2.13 FailureofFunctionallyGradedMaterials

2.13Failure of Functionally GradedMaterialsG.H. PAULINO,Z.-H. JIN, and R.H. DODDS,Jr.University of Illinois, Urbana, USA

2.13.1 INTRODUCTION 608

2.13.1.2 Background 6082.13.1.3 Processing and Manufacturing of FGMs 609

2.13.2 MECHANICS MODELS OF FGMs 610

2.13.2.1 Continuum Model 6102.13.2.2 Micromechanics Models 610

2.13.2.2.1 Self-consistent model 6112.13.2.2.2 Mori–Tanaka model 6112.13.2.2.3 Tamura–Tomota–Ozawa model 6122.13.2.2.4 Hashin–Shtrikman bounds 612

2.13.3 FRACTURE OF FGMs 613

2.13.3.1 Crack-tip Fields 6132.13.3.2 Stress Intensity Factors 613

2.13.3.2.1 Mechanical loads 6132.13.3.2.2 Thermal loads 6152.13.3.2.3 Contact loads 617

2.13.3.3 Crack Growth 6192.13.3.3.1 Crack deflection 6192.13.3.3.2 Fracture toughness and the R-curve 620

2.13.3.4 Dynamic Fracture 6202.13.3.4.1 Crack-tip fields for a propagating crack 6202.13.3.4.2 Dynamic SIFs 622

2.13.3.5 Viscoelastic Fracture 6232.13.3.5.1 Basic equations 6242.13.3.5.2 Correspondence principle, revisited 6242.13.3.5.3 Viscoelastic models 6252.13.3.5.4 A crack in a viscoelastic FGM strip 625

2.13.3.6 Je-integral 627

2.13.4 FINITE ELEMENT MODELING 627

2.13.4.1 Graded Finite Elements 6272.13.4.2 SIF Computation 628

2.13.4.2.1 Displacement correlation technique 6282.13.4.2.2 Modified crack-closure integral method 6292.13.4.2.3 J-integral formulation 6302.13.4.2.4 Examples 631

2.13.4.3 Cohesive Zone Modeling 6322.13.4.3.1 A cohesive zone model 6332.13.4.3.2 Cohesive element 6352.13.4.3.3 Cohesive fracture simulation 636

2.13.5 SOME EXPERIMENTAL ASPECTS 638

2.13.5.1 Metal/Ceramic FGM Fracture Specimen 6382.13.5.2 Precracking of Metal/Ceramic FGMs 6392.13.5.3 Precracking by Reverse Four-point Bending 6392.13.5.4 Test Results 640

607

2.13.6 CONCLUDING REMARKS AND PERSPECTIVES 641

2.13.7 REFERENCES 641

2.13.1 INTRODUCTION

Many future advancements in science andtechnology will rely heavily upon the develop-ment of new material systems that safelywithstand the ever-increasing performancedemands imposed upon them. Functionallygraded materials (FGMs) hold promise forapplications requiring ultrahigh material per-formance such as thermal barrier coatings,bone and dental implants, piezoelectric andthermoelectric devices, optical materials withgraded reflective indices, and high-performancespaceflight structures (including engines). Newapplications are continuously being discovered.

FGMs are characterized by spatially variedmicrostructures created by nonuniform distri-butions of the reinforcement phase with differ-ent properties, sizes and shapes, as well as byinterchanging the role of reinforcement andmatrix (base) materials in a continuous manneras illustrated by Figure 1. This new concept ofengineering the microstructure of the materialmarks the beginning of a paradigm shift in theway we think about materials and structures asit allows one, due to recent advances in materialprocessing, to fully integrate material andstructural design considerations.

2.13.1.2 Background

The initial response to provide materials thataccommodate nonhomogeneous service re-

quirements employed layered structures, e.g.,coatings (Kaczmarek et al., 1984; Suganumaet al., 1984). However, under severe stressgradients, such structures proved susceptible tofailure from spalling triggered by high inter-facial stresses. In contrast, FGMs makepossible stress relaxation under the imposedthermomechanical loads. This is accomplishedby providing a compositional distributionfunction, C(x), consistent with these condi-tions. For instance, such a function can beexpressed as (Hirano et al., 1990)

CðxÞ ¼ ðC2 C1Þx x1

x2 x1

n

ð1Þ

where C1 and C2 denote composition values atopposite ends of the FGM layer correspondingto the distances x1 and x2, respectively. Figure2 illustrates this function. The exponent n inEquation (1) dictates the nature of the desireddistribution function based on stress analyses.For example, in FGMs of TiC/Ni, Ma et al.(1992) report that the optimum value for n liesin the range 0.5–0.7. Thus, to accommodate thedesired distribution function, the method ofFGM manufacturing must have the capabilityto provide the predetermined (desired) expo-nent n (see Equation (1) and Figure 2).

FGMs possess the distinguishing feature ofnonhomogeneity with regard to their thermo-mechanical and strength-related propertiesincluding yield strength, fracture toughness,fatigue, and creep behavior (see, e.g., Erdogan,

Figure 1 Transition in a CrNi/PSZ FGM from CrNi alloy to zirconia (PSZ) (after Ilschner, 1996).

608 Failure of Functionally Graded Materials

1995; Paulino and Jin, 2001a). Numerousinvestigators have developed theoretical andnumerical fracture mechanics models forFGMs (Noda and Jin, 1993; Bao and Wang,1995; Giannakopoulos et al., 1995; Lee andErdogan, 1995; Choi, 1996a, 1996b; Jin andBatra, 1996c; Bao and Cai, 1997; Ozturk andErdogan, 1997; Gu and Asaro, 1997a; Chanet al., 2001). Yang and Shih (1994) havestudied the fracture process along a nonhomo-geneous interlayer between two dissimilarmaterials. Jin and Batra (1996a) have presenteda micromechanics-based model of fracturetoughness. Erdogan (1995) and Markworthet al. (1995) have presented complementaryreviews of modeling techniques applied toFGMs. Noda (1999) has written an extensivereview on thermal stresses in FGMs. Pinderaet al. (1998) have presented a thermomechani-cal analysis of functionally graded thermalbarrier coatings with different microstructuralscales. Jin et al. (2002) presented a cohesivezone model to simulate progressive materialfailure and crack extension in FGMs coupledwith some experimental validation.

This work presents a comprehensive reviewon fracture/failure of FGMs. Several relevantaspects of the above mentioned papers arediscussed in detail. Moreover, the presentationthat follows integrates aspects such as proces-sing, experiments, theory/analysis, and model-ing/simulation.

2.13.1.3 Processing and Manufacturing ofFGMs

Modeling and design of FGMs should beconsidered in conjunction with synthesis tech-niques—thus a few remarks about materialsprocessing are in order. Details of processing

technologies can be found, e.g., in Hirai (1996),and Suresh and Mortensen (1998). The fabri-cation of FGMs include, e.g., vapor phase(e.g., chemical vapor deposition: CVD, physi-cal vapor deposition: PVD), liquid phase (e.g.,sol–gel, plasma, spraying), and solid phase(e.g., self-propagating high-temperature synth-esis: SHS, powder stacking) methods (Hirai,1996). In the SHS method, a variety ofmodifications have been used to obtain denseproducts (Miyamoto et al., 1992; Sata, 1993).An important advantage of the SHS methodrelative to other powder processing methodslies in its short duration. A developed ap-proach, known as field activated combustionsynthesis (FACS) (Munir et al., 1995; Feng andMunir, 1994, 1995a, 1995b; Xue and Munir,1995, 1997; Munir, 1996, 1997), has beenshown to improve the performance of previousSHS methods. In FACS, an electric field isapplied during combustion to synthesize ma-terials which cannot be prepared by SHS, e.g.,SiC, B4C, MoSi2–SiC, etc. (Carrillo-Heianet al., 2001a, 2001b). Xue and Munir (1995)showed that FACS can extend the composi-tional limit in composites. Table 1 provides asummary of fabrication methods and examplesof FGMs fabricated by these methods.

0.0 0.2 0.4 0.6 0.8 1.0x − x1

0.0

0.2

0.4

0.6

0.8

1.0

C(x

) / (

C2_ C

1)

n = 0.2

0.5

1.0

2.0

5.0

Figure 2 Plot of the normalized compositionfunction C(x) versus xx1 for selected values of nwith x1¼ 1.0 and x2¼ 2.0—see Equation (1).

Table 1 Some fabrication methods and examplesof FGMs.

Methods Examples

Vapor phase methodsCVD SiC/C, SiC/TiCCVI SiC/C, TiB2/SiC

Liquid phase methodsElectro-depositing Cu/CuZn, Cu/CuNiElectro-plating CoNiReP(f.c.c)/

CoNiReP(h.c.p.)Plasma spraying PSZ/SS, PSZ/NiCrAlCentrifugal casting SiC/Al

Solid phase methods(a) Powder stackingmethodsCentrifugal ZrO2/NiCrSpraying PSZ/SSFiltration Al2O3/NiSedimentation Al2O3/NiAl, A12O3/W

(b) Sintering methodsSintering, HP and

HIPPSZ/SS, Al2O3/NiAlN/Ni, Si3N4/Mo

Plasma activated PSZ/SS, PSZ/TiFACS MoSi2/SiC, Nb5Si3/NbSHS TiB2/Cu, TiB2/NiSPS ZrO2/SUS401LDiffusion and

reactionPZT/NiNbAl2O3/Sn/Nb/Sn/Al2O3

Introduction 609

2.13.2 MECHANICS MODELS OF FGMs

2.13.2.1 Continuum Model

From the continuum mechanics point ofview, FGMs are nonhomogeneous materials,i.e., the material properties of an FGM, e.g.,Young’s modulus (E), Poisson’s ratio (n), massdensity (r), yield stress (s0), and tangentmodulus (ET), are functions of spatial position.This work focuses on FGMs undergoing elasticdeformations governed by the equations ofnonhomogeneous elastic materials (Fung,1965). Those basic equations include theequation of motion

sij; j ¼ rðxÞ@2ui

@t2ð2Þ

the strain–displacement relation

eij ¼ 12ðui;j þ uj;iÞ ð3Þ

and the constitutive law

eij ¼ SijklðxÞskl ð4Þ

in which sij are stresses, eij are strains, ui aredisplacements, x¼ (x1, x2, x3)¼ (x, y, z),Sijkl(x) are the compliance coefficients, r(x) isthe mass density, the Latin indices have therange 1, 2, 3, and repeated indices imply thesummation convention. Under isotropic con-ditions, Equation (4) reduces to

eij ¼1þ nðxÞ

EðxÞ sij nðxÞEðxÞskkdij ð5Þ

where E(x) is Young’s modulus, n(x) isPoisson’s ratio, and dij is the Kronecker delta.In the continuum analysis, E, n, and othermaterial properties (e.g., shear modulus) arecontinuous functions of the spatial coordinatex. They can be calculated from a micromecha-nics model (see Section 2.13.2.2) or can beassumed to follow some elementary functions.An exponentially varying Young’s modulusand a constant Poisson’s ratio find frequent usein quasistatic crack problems (e.g., Konda andErdogan, 1994). The benefit of using anexponential function for Young’s modulus isthat the final governing equations (e.g., Na-vier–Cauchy equations) for FGMs reduce toconstant coefficient, partial differential equa-tions that may be solved either analytically orsemianalytically. Other commonly used func-tions include linear (e.g., Marur and Tippur,1998; Kim and Paulino, 2002) and hyperbolic–tangent (e.g., Eischen, 1987a).

Now consider quasistatic, two-dimensional(2D) deformations of a nonhomogeneous

material with the following properties:

m mðx; yÞ ¼ m0 expðbx þ gyÞ; n n0 ð6Þ

where m is the space-dependent shear modulus

mðx; yÞ ¼ Eðx; yÞ2ð1þ nÞ ð7Þ

and m0, n0, b, and g are material constants. Theinverse of the parameters b and g defines thelength scale of material nonhomogeneity in thex- and y-directions, respectively. The Navier–Cauchy equations for the material with theproperties Equation (6) are

ðkþ 1Þ@2u

@x2þ ðk 1Þ @

2u

@y2þ 2

@2v

@x@y

þ bðkþ 1Þ@u

@xþ gðk 1Þ @u

@yþ gðk 1Þ@v

@x

þ bð3 kÞ@v

@y¼ 0

ðk 1Þ@2v

@x2þ ðkþ 1Þ@

2v

@y2þ 2

@2u

@x@y

þ gð3 kÞ@u

@xþ bðk 1Þ@u

@yþ bðk 1Þ@v

@x

þ gðkþ 1Þ@v

@y¼ 0

ð8Þ

where u¼ u1, v¼ u2, and k¼ 34n0 for planestrain, and k¼ (3n0)/(1þ n0) for plane stress.Notice that when b-0 and g-0, the aboveequations reduce to the standard equations ofhomogeneous elasticity (Timoshenko andGoodier, 1973). Once the displacement solu-tion is available, stresses are found simply byusing Hooke’s law

sxx ¼ m0 expðbx þ gyÞk 1

ðkþ 1Þ @u

@xþ ð3 kÞ @v

@y

syy ¼ m0 expðbx þ gyÞk 1

ðkþ 1Þ @v

@yþ ð3 kÞ @u

@x

sxy ¼ m0 expðbx þ gyÞ @u

@yþ @v

@x

ð9Þ

2.13.2.2 Micromechanics Models

The material properties of an FGM, such asYoung’s modulus and Poisson’s ratio, mostoften are evaluated from properties of theconstituent materials using micromechanicsmodels. Conventional composite models (Ne-mat-Nasser and Hori, 1993) generally applywhen the volume fraction of one constituent inthe FGM remains much smaller than that ofthe other. However, the validity of such modelscannot be assured over the entire range of


material volume fractions; they rely on thespatial uniformity of constituent distributionsand microstructure of the composite. The mainfeature of FGMs lies in the nonuniformmicrostructure with continuous change involume fractions. In fact, micromechanicsmodels widely applicable to FGMs remainlargely unavailable (Zuiker, 1995). Dvorak andZuiker (1994), Reiter et al. (1997), and Reiterand Dvorak (1998) have indicated that, amongvarious micromechanics models for conven-tional composite materials, Mori–Tanaka andself-consistent models may be used to estimatethe effective properties of graded materialswith reasonable accuracy. However, a theore-tical basis for such applications remainsunclear because the concept of a representativevolume element cannot be unique for FGMs inthe presence of continuously graded properties.To overcome this limitation of standardmicromechanics models in FGM applications,Aboudi et al. (1999) developed a higher-ordermicromechanical theory. The fundamentalframework of the higher-order theory reliesupon volumetric averaging of the variousquantities, satisfaction of the field equationsin a volumetric sense, and imposition in anaverage sense of boundary and interfacialconditions between the subvolumes to char-acterize the graded microstructures. A detaileddescription and discussion of the higher-ordertheory can be found in the review paper byAboudi et al. (1999). Moreover, Section 2.13.2of their paper provides insights on the applica-tion of micromechanics models (originallydeveloped for conventional composites) toFGMs. Motivated by the above exposition,this section first reviews the self-consistent andMori–Tanaka models. Then, a modified rule ofmixture model proposed by Tamura et al.(1973) is discussed. Finally, the popularHashin–Shtrikman bounds (Hashin andShtrikman, 1963) are introduced.

