# American Institute of Mathematical Sciences

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. [2] F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A phenomenological model for ferromagnetism in shape-memory materials,, In preparation, (2011). [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. doi: 10.1002/gamm.201110014. [4] F. Auricchio and J. Lubliner, A uniaxial model for shape-memory alloys,, Internat. J. Solids Structures, 34 (1997), 3601. doi: 10.1016/S0020-7683(96)00232-6. [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. 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. Methods Engrg., 55 (2002), 1255. [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. doi: 10.1002/nme.1086. [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. doi: 10.1002/nme.1087. [9] F. Auricchio, A. Reali and U. Stefanelli, A phenomenological 3D model describing stress-induced solid phase transformations with permanent inelasticity,, in, (2007), 1. [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. doi: 10.1016/j.ijplas.2006.02.012. [11] A.-L. Bessoud and U. Stefanelli, Magnetic shape memory alloys: Three-dimensional modeling and analysis,, Math. Models Meth. Appl. Sci., 21 (2011), 1043. [12] A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys,, Z. Angew. Math. Phys., (). [13] H. Brézis, "Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", Math Studies, (1973). [14] M. Brokate and J. Sprekels, Optimal control of shape memory alloys with solid-solid phase transitions,, in, 280 (1990), 208. [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. doi: 10.1080/01630569808816840. [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). [17] V. Evangelista, S. Marfia and E. Sacco, Phenomenological 3D and 1D consistent models for shape-memory alloy materials,, Comput. Mech., 44 (2009), 405. doi: 10.1007/s00466-009-0381-8. [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. [19] F. Falk, Model free energy, mechanics and thermodynamics of shape memory alloys,, Acta Metallurgica, 28 (1990), 1773. doi: 10.1016/0001-6160(80)90030-9. [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. [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. [22] M. Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag, (2002). [23] M. Frémond and S. Miyazaki, "Shape Memory Alloys,", CISM Courses and Lectures, (1996). [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. [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. doi: 10.1016/S0045-7825(01)00271-7. [26] D. Helm and P. Haupt, Shape memory behaviour: modelling within continuum thermomechanics,, Intern. J. Solids Struct., 40 (2003), 827. doi: 10.1016/S0020-7683(02)00621-2. [27] K.-H. Hoffmann and D. Tiba, Control of a plate with nonlinear shape memory alloy reinforcements,, Adv. Math. Sci. Appl., 7 (1997), 427. [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. doi: 10.1002/(SICI)1099-1476(19980510)21:7<589::AID-MMA904>3.0.CO;2-D. [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. doi: 10.1177/1081286510386935. [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. doi: 10.1051/m2an/2010024. [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. doi: 10.1016/j.mechmat.2005.05.027. [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. doi: 10.1016/j.ijplas.2007.03.011. [33] V. I. Levitas, Thermomechanical theory of martensitic phase transformations in inelastic materials,, Intern. J. Solids Struct., 35 (1998), 889. doi: 10.1016/S0020-7683(97)00089-9. [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. [35] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. Partial Differential Equations, 22 (2005), 73. [36] A. Mielke, Evolution of rate-independent systems,, in, 2 (2005), 461. [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. doi: 10.1137/080726215. [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. doi: 10.1137/090750238. [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, (2010), 185. [40] A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys,, Adv. Math. Sci. Appl., 17 (2007), 667. [41] A. Mielke and F. Rindler, Reverse approximation of energetic solutions to rate-independent processes,, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 17. [42] A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA, 11 (2004), 151. [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), 438. [44] L. Paoli and A. Petrov, Global existence result for phase transformations with heat transfer in shape memory alloys,, Preprint WIAS, (2011). [45] I. Pawłow and A. Żochowski, A control problem for a thermoelastic system in shape memory materials,, Sūrikaisekikenkyūsho Kōkyūroku, (2011), 8. [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. [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. [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. doi: 10.1016/j.ijplas.2007.05.005. [49] F. Rindler, Optimal control for nonconvex rate-independent evolution processes,, SIAM J. Control Optim., 47 (2008), 2773. doi: 10.1137/080718711. [50] F. Rindler, Approximation of rate-independent optimal control problems,, SIAM J. Numer. Anal., 47 (2009), 3884. doi: 10.1137/080744050. [51] T.Roubíček, Models of microstructure evolution in shape memory alloys,, in, (2004), 269. [52] J. Sokołowski and J. Sprekels, Control problems with state constraints for shape memory alloys,, Math. Methods Appl. Sci., 17 (1994), 943. [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. doi: 10.1016/S0997-7538(98)80005-3. [54] U. Stefanelli, Magnetic control of magnetic shape-memory single crystals,, Phys. B, 407 (2012), 1316. [55] P. Thamburaja and L. Anand, Polycrystalline shape-memory materials: effect of crystallographic texture,, J. Mech. Phys. Solids, 49 (2001), 709. doi: 10.1016/S0022-5096(00)00061-2. [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), (2012). [57] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part II: Regularization and differentiability,, Preprint SPP1253-119, (2011), 1253. [58] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part III: Optimality conditions,, Preprint SPP1253-119, (2011), 1253.

