Size effects in periodic materials with low scale separation

In this work, we explore the limits of classical homogenization methods pertaining to the homogenization of periodic materials at low scale separations. Classical homogenization techniques have proven to be very effective for large classes of materials with sufficient scale separation between the scale of the heterogeneity and the macro-scale dimension, but inaccurate at low scale separations. We investigate two classes of problems: linear elastic composite materials subjected to antiplane shear and a nonlinear material which undergoes reversible pattern transformations upon reaching a critical applied load due to local elastic instability.