2.13.2.2.1 Self-consistent model

The self-consistent method was first pro-posed to estimate the elastic properties ofpolycrystalline materials which, in fact, arejust one phase media. Due to the random orpartially random distribution of crystal orien-tation, discontinuities in properties exist acrosscrystal interfaces. In the application to poly-crystalline aggregates, a single anisotropiccrystal is viewed as a spherical or ellipsoidalinclusion embedded in an infinite medium ofthe unknown isotropic properties of theaggregate. Then the system is subjected touniform stress or strain conditions at largedistances from the inclusion. Next the orienta-

tion average of the stress or strain in theinclusion is set equal to the correspondingapplied value of the remote stress or strain.Thus the name ‘‘self-consistent’’ comes fromthis procedure.

When applying the self-consistent method toa two-phase composite, the shear and bulkmoduli m and K of the composite have thefollowing forms (Hill, 1965):

1

K þ ð4=3Þm ¼ V1

K1 þ ð4=3Þmþ V2

K2 þ ð4=3Þm ð10Þ

V1K1

K1 þ ð4=3Þmþ V2K2

K2 þ ð4=3Þm

þ 5V1m2m m2

þ V2m1m m1

þ 2 ¼ 0 ð11Þ

where V1 and V2 are the volume fractions ofphase 1 and phase 2, respectively, m1 and m2 arethe shear moduli of phase 1 and phase 2,respectively, and K1 and K2 are the bulkmoduli of phase 1 and phase 2, respectively.Once the above nonlinear equations are solvedfor m and K, Young’s modulus E and Poisson’sratio n of the composite are then determinedfrom the following elasticity relations:

E ¼ 9mK

mþ 3K; n ¼ 3K 2m

2ðmþ 3KÞ ð12Þ

2.13.2.2.2 Mori–Tanaka model

Like the self-consistent method, the Mori–Tanaka method also uses the average localstress and strain fields in the constituents of acomposite to estimate the effective materialproperties of the composite. The Mori–Tanakamethod (Mori and Tanaka, 1973), however,involves rather complicated manipulations ofthe field variables along with the concepts ofeigenstrain, backstress, etc. Later, Benveniste(1987) provided a more direct and simplifiedderivation of the Mori–Tanaka method.

The Mori–Tanaka estimates of the effectiveshear and bulk moduli m and K of a two-phasecomposite with spherical inclusion were de-rived by Benveniste (1987) as follows:

m ¼ m1 þ V2ðm2 m1Þ

1þ V1ðm2 m1Þ m1 þm1ð9K1 þ 8m1Þ6ðK1 þ 2m1Þ

1( )

ð13Þ

K ¼ K1 þ V2ðK2 K1Þ 1þ V1K2 K1

K1 þ ð4=3Þm1

1

ð14Þ

Mechanics Models of FGMs 611

Once the above equations are evaluated for mand K, Young’s modulus E and Poisson’s ration are given by Equation (12).

2.13.2.2.3 Tamura–Tomota–Ozawa model

Based on a rule of mixtures, Tamura et al.(1973) proposed a simple model (Tamura–Tomota–Ozawa (TTO) model) to describe thestress–strain curves of composite materials.Their model has been employed to studyFGMs by Williamson et al. (1993), Giannako-poulos et al. (1995), and Carpenter et al.(1999). The TTO model couples the uniaxialstress (s) and strain (e) of the composite to thecorresponding average uniaxial stresses andstrains of the two constituent materials by

s ¼ V1s1 þ V2s2; e ¼ V1e1 þ V2e2 ð15Þ

where si and ei (i¼ 1, 2) are the average stressesand strains of the constituent phases, respec-tively, and Vi (i¼ 1, 2) are volume fractions.The TTO model introduces an additionalparameter, q, to represent the ratio of stress-to-strain transfer

q ¼ s1 s2e1 e2

; 0oqoN ð16Þ

It is clear that q¼ 0 and q-N correspond toproperty averaging with equal stress and equalstrain, respectively. Young’s modulus of thecomposite may be obtained from Equations(15) and (16) as

E ¼ V2E2q þ E1

q þ E2þ ð1 V2ÞE1

V2q þ E1

q þ E2þ ð1 V2Þ

1

ð17Þ

where Ei (i¼ 1, 2) are Young’s moduli of thebasic constituent phases.

For applications of the TTO model toceramic/metal (brittle/ductile) composites, theyield stress of the composite, sY, is given by

sYðV2Þ ¼ sY2 V2 þq þ E2

q þ E1

E1

E2ð1 V2Þ

ð18Þ

where sY2 is the yield stress of the metal (phase2). Idealization of the metal as a bilinearmaterial with a tangent modulus H2 leads toa bilinear composite with the following tangentmodulus H

H ¼ V2H2q þ E1

q þ H2þ ð1 V2ÞE1

V2q þ E1

q þ H2þ ð1 V2Þ

1

ð19Þ

Figure 3 shows the schematic of the stress–strain curve of the composite described by theTTO model.

When the metal follows a power-law relationwith an exponent n2, the following nonlinearequations determine the stress–strain curve forthe composite

eeY

¼ V1E

q þ E1

s2sY

þ ðq þ V2E1ÞEðq þ E1ÞE2

sY2

sY

s2sY2

n2

ssY

¼ V2q þ E1

q þ E1

s2sY2

þ V1qE1

ðq þ E1ÞE2

sY2

sY

s2sY2

n2ð20Þ

where eY2¼ sY2/E2 and eY¼ sY/E denote theyield strains of the metal and the composite,respectively. The constant q in the TTO modelgoverns the interaction of the constituents inan FGM. In applications, q may be approxi-mately determined by experimental calibration.

2.13.2.2.4 Hashin–Shtrikman bounds

Because effective properties of a compositematerial are estimated from micromechanicsmodels such as the self-consistent and Mori–Tanaka models discussed above, it is importantto provide bounds for those material properties.For a two-phase composite material, Hashinand Shtrikman (1963) derived the followingbounds for the effective shear and bulk moduli

m1 ¼ m1 þ V21

m2 m1þ 6ðK1 þ 2m1ÞV1

5m1ð3K1 þ 4m1Þ

1

m2 ¼ m2 þ V11

m1 m2þ 6ðK2 þ 2m2ÞV2

5m2ð3K2 þ 4m2Þ

1

K1 ¼ K1 þ V2

1

K2 K1þ 3V1

ð3K1 þ 4m1Þ

1

K2 ¼ K2 þ V1

1

K1 K2þ 3V2

ð3K2 þ 4m2Þ

1

ð21Þ

H

Phase 1 (brittle)

σ

Composite

Phase 2 (ductile)

qE

E1

E2

H2

ε

Figure 3 Schematic of the stress–strain curve forthe TTO model.


where m1 and K1 are the lower bounds, and m2

and K2 are the upper bounds of the shear and

bulk moduli, respectively, when K24K1,m24m1.

2.13.3 FRACTURE OF FGMs

2.13.3.1 Crack-tip Fields

As discussed in Section 2.13.2.1, FGMs arenonhomogeneous materials from the viewpointof continuum mechanics, i.e., Young’s mod-ulus, E, and Poisson’s ratio, n, of an FGM varywith spatial position. Eischen (1987a) studiedthe asymptotic crack-tip fields in generalnonhomogeneous materials by using the tradi-tional eigenfunction expansion technique asemployed by Williams (1957). He has shownthat the stress field angular functions asso-ciated with the first two terms (i.e., r–1/2 and r0)are not affected by the material propertyvariation, and the effect of nonhomogeneityis reflected in higher order terms (i.e., r1/2 andhigher). However, explicit forms of the higher-order terms were not given. Parameswaran andShukla (2002) have provided the first six termsof the stress field for a stationary crack, alignedalong the gradient, in a material with exponen-tially graded E and constant n. They used anasymptotic analysis coupled with Wester-gaard’s stress function approach.

When E and n are sufficiently smoothfunctions, Eischen (1987a) showed that the2D crack-tip elastic fields remain identical tothose in homogeneous materials. When E and nare not sufficiently smooth functions, i.e., theyare continuous but only piecewise continuouslydifferentiable (e.g., for a graded layer bondedto a homogeneous substrate), Jin and Noda(1994a) obtained the same result and furtherpointed out that the result can be extended to3D, anisotropic, and elastic–plastic cases.Therefore, the asymptotic crack-tip stress anddisplacement fields in FGMs can be written as(Eischen, 1987a; Jin and Noda, 1994a)

sabðr; yÞ ¼1ffiffiffiffiffiffiffi2pr

p KI *sð1Þab ðyÞ þ KII *s

ð2Þab ðyÞ

n o; r-0

ð22Þ

uaðr; yÞ ¼1

mtip

ffiffiffiffiffiffir

2p

rKIu

ð1Þa ðyÞ þ KIIu

ð2Þa ðyÞ

n o; r-0

ð23Þ

where a, b¼ 1, 2, sab are stresses, ua aredisplacements, (r, y) are the polar coordinatescentered at the crack tip, KI and KII are mode Iand mode II stress intensity factors (SIFs),respectively, *sð1Þab ðyÞ and *sð2Þab ðyÞ are standard

angular distribution functions of stresses, uð1Þa ðyÞ

and uð2Þa ðyÞ are standard angular distribution

functions of displacements, and mtip is the shearmodulus at the crack tip. Many textbooks onfracture list the standard angular functions, e.g.,Anderson (1995). The singular parts of thecrack-tip stresses in a material with the proper-ties in Equation (6), as used by Erdogan (1995),are also given by Equation (22). Under plane-stress conditions, the crack-tip energy releaserate, G, is related to the SIFs by

G ¼ 1

EtipðK2

I þ K2IIÞ ð24Þ

where 1/Etip is replaced by ð1 n2tipÞ=Etip in theabove equation for plane-strain conditions. InEquation (24), Etip and ntip denote Young’smodulus and Poisson’s ratio at the crack tip,respectively.

2.13.3.2 Stress Intensity Factors

The assessment of crack-like defects anddamage in FGMs requires SIF solutions forvarious crack problems. Extensive efforts havebeen made in recent years to obtain SIFsolutions. This section reviews and discussessome available SIF solutions for cracks innonhomogeneous materials. Emphasis isplaced on analytical/semianalytical techniquesfor solving crack problems of nonhomoge-neous materials. Section 2.13.4 discusses thecomputation of SIF solutions via numericalmethods, e.g., finite element methods (FEMs).

2.13.3.2.1 Mechanical loads

The singular integral equation technique is apowerful analytical/semianalytical method tosolve crack problems of nonhomogeneousmaterials (Chan et al., 2003). A number ofcrack problems in FGMs have been studiedusing this method. For example, Delale andErdogan (1983) studied a crack in an infiniteFGM plate with a shear modulus that variesexponentially in the crack direction, which is aspecial case (y¼ 0) of the problem discussedbelow. For radial multiple-crack problems,Shbeeb et al. (1999a, 1999b) proposed afundamental solution and performed a para-metric study of SIFs. Delale and Erdogan(1988) have also considered a crack in anFGM layer between two dissimilar homoge-neous half planes. Erdogan et al. (1991) studieda crack problem in bonded nonhomogeneousmaterials. Choi et al. (1998a) considered colli-near cracks in a layered half plane with anFGM interlayer. Choi (2001a) studied an

Fracture of FGMs 613

inclined crack in an FGM layer between twodissimilar homogeneous half planes. Erdoganand Wu (1997) considered a crack in a finite-width FGM plate under tension, bending andfixed-grip displacement loads, respectively. Guand Asaro (1997a) calculated SIFs for a semi-infinite crack in an FGM strip. Kadioglu et al.(1998) studied a crack problem for an FGMlayer on an elastic foundation. Erdogan andOzturk (1995), Chen and Erdogan (1996), andJin and Batra (1996c) investigated cracks inFGM coatings. Ozturk and Erdogan (1997,1999) investigated both mode I and mixed-mode crack problems in a nonhomogeneousorthotropic medium. Wang et a1. (1997) studieda penny-shaped crack in a nonhomogeneouslayer between dissimilar materials. Chan et al.(2001) investigated the crack problem forFGMs under antiplane shear loading by meansof a displacement-based formulation leading toa hypersingular integral equation formulation.Besides the integral-equation technique, Fettet al. (2000) used a weight function method tocalculate SIFs for FGMs.

We now describe a singular integral equationapproach to obtain the SIFs for a crack in aninfinite nonhomogeneous medium as shown inFigure 4. The shear modulus and Poisson’sratio follow the functional form

m mðx1Þ ¼ m0 expðdx1Þ; n n0 ð25Þ

where m0, n0, and d are material constants. Theabove expression shows that the shear modulusvaries exponentially along an arbitrary x1-direction. A crack of length 2a exists alongthe x-direction, which is inclined by angle yfrom the x1-direction (see Figure 4). This is amixed-mode crack problem studied by Kondaand Erdogan (1994). In the (x, y) coordinate

system, the shear modulus and Poisson’s ratiohave the form given by Equation (6)

m mðx; yÞ ¼ m0 expðbx þ gyÞ; n n0

where b and g are related to d and y by

b ¼ d sin y; g ¼ d cos y ð26Þ

The Navier–Cauchy equations for the ma-terial with the properties of Equation (25) aregiven in Equation (8). The boundary condi-tions of the crack problem can be formu-lated as

syyðx; þ 0Þ ¼ p1ðxÞ; xj joa

vðx; þ 0Þ ¼ vðx; 0Þ; xj j4að27Þ

sxyðx;þ0Þ ¼ p2ðxÞ; xj joa

uðx;þ0Þ ¼ uðx;0Þ; xj j4að28Þ

syyðx;þ0Þ ¼ syyðx;0Þsxyðx;þ0Þ ¼ sxyðx;0Þ; xj joN

ð29Þ

By means of a Fourier transform, the bound-ary value problem, Equation (8) and Equa-tions (27)–(29), reduces to the followingsystem of singular integral equations

Z 1

1

f2ðsÞs r

þX2j¼1

L1jðr; sÞfjðsÞ" #

ds

¼ 1þ k2mðr; 0Þp1ðrÞ; rj jo1

Z 1

1

f1ðsÞs r

þX2j¼1

L2jðr; sÞfjðsÞ" #

ds

¼ 1þ k2mðr; 0Þp2ðrÞ; rj jo1

ð30Þ

where the unknown density functions f1(r) andf2(r) are the slopes of the crack displacementprofile defined by

f1ðxÞ ¼@

@x½uðx;þ0Þ uðx;0Þ

f2ðxÞ ¼@

@x½vðx;þ0Þ vðx;0Þ

ð31Þ

respectively, the nondimensional coordinate ris

r ¼ x=a ð32Þ

and the Fredholm kernels Lij (r, s) (i, j¼ 1, 2)can be found in Konda and Erdogan (1994). Inaddition, the functions f1(r) and f2(r) satisfythe uniqueness conditionsZ 1

1

f1ðrÞ dr ¼ 0;

Z 1

1

f2ðrÞ dr ¼ 0 ð33Þ

a

y

-a

x

x1

y1µ(x1)

θ

Figure 4 Crack geometry in nonhomogeneousmedium (free space).


