American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020459

An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys

Barbora Benešová 1,2,, , Miroslav Frost 2, , Lukáš Kadeřávek 3,4, , Tomáš Roubíček 1,2, and  Petr Sedlák 2,

1. 

Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, CZ-186 75 Praha 8, Czech Republic
2. 

Institute of Thermomechanics, Czech Academy of Sciences, Dolejškova 5, CZ-182 00 Praha 8, Czech Republic
3. 

Institute of Physics, Czech Acad. Sci., Prague, Czech Republic, Na Slovance, CZ-18121 Praha 8, Czech Republic
4. 

Faculty of Nuclear Sciences and Physical Engineering, Czech Tech. Univ., Trojanova 13, CZ-120 00 Praha 2, Czech Republic

* Corresponding author: Barbora Benešová

Received  March 2020 Revised  June 2020 Early access  November 2020

Fund Project: This research has been partially supported from the grants 18-03834S (especially regarding the focus on shape memory alloys, experiments and modeling), 19-04956S (especially regarding the focus on the dynamic and nonlinear behavior) of Czech Science Foundation and LTAUSA18199 (especially regarding the focus on numerics) of MSMT CR, and also from the institutional support RVO: 61388998. Moreover, T.R. acknowledges a stay at Caltech in 2004 and discussions with Kaushik Bhattacharya about the isothermal variant of this sort of models

A phenomenological model for polycrystalline NiTi shape-memory alloys with a refined dissipation function is here enhanced by a thermomechanical coupling and rigorously analyzed as far as existence of weak solutions and numerical stability and convergence of the numerical approximation performed by a staggered time discretization. Moreover, the model is verified on one-dimensional computational simulations compared with real laboratory experiments on a NiTi wire.

Keywords: Martensitic transformation, Nitinol wire, matematical modelling, existence of weak solutions, staggered time discretization, convergence, computational simulations, experiments.
Mathematics Subject Classification: 35Q74, 65M12, 74N10, 80A17.
Citation: Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020459
References:
[1]

R. Alessi and D. Bernardini, Analysis of localization phenomena in shape memory alloys bars by a variational approach, Int. J. Solids Struct., 73/74 (2015), 113-133.  doi: 10.1016/j.ijsolstr.2015.06.021.  Google Scholar

[2]

J. Arghavani, F. Auricchio, R. Naghdabadi, A. Reali and S. Sohrabpour, A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings, Int. J. Plast., 26 (2010), 976-991.   Google Scholar

[3]

K. M. Armattoe, C. Bouby, M. Haboussi and T. B. Zineb, Modeling of latent heat effects on phase transformation in shape memory alloy thin structures, Int. J. Solids Struct., 88/89 (2016), 283-295.  doi: 10.1016/j.ijsolstr.2016.02.024.  Google Scholar

[4]

K. Armattoe, M. Haboussi and T. B. Zineb, A 2D finite element based on a nonlocal constitutive model describing localization and propagation of phase transformation in shape memory alloy thin structures, Int. J. Solids Struct., 51 (2014), 1208-1220.  doi: 10.1016/j.ijsolstr.2013.11.028.  Google Scholar

[5]

F. Auricchio, D. Fugazza and R. Desroches, Rate-dependent thermo-mechanical modelling of superelastic shape-memory alloys for seismic applications, Journal of Intelligent Material Systems and Structures, 19 (2008), 47-61.  doi: 10.1177/1045389X06073426.  Google Scholar

[6]

A. Baêta-Neves, M. Savi and P. Pacheco, On the Fremond's constitutive model for shape memory alloys, Mech. Res. Commun., 31 (2004), 677-688.   Google Scholar

[7]

Z. Bažant and M. Jirásek, Nonlocal integral formulations of plasticity and damage: Survey of progress, J. Eng. Mech., 128 (2002), 1119-1149.   Google Scholar

[8]

N. J. Bechle and S. Kyriakides, Localization in NiTi tubes under bending, Int. J. Sol, 51 (2014), 967-980.  doi: 10.1016/j.ijsolstr.2013.11.023.  Google Scholar

[9]

B. Benešová and T. Roubíček, Micro-to-meso scale limit for shape-memory-alloy models with thermal coupling, Multiscale Model. Simul, 10 (2012), 1059-1089.  doi: 10.1137/110852176.  Google Scholar

[10]

K. Bhattacharya, P. Purohit and B. Craciun, Mobility of twin and phase boundaries, J. de Physique IV, 112 (2003), 163-166.  doi: 10.1051/jp4:2003856.  Google Scholar

[11]

L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169.  doi: 10.1016/0022-1236(89)90005-0.  Google Scholar

[12]

E. Bonetti, M. Frémond and C. Lexcellent, Global existence and uniqueness for a thermomechanical model for shape memory alloys with partition of the strain, Math. Mech. Solids, 11 (2006), 251-275.  doi: 10.1177/1081286506040403.  Google Scholar

[13]

C. Bouvet, S. Calloch and C. Lexcellent, A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loading, Eur. J. Mech. A, 23 (2004), 37-61.  doi: 10.1016/j.euromechsol.2003.09.005.  Google Scholar

[14]

B.-C. Chang, J. A. Shaw and M. A. Iadicola, Thermodynamics of shape memory alloy wire: Modeling, experiments and application, Continuum Mech. Thermodyn., 18 (2006), 83-118.  doi: 10.1007/s00161-006-0022-9.  Google Scholar

