Fundamental solution and the weight functions of the transient problem on a semi-infinite crack propagating in a half-plane

TL;DR

This paper develops a mathematical framework for analyzing a semi-infinite crack propagating in a half-plane, deriving explicit solutions and weight functions to understand boundary effects and crack propagation dynamics.

Contribution

It introduces a novel method combining Laplace and Fourier transforms with matrix factorization to solve the transient crack problem in a semi-infinite domain.

Findings

Closed-form solution when the crack is far from the boundary

Numerical weight functions for crack propagation analysis

Insights into boundary effects on crack behavior

Abstract

The two-dimensional transient problem that is studied concerns a semi-infinite crack in an isotropic solid comprising an infinite strip and a half-plane joined together and having the same elastic constants. The crack propagates along the interface at constant speed subject to time-independent loading. By means of the Laplace and Fourier transforms the problem is formulated as a vector Riemann-Hilbert problem. When the distance from the crack to the boundary grows to infinity the problem admits a closed-form solution. In the general case, a method of partial matrix factorization is proposed. In addition to factorizing some scalar functions it requires solving a certain system of integral equations whose numerical solution is computed by the collocation method. The stress intensity factors and the associated weight functions are derived. Numerical results for the weight functions are reported and the boundary effects are discussed. The weight functions are employed to describe propagation of a semi-infinite crack beneath the half-plane boundary at piecewise constant speed.