Critical Current and n-Value of Heterogeneously Cracked Superconducting Tapes, Studied by a Monte Carlo Simulation Method Combined with a Model of Current Shunting at Cracks

Abstract

Influences of crack size-distribution and specimen length on the critical current and n-value of heterogeneously cracked superconducting tapes were studied by a Monte Carlo simulation method combined with a model of crack-induced current shunting. In simulation, model specimens, constituted of a series of sections having cracks with different size to each other, were used. It was shown by the present simulation that (i) both of the critical current and n-value decrease with increasing width of crack size distribution, (ii) n-value decreases more sensitively to the increase in width of crack size distribution in comparison with the critical current, and (iii) the features (i) and (ii) stated above are enhanced with increasing specimen length. Also, the experimentally observed feature that the width of distribution of critical current decreases with increasing length in heterogeneously cracked superconducting tapes was realized by the present simulation. This result was in good accordance with the reported feature that local information of critical current values in a specimen is diluted when the voltage tap distance is large.

1. Introduction

In fabrication and in operation, RE(Y, Sm, Dy, Gd,....)Ba₂Cu₃O_7−δ coated superconducting tapes (hereafter noted as REBCO tapes), as well as filamentary Bi₂Sr₂Ca₂Cu₃O_10+x (BSCCO), Nb₃Sn, Nb₃Al and MgB₂ tapes, are subjected to thermal, mechanical and electromagnetic stresses/strains. When the superconducting layers/filaments are cracked by such stresses, the critical current I_c and n-value of both coated^1–9) and filamentary^10–19) tapes are reduced. Cracking of the coated layer/filament takes place heterogeneously. Accordingly, even at a given applied stress/strain, the I_c- and n-values are different from location to location within a specimen^9,14), they are different from specimen to specimen^7,9,13,14) and they are dependent on specimen length^9,14).

In our recent work²⁰⁾, the extreme case where one crack exists in a REBCO-coated specimen was studied for various specimen lengths by a modeling analysis. The experimentally observed feature that the decrease in n-value with decreasing I_c becomes sharper for longer specimen under local crack-evolution⁹⁾ was reproduced. Then, we investigated the I_c and n-value of a REBCO-coated model specimen constituted of a series of three sections having cracks with different size to each other²¹⁾. It was shown that, (a) in both cases of large and small differences in crack size among sections, the section with the largest crack plays a dominant role in determination of the voltage (V) –current (I) curve of the specimen and acts a dominant role to reduce I_c and n-value of the specimen from the original I_c and n-value in the non-cracked state, and (b) under a given size of the largest crack, the I_c becomes higher but the n-value becomes lower with increasing difference in crack size among the sections than the I_c and n-value in the case of no difference in crack size. The features (a) and (b) have been observed experimentally in both REBCO-coated- and BSCCO-filamentary specimens^9,14).

As shown above, the influence of difference in crack size among the sections has been studied to some extent but not systematically. It is required to develop a method to obtain systematic information. In the present work, a Monte Carlo simulation method was employed, with which the influences of crack size distribution and specimen length on I_c and n-value were investigated.

2. Model and Procedure for Simulation

2.1 Model specimen

The configuration of the model specimen employed in the present work is shown in Fig. 1(a). The specimen is constituted of a series of N sections (S(1), S(2),...S(N)) with a length L₀ = 1.5 cm in each. Each section has one crack and the crack size is different among sections. In simulation, N was varied from 1 to 20, namely the length of the specimen L = NL₀ was varied from 1.5 to 30 cm, due to the following reasons. (a) It has been known that local information of I_c-values in a specimen is diluted when the voltage tap distance is large²²⁾. Preliminary simulation study indicated that this feature can be detected in the specimen length-range of L = 1.5 to 30 cm. (b) For materials characterization, experiments to measure I_c and n-value under applied stress in laboratory are usually conducted in the specimen length range less than several tens of centimeters.

Fig. 1

Schematic representation of (a) the model specimen with length L (3～30 cm), composed of a series of sections with a length L₀ (=1.5 cm in this work) and (b) current path in a section with a partial crack.

