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Numerical evaluations of maximum stresses in bulk superconductors
Physica C 392–396 (2003) 575–578 www.elsevier.com/locate/physc
Numerical evaluations of maximum stresses in bulk superconductors M. Tsuchimoto *, K. Murata Hokkaido Institute of Technology, 7-15 Maeda, Teine-ku, Sapporo 006-8585, Japan Received 13 November 2002; accepted 24 February 2003
Abstract Stresses in bulk high-temperature superconductors (HTSs) are numerically evaluated during magnetization process by field cooling. Shielding current distributions are obtained through macroscopic numerical simulation using Maxwell equations and critical state model. Stress distributions are obtained through numerical analysis using finite-element method. Maximum stresses are evaluated from experimental results of destruction of the bulk HTS. Differences between one-dimensional and axisymmetric three-dimensional solutions are discussed for the stress distributions. Values and positions of the maximum stresses are affected by fixed conditions of the bulk HTS. Ó 2003 Elsevier B.V. All rights reserved. PACS: 74.70; 74.72; 07.10.P Keywords: Bulk HTS; Maximum stress; Field-cooled magnetization
1. Introduction A trapped field magnet is one application for melt-processed bulk high-Tc superconductors (HTSs) [1]. In magnetization process by field cooling, stresses are produced by Lorentz force between shielding currents and magnetic fields. Evaluation of the maximum stress during the magnetization is important from the viewpoint of destruction of bulk HTS [2–6]. In this study, maximum stresses in bulk HTSs are numerically evaluated during the magnetization process by field cooling. Shielding current
distributions are obtained through macroscopic numerical simulation using Maxwell equations and critical state model [6,7]. Stress distributions are obtained through numerical analysis using finite-element method [8]. Maximum stresses are discussed with experimental results of destruction of bulk HTSs [4]. Numerical solutions in axisymmetric three-dimensions with boundary conditions are different from one-dimensional analytical solution. The present numerical analysis is useful to study the maximum stresses from the destructive experiments. 2. Numerical formulation
*
Corresponding author. Fax: +81-11-681-3622. E-mail address: [email protected] (M. Tsuchimoto).
Macroscopic electromagnetic phenomena in HTS are described by Maxwell equations:
0921-4534/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4534(03)00844-X
576
M. Tsuchimoto, K. Murata / Physica C 392–396 (2003) 575–578
oB ; ot
3. Results and discussion r B ¼ l0 J;
r B ¼ 0;
ð1Þ
where l0 , E and B are magnetic permeability in air, and electric and magnetic fields, respectively. A type-II superconductor in a quasi-static field is well described by standard critical state model. Constitutive relationships between the shielding current density J SC and the electric field E are obtained from the force balance on a fluxoid [7]: J SC ¼ J c ðjBjÞ oJ SC ¼0 ot
E jEj
ðif jEj 6¼ 0Þ;
ðif jEj ¼ 0Þ:
ð2Þ
When the electric field E is induced in a local region by change of the magnetic field, shielding currents with the critical current density Jc are obtained. If there is no electric field by the shielding effect, the situation of currents is not changed. Although the critical current density Jc has a strong dependence on magnetic field, Bean model is applied to the present analysis to clarify fundamental relationship between the shielding current distribution and the stress distribution. When external magnetic field is changed, selfconsistent currents distributions are obtained stably using numerical techniques [6,7]. Then stress distributions are evaluated at each time step with the shielding current distribution [6]. From principle of virtual work, the following equation is obtained with finite-element method for region V and its surface S [8]: Z Z T T dfeg frg dV dfU g fF g dV V V Z T ð3Þ dfU g fT g dS ¼ 0; S
where {e}, {r}, {U }, {F } and {T } are strain, stress, displacement, body force and surface force vectors, respectively. Boundary conditions for (3), i.e., fixed situation of the bulk HTS in experiments, are given by surface displacements and surface force. Then stress distributions are calculated from obtained displacements. Relations of the stresses, strains and displacements are summarized in the previous papers [5,6].
