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临界电流密度对圆柱状超导体力学特性的影响

程鹏 杨育梅

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临界电流密度对圆柱状超导体力学特性的影响

程鹏, 杨育梅

Effects of critical current density on mechanical properties of cylindrical superconductors

Cheng Peng, Yang Yu-Mei
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  • 高温超导体具有较高的临界温度、高载流能力和低能耗特性, 在电力领域得到了广泛的应用, 其在通有承载电流情况下的力学特性得到了广泛的关注. 研究了承载电流情形下圆柱状超导结构内的磁通钉扎力学响应. 考虑临界电流密度沿径向非均匀分布, 基于临界态Bean模型, 获得了圆柱状超导结构内的感应磁场及电流的分布规律. 结合平面应变方法, 给出了结构内磁通钉扎力、应力及磁致伸缩的解析表达式. 结果表明: 临界电流密度非均匀分布时, 超导结构内的应力变化趋势与均匀分布时一致, 然而临界电流密度的非均匀分布将导致超导结构内的应力和磁致伸缩的极值增大, 并引起结构内局部径向应力大小发生改变以及环向应力分布不连续. 本研究表明临界电流密度非均匀性对超导结构力学性能的影响是显著的, 可为超导体的设计和实际应用提供参考依据.
    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.
      通信作者: 杨育梅, ymyang@lut.cn
    • 基金项目: 国家自然科学基金青年科学基金(批准号: 11402106)资助的课题
      Corresponding author: Yang Yu-Mei, ymyang@lut.cn
    • Funds: Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11402106)
    [1]

    Shinichi M, Kengo N, Hisaki S, Taro M, Ken N, Masafumi O, Tomohisa Y, Yoshiki M, Kazufumi M, Tadakazu M, Hideki S 2017 IEEE Trans. Appl. Supercond. 27 3600804

    [2]

    Chen X Y, Jin J X, Xin Y, Shu B, Tang C L, Zhu Y P, Sun R M 2014 IEEE Trans. Appl. Supercond. 24 3801606

    [3]

    Patel A, Hopkins S C, Baskys A, Kalitka V, Molodyk A, Glowacki B A 2015 Supercond. Sci. Technol. 28 115007Google Scholar

    [4]

    Larbalestier D, Gurevich A, Feldmann D M, Polyanskii A 2001 Nature 414 386Google Scholar

    [5]

    Ren Y T, Weinstein R M, Liu J, Sawh R P, Foster C C 1995 Physica C 251 15Google Scholar

    [6]

    Guan M Z, Hu Q, Gao P F, Wang X Z, Yang T J, Wu W, Xin C J, Wu B M, Ma L Z 2016 Chin. Phys. Lett. 33 58502Google Scholar

    [7]

    Zeng J, Zhou Y H, Yong H D 2010 J. Appl. Phys. 108 033901Google Scholar

    [8]

    Huang C G, Zhou Y H 2014 J. Appl. Phys. 115 033904Google Scholar

    [9]

    Ikuta H, Hirota N, Nakayama Y, Kitazawa K 1993 Phys. Rev. Lett. 70 2166Google Scholar

    [10]

    Ikuta H, Kishio K, Kitazawa K 1994 J. Appl. Phys. 76 4776

    [11]

    Johansen T H 1999 Phys. Rev. B 60 9690Google Scholar

    [12]

    Johansen T H 1999 Phys. Rev. B 59 11187Google Scholar

    [13]

    Johansen T H, Shantsev D V 2003 Supercond. Sci. Technol. 16 1109Google Scholar

    [14]

    Xue C, He A, Yong H D, Zhou Y H 2013 J. Appl. Phys. 113 023901Google Scholar

    [15]

    Huang C G, Yong H D, Zhou Y H 2013 Supercond. Sci. Technol. 26 105007Google Scholar

    [16]

    Haken B T, Eck H J N V, Kate H H J T 2000 Physica C 334 163Google Scholar

    [17]

    Grasso G, Hensel B, Jeremie A, Flükiger R 1995 Physica C 241 45Google Scholar

    [18]

    Lehtonen J R, Ahoranta M, Mikkonen R 2002 Physica C 372−376 1743

    [19]

    Inada R, Nakamura Y, Oota A 2006 Physica C 442 139Google Scholar

    [20]

    Sun J, Watanabe H, Hamabe M, Yamamoto N, Kawahara T, Yamaguchi S 2013 Physica C 494 297Google Scholar

    [21]

    Witanachchi S, Lee S Y, Song L W, Kao Y H, Shaw D T 1990 Appl. Phys. Lett. 57 2133Google Scholar

    [22]

    Noji H 2011 Physica C 471 995Google Scholar

    [23]

    Zheng Y L, Feng W J, Liu Q F 2013 J. Supercond. Novel Magn. 26 2937Google Scholar

  • 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

    Fig. 1.  (a) Schematic diagram of a long cylindrical superconducting structure; (b) schematic diagram of critical current density distribution.

    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

    Fig. 2.  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

    Fig. 3.  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 4  上升场情形下结构内的径向位移沿半径方向的分布

    Fig. 4.  Distribution of radial displacement within the structure along the radial direction in the case of an ascending field.

    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

    Fig. 5.  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

    Fig. 6.  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 7  下降场中结构内的径向位移沿半径方向的分布

    Fig. 7.  Distribution of radial displacement within the structure in the descending field along the radial direction.

