strain at the transition of elastic to plastic deformation Note 1 to entry: The stress Relasticmax and the corresponding strain Aelasticmax refer to point G in Figure A.6 o0f Annex A.5
General
The test machine and the extensometers shall conform to ISO 7500-1 and ISO 9513, respectively The calibration shall obey ISO 376 The special requirements of this standard are presented here.
Testing machine
A tensile machine control system must maintain a constant stroke speed, and the grips should be designed with adequate structure and strength to securely hold the test specimen To prevent slippage during testing, the grip faces should be roughened through filing, knurling, or similar methods Gripping mechanisms can be screw-type, pneumatically, or hydraulically actuated.
Extensometer
The extensometer must weigh 30 g or less, depending on the wire diameter, to avoid impacting the mechanical properties of the brittle reacted superconductive wire It is essential to symmetrically balance the mass of the extensometers around the wire to prevent any misalignment forces Additionally, precautions should be taken to avoid applying bending moments to the test specimen.
The strain measuring method used significantly influences the quantities that can be accurately determined For a conventional single extensometer system, it is advisable to measure the elastic modulus (E U) and the yield strength (R p0,2-U) Conversely, an averaging double extensometer system can measure all relevant quantities effectively, as it compensates for bending effects in the tested sample, ensuring an accurate determination of the modulus of elasticity (E 0).
NOTE Further information is given in Clauses A.2 and A.3
General
The wire should be straightened before heat treatment and should be inserted into a ceramic or quartz tube with slightly larger inner diameter referring to the wire size
The constant temperature zone length of the heat treatment furnace shall be longer than the total length mentioned below in 6.2
Care shall be taken to prevent bending or pre-loading when the reacted specimen is manually handled during removal from the ceramic or quartz tube and mounting.
Length of specimen
The total length of the test specimen is determined by adding the inward distance between grips to the lengths of both grips It is essential that the inward distance between grips is at least 60 mm to accommodate the installation of extensometers.
Removing insulation
Before heat treatment, any insulating coating on the test specimen surface must be removed using either a chemical or mechanical method, ensuring that the specimen surface remains undamaged (refer to Clause A.7).
Determination of cross-sectional area (S 0)
To determine the cross-sectional area of a specimen after removing the insulation coating, a micrometer or similar measuring device must be utilized For round wires, the cross-sectional area is calculated by taking the arithmetic mean of two orthogonal diameters In the case of rectangular wires, the area is derived from the product of thickness and width Any necessary corrections for the corners of the cross-sectional area should be established through discussions among the relevant parties (refer to Clause A.8).
Specimen gripping
When mounting the test specimen on the tensile machine grips, ensure that the specimen and tensile loading axis align in a straight line to minimize any machine/specimen mismatch Refer to Clause A.9 for details on gripping techniques.
Setting of extensometer
When installing the extensometer, it is crucial to avoid deforming the test specimen The device should be positioned centrally between the grips, ensuring that the measurement direction aligns with the axis of the specimen.
During mounting care should be taken not to pre-load the specimen After installation, loading shall be physically zeroed
Double extensometer shall be mounted symmetrically around the cross-section to allow averaging of the strain to compensate the bending effects
To ensure optimal performance of the stress-strain curve for rectangular wires, it is essential to mount the extensometer symmetrically on the narrow sides of the wire for accurate strain measurement.
Testing speed
The tensile tests shall be performed with displacement control The machine crosshead speed is recommended to be set between 0,1 mm/min and 0,5 mm/min.
Test
The tensile machine will be initiated after setting the crosshead speed to a predetermined level Signals from the extensometers and load cell will be recorded, saved, and plotted on the diagram as illustrated in Figures 1 (a) and 1 (b) Once the total strain reaches between 0.3% and 0.4%, the tensile force will be decreased by 30% to 40% while maintaining the crosshead speed This process will continue until the wire is reloaded and ultimately fractures.
Prior to the start of any material test program it is advisable to check the complete test equipment using similar size wires of known elastic properties (See Clause A.14)
Modulus of elasticity (E)
The modulus of elasticity is typically calculated using the formula applied to the straight sections of both the unloading and initial loading curves It is essential to utilize appropriate software for data evaluation to analyze the plotted data effectively, particularly by zooming in on the stress versus strain graph in areas where deviations from linearity are anticipated.
E is the modulus of elasticity;
∆F is the increment of the corresponding force;
∆A is the increment of strain corresponding to ∆F;
The original cross-sectional area of the test specimen is denoted as S₀ The unloading process occurs at the strain indicated by point Aᵁ in Figure 1(a), utilizing the same formula for both the unloading modulus of elasticity (Eᵁ) and the initial loading modulus (E₀) It is advisable to measure the unloading curve starting at point Aᵁ, which should ideally be between 0.3% and 0.4%.
The modulus of elasticity, denoted as \( E_U \), is derived from the unloading curve and is represented by the slope of the line between 0.3% and 0.4% strain, as shown in Figure 1(a) In contrast, the modulus of elasticity from the initial loading curve is referred to as \( E_0 \) and is indicated by the zero offset line.
The initial stress-strain curve's straight portion is notably narrow, as shown in Figure A.6 of Clause A.5 To achieve low relative standard uncertainty in measurements, the averaging double extensometer system is currently the only viable technique Therefore, the quantity of \( E_U \) should be considered representative for this text, while \( E_0 \) should only be reported when measured using the double extensometer system.
After the test, the results shall be examined using the ratio E 0 /E U The ratio shall satisfy the condition as given in Equation 2 in which ∆ = 0,3 (see Clause A.12)
If the test fails to meet the required conditions, it is deemed invalid Consequently, the test must be repeated after reviewing the experimental procedure in accordance with the current testing method.
