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

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Abstract
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.