Nonlinear contact behavior of HTS tapes during pancake coiling and CORC cabling

The REBCO tapes have a width of 2 mm (SCS2030) and 4 mm (SCS4050), which is much larger than their thickness of 45 μm (SCS2030) and 96 μm (SCS4050). As such, the simplified 2D model for the winding process of the pancake coils, as shown in figure 3, is based on the following assumptions and approximations with all parameters used in the equations listed in table 1:

• (a)  
  The dynamic factor in the winding process is ignored, assuming a quasi-static process.
• (b)  
  The width is considered to be much larger than the thickness, assuming that ${ }{\varepsilon _z} = 0$, which is the plane strain hypothesis.
• (c)  
  It is assumed that the multi-layered composite REBCO tape is a homogeneous material, the mechanical parameters are calculated by the volume fraction of each material component, which is the homogenization hypothesis, and only validating the XY plane.
• (d)  
  The friction during the winding process is ignored.
• (e)  
  The end effect of the REBCO tape and the center roller effect are not taken into account, according to the Saint–Venant principle. (The meaning of the Saint–Venant principle is that the difference between the effects of two different but statically equivalent loads becomes very small at sufficiently large distances from loads.)
• (f)  
  It is assumed that the central core is rigid.
• (g)  
  The distance between the reel and the core is considered very long, $d \gg r$.

Zoom In Zoom Out Reset image size

Table 1. List of parameters.

Parameter	Description	Parameter	Description
$F$	Winding pre-tension force	$t$	Tape thickness
$\bar F$	Normalized $\bar F = \frac{{F\rho \sqrt {3\left( {1 - {\nu ^2}} \right)} }}{{2w{t^2}\nu E}}$	$2w$	Tape width
$T$	$T = F/(2w*t)$	$\delta$	Characteristic length
r	Core radius	$l$	Tape length
$\rho$	Curvature radius	$\nu$	Poisson ratio
$M$	Applied moment	$\tau$	Bending deflection
$\theta$	Relaxation angle	$P$	Helix pitch
$b$	Relaxation offset distance	$\alpha$	Winding helix angle
$d$	Distance between reel and core $d \gg r$	$\lambda$	Winding factor $\sqrt[4]{{\frac{{3\left( {1 - {\nu ^2}} \right)}}{{{\rho ^2}{t^2}}}}}$
$p$	Normal contact force	$\bar \lambda$	Normalized $\bar \lambda = \,\lambda w$
$\bar p$	Average normal contact force	$D$	Tape stiffness $D = EI$
$f$	Tangential friction force	$E$	Young's modulus
$q$	Uniform contact load	$I$	Moment of inertia
$\zeta$	$\zeta = \frac{{\sqrt {3\left( {1 - {\nu ^2}} \right)} }}{{2w{t^2}\nu E}}$	$\eta$	$\eta = w\sqrt[4]{{\frac{{3\left( {1 - {\nu ^2}} \right)}}{{{t^2}}}}}$

Based on the above assumptions, considering isotropic stiffness of the tape for winding, the position where the tape starts to contact the core is in reality at point $A$, and not at point ${A_0}$ due to the relaxation effect. This phenomenon always exists in a natural winding process [23]. The relaxation offset distance is defined as the distance at which the pre-tension force extension line is tangent to the core, and the relaxation angle is defined as $\theta$ in figure 3(a). For the force analysis from figure 3, the following can be posed:

$$\begin{equation}{\text{Arc length equation:}}\;{\text{d}}s = \rho {\text{d}}\theta ,\end{equation} \tag{ 1 }$$

$$\begin{equation}{\text{Bending equation:}}\;\;\frac{1}{\rho } = \frac{M}{D},\end{equation} \tag{ 2 }$$

$$\begin{align}&{\text{Force analysis at infinitesimal section}}\;S:\nonumber\\&\quad\quad\quad\quad\quad{\text{d}}M = - F\sin \left[ {\theta - \theta \left( s \right)} \right]{\text{d}}s\end{align} \tag{ 3 }$$