To obtain the numerical solution of thesystem of governing integral Equation (30),the density functions, f1(r) and f2(r), areexpanded into series of Chebyshev polyno-mials as follows:

f1ðrÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 r2p

XNn¼1

AnTnðrÞ; rj jr1

f2ðrÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 r2p

XNn¼1

BnTnðrÞ; rj jr1

ð34Þ

where Tn(r) are Chebyshev polynomials of thefirst kind, and An and Bn are unknownconstants. The functions f1(r) and f2(r) givenby Equations (34) already satisfy the conditionof Equations (33). By substituting Equation(34) into integral Equations (30), we have

XNn¼1

fUn1ðrÞBn þ H11n ðrÞAn þ H12

n ðrÞBng

¼ 1þ k2mðr; 0Þp1ðrÞ; rj jr1

XNn¼1

fUn1ðrÞAn þ H21n ðrÞAn þ H22

n ðrÞBng

¼ 1þ k2mðr; 0Þp2ðrÞ; rj jr1

ð35Þ

where Un(r) are Chebyshev polynomials of thesecond kind and Hij

n ðrÞ ði; j ¼ 1; 2Þ are givenby

Hijn ðrÞ ¼

1

p

Z 1

1

Lijðr; sÞ TnðsÞffiffiffiffiffiffiffiffiffiffiffiffiffi1 s2

p ds; i; j ¼ 1; 2 ð36Þ

To solve the functional Equations (35), theseries on the left-hand side are first truncatedat the Nth term. A collocation technique(Erdogan et al., 1973; Erdogan, 1978) is thenused and the collocation points, ri, are chosenas the roots of the Chebyshev polynomials ofthe first kind

ri ¼ cosð2i 1Þp

2N; i ¼ 1; 2; y; N ð37Þ

which are strategically selected such that morepoints are located closer to the crack tips thanon the midregion of the crack surface. The

SIFs at the crack tip x¼ a are calculated from

KIðaÞ ¼ ffiffiffiffiffiffipa

p 2mða; 0Þ1þ k

XN

n¼1

Bn

KIIðaÞ ¼ ffiffiffiffiffiffipa

p 2mða; 0Þ1þ k

XN

n¼1

An

ð38Þ

and the SIFs at the crack tip x¼ –a are

KIðaÞ ¼ffiffiffiffiffiffipa

p 2mða; 0Þ1þ k

XN

n¼1

ð1ÞnBn

KIIðaÞ ¼ffiffiffiffiffiffipa

p 2mða; 0Þ1þ k

XN

n¼1

ð1ÞnAn

ð39Þ

Table 2 shows the SIFs (normalized bys0

ffiffiffiffiffiffipa

p) for various values of ad (nonhomo-

geneity parameter) and two crack orientations,y¼ 0 and y¼ p/2, under uniform crack surfacepressure s0. The angle y¼ 0 corresponds to acrack aligned with the material gradationdirection. The problem becomes symmetricabout the crack line and thus KII¼ 0. Wheny¼ p/2, the crack is perpendicular to thematerial gradation direction and the problemis symmetric about x¼ 0; hence, KI(–a)¼KI(a)and KII(–a)¼ –KII(a). Table 3 shows thenormalized SIFs for various values of ad andvarious crack orientations under uniformcrack surface pressure s0. These results war-rant careful examination in applications fordefect assessment. For example, one surprisingresult shown in Table 3 is that the maximumvalues of KI and KII do not generallycorrespond to the limiting crack orientationsy¼ 0 and y¼ p/2. Also, the values of ycorresponding to the maximum SIFs dependon the nonhomogeneity parameter ad. Moredetailed SIF results for a single crack undervarious loading conditions can be found inKonda and Erdogan (1994).

2.13.3.2.2 Thermal loads

Thermal loads often have a pronounced rolein industrial applications of FGMs. Thetheoretical treatment of cracks under thermal

Table 2 Normalized SIFs under uniform crack surface pressure.

ad

Normalized SIFs 0.1 0.25 0.5 1.0 2.5

y¼ 0 KI(a) 1.023 1.055 1.103 1.189 1.382KI(a) 0.975 0.936 0.871 0.757 0.536

y¼ p/2 KI(a) 1.008 1.036 1.101 1.258 1.808KII(a) 0.026 0.065 0.129 0.263 0.697

After Konda and Erdogan (1994).


loads is similar to that for cracks undermechanical loads. Under steady-state thermalloading conditions, Noda and Jin (1993)studied a crack parallel to the boundaries ofan infinite strip. Jin and Noda (1994b) con-sidered an edge crack in a nonhomogeneoushalf plane. Erdogan and Wu (1996) investi-gated a crack perpendicular to the boundariesof an infinite strip. Nemat-Alla and Noda(2000) considered an edge crack problem in asemi-infinite FGM plate with a bidirectionalgradation in the coefficient of thermal expan-sion. Under transient thermal loading condi-tions, Jin and Noda (1994c) studied a crackparallel to the boundary of a nonhomogeneoushalf plane. Jin and Batra (1996b) investigatedan edge crack in an FGM strip. Kokini andCase (1997) considered both edge and interfacecracks in functionally graded ceramic coatings.Choi et al. (1998b) investigated collinear cracksin a layered half plane with a graded homo-geneous interfacial zone. Wang et al. (2000)developed a multilayered material model tostudy various crack problems in FGMs underthermal loads. For a comprehensive review onthermal stresses in FGMs and related topics,see the review paper by Noda (1999).

By using a multilayered material model, Jinand Paulino (2001) studied an edge crack in anelastic homogeneous but thermally nonhomo-geneous strip subjected to transient tempera-ture boundary conditions as shown in Figure 5.Here a is the crack length, b is the stripthickness, Nþ 1 is the total number of homo-geneous layers, and T0 and Tb are boundarythermal loads. By means of the Laplace trans-

form and an asymptotic analysis, they obtainedan analytical first-order temperature solutionfor short times and calculated thermal stressintensity factors (TSIF) (normalized byEa0T0

ffiffiffiffiffiffipb

p=ð1 nÞ; where E and n are Young’s

modulus and Poisson’s ratio, respectively, anda0 is the thermal expansion coefficient of thefirst homogeneous layer). Figure 6(a) shows thenormalized TSIFs for edge cracks with lengthsa/b¼ 0.1, 0.3, and 0.5 in a homogeneous strip

-T0 -Tb -T0 -Tb

Ny

x

b

y

x

b(a) (b)

a

0 1 2 0 1 2 N

Figure 5 An FGM strip occupying the region0rXrb and |y|oN. The bounding surfaces ofthe strip are subjected to uniform thermal loads T0

and Tb: (a) a layered material and (b) an edge crackin the layered material.

Table 3 Effects of the nonhomogeneity parameter ad and the crack orientation y on the normalized SIFsunder uniform crack surface pressure.

y/p

Normalized SIFs 0.0 0.1 0.2 0.3 0.4 0.5

ad¼ 0.1 KI(a) 1.023 1.023 1.022 1.020 1.015 1.008KI(a) 0.975 0.977 0.983 0.991 1.003 1.008KII(a) 0.000 0.010 0.019 0.024 0.027 0.026

KII(a) 0.000 0.006 0.012 0.018 0.023 0.026

ad¼ 0.25 KI(a) 1.055 1.057 1.061 1.061 1.053 1.036KI(a) 0.936 0.943 0.962 0.987 1.014 1.036KII(a) 0.000 0.030 0.054 0.067 0.070 0.065

KII(a) 0.000 0.011 0.024 0.039 0.054 0.065

ad ¼ 0.5KI(a) 1.103 1.113 1.133 1.145 1.135 1.101KI(a) 0.871 0.887 0.931 0.990 1.050 1.101KII(a) 0.000 0.073 0.126 0.149 0.148 0.129

KII(a) 0.000 0.013 0.034 0.064 0.099 0.129

ad¼ 1.0KI(a) 1.189 1.222 1.294 1.348 1.336 1.258KI(a) 0.757 0.788 0.878 1.004 0.139 1.258KII(a) 0.000 0.179 0.306 0.352 0.327 0.263

KII(a) 0.000 0.010 0.039 0.099 0.181 0.263

After Konda and Erdogan (1994)


based on the asymptotic solution and thecomplete solution. The TSIFs based on theasymptotic solution show good agreement withthose based on the complete solution for timesup to approximately t¼ 0.1, where t is thenondimensional time given by t¼ tk0/b

2 inwhich t is time and k0 is the thermal diffusivityof the first homogeneous layer. Figure 6(b)shows the normalized TSIFs versus nondimen-sional time t for cracks in both homogeneousand TiC/SiC FGM strips for the volumefraction profiles n¼ 0.2, 1.0, and 2.0, where nis the power index in the SiC volume fractionfunction given by (see Figure 2)

VSiCðXÞ ¼ ðX=bÞn ð40Þ

The peak TSIFs for the FGM are lower thanthat for the homogeneous strip.

2.13.3.2.3 Contact loads

The application of FGMs as wear-resistantcoatings introduces contact loads in the frac-ture analysis. Some crack problems of FGMssubjected to contact loads have been studied byChoi (2001b) and Dag and Erdogan (2002).

Choi (2001b) studied a crack in a semi-infinite homogeneous substrate coated by atwo-layer material system with an FGMinterlayer and a homogeneous top layer, asshown in Figure 7. The top and the FGMlayers have thicknesses of h1 and h2, respec-tively. The substrate contains a crack of length2c which is perpendicular to the coating surfaceand d is the distance between the center of thecrack and the interface of the substrate with theFGM layer. The surface of the top layer isloaded by a moving contact pressure withfriction, which can be approximately described

0.00 0.10 0.20 0.30Nondimensional time

0.00

0.05

0.10

0.15N

orm

aliz

ed T

SIF

completeasymptotic

a/b = 0.1 0.3 0.5

0.00 0.02 0.04 0.06 0.08 0.10Nondimensional time

0.00

0.05

0.10

0.15

Nor

mal

ized

TS

IF

homogeneousn = 0.2n = 1.0n = 2.0

a/b = 0.1

(a) (b)

Figure 6 (a) Normalized TSIFs for a homogeneous strip: asymptotic solution versus complete solution and(b) TSIFs for the FGM strip for various material nonhomogeneity parameter n (after Jin and Paulino, 2001).

c

x

xx

2c

E(x )δ

N(y)

T(y)hh

1

2h

y

3

2

1

2

e

dsE

a

b

δE

Figure 7 Schematic of a coating/substrate system with an FGM interlayer and a crack under Hertziancontact with friction.


by the Hertzian contact load

NðyÞ ¼ p0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 y e

d

2r

TðyÞ ¼ fNðyÞ; jy ejodð41Þ

where N(y) is the normal contact pressure, T(y)the tangential traction, p0 the peak pressure, dthe contact half-width, e the distance betweenthe contact center and the crack, and f thefriction coefficient. The Poisson’s ratio isassumed constant throughout the mediumand Young’s modulus is

E ¼ Ec; 0rx1rh1 ð42Þ

E ¼ Ec expðbx2Þ; 0rx2rh2 ð43Þ

E ¼ Es; 0rx3rN ð44Þ

where xi (i¼ 1, 2, 3) is the local x-coordinate ineach layer. By using a singular integralequation approach, Choi (2001b) solved thecrack problem of Figure 7 and studied effectsof the material nonhomogeneity and contactparameters on SIFs. Figure 8 shows the SIFs(normalized by K0 ¼ p0

ffiffiffic

p) for Ec/Es¼ 0.2,

d/c¼ h/c¼ 1.0, d/c¼ 5.0, and f¼ 0.7. Thethickness ratio is chosen as h2/h¼ 0.1.

When the contact moves from the left side ofthe crack (e/do0) toward the crack (e/do0),the absolute value of the mode I SIF, KI

increases first, reaches a maximum, and thendecreases. KI remains negative for e/dr0. Theabsolute mode II SIF, KII, follows a similarpattern. When the contact moves from thecrack to its right side (e/d40), KI increases andbecomes positive near e/d¼ 0.5 or greater,depending on the friction coefficient f. At the

same time, KII remains negative but |KII|decreases and reaches a peak near e/d¼ 1.0.When f¼ 0, i.e., frictionless contact, KI issymmetric about the crack line and is negativefor all contact locations, while KII is antisym-metric. The negative KI values in Figure 8(a)simply indicate crack closure at those loca-tions, and are shown just to illustrate thiseffect. The absolute values of KI and KII

increase with increasing f when e/do0. Whene/d40, the maximum values of positive KI andKII increase with increasing f. As to the effectof material nonhomogeneity on the SIFs, ageneral observation by Choi (2001b) is thatincreasing the thickness of the FGM layercauses some increase in the magnitude of theSIFs, especially for KI. A thinner FGMinterlayer, to some extent, becomes moreeffective in shielding the crack tip.

Dag and Erdogan (2002) investigated asurface crack in a nonhomogeneous half planeloaded by a sliding rigid stamp. Figure 9 showsthe contact crack problem, where P is thecontact load, f is the friction coefficient, d is thecrack length, and a and b are the y-coordinatesof the left and right ends of the stamp,

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

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

e/δ

KI(a

)/K

0

h2/h=0.1

Ec/E

s = 0.2

-4 -3 -2 -1 0 1 2 3 4-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

e/δ

KII(a

)/K

0

Ec/E

s = 0.2

h2/h=0.1

(a) (b)

Figure 8 Mixed-mode SIFs for Ec/Es¼ 0.2, d/c¼ h/c¼ 1.0, d/c¼ 5.0, and f¼ 0.7: (a) normalized mode I SIFat crack tip x3¼ a and (b) normalized mode II SIF at crack-tip x3¼ a. See Figure 7 to identify the crack-tiplocation (after Choi, 2001b).

fPP

a b

FGMd

y

µ( )x

x

Figure 9 A surface crack in a semi-infinite FGMmedium under a sliding stamp.


respectively. The shear modulus and thePoisson ratio of the graded medium areassumed as

m ¼ m0 expðgxÞ; n ¼ const: ð45Þ

Dag and Erdogan (2002) also used a singularintegral equation approach to solve the con-tact crack problem of Figure 9 and calculatedSIFs versus a/d for various values of thematerial nonhomogeneous parameter gd andfriction coefficient f. Detailed results anddiscussions can be found in Dag and Erdogan(2002).