    图 8  临界电流密度分布不同时超导圆柱体的磁致伸缩

    Fig. 8.  Magnetostriction of a superconducting cylinder with different critical current density distributions.

  • [1]

    Shinichi M, Kengo N, Hisaki S, Taro M, Ken N, Masafumi O, Tomohisa Y, Yoshiki M, Kazufumi M, Tadakazu M, Hideki S 2017 IEEE Trans. Appl. Supercond. 27 3600804

    [2]

    Chen X Y, Jin J X, Xin Y, Shu B, Tang C L, Zhu Y P, Sun R M 2014 IEEE Trans. Appl. Supercond. 24 3801606

    [3]

    Patel A, Hopkins S C, Baskys A, Kalitka V, Molodyk A, Glowacki B A 2015 Supercond. Sci. Technol. 28 115007Google Scholar

    [4]

    Larbalestier D, Gurevich A, Feldmann D M, Polyanskii A 2001 Nature 414 386Google Scholar

    [5]

    Ren Y T, Weinstein R M, Liu J, Sawh R P, Foster C C 1995 Physica C 251 15Google Scholar

    [6]

    Guan M Z, Hu Q, Gao P F, Wang X Z, Yang T J, Wu W, Xin C J, Wu B M, Ma L Z 2016 Chin. Phys. Lett. 33 58502Google Scholar

    [7]

    Zeng J, Zhou Y H, Yong H D 2010 J. Appl. Phys. 108 033901Google Scholar

    [8]

    Huang C G, Zhou Y H 2014 J. Appl. Phys. 115 033904Google Scholar

    [9]

    Ikuta H, Hirota N, Nakayama Y, Kitazawa K 1993 Phys. Rev. Lett. 70 2166Google Scholar

    [10]

    Ikuta H, Kishio K, Kitazawa K 1994 J. Appl. Phys. 76 4776

    [11]

    Johansen T H 1999 Phys. Rev. B 60 9690Google Scholar

    [12]

    Johansen T H 1999 Phys. Rev. B 59 11187Google Scholar

    [13]

    Johansen T H, Shantsev D V 2003 Supercond. Sci. Technol. 16 1109Google Scholar

    [14]

    Xue C, He A, Yong H D, Zhou Y H 2013 J. Appl. Phys. 113 023901Google Scholar

    [15]

    Huang C G, Yong H D, Zhou Y H 2013 Supercond. Sci. Technol. 26 105007Google Scholar

    [16]

    Haken B T, Eck H J N V, Kate H H J T 2000 Physica C 334 163Google Scholar

    [17]

    Grasso G, Hensel B, Jeremie A, Flükiger R 1995 Physica C 241 45Google Scholar

    [18]

    Lehtonen J R, Ahoranta M, Mikkonen R 2002 Physica C 372−376 1743

    [19]

    Inada R, Nakamura Y, Oota A 2006 Physica C 442 139Google Scholar

    [20]

    Sun J, Watanabe H, Hamabe M, Yamamoto N, Kawahara T, Yamaguchi S 2013 Physica C 494 297Google Scholar

    [21]

    Witanachchi S, Lee S Y, Song L W, Kao Y H, Shaw D T 1990 Appl. Phys. Lett. 57 2133Google Scholar

    [22]

    Noji H 2011 Physica C 471 995Google Scholar

    [23]

    Zheng Y L, Feng W J, Liu Q F 2013 J. Supercond. Novel Magn. 26 2937Google Scholar

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出版历程
  • 收稿日期:  2019-05-18
  • 修回日期:  2019-07-02
  • 上网日期:  2019-09-01
  • 刊出日期:  2019-09-20

临界电流密度对圆柱状超导体力学特性的影响

  • 兰州理工大学理学院, 兰州 730050
  • 通信作者: 杨育梅, ymyang@lut.cn
    基金项目: 国家自然科学基金青年科学基金(批准号: 11402106)资助的课题

摘要: 高温超导体具有较高的临界温度、高载流能力和低能耗特性, 在电力领域得到了广泛的应用, 其在通有承载电流情况下的力学特性得到了广泛的关注. 研究了承载电流情形下圆柱状超导结构内的磁通钉扎力学响应. 考虑临界电流密度沿径向非均匀分布, 基于临界态Bean模型, 获得了圆柱状超导结构内的感应磁场及电流的分布规律. 结合平面应变方法, 给出了结构内磁通钉扎力、应力及磁致伸缩的解析表达式. 结果表明: 临界电流密度非均匀分布时, 超导结构内的应力变化趋势与均匀分布时一致, 然而临界电流密度的非均匀分布将导致超导结构内的应力和磁致伸缩的极值增大, 并引起结构内局部径向应力大小发生改变以及环向应力分布不连续. 本研究表明临界电流密度非均匀性对超导结构力学性能的影响是显著的, 可为超导体的设计和实际应用提供参考依据.

