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Ag tapes
Physica C 470 (2010) 257–261
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Influence of tape-specific properties on local stress and strain in Bi-2223/Ag tapes M. Ahoranta a, J. Lehtonen b, T. Tarhasaari a,* a b
Electromagnetics, Tampere University of Technology, P.O. Box 692, Tampere 33101, Finland AER Consulting Oy, P.O. Box 31, Vammala 38201, Finland
a r t i c l e
i n f o
Article history: Received 23 September 2008 Received in revised form 10 September 2009 Accepted 5 October 2009 Available online 8 October 2009 Keywords: Mechanical modelling Mechanical properties Bi-2223/Ag tapes
a b s t r a c t Superconductors are typically fragile, and their superconducting properties are reduced under strain. Moreover, superconductor wires operate under large mechanical loads in most applications. Thus, a careful mechanical analysis is essential when designing superconducting applications. So far, plenty of experimental and computational studies concerning stress and strain in, e.g. Bi-2223/Ag tapes have been carried out. However, Bi-2223 tapes contain several properties that vary between different tapes and are often unknown when designing a specific application. The largest uncertainties are related to the creep at high temperatures, stiffness and anisotropy of filaments and the yield strength of silver. In this paper, the influence of uncertainties on the local stress and strain in Bi-2223/Ag tapes is studied computationally to determine efficient strategies to reduce inaccuracy of the results. The results suggest that the filaments can be described as an isotropic material and the filament stiffness is the most important input parameter to estimate the total stiffness of the tape. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction In applications, superconducting wires are typically under high mechanical loading [1]. Therefore, it is essential to know the mechanical behaviour of the wire. Predicting stress distribution is important when estimating the load tolerance of wires because the electric performance is very sensitive to fracture of superconducting filaments [2]. Unfortunately, ambivalent information on the properties of superconductor wires interferes with their mechanical modelling. In silver sheathed Bi-2223 tapes, the largest uncertainty prevails on the anisotropy and stiffness of filaments, and the yield strength of silver matrix. These properties depend on the details of the manufacture process, and finding them for a particular tape would require a detailed analysis that is rarely available. In addition, stress relaxation may occur with time due to creep at high temperatures [3]. The creep rate depends highly on the details of material structure. Thus, the creep deformation during cooling after the heat treatment is poorly known for the tapes. This paper aims at clarifying the influence of the tape properties on local stresses in a Bi-2223 tape. For modelling purposes, the results give an idea about how large and what sort of errors might be produced if the tape properties are not well known. On the other hand, the parameters whose experimental study would be of pri-
* Corresponding author. Tel.: +358 3115 2008; fax: +358 3115 2160. E-mail address: timo.tarhasaari@tut.fi (T. Tarhasaari). 0921-4534/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.10.003
mary importance in improving the accuracy of mechanical analyses are pointed out. The study is performed computationally with a continuum mechanics model solved with the finite element method (FEM). The model allows taking into account the filament geometry of the tape and the multi-axial stress state in materials.
2. Model The loading situations studied in this paper are the thermal stresses generated in cooling after heat treatment, and axial tension and transversal compression at 77 K. In the models, the tape was assumed to be stress free at a typical heat treatment temperature 1103 K. The thermal stresses calculated at 77 K were used as an initial condition for the axial tension and transversal compression models. Axial tension was modelled up to 0.2% elongation of the tape from its length after cooling. In Bi-2223/Ag tapes, such elongation usually degrades critical current intolerably. Transversal compression was modelled up to 150 MPa, which is notably over the typical benchmark level for tapes in magnets 100 MPa [4]. In the investigation, the schematic tape geometry shown in Fig. 1 was used. It had dimensions 4 mm 0.2 mm and pure silver matrix. The filaments formed 30% of the cross section area. The mechanical models and FEM solutions for thermal stresses, axial tension and transversal compression are presented in [5]. A generalized plane strain condition was assumed in all load cases, where the constant axial strain in the tape was either the driving parameter of the problem or an unknown to be solved with the
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Fig. 1. Schematic tape geometry.