2.13.3.3 Crack Growth

Crack extension in FGMs may follow one oftwo scenarios. In the first case, the crack lies inthe direction perpendicular to the materialgradation. Clearly, the crack growth mode ismixed and the crack deflects from the originalorientation. In the second case, the crackpropagation occurs in the material gradationdirection under mode I loading conditions.This section first discusses the crack deflectionproblem, i.e., the determination of crackpropagation direction. The section concludeswith an initial discussion of fracture toughnessunder stable extension (R-curve).

2.13.3.3.1 Crack deflection

SIFs at a kinked crack tip determine thecrack deflection direction at the load when thecrack starts to propagate. Several crack-kink-ing theories exist to predict the kink direction,e.g., the maximum energy release rate criterion,the maximum circumferential stress theory,and so on. The maximum energy release ratecriterion states that the crack growth occursalong the direction which provides the max-imum energy release rate. The maximumcircumferential stress theory assumes that thecrack propagation direction corresponds to theone at which the circumferential stress reachesthe maximum. The maximum circumferentialstress theory is often used for the study ofcrack growth direction in FGMs. According tothis theory, the crack deflection angle relativeto the original crack (tangent) line, f, is

KI sinf KIIð3 cos f 1Þ ¼ 0 ð46Þ

In general, numerical methods are used tocalculate the SIFs at the tip of a finite kinkedcrack (Becker et al., 2001). For small-kinkcracks, however, the SIFs at the kinked cracktip may be calculated based on the SIFs for themain crack tip. Gu and Asaro (1997b) pre-

sented a crack deflection model for small-kinkconditions. The model postulates the idea of alocal homogeneous core near the main cracktip characterized by the near-tip fields, Equa-tions (22) and (23). In this core region, Young’smodulus and the Poisson ratio are Etip and ntip,respectively, as shown in Figure 10. Underthose assumptions, the small crack kinkingmodel presented by Cotterell and Rice (1980)may be applied to FGMs. Cotterell and Rice(1980) assumed that the SIFs at the kinkedcrack tip could be expressed as (see alsoKarihaloo, 1982)

KI ¼C11ðfÞKI þ C12ðfÞKII

KII ¼C21ðfÞKI þ C22ðfÞKII

ð47Þ

where KI and K

II are the SIFs at the kinkedcrack tip, KI and KII are the SIFs at the maincrack tip, f is the kink angle (see Figure 10),and Cij (i, j¼ 1, 2) are coefficients depending onf. For small kink angle, using the first orderapproximation, Cotterell and Rice (1980)obtained

C11 ¼1

43 cos

f2þ cos

3f2

C12 ¼ 3

4sin

f2þ sin

3f2

C21 ¼1

4sin

f2þ sin

3f2

C22 ¼1

4cos

f2þ cos

3f2

ð48Þ

They showed that, for homogeneous materials,SIFs calculated using Equations (47) and (48)are in good agreement with exact solutions forf up to 401 (the error is less than 5%).

With the SIFs at the kinked crack tip givenby Equation (47), the crack growth directioncan be obtained by fracture criteria such as the

φ

(Etip,νtip)

Main crackKinked crack

Figure 10 Crack-tip kinking.


maximum energy release rate criterion or themaximum circumferential stress criterion. Guand Asaro (1997b) calculated the crack growthangles for a double cantilever beam and a four-point bending specimen by using the abovemodel. The material properties were assumedto vary exponentially in the specimen widthdirection. They observed a strong effect ofmaterial gradient on the kink angle when thecrack tip is at the middle of the FGM, but theeffect becomes small when the crack tip liesnear the boundaries of the FGM.

2.13.3.3.2 Fracture toughness and theR-curve

For FGMs, the fracture toughness generallyvaries with spatial position. For example, astrong fracture toughness gradation exists inceramic/metal FGMs because the toughness ofthe metal greatly exceeds that of the ceramic. Avery simple approach to determine the fracturetoughness of ceramic/metal FGMs adopts arule of mixtures. For example, Jin and Batra(1996a) assumed that the critical energy releaserate for the FGM, GIC(X), may be expressed asfollows:

GICðX Þ ¼ VmetðX ÞGmetIC þ ½1 VmetðX ÞGcer

IC ð49Þ

where GmetIC and Gcer

IC are the critical energyrelease rates of the metal and ceramic, respec-tively, Vmet (X) is the metal volume fraction,and X is the material gradation direction. Notethat

GICðX Þ ¼ 1 n2ðX ÞEðXÞ K2

IC

GmetIC ¼ 1 n2met

EmetðKmet

IC Þ2

GcerIC ¼ 1 n2cer

EcerðKcer

IC Þ2

ð50Þ

where KIC(X), KmetIC ; and Kcer

IC are the fracturetoughnesses of the FGM, metal and ceramic,respectively, E(X), Emet, and Ecer are Young’smoduli of the FGM, metal and ceramic,respectively, and n(X), nmet, and ncer are thePoisson ratios of the FGM, metal and ceramic,respectively. The fracture toughness of theFGM is then obtained as follows (Jin andBatra, 1996a):

KICðXÞ ¼ EðX Þ1 n2ðXÞ

ð1 n2metÞVmetðXÞEmet

ðKmetIC Þ2

þð1 n2cerÞð1 VmetðX ÞÞEcer

ðKcerIC Þ2

1=2

ð51Þ

The above equation indicates that the fracturetoughness of the FGM varies with X. When the

crack grows from the ceramic-rich region intothe metal-rich region, the toughness of theFGM increases significantly since the fracturetoughness of the metal greatly exceeds that ofthe ceramic. However, Equation (51) probablyoverestimates the fracture toughness of theFGM given the imperfections that exist be-tween the ceramic and metal particles in aceramic/metal FGM which are not consideredin such a simple mixture relationship. Toimprove the rule of mixture estimates offracture toughness for FGMs, Jin and Batra(1996a, 1998) proposed a crack-bridging mod-el. In the crack-bridging model, a bridgingzone exists ahead of the original main crackwith the following constitutive law that relatesthe bridging traction, s, to the relative openingdisplacement of the bridging surfaces

s ¼ s0ð1 d=d0Þn ð52Þ

where s0 is the maximum bridging traction, d0is a critical opening displacement at which thebridging traction vanishes, and n is a softeningindex. By using both the rule of mixtureestimate and the crack-bridging model, Jinand Batra (1996a) calculated the fracturetoughness and R-curve for an edge crackedinfinite strip made of an Al2O3/Ni FGM. Theirresult shows that the fracture toughnesssignificantly increases when the crack growsfrom the ceramic-rich region towards themetal-rich region. The rule of mixture estimateof the fracture toughness exceeds that pre-dicted from the crack-bridging model. Cai andBao (1998) also studied crack growth in aceramic/metal graded coating using a moresimplified bridging model with constant brid-ging traction. However, the simple crack-bridging models may not be applicable to theceramic–metal interconnected region (percola-tion region) in an FGM (see midportion ofFigure 1). A phenomenological cohesive mod-el, proposed by Jin et al. (2002), may be usedfor the whole range of an FGM component—see discussion in Section 2.13.4.3.

2.13.3.4 Dynamic Fracture

2.13.3.4.1 Crack-tip fields for a propagatingcrack

Parameswaran and Shukla (1999) havestudied crack-tip stress and deformation fieldsfor cracks propagating in FGMs with either alinearly varying or an exponentially varyingshear modulus. In the case of a linearlyvarying modulus, the mass density remainsconstant, while in the case of an exponentiallyvarying modulus, the mass density follows the


same exponential function. While such as-sumptions do not reflect real engineeringmaterials, they enable analytical solutionsand thus allow one to gain insight into thenature of the dynamic crack-tip fields. Thissection begins with an introduction of thosecrack-tip field results.

(i) Crack-tip fields for linear variation of shearmodulus and constant mass density

Consider a semi-infinite crack propagatingat a constant velocity c in the positive X-direction in an FGM as shown in Figure 11.The shear modulus of the FGM follows theform

m ¼ m0ð1þ aXÞ ð53Þ

where m0 and a are material constants. ThePoisson ratio (n) and the mass density (r)remain constant. The Lame constant l is thengiven by

l ¼ l0ð1þ aX Þ ð54Þ

where l0 is a constant. Under these assump-tions, the governing equations of the dynamicproblem under plane-strain conditions are theequations of motion

@sXX

@Xþ @sXY

@Y¼ r

@2u

@t2;

@sXY

@Xþ @sYY

@Y¼ r

@2v

@t2ð55Þ

the strain–displacement relations

eXX ¼ @u

@X; eYY ¼ @v

@Y; eXY ¼ 1

2

@u

@Yþ @v

@X

ð56Þ

and Hooke’s law

sXX ¼ ð1þ aX Þ½ðl0 þ 2m0ÞeXX þ l0eYY sXX ¼ ð1þ aX Þ½ðl0 þ 2m0ÞeXX þ l0eYY sXY ¼ 2ð1þ aX Þm0eXY

ð57Þ

We introduce the dilatation (D) and rotation(o) by

D ¼ @u

@Xþ @v

@Y; o ¼ @v

@X @u

@Yð58Þ

The equations of motion can then be expressedby D and o as follows:

m0ð1þ aX Þr2Dþ 2m0a@D@X

2m0al0=m0 þ 2

@o@Y

¼ rl0=m0 þ 2

@2D@t2

m0ð1þ aX Þr2oþ 2m0a@o@X

þ 2l0a@D@Y

¼ r@2o@t2

ð59Þ

The above equations reduce to the corres-ponding wave equations of dilatation androtation for homogeneous materials by settinga equal to zero. However, due to the non-homogeneous nature of the material, theequations remain coupled in the two fields Dand o, through the nonhomogeneity parametera. When Equations (59) are written in amoving rectangular coordinate system (x, y)centered at the crack tip as shown in Figure 11,we have

a21@2D@x2

þ @2D@y2

þ bxr2Dþ 2b@D@x

2bl0=m0 þ 2

@o@y

¼ 0

a22@2o@x2

þ @2o@y2

þ bxr2oþ 2b@o@x

þ 2b@d@y

¼ 0ð60Þ

where

a1 ¼ 1 rc2

mtipðl0=m0 þ 2Þ

" #1=2

a2 ¼ 1 rc2

mtip

" #1=2

b ¼ m0a=mtip

ð61Þ

in which mtip is the shear modulus at the cracktip.

By using an asymptotic analysis, Parames-waran and Shukla (1999) obtained the crack-tip solutions for D and o, and found that thecrack-tip stress and velocity fields have thefollowing form:

sab ¼ 1ffiffiffiffiffiffiffi2pr

p fKIðtÞ *Sð1Þab ðy; cÞ þ KIIðtÞ *Sð2Þ

ab ðy; cÞg

r-0 ð62Þ

’ua ¼c

mtipffiffiffiffiffiffiffi2pr

p fKIðtÞVð1Þa ðy; cÞ þ KIIðtÞVð2Þ

a ðy; cÞg

r-0 ð63Þ

which are identical to the fields for homo-geneous materials. Note that Equations (62)and (63) have a similar form to those reportedin Equations (22) and (23) for the quasistaticcase, but with displacements replaced byvelocities including the time dependence of

yY

Xcrack

cx

Figure 11 A propagating crack with constantvelocity c.


SIFs. In the above expressions, sab arestresses, ’ua are velocities, (r, y) are the polarcoordinates centered at the crack tip, KI(t)and KII(t) are mode I and mode II dynamicSIFs, respectively, *Sð1Þ

ab ðy; cÞ and *Sð2Þab ðy; cÞ are

standard angular distribution functions ofstresses for propagating cracks, andV

ð1Þa ðy; cÞ and V

ð2Þa ðy; cÞ are standard angular

distribution functions of velocities, which canbe found in dynamic fracture mechanicsbooks, e.g., Freund (1990). Besides the singu-lar term of the crack-tip asymptotic fields,Parameswaran and Shukla (1999) also ob-tained higher order terms of the D and o fieldswith the solutions up to the third order termgiven by

D ¼A0r1=21 cos

y12þ A1 þ A2r

1=21 cos

y12

b2a21

A0r1=21 cos

3y12

þ 4ba2ða21 a22Þðl0=m0 þ 2ÞB0r

1=22 cos

y22

þ 3ð1 a21Þb16a21

A0r1=21

1

6cos

7y12

þ cos3y12

þ C0r1=21 sin

y12þ C2r

1=21 sin

y12

þ b2a21

C0r1=21 sin

3y12

4ba2ða21 a22Þðl0=m0 þ 2ÞD0r

1=22 sin

y22

3ð1 a21Þb16a21

C0r1=21

1

6sin

7y12

þ sin3y12

ð64Þ

o ¼ B0r1=22 sin

y22þ B2r

1=22 sin

y22

b2a22

B0r1=22 sin

3y22

þ 4ba1ða22 a21Þ

A0r1=21 sin

y12

3ð1 a22Þb16a22

B0r1=22

1

6sin

7y22

þ sin3y22

þ D0r1=22 cos

y22þ D1 þ D2r

1=22 cos

y22

b2a22

D0r1=22 cos

3y22

4ba1ða22 a21Þ

C0r1=21 cos

y12

þ 3ð1 a22Þb16a22

D0r1=22

1

6cos

7y22

þ cos3y22

ð65Þ

where

ri ¼ ðx2 þ a2i y2Þ1=2; yi ¼ arctanaiy

x; i ¼ 1; 2

ð66Þ

A0, B0, C0, and D0 are related to the dynamic

SIFs by

A0 ¼ ð1þ a22Þð1 a21Þ4a1a2 ð1þ a22Þ

2

KI

mtipffiffiffiffiffiffi2p

p

B0 ¼ 2a1ð1 a22Þ1þ a22ð1 a21Þ

A0

C0 ¼ 2a2ð1 a21Þ4a1a2 ð1þ a22Þ

2

KII

mtipffiffiffiffiffiffi2p

p

D0 ¼ 1þ a22ð1 a22Þ2a2ð1 a21Þ

C0

ð67Þ

and A1, A2, B2, C2, D1, and D2 are unknownconstants. Note that the first and second-orderterms are identical to those for homogeneousmaterials but the third-order term differs fromthat for homogeneous materials.

