1. Introduction
Triple-helical structures widely exist in engineering cables and natural or artificial biological tissues, such as collagen hybridizing peptides [1], [2], artificial muscles [3], biomimetic basalt fiber/epoxy helical composite spring [4], carbon nanotube ropes [5], and superconducting cables in International Thermonuclear Experimental Reactor (ITER) [6]. The biological tissues and engineering cables with triple-helical structures usually undergo complex fatigue loading conditions. For example, when runners take part in a marathon, triple-helical collagen in tendons undergoes high-stress cyclic loading about 25,000 times without rest [7], and the collagen triple helix exhibits progressive unfolding and eventual catastrophic failure under fatigue loading, which will cause tendinosis, rotator cuff disease, and bone fragility fractures [8], [9].
Especially, ITER superconducting cables consist of triple-helical cable required to sustain 60,000 electromagnetic loading cycles under the temperature of 4.2 K [10]. The fatigue damage of superconducting cables delayed the ITER fusion machine which has led to a surge in costs to about $22 billion [11], [12]. Therefore, it is unacceptable to face the risk and cost of dramatic performance degradation for ITER [13], [14]. Afterward, Japanese suppliers tested superconducting cables with short twist-pitches through their paces at SULTAN, whose results showed little signs of degradation due to tighter spiral led to much more mechanically stable [15]. However, it remains unclear how the short twist-pitches improve superconducting cables because of the complex multiple stages of superconducting cables [16], [17]. Fortunately, Larson et al. [18] reported that varied twist-pitches can change the properties of helical dielectric elastomer actuator filaments with two-helical structures and architected springy filaments with four-helical structures. Inspired by these simple helical structures, we attempted to study the fatigue damage properties of Nb3Sn triple-helical cables which are the elementary structures in superconducting cables of ITER magnets.
Over the past decades, numerous experiments and numerical models have been carried out to focus on the mechanical properties of Nb3Sn triple-helical structures and other triple-helical structures. For example, Bruzzone et al. [19] measured the average elastic modulus and energy dissipation of Nb3Sn triple-helical cables under tensile loading and in cyclic loading-unloading experiments at the temperature of 77 K. Costello [20] presented the theory of wire rope and derived the relationship between mechanical deformation and helical angle of triple-helical cable under tension and compression. Nakamura et al. [21] tested the mechanical deformation of Nb3Sn triple-helical cables with twist-pitches of 25, 50, and 65 mm, during tensile and cyclic loading, which indicated that their deformation increases with the augment of tensile stress. Zhao et al. [22] presented a bottom-up method to analyze the mechanical behaviors of carbon nanotube ropes with triple-helical structures and derived the relationship between the deformation and stress in the rope under uniaxial tension, torsion, and bending. Furthermore, Yue et al. [23], [24] studied the axial thermal expansion coefficient and effective Young’s modulus of artificial muscle with triple-helical structure by experiments and theoretical modeling, which elucidated that the modulus decreases greatly during the tensile process and increases with the increase of twist-pitches, the axial thermal expansion of the strand with the shortest twist-pitch is over three times of that the untwisted structure. In addition, Zitnay et al. [1] tested the triple-helical collagen under sinusoidal cycling strain cycles between 1 and 5% tensile strain, and found that the stress borne by the tissue as well as the linear modulus during loading decreased with increasing number of loading cycles. Later, Zitnay et al. [8] conducted the triple-helical collagen under sinusoidal cycling strain cycles, and found that strain accumulation showed three stages: initially rapid drop, gradually steady, and accelerated increase stages.
It should be noted that the above-mentioned studies pay little attention to the effects of twist-pitches on fatigue properties in triple-helical structures, and the cycles of fatigue loading are less than ten thousand. Moreover, the triple-helical structures usually work in extreme environments, such as high-frequency fatigue, multiple coupled loading, and extremely low temperatures. This paper aims to reveal the fatigue damage evolution behaviors of Nb3Sn triple-helical structure during hundreds of thousands of fatigue cycles at the temperature of 77 K and investigate the effects of twist-pitches on fatigue damage properties of the triple-helical structure. Moreover, the theoretical model of damage considering the influence of twist-pitches has been proposed to illustrate the mechanisms of fatigue damage for Nb3Sn triple-helical structures.
