Specifics of symmetry conditions in gradient elasticity theories

The conditions of symmetry of the common Mindlin types gradient theories of elasticity, which characterize the specificity of these theories in comparison with the classical theory of elasticity, are investigated. We discuss the symmetry conditions of the tensors of the moduli of elasticity of the sixth rank under the permutation of the differentiation indices, which are the consequence of the fact that the second derivatives of the displacement vector do not depend on the order of differentiation. The gradient distortion model and strain gradient model are considered, and the conditions for "variational equivalence" and the differences between these theories from the point of view of symmetry are established. The variational formulation of gradient elasticity of general form and the role of symmetry conditions in the formulation of boundary conditions are investigated. It is shown that for the correct formulation of applied boundary value problems, it is necessary to use the tensors of the elastic moduli of the sixth rank symmetric with respect to the permutation of the last two indices in each triple of indices, even if this symmetry is absent for formally constructed versions of gradient theories.