Applications of the Trefftz method to the anti-plane fracture of 1D hexagonal piezoelectric quasicrystals

Abstract

In terms of solving fracture mechanics problems of quasicrystals, the boundary element method known as Trefftz Method is used for the first time. In this paper, the collocation Trefftz method as an indirect Trefftz method has been demonstrated to be an excellent numerical method for the anti-plane fracture problems in quasicrystals plate, which can exactly satisfy the governing equations and approximately satisfy the boundary conditions. The algorithms of the collocation Trefftz method and the trial functions for the anti-plane fracture problems of piezoelectric quasicrystals are presented. The effects of the geometry size, the loadings and the materials properties on the fracture results are studied. For the materials with three dimensional constitutive relations, such as piezoelectric quasicrystals, the Trefftz method is an effective numerical method to solve the fracture problems.