Field statistics in nonlinear composites

P. PONTE CASTAÑEDA; M. I. IDIART

College Station

Conferencia; 44th Annual Technical Meeting Society of Engineering Science; 2007

Society of Engineering Science

This work is concerned with the extraction of the statistics of the local fields in nonlinearcomposites from the effective potential of suitably perturbed composites. The idea is tointroduce a parameter in the local potentials, generally a tensor, such that differentiation ofthe corresponding effective potential with respect to the parameter yields the volume averageof the desired quantity. In particular, this provides a generalization to the nonlinear case ofwell-known formulas in the context of linear composites, which express phase averages andsecond moments of the local fields in terms of derivatives of the effective potential. Suchexpressions are useful since they allow the generation of estimates for the field statistics innonlinear composites, directly from homogenization estimates for appropriately defined effectivepotentials. Here, use is made of these expressions in the context of the “variational”,“tangent second-order” and “second-order” homogenization methods, to obtain rigorous estimates for the first and second moments of the fields in nonlinear composites. While the“variational” estimates for these quantities are found to be identical to those proposed inprevious works, the tangent second-order and second-order estimates are found be different.In particular, the new estimates for the first moments given in this work are found to beentirely consistent with the corresponding estimates for the macroscopic behavior. As anexample, we generate estimates for the phase averages and second moments of the fields fortwo-phase, power-law composites with isotropic and transversely isotropic microstructures.The accuracy of these estimates is assessed by confronting them against corresponding exactresults for sequentially laminated composites.