$$\begin{align}{\text{Winding differential equation:}}\nonumber\\ D\frac{{{{\text{d}}^2}\theta }}{{{\text{d}}{s^2}}} = - F\sin \left[ {\theta - \theta \left( s \right)} \right]\end{align} \tag{ 4 }$$

$$\begin{align}{\text{Boundary conditions:}}\nonumber\\\theta \left( 0 \right) = 0,{ }\frac{{{\text{d}}\theta }}{{{\text{d}}s}}\left( 0 \right) = \frac{1}{r} \nonumber\\\theta \left( \infty \right) = \theta ,{ }\frac{{{\text{d}}\theta }}{{{\text{d}}s}}\left( \infty \right) = 0 \end{align} \tag{ 5 }$$

$$\begin{equation}{\text{Solution for:}}\;\;\theta = arc\cos \left( {1 - \frac{D}{{2F{r^2}}}} \right)\end{equation} \tag{ 6 }$$

$$\begin{equation}{\text{Relaxation offset distance:}}\;\;b = \frac{D}{{2Fr}}\end{equation} \tag{ 7 }$$

$$\begin{align}{\text{Tape axial force after winding:}}\nonumber\\ {F_A} = F\cos \theta = F\left( {1 - \frac{D}{{2F{r^2}}}} \right)\end{align} \tag{ 8 }$$

$$\begin{align}{\text{Axial strain in REBCO layer:}}\nonumber\\{\varepsilon _{{\text{ReBCO}}}} = \frac{{ - {t_{{\text{ReBCO}}}}}}{\rho } + \frac{{{F_{\text{A}}}}}{{w{\text{*}}t{\text{*}}E}}\end{align} \tag{ 9 }$$

$$\begin{align}&\quad\quad\quad\quad{\text{Axial stress in the tape:}}\nonumber\\&{F_A} = {F_N} = \frac{{{F_{\text{r}}}}}{2} = \mathop \int \limits_{\;0}^{\;{{\pi }}} \left(p \cdot \frac{{2w}}{2}{\text{d}}\varphi \right){\text{sin}}\varphi = pr\end{align} \tag{ 10 }$$

$$\begin{align}&{\text{Radial average contact force:}}\nonumber\\&\bar p = q = \frac{p}{w} = \frac{{F\left( {1 - \frac{D}{{2F{r^2}}}} \right)}}{{r*w}}.\end{align} \tag{ 11 }$$

In this section, assumption 2 from section 2.1 on the plane strain hypothesis is further elaborated by discussing the influence of the tape width on the winding result in detail. The cabling and coiling process can be described by the general equation of the cylindrical shell proposed by Timoshenko [16]; see figure 4.

Zoom In Zoom Out Reset image size

Due to symmetry, three of the six force equilibrium equations are automatically satisfied. Then, the following can be derived:

$$\begin{align} &{\text{Static equilibrium equation:}}\nonumber\\ &\left\{ {\begin{array}{*{20}{c}} {\frac{{{\text{d}}{N_x}}}{{dx}}\rho {\text{d}}x{\text{d}}\varphi = 0} \\[4pt] {\frac{{{\text{d}}{Q_x}}}{{{\text{d}}x}}\rho {\text{d}}xd\varphi + {N_\varphi }{\text{d}}xd\varphi + p\rho {\text{d}}x{\text{d}}\varphi = 0} \\[4pt] {\frac{{{\text{d}}{M_x}}}{{{\text{d}}x}}\rho {\text{d}}x{\text{d}}\varphi - {Q_x}\rho {\text{d}}x{\text{d}}\varphi = 0} \end{array}} \right.\end{align} \tag{ 12 }$$

$$\begin{equation}{\text{Displacement equation:}}{\mkern 1mu} \left\{ {\begin{array}{l} \displaystyle{{\varepsilon _x} = \frac{{{\text{d}}u}}{{{\text{d}}x}}} \\[3pt] \displaystyle {{\varepsilon _\varphi } = - \frac{\tau }{\rho }} \end{array}} \right.\end{equation} \tag{ 13 }$$

$$\begin{equation}{\text{Physical equation:}}{\mkern 1mu} {\mkern 1mu} \left\{ {\begin{array}{l} \displaystyle{{N_x} = \frac{{Et}}{{1 - {\nu ^2}}}\left( {{\varepsilon _x} + \nu {\varepsilon _\varphi }} \right) = 0} \\ \displaystyle{{N_\varphi } = \frac{{Et}}{{1 - {\nu ^2}}}\left( {{\varepsilon _\varphi } + \nu {\varepsilon _x}} \right)} \\ \displaystyle{{M_x} = - D\frac{{{{\text{d}}^2}\tau }}{{{\text{d}}{x^2}}}} \end{array}.} \right.\end{equation} \tag{ 14 }$$

This equation is a linear elastic equation and does not consider material plasticity.

