Rotational invariance conditions in elasticity, gradient elasticity and its connection to isotropy

TL;DR

This paper explores the conditions for rotational invariance in higher gradient elasticity models, clarifies the distinction between local and global invariance, and connects these concepts to isotropy, providing tools for analyzing and extending elasticity models.

Contribution

It introduces a detailed framework for understanding frame-indifference and isotropy in higher gradient elasticity, including new invariance conditions and their implications for linear and nonlinear models.

Findings

Identifies local versus global SO(3)-invariance in elasticity models.

Shows equivalence of invariance conditions to isotropy in linear models.

Provides criteria for checking isotropy and extending to anisotropy.

Abstract

For homogeneous higher gradient elasticity models we discuss frame-indifference and isotropy requirements. To this end, we introduce the notions of local versus global SO(3)-invariance and identify frame-indifference (traditionally) with global left SO(3)-invariance and isotropy with global right SO(3)-invariance. For specific restricted representations, the energy may also be local left SO(3)-invariant as well as local right SO(3)-invariant. Then we turn to linear models and consider a consequence of frame-indifference together with isotropy in nonlinear elasticity and apply this joint invariance condition to some specific linear models. The interesting point is the appearance of finite rotations in transformations of a geometrically linear model. It is shown that when starting with a linear model defined already in the infinitesimal symmetric strain $\varepsilon = {\rm sym} \, {\rm Grad}[u]$, the new invariance condition is equivalent to isotropy of the linear formulation. Therefore, it may be used also in higher gradient elasticity models for a simple check of isotropy and for extensions to anisotropy. In this respect we consider in more detail variational formulations of the linear indeterminate couple stress model, a new variant of it with symmetric force stresses and general linear gradient elasticity.