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Journal of Materials Science

, Volume 42, Issue 16, pp 6895–6900 | Cite as

Stresses in superconductor during oxygenation

  • Ladislav CenigaEmail author
Article
  • 52 Downloads

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 Phase 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction

The superconductor YBa2Cu3O7–Y2BaCuO5 (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 \({\hbox {YBa}_2\hbox{Cu}_3\hbox{O}_{7-{x_0}}}\) (x0 = 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

As presented in Ref. [2], the oxygenation process, applying to the sample YBCO in a form of the cylindrical pellet shown in Fig. 1a (D = 35, h = 20 [mm]), is realized in cylindrical holes with the radius R1, the length h and the inter-cylinder distance 2R3. With regard to an analytical model, this system with the finite dimension D is replaced by an infinite system (D→∞) divided into cylindrical cells with parameters R1, R3, h (see Fig. 1b). Consequently, the oxygenation-induced stresses are investigated within the cylindrical cell, considering the boundary condition \({\left(\sigma_r\right)_{r={R_3}}=0}\) for the radial stress σr. Resulting from the system infinity, analytical models of the oxygenation-induced stresses in a certain cylindrical cell are identical with those in any cylindrical cell.
Fig. 1

Shape of the sample of the single-grain bulk 123 phase (a) as experimentally investigated in [2], and the system (b) considered regarding the analytical 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 R1 and the inter-particle distance 2R3, where the spherical particles are embedded in an infinite matrix divided into spherical cells with the radius R3. 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

The sample of the single-grain bulk phase 123 is oxygenated at the different temperature T to result in the different oxygen concentration C in a crystalline lattice of the 123 phase. Consequently, the different concentration C results in the different crystalline lattice dimensions a1 and a3 in the plane x1x2 and along the axis x3 (see Fig. 2), respectively. An initial state of the phase 123 is thus characterized by the parameters a10, a30 related to the temperature T0 resulting in C0, where T0 is simultaneously final temperature of a crystal growth process of the phase 123 [2, 4]. Performing the oxygenation process at the temperature T < T0, the parameters a10, a30 are transformed to a1 ≠ a10, a3 ≠ a30 corresponding to C at the radii R1, respectively, where C ≠ C0.
Fig. 2

The hollow cylinders A and B with the radii R1 < R2 < R3, and the axes xr, xφ and z = cx3 in the general point P, along which the radial, tangential and longitudinal stresses σr, σφ and σz act, respectively, and radial cracks with the length h formed in the cylinder A provided that σφ A > 0 and on the condition \({\left(\sigma_{\varphi A}\right)_{r={R_1}} > \sigma_{fr}}\) (see section “Stresses in 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 R1 and R2 < R3. Accordingly, the cylindrical cell consists of the cylinders A and B with the radii R1, R2 and R2, R3 (see Fig. 2), respectively. The cylinders A and B are thus represented by the phases YBa2Cu3O7 and \({\hbox{YBa}_2\hbox{Cu}_3\hbox{O}_{7-{x_0}}}\) (x0 = 0.9) with the crystalline lattice dimensions a1, a3 and a10, a30, respectively, where a1, a3 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

