In this paper we analyze a PDE system modeling (nonisothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as "entropic," where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics, as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time-discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its entropic formulation) and of the a priori estimates performed on it. Our time-discrete analysis could be useful toward the numerical study of this model.

"Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage / E. Rocca, R. Rossi. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 47:4(2015), pp. 2519-2586.

"Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage

E. Rocca
Primo
;
2015

Abstract

In this paper we analyze a PDE system modeling (nonisothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as "entropic," where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics, as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time-discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its entropic formulation) and of the a priori estimates performed on it. Our time-discrete analysis could be useful toward the numerical study of this model.
Damage; Global-in-time weak solutions; Phase transitions; Thermoviscoelasticity; Time-discretization; Analysis; Applied Mathematics; Computational Mathematics
Settore MAT/05 - Analisi Matematica
2015
Article (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/345085
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