2.2 Description of V–I relation of sections with a model of crack-induced current shunting

For description of the V(voltage)–I(current) relation of sections under an existent partial crack, where partial crack means the crack that exists in a part of transverse cross-section of the superconducting phase, the model of Fang et al.¹⁰⁾ was employed in a modified form^{3,5,7,9,13,14,19,21)}. Noting the I_c and n-value of sections in the non-cracked state as I_c0 and n₀, respectively, and the critical electric field for determination of critical current as E_c (=1 μV/cm in this work), the V–I curve of a non-cracked section with a length L₀ is expressed as   

\[V = E_{\rm c} L_0 \left( \frac{I}{I_{\rm c0}} \right)^{n_0}\](1)

Current path in a cracked section is shown schematically in Fig. 1(b)²¹⁾, in which one section in a REBCO tape specimen is representatively drawn. In the transverse cross-section in which a partial crack exists, the cracked part and the ligament part constitute of a parallel electric circuit. We define the ratio of cross-sectional area of cracked part to the total cross-sectional area of the superconducting layer as f. The ligament part with an area ratio 1 − f transports current I_RE. At the cracked part with an area ratio f, current I_s (= I − I_RE) shunts into stabilizer such as Ag and Cu. The electric resistance of the shunting circuit is noted as R_t. The voltage, developed at the ligament part that transports current I_RE, is noted as V_RE. The voltage V_s = I_sR_t, developed at the cracked part by shunting current I_s, is equal to V_RE, since the cracked- and ligament-parts constitute a parallel circuit. Noting the current transfer length as s (<<L₀), we can express the V–I relation of the cracked section (L₀ = 1.5 cm) as^3,5,19,21),   

\[V = E_{\rm c} L_0 \left( \frac{I}{I_{\rm c0}} \right)^{n_0} + V_{\rm RE}\](2)

  

\[I = I_{\rm RE} + I_{\rm S} = I_{\rm c0} (1-f) \left( \frac{L_0}{s} \right)^{1/n_0} \left[ \frac{V_{\rm RE}}{E_{\rm c}L_0} \right]^{1/n_0} + \frac{V_{\rm RE}}{R_{\rm t}}\](3)

The term $(1-f)(L_0 /s)^{1/n_0}$ in eq. (3) is, hereafter, noted as the ligament parameter L_p. This parameter $L_{\rm p}(=\!(1\!-\!f)(L_0/s)^{1/n_0})$ was derived by the authors^3,5,9,13,20) by modifying the formulations of Fang et al.¹⁰⁾ In this work, it was used as a monitor of crack size (the smaller the ligament parameter $(1-f)(L_0/s)^{1/n_0}$, the larger is the crack size f.)

The ligament parameter ${\rm Lp}=(1-f)(L_0/s)^{1/n_0}$ has a physical meaning as I_c/I_c0 (ratio of the critical current in cracked state (I_c) to the critical current in non-cracked state (I_c0)) in the virtual case where the voltage V_RE developed at the crack is equal to the whole voltage V and current shunting is negligible²⁰⁾. In practice, current shunting occurs and also voltage is developed not only at the crack but also at the non-cracked region away from the crack, and hence the I_c/I_c0 is not equal to the ligament parameter. However, it is noted that the ligament parameter still gives a fairly good approximation for I_c/I_c0 in both of REBCO-coated tape^{3,5,7,9,19,21)} and BSCCO filamentary tape^13,14).

In the present work, the value of $L_{\rm p}=(1-f)(L_0/s)^{1/n_0}$ was used directly in calculation of eqs. (2) and (3). It has been shown that, in experimental works, the value of $L_{\rm p}=(1-f)(L_0/s)^{1/n_0}$ can be obtained for each cracked specimen by analysis of the measured V–I curve^3,5,7,9,13). Also, in the simulation study as in the present work, it can be used as a parameter to obtain I_c and n-value of cracked specimen^20,21,23). By the approach that uses the value of $(1-f)(L_0/s)^{1/n_0}$ directly in calculation, the specimen length-dependence of critical current and n-value in the specimens with one crack²⁰⁾, and the competition among cracks with different size in contribution to synthesize the V–I curve of the specimen with plural cracks²¹⁾ were revealed. Also, the experimental results of the correlation between n-value and I_c could be described through the calculation using this value²³⁾. This approach has the following advantages. (i) It can be used even when the values of s and f are unknown. (ii) It is simple and can be a useful tool in practice. However, these advantages, in turn, are disadvantage on the point that the influence of each of s and f on current shunting phenomenon, n-value and I_c is not specified directly. For specification, the values of s and f and their correlation shall be obtained in advance. In the present work, by using $L_{\rm p}=(1-f)(L_0/s)^{1/n_0}$ in calculation, the aim to reveal the influences of crack size-distribution and specimen length on the critical current and n-value of heterogeneously cracked superconducting tapes was achieved satisfactorily, as shown later in Section 3.

