Magnetoelastic buckling properties of type-II superconducting thin strip subjected to perpendicular magnetic field and parallel distributed uniform mechanical load

Abstract

The buckling analyses of type-II superconducting strip under applied perpendicular magnetic field and/or distributed uniform mechanical load are investigated in this paper. Based on the Bean critical state model, the electromagnetic body force is firstly given. Then, based on the classical plate theory and two-point initial value method, the critical buckling states of the superconducting strip with different boundary conditions are analyzed. Numerical results show the effects of both the thickness and boundary conditions of superconducting strip on the corresponding critical buckling loads. The present work should be helpful to the research and application of superconducting thin strips.

Keywords

Superconducting thin strip electromagnetic body force buckling critical buckling load

1. Introduction

As one kind of important electromagnetic intelligent materials and structures, type-II superconducting thin strips are widely used in power transmission cables, magnetic energy-storage devices, transformers, fault current limiters, motors, etc. [1–3]. However, as these superconducting systems are subjected to applied magnetic fields, the mechanical deformations of them caused by the corresponding Lorentz force will occur. And as the applied magnetic field approaches or exceeds a critical value, the structural instability including the buckling is inevitable [4,5]. In the meantime, the buckling deformations of superconducting devices will also lead to significant degradation of superconductivity [6]. Thus, the buckling problems of the superconducting devices mentioned above have become one of the most important issues, and up till now, a lot of great achievements have been made [7–17].

On the other hand, as one of important type-II superconducting bodies, although the study of superconducting strips’ mechanical behaviors has attracted extensive attention of scientific researchers in the fields of mechanics, physics, materials and so on in the past twenty years [2,3,18–30], the research on the buckling of superconducting strips is very limited. As far as we know, up to now, only the buckling instability in the critical state of a long type-II superconducting thin strip with strong pinning was reported [31], where a pair of sides of the rectangular strip is clamped-free (C–F) boundaries, the other two sides are free. It is found from [31] that the elastic instability appeared in high perpendicular magnetic fields and might cause an almost periodic series of flux jumps visible in the magnetization curve.

In this work, the magnetoelastic buckling properties of infinitely long type-II superconducting thin strip subjected to both perpendicular magnetic field and parallel distributed uniform mechanical load are investigated, where different boundary conditions are considered. Firstly, based on the Bean critical state model, the electromagnetic body force of infinitely long superconducting strip is given with the demagnetization effect being considered. Then, the governing equation of buckling problem of the superconducting strip is derived based on the classical thin plate theory and solved by using the two-point initial value method for different constraint conditions. Finally, the critical buckling loads of superconducting strips are calculated and discussed. The results should be beneficial to the engineering application of bonding structures between superconducting film and substrate [31].

2. Problem statement and basic equations

As shown in Fig. 1(a), an infinitely long superconducting strip is placed in an applied magnetic field B _a oriented parallel to the z-axis. The width and thickness of the superconducting strip are 2a and d respectively, and d ≪ 2a holds true. The Cartesian coordinate system (x, y, z) is located in the middle plane of this strip. Besides the external magnetic field, the superconducting strip is also subjected to a distributed uniform pressure load which is parallel to the x-axis along the boundaries |x| = a (see Fig. 1(b)). Due to the superconducting strip is infinite in the y direction, the current in this strip is assumed to depend on x only.

2.1. Electromagnetic body force

As shown in Fig. 1, for the considered model subjected to the applied magnetic field B _a, the distribution of current density and flux density can be given as follows [4] $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle j_{y}(x)=\left\{\begin{array}{@{}ll@{}}{\displaystyle \frac{2j_{c}}{{\pi}}}\arctan {\displaystyle \frac{cx}{\sqrt{|x_{p}^{2}-x^{2}|}}} & |x|\lt x_{p},\\ j_{c}{\displaystyle \frac{x}{|x|}} & x_{p}\lt |x|\lt a,\end{array}\right.\end{eqnarray}$ (1) $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle B_{z}(x)=\left\{ \begin{array}{@{} ll@{}}0 & |x|\lt x_{p},\\ B_{c} \text{arctanh}\left({\displaystyle \frac{\sqrt{|x^{2}-x_{p}^{2}|}}{c|x|}}\right) & x_{p}\lt |x|\lt a,\\ B_{c} \text{arctanh}\left({\displaystyle \frac{c|x|}{\sqrt{|x^{2}-x_{p}^{2}|}}}\right) & |x|>a, \end{array} \right. \end{eqnarray}$ (2) where $c=\sqrt{1-x_{p}^{2}/a^{2}}=\tanh (B_{a}/B_{c})$ , $x_p=a/\cosh(B_a/B_c)$ is the location of flux front, $B_{c}={\mu}_{0}j_{c}a/{\pi}={\mu}_{0}J_{c}/({\pi}\bar{d})$ is an introduced characteristic magnetic field, j _c is the critical current density which is a constant with the Bean critical state model, μ₀ is the vacuum permeability, J _c = j _c d and $\bar{d}=d/a$ .

