Perturbation analysis for an imperfect interface crack problem using weight function techniques

TL;DR

This paper develops a new weight function approach using Fourier and Wiener-Hopf techniques to analyze how small inclusions and interface imperfections affect crack propagation in bi-materials under anti-plane shear.

Contribution

It introduces a novel weight function for imperfect interfaces and derives perturbation formulas to assess the impact of small inclusions on crack tip stresses.

Findings

The weight function accurately captures interface imperfection effects.

Inclusion presence can amplify or shield crack propagation.

Computed constants vary with interface and material properties.

Abstract

We analyse a problem of anti-plane shear in a bi-material plane containing a semi-infinite crack situated on a soft imperfect interface. The plane also contains a small thin inclusion (for instance an ellipse with high eccentricity) whose influence on the propagation of the main crack we investigate. An important element of our approach is the derivation of a new weight function (a special solution to a homogeneous boundary value problem) in the imperfect interface setting. The weight function is derived using Fourier transform and Wiener-Hopf techniques and allows us to obtain an expression for an important constant (which may be used in a fracture criterion) that describes the leading order of tractions near the crack tip for the unperturbed problem. We present computations that demonstrate how this constant varies depending on the extent of interface imperfection and contrast in material stiffness. We then perform perturbation analysis to derive an expression for the change in the leading order of tractions near the tip of the main crack induced by the presence of the small defect, whose sign can be interpreted as the inclusion's presence having an amplifying or shielding effect on the propagation of the main crack.