Material point model and the geometry of the entropy form

TL;DR

This paper explores the geometric structure of the entropy form in material point models, using thermodynamical phase space to describe constitutive surfaces as Legendre submanifolds, linking thermodynamics and geometry.

Contribution

It introduces a geometric framework for material point models using thermodynamical phase space, describing constitutive surfaces as Legendre submanifolds shifted by Reeb flow.

Findings

Constitutive surfaces are Legendre submanifolds in thermodynamical phase space.

Closeness of the entropy form relates to admissible process curves on the constitutive surface.

Entropy production function controls the shift of Legendre submanifolds.

Abstract

In this work we investigate the material point model and exploit the geometrical meaning of the "entropy form" introduced by B.Coleman and R.Owen. We analyze full and partial integrability (closeness) condition of the entropy form for the model of thermoelastic point and for the the deformable ferroelectric crystal media point. We show that the thermodynamical phase space (TPS) introduced by R.Hermann and widely exploited by R. Mrugala with his collaborators and other researchers, extended possibly by time, with its canonical contact structure is an appropriate setting for the development of material point models in different physical situations. This allows us to formulate the model of a material point and the corresponding entropy form in terms similar to those of the homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that the admissible processes curves belongs to the constitutive surface S of the model. Our principal result is the description of the constitutive surfaces S of the material point model as the Legendre submanifolds of the TPS shifted by the flow of Reeb vector field. This shift is controlled, at the points of Legendre submanifold by the entropy production function $\sigma$.