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.   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.  doi: 10.1002/gamm.201110014.  Google Scholar

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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.  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.  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.   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.  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.  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, (2007), 1.   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.  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.   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, (1973).   Google Scholar

[14]

M. Brokate and J. Sprekels, Optimal control of shape memory alloys with solid-solid phase transitions,, in, 280 (1990), 208.   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.  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

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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.  Google Scholar

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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.   Google Scholar

[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.  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.   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.   Google Scholar

[22]

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

[23]

M. Frémond and S. Miyazaki, "Shape Memory Alloys,", CISM Courses and Lectures, (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.   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.  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.  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.   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.  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.  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.  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.  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.  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.  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.   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.   Google Scholar

[36]

A. Mielke, Evolution of rate-independent systems,, in, 2 (2005), 461.   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.  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.  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, (2010), 185.   Google Scholar

[40]

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

[42]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA, 11 (2004), 151.   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), 438.   Google Scholar

[44]

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

[45]

I. Pawłow and A. Żochowski, A control problem for a thermoelastic system in shape memory materials,, Sūrikaisekikenkyūsho Kōkyūroku, (2011), 8.   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.   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.   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.  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.  doi: 10.1137/080718711.  Google Scholar

[50]

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

[51]

T.Roubíček, Models of microstructure evolution in shape memory alloys,, in, (2004), 269.   Google Scholar

[52]

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

[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.  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), (2012).   Google Scholar

[57]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part II: Regularization and differentiability,, Preprint SPP1253-119, (2011), 1253.   Google Scholar

[58]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part III: Optimality conditions,, Preprint SPP1253-119, (2011), 1253.   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.   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.  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.  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.  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.   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.  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.  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, (2007), 1.   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.  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.   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, (1973).   Google Scholar

[14]

M. Brokate and J. Sprekels, Optimal control of shape memory alloys with solid-solid phase transitions,, in, 280 (1990), 208.   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.  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.  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.   Google Scholar

[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.  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.   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.   Google Scholar

[22]

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

[23]

M. Frémond and S. Miyazaki, "Shape Memory Alloys,", CISM Courses and Lectures, (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.   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.  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.  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.   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.  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.  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.  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.  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.  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.  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.   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.   Google Scholar

[36]

A. Mielke, Evolution of rate-independent systems,, in, 2 (2005), 461.   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.  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.  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, (2010), 185.   Google Scholar

[40]

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

[42]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA, 11 (2004), 151.   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), 438.   Google Scholar

[44]

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

[45]

I. Pawłow and A. Żochowski, A control problem for a thermoelastic system in shape memory materials,, Sūrikaisekikenkyūsho Kōkyūroku, (2011), 8.   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.   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.   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.  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.  doi: 10.1137/080718711.  Google Scholar

[50]

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

[51]

T.Roubíček, Models of microstructure evolution in shape memory alloys,, in, (2004), 269.   Google Scholar

[52]

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

[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.  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), (2012).   Google Scholar

[57]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part II: Regularization and differentiability,, Preprint SPP1253-119, (2011), 1253.   Google Scholar

[58]

G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part III: Optimality conditions,, Preprint SPP1253-119, (2011), 1253.   Google Scholar

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Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i

[9]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076

[10]

Diego Grandi, Ulisse Stefanelli. The Souza-Auricchio model for shape-memory alloys. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 723-747. doi: 10.3934/dcdss.2015.8.723

[11]

Linxiang Wang, Roderick Melnik. Dynamics of shape memory alloys patches with mechanically induced transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1237-1252. doi: 10.3934/dcds.2006.15.1237

[12]

Shuji Yoshikawa, Irena Pawłow, Wojciech M. Zajączkowski. A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1093-1115. doi: 10.3934/cpaa.2009.8.1093

[13]

Tomáš Roubíček. Modelling of thermodynamics of martensitic transformation in shape-memory alloys. Conference Publications, 2007, 2007 (Special) : 892-902. doi: 10.3934/proc.2007.2007.892

[14]

Alessia Berti, Claudio Giorgi, Elena Vuk. Free energies and pseudo-elastic transitions for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 293-316. doi: 10.3934/dcdss.2013.6.293

[15]

Ferdinando Auricchio, Elena Bonetti. A new "flexible" 3D macroscopic model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 277-291. doi: 10.3934/dcdss.2013.6.277

[16]

Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649

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Takashi Suzuki, Shuji Yoshikawa. Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 209-217. doi: 10.3934/dcdss.2012.5.209

[18]

Ken Shirakawa. Asymptotic stability for dynamical systems associated with the one-dimensional Frémond model of shape memory alloys. Conference Publications, 2003, 2003 (Special) : 798-808. doi: 10.3934/proc.2003.2003.798

[19]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

[20]

Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585

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