Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation

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Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation Hyung Jip Choi ∗ School of Mechanical Systems Engineering, Kookmin University, Seoul 02707, Republic of Korea

a r t i c l e

i n f o

Article history: Received 20 March 2015 Received in revised form 13 August 2015 Accepted 22 August 2015 Available online xxx Keywords: Bonded dissimilar strips Functionally graded interlayer Interfacial crack Singular integral equation Stress intensity factors

a b s t r a c t This paper provides the solution to the problem of dissimilar, homogeneous semi-infinite strips bonded through a functionally graded interlayer and weakened by an embedded or edge interfacial crack. The bonded system is assumed to be under antiplane deformation, subjected to either traction-free or clamped boundary conditions along its bounding planes. Based on the Fourier integral transform, the problem is formulated in terms of a singular integral equation which has a simple Cauchy kernel for the embedded crack and a generalized Cauchy kernel for the edge crack. In the numerical results, the effects of geometric and material parameters of the bonded system on the crack-tip stress intensity factors are presented in order to quantify the interfacial fracture behavior in the presence of the graded interlayer. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The significant advances have been made in the development of functionally graded materials over the last few decades, owing to their tailoring capability to produce spatial and gradual variations of physical properties, coping with a variety of technological problems in engineering practice. In particular, the deliberate incorporation of graded, nonhomogeneous media as a transitional interlayer to join dissimilar bulk materials such as ceramics and metals was found to be one of the viable methods of alleviating various limitations frequently encountered in the use of conventional bonded materials and structures [1]. These drawbacks may include high stress concentrations, poor bonding strength and consequent vulnerability to failure around the interfacial zone that are likely to be caused by the property mismatch apparent at the junction between two piecewise different phases in such composite bodies. From the fracture mechanics viewpoint, even though the existence of graded interphase possibly enhances the resistance against failure, considerable research efforts have focused on the characterization of singular stress field induced by crack-like defects. In this context, a series of solutions to a variety of benchmark crack problems entailing the graded properties has been obtained by Erdogan and his coworkers [2–8], where a crack was assumed to be aligned parallel to, perpendicular to or along the kink line of spatial

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distributions of elastic moduli between homogeneous and nonhomogeneous constituents. The most noteworthy was the near-tip stress field with the square-root singularity and angular distributions around the crack that are identical to those in homogeneous materials, with the effect of material gradation manifesting itself through the values of stress intensity factors. A number of additional contributions toward the fracture analysis of bonded media that takes the presence of a graded interlayer into account have subsequently been reported in the literature. Among them are, for example, the influence of coating architectures on the interfacial fracture behavior in a functionally graded coating/substrate structure under antiplane shear [9]; the plane problem of an interface crack for a graded strip between homogeneous layers of finite thickness [10]; the antiplane analysis of periodic interface cracks in a graded coating/substrate system [11]; the interfacial cracking in a graded coating loaded by a frictional flat punch [12]; the two parallel interface cracks in bonded dissimilar orthotropic half-planes with a nonhomogeneous interlayer [13]; and the multilayered approach applied to some interface crack problems for graded media with arbitrary distributions of material properties [14,15]. Meanwhile, the problems of a crack at an arbitrary angle to the graded interfacial zone in bonded materials were considered under various loading conditions [16–20], whereas the mixed-mode behavior of an arbitrarily oriented crack crossing the interface in a functionally graded layered structure was dealt with in [21] and that of an inclined crack in the functionally graded plane under impact was investigated in [22]. In recent years, the antiplane and mixed-mode interactions of two offset interfacial