English Abstract

    • 高温超导材料不仅具有独特的零电阻特性、抗磁性及迈斯纳效应, 同时相较低温超导材料而言, 具有更高的临界转变温度和临界电流密度, 因而备受关注, 被广泛应用于磁悬浮、核聚变磁约束、电机及储能等装置中[1-5]. 然而, 超导材料具有明显的脆性特征, 在外加磁场激励下, 结构内产生磁通钉扎力, 该钉扎力在外加下降场中产生的拉应力极易引起超导结构的断裂破坏[6-8]. 因此, 在超导材料的应用中, 对其结构内部力学特性的研究就显得尤为关键. Ikuta等[9]通过测试Bi2Sr2CaCu2O超导体在外部磁场作用下出现的力学变形, 指明磁通钉扎引起的磁致伸缩对这一变形起主导作用[10]. 之后Johansen[11,12]分别对长圆柱和长矩形超导体内的应力分布进行了计算, 讨论了零场冷和场冷两种情形的应力分布, 并给出了应力、应变及位移的解析表达关系式. 随后又研究了薄超导圆盘内的应力分布情况, 给出了受退磁效应影响的应力大小和分布曲线[13]. Xue等[14]研究了薄超导带在垂直磁场中的磁弹性行为, 分别采用Bean模型和Kim模型计算了超导体内的体力、应变和位移. Huang等[15]运用有限元方法研究了有限厚度的II型超导圆柱体和超导圆环结构内的磁致伸缩行为. 上述超导结构力学特性的研究, 均基于临界电流密度均匀分布的假定展开.

      然而, 超导体内的临界电流密度具有明显的非均匀特性[16]. Grasso等[17]研究发现, 超导体内临界电流密度从中心到表面逐渐增加. 其后, Lehtonen等[18]研究了超导材料中临界电流密度轴向(沿超导体的轴向变化)和横向(沿超导体的横向变化)两种非均匀分布形式对交流损耗的影响. Inada等[19]研究了临界电流密度非均匀分布的薄超导带上的交流损耗特性, 并讨论了临界电流密度沿带宽度方向不均匀性对电缆导体中交流损耗的影响. 值得注意的是, 目前还没有考虑到非均匀分布的临界电流密度对超导结构力学性能的影响. 为了更好地解决超导体结构设计和应用中出现的强度和刚度问题, 有必要研究临界电流密度非均匀分布对超导结构力学特性的影响.

      本文基于求解平面应变问题的方法[11]和Bean临界态模型, 考虑临界电流密度沿超导圆柱体半径方向呈非均匀分布状态, 推导了承载电流上升和下降过程中超导结构内部的磁通密度和电流分布, 并给出了结构内径向应力、环向应力和磁致伸缩的解析表达式, 详细讨论了临界电流密度非均匀分布对超导圆柱体力学性能的影响.

    • 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zθ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

      Figure 1.  (a) Schematic diagram of a long cylindrical superconducting structure; (b) schematic diagram of critical current density distribution.

      鉴于超导块体中临界电流密度在中心部分变化较小, 而边缘部分变化比较剧烈[16], 为了更好地模拟超导体内临界电流密度非均匀分布情况, 假定每层的横截面面积为${\text{π}}{R^2}/n$. 当$n = 1$时表示超导结构内临界电流密度均匀分布, 当$n > 1$时表示结构内的临界电流密度非均匀分布. 在研究超导体特性时发现, 2至$n - 1$层之间的相互影响较为近似, 当$n > 3$时, 会使得最外层和最内层非均匀分布的临界电流密度引起的力学性质变化的区分度降低. 且三层模型已具有较高的准确性, 应用广泛[20-22], 下面选取$n = 3$时的情况进行分析.

      此时在超导体圆柱体模型中, 从外到内分为三层, 半径依次为${a_1}$, ${a_2}$, ${a_3}$, 各层半径有如下关系:

      $ {a_1} = R,\;s = \dfrac{{{\text{π}}a_1^2}}{3},\;{a_2} = \sqrt {\dfrac{{2s}}{{\text{π}}}} ,\;{a_3} = \sqrt {\dfrac{s}{{\text{π}}}} . $

      假定第i层内的临界电流密度为${J_i}$, 则第i层内的电流大小为${I_i} = s{J_i}$, 圆柱体内前i层的总电流大小为

      $ {I_{c1}} = {I_1},\;{I_{c2}} = {I_1} + {I_2},\;{I_{c3}} = {I_1} + {I_2} + {I_3}. $

    • 当电流${I_{\rm{a}}}$从零开始逐渐增大时, 基于临界态Bean模型, 超导结构内各层临界电流密度$ {J_1}$, $ {J_2}$, $ {J_3}$为常数. 根据麦克斯韦方程可得到电流和磁场分布的表达式. 当$0 < I < {I_{c1}}$时, 其磁通密度和电流分布为

      $ B = \left\{\!\!\!{\begin{array}{*{20}{l}} {0 ,}&{0 < r < x},\\ {{H_1}\left( {{r^2} - {x^2}} \right),}&{ x < r < {a_1}}, \end{array}} \right. $

      $ J = \left\{\!\!\!{\begin{array}{*{20}{c}} {0,}&{0 < r < x},\\ {{J_1},}&{x < r < {a_1}}, \end{array}} \right. $

      其中${H_1} = \dfrac{{{\mu _0}J{}_1}}{{2r}}$, $x = \sqrt {{a_1}^2 - \dfrac{I}{{{\text{π}}{J_1}}}} $为磁通线穿透深度.

