A rate-independent model for permanent inelastic effects in shape memory materials

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March 2011, 6(1): 145-165. doi: 10.3934/nhm.2011.6.145

A rate-independent model for permanent inelastic effects in shape memory materials

Michela Eleuteri ^1, , Luca Lussardi ^2, and Ulisse Stefanelli ^3,

1.	Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38100 Povo (Trento), Italy
2.	Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany
3.	Istituto di Matematica Applicata e Tecnologie Informatiche – CNR, Via Ferrata 1, 27100 Pavia

Received May 2010 Revised October 2010 Published March 2011

Abstract
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This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape memory polycrystalline materials in presence of permanent inelastic effects. The basic features of the model are recalled and the constitutive and the three-dimensional quasi-static evolution problem are proved to be well-posed. Finally, we discuss the convergence of the model to reduced/former ones by means of a rigorous $\Gamma$-convergence analysis.

Keywords: Shape memory materials, $\Gamma$-convergence., well-posedness, permanent inelastic effects.

Mathematics Subject Classification: Primary: 74C05; 35K8.

Citation: Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks and Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145

References:

[1]	T. Aiki, A model of 3D shape memory alloy materials, J. Math. Soc. Japan, 57 (2005), 903-933. doi: 10.2969/jmsj/1158241940.
[2]	S. Antman, J. L. Ericksen, D. Kinderlehrer and I. Müller, "Metastability and Incompletely Posed Problems," in the IMA Volumes in Mathematics and its Applications, 3, Springer-Verlag, New York, 1987.
[3]	M. Arndt, M. Griebel and T. Roubíček, Modelling and numerical simulation of martensitic transformation in shape memory alloys, Contin. Mech. Thermodyn., 15 (2003), 463-485. doi: 10.1007/s00161-003-0127-3.
[4]	M. Arrigoni, F. Auricchio, V. Cacciafesta, L. Petrini and R. Pietrabissa, Cyclic effects in shape-memory alloys: A one-dimensional continuum model, Journal de Physique IV, 11 (2001), 577-582.
[5]	F. Auricchio, A. Mielke and U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials, Math. Models Meth. Appl. Sci., 18 (2008), 125-164. doi: 10.1142/S0218202508002632.
[6]	F. Auricchio and L. Petrini, Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations, Internat. J. Numer. Meth. Engrg., 55 (2002), 1255-1284. doi: 10.1002/nme.619.
[7]	F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations: Solution algorithm and boundary value problems, Internat. J. Numer. Meth. Engrg., 61 (2004), 807-836. doi: 10.1002/nme.1086.
[8]	F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity, Int. J. Plasticity, 23 (2007), 207-226. doi: 10.1016/j.ijplas.2006.02.012.
[9]	F. Auricchio, A. Reali and U. Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1631-1637. doi: 10.1016/j.cma.2009.01.019.
[10]	F. Auricchio and E. Sacco, A one-dimensional model for superelastic shape-memory alloys with different elastic properties between austenite and martensite, Int. J. Non-Linear Mech., 32 (1997), 1101-1114. doi: 10.1016/S0020-7462(96)00130-8.
[11]	F. Auricchio, R. L. Taylor and J. Lubliner, Shape-memory alloys: Macromodelling and numerical simulations of the superelastic behaviour, Comput. Mech. Appl. Mech. Engrg., 146 (1997), 281-312. doi: 10.1016/S0045-7825(96)01232-7.
[12]	A.-L. Bessoud and U. Stefanelli, A three-dimensional model for magnetic shape memory alloys, Math. Models Meth. Appl. Sci. (2010) to appear.
[13]	Z. Bo and D. C. Lagoudas, Thermomechanical modeling of polycrystalline SMAs under cyclic loading. Part III: Evolution of plastic strains and two-way shape memory effect, Int. J. Eng. Sci., 37 (1999), 1175-1203. doi: 10.1016/S0020-7225(98)00115-3.
