### Abstract

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 F_{0} satisfies h′(det F_{0})|F_{0}|^{3-p} ≤ k, then W is W^{1,p}-quasiconvex at F_{0} 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 F_{0}x and u by the L^{p}-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 language | English |
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Pages (from-to) | 159-176 |

Number of pages | 18 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 8 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Mar 1999 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

^{1,p}-quasiconvexity in nonlinear elasticity.

*Calculus of Variations and Partial Differential Equations*,

*8*(2), 159-176. https://doi.org/10.1007%2Fs005260050121