Macroscopic behavior and field statistics in viscoplastic composites

Most man-made as well as natural materials of interest in engineering and physical sciences are intrinsically heterogeneous. Common examples are particle-reinforced composites, porous materials, and polycrystalline solids such as metals, ice, and many rocks. A fundamental problem in mechanics of materials is the estimation of the macroscopic response of such heterogeneous materials from the properties and geometrical arrangement (microstructure) of their constituents. In addition, incorporating the effect of local processes (e.g., microstructure evolution, damage, work hardening, recrystallization) on the macroscopic response requires statistical information about spatial distribution of the local fields within the material. To this end, we have developed nonlinear homogenization methods capable of delivering estimates not only for the macroscopic behavior but also for the field statistics in viscoplastic composites. These methods are based on suitably designed variational principles, which make use of an optimally chosen \u27linear comparison composite\u27, allowing direct conversion of linear estimates into corresponding estimates for the effective potentials of nonlinear composites. In order to extract estimates for the field statistics from these methods, a novel procedure is proposed, making use of suitably perturbed effective potentials. By means of this procedure, we obtain estimates for the first moments of the local fields in each phase that are entirely consistent with the corresponding estimates for the effective behavior. In addition, unlike earlier approaches, this procedure is not limited to first and second moments, and can be used to estimate higher-order moments as well as the phase average of more general convex functions of the fields. Sample results are given for two-phase composites with random \u27particulate\u27 microstructures exhibiting overall transversely isotropic and isotropic symmetry. Their accuracy is assessed by confronting them with corresponding exact results for nonlinear sequential laminates. Homogenization estimates are found to be in good agreement with the exact results, even for high nonlinearities, when the strain-rate fields are found to become strongly heterogeneous