Keywords: extended Hashin-Shtrikman variational principles, bounds on material of overall structure, eigenparameters, relaxation stress, plastic strains, applications.

Classical Hashin-Shtrikman bounds are based on the assumption that all phases behave purely elastically and isotropically. These principles have recently been extended by the first author. The extension consists of introducing eigenparameters in the formulation. Moreover, these eigenparameters were used in estimations of bounds with the result that elastic strain is in a certain relation with plastic strains or time dependent variables. In the classical approach of Hashin and Shtrikman the energies of the entire structure were compared with local energies (on the micro scale level), and so was the idea of the procedure involving eigenparameters, with plastic strains or relaxation stresses being considered.

In this paper a new variational structure is suggested, which can be applied to the estimation of bounds on nonlinear mechanical modules for composite material. It appears that the variational bounds are more precise than that obtained in a standard way of explicit homogenization. The reason is that new parameters (components of the polarization tensor) are available and the extreme of the energy is influenced by these parameters. Differentiating by them gives better estimation. Moreover, comparing energies of the phases and the homogenized bodies leads to more qualified estimates. In the approach suggested here the main role eigenparameters play, which can stand for various phenomena, involving plasticity, but also many other properties can be represented with them: prestress, quantities defining the large scale of hereditary problems, viscoplasticity, swelling of soil, wetting of material, etc. The bounds in our case are concentrated on a change of shear modules and using simple programming means the graphs are obtained based on the choice of the material properties defining the plasticity.

A large range of real problems can be studied by the approach proposed. The influence of various ratios of phase stiffnesses can be studied and even the effect of voids and pores can be considered.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £140 +P&P)