Symmetry properties and simplification of multiscale second-order gradient elasticity

The symmetry conditions of the gradient distortion models were investigated. We studied the variational significance of order-of-differentiation symmetry conditions of strain gradient elasticity. This symmetry is a necessary and sufficient conditions for the continuity of the first derivatives of displacements (i.e., conditions for the absence of defects in the medium under consideration). We believe that ignoring this symmetry can lead to ill-posedness in boundary value problems and, as a result, to erroneous results in modeling. Thus the problem under consideration is relevant to all applied models of higher order elasticity theory that take into account size effects and are used for modeling in micro and nanomechanics. The discussed symmetry properties must be account for, due to formulation of boundary value problems for applied gradient distortion theories and especially for the vector variants of such models. Otherwise, the boundary value problem may lose correctness and in the result, the energetically insignificant components of the sixth rank tensor can be appeared in the formulation of the boundary conditions. A example is given, indicating the possibility of the occurrence of ill-posedness of the boundary value problem.