An exponential scaling law for the strain dependence of the Nb₃Sn critical current density

In the development of the proposed scaling law, the main assumptions are based on mathematical considerations that take into account the shape of the strain function derived from critical current measurements of strands under applied axial stress. These assumptions can be summarized by the following relationships:

$$\begin{eqnarray} &&s(\boldsymbol{\varepsilon})=\sum _{i=1}^{N}\frac{{\mathrm{e}}^{-{F}_{i}(\varepsilon )}}{N} \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&{F}_{i}(\boldsymbol{\varepsilon})={F}_{i}({I}_{1},{J}_{2},{J}_{3})={F}_{i,1}({I}_{1})+{F}_{i,2}({J}_{2})+{F}_{i,3}({J}_{3}) \end{eqnarray} \tag{ 6 }$$

where (1) ε is the strain tensor of the superconductor; (2) F-functions are equal to zero when the argument is zero and positive in all other cases; (3) I₁ is the first invariant of the strain tensor, and J₂ and J₃ the second and third invariants of the deviator strain tensor. The strain tensor represents the global strain status of the material while the deviatoric strain tensor contains only the components producing a variation of shape. Note that equation (5) is equal to one only if the strain tensor is equal to zero; in all other cases, the strain function value is lower than one (and larger than zero).

By approximating the F_i,j functions with a second order Taylor expansion, we then write

$$\begin{eqnarray} {F}_{i}(\boldsymbol{\varepsilon})&=&{\alpha }_{i,1}{I}_{1}+{\alpha }_{i,2}{I}_{1}^{2}+{\beta }_{i,1}{J}_{2}+{\beta }_{i,2}{J}_{2}^{2}\nonumber\\ &&+~{\gamma }_{i,1}{J}_{3}+{\gamma }_{i,2}{J}_{3}^{2} \end{eqnarray} \tag{ 7 }$$

where α,β and γ are positive parameters.

The strain function is then further simplified by using considerations related to the form of the three invariants. With respect to the principal strains ε₁,ε₂,ε₃, the three invariants can be written as follows:

$$\begin{eqnarray} {I}_{1}&=&{\varepsilon }_{1}+{\varepsilon }_{2}+{\varepsilon }_{3} \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} {J}_{2}&=&\frac{1}{6}[({\varepsilon }_{1}-{\varepsilon }_{2})^{2}+({\varepsilon }_{2}-{\varepsilon }_{3})^{2}+({\varepsilon }_{3}-{\varepsilon }_{1})^{2}] \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} {J}_{3}&=&\left \{ {\varepsilon }_{1}-\frac{{I}_{1}}{3}\right \} \left \{ {\varepsilon }_{2}-\frac{{I}_{1}}{3}\right \} \left \{ {\varepsilon }_{3}-\frac{{I}_{1}}{3}\right \} . \end{eqnarray} \tag{ 10 }$$

By neglecting the terms of fourth order and larger in ε (i.e. J₂² and J₃³) and eliminating the terms that can be negative (I₁ and J₃ in equation (7), which could lead to exponentials larger than one), the strain function can be simplified to

$$\begin{eqnarray} &&s(\boldsymbol{\varepsilon})=\sum _{i=1}^{N}\frac{{\mathrm{e}}^{-\{ {\alpha }_{i}{I}_{1}^{2}+{\beta }_{i}{J}_{2}\} }}{N}. \end{eqnarray} \tag{ 11 }$$

Equation (11) summarizes the original idea regarding the scaling law proposed in this paper. To specialize the scaling function to the case of uni-axial strain applied to a single wire, we need to introduce further simplifications in the way the variants are evaluated. In this case, it will be assumed that (1) the principal strain component ε₁ is along the strand axis (longitudinal component); (2) the other two principal components ε₂, ε₃, which are perpendicular to the strand axis, are equal. Using these assumptions one can then write

$$\begin{eqnarray} {\varepsilon }_{1}&=&{\varepsilon }_{\mathrm{l}0}+{\varepsilon }_{\mathrm{a}} \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} {\varepsilon }_{2}&=&{\varepsilon }_{3}={\varepsilon }_{\mathrm{t}0}-\nu {\varepsilon }_{\mathrm{a}} \end{eqnarray} \tag{ 13 }$$

