Transient response of an edge interfacial crack in bonded dissimilar strips with a functionally graded interlayer under an antiplane shear impact

Accepted Manuscript Transient response of an edge interfacial crack in bonded dissimilar strips with a functionally graded interlayer under an antiplane shear impact Hyung Jip Choi PII: DOI: Reference:

S0167-8442(17)30125-8 http://dx.doi.org/10.1016/j.tafmec.2017.07.026 TAFMEC 1931

To appear in:

Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

14 March 2017 12 July 2017 31 July 2017

Please cite this article as: H.J. Choi, Transient response of an edge interfacial crack in bonded dissimilar strips with a functionally graded interlayer under an antiplane shear impact, Theoretical and Applied Fracture Mechanics (2017), doi: http://dx.doi.org/10.1016/j.tafmec.2017.07.026

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Transient response of an edge interfacial crack in bonded dissimilar strips with a functionally graded interlayer under an antiplane shear impact Hyung Jip Choi School of Mechanical Engineering, Kookmin University, Seoul 02707, Republic of Korea E-mail: [email protected]; Tel: +82 2 910 4682; Fax: +82 2 910 4839

The transient response of an edge interfacial crack in bonded media with a functionally graded interlayer is investigated under the condition of an antiplane shear impact. The graded interlayer is assumed to follow power-law variations of the shear modulus and mass density between two dissimilar, homogeneous semi-infinite strips. Based on the use of Laplace and Fourier integral transforms, the crack problem is formulated in terms of a singular integral equation with a generalized Cauchy kernel, which is solved by the expansion-collocation technique in the Laplace domain. The time-dependent crack-tip behavior is determined through an inverse Laplace transform and the values of the mode III stress intensity factors are obtained as a function of time. The numerical results include the variations of such dynamic stress intensity factors for various combinations of the material and geometric parameters of the bonded system; more specifically, the effects of shear modulus, mass density, layer thickness, and their interactions on the dynamic overshoot characteristics of the transient crack-tip behavior in the presence of the graded interlayer are examined.

Keywords: Bonded dissimilar strips, Functionally graded interlayer, Edge interfacial crack, Singular integral equation, Dynamic stress intensity factors

1. Introduction The utilization of nonhomogeneous media as a transitional interlayer to join dissimilar bulk constituents has proved to be a highly viable application in engineering practice; this has become possible by the progress made in the area of functionally graded materials that feature gradual spatial variations of their thermophysical properties [1]. From the fracture mechanics point of view, however, it is imperative to understand how the joining of such constituents with the insertion of a graded interphase is degraded by crack-like defects that may exist around the interfacial zone under different loading conditions [2–4]. Due to material systems often being exposed to external loadings that are dynamic in nature, various elastodynamic crack problems entailing the properties of functionally graded materials have been also reported upon. Among them are, under an antiplane deformation mode, the impact response of a crack and that of two collinear cracks in a graded layer bonded to dissimilar half-planes [5,6]; the effects of property gradients and their directions on the dynamic behavior of a crack in a functionally 1|Page

graded material [7]; a transient analysis of periodic surface cracking in a graded coating [8]; the stress wave scattering induced by an interfacial crack for a semi-infinite homogeneous strip bonded to a finite graded strip [9]; multiple cracks in an orthotropic strip with a graded coating under timeharmonic excitation [10]; multiple cracks parallel or perpendicular to the boundary in a graded orthotropic half-plane under impact [11]; the dynamic problem of an interfacial crack with a graded coating [12]; the impact behavior of a crack perpendicular or at an arbitrary angle to a graded interfacial zone in bonded media [13–15]; and the torsional impact of a penny-shaped crack in a graded interlayer between two dissimilar half-spaces [16]. For inplane fracture problems, the solutions may include the impact response of a periodic array of cracks and that of an inclined crack in an infinite graded material [17,18] and a dynamic analysis of a crack crossing the interface in a functionally graded layered structure and that of a surface crack in a graded coating [19,20]. Multilayered approaches have also been proposed to tackle some transient crack problems with arbitrarily graded mechanical properties, based on piecewise constant, power, and exponential functions [21–23]. Most of the prior research efforts made to resolve interfacial crack problems, which involve graded properties subjected to time-dependent loading conditions, appear to have focused on embedded cracks. Interfacial failures may arise from a free surface; therefore, the objective of this paper is to investigate the impact response of an edge interfacial crack in bonded dissimilar semi-infinite strips with a functionally graded interlayer. Under the state of antiplane deformation, the shear modulus and mass density of the graded interlayer are assumed to vary in the form of power functions; as such, Laplace and Fourier integral transforms are employed to reduce the crack problem to the solution of a singular integral equation with a generalized Cauchy kernel in the Laplace domain. The mode III stress intensity factors are defined, followed by a Laplace inversion to recover their time-dependence. The values of such transient stress intensity factors are provided as a function of time so that the dynamic load transfer and the ensuing overshoot characteristics of the crack-tip response can be addressed for various combinations of the material and geometric parameters of the bonded system.

