# The limits of hamiltonian structures in three-dimensional elasticity, shells, and rods

- 129 Downloads
- 6 Citations

## 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 and*derive* 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.

## Preview

Unable to display preview. Download preview PDF.

### References

- Abresch, U. [1987] Constant mean curvature tori in terms of elliptic functions.
*J. Reine Angew. Math.***374**, 169–192.MATHMathSciNetGoogle 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.MathSciNetGoogle Scholar - Caflisch, R. and J. H. Maddocks [1984] Nonlinear dynamical theory of the elastica.
*Proc. R. Soc. Edin.***99A**, 1–23.MathSciNetGoogle Scholar - Camassa, R. and D. Holm [1993] An integrable shallow water equation with peaked solitons,
*Phys. Rev. Lett.*,**71**, 1661–1664.MATHMathSciNetCrossRefGoogle Scholar - Ciarlet, P. G. [1980], A justification of the von Kármán equations.
*Arch. Ratl. Mech. Anal.***73**, 349–389.MATHMathSciNetGoogle 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.MATHMathSciNetGoogle 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.MATHMathSciNetGoogle Scholar - Ciarlet, P. G. and B. Miara [1992], Two dimensional shallow shell equations.
*Comm. Pure Appl. Math.***XLV**, 327–360.MathSciNetGoogle Scholar - Destuynder, P. [1985], A classification of thin shell theories.
*Acta Appl. Math.***4**, 15–63.MATHMathSciNetCrossRefGoogle 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.MATHMathSciNetCrossRefGoogle 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.MATHMathSciNetGoogle 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.MATHMathSciNetCrossRefGoogle Scholar - Ge, Z. [1991] Equivariant symplectic difference schemes and generating functions,
*Physica D***49**, 376–386.MATHMathSciNetCrossRefGoogle 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.Google Scholar - Ge, Z. and J. E. Marsden [1988] Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory,
*Phys. Lett. A***133**, 134–139.MathSciNetCrossRefGoogle Scholar - Ge, Z. and C. Scovel [1994] A Hamiltonian truncation of the shallow water equation.
*Lett. Math. Phys.***31**, 1–13.MATHMathSciNetCrossRefGoogle 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.Google Scholar - Koiter, W. T. [1970], On the foundation of the linear theory of thin elastic shells.
*Proc. Kon. Nederl. Akad. Wetensch.***B69**, 1–54.MathSciNetGoogle 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.MATHMathSciNetCrossRefGoogle 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).Google Scholar - 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.MATHMathSciNetGoogle Scholar - Maddocks, J. [1991] On the stability of relative equilibria.
*IMA J. Appl. Math.***46**, 71–99.MATHMathSciNetGoogle Scholar - Marsden, J. E. and T. J. R. Hughes [1994]
*Mathematical Foundations of Elasticity*. Dover, New York; reprint of [1983] Prentice-Hall edition.Google Scholar - 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.MATHMathSciNetGoogle Scholar - Mielke, A. and P. Holmes [1988] Spatially complex equilibira of buckled rods.
*Arch. Ratl. Mech. Anal.*,**101**, 319–348.MATHMathSciNetGoogle 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.CrossRefGoogle 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.MATHMathSciNetGoogle 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.MATHMathSciNetGoogle 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.MathSciNetCrossRefGoogle 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.MATHMathSciNetCrossRefGoogle Scholar