This paper is organized as follows: In Section 2, the strain cycling tests were conducted at liquid nitrogen temperature, and the experimental results are discussed in Section 3. In Section 4, the theoretical model of damage evolution considering the effects of twist-pitches was established, and the comparisons between the theoretical model and experimental results were expounded in Section 5. Finally, the discussion and conclusion were presented in Section 6 and Section 7, respectively.
2. Experiments and methods
2.1. Materials
In this work, Nb3Sn triple-helical cables were twisted by three Nb3Sn superconducting wires manufactured by the Innernal-Tin method [25] from Western Superconducting Technologies Co. Ltd. with the twist-pitches of 22, 30, 45, 60, 75, and 110 mm, respectively. Nb3Sn superconducting strands consisted of 50% Cu matrix and 50% Nb3Sn sub-elements. Whose heat-treated conditions are that, 210 °C×50 h + 340 °C × 25 h + 450 °C × 25 h + 575 °C × 100 h + 650 °C × 100 h. The diameter and effective length of Nb3Sn wires are 0.81 mm and 121 mm, respectively. The Nb3Sn superconducting wire and the Nb3Sn triple-helical cable are shown in Fig. 1
Fig. 1. (a) Nb3Sn superconducting wire, (b) Nb3Sn triple-helical cable. |
2.2. Experimental methods
For static tensile testing, static loading-unloading tests were performed using the electronic universal testing machine at liquid nitrogen temperature. To determine the properties of Nb3Sn triple-helical cables, the strain rate was kept at 0.1 mm/min during loading, and the stress rate was kept at 0.1 N/s during unloading. Before loading, the sample should be cooled in liquid nitrogen for about 20 minutes to ensure that the entire sample reaches liquid nitrogen temperature.
For the fatigue cyclic tests, the strain-controlled tension-tension fatigue tests were conducted with INSTRON 8802 fatigue machine at liquid nitrogen temperature [26], and the sine cyclic waveform was applied at a frequency of 2 Hz during fatigue tests. The fatigue strain ranges for Nb3Sn triple-helical cables vary from 0.67% to 0.72%, which is slightly above the strain at yield point to shorten the fatigue failure time. For each group of tests, at least two samples were performed, and the strain was obtained by directly measuring the displacement from the fatigue machine, which can be calculated from the ratio of displacement to total length. These tests mainly measured the fatigue life, damage factor, and energy dissipation of Nb3Sn triple-helical cables. The energy dissipation can be obtained by integrating the area of stress-strain hysteresis loops. The damage factor indicates the degree to which the material is damaged under load, and can be determined by Eq. (15) in Section 4.1.
3. Experimental results
3.1. Quasi-static mechanical behaviors
Fig. 2 shows the stress-strain curves and energy dissipation of Nb3Sn triple-helical cables at temperature 77 K with twist-pitches of 22, 45, and 110 mm, respectively. From Fig. 2(a), the stress-strain response of Nb3Sn triple-helical cables exhibits significant nonlinearity during the loading-unloading-reloading processes. When the strain is the same, the shorter twist-pitches of the Nb3Sn triple helical cable, the smaller the stress. By integrating the area of stress-strain hysteresis loops, the relationship between the energy dissipation and unloading stress can be obtained (see Fig. 2(b)). It can be seen that the relationship between the energy dissipation and twist-pitches of Nb3Sn triple-helical cables shows positive correlation trends.
Fig. 2. Quasi-static mechanical behaviors of Nb3Sn triple-helical cables at liquid nitrogen temperature (a) stress-strain curves and (b) energy dissipation. |
3.2. Fatigue behaviors
3.2.1. Energy dissipation
Fig. 3 shows the evolution results of energy dissipation under various strain cycling range and twist-pitches during strain cycling of Nb3Sn triple-helical cables at liquid nitrogen temperature. In Fig. 3(a), strain cycling ranges of fatigue tests are 0.67%, 0.68%, and 0.70%, respectively, and twist-pitches of samples are 45 mm. In Fig. 3(b), twist-pitches of samples are 30 mm, 45 mm, and 110 mm, respectively, and strain cycling ranges of fatigue tests are 0.68%. The results show that the energy dissipation of Nb3Sn triple-helical cables increases with the number of cycles, and the rate of energy dissipation increases rapidly at the end of fatigue. As can be seen from Fig. 3(a), the energy dissipation of Nb3Sn triple-helical cables increases with increasing strain cycle ranges. It can be seen from Fig. 3(b), the relationship between the energy dissipation and twist-pitches also shows positive correlation trends, namely, the shorter twist-pitch of the triplet, results in lower energy dissipation.
