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The paper surveys mathematical models of thermo-mechanical evolution of shape memory alloys and related mathematical results. The survey is confined to so-called diffused-interface or phase-field models based on Landau-Ginzburg free energy as a ther-modynamic potential. It includes the well-known models due to Falk, Fremond and Fried-Gurtin. The focus is on a three-dimensional (3-D) generalization of Falk's model based on the linearized strain tensor, absolute temperature and strain tensor gradient. For such model the thermodynamical basis and the recent mathematical results on its well-posedness are presented.
Czasopismo
Rocznik
Tom
Strony
629--658
Opis fizyczny
Bibliogr. 84 poz.
Twórcy
autor
- Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
autor
- current address: ICM Warsaw University, Pawinskiego 5a, 02-106 Warsaw, Poland permanent address (on leave): Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-950 Warsaw, Poland Institute of Mathematics and Operations Research, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland
Bibliografia
- Aiki, T. (2000) Weak solutions for Falk’s model of shape memory alloys. Math. Meth. Appl. Sci., 23, 299–319.
- Aiki, T. and Kenmochi, N. (2001) Some models for shape memory alloys. In: Kenmochi, N. Niezgodka, M., Otani, M., eds., Mathematical Aspects of Modelling Structure Formation Phenomena, Gakuto International Series, Mathematical Sciences and Applications, 17, Gakkotosho, Tokyo, 144– 162.
- Alt, H. W. and Paw low, I. (1996) On the entropy principle of phase transition models with a conserved order parameter. Adv. Math. Sci. Appl., 6, 291–376.
- Andrews, G. (1980) On the existence of solutions to the equation utt = uxxt + σ(ux)x. J. Differential Eqs., 35, 200-231.
- Balandraud, X., Ernst, E. and Soos, E. (2000) Relaxation and creep phenomena in shape memory alloys. Part I: Hysteresis loop and pseudoelastic behavior. Z. angew. Math. Phys., 51, 171–203.
- Barsch, G. R. and Krumhansl, J. A. (1984) Twin boundaries in ferroelastic media without interface dislocations. Phys. Rev. Letters, 53, 11, 1069–1072.
- Barsch, G. R. and Krumhansl, J. A. (1988) Nonlinear and nonlocal continuum model of transformation precursors in martensites. Metall. Trans., 19 A, 4, 761–775.
- Bernardini, D. (2001) On the macrosocopic free energy functions for shape memory alloys. Journal of the Mechanics and Physics of Solids, 49, 813– 837.
- Bonetti, E. (2001) Global solution to Fremond model for shape memory alloys with thermal memory. Nonlinear Anal., 46, 535–565
- Bonetti, E. (2002) Asymptotic analysis of a diffusive model for shape memory alloys with Cattaneo-Maxwell heal flux law. Differential Integral Equations, 15, 527–566.
- Brokate, M. and Sprekels, J. (1991) Optimal control of thermomechanical phase transitions in shape memory alloys: Necessary conditions of optimality. Math. Meth. Appl. Sci., 14, 265–280.
- Brokate, M. and Sprekels, J. (1996) Hysteresis and Phase Transitions. Appl. Math. Sci., 121, Springer, New York.
- Bubner, N., Sokolowski, J. and Sprekels, J. (1998) Optimal boundary control problems for shape memory alloys under state constraints for stress and temperature. Numer. Funct. Anal. Optimiz., 19, 5–6, 489–498.
- Chemetov, N. (1988) Uniqueness results for the full Fremond model of shape memory alloys. Z. Anal. Anwendungen, 17, 4, 877–892.
- Chen, Z. and Hoffmann, K.-H. (1994) On a one-dimensional nonlinear thermoviscoelastic model for structural phase transitions in shape memory alloys. J. Diff. Eqs., 112, 325–350.
- Colli, P. (1995) Global existence for the three-dimensional Fremond model of shape memory alloys. Nonlinear Analysis, 24, 11, 1565–1579.
