  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.
