An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. (English) Zbl 1476.35262
Summary: 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.
MSC:
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 74N10 | Displacive transformations in solids |
| 80A17 | Thermodynamics of continua |
Keywords:
martensitic transformation; Nitinol wire; matematical modelling; existence of weak solutions; staggered time discretization; convergence; computational simulations; experimentsReferences:
| [1] | R. Alessi; D. Bernardini, Analysis of localization phenomena in shape memory alloys bars by a variational approach, Int. J. Solids Struct., 73/74, 113-133 (2015) · doi:10.1016/j.ijsolstr.2015.06.021 |
| [2] | J. Arghavani; F. Auricchio; R. Naghdabadi; A. Reali; S. Sohrabpour, A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings, Int. J. Plast., 26, 976-991 (2010) · Zbl 1452.74088 |
| [3] | K. M. Armattoe; C. Bouby; M. Haboussi; T. B. Zineb, Modeling of latent heat effects on phase transformation in shape memory alloy thin structures, Int. J. Solids Struct., 88/89, 283-295 (2016) · doi:10.1016/j.ijsolstr.2016.02.024 |
| [4] | K. Armattoe; M. Haboussi; 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, 1208-1220 (2014) · doi:10.1016/j.ijsolstr.2013.11.028 |
| [5] | F. Auricchio; D. Fugazza; R. Desroches, Rate-dependent thermo-mechanical modelling of superelastic shape-memory alloys for seismic applications, Journal of Intelligent Material Systems and Structures, 19, 47-61 (2008) · doi:10.1177/1045389X06073426 |
| [6] | A. Baêta-Neves; M. Savi; P. Pacheco, On the Fremond’s constitutive model for shape memory alloys, Mech. Res. Commun., 31, 677-688 (2004) · Zbl 1098.74519 |
| [7] | Z. Bažant; M. Jirásek, Nonlocal integral formulations of plasticity and damage: Survey of progress, J. Eng. Mech., 128, 1119-1149 (2002) |
| [8] | N. J. Bechle; S. Kyriakides, Localization in NiTi tubes under bending, Int. J. Sol, 51, 967-980 (2014) · doi:10.1016/j.ijsolstr.2013.11.023 |
| [9] | B. Benešová; T. Roubíček, Micro-to-meso scale limit for shape-memory-alloy models with thermal coupling, Multiscale Model. Simul, 10, 1059-1089 (2012) · Zbl 1255.74055 · doi:10.1137/110852176 |
| [10] | K. Bhattacharya; P. Purohit; B. Craciun, Mobility of twin and phase boundaries, J. de Physique IV, 112, 163-166 (2003) · doi:10.1051/jp4:2003856 |
| [11] | L. Boccardo; T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87, 149-169 (1989) · Zbl 0707.35060 · doi:10.1016/0022-1236(89)90005-0 |
| [12] | E. Bonetti; M. Frémond; C. Lexcellent, Global existence and uniqueness for a thermomechanical model for shape memory alloys with partition of the strain, Math. Mech. Solids, 11, 251-275 (2006) · Zbl 1143.74021 · doi:10.1177/1081286506040403 |
| [13] | C. Bouvet; S. Calloch; C. Lexcellent, A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loading, Eur. J. Mech. A, 23, 37-61 (2004) · Zbl 1057.74027 · doi:10.1016/j.euromechsol.2003.09.005 |
| [14] | B.-C. Chang; J. A. Shaw; M. A. Iadicola, Thermodynamics of shape memory alloy wire: Modeling, experiments and application, Continuum Mech. Thermodyn., 18, 83-118 (2006) · Zbl 1149.74381 · doi:10.1007/s00161-006-0022-9 |
| [15] | D. Chatziathanasiou; Y. Chemisky; G. Chatzigeorgiou; F. Meragni, Modeling of coupled phase transformation and reorientation in shape memory alloys under non-proportional thermomechanical loading, Int. J. Plast., 82, 192-224 (2016) · doi:10.1016/j.ijplas.2016.03.005 |
| [16] | Y. Chemisky; A. Duval; E. Patoor; T. Ben Zineb, Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation, Mech. Mater., 43, 361-376 (2011) · doi:10.1016/j.mechmat.2011.04.003 |
| [17] | C. Cisse; W. Zaki; T. Ben Zineb, A review of constitutive models and modeling techniques for shape memory alloys, Int. J. Plasticity, 76, 244-284 (2016) · doi:10.1016/j.ijplas.2015.08.006 |
| [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. |
| [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. |
| [20] | P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys, Nonlinear Analysis, Th. Meth. Appl., 24, 1565-1579 (1995) · Zbl 0832.35066 · doi:10.1016/0362-546X(94)00097-2 |
| [21] | P. Colli; M. Frémond; A. Visintin, Thermo-mechanical evolution of shape memory alloys, Quarterly Appl. Math., 48, 31-47 (1990) · Zbl 0693.73003 · doi:10.1090/qam/1040232 |
