September  2014, 3(3): 411-427. doi: 10.3934/eect.2014.3.411

Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials

1. 

Dipartimento di Matematica e Informatica, Università di Firenze, viale Morgagni 67/a, I-50134 Firenze, Italy

2. 

Dipartimento di Matematica e Fisica "N.Tartaglia", Università Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia, Italy

Received  April 2013 Revised  January 2014 Published  August 2014

We address the thermal control of the quasi-static evolution of a polycrystalline shape memory alloy specimen. The thermomechanical evolution of the body is described by means of an extension of the phenomenological Souza-Auricchio model [6,7,8,57] accounting also for permanent inelastic effects [9,11,27]. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the energetic sense. In a similar way as in [28], using results by Rindler [49,50] we prove the existence of optimal controls for a suitably large class of cost functionals.
Citation: Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411
References:
[1]

T. Aiki and N. Kenmochi, Some models for shape memory alloys, Mathematical aspects of modelling structure formation phenomena, GAKUTO Internat. Ser. Math. Sci. Appl., Tokio, 17 (2002), 144-162.  Google Scholar

[2]

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.  Google Scholar

[3]

F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A phenomenological model for the magneto-mechanical response of single-crystal magnetic shape memory alloys, Preprint IMATI-CNR 3PV13/3/0, 2013. Google Scholar

[4]

F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A three-dimensional phenomenological models for magnetic shape memory alloys, GAMM-Mitt., 34 (2011), 90-96. doi: 10.1002/gamm.201110014.  Google Scholar

[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.  Google Scholar

[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. Methods Engrg., 55 (2002), 1255-1284. doi: 10.1002/nme.619.  Google Scholar

[7]

F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations. Part I: Solution algorithm and boundary value problems, Internat. J. Numer. Meth. Engrg., 61 (2004), 807-836. doi: 10.1002/nme.1086.  Google Scholar

[8]

F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations. Part II: Thermomechanical coupling and hybrid composite applications, Internat. J. Numer. Meth. Engrg., 61 (2004), 716-737. doi: 10.1002/nme.1087.  Google Scholar

[9]

F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induces solid phase transformation with permanent inelasticity, Int. J. Plasticity, 23 (2007), 207-226. doi: 10.1016/j.ijplas.2006.02.012.  Google Scholar

[10]

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.  Google Scholar

[11]

N. Barrera, P. Biscari and M. F. Urbano, Macroscopic modeling of functional fatigue in shape memory alloys,, Eur. J. Mech. A/Solids, ().   Google Scholar

[12]

S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion, Math. Modelling Numer. Anal., 45 (2011), 477-504. doi: 10.1051/m2an/2010063.  Google Scholar

[13]

B. Benesova, M. Frost and P. Sedlak, A microscopically motivated constitutive model for shape memory alloys: formulation, analysis and computations,, Preprint NCMM/2013/17, ().   Google Scholar

[14]

A.-L. Bessoud and U. Stefanelli, Magnetic shape memory alloys: Three-dimensional modeling and analysis, Math. Models Meth. Appl. Sci., 21 (2011), 1043-1069. doi: 10.1142/S0218202511005246.  Google Scholar

[15]

A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys, Preprint IMATI-CNR, 23PV10/21/0, 2010. Google Scholar

[16]

A. Berti, C. Giorgi and E. Vuk, Free energies and pseudo-elastic transitions for shape memory alloys, Discrete Cont. Dyn. S. - Series S, 6 (2013), 293-316.  Google Scholar

[17]

V. Berti, M. Fabrizio and D. Grandi, Phase transitions in shape memory alloys: A non-isothermal Ginburg-Landau model, Physica D, 239 (2010), 95-102. doi: 10.1016/j.physd.2009.10.005.  Google Scholar

[18]

H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Math Studies, 5, North-Holland, Amsterdam/New York, 1973.  Google Scholar

[19]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Applied Mathematical Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[20]

M. Brokate and J. Sprekels, Optimal control of shape memory alloys with solid-solid phase transitions, in Emerging applications in free boundary problems (Montreal, 1990), Pitman Res. Notes Math. Ser., 280, Longman Sci. Tech., Harlow, 1993, 208-214.  Google Scholar

[21]

N. Bubner, J. Sokołowski and J. Sprekels, Optimal boundary control problems for shape memory alloys under state constraints for stress and temperature, Numer. Funct. Anal. Optim., 19 (1998), 489-498. doi: 10.1080/01630569808816840.  Google Scholar

