Bounds of third order for the overall response of nonlinear composites

Abstract

Composites whose response can be described in terms of a convex potential function are discussed. Bounds are constructed for the overall, or effective, potential of the composite, given the individual potentials of its constituents. Steady-state creep is considered explicitly but the results apply equally well to physically nonlinear elasticity, or deformation-theory plasticity, if strain-rate is reinterpreted as infinitesimal strain. Earlier work employed a linear “comparison” medium. This permitted the construction of only one bound—either an upper bound or a lower bound—or even in some cases no bound at all. Use of a nonlinear comparison medium removes this restriction but at the expense of requiring detailed exploration of the properties of the trial fields that are employed. The fields used here—and previously—have the property of “bounded mean oscillation”; the use of a theorem that applies to such fields permits the construction of the bounds that were previously inaccessible. Illustrative results, which allow for three-point correlations, are presented for an isotropic two-phase composite, each component of which is isotropic, incompressible and conforms to a power-law relation between equivalent stress and equivalent strain-rate. Generalized Hashin-Shtrikman-type bounds follow by allowing the parameter corresponding to the three-point correlations to take its extreme values.