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cracks in bonded materials with a functionally graded interlayer were also examined in [23–26]. As can be inferred from the foregoing, although a great deal of attempts made to date have resolved various issues and provided insightful results for the interfacial crack problems entailing the graded constituents, these prior studies appear to have mainly been concerned with such cases as containing a crack embedded deep inside the bonded system. It should be reminded that most interfaces intersect a free surface and in consideration of the interfacial failure that often initiates near or from the free surface, the problems of particular technical importance would be that of an embedded interfacial crack interacting with the neighboring free surface and that of an edge interfacial crack breaking the free surface, accounting for both the boundary and size effects. The present paper is, therefore, aimed at further quantifying the interfacial fracture behavior by solving the problem of an embedded or edge interfacial crack in bonded dissimilar semi-infinite strips with a functionally graded interlayer. The state of antiplane deformation is assumed, because it has the practical significance of its own when the third fracture mode is separable and it may also form the informative basis for understanding the more involved inplane counterpart. Both the traction-free and clamped boundary conditions are considered along the bounding planes of the bonded system. The Fourier integral transform method is employed and a singular integral equation is derived with a simple Cauchy kernel for the embedded crack and with a generalized Cauchy kernel for the edge crack, which is solved by the expansion-collocation technique. The mode III stress intensity factors are evaluated, addressing the effects of geometric parameters of the bonded system, in conjunction with that of the material stiffness. 2. Problem statement and formulation The problem configuration under consideration is shown in Fig. 1, where two dissimilar, homogeneous strips of semi-infinite length are bonded through a functionally graded interlayer. The

constituents of this bonded system are distinguished in order from the top with the thickness hj , j = 1, 2, 3, respectively, and an interfacial crack of length 2c = b − a is located along A(a,0) < (x,y) < B(b,0), where a > 0 is for the embedded crack and a = 0 for the edge crack. The shear modulus of the graded, nonhomogeneous interlayer, 2 (y), is represented in terms of an exponential function [2] 2 (y) = 3 eˇy ,

1 ln h2

ˇ=

  1

3

;

0 < y < h2

(1)

which renders the continuous transition of elastic moduli across the nominal interfaces with the top and bottom strips of shear moduli j , j = 1, 3, respectively. Under the state of antiplane deformation, there exist only outof-plane displacement components, wj (x,y), j = 1, 2, 3, with the stress components and governing equations given by jxz = j

∂wj

∇ 2 wj + ˇ

∂x

jyz = j

,

∂wj

= 0;

∂y

∂wj ∂y

;

j = 1, 2, 3

(2)

j = 1, 2, 3

(3)

where ˇ = 0 for the homogeneous constituents (j = 1, 3) and ˇ = / 0 for the interlayer (j = 2). It is assumed that the bonded system is subjected to antiplane shear traction applied solely on the crack surfaces and that the lefthand side flank edges at x = 0 are traction-free. A set of boundary and interface conditions is thus prescribed as 1xz (0, y) = 0;

h2 < y < h1 + h2

(4)

2xz (0, y) = 0;

0 < y < h2

(5)

3xz (0, y) = 0;

− h3 < y < 0

(6)

1yz (x, h2 ) = 2yz (x, h2 ), w1 (x, h2 ) = w2 (x, h2 ); w2 (x, 0) = w3 (x, 0); 3yz (x, 0) = f (x);

2yz (x, 0) = 3yz (x, 0);

x>0

x>0 0 < x < a,

(7) (8)

x>b

(9)

a
(10)

where the function f(x) denotes the crack surface traction, with the bounding planes being either traction-free 1yz (x, h1 + h2 ) = 0,

3yz (x, −h3 ) = 0;

x>0

(11)

or rigidly clamped such that w1 (x, h1 + h2 ) = 0,

w3 (x, −h3 ) = 0;

(12)

x>0

Based on the Fourier integral transformation, the general solutions for the displacements that satisfy the conditions at the flank edges in Eqs. (4)–(6) are readily obtained as wj (x, y) =

w2 (x, y) =

2 

2 



∞

(Aj1 esy + Aj2 e−sy ) cos sx ds;

j = 1, 3

(13)

0

 0

∞

2 

A2k ek y cos sx ds

(14)

k=1

where s is the transform variable, Ajk (s), j = 1, 2, 3, k = 1, 2, are arbitrary unknowns and k (s), k = 1, 2, are written as



Fig. 1. Bonded dissimilar strips with a functionally graded interlayer containing (a) an embedded interfacial crack; (b) an edge interfacial crack.

