MATHEMATICA BOHEMICA, Vol. 126, No. 2, pp. 521-529 (2001)

Rank 1 convex hulls of isotropic functions in dimension 2 by 2

M. Silhavy

M. Silhavy, Mathematical Institute AS CR, Zitna 25, 115 67 Praha 1, Czech Republic, e-mail:

Abstract: Let $f$ be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set $\text {M}^{2\times 2}$ of all $2$ by $2$ matrices. Based on conditions for the rank 1 convexity of $f$ in terms of signed invariants of $\bold A$ (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.

Keywords: rank 1 convexity, relaxation, stored energies

Classification (MSC2000): 49J45, 74N99

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