Thermal Conductivity and Specific Heat Capacity of Austenite and Stress-Induced Martensite in Superelastic NiTi at Ambient Temperature

Abstract

Infrared thermography (IRT) and heat source reconstruction (HSR) were used in the study to measure two thermophysical properties of superelastic nickel-titanium (NiTi) shape memory alloy (SMA) wires, namely thermal conductivity and specific heat capacity. Since the values are potentially phase dependent, the identification was carried out at ambient temperature with and without mechanical loading, i.e., in the stress-induced martensite state and in the austenite state, respectively. A uniaxial testing machine was used to apply constant deformation while allowing temperature measurements by IRT during Joule heating and natural return the thermal equilibrium. The data processing by HSR, including preliminary filtering operations, was preliminarily evaluated from temperatures provided by a model (with added noise). It gave an error of ± 0.5% for the thermal conductivity and ± 1% for the specific heat capacity. The experimental analysis showed that the thermal conductivity of stress-induced martensite is 26% higher than that of austenite at the same temperature (here the ambient temperature). Regarding the specific heat capacity, that of stress-induced martensite is 4.7% lower than that of austenite. Explanations for these differences were proposed from solid-state physics theory. The measured values were also compared to data collected in the literature (obtained at zero stress at different temperatures).

Appendix

The aim of this appendix is to describe the procedure for identifying the thermophysical properties involved in the heat equation, based on the reconstruction of heat sources. To test the robustness of the procedure, thermal data are first created from a model (Section A.1), to serve as input data for processing (Section A.2).

1.1 Creation of Synthetic Thermal Data

Synthetic thermal data were created using known properties of three non-SMA materials, namely glass, titanium and aluminum (see Table 3).These materials were chosen for their wide difference of thermophysical parameter values, although the Joule effect cannot obviously be applied to the glass material.

The simulations rely on the 1D expression of the heat diffusion equation and the experimental trends at the specimen boundaries (at $z=0$ and $z=L)$ observed in Figs. 3 and  4. Also, the trend for the ambient temperature ${T}_{\text{ref}}$ is inspired from the experimental observations. Thus,

$$\left\{\begin{array}{c}{T}_{\text{ref}}\left(t\right)={\theta }_{\text{ref}}\times t+{T}_{\text{ref}}\left(0\right), t>0\\ T\left(0,t\right)=T\left(\text{0,0}\right)+{\Theta }_{1}\times \left(1-\text{exp}\left(-\frac{t}{\tau }\right)\right)+ {\theta }_{\text{ref}}\times t, t\in [0, 300 s] \\ T\left(0,t\right)=T\left(\text{0,0}\right)+{\Theta }_{1}\times \text{exp}\left(-\frac{t}{\tau }\right)+ {\theta }_{\text{ref}}\times t, t\in [300 s, 600 s]\\ T\left(L,t\right)=T\left(L,0\right)+{\Theta }_{2}\times \left(1-\text{exp}\left(-\frac{t}{\tau }\right)\right)+ {\theta }_{\text{ref}}\times t, t\in [0, 300 s]\\ T\left(L,\text{t}\right)=T\left(L,0\right)+{\Theta }_{2}\times \text{exp}\left(-\frac{t}{\tau }\right)+ {\theta }_{\text{ref}}\times t, t\in [300 s, 600 s]\end{array}\right.$$

(12)

where ${\theta }_{\text{ref}}$ is the temperature change of the ambient air with respect to the beginning of the test $[^\circ C]$, ${\Theta }_{1}$ and ${\Theta }_{2}$ are the temperature changes at the specimen ends at the end of the heating stage $[^\circ C]$ and $\tau$ is a characteristic time $[s]$. This latter quantity was considered here to be defined by $\tau =\rho Cr/2h$, where, as above, $\rho$ is the material density $[{\text{kg}}\cdot {m}^{-3}]$, $C$ is the specific heat capacity of the material $[{\text{J}}\cdot {\text{kg}} ^{-1}\cdot {\text{K}}^{-1}]$, $r$ is the specimen radius $[m]$ and $h$ is the heat convection coefficient [${\text{W}}\cdot{\text{m}}^{-2}.{\text{K}}^{-1}$].

The initial condition $T\left(z,0\right)$ was obtained from the numerical solution of Eq. 13 below, where the heat source in the right-hand side is equal to zero, solved using the Matlab function ‘bvp5c’:

$$-\lambda \frac{{\partial }^{2}T\left(z,0\right)}{\partial {z}^{2}}+\frac{2}{r}\left(h\times \left(T\left(z,0\right)-{T}_{\text{ref}}\left(0\right)\right)+\epsilon \times {\sigma }_{\text{SB}}\times \left({T}^{4}\left(z,0\right)-{T}_{\text{ref}}^{4}\left(0\right)\right)\right)=0$$

(13)

Then, the whole temperature evolution $T\left(z,\text{t}\right)$ over the test duration was calculated using Eq. 2. Next, Gaussian noise with a standard deviation of 20 ${\text{mK}}$ is added to mimic experimental thermal data.

