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

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 frameindifference (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 ε = sym 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. 3 tr(X) ½ ∈ sl(3) and we have the orthogonal Cartan-decomposition of the Lie-algebra gl(3) gl(3) = {sl(3) ∩ Sym(3)} ⊕ so(3) ⊕ R·½ , X = dev sym X + skew X + 1 3 tr(X) ½ ,(1.1) simply allowing to split every second order tensor uniquely into its trace-free symmetric part, skew-symmetric 2 Indeed, in Murdoch [39, eq.(16)-(20)] we find the same conclusions regarding isotropy of a material point. 3 This remark refers to eq.(1), Mindlin [37], which reads, in the notation of the present paper, ∂ m 31Here, m is the second-order couple stress tensor and σ 21 − σ 12 = 2 axl( skew σ) represents the skew-symmetric part of the total force stress tensor σ. The equation is given for the planar situation, see also eq.(6.11).

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

“…As explained in detail in [58], isotropy of the curvature energy is tantamount to requiring form-invariance of expression (16) under the transformation (18), i.e. :…”

“…W relax curv must be an isotropic scalar function. We need to highlight the fact that Curl P is not just an arbitrary combination of first derivatives of P (and as such included in the standard Mindlin-Eringen most general anisotropic micromorphic format), but that the formulation in Curl P supports a completely invariant setting, as seen in [58], [64]. Since Curl P is a second order tensor, it allows us to discard the 6 th order tensors of classical Mindlin-Eringen micromorphic elasticity and to work instead with 4 th order tensors whose anisotropy classification is much easier and well-known [7].…”

“…see [58,59]. Based on these transformation laws, we may investigate the anisotropy of the rotational coupling with the representation in terms of the second order tensor C c .…”

Chirality in the plane

“…In searching for a generic objective [28] and chiral term for a chiral energy functional, we begin by recalling objectivity. We call an energy functional objective if it is invariant under global (macroscopic) left-rotations Q with det Q = +1, here Q ∈ SO(3) is a constant orthogonal matrix.…”

On isotropy conditions in second gradient materials

“…The latter might explain why one may be inclined to allow non-constant rotation fields in (9), which is forbidden for higher gradient materials [4].…”