      ${I_{c1}} < I < {I_{c2}}$时, 磁通线逐渐渗透到第二层, 此时结构内的电流及磁通密度的分布如下所示:

      $ B = \left\{\!\!\!\!{\begin{array}{*{20}{c}} {0,} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\;{0 < r < x},\\ {{H_2}\left( {{r^2} - {x^2}} \right),} \quad\quad\quad\quad\quad\quad\quad\quad\; {x < r < {a_2}},\\ {{H_2}\left( {{a_2}^2 - {x^2}} \right) +\! {H_1}\left( {{r^2} - {a_2}^2} \right),}\;\;{{a_2} < r < {a_1}}, \end{array}} \right. $

      $ J = \left\{\!\!\!{\begin{array}{*{20}{c}} {0,}&{0 < r < x},\\ {{J_2},}&{x < r < {a_2}},\\ {{J_1},}&{{a_2} < r < {a_1}}, \end{array}} \right. $

      其中${H_1} = \dfrac{{{\mu _0}J{}_1}}{{2r}}$, ${H_2} = \dfrac{{{\mu _0}J{}_2}}{{2r}}$, $x = \sqrt {\dfrac{{{I_{c1}} - I}}{{{\text{π}}{J_2}}} + {a_2}^2} $.

      ${I_{c2}} < I < {I_{c3}}$时, 磁通线穿透了最外面两层, 开始渗透到第三层, 此时结构内的电流及磁通密度的分布如下:

      $B = \begin{cases} {0,} \qquad\qquad\qquad\quad\; {0 < r < {{x}}},\\ {{H_3}\left( {{r^2} - {x^2}} \right),} \qquad {x < r < {a_3}},\\ {{H_3}\left( {{a_3}^2 - {x^2}} \right) + {H_2}\left( {{r^2} - {a_3}^2} \right),} \\ \qquad\qquad\qquad \qquad\; {{a_3} < r < {a_2}},\\ {H_3}\left( {{a_3}^2 - {x^2}} \right) + {H_2}\left( {{a_2}^2 - {a_3}^2} \right)\\ + {H_1}\left( {{r^2} - {a_2}^2} \right), \;\;\; {{a_2} < r < {a_1}}; \end{cases} $

      $ 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. $

      其中${H_1} = \dfrac{{{\mu _0}J{}_1}}{{2r}}$, ${H_2} = \dfrac{{{\mu _0}J{}_2}}{{2r}}$, ${H_3} = \dfrac{{{\mu _0}J{}_3}}{{2r}}$,

      $x = \sqrt {\dfrac{{{I_{c1}} +{I_{c2}} - I}}{{{\text{π}}{J_3}}} + {a_3}^2}.$

    • 当超导圆柱体中的承载电流从最大值${I_{c3}}$开始下降时, 超导体内的磁通开始从结构外边缘排出, 此时, 结构内的电流及磁通密度按如下方式重新分布, 用x0表示超导体内电流发生反向的位置, 则有:

      ${a_2} < {x_0} < {a_1}$, 即${I_{c3}}-2{I_{c1}} < I < {I_{c3}}$时,

      $ B = \left\{ {\begin{aligned} & {{H_3}{r^2},}\quad\quad\;\quad\quad\quad\quad\quad\quad\quad {0 < r < {a_3}},\\ & {{H_3}{a_3}^2 + {H_2}\left( {{r^2} - {a_3}^2} \right),}\quad\;\; {{a_3} < r < {a_2}},\\ & {{H_3}{a_3}^2 + {H_2}\left( {{a_2}^2 - {a_3}^2} \right) + {H_1}\left( {{r^2} - {a_2}^2} \right),}\\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {{a_2} < r < {x_0}},\\ & {H_3}{a_3}^2 + {H_2}\left( {{a_2}^2 - {a_3}^2} \right) + {H_1}\left( {{x_0}^2 - {a_2}^2} \right) \\ & - {H_1}\left( {{r^2} - {x_0}^2} \right),\quad\quad\quad\quad {{x_0} < r < {a_1}}, \end{aligned}} \right. $

      $ J = \left\{\!\!\!{\begin{array}{*{20}{c}} {{J_3},}&{0 < r < {a_3}},\\ {{J_2},}&{{a_3} < r < {a_2}},\\ {{J_1},}&{{a_2} < r < {x_0}},\\ { - {J_1},}&{{x_0} < r < {a_1}}, \end{array}} \right. $

      其中$ {x_0} = \sqrt {\dfrac{{I - {I_{c3}}}}{{2{\text{π}}{J_1}}} + {a_1}^2} $;

      ${a_3} \!< {x_0} < {a_2}$, 即${I_{c3}}\!-2{I_{c2}} \!< I < {I_{c3}}-2{I_{c1}}$时,

      $ B = \left\{ {\begin{aligned} & {{H_3}{r^2},} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\,{0 < r < {a_3}},\\ & {{H_3}{a_3}^2 + {H_2}\left( {{r^2} - {a_3}^2} \right),}\quad\quad\;\; {{a_3} < r < {x_0}},\\ & {{H_3}{a_3}^2 + {H_2}\left( {{x_0}^2 - {a_3}^2} \right) - {H_2}\left( {{r^2} - {x_0}^2} \right),}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{{x_0} < r < {a_2}},\\ & {H_3}{a_3}^2 + {H_2}\left( {{x_0}^2 - {a_3}^2} \right) - {H_2}\left( {{a_2}^2 - {x_0}^2} \right) \\ & - {H_1}\left( {{r^2} - {a_2}^2} \right),\quad\quad\quad\quad\quad\; {a_2} < r < {a_1}, \end{aligned}} \right. $