From (12)–(14), we get

$$\begin{align} {\frac{{{{\text{d}}^4}\tau }}{{{\text{d}}{x^4}}} + 4{\lambda ^4}\tau = \frac{{p\left( x \right)}}{D}} \nonumber\\ {{\text{ where,}}\,{\lambda ^4} = \frac{{3\left( {1 - {\nu ^2}} \right)}}{{{\rho ^2}{t^2}}}} .\end{align} \tag{ 15 }$$

Equation (15) is a fourth-order non-homogenous differential equation with a general solution:

$$\begin{align}\tau \left( x \right) = & \ {e^{\lambda x}}\left( {{C_1}\cos \lambda x + {C_2}\sin \lambda x} \right) \nonumber\\ &+ {e^{ - \lambda x}}\left( {{C_3}\cos \lambda x + {C_4}\sin \lambda x} \right) + g\left( x \right).\end{align} \tag{ 16 }$$

Here, ${C_1},{C_2},{C_3}\,$ and ${C_4}$ are undetermined constants determined by boundary conditions, and $g\left( x \right)$ is a particular solution, which depends upon the form of applied loading $p\left( x \right)$. When $p\left( x \right)$ is zero, there is no contact between the tape and the core, which represents a pure bending process. Then, the solution of equation (15) is:

$$\begin{equation}\tau \left( x \right) = {C_{\text{a}}}\cos \lambda x*\cosh \lambda x + {C_{\text{b}}}\sin \lambda x*\sinh \lambda x\end{equation} \tag{ 17 }$$

with

$$\begin{equation*}{C_{\text{a}}} = \frac{{\nu t}}{{\sqrt {3\left( {1 - {\nu ^2}} \right)} }}\frac{{\cos \lambda w*\sinh \lambda w - \sin \lambda w*\cosh \lambda w}}{{\sin \lambda 2w + \sinh \lambda 2w}}\end{equation*}$$

$$\begin{equation*}{C_{\text{b}}} = \frac{{\nu t}}{{\sqrt {3\left( {1 - {\nu ^2}} \right)} }}\frac{{\cos \lambda w*\sinh \lambda w + \sin \lambda w*\cosh \lambda w}}{{\sin \lambda 2w + \sinh \lambda 2w}}.\end{equation*}$$

Due to the symmetry in the width direction of the tape and $\lambda w \ll 1$, the approximate simplified form of equation (17) can be obtained as [20]:

$$\begin{equation}\tau \left( x \right) = - \frac{{\nu t}}{{\sqrt {12\left( {1 - {\nu ^2}} \right)} }}{{\text{e}}^{ - \lambda x}}\left( {\cos \lambda x - \sin \lambda x} \right).\end{equation} \tag{ 18 }$$

Figure 5 shows the deflection diagram of the half-width plate after pure bending for different core radii. It presents the obviously anticlastic shape mode of tapes subjected to pure bending.

Zoom In Zoom Out Reset image size

Based on equation (18), we also get the 'ear' lifting length (distance from the edge to the point of maximum contact force) of the tape as:

$$\begin{equation}{\delta _1} = \frac{\pi }{{2\lambda }},\,{\text{from}}\,\,\frac{{{\text{d}}\tau \left( x \right)}}{{{\text{d}}x}} = 0\end{equation} \tag{ 19 }$$

$$\begin{equation}{\delta _2} = \frac{{5\pi }}{{4\lambda }},\,{ }{\delta _3} = \frac{\pi }{{4\lambda }},\,{\text{from}}\,\tau \left( x \right) = 0.\end{equation} \tag{ 20 }$$

and the 'ear' lifting height (height at which the edge is lifted) is:

$$\begin{equation}h = \frac{{\nu t}}{{\sqrt {12\left( {1 - {\nu ^2}} \right)} }}.\end{equation} \tag{ 21 }$$

When $\,p\left( x \right)$ is a non-zero constant, three contact shapes can be obtained according to [22], as shown in figure 6.

