Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials

Abstract

A new micromechanics model, namely, the variational asymptotic method for unit cell homogenization (VAMUCH), is developed to predict the effective properties of periodically heterogeneous materials and recover the local fields. Considering the periodicity as a small parameter, we can formulate a variational statement of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. In comparison to other approaches, VAMUCH does not rely on ad hoc assumptions, has the same rigor as MHT, has a straightforward numerical implementation, and can calculate the complete set of properties simultaneously without using multiple loadings. This theory is implemented using the finite element method and an engineering program, VAMUCH, is developed for micromechanical analysis of unit cells. Many examples of binary composites, fiber reinforced composites, and particle reinforced composites are used to demonstrate the application, power, and accuracy of the theory and the code of VAMUCH.