Fig. 1.

A superconducting strip subjected to perpendicular magnetic field B _a and parallel mechanical load P.

Based on Lorentz’s theorem, the electromagnetic body force can be given as $\begin{eqnarray}\displaystyle \mathbf{f}=\mathbf{j}_{\mathbf{y}}\times \mathbf{B}_{\mathbf{z}}. & & \displaystyle\end{eqnarray}$ (3)

In the present problem, the electromagnetic body force is along the x direction, the value of it can be directly obtained from f _x = −j _y B _z. Meanwhile, the sum of electromagnetic body force in the z direction of the superconducting strip is $\begin{eqnarray}\displaystyle F_{x}=f_{x}d=\left\{\begin{array}{@{}ll@{}}0 & 0<|x|<x_{p},\\ -{\displaystyle \frac{{\mu}_{0}J_{c}^{2}}{{\pi}\bar{d}}}{\displaystyle \frac{x}{|x|}}\text{arctanh}\left({\displaystyle \frac{\sqrt{|x^{2}-x_{p}^{2}|}}{c|x|}}\right) & x_{p}<|x|<a.\end{array}\right. & & \displaystyle\end{eqnarray}$ (4)

2.2. Governing equation for buckling problem

Based on the Kirchhoff plate theory and noting the infinite length of superconducting strip in the y direction, the governing equation for this strip under electromagnetic body force and applied uniform distributed load (see Fig. 1) can be given as [5,32]: $\begin{eqnarray}\displaystyle D{\displaystyle \frac{d^{4}w}{dx^{4}}}-N_{x}{\displaystyle \frac{d^{2}w}{dx^{2}}}+F_{x}{\displaystyle \frac{dw}{dx}}=0, & & \displaystyle\end{eqnarray}$ (5) where $D=Ed^3/[12(1-μ^2)]$ is the flexural rigidity of the superconducting strip, E and μ are respectively the Young’s modulus and Poisson ratio. w is the deflection (i.e. the displacement component parallel to the z-axis). With the combined action of body force and distributed load, the mid-plate internal force N _x can be presented as $\begin{eqnarray}\displaystyle N_{x}=\displaystyle \int _{x}^{a}F_{x}dx-P, & & \displaystyle\end{eqnarray}$ (6) which is parallel to the x-axis. Considering Eq. ((4)), the integral in Eq. ((6)) can be further expressed as follows $\begin{eqnarray}\displaystyle \displaystyle \int _{x}^{a}F_{x}dx=\left\{\begin{array}{@{}ll@{}}{\displaystyle \frac{{\mu}_{0}J_{c}^{2}a}{{\rm\pi}\bar{d}}}\ln \left({\displaystyle \frac{x_{p}}{a}}\right) & 0\lt |x|\lt x_{p},\\ {\displaystyle \frac{{\mu}_{0}J_{c}^{2}a}{2{\rm\pi}\bar{d}}}\left[2\ln \left({\displaystyle \frac{x_{p}}{a}}\right)+{\displaystyle \frac{x}{a}}\ln \left({\displaystyle \frac{cx+\sqrt{x^{2}-x_{p}^{2}}}{cx-\sqrt{x^{2}-x_{p}^{2}}}}\right)\right. & \\ \left.\quad -\ln \left({\displaystyle \frac{ac+\sqrt{x^{2}-x_{p}^{2}}}{ac-\sqrt{x^{2}-x_{p}^{2}}}}\right)\right] & x_{p}\lt |x|\lt a.\end{array}\right. & & \displaystyle\end{eqnarray}$ (7)

For convenience, Eq. (5) can be further normalized as $\begin{eqnarray}\displaystyle {\displaystyle \frac{d^{4}\bar{w}}{d\bar{x}^{4}}}-\bar{N}_{x}{\displaystyle \frac{d^{2}\bar{w}}{d\bar{x}^{2}}}+\bar{F}_{x}{\displaystyle \frac{d\bar{w}}{d\bar{x}}}=0, & & \displaystyle\end{eqnarray}$ (8) in which $\bar{x}=x/a$ , $\bar{w}=w/a$ , $\bar{N}_{x}=a^{2}N_{x}/D$ , $\bar{F}_{x}=a^{3}F_{x}/D$ .

