This paper uses Hamiltonian structures to study the problem of the limit of three-dimensional (3D) 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 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 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 andderive 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.
Shell Model Poisson Bracket Director Field Hamiltonian Structure Frenet Frame
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Ciarlet, P. G.  Mathematical shell theory: recent developments and open problems, inDuration and Change: Fifty years at Oberwolfach, M. Artin, H. Kraft, and R. Remmert eds., Springer-Verlag, New York, 159–176.Google Scholar
Ciarlet, P. G. and V. Lods  Analyse asymptotique des coques linéairement élastiques. III. Une justification du modèle de W. T. Koiter.C. R. Acad. Sci. Paris319 299–304.MATHMathSciNetGoogle Scholar
Ciarlet, P. G., V. Lods, and B. Miara  Analyse asymptotique des coques linéairement élastiques. II. Coques “en flexion”.C. R. Acad. Sci. Paris319, 95–100, 1994.MATHMathSciNetGoogle Scholar
Ciarlet, P. G. and B. Miara , Two dimensional shallow shell equations.Comm. Pure Appl. Math.XLV, 327–360.MathSciNetGoogle Scholar
Marsden, J. E. and T. J. R. Hughes Mathematical Foundations of Elasticity. Dover, New York; reprint of  Prentice-Hall edition.Google Scholar
Marsden, J. E., T. S. Ratiu, and G. Raugel  Equations d’Euler dans une coque sphérique mince (The Euler equations in a thin spherical shell),C. R. Acad. Sci. Paris321, 1201–1206.MATHMathSciNetGoogle Scholar
Mielke, A. and P. Holmes  Spatially complex equilibira of buckled rods.Arch. Ratl. Mech. Anal.,101, 319–348.MATHMathSciNetGoogle Scholar
Naghdi, P. , The theory of shells and plates.Handbuch der Physik Band VIa/2, S. Flügge and C. Truesdell, eds., Springer-Verlag, Berlin, 425–640.Google Scholar
Shi, Y. and J. E. Hearst  The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling.J. Chem. Phys.101, 5186–5200.CrossRefGoogle Scholar
Simo, J. C., M. S. Rifai, and D. D. Fox , On a stress resultant geometrically exact shell models. Part VI: Conserving algorithms for nonlinear dynamics.Comp. Meth. Appl. Mech. Eng.34, 117–164.MATHMathSciNetGoogle Scholar
Simo, J. C., J. E. Marsden, and P. S. Krishnaprasad  The Hamiltonian structure of nonlinear elasticity: The material, spatial, and convective representations of solids, rods, and plates.Arch. Ratl. Mech. Anal.104, 125–183.MATHMathSciNetGoogle Scholar
Simo, J. C., T. A. Posbergh, and J. E. Marsden  Stability of coupled rigid body and geometrically exact rods: block diagonalization and the energy-momentum method,Phys. Rep.193, 280–360.MathSciNetCrossRefGoogle Scholar
Simo, J. C., T. A. Posbergh, and J. E. Marsden  Stability of relative equilibria II: Three dimensional elasticity,Arch. Ratl. Mech. Anal.,115, 61–100.MATHMathSciNetCrossRefGoogle Scholar