[15]

D. Chatziathanasiou, Y. Chemisky, G. Chatzigeorgiou and F. Meragni, Modeling of coupled phase transformation and reorientation in shape memory alloys under non-proportional thermomechanical loading, Int. J. Plast., 82 (2016), 192-224.  doi: 10.1016/j.ijplas.2016.03.005.  Google Scholar

[16]

Y. Chemisky, A. Duval, E. Patoor and T. Ben Zineb, Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation, Mech. Mater., 43 (2011), 361-376.  doi: 10.1016/j.mechmat.2011.04.003.  Google Scholar

[17]

C. Cisse, W. Zaki and T. Ben Zineb, A review of constitutive models and modeling techniques for shape memory alloys, Int. J. Plasticity, 76 (2016), 244-284.  doi: 10.1016/j.ijplas.2015.08.006.  Google Scholar

[18]

C. Cisse, W. Zaki and T. Ben Zineb, A review of modeling techniques for advanced effects in shape memory alloy behavior, Smart Mater. Struct., 25 (2016), 103001. doi: 10.1088/0964-1726/25/10/103001.  Google Scholar

[19]

T. J. Cognata, D. J. Hartl, R. Sheth and C. Dinsmore, A morphing radiator for high-turndown thermal control of crewed space exploration vehicles, in Proc. 23rd AIAA/AHS Adaptive Structures Conf., (2015), 5–9. doi: 10.2514/6.2015-1509.  Google Scholar

[20]

P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys, Nonlinear Analysis, Th. Meth. Appl., 24 (1995), 1565-1579.  doi: 10.1016/0362-546X(94)00097-2.  Google Scholar

[21]

P. Colli, M. Frémond and A. Visintin, Thermo-mechanical evolution of shape memory alloys, Quarterly Appl. Math., 48 (1990), 31-47.  doi: 10.1090/qam/1040232.  Google Scholar

[22]

P. Colli and J. Sprekels, Global existence for a three-dimensional model for the thermo-mechanical evolution of shape memory alloys, Nonlinear Anal., 18 (1992), 873-888.  doi: 10.1016/0362-546X(92)90228-7.  Google Scholar

[23]

P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Part. Diff. Eq., 15 (1990), 737-756.  doi: 10.1080/03605309908820706.  Google Scholar

[24]

F. D. Fischer, J. Svoboda and H. Petryk, Thermodynamic extremal principles for irreversible processes in materials science, Acta Mater., 67 (2014), 1-20.  doi: 10.1016/j.actamat.2013.11.050.  Google Scholar

[25]

M. Frémond, Matériaux à mémoire de forme, C.R. Acad. Sci. Paris Sér.II, 304 (1987), 239-244.   Google Scholar

[26]

M. Frémond and S. Miyazaki, Shape Memory Alloys, Springer, Wien, 1996. Google Scholar

[27]

M. Frost, B. Benešová and P. Sedlák, A microscopically motivated constitutive model for shape memory alloys: Formulation, analysis and computations, Math. Mech. Solids, 21 (2016), 358-382.   Google Scholar

[28]

M. Frost, B. Benešová, H. Seiner, M. Kružík, P. Šittner and P. Sedlák, Thermomechanical model for NiTi-based shape memory alloys covering macroscopic localization of martensitic transformation, Int. J. Solids Struct., (2020). doi: 10.1016/j.ijsolstr.2020.08.012.  Google Scholar

[29]

M. Frost, P. Sedlák, L. Kadeřávek, L. Heller and P. Šittner, Modeling of mechanical response of NiTi shape memory alloy subjected to combined thermal and non-proportional mechanical loading: A case study on helical spring actuator, J. Intel. Mat. Syst. Str., 27 (2016), 1927-1938.   Google Scholar

[30]

M. Frost, P. Sedlák, A. Kruisová and M. Landa, Simulations of self-expanding braided stent using macroscopic model of NiTi shape memory alloys covering R-phase, J. Mater. Eng. Perform., 23 (2014), 2584-2590.  doi: 10.1007/s11665-014-0966-z.  Google Scholar

[31]

C. Grabe and O. T. Bruhns, On the viscous and strain rate dependent behavior of polycrystalline NiTi, Int. J. Solids Struct., 45 (2008), 1876-1895.  doi: 10.1016/j.ijsolstr.2007.10.029.  Google Scholar

[32]

X. Gu, W. Zaki, C. Morin, Z. Moumni and W. Zhang, Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases, Int. J. Solids Struct., 54 (2015), 28-99.  doi: 10.1016/j.ijsolstr.2014.11.005.  Google Scholar

[33]

M. R. Hajidehi and S. Stupkiewicz, Gradient-enhanced model and its micromorphic regularization for simulation of Lüders-like bands in shape memory alloys, Int. J. Solids Struct., 135 (2018), 208-218.   Google Scholar

[34]

B. Halphen and Q. S. Nguyen, Sur les matériaux standard généralisés, J. Mécanique, 14 (1975), 39-63.   Google Scholar

[35]

M. A. Iadicola and J. A. Shaw, Rate and thermal sensitivities of unstable transformation behavior in a shape memory alloy, Int. J. Plast., 20 (2004), 577-605.  doi: 10.1016/S0749-6419(03)00040-8.  Google Scholar

[36]

