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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 |
References:
[1] |
T. Aiki and N. Kenmochi, Some models for shape memory alloys,, Mathematical aspects of modelling structure formation phenomena, 17 (2002), 144.
|
[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.
doi: 10.1007/s00161-003-0127-3. |
[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.
doi: 10.1002/gamm.201110014. |
[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.
doi: 10.1002/nme.619. |
[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 three-dimensional model describing stress-induces solid phase transformation with permanent inelasticity,, Int. J. Plasticity, 23 (2007), 207.
doi: 10.1016/j.ijplas.2006.02.012. |
[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.
doi: 10.1016/j.cma.2009.01.019. |
[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.
doi: 10.1051/m2an/2010063. |
[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.
doi: 10.1142/S0218202511005246. |
[15] |
A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys,, Preprint IMATI-CNR, (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.
|
[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.
doi: 10.1016/j.physd.2009.10.005. |
[18] |
H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,, Math Studies, (1973).
|
[19] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Applied Mathematical Sciences, (1996).
doi: 10.1007/978-1-4612-4048-8. |
[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), 208.
|
[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.
doi: 10.1080/01630569808816840. |
[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.
doi: 10.1016/0362-546X(92)90228-7. |
[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.
doi: 10.1016/j.physb.2011.10.017. |
[25] |
M. Eleuteri, J. Kopfová and P. Krejčí, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam,, Comm. Pure Appl. Anal., 12 (2013), 2973.
doi: 10.3934/cpaa.2013.12.2973. |
[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.
doi: 10.3934/nhm.2011.6.145. |
[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.
|
[29] |
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. |
[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. Google Scholar |
[31] |
F. Falk, Martensitic domain boundaries in shape-memory alloys as solitary waves,, J. Phys. C4 Suppl., 12 (1982), 3. 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.
doi: 10.1088/0953-8984/2/1/005. |
[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. 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.
doi: 10.1007/s00161-011-0221-x. |
[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.
doi: 10.1016/0362-546X(90)90079-V. |
[36] |
K.-H. Hoffmann and D. Tiba, Control of a plate with nonlinear shape memory alloy reinforcements,, Adv. Math. Sci. Appl., 7 (1997), 427.
|
[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.
doi: 10.1002/(SICI)1099-1476(19980510)21:7<589::AID-MMA904>3.0.CO;2-D. |
[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.
|
[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.
doi: 10.1051/m2an/2010024. |
[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.
doi: 10.1177/1081286510386935. |
[41] |
G. A. Maugin, The Thermomechanics of Plasticity and Fracture,, Cambridge Texts in Applied Mathematics. Cambridge University Press, (1992).
doi: 10.1017/CBO9781139172400. |
[42] |
A. Mielke, Evolution of rate-independent systems,, in Handbook of Differential Equations, 2 (2005), 461.
|
[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.
doi: 10.1137/080726215. |
[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.
doi: 10.1137/090750238. |
[45] |
A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys,, Adv. Math. Sci. Appl., 17 (2007), 667.
|
[46] |
A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA, 11 (2004), 151.
doi: 10.1007/s00030-003-1052-7. |
[47] |
L. Paoli and A. Petrov, Global existence result for phase transformations with heat transfer in shape memory alloys,, Preprint WIAS, (2011). Google Scholar |
[48] |
I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials,, Control Cybernet., 29 (2000), 341.
|
[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] |
R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity,, Quaderno 05/2012 del Seminario Matematico di Brescia, (2012), 1. Google Scholar |
[52] |
T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains,, SIAM J. Math. Anal., 42 (2010), 256.
doi: 10.1137/080729992. |
[53] |
T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current,, Zeit. angew. Math. Phys., 61 (2010), 1.
doi: 10.1007/s00033-009-0007-1. |
[54] |
T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis,, Arch. Rat. Mech. Anal., ().
doi: 10.1007/s00205-013-0648-2. |
[55] |
A. Sadjadpour and K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys,, Smart Mater. Struct., 16 (2007), 1751.
doi: 10.1088/0964-1726/16/5/030. |
[56] |
J. Sokołowski and J. Sprekels, Control problems with state constraints for shape memory alloys,, Math. Methods Appl. Sci., 17 (1994), 943.
doi: 10.1002/mma.1670171204. |
[57] |
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. |
[58] |
U. Stefanelli, Magnetic control of magnetic shape-memory single crystals,, Phys. B, 407 (2012), 1316.
doi: 10.1016/j.physb.2011.06.043. |
[59] |
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), 50 (2012), 2836.
doi: 10.1137/110839187. |
[60] |
G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, Part II: regularization and differentiability,, Preprint SPP1253-119, (2011), 1253. Google Scholar |
[61] |
G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, Part III: optimality conditions,, Preprint SPP1253-119, (2011), 1253. Google Scholar |
[62] |
S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, Quasi-linear thermoelasticity system arising in shape memory materials,, SIAM J. Math. Anal., 38 (2007), 1733.