where ε_a is the axial strain applied to the strand during the measurements; ε_l0,ε_t0 are the longitudinal and transverse strains of the Nb₃Sn that are due to differential thermal contraction between the Nb₃Sn and the other materials of the composite wire; and ν is the 'effective' Poisson ratio of Nb₃Sn. The term 'effective' derives from the fact that equations (12) and (13) are the exact solution for the elastic strain of a cylindrical straight beam under pure axial load, but they only represent an approximation for the real strain status of the Nb₃Sn filaments. Indeed, the filaments are not straight (they are generally twisted), and they experience not only a longitudinal load but also a transverse one, that is applied by the matrix in which they are embedded.

Finally, by substituting equations (12) and (13) into (8) and (9), one can obtain

$$\begin{eqnarray} {I}_{1}&=&(1-2 \nu ){\varepsilon }_{\mathrm{a}}+{\varepsilon }_{\mathrm{l}0}+2{\varepsilon }_{\mathrm{t}0} \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} {J}_{2}&=&\frac{1}{3}({\varepsilon }_{\mathrm{l}0}-{\varepsilon }_{\mathrm{t}0}+(1+\nu ){\varepsilon }_{\mathrm{a}})^{2}. \end{eqnarray} \tag{ 15 }$$

From high-energy synchrotron x-ray diffraction measurements on three different types of Nb₃Sn wire (bronze route, restacked rod process internal tin, and powder in tube strands) under axial tensile stress at 4.2 K, the effective Poisson ratio of Nb₃Sn is estimated to be about 0.36 [19]. Details of the measurement set-up, procedures and data analysis are reported in [20]. A similar value of the effective Poisson ratio was also found by other authors [21] using a similar measurement technique.

Starting from the main ideas presented in the previous paragraphs, equation (11) was then simplified by setting the N value at two in order to limit the number of parameters (N = 1 cannot reproduce the typical asymmetrical shape of the strain function).

$$\begin{eqnarray} &&s(\boldsymbol{\varepsilon})=\frac{{\mathrm{e}}^{-\{ {\alpha }_{1}{I}_{1}^{2}+{\beta }_{1}{J}_{2}\} }+{\mathrm{e}}^{-\{ {\alpha }_{2}{I}_{1}^{2}+{\beta }_{2}{J}_{2}\} }}{2}. \end{eqnarray} \tag{ 16 }$$

When fitting the experimental data with equation (16) and equations (14) and (15) (see section 3 for more information about the analyzed data, and the appendix for tables A.1 and A.2 of the B_c2 data used to derive the optimal parameters of the strain function), it was found that α₁ and β₂ could be set equal to zero, and hence the strain function was simplified to

$$\begin{eqnarray} &&s(\boldsymbol{\varepsilon})=\frac{{\mathrm{e}}^{-{C}_{1}{J}_{2}}+{\mathrm{e}}^{-{C}_{1}{C}_{2}{I}_{1}^{2}}}{2} \end{eqnarray} \tag{ 17 }$$

C₁ and C₂ being positive fitting parameters. In order to avoid a second inflection point in the compressive region, the strain function was then slightly modified to

$$\begin{eqnarray} &&s(\boldsymbol{\varepsilon})=\frac{{\mathrm{e}}^{-{C}_{1}\frac{{J}_{2}+{C}_{3}}{{J}_{2}+1}{J}_{2}}+{\mathrm{e}}^{-{C}_{1}{C}_{2}\frac{{I}_{1}^{2}+{C}_{3}}{{I}_{1}^{2}+1}{I}_{1}^{2}}}{2} \end{eqnarray} \tag{ 18 }$$

where C₃ is a parameter larger than one.

