April  2013, 6(2): 369-386. doi: 10.3934/dcdss.2013.6.369

Thermal control of the Souza-Auricchio model for shape memory alloys

1. 

Università degli Studi di Milano, via Saldini 50, I-20133 Milano, Italy

2. 

Università Cattolica del Sacro Cuore, via dei Musei 41, I-25121 Brescia, Italy

3. 

Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes" - CNR, via Ferrata 1, I-27100 Pavia, Italy

Received  July 2011 Revised  December 2011 Published  November 2012

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 the phenomenological SOUZA$-$AURICCHIO model [6,53]. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the energeticsense. By recovering in this context a result by RINDLER [49,50] we prove the existence of optimal controls for a suitably large class of cost functionals and comment on their possible approximation.
Citation: Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. Thermal control of the Souza-Auricchio model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 369-386. doi: 10.3934/dcdss.2013.6.369
References:
[1]

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.  Google Scholar

[2]

F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A phenomenological model for ferromagnetism in shape-memory materials, In preparation, (2011). Google Scholar

[3]

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

[4]

F. Auricchio and J. Lubliner, A uniaxial model for shape-memory alloys, Internat. J. Solids Structures, 34 (1997), 3601-3618. doi: 10.1016/S0020-7683(96)00232-6.  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.  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 phenomenological 3D model describing stress-induced solid phase transformations with permanent inelasticity, in "Topics on Mathematics for Smart Systems," 1-14. World Sci. Publ., Hackensack, NJ, 2007.  Google Scholar

[10]

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

[11]

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

[12]

A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys,, Z. Angew. Math. Phys., ().   Google Scholar

[13]

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

[14]

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), 280 of Pitman Res. Notes Math. Ser., 208-214. Longman Sci. Tech., Harlow, 1993.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

F. Falk, Model free energy, mechanics and thermodynamics of shape memory alloys, Acta Metallurgica, 28 (1990), 1773-1780. doi: 10.1016/0001-6160(80)90030-9.  Google Scholar

[20]

G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math., 595 (2006), 55-91.  Google Scholar

[21]

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

[22]

M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002.  Google Scholar

[23]

M. Frémond and S. Miyazaki, "Shape Memory Alloys," CISM Courses and Lectures, vol. 351, Springer-Verlag, 1996. Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

D. Helm and P. Haupt, Shape memory behaviour: modelling within continuum thermomechanics, Intern. J. Solids Struct., 40 (2003), 827-849. doi: 10.1016/S0020-7683(02)00621-2.  Google Scholar

[27]

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

[28]

K.-H. Hoffmann and A. żochowski, 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

[29]

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

[30]

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

[31]

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. Materials, 38 (2006), 391-429. doi: 10.1016/j.mechmat.2005.05.027.  Google Scholar

[32]

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.  Google Scholar

[33]

V. I. Levitas, Thermomechanical theory of martensitic phase transformations in inelastic materials, Intern. J. Solids Struct., 35 (1998), 889-940. doi: 10.1016/S0020-7683(97)00089-9.  Google Scholar

[34]

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.  Google Scholar

[35]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  Google Scholar

[36]

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

[37]

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

[38]

A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for space-time discretizations of a rate-independent variational inequality, SIAM J. Numer. Anal., 48 (2010), 1625-1646. doi: 10.1137/090750238.  Google Scholar

[39]

A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error bounds for space-time discretizations of a 3d model for shape-memory materials, in "IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials" (editor, K. Hackl), 185-197. Springer, 2010. Proceedings of the IUTAM Symposium on Variational Concepts, Bochum, Germany, Sept.\ 22-26, 2008. Google Scholar

[40]

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

[41]

A. Mielke and F. Rindler, Reverse approximation of energetic solutions to rate-independent processes, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 17-40.  Google Scholar

[42]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA, Nonlinear Diff. Equations Applications, 11 (2004), 151-189.  Google Scholar

[43]

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. Google Scholar

[44]

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

[45]

I. Pawłow and A. Żochowski, A control problem for a thermoelastic system in shape memory materials, Sūrikaisekikenkyūsho Kōkyūroku, No. 1210 (2011), 8-23, Free boundary problems (Japanese) (Kyoto, 2000).  Google Scholar

[46]

{R. Peyroux, A. Chrysochoos, Ch. Licht and M. Löbel}, Phenomenological constitutive equations for numerical simulations of SMA's structures. Effect of thermomechanical couplings, J. Phys. C4 Suppl., 6 (1996), 347-356. Google Scholar

[47]

B. Raniecki and Ch. Lexcellent, $R_L$ models of pseudoelasticity and their specification for some shape-memory solids, Eur. J. Mech. A Solids, 13 (1994), 21-50.  Google Scholar

[48]

S. Reese and D. Christ, Finite deformation pseudo-elasticity of shape memory alloys - Constitutive modelling and finite element implementation, Int. J. Plasticity, 24 (2008), 455-482. doi: 10.1016/j.ijplas.2007.05.005.  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]

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, Dordrecht, 2004, 269-304.  Google Scholar

[52]

J. Sokołowski and J. Sprekels, Control problems with state constraints for shape memory alloys, Math. Methods Appl. Sci., 17 (1994), 943-952.  Google Scholar

[53]

A. C. Souza, E. N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced tranformations, Eur. J. Mech. A Solids, 17 (1998), 789-806. doi: 10.1016/S0997-7538(98)80005-3.  Google Scholar

