An assessment of multiscale asymptotic expansion method for linear static problems of periodic composite structures

Abstract

The decoupled multiscale asymptotic expansion method (MsAEM) is applicable to the analysis of composite structures with periodic microstructures. When using the MsAEM to solve linear static problems, unit-cell influence functions must be solved from the microscale (or unit cell) problem with microscopic periodic boundary conditions, and homogenized displacements and their derivatives must be solved from the macroscale (or homogenized) problem with macrostructural boundary conditions. This paper compares the influence functions solved by using the normalization condition, the Dirichlet homogeneous boundary condition, and the oversampling technique or the super unit cell approach. It is concluded that the oversampling technique is a better way of dealing with periodic boundary conditions compared to the normalization condition. Another contribution of this work is to propose a combination method in which the MsAEM is used inside the structure and the multiscale eigenelement method (MEM) is used dealing with the boundary layer problem, and it is shown that this combination method can achieve accurate microscopic results for structures with any boundary conditions. In addition, the paper reveals the effects of different order expansion terms and gives some practical suggestions on how to determine the expansion order. Numerical comparisons validate the present combination method and all conclusions.