×

zbMATH — the first resource for mathematics

Positivity of the temperature for phase transitions with micro-movements. (English) Zbl 1116.80015
In this paper the thermodynamic consistency of a class of solid liquid phase transition models proposed by Fremond is studied. The derivation of the model based on the two scale, micro and macro, modeling is reviewed in some detail. The thermodynamic consistency of these models is proved to be equivalent with the temperature positively. Under rater weak assumptions on the data it is proved that the temperature is separated uniformly by zero almost everywhere. The proof is based on a uniform bound for the inverse temperature. The result is quite general and may be applied to all dissipative models of that type.

MSC:
80A22 Stefan problems, phase changes, etc.
35B50 Maximum principles in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bonfanti, G.; Frémond, M.; Luterotti, F., Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements, Nonlinear anal. real world appl., 5, 1, 123-140, (2004) · Zbl 1092.80006
[2] Bonfanti, G.; Luterotti, F., Convergence results for a phase transition model with vanishing microscopic acceleration, Math. models methods appl. sci., 14, 3, 375-392, (2004) · Zbl 1076.80002
[3] Caginalp, G., An analysis of a phase field model of a free boundary, Arch. rational mech. anal., 92, 3, 205-245, (1986) · Zbl 0608.35080
[4] Colli, P.; Gilardi, G.; Grasselli, M., Well-posedness of the weak formulation for the phase-field model with memory, Adv. differential equations, 2, 3, 487-508, (1997) · Zbl 1023.45501
[5] Colli, P.; Luterotti, F.; Schimperna, G.; Stefanelli, U., Global existence for a class of generalized systems for irreversible phase changes, Nonlinear differential equations appl., 9, 3, 255-276, (2002) · Zbl 1004.35061
[6] Colli, P.; Sprekels, J., Global solution to the penrose – fife phase-field model with zero interfacial energy and Fourier law, Adv. math. sci. appl., 9, 1, 383-391, (1999) · Zbl 0930.35036
[7] Frémond, M., Non-smooth thermomechanics, (2002), Springer Berlin · Zbl 0990.80001
[8] Frémond, M.; Visintin, A., Dissipation dans le changement de phase, surfusion, changement de phase irréversible, C. R. acad. sci. Paris Sér. II Méc. phys. chim. sci. univers sci. terre, 301, 18, 1265-1268, (1985) · Zbl 0582.73007
[9] Fried, E.; Sellers, S., Microforces and the theory of solute transport, Z. angew. math. phys., 51, 5, 732-751, (2000) · Zbl 0985.76082
[10] Germain, P., Mécanique des milieux continus, (1973), Masson Paris · Zbl 0254.73001
[11] Gurtin, M.E., Generalized ginzburg – landau and cahn – hilliard equations based on a microforce balance, Phys. D, 92, 3-4, 178-192, (1996) · Zbl 0885.35121
[12] Gurtin, M.E., On the plasticity of single crystals: free energy, microforces, plastic-strain gradients, J. mech. phys. solids, 48, 5, 989-1036, (2000) · Zbl 0988.74021
[13] Laurençot, Ph.; Schimperna, G.; Stefanelli, U., Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. math. anal. appl., 271, 2, 426-442, (2002) · Zbl 1004.35058
[14] J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. 1, Springer, New York, Heidelberg, 1972. · Zbl 0227.35001
[15] Luterotti, F.; Schimperna, G.; Stefanelli, U., Global solution to a phase field model with irreversible and constrained phase evolution, Q. appl. math., 60, 2, 301-316, (2002) · Zbl 1032.35109
[16] Luterotti, F.; Schimperna, G.; Stefanelli, U., A generalized phase relaxation model with hysteresis, Nonlinear anal., 55, 4, 381-398, (2003) · Zbl 1028.35005
[17] Errata and Addendum, Z. Anal. Andwendungen 22 (1) (2002) 239-240. · Zbl 1003.80003
[18] Miranville, A., A model of cahn – hilliard equation based on a microforce balance, C. R. acad. sci. Paris Sér. I math., 328, 12, 1247-1252, (1999) · Zbl 0932.35118
[19] A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math. (4) (2003) 165-185. · Zbl 1031.35003
[20] A. Miranville, G. Schimperna, Nonisothermal phase separation based on a microforce balance, Discrete Contin. Dyn. Syst. Ser. B 5(3) (2005) 753-768. · Zbl 1140.80388
[21] Moreau, J.-J., Sur LES lois de frottement, de viscosité et plasticité, C. R. acad. sci. Paris Sér. II Méc. phys. chim. sci. univers sci. terre, 271, 608-611, (1970)
[22] Penrose, O.; Fife, P.C., Thermodynamically consistent models of phase field type for the kinetics of phase transitions, Phys. D, 43, 44-62, (1990) · Zbl 0709.76001
[23] Penrose, O.; Fife, P.C., On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model, Phys. D, 69, 107-113, (1993) · Zbl 0799.76084
[24] G. Schimperna, F. Luterotti, U. Stefanelli, Local solution to Frémond’s full model for irreversible phase transitions, in: Mathematical Models and Methods for Smart Materials, Cortona, 2001, Series on Advances on Mathematical and Applied Sciences, vol. 62, World Scientific Publishing, River Edge, NJ, 2002, pp. 323-328. · Zbl 1049.35096
[25] Visintin, A., Stefan problem with phase relaxation, IMA J. appl. math., 34, 3, 225-246, (1985) · Zbl 0585.35053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.