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##### 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. [2] F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A phenomenological model for ferromagnetism in shape-memory materials,, In preparation, (2011). [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. doi: 10.1002/gamm.201110014. [4] F. Auricchio and J. Lubliner, A uniaxial model for shape-memory alloys,, Internat. J. Solids Structures, 34 (1997), 3601. doi: 10.1016/S0020-7683(96)00232-6. [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. 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. Methods Engrg., 55 (2002), 1255. [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. doi: 10.1002/nme.1086. [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. doi: 10.1002/nme.1087. [9] F. Auricchio, A. Reali and U. Stefanelli, A phenomenological 3D model describing stress-induced solid phase transformations with permanent inelasticity,, in, (2007), 1. [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. doi: 10.1016/j.ijplas.2006.02.012. [11] A.-L. Bessoud and U. Stefanelli, Magnetic shape memory alloys: Three-dimensional modeling and analysis,, Math. Models Meth. Appl. Sci., 21 (2011), 1043. [12] A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys,, Z. Angew. Math. Phys., (). [13] H. Brézis, "Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", Math Studies, (1973). [14] M. Brokate and J. Sprekels, Optimal control of shape memory alloys with solid-solid phase transitions,, in, 280 (1990), 208. [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. doi: 10.1080/01630569808816840. [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). [17] V. Evangelista, S. Marfia and E. Sacco, Phenomenological 3D and 1D consistent models for shape-memory alloy materials,, Comput. Mech., 44 (2009), 405. doi: 10.1007/s00466-009-0381-8. [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. [19] F. Falk, Model free energy, mechanics and thermodynamics of shape memory alloys,, Acta Metallurgica, 28 (1990), 1773. doi: 10.1016/0001-6160(80)90030-9. [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. [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. [22] M. Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag, (2002). [23] M. Frémond and S. Miyazaki, "Shape Memory Alloys,", CISM Courses and Lectures, (1996). [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. [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. doi: 10.1016/S0045-7825(01)00271-7. [26] D. Helm and P. Haupt, Shape memory behaviour: modelling within continuum thermomechanics,, Intern. J. Solids Struct., 40 (2003), 827. doi: 10.1016/S0020-7683(02)00621-2. [27] K.-H. Hoffmann and D. Tiba, Control of a plate with nonlinear shape memory alloy reinforcements,, Adv. Math. Sci. Appl., 7 (1997), 427. [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. doi: 10.1002/(SICI)1099-1476(19980510)21:7<589::AID-MMA904>3.0.CO;2-D. [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. doi: 10.1177/1081286510386935. [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. doi: 10.1051/m2an/2010024. [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. doi: 10.1016/j.mechmat.2005.05.027. [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. doi: 10.1016/j.ijplas.2007.03.011. [33] V. I. Levitas, Thermomechanical theory of martensitic phase transformations in inelastic materials,, Intern. J. Solids Struct., 35 (1998), 889. doi: 10.1016/S0020-7683(97)00089-9. [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. [35] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. Partial Differential Equations, 22 (2005), 73. [36] A. Mielke, Evolution of rate-independent systems,, in, 2 (2005), 461. [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. doi: 10.1137/080726215. [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. doi: 10.1137/090750238. [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, (2010), 185. [40] A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys,, Adv. Math. Sci. Appl., 17 (2007), 667. [41] A. Mielke and F. Rindler, Reverse approximation of energetic solutions to rate-independent processes,, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 17. [42] A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA, 11 (2004), 151. [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), 438. [44] L. Paoli and A. Petrov, Global existence result for phase transformations with heat transfer in shape memory alloys,, Preprint WIAS, (2011). [45] I. Pawłow and A. Żochowski, A control problem for a thermoelastic system in shape memory materials,, Sūrikaisekikenkyūsho Kōkyūroku, (2011), 8. [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. [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. [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. doi: 10.1016/j.ijplas.2007.05.005. [49] F. Rindler, Optimal control for nonconvex rate-independent evolution processes,, SIAM J. Control Optim., 47 (2008), 2773. doi: 10.1137/080718711. [50] F. Rindler, Approximation of rate-independent optimal control problems,, SIAM J. Numer. Anal., 47 (2009), 3884. doi: 10.1137/080744050. [51] T.Roubíček, Models of microstructure evolution in shape memory alloys,, in, (2004), 269. [52] J. Sokołowski and J. Sprekels, Control problems with state constraints for shape memory alloys,, Math. Methods Appl. Sci., 17 (1994), 943. [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. doi: 10.1016/S0997-7538(98)80005-3. [54] U. Stefanelli, Magnetic control of magnetic shape-memory single crystals,, Phys. B, 407 (2012), 1316. [55] P. Thamburaja and L. Anand, Polycrystalline shape-memory materials: effect of crystallographic texture,, J. Mech. Phys. Solids, 49 (2001), 709. doi: 10.1016/S0022-5096(00)00061-2. [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), (2012). [57] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part II: Regularization and differentiability,, Preprint SPP1253-119, (2011), 1253. [58] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part III: Optimality conditions,, Preprint SPP1253-119, (2011), 1253.
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