      $ J = \left\{\!\!\!{\begin{array}{*{20}{c}} {{J_3},}&{0 < r < {a_3}},\\ {{J_2},}&{{a_3} < r < {x_0}},\\ { - {J_2},}&{{x_0} < r < {a_2}},\\ { - {J_1},}&{{a_2} < r < {a_1}}, \end{array}} \right. $

      其中$ {x_0} = \sqrt {\dfrac{{I - {I_{c3}} + 2{I_{{\rm{c1}}}}}}{{2{\text{π}}{{{J}}_2}}} + {a_2}^2} $;

      $0 < {x_0} < {a_3}$, 即$-{I_{c3}} < I < {I_{c3}}-2{I_{c2}}$时,

      $ B = \left\{ {\begin{aligned} &{{H_3}{r^2},}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {0 < r < {x_0}},\\ &{{H_3}{x_0}^2 - {H_3}\left( {{r^2} - {x_0}^2} \right),}\quad\quad {{x_0} < r < {a_3}},\\ &{{H_3}{x_0}^2 - {H_3}\left( {{a_3}^2 - {x_0}^2} \right) - {H_2}\left( {{r^2} - {a_3}^2} \right),}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\, {{a_3} < r < {a_2}},\\ &{H_3}{x_0}^2 - {H_3}\left( {{a_3}^2 - {x_0}^2} \right) - {H_2}\left( {{a_2}^2 - {a_3}^2} \right) \\ & - {H_1}\left( {{r^2} - {a_2}^2} \right),\quad\quad\quad\quad\;\; {{a_2} < r < {a_1}}, \end{aligned}} \right. $

      $ J = \left\{\!\!\!{\begin{array}{*{20}{c}} {{J_3},}&{0 < r < {x_0}},\\ { - {J_3},}&{{x_0} < r < {a_3}},\\ { - {J_2},}&{{a_3} < r < {a_2}},\\ { - {J_1},}&{{a_2} < r < {a_1}}, \end{array}} \right. $

      其中$ {x_0} = \sqrt {\dfrac{{I - {I_{c3}} + 2{I_{{\rm{c2}}}}}}{{2{\text{π}}{{{J}}_3}}} + {a_3}^2} $.

    • 由于超导体俘获磁通线的特性, 进入超导体内的磁通线被钉扎在结构内部的缺陷处, 该钉扎力的大小等于洛伦兹力的大小:

      $ f\left( r \right) = {f_{\rm{L}}} = j\left( r \right) \times B\left( r \right) = - \dfrac{1}{{2{\mu _0}}}\dfrac{{{\rm{d}}{B^2}}}{{{\rm{d}}r}}, $

      根据弹性力学平衡方程则得到:

      $ \dfrac{{{\rm{d}}{\sigma _r}(r)}}{{{\rm{d}}r}} + \dfrac{{{\sigma _r}\left( r \right) - {\sigma _\theta }\left( r \right)}}{r} + f\left( r \right) = 0, $

      由于超导圆柱体是轴对称结构, 所以结构内部的应力仅与半径r有关, 变形只需要用径向位移$u\left( r \right)$来描述, 则几何方程表达为

      $ {\varepsilon _r} = \dfrac{{{\rm{d}}u\left( r \right)}}{{{\rm{d}}r}},\;\;\;\;\;\;{\varepsilon _\theta } = \dfrac{{u\left( r \right)}}{r}. $

      结合物理方程, 应力${\sigma _r}$, ${\sigma _\theta }$, ${\sigma _z}$与应变εr, εθ, εz之间的关系为[11]

      $ \left\{ {\begin{aligned} & {{\sigma _r} = \dfrac{{E\left( {1 - \upsilon } \right)}}{{1 - \upsilon - 2{\upsilon ^2}}}{\varepsilon _r} + \dfrac{{E\upsilon }}{{1 - \upsilon - 2{\upsilon ^2}}}\left( {{\varepsilon _\theta } + {\varepsilon _z}} \right),}\\ & {{\sigma _\theta } = \dfrac{{E\left( {1 - \upsilon } \right)}}{{1 - \upsilon - 2{\upsilon ^2}}}{\varepsilon _\theta } + \dfrac{{E\upsilon }}{{1 - \upsilon - 2{\upsilon ^2}}}\left( {{\varepsilon _r} + {\varepsilon _z}} \right),}\\ & {{\sigma _z} = \dfrac{{E\left( {1 - \upsilon } \right)}}{{1 - \upsilon - 2{\upsilon ^2}}}{\varepsilon _z} + \dfrac{{E\upsilon }}{{1 - \upsilon - 2{\upsilon ^2}}}\left( {{\varepsilon _\theta } + {{\rm{\varepsilon }}_r}} \right),} \end{aligned}} \right. $

      其中E为超导体的杨氏模量, ν为泊松比. 将(17)和(18)式代入方程(16)可得

      ${u{''}} + \dfrac{1}{r}{u{'}} - \dfrac{1}{{{r^2}}}u + \dfrac{{1 - v - 2{v^2}}}{{(1 - v)E}}f\left( r \right) = 0,$

