Int J Fract DOI 10.1007/s10704-013-9806-7 © Springer Science+Business Media Dordrecht 2013

LETTERS IN FRACTURE AND MICROMECHANICS

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ON THE COMPLEX STRESS INTENSITY FACTOR FOR SPLIT TYPE INTERFACE CRACKS BASED ON AN APPROXIMATE METHOD

Huseyin Lekesiz Mechanical Engineering Department, Faculty of Natural Sciences, Engineering and Architecture, Bursa Technical University, Turkey e-mail: huseyin.lekesiz@btu.edu.tr

Abstract. The simple method developed by Kachanov (1985) for multiple interacting cracks in homogenous medium is extended to predict complex stress intensity factor for multiple split type interface cracks. Calculations are implemented for two equal cracks and infinite row of periodic cracks at the interface between two dissimilar isotropic materials. Results for infinite row of cracks are compared against the exact analytical solution provided by Sih (1973). The approximate method leads to the results very close the exact solution for crack density up to 0.90 (relative error is less than 3.8% for real part of stress intensity factor) and material dissimilarity does not have a major influence on the error. For crack densities higher than 0.90, the influence of material dissimilarity is more evident and the error increases as material dissimilarity increases. The promising match between the approximate and exact method proves the capability of the approximate method for solving other interacting interface crack problems, such as multiple penny-shaped interface cracks, in which the solution is not obtained in the literature yet.

Keywords: Interface crack, split crack, crack interaction, stress intensity factor.

1. Introduction. The quality of the bonding at the interface is the primary factor determining the durability of the layered structures. Bonded surfaces in layered structures may be degraded over time due to some mechanical and environmental factors (Lavrantyev and Rokhlin 1994). Damaged interface may exist in the form of bonded/debonded regions in a configuration of slit or penny-shaped cracks. Therefore, the fracture mechanics solutions for these crack types have a significant attraction for researchers. Non-interacting interfacial slit cracks have been investigated in great detail by several researchers (Williams 1959; Erdogan 1965; England 1965; Comninou 1977). A solution for single non-interacting penny-shaped interface crack under uniform tension is provided by Mossakovskii and Rybka (1964) and a more general solution for arbitrary loading is provided by Willis (1972). A fracture mechanics solution for interacting penny-shaped cracks in a homogenous medium is studied by Kachanov and Laures (1989), Fabrikant (1987, 1989) and recently by Lekesiz et al. (2013). However, corresponding problem for interface cracks between dissimilar materials has not been studied in the literature, yet. Interaction between cracks needs to be taken into account because it may influence the interfacial properties significantly. Semi-analytical and numerical methods such as body-force method (Noda and Oda 1997), boundary element-free method (Sun et al. 2006) and contour integral method (Yang and Kuang 1996) are utilized to solve two or more interacting slit type interface cracks. In this study, stress intensity factors (SIFs) for two and infinite row of collinear interface cracks are calculated by extending the approximate method by Kachanov (1985) for interface cracks and accuracy of the approximation is tested for the infinite row of periodic array of collinear interface cracks for which the exact solution is given (Sih 1973). Promising match between approximate and

H. Lekesiz

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exact solutions for most material combinations proves the capacity of the method to attain a solution for interacting interface crack problems which are not elucidated, yet.

2. Two Cracks. Consider two equal cracks with the length of 2a at the interface between two dissimilar, linearly elastic, isotropic half spaces with elastic moduli E1, E2 and Poisson’s ratios 1, 2. The distance between two crack centers is 2b and cracks are subjected to a remote uniform tension, p0 as shown in Figure 1. To obtain mode I and mode II stress intensity factors (SIF) for this configuration, the method developed by Kachanov (1985) for the interacting cracks in homogenous medium is applied for interface cracks. Following the method, first, the problem with the remote loading is replaced by an equivalent problem where crack faces are loaded with uniform tractions and no traction at infinity. Second, the equivalent problem is separated into two problems with each having unknown tractions on crack face resulting from the interactions.

