Beyond the classical Hashin–Shtrikman bounds: the Hashin–Shtrikman principle revisited

International audienceThe theory underlying upscaling of mechanical properties is now wellunderstood. Nonetheless, practical homogenization remains a challengedue to the rather complex boundary value problems that need to besolved.Various techniques have been proposed, ranging from simple(Eshelby-based) mean-field/effective-field to full-field(computationally intensive) approaches. Conceptually, these twofamilies of approaches can be thought of as occupying the two ends ona "complexity scale". Quite interestingly, methods that populate theintermediate range of this scale are quite scarce.In this talk, I will show how the celebrated principle of Hashin andShtrikman [JMPS 10(4), pp. 335—342, 1962] provides a rigorous, uniformframework to explore the whole "complexity scale".I will first briefly recall this variational principle, and discussthe features that make it an attractive alternative to the moreclassical minimum potential/complementary energy principles.It is well known that the Hashin—Shtrikman *bounds* allow to revisitthe Mori—Tanaka scheme. Similarly, I will show that theHashin—Shtrikman *principle* can shed a new light on somehomogenization techniques such as the celebrated FFT-based techniqueof Moulinec and Suquet [CMAME 157(1-2), pp. 69—94, 1998] or theequivalent inclusion method of Moschovidis and Mura[J. App. Mech. 42(4), pp. 847—852, 1975].In the last (and main) part of my talk, I will discuss a possibleapproach allowing to improve on the celebrated Hashin—Shtrikman boundsby means of enriched trial stress-polarization fields. The goal is toproduce bounds/estimates of the effective properties that areanalytical or semi-analytical, while accounting for a wider range ofstatistical descriptors of the microstructure (beyond the volumefractions)