Matrix representations for 3D strain-gradient elasticity

TL;DR

This paper derives explicit matrix representations of the sixth-order elastic tensor for 3D anisotropic materials in strain-gradient elasticity, facilitating modeling of size and non-local effects in complex materials.

Contribution

It provides the first complete, compact matrix forms of the sixth-order elastic tensor for all 3D anisotropic cases in strain-gradient elasticity.

Findings

Explicit matrix representations for all 3D anisotropic cases.

Enables practical application of strain-gradient elasticity to anisotropic materials.

Facilitates computational modeling of size effects in complex materials.

Abstract

The theory of first strain gradient elasticity (SGE) is widely used to model size and non-local effects observed in materials and structures. For a material whose microstructure is centrosymmetric, SGE is characterized by a sixth-order elastic tensor in addition to the classical fourth-order elastic tensor. Even though the matrix form of the sixth-order elastic tensor is well-known in the isotropic case, its complete matrix representations seem to remain unavailable in the anisotropic cases. In the present paper, the explicit matrix representations of the sixth-order elastic tensor are derived and given for all the 3D anisotropic cases in a compact and well-structured way. These matrix representations are necessary to the development and application of SGE for anisotropic materials