problem. In thermal stress model the tape perimeter was free from traction forces and the axial strain unknown. It could be solved by requiring zero net axial force in the tape. In axial tension model the perimeter was again free from forces. The axial strain was now a pre-determined driving parameter of the problem. Transversal compression was modelled with constraints which describe an idealized situation where the tape wide surfaces are pushed towards each other with a rigid object and the friction between compressing object and the tape is infinite. The displacement in compressed direction was constrained to pre-determined value on the wide edges of the tape. The perpendicular displacement there and the axial strain in the whole cross section were also forced zero. The short edges of the cross section were free from forces. The analysis was performed with a self-made code [6] implemented in Matlab environment [7]. Triangular mesh used was generated with GMSH software [8]. The solution was performed by increasing the loading in steps. U-p formulation [9] was used where the variables of the problem consisted of nodal displacements and hydrostatic stresses. Displacements in elements were approximated with quadratic interpolation polynomials whereas the hydrostatic stresses were interpolated with first order polynomials. The filament material was described with the elastic material model and silver with the elasto-plastic material model with linear hardening. In the models, the Young’s modulus of all materials and the yield strength of silver varied with temperature, but the Poisson’s ratio and tangential modulus of silver during plastic deformation were constants. The material parameters and thermal expansion data for silver are reported in [5], and the parameters used for the filaments are clarified more in the next section. In the assessment of the results, most attention was paid to the overall stiffness of the tape and to the fracture probability of the filaments. To study the overall axial stiffness, the axial force needed for elongating the tape in axial tension was examined. Transversal stiffness was derived from the relationship between the transversal force and tape thickness reduction in transversal compression. To remove the influence of tape dimensions, the results are reported as the average axial stress versus applied axial strain and the average transversal compression versus transversal strain curves. The fracture probability was studied with the maximum-principal-stress theory that typically describes well the behaviour of brittle materials [10]. According to it, a material fractures when in some point the first principal stress, rI, exceeds the fracture strength of the material in tension or the third principal stress, rIII, becomes lower than the fracture strength in compression. The first and the third principal stresses are the most tensile and the most compressive normal stresses that act on any surface in a point. Measured tensile fracture strength of filaments has ranged from 10 to 115 MPa [11,12], and it also varies within one filament as discussed in [5]. Measured fracture strengths of Bi-2223 in compression were not found. Typically in brittle materials, the strength in compression is many times larger than the tensile strength, and therefore, the possibility of compressive fracture was here inspected only in transversal compression. The maximum-principal-stress theory is simple, and was chosen because the data on the fracture behaviour of Bi-2223 is scarce. However, the theory ignores possible anisotropy of a material and does not take into ac-
count the stresses in other directions than the principal stress in question.
3. The tape-specific parameters The degree of the filament anisotropy depends on the degree of the grain alignment. Its impact on stresses was investigated by comparing the results of two models where the filaments were strongly anisotropic in one and totally isotropic in the other. The anisotropic properties connected to the width and longitudinal directions of tape were considered to be the same and to differ only in the thin direction. Modelling the anisotropy affected the elastic stiffness matrix and the thermal contraction of the filaments. The anisotropic stiffness matrix was formed based on the matrix measured with a dense, textured, bulk bar [13]. The matrix was multiplied with a first order polynomial of temperature to model the temperature dependence. The polynomial was chosen to produce a desired axial Young’s modulus, Ef, at room temperature, and its slope was determined based on the measurements of the first element of the stiffness matrix at 293 and 1033 K [13]. In the isotropic model, the Young’s modulus was in every direction equal to the axial stiffness in the anisotropic model. The thermal expansion data for anisotropic Bi-2223 shown in Fig. 2 was collected from [11,14,15]. In the isotropic model, the thermal expansion in every direction corresponded to the expansion in the ab plane in the anisotropic model. The filament stiffness decreases with increasing porosity of the filament material. Although the manufacturers aim at as low
Fig. 2. Thermal expansion of Bi-2223 (full line) in different directions and of silver (dashed line).
