Summary
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abresch, U. [1987] Constant mean curvature tori in terms of elliptic functions.J. Reine Angew. Math. 374, 169–192.
Antman, S. S. [1972], The theory of rods,Handbuch der Physik, Band VIa/2, S. Flügge and C. Truesdell, eds., Springer-Verlag, Berlin, 641–703.
Antman, S. S. [1995],Nonlinear Problems of Elasticity, Applied Mathematical Sciences,107, Springer-Verlag, New York.
Antman, S. S. and W. H. Warner [1967] Dynamical theory of hyperelastic rods.Arch. Ratl. Mech. Anal. 23, 135–162.
Caflisch, R. and J. H. Maddocks [1984] Nonlinear dynamical theory of the elastica.Proc. R. Soc. Edin. 99A, 1–23.
Camassa, R. and D. Holm [1993] An integrable shallow water equation with peaked solitons,Phys. Rev. Lett.,71, 1661–1664.
Ciarlet, P. G. [1980], A justification of the von Kármán equations.Arch. Ratl. Mech. Anal. 73, 349–389.
Ciarlet, P. G. [1994] 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.
Ciarlet, P. G. and V. Lods [1994] Analyse asymptotique des coques linéairement élastiques. III. Une justification du modèle de W. T. Koiter.C. R. Acad. Sci. Paris 319 299–304.
Ciarlet, P. G., V. Lods, and B. Miara [1994] Analyse asymptotique des coques linéairement élastiques. II. Coques “en flexion”.C. R. Acad. Sci. Paris 319, 95–100, 1994.
Ciarlet, P. G. and B. Miara [1992], Two dimensional shallow shell equations.Comm. Pure Appl. Math. XLV, 327–360.
Destuynder, P. [1985], A classification of thin shell theories.Acta Appl. Math. 4, 15–63.
do Carmo, M. [1976],Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, N.J..
Foltinek, K. [1994] The Hamilton theory of elastica.Amer. J. Math. 116, 1479–1488.
Fox, D., A. Raoult, and J. C. Simo [1992] Modèles asymptotiques invariants pour des structures minces élastiques.C. R. Acad. Sci. Paris 315, 235–240.
Fox, D., A. Raoult, and J. C. Simo [1993] A justification of nonlinear properly invariant plate theories.Arch. Ratl. Mech. Anal.,124, 157–199.
Ge, Z. [1991] Equivariant symplectic difference schemes and generating functions,Physica D 49, 376–386.
Ge, Z., H. P. Kruse, J. E. Marsden and C. Scovel [1995] Poisson Brackets in the Shallow Water Approximation.Canad. Appl. Math. Quart., to appear.
Ge, Z. and J. E. Marsden [1988] Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory,Phys. Lett. A 133, 134–139.
Ge, Z. and C. Scovel [1994] A Hamiltonian truncation of the shallow water equation.Lett. Math. Phys. 31, 1–13.
John, F. [1971] Refined interior equations for the elastic shells.Comm. Pure Appl. Math. 24, 584–675.
Kato, T. [1985]Abstract Differential Equations and Nonlinear Mixed Problems. Lezioni Fermiane, Scuola Normale Superiore, Accademia Nazionale dei Lincei.
Koiter, W. T. [1970], On the foundation of the linear theory of thin elastic shells.Proc. Kon. Nederl. Akad. Wetensch. B69, 1–54.
Landau, L. D. and E. M. Lifshitz [1959],Theory of Elasticity, Addison-Wesley, Reading, MA.
Langer, J. and R. Perline [1991] Poisson geometry of the filament equation.J. Nonlin. Sci. 1, 71–94.
Le Dret, H. and A. Raoult [1995] The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity.J. Math. Pure Appl. (to appear).
Love, A. E. H. [1944]A Treatise on the Mathematical Theory of Elasticity. Dover, New York.
Maddocks, J. [1984] Stability of nonlinearly elastic rods.Arch. Ratl. Mech. Anal. 85, 311–354.
Maddocks, J. [1991] On the stability of relative equilibria.IMA J. Appl. Math. 46, 71–99.
Marsden, J. E. and T. J. R. Hughes [1994]Mathematical Foundations of Elasticity. Dover, New York; reprint of [1983] Prentice-Hall edition.
Marsden, J. E., T. S. Ratiu, and G. Raugel [1995] Equations d’Euler dans une coque sphérique mince (The Euler equations in a thin spherical shell),C. R. Acad. Sci. Paris 321, 1201–1206.
Mielke, A. and P. Holmes [1988] Spatially complex equilibira of buckled rods.Arch. Ratl. Mech. Anal.,101, 319–348.
Naghdi, P. [1972], The theory of shells and plates.Handbuch der Physik Band VIa/2, S. Flügge and C. Truesdell, eds., Springer-Verlag, Berlin, 425–640.
Shi, Y. and J. E. Hearst [1994] The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling.J. Chem. Phys. 101, 5186–5200.
Simo, J. C., M. S. Rifai, and D. D. Fox [1992], On a stress resultant geometrically exact shell models. Part VI: Conserving algorithms for nonlinear dynamics.Comp. Meth. Appl. Mech. Eng. 34, 117–164.
Simo, J. C., J. E. Marsden, and P. S. Krishnaprasad [1988] The Hamiltonian structure of nonlinear elasticity: The material, spatial, and convective representations of solids, rods, and plates.Arch. Ratl. Mech. Anal. 104, 125–183.
Simo, J. C., T. A. Posbergh, and J. E. Marsden [1990] Stability of coupled rigid body and geometrically exact rods: block diagonalization and the energy-momentum method,Phys. Rep. 193, 280–360.
Simo, J. C., T. A. Posbergh, and J. E. Marsden [1991] Stability of relative equilibria II: Three dimensional elasticity,Arch. Ratl. Mech. Anal.,115, 61–100.
Author information
Authors and Affiliations
Additional information
Communicated by Stephen Wiggins
This paper is dedicated to the memory of Juan C. Simo
This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.
Rights and permissions
About this article
Cite this article
Ge, Z., Kruse, H.P. & Marsden, J.E. The limits of hamiltonian structures in three-dimensional elasticity, shells, and rods. J Nonlinear Sci 6, 19–57 (1996). https://doi.org/10.1007/BF02433809
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02433809