Skip to main content
Log in

The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  • R. Abraham & J. Marsden [1978] Foundations of Mechanics, Second Edition, Addison Wesley.

  • R. Abraham, J. Marsden & T. Ratiu [1983] Manifolds, Tensor Analysis, and Applications, Addison-Wesley, Second Edition, Springer-Verlag, New York, 1988.

    Google Scholar 

  • S. S. Antman [1972] The Theory of Rods, Handbuch der Physik, Vol. VIa/2, Springer, Berlin.

    Google Scholar 

  • S. S. Antman [1974] Kirchhoff's Problem for Nonlinearly Elastic Rods, Quart. J. of Appl. Math. 32, 221–240.

    Google Scholar 

  • S. S. Antman & K. B. Jordan [1975] Qualitative Aspects of the Spatial Deformation of Non-linearly Elastic Rods, Proc. Roy. Soc. Edinburgh Sect. A 73 (5), 85–105.

    Google Scholar 

  • S. S. Antman [1976] Ordinary Differential Equations of Nonlinear Elasticity I: Foundations of the Theories of Non-linearly Elastic Rods and Shells, Arch. Rational Mech. Anal. 61 (4), 307–351.

    Google Scholar 

  • S. S. Antman [1978] Buckled States of Nonlinearly Elastic Plates, Arch. Rational Mech. Anal. 67 (2), 111–149.

    Google Scholar 

  • S. Antman & C. S. Kenney [1981] Large Buckled States of Nonlinearly Elastic Rods Under Torsion, Thrust, and Gravity, Arch. Rational Mech. Anal. 76 (4), 289–338.

    Google Scholar 

  • S. S. Antman [1984] Large Lateral Buckling of Nonlinearly Elastic Beams, Arch. Rational Mech. Anal. 84 (4), 293–305.

    Google Scholar 

  • V. Arnold [1966] Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, Grenoble 16, 319–361.

    Google Scholar 

  • A. Chorin, T.J. R. Hughes, M. F. McCracken & J. E. Marsden [1978], Product formulas and Numerical Algorithms, Comm. Pure. Appl. Math. 31, 205–256.

    Google Scholar 

  • Y. Choquet-Bruhar, C. Dewitt-Morette & M. Dillard-Bleick [1984] Analysis, Manifolds, and Physics, North Holland, second edition.

  • P. Ciarlet [1988] Mathematical Elasticity. Volume 1: Three Dimensional Elasticity, Studies in Mathematics and its Applications, North-Holland, Amsterdam

    Google Scholar 

  • H. Cohen & C. N. DeSilva [1966] Nonlinear Theory of Elastic Directed Surfaces, J. Math. Phys. 7, No. 6, 960–966.

    Google Scholar 

  • D. Ebin & J. E. Marsden [1970] Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, Ann. Math. 92, 102–163.

    Google Scholar 

  • J. L. Ericksen & C. Truesdell [1958] Exact Theory of Stress and Strain in Rods and Shells, Arch. Rational Mech. Anal. 1, 295–333.

    Google Scholar 

  • H. Goldstein [1980] Classical Mechanics, Second Edition, Addison-Wesley.

  • M. Golubitsky & I. Stewart [1987] Generic Bifurcation of Hamiltonian Systems with Symmetry, Physica 24 D, 391–405.

    Google Scholar 

  • A. E. Green, P. M. Naghdi & W. L. Wainwright [1965] A General Theory of a Cosserat Surface, Arch. Rational Mech. Anal. 20, 287–308.

    Google Scholar 

  • A. E. Green & W. Zerna [1968] Theoretical Elasticity, Oxford U. Press.

  • J. Guckenheimer & P. Holmes [1983] Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New York.

    Google Scholar 

  • D. D. Holm & B. A. Kuperschmidt [1983a] Poisson Brackets and Clebsch Representations for Magnetohydrodynamics, Multifluid Plasmas, and Elasticity, Physica 6 D, 347–363.

    Google Scholar 

  • D. D. Holm, J. E. Marsden & T. Ratiu [1986] The Hamiltonian Structure of Continuum Mechanics in the Material, Inverse Material, Spatial, and Convective Representations, Séminaire de Mathématiques Supérieures, 100, Les Presses de l'Université de Montréal.