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porosity as possible, the porosity, and thus the filament Young’s modulus, have been observed to vary in large range. At room temperature, moduli have been measured between 93 GPa and 121 GPa [11,12], and values as low as 40 GPa have been estimated from the tape overall stiffness [16]. To study the impact of filament stiffness, several Ef (293 K) values were tested in the material model. The silver matrix typically yields already during cooling [5], but the yield strength depends on the material purity and the details of the heat treatment and varies between different tapes. At room temperature, values between 11 and 25 MPa have been reported for silver that has been heat treated with similar procedures than the Bi-2223 tapes [16–19]. The impact of the yield strength was studied with the yield strength, YAg, at room temperature as a free parameter. Otherwise, the temperature dependence was a first order polynomial that decreased to zero at the melting point of silver. Creep deformation is notable in metallic materials at temperatures over one third of the melting point, which means approximately 370 K for silver. However, the creep rate is highly dependent on factors such as impurities and grain size, and is therefore, often poorly known. Creep deformation was modelled with the method presented in [5,6]. It was assumed to remove all stresses from the tape over some limit temperature, TL, but at lower temperatures it was ignored. The impact of creep was studied by modelling the tape with several different TL. In the model, the degree of creep deformation enhanced when TL was lowered. 4. Results and discussion The outlines of the stress distributions resembled those reported in [5]. After cooling, filaments were mostly under compression and the matrix under tension. The axial stress was compressive everywhere in the filaments, but the cross sectional stresses were locally slightly tensional. When axial tension was applied after cooling, compression in filaments turned gradually into tension. Under small applied tension, the largest tensile normal stresses were the cross sectional, but at the applied tension about 0.35–0.55%, the axial stress exceeded them. The transversal compression model prohibited the axial deformation, and the whole central part of tape was under hydrostatic compression. At the thin edge of the tape, the silver bulged outwards and became locally under tension in the axial and tape width directions. In filaments closest to the edge, tensional stresses appeared as the applied compression exceeded about 100 MPa. 4.1. Anisotropy The impact of filament anisotropy was studied with Ef (293 K) = 100 GPa, TL = 700 K and YAg (293 K) = 11 MPa. According to the results, the anisotropy affected neither the overall tape stiffness nor the first principal stress in filaments significantly. Probably due to the thin filament shape, the two models even predicted practically the same overall transverse stiffness. Nevertheless, anisotropy had some impact on horizontal normal stress, where local differences of 20–60 MPa appeared between the anisotropic and isotropic models. The difference in the first principal stress between the anisotropic and isotropic models is clarified in Table 1. After cooling, the relative difference between material models was large, but rI was so close to zero that it would hardly cause fracture. In external loading, the difference between the two models was small. In most cases, the anisotropic model predicted slightly larger rI. As the influence of the filament anisotropy was so small, the isotropic model was used in the rest of the tests. The influence of
Table 1 Key figures of first principal stress distribution in filaments for anisotropic and isotropic filament models after cooling, axial tension 0.2% and transversal compression 100 MPa. Loading case
Cooling Axial tension Compression
Median of rI (MPa)
Maximum of rI (MPa)
Anisotropic
Isotropic
Anisotropic
Isotropic
0.17 168 59.7
0.37 171 59.5
2.6 170 11.6
2.7 175 12.0
filament Young’s modulus was inspected with Ef (293 K) ranging from 40 GPa to 130 GPa. Creep deformation was tested with TL in the range 370–1103 K. The impact of silver yield strengths was studied with YAg (293 K) = 11 MPa and YAg (293 K) = 25 MPa. When studying the influence of one of the parameters, the others were fixed to intermediate values. These default values were Ef (293 K) = 100 GPa, TL = 700 K and YAg (293 K) = 11 MPa. 4.2. Filament stiffness As shown in Fig. 3, the filament stiffness had large impact on the axial and transversal stiffness of the tape. In fact, compared to the filament stiffness, the other studied parameters had insignificant influence on the overall tape stiffness. The impact of the filament stiffness on the local stress state in filaments is shown in Fig. 4. Stiffer filaments led to less compressive axial thermal strain, but the axial thermal stress of filaments was almost independent from Ef. In axial tension and transversal compression, increasing filament stiffness increased the rate in which the magnitude of filament stresses grew. This was natural, since both of the models were strain driven. It is important to note that stiffer filaments often have larger fracture strength. Then, it is likely that a modeller overestimates or underestimates both the stiffness and the fracture strength simultaneously, and the total error in the fracture prediction is not as dramatic as Fig. 4 suggests. 4.3. Silver yield strength and creep deformation Creep deformation and the silver yield strength influenced the thermal pre-compression of filaments and, thereby, affected the point where the axial stress turns tensile under external loading. Creep deformation removed thermal stresses and lowered filament compression after cooling. If the yield strength was increased, elastic deformation in silver became larger, its thermal contraction became more dominative, and the thermal compression of filaments increased. In axial tension, the creep deformation and the silver yield strength influenced rI in filaments practically only by shifting the point where the linear growth begins. Changes in YAg and TL did not change the slope of the rI curve during the linear growth. For example, when YAg (293 K) = 11 MPa and Ef (293 K) = 100 GPa, the linear growth began at around 0.05% applied tension without creep deformation, whereas with TL = 370 K it started at 0.03% tension. Increasing the yield strength of silver from 11 MPa to 25 MPa, when TL = 700 K and Ef (293 K) = 100 GPa, changed the turning point from 0.04% to 0.07% applied strain. In transversal compression, the influence of TL and YAg on filament stresses was slightly more complex as shown in Fig. 5. As already mentioned, silver yield strength and creep influenced the tape overall stiffness much less than the filament stiffness, whose impact was seen in Fig. 2. In the results obtained with different YAg (293 K) and TL using Ef (293 K) = 100 GPa, the average axial stress at axial tension of 0.2% varied within 3 MPa and the average transversal compression at 0.1% within 8 MPa.
