Field statistics in nonlinear composites. I. Theory

M. I. IDIART; P. PONTE CASTAÑEDA

PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES

Royal Society Publishing

Año: 2007 vol. 463 p. 183 - 202

1364-5021

This work presents a means for extracting the statistics of the local fields in nonlinear composites from the effective potential of suitably perturbed composites. The idea is to introduce a parameter in the local potentials, generally a tensor, such that differentiation of the corresponding effective potential with respect to the parameter yields the volume average of the desired quantity. In particular, this provides a generalization to the nonlinear case of well-known formulae in the context of linear composites, which express phase averages and second moments of the local fields in terms of derivatives of the effective potential. Such expressions are useful since they allow the generation of estimates for the field statistics in nonlinear composites, directly from homogenization estimates for appropriately defined effective potentials. 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 in previous 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 be entirely consistent with the corresponding estimates for the macroscopic behavior. Sample results for two-phase, power-law composites are provided in part II of this work.