2.3. Boundary conditions

Same with Ref. [18], five kinds of different boundary conditions are evaluated here.

For simply supported-simply supported (S–S) boundaries (Case 1), we have $\begin{eqnarray}\displaystyle \bar{w}|_{_{\bar{x}=\pm 1}}=0,\quad \bar{M}_{x}|_{_{\bar{x}=\pm 1}}=0, & & \displaystyle\end{eqnarray}$ (9) where the normalized moment $\bar{M}_{x}$ is derived as $\bar{M}_{x}={\displaystyle \frac{a}{D}}M_{x}=-{\displaystyle \frac{a}{D}}\int _{-d/2}^{d/2}z{\sigma}_{x}dz=-{\displaystyle \frac{d^{2}\bar{w}}{d\bar{x}^{2}}}$ . For simply supported-clamped (S–C) boundaries (Case 2), we get $\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\bar{w}|_{\bar{x}=-1}=0,\quad \bar{M}_{x}|_{\bar{x}=-1}=0,\\[6.0pt] \bar{w}|_{\bar{x}=1}=0,\quad \left.\displaystyle {\displaystyle \frac{d\bar{w}}{d\bar{x}}}\right|_{\bar{x}=1}=0.\end{array}\right. & & \displaystyle\end{eqnarray}$ (10) For simply supported-free (S–F) boundaries (Case 3), the corresponding boundary conditions are $\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\bar{w}|_{\bar{x}=-1}=0,\quad \bar{M}_{x}|_{\bar{x}=-1}=0,\\[6.0pt] \bar{M}_{x}|_{\bar{x}=1}=0,\quad \left.\left(\bar{Q}_{x}-\bar{P}{\displaystyle \frac{d\bar{w}}{d\bar{x}}}\right)\right|_{\bar{x}=1}=0,\end{array}\right. & & \displaystyle\end{eqnarray}$ (11) where the normalized equivalent transverse shear $\bar{Q}_{x}={\displaystyle \frac{a^{2}}{D}}Q={\displaystyle \frac{a^{2}}{D}}\int _{-d/2}^{d/2}{\tau}_{xz}dz=-{\displaystyle \frac{d^{3}\bar{w}}{d\bar{x}^{3}}}$ , and $\bar{P}={\displaystyle \frac{P}{P_{c}}}\equiv {\displaystyle \frac{a^{2}}{D}}P$ . For clamped–clamped (C–C) boundaries (Case 4), we have $\begin{eqnarray}\displaystyle \bar{w}|_{\bar{x}=\pm 1}=0,\quad \left.{\displaystyle \frac{d\bar{w}}{d\bar{x}}}\right|_{\bar{x}=\pm 1}=0. & & \displaystyle\end{eqnarray}$ (12) Finally, for clamped-free (C–F) boundaries (Case 5), the following relations hold true: $\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\bar{w}|_{\bar{x}=-1}=0,\quad \left.{\displaystyle \frac{d\bar{w}}{d\bar{x}}}\right|_{\bar{x}=-1}=0,\\[6.0pt] M_{x}|_{\bar{x}=1}=0,\quad \left.\left(\bar{Q}_{x}-\bar{P}{\displaystyle \frac{d\bar{w}}{d\bar{x}}}\right)\right|_{\bar{x}=1}=0.\end{array}\right. & & \displaystyle\end{eqnarray}$ (13)

3. Solving method

3.1. The two-point boundary value method

In order to solve Eq. ((8)) together with different boundary conditions (i.e., Eqs (9.1)–(9.5)), the two-point boundary value method [32] is used.