doi: 10.1137/060653159. |
show all references
References:
[1] |
T. Aiki and N. Kenmochi, Some models for shape memory alloys,, Mathematical aspects of modelling structure formation phenomena, 17 (2002), 144.
|
[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.
doi: 10.1007/s00161-003-0127-3. |
[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.
doi: 10.1002/gamm.201110014. |
[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.
doi: 10.1002/nme.619. |
[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 three-dimensional model describing stress-induces solid phase transformation with permanent inelasticity,, Int. J. Plasticity, 23 (2007), 207.
doi: 10.1016/j.ijplas.2006.02.012. |
[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.
doi: 10.1016/j.cma.2009.01.019. |
[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.
doi: 10.1051/m2an/2010063. |
[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.
doi: 10.1142/S0218202511005246. |
[15] |
A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys,, Preprint IMATI-CNR, (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.
|
[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.
doi: 10.1016/j.physd.2009.10.005. |
[18] |
H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,, Math Studies, (1973).
|
[19] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Applied Mathematical Sciences, (1996).
doi: 10.1007/978-1-4612-4048-8. |
[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), 208.
|
[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.
doi: 10.1080/01630569808816840. |
[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.
doi: 10.1016/0362-546X(92)90228-7. |
[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.
doi: 10.1016/j.physb.2011.10.017. |
[25] |
M. Eleuteri, J. Kopfová and P. Krejčí, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam,, Comm. Pure Appl. Anal., 12 (2013), 2973.
doi: 10.3934/cpaa.2013.12.2973. |
[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.
doi: 10.3934/nhm.2011.6.145. |
[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.
|
[29] |
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. |
[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. Google Scholar |
[31] |
F. Falk, Martensitic domain boundaries in shape-memory alloys as solitary waves,, J. Phys. C4 Suppl., 12 (1982), 3. 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.
doi: 10.1088/0953-8984/2/1/005. |
[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. 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.
doi: 10.1007/s00161-011-0221-x. |
[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.
doi: 10.1016/0362-546X(90)90079-V. |
[36] |
K.-H. Hoffmann and D. Tiba, Control of a plate with nonlinear shape memory alloy reinforcements,, Adv. Math. Sci. Appl., 7 (1997), 427.
|
[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.
doi: 10.1002/(SICI)1099-1476(19980510)21:7<589::AID-MMA904>3.0.CO;2-D. |
[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.
|
[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.
doi: 10.1051/m2an/2010024. |
[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.
doi: 10.1177/1081286510386935. |
[41] |
G. A. Maugin, The Thermomechanics of Plasticity and Fracture,, Cambridge Texts in Applied Mathematics. Cambridge University Press, (1992).
doi: 10.1017/CBO9781139172400. |
[42] |
A. Mielke, Evolution of rate-independent systems,, in Handbook of Differential Equations, 2 (2005), 461.
|
[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.
doi: 10.1137/080726215. |
[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.
doi: 10.1137/090750238. |
[45] |
A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys,, Adv. Math. Sci. Appl., 17 (2007), 667.
|
[46] |
A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA, 11 (2004), 151.
doi: 10.1007/s00030-003-1052-7. |
[47] |
L. Paoli and A. Petrov, Global existence result for phase transformations with heat transfer in shape memory alloys,, Preprint WIAS, (2011). Google Scholar |
[48] |
I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials,, Control Cybernet., 29 (2000), 341.
|
[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] |
R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity,, Quaderno 05/2012 del Seminario Matematico di Brescia, (2012), 1. Google Scholar |
[52] |
T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains,, SIAM J. Math. Anal., 42 (2010), 256.
doi: 10.1137/080729992. |
[53] |
T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current,, Zeit. angew. Math. Phys., 61 (2010), 1.
doi: 10.1007/s00033-009-0007-1. |
[54] |
T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis,, Arch. Rat. Mech. Anal., ().
doi: 10.1007/s00205-013-0648-2. |
[55] |
A. Sadjadpour and K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys,, Smart Mater. Struct., 16 (2007), 1751.
doi: 10.1088/0964-1726/16/5/030. |
[56] |
J. Sokołowski and J. Sprekels, Control problems with state constraints for shape memory alloys,, Math. Methods Appl. Sci., 17 (1994), 943.
doi: 10.1002/mma.1670171204. |
[57] |
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. |
[58] |
U. Stefanelli, Magnetic control of magnetic shape-memory single crystals,, Phys. B, 407 (2012), 1316.
doi: 10.1016/j.physb.2011.06.043. |
[59] |
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), 50 (2012), 2836.
doi: 10.1137/110839187. |
[60] |
G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, Part II: regularization and differentiability,, Preprint SPP1253-119, (2011), 1253. Google Scholar |
[61] |
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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