Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators

Abstract: We revisit stress problems in linear elasticity to provide a perspective from the geometrical and functionalanalytic points of view. For the static stress problem of linear elasticity with mixed boundary conditions we write the associated pair of unbounded adjoint operators. The stress solution is found as an intersection of affine translations of the fundamental subspaces of the adjoint operators. In particular, we treat the equilibrium equation in the operator form, involving spaces of traces on a part of the boundary, known as Lions-Magenes spaces. Our analysis of the pair of adjoint operators for the problem with mixed boundary conditions relies on the properties of the analogous pair of operators for the problem with the displacement boundary conditions, which we also include in the paper.