K. Jacobus, H. Sehitoglu and M. Balzer, Effect of stress state on the stress-induced martensitic transformation in polycrystalline Ni-Ti alloy, Metall, 27 (1996), 3066-3073.  doi: 10.1007/BF02663855.  Google Scholar

[37]

J. M. Jani, M. Leary, A. Subic and M. A. Gibson, A review of shape memory alloy research, applications and opportunities, Materials and Design, 56 (2014), 1078-1113.   Google Scholar

[38]

D. Jiang, S. Kyriakides and C. M. Landis, Propagation of phase transformation fronts in pseudoelastic niti tubes under uniaxial tension, Extrem Mech. Letters, 15 (2017), 113-121.  doi: 10.1016/j.eml.2017.06.006.  Google Scholar

[39]

M. Jirásek and S. Rolshoven, Localization properties of strain-softening gradient plasticity models, Part Ⅱ: Theories with gradients of internal variables, Int. J. Solids Struct., 46 (2009), 2239-2254.   Google Scholar

[40]

P. Junker and K. Hackl, About the influence of heat conductivity on the mechanical behavior of poly-crystalline shape memory alloys, Int. J. Structural Changes in Solids, 3 (2011), 49-62.   Google Scholar

[41]

P. Junker, J. Makowski and K. Hackl, The principle of the minimum of the dissipation potential for non-isothermal processes, Continuum Mech. Thermodyn., 26 (2014), 259-268.  doi: 10.1007/s00161-013-0299-4.  Google Scholar

[42]

A. Kelly, A. P. Stebner and K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys that accounts for initiation and saturation of phase transformation, J. Mech. Phys. Solids, 97 (2016), 197-224.  doi: 10.1016/j.jmps.2016.02.007.  Google Scholar

[43]

M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Springer, Cham/Switzerland, 2019. Google Scholar

[44]

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. Mater., 38 (2006), 430-462.  doi: 10.1016/j.mechmat.2005.08.003.  Google Scholar

[45]

D. C. Lagoudas, D. J. Hartl, Y. Chemisky, L. G. Machado and P. Popov, Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys, Int. J. Plast., 32/33 (2012), 155-183.  doi: 10.1016/j.ijplas.2011.10.009.  Google Scholar

[46]

P. Luig and O. T. Bruhns, On the modeling of shape memory alloys using tensorial internal variables, Mater. Sci. Engr. A, 481/482 (2008), 379-383.  doi: 10.1016/j.msea.2007.03.123.  Google Scholar

[47]

G. A. Maugin, The Thermomechanics of Plasticity and Fracture, Cambridge Univ. Press, 1992. doi: 10.1017/CBO9781139172400.  Google Scholar

[48]

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

[49]

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

[50]

A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Springer New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[51]

Q. S. Nguyen, Stability and Nonlinear Solid Mechanics, J.Wiley, Chichester, 2000. Google Scholar

[52]

K. Otsuka and C. M. Wayman, Shape Memory Materials, Cambridge Univ. Press, 1998. Google Scholar

[53]

H. Petryk, Incremental energy minimization in dissipative solids, R. C. Mécanique, 331 (2003), 469-474.  doi: 10.1016/S1631-0721(03)00109-8.  Google Scholar

[54]

E. A. Pieczyska, H. Tobushi and K. Kulasinski, Development of transformation bands in TiNi SMA for various stress and strain rates studied by a fast and sensitive infrared camera, Smart Mater. Struct., 22 (2013), 035007. doi: 10.1088/0964-1726/22/3/035007.  Google Scholar

[55]

M. Razaee-Hajidehi, K. Tůma and S. Stupkiewicz, Gradient-enhanced thermomechanical 3D model for simulation of transformation patterns in pseudoelastic shape memory alloys, Int. J. Plasticity, 128 (2020), 102589. doi: 10.1016/j.ijplas.2019.08.014.  Google Scholar

[56]

B. Reedlunn, C. B. Churchill, E. E. Nelson, J. A. Shaw and S. H. Daly, Tension, compression, and bending of superelastic shape memory alloy tubes, J. Mech. Phys. Solids, 63 (2014), 506-537.  doi: 10.1016/j.jmps.2012.12.012.  Google Scholar

[57]

T. Roubíček, Models of microstructure evolution in shape memory materials, Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, Springer, Dordrecht, 170 (2004), 269–304. doi: 10.1007/1-4020-2623-4_12.  Google Scholar

[58]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013. Google Scholar

[59]

A. Sadjadpour and K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys, Smart Mater. Struct., 16 (2007), 1751-1765.  doi: 10.1088/0964-1726/16/5/030.  Google Scholar

[60]

A. Sadjadpour and K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys: The one-dimensional case, Smart Mater. Struct., 16 (2007), S51–S62. doi: 10.1088/0964-1726/16/1/S06.  Google Scholar

[61]

L. Saint-Sulpice, S. Arbab Chirani and S. Calloch, A 3D super-elastic model for shape memory alloys taking into account progressive strain under cyclic loadings, Mech. Mater., 41 (2009), 12-26.  doi: 10.1016/j.mechmat.2008.07.004.  Google Scholar

[62]

P. Sedlák, M. Frost, B. Benešová, P. Šittner and T. Ben Zineb, Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings, Int. J. Plast., 39 (2012), 132-151.   Google Scholar

[63]

P. Sedmák, J. Pilch, L. Heller, J. Kopeček, J. Wright, P. Sedlák, M. Frost and P. Šittner, Grain-resolved analysis of localized deformation in nickel-titanium wire under tensile load, Science, 353 (2016), 559-562.   Google Scholar

