An isoperimetric estimate and W1,p-quasiconvexity in nonlinear elasticity
Stefan Müller, Jeyabal Sivaloganathan, and Scott J. Spector
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Submission date: 06. Feb. 1998
published in: Calculus of variations and partial differential equations, 8 (1999) 2, p. 159-176
DOI number (of the published article): 10.1007/s005260050121
MSC-Numbers: 73G05, 49K20, 26B10, 73C50
Keywords and phrases: cavitation, condition (inv), distributional jacobian, isoperimetric inequality, monotonicity in the sense of lebesgue, quasiconvexity
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A class of stored energy densities that includes functions of the form with a>0, g and h convex and smooth, and 2<p<3 is considered. The main result shows that for each such W in this class there is a k>0 such that, if a 3 by 3 matrix satisfies , then W is -quasiconvex at on the restricted set of deformations u that satisfy condition (INV) and a.e. (and hence that are one-to-one a.e.). Condition (INV) is (essentially) the requirement that u be monotone in the sense of Lebesgue and that holes created in one part of the material not be filled by material from other parts. The key ingredient in the proof is an isoperimetric estimate that bounds the integral of the difference of the Jacobians of x and u by the -norm of the difference of their gradients. These results have application to the determination of lower bounds on critical cavitation loads in elastic solids.