Optimal Lower Bounds on Local Stress inside Random Media

A methodology is presented for bounding the higher $L^p$ norms, $2\leq p\leq\infty$, of the local stress inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. These results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media.