ˇ 1 = − + 2

ˇ2 + s2 , 4



ˇ 2 = − − 2

ˇ2 + s2 4

(15)

Please cite this article in press as: H.J. Choi, Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation, Mech. Re. Commun. (2015), http://dx.doi.org/10.1016/j.mechrescom.2015.08.006

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For the present interfacial crack problem, an auxiliary function is introduced to replace the mixed conditions in Eqs. (9) and (10)

⎧ ⎨ ∂ [w (x, +0) − w (x, −0)]; a < x < b 2 3 ∂x (x) = ⎩ 0 < x < a,

0;

(16)

a singular integral equation is derived for the embedded (a > 0) or edge interfacial crack (a = 0) as

 b 1 r−x

a

x>b

and the subsequent applications of interface conditions in Eqs. (7) and (8) plus either of the boundary conditions in Eqs. (11) and (12) can determine the unknowns, Ajk (s), j = 1, 2, 3, k = 1,2, in terms of (x), which is to be evaluated from the crack surface condition in Eq. (10).

3

+

2 1 f (x); + k(x, r) (r)dr = r+x 3

a
(25)

where the bounded kernel k(x,r) is obtained as



∞



k(x, r) = 4 0

ˇ 8

ds +



1 ˇ (s) − − sin sr cos sx 2 8s

 |r − x| r−x

+

|r + x| r+x

 (26)

3. Integral equation and stress intensity factors After substituting the expressions of Ajk (s), j = 1, 2, 3, k = 1, 2, obtainable as aforementioned, into Eq. (2) and some algebraic manipulations, the traction component along the cracked interface can be written in the form as lim 3yz (x, y) =

y→0

23 



b

H(x, r)(r)dr;

x>0

(17)

a

where H(x,r) is a kernel function such that



∞

H(x, r) =

(s) sin sr cos sx ds

It should be remarked that when a > 0, the Cauchy kernel 1/(r − x) in Eq. (25) becomes unbounded at x = r and the other two terms inside the bracket remain bounded in the closed interval [a,b]. When a = 0, however, besides the simple Cauchy kernel, the term 1/(r+x) also becomes unbounded at the end point x = r = 0. Hence, the first two terms in the kernels of the integral equation constitute a generalized Cauchy singular kernel [28]. Now that the dominant singular part in the integral equation is due to the Cauchy-type, the auxiliary function (r) for the two crack configurations is expressed as [29]

⎧ ⎪ ⎨

(18)

0

(s) =

(l m1 ∓ l4 m2 − l1 m3 ± l2 m4

3



)(esh3



∓ e−sh3 )

e−sh3 l2 m5 − l4 m6 ± l5 ˇ + esh3 l1 m6 − l3 m5 ∓ l6 ˇ



(r) =

in which the upper and lower signs refer to traction-free and clamped boundary conditions, respectively, with the following contractions: l1 = e(1 h2 +sh1 ) ,

l2 = e(1 h2 −sh1 ) ,

l4 = e(2 h2 −sh1 ) , m1 = 1 − s,

m2 = 1 + s,

m5 = m4 − m1 ,

l6 = l2 − l4

r

x

m3 = 2 − s,

m6 = m2 − m3

(20)

b−a = 2

(21) ( ) =

(s) =

e1 h2 m3 − e2 h2 m1 e2 h2 m5 − e1 h2 m6

(22)

lim (s) =

ˇ 1 + 2 8s

(23)

giving rise to the singular behavior the kernel H(x,r) in Eq. (18) may have. By separating the leading term from the kernel and making use of Fourier representations of generalized functions [27]



