Global solvability of a dissipative Frémond model for shape memory alloys. I. Mathematical formulation and uniqueness
Author:
Elena Bonetti
Journal:
Quart. Appl. Math. 61 (2003), 759-781
MSC:
Primary 74N99; Secondary 35K85, 35Q72, 74H20
DOI:
https://doi.org/10.1090/qam/2019622
MathSciNet review:
MR2019622
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Abstract: The mathematical formulation of a dissipative Frémond model for shape memory alloys is given in terms of an initial and boundary values problem. Uniqueness of sufficiently regular solutions is proved by use of a contracting estimates procedure in the case when quadratic dissipative contributions are neglected in the energy balance. The related existence result is only established while its proof will be detailed by the author in a subsequent paper.
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M. Frémond, Shape memory alloys. A thermomechanical model, in "Free Boundary Problems: theory and applications", vol. I–II (K. H. Hoffmann and J. Sprekels, eds.), Pitman Res. Notes Math. Ser. 185, Longman, London, 1990
M. Frémond, Sur l’inégalité de Clausius-Duhem, C. R. Acad. Sci. Sér. II Paris, 311, 757–762 (1990)
M. Frémond, The principle of virtual power in solid mechanics, in: Continuum thermomechanics: the art and the science of modelling material behavior (Paul Germain anniversary vol.), Kluwer Acad. Press, Boston, 2000
M. Frémond, Nonsmooth thermo-mechanics, Springer-Verlag, Heidelberg, 2001
M. Frémond and S. Miyazaki, Shape memory alloys, in: CISM Courses and Lectures No. 351, Springer-Verlag, New York, 1996
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J. J. Moreau, Sur les lois de frottement, de viscosité et de plasticité, in: C. R. Acad. Sci., 27, 608–611, Paris, 1970
I. Müller, Pseudoelasticity in shape memory alloys, An extreme case of thermoelasticity, in: Proc. termoelasticità finita, Acc. Naz. dei Lincei, 1985
I. Müller and K. Wilmanski, A model for phase transitions in pseudoelastic bodies, Nuovo Cimento B, 57, 283–318 (1980)
M. Niezgódka and J. Sprekels, Existence of a solution for a mathematical model of structural phase transition in shape memory alloys, Math. Methods Appl. Sci., 10, 197–223 (1988)
E. Patoor and M. Berveiller, Les alliages à mémoire de forme, Hermàs, Paris, 1990
- N. Shemetov, Existence result for the full one-dimensional Frémond model of shape memory alloys, Adv. Math. Sci. Appl. 8 (1998), no. 1, 157–172. MR 1623322
- Jürgen Sprekels, Shape memory alloys: mathematical models for a class of first order solid-solid phase transitions in metals, Control Cybernet. 19 (1990), no. 3-4, 287–308 (1991). Optimal design and control of structures (Jabłonna, 1990). MR 1118688
M. Achenback and I. Müller, Model for shape memory, J. Physique, C${_4}$ Suppl. 12, 12, 163–167 (1982)
C. Baiocchi, Sulle equazioni differenziali astratte del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl., (4), 76, 233–304 (1967)
V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff, Leyden, 1976
V. Barbu, P. Colli, G. Gilardi, and M. Grasselli, Existence, uniqueness, and long time behavior for a nonlinear Volterra integrodifferential equation, Differential Integral Equations, 13, 1233–1262 (2000)
D. Blanchard, M. Frémond, and A. Visintin, Phase change with dissipation, in: Thermomechanical in coupling solids, H. D. Bui and Q. S. Nguyen eds., North Holland, 1987, 411–418
E. Bonetti, Global solution to a Frémond model for shape memory alloys with thermal memory, Nonlinear Anal., 46, 535–565 (2001)
E. Bonetti, Global solution to a nonlinear phase transition model with dissipation, Adv. Math. Sci. Appl., 12, 355–376 (2002)