2. Problem statement and formulation Consider two homogeneous, but dissimilar semi-infinite strips bonded through a functionally graded interlayer. As shown in Fig. 1, the constituents of this bonded system are distinguished in order from the top; the constituents have thicknesses of hj, j=1,2,3, and an edge crack of length c is located along the nominal interface at y=h2. After the shear moduli and mass densities of the homogeneous strips are denoted by (j,j), j=1,3, those of the graded interlayer (2,2) can be approximated in terms of power functions [16, 24]

2 ( y)  3 1   y  , 2 ( y)  3 1   y  

 2

(1) 2|Page

where  and  are material gradation parameters specified so as to make the transition of these properties continuous across the nominal interfaces as



1 / 3  1 / 3 h2 1 / 3

, 

2ln( 1 / 3 ) ln( 1 / 3 )  ln( 1 / 3 )

(2)

from which arbitrary nonhomogeneous material combinations can be simulated in the bonded system. However, when an exponential-law approximation is used for the graded properties as in [5–12, 17– 20], both the shear modulus and mass density should retain the identical form of spatial variations such that the ratio 2(y)/2(y) remains constant. This does not appear to be physically representative enough, but just to allow for the corresponding dynamic crack problems to be analytically tractable. With wj(x,y,t), j=1,2,3, referring to the non-vanishing displacement components in the z-direction under the assumption of antiplane deformation, the stress components are given by

 jxz ( x, y, t )   j

w j x

,  jyz ( x, y, t )   j

w j y

; j  1,2,3

(3)

and the governing equations of motion are written as

 j 2 wj  wj  ; j  1,3  j t 2

(4)

3  2 w2  w2  w2   1   y y 3 (1   y)2 t 2

(5)

2

2

where t is the time and 2 is a Laplacian operator in the variables x and y. If the bonded system, which is initially at rest, is suddenly loaded by antiplane tractions applied on the crack surfaces and its left-hand side flanks at x=0 are traction-free, then together with the initial conditions w j ( x, y,0)  0,

w j t

( x, y,0)  0 ; j  1,2,3

(6)

a relevant set of boundary and interface conditions is prescribed as

1xz (0, y, t )  0 ; h2  y  h1  h2

(7)

 2 xz (0, y, t )  0 ; 0  y  h2

(8)

 3 xz (0, y, t )  0 ;  h3  y  0

(9)

1 yz ( x, h1  h2 , t )  0,  3 yz ( x, h3 , t )  0 ; x  0

(10)

1 yz ( x, h2 , t )   2 yz ( x, h2 , t ),  2 yz ( x,0, t )   3 yz ( x,0, t ) ; x  0

(11)

w2 ( x,0, t )  w3 ( x,0, t ) ; x  0

(12)

w1 ( x, h2 , t )  w2 ( x, h2 , t ) ; x  c

(13)

3|Page

(14)

 2 yz ( x, h2 , t )  f ( x) H (t ) ; 0  x  c

where f(x) is the crack surface traction and H(t) is the Heaviside unit step function. Upon defining the Laplace transform over the time variable t such that [25] 

w*j ( x, y, p)   w j ( x, y, t ) e  pt dt ; j  1,2,3

(15)