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Fig. 3. (a) Average axial stress vs. axial strain in axial tension test and (b) average transversal compression vs. transversal strain calculated with Ef (293 K) = 40 GPa (s), Ef (293 K) = 70 GPa (h), Ef (293 K) = 100 GPa (e) and Ef (293 K) = 130 GPa (4).
Fig. 4. (a) Growth of first principal stress in filaments in axial tension and (b) growth of the first (black symbols) and the third (white symbols) principal stress in filaments in transversal compression with Ef (293 K) = 40 GPa (s), Ef (293 K) = 70 GPa (h), Ef (293 K) = 100 GPa (e) and Ef (293 K) = 130 GPa (4). Full lines correspond to median and the dashed lines to most tensile rI or to most compressive rIII in the filament area.
4.4. Suggestions for improving the modelling accuracy As a whole, the results showed that the reliability of predictions on overall stiffness depends mainly on the accuracy of the filament stiffness. However, accurate estimates on the stress state of the filaments require also good knowledge on the degree of the creep deformation and the yield strength of silver. Performing a few basic experiments with a short sample of the tape would increase markedly the reliability of modelling the tape under other types of loading. Due to the strong correlation of the tape and filament stiffness, the filament Young’s modulus can be deduced from the tape stiffness if such has been measured. Finding
out the yield strength of silver or the degree of creep deformation with such a simple test is not self-evident because the two parameters have very similar impact on the stresses. The silver yield strength could be clarified with analyses of the purity and grain size of the matrix metal and measurements of the yield strength of different silver grades. Modelling of creep deformation could be improved if systematic comparisons were performed between measured critical currents and modelled stresses with a large set of tapes to find relevant TL values.
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2223/Ag tapes was studied with finite element models. The investigated loading situations concerned the thermal stresses, axial tension and transversal compression. The results showed that the anisotropy of filaments had only a small impact on the tape overall stiffness or the fracture probability of the filaments. Thereby, knowledge of filament alignment is not necessarily needed to estimate the tape mechanical behaviour. The filament stiffness influenced significantly both axial and transversal stiffness of the tape, but the degree of creep deformation or the silver yield strength did not affect them markedly. However, the filament stiffness, silver yield strength and creep all influenced significantly the filament stresses. The filament stiffness determined mostly the growth rate of stresses under axial or transversal loading. The degree of creep deformation or the silver yield strength affected mostly the pre-compression of the filaments after cooling. Because of their direct connection, measured data on tape stiffness, if available, can be used to extract the filament stiffness for further modelling. To improve the reliability of predicting the filaments stresses, more experimental analysis would be needed on the yield strength of the silver matrix. In addition the creep deformation should be studied with systematic comparisons between experiments and models. References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] Fig. 5. First (black symbols) and third (white symbols) principal stresses in filaments in transversal compression with (a) TL = 370 K (s) and TL = 1103 K (4) and (b) YAg (293 K) = 11 MPa (o) and YAg (293 K) = 25 MPa (4). Full lines show median and dashed lines most tensile rI and most compressive rIII in the filament area.
