Null Lagrangians in Cosserat elasticity

TL;DR

This paper characterizes null Lagrangians in nonlinear Cosserat elasticity, providing conditions for their identification in 3D bodies and shells, and linking them to elasticity tensors in linearized theory.

Contribution

It offers a complete characterization of null Lagrangians in Cosserat elasticity using Olver and Sivaloganathan's theorem, including necessary and sufficient conditions for linearized tensors.

Findings

Complete set of conditions for null Lagrangians in 3D and shells

Application of Olver and Sivaloganathan's theorem to Cosserat elasticity

Necessary and sufficient conditions for null Lagrangians in linearized theory

Abstract

In the framework of nonlinear theory of Cosserat elasticity, also called micropolar elasticity, we provide the complete characterization of null Lagrangians for three dimensional bodies as well as for shells. Using the Gibb's rotation vector for description of the microrotation, this task is possible by an application of a theorem stated by Olver and Sivaloganathan in `{the structure of null Lagrangians}' (Nonlinearity, {1}, 1988, pp. 389-398). A set of necessary and sufficient conditions is also provided for the elasticity tensors to correspond to a null Lagrangian in linearized micropolar theory.