[64]

J. A. Shaw and S. Kyriakides, On the nucleation and propagation of phase transformation fronts in a NiTi alloy, Acta Mater., 45 (1997), 683-700.  doi: 10.1016/S1359-6454(96)00189-9.  Google Scholar

[65]

P. Šittner, Y. Liu and V. Novák, On the origin of Lüders-like deformation of NiTi shape memory alloys, J. Mech. Phys. Solids, 53 (2005), 1719-1746.   Google Scholar

[66]

A. P. Stebner and L. C. Brinson, Explicit finite element implementation of an improved three dimensional constitutive model for shape memory alloys, Comput. Methods Appl. Mech. Eng., 257 (2013), 17-35.  doi: 10.1016/j.cma.2012.12.021.  Google Scholar

[67]

S. Stupkiewicz and H. Petryk, A robust model of pseudoelasticity in shape memory alloys, Int. J. Numer. Meth. Engng., 93 (2013), 747-769.  doi: 10.1002/nme.4405.  Google Scholar

[68]

M. Thomasová, H. Seiner, P. Sedlák, M. Frost, M. Ševčík, I. Szurman, R. Kocich, J. Drahokoupil, P. Šittner and M. Landa, Evolution of macroscopic elastic moduli of martensitic polycrystalline NiTi and NiTiCu shape memory alloys with pseudoplastic straining, Acta Materialia, 123 (2017), 146-156.   Google Scholar

[69]

H. Tobushi, Y. Shimeno, T. Hachisuka and K. Tanaka, Influence of strain rate on superelastic properties of TiNi shape memory alloy, Mech. Mater., 30 (1998), 141-150.  doi: 10.1016/S0167-6636(98)00041-6.  Google Scholar

[70]

J. Uchil, K. P. Mohanchandra, K. Ganesh Kumara, K. K. Mahesh and T. P. Murali, Thermal expansion in various phases of Nitinol using TMA, Physica B, 270 (1999), 289-297.  doi: 10.1016/S0921-4526(99)00186-6.  Google Scholar

[71]

J. Wang, Z. Moumni, W. Zhang, Y. Xu and W. Zaki, A 3D finite-strain-based constitutive model for shape memory alloys accounting for thermomechanical coupling and martensite reorientation, Smart Mater. Struct., 26 (2017), 065006. doi: 10.1088/1361-665X/aa6c17.  Google Scholar

[72]

W. Zaki and Z. Moumni, A three-dimensional model of the thermomechanical behavior of shape memory alloys, J. Mech. Phys. Solids, 55 (2007), 2455-2490.  doi: 10.1016/j.jmps.2007.03.012.  Google Scholar

[73]

X. Zhang, P. Feng, Y. He, T. Yu and Q. Sun, Experimental study on rate dependence of macroscopic domain and stress hysteresis in niti shape memory alloy strips, Int. J. Mech. Sci., 52 (2010), 1660-1670.  doi: 10.1016/j.ijmecsci.2010.08.007.  Google Scholar

show all references

References:
[1]

R. Alessi and D. Bernardini, Analysis of localization phenomena in shape memory alloys bars by a variational approach, Int. J. Solids Struct., 73/74 (2015), 113-133.  doi: 10.1016/j.ijsolstr.2015.06.021.  Google Scholar

[2]

J. Arghavani, F. Auricchio, R. Naghdabadi, A. Reali and S. Sohrabpour, A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings, Int. J. Plast., 26 (2010), 976-991.   Google Scholar

[3]

K. M. Armattoe, C. Bouby, M. Haboussi and T. B. Zineb, Modeling of latent heat effects on phase transformation in shape memory alloy thin structures, Int. J. Solids Struct., 88/89 (2016), 283-295.  doi: 10.1016/j.ijsolstr.2016.02.024.  Google Scholar

[4]

K. Armattoe, M. Haboussi and T. B. Zineb, A 2D finite element based on a nonlocal constitutive model describing localization and propagation of phase transformation in shape memory alloy thin structures, Int. J. Solids Struct., 51 (2014), 1208-1220.  doi: 10.1016/j.ijsolstr.2013.11.028.  Google Scholar

[5]

F. Auricchio, D. Fugazza and R. Desroches, Rate-dependent thermo-mechanical modelling of superelastic shape-memory alloys for seismic applications, Journal of Intelligent Material Systems and Structures, 19 (2008), 47-61.  doi: 10.1177/1045389X06073426.  Google Scholar

[6]

A. Baêta-Neves, M. Savi and P. Pacheco, On the Fremond's constitutive model for shape memory alloys, Mech. Res. Commun., 31 (2004), 677-688.   Google Scholar

[7]

Z. Bažant and M. Jirásek, Nonlocal integral formulations of plasticity and damage: Survey of progress, J. Eng. Mech., 128 (2002), 1119-1149.   Google Scholar

[8]

N. J. Bechle and S. Kyriakides, Localization in NiTi tubes under bending, Int. J. Sol, 51 (2014), 967-980.  doi: 10.1016/j.ijsolstr.2013.11.023.  Google Scholar

[9]

B. Benešová and T. Roubíček, Micro-to-meso scale limit for shape-memory-alloy models with thermal coupling, Multiscale Model. Simul, 10 (2012), 1059-1089.  doi: 10.1137/110852176.  Google Scholar

