Periodic homogenization and material symmetry in linear elasticity

TL;DR

This paper explores how microscopic symmetries in periodic elastic materials influence the macroscopic elasticity tensor, revealing complex relationships and exceptions in symmetry inheritance through homogenization theory.

Contribution

It extends previous homogenization results by considering more general microscale symmetries and provides explicit examples demonstrating nuanced symmetry relationships.

Findings

Microscopic anisotropy can lead to macroscopic symmetry groups beyond simple identity.

Not all macroscopic elastic symmetries are derived from microscale symmetries.

Explicit examples illustrate complex symmetry inheritance in homogenized elastic materials.

Abstract

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demon- strates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.