Considering the system consisted of the hollow cylinders A and B of the 123 phase and with the radii R1 < R2 < R3 (see Fig. 2), the radial, tangential and longitudinal stresses σr, σφ and σz are investigated in the general point P along the axis xr, xφ and z, respectively, where xr, xφx1x2; zx3. With regard to parameters of the phase 123 [1], the isotropic and anisotropic planes x1x2, x1x3 and x2x3, respectively, correspond to the tetragonal lattice, and the coordinate r along the axis xr is related to the cylinders A, B for \({r\in\left\langle R_1, R_2\right\rangle},\)\({r\in\left\langle R_2, R_3\right\rangle},\) respectively. With regard to isotropy of the plane x1x2, we get
$$\frac{\partial u_r}{\partial\varphi}=0,$$
(1)
and consequently the shear strain ɛrφ = ɛrφ = (1/r) (∂ur/∂φ) = 0 [5] results in the shear stress σrφ = σφ r = 0, where ur is displacement of an infinitesimal cylindrical part of the system along the axis xr, and the angle \({\varphi=\angle\left(x_1,OP\right)}.\)
Additionally, the displacements ur and uz, the latter along the axis z, are assumed to be independent on the variables z and r, respectively, as derived by the condition
$$\frac{\partial u_r}{\partial z}=\frac{\partial u_z}{\partial r}=0,$$
(2)
and consequently the shear strain ɛrν = (∂ur/∂z) + (∂uz/∂r) = 0 [5] results in the shear stress σrν = σν r = 0. Finally, along with Eqs. (1), (2), the strains ɛzA, ɛzB along the axis z are connected with the condition
$$\varepsilon_{zA}=\varepsilon_{zB}=\frac{a_3-a_{30}}{a_{30}}.$$
(3)
Considering the dimension changes a1a10, a3a30 to originate in crystalline lattices of the cylinder A only, the cylinders A and B tend to exhibit the radii R2 + R2[(a1a10)/a10] and R2, respectively, where R2/a10 represents a number of crystalline lattices with the dimension a10 related to the distance R2. The radius change R2(a1a10)/a10 induces the compressive or tensile radial stress p2 > 0 or p2 < 0 acting on the AB boundary, respectively. The radial stress p2 results in the radial strains \(\left(\varepsilon_{rA}\right)_{r={R_2}}, {\left(\varepsilon_{rB}\right)_{r={R_2}}}\) in the cylinders A, B, respectively. The AB boundary thus exhibits the radius
$$R_{AB}=R_2+R_2\frac{a_1-a_{10}}{a_{10}}+R_2\left(\varepsilon_{rA}\right)_{r={R_2}}=R_2+R_2\left(\varepsilon_{rB}\right)_{r={R_2}},$$
(4)
and consequently we get the condition
$$\left(\varepsilon_{rB}\right)_{r={R_2}}-\left(\varepsilon_{rA}\right)_{r={R_2}}=\frac{a_1-a_{10}}{a_{10}}$$
(5)
used for the determination of the radial stress p2. The Cauchy’s equations have the forms [5]
$$\varepsilon_r=\frac{\partial u_r}{\partial r},$$
(6)
$$\varepsilon_{\varphi}=\frac{u_r}{r},$$
(7)
$$\varepsilon_z=\frac{\partial u_z}{\partial z},$$
(8)
and consequently the compatibility equation (9) to result from Eqs. 6, 7, and the equilibrium equation (10) related to the axis xr [5] have the forms
$$\varepsilon_r-\varepsilon_{\varphi}-r\frac{\partial\varepsilon_{\varphi}}{\partial r}=0,$$
(9)
$$\sigma_{\varphi}=\sigma_r+r\frac{\partial\sigma_r}{\partial r},$$
(10)