Zoom In Zoom Out Reset image size

Equations (22) and (23) give the expressions of ${\bar F_{\text{a}}}$ and ${\bar F_{\text{b}}}$, which are two critical normalized values when the winding factor $\lambda$ and tape width $w$ change. The contact type I is the single-line contact, the contact type II is the double-line contact, and the contact type III is the surface contact

$$\begin{equation}{\bar F_{\text{a}}} = \frac{{\sin \left( {\lambda w} \right) \cdot \cosh \left( {\lambda w} \right) + \cos \left( {\lambda w} \right)\sinh \left( {\lambda w} \right)}}{{2\lambda w\left[ {{\text{cos}}{{\text{h}}^2}\left( {\lambda w} \right) - {\text{co}}{{\text{s}}^2}\left( {\lambda w} \right)} \right]}}.\end{equation} \tag{ 22 }$$

$$\begin{equation}{\bar F_{\text{b}}} = \frac{1}{{2\lambda w \cdot \cosh \left( {\frac{\pi }{2}} \right)}}.\end{equation} \tag{ 23 }$$

The horizontal axis in figure 6 is basically the tape width divided by the tape thickness and the radius of curvature. The vertical axis is basically the pre-tension force multiplied by the radius of curvature and divided by the thickness of the tape. Therefore, the larger the radius of curvature, the contact shape is either type I (at low tension) or type III (at high tension). Otherwise, the contact shape is either type II (at low tension) or type III (at high tension).

It is worth noting that this method is an approximate solution, which approaches the contact force as a combination of the concentrated and uniform load. For the real case, the contact force distribution $p\left( x \right)$ is a function of the deflection $\tau$. Then, equation (15) has strong nonlinear characteristics and it is difficult to achieve an analytical solution. Therefore, a FEM method is required. In the next section, the contact force distribution is given by the finite element method.

The CORC cabling process goes with a specific helix angle as defined in figure 7.

Zoom In Zoom Out Reset image size

Therefore, the core radius r from above in the CORC cabling process will be replaced with a helical curvature radius ρ:

$$\begin{equation}{\text{Pitch and core radius:}}\;\;{\text{tan}}\alpha = \frac{P}{{2\pi r}}\end{equation} \tag{ 24 }$$

$$\begin{equation}{\text{Radius of helical curvature:}}\;\;\rho = \left( {1 + {\text{ta}}{{\text{n}}^2}\alpha } \right)r\end{equation} \tag{ 25 }$$

$$\begin{align}\begin{array}{l} {\textrm{Average normal contact force of helical surface:}}\\ \quad \quad {\textrm{}}\quad {\textrm{}}\quad {\textrm{}}\quad {\textrm{}}\quad {\textrm{}}\bar p = \frac{{F\left( {1 - \frac{D}{{2F{\rho ^2}}}} \right)}}{\rho } \end{array}\end{align} \tag{ 26 }$$

The description of the CORC cabling process also contains the relaxation and Poisson effects. After replacing r with $\rho$, equations (26) and (19) can approximately describe the relaxation, the contact force and the 'ear' lifting length during the CORC cabling process. The exact distribution can then be obtained from the FEM.

First, the winding process of a pancake coil is simulated. Therefore, the SCS4050 tape is taken as an example, the pre-tension force is set to 50 MPa, the core radius is taken to be 3 mm and the time is 10 s. The result, shown in figure 9, depicts that the contact force between the tape and the core is uniformly along the tape length, symmetrically along the tape width and without contact force on the tape edge. This happens to be the contact type III shown in figure 6. The yellow dashed line indicates where the relaxation area ends. The red dashed line position indicates the area with stress concentration, at which the tape and the core start to make contact. The relaxation effect remains stable during the entire calculation time.