5. Conclusions The impact of anisotropy and stiffness of filaments and yield strength and creep deformation of silver on local stresses in Bi-
H.W. Weijers, J. Schwartz, B. ten Haken, Physica C 372–376 (2002) 1364. M. Sugano, K. Osamura, Physica C 402 (2004) 341. J.T. Boyle, J. Spence, Stress Analysis for Creep, Butterworths, England, 1983. J.W. Ekin, S.L. Bray, N. Cheggour, C.C. Clickner, S.R. Foltyn, P.N. Arendt, A.A. Polyanskii, D.C. Larbalestier, C.N. McCowan, IEEE Trans. Appl. Supercond. 11 (2001) 3389. M. Ahoranta, J. Lehtonen, T. Tarhasaari, Supercond. Sci. Technol. 22 (2009) 015012. M. Ahoranta, J. Lehtonen, T. Tarhasaari, K. Weiss, Supercond. Sci. Technol. 21 (2008) 025005. The Mathworks, Inc., 3 Apple Hill Drive, Natick, MA. Gmsh: a three-dimensional finite element mesh generator with built-in preand post-processing facilities.. R. Passerini, M. Dhallé, E. Giannini, G. Wittz, B. Seeber, R. Flükiger, Physica C 371 (2002) 173. M. Sugano, K. Osamura, M. Hojo, Supercond. Sci. Technol. 16 (2003) 571. K.C. Goretta, D.S. Kupperman, S. Majumdar, M.W. Such, N. Murayama, Supercond. Sci. Technol. 11 (1998) 1409. E. Giannini, E. Bellingeri, R. Passerini, F. Marti, R. Flükiger, Adv. Supercond. XII (2000) 637. N. Yamada, K. Nara, M. Okaji, T. Hikata, T. Kaneko, N. Sadakata, Cryogenics 38 (1998) 397. M. Ahoranta, P. Bukva, P. Kovácˇ, R. Mikkonen, T. Tarhasaari, Physica C 432 (2005) 239. M. Sugano, K. Osamura, A. Nyilas, Supercond. Sci. Technol. 16 (2003) 1064. A. Nyilas, K. Osamura, M. Sugano, Supercond. Sci. Technol. 16 (2003) 1036. M. Hojo, M. Nakamura, T. Matsuoka, M. Tanaka, S. Ochiai, M. Sugano, K. Osamura, Supercond. Sci. Technol. 16 (2003) 1043.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Influence of tape-specific properties on local stress and strain in Bi-2223/Ag tapes M. Ahoranta a, J. Lehtonen b, T. Tarhasaari a,* a b
Electromagnetics, Tampere University of Technology, P.O. Box 692, Tampere 33101, Finland AER Consulting Oy, P.O. Box 31, Vammala 38201, Finland
a r t i c l e
i n f o
Article history: Received 23 September 2008 Received in revised form 10 September 2009 Accepted 5 October 2009 Available online 8 October 2009 Keywords: Mechanical modelling Mechanical properties Bi-2223/Ag tapes
a b s t r a c t Superconductors are typically fragile, and their superconducting properties are reduced under strain. Moreover, superconductor wires operate under large mechanical loads in most applications. Thus, a careful mechanical analysis is essential when designing superconducting applications. So far, plenty of experimental and computational studies concerning stress and strain in, e.g. Bi-2223/Ag tapes have been carried out. However, Bi-2223 tapes contain several properties that vary between different tapes and are often unknown when designing a specific application. The largest uncertainties are related to the creep at high temperatures, stiffness and anisotropy of filaments and the yield strength of silver. In this paper, the influence of uncertainties on the local stress and strain in Bi-2223/Ag tapes is studied computationally to determine efficient strategies to reduce inaccuracy of the results. The results suggest that the filaments can be described as an isotropic material and the filament stiffness is the most important input parameter to estimate the total stiffness of the tape. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction In applications, superconducting wires are typically under high mechanical loading [1]. Therefore, it is essential to know the mechanical behaviour of the wire. Predicting stress distribution is important when estimating the load tolerance of wires because the electric performance is very sensitive to fracture of superconducting filaments [2]. Unfortunately, ambivalent information on the properties of superconductor wires interferes with their mechanical modelling. In silver sheathed Bi-2223 tapes, the largest uncertainty prevails on the anisotropy and stiffness of filaments, and the yield strength of silver matrix. These properties depend on the details of the manufacture process, and finding them for a particular tape would require a detailed analysis that is rarely available. In addition, stress relaxation may occur with time due to creep at high temperatures [3]. The creep rate depends highly on the details of material structure. Thus, the creep deformation during cooling after the heat treatment is poorly known for the tapes. This paper aims at clarifying the influence of the tape properties on local stresses in a Bi-2223 tape. For modelling purposes, the results give an idea about how large and what sort of errors might be produced if the tape properties are not well known. On the other hand, the parameters whose experimental study would be of pri-
* Corresponding author. Tel.: +358 3115 2008; fax: +358 3115 2160. E-mail address: timo.tarhasaari@tut.fi (T. Tarhasaari). 0921-4534/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.10.003
mary importance in improving the accuracy of mechanical analyses are pointed out. The study is performed computationally with a continuum mechanics model solved with the finite element method (FEM). The model allows taking into account the filament geometry of the tape and the multi-axial stress state in materials.
