Morphology-dependent Hashin–Shtrikman bounds on the effective properties of stress-gradient materials

Stress-gradient materials are generalized continua with two generalized stress variables: the Cauchy stress field and its gradient. For homogenization purposes, we introduce an extension to stress-gradient materials of the principle of Hashin and Shtrikman. The variational principle is first stated within the framework of periodic homogenization, then extended to random homogenization. Contrary to the usual derivation of the classical principle, we adopt here a stress-based approach, much better suited to stress-gradient materials. We show that, in many cases of interest, the third-order trial eigenstrain may be discarded, leaving only one (second-order) trial eigenstrain in the functional to optimize. For N-phase material, the bounds are very similar in structure to their classical counterpart. One notable difference is the fact that, even in the case of isotropy, the bounds depend on some additional microstructural parameters (besides the usual volume fractions). •Variational bounds on the effective elastic moduli of stress-gradient materials are derived.•In many cases of interest, the third-order trial strain-polarization needs not be considered.•Resulting bounds are sensitive to the two-point microstructural correlations.