∞

sin s(r − x) ds =



0

0

∞

1 , r−x

 |r − x| 1 sin s(r − x)ds = s 2 r−x

(27) 0
 

+



b+a ; 2

− 1 < ( , ) < 1

⎧ ∞    ⎪ 1 ⎪ ⎪ cn Tn ( );   < 1, for a > 0  ⎪ ⎨ 1 − 2 n=0

∞  ⎪ 1 ⎪ ⎪ cn Tn ( ); ⎪ ⎩ 1 − n=0

    < 1, for a = 0

(28)

(29)

in which cn , n ≥ 0, are unknown coefficients. For the case of an embedded crack, it follows from Eqs. (9) and (16) that the function ( ) must fulfill the single-valuedness:



1

( )d = 0

It is noted that the function (s) in Eq. (19) or (22), which is dependent on the material and geometric parameters of the bonded media as well as on the variable s, possesses the asymptote as s tends to infinity such that

s→∞

a
the solution to the integral equation can be written as the series expansions of the Chebyshev polynomials of the first kind Tn

m4 = 2 + s,

When the top and bottom homogeneous constituents are treated as quarter-planes by letting hj → ∞, j = 1, 3, the regularity conditions are enforced, instead of Eq. (11) or (12), such that w1 (x, +∞) = 0 and w3 (x, −∞) = 0. In this case, it can be shown that the integrand (s) in Eq. (19) is simplified as

;

where g(r) is a bounded function and in the normalized interval

 

l3 = e(2 h2 +sh1 ) ,

l5 = l1 − l3 ,

⎪ ⎩

(19)

g(r) (r − a)(b − r) g(r) ; √ b−r

(30)

−1

and by using the orthogonality of Tn , it can be shown that c0 = 0. In the edge crack problem, such a compatibility condition is not needed in ensuring the unique solution. Upon substituting from Eqs. (28) and (29) into Eq. (25) and via the integral identities [30,31]



1

 −1



1

Tn ( ) d 1 − 2 ( − )

T ( )

n

(24) −1

   − 

1 − 2 −

= Un−1 ( );

d =

n ≥ 1,

    < 1

 2 Un−1 ( ) 1 − 2 ; n

n ≥ 1,

(31)

    < 1 (32)

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1

 −1



Tn ( )d 1 − ( − )

1

T ( ) − Tn ( )

n

=

1 − ( − )

−1

2.0

d

   1 + (1 − )/2    + ln    ; n ≥ 0, 1 −  1 − (1 − )/2      < 1 (33)

1.8

tip A tip B Ozturk and Erdogan [6]

Tn ( )

=

= lim



x→a

⎪ ⎩

2

2

0;

a=0

(B)

KIII = lim

(−1)n cn ;

a>0

0.2 0.4

1.0

1.0 3.0 0.4

0.6

0.8

1.0

2.0 1.8

µ1/µ3=0.1

1.6

⎧  N ⎪ b − a  ⎪ 3 ⎪ cn ; a > 0 − ⎪ ⎨ 2 2 (35) a=0

1.4 1.2

(B)

where KIII and KIII denote, respectively, the stress intensity factors at the left- and right-hand side crack tips. 4. Results and discussion Numerical results are obtained for various combinations of material (1 /3 ) and geometric parameters (c/d, hj /2c; j = 1, 2, 3) of the bonded system, subjected to uniform crack surface traction as f(x) = − o in Eq. (10). No more than thirty-term expansions of the auxiliary function in Eq. (29) are found to suffice in achieving up to three-digit accuracy beyond the decimal point for the problem configurations considered in this study, with the improper integral in Eq. (26) and the other integrals in the functional equation evaluated based on the Gauss-Legendre and Gauss-Chebyshev quadratures, respectively [32]. The resulting values of stress intensity factors are normalized by Ko =  o l1/2 where l = c for the embedded crack and l = 2c = b for the edge crack. The case of bonded quarter-planes (hj /2c = ∞, j = 1, 3) with a graded interlayer is first examined in Figs. 2 and 3 for the embedded (0 < c/d < 1.0) and edge interfacial crack (c/d = 1.0), respectively, together with some data points accessible in Tables 1 and 2. These figures and tabulated results imply that the severity of interfacial cracking is magnified for the smaller shear modulus ratio, 1 /3 , caused by the relieved constraint from the less stiff upper quarter-plane transmitted through the interlayer, but suppressed below that of the homogeneous half-plane when 1 /3 > 1.0. For the interlayer thickness specified as h2 /2c = 0.5, it is also observed from Fig. 2 and Table 1 that when c/d approaches zero, i.e., a crack