| [22] | P. Colli; J. Sprekels, Global existence for a three-dimensional model for the thermo-mechanical evolution of shape memory alloys, Nonlinear Anal., 18, 873-888 (1992) · Zbl 0766.35062 · doi:10.1016/0362-546X(92)90228-7 |
| [23] | P. Colli; A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Part. Diff. Eq., 15, 737-756 (1990) · Zbl 0707.34053 · doi:10.1080/03605309908820706 |
| [24] | F. D. Fischer; J. Svoboda; H. Petryk, Thermodynamic extremal principles for irreversible processes in materials science, Acta Mater., 67, 1-20 (2014) · doi:10.1016/j.actamat.2013.11.050 |
| [25] | M. Frémond, Matériaux à mémoire de forme, C.R. Acad. Sci. Paris Sér.II, 304, 239-244 (1987) |
| [26] | M. Frémond and S. Miyazaki, Shape Memory Alloys, Springer, Wien, 1996. |
| [27] | M. Frost; B. Benešová; P. Sedlák, A microscopically motivated constitutive model for shape memory alloys: Formulation, analysis and computations, Math. Mech. Solids, 21, 358-382 (2016) · Zbl 1370.74012 |
| [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). |
| [29] | M. Frost; P. Sedlák; L. Kadeřávek; L. Heller; 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, 1927-1938 (2016) |
| [30] | M. Frost; P. Sedlák; A. Kruisová; M. Landa, Simulations of self-expanding braided stent using macroscopic model of NiTi shape memory alloys covering R-phase, J. Mater. Eng. Perform., 23, 2584-2590 (2014) · doi:10.1007/s11665-014-0966-z |
| [31] | C. Grabe; O. T. Bruhns, On the viscous and strain rate dependent behavior of polycrystalline NiTi, Int. J. Solids Struct., 45, 1876-1895 (2008) · Zbl 1149.74015 · doi:10.1016/j.ijsolstr.2007.10.029 |
| [32] | X. Gu; W. Zaki; C. Morin; Z. Moumni; W. Zhang, Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases, Int. J. Solids Struct., 54, 28-99 (2015) · doi:10.1016/j.ijsolstr.2014.11.005 |
| [33] | M. R. Hajidehi; S. Stupkiewicz, Gradient-enhanced model and its micromorphic regularization for simulation of Lüders-like bands in shape memory alloys, Int. J. Solids Struct., 135, 208-218 (2018) |
| [34] | B. Halphen; Q. S. Nguyen, Sur les matériaux standard généralisés, J. Mécanique, 14, 39-63 (1975) · Zbl 0308.73017 |
| [35] | M. A. Iadicola; J. A. Shaw, Rate and thermal sensitivities of unstable transformation behavior in a shape memory alloy, Int. J. Plast., 20, 577-605 (2004) · Zbl 1134.74345 · doi:10.1016/S0749-6419(03)00040-8 |
| [36] | K. Jacobus; H. Sehitoglu; M. Balzer, Effect of stress state on the stress-induced martensitic transformation in polycrystalline Ni-Ti alloy, Metall, 27, 3066-3073 (1996) · doi:10.1007/BF02663855 |
| [37] | J. M. Jani; M. Leary; A. Subic; M. A. Gibson, A review of shape memory alloy research, applications and opportunities, Materials and Design, 56, 1078-1113 (2014) |
| [38] | D. Jiang; S. Kyriakides; C. M. Landis, Propagation of phase transformation fronts in pseudoelastic niti tubes under uniaxial tension, Extrem Mech. Letters, 15, 113-121 (2017) · doi:10.1016/j.eml.2017.06.006 |
| [39] | M. Jirásek; S. Rolshoven, Localization properties of strain-softening gradient plasticity models, Part Ⅱ: Theories with gradients of internal variables, Int. J. Solids Struct., 46, 2239-2254 (2009) · Zbl 1217.74024 |
| [40] | P. Junker; 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, 49-62 (2011) |
| [41] | P. Junker; J. Makowski; K. Hackl, The principle of the minimum of the dissipation potential for non-isothermal processes, Continuum Mech. Thermodyn., 26, 259-268 (2014) · Zbl 1341.80008 · doi:10.1007/s00161-013-0299-4 |
| [42] | A. Kelly; A. P. Stebner; 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, 197-224 (2016) · doi:10.1016/j.jmps.2016.02.007 |
| [43] | M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Springer, Cham/Switzerland, 2019. · Zbl 1416.74001 |
| [44] | D. C. Lagoudas; P. B. Entchev; P. Popov; E. Patoor; L. C. Brinson; X. Gao, Shape memory alloys, Part Ⅱ: Modeling of polycrystals, Mech. Mater., 38, 430-462 (2006) · doi:10.1016/j.mechmat.2005.08.003 |
| [45] | D. C. Lagoudas; D. J. Hartl; Y. Chemisky; L. G. Machado; P. Popov, Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys, Int. J. Plast., 32/33, 155-183 (2012) · doi:10.1016/j.ijplas.2011.10.009 |
| [46] | P. Luig; O. T. Bruhns, On the modeling of shape memory alloys using tensorial internal variables, Mater. Sci. Engr. A, 481/482, 379-383 (2008) · doi:10.1016/j.msea.2007.03.123 |
