Fundamental solutions of penny-shaped and half-infinite plane cracks embedded in an infinite space of one-dimensional hexagonal quasi-crystal under thermal loading

Abstract

This paper presents fundamental solutions for an infinite space of one-dimensional hexagonal quasi-crystal medium, which contains a penny-shaped or half-infinite plane crack subjected to two identical thermal loadings on the upper and lower crack lips. In view of the symmetry of the problem with respect to the crack plane, the original problem is transformed to a mixed boundary problem for a half-space, which is solved by means of a generalized method of potential theory conjugated with the newly proposed general solutions. When the cracks are under the action of a pair of point temperature loadings, fundamental solutions in terms of elementary functions are derived in an exact and complete way. Important parameters in crack analyses such as stress intensity factors and crack surface displacements are presented as well. The underlying relations between the fundamental solutions for the two cracks involved in this paper are discovered. The temperature fields associated with these two cracks are retrieved in alternative manners. The obtained solutions are of significance to boundary element analysis, and have an important role in clarifying simplified studies and serving as benchmarks for computational fracture mechanics can be expected to play.