1 Introduction

Bi₂Sr₂Ca₁Cu₂O_8+x(Bi2212) superconducting composite wires have the high critical current density and high critical temperature^[1]. The critical current density can reach 10³ A/mm² in field up to 45 T^[2]. The wires are made in the round wire and multifilament form which can be used in many high field magnets. They can exceed the limitation of about 20 T in Nb₃Sn magnet^[3]. The preparation of Bi2212 wire adopts a powder-in-tube process which undergoes complex heat treatment^[4-5]. The highest temperature of heat treatment is about 890 ℃^[6], while the wire works at very low temperatures of (4 K, 77 K) in the application. When the wire is cooled down to the operation temperature, the filaments will be subject to a pre-compression load^[7-8]. It is well known that the Bi2212 filament is brittle after the reaction, and the critical current is sensitive to the strain^[9]. For the wire working in the such low temperature and high magnetic field, it is necessary to consider the mechanical behavior to enhance the electromagnetic stability.

There are many theoretical or experimental researches which consider the mechanical characteristics in the Bi2212 wire^{[4, 6, 10-13]}. Cheggour et al.^[4] made a systematic study on the critical current degradation. The experimental data show that the critical current will decrease by 26% when the wire undergoes 1% compressive strain. They pointed out that the breakage of filaments (tensile loading) and buckling (compressive loading) leads to the degradation of critical current. Godeke et al.^[6] and Bjoerstad et al.^[10] also studied the effect of mechanical deformation on the critical current. Ochiai et al.^[14] considered the degradation of critical current as the superconductor experiences the thermal cycling between the room temperature and 77 K. The estimation of Young's modulus and the residual strain was accomplished by Shin et al.^[15]. However, the temperature range is from the Bi2212 wire preparation temperature to the room temperature, which may cause the buckling of filaments. It is important to investigate the relationship between the buckling phenomenon and the critical current degradation.

In addition, for the buckling analysis of composite, the classical models are Rosen's extension model and shear model^[16]. The extension model and shear model are studied extensively with the analytical and numerical methods^[17-19]. The Bi2212 wire contains many defects or voids^[20-21] and it is important to consider the buckling behavior in the wire with voids. In this paper, the degradation of current density in the Bi2212 wire is studied based on the previous model^[22]. According to the experimental results, the approximate relationship between the critical current and buckling wavelength is presented. Then, we present some studies on the effect of parameters. Furthermore, the bridges between filaments exist widely in the wire^[23]. Finally, we try to find the relationship between the critical current and the angle of bridge with the finite element method.

2 Basic equations

Consider a unidirectional filament composite where the filaments are aligned in the x-direction, as shown in Fig. 1. The uniform compressive stress is applied to the composite. According to the two-dimensional model, the composite is composed of the filament with a width 2r and the matrix with a width 2a. In the fabrication of Bi2212 wire with high temperature and chemical reaction, the inclusion and gas bubbles are inevitable^{[20-21, 24]}. The inclusion and void will affect the mechanical behavior significantly and should not be neglected in the analytical model^[25]. Here, the width of inclusion or void is expressed as b. Then, the following assumptions are adopted in the analytical model.

(ⅰ) When the buckling occurs, each filament is assumed to deform in a sinusoidal pattern^[22],

(1)

(2)

where l₀ is the buckling wavelength, Y is the displacement of the midline of the filament, and A is the amplitude of the horizontal displacement.

(ⅱ) When the load exceeds the critical value, two filaments which are located on two sides of the matrix may have different amplitudes of filament buckling. In Fig. 1, the filaments with the odd numbers hold different amplitudes of horizontal displacements from the filaments with the even numbers. It can be described as

(3)

where K is the parameter, and the range is [-1, 1].

(ⅲ) The filament is assumed to be infinitely long, and the wavelength is much larger than the filament width. We suppose that the lines which are vertical to the center line in the initial state will always be vertical to the center line during buckling^[26].

(ⅳ) Deboning is neglected in this mode. In other words, the stress in the matrix and the inclusion is continuous and can be described by the Airy stress functions^[27],

(4)

(5)

where the subscripts m and v represent the matrix and inclusion, respectively, and c₁, c₂, …, c₈ are the undetermined coefficients. It can be proven that the stress functions satisfy the biharmonic equation,

(6)

The periodic element is presented in Fig. 1. Due to the symmetry of the periodic element, we take half of the periodic element as the analysis element.

Consider the equilibrium of a filament segment^[22](see Fig. 2), where P is the applied compressive force along the longitudinal direction. The equilibria of the force and moment in the filament segment imply^[22]

(7)

(8)

where S is the transverse shear force, and σ_y and τ_xy are the normal stress and the shear stress at the filament surface.

The relationship between the bending moments M and Y is

(9)

Combining Eqs.(7)-(9), one obtains

(10)

where E_f is Young's modulus of the filament, and I is the moment of the inertia of the filament area.

The micro-buckling of the filament can lead to the nonuniform deformation of the matrix. It is simplified that the matrix is linearly elastic with Young's modulus E_m and Poisson's ratio υ_m, respectively. As Young's modulus of inclusion E_v is very small, the inclusion can be regarded as a void. Poisson's ratio of this part is represented by υ_v.