(ii) Crack-tip fields for exponential variation ofboth shear modulus and mass density

Parameswaran and Shukla (1999) also con-sidered crack-tip fields for a crack propagatingin an FGM with the following spatial variationof shear modulus and mass density

m ¼ m0 expðaXÞ; r ¼ r0 expðaX Þ ð68Þ

where m0, r0, and a are material constants.Poisson’s ratio (n) is again assumed constant.Under these assumptions, the equations for thedilatation D and rotation o in the movingrectangular coordinates (x, y) are

a21@2D@x2

þ @2D@y2

þ a@D@x

a

l0=m0 þ 2

@o@y

¼ 0

a22@2o@x2

þ @2o@y2

þ a@o@x

al0m0

@d@y

¼ 0

ð69Þ

where

a1 ¼ 1 r0c2

l0 þ 2m0Þ

1=2; a2 ¼ 1 r0c2

m0

1=2ð70Þ

The leading terms of the asymptotic crack-tipstress and velocity fields are again foundidentical to those for homogeneous materials.Parameswaran and Shukla (1999) also ob-tained the crack-tip D and o solutions up tothe third-order term.

2.13.3.4.2 Dynamic SIFs

Due to difficulties of analytical modeling,only limited solutions for dynamic stressintensity factors (DSIFs) are available. Babaeiand Lukasiewicz (1998) considered a crack inan FGM layer between two dissimilar halfplanes under anti-plane shear impact load. Itou(2001) studied DSIFs for a crack in a


nonhomogeneous layer sandwiched betweentwo dissimilar homogeneous materials underin-plane stress wave loading. Li et al. (2001)investigated a cylindrical crack in an FGMlayer under torsional dynamic loading.

Meguid et al. (2002) considered a crack offinite length (2a) propagating at a constantvelocity c along the x-direction (see Figure 11)in an infinite FGM medium with the followingYoung’s modulus and mass density

E ¼ E0 expðayÞ; r ¼ r0 expðayÞ ð71Þ

where E0, r0, and a are material constants, andPoisson’s ratio remains constant. Again,although the above assumptions do not reflectreal materials, they are valuable to gain insightinto the nature of DSIFs. Under the aboveassumptions, the Navier–Cauchy equations inthe moving coordinates (x, y) become

b1@2u

@x2þ @2u

@y2þ b2

@2v

@x@yþ b

@u

@yþ @v

@x

¼ 0

b3@2v

@y2þ @2v

@x2þ b4

@2u

@x@yþ b3b n

@u

@xþ @v

@y

¼ 0

ð72Þ

where

b1 ¼2

1 n1 c2

c21

; b2 ¼

1þ n1 n

b3 ¼2

1 n1 c2

c22

1

; b4 ¼1þ n1 n

1 c2

c22

1

c1 ¼E0

ð1 n2Þr0

1=2; c2 ¼

E0

2ð1þ nÞr0

1=2ð73Þ

Meguid et al. (2002) solved these governingequations under both normal crack face load,

s0, and shear load, t0, by using an integralequation method. Figure 12(a) shows themode I DSIF (normalized by s0


p), kI,

versus the nonhomogeneous parameter aafor different crack propagation speeds.kI increases with increasing aa; and withincreasing crack propagation speed, the effectof a increases. Figure 12(b) shows the normal-ized mode II DSIF, kII, versus the nonhomo-geneity parameter aa for different crackpropagation speeds. A significant effect of aaupon kII is observed. More detailed DSIFresults can be found in the paper of Meguidet al. (2002).

2.13.3.5 Viscoelastic Fracture

To date, FGMs have found primary poten-tial application in high-temperature technolo-gies. In general, materials exhibit creepand stress relaxation behavior at high tem-peratures. Viscoelasticity offers a basis tostudy the phenomenological behavior of creepand stress relaxation of polymers andpolymer-based FGMs fabricated by, e.g.,Parameswaran and Shukla (1998), Lambroset al. (1999), and Marur and Tippur (2000b);however, those papers are not concerned withviscoelasticity.

The elastic–viscoelastic correspondence prin-ciple (or elastic–viscoelastic analogy) repre-sents one of the most useful tools inviscoelasticity because the Laplace transformof the viscoelastic solution can be directlyobtained from the corresponding elastic solu-tion. However, the correspondence principledoes not in general hold for FGMs. To avoidthis problem, Paulino and Jin (2001a) haveshown that the correspondence principle can

0 1 2αa

1

1.5

2

2.5

k IV/c2=0.0V/c2=0.2V/c2=0.4V/c2=0.6V/c2=0.7V/c2=0.8

0 1 2 αa

0

1

2

3

k II

V/c2=0.0V/c2=0.2V/c2=0.4V/c2=0.6V/c2=0.7V/c2=0.8

(b)(a)

Figure 12 (a) Normalized mode I DSIF versus nonhomogeneity parameter, aa and (b) normalized mode IIDSIF versus nonhomogeneity parameter, aa (after Meguid et al., 2002).


still be used to obtain the viscoelastic solutionfor a class of FGMs exhibiting relaxation orcreep functions with separable kernels in spaceand time. In the development below, therevisited correspondence principle is presentedand the solution of a crack problem is given toillustrate application of the correspondenceprinciple to viscoelastic FGMs.

Other studies on crack problems ofnonhomogeneous viscoelastic materials di-rectly solve the viscoelastic governing equa-tions. For example, Alex and Schovanec (1996)have considered stationary cracks subjected toantiplane shear loading. Herrmann and Scho-vanec (1990, 1994) have studied quasistatic anddynamic crack propagation in nonhomoge-neous viscoelastic media under antiplane shearconditions. Schovanec and Walton (1987a,1987b) have also considered quasistatic propa-gation of a plane-strain mode I crack in apower-law nonhomogeneous linearly viscoelas-tic body and calculated the correspondingenergy release rate. Although these investiga-tions employed a separable form for therelaxation functions, they did not make useof the correspondence principle. When applic-able, use of the correspondence principlesubstantially simplifies solution of viscoelasticproblems (see Paulino and Jin, 2001a; 2001b).

2.13.3.5.1 Basic equations

The basic equations of quasi-static visco-elasticity of FGMs are

sij; j ¼ 0; ðequilibrium equationÞ ð74Þ

eij ¼ ðui; j þ uj; iÞ=2; ðstrain2displacement relationÞð75Þ

sij ¼ 2R t

0 mðx; t tÞ deij

dtdt

skk ¼ 3R t

0 Kðx; t tÞdekk

dtdt

9>>=>>; ðconstitutive lawÞ

ð76Þ

sij ¼ sij skkdij=3; eij ¼ eij ekkdij=3 ð77Þ

in which sij are stresses, eij are strains, sij and eij

are deviatoric components of stress and straintensors, ui are displacements, dij is the Kro-necker delta, x¼ (x1, x2, x3), m(x, t), and K(x, t)are the appropriate relaxation functions, t isthe time, the Latin indices have the range 1, 2,3, and repeated indices imply the summationconvention. Note that the relaxation functionsalso depend on spatial positions, whereas inhomogeneous viscoelasticity, they are only

functions of time, i.e., m m(t) and KK(t)(Christensen, 1971).

The boundary conditions are given by

sijnj ¼ Si; on Bs ð78Þ

ui ¼ Di; on Bu ð79Þ

where nj are the components of the unitoutward normal to the boundary of the body,Si are the tractions prescribed on Bs, and Di arethe prescribed displacements on Bu. Notice thatBs and Bu are required to remain constant withtime.

2.13.3.5.2 Correspondence principle, revisited

In general, the standard correspondenceprinciple of homogeneous viscoelasticity doesnot hold for FGMs. To circumvent thisproblem, consider a special class of FGMs inwhich the relaxation functions have the follow-ing general form (separable in space and time):

mðx; tÞ ¼ m0 *mðxÞf ðtÞKðx; tÞ ¼ K0KðxÞgðtÞ

ð80Þ

The constitutive law Equation (76) thenreduces to

sij ¼ 2m0 *mðxÞZ t

0

f ðt tÞ deij

dtdt

skk ¼ 3K0KðxÞZ t

0

gðt tÞ dekk

dtdt

ð81Þ

By assuming that the material is initially atrest, the Laplace transforms of the basicEquations (74), (75), (81), and the boundaryconditions Equations (78) and (79) are ob-tained as

%sij; j ¼ 0 ð82Þ

%eij ¼ ð %ui; j þ %uj; iÞ=2 ð83Þ

%sij ¼ 2m0 *mðxÞp %fðpÞ %eij ; %skk ¼ 3K0KðxÞp %gðpÞ%ekk ð84Þ

%sijnj ¼ %Si; on Bs ð85Þ

%ui ¼ %Di; on Bu ð86Þ

where a bar over a variable represents itsLaplace transform, and p is the transformvariable. The set of Equations (82)–(84), andconditions Equations (85) and (86) have aform identical to those of nonhomogeneouselasticity with shear modulus m ¼ m0 *mðxÞ andbulk modulus K ¼ K0KðxÞ provided that thetransformed viscoelastic variables are asso-ciated with the corresponding elastic variables,and m0p %fðpÞ and K0p %gðpÞ are associated with m0and K0, respectively. Therefore, the correspon-dence principle of homogeneous viscoelasticity


still holds for the FGM with the materialproperties given in Equation (80), i.e., theLaplace transformed nonhomogeneous visco-elastic solution can be obtained directly fromthe solution of the corresponding nonhomoge-neous elastic problem by replacing m0 and K0

with m0p %fðpÞ and K0p %gðpÞ; respectively. Thefinal solution is realized upon inverting thetransformed solution.

2.13.3.5.3 Viscoelastic models

Among the various models for gradedviscoelastic materials, we introduce the stan-dard linear solid defined by

mðx; tÞ ¼ mNðxÞ þ ½meðxÞ m

NðxÞ exp½t=tmðxÞ

Kðx; tÞ ¼ KNðxÞ þ ½KeðxÞ KNðxÞ exp½t=tKðxÞð87Þ

the power-law model given by

mðx; tÞ ¼ meðxÞ½tmðxÞ=tq; Kðx; tÞ¼ KeðxÞ½tK ðxÞ=tq; 0oqo1 ð88Þ

and the Maxwell material model expressed by

mðx; tÞ ¼ meðxÞ exp½t=tmðxÞ;Kðx; tÞ ¼ KeðxÞ exp½t=tK ðxÞ

ð89Þ

where tm(x) and tK(x) are the relaxation timesin shear and bulk moduli, respectively, and q isa material constant.

The discussion below indicates the revisionsneeded in the general models so that thecorrespondence principle holds.

(i) Standard linear solid (Equation (87)). Ifthe relaxation times tm and tK are constant, ifme(x) and mN(x) have the same functionalform, and if Ke(x) and KN(x) have the samefunctional form, then the standard linear solidsatisfies assumption Equation (80).

(ii) Power-law model (Equation (88)). If therelaxation times tm and tK are independent ofspatial position, then the assumption Equation(80) is readily satisfied. Moreover, even if therelaxation times do depend on the spatialposition in Equation (88), the correspondenceprinciple may still be applied with somerevision, which consists of taking the corre-sponding nonhomogeneous elastic materialwith the following properties:

m ¼ meðxÞ½tmðxÞq; K ¼ KeðxÞ½tKðxÞq ð90Þ

instead of m¼ me(x) and K¼Ke(x).(iii) Maxwell material (Equation (89)). If the

relaxation times tm and tK are independent ofspatial position, then Equation (80) is satisfied.

2.13.3.5.4 A crack in a viscoelastic FGM strip

Paulino and Jin (2001b, 2001c) have appliedthe revisited correspondence principle to someantiplane shear crack problems. They alsoconsidered (Jin and Paulino, 2002) an infinitenonhomogeneous viscoelastic strip containinga crack of length 2a, as shown in Figure 13.The FGM strip is subjected to a uniformtension s0S(t) in the y-direction along both thelower boundary (y¼h1) and the upperboundary (y¼ h2), where s0 is a constant andS(t) is a nondimensional function of time t.The crack lies on the x-axis, from –a to a, andthe crack surfaces remain traction free.

The extensional relaxation function of thematerial is given by

E ¼ E0 expðby=hÞf ðtÞ ð91Þ

where h is a scale length and f(t) is anondimensional function of time t, e.g.,

f ðtÞ ¼EN=E0 þ ð1 EN=E0Þ expðt=t0Þðlinear standard solidÞ ð92Þ

f ðtÞ ¼ ðt0=tÞq ðpower-lawmaterialÞ ð93Þ

The following form, which is a generalizationof the power-law material model is alsoconsidered

E ¼ E0 expðby=hÞ½t0 expðdy=hÞ=tq ð94Þ

in which the relaxation time depends on yexponentially. In the above expressions, theparameters E0, EN, b, t0; d, q are materialconstants. Poisson’s ratio takes on the form(also separable in space and time)

n ¼ n0ð1þ gy=hÞ expðby=hÞgðtÞ ð95Þ

where n0 and g are material constants, and g(t)is a nondimensional function of time t.

h

h2

h1

-a a

y

x

σ=σ0 Σ(t)

σ=σ0 Σ(t)

Figure 13 A viscoelastic FGM strip occupies theregion |x|oN and –h1ryrh2 with a crack at |x|raand y¼ 0. The boundaries of the strip (y¼ –h1, h2)are subjected to uniform traction s0S(t).


It can be clearly seen from Equations (91)–(95) that the relaxation moduli and Poisson’sratio are separable functions in space and time.This is a necessary condition for applying therevisited correspondence principle of Paulinoand Jin (2001a). This kind of relaxationfunction may be appropriate for an FGM withits constituent materials having differentYoung’s moduli and Poisson’s ratios, buthaving approximately the same viscoelasticrelaxation behavior. Since the FGM is a specialcomposite of its constituents, the viscoelasticrelaxation behavior may remain unchanged ifits constitutents have the same relaxationbehavior. Thus, the relaxation moduli of theFGM would have separable forms in space andtime. Further, the material constants b and gdescribe the spatial gradation in Young’smodulus and the Poisson ratio. Potentially,this kind of FGM may include some poly-meric/polymeric materials such as Propylenehomopolymer/Acetal copolymer. The relaxa-tion behavior of Propylene homopolymer andAcetal copolymer are found to be similar—seefigures 7.5 and 10.3, respectively, of Ogorkie-wicz (1970).

According to the correspondence principle, acrack in an elastic strip of an FGM with thefollowing properties,

E ¼ E0 expðby=hÞ; n ¼ n0ð1þ gy=hÞ expðby=hÞð96Þ

may be considered. The Laplace transform ofthe viscoelastic solution is directly obtainedfrom the elastic solution.