Fig. 3. The relationships between energy dissipation and fatigue cycling numbers at liquid nitrogen temperature (a) at various strain ranges with a twist-pitch of 45 mm and (b) at various twist-pitches with a strain cycling range of 0.68%. |
3.2.2. Damage degradation
Fig. 4 shows the degradation of chord modulus for Nb3Sn triple-helical cables under various strain cycling ranges and twist-pitches at liquid nitrogen temperature, which is defined as the slope of a straight line connecting the starting and ending points of the unloading stress-strain curve. In Fig. 4(a), strain cycling ranges of fatigue tests are 0.67%, 0.68%, and 0.70%, respectively, and twist-pitches of samples are 45 mm. In Fig. 4(b), twist-pitches of samples are 30 mm, 45 mm, and 110 mm, respectively, and strain cycling ranges of fatigue tests are 0.68%. The results show that the chord modulus of Nb3Sn triple-helical cables degrades with the number of cycles, and the degradation rate of the chord modulus is faster in the early fatigue stage. As can be seen from Fig. 4(a), the larger the strain cycling range is, the smaller the chord modulus is in the fatigue process. It can be seen from Fig. 4(b), the shorter the twist-pitch of Nb3Sn triple-helical cables is, the smaller the modulus in the fatigue process. According to the degradation of chord modulus in Fig. 4, the damage factor under various strain cycling ranges and twist-pitches in the strain cycling process at liquid nitrogen temperature can be obtained (see Fig. 5). As shown in Fig. 5(a), the larger the strain cycling range is, the faster the accumulation of damage factor increases. It can be seen from Fig. 5(b), the shorter the twist-pitch is, the slower the accumulation of damage factor will be. The speed of the damage accumulation not only decides the fatigue life of samples but also evaluates whether the performance meets the requirements.
Fig. 4. The relationships between the degradation of chord modulus and fatigue cycling numbers at liquid nitrogen temperature (a) at various strain ranges with twist-pitch of 45 mm, (b) at various twist-pitches with strain cycling range of 0.68%. |
Fig. 5. The relationships between the degradation of damage factor and fatigue cycling numbers at liquid nitrogen temperature (a) at various strain ranges with a twist-pitch of 45 mm and (b) at various twist-pitches with a strain cycling range of 0.68%. |
3.2.3. Fatigue life
Fig. 6 shows the fatigue life of Nb3Sn triple-helical cables under various strain cycling ranges and twist-pitches during strain cycling at liquid nitrogen temperature. As can be seen from Fig. 6(a), the smaller the strain cycling ranges are, the larger the fatigue life will be. It can be seen from Fig. 6(b), the shorter the twist-pitches of Nb3Sn triple-helical cables are, the longer the fatigue life is. The reason that the fatigue life is longer at shorter twist-pitches and smaller strain cycling ranges can be attributed to slower energy dissipation and damage accumulation.
Fig. 6. The fatigue life of Nb3Sn triple-helical cables at liquid nitrogen temperature (a) at various strain ranges with a twist-pitch of 45 mm and (b) at various twist-pitches with a strain cycling range of 0.68%. |
According to the above experimental results, Nb3Sn triple-helical cables with short twist-pitches possess excellent fatigue damage resistance. To clarify the reason for excellent fatigue damage resistance in Nb3Sn triple-helical cables with short twist-pitches, a damage evolution model considering the effect of twist-pitches was proposed in Section 4, and we discussed the relationship between the helical structure of Nb3Sn triple-helical cables and excellent fatigue damage resistance in Section 6.
4. Damage evolution model considering the effects of twist-pitches
For the single Nb3Sn superconducting wire, the energy dissipation shows three stages: initially rapid drop, gradually steady, and accelerated increase stages [26]. However, the energy dissipation of Nb3Sn triple-helical cables increases with the number of cycles, and the rate of energy dissipation increases rapidly at the end of fatigue. Moreover, the fatigue damage evolution model in Ref. [26] cannot reveal the effects of twist-pitches on fatigue behaviors in Nb3Sn triple-helical cables. Therefore, the theoretical model of damage evolution considering the effects of twist pitches has been established in this section.