2.3 Application of Monte Carlo method to give ligament parameter-value and calculation of V–I curve of sections with a length L₀

Since wide distribution of ligament parameter $L_{\rm p}(=(1-f)(L_0/s)^{1/n_0})$ corresponds to wide distribution of f (crack size) and since the standard deviation of 1 − f is the same as that of f, the standard deviation of L_p, ΔL_p, was used as a monitor of the width of crack size distribution; the larger the ΔL_p, the wider the width of crack size distribution. For formulation of distribution of L_p of cracked sections with a length L₀, the normal distribution function was used. Noting the average of L_p as L_p,ave, the cumulative probability F(L_p) was expressed by   

\[F(L_{\rm P}) = \frac{1}{2} \left\{1 + {\rm erf} \left( \frac{L_{\rm p} - L_{\rm p,ave}}{\sqrt 2 \Delta L_{\rm p}} \right) \right\}\](4)

As a value of L_p,ave, 0.670 was chosen as to describe the representative situation, where the cracks reduce the critical current of sections by ≈1/3 on average from the non-cracked state. Six cases of ΔL_p = 0.01, 0.025, 0.05, 0.1, 0.15 and 0.2 were taken up to reveal the influence of the extent of distribution of L_p (=extent of distribution of crack size) on I_c and n-value of sections and specimens. We obtained the L_p-value for each cracked section with a Monte Carlo method by generating a random value R in the range of 0～1, setting F(L_p) = R and substituting the values of L_p,ave and ΔL_p in eq. (4).

The V–I curve of each cracked section with a length L₀ = 1.5 cm was calculated by substituting the L_p-value, given by the Monte Carlo method stated above, and the values of R_t = 2 μΩ, I_c0 = 200 A and n₀ = 40 into eqs. (2) and (3). Here, the values of R_t, I_c0 and n₀ were taken from the experimental result⁷⁾ of a copper-stabilized DyBCO tape test-pieces, whose V–I curves were measured at 77 K in a self-magnetic field for the voltage probe distance L₀ = 1.5 cm. The thickness of the superconducting DyBCO layer of the tape was 2.5 µm. The critical current, I_c0, and n-value, n₀, in the non-cracked state were taken from each average value that were obtained by applying the same criterion as in this work to the V–I curves measured under no applied stress. The value of R_t (2 μΩ) was taken from an average R_t-value obtained by analysis of the V–I curves of the test-pieces in which the superconducting DyBCO layer was cracked under applied tensile stress.

The simulation procedure based on the Monte Carlo method stated above was repeated for 120 times and, as a result, 120 sets of (V–I curve, I_c-value, n-value) for the sections were obtained for each ΔL_p value.

2.4 Calculation of V–I curve, and estimation of critical current and n-value of specimens

As the specimen is constituted of a series electric circuit of sections (Fig. 1(a)), the current of the specimen is the same as that of sections;   

\[I = I_{{\rm S}(i)}(i = 1\ {\rm to} \ N)\](5)

and the voltage of the specimen is the sum of the voltages of all sections;   

\[V = \sum \limits_{i = 1}^N V_{{\rm S}(i)} \](6)

Using the V–I curves of sections (L₀ = 1.5 cm) obtained by eqs. (2) and (3), we calculated the V–I curves of the specimens for L = 3 to 30 cm by eqs. (5) and (6). From the calculated V–I curves, the critical current I_c of the specimen was obtained by the electric field criterion of E = E_c = 1 μV/cm (corresponding to V = V_c = E_cL μV). The n-value of the specimen was obtained by fitting the E–I curve to the form of E ∝ Iⁿ in the electric field range of E = 0.1 − 10 μV/cm, namely by fitting the V–I curve to the form of V ∝ Iⁿ in the voltage range of V = 0.1L − 10L μV.

