Entropy form and the contact geometry of the material point model

In this work we investigate a material point model (MP-model) and exploit the geometrical meaning of the "entropy form" introduced by Coleman and Owen. We show that a modification of the thermodynamical phase space (studied and exploited in numerous works) is an appropriate setting for the development of the MP-model in different physical situations. This approach allows to formulate the MP-model and the corresponding entropy form in terms similar to those of homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that admissible processes curves belong to the (extended) constitutive surfaces foliating the extended thermodynamical phase space P of the model over the space X of basic variables. Extended constitutive surfaces Σ S,κ are described as the Legendre submanifolds Σ S of the space P shifted by the flow of Reeb vector field. This shift is controlled by the entropy production function κ. We determine which contact Hamiltonian dynamical systems ξ K are tangent to the surfaces Σ S,κ, introduce conformally Hamiltonian systems μξ K where conformal factor μ characterizes the increase of entropy along the trajectories. These considerations are then illustrated by applying them to the ColemanOwen model of thermoelastic point.