4.1. Recursive model between the triple-helical cable and wires
In Fig. 7, the triple-helical cables are from three same wires twisting around each other, and the relationship of length between the triple-helical cable and single wire can be written as
$L_{\text {single }}=\sqrt{L_{\text {triplet }}^{2}+\left(2 \pi r_{0}\right)^{2}} \text {, }$
where $L_{\text {triplet }}$ is the twist-pitch of the triple-helical cable, and $L_{\text {single }}$ is the length of a single wire in the triple-helical cable, $r_0$ is the helical radius of the triple-helical cable. Under the tension force $F_{\text {triplet }}$, the twist-pitch of the triple-helical cable $L_{\text {triplet }}$ stretches to $L_{\text {triplet }}+\Delta L_{\text {triplet }}$. Ignoring the changes in cross-section size, the length of wire in the triple-helical cable after deformation can be expressed as follow
$L_{\text {single }}+\Delta L_{\text {single }}=\sqrt{\left(L_{\text {triplet }}+\Delta L_{\text {triplet }}\right)^{2}+\left(2 \pi r_{0}\right)^{2}}$
Fig. 7. (a) Geometric structure of triple-helical cable, (b) the cross-sectional configuration, (c) the geometric relation of elongation and single wire. |
Substituting Eq. (1) into Eq. (2), the relationship of the elongation length between a single wire and the triple-helical cable after deformation can be written as
$\Delta L_{\text {single }}=\sqrt{\left(L_{\text {triplet }}+\Delta L_{\text {triplet }}\right)^{2}+\left(2 \pi r_{0}\right)^{2}}-\sqrt{L_{\text {triplet }}^{2}+\left(2 \pi r_{0}\right)^{2}} \text {. }$
Although Eq. (3) can be used to solve the relationship of elongation length between the triple-helical cable and single wire, its form is too complicated to solve their damage evolution equations. To apply the recursive model to the damage evolution equation, the length of a single wire can be the function of the twist-pitch of the triple-helical cable. According to Eq. (1), the derivatives of the function $L_{\text {single }}$ can be calculated as
$\frac{d L_{\text {single }}}{d L_{\text {triplet }}}=L_{\text {triplet }}\left[L_{\text {triplet }}^{2}+\left(2 \pi r_{0}\right)^{2}\right]^{-\frac{1}{2}}$
Based on Cauchy's Mean Value, when there is a very small increment $\Delta L_{\text {triplet }}$, the increment $\Delta L_{\text {single }}$ can be approximated as
$\Delta L_{\text {single }}=L_{\text {triplet }}\left[L_{\text {triplet }}^{2}+\left(2 \pi r_{0}\right)^{2}\right]^{-\frac{1}{2}} \Delta L_{\text {triplet }}.$
Substituting Eq. (1) into Eq. (5), which can be obtained as
$\frac{\Delta L_{\text {single }}}{L_{\text {single }}}=\frac{L_{\text {triplet }}^{2}}{L_{\text {triglet }}^{2}+\left(2 \pi r_{0}\right)^{2}} \frac{\Delta L_{\text {triplet }}}{L_{\text {triplet. }}}.$
Furthermore, Eq. (6) can also be re-written as
$\varepsilon_{\text {single }}=\varepsilon_{\text {triplet }} \sin ^{2} \alpha,$
where $\alpha=\arctan \frac{L_{\text {bipplef }}}{2 \pi r_{0}}$ denotes the helical angle, $\varepsilon_{\text {triplet }}$ is the strain of Nb3Sn triple-helical cable, $\varepsilon_{\text {triplet }}$ is the strain of a single wire. Eq. (7) can describe the recursive relationship between the triple-helical cable and single wire, which has the simple form to describe the damage evolution of triple-helical cables.
