A new method of determining symmetry types of the fifth-order elastic stiffness tensor in the (first) strain-gradient theory

Abstract

A new method without constraint on orders is proposed in this paper to determine the symmetry types of the fifth-order elastic stiffness tensor in the (first) strain-gradient theory. The odd-order tensor seems more complex than the even-order tensor in terms of symmetry problems. Based on the knowledge of the irreducible decomposition of high-order tensors and the multi-polar representations of deviator, the new method solves symmetry problems by a process of elimination, which is clear and simpler than the methods proposed before. By virtue of the pattern of the unit vector set of deviators, the natural coordinate system and the number of distinct components among the 28 types of the fifth-order elastic stiffness tensor also have been given, which is of great importance to the (first) strain-gradient theory under anisotropy.