, Volume 6, Issue 1, pp 1957
First online:
The limits of hamiltonian structures in threedimensional elasticity, shells, and rods
 Z. GeAffiliated withThe Fields Institute for Research in Mathematical Sciences
 , H. P. KruseAffiliated withInstitut für Angewandte Mathematik, Universität Hamburg
 , J. E. MarsdenAffiliated withControl and Dynamical Systems, California Institute of Technology 10444
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This paper uses Hamiltonian structures to study the problem of the limit of threedimensional (3D) elastic models to shell and rod models. In the case of shells, we show that the Hamiltonian structure for a threedimensional elastic body converges, in a sense made precise, to that for a shell model described by a onedirector 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 onedirector model for shells. Another way of stating the convergence result is that there is an almostPoisson 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 VenantKirchhoff material andderive the corresponding limiting shell and rod theories. The limiting shell model is an interesting Kirchhofflike 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 onedirector Kirchhoff elastic rod model, which reduces to the wellknown Euler elastica if one adds an additional single constraint that the director lines up with the Frenet frame.
 Title
 The limits of hamiltonian structures in threedimensional elasticity, shells, and rods
 Journal

Journal of Nonlinear Science
Volume 6, Issue 1 , pp 1957
 Cover Date
 199601
 DOI
 10.1007/BF02433809
 Print ISSN
 09388974
 Online ISSN
 14321467
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Authors

 Z. Ge ^{(1)}
 H. P. Kruse ^{(2)}
 J. E. Marsden ^{(3)}
 Author Affiliations

 1. The Fields Institute for Research in Mathematical Sciences, 222 College Street, M5T 3J1, Toronto, Ontario
 2. Institut für Angewandte Mathematik, Universität Hamburg, Bundestrasse 55, D20146, Hamburg, Germany
 3. Control and Dynamical Systems, California Institute of Technology 10444, 91125, Pasadena, CA