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An estimate of the $$H^1$$-norm of deformations in terms of the $$L^1$$-norm of their Cauchy-Green tensors. (English) Zbl 1183.74008
Summary: Let $$\Omega$$ be a bounded open connected subset of $$\mathbb R^n$$ with a Lipschitz-continuous boundary and let $$\pmb\Theta\in{\mathcal C}^1(\overline{\Omega};\mathbb R^n)$$ be a deformation of the set $$\overline{\Omega}$$ satisfying $$\det\nabla\pmb\Theta>0$$ in $$\overline{\Omega}$$. It is established that there exists a constant $$C(\pmb\Theta)$$ with the following property: for each deformation $$\pmb\Phi\in H^1(\Omega;\mathbb R^n)$$ satisfying $$\det\nabla\pmb\Phi>0$$ a.e. in $$\Omega$$, there exist an $$n\times n$$ rotation matrix $${\mathbf R}={\mathbf R}(\pmb\Phi,\pmb\Theta)$$ and a vector $${\mathbf b}={\mathbf b}(\pmb\Phi,\pmb\Theta)$$ in $$\mathbb R^n$$ such that
$\big\|\pmb\Phi-({\mathbf b}+{\mathbf R}\pmb\Theta)\big\|_{{\mathbf H}^1(\Omega)}\leq C(\pmb\Theta) \big\|\nabla\pmb\Phi^T\nabla\pmb\Phi-\nabla\pmb\Phi^T\nabla\pmb\Theta \big\|_{{\mathbf L}^1(\Omega)}^{1/2}.$
The proof relies in particular on a fundamental ‘geometric rigidity lemma’, recently proved by G. Friesecke, R. D. James, and S. Müller.

##### MSC:
 74A05 Kinematics of deformation
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