analysis of a transiently propagating crack in functionally graded materials under mode i and ii

International Journal of Engineering Science 47 (2009) 852–865

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International Journal of Engineering Science

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Analysis of a transiently propagating crack in functionally gradedmaterials under mode I and II

Kwang Ho Lee *

School of Mechanical and Automotive Engineering, Kyungpook National University, Sangju, Kyeongbuk 742-711, South Korea

a r t i c l e i n f o

Article history:Received 14 January 2009Received in revised form 29 April 2009Accepted 5 May 2009Available online 5 June 2009

Keywords:Transiently propagating crackAccelerating and decelerating crackDynamic stress intensity factorStress and displacement fieldsCrack kinking angle

0020-7225/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.ijengsci.2009.05.004

* Tel.: +82 54 530 1404.E-mail addresses: [email protected], [email protected]

a b s t r a c t

Crack tip stress and displacement fields for a transiently propagating crack along gradientin functionally graded materials (FGMs) with a linear variation of shear modulus are devel-oped. The higher order terms of the transient stress and displacement fields at crack tipwere obtained by transforming the general partial differential equations of the dynamicequilibrium into Laplace’s equations whose solutions have harmonic functions. Thus, thefields can be expressed very simply. Using these stress components, isochromatics andthe first invariant at crack tip are generated.

The results show that the isochromatics (constant maximum shear stress) for mode Icrack tilt backward around the crack tip with an increase of crack tip acceleration_cðdc=dtÞ, and tilt forward around the crack tip with an increase of rate of change ofdynamic mode I stress intensity factor _K IðdK I=dtÞ. The isochromatics for mixed mode crackmove to upper direction with an increases of _K I and _K II , and lower direction with anincrease of _c. Contours of the first stress invariant for mode I crack enlarge around the cracktip with an increase of _c, and decrease around the crack tip with an increase of _K I . As _K IðIIÞ

decreases at crack initiation, the predicted kinking angles increase. As _c increases, the pre-dicted kinking angles also increase.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) were developed mainly to satisfy the demand of ultra-high-temperature environ-ment and to eliminate the stress singularities [1,2]. But recently, they are also developed for various purposes [3,4] and havethe various types of spatial variation of the material composition to meet the application purposes. Thus, FGM has a mediumwith varying elastic and physical properties and fracture analysis is needed. Considerable amounts of theoretical, numericaland experimental works have been reported on static or steady state dynamic fracture behaviors of FGMs. However, thetransient dynamic fracture of FGMs has received much less attention.

The behavior of propagating cracks in FGMs has also attracted some attention. Following an earlier study by Atkinson [5],several groups [6–9] have investigated the behavior of propagating cracks in FGMs. Jiang and Wang [10] developed the open-ing and sliding displacement for a propagating crack in FGM in which the properties were assumed to vary exponentiallyperpendicular to the crack propagation direction. Meguid et al. [11] provided a theoretical analysis for a finite crack prop-agating in an infinite inhomogeneous medium. Ma et al. [12] analyzed for crack propagating in a functionally graded stripunder the plane loading. Lee et al. [13] developed nonhomogeneity specific terms for stress and displacement fields usingdisplacement potentials under thermo-mechanical loading. In experimental studies for FGMs, Yao et al. [14] and Rousseau

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K.H. Lee / International Journal of Engineering Science 47 (2009) 852–865 853

and Tippur [15] studied the fracture by caustic and CGS method respectively when the crack propagates along the elasticgradient in FGM under mode I loading. Abanto-Bueno and Lambros [16] studied the fracture at crack imitation under mixedmode state and showed that the crack could propagate along gradient in FGM. In all these studies, the fracture analysis wasconducted under assumption that the crack propagates at constant velocity in steady state.

However, the crack propagates at acceleration or deceleration with changing stress intensity factor, especially, the accel-eration or deceleration would be more predominant at crack initiation or stopping. When the cracks propagate at constant,increasing or decreasing velocity, even if their instant velocities are same, the stresses at the propagating crack tip will bedifferent due to characteristics of crack speed. For a propagating transient crack in isotropic materials, many researcheshad been investigated by Tsi [17], Kostrov [18], Freund [19], Yang et al. [20], Nishioka [21] and Ing and Ma [22]. In thesestudies, the propagating crack tip fields with transient parameters which are the rate of change of stress intensity factorðdKD=dt ¼ _KDÞ and the crack tip acceleration ðdc=dt ¼ _cÞ are very complex. Thus, most experimental studies have been usednot the transient fields but the steady state fields ð _KD ¼ 0; _c ¼ 0Þ even if the crack is in the transient state. However, the suit-able crack tip fields with actual situation will have to be used for more efficient fracture analysis. FGMs with a linearly vary-ing elastic property under almost constant Poisson’s ratio [23–25] are developed to meet the application purposes.Nevertheless, the crack tip fields for a transiently propagating crack in FGM with a linearly varying elastic property havenot yet been reported.

In this paper, the stress and displacement fields around a transiently propagating crack tip in FGMs having a linear var-iation of shear modulus are developed. The effect of Poisson’s ratio on the deformation is much less than that of Young’smodulus. Thus, Poisson’s ratio of the plates is assumed to be constant [26]. The transient elastodynamic problem for FGMis formulated in terms of displacement potentials and the solution is obtained through an asymptotic analysis. In this study,the higher order terms of the transient stress and displacement fields at crack tip are obtained by transforming the generalpartial differential equations of the dynamic equilibrium into Laplace’s equations whose solutions have harmonic functions.Thus, the fields can be expressed simply. Using the stress fields developed in this study, effects of acceleration, decelerationand rates of change of stress intensity factors on the isochromatics, the first stress invariant and kinking angle in FGMs arediscussed.

