The limits of hamiltonian structures in three-dimensional elasticity, shells, and rods Z. Ge H. P. Kruse J. E. Marsden Article Received: 01 September 1995 Revised: 15 October 1995 DOI :
10.1007/BF02433809

Cite this article as: Ge, Z., Kruse, H.P. & Marsden, J.E. J Nonlinear Sci (1996) 6: 19. doi:10.1007/BF02433809 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.

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.

References Abresch, U. [1987] Constant mean curvature tori in terms of elliptic functions.

J. Reine Angew. Math.
374 , 169–192.

MATH MathSciNet Google Scholar 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.

Google Scholar Antman, S. S. [1995],

Nonlinear Problems of Elasticity , Applied Mathematical Sciences,

107 , Springer-Verlag, New York.

Google Scholar Antman, S. S. and W. H. Warner [1967] Dynamical theory of hyperelastic rods.

Arch. Ratl. Mech. Anal.
23 , 135–162.

MathSciNet Google Scholar Caflisch, R. and J. H. Maddocks [1984] Nonlinear dynamical theory of the elastica.

Proc. R. Soc. Edin.
99A , 1–23.

MathSciNet Google Scholar Camassa, R. and D. Holm [1993] An integrable shallow water equation with peaked solitons,

Phys. Rev. Lett. ,

71 , 1661–1664.

MATH MathSciNet CrossRef Google Scholar Ciarlet, P. G. [1980], A justification of the von Kármán equations.

Arch. Ratl. Mech. Anal.
73 , 349–389.

MATH MathSciNet Google Scholar Ciarlet, P. G. [1994] Mathematical shell theory: recent developments and open problems, in

Duration 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 [1994] Analyse asymptotique des coques linéairement élastiques. III. Une justification du modèle de W. T. Koiter.

C. R. Acad. Sci. Paris
319 299–304.

MATH MathSciNet Google Scholar 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.

MATH MathSciNet Google Scholar Ciarlet, P. G. and B. Miara [1992], Two dimensional shallow shell equations.

Comm. Pure Appl. Math.
XLV , 327–360.

MathSciNet Google Scholar Destuynder, P. [1985], A classification of thin shell theories.

Acta Appl. Math.
4 , 15–63.

MATH MathSciNet CrossRef Google Scholar do Carmo, M. [1976],

Differential Geometry of Curves and Surfaces , Prentice-Hall, Englewood Cliffs, N.J..

Google Scholar Foltinek, K. [1994] The Hamilton theory of elastica.

Amer. J. Math.
116 , 1479–1488.

MATH MathSciNet CrossRef Google Scholar 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.

MATH MathSciNet Google Scholar Fox, D., A. Raoult, and J. C. Simo [1993] A justification of nonlinear properly invariant plate theories.

Arch. Ratl. Mech. Anal. ,

124 , 157–199.

MATH MathSciNet CrossRef Google Scholar Ge, Z. [1991] Equivariant symplectic difference schemes and generating functions,

Physica D
49 , 376–386.

MATH MathSciNet CrossRef Google Scholar 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.

MathSciNet CrossRef Google Scholar Ge, Z. and C. Scovel [1994] A Hamiltonian truncation of the shallow water equation.

Lett. Math. Phys.
31 , 1–13.

MATH MathSciNet CrossRef Google Scholar John, F. [1971] Refined interior equations for the elastic shells.

Comm. Pure Appl. Math.
24 , 584–675.

Google Scholar 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.

MathSciNet Google Scholar Landau, L. D. and E. M. Lifshitz [1959],

Theory of Elasticity , Addison-Wesley, Reading, MA.

Google Scholar Langer, J. and R. Perline [1991] Poisson geometry of the filament equation.

J. Nonlin. Sci.
1 , 71–94.

MATH MathSciNet CrossRef Google Scholar 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.

Google Scholar Maddocks, J. [1984] Stability of nonlinearly elastic rods.

Arch. Ratl. Mech. Anal.
85 , 311–354.

MATH MathSciNet Google Scholar Maddocks, J. [1991] On the stability of relative equilibria.

IMA J. Appl. Math.
46 , 71–99.

MATH MathSciNet Google Scholar 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.

MATH MathSciNet Google Scholar Mielke, A. and P. Holmes [1988] Spatially complex equilibira of buckled rods.

Arch. Ratl. Mech. Anal. ,

101 , 319–348.

MATH MathSciNet Google Scholar 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.

Google Scholar Shi, Y. and J. E. Hearst [1994] The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling.

J. Chem. Phys.
101 , 5186–5200.

CrossRef Google Scholar 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.

MATH MathSciNet Google Scholar 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.

MATH MathSciNet Google Scholar 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.

MathSciNet CrossRef Google Scholar 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.

MATH MathSciNet CrossRef Google Scholar © Springer-Verlag New York Inc. 1996

Authors and Affiliations Z. Ge H. P. Kruse J. E. Marsden 1. The Fields Institute for Research in Mathematical Sciences Toronto 2. Institut für Angewandte Mathematik Universität Hamburg Hamburg Germany 3. Control and Dynamical Systems California Institute of Technology 104-44 Pasadena