,9], Fig. 4. The interfaces inthis study are taken in the same context of ®nite thick-ness functionally graded interfaces. The heterogeneousmedia moduli variation may be represented as a Fourierseries of spatial coordinates. The rectangular wave formof the moduli inhomogeneity may be expressed in aFourier series (Appendix A) as, Fig. 5,

�~ �x� � �o � �AXn�1n�1

�n sin�n!x� �22a�

where the Fourier sine coe�cients are,

�n � 1

�2n2l�

1� �ÿ1�n�1ÿ �sin�2�n�

l� �22b�

and ! � 2�=l,

l =bimaterial wavelength�o =the mean (average) value of the shear moduli�A =the amplitude of the shear moduli variation� =interface thickness

As the crack advances through this layered medium,consider the crack tip to be at location x. Eq. (20) with� de®ned by Eq. (22) may be simpli®ed to [19],

K�CI �x�Ko

I

� 1� C3

Xn�1n�1

�1� �ÿ1�n�1�n2

sin�2�n�

l�

� 2 sin�n!x� ÿ cos�n!x��

where ! � 2�=l and C3 � �A

�o

�3ÿ 4��8�2�1ÿ ��

l�

Here, K�CI �x=l� is understood as the stress intensity

factor for the crack in an inhomogeneous media, withthe tip at location x and l being the material wave-length. Ko

I is understood to be the apparent stressIntensity factor when the medium is considered to behomogeneous with shear modulus equal to �o. Fig. 6shows the e�ect of the moduli inhomogeneity onKtip

I �x=l�, for the interfacial zone thickness �=l � 0:1and for single term and eight term Fourier expansions

Fig. 5. Fourier representation of the periodic moduli inhomogeneity.

Fig. 4. Crack normal to bimaterial multilayered media with graded

interfaces.


of Eq. (23). The single term expansion produces apurely sinusoidal e�ect on near tip stress intensity fac-tor, on the other hand the eight term expansion revealsthat the peak shielding (anti-shielding) occurs close tothe end of the lower (higher) modulus phase. This issimilar to the shielding e�ect when a crack in softermedium is perpendicular to the interface with a hardermaterial [6]. Note that zero corresponds to zero shield-

ing and positive and negative values correspond to anti-shielding and shielding respectively. Note, the near tipstress intensity factor is not in phase with the modulusvariation, Fig. 6. It su�ers a phase shift of about 26�

with respect to the modulus variation. Eq. (23) revealsthat the amplitude of modulus variation �A=�o comesout in terms of a linear multiplicative constant. Further,for very small interfacial zones, i.e. l=� ! 1, the

Fig. 6. E�ect of moduli inhomogeneity on crack tip stress intensity factor.

Fig. 7. E�ect of sinusoidally varying moduli inhomogeneity on crack tip stress intensity factor.


interfacial zone size term cancels out with the sineterm, and hence for very ®ne interfaces the near tipstress intensity factor is una�ected. For small values ofl=� the dependence is more complicated, given by a

sin�=l� ��=l� � type form. However, this solution is not valid

for the limit case of interface thickness �=l � 0.Fig. 7 shows the e�ect of sinusoidal inhomogeneity

(one term expansion) on near tip stress intensity factoras a function of the extent of inhomogeneity. Poisson'sratio is considered to be homogeneous, therefore,�A=�o � EA=Eo. Comparing curves 1 and 3 where theinhomogeneity increases from 10 to 50% of the averagemodulus, the peak shielding (or anti- shielding) is seento increase almost 5 times. Clearly near tip moduliinhomogeneity creates a strong dependence of the cracktip driving force on the location of the crack tip relativeto the inhomogeneity, an aspect generally not capturedin homogenized models of layered materials.Fig. 8 shows the e�ect of the modi®ed step moduli

inhomogeneity, as present in layered bimaterial media.Comparing the curves in Fig. 8 with correspondingcurves in Fig. 7 it can be seen that though the peaks inshielding and anti-shielding regimes are within 5% forthe same level of inhomogeneity, the shape of the curvesis substantially di�erent. Instead of the peak shielding(or anti-shielding) close to the center of the layers forsinusoidal inhomogeneity, the peaks in case of themodi®ed step inhomogeneity occur close to the end oflower (or higher) modulus layers.

5. Conclusions

A moduli perturbation based analytical solution ispresented for the study of e�ect of the moduli inhomo-geneity, residual stress and inelastic strains, on a crackpropagating through a bimaterial multilayered media.It is shown that to the ®rst order, the e�ects of moduliinhomogeneity, residual stresses and inelastic strain®eld on the crack tip driving force are superposable.This is a rather signi®cant result as it greatly simpli®esthe procedure obtaining detailed solutions for thermo-elastic fracture problems in inhomogeneous media withgraded interfaces. Further, these e�ects must beaccounted for in the mesomechanical zone near themacrocrack tip for an accurate description of processzone e�ects in inhomogeneous media [19]. This e�ect ofperiodic moduli inhomogeneity is combined with theresults for interaction of microcracks with macrocracksin predicting the R-curve response of cracks propagat-ing through multilayered bimaterial media with pre-ferential microcracking in one of the constituents (Ref.[1]).

