Differential Complexes in Continuum Mechanics

TL;DR

This paper explores differential complexes involving tensors in continuum mechanics, linking kinematics and kinetics, and deriving compatibility conditions for motions and stresses in various geometries.

Contribution

It introduces a tensorial complex framework that unifies continuum mechanics kinematics and kinetics, and connects it to the de Rham complex for compatibility analysis.

Findings

Derived compatibility conditions for displacement and stress functions.

Established local compatibility equations for 2D and 3D motions.

Linked tensorial complexes to de Rham complex for non-contractible bodies.

Abstract

We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motions of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of the displacement gradient and the existence of stress functions on non-contractible bodies. We also derive the local compatibility equations in terms of the Green deformation tensor for motions of $2$D and $3$D bodies, and shells in curved ambient spaces with constant curvatures.