Phase Transitions of Elastic Solid Materials

Abstract

In recent years the analysis of phase transitions for mixtures of two or more non-interacting fluids has been successfully undertaken within the Van der Waals-Cahn-Hilliard gradient theory of phase transitions (see Baldo [1], Fonseca & Tartar [5], Gurtin [9], Kohn & Sternberg [11], Modica [12], Owen [13], Sternberg [15]). If the nonnegative Gibbs free energy W vanishes only at two points a and b, this theory permits us to select among all the minimizers of $$\int\limits_\Omega\ \ W(v(x))\ \ dx$$ with prescribed total mass $$m=\int\limits_\Omega\ \ v(x)\ dx={\rm meas}\ (\Omega)\ (\theta_{a}+(1-\theta)b),\ \ \ \ {\rm with}\ \theta \in\ (0,1),$$ those that have minimal interfacial area, i.e. it singles out those solutions v ∈ {a, b} a.e. such that the set {v = a} minimizes Per _Ω(ω) among all subsets ω of Ω with meas (ω) = θ = meas ({v = a}).