For Case 1, introduce two deflection functions $\bar{w}_{1}$ and $\bar{w}_{2}$ , which respectively satisfies $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\bar{w}_{1}(-1)=0,\quad \bar{w}_{1}^{\prime }(-1)=1,\quad \bar{w}_{1}^{\prime \prime }(-1)=0,\quad \bar{w}_{1}^{\prime \prime \prime }(-1)=0,\\[6.0pt] \bar{w}_{2}(-1)=0,\quad \bar{w}_{2}^{\prime }(-1)=0,\quad \bar{w}_{2}^{\prime \prime }(-1)=0,\quad \bar{w}_{2}^{\prime \prime \prime }(-1)=1.\end{array} & & \displaystyle\end{eqnarray}$ (14) By using the Runge–Kutta method, the numerical solutions to Eq. ((8)) (i.e., $\bar{w}_{1}$ and $\bar{w}_{2}$ ) together with Eq. (10) can be easily obtained. Further assuming $\bar{w}=c_{1}\bar{w}_{1}+c_{2}\bar{w}_{2}$ to satisfy Eq. ((8)) and the boundary condition at $\bar{x}=-1$ , and substituting $\bar{w}$ into boundary condition at $\bar{x}=1$ yields $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}cc@{}}\bar{w}_{1}(1) & \bar{w}_{2}(1)\\[6.0pt] \bar{w}_{1}^{\prime \prime }(1) & \bar{w}_{2}^{\prime \prime }(1)\end{array}\right]\left(\begin{array}{@{}l@{}}c_{1}\\ c_{2}\end{array}\right)=0. & & \displaystyle\end{eqnarray}$ (15)

To get the non-zero solutions to both c ₁ and c ₂, the coefficient determinant of Eq. (11) must be zero. Therefore, for the Case 1, the following equation holds true $\begin{eqnarray}\displaystyle \bar{w}_{1}(1)\bar{w}_{2}^{\prime \prime }(1)-\bar{w}_{2}(1)\bar{w}_{1}^{\prime \prime }(1)=0. & & \displaystyle\end{eqnarray}$ (16)

Similarly, for Cases 2 and 3, we have $\begin{eqnarray}\displaystyle \bar{w}_{1}(1)\bar{w}_{2}^{\prime }(1)-\bar{w}_{2}(1)\bar{w}_{1}^{\prime }(1)=0, & & \displaystyle\end{eqnarray}$ (17) and $\begin{eqnarray}\displaystyle \bar{w}_{2}^{\prime \prime }(1)\times [\bar{w}_{1}^{\prime \prime \prime }(1)+\bar{P}\times \bar{w}_{1}^{\prime }(1)]-\bar{w}_{1}^{\prime \prime }(1)\times [\bar{w}_{2}^{\prime \prime \prime }(1)+\bar{P}\times \bar{w}_{2}^{\prime }(1)]=0, & & \displaystyle\end{eqnarray}$ (18) respectively.

Referring to Ref. [19], we can further obtain that for C–C boundary case (i.e., Case 4), $\begin{eqnarray}\displaystyle \bar{w}_{1}(1)\bar{w}_{2}^{\prime }(1)-\bar{w}_{2}(1)\bar{w}_{1}^{\prime }(1)=0 & & \displaystyle\end{eqnarray}$ (19) holds true, and for Case 5, $\begin{eqnarray}\displaystyle \bar{w}_{1}^{\prime \prime }(1)\times [\bar{w}_{2}^{\prime \prime \prime }(1)+\bar{P}\times \bar{w}_{2}^{\prime }(1)]-\bar{w}_{2}^{\prime \prime }(1)\times [\bar{w}_{1}^{\prime \prime \prime }(1)+\bar{P}\times \bar{w}_{2}^{\prime }(1)]=0 & & \displaystyle\end{eqnarray}$ (20) holds true. However, it should be pointed out that $\bar{w}_{1}$ and $\bar{w}_{2}$ in Eqs (12.4) and ((20)) are the two solutions to Eq. ((8)) and satisfy the following conditions at $\bar{x}=-1$ , $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\bar{w}_{1}(-1)=0,\quad \bar{w}_{1}^{\prime }(-1)=0,\quad \bar{w}_{1}^{\prime \prime }(-1)=1,\quad \bar{w}_{1}^{\prime \prime \prime }(-1)=0,\\[6.0pt] \bar{w}_{2}(-1)=0,\quad \bar{w}_{2}^{\prime }(-1)=0,\quad \bar{w}_{2}^{\prime \prime }(-1)=0,\quad \bar{w}_{2}^{\prime \prime \prime }(-1)=1.\end{array} & & \displaystyle\end{eqnarray}$ (21)

It is remarkable that for Case 1, Case 2 and Case 3, the boundary conditions at $\bar{x}=-1$ are the same, so the forms of $\bar{w}_{1}$ and $\bar{w}_{2}$ with these three cases are the same. Similarly, the forms of $\bar{w}_{1}$ and $\bar{w}_{2}$ for Case 4 and Case 5 are the same as well. Certainly, they are different with the first three cases and the other two cases.

