A mathematical framework for reducing the domain in the mechanical analysis of periodic structures

TL;DR

This paper introduces a mathematical framework for deriving Periodic Boundary Conditions on reduced analysis domains called rUCs, using symmetries and translations, applicable to various periodic structures like woven and honeycomb materials.

Contribution

It develops a general theoretical approach for domain reduction in periodic structure analysis, including the novel concept of Offset-reduced Unit Cells (OrUCs), without loading restrictions.

Findings

Framework successfully applied to 3D woven structures

Framework applicable to honeycomb structures

Enables analysis of smaller domains without loss of accuracy

Abstract

A theoretical framework is developped leading to a sound derivation of Periodic Boundary Conditions (PBCs) for the analysis of domains smaller then the Unit Cells (UCs), named reduced Unit Cells (rUCs), by exploiting non-orthogonal translations and symmetries. A particular type of UCs, Offset-reduced Unit Cells (OrUCs) are highlighted. These enable the reduction of the analysis domain of the traditionally defined UCs without any loading restriction. The relevance of the framework and its application to any periodic structure is illustrated through two practical examples: 3D woven and honeycomb.