  • D. D. Holm, J. E. Marsden, T. S. Ratiu & A. Weinstein [1985] Nonlinear Stability of Fluid and Plasma Equilibria, Phys. Rep. 123, 1–116.

    Google Scholar 

  • P. Holmes & J. Marsden [1983] Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups, Indiana Univ. Math. J. 32, 273–310.

    Google Scholar 

  • T. J. R. Hughes & J. Winget [1980] Finite Rotation Effects in Numerical Integration of Rate Constitutive Equations Arising in Large-Deformation Analysis, Int. J. Num. Meth. Engng. 15, 1862–1867.

    Google Scholar 

  • T. J. R. Hughes, W. K. Liu & P. Caughy [1978] Transient Finite Element Formulations that Preserve Energy, J. Applied Mechanics 45, 366–370.

    Google Scholar 

  • Z. R. Iwinski & L. A. Turski [1976] Canonical Theories of Systems Interacting Electromagnetically, Lett. in Appl. and Eng. Sci. 4, 179–191.

    Google Scholar 

  • P. S. Krishnaprasad [1985] Lie Poisson Structures, Dual Spin Spacecraft and Asymptotic Stability, Nonlinear Analysis, Theory, Methods, and Appl. 9, 1011–1035.

    Google Scholar 

  • P. S. Krishnaprasad & J. E. Marsden [1987] Hamiltonian Structure and Stability for Rigid Bodies with Flexible Attachments, Arch. Rational Mech. Anal. 98, 71–93.

    Google Scholar 

  • P. S. Krishnasaprad, J. E. Marsden, T. Posbergh & J. C. Simo [1988] Nonlinear Stability of Coupled Rigid Body, Rod and Plate Structures (in preparation).

  • D. Lewis, J. E. Marsden, R. Montgomery & T. Ratiu [1986] The Hamiltonian Structure for Dynamic Free Boundary Problems, Physica 18 D, 391–404.

    Google Scholar 

  • D. Lewis, J. E. Marsden & T. Ratiu [1986a] Formal Stability of Liquid Drops with Surface Tension, in Perspectives in Nonlinear Dynamics, ed. by M. F. Schlessinger and others, World Scientific, 71–83.

  • D. Lewis, J. E. Marsden & T. Ratiu [1987] Stability and bifurcation of a rotating planar liquid drop (J. Math. Phys.).

  • A. Libai & J. G. Simmonds [1983] Nonlinear Elastic Shell Theory. Advances in Applied Mechanics, 23, 271–371, edited by J. Hutchinson and T. Wu.

  • A. E. H. Love [1944] The Mathematical Theory of Elasticity, 4th edition, Dover, New York.

    Google Scholar 

  • J. E. Marsden [1982] A Group Theoretical Approach to the Equations of Plasma Physics, Canadian Math. Bull. 25, 129–142.

    Google Scholar 

  • J. E. Marsden [1988] Lie-Poisson Hamilton Jacobi theory and Lie-Poisson Integrators, Phys. Lett. A. (to appear).

  • J. E. Marsden & T. Hughes [1983] Mathematical Foundations of Elasticity, Prentice Hall.

  • J. E. Marsden & T. Ratiu [1986] Reduction of Poisson Manifolds, Letters in Math. Phys. 11, 161–169.

    Google Scholar 

  • J. E. Marsden, T. Ratiu & A. Weinstein [1984a] Semi-direct Products and Reduction in Mechanics, Trans. Am. Math. Soc. 281, 147–177.

    Google Scholar 

  • J. E. Marsden, T. Ratiu & A. Weinstein [1984b] Reduction and Hamiltonian Structures on Duals of Semidirect Product Lie Algebras, Cont. Math. AMS 28, 55–100.

    Google Scholar 

  • J. E. Marsden & A. Weinstein [1974] Reduction of Symplectic Manifolds with Symmetry, Rep. Math. Phys. 5, 121–130.

    Google Scholar 

  • J. E. Marsden & A. Weinstein [1982] The Hamiltonian Structure of the Maxwell Vlasov Equations, Physica D 4, 394–406.