[14]	M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,'' vol. 121 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996.
[15]	N. Chemetov, Well-posedness of two-shape memory model, Math. Methods Appl. Sci., 29 (2006), 209-233. doi: 10.1002/mma.672.
[16]	P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys, Nonlinear Anal., 24 (1995), 1565-1579. doi: 10.1016/0362-546X(94)00097-2.
[17]	P. Colli and J. Sprekels, Global existence for a three-dimensional model for the thermodynamical evolution of shape memory alloys, Nonlinear Anal., 18 (1992), 873-888. doi: 10.1016/0362-546X(92)90228-7.
[18]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence,'' Birkhäuser-Boston, 1993.
[19]	G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics,'' Springer-Berlin, 1976.
[20]	F. Falk, Martensitic domain boundaries in shape-memory alloys as solitary waves, J. Phys. C4 Suppl., 12 (1982), 3-15.
[21]	F. Falk and P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape-memory alloys, J. Phys. Condens. Matter, 2 (1990), 61-77. doi: 10.1088/0953-8984/2/1/005.
[22]	M. Frémond, Matériaux à mémoire de forme, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 304 (1987), 239-244.
[23]	M. Frémond, "Non-Smooth Thermomechanics,'' Springer-Verlag, 2002.
[24]	M. Frémond and S. Miyazaki, "Shape Memory Alloys,'' Springer-Verlag, 1996.
[25]	E. Fried and M. E. Gurtin, Dynamic solid-solid transitions with phase characterized by an order parameter, Phys. D, 72 (1994), 287-308. doi: 10.1016/0167-2789(94)90234-8.
[26]	S. Govindjee and C. Miehe, A multi-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Methods Appl. Mech. Engrg., 191 (2001), 215-238. doi: 10.1016/S0045-7825(01)00271-7.
[27]	S. Govindjee and E. P. Kasper, A shape memory alloy model for Uranium-Niobium accounting for plasticity, J. Intell. Mater. Syst. Struct., 8 (1997), 815-823. doi: 10.1177/1045389X9700801001.
[28]	D. Helm and P. Haupt, Shape memory behaviour: Modelling within continuum thermomechanics, Internat. J. Solids Structures, 40 (2003), 827-849. doi: 10.1016/S0020-7683(02)00621-2.
[29]	K. H. Hoffmann, M. Niezgódka and S. Zheng, Existence and uniqueness to an extended model of the dynamical developments in shape memory alloys, Nonlinear Anal., 15 (1990), 977-990. doi: 10.1016/0362-546X(90)90079-V.
[30]	Y. Huo, I. Müller and S. Seelecke, Quasiplasticity and pseudoelasticity in shape memory alloys, in Phase transitions and hysteresis, in Lecture Notes in Math., Vol. 1584, eds. M. Brokate et al (Springer 1994), 87-146.
[31]	P. Krejčí and U. Stefanelli, Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires, Math. Mech. Solids (2010), to appear.
[32]	P. Krejčí and U. Stefanelli, Well-posedness of a thermo-mechanical model for shape memory alloys under tension, M2AN Math. Model. Anal. Numer., (2010), to appear.
[33]	M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructures and its evolution in shape-memory-alloy single cristals, in particular in CuAlNi, Meccanica, 40 (2005), 389-418. doi: 10.1007/s11012-005-2106-1.
[34]	M. Kružík and J. Zimmer, A model of shape memory alloys accounting for plasticity, IMA J. Appl. Math., (2010), to appear.
[35]	D. C. Lagoudas, P. B. Entchev, P. Popov, E. Patoor, L. C. Brinson and X. Gao, Shape memory alloys, Part II: Modeling of polycrystals, Mech. Mater., 38 (2006), 391-429. doi: 10.1016/j.mechmat.2005.08.003.
[36]	D. C. Lagoudas and P. B. Entchev, Modeling of transformation-induced plasticity and its effect on the behavior of porous shape memory alloys. Part I: Constitutive model for fully dense SMAs, Mech. Mater., 36 (2004), 865-892.
[37]	A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.
[38]	G. A. Maugin, "The Thermomechanics of Plasticity and Fracture,'' Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1992.
[39]	A. Mielke, Evolution of rate-independent systems, In C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Elsevier, (2005), 461-559.
