An isoperimetric estimate and W1,p-quasiconvexity in nonlinear elasticity

Stefan Müller, Jeyabal Sivaloganathan, Scott J. Spector

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A class of stored energy densities that includes functions of the form JV(F) = a|F|p+g(F, adj F)+h(det F) 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 F0 satisfies h′(det F0)|F0|3-p ≤ k, then W is W1,p-quasiconvex at F0 on the restricted set of deformations u that satisfy condition (INV) and det Δu > 0 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 F0x and u by the Lp-norm of the difference of their gradients. These results have application to the determination of lower bounds on critical cavitation loads in elastic solids.

Original languageEnglish
Pages (from-to)159-176
Number of pages18
JournalCalculus of Variations and Partial Differential Equations
Issue number2
Publication statusPublished - 1 Mar 1999


ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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