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The limits of Hamiltonian structures in three-dimensional elasticity, shells, and rods

Ge, Zhong and Kruse, Hans Peter and Marsden, Jerrold E. (1996) The limits of Hamiltonian structures in three-dimensional elasticity, shells, and rods. Journal of Nonlinear Science, 6 (1). pp. 19-57. ISSN 0938-8974. doi:10.1007/BF02433809.

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This paper uses Hamiltonian structures to study the problem of the limit of three-dimensional (3D) elastic models to shell and rod models. In the case of shells, we show that the Hamiltonian structure for a three-dimensional elastic body converges, in a sense made precise, to that for a shell model described by a one-director Cosserat surface as the thickness goes to zero. We study limiting procedures that give rise to unconstrained as well as constrained Cosserat director models. The case of a rod is also considered and similar convergence results are established, with the limiting model being a geometrically exact director rod model (in the framework developed by Antman, Simo, and coworkers). The resulting model may or may not have constraints, depending on the nature of the constitutive relations and their behavior under the limiting procedure. The closeness of Hamiltonian structures is measured by the closeness of Poisson brackets on certain classes of functions, as well as the Hamiltonians. This provides one way of justifying the dynamic one-director model for shells. Another way of stating the convergence result is that there is an almost-Poisson embedding from the phase space of the shell to the phase space of the 3D elastic body, which implies that, in the sense of Hamiltonian structures, the dynamics of the elastic body is close to that of the shell. The constitutive equations of the 3D model and their behavior as the thickness tends to zero dictates whether the limiting 2D model is a constrained or an unconstrained director model. We apply our theory in the specific case of a 3D Saint Venant-Kirchhoff material andderive the corresponding limiting shell and rod theories. The limiting shell model is an interesting Kirchhoff-like shell model in which the stored energy function is explicitly derived in terms of the shell curvature. For rods, one gets (with an additional inextensibility constraint) a one-director Kirchhoff elastic rod model, which reduces to the well-known Euler elastica if one adds an additional single constraint that the director lines up with the Frenet frame.

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Additional Information:© 1996 Springer-Verlag New York Inc. Received September 1, 1995 and in revised form October 15, 1995. Communicated by Stephen Wiggins. This paper is dedicated to the memory of Juan C. Simo. The authors wish to especially thank the late Juan-Carlos Simo who inspired and helped initiate this work. We also thank John Maddocks and Annie Raoult for useful comments. Zhong Ge and Hans-Peter Kruse thank Jerry Marsden and the University of California for their hospitality during their visits. The authors wish to acknowledge the following research support: the Ministry of Colleges and Universities of Ontario and the Natural Sciences and Engineering Research Council of Canada (ZG), partial support by the Humboldt Foundation during a stay at the Department of Mathematics, University of California, Berkeley, CA 94720 and by the DFG under contract Sch 233/3-1 (HPK) and partial support by the Fields Institute (JEM).
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Ontario Ministry of Colleges and UniversitiesUNSPECIFIED
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Alexander von Humboldt FoundationUNSPECIFIED
Department of Energy (DOE)DE-FG03-95ER-25251
Deutsche Forschungsgemeinschaft (DFG)Sch 233/3-1
Fields Institute for Research in the Mathematical SciencesUNSPECIFIED
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ID Code:19340
Deposited By: Ruth Sustaita
Deposited On:10 Aug 2010 20:59
Last Modified:08 Nov 2021 23:51

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