Closed-form asymptotic micromechanics model of fiber reinforced polymer and metal matrix composites

Abstract

This work presents an analytical asymptotically-correct micromechanics model that helps to predict the effective material properties of a unidirectional composite material. The conventional and numerical approaches estimate the homogenized material properties of composites for their defined component volume fractions, their constituent properties and configurational geometry. Presently these approaches are based on kinematic assumptions such as having displacement or stress components vary through the cross section for beam like structures or through the thickness for plate like structures according to certain predefined functions that doesn’t always logically follow the 3D analysis. In the present formulation, the micromechanics model is developed by accommodating all possible deformations without assuming the displacement function or stress components. These are derived by minimizing the potential energy in terms of generalized strain measures. In the present formulation, Berdichevsky’s Variational Asymptotic Method (VAM) is employed as a mathematical tool to accomplish the homogenization procedure. The Hashin-Rosen model popularly referred to as the Concentric Cylinder Model (CCM) serves as the framework to estimate all the relevant homogenized elastic moduli and coupling coefficients. The derived quantities of interest are obtained as closed form expressions which are functions of the properties of the reinforcement material, the matrix material, their volumes fraction and the geometry of their relative arrangement. These expressions are arrived following the 3D elasticity governing rules by satisfying the interfacial displacement continuity and transverse stress equilibria conditions at the reinforcement and matrix materials interface. The developed expressions for the elastic moduli, shear moduli and Poisson’s ratios of few typical polymer and metal matrix composite materials are validated with some of the relevant results available in the literature.