3. Results and Discussion

3.1 Influence of distribution of crack size on the difference in V–I curve among the sections and on the V–I curve, critical current I_c and n-value of specimens

Figure 2 shows six examples of the simulated V–I curves of specimens with a length L = 9 cm and sections with a length L = L₀ = 1.5 cm, obtained for the random values generated in the computer R = 0.063, 0.514, 0.667, 0.203, 0.951, 0.641 for the section S1 to S6, respectively. To obtain L_p-values of sections, these random values were used commonly for ΔL_p = (a) 0.01, (b) 0.025, (c) 0.05, (d) 0.1, (e) 0.15 and (f) 0.2. The obtained L_p-values of the sections in the examples in Fig. 2 had the following features. (i) The average values were 0.675～0.680 in all examples, (ii) the values of the standard deviation ΔL_p,Ex of the obtained ligament parameter values for the examples shown in Fig. 2(a) to (f) were 0.011, 0.028, 0.056, 0.111, 0.165 and 0.215, respectively, and accordingly, (iii) the examples (a) to (f) had almost the same average crack size but the width of crack size distribution was different from example to example; it was small in the example (a) and it became larger in the order of the examples (a) to (f).

Fig. 2

Examples of the V–I curves of the specimens (L = 9 cm) and sections (L = L₀ = 1.5 cm), obtained for the computer-generated random values R = 0.063, 0.514, 0.667, 0203, 0.951 and 0.641 for the sections S1 to S6, respectively. With these random values, the average ligament parameter of sections was almost common in all examples but the standard deviation of the ligament parameter of sections ΔL_p,Ex, was varied from 0.011 (a) to 0.215 (f).

From the V–I curves in Fig. 2, the I_c- and n-values of sections and specimens were obtained with the criteria stated in Subsection 2.4. The average critical current, I_c,ave, and average n-value, n_ave, of the sections, and I_c and n-value of the specimen are shown for each example in Fig. 2, for comparison.

Figure 3(a) shows the obtained I_c- and I_c,ave-values of the sections and I_c-values of the specimens for the 6 examples, plotted against ΔL_p,Ex–value that refers the difference in crack size among the sections. The changes in I_c,ave of the sections and I_c of specimen with increasing ΔL_p,Ex are shown with the broken and solid curves, respectively. Figure 3(b) shows the n-values of the sections and specimens, and n_ave-values of the sections, plotted against ΔL_p,Ex–value, together with the changes in n_ave-value of the sections (broken curve) and n-value of specimen (solid curve) with increasing ΔL_p,Ex.

Fig. 3

Critical current I_c and n-value of sections and specimens obtained from the V–I curves of the examples (a) to (f) in Fig. 2, plotted against the standard deviation of ligament parameter ΔL_p,Ex of sections in the examples. (a) Change in I_c-values of the sections and specimens, and the average critical current (I_c,ave) of sections with ΔL_p,Ex. (b) Change in n-value of sections and specimens, and the average n-value (n_ave) of sections with ΔL_p,Ex. (c) Comparison of the change in I_c/I_c(ΔL_p = 0) and n/n(ΔL_p = 0) with ΔL_p,Ex, where I_c/I_c(ΔL_p = 0) and n/n(ΔL_p = 0) refer to the retention of critical current I_c and n-value of the specimen in comparison with the critical current, I_c(ΔL_p = 0), and n-value, n(ΔL_p = 0) in the case of ΔL_p = 0 where the ligament parameter (crack size) is the same in all sections.