Finally, by solving Eqs (12), the critical buckling states with five kinds of different boundary conditions can be obtained.

4. Numerical results and discussions

In the following numerical examples, the buckling analyses of superconducting strips under different constraints and loading states are carried out. According to Refs [33,34], the material constants are adopted as E = 125 GPa, v = 0.3, μ₀ = 4π ×10⁻⁷ N/A, and j _c = 10¹⁰ A/m².

4.1. Under only uniform distributed mechanical load

In order to verify the validity of the present study, the critical buckling load of the superconducting strip only subjected to uniform distributed mechanical load is firstly calculated. The results are plotted in Table 1. It is noted that the obtained critical bucking loads $\bar{P}_{cr}$ are nearly the same as those calculated for the corresponding buckling problem of an infinitely long elastic strip under the same load conditions [32], which implies that the present methods including results obtained here are correct. It is also interest to note that in this case the normalized $\bar{P}_{cr}$ only depends on boundary conditions and is independent of either material parameters or strip’s thickness. Besides, it is found that the corresponding critical buckling loads decrease in turn according to different boundary conditions of Case 4, Case 2, Case 1, Case 5 and Case 3. Additionally, as pointed out in Ref. [19], due to the S–F boundary belongs to non-statically determinate constraints, Case 3 is not considered latter.

Table 1
The values of normalized critical buckling load $\bar{P}_{cr}$ under different boundary conditions as B _a = 0

S–S S–C C–C C–F S–F

Present 9.8696 20.1907 39.4784 2.4674 0

Ref. [19] 9.8696 20.1900 39.4780 2.4674 0

	S–S	S–C	C–C	C–F	S–F
Present	9.8696	20.1907	39.4784	2.4674	0
Ref. [19]	9.8696	20.1900	39.4780	2.4674	0

4.2. Under only applied magnetic field

In this part, the critical buckling state of superconducting strip under only applied perpendicular magnetic field is evaluated. Because the compressive electromagnetic body force generally increases with the increasing of the applied magnetic field, the critical bucking magnetic field is only investigated in the process of increasing the applied magnetic field. Figure 2 describes the effects of normalized thickness $\bar{d}$ on the critical buckling magnetic field $\bar{B}_{\text{acr}}=B_{\text{acr}}/B_{c}$ for four kinds of different boundary conditions. Obviously, for one given kind of boundary conditions, the critical buckling magnetic field increases with the increasing of normalized thickness. The reason for this is perhaps that as shown in Fig. 3, for given boundary conditions and fixed applied magnetic field, with the increasing of $\bar{d}$ , x _p increases, while the distribution area of electromagnetic body force decreases. Besides, similar to the case of purely mechanical load applied at $\bar{x}=\pm 1$ , for a fixed $\bar{d}$ , the critical buckling magnetic fields still decrease in the order of Case 4, Case 2, Case 1 and Case 5, which indicates that for the bucking problem of superconducting strip in applied magnetic field, boundary restraint is still one of the most important factors affecting its buckling.

Fig. 2.

Normalized critical magnetic field $\bar{B}_{\text{acr}}$ versus characteristic scale $\bar{d}$ for different boundary conditions.

Fig. 3.

The distributions of normalized magnetic field $\bar{B}_{z}$ in the case of minimum critical buckling magnetic field (i.e., corresponding to $\bar{d}=∼0.008$ ) for different characteristic scales under C–C boundary condition.

Figure 4 shows the buckling modes of the superconducting strip corresponding to different boundary conditions. It is seen from Fig. 4 that in terms of buckling resistance, clamped constraint is generally better than simply supported constraint. Certainly, the free end is the weakest. In fact, this perhaps further explains why the critical magnetic field is the largest under C–C constraints.

Fig. 4.

The buckling modes of superconducting strip for different boundary conditions.

