The limits of hamiltonian structures in three-dimensional elasticity, shells, and rods
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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.
- Abresch, U. (1987) Constant mean curvature tori in terms of elliptic functions. J. Reine Angew. Math. 374: pp. 169-192
- Antman, S. S. The theory of rods. In: Flügge, S., Truesdell, C. eds. (1972) Handbuch der Physik. Springer-Verlag, Berlin, pp. 641-703
- Antman, S. S. (1995) Nonlinear Problems of Elasticity. Springer-Verlag, New York
- Antman, S. S., Warner, W. H. (1967) Dynamical theory of hyperelastic rods. Arch. Ratl. Mech. Anal. 23: pp. 135-162
- Caflisch, R., Maddocks, J. H. (1984) Nonlinear dynamical theory of the elastica. Proc. R. Soc. Edin. 99A: pp. 1-23
- Camassa, R., Holm, D. (1993) An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71: pp. 1661-1664 CrossRef
- Ciarlet, P. G. (1980) A justification of the von Kármán equations. Arch. Ratl. Mech. Anal. 73: pp. 349-389
- Ciarlet, P. G. Mathematical shell theory: recent developments and open problems. In: Artin, M., Kraft, H., Remmert, R. eds. (1994) Duration and Change: Fifty years at Oberwolfach. Springer-Verlag, New York, pp. 159-176
- Ciarlet, P. G., Lods, V. (1994) Analyse asymptotique des coques linéairement élastiques. III. Une justification du modèle de W. T. Koiter. C. R. Acad. Sci. Paris 319: pp. 299-304
- Ciarlet, P. G., Lods, V., Miara, B. (1994) Analyse asymptotique des coques linéairement élastiques. II. Coques “en flexion”. C. R. Acad. Sci. Paris 319: pp. 95-100
- Ciarlet, P. G., Miara, B. (1992) Two dimensional shallow shell equations. Comm. Pure Appl. Math. XLV: pp. 327-360
- Destuynder, P. (1985) A classification of thin shell theories. Acta Appl. Math. 4: pp. 15-63 CrossRef
- 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: pp. 1479-1488 CrossRef
- Fox, D., Raoult, A., Simo, J. C. (1992) Modèles asymptotiques invariants pour des structures minces élastiques. C. R. Acad. Sci. Paris 315: pp. 235-240
- Fox, D., Raoult, A., Simo, J. C. (1993) A justification of nonlinear properly invariant plate theories. Arch. Ratl. Mech. Anal. 124: pp. 157-199 CrossRef
- Ge, Z. (1991) Equivariant symplectic difference schemes and generating functions. Physica D 49: pp. 376-386 CrossRef
- Ge, Z., H. P. Kruse, J. E. Marsden and C. Scovel  Poisson Brackets in the Shallow Water Approximation.Canad. Appl. Math. Quart., to appear.
- Ge, Z., Marsden, J. E. (1988) Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory. Phys. Lett. A 133: pp. 134-139 CrossRef
- Ge, Z., Scovel, C. (1994) A Hamiltonian truncation of the shallow water equation. Lett. Math. Phys. 31: pp. 1-13 CrossRef
- John, F. (1971) Refined interior equations for the elastic shells. Comm. Pure Appl. Math. 24: pp. 584-675
- Kato, T. 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: pp. 1-54
- Landau, L. D., Lifshitz, E. M. (1959) Theory of Elasticity. Addison-Wesley, Reading, MA
- Langer, J., Perline, R. (1991) Poisson geometry of the filament equation. J. Nonlin. Sci. 1: pp. 71-94 CrossRef
- Le Dret, H. and A. Raoult  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: pp. 311-354
- Maddocks, J. (1991) On the stability of relative equilibria. IMA J. Appl. Math. 46: pp. 71-99
- Marsden, J. E. and T. J. R. Hughes Mathematical Foundations of Elasticity. Dover, New York; reprint of  Prentice-Hall edition.
- Marsden, J. E., Ratiu, T. S., Raugel, G. (1995) Equations d’Euler dans une coque sphérique mince (The Euler equations in a thin spherical shell). C. R. Acad. Sci. Paris 321: pp. 1201-1206
- Mielke, A., Holmes, P. (1988) Spatially complex equilibira of buckled rods. Arch. Ratl. Mech. Anal. 101: pp. 319-348
- Naghdi, P. The theory of shells and plates. In: Flügge, S., Truesdell, C. eds. (1972) Handbuch der Physik. Springer-Verlag, Berlin, pp. 425-640
- Shi, Y., Hearst, J. E. (1994) The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling. J. Chem. Phys. 101: pp. 5186-5200 CrossRef
- Simo, J. C., Rifai, M. S., Fox, D. D. (1992) On a stress resultant geometrically exact shell models. Part VI: Conserving algorithms for nonlinear dynamics. Comp. Meth. Appl. Mech. Eng. 34: pp. 117-164
- Simo, J. C., Marsden, J. E., Krishnaprasad, P. S. (1988) The Hamiltonian structure of nonlinear elasticity: The material, spatial, and convective representations of solids, rods, and plates. Arch. Ratl. Mech. Anal. 104: pp. 125-183
- Simo, J. C., Posbergh, T. A., Marsden, J. E. (1990) Stability of coupled rigid body and geometrically exact rods: block diagonalization and the energy-momentum method. Phys. Rep. 193: pp. 280-360 CrossRef
- Simo, J. C., Posbergh, T. A., Marsden, J. E. (1991) Stability of relative equilibria II: Three dimensional elasticity. Arch. Ratl. Mech. Anal. 115: pp. 61-100 CrossRef
- The limits of hamiltonian structures in three-dimensional elasticity, shells, and rods
Journal of Nonlinear Science
Volume 6, Issue 1 , pp 19-57
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- 1. The Fields Institute for Research in Mathematical Sciences, 222 College Street, M5T 3J1, Toronto, Ontario
- 2. Institut für Angewandte Mathematik, Universität Hamburg, Bundestrasse 55, D-20146, Hamburg, Germany
- 3. Control and Dynamical Systems, California Institute of Technology 104-44, 91125, Pasadena, CA