Finally, filtering operations are performed to smooth the data, as it is necessary from experimental data provided by an IR camera:

• Step 1: time filtering is performed by sliding average operation over 10 values. This reduces the noise by $\sqrt{10}$, leading to a standard deviation of $20/\sqrt{10}=6.3$ mK while penalizing the temporal resolution from 0.1 ${\text{s}}$ (experimental acquisition frequency of $10 {\text{Hz}}$) to 1 ${\text{s}}$;

• Step 2: spatial filtering is then performed by fitting the data by a polynomial function of order 6 at each time step. This order was defined after several tests, not reported here. It provides the best results for the identification of the thermophysical parameters in the next section.

Figure 8 shows an example of synthetic temperature evolution obtained by this procedure, for titanium. The filtered data are used in the following for the processing by HSR.

Fig. 8

Same as Fig. 3, but with synthetic thermal data provided by a model, for titanium. Note that noise was added to mimic experimental data, followed by filtering operations before processing by HSR

1.2 Identification Procedure of ${{C}}$, ${{\lambda}}$ and ${{h}}$ from the Thermal Data

The identification of parameters from experimental data is an inverse problem that is ill-posed in the sense of Hadamard [69] due to unavoidable noise in the inputs. It is here proposed to exploit the transient state (Eq. 1) and the steady state (Eq. 2) problems to reduce the computational burden and the number of “design” variables. A steady state 1D profile was created considering the time average from 125 ${\text{s}}$ to 300 ${\text{s}}$ corresponding to SS1 in Fig. 8a (while the heat storage term is equal to zero). This additional average operation further decreases the noise in the steady state data.

1.2.1 Identification of the Specific Heat Capacity $C$

The specific heat capacity $C$ is estimated from Eq. 1 at the beginning of the heating stage: see Eq. 14. At the temporal resolution of the smoothed data recording (time step of 1 ${\text{s}}$), the evolution can be considered as adiabatic, justifying the presence of the heat storage term only in the left-hand side of the equation. The time derivatives are estimated by finite differences.

$$C(z)=\frac{{S}_{\text{el}}}{\rho \times \frac{\partial T\left(z,0\right)}{\partial t}}=\frac{{S}_{el}}{\rho \times \left(\frac{T\left(z,1\right)-T\left(z,0\right)}{\Delta t}\right)}$$

(14)

Figure 9 shows the 1D profile of specific heat capacity obtained along the wire. It can be seen that the values are nearly constant in the central part of the wire. The higher values at both ends can be explained by a lack of adiabaticity due to contact with the steel jaws (boundary conditions have been defined on the basis of experiments, see section A.1). It can be also explained by the fitting by a polynomial function of degree 6, as described in Section A.1; see also Section A.2.3 for additional information. The values given in Tables 1, 2, and 3 for the specific heat capacity $C$ are the average over the range $[10 mm;30 mm]$; see the red line in the graph.

Fig. 9

Example of reconstructed specific heat capacity $\text{C}(\text{z})$ along a titanium wire. Values are averaged over the range $[10\,{\text{ mm}};30\,{\text{ mm}}]$ to extract the specific heat capacity of the material

1.2.2 Identification of $\lambda$ and $h$

To facilitate the calculation of the two remaining design variables, $\lambda$ and $h$, a system of equations describing the steady state and the transient state was used. We thus obtained three Eqs. (15) for two design variables ($\lambda$ and $h$).

$$\left\{\begin{array}{c}\text{min}\left(\left|{S}_{\text{el}}\left(t\right)- \rho C\frac{\partial T\left(z,t\right)}{\partial t}+\lambda \frac{{\partial }^{2}T\left(z,t\right)}{\partial {z}^{2}}-\frac{2}{r}\left(h\times \left(T\left(z,t\right)-{T}_{\text{ref}}\left(t\right)\right)+\epsilon {\sigma }_{\text{SB}}\times \left({T}^{4}\left(z,t\right)-{T}_{\text{ref}}^{4}\left(t\right)\right)\right)\right|\right), t\in \left[0, 300 s\right]\\ \text{min}\left(\left|- \rho C\frac{\partial T\left(z,t\right)}{\partial t}+\lambda \frac{{\partial }^{2}T\left(z,t\right)}{\partial {z}^{2}}-\frac{2}{r}\left(h\times \left(T\left(z,t\right)-{T}_{\text{ref}}\left(t\right)\right)+\epsilon {\sigma }_{\text{SB}}\times \left({T}^{4}\left(z,t\right)-{T}_{\text{ref}}^{4}\left(t\right)\right)\right)\right|\right), t\in [301 s, 600 s]\\ \text{min}\left(\left|{S}_{\text{el}}\left(t\right)+\lambda \frac{{\partial }^{2}T\left(z,t\right)}{\partial {z}^{2}}-\frac{2}{r}\left(h\times \left(T\left(z,t\right)-{T}_{\text{ref}}\left(t\right)\right)+\epsilon {\sigma }_{\text{SB}}\times \left({T}^{4}\left(z,t\right)-{T}_{\text{ref}}^{4}\left(z,t\right)\right)\right)\right|\right), t\in \left[250 s, 300 s\right]\end{array}\right.$$