    Google Scholar 

  • J. E. Marsden & A. Weinstein [1983] Coadjoint Orbits, Vortices and Clebsch Variables for Incompressible Fluids, Physica 8 D, 305–323.

    Google Scholar 

  • J. E. Marsden, A. Weinstein, T. Ratiu, R. Schmid & R. G. Spencer [1983] Hamiltonian Systems with Symmetry, Coadjoint Orbits and Plasma Physics, Proc. IUTAMISIMM Symposium on Modern Developments in Analytical Mechanics, Torino, June 7–11, 1982, Atti della Academia delle Scienze di Torino 117, 289–340.

  • R. Montgomery, J. Marsden & T. Ratiu [1984] Gauged Lie-Poisson Structures, Contemporary Mathematics, AMS 28, 101–114.

    Google Scholar 

  • P. J. Morrison & J. M. Greene [1980] Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics, Phys. Rev. Lett. 45, 790–794.

    Google Scholar 

  • P. M. Naghdi [1972] The Theory of Plates and Shells, in Handbuch der Physik, Vol. VIa/2, Springer, Berlin.

    Google Scholar 

  • P. M. Naghdi [1980] Finite Deformations of Elastic Rods and Shells, in Proceedings IUTAM Symposium on Finite Elasticity, Lehigh University, Bethlehem.

    Google Scholar 

  • R. Reissner [1973] On a one-dimensional, large-displacement, finite-strain beam-theory, Stud. Appl. Math. 52, 87–95.

    Google Scholar 

  • E. Reissner [1981] On Finite Deformations of Space-Curve Beams, ZAMP 32, 734–744.

    Google Scholar 

  • J. C. Simo & J. E. Marsden [1984] On the Rotated Stress Tensor and the Material Version of the Doyle-Ericksen formula, Arch. Rational Mech. Analysis 86, 213–231.

    Google Scholar 

  • J. C. Simo [1985] A Finite Strain Beam Formulation. The Three Dimensional Dynamic Problem. Part I. Comp. Meth. Appl. Mech. Engng. 49, 55–70.

    Google Scholar 

  • J. C. Simo & L. Vu-Quoc [1986a] A Three-Dimensional Finite Strain Rod Model. Part II: Computational Aspects. Comp. Meth. Appl. Mech. Engng. 58, 79–116.

    Google Scholar 

  • J. C. Simo & L. Vu-Quoc [1986b] On the Dynamics of Flexible Beams Under Large Overall Motions—The plane Case; Parts I and II, J. Appl. Mech. 54, No. 3.

  • J. C. Simo & L. Vu-Quoc [1987] On the Dynamics in Space of Rods Urdegoing Large Overall Motions, Comp. Meth. Appl. Mech. Engng. (to appear).

  • J. C. Simo & D. D. Fox [1988] On a Stress Resultant, Geometrically Exact Shell Model. Part I: Formulation and Optimal Parametrization, Comp. Meth. Appl. Mech. Engng. (to appear).

  • J. C. Simo [1987] On a Fully Three Dimensional Finite-Strain Viscoelastic Damage Model: Formulation and Computational Aspects, Comp. Meth. Appl. Mech. Engng. 60, 137–158.

    Google Scholar 

  • J. C. Simo & L. Vu-Quoc [1986c] The role of Nonlinear Theories in Transient Dynamic Analysis of Flexible Structures, J. Sound and Vibration, (To appear).

  • J. C. Simo, D. D. Fox & M. S. Rifai, [1988a] On a Stress Resultant Geometrically Exact Shell Model. Part II: The Linear Theory, Computational Aspects, Comp. Meth. Appl. Mech. Engng. (to appear)

  • J. C. Simo, D. D. Fox & M. S. Rifai, [1988b] On a Stress Resultant Geometrically Exact Shell Model. Part III: Computational Aspects of the Nonlinear Theory, Comp. Meth. Appl. Mech. Engng. (to appear).

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by S. Antman

Rights and permissions

Reprints and permissions

About this article

Cite this article

Simo, J.C., Marsden, J.E. & Krishnaprasad, P.S. The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates. Arch. Rational Mech. Anal. 104, 125–183 (1988). https://doi.org/10.1007/BF00251673

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00251673

Keywords

Navigation