Compatibility conditions of continua using Riemann-Cartan geometry

TL;DR

This paper explores the geometric compatibility conditions in continuum mechanics using Riemann-Cartan geometry, revealing that key conditions relate to the Einstein tensor, thus unifying different theories under a common geometric framework.

Contribution

It demonstrates that Vallée's and Nye's compatibility conditions are equivalent to the vanishing of the Einstein tensor in Riemann-Cartan geometry, providing a new geometric perspective.

Findings

Vallée's compatibility condition equals Einstein tensor vanishing.

Nye's tensor compatibility also relates to Einstein tensor.

The approach extends to micro-continuum theories.

Abstract

The compatibility conditions for generalised continua are studied in the framework of differential geometry, in particular Riemann-Cartan geometry. We show that Vall\'{e}e's compatibility condition in linear elasticity theory is equivalent to the vanishing of the three dimensional Einstein tensor. Moreover, we show that the compatibility condition satisfied by Nye's tensor also arises from the three dimensional Einstein tensor which appears to play a pivotal role in continuum mechanics not mentioned before. We discuss further compatibility conditions which can be obtained using our geometrical approach and apply it to the micro-continuum theories.