Figure 14 shows SIFs, KI and KII (normal-ized by s0SðtÞ


p), versus nondimensio-

nal crack length 2a/h considering variousdimensionless nonhomogeneity parameter b

for the linear standard solid and the power-law model with constant relaxation time (seeEquations (92) and (93)). The SIFs areidentical for both models. The crack is locatedin the middle of the strip, i.e., h2¼ 0.5h.Figure 14(a) shows the results for mode ISIFs. The mode I SIF increases with increas-ing ratio 2a/h for all b values considered here.The SIF is higher than that of the correspond-ing homogeneous material (b¼ 0) and is aneven function of b. However, this symmetryremains valid only for the crack located in thecenter of the strip. Figure 14(b) shows theresults for mode II SIFs. The mode II SIF isasymmetric about b¼ 0 (again, this antisym-metry is valid only for the crack located in thecenter of the strip). The absolute value of themode II SIF increases with increasing 2a/h. Asexpected, the mode II SIF is zero for thiscrack geometry in a homogeneous strip(b¼ 0).

Figure 15 shows normalized SIFs versus thenondimensional crack length 2a/h for b¼ 2 andvarious values of d for the power-law materialwith position-dependent relaxation time (seeEquation (94)). The crack is located in themiddle of the strip. The effect of spatialposition dependence of the relaxation time onthe SIFs is reflected through the dimensionlessparameter d (the parameter q is taken as 0.4 inall calculations). Thus the curves for d¼71may be obtained from the curve d¼ 0 byshifting this curve by b¼80.4. Figure 15makes clear that, with respect to the corre-sponding model for constant relaxation time(i.e., d¼ 0), a positive d increases the SIFswhen b40. When |b| becomes relatively smallcompared with |qd|, the variation of SIFs withd follows different paths depending on thevalue of bþ qd.

0.0 0.5 1.0 1.5 2.0Nondimensional crack length 2a/h

1.0

2.0

3.0

4.0

Nor

mal

ized

Mod

e I

SIF |β| = 2.0

|β| = 1.0|β| = 0.0

h_2 / h = 0.5


-1.0

-0.5

0.0

0.5

1.0

Nor

mal

ized

Mod

e II

SIF β = 2.0

β = 1.0

β = 0.0

β = −1.0

β = −2.0

h_2 / h = 0.5

(b)(a)

Figure 14 Normalized SIFs versus nondimensional crack length, 2a/h, for various material nonhomogeneityparameter b considering the linear standard solid and the power-law material with constant relaxation time(h2¼ 0.5h): (a) mode I SIF and (b) mode II SIF (after Jin and Paulino, 2002).


2.13.3.6 Je-integral

Path independent integrals, especially theJ-integral (Eshelby, 1951; Cherepanov, 1967,1968; Rice, 1968), play an important role inpredicting fracture of materials. In general, theJ-integral becomes path-dependent for nonho-mogeneous materials. Honein and Herrmann(1997) extended the J-integral to certain classesof nonhomogeneous elastic materials withvarying Young’s modulus in the crack-linedirection. For instance, consider a nonhomo-geneous material with Poisson’s ratio inde-pendent of x1 (crack-line direction) and theshear modulus varying exponentially in the x1

direction

mðx1; x2Þ ¼ m0ðx2Þ expðbx1Þ ð97Þ

where m0(x2) is an arbitrary function of x2 andb is a material constant. The path-independentintegral for this nonhomogeneous material is(Honein and Herrmann, 1997)

Je ¼ZG

Wn1 sijnj

@ui

@x1

b

2sijnjui

ds ð98Þ

where G is a contour enclosing the crack tip, n1is the first component of the unit outwardnormal to G, sijnj¼Si are the components oftractions along G, ds is an infinitesimal lengthelement along the contour G, and W is thestrain energy density. The extra term inEquation (98), which appears outside the innerparentheses, arises from the material nonho-mogeneity. This extra term maintains the pathindependence of the Je-integral for the non-homogeneous material, Equation (97).

Next consider a nonhomogeneous materialwith constant Poisson ratio and the following

shear modulus

mðx1Þ ¼ m0ðbx1 þ 1Þa ð99Þ

where b and a are material constants. Thepath-independent integral, derived by Honeinand Herrmann (1997), for this nonhomoge-neous material is

Je ¼ZG

Wn1 sijnj

@ui

@x1

þb Wxjnj sijnjxk

@ui

@xk

a2sijnjui

ds ð100Þ

Again, note the extra term within parenthesesis due to the material nonhomogeneity. Final-ly, Honein and Herrmann (1997) showed thatthe Je-integral is related to SIFs by

Je ¼1

E0tip

ðK2I þ K2

IIÞ ð101Þ

where E 0tip ¼ Etip for plane stress and E 0

tip ¼Etip=ð1 n2tipÞ for plane strain.

2.13.4 FINITE ELEMENT MODELING

2.13.4.1 Graded Finite Elements

The standard parametric finite element (FE)formulation interpolates displacements andcoordinates from the element nodal values.Similarly, material properties can also beinterpolated from the element nodal valuesusing shape functions. The general approach is


1.0

2.0

3.0

4.0

Nor

mal

ized

Mod

e I

SIF

δ = 1.0δ = 0.0δ = −1.0

h_2 / h = 0.5

β = 2.0


0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

Mod

e II

SIF

δ = 1.0δ = 0.0δ = −1.0

h_2 / h = 0.5

β = 2.0

(b)(a)

Figure 15 Normalized SIFs versus nondimensional crack length, 2a/h, for various material nonhomogeneityparameter d considering the power-law material with position-dependent relaxation time (b¼ 2.0, q¼ 0.4,h2¼ 0.5h): (a) mode I SIF and (b) mode II SIF (after Jin and Paulino, 2002).

Finite Element Modeling 627

illustrated as follows:

Displacements ðu; v;wÞ: e:g:; u ¼X

i

Niui ð102Þ

Coordinates ðx; y; zÞ: e:g:; x ¼X

i

Nixi ð103Þ

Material properties (E, n): e.g.,

E ¼X

i

%NiEi; n ¼X

i

Nini ð104Þ

where Ni, Ni; %Ni; and Ni are shape functionscorresponding to node i, and the summation isdone over the element nodal points. Kim andPaulino (2002) have proposed a fully isopara-metric element formulation, i.e.,

N ¼ N ¼ %N ¼ N ð105Þ

This approach interpolates material propertiesat each Gaussian integration point from thenodal material properties of the element usingisoparametric shape functions, which are thesame for spatial coordinates and displacements(see Figure 16(a)). An alternative approach isalso shown in Figure 16(b) where the materialproperties are evaluated directly at the Gausspoints (Santare and Lambros, 2000). The dis-cussion here presents the generalized isopara-metric formulation by Kim and Paulino (2002).

The element stiffness matrix is given by(Cook et al., 1989)

ke ¼ZOe

BeTDeðxÞBe dOe ð106Þ

where ue is the nodal displacement vector, Be

is the strain–displacement matrix which con-tains gradients of the interpolating functions,De(x) is the constitutive matrix, and Oe is thedomain of element (e). The elasticity matrixDe(x)¼De(x, y) becomes a function of spatialcoordinates.

The integral in Equation (106) is evaluatedby Gaussian quadrature with the matrix De(x)specified at each Gaussian integration point.Thus, for 2D problems, the resulting integralbecomes

ke ¼Xn

i¼1

Xn

j¼1

BeT

ij DeijB

eijJijwiwj ð107Þ

where the subscripts i and j refer to theGaussian integration points, Jij is the determi-nant of the Jacobian matrix, and wi are theGaussian weights. The performance of thegraded elements has been investigated byPaulino and Kim (2003) within the context ofthe weak patch test. The remainder of Section2.13.4 focuses on 2D problems.

2.13.4.2 SIF Computation

2.13.4.2.1 Displacement correlation technique

The displacement correlation technique(DCT) represents one of the simplest methodsto evaluate SIFs. It consists of correlatingnumerical results for displacement at specificlocations on the crack with available analyticalsolutions. For quarter-point singular elements,the crack opening displacement (COD) profile

x

P(x,y)

z

y

P(x,y)

x

z

y

(a)

(b)

Figure 16 Graded element: (a) generalized isoparametric formulation and (b) direct Gaussian integrationformulation. P denotes a generic material property.


at x¼r is given by (Shih et al., 1976)

CODðrÞ ¼ ð4u2;i1 u2;i2Þffiffiffiffiffiffir

Da

rð108Þ

where u2,i1 and u2,i2 are the displacementsrelative to the crack tip in the x2-direction atlocations (i1) and (i2), r is the distance fromthe crack tip along the x1-direction, and Da is acharacteristic length associated with the crack-tip elements (see Figure 17).

For FGMs, material properties need to beconsidered at the crack-tip location. Thus,consistent with Equation (23), the analyticalexpression for COD(r), neglecting higherorder terms, can be written as

CODðrÞ ¼ KIkþ 1

m

tip

ffiffiffiffiffiffir

2p

rð109Þ

where ktip¼ 34ntip for plane strain,ktip¼ (3ntip)/(1þ ntip) for plane stress, and ntipdenotes Poisson’s ratio at the crack-tip loca-tion. By correlating Equations (108) and (109),the SIF for mode I can be evaluated by

KI ¼ffiffiffiffiffiffi2pDa

r4

mkþ 1

tip

u2;i1 m

kþ 1

tip

u2;i2

" #

ð110Þ

For mode II, the crack sliding displacement(CSD) replaces the COD and the followingexpression is readily obtained

KII ¼ffiffiffiffiffiffi2pDa

r4

mkþ 1

tip

u1;i1 m

kþ 1

tip

u1;i2

" #

ð111Þ

Note that in the above expressions for SIFs,the material properties (m m(x) and k k(x))have been taken at the crack-tip location

consistent with Equation (23). This approachwas presented by Kim and Paulino (2002).

2.13.4.2.2 Modified crack-closure integralmethod

The modified crack-closure integral method(MCC) was proposed by Rybicki and Kanni-nen (1977) based on Irwin’s virtual crack-closure method (Irwin, 1957) using the stressesahead of the crack tip and the displacementsbehind the crack tip. The energy release ratescan be obtained for modes I and II separatelyby this method, which utilizes only a single FEanalysis. No assumption of isotropy or homo-geneity around the crack is necessary. Thus themethod is ideally suited for FGMs. The energyrelease rate is estimated only in terms of thework done by the stresses (or equivalent nodalforces) over the displacements produced by theintroduction of a virtual crack extension. Theexpression for GI (strain energy release rate formode I) and GII (strain energy release rate formode II) may be obtained according to Irwin(1957) as

GI ¼ limda-0

2

da

Z x1¼da

x1¼0

1

2s22ðr ¼ x1; y ¼ 0; aÞ

u2ðr ¼ da x1; y ¼ p; a þ daÞ dx1 ð112Þ

GII ¼ limda-0

2

da

Z x1¼da

x1¼0

1

2s12ðr ¼ x1; y ¼ 0; aÞ

u1ðr ¼ da x1; y ¼ p; a þ daÞ dx1 ð113Þ

where s12 sxy and s22 syy are shear andnormal stresses ahead of the crack tip, andu1 ux and u2 uy are the displacementsrelative to the crack tip coordinates, respec-tively. Figure 18 illustrates a self-similar virtualcrack extension da and the distribution ofnormal stress ahead of the crack-tip.

For FGMs, SIFs can be related to the valuesof the potential energy release rates throughthe following expressions (Jin and Batra,1996a):

GI ¼kþ 1

8m

tip

K2I and GII ¼

kþ 1

8m

tip

K2II

ð114Þ

Ramamurthy et al. (1986) and Raju (1987)have shown that the values of GI and GII can bewritten in terms of the equivalent nodal forcesF2Fy and F1Fx, and the relative nodaldisplacements u2 and u1 when employingquarter-point singular elements around thecrack tip (see Figure 17). They provided initialexpressions for GI and GII valid only for pure

Figure 17 Crack-tip rosette of singular quarter-point (1st ring) and regular (2nd ring) FEs.


mode, homogeneous problems. In general, formixed-mode problems, the deformation isneither symmetric nor antisymmetric aboutthe local x1 axis. Thus Raju (1987) proposedthe corrected formulae, later employed by Kimand Paulino (2002).

2.13.4.2.3 J-integral formulation

Consider 2D deformation cases. The follow-ing path-independent J

k -integral for generalnonhomogeneous materials is established as(Eischen, 1987a; Eischen, 1987b; Kim andPaulino, 2002)

Jk ¼ lim

Ge-0

ZG0

Wnk sijnjui:k

dG

ZO0

ðW; kÞexpl dO

þZGc

ð½Wþ Wnþk ½Sþi uþi;k S

i ui;kÞ dG

ð115Þ

where W¼W(eij, xi) is the strain energy density

W ¼Z eij

0

sij deij ð116Þ

(W,k)expl is the ‘‘explicit’’ derivative of Wrespective to xk

ðW ;k Þexpl ¼@

@xk

W ðeij ; xiÞjeij¼const:; xm¼const: for mak

ð117Þ

Si¼ sijnj, G0; Gc ¼ Gþc þ G

c ; and O0 are illu-strated in Figure 19, superscripts (þ ) and ()refer to the upper and lower crack faces, andnþ

k ¼ nk is the outward unit normal vector to

Gþc : The notation [WþW] denotes the

discontinuity (or jump) in the strain energy

density across the crack opening. Notice thatthe material nonhomogeneity affects the stan-dard J-integral (Rice, 1968) by adding an areaintegral term involving the derivative of thestrain energy density. This term must beincluded to allow computation over relativelylarge regions to evaluate the J-integral usingthe FEM.

By introducing the equivalent domain inte-gral (EDI) (Raju and Shivakumar, 1990; Guet al., 1999), Kim and Paulino (2002) convertedthe closed contour integral of Equation (115)to the following expression if the crack surfacesremain traction free:

Jk ¼

ZA

½sijui;k Wdkj q;j dA Z

A

W ; kð Þexplq dA

þZGc

½Wþ Wqnþk dG ð118Þ

where q is a smooth function which has thevalue of unity on Ge and zero on G0. The twooften-used shapes of the q function are thepyramid function and the plateau function,the latter being illustrated by Figure 20. All the

-

=

r = = a+δa )δa ,

a

-( ,

δaa

r

( rσ

u

= ,a )22

2

1

11

1

x1

yx2

σ ,22

x

0

θ π

x

x

xx

θ,

δ

Figure 18 Self-similar crack extension and normal stress distribution.

Γ

Γ Γc+

r

ε

0

x

x

crack

θ

Γ

c-

Γ

2

2

1 u

Ωε

Ω

Γτ

Γ Γ Γ Γ Γ+

0 c ε-c

+ = Ω= + + +

ΩΩ ε0

x x1

Figure 19 Schematic of a cracked body.


details about the computational implementa-tion of J

k -integral can be found in the paper byKim and Paulino (2002).

The relation among the two components ;ofthe J

k -integral and the mode I and II SIFs isestablished for plane stress as (Eischen, 1987a)

J1 ¼ K2

I þ K2II

Etipð119Þ

J2 ¼ 2KIKII

Etipð120Þ

For plane strain, the Young modulus at thecrack-tip Etip must be divided by ð1 n2tipÞ;where ntip denotes Poisson’s ratio at thecrack tip.

2.13.4.2.4 Examples

The FEM has been used to calculate SIFsfor FGMs in a number of publications by, forexample, Anlas et al. (2000), Bao and Wang(1995), Marur and Tippur (2000a, 2000b), andRousseau and Tippur (2000, 2001). Here weonly describe a few examples by Kim andPaulino (2002).

(i) Three-point bending specimen with crackperpendicular to material gradation

Marur and Tippur (1998) have fabricatedFGM specimens using a gravity assistedcasting technique with two-part slow curingepoxy and uncoated solid glass sphere fillers.They have also analyzed a three-point bendingspecimen with a crack normal to the elasticgradient using both experimental (static frac-ture tests) and numerical (FEM) techniques(Marur and Tippur, 2000b). Figure 21(a)shows the specimen geometry and boundaryconditions (BCs). Figure 21(b) shows linearvariation of Young’s modulus in the materialgradient region from 10.79GPa to 3.49GPa,and the linear variation of Poisson’s ratio from

( )

x1

2x

q x , x 1 2

Figure 20 Plateau weight function (q function)used in the EDI implementation of J

k :

x38 21 17

95

5.8

22

100 N

6.6

glass-richepoxy

−15 −5 5 15 25 353

4

5

6

7

8

9

10

11 0.36

0.28

0.29

0.30

0.31

0.32

0.33

0.34

0.35

νE

x (mm)

νE

(a)

(b)

Figure 21 Three-point bending specimen with crack perpendicular to the material gradation: (a) specimengeometry (length in mm) and (b) variation of Young’s modulus and Poisson’s ratio (all dimensions in mm).


0.348 on the epoxy side to 0.282 on the glass-rich side. Figure 22 shows the detail of themesh using 16 sectors (S16) and 4 rings (R4)around the crack tip. Marur and Tippur(2000b) used the experimental strain data tocompute jK j ¼ 0:65MPam1=2 and c¼3.451,while their FEM results are jK j ¼0:59MPam1=2 and c¼ –3.241, where c¼tan–1(KII/KI) is the mode-mixity parameter.Table 4 lists these numerical results togetherwith the present ones obtained using MCC,DCT, and J

k -integral, which are in reasonableagreement with each other.

(ii) Multiple interacting cracks in a plate

Figure 23(a) shows two cracks each of length2a located with the angle yI (y1¼ 301, y2¼ 601)in a finite 2D plate. Figure 23(b) shows thecomplete FEM mesh, and Figures 23(c) and23(d) show details of the center region andaround crack tips, respectively. The distancefrom the origin to the two crack tips is 1.0.Shbeeb et al. (1999a, 1999b) have providedsemianalytical solutions for this example ob-tained with the integral equation method.However, the graphical results given in theirpaper makes accurate verification difficult. The

applied load corresponds to s22(x1,10)¼ s0¼ 1.0. This stress distribution wasobtained by applying nodal forces along thetop edge of the mesh. The displacementboundary condition is prescribed such thatu2¼ 0 along the lower edge and, in addition,u1¼ 0 for the node at the left-hand side.Young’s modulus is an exponential functionof x2, while Poisson’s ratio remains constantequal to zero in this numerical analysis. Thepaper by Shbeeb et al. (1999b) did not specifythe actual value of Poisson’s ratio. Eight-nodeserendipity elements (Q8) are used over most ofthe mesh, while at each crack tip, quarter-pointsix-node triangles (T6qp) were used. The meshhas 2287 Q8 and 214 T6 with a total of 2501elements and 7517 nodes. The following dataare used for the FE analysis: 2a¼ 2; L/W¼ 1.0;Eðx2Þ ¼ %Eebx2 ; %E ¼ 1:0; ba¼ (0.0, 0.25, 0.5,0.75, 1.0); n¼ 0.0; generalized plane stress; and2 2 Gauss quadrature. With this geometricalconfiguration, the material property E growsquite large as the plate size increases. However,this is just an artifact of the modeling, whichoccurs because the nonhomogeneity parameterb introduces a geometrical length scale (1/b)into the problem. The relevant material varia-tion is that around the crack region

The geometrical configuration for this pro-blem approximates the infinite plate as origin-ally conceived by Shbeeb et al. (1999b). Table 5shows a comparison of the normalized SIFs atthe two crack tips for the lower crack with theangle y¼ 301 computed by using the DCT,MCC, and the J

k -integral with those reportedby Shbeeb et al. (1999b). In this table KI(a

–) andKI(a

þ ) refer to the SIFs at the left and rightcrack tips, respectively. Similar notation isadopted for KII. Both the DCT and J

k -integralbased values agree reasonably well with theresults by Shbeeb et al. (1999b), and the MCCprovides the best estimates for the SIFs.

2.13.4.3 Cohesive Zone Modeling

Cohesive fracture models have been widelyused to simulate and analyze crack growth inductile and quasibrittle materials. In a cohesivefracture model, a narrow band termed a

Table 4 FEM SIFs for three-point bending specimen with crack perpendicular to material gradation.

Kim and Paulino (2002)

Parameters MCC Jk -integral DCT Marur and Tippur (2000b)

KI 0.557 0.557 0.558 0.589KII –0.028 –0.026 –0.026 –0.033|K| 0.5575 0.5576 0.5580 0.59c 2.871 –2.671 –2.641 –3.241

Figure 22 FE mesh for three-point bending speci-men with crack perpendicular to the materialgradation: detail around the crack tip showing 16sectors and 4 rings of elements.


cohesive zone, or process zone, exists ahead ofthe crack front. Material behavior in thecohesive zone follows a cohesive constitutivelaw which relates the cohesive traction to therelative displacements of the adjacent surfaces.Crack growth then occurs by progressivedecohesion of the cohesive surfaces. Thoughcohesive fracture models have been successfullyemployed to simulate failure processes inhomogeneous materials and conventional com-posites, few studies have extended the conceptto FGMs. Generalization of the cohesive zoneconcept to model fracture in FGMs representsa challenging task in view of the differentfailure mechanisms present in an FGM. In atypical ceramic/metal FGM, the ceramic-richregion may be regarded as a metal-particle-reinforced ceramic-matrix composite, whereasthe metal-rich region may be treated as aceramic-particle-reinforced metal-matrix com-posite (see discussion on micromechanics mod-els in Section 2.13.2.2). Though models for thefailure mechanisms of conventional composites

may be adopted to study the fracture processesin the ceramic-rich or metal-rich region, thefailure mechanisms operative in the intercon-necting region which has no distinct matrix andinclusion phases remain unknown. This sectiondescribes a new volume-fraction-based, phe-nomenological cohesive fracture model (Jinet al., 2002) and crack-growth analysis resultsobtained using the model.

2.13.4.3.1 A cohesive zone model

Consider general 3D mixed-mode fractureproblems. In a 3D setting, Camacho and Ortiz(1996) have introduced an effective displace-ment jump across the cohesive surfaces, deff, andan effective cohesive traction, seff, as follows:

deff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2n þ Z2d2s

qð121Þ

seff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2n þ Z2s2s

qð122Þ

( )

2

E−=2 eEυ

2 W

2 L

1 = 1.0L,σ

1

2a

2β

= 0.0

x

x

x

x

x( )

(a) (b)

(d)(c)

Figure 23 Two interacting cracks in a plate: (a) geometry and BCs; (b) FE mesh; (c) mesh detail of centerregion and (d) mesh detail around crack tips showing 16 sectors (S16) and four rings (R4) of elements aroundeach tip.


where dn and ds are the normal and tangentialdisplacement jumps across the cohesive sur-faces, sn and ss are the normal and shearcohesive tractions, and the parameter Z assignsdifferent weights to the opening and slidingdisplacements (Z is usually taken as

ffiffiffi2

p). Here

we assume that resistance of the cohesivesurfaces to relative sliding is isotropic in thecohesive (tangent) plane so that

ds ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2s1 þ d2s2

qð123Þ

ss ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2s1 þ s2s2

qð124Þ

in which ds1 and ds2 are the two relative slidingdisplacements across the cohesive surfaces,and ss1 and ss2 are the two shear cohesivetractions.

With the introduction of the above effectivetraction and displacement, a free energypotential is assumed to exist in the followingvolume-fraction-based form (Jin et al., 2002)

fðx; deff ; qÞ ¼ VmetðxÞVmetðxÞ þ bmet½1 VmetðxÞ

fmetðdeff ; qÞ

þ 1 VmetðxÞ1 VmetðxÞ þ bcerVmetðxÞ

fcerðdeff ; qÞ

ð125Þ

where Vmet(x) denotes the volume fraction ofthe metal, bmet (Z1) and bcer (Z1) representthe two cohesive gradation parameters, x¼ (x1,x2, x3), q is an internal variable describing theirreversible processes of decohesion, and fmet

and fcer are the free energy potentials forthe metal and ceramic phases, respectively,

given by

fmetðdeff ; qÞ ¼

escmetdcmet 1 1þ deff

dcmet

exp deff

dcmet

ð126Þ

fcerðdeff ; qÞ ¼

esccerdccer 1 1þ deff

dccer

exp deff

dccer

ð127Þ

where e¼ exp(1), scmet is the maximum cohesivetraction of the metal phase, dcmet the valueof deff at seff ¼ scmet; sccer the maximum cohe-sive traction of the ceramic phase, and dccer thevalue of deff at seff ¼ sccer; as illustrated byFigure 24.

The effective cohesive traction is thenobtained from the above potential as follows:

seff ¼@f@deff

¼ VmetðxÞVmetðxÞ þ bmet½1 VmetðxÞ

escmet

deffdcmet

exp deffdcmet

þ 1 VmetðxÞ1 VmetðxÞ þ bcerVmetðxÞ

esccerdeffdccer

exp deffdccer

if deff ¼ dmax

eff and ’deffZ0 ð128Þ

for the loading case, and

seff ¼smaxeff

dmaxeff

deff ; if deffodmax

eff or ’deffo0 ð129Þ

Table 5 Normalized SIFs for lower crack ðK0 ¼ s22ffiffiffiffiffiffipa

pÞ:

Method b KI(a)/K0 KII(a

)/K0 KI(aþ )/K0 KII(a

þ )/K0

Shbeeb et al. (1999b) 0.00 0.59 0.43 0.78 0.42(approximate) 0.25 0.62 0.39 0.82 0.48

0.50 0.66 0.36 0.88 0.570.75 0.69 0.34 0.98 0.685

MCC 0.00 0.589 0.423 0.804 0.4080.25 0.626 0.385 0.816 0.4740.50 0.662 0.346 0.842 0.5460.75 0.696 0.312 0.880 0.625

Jk -integral 0.00 0.603 0.431 0.801 0.431

0.25 0.627 0.401 0.811 0.4950.50 0.662 0.363 0.841 0.5580.75 0.692 0.347 0.895 0.617

DCT 0.00 0.598 0.413 0.812 0.3990.25 0.632 0.375 0.818 0.4630.50 0.672 0.336 0.838 0.5330.75 0.712 0.302 0.869 0.610

After Kim and Paulino (2002).


for the unloading case, where smaxeff is the value

of seff at deff ¼ dmaxeff calculated from Equation

(128). Here, the internal variable q is chosen asdmaxeff ; the maximum value of deff attained. The

cohesive law for general 3D deformations isthen formulated as follows:

sn ¼ @f@dn

¼ @f@deff

@deff@dn

¼ seffdeff

dn

ss ¼@f@ds

¼ @f@deff

@deff@ds

¼ Z2seffdeff

ds

ð130Þ

Compared with the formulation for homoge-neous materials, the FGM cohesive fracturemodel, given by Equations (128)–(130), in-creases the number of material-dependentparameters by two (bmet, bcer). Values for thelocal separation energies and peak cohesivetractions related to the pure ductile and brittlephases are obtained using standard proceduresfor homogeneous materials (see Roy andDodds (2001) for example). The material-dependent parameters bmet and bcer describeapproximately the overall effect of cohesivetraction reduction (from the level predicted bythe rule of mixtures) and the transition betweenthe fracture mechanisms of the metal andceramic phases. The preliminary analyses ofcrack growth in ceramic/metal FGMs by Jinet al. (2002) indicate that bmet plays a far moresignificant role than bcer, which can be simplyset to unity. It is anticipated that the parameterbmet may be experimentally calibrated by twodifferent procedures. The first procedure de-termines bmet by matching the predicted andmeasured crack growth responses in standardfracture mechanics specimens of FGMs. In-stead of using FGM specimens, the secondprocedure employs fracture specimens made ofa monolithic composite each with a fixedvolume fraction of the constituents. This opens

the potential to calibrate bmet for each volumefraction level of metal and ceramic, whichcomprise the FGM specimens, i.e., bmet canbecome a function of Vmet in the presentmodel.

2.13.4.3.2 Cohesive element

Figure 25 illustrates the 3D interface cohe-sive element where (x1, x2, x3) are the globalcartesian coordinates, (s1, s2, n) are theelement-specific, local Cartesian coordinates,and (Z, z) are the parametric coordinates. Theinterface-cohesive element consists of twofour-node bilinear isoparametric surfaces.Nodes 1–4 lie on one surface of the elementwith nodes 5–8 on the opposite surface. Thetwo surfaces initially occupy the same location.When the whole body deforms, the twosurfaces undergo both normal and tangentialdisplacements relative to each other. Thecohesive tractions corresponding to the relativedisplacements follow the constitutive relationsEquations (128)–(130), and thus maintain thetwo surfaces in a ‘‘cohesive’’ state.

The tangent stiffness matrix of the cohesiveelement is given by (Roy and Dodds, 2001)

KT ¼Z 1

1

Z 1

1

BTcohDcohBcohJ0 dZ dz ð131Þ

where Bcoh extracts the relative displacementjumps within the cohesive element from thenodal displacements, J0 is the Jacobian of thetransformation between parametric (Z, z) andCartesian coordinates (s1, s2) in the tangentplane of the cohesive element, and Dcoh is thetangent modulus matrix of the cohesive lawEquations (128)–(130) (Jin et al., 2002). ForFGMs, the Dcoh matrix depends on spatialposition through the graded volume fraction of

0 1 2 3 4 5 6Nondimensional separation

0.0

0.2

0.4

0.6

0.8

1.0

Non

dim

ensi

onal

coh

esiv

e tr

actio

n

unloading

maxmax(δ σ ),

loading

0 1 2 3 4 5 6Nondimensional separation

0.0

0.2

0.4

0.6

0.8

1.0

Non

dim

ensi

onal

coh

esiv

e tr

actio

n

δ / δ = 0.06δ / δ = 0.15δ / δ = 0.24

cer

cer

cer

met

met

met

c

c

c

c

c

c

( = separation at peak traction for ceramic, = separation at peak traction for metal )

δ

δ

cer

met

c

c

(b)(a)

Figure 24 Normalized cohesive traction versus nondimensional separation displacement: (a) for metal,smet=scmet versus d=dcmet and (b) for ceramic, scer=scmet versus d=dcmet (where metal/ceramic strength ratio,scmet=s

ccer; is taken to be 3).


the metal phase, Vmet, in a ceramic/metalFGM. Vmet is approximated by the standardinterpolation

Vmet ¼X4i¼1

NiVimet ð132Þ

where Vimetði ¼ 1; 2; 3; 4Þ are the values of

Vmet at the nodal points of the interface-cohesive elements and Ni(x, Z, z) (i¼ 1,2,y,n) the standard shape functions. Thepresent formulation is fully isoparametric(Kim and Paulino, 2002).