To effectively execute the unloading-reloading procedure, the loading curve should reach a strain of A U, which falls between 0.3% and 0.4% At this point, the stress must be decreased to r umin, which is the minimum stress level from the maximum stress (r umax) where unloading commenced Subsequently, the wire is reloaded The slope of the unloading curves is determined from the linear segment between the stress levels r umax and r umin.
The typical range of \( r_{\text{umax}} \) is 99% of the maximum stress, which indicates the point where unloading begins In contrast, the range of \( r_{\text{umin}} \) is set at 90%, corresponding to the onset of unloading stress, as illustrated in Figure 1(b).
The 0,2 % proof strength of the composite is determined in two ways from the unloading/reloading and initial loading part of the stress-strain curve as shown in Figures 1(a) and 1(b)
The 0.2% proof strength of the composite under unloading, denoted as R\(_{p0.2-U}\), is determined by adjusting the linear portion of the unloading slope parallel to the origin of the fitted curve, which may include negative strain values A parallel line is then shifted to 0.2% on the abscissa from this strain point The intersection of this line with the stress-strain curve identifies point C, which is defined as the 0.2% proof strength The unloading line, such as U\(_{0.35}\) shown in Fig 1(a), is crucial for determining R\(_{p0.2-U}\).
The 0.2% proof strength under loading, denoted as R\(_{p0.2}\), is determined by shifting the initial linear portion of the stress-strain curve 0.2% along the strain axis The intersection point, referred to as point A, between this adjusted linear line and the stress-strain curve defines the 0.2% proof strength under loading.
Each of 0,2 % proof strength value shall be calculated using the formula (3) given below:
R p0,2-i is the 0,2 % proof strength (MPa) at each point;
F i is the force (N) at each point; as i = 0 or U
Measurements must be conducted at temperatures ranging from 283 K to 308 K A force measuring cell with a relative standard uncertainty of less than 0.1% is required, applicable from zero to the load cell's maximum force capacity Extensometers should maintain a relative standard uncertainty of strain below 0.05% Additionally, the displacement measuring transducer, such as an LVDT (linear variable differential transformer), must have a relative standard uncertainty of less than 0.01% for calibration purposes.
Table A.1 presents the relative standard uncertainty values for the measured moduli of elasticity, E₀ and Eᵤ, along with the proof strengths, Rₚ₀,₂₋₀ and Rₚ₀,₂₋ᵤ, as determined by an international round robin test involving eleven representative research groups.
The international round robin test indicated a relative standard uncertainty of 1.4% for E₀ based on an average of 17 test data points following the qualification check Additionally, the relative standard uncertainty was reported as 1.3% for E₁ (N = 15), 1.5% for Rₚ₀.₂₋₀ (N = 17), and 2.5% for the other parameters.
Specimen
The following information shall be reported: a) Name of the manufacturer of the specimen b) Classification and/or symbol c) Lot number
The report should include details on the raw materials and their chemical composition, the cross-sectional shape and dimensions of the wire, the diameter of the filaments, the total number of filaments, and the ratio of copper to non-copper materials.
Results
Results of the following mechanical properties shall be reported a) Modulus of elasticity (E 0 and E U ) b) 0,2 % proof strengths (R p0,2-0 and R p0,2-U )
The following information shall be reported if required: c) Tensile stress R elasticmax d) Strain A elasticmax e) Tensile strength (R m ) f) Percentage elongation after fracture (A) g) 0,2 % proof strength determined by means of function fitting method (R p0,2-F )
Test conditions
The following information shall be reported: a) Crosshead speed b) Distance between grips c) Temperature d) Manufacturer and model of testing machine e) Manufacturer and model of extensometers f) Gripping method
The Figure 1(a) shows the over-all relation between stress and strain; (b) is the enlarged view indicating the unload and reload procedure
U: Computed unloading line of U between 0,3 % and 0,4 % strain using 1st order regression line in Figure
The experimental results indicate a 0.2% strain shift from the initial origin of the loading line, referred to as Point A, while Point C represents a 0.2% strain shift from the origin of the fit curve, determined by the slope of the unloading line, such as U0.35.
Point H: Final fracture point of the wire
The slope of the initial loading line is typically less steep than that of the unloading lines To determine the 0.2% proof strength (R p0,2-0) of the composite, the line must be drawn from the 0.2% offset point on the abscissa, reflecting the yielding of ductile components like copper and bronze, which is represented by point A derived from the initial loading line.
Point C is determined using the unloading line, where the slope between 0.3% and 0.4% is adjusted to align with the origin of the fitted curve, potentially involving a negative strain shift The parallel 0.2% strain shift of this slope intersects the fitted curve at point C, which represents the 0.2% proof strength of the composite, denoted as \( R_{p0.2-U} \).
The graph in Figure 1(b) shows the raw data of the unloading region The slope should be determined between
99 % of maximum stress at the onset of unloading and 90 % stress of the maximum stress as indicated (see 8.1)
Figure 1 – Stress-strain curve and definition of modulus of elasticity and 0,2 % proof strengths for Cu/Nb 3 Sn wire
Additional information relating to Clauses 1 to 10
Scope
This annex gives reference information on the variable factors that may affect the tensile test methods All items described in this annex are informative.
Extensometer
Double extensometer
Any extensometer that includes two individual extensometers for recording separate signals, which can then be averaged by software, or one signal that is pre-averaged by the extensometer system, is suitable for use.
In Figures A.1 and A.2 typical advanced light weight extensometers are shown
The extensometer has a gauge length of ~ 12 mm (total mass ~ 0,5 g) The two extensometers are wired together into a single type extensometer, thus averaging the two displacement records electrically.