[54]

U. Stefanelli, Magnetic control of magnetic shape-memory single crystals, Phys. B, 407 (2012), 1316-1321. Google Scholar

[55]

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.  Google Scholar

[56]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, Part I: Existence and discretization in time, SIAM Journal on Control and Optimization (SICON), to appear (2012). Google Scholar

[57]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part II: Regularization and differentiability, Preprint SPP1253-119, 2011. Google Scholar

[58]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part III: Optimality conditions, Preprint SPP1253-119, 2011. Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A phenomenological model for ferromagnetism in shape-memory materials, In preparation, (2011). Google Scholar

[3]

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

[4]

F. Auricchio and J. Lubliner, A uniaxial model for shape-memory alloys, Internat. J. Solids Structures, 34 (1997), 3601-3618. doi: 10.1016/S0020-7683(96)00232-6.  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.  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 phenomenological 3D model describing stress-induced solid phase transformations with permanent inelasticity, in "Topics on Mathematics for Smart Systems," 1-14. World Sci. Publ., Hackensack, NJ, 2007.  Google Scholar

[10]

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

[11]

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

[12]

A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys,, Z. Angew. Math. Phys., ().   Google Scholar

[13]

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

[14]

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), 280 of Pitman Res. Notes Math. Ser., 208-214. Longman Sci. Tech., Harlow, 1993.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

F. Falk, Model free energy, mechanics and thermodynamics of shape memory alloys, Acta Metallurgica, 28 (1990), 1773-1780. doi: 10.1016/0001-6160(80)90030-9.  Google Scholar

[20]

G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math., 595 (2006), 55-91.  Google Scholar

[21]

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

[22]

M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002.  Google Scholar

[23]

M. Frémond and S. Miyazaki, "Shape Memory Alloys," CISM Courses and Lectures, vol. 351, Springer-Verlag, 1996. Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

D. Helm and P. Haupt, Shape memory behaviour: modelling within continuum thermomechanics, Intern. J. Solids Struct., 40 (2003), 827-849. doi: 10.1016/S0020-7683(02)00621-2.  Google Scholar

[27]

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

[28]

K.-H. Hoffmann and A. żochowski, 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

[29]

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

[30]

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

[31]

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. Materials, 38 (2006), 391-429. doi: 10.1016/j.mechmat.2005.05.027.  Google Scholar

[32]

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.  Google Scholar

[33]

V. I. Levitas, Thermomechanical theory of martensitic phase transformations in inelastic materials, Intern. J. Solids Struct., 35 (1998), 889-940. doi: 10.1016/S0020-7683(97)00089-9.  Google Scholar

[34]

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.  Google Scholar

[35]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  Google Scholar

[36]

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

[37]

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

[38]

A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for space-time discretizations of a rate-independent variational inequality, SIAM J. Numer. Anal., 48 (2010), 1625-1646. doi: 10.1137/090750238.  Google Scholar

[39]

A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error bounds for space-time discretizations of a 3d model for shape-memory materials, in "IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials" (editor, K. Hackl), 185-197. Springer, 2010. Proceedings of the IUTAM Symposium on Variational Concepts, Bochum, Germany, Sept.\ 22-26, 2008. Google Scholar

[40]

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

[41]

A. Mielke and F. Rindler, Reverse approximation of energetic solutions to rate-independent processes, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 17-40.  Google Scholar

[42]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA, Nonlinear Diff. Equations Applications, 11 (2004), 151-189.  Google Scholar

[43]

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. Google Scholar

[44]

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

[45]

I. Pawłow and A. Żochowski, A control problem for a thermoelastic system in shape memory materials, Sūrikaisekikenkyūsho Kōkyūroku, No. 1210 (2011), 8-23, Free boundary problems (Japanese) (Kyoto, 2000).  Google Scholar

[46]

{R. Peyroux, A. Chrysochoos, Ch. Licht and M. Löbel}, Phenomenological constitutive equations for numerical simulations of SMA's structures. Effect of thermomechanical couplings, J. Phys. C4 Suppl., 6 (1996), 347-356. Google Scholar

[47]

B. Raniecki and Ch. Lexcellent, $R_L$ models of pseudoelasticity and their specification for some shape-memory solids, Eur. J. Mech. A Solids, 13 (1994), 21-50.  Google Scholar

[48]

S. Reese and D. Christ, Finite deformation pseudo-elasticity of shape memory alloys - Constitutive modelling and finite element implementation, Int. J. Plasticity, 24 (2008), 455-482. doi: 10.1016/j.ijplas.2007.05.005.  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]

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, Dordrecht, 2004, 269-304.  Google Scholar

[52]

J. Sokołowski and J. Sprekels, Control problems with state constraints for shape memory alloys, Math. Methods Appl. Sci., 17 (1994), 943-952.  Google Scholar

[53]

A. C. Souza, E. N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced tranformations, Eur. J. Mech. A Solids, 17 (1998), 789-806. doi: 10.1016/S0997-7538(98)80005-3.  Google Scholar

[54]

U. Stefanelli, Magnetic control of magnetic shape-memory single crystals, Phys. B, 407 (2012), 1316-1321. Google Scholar

[55]

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.  Google Scholar

[56]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, Part I: Existence and discretization in time, SIAM Journal on Control and Optimization (SICON), to appear (2012). Google Scholar

[57]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part II: Regularization and differentiability, Preprint SPP1253-119, 2011. Google Scholar

[58]

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