Finally, when deriving the optimal fitting parameters, it was noticed that C₂ and C₃ could be set equal to constant values that depend on the Nb₃Sn 'effective' Poisson ratio. In particular, with an effective Poisson ratio equal to 0.36, C₂ and C₃ can be set respectively equal to about 1 and 3. Therefore, the strain function assumes the following final form:

$$\begin{eqnarray} &&s(\boldsymbol{\varepsilon})=\frac{{\mathrm{e}}^{-{C}_{1}\frac{{J}_{2}+3}{{J}_{2}+1}{J}_{2}}+{\mathrm{e}}^{-{C}_{1}\frac{{I}_{1}^{2}+3}{{I}_{1}^{2}+1}{I}_{1}^{2}}}{2}. \end{eqnarray} \tag{ 19 }$$

In conclusion, assuming an effective Poisson ratio equal to 0.36, the proposed strain function is equation (19), where, in the case of applied axial stress acting on a wire, the invariants J₂ and I₁ can be calculated using equations (14) and (15). This strain function has originally three fitting parameters C₁,ε_l0 and ε_t0. Nevertheless, during the calculation of the optimal fitting parameters for the datasets analyzed (see appendix), it was found that ε_t0 can be written as a function of ε_l0 in the following way:

$$\begin{eqnarray} &&{\varepsilon }_{\mathrm{t}0}=-\nu {\varepsilon }_{\mathrm{l}0}+K \end{eqnarray} \tag{ 20 }$$

where ε_l0,ε_t0 are expressed as percentages (as are all the strain values in this paper) and K is equal to 0.1. Hence the fitting parameters are reduced to two: C₁ and ε_l0.

Simplifying Nb₃Sn as an isotropic linear elastic material, one can write its elastic energy U associated with a strain state ε, as follows [22]:

$$\begin{eqnarray} U(\boldsymbol{\varepsilon})&=&\frac{E}{2(1+\nu )}\left \{ \frac{\nu }{1-2 \nu }({\varepsilon }_{1}+{\varepsilon }_{2}+{\varepsilon }_{3})^{2}\right .\nonumber\\ &&\left .\vphantom {\frac {\nu }{1-2\nu }} +~({{\varepsilon }_{1}}^{2}+{{\varepsilon }_{2}}^{2}+{{\varepsilon }_{3}}^{2})\right \} \nonumber\\ &=&\frac{1}{6}\frac{E}{1-2 \nu }{{I}_{1}}^{2}+\frac{E}{1+\nu }{J}_{2}\equiv {U}_{{I}_{1}}+{U}_{{J}_{2}} \end{eqnarray} \tag{ 21 }$$

where v and E are respectively the Poisson ratio and the Young modulus of Nb₃Sn. The elastic energy is hence composed of only two terms: one, U_I1, associated with the hydrostatic invariant and the other, U_J2, with the second deviatoric invariant. In the case where J₂ and I₁² values are much smaller or much larger than one, the scaling function (equation (19)) can then be written as

$$\begin{eqnarray} &&s(\boldsymbol{\varepsilon})=\frac{{\mathrm{e}}^{{-{C}_{1}}^{\prime}{U}_{{J}_{2}}}+{\mathrm{e}}^{{-{C}_{1}}^{\prime}\frac{6(1-2 \nu )}{1+\nu }{U}_{{I}_{1}}}}{2} \end{eqnarray} \tag{ 22 }$$

where C₁' = 3C₁(1 + ν)/E when J₂ and I₁² are much smaller than one or C₁' = C₁(1 + ν)/E when the invariants are much larger than one. Assuming an effective Poisson ratio equal to 0.36 as described above, equation (21) becomes

$$\begin{eqnarray} &&\fl s(\boldsymbol{\varepsilon})=\frac{{\mathrm{e}}^{{-{C}_{1}}^{\prime}{U}_{{J}_{2}}}+{\mathrm{e}}^{{-{C}_{1}}^{\prime}{1.2 3 5 U}_{{I}_{1}}}}{2}\approx \frac{{\mathrm{e}}^{{-{C}_{1}}^{\prime}{U}_{{J}_{2}}}+{\mathrm{e}}^{{-{C}_{1}}^{\prime}{U}_{{I}_{1}}}}{2}. \end{eqnarray} \tag{ 23 }$$

This justifies equation (18), suggesting that the proper quantity for scaling is the elastic energy.