Figure A.1 – Light weight ultra small twin type extensometer
The extensometer has a gauge length of ~ 26 mm (total mass ~ 3 g) Each of the two extensometers is a single type extensometer, the averaging should be carried out by software.
Figure A.2 – Low mass averaging double extensometer
Single extensometer
A single extensometer weighing 31 g, along with a balance weight, was utilized in a RRT for Cu/Nb-Ti wires in Japan, yielding reliable results that contributed to the establishment of the international standard IEC 61788-6.
1 Figures in square brackets in this annex refer to the Reference documents listed in Clause A.15
35 Frame Gauge length setting hole
Figure A.3 – An example of the extensometer provided with balance weight and vertical specimen axis
Optical extensometers
Any type of optical extensometer can be used if it is based on two single optical beams, where the signal can be recorded and averaged
Non-contact systems can be utilized similarly to an averaging double extensometer, employing either two laser beams or alternative optical systems.
Figure A.4(a) illustrates the schematic scan of stripes at a frequency of 50 Hz using a rotating deflector The software analyzes the minor displacement changes that occur during the specimen loading process Additionally, Figure A.5(b) depicts the double mirror configuration of a typical advanced double laser beam system.
(a) Schematic illustration (b) Overview of the present extensometer
(c) The results of a reacted Nb 3 Sn wire with 0,81 mm ỉ measured with a double beam laser extensometer.
Figure A.4 – Double beam laser extensometer
Requirements of high resolution extensometers
The requirements for extensometers can be determined from Figure A.5, emphasizing the need for low relative standard uncertainty in recorded values, particularly between zero % strain and 0.01 % strain In this range, the total displacement is 2.5 µm for a 25 mm gauge length and 1.2 µm for shorter lengths.
12 mm gauge length In fact, the signals should be acquired with a low noise around
To achieve stable records within the required strain range, it is crucial to maintain a calibration factor of 10 V per 1 mm displacement using a 12 mm gauge length extensometer Ensuring a V pp of the signal below 1 mV is essential for achieving low relative standard uncertainty This can be accomplished with advanced signal conditioners, shielded and twisted cables, and high-resolution data acquisition systems exceeding 16-bit resolution The original raw data from the reacted Nb 3 Sn measurement is presented in a load versus displacement graph, as shown in Figure A.5 A high signal-to-noise ratio is necessary to minimize data scatter and accurately resolve the curve well below the 1 µm range.
Achieving a zero offset gradient with low relative standard uncertainty is essential for accurately assessing the modulus of elasticity This requires the use of high-resolution extensometers that have an extremely low noise-to-signal ratio.
The double extensometer system, utilizing either two mechanical extensometers, two laser beams, or other optical systems symmetrically positioned within a 180° sector, effectively ensures compensation for bending.
This figure shows the necessary low relative standard uncertainty with respect to the displacement resolution The data are taken from the measurement of the sample as shown in Figure A.1
Figure A.5 – Load versus displacement record of a reacted Nb 3 Sn wire
Tensile stress R elasticmax and strain A elasticmax
The tensile stress at which the transition of elastic to plastic deformation occurs is calculated in general using the following formula (Figure A.6)
R elasticmax is the tensile stress (MPa) at the transition of elastic to plastic deformation;
F elasticmax is the force (N) at the transition of elastic to plastic deformation
The strain at which the transition of elastic to plastic deformation occurs (Figure A.6) referred to the stress R elasticmax is defined as follows:
2 Figures in square brackets in this annex refer to the Reference documents listed in Clause A.15 where total
∆A is the total strain increment referring to zero offset strain and to strain where the transition of elastic to plastic deformation occurs;
A max is the observed value of strain referred to the stress R elasticmax ;
A 0 is the zero offset strain
The values R elasticmax and A elasticmax are treated as being of informative character
The enlarged figure illustrates the transition region of elastic plastic deformation at point G, with large circles indicating the initial straight portion of the record A linear first-order regression analysis yields an elasticity modulus of \$E_0 = 134.7 \, \text{GPa}\$ and a regression coefficient square that must exceed 0.99 to confirm linearity The computed value of \$0.188 \, \text{MPa}\$ results from the regression line analysis, accounting for an ordinate offset due to plot scatter The stress \$R_{\text{elastic max}}\$ and corresponding strain \$A_{\text{elastic max}}\$ mark the transition region, with \$R_{\text{elastic max}}\$ being crucial for evaluating \$E_0\$ A low \$R_{\text{elastic max}}\$ (e.g., < 5 MPa) suggests greater uncertainty in \$E_0\$ due to a limited linear region, which can lead to high uncertainty from data scatter and may necessitate repeating the measurement.
Figure A.6 – Stress-strain curve of a reacted Nb 3 Sn wire
Functional fitting of stress-strain curve obtained by single extensometer
The functional fitting method is effective for determining the 0.2% proof strength using a single extensometer In Cu/Nb₃Sn wire, the copper and bronze materials yield during cooling from heat treatment to room temperature, resulting in a curved stress-strain curve that complicates the evaluation of the initial modulus of elasticity Additionally, the non-linear shape of the specimen due to heat treatment and pre-straining during setup leads to a bent stress-strain curve, making it challenging to identify the intrinsic zero strain point The functional fitting method effectively eliminates these strains from the experimental data, allowing for a more accurate approximation of the stress-strain curves using an exponential function.
The equation \$F_0 = -A\$ represents the relationship between load (F), cross section (S_0), and strain (A) obtained from testing The parameters a, b, and n are determined through non-linear least-mean-square fitting To maintain precision during calculations, A is expressed as a percentage, with an upper fitting limit of 0.5% and a lower limit that is adjusted until convergence of the three parameters is achieved.