If all sections have a same ligament parameter (same crack size), namely if ΔL_p = 0, all sections and the specimen have a same I_c and a same n-value. When the I_c and n-value of specimen in a case of ΔL_p = 0 are noted as I_c(ΔL_p = 0) and n(ΔL_p = 0), respectively, the values of I_c/I_c(ΔL_p = 0) and n/n(ΔL_p = 0) show the retention of critical current and n-value in comparison with those in the case of ΔL_p = 0, respectively. Figure 3(c) shows the change in I_c/I_c(ΔL_p = 0) (solid curve) and n/n(ΔL_p = 0) (broken curve) with increasing ΔL_p,Ex.

The following features are read from the results in Figs. 2 and 3.

• (1)   Reflecting the feature that the examples (a) to (f) had almost the same average crack size, the average critical current, I_c,ave, and also the average n-value, n_ave, of the sections were 135～136 A and 26.5～27.6, which were almost the same in all examples (a) to (f). Reflecting the feature that the width of distribution of crack size increased in the order of the examples (a) to (f), both of I_c and n-value of the specimen decreased more in the order of (a) to (f), showing that the I_c and n-value of the specimen are strongly dependent on the extent of the difference in crack size among sections.
• (2)   In any width of distribution of crack size in sections, the V–I curve of the specimen exists near to that of the lowest I_c-section that has the largest crack among the sections. The lowest I_c-section most contributes to the synthesis of the V–I curve of the specimen. The sections having the second largest, third largest,... cracks (namely, having the second lowest, third lowest, ... I_c-values) also contribute to the synthesis of the V–I curve of the specimen when the interspacing between their V–I curves and the V–I curve of the lowest I_c-section is small (when the difference in size between the second, third,...largest cracks and the largest crack is small) but not when the interspacing is large (difference in crack size is large), as shown below in detail.
• (3)   When the difference in crack size among the sections is very small (ΔL_p,Ex = 0.011) as shown in Fig. 2(a), the V–I curves of sections exist very near to each other. In the voltage range of V = 0.9 to 90 μV in which I_c and n-value of the specimen are determined, the voltages developed at all sections contribute to the voltage of the specimen. Thus, the V–I curve and hence I_c and n-value of specimen are affected by all sections when the difference in crack size among the sections is very small. As a result, the I_c and n-value of the specimen at ΔL_p,Ex = 0.011 in Fig. 3(a) and 3(b) are similar to those in the case of ΔL_p = 0.
• (3)   When the difference in crack size among the sections is very large (ΔL_p,Ex = 0.215) as shown in Fig. 2(f), the V–I curve of the section with the largest crack and hence with the lowest I_c is in the far lower current range than the V–I curves of the other sections. As a result, the voltage in the range of 0.9 to 90 μV of the specimen is developed almost at the lowest I_c-section with the largest crack, and, accordingly, the V–I curve of the specimen is similar to that of the lowest I_c-section. In this situation, the I_c of the specimen determined at V = 9 μV is higher than the I_c of the lowest I_c-section determined at V = 1.5 μV, as seen in Fig. 2(f). Furthermore, the shunting current at the largest crack increases with increasing voltage^3,5,9,19,21), which causes large curvature in the V–I curve of the specimen as seen in Fig. 2(f). Due to the enhanced shunting current at higher voltage, the n-value of the specimen obtained in the range of V = 0.9～90 μV is lower than that of the lowest I_c-section obtained in the range of V = 0.15～15 μV. In this way, when one of the cracks is far larger than the others, the section with the largest crack tends to determine the V–I curve of the specimen, and the I_c and n-value of the specimen are higher and lower than those of the lowest I_c-section, respectively.
• (4)   As shown in Fig. 2(a) to (f), the interspacing among the V–I curves of sections increases with increasing ΔL_p,Ex and the number of sections that contribute to the synthesis of the V–I curve of the specimen decreases. Thus, with increasing ΔL_p,Ex, the feature of the synthesized V–I curve and hence the I_c and n-value of the specimen turn from (2) to (3) stated above.
• (5)   The value of I_c of specimen is within the highest and lowest values of I_c of sections. In more detail, it is within the range of the lowest and average values of I_c of sections for any ΔL_p,Ex (Fig. 3(a)). On the other hand, n-value of the specimen is nearly the same to that of sections when ΔL_p,Ex is small (0.011), but it becomes lower than the lowest n-value among the sections with increasing ΔL_p,Ex. (Fig. 3(b)).
• (6)   The n/n(ΔL_p = 0) of specimen decreases with increasing ΔL_p,Ex more significantly than I_c/I_c(ΔL_p = 0) (Fig. 3(c)); the n-value is more sensitive to the distribution of crack size than I_c-value.

