A Phase Field Model With Thermal Memory Governed by the Entropy Balance

Abstract

We introduce a thermomechanical model describing dissipative phase transitions with thermal memory in terms of the entropy balance and the principle of virtual power written for microscopic movements. The thermodynamical consistence of this model is verified and existence of solutions is proved for a related initial and boundary value problem.

Solid-liquid phase changes with different densities

“…The proof of Theorem 3.7 is based on the following scheme, which follows the idea of [6]: we first solve an approximating problem in which γ is substituted by a Lipschitzcontinuous function, and then we make a priori estimates (independent of the approximation parameter), which will allow us to pass to the limit by means of compactness and monotonicity arguments. In view of these considerations, let us state here a preliminary result (whose proof will be given in Section 4).…”

“…Let us note that within the small perturbations assumption the entropy balance and the classical heat equation are equivalent in mechanical terms (cf. [6,7]). …”

“…In that case the proofs mainly rely on the fact that β 1 + β 2 = 1; hence the mass balance equation as well as the effects of the pressure can be neglected and so the system reduces to the coupling of equation (1.2) (with only one phase variable, say χ = β 1 − β 2 ) and (1.3). More in particular, in [6] the model coupling these two equations and accounting also for some memory effects in (1.2) is studied and an existence (of a weak solution) result is proved in case ϕ = I [0,1] , while uniqueness still remains an open problem. In [7] and [4,5] the uniqueness of a solution is established and also some investigation on the long-time behaviour of solutions is performed respectively in case ν = 0 and without any thermal memory for a system in which a different heat flux law in (1.2) (leading to an internal energy balance with ϑ instead of log ϑ inside the Laplacian) is taken into account.…”

Convergence of phase field to phase relaxation models governed by an entropy equation with memory

“…Moreover, we quote References [2,4,5], where the authors study a PDE system analogous to (1)-(2) with a different heat flux law (in particular without memory terms) and a linear growth for . The above papers deal with well-posedness and long time behaviour for ε 0.…”

“…In Reference [21] the authors treat the same system assuming k 0 = 0 and , to be linear, but allowing more convolution contributions in Equation (1) and a general maximal monotone graph in (2).…”