Complete symmetry classification and compact matrix representations for 3D strain gradient elasticity

Strain Gradient Elasticity (SGE) is now often used in mechanics and physics, owing to its capability to model some non-classical phenomena, such as size effects, in materials and structures. However, certain fundamental questions about it have not yet received complete responses. In its linear setting, the constitutive law of SGE is characterized by a fourth-order elasticity tensor, a fifth-order one and a sixth-order one. Even if the matrix representations for the 3D fourth- and sixth-order elasticity tensors are available for all possible symmetry classes, the counterparts for the 3D fifth-order tensor, whose presence is unavoidable for materials with non-centrosymmetric microstructure, are still lacking. In addition, although the symmetry classes for each of the fourth-, fifth- and sixth-order elasticity tensor spaces are known, the symmetry classes for these tensor spaces as a whole have never been reported and clarified in the literature. The present work solves these two fundamental problems preventing the full understanding and exploitation of the linear constitutive law of SGE. Precisely, the matrix representations of the fifth-order tensor for all of its 29 symmetry classes are provided in a compact and well-structured way. Further, linear SGE is shown to possess 48 symmetry classes, and for each of these symmetry classes, the matrix representations of its fourth-, fifth- and sixth-order tensors are now available. (C) 2018 Elsevier Ltd. All rights reserved.