      求解方程(19)得

      $\begin{split}u =\, &\dfrac{{1 - v - 2{v^2}}}{{2(1 - v)E}}\\ &\times r\left[ {\dfrac{1}{{{r^2}}}\int_0^r {{t^2}f(t){\rm{d}}t - \int_0^r {f(t){\rm{d}}t} } - C - \dfrac{D}{{{r^2}}}} \right].\end{split}$

      对于本文所研究的问题, 需要注意的是在相邻两层之间相邻部分的径向应力${\sigma _r}$和径向位移$u\left( r \right)$是相等的, 则边界条件可表示为

      $ \left\{ {\begin{aligned} & {{\sigma _{{{r}}1}}\left( {{a_2}} \right) = {\sigma _{{{r}}2}}\left( {{a_2}} \right),}\\ & {{\sigma _{{{r}}2}}\left( {{a_3}} \right) = {\sigma _{{{r}}3}}\left( {{a_3}} \right),}\\ & {{u_1}\left( {{a_2}} \right) = {u_2}\left( {{a_2}} \right),}\\ & {{u_2}\left( {{a_3}} \right) = {u_3}\left( {{a_3}} \right),}\\ & {{\sigma _{r1}}\left( {{a_1}} \right) = 0,}\\ & {{u_n}\left( 0 \right) = 0.} \end{aligned}} \right. $

      结合(16)—(20)式, 通过平面应变方法[11], 可得到圆柱超导体内的径向应力分布如下:

      $ \left\{ {\begin{aligned} & {{\sigma _{r3}} = {P_3}{M_{r0}} + {P_3}{C_3} - \dfrac{{{Q_3}}}{{{r^2}}}{N_{r0}} + \dfrac{{{Q_3}}}{{{r^2}}}{D_3},}\\& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {0 < r < {a_3},}\\ & {{\sigma _{r2}} = {P_2}{M_{r3}} + {P_2}{C_2} - \dfrac{{{Q_2}}}{{{r^2}}}{N_{r3}} + \dfrac{{{Q_2}}}{{{r^2}}}{D_2},}\\&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {{a_3} < r < {a_2},}\\ & {{\sigma _{r1}} = {P_1}{M_{r2}} + {P_1}{C_1} - \dfrac{{{Q_1}}}{{{r^2}}}{N_{r2}} + \dfrac{{{Q_1}}}{{{r^2}}}{D_1},}\\&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {{a_2} < r < {a_1},} \end{aligned}} \right. $

      环向应力的表达式为

      $ \left\{ {\begin{aligned} & {{\sigma _{\theta 3}} = {P_3}{M_{r0}} + {P_3}{C_3} + \dfrac{{{Q_3}}}{{{r^2}}}{N_{r0}} - \dfrac{{{Q_3}}}{{{r^2}}}{D_3},}\\& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {0 < r < {a_3},}\\ & {{\sigma _{\theta 2}} = {P_2}{M_{r3}} + {P_2}{C_2} + \dfrac{{{Q_2}}}{{{r^2}}}{N_{r3}} - \dfrac{{{Q_2}}}{{{r^2}}}{D_2},}\\& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {{a_3} < r < {a_2},}\\ & {{\sigma _{\theta 1}} = {P_1}{M_{r2}} + {P_1}{C_1} + \dfrac{{{Q_1}}}{{{r^2}}}{N_{r2}} - \dfrac{{{Q_1}}}{{{r^2}}}{D_1},}\\& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {{a_2} < r < {a_1},} \end{aligned}} \right. $

      径向位移的表达式为

      $ \left\{ {\begin{aligned} & {u_3} = - {X_3}r\left( {{M_{r0}} - \dfrac{{{N_{r0}}}}{{{r^2}}} + {C_3} + \dfrac{{{D_3}}}{{{r^2}}}} \right)\\ & \quad\quad\, - \dfrac{{{v_3}}}{E}r\left[ { - 2v{P_3}\left( {{M_{30}} - \dfrac{{{N_{30}}}}{{{a_3}^2}} + {C_3}} \right)} \right],\\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {0 < r < {a_3},}\\ & {u_2} = - {X_2}r\left( {{M_{r3}} - \dfrac{{{N_{r3}}}}{{{r^2}}} + {C_2} + \dfrac{{{D_2}}}{{{r^2}}}} \right) \\ &\quad\quad\, - \dfrac{{{v_2}}}{E}r\left[ { - 2v{P_2}\dfrac{{{a_2}^2}}{{{a_3}^2}}\left( {{M_{23}} - \dfrac{{{N_{23}}}}{{{a_2}^2}} + {C_2}} \right)} \right],\\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {{a_3} < r < {a_2},}\\ & {u_1} = - {X_1}r\left( {{M_{r2}} - \dfrac{{{N_{r2}}}}{{{r^2}}} + {C_1} + \dfrac{{{D_1}}}{{{r^2}}}} \right) \\ &\quad\quad\, - \dfrac{{{v_1}}}{E}r\left[ { - 2v{P_1}\dfrac{{{a_1}^2}}{{{a_3}^2}}\left( {{M_{12}} - \dfrac{{{N_{12}}}}{{{a_1}^2}} + {C_1}} \right)} \right],\\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {{a_2} < r < {a_1}.} \end{aligned}} \right. $

      此时, 圆柱体内的磁致伸缩大小为

      $\begin{split} \dfrac{{\Delta u}}{{{a_1}}} =\; & - {X_1}\left( {{M_{12}} - \dfrac{{{N_{12}}}}{{{r^2}}} + {C_1} + \dfrac{{{D_1}}}{{{r^2}}}} \right) \\ &- \dfrac{v}{E}\left[ { - 2v{P_1}\dfrac{{{a_1}^2}}{{{a_3}^2}}\left( {{M_{12}} - \dfrac{{{N_{12}}}}{{{a_1}^2}} + {C_1}} \right)} \right],\end{split}$

      其中关于$P,\;Q,\;X,\;M,\;N,\;{C_1},\;{C_2},\;{C_3},\;$${D_1},\;{D_2},\;{D_3}$详细见附录A.

