第二类超导体力—磁耦合基本特性的理论研究
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摘要
超导体的力-磁耦合基本特性以及断裂问题是超导理论和超导技术中关注的基础性课题,本学位论文针对非理想第二类超导体在电磁场作用下的力学行为以及断裂特性进行理论研究,同时分析了理想第二类超导体的力-磁耦合特性,讨论了力学变形对于超导特性的影响。
首先,分析了高温超导体在磁化过程中由于电磁力的作用而引起的力学变形问题,在基于临界态模型建立的超导体应力应变分析模式的基础上,考虑了随励磁速率变化的磁通流动和蠕动这两个物理过程,模拟了超导体磁化过程中的力学问题。对于超导体磁致伸缩随时间对数衰减的实验现象,采用Anderson磁通蠕动模型将电流密度随时间的变化关系考虑到电磁体力的计算中,模拟结果与Ikuta等人的实验很好的吻合。其中,成功地模拟了磁致伸缩随时间的对数衰减以及在磁场上升阶段中出现的先增大后减小的非单调衰减现象,并可利用本模型预测最大值出现的时刻以及峰值。此外,对于交变场的励磁过程,本文首次同时考虑磁通流动和磁通蠕动效应,讨论了不同励磁速率下超导体的磁致伸缩和应力分布。
其次,提出了一套研究高温超导体裂纹尖端临界电流奇异性以及俘获磁场减小这一问题的理论方法。首先针对比较常见的含中心椭圆孔洞的长圆柱形超导体,通过复变函数和保角变换的方法将问题简化为一个轴对称问题。分别基于临界态Bean模型和Kim模型讨论了临界电流集中性随椭圆孔洞形状因子的变化关系,并得到了线型裂纹尖端的临界电流奇异性。文中首次定义并给出了电流密度强度因子KJ,揭示出临界电流密度的-1次奇异性,且与超导圆柱的半径并无关系,其中Bean模型的结果仅由裂纹尺寸a以及临界电流密度Jc决定,因而具有一定普适性。同时,还获得了裂纹周围俘获磁场的分布,并发现由Bean模型和Kim模型得到的结果变化并不大,裂纹降低了超导体中心的俘获磁场,这与已有的实验相吻合。
最后,研究了理想第二类超导体的力-磁耦合特性。基于修正的GL方程,分析了超导体波函数以及相干长度的应变效应,分别获得了一维问题的解析解和二维问题的数值解,其中数值解采用了傅里叶级数展开以及迭代算法。除此之外,还计算了理想第二类超导体在磁场中的表面变形。当磁通进入超导体并形成Abrikosov磁通晶格后,超导体受到由静电势引起的体力以及表面偶极子导致的面力作用,因而超导体的表面变形也具有一定的周期性,通过数值方法求解了GL方程并讨论了表面偶极子在总的表面变形中所占的比例。
The crack problem and the Magneto-Elastic coupling properties of the superconductors have become the fundamental subjects in both superconducting applications and theories. This thesis presents a theoretical investigation on the mechanical behaviors and fracture properties for the high temperature superconductors. Besides this, the Magneto-Elastic coupling properties of the ideal type-II superconductors are also investigated in this work.
Firstly, the mechanical deformation of the high temperature superconductors induced by the electromagnetic forces in the magnetization process is analyzed. Based on the model proposed by Johansen, which is restricted to the critical state models, we consider the effect of the flux creep and flux flow on the stress and magnetostriction of the superconductors. The Anderson's flux creep model is used to model the logarithmic decrement of the magnetostriction, in which the decrease of the current density with the time is considered in the calculation of the electromagnetic body force. The obtained results are consistent with the experiments of Ikuta. It is noted that our model successfully simulate the non-monotonic decrease of the magnetostriction in the ascent branch of the applied field, the time where the maximum value occurs can also be predicted. When the magnetization field is time-dependent and sawtooth-like, the effect of flux flow should be also considered. Both the effects of flux creep and flux flow on the stress and magnetostriction are discussed in our model.
Secondly, the singularity of current distribution around the crack in a long cylindrical superconductor and the decrement of the trapped field are investigated analytically. A general model that a long cylindrical superconductor containing an internal elliptical hole placed in a magnetic field is considered. After a simple conformal mapping is employed to the case that the superconductor is fully penetrated, the current streamlines, the current density and the trapped field around the crack in the superconductor without deformation are obtained. On the basis of the Bean model and the Kim model, the dependence of the current concentration with the shape factor of the elliptical hole is discussed. Also, the singularity of current distribution around the crack is obtained. The current density intensity factor KI is defined and found to be independent on the radius of the superconductor and only determined by the crack length α and the critical current density without the crack J1. Thus, the result can be suitable to the generalized case of a crack tip. We also obtain the trapped magnetic field of the superconductor with a crack, the obvious decrement agrees with the reported experimental results.
Finally, the Magneto-Elastic coupling properties of the ideal type-II superconductors are studied theoretically. Based on the revised GL equations, the strain effect on the superconducting properties such as wave functions and coherence length is analyzed. We obtain the analytical solution of the ID problem in which the applied magnetic field is assumed to be very small and the second GL equation can thus be omitted. However, for the2D problem in which the magnetic field cannot be ignored we can only solved by numerical methods. In light of the periodicity of the wave functions, we use the Fourier series expansion and iterative algorithm and obtain the wave functions of the superconductors with prestrains. In addition, we also analyze the surface deformation of the ideal type-II superconductors caused by the Abrikosov vortex lattice. It is known that superconductor undertakes a long-range body force caused by the gradient of the electrostatic potential and a surface force induced by the surface dipole. We use the same numerical methods which are used to solve the coupled nonlinear GL equations and obtain the surface deformation based on the theory of linear elasticity.