(15)

The first and second equations correspond the heating and cooling stages of the test, the term ${S}_{\text{el}}$ being null in the latter case (natural cooling). The third equation corresponds to the steady regime of the heating stage, the heat storage term being null in this case. The optimization problem was carried out using the GlobalSearch solver of the Matlab Global Optimization toolbox to apply global optimization. In addition, the Matlab optimization algorithm fmincon with large bounds was used to obtain first estimates for $\lambda$ and $h$. Table 3 gives the results of the identification for the three materials considered. Following comments can be made from these results:

• the relative error on the convection coefficient $h$ is small in the three cases of material: less than 2%;

• the relative error in the identification of the thermal conductivity $\lambda$ decreases when $\lambda$ increases. While it is about 0.05% for aluminum, that for titanium is about 0.32% and that for glass is 3.84%;

• the relative error in the identification of the specific heat capacity $C$ is low for glass and titanium, with relative errors of + 3.85% and + 0.65%, respectively. The largest errors was found for aluminum: + 13.10%.

As a conclusion, since the thermophysical parameters of NiTi are a priori close to those of titanium (according to the literature), the following errors can be reasonably declared for our experiments: ± 0.5% for the thermal conductivity $\lambda$; and ± 1% for the specific heat capacity $C$.

Table 3 Results of HSR identification from model data (with added noise and filtering operations to reproduce an experimental situation). Three materials with very different properties are considered to assess the robustness of the processing between extreme cases

1.2.3 Additional Information

Figure 10a presents the evolution of the spatially averaged contributions of the left-hand side of Eq. 1 as a function of time, for titanium: heat storage term ${\phi }_{\text{hs}}$, conduction term ${\phi }_{\text{cd}}$, convection term ${\phi }_{\text{cv}}$, radiation term ${\phi }_{\text{rd}}$. The reconstructed heat source ${S}_{\text{th}}={\phi }_{\text{hs}}+{\phi }_{\text{cd}}+{\phi }_{\text{cv}}+{\phi }_{\text{rd}}$ and input heat source ${S}_{\text{el}}$ are also displayed, confirming the good reconstruction. The impact of the value of $C$ on the heat storage term ${\phi }_{\text{hs}}$ along the 50 first seconds of the heating is shown in Fig. 10b. It can be clearly seen that the slope for $C=528 {\text{J}}\cdot{\text{K}}^{-1}\cdot {\text{kg}^{-1}}$ (the input data) tends to the heat source ${S}_{\text{el}}$ when ${\phi }_{\text{hs}}$ are extended to $t=0 s$ (time considered for the identification of $C$; see Eq. 14).

Considering only spatially averaged quantities for the contributions of the left-hand side of Eq. 1 makes the analysis partial. For this reason, we used the steady state equation (Eq. 2) during the heating stage to visualize the contributions along the spatial dimension: see Fig. 10c. It can be seen that, while the temperature profile is convex, the convection and radiation terms are convex whereas the conduction term ${\phi }_{\text{cd}}$ is concave. In other words, they are opposite and complementary. The use of a polynomial expression of order 6 for the fitting of the noisy temperature profiles (see Section A.1) leads to a small discrepancy between ${S}_{\text{th}}$ and ${S}_{\text{el}}$ near the ends.

Fig. 10

Results of the identification from synthetic data provided by a model, for titanium: (a) variation of the heat source contributions. ${\phi }_{\text{hs}}$ is the heat storage term; ${\phi }_{\text{cd}}$ is the conduction term; ${\phi }_{\text{cv}}$ is the convection term; ${\phi }_{\text{rd}}$ is the radiation term; ${S}_{\text{th}}$ is the reconstructed heat source, i.e., the sum of ${\phi }_{\text{hs}}$, ${\phi }_{\text{cd}}$, ${\phi }_{\text{cv}}$, and ${\phi }_{\text{rd}}$; and ${S}_{\text{el}}$ is the input electric heat source; (b) effect of the specific heat capacity value in time, c) contribution of the different heat sources in steady state at the end of the heating stage (300 $\text{s}$)