Author(s)

A. N. Galybin

Abstract

This study is aimed at the development of a Trefftz-type method for solving plane elastic boundary value problems for open contours, which models crack propagation in brittle materials. The idea of the approach is as follows. Complex potentials are sought as linear combinations of independent holomorphic functions with the weights addressing singularities at the crack tips. Then the collocation method is applied to satisfy boundary conditions, which leads to a linear system for determination of unknown coefficients in the representation for complex potentials. The system is, in general, overdetermined and, thus, the SVD regularisation is applied to find its approximate solution. Two examples are presented. Keywords: cracks, complex potentials, Trefftz method, collocations, ill-posed problems. 1 Introduction Common technique for solving plane elastic problems with cracks assumes determination of two holomorphic functions (complex potentials, see Muskhelishvili [1]) that, in general, have square root singularities at crack tips. Usually, complex potentials are found by solving singular integral equation, SIE, (or a system of SIEs) with respect to unknown generalised crack opening displacements or their densities. This approach requires discretisation of the boundary and application of special quadratures for singular integrals, which represent two common steps in boundary integral methods. The present study is aimed to avoid these two steps and to apply directly the Trefftz approach by representing complex potentials as linear combinations of known holomorphic functions.

Keywords

cracks, complex potentials, Trefftz method, collocations, ill-posed problems.