Variational principles for linear and nonlinear mixtures: New derivations and application to bilinear materials and yield behavior.

摘要

We study in this paper the effective energy potential of nonlinear composites in terms of the corresponding energy potential for linear material. We study this for power law material using Suquet's variational principle, for bilinear material using Ponte Castaneda variational principle, and we studied the yield behavior as limiting cases of both.; We show that three established variational principles can be derived from the Cauchy-Buniakowski-Schwartz inequality; these variational principles (bounds) are: (a) bounds on yield behavior of mixtures, (b) The Hashin-Shtrikman variational principle for linear materials, (c) The Debotton and Ponte Castaneda bound for power-law polycrystals.; We also compute the actual stress and strain fields for laminate material for two choices of the average strain, and compute a bound on the effective potential for spherical inclusions using Hashin's linear bound, as well as for spheroidal inclusions using Willis's linear bound. We obtained bounds sharper than the established bounds for both laminate and aligned ellipsoids geometry, and we investigate also how the bound depends on the specified geometry and on specific parameters of the problem.