[10]

K. Bhattacharya, P. Purohit and B. Craciun, Mobility of twin and phase boundaries, J. de Physique IV, 112 (2003), 163-166.  doi: 10.1051/jp4:2003856.  Google Scholar

[11]

L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169.  doi: 10.1016/0022-1236(89)90005-0.  Google Scholar

[12]

E. Bonetti, M. Frémond and C. Lexcellent, Global existence and uniqueness for a thermomechanical model for shape memory alloys with partition of the strain, Math. Mech. Solids, 11 (2006), 251-275.  doi: 10.1177/1081286506040403.  Google Scholar

[13]

C. Bouvet, S. Calloch and C. Lexcellent, A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loading, Eur. J. Mech. A, 23 (2004), 37-61.  doi: 10.1016/j.euromechsol.2003.09.005.  Google Scholar

[14]

B.-C. Chang, J. A. Shaw and M. A. Iadicola, Thermodynamics of shape memory alloy wire: Modeling, experiments and application, Continuum Mech. Thermodyn., 18 (2006), 83-118.  doi: 10.1007/s00161-006-0022-9.  Google Scholar

[15]

D. Chatziathanasiou, Y. Chemisky, G. Chatzigeorgiou and F. Meragni, Modeling of coupled phase transformation and reorientation in shape memory alloys under non-proportional thermomechanical loading, Int. J. Plast., 82 (2016), 192-224.  doi: 10.1016/j.ijplas.2016.03.005.  Google Scholar

[16]

Y. Chemisky, A. Duval, E. Patoor and T. Ben Zineb, Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation, Mech. Mater., 43 (2011), 361-376.  doi: 10.1016/j.mechmat.2011.04.003.  Google Scholar

[17]

C. Cisse, W. Zaki and T. Ben Zineb, A review of constitutive models and modeling techniques for shape memory alloys, Int. J. Plasticity, 76 (2016), 244-284.  doi: 10.1016/j.ijplas.2015.08.006.  Google Scholar

[18]

C. Cisse, W. Zaki and T. Ben Zineb, A review of modeling techniques for advanced effects in shape memory alloy behavior, Smart Mater. Struct., 25 (2016), 103001. doi: 10.1088/0964-1726/25/10/103001.  Google Scholar

[19]

T. J. Cognata, D. J. Hartl, R. Sheth and C. Dinsmore, A morphing radiator for high-turndown thermal control of crewed space exploration vehicles, in Proc. 23rd AIAA/AHS Adaptive Structures Conf., (2015), 5–9. doi: 10.2514/6.2015-1509.  Google Scholar

[20]

P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys, Nonlinear Analysis, Th. Meth. Appl., 24 (1995), 1565-1579.  doi: 10.1016/0362-546X(94)00097-2.  Google Scholar

[21]

P. Colli, M. Frémond and A. Visintin, Thermo-mechanical evolution of shape memory alloys, Quarterly Appl. Math., 48 (1990), 31-47.  doi: 10.1090/qam/1040232.  Google Scholar

[22]

P. Colli and J. Sprekels, Global existence for a three-dimensional model for the thermo-mechanical evolution of shape memory alloys, Nonlinear Anal., 18 (1992), 873-888.  doi: 10.1016/0362-546X(92)90228-7.  Google Scholar

[23]

P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Part. Diff. Eq., 15 (1990), 737-756.  doi: 10.1080/03605309908820706.  Google Scholar

[24]

F. D. Fischer, J. Svoboda and H. Petryk, Thermodynamic extremal principles for irreversible processes in materials science, Acta Mater., 67 (2014), 1-20.  doi: 10.1016/j.actamat.2013.11.050.  Google Scholar

[25]

M. Frémond, Matériaux à mémoire de forme, C.R. Acad. Sci. Paris Sér.II, 304 (1987), 239-244.   Google Scholar

[26]

M. Frémond and S. Miyazaki, Shape Memory Alloys, Springer, Wien, 1996. Google Scholar

[27]

M. Frost, B. Benešová and P. Sedlák, A microscopically motivated constitutive model for shape memory alloys: Formulation, analysis and computations, Math. Mech. Solids, 21 (2016), 358-382.   Google Scholar

[28]

M. Frost, B. Benešová, H. Seiner, M. Kružík, P. Šittner and P. Sedlák, Thermomechanical model for NiTi-based shape memory alloys covering macroscopic localization of martensitic transformation, Int. J. Solids Struct., (2020). doi: 10.1016/j.ijsolstr.2020.08.012.  Google Scholar

[29]

M. Frost, P. Sedlák, L. Kadeřávek, L. Heller and P. Šittner, Modeling of mechanical response of NiTi shape memory alloy subjected to combined thermal and non-proportional mechanical loading: A case study on helical spring actuator, J. Intel. Mat. Syst. Str., 27 (2016), 1927-1938.   Google Scholar

[30]

M. Frost, P. Sedlák, A. Kruisová and M. Landa, Simulations of self-expanding braided stent using macroscopic model of NiTi shape memory alloys covering R-phase, J. Mater. Eng. Perform., 23 (2014), 2584-2590.  doi: 10.1007/s11665-014-0966-z.  Google Scholar

[31]

C. Grabe and O. T. Bruhns, On the viscous and strain rate dependent behavior of polycrystalline NiTi, Int. J. Solids Struct., 45 (2008), 1876-1895.  doi: 10.1016/j.ijsolstr.2007.10.029.  Google Scholar

