On microcontinuum field theories: the Eshelby stress tensor and incompatibility conditions

TL;DR

This paper explores advanced microcontinuum elasticity theories, deriving incompatibility conditions, Bianchi identities, and calculating the Eshelby stress tensor to understand forces driving defects and inhomogeneities.

Contribution

It provides a comprehensive analysis of incompatibility conditions and the Eshelby stress tensor in various incompatible microcontinuum elasticity theories.

Findings

Derived incompatibility conditions and Bianchi identities.

Calculated Eshelby stress tensor for inhomogeneous media.

Identified configurational forces driving defects.

Abstract

We investigate linear theories of incompatible micromorphic elasticity, incompatible microstretch elasticity, incompatible micropolar elasticity and the incompatible dilatation theory of elasticity (elasticity with voids). The incompatibility conditions and Bianchi identities are derived and discussed. The Eshelby stress tensor (static energy momentum) is calculated for such inhomogeneous media with microstructure. Its divergence gives the driving forces for dislocations, disclinations, point defects and inhomogeneities which are called configurational forces.