Ge, Z., H. P. Kruse and J. E. Marsden
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 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.