- Colli, P., Fremond, M. and Visintin, A. (1990) Thermo-mechanical evolution of shape memory alloys. Quart. Appl. Math., XLVIII, 1, 31–47.
- Colli, P., Laurencot, P. and Stefanelli, U. (2000) Long-time behavior for the full one-dimensional Fremond model of shape memory alloys. Contin. Mech. Thermodyn., 12, 6, 423–433.
- Colli, P. and Sprekels, J.(1992) Global existence for a three-dimensional model for the thermo-mechanical evolution of shape memory alloys. Nonlinear Anal., 18, 873–888.
- Colli, P. and Sprekels, J. (1993) Positivity of temperature in the general Fremond model for shape memory alloys. Continuum Mech. Thermodyn., 5, 255–264.
- Colli, P. and Sprekels, J. (1995) Global solution to the full one-dimensional Fremond model for shape memory alloys. Math. Methods Appl. Sci., 18, 371–385.
- Ericksen, J. L. (1986) Constitutive theory for some constrained elastic crystals. Int. J. Solids and Structures, 22, 9, 951–964.
- Falk, F. (1980) Model free energy, mechanics and thermodynamics of shape memory alloys. Acta Metall., 28, 1773–1780.
- Falk, F. (1982) Landau theory and martensitic phase transitions. J. Phys., C4, 43, 3–15.
- Falk, F. (1983) One-dimensional model of shape memory alloys. Arch. Mech., 35, 63–84.
- Falk, F. (1990) Elastic phase transitions and nonconvex energy functions. In: Hoffmann, K.-H., Sprekels, J., eds., Free Boundary Problems: Theory and Applications, vol. I, Pitman Research Notes Math. Ser. 185, Longman, 45–59.
- Falk, F. and Konopka, P. (1990) Three-dimensonal Landau theory describing the martensitic phase transformation of shape memory alloys. J. Phys.: Condens. Matter, 2, 61–77.
- Fremond, M. (1987) Materiaux ´a memoire de forme. C. R. Acad. Sci Paris, Ser. II Mec. Phys. Chim. Sci. Univers Sci. Terre, 304, 7, 239–244.
- Fremond, M. (1990) Shape memory alloys. A thermomechanical model. In: Hoffmann, K.-H., Sprekels, J., eds., Free Boundary Problems: Theory and Applications. Pitman Research Notes Math. Ser., 185, Longman, London, 295–306.
- Fremond, M. (2002) Non-Smooth Thermomechanics. Springer, Berlin.
- Fremond, M. and Miyazaki, S. (1996) Shape Memory Alloys. CISM Courses and Lecture, 351, Springer.
- Fried, E. and Grach, G. (1997) An order-parameter-based theory as a regularization of a sharp-interface theory for solid-solid phase transitions. Arch. Rational Mech. Anal., 138, 355–404.
- Fried, E. and Gurtin, M. E. (1993) Continuum theory of thermally induced phase transitions based on an order parameter. Physica D, 68, 326–343.
- Fried, E. and Gurtin, M. E. (1994) Dynamic solid-solid transitions with phase characterized by an order parameter. Physica D, 72, 287–308.
- Fried, E. and Gurtin, M. E. (1999) Coherent solid-state phase transitions with atomic diffusion: A thermomechanical treatment. Journal of Statistical Physics, 95, 5–6, 1361–1427.
- Gurtin, M. E. (1996) Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D, 92, 178–192.
- Hoffmann, K.-H., Niezgodka, M. and Zheng, S. (1990) Existence and uniqueness of global solutions to an extended model of the dynamical developments in shape memory alloys. Nonlinear Analysis, 15, 10, 977-990.
- Hoffmann, K.-H. and Sprekels, J. (1987) Phase transitions in shape memory alloys I: Stability and optimal control. Numer. Funct. Anal. Optimiz., 9, 7-8, 743–760.
- Hoffmann, K.-H. and Tiba, D. (1997) Control of a plate with nonlinear shape memory alloy reinforcement. Adv. Math. Sci. Appl., 7, 1, 427–436.