2. Stress and displacement fields for a transient crack propagating under inplane loading

2.1. Formation of equilibrium equations

Shear modulus l a FGM are assumed to vary linearly as l ¼ l0ð1þ 1XÞ, whereas Poisson’s ratio m and density q is as-sumed to be constant. Then, the relationship between stresses and strains can be written as

rXX ¼ ½ðk0 þ 2l0ÞeXX þ k0eYY �ð1þ 1XÞ; rYY ¼ ½k0eX þ ðk0 þ 2l0ÞeYY �ð1þ 1XÞ; sXY ¼ l0cXY ð1þ 1XÞ ð1Þ

where X is the reference coordinate, rij and eij the inplane stress and strain components, k0 and l0 denote Lame’s constants atX ¼ 0 and 1 is a nonhomogeneous parameter. The displacements u and v which are obtained from dilatational and shearwave potentials U and W can be expressed by Eq. (2)

u ¼ oUoXþ oW

oY; v ¼ oU

oY� oW

oXð2Þ

The equilibrium in the dynamic state is given by Eq. (3)

orXX

oXþ osXY

oY¼ q

o2uot2 ;

osXY

oXþ orYY

oY¼ q

o2vot2 ð3Þ

Substituting Eq. (2) into Eq. (1), and substituting Eq. (1) into Eq. (3), the equations for the dynamic state can be obtained as

o

oXð1þ 1XÞðkþ 2Þr2U� q

l0

o2Uot2

( )þ o

oYð1þ 1XÞr2W� q

l0

o2Wot2

( )� 21

o2U

oY2 �o2WoX oY

( )¼ 0 ð4aÞ

o

oYð1þ 1XÞðkþ 2Þr2U� q

l0

o2Uot2

( )� o

oXð1þ 1XÞr2W� q

l0

o2Wot2

( )þ 21

o2W

oY2 þo2U

oX oY

( )¼ 0 ð4bÞ

where k ¼ k0=l0. As shown in Fig. 1, when a crack propagates at speed cðtÞ in fixed coordinates ðX;YÞ, the moving crack tipcoordinates ðx; yÞ are x ¼ X � aðtÞ; y ¼ Y . Where aðtÞ is the half crack length of a center crack or the crack length of an edgecrack. The equations for the dynamic state can be obtained as

a2lo2Uox2 þ

o2Uoy2 þ bx

o2Uox2 þ

o2Uoy2

!þ 2b

kþ 2oWoyþ q

loðkþ 2Þ_coUoxþ 2c

o2Uoxot

� o2Uot2

!¼ 0 ð5aÞ

a2so2Wox2 þ

o2Woy2 þ bx

o2Wox2 þ

o2Woy2

!� 2b

oUoyþ q

lo

_coWoxþ 2c

o2Woxot

� o2Wot2

!¼ 0 ð5bÞ

0.00 0.05 0.10 0.15 0.200.0

0.5

1.0

1.5

2.0

2.5

3.0

ζ = 0/m

μ=1.32(1+ζX)

: ζ = 4/m : ζ = -4/m

She

ar m

odul

us, μ

(GPa

)

X Location from left side of specimen (m)

Fig. 1. Variation of shear modulus l with X location.

854 K.H. Lee / International Journal of Engineering Science 47 (2009) 852–865

where al ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðc=clÞ2

q;as ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðc=csÞ2

q; cs ¼

ffiffiffiffiffiffiffiffiffiffiffilc=q

p;lc ¼ l0ð1þ a1Þ; bða; 1Þ ¼ 1l0=lc ¼ 1=ð1þ a1Þ. cl ¼ cs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1� mÞ=ð1� 2mÞ

pfor plane strain and cl ¼ cs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=ð1� mÞ

pfor plane stress. lc is shear modulus at crack tip. c; cl and cs are the crack propagation

velocity, elastic dilatational wave velocity and elastic shear wave velocity at the crack tip. To obtain an asymptotic expansionof the fields around the crack tip, we assume the general solutions of Eq. (5) for Uðzl; tÞ and Wðzs; tÞ as follows

Unðzl; tÞ ¼ �Z

/nðzl; tÞdzl; Wnðzs; tÞ ¼ �Z

wnðzs; tÞdzs ð6Þ

Analytical function /nðzl; tÞ and wnðzs; tÞ can be expended by power series as

/nðzl; tÞ ¼X1n¼1

Anzn=2l ; wnðzs; tÞ ¼

X1n¼1

Bnzn=2s ð7Þ

where An ¼ Aon þ iA�n; Bn ¼ Bo

n þ iB�n; zlðsÞ ¼ xþmlðsÞy. An and Bn are complex constants to be determined from boundary condi-tions, and ml and ms are roots of characteristic equations (see Eqs. (10) and (19)) which show the characteristics of physicalproperties, nonhomogeneity and transients. The concept of characteristic roots was introduced first by this author. Substi-tuting Eqs. (6) and (7) into Eq. (5), the structure of Eq. (5) can be expressed as Eq. (8)

a2lo2Un

ox2 þo2Un

oy2 þ bxo2Un

ox2 þo2Un

oy2

!¼ � 2b

kþ 2oWn�2

oy� 2c1=2

c2l

o

otc1=2 oUn�2

ox

� �þ 1

c2l

o2Un�4

ot2 ð8aÞ

a2so2Wn

ox2 þo2Wn

oy2 þ bxo2Wn

ox2 þo2Wn

oy2

!¼ 2b

oUn�2

oy� 2c1=2

c2s

o

otc1=2 oWn�2

ox

� �þ 1

c2s

o2Wn�4

ot2 ð8bÞ

when n < 0;Un ¼ Wn ¼ 0.

2.2. Stress and displacement fields for n ¼ 1; 2

From Eq. (8), the equilibrium equation for n ¼ 1;2 can be written as

a2lo2Un

ox2 þo2Un

oy2 þ bxo2Un

ox2 þo2Un

oy2

!¼ 0 ð9aÞ

a2so2Wn

ox2 þo2Wn

oy2 þ bxo2Wn

ox2 þo2Wn

oy2

!¼ 0 ð9bÞ

Substituting Eqs. (6) and (7) into Eq. (9), Eq. (10) can be obtained as

n2½a2

l þm2l þ bxð1þm2

l Þ�Anzn=2�1l

n o¼ 0;

n2½a2

s þm2s þ bxð1þm2

s Þ�Bnzn=2�1s

n o¼ 0 ð10Þ


Thus, ml and ms in Eq. (10) can be obtained as follows

ml ¼ ial; ms ¼ ias ð11Þ

In the case of n ¼ 1;U1 and W1 are the displacement potentials very close the crack tip ðr ! 0Þ. Thus,

For n ¼ 1 : al ¼ al; as ¼ as ð12Þ

For n ¼ 2 : al ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2

l þ bxÞ=ð1þ bxÞq

; as ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2

s þ bxÞ=ð1þ bxÞq

ð13Þ

When n ¼ 1, Eq. (9) is Laplace’s equation in complex domain zlðsÞ ¼ xþ ialðsÞy and the same as that for a homogeneous mate-rial and can be rewritten as ða2

l þm2l ÞU

00nðzl; tÞ ¼ 0 and ða2

s þm2s ÞW

00nðzs; tÞ ¼ 0. Thus, the solutions for Uðzl; tÞ and Wðzl; tÞ in Eq.

