Null Lagrangians in linear theories of micropolar type and few other generalizations of elasticity

TL;DR

This paper investigates null Lagrangians within linear generalized elasticity theories, including micropolar media and quasicrystals, revealing conditions under which null Lagrangians exist and their relation to symmetry properties of elasticity tensors.

Contribution

It introduces a method to identify null Lagrangians in various elasticity models and explores their properties and implications for energy decomposition.

Findings

Non-zero null Lagrangians can be found in some cases.

Null Lagrangians vanish under certain symmetry conditions.

Energy can be split into null Lagrangian and remainder.

Abstract

In the context of linear theories of generalized elasticity including those for homogeneous micropolar media, quasicrystals, piezoelectric and piezomagnetic media, we explore the concept of null Lagrangians. For obtaining the family of null Lagrangians we employ the sufficient conditions of H. Rund. In some cases a non-zero null Lagrangian is found and the stored energy admits a split into a null Lagrangian and a remainder. However, the null Lagrangian vanishes whenever the relevant elasticity tensor obeys certain symmetry conditions which can be construed as an analogue of the Cauchy relations.