- Hoffmann, K.-H. and Zheng, S. (1988) Uniqueness for structural phase transitions in shape memory alloys. Math. Methods Appl. Sci., 10, 145–151.
- Hoffmann, K.-H. and Zochowski, A. (1992) Existence of solutions to some nonlinear thermoelastic systems with viscosity. Math. Meth. Appl. Sci., 15, 187–204.
- Hoffmann, K.-H. and Zochowski A. (1992) Analysis of the thermoelastic model of a plate with non-linear shape memory alloy reinforcements. Math. Meth. Appl. Sci., 15, 631–645.
- Hoffmann, K.-H. and Zochowski, A. (1998) Control of the thermoelastic model of a plate activated by shape meory alloy reinforcements. Math. Meth. Appl. Sci., 21, 7, 589–603.
- Kloucek, P. and Luskin, M. (1994) The computation of the dynamics of the martensitic transformation. Continuum Mech. Thermodyn., 6, 209– 240.
- Ladyzhenskaya, O.A., Solonnikov, V.A. and Uraltseva, N.N. (1967) Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow (in Russian).
- Morin, P. and Spies, R. D. (1997) Identifiability of the Landau-Ginzburg potential in a mathematical model of shape memory alloys. J. Math. Anal. Appl., 212, 292–315.
- Muller, I. and Seelecke, S. (2002) Thermodynamic aspects of shape memory alloys. Topics in the mathematical modelling of smart materials. Math. Comput. Modelling, 34, 12–13, 1307–1355.
- Niezgodka, M. and Sprekels, J. (1988) Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys. Math. Meth. Appl. Sci., 10, 197–223.
- Niezgodka, M., Zheng, S. and Sprekels, J. (1988) Global solutions to a model of structural phase transitions in shape memory alloys. J. Math. Anal. Appl., 130, 39–54.
- Pagano, S., Alart, P. and Maisonneuve, O. (1998) Solid-solid phase transition modelling. Local and global minimizations of non-convex and relaxed potentials. Isothermal case for shape memory alloys. Internat. J. Engrg. Sci., 36, 10, 1143–1172.
- Pawlow, I. (2000a) Thermodynamically consistent models for media with microstructures. Adv. Math. Sci. Appl., 10, 1, 265–303.
- Pawlow, I. (2000b) Three-dimensional model of thermomechanical evolution of shape memory materials. Control Cybernet., 29, 1, 341–365.
- Pawlow, I. and Zajączkowski, W. M. (2002a) Unique global solvability in two-dimmensional nonlinear thermoelasticity. Submitted.
- Pawlow, I. and Zajączkowski, W. (2002b) Global existence to a three-dimensional nonlinear thermoelasticity system arising in shape memory materials. Submitted.
- Pawlow, I. and Zochowski, A. (2000) Nonlinear thermoelastic system with viscosity and nonlocality. In: Kenmochi, N., ed., I Proceedings Free Boundary Problems. Theory and Applications I. Gakuto Internat. Ser. Math. Sci. Appl., 13, 251–265.
- Pawlow, I. and Zochowski, A. (2002a) Existence and uniqueness of solutions for a three-dimensional thermoelastic system. Dissertationes Mathematicae, 406.
- Pawlow, I. and Zochowski, A. (2002b) Control problem for a nonlinear thermoelasticity system. To appear in Math. Methods Appl. Sci.
- Racke, R. and Zheng, S. (1997) Global existence and asymptotic behavior in nonlinear thermoviscoelasticity. J. Diff. Eqs., 134, 46–67.
- Roubıcek (1999) Dissipative evolution of microstructure in shape memory alloys. In: Bungartz, H. J., Hoppe, R. W., Zenger, C., eds., Lectures on Applied Mathematics. Springer, Berlin, 45–63
- Rybka, P. (1992) Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions. Proc. Roy. Soc. Edinburgh, 121 A, 101–138.
- Rybka, P. (1997) The viscous damping prevents propagation of singularities in the system of viscoelasticity. Proc. Roy. Soc. Edinburgh, 127 A, 1067–1074.
