Weak solutions to anti-plane boundary value problems in a linear theory of elasticity with microstructure

Abstract

In this paper we formulate the interior and exterior Dirichlet and Neumann boundary value problems of anti-plane micropolar elasticity in a weak (Sobolev) space setting, we show that these problems are well-posed and the corresponding weak solutions depend continuously on the data. We show that the problem of torsion of a micropolar beam of (non-smooth) arbitrary cross-section can be reduced to an interior Neumann boundary value problem in antiplane micropolar elasticity and consider an example which demonstrates the significance of material microstructure.