[40]	A. Mielke, L. Paoli and A. Petrov, On existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys, SIAM J. Math. Anal., 41 (2009), 1388-1414. doi: 10.1137/080726215.
[41]	A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for discretizations of a rate-independent variational inequality, SIAM J. Numer. Anal., 48 (2010), 1625-1646. doi: 10.1137/090750238.
[42]	A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error control for space-time discretizations of a 3D model for shape-memory materials, Proc. of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (Bochum 2008), IUTAM Bookseries, Springer, 2009.
[43]	A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl., 17 (2007), 160-182.
[44]	A. Mielke and T. Roubíček, A rate independent model for inelastic behaviour of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597. doi: 10.1137/S1540345903422860.
[45]	A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.
[46]	A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, Proc. of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, eds. H.-D Alber, R. Balean and R. Farwig (Shaker-Verlag, 1999), 117-129.
[47]	A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rational Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194.
[48]	I. Müller, Thermodynamics of ideal pseudoelasticity, J. Phys. IV, C2-5 (1995), 423-431.
[49]	A. Paiva, M. A. Savi, A. M. B. Braga and P. M. C. L. Pacheco, A constitutive model for shape memory alloys considering tensile-compressive asymmetry and plasticity, Internat. J. of Solid. Struct., 42 (2005), 3439-3457. doi: 10.1016/j.ijsolstr.2004.11.006.
[50]	I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.
[51]	B. Peultier, T. Ben Zineb and E. Patoor, Macroscopic constitutive law for SMA: Application to structure analysis by FEM, Materials Sci. Engrg. A, 438-440 (2006), 454-458. doi: 10.1016/j.msea.2006.01.104.
[52]	P. Popov and D. C. Lagoudas, A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite, Int. J. Plasticity, 23 (2007), 1679-1720. doi: 10.1016/j.ijplas.2007.03.011.
[53]	B. Raniecki and Ch. Lexcellent, $R_L$ models of pseudoelasticity and their specification for some shape-memory solids, European J. Mech. A Solids, 13 (1994), 21-50.
[54]	S. Reese and D. Christ, Finite deformation pseudo-elasticity of shape memory alloys - Constitutive modelling and finite element implementation, Int. J. Plasticity, 28 (2008), 455-482. doi: 10.1016/j.ijplas.2007.05.005.
[55]	T. Roubíček, Evolution model for martensitic phase transformation in shape-memory alloys, Interfaces Free Bound., 4 (2002), 111-136. doi: 10.4171/IFB/55.
[56]	T. Roubíček, Models of microstructure evolution in shape memory alloys, in Nonlinear Homogenization and its Appl. to Composites, Polycrystals and Smart Materials, eds. P. Ponte Castaneda, J. J. Telega and B. Gambin, NATO Sci. Series II/170 (Kluwer, 2004), 269-304.
[57]	A. C. Souza, E. N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induces transformations, Eur. J. Mech. A/Solids, 17 (1998), 789-806. doi: 10.1016/S0997-7538(98)80005-3.
[58]	P. Thamburaja and L. Anand, Polycrystalline shape-memory materials: Effect of crystallographic texture, J. Mech. Phys. Solids, 49 (2001), 709-737. doi: 10.1016/S0022-5096(00)00061-2.
[59]	F. Thiebaud, Ch. Lexcellent, M. Collet and E. Foltete, Implementation of a model taking into account the asymmetry between tension and compression, the temperature effects in a finite element code for shape memory alloys structures calculations, Comput. Materials Sci., 41 (2007), 208-221. doi: 10.1016/j.commatsci.2007.04.006.
[60]	A. Visintin, "Models of Phase Transitions,'' Progress in Nonlinear Differential Equations and their Applications, 28. Birkhäuser Boston, MA, 1996.
[61]	S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, Quasi-linear thermoelasticity system arising in shape memory materials, SIAM J. Math. Anal., 38 (2007), 1733-1759. doi: 10.1137/060653159.