(9) can be expressed as U1ðzl; tÞ ¼ �ReR

/1ðzl; tÞdzl and W1ðzs; tÞ ¼ �ImR

w1ðzs; tÞdzs, respectively. Substituting the differen-tiation of U1 and W1 into Eq. (2), the displacements u and v for n ¼ 1 can be expressed as Eq. (14)

u ¼ �Re /1ðzl; tÞ þ asw1ðzs; tÞf g; v ¼ Imfal/1ðzl; tÞ þ w1ðzs; tÞg ð14Þ

Substituting the differentiation of Eq. (14) into Eq. (1), the stress rij for n ¼ 1 can be expressed as Eq. (15)

rxx1 ¼ �lRe ð1þ 2a2l � a2

s Þ/01ðzl; tÞ þ 2asw

01ðzs; tÞ

� �; ryy1 ¼ lRe ð1þ a2

s Þ/01ðzl; tÞ þ 2asw

01ðzs; tÞ

� �;

sxy1 ¼ lIm 2al/01ðzl; tÞ þ ð1þ a2

s Þw01ðzs; tÞ

� �ð15Þ

Substituting Eq. (7) into Eq. (15), applying traction free boundary conditions on the crack surface, A1 and B1 can be obtainedas

Ao1ðtÞ ¼ �

2lc

ffiffiffiffiffiffiffi2pp BIðcÞKo

nðtÞ; A�1ðtÞ ¼2

lc

ffiffiffiffiffiffiffi2pp BIIðcÞK�nðtÞ; Bo

1ðtÞ ¼ ho1Ao

1; B�1ðtÞ ¼ �h�1A�1ðtÞ ð16Þ

where BIðcÞ ¼ 1þa2s

4alas�ð1þa2s Þ

2 ;BIIðcÞ ¼ 2as

4alas�ð1þa2s Þ

2 ;ho1 ¼

2al1þa2

s;h�1 ¼

1þa2s

2asand lc is the shear modulus at crack tip. Ko

1ðtÞ and K�1ðtÞ are

stress intensity factors K IðtÞ and K IIðtÞ, respectively. The fields for n ¼ 2 can be obtained using the same method as those ofn ¼ 3;4, and the structure of stress and displacement fields are the same as Eqs. (28) and (29) in next section. In Eqs. (28) and

(29), ho2ðc; _c; tÞ ¼ 2ð1�a2

lÞ�ð1�a2

s Þð1�a2lÞ

2asð1�a2lÞ ;h�2ðc; _c; tÞ ¼ 2al

ð1þa2s Þ

, where alðsÞ is shown in Eq. (13).

2.3. Stress and displacement fields for n ¼ 3; 4

For n P 3 in Eq. (8), the Unðzl; tÞ and Wnðzs; tÞ have nonhomogeneous and transient terms, and n = 3 and 4 are consideredto generate the fields in this study. The relation between Un�2ðzl; tÞ and Wn�2ðzs; tÞ in Eq. (8) can be written as [27]

o

oyWn�2ðzs; tÞ ¼ �ðkþ 2Þ o

oxUn�2ðzl; tÞ;

1kþ 2

o

oxWn�2ðzs; tÞ ¼

o

oyUn�2ðzl; tÞ ð17Þ

Substituting Eq. (17) into Eqs. (8a) and (8b), Eqs. (8a) and (8b) can be written as

a2lo2Un

ox2 þo2Un

oy2 þ bxo2Un

ox2 þo2Un

oy2

!¼ 2b

oUn�2

ox� 2c1=2

c2l

o

otc1=2 oUn�2

ox

� �; n ¼ 3;4 ð18aÞ

a2so2Wn

ox2 þo2Wn

oy2 þ bxo2Wn

ox2 þo2Wn

oy2

!¼ 2b

kþ 2oWn�2

ox� 2c1=2

c2s

o

otc1=2 oWn�2

ox

� �; n ¼ 3;4 ð18bÞ

Substituting Eq. (7) into Eq. (18), Eq. (18) can be expressed as Eq. (19).

n2½a2

l þm2l þ bxð1þm2

l Þ�Anzn=2�1l ¼ 2bAn�2zn=2�1

l � 2c1=2

c2l

o

otc1=2An�2zn=2�1

l

� � ; n ¼ 3;4 ð19aÞ

n2½a2

s þm2s þ bxð1þm2

s Þ�Bnzn=2�1s ¼ 2b

kþ 2Bn�2zn=2�1

s � 2c1=2

c2s

o

otc1=2Bn�2zn=2�1

s

� � ; n ¼ 3;4 ð19bÞ

Thus, when n ¼ 3;4;ml and ms can be expressed as Eq. (20)

