On Dual Configurational Forces

Abstract

The dual conservation laws of elasticity are systematically re-examined by using both Noether's variational approach and Coleman–Noll–Gurtin's thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy–momentum tensor) $\mathcal{P}$ and the dual configuration force (dual energy–momentum tensor) ${\mathcal{L}}$,

$$\mathcal{P} - \mathcal{L} = ({\bf P}: {\bf F}) {\bf 1} - \nabla ({\bf P} \cdot {\bf x})~. \nonumber$$

This and other results derived in this paper may lead to a better understanding of configurational mechanics and therefore of mechanics of defects.