Stresses in superconductor during oxygenation
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Abstract
The paper deals with an analytical model of stresses acting in the superconductor YBCO during an oxygenation process to transform the tetragonal lattice of the non-superconductive phase \({\hbox{YBa}_2\hbox{Cu}_3\hbox{O}_{7-{x_0}} (x_{0}=0.9)}\) to the orthorhombic lattice of the superconductive phase \({\hbox{YBa}_2\hbox{Cu}_3\hbox{O}_7}.\) Accordingly, the oxygenation-induced stresses originate as a consequence of the difference in dimensions of the crystalline lattices. Additionally, critical temperature of the oxygenation process with regard to a crack formation in the superconductor YBCO is derived.
Keywords
Critical Temperature Crack Formation Radial Stress Crystalline Lattice Superconductive PhaseIntroduction
The superconductor YBa_{2}Cu_{3}O_{7}–Y_{2}BaCuO_{5} (123–211) represents a matrix–particle system acted by stresses originating during a cooling process as a consequence of the difference in thermal expansion coefficients of the phases 123 and 211 [1], as well as during an oxygenation process of the single-grain bulk phase 123. The oxygenation process transforms the tetragonal lattice of the non-superconductive phase \({\hbox {YBa}_2\hbox{Cu}_3\hbox{O}_{7-{x_0}}}\) (x_{0} = 0.9) to the orthorhombic lattice of the superconductive phase \({\hbox{YBa}_2\hbox{Cu}_3\hbox{O}_7}.\) The superconductive phase is required because of its zero resistance against electric current. Accordingly, the stresses originating during the oxygenation process are a consequence of different dimensions of the tetragonal and orthorhombic lattices, as experimentally investigated in [2]. Finally, this paper represents an analytical contribution to the experimental results.
Analytical model
Cell model
In spite of the fact that an influence of the matrix between the cylindrical cells is not considered, the same approach, presented in [3], is used in case of an analytical model of thermal stresses originating in a matrix–particle system consisted of periodically distributed spherical particles with the radius R_{1} and the inter-particle distance 2R_{3}, where the spherical particles are embedded in an infinite matrix divided into spherical cells with the radius R_{3}. Consequently, an influence of the matrix between the spherical cells is not considered, and the same boundary condition \({\left(\sigma_r\right)_{r={R_3}}=0}\) is used to result in more than satisfactory theoretical results compared with experimental observation.
Cylinders A and B
Additionally, with regard to simplification of the analytical model, the oxygen concentration C is assumed to be constant within the cylinder with the radii R_{1} and R_{2} < R_{3}. Accordingly, the cylindrical cell consists of the cylinders A and B with the radii R_{1}, R_{2} and R_{2}, R_{3} (see Fig. 2), respectively. The cylinders A and B are thus represented by the phases YBa_{2}Cu_{3}O_{7} and \({\hbox{YBa}_2\hbox{Cu}_3\hbox{O}_{7-{x_0}}}\) (x_{0} = 0.9) with the crystalline lattice dimensions a_{1}, a_{3} and a_{10}, a_{30}, respectively, where a_{1}, a_{3} are assumed not to be functions of the spherical variables \({r\in\left\langle R_1, R_2\right\rangle},\)\({\varphi\in\left\langle 0, 2\pi\right\rangle}.\)
Stresses in cylinders A and B
Critical temperature of oxygenation process
Oxygenation-induced stresses in phase 123
E_{1} (GPa) | E_{3} (GPa) | μ_{1} | μ_{3} | R_{1} (10^{−3} m) | R_{2} (10^{−3} m) | R_{3} (10^{−3} m) | k_{1} (10^{−15} m T^{−1}) | k_{3} (10^{−15} m T^{−1}) | q_{1} (10^{−10} m) | q_{3} (10^{−10} m) |
---|---|---|---|---|---|---|---|---|---|---|
182 | 143 | 0.255 | 0.255 | 0.5 | 1 | 1.5 | 1.81596 | 27.3978 | 3.8488 | 11.5869 |
To avoid the crack formation, the critical temperature T_{op} is required to be as small as possible in comparison with the initial temperature T_{0}. Accordingly, a state of the sample YBCO with small radius R_{1} (for r_{12}→ 0) and high density of the cylindrical holes (for r_{32}→ 1) is required, what results in small critical temperature T_{op} as presented in Fig. 7.
Conclusions
- 1.
The sample with the finite diameter \({D\nrightarrow\infty}\) and the height h, containing cylindrical holes with the radius R_{1}, the length h and the inter-cylinder distance 2R_{3}, as shown in Fig. 1a and experimentally investigated in [2], is replaced by the system with D→∞ shown in Fig. 1b,
- 2.
considering the same approach as presented in [3] to result in sufficiently exact theoretical results, the system is divided into cylindrical cells with the radii R_{1}, R_{3}, and consequently the oxygenation-induced stresses are investigated within the cell (see section “Cell model”),
- 3.
as presented in section “Cylinders A and B”, with regard to simplification of the analytical model, the cylinder with the radii R_{1}, R_{3} is divided into the cylinders A and B with the radii R_{1}, R_{2} < R_{3} and R_{2}, R_{3}, and with the constant oxygen concentration C and C_{0} ≠ C to result in the crystalline lattice dimensions a_{1}, a_{3} and a_{10} ≠ a_{1}, a_{30} ≠ a_{3} in the plane x_{1}x_{2}, along the axis x_{3} (see Fig. 2), respectively,
- 4.
the analytical model of the oxygenation-induced stresses and the condition (30) to avoid the crack formation in the layer A acted by the tensile tangential stress σ_{φ A} > 0 (see Eq. 22) are presented in sections “Stresses in cylinders A and B” and “Critical temperature of oxygenation process”, along with the critical temperature of the oxygenation process, T_{op}, regarding the crack formation,
- 5.
an application of the analytical model to phase 123 is presented in section “Oxygenation-induced stresses in phase 123”, along with nomograms of the tangential, radial stresses in the cylinder A and the critical temperature, σ_{φ A}, σ_{rA} and T_{op}, respectively.
Notes
Acknowledgements
This work was supported by APVV No. COST-0022-06, by the Slovak Grant Agency VEGA (1/1111/04, 2/4062/04, 2/4173/04, 2/4175/04), by NANOSMART, Centre of Excellence, Slovak Academy of Sciences, by Science and Technology Assistance Agency under the Contracts APVT-51-049702, APVV-20-061505, APVV-20-024405, by SICMAC RTN2-2001-00488, by 2003 SO 51/03R8 06 00/03R 06 03-2003, by NENAMAT INCO-CT-2003-510363, by EU 5th FP Project No. GRD1-2000-25352 “Smart Weld”, by COST Action 536 and COST Action 538, by János Bolyai Research Grant NSF-MTA-OTKA grant—MTA: 96/OTKA: 049953, OTKA 63609.
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