show all references

References:

[1]	T. Aiki, A model of 3D shape memory alloy materials, J. Math. Soc. Japan, 57 (2005), 903-933. doi: 10.2969/jmsj/1158241940.
[2]	S. Antman, J. L. Ericksen, D. Kinderlehrer and I. Müller, "Metastability and Incompletely Posed Problems," in the IMA Volumes in Mathematics and its Applications, 3, Springer-Verlag, New York, 1987.
[3]	M. Arndt, M. Griebel and T. Roubíček, Modelling and numerical simulation of martensitic transformation in shape memory alloys, Contin. Mech. Thermodyn., 15 (2003), 463-485. doi: 10.1007/s00161-003-0127-3.
[4]	M. Arrigoni, F. Auricchio, V. Cacciafesta, L. Petrini and R. Pietrabissa, Cyclic effects in shape-memory alloys: A one-dimensional continuum model, Journal de Physique IV, 11 (2001), 577-582.
[5]	F. Auricchio, A. Mielke and U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials, Math. Models Meth. Appl. Sci., 18 (2008), 125-164. doi: 10.1142/S0218202508002632.
[6]	F. Auricchio and L. Petrini, Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations, Internat. J. Numer. Meth. Engrg., 55 (2002), 1255-1284. doi: 10.1002/nme.619.
[7]	F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations: Solution algorithm and boundary value problems, Internat. J. Numer. Meth. Engrg., 61 (2004), 807-836. doi: 10.1002/nme.1086.
[8]	F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity, Int. J. Plasticity, 23 (2007), 207-226. doi: 10.1016/j.ijplas.2006.02.012.
[9]	F. Auricchio, A. Reali and U. Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1631-1637. doi: 10.1016/j.cma.2009.01.019.
[10]	F. Auricchio and E. Sacco, A one-dimensional model for superelastic shape-memory alloys with different elastic properties between austenite and martensite, Int. J. Non-Linear Mech., 32 (1997), 1101-1114. doi: 10.1016/S0020-7462(96)00130-8.
[11]	F. Auricchio, R. L. Taylor and J. Lubliner, Shape-memory alloys: Macromodelling and numerical simulations of the superelastic behaviour, Comput. Mech. Appl. Mech. Engrg., 146 (1997), 281-312. doi: 10.1016/S0045-7825(96)01232-7.
[12]	A.-L. Bessoud and U. Stefanelli, A three-dimensional model for magnetic shape memory alloys, Math. Models Meth. Appl. Sci. (2010) to appear.
[13]	Z. Bo and D. C. Lagoudas, Thermomechanical modeling of polycrystalline SMAs under cyclic loading. Part III: Evolution of plastic strains and two-way shape memory effect, Int. J. Eng. Sci., 37 (1999), 1175-1203. doi: 10.1016/S0020-7225(98)00115-3.
[14]	M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,'' vol. 121 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996.
[15]	N. Chemetov, Well-posedness of two-shape memory model, Math. Methods Appl. Sci., 29 (2006), 209-233. doi: 10.1002/mma.672.
[16]	P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys, Nonlinear Anal., 24 (1995), 1565-1579. doi: 10.1016/0362-546X(94)00097-2.
[17]	P. Colli and J. Sprekels, Global existence for a three-dimensional model for the thermodynamical evolution of shape memory alloys, Nonlinear Anal., 18 (1992), 873-888. doi: 10.1016/0362-546X(92)90228-7.
[18]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence,'' Birkhäuser-Boston, 1993.
[19]	G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics,'' Springer-Berlin, 1976.
[20]	F. Falk, Martensitic domain boundaries in shape-memory alloys as solitary waves, J. Phys. C4 Suppl., 12 (1982), 3-15.
[21]	F. Falk and P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape-memory alloys, J. Phys. Condens. Matter, 2 (1990), 61-77. doi: 10.1088/0953-8984/2/1/005.
[22]	M. Frémond, Matériaux à mémoire de forme, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 304 (1987), 239-244.
[23]	M. Frémond, "Non-Smooth Thermomechanics,'' Springer-Verlag, 2002.
[24]	M. Frémond and S. Miyazaki, "Shape Memory Alloys,'' Springer-Verlag, 1996.
[25]	E. Fried and M. E. Gurtin, Dynamic solid-solid transitions with phase characterized by an order parameter, Phys. D, 72 (1994), 287-308. doi: 10.1016/0167-2789(94)90234-8.
[26]	S. Govindjee and C. Miehe, A multi-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Methods Appl. Mech. Engrg., 191 (2001), 215-238. doi: 10.1016/S0045-7825(01)00271-7.
[27]	S. Govindjee and E. P. Kasper, A shape memory alloy model for Uranium-Niobium accounting for plasticity, J. Intell. Mater. Syst. Struct., 8 (1997), 815-823. doi: 10.1177/1045389X9700801001.
[28]	D. Helm and P. Haupt, Shape memory behaviour: Modelling within continuum thermomechanics, Internat. J. Solids Structures, 40 (2003), 827-849. doi: 10.1016/S0020-7683(02)00621-2.
[29]	K. H. Hoffmann, M. Niezgódka and S. Zheng, Existence and uniqueness to an extended model of the dynamical developments in shape memory alloys, Nonlinear Anal., 15 (1990), 977-990. doi: 10.1016/0362-546X(90)90079-V.
[30]	Y. Huo, I. Müller and S. Seelecke, Quasiplasticity and pseudoelasticity in shape memory alloys, in Phase transitions and hysteresis, in Lecture Notes in Math., Vol. 1584, eds. M. Brokate et al (Springer 1994), 87-146.