0.2 0.4

1.0

n=0 (A)

0.2

(34)

2(x − b)3yz (x, 0)

N ⎪ 3   ⎪ ⎪ − b cn ; ⎪ ⎩ 2

1.2

Fig. 2. Variations of stress intensity factors KIII /Ko versus c/d for an embedded interfacial crack in bonded quarter-planes with a graded interlayer for different values of 1 /3 (hj /2c = ∞, j = 1,3, h2 /2c = 0.5 and Ko =  o c1/2 ).

n=1

n=1

µ1/µ3=0.1

c/d



x→b

=

2(a − x)3yz (x, 0)

⎧  N ⎪ ⎨ 3 b − a 

1.4

0.8 0.0

KIII/Ko

(A) KIII

KIII/Ko

where Un are the Chebyshev polynomials of the second kind, the singular integral equation is regularized. The resulting functional equation can be solved for the coefficients cn by truncating the series expansions in Eq. (29) with a finite number of terms at n = N and applying the roots of the Chebyshev polynomials of the first kind as collocation points [28]. Once the values of the coefficients cn are determined, the lefthand side of the integral equation in Eq. (25) provides the singular tractions ahead of the tips of the interfacial crack. In consequence, the criticality of such local stress intensifications in the near-tip regions can be extracted by defining and evaluating the mode III stress intensity factors as

1.6

1.0

0.8

3.0

0.6 0.0

0.5

1.0

1.5 2.0 h2/2c

2.5

3.0

Fig. 3. Variations of stress intensity factors KIII /Ko versus h2 /2c for an edge interfacial crack in bonded quarter-planes with a graded interlayer for different values of 1 /3 (c/d = 1.0, hj /2c = ∞, j = 1, 3, and Ko =  o b1/2 ).

embedded deep inside the bonded media, the current solutions tend to those for bonded half-planes with a graded interlayer [6], while the stress intensification is amplified when the crack is located close to the flank edges at x = 0, with c/d approaching unity. In addition, from Fig. 3 where the results for the edge crack are illustrated versus h2 /2c, the more remarkable influence of h2 /2c is noted for 1 /3 less than unity, accompanied by substantial attenuation of the crack-tip state with increasing h2 /2c. When 1 /3 = 3.0, however, the stress intensification becomes somewhat enlarged as h2 /2c increases. To be mentioned now is that the results for 1 /3 = 1.0 are in agreement with those in [33]. The bonded system of finite thickness containing an embedded interfacial crack is next considered where the variations of stress intensity factors are plotted in Figs. 4 and 5 as a function of c/d. With the thickness of homogeneous strips as hj /2c = h/2c = 0.5, j = 1, 3, the results in Fig. 4a and b correspond to those obtained under the traction-free and clamped boundary conditions, respectively, showing the effect of interlayer thickness, h2 /2c, in conjunction

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Table 1 Normalized stress intensity factors KIII /Ko for an embedded interfacial crack in bonded quarter-planes with a graded interlayer for different values of ␮1 /3 and c/d (hj /2c = ∞, j = 1, 3, h2 /2c = 0.5 and Ko =  o c1/2 ). c/d