    • 根据以上理论推导, 得到了超导圆柱体内径向应力、环向应力、径向位移和磁致伸缩的解析表达式. 为便于结果讨论, 对以下参数归一化:

      $ \begin{split} & {I_0} = {\text{π}}{j_c}{a_1}^2,\; {{\rm{\sigma }}_0} = {\mu _0}{j_c}{a_1}^2,\\ & {{\beta _1} = {{{j_{c1}}}}/{{{j_c}}} = 0.8,}\quad {{\beta _2} = {{{j_{c2}}}}/{{{j_c}}} = 0.9,}\\ & {\beta _3} = {{{j_{c3}}}}/{{{j_c}}} = 1,\quad i = {{{I_a}}}/{{{I_0}}}. \end{split} $

      图2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$. 图2(a)图3(a)显示了$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

      Figure 2.  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

      Figure 3.  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)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

      Figure 4.  Distribution of radial displacement within the structure along the radial direction in the case of an ascending field.

    • 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图3相同的计算参数. 图5(a)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

      Figure 5.  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

      Figure 6.  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)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

      Figure 7.  Distribution of radial displacement within the structure in the descending field along the radial direction.

    • 图8展示了承载电流i从0.9降低到–0.9过程中, $n = 1$$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

      Figure 8.  Magnetostriction of a superconducting cylinder with different critical current density distributions.

    • 本文考虑超导体内临界电流密度的非均匀分布, 基于Bean临界态模型和平面应变方法, 研究了临界电流密度在结构内非均匀分布时的应力、位移和磁致伸缩特性. 与均匀分布情况对比可以得到, 上升场时, 临界电流密度的分布不影响超导结构内应力和位移的变化趋势, 但局部应力的数值会随着临界电流密度的非均匀分布而发生变化, 在局部临界电流密度较大处压应力数值较大. 且临界电流密度的非均匀分布导致超导体内应力出现更大的极值, 因此发生破坏的风险较均匀分布情况下更高, 对强度有着更高的要求. 而在下降场时, 临界电流密度明显影响拉应力和磁致伸缩特性, 使得非均匀分布时拉应力峰值和磁致伸缩数值大于均匀分布时的数值, 而环向应力不仅拉应力和压应力同时存在而且在临界电流密度不连续处显示出了不连续的特点. 这就导致对超导结构抗剪强度有着更高的要求. 而磁致伸缩的程度也比均匀分布情况下更高, 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, 所以相应的刚度要求也会发生变化. 因此, 在高温超导体实际应用中, 必须要充分考虑临界电流密度非均匀分布导致的超导材料结构强度降低, 以保证其在工程应用中的安全性.

    • (22)—(25)式中$ P,\;Q,\;X,\;M,\;N,\;{C_1},\;{C_2},\;{C_3},\;{D_1},\;$${D_2},\;{D_3}$的表达式

      $\begin{split} & {P_i} = - \dfrac{1}{{2\left( {1 - {v_{}}} \right)}},\;\;{Q_i} = - \dfrac{{1 - 2{v_{}}}}{{2\left( {1 - {v_{}}} \right)}},\\ &{X_i} = \dfrac{{1 - {v_{}} - 2{v_{}}^2}}{{2\left( {1 - {v_{}}} \right){E_{}}}},\;\;i = 1,\;2,\;3,\end{split}\tag{A1}$

      $\begin{split} & {M_{12}} = \int_{{a_2}}^{{a_1}} {f\left( t \right){\rm{d}}t} ,\;\;\;\;\;{M_{23}} = \int_{{a_3}}^{{a_2}} {f\left( t \right){\rm{d}}t} ,\\ & {M_{30}} = \int_0^{{a_3}} {f\left( t \right){\rm{d}}t},\end{split}\tag{A2}$

      $\begin{split} & {N_{12}} = \int_{{a_2}}^{{a_1}} {{t^2}f\left( t \right){\rm{d}}t} , \;\;\;\;{N_{23}} = \int_{{a_3}}^{{a_2}} {{t^2}f\left( t \right){\rm{d}}t} , \\ &{N_{30}} = \int_0^{{a_3}} {{t^2}f\left( t \right){\rm{d}}t} ,\end{split}\tag{A3}$

      ${{A}} = \left( {\begin{array}{*{20}{c}} {{{{A}}_{11}}}&{{{{A}}_{12}}} \\ {{{{A}}_{21}}}&{{{{A}}_{22}}} \end{array}} \right) = \left( {{\delta _1}}\quad {{\delta _2}}\quad {{\delta _3}}\quad {{\delta _4}}\quad {{\delta _5}}\right),\tag{A4}$