0

where p is the Laplace variable, the time-dependence can be eliminated from the formulation of the problem. The Fourier transform is then applied over the variable x to yield general solutions for the displacements in the Laplace domain [26]

w*j ( x, y, p)  w2* ( x, y, p) 

1  2

2

1   y  

A  2



0





0

j1

e

j y

 s

A21I 

 

 Aj 2 e

 j y

 cos sx ds ;

j  1,3



 s



 

(1   y )   A22 K 

(16) 

(1   y )  cos sx ds 

(17)

in which s is the Fourier variable, Ajk(s,p), j=1,2,3, k=1,2, are arbitrary unknowns, and I[ ] and K[ ] are the modified Bessel functions of the first and second kind, respectively, with j(s,p), j=1,3, and

(p):

 j ( s, p )  s 2 

j 2 p ; j  1,3 j

3  p  1    ( p)        2    3  2

(18)

2

(19)

and it can be shown from Eqs. (3), (16), and (17) that the conditions in Eqs. (7)-(9) are satisfied. To account for the existence of edge interfacial cracking, an auxiliary function is introduced in the Laplace domain to replace the mixed conditions in Eqs. (13) and (14) as

 * ( x, p) 

 *  w1 ( x, h2 , p)  w2* ( x, h2 , p)  ; x  0 x

(20)

and through subsequent applications of Eqs. (10)-(12) and (20), the expressions of Ajk(s,p), j=1,2,3, k=1,2, can be obtained in terms of *(x,p), which is to be evaluated from the remaining crack surface condition as in Eq. (14).

3. Integral equation and dynamic stress intensity factors After substituting the expressions of Ajk(s,p), j=1,2,3, k=1,2, into Eq. (3), accompanied by some algebraic manipulations and an inverse Fourier transform, the traction component along the cracked interface can be written in the form as

lim  2* yz ( x, y, p) 

y  h2

21





c

0

L( x, r , p)  * (r , p) dr ; x  0

(21)

where L(x,r,p) is a kernel function such that 4|Page



L( x, r , p)   ( s, p)sin sr cos sx ds 0

 ( s, p ) 

1  X 1 (s, p)22 (s, p)  Y1 ( s, p)21 ( s, p)  2 1h1 )   (1  e s  11 ( s, p)22 ( s, p)  12 ( s, p)21 ( s, p) 

(22) (23)

in which Δlk(s,p), l,k=1,2, are given by

11 (s, p)  11 (s, p)   21 (s, p) e21h1

(24)

12 (s, p)  12 (s, p)   22 (s, p) e21h1

(25)

21 (s, p)  31 (s, p) e23h3   41 (s, p)

(26)

22 (s, p)  32 (s, p) e23h3   42 (s, p)

(27)

with the contractions for lk(s,p), l=1,2,3,4, k=1,2, as

11 (s, p)  X1 (s, p)  1 (1   h2 ) I (s1 )

(28)

12 (s, p)  Y1 (s, p)  1 (1   h2 ) K (s1 )

(29)

 21 (s, p)   X1 (s, p)  1 (1   h2 ) I (s1 )

(30)

 22 (s, p)  Y1 (s, p)  1 (1   h2 ) K (s1 )

(31)

31 (s, p)  X 2 (s, p)  3 I (s2 )

(32)

32 (s, p)  Y2 (s, p)  3 K (s2 )

(33)

 41 (s, p)   X 2 (s, p)  3 I (s2 )

(34)

 42 (s, p)  Y2 (s, p)  3 K (s2 )

(35)

X j (s, p)   I (s j )   s j I 1 (s j ) ; j  1,2

(36)

Yj (s, p)   K ( s j )   s j K 1 ( s j ) ; j  1,2

(37)

where =α(1−β+2ν)/2, s1=s(1+αh2)/α, and s2=s/α. For dissimilar quarter planes (h1=h3=∞) bonded with a graded interlayer, boundedness conditions are imposed instead of traction-free conditions in Eq. (10) such that w1(x,+∞,t)=0 and w3(x,−∞,t)=0. In this case, the integrand (s,p) in Eq. (23) can be simplified to the following expression:

 ( s, p ) 

1  X1 ( s, p) 42 ( s, p)  Y1 ( s, p) 41 ( s, p)    s  11 ( s, p) 42 ( s, p)  12 ( s, p) 41 ( s, p) 

(38)

