An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy-Green tensors

Let Omega be a bounded open connected subset of R-n with a Lipschitz-continuous boundary and let Theta is an element of C-1((Omega) over barR(n)) be a deformation of the set (Omega) over bar satisfying detdelTheta > 0 in (Omega) over bar. It is established that there exists a constant C(Theta) with the following property: for each deformation Phi is an element of H-1 (Omega; R-n) satisfying det delPhi > 0 a.e. in Omega, there exist an n x n rotation matrix R = R (Phi, Theta) and a vector b = b(Phi, Theta) in R-n such that \\Phi - (b + RTheta)\\(1)(H)((Omega)) less than or equal to C(Theta)\\delPhi(T)delPhi - delTheta(T)delTheta\\(1)(1/2)(L)((Omega)). The proof relies in particular on a fundamental 'geometric rigidity lemma', recently proved by G. Friesecke, R.D. James, and S. Muller. (C) 2004 Academie des scicnces. Published by Elsevier SAS. All rights reserved.