Weight function approach to a crack propagating along a bimaterial interface under arbitrary loading in an anisotropic solid

Abstract

The focus of the paper is on the study of the dynamic steady-state propagation of interfacial cracks in anisotropic bimaterials under general, non-symmetric loading conditions. Symmetric and skew-symmetric weight functions, defined as singular non-trivial solutions of a homogeneous traction-free crack problem, have been recently derived for a quasi-static semi-infinite crack at the interface between two dissimilar anisotropic materials. In this paper, the expressions for the weight functions are generalised to the case of a dynamic steady-state crack between two anisotropic media. A functional matrix equation, through which it is possible to evaluate stress intensity factors and the energy release rate at the crack tip, is obtained. A general method for calculating asymptotic coefficients of the displacement and traction fields, without any restriction regarding the loading applied on the crack faces, is developed. The proposed approach is applied for computing stress intensity factors and higher order asymptotic terms corresponding to two different examples of loading configurations acting on the crack faces in an orthotropic bimaterial. arXiv:1305.0486v4 [physics.class-ph]

A general approach for calculations of weight functions and stress intensity factors

“…The weight function method has been widely applied to determine the SIFs of cracked structures since it is able to take complex loading conditions into consideration. 17,18 It has been shown that, if the SIF K ( a ) (1) and the crack face displacement u (1) ( x , a ) of any linearly elastic cracked solid are known as functions of the crack length a for a symmetrical load system (1), then for the same cracked solid subjected to any other symmetrical load system (2) in mode I loading conditions, the SIF K ( a ) (2) can be obtained by the simple integration of the weight function h ( x , a ) and the stress distribution function σ (2) ( x ) 15,16,21,22 where the weight function, independent of σ (2) ( x ), is defined as In equations (1) and (2), a is the half or full crack length for edge cracks and center cracks, respectively; H is a material constant, H = E for plane stress condition and H = E /(1 − v 2 ) for plane strain condition with E the Young’s modulus and v the Poisson’s ratio; K ( a ) (1) and u (1) ( x , a ) are, respectively, the known reference SIF and the crack face displacement in mode I loading conditions for the known load system (1); and σ (2) ( x ) is the stress distribution function across the plane of the crack in the crack free solid subjected to the load system (2).…”

“…The weight function method has been widely applied to determine the SIFs of cracked structures since it is able to take complex loading conditions into consideration. 17,18 It has been shown that, if the SIF K(a) (1) and the crack face displacement u (1) (x, a) of any linearly elastic cracked solid are known as functions of the crack length a for a symmetrical load system (1), then for the same cracked solid subjected to any other symmetrical load system (2) in mode I loading conditions, the SIF K(a) (2) can be obtained by the simple integration of the weight function h(x, a) and the stress distribution function (2) (x) 15,16,21,22 KðaÞ ð2Þ ¼…”

“…The weight function method provides a powerful and reliable method to calculate the SIF around a crack tip in a 2D linearly elastic body subjected to any arbitrarily chosen load systems. [15][16][17][18][19][20] Petroski and Achenbach 21 developed a judicious representation of the crack face displacement for the purpose of computing the weight function of edge cracks from only one reference load configuration. It was shown that the crack face displacements obtained by their representation were in excellent agreement with those given by the analytical methods.…”

Interfacial cracks in piezoelectric bimaterials: an approach based on weight functions and boundary integral equations

“…An alternative formulation where the weight functions are defined as the singular displacement field of the homogeneous traction free problem was proposed by Willis and Movchan [16]. Recently, this weight functions definition has been used in the derivation of stress intensity factors for both static and dynamic crack problems in isotropic and anisotropic bimaterials [8,17,18] and in thermodiffusive elastic media [19]. These weight functions have also been used in the derivation of singular integral equations relating interfacial tractions and crack displacement to the applied loadings on the crack faces [9,20,21].…”

Integral identities for fracture along imperfectly joined anisotropic ceramic bimaterials

“…Expressions were found for the stress intensity factors at the crack tip under the restriction of symmetric loading on the crack faces. Using weight function techniques introduced by Bueckner (1985) and developed further by Willis and Movchan (1995), an approach was developed to find stress intensity factors for an interfacial crack along a perfect interface under asymmetric loading for both the static and dynamic cases, see Morini et al (2013b) and Pryce et al (2013) respectively. More widely, weight functions are well developed in the literature for a wide range of fractured body geometries and allow for the evaluation of important constants that may act as fracture criteria.…”