Analysis of interfacial crack problems using the weight functions technique

A general procedure for the analysis of interfacial crack problems based on weight functions theory and reciprocity theorems the derivation is proposed. The weight functions are defined as a non-trivial singular solutions of the homogeneous boundary value problem for a solid with a crack. For a semi-infinite crack in isotropic bi-materials, the problem can be reduced to solving a matrix Wiener-Hopf functional equation. Instead, for interfacial cracks between anisotropic media, the Stroh matrix representation of displacements and tractions, combined with a Riemann-Hilbert formulation, is used to obtain an algebraic eigenvalue problem, that is solved in a closed form. The proposed general method is applied to the case of a quasi-static semi-infinite crack propagation between two dissimilar orthotropic media: explicit expressions for the weight matrix functions are evaluated and then used together with Betti integral formula in the computation of complex stress intensity factor corresponding to an asymmetric load acting on the crack faces. Moreover, weight function matrices are used together with Betti's reciprocity theorems in order to formulate the fracture problems in terms of singular integral equations relating the applied loading and the resulting crack opening. The derived compact formulation can be applied for modeling many multiphysics phenomena, where the elastic problem is coupled with other concurrent physical phenomena.