Radius and thickness of a disk HTS in Fig. 1 are R ¼ 23:0 mm and H ¼ 15:0 mm in experiments [4]. The bulk HTS was cooled after external field 8.0 T was applied, then the external field was reduced to 0.0 T. The bulk was divided in two parts when applied field was 5.37 T [4]. In the present analysis, PoissonÕs ratio and YoungÕs modulus are set to 0.3 and 1.0, respectively. Critical current density Jc is set to a constant value of 3.0 108 A/m2 in Bean model. One-dimensional maximum trapped field is about 8.7 T and that in three-dimensions is about 4.2 T in full magnetization. Though radial, hoop, axial and shear stresses are obtained in the axisymmetric analysis [5,6], the most important maximum hoop stresses are shown in Fig. 2. Axisymmetric three-dimensional solutions with open
Fig. 1. Disk model in axisymmetric analysis with radius R and thickness H .
1D analytical fixed side-surface open boundary
35
Maximum hoop stress [Mpa]
rE ¼
30 25 20 15 10 5 0
8.0 7.4 6.8 6.2 5.6 5.0 4.4 3.8 3.2 2.6 2.0 1.4 0.8 0.2 External field [T]
Fig. 2. Transition of maximum hoop stresses in axisymmetric three-dimensional and in one-dimensional analyses, where external field is reduced from 8.0 to 0.0 T. The maximum stresses at 5.37 T are 27.8, 14.5, 12.9 MPa, respectively.
M. Tsuchimoto, K. Murata / Physica C 392–396 (2003) 575–578
boundary condition and with fixed side-surface condition are compared with one-dimensional so-
3.0E+08 2.5E+08 2.0E+08 1.5E+08 1.0E+08 5.0E+07 0.0E+00
+H/2 Axis, z -H/2
R
R/2
0
Radius, r
577
lution. Large hoop stresses are obtained under the open boundary condition. The maximum hoop stresses at 5.37 T are 27.8, 14.5, 12.9 MPa, respectively. Shielding currents distribute all cross-section when the external field is reduced to 3.8 T in axisymmetric three-dimensional analysis. Fig. 3 shows shielding current distribution at 5.37 T of a cross-section in Fig. 1. Fig. 4(a) and (b) also show hoop stress distributions at 5.37 T for open boundary and fixed side-surface conditions. The maximum stresses are obtained at upper and lower surfaces for both cases. The stress distributions in Fig. 4 strongly depend on the shielding current distribution in Fig. 3. The stress
Fig. 3. Shielding current distribution in cross-section in Fig. 1, where external field is reduced to 5.37 T in axisymmetric threedimensional analysis.
Hoop stress [MPa]
3.0E+08 2.5E+08 30
2.0E+08
25
1.5E+08
20
1.0E+08
15
5.0E+07
10 5
0.0E+00 +H/2
0 0
R/2
-H/2
R /2
0
Radius, r
Axis, z
R
R
Shielding current
(a)
Radius, r
(a)
Open boundary condition
Stress [MPa]
Hoop stress [MPa]
15 10 5 0 -5
0
R/2 Radius, r
(b)
R
+H/2 Axis, z -H/2
Fixed side-surface condition (b)
Fig. 4. Hoop stress distributions in cross-section in Fig. 1, where external field is reduced to 5.37 T in axisymmetric threedimensional analysis: (a) open boundary condition and (b) fixed side-surface condition.
14 12 10 8 6 4 2 0
0
R /2 Radius, r
R
Hoop stress
Fig. 5. Shielding current hoop stress and distributions in onedimensional analysis where external field is reduced to 5.37 T: (a) shielding current and (b) hoop stress.