The proposed strain function has a form that recalls the formula of the critical temperature developed in the framework of Migdal–Eliashberg theory (which extends the results of BCS theory to the strong coupling regime of the electron–phonon interaction potential [23]):

$$\begin{eqnarray} &&{T}_{\mathrm{c}}=\frac{\left \langle \omega \right \rangle }{1.2 0}{\mathrm{e}}^{-\frac{1.0 4(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.6 2 \lambda )}} \end{eqnarray} \tag{ 24 }$$

where λ is the dimensionless electron–phonon coupling constant, measuring the strength of the interaction between electron and phonons in the system, μ* is the renormalized pseudo-potential, recounting the effective strength of the Coulomb repulsion, and 〈ω〉 is the average phonon frequency. Equation (24) is known as McMillan's formula [24, 25], later corrected by Dynes as concerns the multiplicative pre-factor [25, 26]. According to McMillan's formula the critical temperature decreases exponentially to zero on decreasing the electron–phonon coupling constant; on the other hand, from equation (2) and from the proposed strain function, the critical temperature decreases exponentially to zero on increasing the strain in the material. This analogy might suggest that a formulation of the strain function similar to equation (18) might be derived directly from the Migdal–Eliashberg theory once the proper correlation between the electron–phonon coupling constant and the strain is calculated.

In the proposed exponential function, the strain dependence is ruled by only the two components (hydrostatic and deviatory) of the elastic energy associated with a certain strain. In fact, in equation (19), the dependence of the strain function on one invariant is not affected by the value of the other invariant. This might be a problem in the case of stress loads that modify only one invariant. For example, in the case of increasing pressure the strain function in equation (19) would never go to zero. In order to include this possibility without substantially changing the results discussed so far, one option could be to modify equation (19) as follows:

$$\begin{eqnarray} &&s(\boldsymbol{\varepsilon})={\mathrm{e}}^{-{C}_{\mathrm{4}}({J}_{2}+{I}_{1}^{2})}\frac{{\mathrm{e}}^{-{C}_{1}\frac{{J}_{2}+3}{{J}_{2}+1}{J}_{2}}+{\mathrm{e}}^{-{C}_{1}\frac{{I}_{1}^{2}+3}{{I}_{1}^{2}+1}{I}_{1}^{2}}}{2} \end{eqnarray} \tag{ 25 }$$

where C₄ is a positive parameter much smaller than C₁. Nevertheless, at present the recommended strain function is equation (19).

From the critical current measurements at different magnetic fields and strain, the self-field correction and the pinning force were calculated first. Then, in order to determine the B_c2(4.2 K,ε) values, the experimental data were fit with the following pinning force scaling law:

$$\begin{eqnarray} &&{F}_{p}(4.2~\mathrm{K},\varepsilon )=C(4.2~\mathrm{K},\varepsilon ){b}^{p}(1-b)^{q}, \end{eqnarray} \tag{ 26 }$$

where C(4.2 K,ε),B_c2(4.2 K,ε),q were free parameters and p was kept constant at 0.5.

During this iterative fit procedure, the critical current data measured at a normalized upper critical field b lower than 0.4 or higher than 0.8 were excluded. Figure 1 shows the results of the fitting procedure for the Furukawa strand tested by Durham University. From the figure one can notice that there are no experimental data in the region around the pinning force maximum (the same was true for all the other datasets analyzed). Because of this and because the p parameter mainly defines the shape of the pinning force in the low field region, p was set equal to 0.5 (that is the value proposed in the Kramer model). In the B_c2 calculation, critical current measurements at normalized fields below 0.4 were excluded to avoid the field region where the pinning force curve might be affected by the use of a non-optimal p parameter. On the other hand, the data at normalized fields above 0.8 were trimmed because at very high field the current can be mostly carried by a limited fraction of the superconductor that has a negligible effect on the transport current properties at lower fields. Indeed, because of composition gradients of Nb₃Sn in technical wires, it is known that filaments may contain a relatively small portion of superconducting phase whose upper critical field is higher than the bulk. This portion is generally associated with large grains and composition close to stoichiometry. The fraction of current carried by this superconducting phase is relevant only in the vicinity of B_c2, but decreases to a negligible value for fields significantly lower than B_c2. The effect of non-homogeneous composition is visible in figure 1 as a deviation of the data from the fit at high fields [27, 28].

