Compatibility conditions of continua using Riemann-Cartan geometry

Understand

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.