Linear Comparison Methods for Nonlinear Composites

TL;DRAbstract

Recently developed methods for estimating the effective behavior of nonlinear composites are reviewed. The methods follow from variational principles expressing the effective behavior of the given nonlinear composites in terms of the behavior of suitably chosen “linear comparison” composites. This allows the use of classical bounds and estimates (e.g., Hashin-Shtrikman, self consistent approximations) for linear materials to generate corresponding bounds and estimates for nonlinear ones. The first method (Ponte Castañeda 1991) makes use of the “secant” moduli of the phases, evaluated at the second moments of the strain field over the phases, and delivers bounds, but these bounds are exact only to first order in the heterogeneity contrast. The second method (Ponte Castañeda 1996) makes use of the “tangent” moduli, evaluated at the phase averages (or first moments) of the strain field, and yields estimates that are exact to second-order in the contrast, but that can violate the bounds in

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