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hyperelastic material; deformation gradient; strain tensor; matrix and spectral norms; bi-Lipschitzian map
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 $\vec{\phi }\big |_{\partial \Omega } \ne \vec{\hspace{0.7pt}\mathop {\mathrm {id}}}$.
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