The 0.2% proof strength R\(_{p0.2-F}\) of the composite, resulting from the yielding of copper and bronze components, is determined using a function fitting method This involves shifting the linear portion under unloading parallel to the 0.2% offset point relative to the zero strain point defined by parameter b The intersection of this line with the fitted stress-strain curve identifies point C, which represents the 0.2% proof strength Excluding parameter b in the simplified equation leads to neglecting pre-strain, resulting in a larger proof stress value closer to R\(_{p0.2-F}\) It has been reported that commercially available non-linear least-square fitting software can yield results nearly identical to experimental data for the same parameters, provided the allowable error is set to less than 0.1 Confirmation of the alignment between data points and the fitting curve is also established.
Removing insulation
To prepare the test specimen, it is essential to remove the coating using a suitable method Typically, Nb 3 Sn conductors are insulated with glass or ceramic fibers, which can be easily stripped or peeled away For other types of insulation, either an organic solvent or a mechanical method should be employed Performing this removal prior to the heat treatment reaction helps prevent any damage to the specimen's surface.
The coating is not intended to serve as a structural element Analyzing the measurement of a multi-component composite, which includes insulation, is overly complex Consequently, this test method focuses solely on the bare reacted wire to preserve its mechanical properties.
Cross-sectional area determination
To achieve a smaller relative standard uncertainty, the cross-sectional area can be determined by adjusting the radius of the corner of the rectangular wire as specified in the manufacturing guidelines In cases of rolling or Turk's-head finishes, where the corner radius is not regulated, a correction is applied using a macro photograph of the cross-section.
Fixing of the reacted Nb3Sn wire to the machine by two gripping
To ensure proper gripping of the specimen during testing, it can be soft soldered to a metallic sleeve at the grip region, which should securely attach to the machine's pull rods Alternatively, the wire can be directly chucked, but care must be taken to avoid mechanical damage to the wire The test specimen must be mounted using the tensile machine's grips, ensuring that the specimen and tensile loading axis are aligned to prevent any mismatching It is crucial to avoid bending or deformation during the sample mounting process.
The international round robin test results indicate that multiple gripping techniques can yield accurate test outcomes Consequently, it is advised that the gripping technique outlined in the current standard be considered among various acceptable methods.
3 Figures in square brackets in this annex refer to the Reference documents listed in Clause A.15
Figure A.7 – Two alternatives for the gripping technique
The left image illustrates the gripping of soldered Nb₃Sn wire into an M6 brass thread, secured by an aluminum sleeve that facilitates the pulling action The combined mass of the aluminum sleeve and M6 thread is approximately 12 grams.
The image depicts the clamping of bare Nb₃Sn wire within a V-groove of a lightweight aluminum block, which has a total mass of approximately 7 grams This block is secured in a small frame, serving as a precise fixture for the pulling action.
Nb 3 Sn wire soldered into the M6 brass thread Nb 3 Sn wire fixed inside the Al-block using two M3 screws
The illustrations depict two methods for securing the wire to the machine: soldering and clamping Before measurements commence, it is crucial that the wire is not subjected to any pre-load Additionally, the lower end of the wire, where it connects to the fixing block, must remain free from contact with the machine to prevent any pre-loading.
Figure A.8 – Details of the two alternatives of the wire fixing to the machine
Tensile strength (R m)
The tensile strength at which the fracture occurs is calculated using the following formula
R m is the tensile stress (MPa) at the fracture;
F max is the maximum force (N) at the fracture
In wires with a low copper to non-copper volume ratio, premature fractures at the grips lead to reduced tensile strength and lower percentage elongation after fracture These properties are crucial for understanding the mechanical characteristics of composite materials and serve as important metrics for test validity However, due to significant variances and the limited strain region of interest for the wire, these values are primarily utilized as reference points.
Percentage elongation after fracture (A)
The measurement of elongation after fracture serves as a reference point, and the cross-head movement can help estimate this value To apply this method, it is essential to record the cross-head position at the moment of fracture The elongation after fracture can be calculated using a specific formula, expressed as a percentage.
A f is the percentage elongation after fracture;
L u is the distance between grips after fracture
L g is the initial inward distance between grips.
Relative standard uncertainty
Owing to the nature of any measurement, the obtained test results have a scatter in all cases
To evaluate the quality of measured data, the concept of uncertainty provides a solid foundation for independent assessment Detailed information regarding the uncertainty of a measurand can be found in Annexes B and C.
Significant insights were gained from the international round robin tests involving Nb 3 Sn wire measurements conducted by 11 research groups The evaluation of the data provided crucial information regarding the modulus of elasticity, particularly at the zero offset line and from the unloading line within the strain range of 0.3% to 0.4% Generally, the moduli of elasticity derived from the initial loading line exhibited greater variability compared to those from the unloading line A detailed analysis in reference [8] indicates that the ratio of \(E_0\) to \(E_U\) varies from unity when all results are aggregated This variation can be expressed by the relation \(1 - \Delta < \frac{E_0}{E_U} < 1 + \Delta\), where \(\Delta\) represents the deviation from unity To minimize scatter in the initial loading line and eliminate disqualified data, a value of \(\Delta = 0.3\) has been identified as appropriate.
Table A.1 summarizes key quantities, including the modulus of elasticity derived from the initial loading line and the proof strengths The standard uncertainties associated with these results indicate that excluding data beyond ∆ = 0.3 reduces the standard uncertainties by at least a factor of 2.