Note that (s,p) in Eq. (23) or (38), which is not dependent solely on the transform variables s and p but also on the physical parameters of the bonded system, possesses an asymptote as s tends to infinity such that

lim ( s, p)  s 

1   2 8s

(39)

which gives rise to the singular behavior of the kernel L(x,r,p) in Eq. (22). 5|Page

As a result, upon separating the leading term in the kernel and making use of Fourier representations of generalized functions [27]

 

 0

 0

sin s(r  x) ds 

1 rx

(40)

1  rx sin s(r  x) ds  s 2 rx

(41)

one can derive a singular integral equation in the Laplace domain as



c

1

1

   0 r  x rx 

 rx  2 f ( x)  ; 0 xc    k ( x, r , p)   * (r , p) dr    8  rx rx  1 p 

  r  x

(42)

where k(x,r,p) is a bounded kernel such that  1   k ( x, r , p)  4  ( s, p)   sin sr cos sx ds 0 2 8s  

(43)

and the simple Cauchy kernel, 1/(rx), plus the term 1/(rx) that also becomes unbounded at the end point x=r=0 constitute a generalized Cauchy singular kernel [28]. Because the dominant part of the integral equation is yet owing to a simple Cauchy kernel, the square-root singular nature at the tip of an edge interfacial crack can be preserved in the Laplace domain by expressing the auxiliary function *(r, p) as [29]

 * (r , p) 

g (r , p) ; 0r c cr

(44)

where g(r,p) is an unknown function, and in the normalized interval x=c(1+ξ)/2, r=c(1+η)/2, and −1<(ξ, η)<1, the solution to the integral equation can be expanded into a series of Chebyshev polynomials of the first kind Tn such that

 * ( , p) 

1 p 1 



a n 0

n

Tn ( ) ;   1

(45)

in which an, n0, are the coefficients to be determined. After substituting Eq. (45) into Eq. (42) and using the following integral identity [30]:



1

1

Tn ( )d 1   (   )



1

1

Tn ( )  Tn ( ) 1   (   )

d 

Tn ( ) 1

ln

2  1 2  1

; n  0,   1

(46)

the Cauchy-type singular integral is regularized and the resulting functional equation can be recast into a system of linear algebraic equations for an, 0≤n≤N by truncating the series expansion at n=N and applying the zeros of TN+1(ξ) as a set of collocation points [28]

   2k  1   TN 1 (k )  0, k  cos    ; 0  k  N  2  N  1 

(47)

together with the involved bounded integrals treated via the appropriate Gaussian quadratures [31]. The integral equation in Eq. (42) can thereafter provide the singular traction ahead of the crack tip 6|Page

in the Laplace domain. To measure the degree of severity of such local stress intensification, the stress intensity factor is first defined and can be evaluated as

K III* ( p)  lim 2( x  c)  2* yz ( x, h2 , p)   x c

1 2p

N

c

a

n

(48)

n 0

and the inverse Laplace transform to recover its time-dependence can then be numerically implemented based on the algorithm developed by Stehfest [32]:

K III (t ) 

ln 2 M m  Vm K III*  ln 2  ; t  0  t m 1 t 

(49)

where KIII(t) is the mode III dynamic stress intensity factor for a specific time instant t, M is an even integer, and Vm is given by Vm  (1)m  M / 2

min( m , M / 2)



k  (1 m )/ 2

k M / 2 (2k )! ( M / 2  k )!k !(k  1)!(m  k )!(2k  m)!