Based on the stress functions (4) and (5), the stresses in the matrix and inclusion are derived as follows:

(11)

(12)

(13)

(14)

(15)

(16)

Combining Eqs.(11)-(16), the displacements of the matrix and inclusion can be expressed as follows:

(17)

(18)

(19)

(20)

where u_m(x, y) and v_m (x, y) are the displacements of the matrix in x-and y-directions, and u_v (x, y) and v_v(x, y) are the displacements of the inclusion.

Then, c₁, c₂, …, c₈ could be obtained by using the stress and displacement boundary conditions at the filament-matrix and filament-inclusion interfaces, i.e.,

(21)

(22)

(23)

(24)

Substituting c₁, c₂, …, c₈ into Eqs.(10)-(16), the critical load of buckling can be obtained. The parameter K is a function of the wavelength. For a given mechanical loading, the deformation mode should be fixed. Thus, the wavelength corresponds to the unique value of K.

3 Residual strain and wavelength 3.1 Estimation of residual strain of Bi2212 filaments

During the preparation of Bi2212 wire, the highest temperature is more than 880 ℃^[4]. The Bi2212 wire consists of filaments, Ag and Ag alloy, which have different thermal expansion coefficients. Therefore, the residual strain is remarkable as the wire is cooled down to the working temperature (4.2 K). For the buckling analysis, it is necessary to consider the residual strain. It is well known that the Bi2212 wire exhibits the plastic behavior in the larger strain^{[12, 28]}. This is due to the reason that Ag and Ag alloy are elasto-plastic materials which will yield for a larger strain or stress. Here, we use the bilinear model to describe the stress-strain relationship between Ag and Ag alloy. For Bi2212 filaments, they always behave elastically. The material parameters are presented in Table 1.

Based on the difference of material parameters at different deformation stages, the calculation can be divided into three stages as follows:

(Ⅰ) For the first stage, the composite is cooled from 888 ℃ to the yielding of Ag. In this stage, all materials are elastic. The residual strain can be calculated by the following equations^{[8, 15]}:

(25)

(26)

(27)

(28)

(29)

(30)

(31)

where ε_Alloy⁽¹⁾ and ε_Bi⁽¹⁾ are the residual strains of each component, and ε_Ag⁽¹⁾ is the yielding strain of Ag. σ_Ag⁽¹⁾, σ_Alloy⁽¹⁾, and σ_Bi⁽¹⁾ are the residual stresses. α₁ is the thermal expansion coefficient of the composite, and ∆T₁ is the temperature reduction in the first stage. After combining Eqs.(25)-(31), the unknown variables can be solved analytically. It is found that α₁ is about 18.32× 10^-6 K^-1, and ∆T₁ is about -49 K.

(Ⅱ) In the second stage, Ag alloy will change from the elastic deformation to the plastic yielding. Bi and Ag alloy still show linearly elastic behavior in this stage, while Ag will experience the plastic deformation. In this stage, Eqs.(26)-(29) should take the form

(32)

(33)

(34)

(35)

where ε_Ag⁽²⁾ and ε_Bi⁽²⁾ are the residual strains of each component, and the superscript (2) represents the second stage. In the second stage, ε_Alloy⁽²⁾ is the yield strain of Ag alloy. The subscripts "yield" and "reduced" stand for the material stress states, and the yield strain and reduced modulus of Ag can be found in Table 1. Here, α₂ is about 15.36× 10^-6 K^-1, and ∆T₂ is about -741 K.

(Ⅲ) The last stage is from the yielding of Ag alloy to 4.2 K. The plastic deformation will occur in both Ag and Ag alloy. In this stage, ∆T₃ is -366 K. ε_Bi⁽³⁾ is about -0.003 68, and σ_Bi⁽³⁾ is about-244 MPa.

3.2 Compression load and micro-buckling wavelength

For the Bi2212/Ag/Ag alloy wire, the strain which leads to filament buckling includes two parts, the residual strain and the applied strain,

(36)

(37)

Then, we have obtained the total compression loading applied on the Bi filament. The wavelength is computed as the applied strain changes from 0 to -0.008. In terms of the experimental test, the widths of the Ag matrix and Bi filaments are assumed to be 14 μm. The volume fraction of void is 5%. The variation of critical current with the applied strain is given by the experiment^[4], and the critical current without loading is about 114 A. The variation of wavelength with the applied strain is shown in Fig. 3.

Based on the above results, the critical load of filament buckling can be obtained as 232 MPa for K=1, which is very close to the residual stress 244 MPa. This means that the buckling occurs even when the applied loading vanishes.

3.3 Fit with experimental results

In the above section, the relationship between the applied strain and the wavelength is obtained. As pointed out in Ref.[4], the reduction of critical current density is dependent on the buckling of Bi2212 filaments. It is necessary to find the wavelength dependence of the critical density. Then, we try to determine the relationship based on the experimental results^[4]. By considering the stress-strain behavior, the variation of wavelength with the critical current is plotted in Fig. 4.