3.2 Existence of plural n-values for one I_c-value in heterogeneously cracked specimen

Two examples, showing the influence of crack size distribution of sections on the n-value of the specimen, are presented in Fig. 4. These examples were taken up from many simulation results as to show that n-value is not uniquely determined by the critical current value; namely, plural n-values can exists for one critical current value in heterogeneously cracked specimens. In the example shown in Fig. 4(a), the V–I curves of sections exist rather densely, corresponding to the case where the difference in crack size among sections is relatively small. On the other hand, in the example in (b), the V–I curves of sections exist rather coarsely, corresponding the case where the difference in crack size among sections is large. The I_c values of the specimen in (a) and (b) were the same (122 A) but n-values in (a) and (b) were different; 18.6 and 13.6, respectively. In this way, even when the specimens have a same I_c-value, plural n-values can exist, depending on the difference in crack size among the sections. This result shows that I_c and n-value are not in one to one relation under the occurrence of heterogeneous cracking of superconducting layer.

Fig. 4

Examples showing that n-value of the specimen is different (n = 18.6 in the example in (a) but 13.6 in the example in (b)), while the critical current is the same (122 A in both examples). In comparison with the example in (b), the example in (a) is characterized by the smaller distance of the V–I curves of sections (namely smaller difference in crack size among sections), and hence characterized by more number of sections that contribute to synthesize the voltage of the specimen, acing to raise n-value of the specimen.

Figure 5 shows an example of the plot of n-value against I_c-value. In this example, the results of the 18 specimens with a length L = 9 cm simulated for ΔL_p = 0.01, 0.025, 0.05, 0.1, 0.15 and 0.2 are presented. The (I_c, n) value for each specimen is shown with open symbols. The closed symbols show the average values (I_c,ave, n_ave) of 18 specimens, for each value of ΔL_p. The tendency that the decrease in n-value with decreasing I_c is clearly found. However, as indicated above, different n-values (I_c-values) can exist for a given I_c-value (n-value). The results in Fig. 5 clearly show that, as the extent of difference in crack size among the sections is different from specimen to specimen, the relation of n-value to I_c in specimens is not uniquely determined. For description of the experimentally measured correlation between I_c and n-value of cracked superconducting tape, an approach to surround the n vs I_c plot by the upper bound and the lower bound²³⁾ may be useful.

Fig. 5

Plot of n-value against critical current value of the specimens with a length L = 9 cm simulated for ΔL_p = 0.01 to 0.2. The (I_c, n) value for each specimen is shown with open symbol. The closed symbols show the average values of 18 specimens for each value of ΔL_p.

3.3 Influences of distribution of crack size and specimen length on distribution and average of critical current I_c and n-value

Figures 6 and 7 show the obtained I_c- and n-values, plotted against the standard deviation of ligament parameter, ΔL_p, for representative specimen length L = (a) 1.5 cm, (b) 3 cm, (c) 6 cm, (d) 9 cm and (e) 15 cm. The results for L = 1.5 cm were taken from the results for the sections with L = L₀ = 1.5 cm. Figures 8 and 9 show the average critical current, I_c,ave and average n-value, n_ave, plotted against ΔL_p and L, respectively. The following features are read from Figs. 6 to 9.

Fig. 6

Simulated critical current (I_c)-values for specimen length L = (a) 1.5 cm, (b) 3.0 cm, (c) 6 cm, (d) 9 cm and (e) 15 cm, plotted against the standard deviation of the ligament parameter ΔL_p.

Fig. 7

Simulated n-values for specimen length L = (a) 1.5 cm, (b) 3.0 cm, (c) 6 cm, (d) 9 cm and (e) 15 cm, plotted against the standard deviation of the ligament parameter ΔL_p.

Fig. 8

Change in (a) average critical current, I_c,ave and (b) average n-value, n_ave, with increasing standard deviation of ligament parameter ΔL_p for L = 1.5～30 cm.