(50)

For each time instant (t≠0), a total of M problems should be solved over the Laplace variable p=mln2/t, m=1, …, M. Note that, at large times, the values of such transient stress intensity factors would converge to the elastostatic solutions; these static limits can be obtained from the final-value theorem as [25]

lim KIII (t )  ( KIII )static  lim p KIII* ( p) t 

(51)

p 0

4. Results and discussion Under the condition of impact loading applied uniformly on the crack surfaces as f(x)=o in Eq. (14), the evolutions of the dynamic stress intensity factors versus dimensionless time t*=cst/c are provided for various combinations of both the material (1/3, 1/3) and geometric parameters (hj/c, j=1,2,3) of the bonded system, where cs=(3/3)1/2 is the shear wave speed in the lower semi-infinite homogeneous strip. A thirty-term expansion of the auxiliary function in Eq. (45) over each value of the Laplace variable p is found to be sufficient for achieving the desired level of accuracy for the problem configurations considered in this study. To check the convergence of the numerical results with M in the Laplace inversion formula of Eq. (49), the normalized dynamic stress intensity factors, KIII(t)/oc1/2, for an edge-cracked homogeneous half-plane (h1=h3=∞) are first generated for different values of M and compared with the following analytical solution available for initial times [33]: K III (t )

o c

4  t* 1  t*  t* / 2  (t / 2   )(  1)   H   1   *     2   2  1   1 

1/ 2



d    ; 0  t*  4  

(52)

where the graded interlayer is taken to be nearly homogeneous with 1/3=1.0 and 1/3=1.001 so as to circumvent division by zero in Eq. (2). Fig. 2 shows that as M increases up to 12, the current results begin to correspond more with those evaluated from Eq. (52); the difference between them is less than 7|Page

4% for the maximum dynamic overshoot of 1.228 against the exact peak of 4/π at t*=2.0. Note that further increases in M lead to curves that are indistinguishable from the one for M=12 or unstable; this holds true in the other numerical illustrations that follow. To put the results in Figs. 3-7 into perspective, certain qualitative features that are generic and typical of impact-induced transient crack-tip responses should be recalled; namely, immediately after the impact, the dynamic stress intensity factors rise rapidly with time before reaching peak values; subsequently, they decrease with some oscillations before finally settling down to the elastostatic values that are added by the small circles in each of these figures. The resultant fluctuating behavior can be attributed to the interactions between the waves scattered from the crack and those reflected from the boundaries and/or interfaces in layered composite media [34]. The variations of the dynamic stress intensity factors for the edge crack in a homogeneous material (1/3=1.0 and 1/3=1.001) of finite thickness are illustrated in Fig. 3 for different values of h/c, where h=h1=h2+h3. As expected, the greater magnitude of the dynamic stress intensification is shown to exist for the smaller strip thicknesses and the results for h/c=5.0 are predicted to closely match those obtained from Eq. (52). The effect of the shear modulus ratio 1/3 is examined in Fig. 4 for the edge interfacial crack in bonded dissimilar strips that have a graded interlayer; in this figure, the other parameters are set to

1/3=1.0, h1/c=h2/c=1.0, and h3/c=5.0. It is observed that the dynamic stress intensity factors, especially their peaks, are greater in proportion to the values of 1/3, but are suppressed below that of the homogeneous medium when 1/3 is less than unity. This dependence on 1/3 can be described in terms of relieved or augmented constraints, respectively, on the near-tip deformation from the less stiff or stiffer lower substrate, transmitted through the graded interlayer. Note that it takes longer for the stress intensity factors to climb to their peaks as 1/3 decreases. The results in Fig. 5 depict how the impact response of the edge crack is affected by the mass density ratio 1/3 for different values of 1/3, where h1/c=h2/c=1.0 and h3/c=5.0. Specifically, when

1/3 is greater than unity, the increase in 1/3 tends to enlarge the peak values of the dynamic stress intensity factors, but when 1/3 is less than unity, the opposite occurs such that the overshoot behavior is somewhat suppressed by the increase in 1/3. To be more specific, the increase in mass density on the less stiff side of the bonded system is found to alleviate the severity of the near-tip state under impact, which appears to be more remarkable when 1/3=5.0. After the impact, the transient stress intensification becomes less affected by 1/3 as time increases, and a common elastostatic limit for each value of 1/3 is approached; this limit can be seen to occur earlier when 1/3 is greater than unity. The variations of the dynamic stress intensity factors for different values of h1/c and 1/3 are plotted in Fig. 6, where 1/3=1.0, h2/c=1.0, and h3/c=5.0. For these sets of values, the overshoot 8|Page

behavior is found to be substantially less as h1/c increases; notably, the results for h1/c=5.0 approximate those of an edge interfacial crack in bonded quarter planes that have a graded interlayer. The effect of graded interlayer thickness, h2/c, can be seen in Fig. 7 in conjunction with that of

1/3 for values of 1/3=1.0, h1/c=1.0, and h3/c=5.0. When 1/3=5.0, the values of the dynamic stress intensity factors are shown to decrease as h2/c increases, which suggests that the interlayer plays a role in relaxing the crack-tip severity under impact. However, when 1/3=0.2, the overshoot behavior is shown to become rather enlarged with the increase in h2/c.