ml ¼ ial; ms ¼ ias; ð20Þ


where

al ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

l þ bx� koð�Þn

2n

2b� cc2

l

_ccþ 2

_Aoð�Þn�2

Aoð�Þn�2

þ dlðnÞ" #" # !,

ð1þ bxÞ

vuutas ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

s þ bx� koð�Þn

2n

2bkþ 2

� cc2

s

_ccþ 2

_Boð�Þn�2

Bon�2þ dsðnÞ

" #" # !,ð1þ bxÞ

vuut
For mode I
kon ¼ Ao

n�2=Aon ¼ Bo

n�2=Bon; dlð3Þ ¼

2 _al

alsin2 hl

2; dsð3Þ ¼

2 _as

ascos2 hs

2; dlð4Þ ¼ 0; dsð4Þ ¼

2 _as

as

_Aon�2 ¼ �

2lc

ffiffiffiffiffiffiffi2pp 2as _asD� ð1þ a2

s Þ _D

D2 Kon�2 þ

1þ a2s

D_Ko

n�2

" #

_Bon�2 ¼

2lc

ffiffiffiffiffiffiffi2pp 2 _alD� 2al

_D

D2 Kon�2 þ

2al

D_Ko

n�2

" #; kþ 2 ¼ ð1� a2

s Þ=ð1� a2l Þ

For mode II

k�n ¼ A�n�2=A�n ¼ B�n�2=B�n; dlð3Þ ¼2 _al

alcos2 hl

2; dsð3Þ ¼

2 _as

assin2 hs

2; dlð4Þ ¼

2 _al

al; dsð4Þ ¼ 0

_A�n�2 ¼2

lc

ffiffiffiffiffiffiffi2pp 2 _asD� 2as

_D

D2 K�n�2 þ2as

D_K�n�2

" #

_B�n�2 ¼ �2

lc

ffiffiffiffiffiffiffi2pp 2as _asD� ð1þ a2

s Þ _D

D2 K�n�2 þ1þ a2

s

D_K�n�2

" #

D ¼ 4alas � ð1þ a2s Þ

2; _D ¼ �4c _c

as

alc2l

þ al

asc2s� 1þ a2

s

c2s

�_c ¼ dc=dt; _as ¼

�c _casc2

s; _al ¼

�c _calc2

l

; _Koð�Þn ¼ dKoð�Þ

n =dt; Ko1 ¼ Kd

I ; K�1 ¼ KdII

The unit of kon and k�n is length. mjð¼ iajÞ in Eq. (20) is dependent on crack propagation velocity ðcÞ, acceleration ð _cÞ, rate of

change of the stress intensity factor ½ _KðdK=dtÞ�, physical properties ðcl; csÞ, nonhomogeneity ð1Þ and angle ðhÞ at the crack tip.In this stage, one assumes that mj is independent of x and y around crack tip ðr ! 0Þ. When ml and ms are the same as Eq.(20), Eq. (18) can expressed as Eq. (21)

a2lo2Unðzl; tÞ

ox2 þ o2Unðzl; tÞoy2 ¼ 0; a2

so2Wnðzs; tÞ

ox2 þ o2Wnðzs; tÞoy2 ¼ 0 ð21Þ

where zj ¼ xþ iajy and aj is dependent on transients in addition to physical properties and crack propagation velocity. Eq.(21) is also Laplace’s equation in complex domain zj and can be rewritten as ða2

l þm2l ÞU

00nðzlÞ ¼ 0 and ða2

s þm2s ÞW

00nðzsÞ ¼ 0.

Thus ml ¼ ial and ms ¼ ias. When n > 3, Eq. (18) can be transformed into Laplace’s equations as Eq. (21), and the solutionsof Laplace’s equations have harmonic functions. Thus, Unðzl; tÞ and Wnðzs; tÞ can be written as

Unðzl; tÞ ¼ �ReZ

/nðzl; tÞdzl; Wnðzs; tÞ ¼ �ImZ

wnðzs; tÞdzs ð22Þ

/nðzlÞ and wnðzsÞ are dependent on the acceleration (dc/dt) and the rate of change of stress intensity factor ðdK=dtÞ in additionto the crack propagation velocity and the physical properties. Substituting the differentiation of UnðzlÞ and WnðzsÞ in Eq. (22)into Eq. (2), the displacements u and v for n ¼ 3;4 can be expressed as Eq. (23)

u ¼ �Re /nðzl; tÞ þ aswnðzs; tÞf g; v ¼ Im al/nðzl; tÞ þ wnðzs; tÞf g ð23Þ

Substituting the differentiation of Eq. (23) into Eq. (1), the stress rij for n ¼ 3;4 can be expressed as Eq. (25)

rxxn ¼ �lRe1� a2

s

1� a2l

ð1� a2l Þ þ 2a2

l

�/0nðzl; tÞ þ 2asw

0nðzs; tÞ

ryyn ¼ lRe �1� a2

s

1� a2l

ð1� a2l Þ þ 2

�/0nðzl; tÞ þ 2asw

0nðzs; tÞ

sxyn ¼ lIm 2al/

0nðzl; tÞ þ ð1þ a2

s Þw0nðzs; tÞ

� �ð24Þ

The /nðzl; tÞ and wnðzl; tÞ can be written as


/nðzl; tÞ ¼X4

n¼3

½bAonðtÞ þ bA�nðtÞ�zn=2

l ; wnðzl; tÞ ¼X4

n¼3

½bBonðtÞ þ bB�nðtÞ�zn=2

s ð25Þ

Substituting Eq. (25) into Eq. (24), the stress fields become

rxxn ¼ �X4

n¼3

ln2

bAonðtÞ

1� a2s

1� a2l


l

�rðn�2Þ=2

l cosn� 2

2

� �hl þ 2bBo

nðtÞasrðn�2Þ=2s cos

n� 22

� �hs

þX4

n¼3

l n2

bA�nðtÞ 1� a2s

1� a2l


l

�rðn�2Þ=2

l sinn� 2

2

� �hl þ 2bB�nðtÞasrðn�2Þ=2

s sinn� 2

2

� �hs

ryyn ¼X4

n¼3

l n2

bAonðtÞ �

1� a2s

1� a2l

ð1� a2l Þ þ 2

�rðn�2Þ=2

l cosn� 2

2

� �hl þ 2bBo

nðtÞasrðn�2Þ=2s cos

n� 22

� �hs

�X4

n¼3

l n2

bA�nðtÞ �1� a2s

1� a2l

ð1� a2l Þ þ 2

�rðn�2Þ=2

l sinn� 2

2

� �hl þ 2bB�nðtÞasrðn�2Þ=2

s sinn� 2

2

� �hs

sxyn ¼X4

n¼3

l n2

bAonð2alÞrðn�2Þ=2

l sinn� 2

2

� �hl þ bBo

nðtÞð1þ a2s Þrðn�2Þ=2

s sinn� 2

2

� �hs

þX4

n¼3

l n2

bA�nð2alÞrðn�2Þ=2l cos

n� 22

� �hl þ bB�nðtÞð1þ a2

s Þrðn�2Þ=2s cos

n� 22

� �hs

ð26Þ

Applying traction free boundary conditions on the crack surface, bAonðtÞ; bBo

nðtÞ; bA�nðtÞ and bB�nðtÞ can be obtained as

bAonðtÞ ¼ �

2lc

ffiffiffiffiffiffiffi2pp bBIðc; _c; tÞbK o

nðtÞ; bA�nðtÞ ¼ 2lc

ffiffiffiffiffiffiffi2pp bBIIðc; _c; tÞbK �nðtÞ; bBo

nðtÞ ¼ �honðc; _c; tÞbAo

nðtÞ; bB�nðtÞ ¼ �h�nðc; _c; tÞbA�nðtÞð27Þ

where

bBIðc; _c; tÞ ¼ ð1þ a2s Þð1� a2

l Þ4alasð1� a2

l Þ þ ð1þ a2s Þ½ð1� a2

s Þð1� a2l Þ � 2ð1� a2

l Þ�;