[22]

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.  Google Scholar

[23]

T. W. Duerig and A. R. Pelton editors, SMST-2003 Proceedings of the International Conference on Shape Memory and Superelastic Technology Conference, ASM International, 2003. Google Scholar

[24]

M. Eleuteri, J. Kopfová and P. Krejčí, A thermodynamic model for material fatigue under cyclic loading, Physica B: Condensed Matter, 407 (2012), 1415-1416. doi: 10.1016/j.physb.2011.10.017.  Google Scholar

[25]

M. Eleuteri, J. Kopfová and P. Krejčí, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam, Comm. Pure Appl. Anal., 12 (2013), 2973-2996. doi: 10.3934/cpaa.2013.12.2973.  Google Scholar

[26]

M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue accumulation in a thermo-visco-elastoplastic plate, Discrete Cont. Dyn. S. Series B, 19 (2014). Google Scholar

[27]

M. Eleuteri, L. Lussardi and U. Stefanelli, A rate-independent model for permanent inelastic effects in shape memory materials, Netw. Heterog. Media, 6 (2011), 145-165. doi: 10.3934/nhm.2011.6.145.  Google Scholar

[28]

M. Eleuteri, L. Lussardi and U. Stefanelli, Thermal control of the Souza-Auricchio model for shape memory alloys, Discrete Cont. Dyn. S. - Series S, 6 (2013), 369-386.  Google Scholar

[29]

V. Evangelista, S. Marfia and E. Sacco, Phenomenological 3D and 1D consistent models for shape-memory alloy materials, Comput. Mech., 44 (2009), 405-421. doi: 10.1007/s00466-009-0381-8.  Google Scholar

[30]

V. Evangelista, S. Marfia and E. Sacco, A 3D SMA constitutive model in the framework of finite strain, Internat. J. Numer. Methods Engrg., 81 (2010), 761-785. Google Scholar

[31]

F. Falk, Martensitic domain boundaries in shape-memory alloys as solitary waves, J. Phys. C4 Suppl., 12 (1982), 3-15. Google Scholar

[32]

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.  Google Scholar

[33]

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. Google Scholar

[34]

S. Frigeri and U. Stefanelli, Existence and time-discretization for the finite-strain Souza-Auricchio constitutive model for shape-memory alloys, Contin. Mech. Thermodyn., 24 (2012), 63-77. doi: 10.1007/s00161-011-0221-x.  Google Scholar

[35]

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.  Google Scholar

[36]

K.-H. Hoffmann and D. Tiba, Control of a plate with nonlinear shape memory alloy reinforcements, Adv. Math. Sci. Appl., 7 (1997), 427-436.  Google Scholar

[37]

K.-H. Hoffmann and A. Z. Ochowski, Control of the thermoelastic model of a plate activated by shape memory alloy reinforcements, Math. Methods Appl. Sci., 21 (1998), 589-603. doi: 10.1002/(SICI)1099-1476(19980510)21:7<589::AID-MMA904>3.0.CO;2-D.  Google Scholar

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O. Klein, Stability and uniqueness results for a numerical appproximation of the thermomechanical phase transitions in shape memory alloys, Adv. in Math. Sci. and Appl., 5 (1995), 91-116.  Google Scholar

[39]

P. Krejčí and U. Stefanelli, Well-posedness of a thermo-mechanical model for shape memory alloys under tension, M2AN Math. Model. Numer. Anal., 44 (2010), 1239-1253. doi: 10.1051/m2an/2010024.  Google Scholar

[40]

P. Krejčí and U. Stefanelli, Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires, Math. Mech. Solids, 16 (2011), 349-365. doi: 10.1177/1081286510386935.  Google Scholar

[41]

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A. Mielke, Evolution of rate-independent systems, in Handbook of Differential Equations, Evolutionary Equations, (editors, C. Dafermos and E. Feireisl), 2 (2005), 461-559.  Google Scholar

[43]

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.  Google Scholar

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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.  Google Scholar

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A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl., 17 (2007), 667-685.  Google Scholar

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A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA, Nonlinear Diff. Equations Applications, 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7.  Google Scholar

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L. Paoli and A. Petrov, Global existence result for phase transformations with heat transfer in shape memory alloys, Preprint WIAS, 1608, (2011). Google Scholar

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I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.  Google Scholar

