Effective spring stiffness for non-interacting penny-shaped cracks at an interface between two dissimilar,isotropic,linearly elastic materials

Abstract

A quasi-static approximation has been widely used to model ultrasonic wave interactions at imperfect interfaces between two linearly elastic materials. To simulate the reduction of static stiffness of the overall structure due to compromised interfaces (micro-cracks or micro-disbonds), the compliance of an imperfect interface to a loading is represented by continuous, uniform distributions of linearly elastic interfacial springs. In this work, a representation by normal and transverse effective spring stiffnesses of an array of non-interacting penny-shaped cracks at the interface between two dissimilar, isotropic, linearly elastic materials is obtained based on classical fracture mechanics. The results obtained are useful in estimating the disbonded area, which is critical in assessing the bond integrity and estimating the remaining life. Special care is taken to avoid crack surface interpenetration for transverse loading, and the valid loading range is obtained to assure negligibility of crack surface interpenetration for all possible ranges of isotropic, linearly elastic material combinations. For linear ultrasound applications, it is shown that the expression obtained for transverse springs can be used for most isotropic, linearly elastic material combinations, if the initial maximum crack opening displacement is more than 10⁻⁵ of the crack radius. The obtained expressions are applied to estimate the accuracy of existing approximate models based on the analysis of penny-shaped cracks in a homogeneous material, rules of mixture, and Hertzian-based effective moduli. It is shown that for most practical material combinations the error is below 5%.

Keywords

penny-shaped cracks dissimilar materials interface spring stiffness ultrasound

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References

Baik, J. , and Thompson, RB Ultrasonic scattering from imperfect interfaces: a quasi-static model. J Nondestr Eval 1984; 4: 177-196.

Margetan, FJ , Thompson, RB , and Gray, TA Interfacial spring model for ultrasonic interactions with imperfect interfaces: theory of oblique incidence and application to diffusion-bonded butt joints. J Nondestr Eval 1988; 7: 131-152.

Angel, YC , and Achenbach, JD Reflection and transmission of elastic-waves by a periodic array of cracks . J Appl Mech Trans ASME 1985; 52: 33-41.

Nakagawa, S. , Nihei, KT , and Myer, LR Plane wave solution for elastic wave scattering by a heterogeneous fracture. J Acoust Soc Am 2004; 115: 2761-2772.

Kim, J. , Baltazar, A. , and Rokhlin, SI Ultrasonic assessment of rough surface contact between solids from elastoplastic loading-unloading hysteresis cycle. J Mech Phys Solids 2004; 52: 1911-1934.

Palmer, DD , Rehbein, DK , Smith, JF , and Buck, O. Nondestructive characterization of mechanical strength of diffusion bonds, II. Application of a quasi-static spring model . J Nondestr Eval 1988; 7: 167-174.

Thompson, RB , and Thompson, DO Past experiences in the development of tests for adhesive bond strength. J Adhes Sci Technol 1991; 5: 583-599.

Rokhlin, SI , Xie, B. , and Baltazar, A. Quantitative ultrasonic characterization of environmental degradation of adhesive bonds. J Adhes Sci Technol 2004; 18: 327-360.

Pecorari, C. , and Kelly, PA The quasi-static approximation for a cracked interface between a layer and a substrate. J Acoust Soc Am 2000; 107: 2454-2461.

10.

Lavrentyev, AI , and Rokhlin, SI Ultrasonic spectroscopy of imperfect interfaces between layer and two solids. J Acoust Soc Am 1998; 103: 657-644.

11.

Baltazar, A. , Wang, L. , Xie, B. , and Rokhlin, SI Inverse ultrasonic determination of imperfect interfaces and bulk properties of a layer between two solids. J Acoust Soc Am 2003; 114: 1424-1434.

12.

Wang, H. , Qian, ML , and Liu, W. Laser ultrasonic characterization of adhesive bonds between epoxy coating and aluminum substrate. Ultrasonics 2006; 44: E1349-E1353.

13.

Bostrom, A. , and Golub, M. Elastic SH wave propagation in a layered anisotropic plate with interface damage modeled by spring boundary conditions. Q J Mech Appl Math 2008; 62: 39-52.

14.

Johnson, KL Contact mechanics. Cambridge [Cambridgeshire]; New York : Cambridge University Press, 1987 , Chapter 4.

15.

Lavrentyev, AI , and Rokhlin, SI Models for ultrasonic characterization of environmental degradation of interfaces in adhesive joints. J Appl Phys 1994; 76: 4643-4650.

16.

Willis, JR Penny-shaped crack on an interface. Q J Mech Appl Math 1972; 25: 367-385.

17.

Mossakovskii, VI , and Rybka, MT Generalization of the Griffith-Sneddon criterion for the case of a non-homogeneous body. Prikl Mat Mekh 1964; 28: 1061-1069.

18.

Tada, H. , Paris, PC , and Irwin, GR The stress analysis of cracks handbook. New York: ASME Press, 2000.

19.

Dundurs, J. , and Bogy, D. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading . J Appl Mech 1969; 36: 650-652.

20.

Fabrikant, VI Close interaction of coplanar circular cracks in an elastic medium. Acta Mech 1987; 67: 39-59.

21.

Kachanov, M. , and Laures, JP Strong 3-dimensional interactions of several arbitrarily located penny-shaped cracks. Int J Fract Mech 1988; 37: R63-R68.

22.

Lekesiz, H. , Katsube, N. , Rokhlin, S. , and Seghi, R. Effective spring stiffness for a planar periodic array of collinear cracks at an interface between two dissimilar isotropic materials, In print in Int. J. Mech. Mater. (2010). doi:10.1016/j.mechmat.2010.12.004.

23.

Suga, T. , Elssner, G. , and Schmauder, S. Composite parameters and mechanical compatibility of material joints. J Compos Mater 1988; 22: 917-934.

24.

Comninou, M. Interface crack. J Appl Mech Trans ASME 1977 ; 44: 631-636.

25.

Comninou, M. Interface crack in a shear field. J Appl Mech Trans ASME 1978; 45: 287-290.

26.

Comninou, M. , and Schmueser, D. Interface crack in a combined tension-compression and shear field. J Appl Mech Trans ASME 1979 ; 46: 345-348.

27.

Rice, JR Elastic fracture-mechanics concepts for interfacial cracks. J Appl Mech Trans ASME 1988; 55: 98-103.

28.

Hills, DA , and Barber, JR Interface cracks. Int J Mech Sci 1993; 35: 27-37.

29.

Graciani, E. , Mantic, V. , and Paris, F. On the estimation of the first interpenetration point in the open model of interface cracks. Int J Fract 2007; 143: 287-290.

30.

Gorbatikh, L. On elastic compliances of interfacial cracks . Int J Fract 2004; 127: L141-L148.