New integral estimates for deformations in terms of their nonlinear strains

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Abstract

If u is a bi-Lipschitzian deformation of a bounded Lipschitz domain Ω in ℓn (n≧2), we show that the LP norm (p≧1, p≠n) of a certain "nonlinear strain function" e(u) associated with u dominates the distance in Lq (q= np/(n-p) if p<n, q=∞ if p>n) from u to a suitably chosen rigid motion of ℝn. 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 L2. These estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u. In ℝ3 the integral {Mathematical expression}e(u)2dℋ3 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 languageEnglish (US)
Pages (from-to)131-172
Number of pages42
JournalArchive for Rational Mechanics and Analysis
Volume78
Issue number2
DOIs
StatePublished - Jun 1982

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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