2.13.4.3.3 Cohesive fracture simulation

Numerical analyses of crack growth for bothcompact tension, C(T), and single-edge notchbend, SE(B), specimens of a titanium/titaniummonoboride (Ti/TiB) FGM have been per-formed in Jin et al. (2002). The materialproperties for the Ti and TiB are summarizedin Table 6. The Young’s modulus and Pois-son’s ratio of the Ti/TiB FGM are estimated bythe self-consistent method (Hill, 1965).

Figure 26 illustrates the SE(B) specimengeometry and Table 7 provides the geometricparameters for the SE(B) specimen. A layeredFGM version of the SE(B) specimen has been

tested in Carpenter et al. (1999). From amodeling point of view, the FGM compositionvaries from 100% TiB at the cracked surface to100% Ti at the uncracked surface. Thus thevolume fraction of Ti varies from zero at thecracked surface to one at the uncrackedsurface.

The finite element models consist of eight-node isoparametric solid elements and theeight-node interface-cohesive elements. Due to

Table 6 Material properties of Ti and TiB.

MaterialsE

(GPa) nJc

(KJ/m2)scmet

(MPa)dcmet

(mm)sccer

(MPa)dccer(mm)

Ti 107 0.34 150 620 0.089TiB 375 0.14 0.11 4.0 0.01

Figure 25 Eight-node bilinear interface element.

R

B

W

L / 2L / 2

ab0

0

P, ∆

Figure 26 SE(B) specimen geometry.

Table 7 Geometric parameters of SE(B) specimen.

SpecimenL

(mm)W

(mm)B

(mm) a0/WR

(mm)

SE(B) 79.4 14.7 7.4 0.3 10.2


symmetry considerations, only one-quarter ofthe specimen is considered. Interface-cohesiveelements are placed only over the initialuncracked ligament and have a uniform lengthof 0.1 mm for the SE(B) specimen. The finiteelement model has 10 uniform layers ofelements over the half thickness for theSE(B) specimen. Figure 27 shows the frontview of the finite element mesh for the SE(B)specimen.

The SE(B) specimen is loaded by openingdisplacements applied uniformly through thethickness at the specimen center plane. In thelayered FGM SE(B) specimen tested (Carpen-ter et al., 1999), the first layer of the specimenconsists of 15% Ti and 85% TiB, while the lastlayer (7th layer) consists of 100% Ti. Crackinitiation occurred at a measured load of 920Newtons (N). The experimental results showthat load increases with crack extension duringthe initial growth and then decreases withfurther crack extension. The measured loadcorresponding to a crack growth of 5mm isB1200N. Figure 28(a) shows the volume

fraction of Ti in this TiB/Ti specimen. Thedotted (stepped) line shows the propertygradation in the tested specimen. A least-squares approximation yields the power ex-ponent n¼ 0.84 in the metal volume fractionfunction used in the analysis

VmetðYÞ ¼ ðY=WÞn ð133Þ

Figure 28(b) shows the numerical results of theload versus crack extension responses for theSE(B) specimen with bmet¼ 16 and n¼ 0.84.For the bmet selected, the crack initiation loadagrees quite closely with the experimentallymeasured value. Compared with the experi-mental observations after the crack initiation(Carpenter et al., 1999), the discrepancy in thetrend of load versus crack extension responsearises because the finite element analysis doesnot consider plasticity in the backgroundmaterial. When the plasticity effect is takeninto account, it is expected that the trend ofthe load versus crack extension will be moreconsistent with the experimental observations

b0

symmetry plane

P, ∆

Figure 27 Typical mesh for analyses of SE(B) specimen (longitudinal center plane of the quarter-symmetric3D model).

0.0 0.2 0.4 0.6 0.8 1.0Y / W

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Vol

ume

frac

tion

of T

i

layered continuously graded(n = 0.84)

(Y/W) n

0 1 2 3 4 5Crack extension (mm)

0

1

2

3

Load

(kN

)

plane stress3dplane strain

n = 0.84

crack initiation load [25]

β = 16met

(a) (b)

Figure 28 (a) Volume fraction of Ti in the TiB/Ti FGM and (b) load-crack extension response for the SE(B)Ti/TiB specimen with a0/W¼ 0.3, B¼ 7.4mm, n¼ 0.84 (after Jin et al., 2002).


(the calibrated value of bmet may be largerthan 16). Figure 28(b) also shows the numer-ical results of the load versus crack extensionfor plane-stress and plane-stress models.Although there are no differences betweenthe 2D and 3D responses, it is expected thatsignificant differences may develop with plas-ticity in the background material due tovariations in crack front constraint and crackfront tunneling.

Figure 29 shows the effect of bmet and n onthe load versus crack extension responses forthe SE(B) specimen. The power exponent n(shape index of the metal volume fraction) is0.5 in Figure 29(a) and 1.0 in Figure 29(b). Fora given bmet, the load decreases steadily withcrack extension. For a given crack extension, alarger bmet reduces the load. This is expectedbecause a larger bmet corresponds to a lowercohesive traction. Finally, the load becomeslower for larger n. This is because a larger nmeans a lower metal volume fraction, andtherefore, a lower cohesive traction. Becausebmet has a pronounced effect on the load versuscrack extension responses, one may expect tocalibrate the values of bmet from experimentalobservations.

2.13.5 SOME EXPERIMENTALASPECTS

Fracture experiments play a critical role todevelop an understanding of the influence ofproperty gradients on the mechanisms of crackgrowth and on the fracture toughness of thesematerials. In addition, carefully designed frac-ture experiments are needed to calibrate theparameters used in numerical methods (see,e.g., Section 2.13.4.3.3). The presentationbelow focuses on fracture testing of engineer-ing FGMs (e.g., metal/ceramic) with particular

emphasis on an effective technique to generatea sharp precrack in those materials.

2.13.5.1 Metal/Ceramic FGM FractureSpecimen

A functionally graded Ti/TiB material wasprepared by Cercom Incorporated, Vista, CA,using a commercially pure (CP) titanium plateand six tape-cast mixtures of titanium (Ti) andtitanium diboride (TiB2) powders (Nelson andEzis, 1996). The assembled green material wasplaced in a graphite die in an induction furnaceand heated to 5001C in vacuum to removeorganic tape materials. The material was thenheated further to 1,1001C to remove surface-absorbed gases from TiB2 and then heated to1,3051C under a pressure of 13.8MPa. Sub-sequent X-ray diffraction revealed that reac-tions between Ti and TiB2 during the 36 hpressing and cooling time resulted in theformation of titanium monoboride (TiB) withless than 1% of residual TiB2. The composi-tions of the layers in the FGM varied from15%Ti/85%TiB on the brittle side to pure Tion the ductile side. The intermediate composi-tions of the FGM were chosen to reduce thethermal residual stress during cooling from thesintering temperature as described by Goochet al. (1999). Single-edged-notched bend speci-mens (SE(B)) of the FGM were cut usingelectro-discharge machining (EDM) with themachined notch on the side containing thehighest concentration of TiB. The notch wasoriented for crack propagation through thelayers toward the CP–Ti. The specimens weremachined to a width of 14.73mm, a thicknessof 7.37mm and a length of 82.55mm, with anotch 5.08mm deep (initial a/W¼ 0.345) withintegral clip gage knife-edges in accordancewith ASTM E1820-96 (ASTM, 1996), asillustrated by Figure 30.


0

2

4

6

Load

(kN

)

β = 1β = 3β = 5

n = 0.5

met

met

met


0

2

4

6

Load

(kN

)

β = 1β = 3β = 5

n = 1.0

met

met

met

(a) (b)

Figure 29 Load-crack extension response for the SE(B) Ti/TiB specimen with a0/W¼ 0.3, B¼ 7.4mm: (a)n¼ 0.5 and (b) n¼ 1.0 (after Jin et al., 2002).


2.13.5.2 Precracking of Metal/CeramicFGMs

In metal/ceramic FGMs, cracks originatingin the brittle, ceramic side are prevented fromcatastrophic propagation through the increasein fracture toughness as the metal contentincreases. Fracture toughness testing of me-tallic/ceramic FGM SE(B), with the startingnotch in the ceramic-rich side, requires theinitiation of a sharp, short precrack at aspecific desired location in a brittle material.A review of fracture toughness testing ofbrittle materials by Sakai and Bradt (1993)notes that creation of a starter crack of propersharpness and length is critical for accuratefracture toughness measurements. The stan-dard for fracture toughness testing of ad-vanced ceramic materials describes therequirement of a sharp, well-characterizedprecrack as well as the difficulty in obtainingone (ASTM, 1997). Use of machined notchesor blunt cracks can lead to fracture toughnessvalues much larger than those obtained fromspecimens with sharp precracks. The single

edge precracked beam (SEPB) method, basedon standard four-point bending with the notchtip in tension, can lead to longer than desiredpopped-in precracks in multilayered FGMs(Carpenter et al., 2000) even when a rigidtesting machine limits displacement. Longprecracks can also give values of fracturetoughness that are too high due to crack wakeeffects. The method of axial compressionprecracking developed by Ewart and Suresh(1986) for ceramic materials can lead toextensive microstructural damage in metallic/ceramic FGMs and can give low values offracture toughness (Carpenter, 1999). Belowwe describe a method for generating a short,sharp precrack at the root of a machined notchin the brittle side of an FGM due to Carpenteret al. (2000), and the technique is illustratedwith a Ti/TiB FGM.

2.13.5.3 Precracking by Reverse Four-pointBending

The SE(B) specimens were cyclically loadedin reverse four-point bending using a 6,800N

150.00

150.

00

8.00

4.00

4.00

Ti

TiB richGraded Layers

(a) (b)79.38

14.73

10.2210.22

Layer 1

39.69 39.69

1.52

30 deg6.35

1.27

Layer 7

60 deg

P

Figure 30 Seven-layer Ti/TiB FGM: (a) plate provided by CERCOM Inc. and (b) fracture specimen(7.37mm thick) cut from the plate for three-point bending test.

P/2 P/2

14.73

16.05

15% Ti/85% TiB

CP Ti

10.22 10.22

79.38

Figure 31 Seven-layer Ti/TiB reverse four-point bending specimen. The composition for each layer is (1) 15/85, (2) 21/79. (3) 38/62, (4) 53/47, (5) 68/32, (6) 85/15, and (7) 100/0 where the number in parentheses indicatesthe layer number (all dimensions in mm, after Carpenter et al., 2000).

Some Experimental Aspects 639

load, resulting in initiation of a controlled,short, sharp crack at the machine notch. Theloading configuration of the specimens isshown in Figure 31. The load was applied witha compressive load ratio (R¼ 10) at a cyclicfrequency of 5Hz. The load was chosen todevelop the precrack in B5,000 cycles. Thetarget crack length was 0.2mm from thestarting notch which gave the desired a/W¼ 0.358 for the fracture experiments. Thestarting notch and resulting precrack devel-oped by reverse four-point bending fatigue areshown by optical micrographs, Figures 32(a)and (b), respectively. Average modulus values,

measured before and after the precrackingwere 145GPa, indicating that the overallmicrostructure of the FGM was undamagedby reverse four-point bending. After heattinting at 4001C, the specimens were cooledovernight to room temperature and fracturetested in three-point bending according toASTM E1820-96 (ASTM, 1996).

2.13.5.4 Test Results

The load versus CMOD data for J–R tests ofthe beams precracked by uniform axial com-pression, reverse four-point bending and anuncracked specimen are shown in Figure 33.The beam without a precrack exhibited rapid,unstable growth when loads were increased tothe point of crack initiation and gave no usefuldata for the FGM layers near the machinednotch.

The specimen precracked by uniform axialcompression showed stable crack growth withno pop-in during the J–R test indicating that asharp crack was initiated during compressiveloading. The load values and the resultant Jvalues, however, were only 70% of thoseobtained by the specimen that was precrackedwith reverse four-point bending. The lowervalues for that specimen are due to extensivemicrocracking during application of the axialcompressive loads. Figure 34 shows the Jversus crack extension result for the reversefour-point bend precracked specimen (Carpen-ter et al., 2000). The result demonstrates arising R-curve behavior as the metal contentincreases in successive layers of the FGM.

0

500

1000

1500

2000

0 0.02 0.04 0.06 0.08 0.1

CMOD (mm)

Lo

ad (

N)

Axial Compression

Reverse 4-point Bending

Not Pre-cracked

Figure 33 Load versus CMOD plot for Ti/TiB J–R tests (after Carpenter et al., 2000).

50µm

50µm

(a)

(b)

Figure 32 Face of FGM SE(B) specimen: (a) beforeprecracking and (b) precracked in reverse four-pointbending fatigue (after Carpenter et al., 2000).


2.13.6 CONCLUDING REMARKS ANDPERSPECTIVES

Scientific research in FGMs considers, in alarge sense, functions of gradients in materialscomprising thermodynamic, chemical, mechan-ical, optical, electromagnetic, and/or biologicalaspects. The present work suggests a balancedapproach involving modeling, synthesis, andexperiments. It shows that our understandingon failure of FGMs is increasingly clear,however, it also indicates areas for futuredevelopment such as investigation of con-straint effects in fracture, full (i.e., for theentire range of material composition) experi-mental characterization of engineering FGMsunder static and dynamic loading, developmentof fracture criteria with predictive capability,multiscale (space and time) failure considera-tions, and connection of research with actualindustrial applications.

ACKNOWLEDGMENTS

This work was supported by the NASA-Ames ‘‘Engineering for Complex System Pro-gram,’’ and the NASA-Ames Chief Engineer(Dr. Tina Panontin) through grant NAG 2-1424. Additional support for this work wasprovided by the National Science Foundation(NSF) under grant CMS-0115954 (Mechanicsand Materials Program).

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0 2 5

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2.13 FailureofFunctionallyGradedMaterials

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