The Structure of Overall Constitutive Relations for a Class of Nonlinear Composites

Further implications of a variational structure for a class of nonlinear composites, previously discussed, are explored by introducing explicitly a variable corresponding to external loading. A natural dual loading variable emerges by extending duality to the loading variable. Dual variational principles, involving the dual loading, are derived, which define precisely the same overall behaviour as the primal problem which is driven by the original loading. The two loading variables are related through overall constitutive relations which are associated in a natural way with an overall energy function. Similar relations are demonstrated to hold for approximations based on the use of a variational structure. The structure is illustrated through application to a problem for a nonlinear dielectric and to a nonlinear diffusion problem, for which energy bounds and other estimates have recently been discussed elsewhere.