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

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