4 Figures in square brackets in this annex refer to the Reference documents listed in Clause A.15
The variability in experimental data from RRT arises from both intra- and inter-laboratory contributions To determine if the data from different laboratories belong to the same population, an analysis of variance (F-test) was conducted, following the guidelines outlined in GUM H.5.2.1 A typical case is illustrated by the analysis of experimental data on E₀, with the results presented in Table A.1.
To ensure accurate comparisons among laboratories, it is essential to standardize the number of data points used, with three being the chosen amount Consequently, three data points were randomly selected from each laboratory that had three or more data points, while any laboratory with fewer than three data points was excluded from the analysis.
The GUM guidance indicates that the ratio of inter-laboratory variance (\$s_a^2\$) to intra-laboratory variance (\$s_b^2\$) was calculated as \$F_{exp} = \frac{s_a^2}{s_b^2}\$ and compared to the theoretical function \$F_{N J - J 1}(\alpha)\$ When \$F_{exp} < F_{N J - J 1}(\alpha)\$, the data is considered to belong to the same population at a significance level of \$\alpha\%\$ In instances where all data were used for the F-test, the hypothesis did not hold for samples E3, E4, H, and M However, when applying a qualification threshold of \$\Delta = 0.3\$, the hypothesis was valid for all samples at a significance level of 1\% Thus, the F-test confirms the validity of the qualification check regarding the ratio of moduli of elasticity as stated in section 8.1 of the main text, and a more stringent qualification condition can be assessed using a higher significance level.
Table A.1 – Standard uncertainty value results achieved on different Nb 3 Sn wires during the international round robin tests
Sample Property(X) All Data Qualified Data
E2 E 0 , [GPa] 35 113,5 [GPa] 2,8[GPa] 2,5[%] 22 114 [GPa] 1,4 [GPa] 0,3[%]
R p0,2-0 , [MPa] 35 187,1 [MPa] 3,0[MPa] 1,6[%] 22 181,7 [MPa] 1,3 [MPa] 0,7[%] E3 E 0 , [GPa] 33 119,2[GPa] 3,2 [GPa] 2,7[%] 21 121[GPa] 1,5[GPa] 1,3[%]
R p0,2-0 , [MPa] 34 192,1[MPa] 2,3 [MPa] 1,2[%] 21 191,8 [MPa] 1,6 [MPa] 0,8[%] E4 E 0 , [GPa] 36 90,3[GPa] 3,9[GPa] 4,3[%] 14 109,2[GPa] 2,0[GPa] 1,9[%]
R p0,2-0 , [MPa] 37 113,6[MPa] 2,2 [MPa] 1,9[%] 14 111,9 [MPa] 3,7 [MPa] 3,3[%]
H E 0 , [GPa] 33 88,5[GPa] 4,2 [GPa] 4,8[%] 9 109,8 [GPa] 2,0 [GPa] 1,8[%]
R p0,2-0 , [MPa] 33 118,8[MPa] 2,3[MPa] 1,9[%] 9 110,3[MPa] 2,9[MPa] 2,6[%]
K E 0 , [GPa] 34 116,2[GPa] 2,8 [GPa] 2,4[%] 25 115,6 [GPa] 1,1 [GPa] 1,0[%]
R p0,2-0 , [MPa] 33 182[MPa] 2,6 [MPa] 1,4[%] 25 179,7 [MPa] 2,6 [MPa] 1,4[%]
M E 0 , [GPa] 29 88,6[GPa] 5,8 [GPa] 6,6[%] 9 120,4[GPa] 3,0[GPa] 2,5[%]
R p0,2-0 , [MPa] 28 118 [MPa] 4,0 [MPa] 3,4[%] 9 109,2[MPa] 1,7[MPa] 1,5[%]
N: number of total tested wires,
X: modulus of elasticity or proof strength,
SU: standard uncertainty for total data, and
This table presents results of standard uncertainty values achieved on different Nb3Sn wires during the international round robin tests carried out by 11 different research laboratories
Consequently, the average value of relative standard uncertainty (U RSU ) for all samples is given by the equation
RSU (A6) where M indicates the number of different wires and the average number of samples is given,
The U RSU was determined to be 1.5% for E o for N ' based on the average value of all samples following the qualification check, as shown in Table A.1 Additionally, the U RSU was assessed for E u, R p0.20, and R p0.2U, with the findings detailed in Clause 9 of the main text.
Table A.2 – Results of ANOVA (F-test) for the variations of E 0
Sample All data Qualified data
The hypothesis was verified using a significance level of 1% Each laboratory data set consists of three values (n = 3), leading to the relationship \(N = nJ\), where \(J\) represents the number of laboratories, \(N\) is the total number of wires, \(J'\) denotes the number of qualified laboratories, and \(N'\) indicates the number of qualified wires.
Determination of modulus of elasticity E 0
To accurately determine the modulus of elasticity \( E_0 \), a data acquisition system must be utilized that maintains a low relative standard uncertainty in the zero-offset region of the stress-strain record It is essential to evaluate the recorded stress and strain data following a recommended procedure to ensure the data remains unbiased.
To determine the modulus of elasticity \( E_0 \) from the stress versus strain record, one must calculate the first-order linear regression line and the square of the regression coefficient between 0 MPa and 50 MPa The control parameter, the square of the regression coefficient, should exceed 0.99 By gradually reducing the stress from 50 MPa, the linear slope that meets this criterion can be identified The intersection of this slope with the abscissa establishes a new origin for the stress versus strain graph, which is essential for the 0.2% parallel shift used in estimating the \( R_{p0.2-0} \) value (refer to Figure 1).