[32]

X. Gu, W. Zaki, C. Morin, Z. Moumni and W. Zhang, Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases, Int. J. Solids Struct., 54 (2015), 28-99.  doi: 10.1016/j.ijsolstr.2014.11.005.  Google Scholar

[33]

M. R. Hajidehi and S. Stupkiewicz, Gradient-enhanced model and its micromorphic regularization for simulation of Lüders-like bands in shape memory alloys, Int. J. Solids Struct., 135 (2018), 208-218.   Google Scholar

[34]

B. Halphen and Q. S. Nguyen, Sur les matériaux standard généralisés, J. Mécanique, 14 (1975), 39-63.   Google Scholar

[35]

M. A. Iadicola and J. A. Shaw, Rate and thermal sensitivities of unstable transformation behavior in a shape memory alloy, Int. J. Plast., 20 (2004), 577-605.  doi: 10.1016/S0749-6419(03)00040-8.  Google Scholar

[36]

K. Jacobus, H. Sehitoglu and M. Balzer, Effect of stress state on the stress-induced martensitic transformation in polycrystalline Ni-Ti alloy, Metall, 27 (1996), 3066-3073.  doi: 10.1007/BF02663855.  Google Scholar

[37]

J. M. Jani, M. Leary, A. Subic and M. A. Gibson, A review of shape memory alloy research, applications and opportunities, Materials and Design, 56 (2014), 1078-1113.   Google Scholar

[38]

D. Jiang, S. Kyriakides and C. M. Landis, Propagation of phase transformation fronts in pseudoelastic niti tubes under uniaxial tension, Extrem Mech. Letters, 15 (2017), 113-121.  doi: 10.1016/j.eml.2017.06.006.  Google Scholar

[39]

M. Jirásek and S. Rolshoven, Localization properties of strain-softening gradient plasticity models, Part Ⅱ: Theories with gradients of internal variables, Int. J. Solids Struct., 46 (2009), 2239-2254.   Google Scholar

[40]

P. Junker and K. Hackl, About the influence of heat conductivity on the mechanical behavior of poly-crystalline shape memory alloys, Int. J. Structural Changes in Solids, 3 (2011), 49-62.   Google Scholar

[41]

P. Junker, J. Makowski and K. Hackl, The principle of the minimum of the dissipation potential for non-isothermal processes, Continuum Mech. Thermodyn., 26 (2014), 259-268.  doi: 10.1007/s00161-013-0299-4.  Google Scholar

[42]

A. Kelly, A. P. Stebner and K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys that accounts for initiation and saturation of phase transformation, J. Mech. Phys. Solids, 97 (2016), 197-224.  doi: 10.1016/j.jmps.2016.02.007.  Google Scholar

[43]

M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Springer, Cham/Switzerland, 2019. Google Scholar

[44]

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. Mater., 38 (2006), 430-462.  doi: 10.1016/j.mechmat.2005.08.003.  Google Scholar

[45]

D. C. Lagoudas, D. J. Hartl, Y. Chemisky, L. G. Machado and P. Popov, Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys, Int. J. Plast., 32/33 (2012), 155-183.  doi: 10.1016/j.ijplas.2011.10.009.  Google Scholar

[46]

P. Luig and O. T. Bruhns, On the modeling of shape memory alloys using tensorial internal variables, Mater. Sci. Engr. A, 481/482 (2008), 379-383.  doi: 10.1016/j.msea.2007.03.123.  Google Scholar

[47]

G. A. Maugin, The Thermomechanics of Plasticity and Fracture, Cambridge Univ. Press, 1992. doi: 10.1017/CBO9781139172400.  Google Scholar

[48]

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

[49]

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

[50]

A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Springer New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[51]

Q. S. Nguyen, Stability and Nonlinear Solid Mechanics, J.Wiley, Chichester, 2000. Google Scholar

[52]

K. Otsuka and C. M. Wayman, Shape Memory Materials, Cambridge Univ. Press, 1998. Google Scholar

[53]

H. Petryk, Incremental energy minimization in dissipative solids, R. C. Mécanique, 331 (2003), 469-474.  doi: 10.1016/S1631-0721(03)00109-8.  Google Scholar

[54]

E. A. Pieczyska, H. Tobushi and K. Kulasinski, Development of transformation bands in TiNi SMA for various stress and strain rates studied by a fast and sensitive infrared camera, Smart Mater. Struct., 22 (2013), 035007. doi: 10.1088/0964-1726/22/3/035007.  Google Scholar

[55]

M. Razaee-Hajidehi, K. Tůma and S. Stupkiewicz, Gradient-enhanced thermomechanical 3D model for simulation of transformation patterns in pseudoelastic shape memory alloys, Int. J. Plasticity, 128 (2020), 102589. doi: 10.1016/j.ijplas.2019.08.014.  Google Scholar

[56]

B. Reedlunn, C. B. Churchill, E. E. Nelson, J. A. Shaw and S. H. Daly, Tension, compression, and bending of superelastic shape memory alloy tubes, J. Mech. Phys. Solids, 63 (2014), 506-537.  doi: 10.1016/j.jmps.2012.12.012.  Google Scholar

[57]

T. Roubíček, Models of microstructure evolution in shape memory materials, Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, Springer, Dordrecht, 170 (2004), 269–304. doi: 10.1007/1-4020-2623-4_12.  Google Scholar