5. Closing remarks The transient response of an edge interfacial crack in bonded dissimilar strips was investigated in this study with a functionally graded interlayer under the action of antiplane shear impact. The corresponding evolutions of the time-dependent stress intensity factors were illustrated for various combinations of both the material and geometric parameters of the bonded system. More specifically, it was shown that, along with the effect of the shear modulus ratio, increasing the mass density of the less stiff side of a bonded medium is likely to mitigate the dynamic overshoot behavior around a crack. It was also demonstrated that the effect of graded interlayer thickness is dependent on the value of the shear modulus ratio, while increases in the thicknesses of homogeneous substrates tend to attenuate the overshoot behavior for all material combinations.

References [1] Y. Miyamoto, W.A. Kaysser, B.H. Rabin, A. Kawasaki, R.G. Ford (Eds.), Functionally Graded Materials: Design, Processing, and Applications, Kluwer Academic Publishers, MA, 1999. [2] F. Erdogan, Crack problems in nonhomogeneous materials, in: G.P. Cherepanov (Ed.), Fracture, A Topical Encyclopedia of Current Knowledge, Krieger Publishers, FL, 1998, pp. 72-98. [3] N. Noda, Thermal stresses in functionally graded materials, J. Therm. Stress. 22 (1999) 477-512. [4] Z.-H. Jin, Fracture mechanics of functionally graded materials, in: D.Y. Gao, R.W. Ogden (Eds.), Advances in Mechanics and Mathematics, vol. 2, Kluwer Academic Publishers, The Netherlands, 2003, pp. 1-108. [5] R. Babaei, S.A. Lukasiewicz, Dynamic response of a crack in a functionally graded material between two dissimilar half planes under anti-plane shear impact load, Eng. Fract. Mech. 60 (1998) 479-487. [6] L. Ma, L.-Z. Wu, L.-C. Guo, Dynamic behavior of two collinear anti-plane shear cracks in a functionally graded layer bonded to dissimilar half planes, Mech. Res. Commun. 29 (2002) 207215. [7] Ch. Zhang, J. Sladek, V. Sladek, Effects of material gradients on transient dynamic mode-III stress intensity factors in a FGM, Int. J. Solids Struct. 40 (2003) 5251-5270. 9|Page

