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

TL;DR

This paper offers a geometric and functional analytic approach to solving static linear elasticity stress problems with mixed boundary conditions by analyzing adjoint operators and their fundamental subspaces.

Contribution

It introduces a novel operator-theoretic framework for stress analysis in linear elasticity with mixed boundary conditions, emphasizing the role of adjoint operators and Lions-Magenes spaces.

Findings

Stress solutions characterized as intersections of affine subspaces.

Operator framework unifies boundary condition treatments.

Analysis extends to problems with displacement boundary conditions.

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.