The limits of Hamiltonian structures in three-dimensional elasticity, shells, and rods (Q1911762)

In this important article, the authors lay down a theory of shells and rods in dynamics starting from the Hamilton's equations. Using the three-dimensional continuum mechanics, the theory is based on the Cosserat director method, with a linear approximation of the three-dimensional displacement as a function of the thickness. Hence the form of the director vector. The method utilizes the Hamilton theory as a space phase extension, and the dynamic equations are derived in the form of Poisson's brackets approximated by using an ``almost embedding'' theorem. Beginning with the case of unconstrained systems, the weak convergence is shown for the thickness tending to zero. The main difficulty comes from the case of a constrained director model. Various examples of constrained models are given for the case of constant thickness, for linearized Saint-Venant's models, for Kirchhoff membrane models, for inextensible Kirchhoff shells, etc. The authors wish to extend the results to strong convergence results. All the paper is deeply and carefully detailed; however, this article calls for some remarks. First of all, the three-dimensional primal density energy is supposed to be a function not only of strains, but also of the boundary terms, which is new. This remarkable fact confirms, to our belief, the necessity of a constitutive law for the body boundaries, due to variational arguments. It must be observed also that a necessary constraint, classically called Jacobi's relation, is not apparently taken into account, which is a matter of reflexion. Another remark comes from the fact that the unsolved difficulty due to finding and expressing the constitutive equations with director models seems to be overcome thanks again to the main director approximation. In conclusion, this paper, which extends preceding results by Fox, Roult and Simo, is most interesting, though a little intricate, and deserves much attention due to some original theoretical results on shells (and rods) structures.