1 /3 = 0.1

1 /3 = 0.2

Tip A a

0.0 0.01 0.1 0.3 0.5 0.7 0.9 a

Tip B

Tip A

1.2942 1.2946 1.2983 1.3087 1.3350 1.4102

1.1999 1.2006 1.2080 1.2346 1.3246 1.7318

1.2949 1.2942 1.2946 1.2998 1.3219 1.4092 1.8462

1 /3 = 0.4 Tip B

Tip A

1.1999 1.2005 1.2058 1.2189 1.2474 1.3199

1.1079 1.1089 1.1186 1.1492 1.2386 1.6086

1.2004

1 /3 = 1.0 Tip B

Tip A

1.1079 1.1088 1.1158 1.1313 1.1611 1.2284

1.0000 1.0013 1.0138 1.0480 1.1333 1.4539

1.1082

1 /3 = 3.0 Tip B

Tip A

1.0000 1.0012 1.0102 1.0280 1.0579 1.1174

0.9042 0.9058 0.9200 0.9563 1.0375 1.3158

1.0000

Tip B 0.9039 0.9042 0.9056 0.9158 0.9348 0.9644 1.0176

Ozturk and Erdogan [6].

(b)

(a) 1.6

2.4 tip A tip B

2.2

1.4

h2/2c increasing

2.0

tip A tip B h /2c increasing 2 (0.1, 0.2, 0.3, 0.5, 1.0)

(0.1, 0.2, 0.3, 0.5, 1.0)

1.2

µ1/µ3=0.2

1.6

KIII/Ko

KIII/Ko

1.8

1.4

1.0

µ1/µ3=0.2

0.8

1.2 1.0 0.8

µ1/µ3=3.0

0.6 0.0

0.6 µ /µ =3.0 1 3

h2/2c increasing (0.1, 0.2, 0.3, 0.5, 1.0)

0.2

0.4

0.6

0.8

h2/2c increasing (0.1, 0.2, 0.3, 0.5, 1.0)

0.4 0.0

1.0

0.2

0.4

0.6

0.8

1.0

c/d

c/d

Fig. 4. Variations of stress intensity factors KIII /Ko versus c/d for an embedded interfacial crack in bonded strips with a graded interlayer for different values of 1 /3 and h2 /2c: (a) traction-free boundaries; (b) clamped boundaries (hj /2c = h/2c = 0.5, j = 1, 3, and Ko =  o c1/2 ).

with that of 1 /3 . As expected, the imposition of the tractionfree condition leads to noticeable elevation of the near-tip stress state in comparison with that of the clamped condition. A common feature for the two external conditions is that the stress intensity

factors are reduced with the increase in h2 /2c when 1 /3 = 0.2, with the reverse behavior of enlarged stress intensification prevailing for the greater h2 /2c when 1 /3 = 3.0. The effect of thickness of homogeneous strips, hj /2c = h/2c, j = 1, 3, is predicted in Fig. 5a

(a)

2.2 tip A tip B

2.0

1.6 h/2c increasing 1.4

KIII/Ko

KIII/Ko

h/2c increasing (0.3, 0.4, 0.5, 0.7, 1.0, inf) µ1/µ3=0.2

1.4

1.0

1.0

0.8

0.8 h/2c increasing

µ1/µ3=3.0

0.4

0.6 c/d

µ1/µ3=0.2

µ1/µ3=3.0

0.6 h/2c increasing

(0.3, 0.4, 0.5, 0.7, 1.0, inf)

0.2

(0.4, 0.5, 0.7, 1.0, 1.5, inf)

1.2

1.2

0.6 0.0

tip A tip B

1.8

1.8 1.6

(b)

2.0

(0.4, 0.5, 0.7, 1.0, 1.5, inf)

0.8

1.0

0.4 0.0

0.2

0.4

0.6

0.8

1.0

c/d

Fig. 5. Variations of stress intensity factors KIII /Ko versus c/d for an embedded interfacial crack in bonded strips with a graded interlayer for different values of 1 /3 and hj /2c = h/2c, j = 1, 3: (a) traction-free boundaries; (b) clamped boundaries (h2 /2c = 0.5 and Ko =  o c1/2 ).

