# American Institute of Mathematical Sciences

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

 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 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.

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
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Dependence of stress on strain for one isothermal simulation and adiabatic simulations at three different strain rates; see text for details
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
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 $v_1$
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
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, $v_{\rm f} = 10^{-1}\, {\rm s}^{-1}$
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
 $\varOmega$ 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
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)]
 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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