On the homogenization of a new class of locally periodic microstructures in linear elasticity with residual stress

TL;DR

This paper develops a homogenization framework for a broad class of locally periodic microstructures in linear elasticity, accounting for residual stresses, and reveals that in some cases, a single unit cell solution suffices to determine effective properties.

Contribution

It introduces a new homogenization approach for locally periodic microstructures with residual stress, simplifying the computation of effective elasticity tensors.

Findings

Effective residual stress and elasticity tensor are derived from unit cell problems.

For certain microstructures, solving one unit cell problem suffices for the entire domain.

The framework extends homogenization to nonperiodic, anisotropic materials with residual stresses.

Abstract

Many biological and engineering materials have nonperiodic microstructures for which classical periodic homogenization results do not apply. Certain nonperiodic microstructures may be approximated by locally periodic microstructures for which homogenization techniques are available. Motivated by the consideration that such materials are often anisotropic and can posses residual stresses, a broad class of locally periodic microstructures is considered and the resulting effective macroscopic equations are derived. The effective residual stress and effective elasticity tensor are determined by solving unit cell problems at each point in the domain. However, it is found that for a certain class of locally periodic microstructures, solving the unit cell problems at only one point in the domain completely determines the effective elasticity tensor.