Please cite this article in press as: H.J. Choi, Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation, Mech. Re. Commun. (2015), http://dx.doi.org/10.1016/j.mechrescom.2015.08.006

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(a)

(b)

2.2

1.6 µ1/µ3=0.2

2.0

µ1/µ3=3.0

1.4

1.8

h/2c increasing (0.3, 0.4, 0.5, 0.7, 1.0, 1.5, inf)

1.2

h/2c increasing (0.3, 0.4, 0.5, 0.7, 1.0, 1.5, inf)

KIII/Ko

KIII/Ko

1.6 1.4

1.0 0.8

1.2 0.6

1.0 0.8 0.6 0.0

0.4

h/2c increasing (0.3, 0.4, 0.5, 0.7, 1.0, 1.5, inf)

0.5

1.0

1.5 2.0 h2/2c

2.5

3.0

0.2 0.0

µ1/µ3=0.2 µ1/µ3=3.0 h/2c increasing (0.3, 0.4, 0.5, 0.7, 1.0, 1.5, inf)

0.5

1.0

1.5 2.0 h2/2c

2.5

3.0

Fig. 6. Variations of stress intensity factors KIII /Ko versus h2 /2c for an edge interfacial crack in bonded strips with a graded interlayer for different values of 1 /3 and hj /2c = h/2c, j = 1, 3: (a) traction-free boundaries; (b) clamped boundaries (c/d = 1.0 and Ko =  o b1/2 ).

Table 2 Normalized stress intensity factors KIII /Ko for an edge interfacial crack in bonded quarter-planes with a graded interlayer for different values of 1 /3 and h2 /2c (c/d = 1.0, hj /2c = ∞, j = 1, 3, and Ko =  o b1/2 ). h2 /2c

1 /3 = 0.1

1 /3 = 0.2

1 /3 = 0.4

1 /3 = 1.0

1 /3 = 3.0

0.25 0.5 1.0 1.5 2.0 2.5 3.0

1.7432 1.4954 1.3010 1.2157 1.1677 1.1371 1.1159

1.4614 1.3245 1.2041 1.1480 1.1157 1.0949 1.0804

1.2256 1.1675 1.1100 1.0812 1.0641 1.0528 1.0449

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.8456 0.8706 0.9023 0.9223 0.9358 0.9454 0.9526

and b subjected to traction-free and clamped boundary conditions, respectively, for different values of 1 /3 , where the interlayer thickness is fixed as h2 /2c = 0.5. In Fig. 5a, the increase in h/2c is shown to relax the crack-tip criticality, regardless of material combinations, with the results for the bonded quarter-planes being the lower bound. On the other hand, under the clamped condition, as demonstrated in Fig. 5b, the opposite trend of greater stress intensification is indicated for the increased h/2c, with the quarter-plane solutions as the upper bound. In Fig. 6a and b, the results for the edge interfacial crack under the traction-free and clamped boundary conditions are provided, respectively, as a function of interlayer thickness, h2 /2c, for different values of 1 /3 and hj /2c = h/2c, j = 1, 3. In these figures, the increase in h2 /2c is also shown to enlarge the stress intensity factors when 1 /3 = 3.0, whereas the opposite is generally true when 1 /3 = 0.2. For the traction-free boundaries, as plotted in Fig. 6a, the increase in h/2c reduces the stress intensity factors. Under the clamped condition, the results in Fig. 6b further illustrate that the increase in h/2c counteracts the constraint effect from the rigid boundaries, giving rise to the more intensified behavior of edge interfacial cracking for the given values of 1 /3 . 5. Closing remarks The problem of bonded dissimilar, homogeneous strips with a functionally graded interlayer containing an embedded or edge interfacial crack has been investigated under antiplane deformation. The bonded system was subjected to either traction-free

or clamped conditions along its bounding planes. The variations of mode III stress intensity factors were presented as a function of geometric parameters of the problem for different values of the shear modulus ratio. Of particular interest were the effect of thickness of the graded interlayer that is different depending on the material combination and that of the homogeneous strips that is controlled by the external boundary conditions.

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Please cite this article in press as: H.J. Choi, Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation, Mech. Re. Commun. (2015), http://dx.doi.org/10.1016/j.mechrescom.2015.08.006