Fund Project:Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11402106)
Received Date:18 May 2019
Accepted Date:02 July 2019
Available Online:01 September 2019
Published Online:20 September 2019
Abstract:High-temperature superconductor has high critical temperature, high transport current capacity and low energy consumption, which correspondingly offer the wide applications in the field of electric power. As an important concern, the mechanical properties of superconductor carried with transport current have received extensive attention. Still, its mechanical properties in various electromagnetic environments are under study. Most of previous studies are based on the assumption of uniform distribution of critical current density, and only few researches based on the non-uniform distribution of critical current density are carried out. In this work, the mechanical flux pinning response of cylindrical superconducting structures is studied. Considering the non-uniform features of critical current density along the radial direction, the distribution law of induced magnetic field and current for the cylindrical superconducting structure is obtained based on the Bean model. Combined with the plane strain method, the analytical expression of magnetic flux pinning force, stress and magnetostriction in the superconducting structure are obtained. The results show that the uneven distribution of critical current density causes the flux pinning force to change, which further leads the superconductor`s local radial stress to vary with the critical current density. When the transport current flowing through the superconductor is increased in the ascending field, the radial stress and the hoop stress both appear compressive. The non-uniform distribution of critical current density has no significant effect on the overall trend of the internal stress, but displays an obvious influence on the stress distribution, and the superconducting structure is compressed and deformed. The results are consistent with those in the uniform case. When the transport current decreases during field descending, the critical current starts to reverse from the outermost part, then the compressive stress and tensile stress exist simultaneously. The hoop stress has a discontinuous point at the discontinuous portion of the critical current density, thus the damage probability is higher than that of the uniform distribution. In other words, the shear strength of superconductor is required to be higher for application. Also, the degree of magnetostriction is higher when the distribution of critical current density is set to be uniform, that is, the non-uniform distribution of the critical current density causes the superconducting structure to undergo greater deformation. Therefore, in engineering applications, the structural strength of the superconducting material must be enhanced to cope with the challenge from the uneven distribution of critical current density. Keywords:cylindrical superconducting structure/ current carrying/ flux pinning force/ stress strain
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2.模型建立及理论推导考虑一半径为R的无限长超导圆柱体, 承载沿z方向的电流Ia, 如图1(a)所示. 由于临界电流密度横向非均匀分布对材料性能的影响更加显著[18], 假定超导体内临界电流密度在z和θ方向的大小和分布是均匀的, 沿半径方向非均匀分布. 如图1(b)所示, 将垂直于z轴圆柱状超导体横截面延半径方向分为n层, 每层临界电流密度不同. 从外到内临界电流密度分别为J1, J2, J3, ···, Ji, ···, Jn–1, Jn, 半径分别为a1, a2, a3, ···ai, ···, an–1, an. 图 1 (a)长圆柱状超导结构示意图; (b)临界电流密度分布示意图 Figure1. (a) Schematic diagram of a long cylindrical superconducting structure; (b) schematic diagram of critical current density distribution.
$J = \left\{\!\!\!{\begin{array}{*{20}{c}}{0,}&{0 < r < x},\\{{J_3},}&{x < r < {a_3}},\\{{J_2},}&{{a_3} < r < {a_2}},\\{{J_1},}&{{a_2} < r < {a_1}},\end{array}} \right.$
图2和图3分别为承载电流大小从$i = 0$增大到$i = 0.9$时, 超导结构内的径向和环向应力沿半径方向的分布图. $n = 1$时, 结构内临界电流密度${\beta _1} = 0.9$; $n = 3$时, 各层的临界电流密度为${\beta _3} = 1,$${\beta _2}= 0.9,\;{\beta _1}= 0.8$. 图2(a)和图3(a)显示了$n = 3$, i取0.25, 0.45, 0.6, 0.75, 0.9时结构内的径向和环向应力分布情况. 可以看出, 在整个变化过程中径向应力都是负的, 即为压应力, 其大小沿着半径指向中心方向逐渐增大并趋于稳定, 这是因为在感应磁场未穿透的区域结构内部的体力为零, 该区域应力分布的均匀性得到保持. 对于较小的承载电流, 应力增加相对比较缓慢, 而且这两种应力分别以不同的方式沿指向表面方向减小. 径向应力在表面处减小到零, 而环向应力减小到一个不为零的有限值. 这些特点与Johansen[11]的研究结果一致. 说明临界电流密度的非均匀分布对超导体内应力整体的变化趋势影响不显著. 但是由于相邻部位之间的临界电流密度的差异而导致结构内的钉扎力分布是不连续的, 所以环向应力出现了不连续的现象. 在Zheng等[23]关于圆柱形复合超导材料的研究中观察到类似的现象, 但是本研究排除了弹性模量的差异对结果的影响, 证明了临界电流密度的非均匀分布会导致这样的结果出现. 图 2 上升场情形下结构内的径向应力的分布 (a) n = 3, i取不同值; (b) i = 0.9, n取不同值 Figure2. Distribution of radial stress in the structure under the ascending field: (a) n = 3, i takes different values; (b) i = 0.9, n takes different values.
