## Abstract

If u is a bi-Lipschitzian deformation of a bounded Lipschitz domain Ω in ℓ^{n} (n≧2), we show that the L^{P} norm (p≧1, p≠n) of a certain "nonlinear strain function" e(u) associated with u dominates the distance in L^{q} (q= np/(n-p) if p<n, q=∞ if p>n) from u to a suitably chosen rigid motion of ℝ^{n}. This work extends that of F. John, who proved corresponding estimates for p}>1 under the hypothesis that u has "uniformly small strain". We also obtain a bound for the oscillation of Du in L^{2}. These estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u. In ℝ^{3} the integral {Mathematical expression}e(u)^{2}dℋ^{3} is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber; thus the case n=3, p=2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work.

Original language | English (US) |
---|---|

Pages (from-to) | 131-172 |

Number of pages | 42 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 78 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1982 |

## ASJC Scopus subject areas

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering