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Conference Papers Year : 2017

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

Sébastien Brisard


The theory underlying upscaling of mechanical properties is now well understood. Nonetheless, practical homogenization remains a challenge due to the rather complex boundary value problems that need to be solved. Various techniques have been proposed, ranging from simple (Eshelby-based) mean-field/effective-field to full-field (computationally intensive) approaches. Conceptually, these two families of approaches can be thought of as occupying the two ends on a "complexity scale". Quite interestingly, methods that populate the intermediate range of this scale are quite scarce. In this talk, I will show how the celebrated principle of Hashin and Shtrikman [JMPS 10(4), pp. 335—342, 1962] provides a rigorous, uniform framework to explore the whole "complexity scale". I will first briefly recall this variational principle, and discuss the features that make it an attractive alternative to the more classical minimum potential/complementary energy principles. It is well known that the Hashin—Shtrikman *bounds* allow to revisit the Mori—Tanaka scheme. Similarly, I will show that the Hashin—Shtrikman *principle* can shed a new light on some homogenization techniques such as the celebrated FFT-based technique of Moulinec and Suquet [CMAME 157(1-2), pp. 69—94, 1998] or the equivalent 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 possible approach allowing to improve on the celebrated Hashin—Shtrikman bounds by means of enriched trial stress-polarization fields. The goal is to produce bounds/estimates of the effective properties that are analytical or semi-analytical, while accounting for a wider range of statistical descriptors of the microstructure (beyond the volume fractions).
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hal-01513354 , version 1 (25-04-2017)




  • HAL Id : hal-01513354 , version 1


Sébastien Brisard. Beyond the classical Hashin–Shtrikman bounds: the Hashin–Shtrikman principle revisited. Séminaire du Laboratoire de Mécanique des Solides, École Polytechnique, Apr 2017, 91128 Palaiseau Cedex, France. ⟨hal-01513354⟩
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