Figure 1. Two strip cracks at the interface between two dissimilar, linearly elastic, isotropic half spaces with elastic moduli

E1, E2 and Poisson’s ratios 1, 2

The normal and shear tractions, p1 and t1 on the crack #1 can be written in complex form as follows

( ) ( )[ ]xyyy itippitp σσ +−><+><+=+ 122011 (1)

where <p2> and <t2> are the average of the normal and shear tractions on crack #2 face. Different than the cracks in homogenous medium, applied tensile stress generates both normal and shear tractions on interface cracks and therefore normal and shear tractions are interrelated as seen in the Eq. (1). The normal and shear stresses, xyyy σσ , , induced by uniform unit tension applied on crack#2 surface, are given as in

εεσσ

i

xyyy axax

ax

iaxi−+

−

−=+2

222

2

2 2

,

( )bxx 212 −= (2)

(Hills et al. 1996). is the elastic mismatch parameter and given by

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On the complex stress intensity factor for split type interface cracks

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

ββ

πε

11log

21

in which ( ) ( )

( ) ( )1111

2112

2112

+++−−−=

κκκκβ

GGGG

(3) G1 and G2 are shear moduli of material 1 and 2, respectively, 11 43 νκ −= , 22 43 νκ −= for plane strain and ( ) ( )111 13 ννκ +−= , ( ) ( )222 13 ννκ +−= for plane stress. The range of for all physically admissible material combinations is [0, ~0.1748]. When the two materials are switched, only sign of the changes but absolute value is reserved. For most practical applications of material pairs, remains in the range of [0, 0.1]. values for several material combinations can be obtained from Suga et al. (1988).

By taking the average of 11 itp + over the crack #1 region on Eq. (1), we have,

( )( )212122011 xyyy itipptip Λ+Λ><+><+>=<+>< (4)

where ∫−

−=Λa

ayyyy dx

a1

21

121 σ and ∫

−

=Λa

axyxy dx

a 121

21 σ . (5)

21yyΛ represents the average of normal stress over crack #1 region due to unit tension applied on

crack #2 surface. Similarly, 21xyΛ represents the average of shear stress over crack #1 region due to

unit tension applied on crack #2 surface. Tensile stresses on crack #1 and #2 are symmetric and shear stresses are inversely symmetric about the axis passing at x1=b (x2=-b) (based on Eq. (2)), therefore,

>>=<>=<<−>>=<>=<< tttppp 2121 , (6)

and thus Eq. (4) leads to

( )

( ) ( ) ( ) ( ) 0221221

21

0221221

21

1,

1

1ptpp

xyyy

xy

xyyy

yy

Λ−Λ−

Λ−>=<

Λ−Λ−

Λ+>=< . (7)

The complex interfacial stress intensity factors for the inner and outer tips for the crack #1 is given as

( ) ( )( ) ( ) 111

2/1

1

12/12/1

21 2cosh2 dxitpxaxaaiKKaK

a

a

ii +⎟⎟⎠

⎞⎜⎜⎝

⎛±

⎟⎠⎞⎜

⎝⎛=+= ∫

−

+−−

εεπε

π (8)

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H. Lekesiz

(Suo, 1990). By inserting Eq. (7) into Eq. (1) and then into Eq. (8), K can be calculated by utilizing numerical integral algorithms. In this study, both Eq. (5) and (8) are evaluated numerically using built-in ‘quad’ function in a commercial computation program, Matlab®. For convenience, a normalized K is defined as ( )επ iapKikkk −=+= 2/1

021 . Real and imaginary parts of the normalized stress intensity factor is plotted as function of a/b and in Figure 2 (a) and (b), respectively. The figures are plotted in the range of a/b [0, 0.99] because a/b larger than 0.99 practically indicates separation of the half spaces. Inner tip is indicated by white surface and outer tip is indicated by gray surface in both graphs. For both inner and outer tips, k1 increases as cracks gets closer and increases. =0 corresponds to the homogeneous case and values for this case given by Kachanov (1985) are indicated by black circular dots in the Figure 2(a). The influence of material dissimilarity ( ) is more evident for higher a/b values. For a/b=0.99, k1 varies between 2.4906 ( =0) and 2.9001 ( =0.1748) while it varies between 1.0000 ( =0) and 1.0349 ( =0.1748) for a/b=0 (single crack case).

(a) (b)

Figure 2 (a) k1 and (b) k2 for two equal cracks at the interface between dissimilar materials

As can be seen in figure 2 (b), k2 value sharply increases as increases for inner tip while the increase is even sharper for high a/b values especially for a/b>0.9. For outer tip, normalized k2 value remains almost constant in the all range of a/b for a given and a slight decrease is observed as a/b approaches 0.99.

3. Infinite row of cracks Consider infinite series of equal cracks with the length of 2a at the interface between two dissimilar, linearly elastic, isotropic half spaces with elastic moduli E1, E2 and Poisson’s ratios 1, 2 . The cracks are spaced at constant interval of 2b and cracks are subjected to a remote uniform tension, p0 as shown in Figure 3.