where ∂σφ/∂φ = 0 and ∂σz/∂z = 0 results from equilibrium equations related to the axes xφ and z due to σrφ = σφ r = 0 and σrν = σν r = 0, respectively. Finally, the Hooke’s laws corresponding to the tetragonal lattice are derived as [5]
$$\varepsilon_r=s_{11}\sigma_r+s_{12}\sigma_{\varphi}+s_{13}\sigma_z,$$
(11)
$$\varepsilon_{\varphi}=s_{12}\sigma_r+s_{11}\sigma_{\varphi}+s_{13}\sigma_z,$$
(12)
$$\varepsilon_z=s_{31}\left(\sigma_r+\sigma_{\varphi}\right)+s_{33}\sigma_z,$$
(13)
and the elastic modulus sij (i, j = 1, 2, 3) [5] has the form
$$s_{ij}= \frac{\delta_{ij}-\mu_j\left(1-\delta_{ij}\right)}{E_i},\quad i, j=1, 2, 3,$$
(14)
where Ei and μj are the Young’s modulus and the Poisson’s number, and δij = 0, 1 for i ≠ j, i = j is the Kronecker’s symbol, respectively. With regard to Eqs. 10, 13, the stress σz is derived as
$$\sigma_z=\frac{1}{s_{33}}\left[\varepsilon_z-s_{31}\left(2\sigma_r+r\frac{\partial\sigma_r}{\partial r}\right)\right].$$
(15)
With regard to \({\partial\varepsilon_z/\partial r=\left(\partial/\partial r\right)\left(\partial u_z/\partial z\right) =\left(\partial/\partial z\right)\left(\partial u_z/\partial r\right)=0}\) (see Eqs. 2, 8), considering Eqs. 10–12, 15, the compatibility equation (9) is transformed to the form
$$r\frac{\partial^2\sigma_r}{\partial r^2}+3\frac{\partial\sigma_r}{\partial r}=0,$$
(16)
and consequently, assuming the radial stress in the form σr = Crλ, we get
$$\sigma_r=C_1+\frac{C_2}{r^2},$$
(17)
where the coefficients C1, C2 are determined from boundary conditions. Considering the boundary conditions
$$\left(\sigma_{rA}\right)_{r={R_1}}=0,$$
(18)
$$\left(\sigma_{rA}\right)_{r=R_2}=\left(\sigma_{rB}\right)_{r={R_2}}=-p_2,$$
(19)
$$\left(\sigma_{rB}\right)_{r=R_3}=0,$$
(20)
the stresses in the cylinders A and B have the forms
$$\sigma_{rA}=-\frac{p_2}{1-r_{12}^2}\left[1-\left(\frac{R_1}{r}\right)^2\right],\quad r_{12}=\frac{R_1}{R_2}\in\left\langle 0,1\right),$$
(21)
$$\sigma_{\varphi A}=-\frac{p_2}{1-r_{12}^2}\left[1+\left(\frac{R_1}{r}\right)^2\right],$$
(22)
$$\sigma_{zA}= \frac{1}{s_{33A}} \left(\frac{2s_{31A}p_2}{1-r_{12}^2}+\varepsilon_{zA}\right),$$
(23)
$$\sigma_{rB}=-\frac{p_2}{1-r_{32}^2}\left[1-\left(\frac{R_3}{r}\right)^2\right],\quad r_{32}=\frac{R_3}{R_2} > 1,$$
(24)
$$\sigma_{\varphi B}=-\frac{p_2}{1-r_{32}^2}\left[1+\left(\frac{R_3}{r}\right)^2\right],$$
(25)
$$\sigma_{zB}=\frac{1}{s_{33B}}\left( \frac{2s_{31B}p_2}{1-r_{32}^2}+\varepsilon_{zB}\right).$$
(26)
With respect to Eqs. 5, 12, 21–26, the radial stress p2 is derived as
$$p_2=\frac{1}{c_1-c_2}\left[\frac{a_1-a_{10}}{a_{10}}+\frac{a_3-a_{30}}{a_{30}}\left(\frac{s_{13A}}{s_{33A}}-\frac{s_{13B}}{s_{33B}}\right)\right].$$
(27)
and the coefficients c1, c2 have the forms
$$c_1=s_{12A}+\frac{1}{1-r_{12}^2} \left[s_{11A}\left(1+r_{12}^2\right)-\frac{2s_{13A}s_{31A}}{s_{33A}}\right],$$
(28)
$$c_2=s_{12B}+\frac{1}{1-r_{32}^2}\left[s_{11B}\left(1+r_{32}^2\right)-\frac{2s_{13B}s_{31B}}{s_{33B}}\right],$$
(29)
where a10, a1 and a30, a3 are lattice parameters along the axis x1 and x3, related to the temperature T0, TT0, respectively.