[58]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013. Google Scholar

[59]

A. Sadjadpour and K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys, Smart Mater. Struct., 16 (2007), 1751-1765.  doi: 10.1088/0964-1726/16/5/030.  Google Scholar

[60]

A. Sadjadpour and K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys: The one-dimensional case, Smart Mater. Struct., 16 (2007), S51–S62. doi: 10.1088/0964-1726/16/1/S06.  Google Scholar

[61]

L. Saint-Sulpice, S. Arbab Chirani and S. Calloch, A 3D super-elastic model for shape memory alloys taking into account progressive strain under cyclic loadings, Mech. Mater., 41 (2009), 12-26.  doi: 10.1016/j.mechmat.2008.07.004.  Google Scholar

[62]

P. Sedlák, M. Frost, B. Benešová, P. Šittner and T. Ben Zineb, Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings, Int. J. Plast., 39 (2012), 132-151.   Google Scholar

[63]

P. Sedmák, J. Pilch, L. Heller, J. Kopeček, J. Wright, P. Sedlák, M. Frost and P. Šittner, Grain-resolved analysis of localized deformation in nickel-titanium wire under tensile load, Science, 353 (2016), 559-562.   Google Scholar

[64]

J. A. Shaw and S. Kyriakides, On the nucleation and propagation of phase transformation fronts in a NiTi alloy, Acta Mater., 45 (1997), 683-700.  doi: 10.1016/S1359-6454(96)00189-9.  Google Scholar

[65]

P. Šittner, Y. Liu and V. Novák, On the origin of Lüders-like deformation of NiTi shape memory alloys, J. Mech. Phys. Solids, 53 (2005), 1719-1746.   Google Scholar

[66]

A. P. Stebner and L. C. Brinson, Explicit finite element implementation of an improved three dimensional constitutive model for shape memory alloys, Comput. Methods Appl. Mech. Eng., 257 (2013), 17-35.  doi: 10.1016/j.cma.2012.12.021.  Google Scholar

[67]

S. Stupkiewicz and H. Petryk, A robust model of pseudoelasticity in shape memory alloys, Int. J. Numer. Meth. Engng., 93 (2013), 747-769.  doi: 10.1002/nme.4405.  Google Scholar

[68]

M. Thomasová, H. Seiner, P. Sedlák, M. Frost, M. Ševčík, I. Szurman, R. Kocich, J. Drahokoupil, P. Šittner and M. Landa, Evolution of macroscopic elastic moduli of martensitic polycrystalline NiTi and NiTiCu shape memory alloys with pseudoplastic straining, Acta Materialia, 123 (2017), 146-156.   Google Scholar

[69]

H. Tobushi, Y. Shimeno, T. Hachisuka and K. Tanaka, Influence of strain rate on superelastic properties of TiNi shape memory alloy, Mech. Mater., 30 (1998), 141-150.  doi: 10.1016/S0167-6636(98)00041-6.  Google Scholar

[70]

J. Uchil, K. P. Mohanchandra, K. Ganesh Kumara, K. K. Mahesh and T. P. Murali, Thermal expansion in various phases of Nitinol using TMA, Physica B, 270 (1999), 289-297.  doi: 10.1016/S0921-4526(99)00186-6.  Google Scholar

[71]

J. Wang, Z. Moumni, W. Zhang, Y. Xu and W. Zaki, A 3D finite-strain-based constitutive model for shape memory alloys accounting for thermomechanical coupling and martensite reorientation, Smart Mater. Struct., 26 (2017), 065006. doi: 10.1088/1361-665X/aa6c17.  Google Scholar

[72]

W. Zaki and Z. Moumni, A three-dimensional model of the thermomechanical behavior of shape memory alloys, J. Mech. Phys. Solids, 55 (2007), 2455-2490.  doi: 10.1016/j.jmps.2007.03.012.  Google Scholar

[73]

X. Zhang, P. Feng, Y. He, T. Yu and Q. Sun, Experimental study on rate dependence of macroscopic domain and stress hysteresis in niti shape memory alloy strips, Int. J. Mech. Sci., 52 (2010), 1660-1670.  doi: 10.1016/j.ijmecsci.2010.08.007.  Google Scholar