• (1)   In this simulation, the standard deviation of L_p, ΔL_p, was varied (0～0.2) and the average of L_p, L_p,ave, was fixed (0.670) for sections (L₀ = 1.5 cm), as to give the same I_c,ave and n_ave for sections in any value of ΔL_p. In the simulation results, as planned, the larger the ΔL_p, namely the wider the distribution of crack size, the wider became the distributions of both I_c- (Fig. 6(a)) and n-values (Fig. 7(a)) of sections, and the I_c,ave and n_ave of the sections were kept almost constant for any ΔL_p (Fig. 6(a), Fig. 7(a) and the results for L = 1.5 cm in Fig. 8).
• (2)   The I_c and n-value of specimens with L ≥ 3.0 cm were estimated from the V–I curves of specimen synthesized from the V–I curves of sections with eqs. (5) and (6). As well as the results for sections with L = 1.5 cm (Fig. 6(a) and Fig. 7(a)), the widths of distributions of I_c and n-value of specimens for L ≥ 3.0 cm (Figs. 6(b)–(e), Fig. 7(b)–(e)) increased with increasing ΔL_p for any length.
• (3)   Under the condition where the I_c,ave and n_ave of sections (L = 1.5 cm) kept constant for all values of ΔL_p (Fig. 6(a), Fig. 7(a) and the result for L = 1.5 cm in Fig. 8)), the I_c,ave and n_ave of specimens decreased with increasing ΔL_p (Fig. 8) in the whole range of the specimen length investigated (L = 3～30 cm). It is noted that the decrease in I_c,ave and n_ave of specimens (L ≥ 3.0 cm) with increasing ΔL_p was enhanced with increasing specimen length L (Fig. 9).
  

  Fig. 9

  Change in (a) average critical current, I_c,ave and (b) average n-value, n_ave, with increasing specimen length L for standard deviation of ligament parameter ΔL_p = 0.01 to 0.2.

3.4 Influences of specimen length on the decrease in average n-value, n_ave, with decreasing average critical current, I_c,ave

As has been shown in Fig. 5, the relation of n-value to I_c of each specimen was complex, depending on the extent of difference in crack size among sections. However, despite the complexity, the tendency that the n-value decreases with decreasing I_c was clearly detected by plotting the average n-value, n_ave, against average I_c, I_c,ave. In the example in Fig. 5, the specimen length was limited to 9 cm. As a next step, to reveal the influences of specimen length on the change in n_ave with I_c,ave, the values of (I_c,ave, n_ave) were read from the I_c,ave-L and n_ave-L relations in Fig. 9, and n_ave was plotted against I_c,ave for L = 3～30 cm, as shown in Fig. 10.

Fig. 10

Influences of specimen length L on the change in average n-value, n_ave, with decreasing average critical current, I_c,ave in the range of ΔL_p = 0～0.2 and L = 3～30 cm.

When ΔL_p = 0, values of (I_c,ave, n_ave) do not change with L. Both I_c,ave and n_ave decrease with increase in ΔL_p (Figs. 8 and 10) and this phenomenon is enhanced in longer specimens (Fig. 9). It is important that the level of n_ave of n_ave–I_c,ave relation becomes lower with increasing L (Fig. 10), reflecting the feature that n-value is reduced more than the critical current by cracks (Fig. 3(c)). It has been reported that, when the values of (I_c, n) of the coated- and filamentary-tape with stress-induced cracks were measured for short and long specimens, the level of n_ave of the n_ave-I_c,ave correlation diagram of longer specimen was lower than that of shorter specimen^9,14). This experimentally observed feature of the length dependence of the n_ave-I_c,ave relation is reproduced by the present simulation method, as in Fig. 10, from a viewpoint of heterogeneous cracking.

