Preprint 1/1998

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
Pages: 23
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 tex2html_wrap_inline22 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 tex2html_wrap_inline40 satisfies tex2html_wrap_inline42, then W is tex2html_wrap_inline46-quasiconvex at tex2html_wrap_inline40 on the restricted set of deformations u that satisfy condition (INV) and tex2html_wrap_inline50 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 tex2html_wrap_inline40x and u by the tex2html_wrap_inline54-norm of the difference of their gradients. These results have application to the determination of lower bounds on critical cavitation loads in elastic solids.

23.06.2018, 02:10