An approximate homogenization scheme for nonperiodic materials

Recently in [1], Briane announced a new homogenization method for certain nonperiodic materials in which the H-limit of a highly oscillatory but nonperiodic matrix A is obtained by comparing to a locally-periodic matrix B in domains whose size α( ) → 0 as → 0 but slower than ε. The H-limit of B is a function of every point in the material, and so theoretically, in order to homogenize A , 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 (α) by proving that the difference of two nearby cell solutions (within a cube of side length α) is (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.