On singularities of solution of the elasticity system in a bounded domain with angular corner points

Abstract

This paper aims to give a mathematically rigorous description of the corner singularities of the weak solutions for the plane linearized elasticity system in a bounded planar domain with angular corner points on the boundary. The qualitative properties of the solution including its regularity depend crucially on these corner points or such types of boundary conditions. In particular, the resulting expansion of the solutions of the underlying problem involves singular vector functions, inlines, depending on a certain parameter ξμ. We derive the transcendental equations for all 10 possible cases of combinations of the boundary conditions generated by the basic four ones in classical elasticity proposing in the two natural directions of the boundary, that is, tangential and normal direction, respectively, which depends on ξμ. So a MATLAB program is developed whereby ξμ can be computed, and figures showing their distributions are presented. The leading singular exponents are computed for these combinations of the boundary conditions, wherein critical angles ωcritical are listed such that for interior angles ω < ωcritical the H2‐regularity of solution can be guaranteed. Moreover, the characterization of stress singularities in terms of the inner angle of a corner point is studied, and the regularity results are given.