Consistent Hashin-Shtrikman bounds on the effective properties of periodiccomposite materials

In this paper the four classical Hashin-Shtrikman variational principles, a pplied to the homogenization problem for periodic composites with a nonline ar hyperelastic constitutive behavior, are analyzed. It is proved that two of them are indeed minimum principles while the other two are saddle point principles. As a consequence, every approximation of the former ones provid e bounds on the effective properties of composite bodies, while approximati ons of the latter ones may supply inconsistent bounds, as it is shown by tw o numerical examples. Nevertheless, the approximations of the saddle point principles are expected to provide better estimates than the approximations of the minimum principles.