Figure 1.  Dependence of stress on strain for one isothermal simulation and adiabatic simulations at three different strain rates; see text for details
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Figure 2.  Evolution of temperature (left vertical axis) and volume fraction of martensite (right vertical axis) with relative progress of the stretching (horizontal axis: 0 = start of stretching, 0.5 = maximum tension, 1 = complete unstretching) at three different strain rates
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Figure 3.  Evolution of cumulative sum of three respective contributions to the total heat – latent heat, energy dissipated in rate-independent (RI) processes and in rate-dependent (RD) processes, respectively – integrated over the whole wire at three different strain rates. Relative progress of stretching on the horizontal axis (0 denotes start of stretching, 0.5 maximum tension, 1 complete unstretching). The RI contribution for $ v_2 $ basically coincides with the RI contribution of $ v_1 $
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Figure 4.  Comparison of in-situ experimental (left) and computational (right) evolution of spatial distribution of strain during loading at the total-strain rate $ v_{\rm s} = 10^{-3}\, {\rm s}^{-1} $ of a thin NiTi wire
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Fig. 4 but under loading at the higher speed, $ v_{\rm f} = 10^{-1}\, {\rm s}^{-1} $">Figure 5.  The same as in Fig. 4 but under loading at the higher speed, $ v_{\rm f} = 10^{-1}\, {\rm s}^{-1} $
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Table 1.  Main nomenclature for variables and for the data of the model
$ \varOmega $ a domain in $ \mathbb{R}^d $ (a specimen) $ K $ bulk modulus
$ \varGamma $ the boundary of $ \varOmega $ $ G=G(\xi) $ (a part of) shear modulus
$ u $ displacement $ \mathcal{G}=\mathcal{G}(\xi, e) $ shear strain energy
$ \varepsilon(u) $ total (small) strain $ =\frac12(\nabla u)^\top\!{+}\frac12\nabla u $ $ c_{\rm v} $ heat capacity,
$ e $, $ \pi $ elastic and inelastic strains $ \mathbb{K} $ heat-transfer tensor
$ \xi $ volume fraction of martensite $ \theta_{_{\rm D}} $ prescribed outer temperature
$ \theta $ temperature $ u_{_{\rm D}} $ prescribed boundary displacement
$ s $ entropy $ s_\text{AM} $ entropy of A/M-transformation
$ w $ heat part of the internal energy $ \theta_\mathrm{tr} $ the transformation temperature
$ \eta $ a regularization parameter for approximation $ \tau $ a time step for time discretisation
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$ \varOmega $ a domain in $ \mathbb{R}^d $ (a specimen) $ K $ bulk modulus
$ \varGamma $ the boundary of $ \varOmega $ $ G=G(\xi) $ (a part of) shear modulus
$ u $ displacement $ \mathcal{G}=\mathcal{G}(\xi, e) $ shear strain energy
$ \varepsilon(u) $ total (small) strain $ =\frac12(\nabla u)^\top\!{+}\frac12\nabla u $ $ c_{\rm v} $ heat capacity,
$ e $, $ \pi $ elastic and inelastic strains $ \mathbb{K} $ heat-transfer tensor
$ \xi $ volume fraction of martensite $ \theta_{_{\rm D}} $ prescribed outer temperature
$ \theta $ temperature $ u_{_{\rm D}} $ prescribed boundary displacement
$ s $ entropy $ s_\text{AM} $ entropy of A/M-transformation
$ w $ heat part of the internal energy $ \theta_\mathrm{tr} $ the transformation temperature
$ \eta $ a regularization parameter for approximation $ \tau $ a time step for time discretisation
Table 2.  Material parameters used in simulations
Parameter Value Unit Parameter Value Unit
$ l $ $ 25 $ [mm] $ \rho $ $ 0.1 $ [mm]
$ f^{\rm tens} $ $ 0.08 $ [1] $ K $ $ 148 $ [GPa]
$ G_{\rm A} $ $ 25 $ [GPa] $ G_{\rm M} $ $ 12 $ [GPa]
$ c_{\rm v} $ $ 5 $ [MPa/$ ^\circ $C] $ E^{\rm nl} $ $ 80 $ [MPa]
$ E^{\rm int} $ $ 10 $ [MPa] $ s_{\rm AM}^{} $ $ 360 $ [kPa/$ ^\circ $C]
$ \beta $ $ 5 $ [1] $ \mu $ $ 100 $ [kPa$ \, $s]
$ \sigma^{\rm reo}_{\rm tr} $ $ 90 $ [MPa] $ \Sigma^{\rm reo} $ $ -10 $ [kPa/$ ^\circ $C]
$ A_{\rm s} $ $ -15 $ [$ ^\circ $C] $ A_{\rm f} $ $ -5 $ [$ ^\circ $C]
$ M_{\rm s} $ $ -20 $ [$ ^\circ $C] $ M_{\rm f} $ $ -25 $ [$ ^\circ $C]
$ T_0 $ $ -20 $ [$ ^\circ $C] $ \mathbb{K}_{11} $ $ 90 $ [W/(m$ ^\circ $C)]
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Parameter Value Unit Parameter Value Unit
$ l $ $ 25 $ [mm] $ \rho $ $ 0.1 $ [mm]
$ f^{\rm tens} $ $ 0.08 $ [1] $ K $ $ 148 $ [GPa]
$ G_{\rm A} $ $ 25 $ [GPa] $ G_{\rm M} $ $ 12 $ [GPa]
$ c_{\rm v} $ $ 5 $ [MPa/$ ^\circ $C] $ E^{\rm nl} $ $ 80 $ [MPa]
$ E^{\rm int} $ $ 10 $ [MPa] $ s_{\rm AM}^{} $ $ 360 $ [kPa/$ ^\circ $C]
$ \beta $ $ 5 $ [1] $ \mu $ $ 100 $ [kPa$ \, $s]
$ \sigma^{\rm reo}_{\rm tr} $ $ 90 $ [MPa] $ \Sigma^{\rm reo} $ $ -10 $ [kPa/$ ^\circ $C]
$ A_{\rm s} $ $ -15 $ [$ ^\circ $C] $ A_{\rm f} $ $ -5 $ [$ ^\circ $C]
$ M_{\rm s} $ $ -20 $ [$ ^\circ $C] $ M_{\rm f} $ $ -25 $ [$ ^\circ $C]
$ T_0 $ $ -20 $ [$ ^\circ $C] $ \mathbb{K}_{11} $ $ 90 $ [W/(m$ ^\circ $C)]
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