2. Model The loading situations studied in this paper are the thermal stresses generated in cooling after heat treatment, and axial tension and transversal compression at 77 K. In the models, the tape was assumed to be stress free at a typical heat treatment temperature 1103 K. The thermal stresses calculated at 77 K were used as an initial condition for the axial tension and transversal compression models. Axial tension was modelled up to 0.2% elongation of the tape from its length after cooling. In Bi-2223/Ag tapes, such elongation usually degrades critical current intolerably. Transversal compression was modelled up to 150 MPa, which is notably over the typical benchmark level for tapes in magnets 100 MPa [4]. In the investigation, the schematic tape geometry shown in Fig. 1 was used. It had dimensions 4 mm 0.2 mm and pure silver matrix. The filaments formed 30% of the cross section area. The mechanical models and FEM solutions for thermal stresses, axial tension and transversal compression are presented in [5]. A generalized plane strain condition was assumed in all load cases, where the constant axial strain in the tape was either the driving parameter of the problem or an unknown to be solved with the
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Fig. 1. Schematic tape geometry.
problem. In thermal stress model the tape perimeter was free from traction forces and the axial strain unknown. It could be solved by requiring zero net axial force in the tape. In axial tension model the perimeter was again free from forces. The axial strain was now a pre-determined driving parameter of the problem. Transversal compression was modelled with constraints which describe an idealized situation where the tape wide surfaces are pushed towards each other with a rigid object and the friction between compressing object and the tape is infinite. The displacement in compressed direction was constrained to pre-determined value on the wide edges of the tape. The perpendicular displacement there and the axial strain in the whole cross section were also forced zero. The short edges of the cross section were free from forces. The analysis was performed with a self-made code [6] implemented in Matlab environment [7]. Triangular mesh used was generated with GMSH software [8]. The solution was performed by increasing the loading in steps. U-p formulation [9] was used where the variables of the problem consisted of nodal displacements and hydrostatic stresses. Displacements in elements were approximated with quadratic interpolation polynomials whereas the hydrostatic stresses were interpolated with first order polynomials. The filament material was described with the elastic material model and silver with the elasto-plastic material model with linear hardening. In the models, the Young’s modulus of all materials and the yield strength of silver varied with temperature, but the Poisson’s ratio and tangential modulus of silver during plastic deformation were constants. The material parameters and thermal expansion data for silver are reported in [5], and the parameters used for the filaments are clarified more in the next section. In the assessment of the results, most attention was paid to the overall stiffness of the tape and to the fracture probability of the filaments. To study the overall axial stiffness, the axial force needed for elongating the tape in axial tension was examined. Transversal stiffness was derived from the relationship between the transversal force and tape thickness reduction in transversal compression. To remove the influence of tape dimensions, the results are reported as the average axial stress versus applied axial strain and the average transversal compression versus transversal strain curves. The fracture probability was studied with the maximum-principal-stress theory that typically describes well the behaviour of brittle materials [10]. According to it, a material fractures when in some point the first principal stress, rI, exceeds the fracture strength of the material in tension or the third principal stress, rIII, becomes lower than the fracture strength in compression. The first and the third principal stresses are the most tensile and the most compressive normal stresses that act on any surface in a point. Measured tensile fracture strength of filaments has ranged from 10 to 115 MPa [11,12], and it also varies within one filament as discussed in [5]. Measured fracture strengths of Bi-2223 in compression were not found. Typically in brittle materials, the strength in compression is many times larger than the tensile strength, and therefore, the possibility of compressive fracture was here inspected only in transversal compression. The maximum-principal-stress theory is simple, and was chosen because the data on the fracture behaviour of Bi-2223 is scarce. However, the theory ignores possible anisotropy of a material and does not take into ac-
count the stresses in other directions than the principal stress in question.
3. The tape-specific parameters The degree of the filament anisotropy depends on the degree of the grain alignment. Its impact on stresses was investigated by comparing the results of two models where the filaments were strongly anisotropic in one and totally isotropic in the other. The anisotropic properties connected to the width and longitudinal directions of tape were considered to be the same and to differ only in the thin direction. Modelling the anisotropy affected the elastic stiffness matrix and the thermal contraction of the filaments. The anisotropic stiffness matrix was formed based on the matrix measured with a dense, textured, bulk bar [13]. The matrix was multiplied with a first order polynomial of temperature to model the temperature dependence. The polynomial was chosen to produce a desired axial Young’s modulus, Ef, at room temperature, and its slope was determined based on the measurements of the first element of the stiffness matrix at 293 and 1033 K [13]. In the isotropic model, the Young’s modulus was in every direction equal to the axial stiffness in the anisotropic model. The thermal expansion data for anisotropic Bi-2223 shown in Fig. 2 was collected from [11,14,15]. In the isotropic model, the thermal expansion in every direction corresponded to the expansion in the ab plane in the anisotropic model. The filament stiffness decreases with increasing porosity of the filament material. Although the manufacturers aim at as low
Fig. 2. Thermal expansion of Bi-2223 (full line) in different directions and of silver (dashed line).
