The Entropy Rate Admissibility Criterion for a Phase Transition Problem

Abstract

We summarize the recent results concerning the entropy rate admissibility criterion applied to a hyperbolic-elliptic mixed type system describing a phase transition problem. The system we discuss is given by

$$ \begin{gathered} v_t - u_x = 0, \hfill \\ u_t - f(v)_x = 0, \hfill \\ \end{gathered} $$

(1)

where v, u, and are strain, velocity, and stress, respectively. The first and second equations express the conservation of mass and linear momentum, respectively. We assume that is a nonmonotone function of v as depicted in Figure 1.1.. It is important to note that if is nonnegative, the system is hyperbolic and if is negative, the system is elliptic. In our case there are two intervals (0,a] and [ß, oo) where the system is hyperbolic. These two intervals correspond to two different phases of a material. These two phases are called the a-and,ß-phase. The Maxwell stress is the value of stress for which the areas A and are equal. The strains for the Maxwell stress in the a-and 0-phase are denoted by v_c, and vo, respectively. This is a hyperbolic-elliptic mixed system capable of describing the phase transition problem. A very well-known example is the van der Waals fluid where -f is the pressure and v is the specific volume (the reciprocal of density). Another well-known example is the martensite-austenite phase transition in elasticity.