578
M. Tsuchimoto, K. Murata / Physica C 392–396 (2003) 575–578
distributions also depend on boundary conditions and larger stresses are obtained for the open boundary condition in Fig. 4(a). Fig. 5(a) and (b) shows one-dimensional shielding current and hoop stress distributions at 5.37 T. Though values of the maximum hoop stresses at 5.37 T in Fig. 2 are nearly equal for both the fixed side-surface condition and the one-dimensional solution, the positions of the maximum hoop stresses are different. The radius of the position is 1.7 mm for the axisymmetric three-dimensional analysis in Fig. 4(b), and 15.8 mm for one-dimensional analysis in Fig. 5(b).
4. Conclusion Properties of maximum hoop stresses are numerically evaluated with Bean model during the magnetization process by field cooling in a bulk high-temperature superconductor. Large hoop stresses are obtained for the open boundary condition and the maximum stresses are obtained at upper and lower surfaces in the axisymmetric
three-dimensional analyses. When and where the bulk is divided during the destructive experiments will be discussed more precisely by the numerical analysis with dependence on magnetic field of critical current density and fixed conditions of the bulk HTS in experiments.
References [1] M. Murakami, Melt Processed High-Temperature Superconductors, World Scientific, Singapore, 1992. [2] Y. Ren, R. Weinstein, J. Liu, R.P. Sawh, C. Foster, Physica C 251 (1995) 15. [3] T.H. Johansen, Phys. Rev. B 60 (1999) 9690. [4] M. Morita, K. Nagashima, S. Takebayashi, M. Murakami, T. Kaneko, T. Tokunaga, Proc. 1998 Int. Workshop Supercond. (ISTEC), 1998, p. 115. [5] M. Tsuchimoto, H. Takashima, Jpn. J. Appl. Phys. 39 (2000) 5816. [6] H. Takashima, M. Tsuchimoto, T. Onishi, Jpn. J. Appl. Phys. 40 (2001) 3171. [7] T. Sugiura, H. Hashizume, K. Miya, Int. J. Appl. Electromagn. Mater. 2 (1991) 183. [8] G. Yagawa, N. Miyazaki, Analysis of Thermal Stress, Creep and Heat Conduction, Science Co, Tokyo, 1985.
Numerical evaluations of maximum stresses in bulk superconductors M. Tsuchimoto *, K. Murata Hokkaido Institute of Technology, 7-15 Maeda, Teine-ku, Sapporo 006-8585, Japan Received 13 November 2002; accepted 24 February 2003
Abstract Stresses in bulk high-temperature superconductors (HTSs) are numerically evaluated during magnetization process by field cooling. Shielding current distributions are obtained through macroscopic numerical simulation using Maxwell equations and critical state model. Stress distributions are obtained through numerical analysis using finite-element method. Maximum stresses are evaluated from experimental results of destruction of the bulk HTS. Differences between one-dimensional and axisymmetric three-dimensional solutions are discussed for the stress distributions. Values and positions of the maximum stresses are affected by fixed conditions of the bulk HTS. Ó 2003 Elsevier B.V. All rights reserved. PACS: 74.70; 74.72; 07.10.P Keywords: Bulk HTS; Maximum stress; Field-cooled magnetization
1. Introduction A trapped field magnet is one application for melt-processed bulk high-Tc superconductors (HTSs) [1]. In magnetization process by field cooling, stresses are produced by Lorentz force between shielding currents and magnetic fields. Evaluation of the maximum stress during the magnetization is important from the viewpoint of destruction of bulk HTS [2–6]. In this study, maximum stresses in bulk HTSs are numerically evaluated during the magnetization process by field cooling. Shielding current
distributions are obtained through macroscopic numerical simulation using Maxwell equations and critical state model [6,7]. Stress distributions are obtained through numerical analysis using finite-element method [8]. Maximum stresses are discussed with experimental results of destruction of bulk HTSs [4]. Numerical solutions in axisymmetric three-dimensions with boundary conditions are different from one-dimensional analytical solution. The present numerical analysis is useful to study the maximum stresses from the destructive experiments. 2. Numerical formulation
*
Corresponding author. Fax: +81-11-681-3622. E-mail address: [email protected] (M. Tsuchimoto).