When the triple-helical cable is under strain $\varepsilon_{\text {triplet }}$, each wire exist tension force $f_T$ and transverse Contact force $f_B$. $f_T$ and $f_B$ can be projected along the axis direction, and the total force of the triple-helical cables can be expressed as [22]
$F_{\text {tripiet }}=n\left(f_{T} \sin \alpha+f_{B} \cos \alpha\right).$
Here, n is the number of wires in the triple-helical cables. Because the transverse contact force $f_B$ is small under tension processes, which can be negligible, Eq. (8) can be written as
$F_{\text {tripiet }}=n f_{T} \sin \alpha.$
Furthermore, Eq. (9) can also be re-written as
$\sigma_{\text {triplet }} \cdot n \cdot S=n \cdot \sigma_{\text {single }} \sin \alpha \cdot S \text {, }$
where $\sigma_{\text {triplet }}$ is the $\sigma_{\text {single }}$ of triple-helical cable, and σsingle is the stress of single wire, $S$ is the cross-sectional area of a single wire. According to Eq. (10), the stress relationship of stress between the triple-helical cables and single wire can be written as
$\sigma_{\text {single }}=\frac{\sigma_{\text {triplet }}}{\sin \alpha}.$
In this work, the degradation of chord modulus is used to describe the fatigue damage, and the chord modulus for the triple-helical cable can be defined as
$E_{\text {triplet }}=\frac{\Delta \sigma_{\text {triplet }}}{\Delta \varepsilon_{\text {tripiet }}}$
According to Eq. (7), (11), and (12), the chord modulus for the triple-helical cable can be written as
$E_{\text {triplet }}=\frac{\Delta \sigma_{\text {single }} \sin ^{3} \alpha}{\Delta \varepsilon_{\text {single }}}$
From Eq. (13), the chord modulus relationship between the triple-helical cables and single wire can be described as
$E_{\text {triplet }}=E_{\text {single }} \sin ^{3} \alpha.$
Based on the stiffness degradation rules, the damage factor of the triple-helical cable and single wire can be defined as [27]
$D_{i}(N)=\frac{E_{i}(0)-E_{i}(N)}{E_{i}(0)-E_{i}\left(N_{f}\right)}.$
where $i=\text { triplet, single }$ denotes the triple-helical cables and single wire; $E_{i}(0)$ is initial chord modulus; $E_{i}( N_f)$ is chord modulus at failure, $ E_{\text {triplet }}(N) $ and $ E_{\text {single }}(N) $ are chord modulus, N is cycling number.
4.2. Fatigue damage evolution model
$ \frac{d D_{\text {single }}(N)}{d N}=\frac{A\left(\Delta \varepsilon_{\text {single }}\right)^{c}}{\left[1-D_{\text {single }}(N)\right]^{b}} $
where A, b, c are three material fatigue constants to be obtained from experiments.
In this work, the variable of helical angle which describes helical structure is introduced into the classic fatigue damage model to reflect the effect of helical angle on the damage evolution characteristics of the triple-helical cables. According to Eqs. (7), (15), a damage evolution model considering the effects of helical angle for the triple-helical cables can be obtained
$ \frac{d D_{\text {triplet }}(N)}{d N}=\frac{A\left(\Delta \varepsilon_{\text {triplat }} \sin ^{2} \alpha\right)^{c}}{\left[1-D_{\text {triplet }}(N)\right]^{b}} $
Eq. (17) can also be re-written as
$ \left[1-D_{\text {triplet }}(N)\right]^{b} d D_{\text {triplet }}(N)=A\left(\Delta \varepsilon_{\text {triplet }} \sin ^{2} \alpha\right)^{c} d N $
Integrating Eq. (18) yields
$ \frac{-1}{b+1}\left[1-D_{\text {triplet }}(N)\right]^{b+1}=A\left(\Delta \varepsilon_{\text {triplet }} \sin ^{2} \alpha\right)^{c} N+F $
where, F is a constant. When N=0, $ D_{\text {triplet }}(0)=0 $, we can obtain as follow
$ F=-\frac{1}{b+1}. $
When $ N=N_{f}, D_{\text {triplet }}\left(N_{f}\right)=1 $, the influences of strain cycling range and helical angle on fatigue life can be obtained as
$ N_{f}=\frac{1}{A(b+1)}\left[\Delta \varepsilon_{\text {triplet }} \sin ^{2} \alpha\right]^{-c} $
Substituting Eqs. (20), (21) into Eq. (19), the damage factors considering the effects of the helical angle for the triple-helical cable can be obtained as
$ D_{\text {triplet }}(N)=1-\left(1-\frac{N}{N_{f}}\right)^{\frac{1}{b+1}} \text {. } $
4.3. Evolution model of energy dissipation
The fatigue damage is often accompanied by energy dissipation, the relationship between damage evolution and energy dissipation can be expressed as [26]
$ \frac{d D_{\text {triglpet }}(N)}{d N}=[K \Delta W(N)]^{M} $
where K and M are material fatigue constants. Substituting Eq. (22) into Eq. (23) yields
$ \frac{1}{(b+1) N_{f}}\left(1-\frac{N}{N_{f}}\right)^{-\frac{b}{b+1}}=[K \Delta W(N)]^{M} $
Eq. (24) can be written as
$ \Delta W(N)=\frac{1}{K}\left\{\frac{1}{(b+1) N_{f}}\left(1-\frac{N}{N_{f}}\right)^{-\frac{b}{b+1}}\right\}^{\frac{1}{M}}. $
When $ N=0 $ and $ \Delta W(0)=\frac{1}{K}\left[\frac{1}{(b+1) N_{f}}\right]^{\frac{1}{M}} $, the $ \Delta W(N) $ can be derived as
$ \Delta W(N)=\Delta W(0)\left(1-\frac{N}{N_{j}}\right)^{-\frac{b}{M(b+1)}}. $
For the triple-helical cables, Eq. (26) can be used to solve the energy dissipation considering the effects of the helical angle.
