Beyond the classical Hashin–Shtrikman bounds: the Hashin–Shtrikman principle revisited
Abstract
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).
Origin : Files produced by the author(s)