bBIIðc; _c; tÞ ¼ 2asð1� a2l Þ

4alasð1� a2l Þ þ ð1þ a2

s Þ½ð1� a2s Þð1� a2

l Þ � 2ð1� a2l Þ�;

ho3ðc; _c; tÞ ¼ h�4ðc; _c; tÞ ¼ 2al

ð1þ a2s Þ; h�3ðc; _c; tÞ ¼ ho

4ðc; _c; tÞ ¼ 2ð1� a2l Þ � ð1� a2

s Þð1� a2l Þ

2asð1� a2l Þ

Substituting Eq. (27) into Eq. (26), applying bAonðtÞ ¼ go

n�2Aon�2ðtÞ=ko

nðtÞ and bA�nðtÞ ¼ g�n�2A�n�2=k�n, the stress fields rijn can be ob-tained as Eq. (28)

rxxn ¼ ð1þ bxÞX4

n¼3

nKonðtÞffiffiffiffiffiffiffi2pp BIðcÞ

1� a2s

1� a2l


l

�r

n�22

l cosn� 2

2

� �hl � 2ho

nasrn�2

2s cos

n� 22

� �hs

þ ð1þ bxÞX4

n¼3

nK�nðtÞffiffiffiffiffiffiffi2pp BIIðcÞ

1� a2s

1� a2l


l

�r

n�22

l sinn� 2

2

� �hl � 2h�nasr

n�22

s sinn� 2

2

� �hs

ryyn ¼ ð1þ bxÞX4

n¼3

nKonðtÞffiffiffiffiffiffiffi2pp BIðcÞ

1� a2s

1� a2l

ð1� a2l Þ � 2

�r

n�22

l cosn� 2

2

� �hl þ 2ho

nasrn�2

2s cos

n� 22

� �hs

þ ð1þ bxÞX4

n¼3

nK�nðtÞffiffiffiffiffiffiffi2pp BIIðcÞ

1� a2s

1� a2l

ð1� a2l Þ � 2

�r

n�22

l sinn� 2

2

� �hl þ 2h�nasr

n�22

s sinn� 2

2

� �hs

sxyn ¼ ð1þ bxÞX4

n¼3

nKonðtÞffiffiffiffiffiffiffi2pp BIðcÞ �2al r

n�22

l sinn� 2

2

� �hl þ ð1þ a2

s Þhonr

n�22

s sinn� 2

2

� �hs

þ ð1þ bxÞX4

n¼3

nK�nðtÞffiffiffiffiffiffiffi2pp BIIðcÞ 2alr

n�22

l cosn� 2

2

� �hl � ð1þ a2

s Þh�nr

n�22

s cosn� 2

2

� �hs

ð28Þ

where Ko3ðtÞ ¼ go

1K IðtÞ=ko3;K

o4ðtÞ ¼ go

2Ko2ðtÞ=ko

4;K�3ðtÞ ¼ g�1K IIðtÞ=k�3;K

�4ðtÞ ¼ g�2K�2ðtÞ=k�4,

rl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ ðalyÞ2

q; rs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ ðasyÞ2

q; hl ¼ tan�1½aly=x�; hs ¼ tan�1½asy=x�


Substituting Eq. (25) into Eq. (23), the displacement fields can be obtained as

un ¼X4

n¼3

KonðtÞBIðcÞ

lo expð1aÞ

ffiffiffiffi2p

rr

n2l cos

n2

� �hl � ash

onr

n2s cos

n2

� �hs

n oþX4

n¼3

K�nðtÞBIIðcÞlo expðaxÞ

ffiffiffiffi2p

rr

n2l sin

n2

� �hl � ash

�nr

n2s sin

n2

� �hs

n ovn ¼

X4

n¼3

KonðtÞBIðcÞ

lo expð1aÞ

ffiffiffiffi2p

r�al r

n2l sin

n2

� �hl þ ho

3rn2s sin

n2

� �hs

n oþX4

n¼3

K�nðtÞBIIðcÞlo expð1aÞ

ffiffiffiffi2p

ralr

n2l cos

n2

� �hl � h�nr

n2s cos

n2

� �hs

n oð29Þ

Substituting alðsÞ of Eqs. (12) and (13) into Eqs. (28) and (29), Eqs. (28) and (29) become the fields for n ¼ 1 and 2. Finally, thefields rij and uðvÞ for a propagating transient crack in FGM under inplane loading are given in Eq. (30)

rij ¼X4

n¼1

rijn; uðvÞ ¼X4

n¼1

unðvnÞ ð30Þ

The higher orders terms of n P 2 are affected by terms of the nonhomogeneous parameter 1. The effects of the crack tipacceleration _c and the rate of change of stress intensity factor _K do not appear in the singular and remote stress termsðn ¼ 1;2Þ, but they appear in the higher order terms greater than or equal the third order ðn P 3Þ. In results [20,21] for tran-sient crack in isotropic materials, the higher order terms ðn P 3Þ in the fields are very complex and the steady and transientterms are mixed. However, the higher order terms ðn P 3Þ in this study have only transient terms and are much simpler thantheirs [20,21] by transforming the general partial differential equations of the dynamic equilibrium into Laplace’s equationswhose solutions have harmonic functions.

3. Discussion on solutions

3.1. Effects of transient factors _c; _K on isochromatics

In order to investigate the effects of the transient terms on a dynamic fracture, contours of constant maximum shearstress (isochromatics) are generated for the opening and mixed mode using the terms n = 1 and 3 in Eq. (30). Isochromaticsat each point around crack tip are generated by the stress optic law (Eq. (31)) combined with stress fields.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðrxx � ryyÞ2 þ 4s2xy

q¼ Nfr=h ð31Þ

where N is the fringe order, h the plate thickness and fr the material fringe constant. The material thickness ðhÞ used in gen-erating the contours is 9.5 mm and material fringe constant fr = 15 (kN/m-fringe). The assumed physical propertieslðXÞ ¼ 1:32ð1þ 1XÞ ðGPaÞ;v ¼ 0:33;qðXÞ ¼ 1200 ð kg=m3Þ and 1 ¼ 4=m. The variation of elastic modulus for this FGM isshown in Fig. 1.

