Models of corner and crack singularity of linear elastostatics and their numerical solutions

The singular solutions for linear elastostatics at corners are essential in both theory and computation. In this paper we seek the singular solutions for corners with the clamped and the free stress boundary conditions, and explore corner singularity in detail. In this paper the singular solutions of linear elastostatics are derived, and two new models of interior crack singularity are proposed. The collocation Trefftz methods are used to obtain highly accurate solutions, where the leading coefficient has 14 (or 12) significant digits by the computation with double precision. Such solutions are useful to examine other numerical methods for singularity problems in linear elastostatics. Also the explicit singular solutions can be adapted to design and develop efficient numerical methods for singularity problems, such as the combined method (Li, 1998, 2008 [19] and [20]) and the Trefftz methods which include the boundary approximation method (Li, 1990, Li et al., 1987 [18] and [26]), the collocation Trefftz method (Li et al., 2008 [24]), the hybrid Trefftz method (Qin, 2000 [36]), the boundary collocation techniques (Kolodziej and Zielinski, 2009 [16]), etc. This paper also explores a systematic analysis for singularity properties and explicit singular solutions for corners of linear elastostatics.