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# Article

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Keywords:
hyperelastic material; deformation gradient; strain tensor; matrix and spectral norms; bi-Lipschitzian map
Summary:
As a measure of deformation we can take the difference $D\vec{\phi }-R$, where $D\vec{\phi }$ is the deformation gradient of the mapping $\vec{\phi }$ and $R$ is the deformation gradient of the mapping $\vec{\gamma }$, which represents some proper rigid motion. In this article, the norm $\Vert D\vec{\phi }-R\Vert _{L^p(\Omega )}$ is estimated by means of the scalar measure $e(\vec{\phi })$ of nonlinear strain. First, the estimates are given for a deformation $\vec{\phi }\in W^{1,p}(\Omega )$ satisfying the condition $\vec{\phi }\big |_{\partial \Omega } = \vec{\hspace{0.7pt}\mathop {\mathrm {id}}}$. Then we deduce the estimate in the case that $\vec{\phi }(x)$ is a bi-Lipschitzian deformation and $\vec{\phi }\big |_{\partial \Omega } \ne \vec{\hspace{0.7pt}\mathop {\mathrm {id}}}$.
References:
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[2] F. John: Bounds for Deformations in Terms of Average Strains. In: Inequalities III (O. Shisha, ed.), Academic Press, New York, 1972. MR 0344392 | Zbl 0292.53003
[3] R. V. Kohn: New integral estimates for deformation in terms of their nonlinear strains. Arch. Rational Mech. Anal. 78 (1982), 131–172. MR 0648942
[4] A. I. Koshelev: The weighted Korn inequality and some iteration processes for quasilinear elliptic systems. Dokl. Akad. Nauk SSSR 271 (1983), 1056–1059. MR 0722019
[5] J. Nečas, I. Hlaváček: Introduction to Mathematical Theory of Elastic and Elastoplastic Bodies. SNTL, Praha, 1983 (in Czech).

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