      ${{{A}}_{11}} = \left| {\begin{array}{*{20}{c}} { - {P_3}}&{{P_2}}&0 \\ 0&{ - {P_2}}&{ - {P_1}} \\ 0&0&{ - {P_1}} \end{array}} \right|,\tag{A5}$

      ${{{A}}_{12}}= \left| {\begin{array}{*{20}{c}} {\dfrac{{{Q_2}}}{{{a_3}^2}}}&0 \\ { - \dfrac{{{Q_2}}}{{{a_2}^2}}}&{\dfrac{{{Q_1}}}{{{a_2}^2}}} \\ 0&{ - \dfrac{{{Q_1}}}{{{a_1}^2}}} \end{array}} \right|,\tag{A6}$

      ${{{A}}_{21}} = \left| {\begin{array}{*{20}{c}} {{a_3}\left( {{X_3} - \dfrac{{2{a_3}^2{v_3}^2{P_3}}}{{{a_n}^2{E_3}}}} \right)}&{ - {a_3}\left( {{X_2} - \dfrac{{2{a_2}^2{v_2}^2{P_2}}}{{{a_n}^2{E_2}}}} \right)}&0 \\ 0&{{a_2}\left( {{X_2} - \dfrac{{2{a_2}^2{v_2}^2{P_2}}}{{{a_n}^2{E_2}}}} \right)}&{{a_2}\left( {\dfrac{{2{a_1}^2{v_1}^2{P_1}}}{{{a_n}^2{E_1}}} - {X_1}} \right)} \end{array}} \right|,\tag{A7}$

      ${{{A}}_{22}} = \left( {\begin{array}{*{20}{c}} {\dfrac{{ - {X_2}}}{{{a_3}}}}&0 \\ {\dfrac{{{X_2}}}{{{a_2}}}}&{\dfrac{{ - {X_1}}}{{{a_2}}}} \end{array}} \right),\tag{A8}$

      ${{\theta}} = \left| {\begin{array}{*{20}{c}} {{P_3}{M_{30}} - {Q_3}\dfrac{1}{{{a_3}^2}}{N_{30}}} \\ {{P_2}{M_{23}} - {Q_2}\dfrac{1}{{{a_2}^2}}{N_{23}}} \\ {{P_1}{M_{12}} - {Q_1}\dfrac{1}{{{a_1}^2}}{N_{12}}} \\ {{X_3}{a_3}\left( { - {M_{30}} + \dfrac{{{N_{30}}}}{{{a_3}^2}}} \right) + \dfrac{{ - 2{a_2}^2{v_2}^2{P_2}}}{{{a_3}{E_2}}}\left( {{M_{23}} - \dfrac{{{N_{23}}}}{{{a_2}^2}}} \right) + \dfrac{{2{a_3}{v_3}^2{P_3}}}{{{E_3}}}\left( {{M_{30}} - \dfrac{{{N_{30}}}}{{{a_3}^2}}} \right)} \\ {{X_2}{a_2}\left( { - {M_{23}} + \dfrac{{{N_{23}}}}{{{a_2}^2}}} \right) + \dfrac{{ - 2{a_2}{a_1}^2{v_1}^2{P_1}}}{{{a_3}^2{E_1}}}\left( {{M_{12}} - \dfrac{{{N_{12}}}}{{{a_1}^2}}} \right) + \dfrac{{2{a_2}^3{v_2}^2{P_2}}}{{{a_3}^2{E_2}}}\left( {{M_{23}} - \dfrac{{{N_{23}}}}{{{a_2}^2}}} \right)} \end{array}} \right|,\tag{A9}$

      ${{{A}}_1} = \left| {\begin{array}{*{20}{c}} \theta &{{\delta _2}}&{{\delta _3}}&{{\delta _4}}&{{\delta _5}} \end{array}} \right|,\tag{A10}$

      ${{{A}}_2} = \left| {\begin{array}{*{20}{c}} {{\delta _1}}&\theta &{{\delta _3}}&{{\delta _4}}&{{\delta _5}} \end{array}} \right|,\tag{A11}$

      ${{{A}}_3} = \left| {\begin{array}{*{20}{c}} {{\delta _1}}&{{\delta _2}}&\theta &{{\delta _4}}&{{\delta _5}} \end{array}} \right|,\tag{A12}$

      ${{{A}}_4} = \left| {\begin{array}{*{20}{c}} {{\delta _1}}&{{\delta _2}}&{{\delta _3}}&\theta &{{\delta _5}} \end{array}} \right|,\tag{A13}$

      ${{{A}}_5} = \left| {\begin{array}{*{20}{c}} {{\delta _1}}&{{\delta _2}}&{{\delta _3}}&{{\delta _4}}&\theta \end{array}} \right|,\tag{A14}$

      $ {C_3} = \left| {\dfrac{{{{{A}}_1}}}{{{A}}}} \right| ,\;\;\;\;\;\;\;{C_2} = \left| {\dfrac{{{{{A}}_2}}}{{{A}}}} \right| ,\;\;\;\;\;\;\; {C_1} = \left| {\dfrac{{{{{A}}_3}}}{{{A}}}} \right|, \tag{A15}$

      $ {D_3} = 0,\;\;\;\;\;{D_2} = \left| {\dfrac{{{{{A}}_4}}}{{{A}}}} \right|,\;\;\;\;\;{D_1} = \left| {\dfrac{{{{{A}}_5}}}{{{A}}}} \right| .\tag{A16}$

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