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

References:

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References:

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[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
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[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
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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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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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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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