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On the complex stress intensity factor for split type interface cracks

Figure 3 Infinite number of periodic strip cracks at the interface between two dissimilar, linearly elastic, isotropic half

spaces with elastic moduli E1, E2 and Poisson’s ratios 1, 2

Cracks located on the left side of crack #1 are numbered with negative sign, and the cracks located on the right side are numbered with positive sign. Similar to Eq. (1), we can write traction on crack #1 as follows:

( )( ) ( )( )

( )( ) ( )( ) ++><+><++><+><++

+><+><++><+><+=+−−−−−−

3332220

22233311

xyyyxyyy

xyyyxyyy

itipitipp

itipitipitp

σσσσ

σσσσ (9)

By taking the average of both sides over crack #1 region, we have

( )( ) ( )( )( )( ) ( )( )+Λ+Λ><+><+Λ+Λ><+><++

Λ+Λ><+><+Λ+Λ><+><+

>=<+><−−

−−−−

−−

313133

2121220

212122

313133

11

xyyyxyyy

xyyyxyyy

itipitipp

itipitip

tip

(10)

The transmission factors, 11 , jxy

jyy ΛΛ are defined in the same manner as in Eq. (5). For a periodic

configuration of infinite cracks as in Figure 3, the normal and shear tractions are identical for all cracks, and due to Eq. (2), average of shear traction is zero. Therefore, we can write

0

,

32123

32123

>=>=<>=<>=<>=<<>>=<>=<>=<>=<>=<<

−−

−−

tttttpppppp

(11)

and, 11 jyy

jyy Λ=Λ− ; 11 j

xyjxy Λ−=Λ (12)

By inserting Eqs. (11) and (12) into Eq. (10), we obtain

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H. Lekesiz

,21 0

1

2

1 ppj

jyy

−∞

=⎟⎟⎠

⎞⎜⎜⎝

⎛Λ−>=< ∑ (13)

For the calculation of SIF of crack #1, we insert Eq. (9) into Eq. (8) along with Eq. (13). The calculations are conducted using Matlab and ‘quad’ function is used for the numerical integrations. k1 for the infinite crack case is plotted as function of a/b and in Figure 4 (a). The values produced in this study are shown as a white surface and exact values are shown with gray surface. As cracks get closer, k1 values increase for any value. For a/b around 0.99, say a/b=0.97, both approximate and exact values become smaller as increases. Relative percent difference between exact and approximate k1 values is plotted in Figure 4 (b). The maximum difference occurs at a/b=0.99 and =0.1748 which is around 35%. For the homogenous case ( =0), the error reaches around 25% for

a/b=0.99. The error remains smaller than 3.8% for any a/b smaller than 0.90.

(a) (b) Figure 4 (a) k1 for infinite number of periodic cracks at the interface between two dissimilar materials (b) Relative percent error between the approximate method based on Kachanov(1985) and exact solution given by Rice and Sih (1965)

k2 values based on approximate (white surface) and exact (gray surface) solutions are plotted as function of a/b and in Figure 5 (a) and the relative percent difference between approximate and exact methods is depicted in Figure 5 (b). Different than Figure 4 (a), k2 values increases as increases for any a/b value. Relative percent difference reaches maximum as a/b approaches 0.99. Maximum difference is about 70% when =0.1748, however, it remains smaller than 8.4% when a/b is smaller than 0.90. As can be seen from Figure 4 and 5, Kachanov’s method can be applied to interface cracks with a very good approximation. For a/b values up to 0.90, the error in the approximate method remains almost same with the homogenous case independent of material dissimilarity. For a/b values larger than 0.90, the error in general increases as the material dissimilarity ( ) increases. The small error in approximate method which is mostly independent of material dissimilarity proves that approximate

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On the complex stress intensity factor for split type interface cracks

method developed for homogenous cracks can be applied to interface cracks with no additional difficulty.

(a) (b) Figure 5 (a) k2 for infinite number of periodic cracks at the interface between two dissimilar materials (b) Relative percent error between the approximate method based on Kachanov(1985) and exact solution given by Rice and Sih (1965)

4. Interpenetration and contact zone The open model for single interface cracks (England 1965; Rice and Sih 1965) used in this paper exhibits an oscillatory stress behavior as the distance from the crack tip approaches zero. It was remarked by several researchers that crack opening displacement also oscillates in the region close to the crack tip. This leads to negative displacement that can be interpreted as interpenetrations of two materials which is physically not possible. Comninou (1977) resolved this issue by defining contact regions at the crack tips. Rice (1988) and Hills and Barber (1993) provided an extensive discussion on where interpenetrations or contact zone is small so that open model can be used. For the case of pure tension, interpenetration zone size is very small and therefore open model is valid for even highly dissimilar material couples, i.e. =0.1748. To ensure that interpenetration zone size (ai) is still very small with the approximate method presented, Eq. (15) in Hills and Barber (1993) is employed using the stress intensity factors calculated in this paper. For infinite row of collinear interface cracks presented in Section 4, interpenetration zone size normalized by half crack length, ai/a is obtained as 1.031(10-5) for =0.1748 and a/b=0.99. Even in such extreme material dissimilarity and very close crack interactions, ai value remains very small and therefore, open model can be used and combined with the approximate method without leading large interpenetration zones in which open model cannot be used.