4.3. Under both uniform distributed load and applied magnetic field

In fact, in practical applications, superconductors are often affected by both external magnetic fields and mechanical loads simultaneously. Thus, the relations between normalized critical buckling magnetic fields versus the normalized applied mechanical loads of superconducting strip with different $\bar{d}$ and different boundary conditions are firstly investigated in this section. Numerical results are plotted in Figs 5–8. As shown in these figures, no matter which kind of constraint conditions, for a fixed $\bar{d}$ , $\bar{B}_{\text{acr}}$ will decrease with the increasing of $\bar{P}$ . Of course, there is no linear dependency relationship between them. In fact, for one kind of definite constraint conditions, the regions under different critical buckling curves shown in corresponding figures can be respectively considered as different steady-state regions corresponding to different $\bar{d}$ . From Figs 5–8, it is further known that for a fixed $\bar{d}$ , the steady-state regions still decrease in the order of Case 4, Case 2, Case 1 and Case 5. Additionally, it is also noted that for one given kind of constraint conditions, the curves for different $\bar{d}$ are not intersecting.

Fig. 5.

$\bar{B}_{\text{acr}}$ versus $\bar{P}$ under S–S condition for different characteristic scales (Case 1).

Fig. 6.

$\bar{B}_{\text{acr}}$ versus $\bar{P}$ under S–C condition for different characteristic scales (Case 2).

Fig. 7.

$\bar{B}_{\text{acr}}$ versus $\bar{P}$ under C–C condition for different characteristic scales (Case 4).

Fig. 8.

$\bar{B}_{\text{acr}}$ versus $\bar{P}$ under C–F condition for different characteristic scales (Case 5).

Then, in order to evaluate the influence of applied magnetic field on critical mechanical load, some typical curves between $\bar{P}_{cr}$ and $\bar{B}_{a}$ under the four kinds of given constraint conditions for a definite thickness of the strip are plotted in Fig. 9. Obviously, for a fixed $\bar{d}$ , the steady-state regions between the critical buckling mechanical load and applied magnetic field is reduced in the order of Case 4, Case 2, Case 1 and Case 5 as well.

Fig. 9.

$\bar{P}_{cr}$ versus $\bar{B}_{a}$ for different boundary conditions as $\bar{d}=$ 0.02.

Finally, for clarity and intuition, the critical buckling loads P _cr (unit: N ⋅ m⁻¹) as different magnetic fields B _a (unit: T) being applied and the critical buckling loads B _acr subjected to different mechanical loads P are respectively shown in Tables 2 and 3, where a = 1 ×10⁻² m. It is interesting to note that even if for a small B _a, e.g., as B _a = 0.1 T, the effects of thickness d on P _cr is very big. Also, it is known from Table 2 that as B _a = 4 T, even if for Case 4, for a thin superconducting strip such as d ≤ 2.0 × 10⁻⁴ m, the buckling has occurred under only the applied magnetic fields. In addition, Table 3 shows that for a fixed mechanical load, even if the applied magnetic field is not large, the buckling can still occur, which implies that in the case of the combined action of mechanical load and external magnetic field, the effects of both external magnetic field and mechanical load must be considered.

Table 2

The critical buckling loads P _cr (N ⋅ m⁻¹) under different thicknesses d (10⁻⁴ m) and applied magnetic fields B _a (T) for different Cases (a =1 ×10⁻² m)

d	B _a = 0.1				B _a = 4
	S–S	S–C	C–C	C–F	S–S	S–C	C–C	C–F
0.8	0.91 × 10²	6.84 × 10²	18.2 × 10²	—	—	—	—	—
2.0	0.86 × 10⁴	1.87 × 10⁴	3.52 × 10⁴	0.18 × 10⁴	—	—	—	—
12	1.95 × 10⁶	3.98 × 10⁶	7.87 × 10⁶	0.49 × 10⁶	1.36 × 10⁶	3.30 × 10⁶	6.98 × 10⁶	—

Note: – representing the buckling having occurred under only the applied magnetic fields.

Table 3

The critical buckling loads B _acr (T) under different thicknesses d (10⁻⁴ m) and mechanical loads P (N ⋅ m⁻¹) for different Cases (a =1 ×10⁻² m)

d	P = 5 ×10²				P = 5 ×10⁵
	S–S	S–C	C–C	C–F	S–S	S–C	C–C	C–F
0.8	0.0375	0.1205	0.1962	–	–	–	–	–
2.0	0.4439	0.6892	0.9663	0.1895	–	–	0.9393	–
12	11.3101	19.9717	30.7063	3.3603	8.2566	17.492	28.898	–

Note: – representing the buckling having occurred under only the mechanical loads.