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F. Rindler, Optimal control for nonconvex rate-independent evolution processes, SIAM J. Control Optim., 47 (2008), 2773-2794. doi: 10.1137/080718711.  Google Scholar

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F. Rindler, Approximation of rate-independent optimal control problems, SIAM J. Numer. Anal., 47 (2009), 3884-3909. doi: 10.1137/080744050.  Google Scholar

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show all references

References:
[1]

T. Aiki and N. Kenmochi, Some models for shape memory alloys, Mathematical aspects of modelling structure formation phenomena, GAKUTO Internat. Ser. Math. Sci. Appl., Tokio, 17 (2002), 144-162.  Google Scholar

[2]

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.  Google Scholar

[3]

F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A phenomenological model for the magneto-mechanical response of single-crystal magnetic shape memory alloys, Preprint IMATI-CNR 3PV13/3/0, 2013. Google Scholar

[4]

F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A three-dimensional phenomenological models for magnetic shape memory alloys, GAMM-Mitt., 34 (2011), 90-96. doi: 10.1002/gamm.201110014.  Google Scholar

[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.  Google Scholar

[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. Methods Engrg., 55 (2002), 1255-1284. doi: 10.1002/nme.619.  Google Scholar

[7]

F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations. Part I: Solution algorithm and boundary value problems, Internat. J. Numer. Meth. Engrg., 61 (2004), 807-836. doi: 10.1002/nme.1086.  Google Scholar

[8]

F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations. Part II: Thermomechanical coupling and hybrid composite applications, Internat. J. Numer. Meth. Engrg., 61 (2004), 716-737. doi: 10.1002/nme.1087.  Google Scholar

[9]

F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induces solid phase transformation with permanent inelasticity, Int. J. Plasticity, 23 (2007), 207-226. doi: 10.1016/j.ijplas.2006.02.012.  Google Scholar

[10]

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.  Google Scholar

[11]

N. Barrera, P. Biscari and M. F. Urbano, Macroscopic modeling of functional fatigue in shape memory alloys,, Eur. J. Mech. A/Solids, ().   Google Scholar

[12]

S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion, Math. Modelling Numer. Anal., 45 (2011), 477-504. doi: 10.1051/m2an/2010063.  Google Scholar

[13]

B. Benesova, M. Frost and P. Sedlak, A microscopically motivated constitutive model for shape memory alloys: formulation, analysis and computations,, Preprint NCMM/2013/17, ().   Google Scholar

[14]

A.-L. Bessoud and U. Stefanelli, Magnetic shape memory alloys: Three-dimensional modeling and analysis, Math. Models Meth. Appl. Sci., 21 (2011), 1043-1069. doi: 10.1142/S0218202511005246.  Google Scholar

[15]

A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys, Preprint IMATI-CNR, 23PV10/21/0, 2010. Google Scholar

[16]

A. Berti, C. Giorgi and E. Vuk, Free energies and pseudo-elastic transitions for shape memory alloys, Discrete Cont. Dyn. S. - Series S, 6 (2013), 293-316.  Google Scholar

[17]

V. Berti, M. Fabrizio and D. Grandi, Phase transitions in shape memory alloys: A non-isothermal Ginburg-Landau model, Physica D, 239 (2010), 95-102. doi: 10.1016/j.physd.2009.10.005.  Google Scholar

[18]

H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Math Studies, 5, North-Holland, Amsterdam/New York, 1973.  Google Scholar

[19]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Applied Mathematical Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[20]

M. Brokate and J. Sprekels, Optimal control of shape memory alloys with solid-solid phase transitions, in Emerging applications in free boundary problems (Montreal, 1990), Pitman Res. Notes Math. Ser., 280, Longman Sci. Tech., Harlow, 1993, 208-214.  Google Scholar

[21]

N. Bubner, J. Sokołowski and J. Sprekels, Optimal boundary control problems for shape memory alloys under state constraints for stress and temperature, Numer. Funct. Anal. Optim., 19 (1998), 489-498. doi: 10.1080/01630569808816840.  Google Scholar

[22]

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.  Google Scholar

[23]

T. W. Duerig and A. R. Pelton editors, SMST-2003 Proceedings of the International Conference on Shape Memory and Superelastic Technology Conference, ASM International, 2003. Google Scholar

[24]

M. Eleuteri, J. Kopfová and P. Krejčí, A thermodynamic model for material fatigue under cyclic loading, Physica B: Condensed Matter, 407 (2012), 1415-1416. doi: 10.1016/j.physb.2011.10.017.  Google Scholar

