On isotropy conditions in second gradient materials

In gradient elasticity, isotropy and frame-indifference requirements are sensitive to the homogeneity of the applied rotation field Q ∈ SO(3). This is in contrast to standard elasticity, where only first gradients of the deformation are under consideration. We use a diffeomorphism to show the effect of inhomogeneous coordinate transformation to the form-invariance requirement of elastic energy. From a classical geometric rigidity result follows that the appearance of a right-local SO(3)-invariance condition is not the general condition for isotropy. The correct statement for isotropy in second gradient elasticity should be a right-global SO(3)-invariance condition. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)