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Table 1 summarizes some relevant results obtained from the calculation of the B_c2 for the datasets analyzed. For each strand are reported (1) the calculated q parameter, (2) the number of different values of applied strain at which the tests were performed and (3) the minimum, maximum and average number of critical current measurements used in calculating B_c2 (for different strain values, different numbers of critical current measurements were carried out in the range 0.4 < b < 0.8). Taking the Furukawa wire as an example, 14 different strains were applied, and for a certain ε_a only five I_c measurements were available for calculating the respective B_c2, while for another ε_a there were up to 14 I_c measurements. Over the 14 different applied strain values, on average, 10.4 I_c measurements were used to calculate the B_c2.

The B_c2 values derived from the eight datasets were fit using equations (2) and (3) and the new exponential scaling law, equation (19). In the calculation, B_c20,C₁,ε_l0 were fitting parameters and T_c0 was set equal to 17 K; note that all the strain values are expressed as percentages (also the invariants refer to the strain as a percentage). Figure 2 shows the results obtained from the strand dataset with the largest range of applied strain [15], while figure 3 shows the exponential strain function derived from the same B_c2 fit with equation (19). In figure 3 the two terms that compose the strain function, the deviatoric and the hydrostatic ones, are also reported. The two components reach their maxima at two different applied strains ε_D and ε_H when, respectively, the second deviatoric and the hydrostatic invariants are equal to zero. From equations (14), (15) and (20), one can find that

$$\begin{eqnarray} {\varepsilon }_{\mathrm{D}}&=&\frac{K}{1+\nu }-{\varepsilon }_{\mathrm{l}0} \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} {\varepsilon }_{\mathrm{H}}&=&-\frac{2 K}{1-2 \nu }-{\varepsilon }_{\mathrm{l}0}. \end{eqnarray} \tag{ 28 }$$

It is interesting to notice that setting K equal to a constant (in this paper the value 0.1 is recommended) also fixes the difference Δε between ε_D and ε_H equal to a constant:

$$\begin{eqnarray} &&\Delta \varepsilon ={\varepsilon }_{\mathrm{D}}-{\varepsilon }_{\mathrm{H}}=\frac{3 K}{(1+\nu )(1-2 \nu )}. \end{eqnarray} \tag{ 29 }$$

The positive value of Δε together with the different concavities of the two strain function components (see figure 3) is the cause of the typical asymmetric shape of s(ε) with respect to its maximum. Note that, as long as Δε is not zero, the strain function is always lower than one in the case of applied axial strain. Indeed, for any applied strain, there will never be a strain free condition: the hydrostatic and the second deviatoric invariant will never be equal to zero at the same time.

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Furthermore, from equations (27) and (28) and figure 3, one can also conclude that ε_l0 establishes the position of ε_D and ε_H, and C₁ the position of the strain function maximum between ε_D and ε_H.

Figure 4 presents a comparison between the different datasets analyzed with their respective fits. In the figure, for the sake of clarity, the B_c2 data were plotted as a function of the difference between the applied strain ε_a and the applied strain value at which the fitting function has its maximum ε_Max.

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The main results of the analysis carried out on the eight datasets are reported in table 2: the upper critical field B_c20; the fitting parameters of the strain function C₁ and ε_l0; the applied strain value ε_Max where the fitting function has its maximum; the root mean square error between the fit and all the experimental data; and the range of the applied strain where the critical current measurements were carried out.

It is interesting to notice that (1) the B_c20 of all the ITER type wires (the first seven in table 2) differ from each other by no more than 0.7 T while the PIT conductor (developed for much higher critical current density [18]) has a significantly higher value as expected; (2) ε_Max ∼− ε_l0 (we recall that for an applied strain equal to −ε_l0 both the deviatoric and the hydrostatic components are not zero; see equations (27) and (28)).

Finally, in order to assess the stability of the exponential scaling function compared to previous scaling functions, the root mean square error between the fit and all the experimental data was calculated as a function of the number of experimental data points used for deriving the fit. We considered the strain functions that are most widely used for this task.

