×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0314.46030
[2] Antman, S.S., Ordinary differential equations of nonlinear elasticity I: foundations of the theories of non-linearly elastic rods and shells, Arch. rational mech. anal., 61, 307-351, (1976) · Zbl 0354.73046
[3] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. rational mech. anal., 63, 337-403, (1977) · Zbl 0368.73040
[4] Ciarlet, P.G., Continuity of a surface as a function of its two fundamental forms, J. math. pures appl., 82, 253-274, (2002) · Zbl 1042.53003
[5] Ciarlet, P.G.; Laurent, F., Continuity of a deformation as a function of its cauchy – green tensor, Arch. rational mech. anal., 167, 255-269, (2003) · Zbl 1030.74003
[6] Ciarlet, P.G.; Mardare, C., Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. math. pures appl., (2004), in press · Zbl 1088.74014
[7] P.G. Ciarlet, C. Mardare, A surface as a function of its two fundamental forms, in preparation · Zbl 1083.53007
[8] P.G. Ciarlet, C. and Mardare, Continuity of a deformation in H1 as a function of its Cauchy-Green tensor in L1, in preparation · Zbl 1084.53063
[9] Friesecke, G.; James, R.D.; Müller, S., A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. pure appl. math., 55, 1461-1506, (2002) · Zbl 1021.74024
[10] Grisvard, P., Elliptic problems in nonsmooth domains, (1985), Pitman Boston · Zbl 0695.35060
[11] John, F., Rotation and strain, Comm. pure appl. math., 14, 391-413, (1961) · Zbl 0102.17404
[12] John, F., Bounds for deformations in terms of average strains, (), 129-144
[13] Kohn, R.V., New integral estimates for deformations in terms of their nonlinear strains, Arch. rational mech. anal., 78, 131-172, (1982) · Zbl 0491.73023
[14] Nečas, J., LES Méthodes directes en théorie des equations elliptiques, (1967), Masson Paris · Zbl 1225.35003
[15] Reshetnyak, Y.G., Mappings of domains in \(R\^{}\{n\}\) and their metric tensors, Siberian math. J., 44, 332-345, (2003)
[16] Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. amer. math. soc., 36, 63-89, (1934) · JFM 60.0217.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.