Fig. 2 shows the stress components in the vicinity of the crack tip propagating at acceleration, deceleration or constantvelocity. The crack tip acceleration will occur greatly at crack initiation. After accelerating, the crack will propagate almostconstant velocity and will be stopping (deceleration). Thus, the normalized crack velocity of Mðc=csÞ 6 0:4 is selected in otherto evaluate the effect of acceleration or deceleration. In Refs. [14,15], the crack can propagates along property gradient underopening mode, even if the crack is mixed mode state, the crack can propagate along gradient under a low ratio of K II=K I [16].Fig. 12b for single edge crack tension fracture experiments in this study also shows that the crack can propagate along gra-dient direction, along this path K II is not equal to zero.

Fig. 3 shows the effect of the rate of change of mode I stress intensity factor _K I for a stationary crack tip (M ¼ 0:02Þ under_c ¼ 0; 1 ¼ 4=m;ko

3 ¼ 10�3 m and Ko3ðtÞ ¼ 0:1K IðtÞ=m. The crack length a ¼ 0:01 m. Even if _K I ¼ 0, the contours, due to nonho-

mogeneity away from crack tip, tilt away the crack face. Generally, when 1 > 0 (modulus increases ahead of the crack), the

Fig. 2. Stress components in the vicinity of the propagating crack tip.

Fig. 3. Effect of rate of change of mode-I SIF for a stationary crack tip (M = 0.02) under K IðtÞ ¼ 1:0 MPaffiffiffiffiffimp

; _c ¼ 0; 1 ¼ 4=m; ko3 ¼ 0:001 m and

Ko3ðtÞ ¼ 0:1K IðtÞ=m.


fringes tilts forward at crack tip where as for 1 <0, the fringes tilts backward. As _K I increases, the fringes tilt more forwardaround the crack tip, and the size of fringes away from crack tip increases.

Fig. 4 shows the effect of crack tip acceleration _c for a crack tip propagating with M = 0.2 under _K I ¼ �105 MPaffiffiffiffiffimp

=s,1 ¼ 4=m; ko

3 ¼ 0:01 m and Ko3ðtÞ ¼ 0:5K IðtÞ=m. When a rate of change of velocity compared to crack tip speed is too high,

Eq. (20) cannot be applied. However, when the low value of ko3 is used in Eq. (20), a higher _c can be applied in Eq. (20). It

can be seen in Fig. 4 that the fringes tilt backward around the crack tip with increasing crack tip acceleration. This variationoccurs greatly at a high rate of change of velocity. Generally, the isochromatics of mode I at propagating crack tip obtained bysteady state fields tilt more towards the crack face (backward) compared to those for a stationary crack. It seems that thebehavior occurs due to the increase of crack tip speed, that is, the crack tip acceleration. Thus, it is natural that the isochro-matics of mode I obtained by transient fields should tilt more backward around the crack tip as _c increases. On the otherhand, if a crack propagates at deceleration, the isochromatic fringes tilt forward around the crack tip.

From Figs. 3 and 4, one can knows that the isochromatics are affected by crack tip speed c, crack tip acceleration _c and rateof change of dynamic stress intensity factor ð _K IÞ. The isochromatics of mode I tilt forward around the crack tip as _K I increasesand tilt backward around the crack tip as _c increases. These phenomena are the same as Nishioka’s [21] for a transientlypropagating crack in homogeneous materials.

Fig. 5 shows the effect of nonhomogeneity for a crack tip propagating with M ¼ 0:2 under _K IðtÞ ¼ �105 MPaffiffiffiffiffimp

= s; _c ¼�2� 107 m=s2. This case is that a crack is stopping. When 1 ¼ �10=m, the fringes tilts backward around the crack tip, as1 increases, the fringes tilt more forward.

Fig. 6 shows the effect of the rate of change of mixed stress intensity factors _K IðtÞ and _K IIðtÞ for a propagating cracktip ðM ¼ 0:4Þ under _c ¼ 0;K IðtÞ ¼ 0:76 MPa

ffiffiffiffiffimp

;K IIðtÞ ¼ 0:18 MPaffiffiffiffiffimp

;Ko2ðtÞ ¼ �0:5K IðtÞ=

ffiffiffiffiffiffiffim

p;Koð�Þ

3 ðtÞ ¼ K IðIIÞðtÞ=m;koð�Þ3 ¼

0:01 m and 1 ¼ 4=m. In experiment of Ref. [16], when the crack propagates along the inclined direction to elastic gradientunder asymmetric external loading, the crack is continually curved. This means that the propagating crack is a mixed mode

Fig. 4. Effect of crack tip acceleration for a crack propagating with M ¼ 0:2 under K IðtÞ ¼ 1:0 MPaffiffiffiffiffimp

; _K IðtÞ ¼ �105 MPaffiffiffiffiffimp

=s; 1 ¼ 4=m; ko3 ¼ 0:01 m and

Ko3ðtÞ ¼ 0:5K IðtÞ=m.

Fig. 5. Effect of nonhomogeneity for a crack propagating with M ¼ 0:2 under K IðtÞ ¼ 1:0 MPaffiffiffiffiffimp

; _K IðtÞ ¼ �105 MPaffiffiffiffiffimp

=s; _c ¼ �2� 107 m=s2; ko3 ¼ 0:01 m

and Ko3ðtÞ ¼ 0:5K IðtÞ=m.

Fig. 6. Effect of crack tip acceleration for a crack propagating with M ¼ 0:4 under _c ¼ 0; K IðtÞ ¼ 0:76MPaffiffiffiffiffimp

; K IIðtÞ ¼ 0:18 MPaffiffiffiffiffimp

; Ko2ðtÞ ¼ �0:5K IðtÞ=ffiffiffiffiffi

mp

; Koð�Þ3 ðtÞ ¼ K IðIIÞðtÞ=m; koð�Þ

3 ¼ 0:01 m and 1 ¼ 4=m.


state and cannot propagate to elastic gradient direction. Actually, Eq. (30) can be applied when the crack propagates alongthe elastic gradient direction. However, if the value of 1a(1 times a) and the angle x between crack direction and elastic gra-dient one are small, the fields for the crack propagating along property gradient can be applied sufficiently for fracture anal-ysis because the effect of x on stress fields around crack tip and stress intensity factor K IðIIÞ is very small [28]. The value of 1ain Figs. 6–8 is 0.04 and �0:1 (a ¼ 0:01 m;1 ¼ 4=m or �10=m). Even if 1a ¼ �0:1, Eq. (30) has sufficient accuracy in0� < a < 25�. Under the mixed mode loading, the crack propagates with having the stress intensity factor K II. Even if K II com-pared to K I is small, it cannot be ignored. As known in Fig. 6, as _K IðIIÞðtÞ increases, the fringes move to upper direction.