| [47] | G. A. Maugin, The Thermomechanics of Plasticity and Fracture, Cambridge Univ. Press, 1992. · Zbl 0753.73001 |
| [48] | A. Mielke; L. Paoli; A. Petrov, On existence and approximation for a 3D model of thermally induced phase transformations in shape-memory alloys, SIAM J. Math. Anal., 41, 1388-1414 (2009) · Zbl 1201.49011 · doi:10.1137/080726215 |
| [49] | A. Mielke; A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl., 17, 667-685 (2007) · Zbl 1138.49014 |
| [50] | A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Springer New York, 2015. · Zbl 1339.35006 |
| [51] | Q. S. Nguyen, Stability and Nonlinear Solid Mechanics, J.Wiley, Chichester, 2000. · Zbl 0949.74001 |
| [52] | K. Otsuka and C. M. Wayman, Shape Memory Materials, Cambridge Univ. Press, 1998. |
| [53] | H. Petryk, Incremental energy minimization in dissipative solids, R. C. Mécanique, 331, 469-474 (2003) · Zbl 1177.74171 · doi:10.1016/S1631-0721(03)00109-8 |
| [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. |
| [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. |
| [56] | B. Reedlunn; C. B. Churchill; E. E. Nelson; J. A. Shaw; S. H. Daly, Tension, compression, and bending of superelastic shape memory alloy tubes, J. Mech. Phys. Solids, 63, 506-537 (2014) · doi:10.1016/j.jmps.2012.12.012 |
| [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. |
| [58] | T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013. · Zbl 1270.35005 |
| [59] | A. Sadjadpour; K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys, Smart Mater. Struct., 16, 1751-1765 (2007) · doi:10.1088/0964-1726/16/5/030 |
| [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. |
| [61] | L. Saint-Sulpice; S. Arbab Chirani; S. Calloch, A 3D super-elastic model for shape memory alloys taking into account progressive strain under cyclic loadings, Mech. Mater., 41, 12-26 (2009) · doi:10.1016/j.mechmat.2008.07.004 |
| [62] | P. Sedlák; M. Frost; B. Benešová; P. Šittner; 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, 132-151 (2012) |
| [63] | P. Sedmák; J. Pilch; L. Heller; J. Kopeček; J. Wright; P. Sedlák; M. Frost; P. Šittner, Grain-resolved analysis of localized deformation in nickel-titanium wire under tensile load, Science, 353, 559-562 (2016) |
| [64] | J. A. Shaw; S. Kyriakides, On the nucleation and propagation of phase transformation fronts in a NiTi alloy, Acta Mater., 45, 683-700 (1997) · doi:10.1016/S1359-6454(96)00189-9 |
| [65] | P. Šittner; Y. Liu; V. Novák, On the origin of Lüders-like deformation of NiTi shape memory alloys, J. Mech. Phys. Solids, 53, 1719-1746 (2005) · Zbl 1120.74402 |
| [66] | A. P. Stebner; L. C. Brinson, Explicit finite element implementation of an improved three dimensional constitutive model for shape memory alloys, Comput. Methods Appl. Mech. Eng., 257, 17-35 (2013) · Zbl 1286.74071 · doi:10.1016/j.cma.2012.12.021 |
| [67] | S. Stupkiewicz; H. Petryk, A robust model of pseudoelasticity in shape memory alloys, Int. J. Numer. Meth. Engng., 93, 747-769 (2013) · Zbl 1352.74227 · doi:10.1002/nme.4405 |
| [68] | M. Thomasová; H. Seiner; P. Sedlák; M. Frost; M. Ševčík; I. Szurman; R. Kocich; J. Drahokoupil; P. Šittner; M. Landa, Evolution of macroscopic elastic moduli of martensitic polycrystalline NiTi and NiTiCu shape memory alloys with pseudoplastic straining, Acta Materialia, 123, 146-156 (2017) |
| [69] | H. Tobushi; Y. Shimeno; T. Hachisuka; K. Tanaka, Influence of strain rate on superelastic properties of TiNi shape memory alloy, Mech. Mater., 30, 141-150 (1998) · doi:10.1016/S0167-6636(98)00041-6 |
| [70] | J. Uchil; K. P. Mohanchandra; K. Ganesh Kumara; K. K. Mahesh; T. P. Murali, Thermal expansion in various phases of Nitinol using TMA, Physica B, 270, 289-297 (1999) · doi:10.1016/S0921-4526(99)00186-6 |
| [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. |
| [72] | W. Zaki; Z. Moumni, A three-dimensional model of the thermomechanical behavior of shape memory alloys, J. Mech. Phys. Solids, 55, 2455-2490 (2007) · Zbl 1171.74032 · doi:10.1016/j.jmps.2007.03.012 |
| [73] | X. Zhang; P. Feng; Y. He; T. Yu; Q. Sun, Experimental study on rate dependence of macroscopic domain and stress hysteresis in niti shape memory alloy strips, Int. J. Mech. Sci., 52, 1660-1670 (2010) · doi:10.1016/j.ijmecsci.2010.08.007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
