On the Strain Hardening Exponent
Definition and its influence
within SINTAP
UNIVERSITY OF CANTABRIA
Report/SINTAP/UC/07
April 1998
J. Ruiz Ocejo
F. Gutiérrez-Solana
Departamento de Ciencia e Ingeniería del Terreno y de los Materiales
E.T.S. de Ingenieros de Caminos, Canales y Puertos
Universidad de Cantabria
Avda de los Castros s/n
39005, Santander (Spain)
Tel. 34-942-201819, Fax 34-942-201818
Report/SINTAP/UC/07
University of Cantabria
Page 1 of 26
1. INTRODUCTION
During the last months, a specific group formed by British Steel, GKSS and the
University of Cantabria has been working on different aspects of the SINTAP Procedure
related to the Y/T ratio as the following:
•
Estimation of Y/T from YS.
•
Definition of N and estimation from Y/T.
•
∗
for Lr
∗
for Lr > 1.
1.
Treatment of yield plateau.
The University of Cantabria has carried out the analysis of the second point. During a
meeting held in Paris in last February, the first results were presented (ref: Report
SINTAP/UC/06) about the treatment for Lr
1. There, it was agreed that further work had to
be done on the different definitions of N.
2. WORK DONE
In order to clarify the use of N within SINTAP, the University of Cantabria was told to
reanalyse stress-strain curves and to calculate the strain hardening exponent through
different ways.
It was decided to perform the calculations by varying the last point considered in the
statistical fit. Therefore, four different intervals have been considered:
•
From YS up to UTS -noted in further figures as N (Y - U)-.
•
From YS up to the mean value of flow stress and UTS -noted in further figures as N (Y
- (F+U)/2)-.
•
From YS up to flow stress -noted in further figures as N (Y - F)-.
•
From YS up to the mean value of yield stress and flow stress -noted in further figures
as N (Y - (Y+F)/2)-.
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University of Cantabria
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Also, for all intervals, two types of fits have been done depending on whether or not
the yield point is fixed. Therefore, eight N values have been determined for each material
studied.
The objective of such analysis is to evaluate the differences between these definitions
of N and to guarantee conservatism when it is predicted by means of the Y/T ratio in order
to be used in a FAD or in a CDF.
3. RESULTS
In the following pages (4 to 22), the results corresponding to the stress-strain curves of
the nineteen studied materials which follow are presented:
•
4Y14A2 S275 JR Steel -provided by British Steel-.
•
4Y17A2 S355 J2 Steel -provided by British Steel-.
•
Y6T8D 355 EMZ Steel -provided by British Steel-.
•
Y6T26H 450 EMZ Steel -provided by British Steel-.
•
4Y18A2 450 EMZ Steel -provided by British Steel-.
•
Microalloyed Steel E500.
•
Y6A22D2C StE690 Steel -provided by British Steel-.
•
Y6A4A4D StE690 Steel -provided by British Steel-.
•
Microalloyed Steel E690 (1).
•
Microalloyed Steel E690 (2).
•
Microalloyed Steel E690 (3).
•
Microalloyed Steel E690 (4).
•
Microalloyed Steel E500.
•
Normalised Steel 4135A (1).
•
Normalised Steel 4135A (2).
•
Quenched Steel 4135B.
Report/SINTAP/UC/07
University of Cantabria
Page 3 of 26
•
Austenitic Steel.
•
Aged Stainless Steel.
•
Stainless Weld Steel.
•
Aluminium.
4. ANALYSIS
In pages 23 to 25 different comparisons between the results are shown. The graphics
on the left hand side of the page compare the values directly and a reference line (N=N) is
also plotted; the graphics on the right hand side compare the values relatively and the
reference line drawn corresponds to (N/N=1).
Pages 23 and 24 present comparisons between N values where the difference is the
last point considered in the mathematical fit for both yield point fixed or not. All values are
compared to the N value calculated up to UTS which corresponds to the common use. It can
be seen that the less points considered, the higher the relative difference is, specially when
the value is obtained only up to the mean value of YS and flow stress. Another remarkable
point could be that, for low N values, these different N definitions lead to lower N values
than the usual one, but for higher ones, it could lead to even higher figures. This change can
be placed at about 0.3.