5. Conclusions

This paper carries out the buckling analyses of type-II superconducting strip under applied perpendicular magnetic field and uniform distributed mechanical load at the boundaries. Based on the Bean critical state model and the Kirchhoff plate theory, the basic equations are firstly established. By utilizing two-point initial value method, the critical buckling loads including both the applied magnetic field and mechanical load of the superconducting strip with different geometry scales and boundary conditions are analyzed. The following conclusions are drawn: (1) For the case of the combined action of mechanical load and external magnetic field of a superconducting strip, both the applied external magnetic field and mechanical load have important effects on the buckling behaviors of the superconducting strip. (2) Whatever the applied load is only uniform distributed mechanical load, perpendicular magnetic field or the combined loads of them, increasing the thickness of superconducting strip can effectively impede structure buckling. (3) For a fixed thickness of the superconducting strip, with the selections of C–C, S–C, S–S and C–F constraint conditions, the superconducting strip becomes more and more vulnerable to instability.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11872257 and 11572358) and a project for innovation by graduate students in Hebei Province, China (Grant No. CXZZBS2018148).

References

Larbalestier

Gurevich

Feldmann

D.M.

, High-T _c superconducting materials for electric power applications, Nature 414 (2001), 368–377.

Suhir

, Approximate evaluation of the elastic interfacial stresses in thin films with application to high-T_c superconducting ceramics, International Journal of Solids and Structures 27 (1991), 1025–1034.

Tamulevičius

, Stress and strain in the vacuum deposited thin films, Vacuum 51 (1998), 127–139.

Brandt

E.H.

and Indenbom

, Type-II-superconductor strip with current in a perpendicular magnetic field, Physical Review B 48 (1993), 12893.

Timoshenko

S.P.

and Gere

J.M.

, Theory of Elastic Stability, McGraw-Hill Book Company, Inc., 1961.

Yue

Zhang

and Zhou

Y.H.

, Buckling behavior of Nb₃Sn strand caused by electromagnetic force and thermal mismatch in ITER cable-in-conduit conductor, IEEE Transactions on Applied Superconductivity 27 (2017), 8400911.

Chattopadhyay

, Buckling behavior of a superconducting magnet coil, International Journal of Solids and Structures 43 (2006), 5158–5167.

van de Ven

A.A.F.

and van Bree

L.G.F.C.

, Buckling of superconducting structures: A variational approach using the law of Biot and Savart, International Journal of Applied Electromagnetics in Materials 3 (1992), 111–137.

Zhou

Y.H.

Zheng

X.J.

and Miya

, Magnetoelastic bending and buckling of three-coil superconducting partial torus, Fusion Engineering and Design 30 (1995), 275–289.

10.

Miya

Takagi

Uesaka

, Finite element analysis of magnetoelastic bucking of eight-coil superconducting full torus, Journal of Applied Mechanics, Transactions ASME 49 (1982), 180–186.

11.

Moon

F.C.

, Bucking of a superconducting coil nested in a three-coil toroidal segment, Journal of Applied Physics 47 (1976), 920–921.

12.

van Lieshout

P.H.

Rongen

P.M.J.

and van de Ven

A.A.F.

, Variational approach to magneto-elastic buckling problems for systems of ferromagnetic or superconducting beams, Journal of Engineering Mathematics 22 (1988), 143–176.

13.

van Lieshout

P.H.

and van de Ven

A.A.F.

, Variational approach to the magneto-elastic buckling problem of an arbitrary number of superconducting beams, Journal of Engineering Mathematics 25 (1991), 353–374.

14.

Nawa

Kajita

Piao

, The mechanism of buckling in the high-strength Bi-2223 conductor due to release of bending strain, IEEE Transactions on Applied Superconductivity 27 (2017), 8400404.

15.

Lin

M.C.

Wang

Yang

T.T.

, Elastoplastic buckling on the bent waveguide of CESR-type SRF cavity, IEEE Transactions on Applied Superconductivity 17 (2007), 1281–1284.

16.

Moon

F.C.

, Experiments on magnetoelastic bucking in a superconducting torus, Journal of Applied Mechanics Transactions ASME 46 (1979), 145–150.

17.

Moon

F.C.