- Shemetov, N. (1998) Existence result for the full one-dimensional Fremond model of shape memory alloys. Adv. Math. Sci. Appl., 1, 8, 157–172.
- Shen, W. and Zheng, S. (1993) On the coupled Cahn-Hilliard equations. Comm. Partial Differential Equations, 18, 701–727.
- Shen, W., Zheng, S. and Zhu, P. (1999) Global existence and asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions. Quart. Appl. Math., LVII, 1, 93–110.
- Sikora, J., Cusumano, J. P. and Jester, W. A. (1998) Spatially periodic solutions in a 1D model of phase transitions with order parameter. Physica D, 121, 275–294.
- Silhavy, M. (1985) Phase transitions in non-simple bodies. Arch. Rational Mech. Anal., 88, 2, 135–161.
- Sokolowski, J. and Sprekels, J. (1994) Control problems with state constraints for shape memory alloys. Math. Meth. Appl. Sci., 17, 943–952.
- Spies, R. D. (1994) Results on a mathematical model of thermomechanical phase transitions in shape memory materials. Smart Mater. Struct., 3, 459–469.
- Spies, R. D. (1995) A state-space approach to a one-dimensional mathematical model for the dynamic of phase transitions in pseudoelastic materials. J. Math. Anal. Appl., 190, 58–100.
- Sprekels, J. (1989a) Global existence for thermomechanical processes with noncouvex free energies of Ginzburg-Landau form. J. Math. Anal. Appl., 141, 333–348.
- Sprekels,J.(1989b) Stability and optimal control of thermomechanical processes with nonconvex free energies of Ginzburg-Landau type. Math. Meth. Appl. Sci., 11, 687–696.
- Sprekels, J. (1990) Shape memory alloys: Mathematical models for a class of first order solid-solid phase transitions in metals. Control Cybernet., 19, 287–308.
- Sprekels, J. and Zheng, S. (1989) Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys. Physica D, 39, 59–76.
- Sprekels, J. and Zheng, S. (1998) Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys. Physica D, 121, 252–262.
- Sprekels, J., Zheng, P. and Zhu, P. (1998) Asymptotic behavior of the solutions to a Landau-Ginzburg system with viscosity for marterisitic phase transitions in shape memory alloys. SIAM J. Math. Anal., 29, 1, 69–84.
- Swart, P. J. and Holmes, P. J. (1992) Energy minimization and the formation of microstructure in dynamic anti-plane shear. Arch. Rational Mech. Anal., 121, 37–85.
- Timofte, A. and Timofte, V. (2001a) Uniqueness theorem for a thermomechanical model of shape memeory alloys. Math. Mech. Solids, 6, 4, 447–466.
- Timofte, A. and Timofte, V. (2001b) Existence theorem for a thermomechanical model of shape memory alloys. Math. Mech. Solids, 6, 5, 541– 545.
- Qin, Y. (2001) Global existence and asymptotic behaviour of the solution to the system in one-dimensional nonlinear thermoviscoelasticity. Quart. Appl. Math., LIX, 1, 113–142.
- Watson, S. J. (2000a) Unique global solvability for initial-boundary value problems in one-dimensional nonlinear thermoviscoelasticity. Arch. Rational Mech. Anal., 153, 1–37.
- Watson, S. J. (2000b) A priori bounds in one-dimensional nonlinear thermoviscoelasticity. Contemporary Mathematics, 255, 229–238.
- Zheng, S. (1989) Global solutions to thermomechanical equations with nonconvex Landau-Ginzburg free energy. J. Appl. Math. Phys. (ZAMP), 40, 111–127.
- Zheng, S. (1995) Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems. Pitman Monographs and Surveys in Pure and Applied Mathematics, 76, Longman.
- Zochowski, A. (1992) ˙ Mathematical Problems in Shape Optimization and Shape Memory Materials. Methoden Verfahren Math. Physik, 38, Peter Lang Verlag
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Bibliografia
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