The Limits of Hamiltonian Structures in Three Dimensional Elasticity, Shells, and Rods

This paper uses Hamiltonian structures to study the problem of the limit of three dimensional 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 have constraints or not, 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 of 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 and derive 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 onedirector 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. ∗The Fields Institute for Research in Mathematical Sciences, 185 Columbia St. West, Waterloo, Ontario N2L 5Z5, email: [email protected] †Zentrum Mathematik, TU München, Arcisstrasse 21, D-80290 München, Germany, email: [email protected] ‡Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, email: [email protected]