Appendix A

Fourier series expansion of moduli inhomogeneityThe shear modulus of the multilayered media with

®nite thickness interfaces and smoothly varying proper-ties, Fig. 3, may be written as,

Fig. 8. E�ect of multilayered moduli variation on crack tip stress intensity factor.


�~ �x� � �o

�

�A x

�04x4�

�A �4x4l2ÿ�

�A ÿ �A�x�ÿ l2�� 1� l

2ÿ�4x4

l2��

ÿ�A l2��4x4lÿ�

ÿ�A � �A�x�ÿ l

�� 1� lÿ�4xl

8>>>>>>>>>>>><>>>>>>>>>>>>:�A1�

This periodic moduli inhomogeneity may be repre-sented via a Fourier series, i.e.,

�~ �x� � �o � �AX1n�1

bn sin�n!x� �A2�

where ! � 2�=l and, the Fourier coe�cients are de®nedas,

bn � 2

l

�l0

�~ �x� sin�n!x�dx �A3�

Substituting Eq. (A1) into Eq. (A3), the Fouriercoe�cients may be written as,

bn � 1

�2n2l�

1� �ÿ1�n�1ÿ �sin�2�n�

l�; n

� 1; 2; 3; . . . �A4�

Thus the periodically inhomogeneous representationof the shear moduli for the multilayered media is, Fig. 5,

�~ �x� � �o � �A

�2l�

X1n�1

1

n21� �ÿ1�n�1ÿ �

sin�2�n�

l�

� sin�n!x� �A5�

References

[1] Muju S, Anderson PM, Mendelsohn DA. Microcrack toughen-

ing in two-phase multilayered media. Acta Materialia 1998;46:

5385±5397.

[2] Suresh S. Modeling and design of multilayered and graded

materials. Progress in Materials Science 1997;42:243±51.

[3] Cook TS, Erdogan F. Stresses in bonded materials with a crack

perpendicular to the interface. International Journal of Engi-

neering Science 1972;10:677±97.

[4] Zak AR, Williams ML. Crack point stress singularities at a bi-

material interface. Journal of Applied Mechanics, March 1963.

[5] Bogy DB. On the plane elastostatics problem of a loaded crack

terminating at a material interface. Journal of Applied Mechan-

ics, December 1971.

[6] Delale F, Erdogan F. The crack problem for a nonhomogeneous

plane. Journal of Applied Mechanics 1983;50:609±14.

[7] Konda N, Erdogan F. The mixed mode crack problem in a non-

homogeneous elastic medium. Engineering Fracture Mechanics

1994;47(4):533±45.

[8] Noda N, Jin Z. Thermal stress intensity factors for a crack in a

strip of a functionally gradient material. International Journal of

Solids and Structures 1993;30(8):1039±56.

[9] Koizumi M. In: Holt JB, Koizumi M, Hirai T, Munir ZA, edi-

tors. The Concept of FGM, Functionally gradient materials. The

American Ceramic Society, 1993. pp. 3±10.

[10] Eischen JW. Fracture of nonhomogeneous materials. Interna-

tional Journal of Fracture 1987;34:3±22.

[11] Rice JR. In: Wei RP, Ganglo� RP, editors. Weight function the-

ory for three-dimensional elastic crack analysis. Fracture

Mechanics: Perspectives and Directions, ASTM STP 1020, 1939.

pp. 29-57.

[12] Bueckner HF. Weight functions and fundamental ®elds for the

penny-shaped and the half-plane crack in three-space. Interna-

tional Journal of Solids and Structures 1987;23(1):57±93.

[13] Rice JR. Three dimensional elastic crack tip interaction with

transformation strains and dislocations. International Journal of

Solids and Structures 1985;1(7):781±91.

[14] Gao H. Application of 3-D weight functions-I. Formulations

of crack interactions with transformation strains and disloca-

tions. Journal of Mechanics and Physics of Solids ;37(2)

1989:133±53.

[15] Gao H. Fracture analysis of nonhomogeneous materials via a

moduli perturbation approach. International Journal of Solids

and Structures 1991;27(13):1663±82.

[16] Eshelby JD. Energy relations and the energy momentum tensor

in continuum mechanics. In: Kanninen MF. Inelastic Behavior of

Solids, 1970. pp. 77-115, 1970.

[17] McMeeking RM, Evans AG. Mechanics of transformation

toughening in brittle materials. Journal of American Ceramic

Society, 1982; 65(5).

[18] Hutchinson JW. Crack tip shielding by microcracking in brittle

solids. Acta Metallurgica 1987;35(7):1605±19.

[19] Muju S. Continuum and micromechanics based damage model-

ing in layered media: e�ect on fracture toughness. Ph.D. thesis,

Ohio State University, 1996.

[20] Steif PS. A semi-in®nite crack penetrating a circular inclusion.

Journal of Applied Mechanics 1987;54:87±92.

[21] Muju S, Anderson PM, Mendelsohn DA. Shielding due to

aligned microcracks in anisotropic media. Mechanics of Materi-

als 1996;22:203±17.


crack propogation in bimaterial multilayered periodically microcracking composite media

Documents