Critical temperature of oxygenation process

Provided that the tangential stress σφ A is tensile, then for σφ A >  0; on the condition \({\left(\sigma_{\varphi A}\right)_{r={R_1}} > \sigma_{fr}};\) and with regard to σφ Ar representing a decreasing function of the variable \({r\in\left\langle R_1,R_2\right\rangle};\) radial cracks with the length h in the cylinder A are formed from a surface with the radius R1 during the oxygenation process, where σfr is critical stress with respect to crack formation. To avoid the crack formation, the condition
$$\left(\sigma_{\varphi A}\right)_{r={R_1}}\leq\sigma_{fr},$$
(30)
being accordingly required to be fulfilled, is transformed, after substitution of Eqs. 22, 27 to Eq. 30, to the temperature condition
$$T\geq T_{op}.$$
(31)
Consequently, assuming the linear temperature dependence
$$a_{1+2i}=k_{1+2i}T+q_{1+2i},\quad i=0,1,$$
(32)
the critical temperature of the oxygenation process, Top, related to the temperature T0, has the form
$$T_{op}=\frac{c_3c_4-c_5}{c_6}$$
(33)
and the coefficients c3c6 are derived as
$$c_3=\left(k_1T_0+q_1\right)\left(k_3T_0+q_3\right),$$
(34)
$$c_4=1+\frac{s_{13A}}{s_{33A}}-\frac{s_{13B}}{s_{33B}}-\frac{\sigma_{fr}}{2}\left(c_1-c_2\right)\left(1-r_{12}^2\right),$$
(35)
$$c_5=q_3\left(k_1T_0+q_1\right)\left(\frac{s_{13A}}{s_{33A}}-\frac{s_{13B}}{s_{33B}}\right)+q_1\left(k_3T_0+q_3\right),$$
(36)
$$c_6=k_3\left(k_1T_0+q_1\right)\left(\frac{s_{13A}}{s_{33A}}-\frac{s_{13B}}{s_{33B}}\right)+k_1\left(k_3T_0+q_3\right),$$
(37)
where a1+2i0 = (a1+2i)T=T_0 (i = 0,1).

Oxygenation-induced stresses in phase 123

With regard to the temperature dependence (32), the crystalline lattice dimensions a1 and a3 of the phase 123 in the plane x1x2 and along the axis x3, respectively, are shown in Fig. 3 [4] as functions of the oxygenation temperature T, where a1, a3 are measured at the room temperature Tr = 20 °C, and consequently changes of a1, a3 due to thermal expansion in the temperature interval \({\left\langle T_r, T\right\rangle}\) are neglected.
Fig. 3

The crystalline lattice dimensions a1 (a) and a3 (b) of the phase 123 in the plane x1x2 and along the axis x3 (see Fig. 2), respectively, as functions of the oxygenation temperature T, where a1, a3 are measured at the room temperature T = 20 °C [4]

Considering material and lattice parameters of the phase 123 presented in Table 1, the radial, tangential, axial stresses, σrA, σφ A, σzA and σrB, σφ B, σzB (see Eqs. 21–26) in the layers A and B, respectively, for the radii R1 =  0.5, R2 =  1, R3 =  1.5, for the initial temperature T0 =  900 °C and the oxygenation temperature T =  400 °C are shown in Fig. 4. The temperature T0 =  900 °C and T =  400 °C to result in a10 =  3.865, a30 =  11.833 and a1 =  3.856, a3 =  11.696 [10−10 m] (see Eq. 32; Table 1) is the same as considered within the experimental results published in [2, 4], respectively. Additionally, the nomograms σφ Ar12r32, σrAr12r32 at the radii r = R1, r = R2 in the layer A to exhibit tendency to release the oxygenation-induced stress loading by radial crack formation (see Figs. 2, 7), are presented in Figs. 5, 6. Finally, the dependencies in Figs. 5, 6 are not asymptotic for r12→ 1 due to reduction of the term 1−r 12 2 in fractions of Eqs. 21, 22, 27–29.
Fig. 4

The radial and tangential stresses σrA, σrB and σφ A, σφ B, along with the axial stresses σzA, σzB, acting in the cylinders A, B (see Eqs. 21–26), respectively, with the radii R1 = 0.5 mm, R2 = 1 mm, R3 = 1.5 mm for the temperature T0 = 900, T = 400 [°C] (see Eq. 32), where σ > 0 or σ < 0 represents tensile or compressive stress, respectively, and p2 is a radial stress acting on the AB boundary (see Eq. 27)

Fig. 5

Nomograms of the tensile tangential stress \({\left(\sigma_{\varphi A}\right)_{r={R_1}} > 0},\) acting at the radius r = R1, as functions of the parameters \({r_{12}\in\left\langle0,1\right\rangle}\) and r32 > 1 (see Eqs. 21, 22, 24) for the temperature T0 = 900, T = 400 [°C] (see Eq. 32), where \({\left(\sigma_{rA}\right)_{r={R_{1}}}=0}\) (see Eq. 18). The dependence \({\left(\sigma_{\varphi A}\right)_{r={R_{1}}}-r_{12}-r_{32}}\) for r32 > 5 is approximately identical to the curve 27 (b)

