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

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.

Acknowledgement

M.L. has been supported by an Emmy-Noether grant of the Deutsche Forschungsgemeinschaft (Grant No. La1974/1-2). G.A.M. benefits from a Max-Planck Award for international cooperation (2002–2006).

Notes

†We use the notation of Schouten 21. Symmetrization over two indices is denoted by parentheses, , antisymmetrization by brackets, .

†Sometimes the third Bianchi identity is called the zeroth Bianchi identity, e.g. Hehl et al. 27, Gronwald and Hehl 28.