[8] J. Chen, Anti-plane problem of periodic cracks in a functionally graded coating-substrate structure, Arch. Appl. Mech. 75 (2006) 138-152. [9] M.S. Matbuly, Stress wave scattering induced by a mode III interfacial crack in nonhomogeneous strips, Acta Mech. 199 (2008) 217-226. [10] M.M. Monfared, M. Ayatollahi, Dynamic stress intensity factors of multiple cracks in an orthotropic strip with FGM coating, Eng. Fract. Mech. 109 (2013) 45-57. [11] H. Haghiri, A.R. Fotuhi, A.R. Shafiei, Elastodynamic analysis of mode III multiple cracks in a functionally graded orthotropic half-plane, Theor. Appl. Fract. Mech. 80 (2015) 155-169. [12] M. Li, Y.L. Tian, P.H. Wen, M.H. Aliabadi, Anti-plane interfacial crack with functionally graded coating: static and dynamic, Theor. Appl. Fract. Mech. 86 (2016) 250-259. [13] H.J. Choi, Impact response of a surface crack in a coating/substrate system with a functionally graded interlayer: antiplane deformation, Int. J. Solids Struct. 41 (2004) 5631-5645. [14] H.J. Choi, Elastodynamic analysis of a crack at an arbitrary angle to the graded interfacial zone in bonded half-planes under antiplane shear impact, Mech. Res. Commun. 33 (2006) 636-650. [15] H.J. Choi, Impact behavior of an inclined edge crack in a layered medium with a graded nonhomogeneous interfacial zone: antiplane deformation, Acta Mech. 193 (2007) 67-84. [16] C. Li, G.J. Weng, Dynamic fracture analysis for a penny-shaped crack in an FGM interlayer between dissimilar half spaces, Math. Mech. Solids 7 (2002) 149-163. [17] B.-L. Wang, Y.-W. Mai, A periodic array of cracks in functionally graded materials subjected to transient loading, Int. J. Eng. Sci. 44 (2006) 351-364. [18] S.-H. Ding, X. Li, The fracture analysis of an arbitrarily oriented crack in the functionally graded material under in-plane impact loading, Theor. Appl. Fract. Mech. 66 (2013) 26-32. [19] L.-C. Guo, N. Noda, Dynamic investigation of a functionally graded layered structure with a crack crossing the interface, Int. J. Solids Struct. 45 (2008) 336-357. [20] M. Gharbi, S. El-Borgi, M. Chafra, A surface crack in a graded coating bonded to a homogeneous substrate under dynamic loading conditions, Int. J. Eng. Sci. 49 (2011) 677-693. [21] S. Itou, Dynamic stress intensity factors for two parallel interface cracks between a nonhomogeneous bonding layer and two dissimilar elastic half-planes subject to an impact load, Int. J. Solids Struct. 47 (2010) 2155-2163. [22] K. Di, Y.-C. Yang, Modeling method for the crack problem of a functionally graded interfacial zone with arbitrary material properties, Acta Mech. 223 (2012) 2609-2620. [23] X.-M. Bai, L.-C. Guo, Z.-H. Wang, S.-Y. Zhong, A dynamic piecewise-exponential model for transient crack problems of functionally graded materials with arbitrary mechanical properties, Theor. Appl. Fract. Mech. 66 (2013) 41-51. [24] T.-C. Chiu, F. Erdogan, One-dimensional wave propagation in a functionally graded elastic medium, J. Sound Vibrat. 222 (1999) 453-487. 10 | P a g e

[25] R.V. Churchill, Operational Mathematics, third ed., McGraw-Hill, New York, 1981. [26] C.R. Wylie, L.C. Barrett, Advanced Engineering Mathematics, sixth ed., McGraw-Hill, New York, 1995. [27] B. Friedman, Lectures on Applications-Oriented Mathematics, John Wiley & Sons, New York, 1991. [28] F. Erdogan, Mixed boundary value problems in mechanics, in: S. Nemat-Nasser (Ed.), Mechanics Today, vol. 4, Pergamon Press, New York, 1978, pp. 1-86. [29] N.I. Muskhelishvili, Singular Integral Equations, Dover Publications Inc., New York, 1992. [30] A.C. Kaya, Applications of Integral Equations with Strong Singularities in Fracture Mechanics, Ph.D. Dissertation, Lehigh University, 1984. [31] P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, second ed., Academic Press, New York, 1984. [32] H. Stehfest, Numerical inversion of Laplace transforms, Commun. ACM. 13 (1970) 47-49, 624. [33] J.W. Morrissey, P.H. Geubelle, A numerical scheme for mode III dynamic fracture problems, Int. J. Num. Meth. Eng. 40 (1997) 1181-1196. [34] G.C. Sih, E.P. Chen, Dynamic response of dissimilar materials with cracks, in: G.C. Sih (Ed.), Mechanics of Fracture, Cracks in Composite Materials, vol. 6, Martinus Nijhoff Publishers, The Hague, 1981, pp. 277-440.

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List of Figure Captions

Fig. 1. Bonded dissimilar semi-infinite strips containing a functionally graded interlayer and an edge interfacial crack.

Fig. 2. Convergence of the dynamic stress intensity factors, KIII(t)/Ko, for increasing values of M in the Laplace inversion formula for an edge crack in a homogeneous half-plane (1/3=1.0, 1/3=1.001, h1=h3=∞, and Ko=oc1/2).

Fig. 3. Dynamic stress intensity factors, KIII(t)/Ko, as a function of dimensionless time, cst/c, for an edge crack in a homogeneous strip for different values of h/c (1/3=1.0, 1/3=1.001, h=h1=h2+h3, and Ko=oc1/2).

