One considers a linear elastic random structure composite material (CM) with a homogeneous matrix. The idea of the effective field hypothesis (EFH, H1) dates back to Faraday, Poisson, Mossotti, Clausius, and Maxwell (1830–1870, see for references and details [1], [2]) who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. It is proved that a concept of the EFH (even if this term is not mentioned) is a (first) background of all four groups of analytical methods in physics and mechanics of heterogeneous media (model methods, perturbation methods, self-consistent methods, and variational ones, see for refs. [1]). New GIEs essentially define the new (second) background (which does not use the EFH) of multiscale analysis offering the opportunities for a fundamental jump in multiscale research of random heterogeneous media with drastically improved accuracy of local field estimations (with possible change of sign of predicted local fields).

Estimates of the Hashin-Shtrikman (H-S) type are developed by extremizing of the classical variational functional involving either a classical GIE [1] or a new one. In the classical approach by Willis (1977), the H-S functional is extremized in the class of trial functions with a piece-wise constant polarisation tensors while in the current work we consider more general class of trial functions with a piece-wise constant effective fields. One demonstrates a better quality of proposed bounds, that is assessed from the difference between the upper and lower bounds for the concrete numerical examples.

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