The 'Twente' strain function [7]:

$$\begin{eqnarray*} &&\fl s({\varepsilon }_{\mathrm{a}})=\frac{1}{1-{C}_{\mathrm{a},1}{\varepsilon }_{0,\mathrm{a}}}\left \{ {C}_{\mathrm{a},1}\left [\sqrt{{\varepsilon }_{\mathrm{sh}}^{2}+{\varepsilon }_{0,\mathrm{a}}^{2}}\right .\right .\nonumber\\ &&\left .\left .-~\sqrt{({\varepsilon }_{\mathrm{a}-{\varepsilon }_{\mathrm{sh}}})^{2}+({\varepsilon }_{\mathrm{0,a}})^{2}}\right ]-{C}_{\mathrm{a,}2}{\varepsilon }_{\mathrm{a}}\right \} +1\nonumber\\ &&\fl {\varepsilon }_{\mathrm{a}}={\varepsilon }_{\mathrm{applied}}+{\varepsilon }_{\mathrm{m}}\nonumber\\ &&\fl {\varepsilon }_{\mathrm{sh}}=\frac{{C}_{\mathrm{a},2}{\varepsilon }_{0,\mathrm{a}}}{\sqrt{{C}_{\mathrm{a,1}}^{2}-{C}_{\mathrm{a,2}}^{2}}} \end{eqnarray*}$$

where C_a,1,C_a,2,ε_0,a and ε_m are fitting parameters.

The 'Twente modified' strain function [8]:

$$\begin{eqnarray*} &&s({\varepsilon }_{I})=\frac{1-{C}_{\mathrm{a},1}\sqrt{{\varepsilon }_{I}^{2}+{\varepsilon }_{0,\mathrm{a}}^{2}}-{C}_{\mathrm{a},2}({\varepsilon }_{I}^{3}-3{\varepsilon }_{0,\mathrm{a}}^{2}{\varepsilon }_{I})}{1-{C}_{\mathrm{a},1}{\varepsilon }_{0,\mathrm{a}}}\nonumber\\ &&{\varepsilon }_{I}={\varepsilon }_{\mathrm{a}}-{\varepsilon }_{\mathrm{m}} \end{eqnarray*}$$

where C_a,1,C_a,2,ε_0,a and ε_m are fitting parameters. It has been proposed to simplify the 'Twente modified' strain function by using the following relationship to calculate C_a,2 [8]:

$$\begin{eqnarray*} &&{C}_{\mathrm{a},2}=1.0 3 4\times {1 0}^{3}{C}_{\mathrm{ a},1} \end{eqnarray*}$$

which reduces the number of fitting parameters to three.

The 'Markiewicz' strain function [10]:

$$\begin{eqnarray*} &&s({\varepsilon }_{0})=\frac{1}{(1+{c}_{2}{\varepsilon }_{0}^{2}+{c}_{3}{\varepsilon }_{0}^{3}+{c}_{\mathrm{4}}{\varepsilon }_{0}^{4})}\nonumber\\ &&{\varepsilon }_{0}={\varepsilon }_{\mathrm{a}}-{\varepsilon }_{\mathrm{m}} \end{eqnarray*}$$

where C₂,C₃,C₄ and ε_m are fitting parameters.

The stability analysis was carried out following these steps.

• (1)  
  Define a data window containing a minimum number of B_c2 data points around B_c2 Max.
• (2)  
  By using the pre-defined data window, compute different fits using different scaling functions (figure 5 shows an example for the Furukawa dataset by taking only six points around B_c2 Max).
• (3)  
  Calculate the RMS error between the fitted curve (determined with only a limited number of data points) and the actual data points (all 14 points in the case of the Furukawa dataset).
• (4)  
  The data window (if not containing already all available data) is enlarged by including one more data point, the one closer to the data window, and the calculation restart from step 2.

Tables 3–5 show the fitting parameters of the different scaling functions when applied to the central six points of the Furukawa dataset (see figure 5) and to the whole dataset (14 points).

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Figure 6 summarizes the results obtained using this procedure in the case of the three datasets with the largest range of applied strain. One can notice that the exponential scaling function is the only one that has a reduced RMS value even when using a limited strain range for calculating the fit. This indicates that the fitting parameters of the exponential scaling function are much more stable than those of other scaling functions. Thanks to the improved stability, the exponential scaling function can be used not only as a fitting function, as for the others proposed in the literature, but also for estimating the conductor performance in a strain range where experimental results are not available.

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