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porosity as possible, the porosity, and thus the filament Young’s modulus, have been observed to vary in large range. At room temperature, moduli have been measured between 93 GPa and 121 GPa [11,12], and values as low as 40 GPa have been estimated from the tape overall stiffness [16]. To study the impact of filament stiffness, several Ef (293 K) values were tested in the material model. The silver matrix typically yields already during cooling [5], but the yield strength depends on the material purity and the details of the heat treatment and varies between different tapes. At room temperature, values between 11 and 25 MPa have been reported for silver that has been heat treated with similar procedures than the Bi-2223 tapes [16–19]. The impact of the yield strength was studied with the yield strength, YAg, at room temperature as a free parameter. Otherwise, the temperature dependence was a first order polynomial that decreased to zero at the melting point of silver. Creep deformation is notable in metallic materials at temperatures over one third of the melting point, which means approximately 370 K for silver. However, the creep rate is highly dependent on factors such as impurities and grain size, and is therefore, often poorly known. Creep deformation was modelled with the method presented in [5,6]. It was assumed to remove all stresses from the tape over some limit temperature, TL, but at lower temperatures it was ignored. The impact of creep was studied by modelling the tape with several different TL. In the model, the degree of creep deformation enhanced when TL was lowered. 4. Results and discussion The outlines of the stress distributions resembled those reported in [5]. After cooling, filaments were mostly under compression and the matrix under tension. The axial stress was compressive everywhere in the filaments, but the cross sectional stresses were locally slightly tensional. When axial tension was applied after cooling, compression in filaments turned gradually into tension. Under small applied tension, the largest tensile normal stresses were the cross sectional, but at the applied tension about 0.35–0.55%, the axial stress exceeded them. The transversal compression model prohibited the axial deformation, and the whole central part of tape was under hydrostatic compression. At the thin edge of the tape, the silver bulged outwards and became locally under tension in the axial and tape width directions. In filaments closest to the edge, tensional stresses appeared as the applied compression exceeded about 100 MPa. 4.1. Anisotropy The impact of filament anisotropy was studied with Ef (293 K) = 100 GPa, TL = 700 K and YAg (293 K) = 11 MPa. According to the results, the anisotropy affected neither the overall tape stiffness nor the first principal stress in filaments significantly. Probably due to the thin filament shape, the two models even predicted practically the same overall transverse stiffness. Nevertheless, anisotropy had some impact on horizontal normal stress, where local differences of 20–60 MPa appeared between the anisotropic and isotropic models. The difference in the first principal stress between the anisotropic and isotropic models is clarified in Table 1. After cooling, the relative difference between material models was large, but rI was so close to zero that it would hardly cause fracture. In external loading, the difference between the two models was small. In most cases, the anisotropic model predicted slightly larger rI. As the influence of the filament anisotropy was so small, the isotropic model was used in the rest of the tests. The influence of
Table 1 Key figures of first principal stress distribution in filaments for anisotropic and isotropic filament models after cooling, axial tension 0.2% and transversal compression 100 MPa. Loading case
Cooling Axial tension Compression
Median of rI (MPa)
Maximum of rI (MPa)
Anisotropic
Isotropic
Anisotropic
Isotropic
0.17 168 59.7
0.37 171 59.5
2.6 170 11.6
2.7 175 12.0
filament Young’s modulus was inspected with Ef (293 K) ranging from 40 GPa to 130 GPa. Creep deformation was tested with TL in the range 370–1103 K. The impact of silver yield strengths was studied with YAg (293 K) = 11 MPa and YAg (293 K) = 25 MPa. When studying the influence of one of the parameters, the others were fixed to intermediate values. These default values were Ef (293 K) = 100 GPa, TL = 700 K and YAg (293 K) = 11 MPa. 4.2. Filament stiffness As shown in Fig. 3, the filament stiffness had large impact on the axial and transversal stiffness of the tape. In fact, compared to the filament stiffness, the other studied parameters had insignificant influence on the overall tape stiffness. The impact of the filament stiffness on the local stress state in filaments is shown in Fig. 4. Stiffer filaments led to less compressive axial thermal strain, but the axial thermal stress of filaments was almost independent from Ef. In axial tension and transversal compression, increasing filament stiffness increased the rate in which the magnitude of filament stresses grew. This was natural, since both of the models were strain driven. It is important to note that stiffer filaments often have larger fracture strength. Then, it is likely that a modeller overestimates or underestimates both the stiffness and the fracture strength simultaneously, and the total error in the fracture prediction is not as dramatic as Fig. 4 suggests. 