Homogenization of balanced plain weave composites with imperfect microstructure: Part I––Theoretical formulation

Abstract

A mathematically sound computational framework is presented for the determination of a representative volume element (RVE) of plain weave fabric composites with reinforcement imperfections. Although treated as periodic two-phase composites the imperfect geometry of such material systems often precludes a straightforward representation of the RVE in terms of a simple periodic unit cell (PUC). To circumvent various limitations of simple unit cells when applied to real material systems the present contribution suggests a rigorous theoretical approach for the derivation of a reliable PUC exploiting the knowledge of the real geometry of a composite supplied by the manufacturer. It is expected that sufficient geometrical information about the real microstucture can be obtained from digitized micrographs of plain weave cross-sections that reveal the most critical imperfections caused by manufacturing processes. No attempt is therefore made to address a specific type of imperfection but rather treat them all in the combined manner. It has been demonstrated that the morphology of imperfect material systems is well-described by an appropriate set of statistical descriptors. Introducing these descriptors into a suitable optimization environment then provides a tool for the derivation of a PUC that statistically resembles the actual composite system as close as possible. In particular the parameters of the idealized unit cells follow from minimization of the objective function defined as the least square difference of the statistical descriptor related to the original microstructure and to the idealized unit cell, respectively. Once the desired geometrical parameters are determined, the finite element model of a woven composite is formulated and used to predict the overall response of the composite by the numerical homogenization method. The quality of the resulting unit cells is addressed from the point of view of effective elastic properties to examine the applicability and limitations of this procedure and to provide modeling strategy for the analysis of real-world material systems. Although applied to plain weave fabric composites the theoretical formulation presented in this paper is rather general and can be applied to any material systems with disordered or imperfect microstructures.