Zoom In Zoom Out Reset image size

As shown in figure 10, a test platform is prepared to verify the theoretical and FEM simulations. Figure 10(d) shows how to measure the relaxation offset distance b and the relaxation angle $\theta$; a micrometer and a protractor are used in this step. Figure 11 compares the relaxation offset angle and relaxation offset distance under different winding pre-tension forces through theoretical calculations, FEM simulations and experimental tests. The tapes were prepared from SCS4050 by SuperPower Inc. The experimental test is performed by hanging different weights (50, 100, 200, 300, 500, 700, 900, 1000, 1500, 2000, 2500, 3000 and 4000 g) acting as the winding pre-tension force, and then slowly rotating the aluminum roller ($r = 10\,$ mm). The measured results show that as the winding pre-tension force increases, the relaxation effect becomes weaker. The results obtained from the theoretical calculations, FEM simulations and winding tests are consistent. However, the FEM model is smaller than the theoretical calculation results and closer to the test data because of the plasticity.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The next step is to calculate the contact behavior between the tape and the core with the FEM model. Figure 12 shows the contact stress distribution during the coiling process for two core radii. When the winding pre-tension force is small, the relaxation effect is more pronounced. For a winding pre-tension force less than $\frac{D}{{2{r^2}}}$, the winding fails (see figure 12(a), when T = 5 MPa, the tape is buckling). Detailed results are gathered in figure 13, showing that the contact force is highest at position δ from the edge of the tape and decreases rapidly to zero at the edge of the tape. This distribution indicates that the tape produces an anticlastic shape due to the Poisson effect during the winding process. According to equation (19), the 'ear' lifting occurs at the position $\,\delta = {\delta _1} = \frac{\pi }{{2\lambda }}$ from the tape edge, where the tape first contacts the core during winding. Finally, after the winding is completed, the contact force at this position is the largest. Interestingly, the 'ear' lifting length $\delta$ is only related to the winding factor $\lambda$ and not to the pre-tension force $F$ and tape width $w$. Following the FEM, with results presented in figure 13, also leads to a similar outcome. Using equation (21) the lifting height $h$ at the edge of the tape is calculated and obviously this part of the tape has no contact with the core during winding and the contact force is zero. Using equation (11), we can approximate the average contact force between the tape and the core, as shown by the dashed lines in figure 13. The contact force is inversely proportional to the radius and directly proportional to the winding pre-tension force. When the winding pre-tension force gradually increases, the contact type changes from double-line contact (contact type II) to single-line contact (contact type I), as shown by the solid lines in figure 13(a). When the core radius gradually increases, the contact type is type III and never changes, but the contact force value is averaging progressively.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For the helical winding used for CORC cabling, the SCS4050 tape is taken as an example. The pre-tension force is 100 MPa, the helix angle is 45°, the core radius is 3 mm and the time is 10 s. The result is shown in figure 14, with the yellow dashed line marking the relaxation area. The red dashed line is the position at which the tape and the core start to make contact. The results show that the helical winding is different from the pancake coiling. During the helical winding process, the relaxation on both sides of the tape is different, while symmetrical for the pancake winding process. For further explanation, the non-relaxation side is referred to as the A-side and the relaxation side as the B-side. Dividing the whole helical wound tape into two sections, the forward helix is the first half of the wound tape length, and the backward helix is the rear half. In the forward helix, the A-side's contact force is greater than that of the B-side, and in the backward helix, the A-side's contact force is less than that of the B-side. The relaxation on both sides of the tape is different during helical winding, which causes the contact force between the tape and the core to be asymmetric in the width direction. In addition, no contact force on the tape edge indicates that the edge 'ear' lifting exists during the helical winding, affecting the distribution of the axial strain in the REBCO layer. The 'ear' lifting length can also be estimated by equation (19). Figure 15 displays the axial strain of the REBCO layer along the helix direction. It can be seen that the axial strain of the REBCO layer also has the 'ear' at the edge of the tape. In the forward helix region, the absolute value of the A-side's axial strain is smaller than that of the B-side.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

This is because the REBCO layer is below the strain neutral axis. The bending deformation causes a compressive strain, the pre-tension force results in a tensile strain, and the contact force is proportional to the winding pre-tension force. Therefore, combining equations (9) and (10) indicates that the greater the contact force in the area, the larger the tensile axial strain, but the overall level of axial strain decreases within this practical range of stress.

