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

Barbora Benešová; Miroslav Frost; Lukáš Kadeřávek; Tomáš Roubíček; Petr Sedlák

doi:10.3934/dcdss.2020459

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November 2021, 14(11): 3925-3952. 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 Published November 2021 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 and Continuous Dynamical Systems - S, 2021, 14 (11) : 3925-3952. doi: 10.3934/dcdss.2020459

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show all references

References:

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[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.
[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.
[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.
[7]	Z. Bažant and M. Jirásek, Nonlocal integral formulations of plasticity and damage: Survey of progress, J. Eng. Mech., 128 (2002), 1119-1149.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	M. Frémond, Matériaux à mémoire de forme, C.R. Acad. Sci. Paris Sér.II, 304 (1987), 239-244.
[26]	M. Frémond and S. Miyazaki, Shape Memory Alloys, Springer, Wien, 1996.
[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.
[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.
[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.
[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.
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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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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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American Institute of Mathematical Sciences

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

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

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