Assessment on the reliability of the test equipment
To ensure the reliability of tensile testing equipment, including the tensile testing unit, load cell, and extensometer system, it is essential to use welding wires of similar sizes and known elastic properties Welding wires with a diameter of around one mm, specifically made from aluminium and pure commercial titanium, are ideal as they fall within the modulus of elasticity range of 70 GPa to 100 GPa It is highly recommended that test laboratories periodically verify the reliability of their tensile setups by measuring the elastic properties of these wires before conducting any measurements These wires are readily available from vendors and should be handled similarly to superconducting heat-treated wires, allowing for loading and unloading in the elastic regime up to 100 MPa without compromising their elastic properties.
Overview
In 1995, international standards organizations, including the IEC, unified the terminology for statistical expressions by adopting the term "uncertainty" for all quantitative measures, while discontinuing the quantitative use of "precision" and "accuracy." However, "accuracy" and "precision" may still be used in a qualitative context The standardized methods for evaluating uncertainty are detailed in the Guide to the Expression of Uncertainty in Measurement (GUM).
Each Technical Committee (TC) had the discretion to modify existing and future standards to align with the new unified approach, a process that can be challenging and may lead to confusion, particularly for those unfamiliar with statistics and the concept of uncertainty During the TC 90 meeting in Kyoto in June 2006, it was resolved to adopt these changes in upcoming standards.
To convert "accuracy" and "precision" values into corresponding "uncertainty" figures, it is essential to understand the source of these values The coverage factor associated with the original number can vary, typically being 1, 2, 3, or another value Additionally, when a manufacturer's specification is represented by a rectangular distribution, the conversion factor is often 3.
The appropriate coverage factor was applied to convert the original number into the equivalent standard uncertainty This conversion process is not a compliance requirement for TC 90 standards; it is provided solely to inform users about the changes made during this process Importantly, this transition to uncertainty terminology does not change the user's obligation to assess their measurement uncertainty to ensure compliance with the standard's criteria.
The TC 90 measurement standards aim to minimize uncertainty in measurements by utilizing the Convener's engineering judgment and error propagation analysis These standards establish straightforward limits for certain influencing quantities, alleviating the need for users to assess their uncertainty Ultimately, the overall uncertainty of a standard is validated through interlaboratory comparisons.
Definitions
Statistical definitions are sourced from the GUM, the International Vocabulary of Basic and General Terms in Metrology (VIM), and the NIST Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results While the GUM includes various statistical terms, not all are explicitly defined within it; for instance, "relative standard uncertainty" and "relative combined standard uncertainty" are mentioned but not formally defined.
Consideration of the uncertainty concept
Statistical evaluations historically relied on the coefficient of variation (COV), defined as the ratio of the standard deviation to the mean The COV is commonly referred to as the relative measure of variability.
The standard deviation and standard uncertainty (SU) are crucial for evaluating the precision of measurements and the consistency of repeated tests The SU is influenced more by the number of repetitions than by the mean, often providing a clearer representation of data variability An example illustrates this with electronic drift and creep voltage measurements from two nominally identical extensometers, both utilizing the same signal conditioner and data acquisition system Ten data pairs were randomly selected from a spreadsheet containing 32,000 cells, with extensometer one (E1) positioned at zero offset and extensometer two (E2) deflected to 1 mm, measuring output signals in volts.
Table B.1 – Output signals from two nominally identical extensometers
Table B.2 – Mean values of two output signals
Table B.3 – Experimental standard deviations of two output signals
Table B.4 – Standard uncertainties of two output signals
Table B.5 – Coefficient of Variations of two output signals
The standard uncertainty is very similar for the two extensometer deflections In contrast the coefficient of variation COV is nearly a factor of 2 800 different between the two data sets
This shows the advantage of using the standard uncertainty which is independent of the mean value.
Uncertainty evaluation example for TC 90 standards
The observed value of a measurement typically does not match the true value of the measurand, serving instead as an estimate Measurement uncertainty is an inherent aspect of any measurement, reflecting the metrological quality and enhancing understanding of the measurement process Each physical measurement result comprises an estimate of the true value and the associated uncertainty The GUM provides a standardized framework for documenting measurement procedures transparently To ascertain the true value, one can measure the best estimate and evaluate uncertainties, which are categorized into Type A (derived from repeated laboratory measurements, often represented by Gaussian distributions) and Type B (based on prior experiments, literature, or manufacturer data, typically expressed as rectangular distributions).
The GUM procedure for calculating uncertainty begins with the user developing a mathematical measurement model that expresses the identified measurand as a function of all input quantities For instance, an example of this model can be seen in the uncertainty calculation of a force measurement, \( F_{LC} \), using a load cell.
The formula for F LC is expressed as F LC = W + d W + d R + d Re, where W denotes the expected standard weight, d W represents the manufacturer's data, d R indicates the repeated checks of standard weight per day, and d Re reflects the reproducibility of checks conducted on different days.
The input quantities for the analysis include the measured weight of standard weights using various balances (Type A), manufacturer’s data (Type B), repeated test results from a digital electronic system (Type B), and the reproducibility of final values measured on different days (Type B) It is essential for the user to identify the distribution type for each input quantity, such as Gaussian distributions for Type A measurements and rectangular distributions for Type B measurements The standard uncertainty of Type A measurements can be evaluated using the formula \( n u_A = s \), where \( s \) represents the experimental standard deviation and \( n \) is the total number of measured data points Additionally, the standard uncertainties of the Type B measurements should also be assessed.