In page 25, the comparison is between N values with the same points considered but
fixing or not the yield point. The graphics show that there are small differences between
them. It also can be seen that, generally, N values with yield point fixed are lower than N
values with yield point not fixed.
Finally, in page 26, a full diagram is presented where N is plotted versus Y/T ratio.
Different reference lines are also drawn.
Within SINTAP, a lower bound with the simplest possible mathematical expression
has been tried to be found in order to assure conservative assessment diagrams for Lr >1.
Based on several studies, a very simple function has been suggested: 0.5*(1 - Y/T).
It can be seen through this figure and the following one (detail) that this line cannot be
considered anymore as it could lead to non-conservative results, specially when N is
Report/SINTAP/UC/07
University of Cantabria
Page 4 of 26
analysed by fixing the yield point and only up to the flow stress or the average between this
and the yield stress.
These values recommend a lower bound between 0.3 and 0.4 times (1 - Y/T). Then, a
final function should be defined within Consortium.
Report/SINTAP/UC/07
University of Cantabria
Page 5 of 26
4Y14A2 S275 JR Steel
700
600
True stress
500
400
300
200
100
0
0.00
0.05
0.10
0.15
0.20
0.25
True strain
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
367.9
730.7
576.8
0.6378
700
700
600
600
500
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
400
0.02
0.04
0.06
0.08
0.1
Log true strain
Logarithmic least square fits
Yield point not fixed
Log true stress
Log true stress
Stress-strain curve
500
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
400
0.02
0.04
0.06
0.08
0.1
Log true strain
Logarithmic least square fits
Yield point fixed
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University of Cantabria
Page 6 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
20
0.29888
0.9958
0.27654
0.9924
14
0.31022
0.9921
0.27345
0.9836
10
0.30366
0.9834
0.26148
0.9717
4
0.21853
0.9658
0.19559
0.9580
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University of Cantabria
Page 7 of 26
4Y17A2 S355 J2 Steel
800
700
True stress
600
500
400
300
200
100
0
0.00
0.05
0.10
0.15
True strain
0.20
0.25
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
423.9
795.9
625.9
0.6773
800
800
700
700
600
500
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
400
Log true stress
Log true stress
Stress-strain curve
600
500
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
400
0.02
0.04
0.06
0.08
0.1
Log true strain
Logarithmic least square fits
Yield point not fixed
0.02
0.04
0.06
0.08
0.1
Log true strain
Logarithmic least square fits
Yield point fixed
Report/SINTAP/UC/07
University of Cantabria
Page 8 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
17
0.26793
0.9972
0.25555
0.9958
12
0.28266
0.9955
0.25673
0.9899
9
0.28505
0.9894
0.24839
0.9786
5
0.22290
0.9717
0.19893
0.9641
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University of Cantabria
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Y6T8D 355 EMZ Steel
700
600
True stress
500
400
300
200
100
0
0.00
0.05
0.10
0.15
0.20
0.25
True strain
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
408.3
706.9
542.7
0.7524
700
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
600
Log true stress
Log true stress
700
500
400
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
600
500
400
0.04
0.06
0.08
0.1
Log true strain
Logarithmic least square fits
Yield point not fixed
0.04
0.06
0.08
0.1
Log true strain
Logarithmic least square fits
Yield point fixed
Report/SINTAP/UC/07
University of Cantabria
Page 10 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
16
0.26052
0.9974
0.24013
0.9937
11
0.26094
0.9947
0.23514
0.9885
7
0.25231
0.9866
0.22197
0.9771
4
0.19617
0.9728
0.17827
0.9670
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Y6T26H 450 EMZ Steel
800
700
True stress
600
500
400
300
200
100
0
0.00
0.05
0.10
0.15
True strain
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
540.5
771.6
656.3
0.8236
700
800
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
Log true stress
Log true stress
800
600
500
0.01
Log true strain
Logarithmic least square fits
Yield point not fixed
0.1
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
700
600
500
0.01
Log true strain
0.1
Logarithmic least square fits
Yield point fixed
Report/SINTAP/UC/07
University of Cantabria
Page 12 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
21
0.15588
0.9889
0.13230
0.9751
14
0.13937
0.9848
0.11940
0.9720
10
0.12201
0.9857
0.10687
0.9757
5
0.08932
0.9875
0.08150
0.9818
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University of Cantabria