In order to avoid the complex expression, here, a simple fitting function is given by

(38)

where α and β are fitting parameters, and α is the critical current without the applied strain. It is interesting to find that the critical current is dependent on l₀⁻². Buckling can lead to the irreversible damage in the Bi2212 wire and cause the reduction of critical current. Some studies on the effect of material parameters are carried out with Eq.(38). Since Eq.(38) is only an approximate fitting, the qualitative results are presented.

4 Results and discussion 4.1 Filament width

In this section, we will consider the effect of filament width on the critical current and wavelength. According to Section 2, the increasing of filament width leads to a longer wavelength. In addition, a longer wavelength results in the higher critical current. The results are shown in Figs. 5 and 6. Three different filament radii are selected, that is, 6 μm, 7 μm, and 8 μm. In Fig. 5, it can be expected that the wavelength increases with the increasing of filament width. Then, the critical current will also increase with the filament width (see Fig. 6). Here, we neglect the effect of self-field on the critical current^[30-32]. In other words, a larger filament width corresponds to the higher critical current under the same applied loading. The result means that the wire with the larger filament width has better electromagnetic stability.

4.2 Volume fraction of void

It is well known that there are many inclusions and gas bubbles in the Bi2212 wire during the fabrication^[20]. The void can affect the mechanical and superconducting properties. The impact of void on the critical current of Bi2212 wire is discussed.

The void will change both the critical stress or strain and the buckling wavelength. Three different void widths are assumed, 2.5%, 5%, and 7.5% of the total width. The variations of wavelength and critical current with the applied strain are shown in Figs. 7 and 8, respectively. The lower void width leads to a higher critical strain. In other words, a lower void width makes the buckling harder. In addition, the buckling does not appear at the smaller strain for 2.5% void (see the line with the rectangle in Fig. 7), and the critical current will not degrade. The reason is that when the volume fraction of the void is 2.5% (see the line with the rectangle in Fig. 8), the residual strain is not large enough to reach the critical buckling strain. Thus, there is no degradation when the applied compressive strain is smaller than 0.002. When the applied strain reaches about -0.002, the buckling occurs, and the critical current begins to degrade. However, for the larger void, the critical current is still reduced in the case of the zero applied strain. In other words, decreasing the void size can avoid the buckling of filament which can enhance the critical current. The results may explain why buckling does not appear in some experiments^[6].

4.3 Young's modulus of Ag alloy

When Young's modulus of Ag alloy changes, the effective modulus of composite will be different. Thus, the compressive loading supported by the filaments is dependent on Young's modulus of Ag alloy, which is given by

(39)

where 0.19, 0.49, and 0.32 represent the volume fractions of Ag alloy, Ag, and Bi^[15]. The impact of Young's modulus of Ag alloy includes two parts. Firstly, the residual strain in the composite is different; the other part is the equivalent Young's modulus. As shown in Fig. 9, with Young's modulus of Ag alloy decreasing, the critical current increases for |σ| 165 MPa. However, the trend is opposite for the larger stress. This is due to the fact that the smaller Young's modulus corresponds to the smaller residual strain, while under the same mechanical loadings, the wire with the smaller Young's modulus has the larger strain.

4.4 Bi filament volume fraction

Since the critical current will change with the volume fraction of Bi filament, the critical current in this section can be expressed as

(40)

where V_f^* is 0.5, and I_c⁽¹⁾ could be gotten by Eq.(38). Figure 10 shows the relationship between the Bi filament volume fraction and the critical current under a fixed loading (ε =-0.004). It is interesting to find that the critical current increases with the Bi filament volume fraction. The relationship between the critical current and the volume fraction is not linear. The reason is that the wavelength changes with the Bi filament volume fraction.

5 Numerical study of filament bridging

There are many bridges between filaments in the Bi2212 wire^[23]. However, it is difficult to derive the analytical results for the wire with the filament bridge. Thus, a numerical study is carried out with the finite element software ABAQUS in this section. The numerical model is shown in Fig. 11, where c is the width of bridging, and β is the angle between the bridge direction and the horizontal direction. The material parameters are listed in Table 2^[23].

During the numerical calculation, the fixed boundary compression displacement is applied. The result is plotted in Fig. 12. The buckling strain of the first mode is -0.012 for the wire without bridging. It can be seen that the strain of the first mode increases with the angle of filament bridge slightly, which means that the buckling of filament will become difficult as the angle of filament bridge increases. In the real Bi2212 wire, the reduction of current density under the compressive loading can be suppressed by the filament bridges.

6 Conclusions

In this paper, the effect of compressive loadings on the critical current of Bi2212 wire is studied by considering the micro-buckling of filament. Based on the approximately numerical and experimental results, the relationship between the critical current and the wavelength is presented. In addition, the effects of different material parameters on the critical current are discussed. The results show that increasing the filament width and Young's modulus of the alloy could improve the critical current under the same mechanical loading. The void can lead to a smaller critical strain and decrease the critical current. It is necessary to reduce the volume fraction of void during the fabrication. Finally, the filament bridging could improve the critical current in the Bi2212 wire slightly by preventing the buckling behavior.