A variational-asymptotic cell method for periodically heterogeneous materials

A new cell method, variational-asymptotic cell method (VACM), is developed to homogenize periodically heterogenous anisotropic materials based on the variational asymptotic method. The variational asymptotic method is a mathematical technique to synthesize both merits of variational methods and asymptotic methods by carrying out the asymptotic expansion of the functional governing the physical problem. Taking advantage of the small parameter (the periodicity in this case) inherent in the heterogenous solids, we can use the variational asymptotic method to systematically obtain the effective material properties. The main advantages of VACM are that: a) it does not rely on ad hoc assumptions; b) it has the same rigor as mathematical homogenization theories; c) its numerical implementation is straightforward because of its variational nature; d) it can calculate different material properties in different directions simultaneously without multiple analyses. To illustrate the application of VACM, a binary composite with two orthotropic layers are studied analytically, and a closed-form solution is given for effective stiffness matrix and the corresponding effective engineering constants. It is shown that VACM can reproduce the results of a mathematical homogenization theory.