SOME SIMPLE EXPLICIT BOUNDS FOR THE OVERALL BEHAVIOR OF NONLINEAR COMPOSITES

Variational expressions developed over the last few years provide bounds for the overall energy functions of a range of nonlinear composite materials. The evaluation of a bound requires the solution of a system of nonlinear algebraic equations and this generally involves a computation. There are, however, certain simple composites, comprising a nonlinear matrix containing either rigid inclusions or cavities, for which very simple explicit formulae can be given. These formulae are displayed here, at a level of generality greater than in any previous presentation. The energy density function of the matrix is arbitrary and the microgeometry of the composite appears through an expression which bounds the energy of a linear composite with the same geometry. To the extent that such linear bounds can be developed making allowance for any amount of statistical information on the composite, the new nonlinear bounds reflect this. New results, at the level of employing bounds of Hashin-Shtrikman type for the linear problem, are given for an incompressible matrix reinforced by aligned rigid platelets or weakened by aligned cracks. In the course of the work, a recently derived formula, more general than any available previously, is presented and developed explicitly for an two-phase composite.