Macroscopic electromagnetic phenomena in HTS are described by Maxwell equations:
0921-4534/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4534(03)00844-X
576
M. Tsuchimoto, K. Murata / Physica C 392–396 (2003) 575–578
oB ; ot
3. Results and discussion r B ¼ l0 J;
r B ¼ 0;
ð1Þ
where l0 , E and B are magnetic permeability in air, and electric and magnetic fields, respectively. A type-II superconductor in a quasi-static field is well described by standard critical state model. Constitutive relationships between the shielding current density J SC and the electric field E are obtained from the force balance on a fluxoid [7]: J SC ¼ J c ðjBjÞ oJ SC ¼0 ot
E jEj
ðif jEj 6¼ 0Þ;
ðif jEj ¼ 0Þ:
ð2Þ
When the electric field E is induced in a local region by change of the magnetic field, shielding currents with the critical current density Jc are obtained. If there is no electric field by the shielding effect, the situation of currents is not changed. Although the critical current density Jc has a strong dependence on magnetic field, Bean model is applied to the present analysis to clarify fundamental relationship between the shielding current distribution and the stress distribution. When external magnetic field is changed, selfconsistent currents distributions are obtained stably using numerical techniques [6,7]. Then stress distributions are evaluated at each time step with the shielding current distribution [6]. From principle of virtual work, the following equation is obtained with finite-element method for region V and its surface S [8]: Z Z T T dfeg frg dV dfU g fF g dV V V Z T ð3Þ dfU g fT g dS ¼ 0; S
where {e}, {r}, {U }, {F } and {T } are strain, stress, displacement, body force and surface force vectors, respectively. Boundary conditions for (3), i.e., fixed situation of the bulk HTS in experiments, are given by surface displacements and surface force. Then stress distributions are calculated from obtained displacements. Relations of the stresses, strains and displacements are summarized in the previous papers [5,6].
Radius and thickness of a disk HTS in Fig. 1 are R ¼ 23:0 mm and H ¼ 15:0 mm in experiments [4]. The bulk HTS was cooled after external field 8.0 T was applied, then the external field was reduced to 0.0 T. The bulk was divided in two parts when applied field was 5.37 T [4]. In the present analysis, PoissonÕs ratio and YoungÕs modulus are set to 0.3 and 1.0, respectively. Critical current density Jc is set to a constant value of 3.0 108 A/m2 in Bean model. One-dimensional maximum trapped field is about 8.7 T and that in three-dimensions is about 4.2 T in full magnetization. Though radial, hoop, axial and shear stresses are obtained in the axisymmetric analysis [5,6], the most important maximum hoop stresses are shown in Fig. 2. Axisymmetric three-dimensional solutions with open
Fig. 1. Disk model in axisymmetric analysis with radius R and thickness H .
1D analytical fixed side-surface open boundary
35
Maximum hoop stress [Mpa]
rE ¼
30 25 20 15 10 5 0
8.0 7.4 6.8 6.2 5.6 5.0 4.4 3.8 3.2 2.6 2.0 1.4 0.8 0.2 External field [T]
Fig. 2. Transition of maximum hoop stresses in axisymmetric three-dimensional and in one-dimensional analyses, where external field is reduced from 8.0 to 0.0 T. The maximum stresses at 5.37 T are 27.8, 14.5, 12.9 MPa, respectively.
