THE LIMITS OF HAMILTONIAN STRUCTURES IN 3-DIMENSIONAL ELASTICITY, SHELLS, AND RODS

This paper uses Hamiltonian structures to study the problem of the lim it of three-dimensional (3D) elastic models to shell and rod models. I n the case of shells, we show that the Hamiltonian structure for a thr ee-dimensional elastic body converges, in a sense made precise, to tha t for a shell model described by a one-director Cosserat surface as th e 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 e stablished, with the limiting model being a geometrically exact direct or rod model (in the framework developed by Antman, Simo, and coworker s). 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 measure d 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 con vergence 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, wh ich implies that, in the sense of Hamiltonian structures, the dynamics of the elastic body is close to that of the shell. The constitutive e quations of the 3D model and their behavior as the thickness tends to zero dictates whether the limiting 2D model is a constrained or an unc onstrained director model. We apply our theory in the specific case of a 3D Saint Venant-Kirchhoff material and derive the corresponding lim iting shell and rod theories. The limiting shell model is an interesti ng Kirchhoff-like shell model in which the stored energy function is e xplicitly derived in terms of the shell curvature. For rods, one gets (with an additional inextensibility constraint) a one-director Kirchho ff elastic rod model, which reduces to the well-known Euler elastica i f one adds an additional single constraint that the director lines up with the Frenet frame.