Curve fitting of stress-strain curve and RRT of Ag/Bi-2212 round wires

Presented by H.W. WeijersMeasurements performed by Bob Walsh and Dustin McRea

Presented at Andong National UniversityAndong, July 15th 2009

Outline• Overview of recent testing• Curve fitting• RRT on Bi-2212• Conclusions

Recent testing

• Student ran measurements with support• Hydraulic rig with calibrated load cell• Used both Shepic (19 g, not balanced) and

Nyilas type (4.2 g) extensometer• 90 mm between grips• Drill chucks for round wire• Plate clamps for tape• Grips:128 g (zeroed out), pins: 76.5g each

(lower pin not zeroed out)

Recent testinggrips and extensometers

Nyilas extensometer close up

Load cell

Bi-2223 3-ply Brass“raw” data

1,2 Shepic3,3 Nyilas

Bi-2223 3-ply stainless steel“raw” data

1,2 Shepic3,3 Nyilas

Bi2212 round wire“raw” data

1,2 Shepic3,4 NyilasReturn line slopes more useful than initial slope

Curve fitting of initial slope

• Purposes– Provide fit of data

• Section Annex 3

– Method to determine initial slope E0

• Section Annex 10

• Vary range to gain insight in quality of fit

Linear and 2nd order poly fit comparison

to 20 MPa to 40 MPa to 60 MPa to 80 MPa to 100 MPaMaterial linear polynomial linear polynomial linear polynomial linear polynomial linear polynomial

 E

GPa R2E

GPa R2E

GPa R2E

GPa R2E

GPa R2E

GPa R2E

GPa RE

GPa RE

GPa RE

GPa R                       

3U19-1 105 0.9915 93 0.9927 100 0.9983 111 0.9990 89 0.9949 114 0.9997 82 0.9939 106 0.9996      3U19-2 103 0.9971 93 0.9978 100 0.9990 105 0.9992 98 0.9994 105 0.9998 96 0.9994 105 0.9999 93 0.9986 106 0.9999

3U19-3 95 0.9850 98 0.9850 96 0.9962 93 0.9963 96 0.9987 96 0.9987 94 0.9991 100 0.9994 93 0.9991 101 0.99973U19-4 104 0.9926 91 0.9940 101 0.9433 106 0.9984 98 0.9990 106 0.9994 95 0.9987 107 0.9998 92 0.9983 106 0.9999

                             3B15-1 93 0.9964 91 0.9964 94 0.9995 93 0.9995 94 0.9998 96 0.9998 92 0.9996 98 0.9999 91 0.9996 97 0.99993B15-2 90 0.9976 94 0.9980 90 0.9993 89 0.9993 91 0.9997 88 0.9997 91 0.9998 89 0.9999 91 0.9999 91 0.99993B15-3 96 0.9927 87 0.9939 100 0.9983 93 0.9987 99 0.9992 101 0.9993 97 0.9994 103 0.9996 95 0.9992 104 0.99983B15-4 152 0.9711 140 0.9716 152 0.9974 106 0.9975 145 0.9982 110 0.9993 140 0.9985 107 0.9996 92 0.9986 104 0.9997

                             Bi 2212-1 63 0.9943 84 0.9990 56 0.9939 69 0.9988 50 0.9917 65 0.9995 44 0.9870 62 0.9996      Bi 2212-2 49 0.9990 45 0.9994 51 0.9990 47 0.9997 51 0.9996 50 0.9996 48 0.9967 56 0.9989      Bi 2212-3 79 0.9978 88 0.9987 68 0.9937 83 0.9986 59 0.9876 80 0.9994 49 0.9788 74 0.9993      Bi 2212-4* 87 1.0000 87 1.0000 83 0.9998 82 0.9988 78 0.9974 91 0.9995            

Poly fit has almost always higher R2 value, otherwise equal

Fitting of initial curvewith increasing stress range

• Linear fit: No convergence of slope and R2

• 2nd order poly: Convergence, ~ same Eo as linear fits extrapolated to zero

Curve fitting of initial slope• 2nd order polynomial fit data better than 1st order

(linear) fit– Not surprising, but both fit with R2 > 0.99– Clearly higher R2 values; linear fit trends down with

increasing range for 2nd order fit doesn’t.– Comparable scatter in E0?

• Propose ?– For reinforced conductor, linear and 2nd order poly fits

could be used, but “range” and convergence criteria need to de defined

Curve fitting over larger rangeR2 values fit to 0.3%

power poly

3U19-1

3U19-2 0.9992 0.9999

3U19-3 0.9954 0.9999

3U19-4 0.9972 0.9999

3B15-1 0.9957 1

3B15-2 0.9936 0.9999

3B15-3 0.9964 1

3B15-4 0.9941 0.9999

range 0.9936-0.9992 0.9999-1

Bi 2212-1 0.9765 0.9989

Bi 2212-2 0.9131 0.9916

Bi 2212-3 0.9933 0.9974

Bi 2212-4 0.9711 0.9862

Data for increasing strain only (return lines removed)

Curve fitting of data 0 to 0.3%• 2nd order polynomial fit data better than power fit

– Consistently higher R2

– Increasing lower bound of range above zero for power fit (as per Standard (A-10) does not necessarily improve either fit

– Trendline power fit in Excel sometimes fails to find proper fit: solver works better

• Propose for Standard– “Range” and convergence criteria need to de defined

• Upper bound of range of fit of 0.5% (A-10) is too high• 0.3% is more reasonable, or fraction (80-90%?) of strain where

curve “kinks” (Relasticmax, elasticmax)

– Skip “a(-b)n” from Standard and replace with poly

Conclusions• Data with Nyilas and Shepic extensometers very

comparable• Unreinforced wire sensitive to handling and/or sample-

sample variation– Chucks are suitable, require careful handling– Value of round-robin TBD

• 2nd order polynomial fit of initial slope– Fits better than linear or a(-b)n

– Not necessarily better predictor of Eo compared to linear– Discussion needed to define criterion,

• 2nd order polynomial fit of slope to 0.3%– Fits better than a(-b)n

– Clearly a better choice– 0 to 0.5% is too large a range for BSCCO

• Some proposals made to adapt Standard

Miscellaneous proposals

• Test report section 10.2, Optional results– When reporting % elongation to failure, add

location of failure (at grips, within extensometer) to report