5. Conclusions The approximate method developed by Kachanov (1985, 1989) for interacting cracks in homogenous medium can be easily extended to predict stress intensity factor for interacting interface cracks. For the case of periodic array of collinear slit cracks, the error in this approximation is almost same as in homogenous case for a/b smaller than 0.90. For higher a/b

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H. Lekesiz

values, as the material dissimilarity approaches the maximum value, =0.1748, the error increases and positive sign of error indicates that the approximate method overestimates stress intensity factor for strongly dissimilar materials. Real part of the stress intensity factor which becomes mode I SIF when =0 (homogenous case) can be predicted more accurately compared to imaginary part which corresponds to mode II SIF when =0.

Acknowlegments The author wish to thank Professor Stanislav Rokhlin and Professor Noriko Katsube for their encouraging and helpful suggestions on crack interaction problems. References Comninou M (1977) The interface Crack. Journal of Applied Mechanics-Transactions of the ASME 44(4):631-636. England AH (1965) A crack between dissimilar media. Journal of Applied Mechanics 32:400-402. Erdogan F (1965) Stress distribution in bonded dissimilar materials with cracks. Journal of Applied Mechanics 32(2):403-410. Fabrikant VI (1987) Close interaction of coplanar circular cracks in an elastic medium. Acta Mechanica 67(1-4):39-59. Fabrikant VI (1989) Flat crack under shear loading. Acta Mechanica 78(1-2):1-31. Hills DA and Barber JR (1993) Interface cracks. International Journal of Mechanical Sciences 35(1): 27-37. Hills DA, Kelly PA, Dai DN, Korsunsky AM (1996) Solution of crack problems: The distributed dislocation technique. Kluwer Academic Publishers Kachanov M (1985) A simple technique of stress-analysis in elastic solids with many cracks. International Journal of Fracture 28(1):R11-R19. Kachanov M, Laures JP (1989) 3-Dimensional problems of strongly interacting arbitrarily located penny-shaped cracks. International Journal of Fracture 41(4):289-313. Lavrentyev AI and Rokhlin SI (1994) Models for ultrasonic characterization of environmental degradation of interfaces in adhesive joints. Journal of Applied Physics 76(8):4643-4650. Lekesiz H, Katsube N, Rokhlin SI, Seghi RR (2013) The stress intensity factors for a periodic array of interacting coplanar penny-shaped cracks. International Journal of Solids and Structures 50(1):186-200 MossakovskiiVI, Rybka MT (1964) Generalization of the Griffith-Sneddon criterion for the case of a non-homogenous body. Prikl. Mat. Mekh 28:1061-1069. Noda NA, Oda K (1997) Interaction effect of stress intensity factors for any number of collinear interface cracks. International Journal of Fracture 84(2):117-128 Rice JR (1988) Elastic fracture mechanics concepts for interfacial cracks. Journal of Applied Mechanics-Transactions of the ASME 55(1):98-103. Rice JR and Sih GC (1965) Plane problems of cracks in dissimilar media. Journal of Applied Mechanics 32(2): 418-423. Sih GC (1973) Handbook of stress-intensity factors for researchers and engineers. Institute of Fracture and Solid Mechanics, Lehigh University Suga T, Elssner G, Schmauder S (1988) Composite parameters and mechanical compatibility of material joints. Journal of Composite Materials 22:917-934 Sun Y, Zhang Z, Kitipornchai S, Liew KM (2006) Analyzing the interaction between collinear interfacial cracks by an efficient boundary element-free method. International Journal of Engineering Science 44:37-48. Williams ML (1959) The stress around a fault or crack in dissimilar media. Bulletin of the Seismological Society of America 49:199-204. Willis JR (1972) Penny-shaped crack on an interface. The Quarterly Journal of Mechanics and Applied Mathematics 25:367-385. Yang XX, Kuang ZB (1996) Contour integral method for stress intensity factors of interface crack. International Journal of Fracture 78:299-313.