Let Ω be a bounded open connected subset of Rn with a Lipschitz-continuous boundary and let Θ ∈ C^1(Ω;R^n) be a deformation of the set Ω satisfying det∇Θ > 0 in Ω. It is established that there exists a constant C(Θ) with the following property: For each deformation Φ ∈ H^1(Ω; R^n) satisfying det ∇Φ > 0 a.e. in Ω, there exist an n×n rotation matrix R=R(Φ,Θ) and a vector b=b(Φ,Θ) in R^n such that ||Φ−(b+RΘ)||_{H^1(Ω)}≤ C(Θ) ||∇Φ^T ∇Φ − ∇Θ^T ∇Θ||_{L^1(Ω)}^{1/2}. The proof relies in particular on a fundamental “geometric rigidity lemma”, recently proved by G. Friesecke, R.D. James, and S. Muller.
@article{hal-00018255,
author = {Mardare, Cristinel and Ciarlet, Philippe G.},
title = {An estimate of the $H^1$-norm of deformations in terms of the $L^1$-norm of their Cauchy-Green tensors},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00018255}
}
Mardare, Cristinel; Ciarlet, Philippe G. An estimate of the $H^1$-norm of deformations in terms of the $L^1$-norm of their Cauchy-Green tensors. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00018255/