New variational principles in plasticity and their application to composite materials

I , a variational method for bounding the effective properties of nonlinear composites with isotropic phases, proposed recently by ponte castañeda (J. Mech. Phys. Solids 39, 45, 1991), is given full variational principle status. Two dual versions of the new variational principle are presented and their equivalence to each other, and to the classical variational principles, is demonstrated. The variational principles are used to determine bounds and estimates for the effective energy functions of nonlinear composites with prescribed volume fractions in the context of the deformation theory of plasticity. The classical bounds of Voigt and Reuss for completely anisotropic composites are recovered from the new variational principles and are given alternative, simpler forms. Also, use of a novel identity allows the determination of simpler forms for nonlinear Hashin-Shtrikman bounds, and estimates, for isotropic, particle-reinforced composites, as well as for transversely isotropic, fiber-reinforced composites. Additionally, third-order bounds of the Beran type are determined for the first time for nonlinear composites. The question of the optimality of these bounds is discussed briefly.