3 - Closed form micromechanics

Abstract

This chapter presents the fundamental concepts related to micromechanics, most importantly the strain and stress concentration tensors relating the local fields to the applied composite-level loading. Then four classical micromechanics theories (the Voigt approximation, Reuss approximation, Mori-Tanaka (MT) method, and method of cells (MOC)) are presented in terms of their unique concentration tensors, from which all effective mechanical properties can be determined. While the Voigt, Reuss, and MT theories provide average fields for each constituent, the MOC enables prediction of the variations of the stress and strain fields within the matrix constituent. Furthermore, in contrast to MT and MOC, the simple Voigt and Reuss approaches, while illustrative from a theoretical standpoint, do not provide good approximations of the effective composite properties nor the local fields. All four theories are extended to accommodate thermomechanical loading, wherein the effective coefficients of thermal expansion (CTEs) are predicted, along with thermal strain concentration tensors, which enable the calculation of local fields in the presence of thermal loading. The MATLAB implementation of these classical micromechanics theories (for solving stand-alone micromechanics and multiscale (micromechanics-based) laminate analysis problems), along with associated flowcharts, are provided and discussed. Finally, a number of example problems comparing the micromechanics theory results are presented for both stand-alone micromechanics and micromechanics-based laminate simulations.