Inherent Symmetry and Microstructure Ambiguity in Micromechanics

TL;DR

This paper introduces an eighth symmetric formulation of the Generalized Method of Cells for triply periodic microstructures, revealing inherent symmetries and microstructure ambiguities that impact micromechanical modeling accuracy.

Contribution

It derives a new symmetric formulation and uncovers inherent symmetries and ambiguities in microstructure representations, highlighting limitations of first-order theories.

Findings

All repeating unit cells may be quarter symmetric representations.

Swapping columns of subcells does not change the fields.

Higher-order theories better model complex microstructures.

Abstract

The computational cost of micromechanics for heterogeneous materials can be reduced in certain cases where symmetric boundary conditions are applicable. We derived an eighth symmetric formulation of the Generalized Method of Cells for triply periodic microstructures. During this endeavor, an inherent symmetry was discovered. This implied that all repeating unit cells may be quarter symmetric representations of other microstructures. Additionally, it was discovered that a repeating unit cell can have columns of subcells swapped with no changes to the local or global fields. We concluded that first-order micromechanics are not well suited for capturing detailed or complex microstructures; however, higher-order theories, such as High Fidelity Generalized Method of Cells, can adequately model these microstructures.