Fig. 7 shows the effect of crack tip acceleration _c for a crack tip propagating with M = 0.4 under, _K IðIIÞ ¼104 MPa

ffiffiffiffiffimp

=s;1 ¼ 4=m;koð�Þ3 ¼ 0:01 m and Koð�Þ

3 ðtÞ ¼ K IðtÞ=m. As the acceleration increases, the fringes move to lowerdirection.

Fig. 8 shows the effect of nonhomogeneity for a crack propagating with M ¼ 0:4 under _c ¼ �4� 107 m=s2;_K IðIIÞðtÞ ¼ �104 MPa

ffiffiffiffiffimp

=s. As known in Fig. 8, the isochromatics of mixed mode are also affected by nonhomogeneity. When1 ¼ �10=m, the fringes move backward around the crack tip. As 1 increases, the fringes move more forward of crack tip.

3.2. Effects of transient factors _c; _K on the first stress invariant

Figs. 9 and 10 show the effects of the transient terms on the contours of first invariant stress ½ðrxx þ ryyÞ;MPa� undermode I loading. As known in Figs. 9 and 10, as _K IðtÞ increases, the sizes of contours decrease in front of the crack tip, andas _c increases, they increase.

Fig. 11 shows the effect of nonhomogeneity for a crack tip propagating with M ¼ 0:4 under_K IðtÞ ¼ �105 MPa

ffiffiffiffiffimp

= s; _c ¼ �4� 107 m=s2. As 1 increases, the size of fringes increases around the crack tip.

Fig. 7. Effect of crack tip acceleration for a crack propagating with under _K IðIIÞðtÞ ¼ 104 MPaffiffiffiffiffimp

=s; K IðtÞ ¼ 0:76 MPaffiffiffiffiffimp


; Ko2ðtÞ ¼

�0:5K IðtÞ=ffiffiffiffiffimp

; Koð�Þ3 ðtÞ ¼ K IðIIÞðtÞ=m; koð�Þ

3 ¼ 0:01m and 1 ¼ 4=m.

Fig. 8. Effect of nonhomogeneity for a crack propagating with M ¼ 0:4 under K IðtÞ ¼ 0:76 MPaffiffiffiffiffimp


; Ko2ðtÞ ¼ �0:5K IðtÞ=

ffiffiffiffiffimp

;_K IðIIÞðtÞ ¼ �104 MPa

ffiffiffiffiffimp

=s; Koð�Þ3 ðtÞ ¼ K IðIIÞðtÞ=m; _c ¼ �1� 107 m=s2 and koð�Þ

3 ¼ 0:01 m.


; _c ¼ 0; 1 ¼ 4=m; ko3 ¼ 0:01 m and Ko

3ðtÞ ¼ K IðtÞ= m.


3.3. Effects of transient factors _c and _K on the direction of crack propagation

The direction of crack propagation has also attracted some attention. Following an earlier study by Griffith [29], severalgroups have investigated the direction of crack propagation. Maximum tangential stress ðrhhÞmax [30], maximum energy


; _K IðtÞ ¼ 105 MPaffiffiffiffiffimp

=s; 1 ¼ 4=m; ko3 ¼ 0:01 m and

Ko3ðtÞ ¼ K IðtÞ=m.

Fig. 11. Effect of nonhomogeneity for a crack propagating with M ¼ 0:4 under K IðtÞ¼1:0MPaffiffiffiffiffimp

; _K IðtÞ¼�105 MPaffiffiffiffiffimp

=s; _c¼�4�107m=s2; ko3¼0:01m and

Ko3ðtÞ¼K IðtÞ=m.


release rate ðGÞmax [31], corrected maximum tangential stress [32] included roxðrxx2or T-stress) and minimum strain energydensity ðSÞmin [33] criterions have been used for study on the direction of crack propagation. In all these studies, the rate ofchange of dynamic stress intensity factor _K IðIIÞ and the crack tip acceleration _c were not considered. In homogeneous mate-rials, the direction of crack propagation is affected by stress intensity factors K I;K II and T-stress roxðrxx2Þ. Specially, the role ofrox is important to predict the kinking angle [34] under mixed mode loading. However, if the direction of crack propagationis analyzed at crack tip ðr ¼ 0Þ, it is not affected by rox. Thus, the direction of crack propagation will be analyzed around thecrack tip where r – 0. Where r can only be determined by comparison with experiments. In nonhomogeneous materials, thedirection of crack propagation is affected by K I;K II;rox and local physical property. The property in FGMs varies around thecrack tip. If the material gradient is very steep, the crack would be forced to grow along a direction where the tangentialstress rhh and energy release rate G are not maximum or strain energy density S is not minimum. In such a case, the criterionsðrhhÞmax; ðGÞmax and ðSÞmin would be comparison of available stress quantities with the local toughness values. However, inFig. 12, the properties on the two crack faces are almost the same. After initiating, the crack will propagates continuouslyalong lower toughness direction without deviation if _c or _K IðIIÞ is constant during crack propagating. In this case, the kinkingangle can be predicted by maximum rhh [16].

Table 1 shows the kinking angles of Fig. 12a conducted in Abanto-Bueno and Lambros’ experiment [16]. The values of K IðIIÞ

and rox (T-stress) at the instant just before crack initiation obtained by the experiment are used for obtaining the crack kink-ing angle a in this study. _K IðIIÞ and _c are also used for obtaining a. As known in Table 1, the kinking angles are dependent of_K IðIIÞ and _c in addition to K IðIIÞ and rox (T-stress). r in this study is 0.002 m. When r < 0:002 m;r ¼ rc which is core region andexpected plastic zone. When _c and _K IðIIÞ are zero, the predicted kinking angle a is from�23:8�ðrc ¼ 0Þ to�24:2�ðrc ¼ 4 mmÞ in0 < rc < 4 mm. When rc ¼ 11 mm;a ¼ �28:3� and it approaches actual kinking angle a ¼ �28� � 1:5� in the experiment. Ifrc ¼ 11 mm, it is too long when one considers that the core region is near at crack tip. Thus, we cannot but consider _c

Fig. 12. Mixed mode fracture specimen under uniform displacement loading Vo .

