Preprint no. 142
Mathematical Institute of the Academy of Sciences of the Czech Republic

Relaxation in a class of SO(n)--invariant energies related to nematic elastomers

Miroslav Silhavy

Mathematical Institute, Acad. Sci.of the Czech Republic, 115 67 PRAHA 1, Zitna 25, Czech Republic, e-mail:


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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