3.5 Variation of critical current value with position along the length, affected by voltage tap distance

It has been shown experimentally that the variations in the local critical current (I_c) along the specimen length depend strongly on the voltage tap distance; larger tap distance can give the appearance of higher uniformity for I_c than is actually present²²⁾. This phenomenon was reproduced well by the present approach as follows. The section- and specimen- length L (1.5 to 30 cm) used in the simulation study was regarded as the voltage tap distance. Taking the total length of a specimen as 180 cm, and arranging the simulation results of I_c-values to position order along the specimen length, we had Fig. 11, in which the variation of I_c with position along the length for voltage tap distance L = (a) 1.5, (b) 3, (c) 6, (d) 15 and (e) 30 cm under a condition of ΔL_p = 0.1 were representatively presented. The I_c,ave and ΔI_c for each L refer to the average and standard deviation of I_c-values, respectively. In the case where the I_c values were estimated in a step of 1.5 cm, the ΔI_c was high as 21.0 A. When the I_c values were estimated in step of 3, 6, 15 and 30 cm, ΔI_c decreased to 17.2, 16.0, 10.3 and 5.73 A, respectively. The reduction in ΔI_c with increasing voltage tap distance observed in the present work reproduces well the reported important warning that local information of critical current values in a specimen is diluted when the tap distance is large²²⁾. Average and standard deviation of I_c values tend to decrease with increasing L under the distributed crack size, due to the higher probability to contain larger crack in longer specimens (larger voltage tap distances).

Fig. 11

Variation of critical current I_c with position in 180 cm-specimen, obtained for the voltage tap distance of L = (a) 1.5 cm, (b) 3 cm, (c) 6 cm, (d) 15 cm and (e) 30 cm, simulated under the condition of L_p,ave = 0.67 and ΔL_p = 0.1. The I_c,ave and ΔI_c refer to the average and standard deviation of I_c-values. The lower bound of I_c for each L, corresponding to the lowest I_c among 1.5 cm-sections contained in the voltage tap distance L, is shown with rectangles, for reference.

As has been shown in Fig. 2 and Fig. 3(a), the largest crack-section (lowest I_c section) plays a dominant role in determination of I_c of specimens. If all sections have the same crack size as the largest one, the V of the V–I curve of specimen increases most rapidly. Accordingly, the lower bound of the critical current of the specimen is given by this situation where the I_c value of the specimen is the same as that of the lowest I_c section. Thus the lower bound of I_c of specimen is given by the lowest I_c value among the sections. In the cases of 3, 6, 15 and 30 cm- voltage tap distances, the lower bounds of I_c are given by the lowest I_c value among 2, 4, 10 and 20 sections existing between the taps. The rectangle shows the lower bound. The shape of the simulated variation of I_c-value with position is well described qualitatively, confirming that the lowest I_c section with the largest crack plays a significant role in determination of I_c of the specimen as an assemble of sections.

4. Conclusions

Influences of crack size-distribution and specimen length on critical current and n-value of heterogeneously cracked superconducting tape were studied by means of a Monte Carlo simulation in combination with a model of crack-induced current shunting. In simulation, model specimen that is constituted of a series of sections having different size-crack to each other was used. The crack size-distribution was monitored by the ligament parameter-distribution. The standard deviation of ligament parameter was varied from 0.01, corresponding to very narrow crack size-distribution, to 0.2, corresponding to very wide one. Specimen length was varied in the range of 1.5～30 cm. Main results are summarized as follows.

• (1)   The critical current value of specimen is within the range of the lowest and average values of critical current of sections for any width of the distribution of ligament parameter, namely for any width of the distribution of crack size. On the other hand, n-value of the specimen is nearly the same to that of sections when width of the crack size-distribution is small, but it becomes lower than the lowest n-value among the sections with increasing width of crack size-distribution.
• (2)   The widths of distribution of critical current and n-value increase with increasing width of crack size distribution for any specimen length.
• (3)   The average values of critical current and n-value decrease with increasing width of crack size distribution. This feature is enhanced more for longer specimens.
• (4)   n-value decreases with increasing width of crack size distribution and with increasing specimen length more sensitively than critical current.
• (5)   n-value is not uniquely determined by the critical current value; namely, plural n-values can exist for one critical current value in heterogeneously cracked specimens.
• (6)   The width of critical current distribution decreases with increasing specimen length. This result is in good accordance with the experimental results showing that local information of critical current values in a specimen is diluted when the voltage tap distance is large.

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