Abstract
The forms that the convexity, polyconvexity, and rank-one convexity inequalities take when the strain energy is required to be a function of the strain G are studied. It is shown in particular that W(G) must be an increasing function of G, in the sense that W(G′)≧W(G) if G′ − G is non-negative definite. Relatively simple sufficient conditions in terms of G alone are given. Necessary and sufficient conditions in terms of G alone are found to be rather complex.
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Communicated by C. M. Dafermos
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Pipkin, A.C. Convexity conditions for strain-dependent energy functions for membranes. Arch. Rational Mech. Anal. 121, 361–376 (1993). https://doi.org/10.1007/BF00375626
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DOI: https://doi.org/10.1007/BF00375626