M. Tsuchimoto, K. Murata / Physica C 392–396 (2003) 575–578
boundary condition and with fixed side-surface condition are compared with one-dimensional so-
3.0E+08 2.5E+08 2.0E+08 1.5E+08 1.0E+08 5.0E+07 0.0E+00
+H/2 Axis, z -H/2
R
R/2
0
Radius, r
577
lution. Large hoop stresses are obtained under the open boundary condition. The maximum hoop stresses at 5.37 T are 27.8, 14.5, 12.9 MPa, respectively. Shielding currents distribute all cross-section when the external field is reduced to 3.8 T in axisymmetric three-dimensional analysis. Fig. 3 shows shielding current distribution at 5.37 T of a cross-section in Fig. 1. Fig. 4(a) and (b) also show hoop stress distributions at 5.37 T for open boundary and fixed side-surface conditions. The maximum stresses are obtained at upper and lower surfaces for both cases. The stress distributions in Fig. 4 strongly depend on the shielding current distribution in Fig. 3. The stress
Fig. 3. Shielding current distribution in cross-section in Fig. 1, where external field is reduced to 5.37 T in axisymmetric threedimensional analysis.
Hoop stress [MPa]
3.0E+08 2.5E+08 30
2.0E+08
25
1.5E+08
20
1.0E+08
15
5.0E+07
10 5
0.0E+00 +H/2
0 0
R/2
-H/2
R /2
0
Radius, r
Axis, z
R
R
Shielding current
(a)
Radius, r
(a)
Open boundary condition
Stress [MPa]
Hoop stress [MPa]
15 10 5 0 -5
0
R/2 Radius, r
(b)
R
+H/2 Axis, z -H/2
Fixed side-surface condition (b)
Fig. 4. Hoop stress distributions in cross-section in Fig. 1, where external field is reduced to 5.37 T in axisymmetric threedimensional analysis: (a) open boundary condition and (b) fixed side-surface condition.
14 12 10 8 6 4 2 0
0
R /2 Radius, r
R
Hoop stress
Fig. 5. Shielding current hoop stress and distributions in onedimensional analysis where external field is reduced to 5.37 T: (a) shielding current and (b) hoop stress.
578
M. Tsuchimoto, K. Murata / Physica C 392–396 (2003) 575–578
distributions also depend on boundary conditions and larger stresses are obtained for the open boundary condition in Fig. 4(a). Fig. 5(a) and (b) shows one-dimensional shielding current and hoop stress distributions at 5.37 T. Though values of the maximum hoop stresses at 5.37 T in Fig. 2 are nearly equal for both the fixed side-surface condition and the one-dimensional solution, the positions of the maximum hoop stresses are different. The radius of the position is 1.7 mm for the axisymmetric three-dimensional analysis in Fig. 4(b), and 15.8 mm for one-dimensional analysis in Fig. 5(b).
4. Conclusion Properties of maximum hoop stresses are numerically evaluated with Bean model during the magnetization process by field cooling in a bulk high-temperature superconductor. Large hoop stresses are obtained for the open boundary condition and the maximum stresses are obtained at upper and lower surfaces in the axisymmetric
three-dimensional analyses. When and where the bulk is divided during the destructive experiments will be discussed more precisely by the numerical analysis with dependence on magnetic field of critical current density and fixed conditions of the bulk HTS in experiments.
References [1] M. Murakami, Melt Processed High-Temperature Superconductors, World Scientific, Singapore, 1992. [2] Y. Ren, R. Weinstein, J. Liu, R.P. Sawh, C. Foster, Physica C 251 (1995) 15. [3] T.H. Johansen, Phys. Rev. B 60 (1999) 9690. [4] M. Morita, K. Nagashima, S. Takebayashi, M. Murakami, T. Kaneko, T. Tokunaga, Proc. 1998 Int. Workshop Supercond. (ISTEC), 1998, p. 115. [5] M. Tsuchimoto, H. Takashima, Jpn. J. Appl. Phys. 39 (2000) 5816. [6] H. Takashima, M. Tsuchimoto, T. Onishi, Jpn. J. Appl. Phys. 40 (2001) 3171. [7] T. Sugiura, H. Hashizume, K. Miya, Int. J. Appl. Electromagn. Mater. 2 (1991) 183. [8] G. Yagawa, N. Miyazaki, Analysis of Thermal Stress, Creep and Heat Conduction, Science Co, Tokyo, 1985.