[31]	P. Krejčí and U. Stefanelli, Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires, Math. Mech. Solids (2010), to appear.
[32]	P. Krejčí and U. Stefanelli, Well-posedness of a thermo-mechanical model for shape memory alloys under tension, M2AN Math. Model. Anal. Numer., (2010), to appear.
[33]	M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructures and its evolution in shape-memory-alloy single cristals, in particular in CuAlNi, Meccanica, 40 (2005), 389-418. doi: 10.1007/s11012-005-2106-1.
[34]	M. Kružík and J. Zimmer, A model of shape memory alloys accounting for plasticity, IMA J. Appl. Math., (2010), to appear.
[35]	D. C. Lagoudas, P. B. Entchev, P. Popov, E. Patoor, L. C. Brinson and X. Gao, Shape memory alloys, Part II: Modeling of polycrystals, Mech. Mater., 38 (2006), 391-429. doi: 10.1016/j.mechmat.2005.08.003.
[36]	D. C. Lagoudas and P. B. Entchev, Modeling of transformation-induced plasticity and its effect on the behavior of porous shape memory alloys. Part I: Constitutive model for fully dense SMAs, Mech. Mater., 36 (2004), 865-892.
[37]	A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.
[38]	G. A. Maugin, "The Thermomechanics of Plasticity and Fracture,'' Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1992.
[39]	A. Mielke, Evolution of rate-independent systems, In C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Elsevier, (2005), 461-559.
[40]	A. Mielke, L. Paoli and A. Petrov, On existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys, SIAM J. Math. Anal., 41 (2009), 1388-1414. doi: 10.1137/080726215.
[41]	A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for discretizations of a rate-independent variational inequality, SIAM J. Numer. Anal., 48 (2010), 1625-1646. doi: 10.1137/090750238.
[42]	A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error control for space-time discretizations of a 3D model for shape-memory materials, Proc. of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (Bochum 2008), IUTAM Bookseries, Springer, 2009.
[43]	A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl., 17 (2007), 160-182.
[44]	A. Mielke and T. Roubíček, A rate independent model for inelastic behaviour of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597. doi: 10.1137/S1540345903422860.
[45]	A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.
[46]	A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, Proc. of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, eds. H.-D Alber, R. Balean and R. Farwig (Shaker-Verlag, 1999), 117-129.
[47]	A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rational Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194.
[48]	I. Müller, Thermodynamics of ideal pseudoelasticity, J. Phys. IV, C2-5 (1995), 423-431.
[49]	A. Paiva, M. A. Savi, A. M. B. Braga and P. M. C. L. Pacheco, A constitutive model for shape memory alloys considering tensile-compressive asymmetry and plasticity, Internat. J. of Solid. Struct., 42 (2005), 3439-3457. doi: 10.1016/j.ijsolstr.2004.11.006.
[50]	I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.
[51]	B. Peultier, T. Ben Zineb and E. Patoor, Macroscopic constitutive law for SMA: Application to structure analysis by FEM, Materials Sci. Engrg. A, 438-440 (2006), 454-458. doi: 10.1016/j.msea.2006.01.104.
[52]	P. Popov and D. C. Lagoudas, A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite, Int. J. Plasticity, 23 (2007), 1679-1720. doi: 10.1016/j.ijplas.2007.03.011.
[53]	B. Raniecki and Ch. Lexcellent, $R_L$ models of pseudoelasticity and their specification for some shape-memory solids, European J. Mech. A Solids, 13 (1994), 21-50.
[54]	S. Reese and D. Christ, Finite deformation pseudo-elasticity of shape memory alloys - Constitutive modelling and finite element implementation, Int. J. Plasticity, 28 (2008), 455-482. doi: 10.1016/j.ijplas.2007.05.005.
[55]	T. Roubíček, Evolution model for martensitic phase transformation in shape-memory alloys, Interfaces Free Bound., 4 (2002), 111-136. doi: 10.4171/IFB/55.
[56]	T. Roubíček, Models of microstructure evolution in shape memory alloys, in Nonlinear Homogenization and its Appl. to Composites, Polycrystals and Smart Materials, eds. P. Ponte Castaneda, J. J. Telega and B. Gambin, NATO Sci. Series II/170 (Kluwer, 2004), 269-304.
[57]	A. C. Souza, E. N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induces transformations, Eur. J. Mech. A/Solids, 17 (1998), 789-806. doi: 10.1016/S0997-7538(98)80005-3.
[58]	P. Thamburaja and L. Anand, Polycrystalline shape-memory materials: Effect of crystallographic texture, J. Mech. Phys. Solids, 49 (2001), 709-737. doi: 10.1016/S0022-5096(00)00061-2.
[59]	F. Thiebaud, Ch. Lexcellent, M. Collet and E. Foltete, Implementation of a model taking into account the asymmetry between tension and compression, the temperature effects in a finite element code for shape memory alloys structures calculations, Comput. Materials Sci., 41 (2007), 208-221. doi: 10.1016/j.commatsci.2007.04.006.
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