首先,分析了高温超导体在磁化过程中由于电磁力的作用而引起的力学变形问题,在基于临界态模型建立的超导体应力应变分析模式的基础上,考虑了随励磁速率变化的磁通流动和蠕动这两个物理过程,模拟了超导体磁化过程中的力学问题。对于超导体磁致伸缩随时间对数衰减的实验现象,采用Anderson磁通蠕动模型将电流密度随时间的变化关系考虑到电磁体力的计算中,模拟结果与Ikuta等人的实验很好的吻合。其中,成功地模拟了磁致伸缩随时间的对数衰减以及在磁场上升阶段中出现的先增大后减小的非单调衰减现象,并可利用本模型预测最大值出现的时刻以及峰值。此外,对于交变场的励磁过程,本文首次同时考虑磁通流动和磁通蠕动效应,讨论了不同励磁速率下超导体的磁致伸缩和应力分布。
其次,提出了一套研究高温超导体裂纹尖端临界电流奇异性以及俘获磁场减小这一问题的理论方法。首先针对比较常见的含中心椭圆孔洞的长圆柱形超导体,通过复变函数和保角变换的方法将问题简化为一个轴对称问题。分别基于临界态Bean模型和Kim模型讨论了临界电流集中性随椭圆孔洞形状因子的变化关系,并得到了线型裂纹尖端的临界电流奇异性。文中首次定义并给出了电流密度强度因子KJ,揭示出临界电流密度的-1次奇异性,且与超导圆柱的半径并无关系,其中Bean模型的结果仅由裂纹尺寸a以及临界电流密度Jc决定,因而具有一定普适性。同时,还获得了裂纹周围俘获磁场的分布,并发现由Bean模型和Kim模型得到的结果变化并不大,裂纹降低了超导体中心的俘获磁场,这与已有的实验相吻合。
最后,研究了理想第二类超导体的力-磁耦合特性。基于修正的GL方程,分析了超导体波函数以及相干长度的应变效应,分别获得了一维问题的解析解和二维问题的数值解,其中数值解采用了傅里叶级数展开以及迭代算法。除此之外,还计算了理想第二类超导体在磁场中的表面变形。当磁通进入超导体并形成Abrikosov磁通晶格后,超导体受到由静电势引起的体力以及表面偶极子导致的面力作用,因而超导体的表面变形也具有一定的周期性,通过数值方法求解了GL方程并讨论了表面偶极子在总的表面变形中所占的比例。
The crack problem and the Magneto-Elastic coupling properties of the superconductors have become the fundamental subjects in both superconducting applications and theories. This thesis presents a theoretical investigation on the mechanical behaviors and fracture properties for the high temperature superconductors. Besides this, the Magneto-Elastic coupling properties of the ideal type-II superconductors are also investigated in this work.
Firstly, the mechanical deformation of the high temperature superconductors induced by the electromagnetic forces in the magnetization process is analyzed. Based on the model proposed by Johansen, which is restricted to the critical state models, we consider the effect of the flux creep and flux flow on the stress and magnetostriction of the superconductors. The Anderson's flux creep model is used to model the logarithmic decrement of the magnetostriction, in which the decrease of the current density with the time is considered in the calculation of the electromagnetic body force. The obtained results are consistent with the experiments of Ikuta. It is noted that our model successfully simulate the non-monotonic decrease of the magnetostriction in the ascent branch of the applied field, the time where the maximum value occurs can also be predicted. When the magnetization field is time-dependent and sawtooth-like, the effect of flux flow should be also considered. Both the effects of flux creep and flux flow on the stress and magnetostriction are discussed in our model.
Secondly, the singularity of current distribution around the crack in a long cylindrical superconductor and the decrement of the trapped field are investigated analytically. A general model that a long cylindrical superconductor containing an internal elliptical hole placed in a magnetic field is considered. After a simple conformal mapping is employed to the case that the superconductor is fully penetrated, the current streamlines, the current density and the trapped field around the crack in the superconductor without deformation are obtained. On the basis of the Bean model and the Kim model, the dependence of the current concentration with the shape factor of the elliptical hole is discussed. Also, the singularity of current distribution around the crack is obtained. The current density intensity factor KI is defined and found to be independent on the radius of the superconductor and only determined by the crack length α and the critical current density without the crack J1. Thus, the result can be suitable to the generalized case of a crack tip. We also obtain the trapped magnetic field of the superconductor with a crack, the obvious decrement agrees with the reported experimental results.
Finally, the Magneto-Elastic coupling properties of the ideal type-II superconductors are studied theoretically. Based on the revised GL equations, the strain effect on the superconducting properties such as wave functions and coherence length is analyzed. We obtain the analytical solution of the ID problem in which the applied magnetic field is assumed to be very small and the second GL equation can thus be omitted. However, for the2D problem in which the magnetic field cannot be ignored we can only solved by numerical methods. In light of the periodicity of the wave functions, we use the Fourier series expansion and iterative algorithm and obtain the wave functions of the superconductors with prestrains. In addition, we also analyze the surface deformation of the ideal type-II superconductors caused by the Abrikosov vortex lattice. It is known that superconductor undertakes a long-range body force caused by the gradient of the electrostatic potential and a surface force induced by the surface dipole. We use the same numerical methods which are used to solve the coupled nonlinear GL equations and obtain the surface deformation based on the theory of linear elasticity.
引文
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