[25]

M. Eleuteri, J. Kopfová and P. Krejčí, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam, Comm. Pure Appl. Anal., 12 (2013), 2973-2996. doi: 10.3934/cpaa.2013.12.2973.  Google Scholar

[26]

M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue accumulation in a thermo-visco-elastoplastic plate, Discrete Cont. Dyn. S. Series B, 19 (2014). Google Scholar

[27]

M. Eleuteri, L. Lussardi and U. Stefanelli, A rate-independent model for permanent inelastic effects in shape memory materials, Netw. Heterog. Media, 6 (2011), 145-165. doi: 10.3934/nhm.2011.6.145.  Google Scholar

[28]

M. Eleuteri, L. Lussardi and U. Stefanelli, Thermal control of the Souza-Auricchio model for shape memory alloys, Discrete Cont. Dyn. S. - Series S, 6 (2013), 369-386.  Google Scholar

[29]

V. Evangelista, S. Marfia and E. Sacco, Phenomenological 3D and 1D consistent models for shape-memory alloy materials, Comput. Mech., 44 (2009), 405-421. doi: 10.1007/s00466-009-0381-8.  Google Scholar

[30]

V. Evangelista, S. Marfia and E. Sacco, A 3D SMA constitutive model in the framework of finite strain, Internat. J. Numer. Methods Engrg., 81 (2010), 761-785. Google Scholar

[31]

F. Falk, Martensitic domain boundaries in shape-memory alloys as solitary waves, J. Phys. C4 Suppl., 12 (1982), 3-15. Google Scholar

[32]

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.  Google Scholar

[33]

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. Google Scholar

[34]

S. Frigeri and U. Stefanelli, Existence and time-discretization for the finite-strain Souza-Auricchio constitutive model for shape-memory alloys, Contin. Mech. Thermodyn., 24 (2012), 63-77. doi: 10.1007/s00161-011-0221-x.  Google Scholar

[35]

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.  Google Scholar

[36]

K.-H. Hoffmann and D. Tiba, Control of a plate with nonlinear shape memory alloy reinforcements, Adv. Math. Sci. Appl., 7 (1997), 427-436.  Google Scholar

[37]

K.-H. Hoffmann and A. Z. Ochowski, Control of the thermoelastic model of a plate activated by shape memory alloy reinforcements, Math. Methods Appl. Sci., 21 (1998), 589-603. doi: 10.1002/(SICI)1099-1476(19980510)21:7<589::AID-MMA904>3.0.CO;2-D.  Google Scholar

[38]

O. Klein, Stability and uniqueness results for a numerical appproximation of the thermomechanical phase transitions in shape memory alloys, Adv. in Math. Sci. and Appl., 5 (1995), 91-116.  Google Scholar

[39]

P. Krejčí and U. Stefanelli, Well-posedness of a thermo-mechanical model for shape memory alloys under tension, M2AN Math. Model. Numer. Anal., 44 (2010), 1239-1253. doi: 10.1051/m2an/2010024.  Google Scholar

[40]

P. Krejčí and U. Stefanelli, Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires, Math. Mech. Solids, 16 (2011), 349-365. doi: 10.1177/1081286510386935.  Google Scholar

[41]

G. A. Maugin, The Thermomechanics of Plasticity and Fracture, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781139172400.  Google Scholar

[42]

A. Mielke, Evolution of rate-independent systems, in Handbook of Differential Equations, Evolutionary Equations, (editors, C. Dafermos and E. Feireisl), 2 (2005), 461-559.  Google Scholar

[43]

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.  Google Scholar

[44]

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.  Google Scholar

[45]

A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl., 17 (2007), 667-685.  Google Scholar

[46]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA, Nonlinear Diff. Equations Applications, 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7.  Google Scholar

[47]

L. Paoli and A. Petrov, Global existence result for phase transformations with heat transfer in shape memory alloys, Preprint WIAS, 1608, (2011). Google Scholar

[48]

I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.  Google Scholar

[49]

F. Rindler, Optimal control for nonconvex rate-independent evolution processes, SIAM J. Control Optim., 47 (2008), 2773-2794. doi: 10.1137/080718711.  Google Scholar

[50]

F. Rindler, Approximation of rate-independent optimal control problems, SIAM J. Numer. Anal., 47 (2009), 3884-3909. doi: 10.1137/080744050.  Google Scholar

[51]

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