A test is performed to analyze the helical winding process and the traces of the contact between the tape and the former; the result is shown in figure 16. The experimental procedure starts with wrapping the core with white paper and covering the tape's inside with paint. After completing the helical winding process, the pressed white paper with paint shows relaxation only at the B-side, which agrees with the FEM result. In summary, the relaxation and Poisson effects, both occurring during the helical winding process, can be described well by the given models.

Zoom In Zoom Out Reset image size

4.2.1. Winding pre-tension force.

The contact force distribution between the tape and the core is directly related to the imposed winding pre-tension force in the helical winding process (see figure 17). First, as the winding pre-tension force increases, the contact force between the tape and the core increases, which agrees with the results calculated by equation (26). Second, during the helical winding process, the winding pre-tension force does not affect the 'ear' lifting length as resulted from equation (19). Furthermore, because of the helical winding, the tape's relaxation condition is different on both sides. The A-side's contact force is higher than that of the B side in the forward helix, while it is the opposite in the backward helix.

Zoom In Zoom Out Reset image size

4.2.2. Winding helix angle.

The helix angle also has a significant influence on the contact stress, and the results of these calculations are shown in figure 18. As the winding helix angle increases, the contact force between the tape and the core decreases. When the helix angle is 0°, corresponding to the pancake winding, the A-side and the B-side will relax simultaneously, while the relaxation only occurs on the B-side at other angles. Moreover, as the angle increases, the relaxation on the B-side also increases. As the winding angle increases, the 'ear' lifting length also gradually decreases until the angle becomes larger than 60°. The contact stress distribution, along with the tape width, changes from contact type II (between 0°and 35°) to contact type I (larger than 55°), which is confirmed by the results from equation (19).

Zoom In Zoom Out Reset image size

4.2.3. Radius of winding core.

Like the other geometrical parameters, the core radius of a CORC also significantly affects the contact stress (figure 19). As the core radius increases, the contact force between the tape and the core decreases. It is also observed that the A-side's contact force is greater in the forward helix than that at the B-side. However, when the core radius is reduced to 1.5 mm, at the same position, the A-side's contact force becomes less than that at the B-side. This is because the stress concentration at the red dashed line position in figure 19(a) causes a gap between the tape and the core. As the core radius increases, the relaxation effect gradually weakens. It can be seen from the yellow and red dashed lines' relative position in figure 19(a). The 'ear' lifting length also increases as the core radius increases.

Zoom In Zoom Out Reset image size

In engineering practice, the winding pre-tension force requires a suitable range. In equation (8), ${F_{\text{A}}}$ is the axial force in the tape. When ${F_A} \leqslant 0$, obviously the tape cannot be tightly wound on the core, and causes winding failure (the tape is buckling). When the winding pre-tension force is taken too large, the axial strain in the REBCO layer can exceed the irreversibility strain limit $\left( {{\varepsilon _{{\text{irr}}}}} \right)$ and as a result, the tape's current transport performance will be degraded irreversibly. The minimum winding pre-tension force for pancake coils and CORC cables is $\frac{D}{{2{r^2}}}$. Figure 20 shows the winding pre-tension force of SCS4050 and SCS2030 for different winding radii. The yellow area represents 60%–95% of the critical current, and the orange area is more than 95% of the critical current after winding. Point A is the minimum winding radius limit. Point D is the minimum winding radius that retains 95% of the critical current transport performance. Although evident, the results show that SCS2030 tape can make smaller pancake coils and thinner CORC cables.

