Bounding Envelopes in Multiphase Material Design

In practice, whenever determining macroscopic effective mechanical properties of materials possessing irregular heterogeneous microstructure, one can only test, numerically or experimentally, finite sized samples. Such materials are exemplified by randomly distributed particles suspended in a homogeneous binding matrix. If one were to compute the effective responses of various equal finite sized samples, with the only mutually distinguishing feature being the various random distributions of the particulate matter, fluctuations would occur. While such fluctuations can be small for large samples, their effects become amplified when computing design sensitivities, such as gradients and Hessians, for macroscopic effective property optimization strategies. Concisely stated, these fluctuations can severely impair the performance of such approaches by destroying the quality of the derivatives. A natural way of eliminating the negative effects of such fluctuations is by ensemble averaging the response of multiple samples until the results stabilize, and then to construct the sensitivities with the stabilized results. The focus of this work is to interpret such an ensemble regularization process, in particular when it is incorporated into effective property design procedures. It is shown that, under certain conditions, this type of regularization produces upper and lower bounding envelopes for objective functions representing desired macroscopic effective responses associated with idealized, fluctuation free, material samples of infinite size.