Fig. 4. Dynamic stress intensity factors, KIII(t)/Ko, as a function of dimensionless time, cst/c, for an edge interfacial crack in bonded dissimilar strips for different values of 1/3 (1/3=1.0, h1/c=h2/c=1.0, h3/c=5.0, and Ko=oc1/2).

Fig. 5. Dynamic stress intensity factors, KIII(t)/Ko, as a function of dimensionless time, cst/c, for an edge interfacial crack in bonded dissimilar strips for different values of 1/3 and 1/3 (h1/c=h2/c=1.0, h3/c=5.0, and Ko=oc1/2).

Fig. 6. Dynamic stress intensity factors, KIII(t)/Ko, as a function of dimensionless time, cst/c, for an edge interfacial crack in bonded dissimilar strips for different values of h1/c and 1/3 (1/3=1.0, h2/c=1.0, h3/c=5.0, and Ko=oc1/2).

Fig. 7. Dynamic stress intensity factors, KIII(t)/Ko, as a function of dimensionless time, cst/c, for an edge interfacial crack in bonded dissimilar strips for different values of h2/c and 1/3 (1/3=1.0, h1/c=1.0, h3/c=5.0, and Ko=oc1/2).

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y

h1 h2

           

c

μ1, 1 μ2 (y), 2 (y) μ3 , 3

h3

Fig. 1

x

1.3 : ref. [33]

1.2 1.1

K(t)/Ko

1.0 M increasing (2, 4, 6, 8, 10, 12)

0.9 0.8 0.7 0.6

=1.0, =1.001

0.5

h1=h3=inf., h2/c=1.0

0.4 0.0 0

1

2

3

4

5 cst/c

Fig. 2

6

7

8

9 10

4.0 =1.0, =1.001

3.6

h=h1=h2+h3

3.2

K(t)/Ko

2.8 2.4 h/c=0.2

2.0

0.3 0.5 1.0 5.0

1.6 1.2 0.8 : ref. [33] : elastostatic sol.

0.4 0.0 0

2

4

6

8 cst/c

Fig. 3

10

12 14 16

1.8 : elastostatic sol. 1.6 1.4 =5.0

K(t)/Ko

1.2 1.0

1.0 0.5

0.8

0.2

2.0

0.6 =1.0

0.4

h1/c=h2/c=1.0, h3/c=5.0

0.0 0

4

8

12

16

cst/c

Fig. 4

20

24

28

2.0  increasing

1.8

(0.5, 1.0, 2.0)

1.6  increasing

1.4 K(t)/Ko

(0.5, 1.0, 2.0) 1.2

=5.0

1.0 0.8

=0.2

0.6 : elastostatic sol.

0.4

h1/c=h2/c=1.0, h3/c=5.0 0.0 0

4

8

12

16

cst/c

Fig. 5

20

24

28

3.0 2.7

: elastostatic sol. =1.0

2.4

h2/c=1.0, h3/c=5.0

K(t)/Ko

2.1 =5.0

1.8 1.5 1.2 0.9

=0.2

0.6

h1/c increasing

0.3

(0.2, 0.3, 0.5, 1.0, 5.0)

0.0 0

4

8

12

16

cst/c

Fig. 6

20

24

28

2.2 : elastostatic sol.

2.0 1.8

h2/c increasing

1.6

(0.2, 0.5, 1.0, 2.0)

=5.0

K(t)/Ko

1.4 1.2 1.0 =0.2

0.8 0.6

h2/c increasing

0.4

(0.2, 0.5, 1.0, 2.0) =1.0

0.2

h1/c=1.0, h3/c=5.0

0.0 0

4

8

12

16

cst/c

Fig. 7

20

24

28

Highlights 

An edge interfacial crack in bonded dissimilar strips with a graded interlayer under antiplane shear impact.



Power-law variations of the shear modulus and mass density of the interlayer.



Dynamic stress intensity factors vs. the material and geometric parameters of the problem.



Effect of mass density ratio in conjunction with that of shear modulus ratio.



Effect of geometric parameters for various material combinations.

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