Luc TARTAR Compensated Compactness with more ... - ICMS

It would be more useful to understand the effective properties
of mixtures for a few properties at the same time, like diffusion
of heat and electricity, magnetic and (linear) elastic properties,
so that it could be used for creating new efficient materials by
selecting adapted microstructures. This seems to require an
improvement of Compensated Compactness, or H-measures.
In the summer of 1977, at a conference in Rio de Janeiro, I
reported that I had not found a reasonable class of constitutive
relations for Homogenization in “nonlinear” elasticity,
and recalled that since the evolution problem is hyperbolic,
even the stationary solution must satisfy a little more than
the “Rankine–Hugoniot condition”, in the sense that some E-
condition must be imposed. Clifford TRUESDELL (1919–2000)
had disagreed with my idea that constitutive relations should
be stable under weak convergence, which I thought was obvious
(since one would not call the effective material elastic without
this property), but 10 years later, Owen RICHMOND (1928–
2001) made an observation about higher order gradients for an
effective behaviour (of perforated aluminum plates).
I often say that Γ-convergence is not Homogenization, and that
it deals with non-physical questions; in particular, mentioning
only stored energy cannot tell if a material is elastic or not.
The evolution equation for (nonlinear) elasticity is an hyperbolic
system of conservation laws: one must pay attention to
the formation of discontinuities, and to the (physically) admissible
ones; it led Peter LAX to talk about “entropy conditions”,
better called E-conditions after Costas DAFERMOS (since they
often are not related to thermodynamical entropy).
Homogenization teaches to be careful with EOS, since they
depend upon which microstructure the material uses.
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