Mathematical Institute, Acad. Sci.of the
Czech Republic, 115 67 PRAHA 1, Zitna 25, Czech Republic, e-mail: silhavy@math.cas.cz
Summary:
A class of isotropic energy functions $W$ is determined which admit explicit
relaxation. Within the class, the rank 1 convex, quasiconvex, and polyconvex
hulls coincide and reduce to the ``Baker--Ericksen hull'' $W^{be},$ i.e.,
the largest function below $W$ satisfying the Baker--Ericksen inequalities.
The construction of $W^{be}$ is based on the monotonicity of $SO(n)$--invariant
rank 1 convex functions and on the classical ordered--forces inequalities
for symmetric convex functions. The class includes compressible and incompressible
energies of nematic elastomers. The relaxed energy leads to a phase diagram
which displays the original solid phase, a liquid phase, and one or two intermediate
solid--liquid (smectic) phases.
Keywords: relaxation, quasiconvexity, nematic elastomers, microstructure
Mathematics Subject Classification
2000: 49J45, 74N15
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