Higher-order asymptotic homogenization of periodic materials with low scale separation

In this work, we investigate the limits of classical homogenization theories pertaining to homogenization of periodic linear elastic composite materials at low scale separations and demonstrate the effectiveness of higher-order periodic homogenization in alleviating this limitation. Classical homogenization techniques are known to be very effective for materials with large scale separation between the scale of the heterogeneity and the macro-scale dimension, but inaccurate at low scale separations. Literature suggests that asymptotic homogenization is capable of pushing the limit to smaller scale separation by taking on board higher-order terms of the asymptotic expansion. We studied infinite two-dimensional elastic two-phase composite materials consisting of stiff inclusions in a soft matrix, subjected to a periodic body force, for various scale ratios between the period of the body force and that of the inclusions. We created reference solution using direct numerical simulation and used ensemble averaging for the complete family of all possible microstructures to obtain the reference homogenized solution. We show that the response predicted using zeroth order classical homogenization deviates from this reference homogenized solution for scale ratios below 10. The higher-order asymptotic homogenization solution still gives a very good approximation even in the low scale separation regime and it becomes better as more higher-order terms are included. The higher-order theory results in a size-dependent macroscopic model, which indeed allows one to push the limitations of homogenization in the direction of less scale separation.