1 where, d W is the range of rectangular distributed values e) Calculate the combined standard uncertainty for the measurand by combining all the standard uncertainties using the expression:
In this scenario, it is assumed that there is no correlation between input quantities When the model equation includes products or quotients, the evaluation of combined standard uncertainty becomes more complex, requiring the use of partial derivatives and sensitivity coefficients To enhance the probability that the measurand falls within a specified interval, the combined standard uncertainty can be multiplied by a coverage factor (e.g., 1 for 68%, 2 for 95%, or 3 for 99%) The final result should be reported as the estimate of the measurand ± the expanded uncertainty, including the unit of measurement, and must state the coverage factor used and the estimated coverage probability.
To facilitate the computation and standardize the procedure, use of appropriate certified commercial software is a straightforward method that reduces the amount of routine work [6,
7] In particular, the indicated partial derivatives can be easily obtained when such a software tool is used Further references for the guidelines of measurement uncertainties are given in
Overview
This article provides specific examples to demonstrate techniques for estimating uncertainty While these examples are illustrative, they do not require users to perform identical analyses to adhere to the standard However, users must assess the uncertainty estimates for each individual influence quantity—such as load, displacement, wire diameter, and gauge length—to ensure they align with the specified uncertainty limits outlined in the standard.
While the two examples provided highlight key sources of error, they are not comprehensive Other potential factors, including friction, wire deformation, insulation removal, misaligned grips, and strain rate, may also contribute to inaccuracies and should be considered, as their impact can vary.
Uncertainty of the modulus of elasticity
The original stress versus strain data for a Nb 3 Sn wire with a diameter of 0.768 mm, collected during an international round robin test in 2006, is presented in Figure C.1 Figure C.1 (a) illustrates the wire's loading process until fracture, while Figure C.1 (b) highlights the initial loading points up to 16 MPa along with the corresponding trend line The computed slope of this trend line is 132069 MPa, expanded by a factor of 100 due to the percentage unit of the abscissa, and it features a squared correlation coefficient of 0.9899.
The stress versus strain curve for a 0.783 mm diameter superconducting wire is illustrated in Graph (a) Graph (b) focuses on the initial segment of this curve, accompanied by a regression analysis to calculate the modulus of elasticity To obtain the modulus of elasticity in MPa, the slope of the line must be multiplied by 100 to convert percentage strain into strain.
Figure C.1 – Measured stress-strain curve
The standard uncertainty estimation of the modulus of elasticity for this wire involves analyzing five variables, each contributing its own specific uncertainty The modulus of elasticity, determined during mechanical loading, is influenced by these variables.
∆L = deflected length of extensometer in zero offset region for the selected load portion, mm
L G = length of extensometer at start of the loading, mm b = an estimate of deviation from the experimentally obtained modulus of elasticity, MPa
To calculate standard uncertainty, actual experimental values are essential By analyzing the data from Figure C.1 (b), the deflected extensometer length can be estimated For a selected stress of 15 MPa, the calculated modulus of elasticity from Figure C.1 (b) allows for the determination of the value of ∆L using the appropriate equations.
P =π⋅D ⋅ (C.4) the force P can be calculated as P = 7,223 N.
Evaluation of sensitivity coefficients
The combined standard uncertainty associated with model Equation (2) is:
The partial differential terms are the so-called sensitivity coefficients By substituting the experimental values in each derivative, the sensitivity coefficients c i can be calculated as follows:
Sensitivity coefficient c 5 is unity (1) owing to the differentiation of Equation 2 with respect to quantity b
Using the above sensitivity coefficients, the combined standard uncertainty u c is finally given by:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 2 2 2 2 3 2 3 2 4 2 4 2 5 2 5 2 c c u c u c u c u c u u = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ (C.10) where the square of each sensitivity coefficient is multiplied by the square of the standard uncertainty of individual variables as given in the model Equation (C.2).
Combined standard uncertainties of each variable
The combined standard uncertainties \( u_i \) in Equation (C.10) encompass the uncertainties associated with force (P), deflected length (\( \Delta L \)), wire diameter (D), and gauge length (\( L_G \)) This section will provide estimates for each combined standard uncertainty based on the available data.
The combined standard uncertainty \( u_1 \) for force \( P \) is derived from both Type A and Type B statistical distributions Force measurements are typically conducted using commercially available load cells, but most manufacturers do not provide uncertainty information in their specifications To determine the combined standard uncertainty \( u_1 \), the provided accuracies and other data sheet information must first be converted into standard uncertainties These specifications are generally treated as limits of a rectangular distribution of errors, with the standard uncertainty for this distribution calculated as the limit divided by 3.
For the measurements given in Figure C.1, the following information for the load cell was available
Table C.1 – Load cell specifications according to manufacturer’s data sheet
Data must be converted to standard uncertainty values prior to combination, classified as Type B uncertainties A temperature range of 303 K to 283 K (∆T = 20 K) has been established to represent permissible laboratory conditions.
The variables are as follows:
Temperature coefficient of zero balance: T coefzero = (0,25 × 20) %
Temperature coefficient of sensitivity: T coefsens = (0,07 ×20) %
The following equation describes the measurement of load and includes the four sources of error from Table C.1:
P = u P + T class + T coefzero + T coefsens + T creep (C.11) where u P is the true value of load
The percentage specifications are converted to load units based on the measured value of
The load cell exhibits a combined standard uncertainty of P = 7,223 N, derived from the stress versus strain curve The values are converted to standard uncertainties under the assumption of a rectangular distribution, incorporating factors such as creep, coefficients, and class.
