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

Simo, J. C.; Marsden, Jerrold E.; Krishnaprasad, P. S.

doi:10.1007/bf00251673

Cited by 215 publications

(154 citation statements)

References 34 publications

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“…This completes the formulation of the time-reversible part of the model, i.e., of non-linear elasticity, the Hamiltonian structure of which has been examined earlier from different perspective [29]. Except for heat conduction, the reversible part of the GENERIC embodies the thermoelastic behavior.…”

Section: Poisson Operator and Reversible Partmentioning

confidence: 61%

Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Hütter

Svendsen

2012

Continuum Mech. Thermodyn.

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Thermodynamic models for viscoplastic solids are often formulated in the context of continuum thermodynamics and the dissipation principle. The purpose of the current work is to show that models for such material behavior can also be formulated in the form of a General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC), see, e.g., Grmela and Öttinger (Phys Rev E, 56:6620-6632, 1997), Öttinger and Grmela (Phys Rev E, 56:6633-6655, 1997), Grmela (J Non-Newtonian Fluid Mech, 165:980-986, 2010). A GENERIC combines Hamiltonian-dynamics-based modeling of time-reversible processes with Onsager-Casimir-based modeling of time-irreversible processes. The result is a model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to model complex fluids, it has recently been applied to anisotropic inelastic solids in Eulerian (Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:45-52, 2008; Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:53-65, 2008; Hütter and Tervoort, in Adv Appl Mech, 42:254-317, 2008) and Lagrangian (Hütter and Svendsen, in J Elast 104:357-368, 2011) settings, as well as to damage mechanics. For simplicity, attention is focused in the current work on the case of thermoelastic viscoplasticity. Central to this formulation is a GENERIC-based form for the viscoplastic flow rule. A detailed comparison with the formulation based on continuum thermodynamics and the dissipation principle is given.

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Section: Poisson Operator and Reversible Partmentioning

confidence: 61%

Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Hütter

Svendsen

2012

Continuum Mech. Thermodyn.

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“…In geometrically exact beam models (Simó [123], Simó et al [124]), the configuration of a beam is described by the line of centroids r and by a moving orthonormal basis…”

Section: Numerical Testsmentioning

confidence: 99%

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Demoures

Gay–Balmaz

Raţiu

2016

Forum of Mathematics, Sigma

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This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

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“…For any two functionals, F : S B → R and G : S B → R, we define the Poisson bracket to be identical to the canonical bracket of Hamiltonian mechanics (see [16]):…”

Section: A Infinite Dimensional Example: Finite Strain Thermoelasticitymentioning

confidence: 99%

A Characterization of Conserved Quantities in Non-Equilibrium Thermodynamics

Romero

2013

Entropy

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Abstract:The well-known Noether theorem in Lagrangian and Hamiltonian mechanics associates symmetries in the evolution equations of a mechanical system with conserved quantities. In this work, we extend this classical idea to problems of non-equilibrium thermodynamics formulated within the GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling) framework. The geometric meaning of symmetry is reviewed in this formal setting and then utilized to identify possible conserved quantities and the conditions that guarantee their strict conservation. Examples are provided that demonstrate the validity of the proposed definition in the context of finite and infinite dimensional thermoelastic problems.

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

Cited by 215 publications

References 34 publications

Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

A Characterization of Conserved Quantities in Non-Equilibrium Thermodynamics

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