The above damage evolution model considering the effects of twist-pitches is summarized here. Firstly, a recursive model of mechanical properties between the triple-helical cables and single wire was derived. Then, the recursive model was introduced into the fatigue damage model, and obtained an important damage evolution equation that could solve the fatigue problem of helical structure (see Eq. (17)). Finally, the fatigue life, damage factor, and energy dissipation were obtained by deriving the damage evolution equation. The theoretical model can explain the mechanism of the effects of twist-pitch on fatigue damage for the triple-helical cable, and Eqs. (21), (22), and (26) can solve the fatigue life, damage factor, and energy dissipation, respectively, where the helical angle can be defined as $ \alpha=\arctan \left(\frac{L_{\text {trrpplet }}}{2 \pi r_{0}}\right) $. The theoretical model will be applied to the fatigue properties of Nb3Sn triple-helical cable.
5. Comparison of theoretical prediction and experimental results
5.1. Fatigue life
To determine the parameters $ \frac{1}{A(b+1)} $ and -c in Eq. (21), we used Eq. (21) to fit the relationship between fatigue life and strain cycling ranges from Fig. 6(a). The experimental data and fitting process were shown in Fig. 8(a). According to the parameters obtained in Fig. 8(a), Eq. (21) could be used to predict fatigue life. The Fig. 8(b) shows the comparison of fatigue life under various twist-pitches between theory results and experimental results from Fig. 6(b), which indicated that theoretical results agree with the experimental data, indicating that Eq. (21) can accurately predict the effects of twist-pitch on fatigue life.
Fig. 8. (a) The experimental data were used to fit the parameters of fatigue life, (b) comparison of theoretical results and experimental data with various twist-pitches. |
5.2. Damage factor
To determine the parameters $ \frac{1}{(b+1)} $ in Eq. (22), we used Eq. (22) to fit the relationship between the damage factor and strain cycling range with a twist-pitch of 45 mm and the strain of 0.68% in Fig. 5(a). The experimental data and fitting process were shown in Fig. 9(a). According to the parameters obtained in Fig. 9(a), Eq. (22) can be used to predict damage factors. Fig. 9(b) and (c) show the comparison of damage factor between theoretical results and experimental data from Fig. 5, and the theoretical results agree with the experimental data under various strain cycling ranges and twists pitches, indicating that Eq. (22) can accurately predict the effects of strain cycling ranges and twist-pitches on damage factor.
Fig. 9. (a) The experimental data were used to fit the parameters of damage factors, comparison of theoretical results and experimental data (b) at various strain ranges with the twist-pitch of 45 mm and (c) at various twist-pitches with a strain cycling range of 0.68%. |
5.3. Energy dissipation
To determine the parameters A and $ -\frac{b}{M(b+1)} $ in Eq. (26), we used Eq. (26) to fit the relationship between energy dissipation and strain cycling range with the twist-pitch of 45 mm and the strain of 0.68% in Fig. 3(a). The experimental data and fitting process were shown in Fig. 10(a). According to the parameters obtained in Fig. 10(a), Eq. (26) could be used to predict energy dissipation. Fig. 10(b) and (c) show the comparison of energy dissipation between theory results and experimental data from Fig. 2, and the theoretical results agree with the experimental data under various strain cycling ranges and twist-pitches, indicating that Eq. (26) can accurately predict the influence of strain cycling ranges and twist-pitches on energy dissipation.