, Bucking of a superconducting ring in a toroidal magnetic field, Journal of Applied Mechanics Transactions ASME 46 (1979), 151–155.

18.

Abrecht

Ariosa

Cloetta

, Strain and high temperature superconductivity: Unexpected results from direct electronic structure measurements in thin films, Physical Review Letters 91 (2003), 057002.

19.

Wang

Peng

, Investigation of the strain relaxation and ageing effect of YBa₂Cu₃O_7−δ thin films grown on silicon-on-insulator substrates with yttria-stabilized zirconia buffer layers, Superconductor Science and Technology 19 (2006), 51.

20.

Yang

Y.M.

and Wang

X.Z.

, Flux-pinning-induced stress and magnetostriction in a superconducting strip under combination of transport current and magnetic field, Journal of Applied Physics 122 (2017), 115103.

21.

Guillamón

Suderow

Vieira

, Direct observation of stress accumulation and relaxation in small bundles of superconducting vortices in tungsten thin films, Physical Review Letters 106 (2011), 077001.

22.

Tan

S.Y.

Zhang

Xia

, Interface-induced superconductivity and strain-dependent spin density waves in FeSe/SrTiO₃ thin films, Nature Materials 12 (2013), 634.

23.

Zhang

X.Y.

Liu

Zhou

, Nonuniform magnetic stresses in high temperature superconducting thin films, Journal of Applied Physics 115 (2014), 157–201.

24.

Xue

Yong

H.D.

, Fracture behaviors of thin superconducting films with field-dependent critical current density, Physica C-Superconductivity and its Applications 492 (2013), 25–31.

25.

Yong

H.D.

Xue

and Zhou

Y.H.

, Thickness dependence of fracture behaviour in a superconducting strip, Superconductor Science and Technology 26 (2013), 055003.

26.

Yong

H.D.

Jing

and Zhou

Y.H.

, Crack problem for superconducting strip with finite thickness, International Journal of Solids and Structures 51 (2014), 886–893.

27.

Yong

H.D.

and Zhou

Y.H.

, Interface crack between superconducting film and substrate, Journal of Applied Physics 110 (2011), 063924.

28.

Feng

W.J.

Liu

Q.F.

and Han

, Crack problem for a functionally graded thin superconducting film with field dependent critical currents, Mechanics Research Communications 61 (2014), 36–40.

29.

Liu

Q.F.

Feng

W.J.

and Su

R.K.L.

, Inclined crack through a rhombic thin superconducting strip with transport current, International Journal of Applied Electromagnetics and Mechanics 49 (2015), 435–442.

30.

Feng

W.J.

Liu

Q.F.

and Su

R.K.L.

, Fracture behaviors of a functionally graded thin superconducting film with transport currents based on the strain energy density theory, Theoretical and Applied Fracture Mechanics 74 (2014), 73–78.

31.

Mints

R.G.

and Brandt

E.H.

, Buckling instability in type-II superconductors with strong pinning, Physical Review B 61 (1999), 11700–11703.

32.

Wang

C.Y.

, Buckling of a heavy standing plate with top load, Steel Construction 48 (2010), 127–133.

33.

Mikheenko

P.N.

and Kuzovlev

Y.E.

, Inductance measurements of HTSC films with high critical currents, Physica C 204 (1993), 229–236.

34.

Babaei Brojeny

A.A.

and Clem

J.R.

, Self-field effects upon the critical current density of flat superconducting strips, Superconductor Science and Technology 18 (2005), 888–895.

Magnetoelastic buckling properties of type-II superconducting thin strip subjected to perpendicular magnetic field and parallel distributed uniform mechanical load

Abstract

Keywords

1. Introduction

2. Problem statement and basic equations

2.1. Electromagnetic body force

3.1. The two-point boundary value method

4.1. Under only uniform distributed mechanical load

Table 1 The values of normalized critical buckling load P ¯ c r under different boundary conditions as B a = 0 S–S S–C C–C C–F S–F Present 9.8696 20.1907 39.4784 2.4674 0 Ref. [19] 9.8696 20.1900 39.4780 2.4674 0

Footnotes

Acknowledgements

References

Table 1
The values of normalized critical buckling load $\bar{P}_{cr}$ under different boundary conditions as B _a = 0

S–S S–C C–C C–F S–F

Present 9.8696 20.1907 39.4784 2.4674 0

Ref. [19] 9.8696 20.1900 39.4780 2.4674 0