A quantitative assessment of the scale separation limits of classical and higher-order asymptotic homogenization

Classical homogenization techniques are known to be effective for materials with large scale separation between the size and spacing of their underlying heterogeneities on the one hand and the structural problem dimensions on the other. For low scale separation, however, they generally become inaccurate. This paper assesses the scale separation limit of classical asymptotic homogenization applied to periodic linear elastic composite materials and demonstrates the effectiveness of higher-order homogenization in stretching this limit. A quantitative assessment is performed on a two-dimensional elastic two-phase composite consisting of stiff circular particles in a softer matrix material and subjected to anti-plane shear, as introduced by Smyshlyaev and Cherednichecko (J. Mech. Phys. Solids 48:1325–1358, 2000). Reference solutions are created rigorously using full-scale numerical simulations in which a family of translated microstructures is considered and the ensemble average of their solutions is defined as the homogenized solution. This solution is used as a reference, which is compared with the periodic homogenization solution for a range of scale ratios. It is shown that the zeroth-order classical homogenization solution significantly deviates from the exact solution below a certain scale ratio for a given microstructure. Below this limit, the higher-order solutions provide a clear improvement of the match. Further, the performance of classical and higher-order asymptotic homogenization solution are evaluated for varying stiffness contrast ratio between the two phases of the microstructure and error contours are presented by comparison with full-scale numerical simulations.