Fig. 10. (a) The experimental data were used to fit the parameters of energy dissipation, comparison of theoretical results and experimental data (b) at various strain ranges with the twist-pitch of 45 mm and (c) at various twist-pitches with a strain cycling range of 0.68%. |
6. Discussion
For the single Nb3Sn superconducting wire, the energy dissipation shows three stages: initially rapid drop, gradually steady, and accelerated increase stages, which is caused by the damage of the Cu matrix and Nb3Sn sub-elements. The energy dissipation of Nb3Sn triple-helical structures increases with the number of cycles. The difference between the two phenomena may be attributed to the helical structure and interaction among the Nb3Sn superconducting wires.
According to the damage evolution model considering the effects of twist- pitches presented in Section 4, one can see that when the Nb3Sn triple-helical structure with short twist-pitches is under axial strain, each wire in triple-helical structure will occur smaller deformation based on Eq. (7). The tensile strain of the Nb3Sn triple-helical structure consists of the deformation of wire and elongation of the helical structure. There is a larger fraction of helical structure elongation and a smaller fraction of deformation of wire under tensile strain in Nb3Sn triple-helical structures with short twist-pitches. When the twist-pitch is infinitely long or the helical angle is 90°, three wires are straight in a triple-helical structure. The deformation of the triple-helical structure with infinitely long twist-pitch is the same as wire under tensile strain, where there is no elongation of the helical structure and the tensile strain of the triple-helical structure is only concluding the deformation of wire. During strain cycling, the elongation of Nb3Sn triple-helical structures cannot almost cause fatigue damage, and fatigue damage is nearly caused by the cycling deformation of superconducting wires. Therefore, Nb3Sn triple-helical structures with short twist-pitches possess better resistance to fatigue damage degradation than short twist-pitches samples. This theoretical model can predict better performance effectively in Nb3Sn triple-helical structures with short twist-pitches, which is opposite to other mechanical models in Refs. [31], [32], [33], which indicated that superconducting cables with long twist-pitches possess better performance.
Because the yield points of Nb3Sn superconducting wire are higher at lower temperatures which can contribute to smaller nonlinearity of stress-strain responses [34], there are smaller energy dissipation, slower damage degradation, and longer fatigue life for Nb3Sn superconducting wire during fatigue stage at a temperature of 77 K than that of 4.2 K. Namely, the Nb3Sn superconducting cables can have more safety margin when the fatigue damage model at a temperature of 77 K was applied to actual working conditions with 4.2 K.
It should be emphasized that the energy dissipation for Nb3Sn triple-helical structures with short twist-pitches is smaller than that of samples with a long twist-pitch, which will generate less heat during fatigue, and reduce the risk of quenching for the magnet system of ITER. These phenomena for Nb3Sn triple-helical structures with short twist-pitches are consistent with better performance of superconducting cables with a short twist-pitches of ITER in Ref. [15]. In the early fatigue stage, the damage degradation of Nb3Sn triple-helical structures grows rapidly, and Nb3Sn sub-elements occur damage seriously, and the degradation of superconducting properties is obvious [11], [26]. Moreover, Nb3Sn triple-helical structures with short twist-pitches can reduce the speed of damage degradation at the early fatigue stage. These findings can provide a better understanding of the performance damage of superconducting cables in ITER and are helpful to design the cabling strategies for large superconducting cables.
7. Conclusion
In this work, the fatigue damage properties of the Nb3Sn triple-helical structures have been investigated by experimental methods at liquid nitrogen temperature. The theoretical damage evolution model has been proposed to reveal the effect of twist-pitches on the fatigue properties of triple-helical structures, applied to Nb3Sn triple-helical cable. The main conclusions can be summarized as follows:
1.The Nb3Sn triple-helical structures with short twist-pitches have a longer fatigue life, slower damage degradation, and smaller energy dissipation. Moreover, the energy dissipation and accumulation of damage factors are positively correlated with the range of strain cycles.
2.The theoretical model considering the twist-pitches is in good agreement with the experimental results, which can accurately predict the effects of twist-pitches and strain cycling ranges on fatigue life, damage degradation, and energy dissipation at liquid nitrogen temperature.
3.The theoretical model reveals that there is a larger elongation of the helical structure and smaller cycling deformation of wires in triple-helical cables with short twist-pitches, which can contribute to better fatigue damage properties during the strain cycling process.
These findings can provide a better understanding of the fatigue behaviors of the superconducting cable of ITER, and the theoretical model of damage evolution for triple-helical structures is helpful to design engineering cables and artificial biological tissues.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 12232005，U2241267), and the Natural Science Foundation of Gansu Province of China (No. 23JRRA1118).
