General Techniques for Exploiting Periodicity and Symmetries in Micromechanics Analysis of Textile Composites

General formulas for obtaining boundary conditions for micromechanics analysis of textile composites are developed based on the concept of equivalent subcell, which is the smallest region that has to be modeled. The equivalent subcell is identified by exploiting the periodicity and symmetries exhibited in the material microstructures and loading. The conventional formulas for the full unit cell of periodic structures and the formulas for the subcell are unified by introducing the “subcell vector” d and a constant vector R that accounts for the mismatch in the local displacement perturbations between two equivalent subcells. The usually lengthy derivation process required to obtain boundary conditions has been significantly simplified by shifting the effort from “deriving” to substituting the parameters established for the equivalent subcells into the formulas. The applications of these formulas are illustrated by generating boundary conditions for the smallest subcells of 4-harness satin weave and plain weave.