Zoom In Zoom Out Reset image size

In addition, we discuss the contact shape between the tape and the core after the winding process under different winding pre-tension forces and different winding factors. There are three shapes of contact (figure 6), identified as respectively: single-line contact (contact type I), double-line contact (contact type II) and surface contact (contact type III). These three different shapes are described by equations (22) and (23). For the winding process, the engineering concern is the winding pre-tension force $F$ and winding factor $\lambda$. Therefore, substituting $\bar F = \frac{{Fr\sqrt {3\left( {1 - {\nu ^2}} \right)} }}{{2w{t^2}\nu E}}$ and $\bar \lambda = w\sqrt[4]{{\frac{{3\left( {1 - {\nu ^2}} \right)}}{{{\rho ^2}{t^2}}}}}$ into equations (22) and (23) results in:

$$\begin{equation}{F_{\text{c}}} = \frac{{\sin \left( {\eta {r^{ - 0.5}}} \right) \cdot \cosh \left( {\eta {r^{ - 0.5}}} \right) + \cos \left( {\eta {r^{ - 0.5}}} \right)\sinh \left( {\eta {r^{ - 0.5}}} \right)}}{{2\eta {r^{0.5}}\left[ {{\text{cos}}{{\text{h}}^2}\left( {\eta {r^{ - 0.5}}} \right) - {\text{co}}{{\text{s}}^2}\left( {\eta {r^{ - 0.5}}} \right)} \right]\zeta }},\end{equation} \tag{ 27 }$$

$$\begin{equation}{F_{\text{d}}} = \frac{1}{{2\eta {r^{0.5}} \cdot \cosh \left( {\frac{\pi }{2}} \right)\zeta }},\end{equation} \tag{ 28 }$$

Here, the parameters $\eta$ and $\zeta$ are constants and only depend on the tape selection (see table 1). In this case, the contact shape diagram is obtained using the SCS2030 and SCS4050 tapes as examples; see figure 21. The black line is calculated by equation (27), the red line is obtained by equation (28) and the blue line is the minimum winding pre-tension force after considering the relaxation effect. The contact shape is also calculated by the FEM model, for both the SCS2030 tape (F = 0.9 or 2.7 N; $\rho$ = 3 or 30 mm) and the SCS4050 tape (F = 5 or 20 N; $\rho$ = 3 or 40 mm). The contact force distribution is consistent with the location of the black spot. Usually, the core radius of pancake coils is larger. When the winding curvature is larger than 15 mm (SCS2030) or 25 mm (SCS4050), the contact shape is either type II (at low tension) or type III (at high tension). For the CORC cable, the helical curvature radius is smaller. When the helical curvature radius is smaller than 10 mm, the contact shape is either type I (at low tension) or type III (at high tension).

Zoom In Zoom Out Reset image size

Interesting to note is the detailed contact behavior of SCS2030 and SCS4050 tapes after the winding process. It is worth mentioning that it is assumed in the calculations that the tape surfaces are flat and plane-parallel in the initial condition. However, the copper surface will have its specific roughness and profile in reality. In figure 22 the results are presented for the tape's contact stress distribution during pancake coiling (tape SCS2030 and SCS4050, r= 3 mm, 2w= 2 or 4 mm, t= 0.045, or 0.096 mm, α = 0°). The dotted lines are the calculation results of equation (26). The results show that the SCS2030 tape also undergoes the edge 'ear' lifting due to the Poisson effect. As previously noted, the contact force also increases as the pre-tension force increases. Compared to the SCS4050 tape, the contact force in the SCS2030 tape is small for the same winding parameters (pre-tension force, core radius, helix angle). The corresponding axial stress in the REBCO layer and the critical current degradation are also small. The 'ear' lifting length for the SCS2030 tape is shorter than for the SCS4050 tape. When the pre-tension force is 10 MPa, the core radius is 3 mm, and the helix angle is zero degrees, both contact shapes of SCS2030 and SCS4050 tapes are of type II, which is a double-line contact.

Zoom In Zoom Out Reset image size

In summary, for both pancake coiling and CORC cabling, the relaxation and Poisson effects need to be considered throughout the above analyses. The non-linear contact behavior between the tape and the core is mainly described by the contact force and the edge 'ear' lifting length. Equation (26) approximates the average force. For a selected tape, the relation between contact shape, winding pre-tension force and core radius can be assessed from figure 21. The FEM model can accurately calculate the contact stress between the tape and the core during pancake coiling and CORC cabling. The calculation results indicate that the SCS2030 tape is indeed more flexible (thinner and narrower) than the SCS4050 tape and more suitable for making pancake coils and CORC cables.