Entropy form and the contact geometry of the material point model (Q2893478)

The authors investigate a material point model (MP-model) and exploit the geometrical meaning of the entropy form introduced by Coleman and Owen. Roughly speaking, they develop a geometrical framework for systems of ``material point'' type and investigate their dynamical properties. They construct the entropy form for a continuum thermodynamical system starting with local balance equations of energy and entropy and using the Fourier or Cattaneo heat propagation laws. They define the state space \(P\) of the entropy form and the class of thermodynamically admissible processes in the space \(P\). They present the examples of the Coleman-Owen thermoelastic point model and of the Cattaneo heat propagation.NEWLINENEWLINEThe authors discuss constitutive properties of thermodynamical systems and the properties of the entropy form in the space \(P\) and the relationship between them. A comparison of the MP-model with the formalism of homogeneous thermodynamics and the formulation of the problem of the modeling of dynamical processes in the MP-model are presented. Then, extended constitutive surfaces \(\Sigma_{\varrho,k}\) are described as Legendre submanifolds \(\Sigma_\varrho\) of the space \(P\) shifted by the flow of Reeb vector field. They prove that this shift is controlled by the entropy production function. They determine which contact Hamiltonian dynamical systems \(\xi_k\) are tangent to the surface \(\Sigma_{\varrho,k}\) introducing conformal Hamiltonian systems \(\mu\xi_k\), where the conformal factor \(\mu\) characterizes the increase of entropy along the trajectories. Finally, they describe the singular points of the vector fields \(\mu\xi_k\). They show that the points on the extended constitutive surface \(\Sigma_{\varrho,k}\) where \(\mu=0\) are the equilibrium points of the dynamical system \(\mu\xi_k\) transversally stable in the domain, where \(\mu>0\). At last, they prove that the entropy form \(\eta\) can be interpreted as the 1-form corresponding to the flat connection on the Gibbs bundle.