Table 1The predicted kinking angle a at crack initiation ðM ¼ 0:02Þ in Fig. 12a.

_c ¼ 0 _c ¼ 3� 106 ðm=s2Þ _c ¼ 5� 106 ðm=s2Þ_K IðIIÞ104 ðMPa

ffiffiffiffiffimp

=sÞ a ð�Þ _K IðIIÞ104 ðMPaffiffiffiffiffimp


=sÞ a ð�Þ

0 � �8 �24:5� 0:5 0 � �12 �27:5� 0:5 0 � �1 �28:5� 0:5�9 � �20 �25:5� 0:5 �13 � �30 �28:5� 0:5 �2 � �20 �29:5� 0:5�21 � �34 �26:5� 0:5 �31 � �52 �29:5� 0:5 �21 � �44 �30:5� 0:5�35 � �51 �27:5� 0:5 �53 � �83 �30:5� 0:5 �45 � �79 �31:5� 0:5

Fig. 12a [16]: K IðtÞ ¼ 0:755 MPaffiffiffiffiffimp


; roxðtÞ ¼ �0:069 MPa; Koð�Þ3 ðtÞ ¼ 0:1K IðIIÞðtÞ=m; ko

3 ¼ 10�3 m; 1 ¼ �5:4=m; m ¼ 0:45; lc ¼ 0:12GPa; q ¼ 910 kg=m3.Initial crack length ao ¼ 2:6 cm; rc ¼ 2 mm, Wide = 7 cm, Height = 9 cm, Thickness = 0.406 mm. Actual kinking angle a ¼ �28� � 1:5� .

Table 2The predicted kinking angle a at crack initiation ðM ¼ 0:02Þ in Fig. 12b.

_c ¼ 0 _c ¼ 5� 106m=s2 _c ¼ 1:1� 107m=s2

_K IðIIÞ104 ðMPaffiffiffiffiffimp



=sÞ a ð�Þ

0 � �16 �18:5� 0:5 0 � �20 �19:5� 0:5 0 � �20 �20:5� 0:5�17 � �39 �19:5� 0:5 �21 � �44 �20:5� 0:5 �21 � �46 �21:5� 0:5�40 � �64 �20:5� 0:5 �45 � �71 �21:5� 0:5 �47 � �74 �22:5� 0:5�65 � �92 �21:5� 0:5 �72 � �102 �22:5� 0:5 �75 � �107 �23:5� 0:5

Fig. 12b: K IðtÞ ¼ 0:995 MPaffiffiffiffiffimp

;KðIItÞ ¼ 0:21 MPaffiffiffiffiffimp

;roxðtÞ ¼ �1:025 MPa;1 ¼ �4=m. ko3 ¼ 10�3 m;m ¼ 0:39;lc ¼ 0:82 GPa;q ¼ 1200 kg=m3: ao ¼ 2:3 cm;

rc ¼ 2 mm.Wide = 10 cm, Height = 17 cm, Thickness = 7.05 mm. Actual kinking angle a ¼ �20:5� � 0:5� .


and _K IðIIÞ at crack initiation. Actually, at dynamic crack initiation, _c is very high plus value and _K IðIIÞ is very high minus one.However, the crack initiation in the experiment is almost quasi-static state. Thus, it seems that _c and _K IðIIÞ have low values atinitiation. Anyway, when _c and _K IðIIÞ are considered, the predicted kinking angle more approaches actual one. As _K IðIIÞ de-creases, that is, when the dynamic stress intensity factors decrease more suddenly at crack initiation, the predicted kinkingangles increase. As _c increases, the predicted kinking angles also increase.

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

: μ(X) = 0.9(1-4X)

E(X) = -9.75X+2.49

ν(X) = 0.385X+0.38 Sh

ear m

odul

us (G

pa)

X location from left side of specimen (m)

Fig. 13. Variation of shear modulus l with X location in the epoxy FGM plate.


Table 2 shows the kinking angle of Fig. 12b conducted under displacement loading of 10 cm/min in this dynamic exper-iment. The functionally graded material was developed in previous study [35]. The variation of shear modulus from left sidein Fig. 12b, lðXÞ ¼ 0:9ð1þ 1XÞ ðGPaÞ, where 1 ¼ �4=m. As shown in Fig. 13, the shear modulus varies linearly. 1ao (ao is ini-tial crack length) is �0.092 and x is 20�. In this case, the stress fields in this study can be applied sufficiently for fractureanalysis. The fracture parameters K IðtÞ;K IIðtÞ and roxðtÞ at just before crack initiation are obtained by Newton–Raphsonand least square method in the photoelasticity. When the transient terms _K IðIIÞ ¼ ð�2 � �4Þ � 105 MPa

ffiffiffiffiffimp

=s and_c ¼ 5� 106 m=s at crack initiation, the predicted kinking angles more approach the actual ones. Most researches have beenconsidered not the transient fields but the steady state fields ð _KD ¼ 0; _c ¼ 0Þ even if the crack is a transient state. To moreaccurate estimates of dynamic fracture, the transient terms included _c and _K IðIIÞ must be considered.

4. Conclusion

In this study, the fields for a transiently propagating crack in FGM with a linearly varying elastic property are developed.Using the transient stress components, the contours of the isochromatics and the first invariant stress around the propagat-ing crack are generated for crack speeds, accelerations, rates of change of stress intensity factors and nonhomogeneity. Thecrack kinking angles under mixed mode loading are obtained from maximum rhh. The results are as follows;

(1) The isochromatic fringes of mode I tilt backward around the crack tip with increase of crack tip acceleration _c and tiltforward around the crack tip with an increase of rate of change of dynamic mode I stress intensity factor ð _K IÞ. The iso-chromatic fringes of mixed mode move to upper direction with increases of _K I and _K II, and lower direction with anincrease of _c.

(2) Contours of the first stress invariant of mode I state enlarge around the crack tip with an increase of _c and decreasearound the crack tip with an increase of _K I.

(3) As 1 increases, the contours of the isochromatics tilt or move to ahead of the crack tip and the size of the first stressinvariant increases.

(4) As _K IðIIÞ decreases, that is, when dynamic stress intensity factors decrease more suddenly at crack initiation, the pre-dicted kinking angles increase. As _c increases, the predicted kinking angles also increase. To more accurate estimatesof dynamic fracture, the transient terms included _c and _K IðIIÞ must be considered.

References

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analysis of a transiently propagating crack in functionally graded materials under mode i and ii

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