4.3. Silver yield strength and creep deformation Creep deformation and the silver yield strength influenced the thermal pre-compression of filaments and, thereby, affected the point where the axial stress turns tensile under external loading. Creep deformation removed thermal stresses and lowered filament compression after cooling. If the yield strength was increased, elastic deformation in silver became larger, its thermal contraction became more dominative, and the thermal compression of filaments increased. In axial tension, the creep deformation and the silver yield strength influenced rI in filaments practically only by shifting the point where the linear growth begins. Changes in YAg and TL did not change the slope of the rI curve during the linear growth. For example, when YAg (293 K) = 11 MPa and Ef (293 K) = 100 GPa, the linear growth began at around 0.05% applied tension without creep deformation, whereas with TL = 370 K it started at 0.03% tension. Increasing the yield strength of silver from 11 MPa to 25 MPa, when TL = 700 K and Ef (293 K) = 100 GPa, changed the turning point from 0.04% to 0.07% applied strain. In transversal compression, the influence of TL and YAg on filament stresses was slightly more complex as shown in Fig. 5. As already mentioned, silver yield strength and creep influenced the tape overall stiffness much less than the filament stiffness, whose impact was seen in Fig. 2. In the results obtained with different YAg (293 K) and TL using Ef (293 K) = 100 GPa, the average axial stress at axial tension of 0.2% varied within 3 MPa and the average transversal compression at 0.1% within 8 MPa.
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Fig. 3. (a) Average axial stress vs. axial strain in axial tension test and (b) average transversal compression vs. transversal strain calculated with Ef (293 K) = 40 GPa (s), Ef (293 K) = 70 GPa (h), Ef (293 K) = 100 GPa (e) and Ef (293 K) = 130 GPa (4).
Fig. 4. (a) Growth of first principal stress in filaments in axial tension and (b) growth of the first (black symbols) and the third (white symbols) principal stress in filaments in transversal compression with Ef (293 K) = 40 GPa (s), Ef (293 K) = 70 GPa (h), Ef (293 K) = 100 GPa (e) and Ef (293 K) = 130 GPa (4). Full lines correspond to median and the dashed lines to most tensile rI or to most compressive rIII in the filament area.
4.4. Suggestions for improving the modelling accuracy As a whole, the results showed that the reliability of predictions on overall stiffness depends mainly on the accuracy of the filament stiffness. However, accurate estimates on the stress state of the filaments require also good knowledge on the degree of the creep deformation and the yield strength of silver. Performing a few basic experiments with a short sample of the tape would increase markedly the reliability of modelling the tape under other types of loading. Due to the strong correlation of the tape and filament stiffness, the filament Young’s modulus can be deduced from the tape stiffness if such has been measured. Finding
out the yield strength of silver or the degree of creep deformation with such a simple test is not self-evident because the two parameters have very similar impact on the stresses. The silver yield strength could be clarified with analyses of the purity and grain size of the matrix metal and measurements of the yield strength of different silver grades. Modelling of creep deformation could be improved if systematic comparisons were performed between measured critical currents and modelled stresses with a large set of tapes to find relevant TL values.
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2223/Ag tapes was studied with finite element models. The investigated loading situations concerned the thermal stresses, axial tension and transversal compression. The results showed that the anisotropy of filaments had only a small impact on the tape overall stiffness or the fracture probability of the filaments. Thereby, knowledge of filament alignment is not necessarily needed to estimate the tape mechanical behaviour. The filament stiffness influenced significantly both axial and transversal stiffness of the tape, but the degree of creep deformation or the silver yield strength did not affect them markedly. However, the filament stiffness, silver yield strength and creep all influenced significantly the filament stresses. The filament stiffness determined mostly the growth rate of stresses under axial or transversal loading. The degree of creep deformation or the silver yield strength affected mostly the pre-compression of the filaments after cooling. Because of their direct connection, measured data on tape stiffness, if available, can be used to extract the filament stiffness for further modelling. To improve the reliability of predicting the filaments stresses, more experimental analysis would be needed on the yield strength of the silver matrix. In addition the creep deformation should be studied with systematic comparisons between experiments and models. References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] Fig. 5. First (black symbols) and third (white symbols) principal stresses in filaments in transversal compression with (a) TL = 370 K (s) and TL = 1103 K (4) and (b) YAg (293 K) = 11 MPa (o) and YAg (293 K) = 25 MPa (4). Full lines show median and dashed lines most tensile rI and most compressive rIII in the filament area.
5. Conclusions The impact of anisotropy and stiffness of filaments and yield strength and creep deformation of silver on local stresses in Bi-
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