Stroh formalism in analysis of skew-symmetric and symmetric weight functions for interfacial cracks

TL;DR

This paper demonstrates how the Stroh formalism can be effectively used to analyze skew-symmetric and symmetric weight functions for interfacial cracks in anisotropic solids, providing explicit solutions and applications to crack propagation.

Contribution

It introduces a novel algebraic eigenvalue approach using the Stroh formalism and Riemann-Hilbert problem to derive explicit weight functions for interfacial cracks in anisotropic materials.

Findings

Explicit expressions for weight matrix functions are derived.

The method is applied to compute stress intensity factors for asymmetric loads.

The approach simplifies solving complex crack problems in anisotropic media.

Abstract

The focus of the article is on analysis of skew-symmetric weight matrix functions for interfacial cracks in two dimensional anisotropic solids. It is shown that the Stroh formalism proves to be an efficient approach to this challenging task. Conventionally, the weight functions, both symmetric and skew-symmetric, can be identified as a non-trivial singular solutions of the 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, 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 in the computation of complex stress intensity factor corresponding to an asymmetric load acting on the crack faces.