图 3 上升场情形下结构内的环向应力的分布 (a) n = 3, i取不同值; (b) i = 0.9, n取不同值 Figure3. Distribution of hoop stress in the structure under the ascending field: (a) n = 3, i takes different values; (b) i = 0.9, n takes different values.
图2(b)和图3(b)显示了$i = 0.9$, $n = 1$和3时, 超导结构沿半径方向的径向和环向应力分布. 可以看出, 径向和环向的压应力都是从中心到表面方向逐渐减小, 并在接近半径中心处发生交叉. 由于$n = 3$时, 临界电流密度分布不均匀, 在最外层时${\beta _1} < 0.9$, 在最里层${\beta _1} > 0.9$, 所以超导结构中心部分$n = 3$的压应力在数值比$n = 1$时大, 而靠近表面部分$n = 1$时应力较大. 这充分说明了临界电流密度的分布情况对超导结构内的应力分布产生显著影响. 图4给出了径向位移沿半径方向的分布特性, 在整个过程中超导结构的径向位移都是负的, 结构整体是压缩变形, 这与临界电流密度均匀分布时的变化规律一致. 图 4 上升场情形下结构内的径向位移沿半径方向的分布 Figure4. Distribution of radial displacement within the structure along the radial direction in the case of an ascending field.
23.2.下降场情形 -->
3.2.下降场情形
当超导结构内的承载电流从最大值${I_{c3}}$开始降低, 在圆柱体外部的临界电流方向开始反向. 在这个再磁化区域中, 体力方向沿着对称轴心向外, 然而内部的体力方向仍然保持不变. 接下来将仔细研究下降场时超导结构应力的变化情况. 图5和图6分别为承载电流从最大值$i = 0.9$开始减小时, 超导结构内的径向和环向应力沿半径方向的分布特性. 其中, 临界电流密度采用与图2和图3相同的计算参数. 图5(a)和图6(a)显示了$n = 3$, i分别取0.6, 0.3, 0, –0.3, –0.6, –0.9时结构内的径向和环向应力分布情况. 可以看出, 承载电流下降阶段, 应力的大小沿着半径指向中心方向先上升后降低. 而产生的极值在数值上先增大后减小并在$i = 0$附近取得最大值. 这是由于在下降场阶段, 超导结构内的临界电流从最外部开始反向, 这导致了结构内部拉应力和压应力同时存在. 这种情况下超导体发生破坏的风险最高, 而环向上不仅拉应力和压应力同时存在而且在临界电流密度有差异处呈不连续的分布, 这就要求将此类超导材料应用于对抗剪强度有要求的结构时需要慎重. 图 5 下降场中结构内的径向应力沿半径方向的分布 (a) n = 3, i取不同值; (b) i = 0, n取不同值 Figure5. Distribution of radial stress in the structure in the falling field along the radial direction: (a) n = 3, i takes different values; (b) i = 0, n takes different values.
图 6 下降场中结构内的环向应力沿半径方向的分布 (a) n = 3, i取不同值; (b) i = 0, n取不同值 Figure6. Distribution of the hoop stress in the structure in the falling field along the radial direction: (a) n = 3, i takes different values; (b) i = 0, n takes different values.
图5(b)和图6(b)分别显示了$i = 0$, $n = 1, 3$时, 超导结构内的径向和环向应力沿半径方向的分布情况. 可以看出, 由于$n = 3$时临界电流密度分布不均匀, 导致超导体中心部分$n = 3$时的应力较$n = 1$时更大, 而靠近表面处$n = 1$时的应力较大. 从图5(b)可以看到临界电流密度的非均匀分布导致拉应力的峰值明显大于均匀分布时的数值. 而这一拉应力极易引起超导体发生断裂或者己有裂纹的扩展. 这说明临界电流密度的非均匀分布可能会导致超导体更容易发生破坏. 图7给出了超导结构沿半径方向的径向位移分布特性. 从径向位移的变化趋势可以看出, 下降场情形中径向位移的变化规律变得复杂, 结构内部会出现了拉应变和压应变共同存在的情况. 比如$i = - 0.3$时, 甚至出现超导结构中心和边缘处产生压应变而中间处产生拉应变. 这是下降阶段超导结构内径向拉应力和压应力同时存在并且临界电流密度非均匀分布共同作用的结果. 图 7 下降场中结构内的径向位移沿半径方向的分布 Figure7. Distribution of radial displacement within the structure in the descending field along the radial direction.
23.3.磁致伸缩 -->
3.3.磁致伸缩
图8展示了承载电流i从0.9降低到–0.9过程中, $n = 1$和$n = 3$时的磁致伸缩分布图. 和预期一致[23], 均匀材料的磁致伸缩不存在磁滞回线. 但观察临界电流密度非均匀分布时可以发现, 产生的磁致伸缩在整个变化过程中都大于均匀分布的情况. 并且在承载电流取最大值时差值最大, 即临界电流密度的非均匀分布导致超导结构产生更大的变形. 因此为了提升超导材料应用的可靠性, 需要密切关注临界电流密度的分布性质. 图 8 临界电流密度分布不同时超导圆柱体的磁致伸缩 Figure8. Magnetostriction of a superconducting cylinder with different critical current density distributions.