Non-canonical Poisson bracket for nonlinear elasticity with extensions to viscoelasticity

The authors introduce a generalized bracket which is capable of generating the dynamical equations governing the flow of both elastic and viscoelastic media. This generalized bracket is divided into two parts: a noncanonical Poisson bracket and a new dissipation bracket. The non-canonical Poisson bracket is the Eulerian equivalent of the canonical Lagrangian Poisson bracket corresponding to an ideal (non-dissipative) continuum. It is derived for a nonlinear elastic medium, and then it is used to obtain the Eulerian equations of motion in nonlinear elasticity valid for large deformations. It is shown that the proposed non-canonical bracket naturally leads to a materially objective relation involving the upper-convected time derivative of a strain tensor, as suggested by Oldroyd 40 years ago. The dissipation bracket for linear irreversible thermodynamics is next proposed in a general, phenomenological circumstance, so that the dissipation processes occurring in real systems (viscous and relaxation phenomena) can be incorporated into the Hamiltonian formalism. This bracket diverges from previously proposed dissipation brackets in that it uses the same generating functional (i.e. the Hamiltonian) as the Poisson bracket, rather than an entropy functional or a dissipative potential. It is shown that, in combination with the choice of an appropriate Hamiltonian functional, the generalized bracket proposed here can generate the governing equations for many viscoelastic media, including the Voigt solid and the Maxwell viscoelastic fluid.

[1]  Darryl D. Holm,et al.  Poisson brackets and clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity , 1983 .

[2]  Dissipation conditions in a system of coupled oscillators , 1988 .

[3]  Miroslav Grmela,et al.  Bracket formulation of dissipative time evolution equations , 1985 .

[4]  M. Grmela.  Hamiltonian dynamics of incompressible elastic fluids , 1988 .

[5]  R. L. Seliger,et al.  Variational principles in continuum mechanics , 1968, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[6]  Jerrold E. Marsden,et al.  The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates , 1988 .

[7]  M. Grmela.  Hamiltonian mechanics of complex fluids , 1989 .

[8]  A. Beris,et al.  Poisson bracket formulation of incompressible flow equations in continuum mechanics , 1990 .

[9]  J. Marsden,et al.  Semidirect products and reduction in mechanics , 1984 .

[10]  J. Oldroyd.  On the formulation of rheological equations of state , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[11]  Harris,et al.  Quantum theory of the damped harmonic oscillator. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[12]  Allan N. Kaufman,et al.  DISSIPATIVE HAMILTONIAN SYSTEMS: A UNIFYING PRINCIPLE , 1984 .

[13]  H. Abarbanel,et al.  Hamiltonian formulation of inviscid flows with free boundaries , 1988 .

[14]  D. Allen-Booth,et al.  Classical Mechanics 2nd edn , 1974 .

[15]  Jerrold E. Marsden,et al.  The Hamiltonian structure of the Maxwell-Vlasov equations , 1982 .

[16]  Miroslav Grmela,et al.  Bracket formulation of dissipative fluid mechanics equations , 1984 .

[17]  Clifford Ambrose Truesdell,et al.  The mechanical foundations of elasticity and fluid dynamics , 1952 .

[18]  Miroslav Grmela,et al.  Generalized constitutive equation for polymeric liquid crystals Part 1. Model formulation using the Hamiltonian (poisson bracket) formulation , 1990 .

[19]  B. Edwards,et al.  Poisson bracket formulation of viscoelastic flow equations of differential type: A unified approach , 1990 .

[20]  V. Arnold.  Variational principle for three-dimensional steady-state flows of an ideal fluid , 1965 .

[21]  G. W. Ford,et al.  Statistical Mechanics of Assemblies of Coupled Oscillators , 1965 .

[22]  J. Herivel,et al.  The derivation of the equations of motion of an ideal fluid by Hamilton's principle , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  L. C. Woods,et al.  The thermodynamics of fluid systems , by L. C. Woods. Pp 359. £12·50. 1985. ISBN 0-19-856180-6 (Oxford University Press) , 1987, The Mathematical Gazette.

[24]  Jerrold E. Marsden,et al.  Nonlinear stability of fluid and plasma equilibria , 1985 .

[25]  P. Morrison,et al.  Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics. , 1980 .