Fig. 6

Nomograms of the tensile tangential and radial stresses, \({\left(\sigma_{\varphi A}\right)_{r={R_2}} > 0}\) (a, b) and \({\left(\sigma_{rA}\right)_{r={R_2}}=-p_2 > 0}\) (c,d) (see Eq. 19), respectively, acting at the radius r = R2, as functions of the parameters \({r_{12}\in\left\langle 0,1\right\rangle}\) and r32 > 1 (see Eqs. 21, 22, 24, 27) for the temperature T0 = 900, T = 400 [°C] (see Eq. 32). The dependencies \({\left(\sigma_{\varphi A}\right)_{r={R_2}}-r_{12}-r_{32}}\) and \({\left(\sigma_{rA}\right)_{r={R_2}}-r_{12}-r_{32}}\) for r32 > 5 are approximately identical to the curve 27 (b, d)

Table 1

Material and lattice parameters of the 123 phase (see Fig. 3a, b; Eq. 32) [1, 4]

E1 (GPa)

E3 (GPa)

μ1

μ3

R1 (10−3 m)

R2 (10−3 m)

R3 (10−3 m)

k1 (10−15 m T−1)

k3 (10−15 m T−1)

q1 (10−10 m)

q3 (10−10 m)

182

143

0.255

0.255

0.5

1

1.5

1.81596

27.3978

3.8488

11.5869

The nomogram Topr12r32 for the initial temperature T0 = 900 °C, and for the critical stress σfr = 25 MPa [4] regarding the radial crack formation is presented in Fig. 7. Resulting from R2 = R2(t) as an increasing time-dependent function for (R2)t=0 = R1 [6], the critical temperature Top for t = 0 and t > 0 is considered for r12 = 1 and \({r_{12}\in\left\langle 0,1\right)}\) (see Eq. 21), respectively, with regard to r32 ≥ 1.
Fig. 7

Nomograms of the temperature Top as a function of the parameters \({r_{12}\in\left\langle 0,1\right\rangle}\) and r32 > 1 (see Eqs. 21, 24, 33), for material and lattice parameters of the 123 phase (see Table 1). The dependence Topr12r32 for r32 > 1.2 is approximately identical to the curve 8

To avoid the crack formation, the critical temperature Top is required to be as small as possible in comparison with the initial temperature T0. Accordingly, a state of the sample YBCO with small radius R1 (for r12→ 0) and high density of the cylindrical holes (for r32→ 1) is required, what results in small critical temperature Top as presented in Fig. 7.

Conclusions

Representing a continuation of the paper [2] with experimental results to concern the oxygenation process resulting in a transformation of the tetragonal lattice of the non-superconductive phase YBa2Cu3O\({_{7-{x_0}}}\) (x0 = 0.9) to the orthorhombic lattice of the superconductive phase YBa2Cu3O7, the latter exhibiting zero resistance against electric current and both denoted as the phase 123, main results concerning the analytical model of the oxygenation-induced stresses presented in this paper are as follows:
  1. 1.

    The sample with the finite diameter \({D\nrightarrow\infty}\) and the height h, containing cylindrical holes with the radius R1, the length h and the inter-cylinder distance 2R3, as shown in Fig. 1a and experimentally investigated in [2], is replaced by the system with D→∞ shown in Fig. 1b,

     
  2. 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 R1, R3, and consequently the oxygenation-induced stresses are investigated within the cell (see section “Cell model”),

     
  3. 3.

    as presented in section “Cylinders A and B”, with regard to simplification of the analytical model, the cylinder with the radii R1, R3 is divided into the cylinders A and B with the radii R1, R2 < R3 and R2, R3, and with the constant oxygen concentration C and C0 ≠ C to result in the crystalline lattice dimensions a1, a3 and a10 ≠ a1, a30 ≠ a3 in the plane x1x2, along the axis x3 (see Fig. 2), respectively,

     
  4. 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 σφ 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, Top, regarding the crack formation,

     
  5. 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, σφ A, σrA and Top, 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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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institute of Materials ResearchSlovak Academy of SciencesKošiceSlovak Republic

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