Abstract
New prescriptions are proposed for the ‘reference’ fields in the context of the ‘second-order’ nonlinear homogenization method [P. Ponte Castañeda, Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—Theory, J. Mech. Phys. Solids 50 (2002) 737–757], and are used to generate estimates for the effective behavior and first moments of the local fields in nonlinear composites. The new prescriptions yield simple, analytical expressions not only for the effective potentials, but also for the macroscopic stress-strain relation, as well as for the phase averages of the strain and stress fields. For illustrative purposes, ‘second-order’ estimates of the Hashin–Shtrikman type are provided for two-phase, transversely-isotropic composites with power-law phases, and are compared with exact results available for power-law, multiple-rank, sequential laminates. The agreement is found to be quite good for all ranges of nonlinearities and inclusion concentrations considered. To cite this article: M.I. Idiart et al., C. R. Mecanique 334 (2006).
Highlights
This work is concerned with the problem of estimating the effective behavior of nonlinear composites [1]
A fairly general nonlinear homogenization method has been introduced by Ponte Castañeda [2], which delivers estimates for the effective potentials W and U which are exact to second order in the heterogeneity contrast
It has been shown that the use of the macroscopic strain and stress as references in the context of the ‘secondorder’ homogenization method leads to simple and accurate estimates for the effective response and field averages in the phases of nonlinear composites, even at large heterogeneity contrast and nonlinearity
Summary
This work is concerned with the problem of estimating the effective (or homogenized) behavior of nonlinear composites [1]. Simple expressions of practical importance are provided for the resulting effective stress-strain relations and the first moments of the local fields in each constituent. The accuracy of these estimates is assessed by comparing them with exact results available for a special class of nonlinear composites. We are concerned with the problem of finding the effective behavior of the composite, which is defined as the relation between the average stress σ = σ and the average strain ε = ε , and can be characterized [1] by an effective strain potential W , such that. The problem of estimating the effective behavior of the composite reduces to that of estimating the effective potentials W or U
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