N. Chemetov, Uniqueness results for the full Frémond model of shape memory alloys, Z. Anal. Anwendungen, 17, 877–892 (1998)
P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity, in: Studies in Mathematics and its Applications, North-Holland, Amsterdam, 1988
P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys, Nonlinear Anal., 24, 1565–1579, (1995)
P. Colli, M. Frémond, and A. Visintin, Thermo-mechanical evolution of shape memory alloys, Quart. Appl. Math., 48, 31–47 (1990)
P. Colli and J. Sprekels, Positivity of temperature in the general Frémond model for shape memory alloys, Contin. Mech. Thermodyn., 5, 255–264 (1993)
P. Colli and J. Sprekels, Global solution to the full one-dimensional Frémond model for shape memory alloys, Math. Methods Appl. Sci., 18, 371–385 (1995)
P. Colli and J. Sprekels, Remarks on the existence for the one-dimensional Frémond model of shape memory alloys, Z. Angew. Math. Mech., 76, Suppl. 2, 413–416 (1996)
G. Duvaut and J. L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin-New York, 1976
F. Falk, Elastic phase transitions and nonconvex energy functions, in: "Free Boundary Problems: theory and applications", vol. I–II (K. H. Hoffmann and J. Sprekels, eds.); Pitman Res. Notes Math. Ser. 185, Longman, London, 1990
M. Frémond, Matériaux a mémoire de forme, C. R. Acad. Sci. Paris. Sér. II Méc. Phys. Chim. Sci. Univers. Sci. Terre, 304, 239–244 (1987)
M. Frémond, Shape memory alloys. A thermomechanical model, in "Free Boundary Problems: theory and applications", vol. I–II (K. H. Hoffmann and J. Sprekels, eds.), Pitman Res. Notes Math. Ser. 185, Longman, London, 1990
M. Frémond, Sur l’inégalité de Clausius-Duhem, C. R. Acad. Sci. Sér. II Paris, 311, 757–762 (1990)
M. Frémond, The principle of virtual power in solid mechanics, in: Continuum thermomechanics: the art and the science of modelling material behavior (Paul Germain anniversary vol.), Kluwer Acad. Press, Boston, 2000
M. Frémond, Nonsmooth thermo-mechanics, Springer-Verlag, Heidelberg, 2001
M. Frémond and S. Miyazaki, Shape memory alloys, in: CISM Courses and Lectures No. 351, Springer-Verlag, New York, 1996
M. Frémond and A. Visintin, Dissipation dans le changement de phase. Surfusion. Changement de phase irréversible, C. R. Acad. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 301, 1265–1268 (1985)
P. Germain, Cours de mécanique des milieux continus, Masson, Paris, 1973
P. Germain, La méthode des puissances en mécanique des milieux continus-1$^{0}$ partie, Téorie du second gradient, J. Mécanique, 12, 235–274 (1973)
G. Guénin, Alliages à memoire de forme, Techniques de l’ingénieur, M530, Paris, 1986
J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I, Springer-Verlag, Berlin, 1972
J. J. Moreau, Sur les lois de frottement, de viscosité et de plasticité, in: C. R. Acad. Sci., 27, 608–611, Paris, 1970
I. Müller, Pseudoelasticity in shape memory alloys, An extreme case of thermoelasticity, in: Proc. termoelasticità finita, Acc. Naz. dei Lincei, 1985
I. Müller and K. Wilmanski, A model for phase transitions in pseudoelastic bodies, Nuovo Cimento B, 57, 283–318 (1980)
M. Niezgódka and J. Sprekels, Existence of a solution for a mathematical model of structural phase transition in shape memory alloys, Math. Methods Appl. Sci., 10, 197–223 (1988)
E. Patoor and M. Berveiller, Les alliages à mémoire de forme, Hermàs, Paris, 1990
N. Shemetov, Existence result for the full one-dimensional Frémond model of shape memory alloys, Adv. Math. Sci. Appl., 8, 157–172 (1998)
J. Sprekels, Shape memory alloys: mathematical models for a class of first order solid-solid phase transitions in metals, Control Cybernet., 19, 287–308 (1990)
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