An approximate homogenization scheme for nonperiodic materials

AbstractRecently in [1], Briane announced a new homogenization method for certain nonperiodic materials in which the H-limit of a highly oscillatory but nonperiodic matrix Aε is obtained by comparing to a locally-periodic matrix Bε in domains whose size α(ε) → 0 as ε → 0 but slower than ϵ. The H-limit of Bε is a function of every point in the material, and so theoretically, in order to homogenize Aε, the solution to the usual periodic cell problem must be obtained for every point in the material. Computationally this is not feasible, so we approximate the homogenization method by keeping α fixed. We show that this approximation is O(α) by proving that the difference of two nearby cell solutions (within a cube of side length α) is O(α) in the H1-norm. This result requires that we show a uniform bound exists for the gradients of the periodic cell solutions in Lp. We then apply our approximate homogenization theory to the analysis of certain defects in fiber-reinforced composites. In particular, we show that when unexpected local spreading of the fibers occurs in a small region of the material, constituent stress concentrations of nearly three can arise