Tables C.2-C.4 summarize uncertainty calculations for displacement, wire diameter, and gauge length These calculations are similar to those previously demonstrated for force
Table C.2 – Uncertainties of displacement measurement
Type A Gaussian distribution Creep and noise contribution n / s u A = according Clause B.3
Type B distribution obtained from data scatter of Figure 1(b) w 3
B d / u = according Clause B.4 with d w of 0,00003 mm
Table C.3 – Uncertainties of wire diameter measurement
Wire diameter, mm Type A Gaussian distribution Five repeated measurements with micrometer device n / s u A = (0,0013)/ 5 mm
Half width of rectangular distribution according manufacture data sheet accuracy of ± 4 àm u B= d w/ 3 mm
To measure the gauge length of the extensometer, a stereo microscope was used with a resolution of 20 àm
Table C.4 – Uncertainties of gauge length measurement
Gauge length, mm Type A Gaussian distribution Five repeated measurements with micrometer device u A= s/ n
Half width of rectangular distribution according manufacture data sheet accuracy of +/-20 àm u B= d w/ 3 mm
The uncertainty in the slope of the fitted stress versus strain curve, as shown in Figure C.1 (b), is estimated to have a maximum half-width difference of ±0.822 MPa between the measured and calculated stress values By utilizing this value along with the gauge length (L G mm) and an extensometer deflection of ∆L = 0.001363 mm, a Type B uncertainty for the modulus of elasticity can be determined Rearranging Equation (C.3) leads to a simplified equation for this estimation.
The Type B uncertainty of the measured modulus of elasticity of the Figure C.1 (b) is
The final combined standard uncertainty, taking into account the result of Equation (C.12) and using the sensitivity coefficients for the four variables in Equation (C.10), results in:
Uncertainty of 0,2 % proof strength R p0,2
To determine the 0.2% proof strength (R p0.2), the modulus of elasticity zero offset line must be parallel shifted to the 0.2% strain position along the abscissa The intersection of this line with the original stress versus strain curve is then computed Additionally, if the fitted modulus of elasticity line has a different origin than zero, this offset must also be taken into account The regression equation illustrated in Figure C.1 (b) indicates an x-axis offset.
29719 offset =− 0 =− (C.19) where A offset indicates offset strain at zero stress
The shifted position of the line along the abscissa is approximately 0.19977% instead of the expected 0.20000% Table C.5 illustrates the stress calculations using the regression line, both with and without considering the uncertainty contribution from Equation (C.18).
Table C.5 – Calculation of stress at 0 % and at 0,1 % strain using the zero offset regression line as determined in Figure C.1 (b)
Description Regression line equation with uncertainty contribution at ε % strain
Baseline modulus of elasticity 1320,692ãA + 0,29719 0,297 132,37
Modulus of elasticity with − 7,6 GPa uncertainty contribution
Graph (a) illustrates the 0.2% offset shifted regression line alongside two additional lines representing the uncertainty contributions relative to the baseline To construct these three lines, four points are required: one common point at zero stress and three calculated stress values at 0.1% strain, as detailed in Table C.5, with the corresponding strain values adjusted by 0.2% The raw stress versus strain curve is also depicted in graph (a) around the intersection region of the three lines with the raw data Meanwhile, graph (b) provides an enlarged view of the original raw data of stress versus strain, along with the shifted lines based on the computations from Table C.5, and includes the linear regression equations for all four functions.
In Table C.5, the selected stresses at 0% strain and 0.1% strain are chosen to establish two distinct points for determining the shifted lines in Figure C.2 The offset shift value derived from Equation (C.19) is then added to the values corresponding to 0% strain and 0.1% strain.
Table C.6 lists the linear regression equations after shifting the lines as determined in Figure C.2 (b)
Table C.6 – Linear regression equations computed for the three shifted lines and for the stress–strain curve in the region where the lines intersect
Description of equations Linear regression equation × is here the strain in % and y the stress in MPa
Linear part of stress versus strain curve (see Figure
Shifted modulus of elasticity baseline y 17,7ãx − 262,93
Modulus of elasticity with + 7,6 GPa uncertainty contribution
Modulus of elasticity with − 7,6 GPa uncertainty contribution
Using the equations from Table C.6, we calculate the three intersection points and determine the stresses at these locations The computations and resulting intersection values are presented in Table C.7 The proof strength is defined as the stress at the intersection of the first line (with a shifted zero offset) and the stress versus strain curve The other two stress values at the intersections indicate estimated error bounds for the proof strength, which are derived from the uncertainty in the modulus of elasticity slope as outlined in Equation (C.18).
Table C.7 – Calculation of strain and stress at the intersections of the three shifted lines with the stress–strain curve
Description Equation set for strain and stress calculation at intersections Strain at intersection, % Stress at intersection, MPa Shifted baseline (mean) (43,546+262,93) / (1317,7-244,08) 0,285463
The standard uncertainty of the proof strength is a Type B determination, and can be estimated using:
The raw data scatter illustrated in Figure C.2 (b) is crucial for the final uncertainty estimate Table C.8 presents the measured stress versus strain data corresponding to Figure C.2 (b) Additionally, columns 3 and 7 of Table C.8 display the computed stress derived from the linear fit within the relevant data region, while columns 4 and 8 highlight the discrepancies between the measured and computed values.
Table C.8 – Measured stress versus strain data and the computed stress based on a linear fit to the data in the region of interest
The extreme differences between the computed and measured stress from the 4 th and 8 th columns of Table C.8 are:
The extreme differences represent observed limits to random error which can be converted to a standard uncertainty using:
Combined standard uncertainty for 0, 2 % proof strength is given:
Thereafter, the 0,2 % proof strength result is given as:
0,2 offset proof strength: R p 0 , 2 3,2MPa +/− 0,96MPa (C.24)
ASTM E 83-85, Standard Practice for Verification and Classification of Extensometers
ASTM E 111-82, Standard Test Method for Young’s Modulus, Tangent Modulus, and Chord Modulus