Abstract
The focus of the paper is on the analysis of skew-symmetric weight functions for interfacial cracks in two-dimensional anisotropic solids. It is shown that the Stroh formalism proves to be an efficient tool for this challenging task. Conventionally, the weight functions, both symmetric and skew-symmetric, can be identified as non-trivial singular solutions of a homogeneous boundary-value problem for a solid with a crack. For a semi-infinite crack, the problem can be reduced to solving a matrix Wiener–Hopf functional equation. Instead, the Stroh matrix representation of displacements and tractions, combined with a Riemann–Hilbert formulation, is used to obtain an algebraic eigenvalue problem, which is solved in a closed form. The proposed general method is applied to the case of a quasi-static semi-infinite crack propagating between two dissimilar orthotropic media: explicit expressions for the weight functions are evaluated and then used in the computation of the complex stress intensity factor corresponding to a general distribution of forces acting on the crack faces.
Highlights
Evaluation of coefficients in asymptotic representations of displacements and stress fields represents an important issue for addressing vector problems of crack propagation in elastic materials (Bercial-Velez et al, 2005; Mishuris & Kuhn, 2001)
The problem is generally reduced to a functional equation of Wiener-Hopf type, and its solution gives the symmetric weight function matrix (Antipov, 1999), while the skew-symmetric component is obtained by the construction of the corresponding full-field singular solution of the elasticity boundary value problem discussed in Piccolroaz et al (2009)
The developed general approach for the derivation of symmetric and skew-symmetric weight function matrices for interfacial plane cracks between dissimilar anisotropic materials, based on Stroh formulation of displacements and stress fields, have been discussed in details and tested by means of the application to the case of a crack placed at the interface between two orthotropic materials under plane stress
Summary
Evaluation of coefficients in asymptotic representations of displacements and stress fields represents an important issue for addressing vector problems of crack propagation in elastic materials (Bercial-Velez et al, 2005; Mishuris & Kuhn, 2001). The explicit derivation of weight functions is fundamental for the evaluation of stress intensity factors corresponding to a general distribution of forces acting on the crack faces, as well as for the calculation of higher order coefficients in the asymptotic expressions of the fields. It is shown that the challenging analysis of the matrix functional Wiener-Hopf equation can be replaced by solving a matrix eigenvalue problem deduced via an equivalent formulation, based on Stroh representation of displacement and stress fields (Stroh, 1962) By means of this new approach, general expressions for weight functions matrices, valid for plane interfacial cracks problems between any anisotropic media, are obtained. The perfect agreement detected between the expressions derived by means of two alternative formulations represents an important benchmarking for the correctness of the obtained results
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