Page 13 of 26
4Y18A2 450 EMZ Steel
800
700
True stress
600
500
400
300
200
100
0
0.00
0.05
0.10
0.15
True strain
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
572.6
802.7
670.7
0.8537
800
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
700
Log true stress
Log true stress
800
600
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
700
600
0.02
0.04
0.06
0.08 0.1
Log true strain
Logarithmic least square fits
Yield point not fixed
0.02
0.04
0.06
0.08 0.1
Log true strain
Logarithmic least square fits
Yield point fixed
Report/SINTAP/UC/07
University of Cantabria
Page 14 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
14
0.14848
0.9840
0.12067
0.9626
11
0.13778
0.9772
0.11154
0.9549
7
0.10689
0.9663
0.08928
0.9492
5
0.08084
0.9694
0.07250
0.9625
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University of Cantabria
Page 15 of 26
Microalloyed Steel E500
700
600
True stress
500
400
300
200
100
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
True strain
Stress-strain curve
Log true stress
640
620
σu (MPa)
UTS (MPa)
YS/UTS
540.0
676.5
636.0
0.8491
660
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
600
580
620
600
580
560
560
540
540
0.004
0.006 0.008 0.01
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
640
Log true stress
660
σy ≈ YS (MPa)
0.02
Log true strain
Logarithmic least square fits
Yield point not fixed
0.04
0.06
0.004
0.006 0.008 0.01
0.02
0.04
Log true strain
Logarithmic least square fits
Yield point fixed
0.06
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University of Cantabria
Page 16 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
88
0.08995
0.9858
0.07622
0.9721
64
0.07944
0.9818
0.06951
0.9726
46
0.06619
0.9858
0.06259
0.9841
23
0.05196
0.9915
0.05650
0.9868
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University of Cantabria
Page 17 of 26
Y6A22D2C StE690 Steel
1000
True stress
800
600
400
200
0
0.00
0.02
0.04
0.06
0.08
True strain
0.10
0.12
0.14
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
680.0
889.9
770.8
0.8822
900
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
Log true stress
Log true stress
900
800
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
800
700
700
0.01
0.1
Log true strain
Logarithmic least square fits
Yield point not fixed
0.01
Log true strain
Logarithmic least square fits
Yield point fixed
0.1
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University of Cantabria
Page 18 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
17
0.10546
0.9755
0.08282
0.9485
12
0.09203
0.9679
0.07306
0.9424
8
0.07217
0.9635
0.05949
0.9443
5
0.05171
0.9692
0.04554
0.9596
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Y6A4A4D StE690 Steel
1000
True stress
800
600
400
200
0
0.00
0.02
0.04
0.06
True strain
0.08
0.10
0.12
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
734.6
946.5
841.0
0.8735
900
1000
True values
N (Y -U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
Log true stress
Log true stress
1000
800
700
900
True values
N (Y -U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
800
700
0.01
Log true strain
Logarithmic least square fits
Yield point not fixed
0.1
0.01
Log true strain
Logarithmic least square fits
Yield point fixed
0.1
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University of Cantabria
Page 20 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
14
0.07153
0.9610
0.05847
0.9419
10
0.06079
0.9603
0.05160
0.9466
6
0.04371
0.9772
0.04044
0.9735
4
0.03472
0.9939
0.03439
0.9939
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Microalloyed Steel E690 (1)
1000
True stress
800
600
400
200
0
0.00
0.01
0.02
0.03
True strain
0.04
0.05
0.06
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
840.4
952.1
904.0
0.9296
940
True values
N (Y -U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
920
Log true stress
Log true stress
940
900
880
920
900
880
860
860
840
840
0.006
0.008 0.01
0.02
Log true strain
Logarithmic least square fits
Yield point not fixed
0.04
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
0.006
0.008 0.01
0.02
Log true strain
Logarithmic least square fits
Yield point fixed
0.04
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University of Cantabria
Page 22 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
68
0.05169
0.9837
0.04825
0.9801
56
0.04383
0.9829
0.04224
0.9818
49
0.03689
0.9923
0.03759
0.9920
38
0.03449
0.9889
0.03710
0.9843
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University of Cantabria
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Microalloyed Steel E690 (2)
1000
True stress
800
600
400
200
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
True strain
Stress-strain curve
Log true stress
900
σu (MPa)
UTS (MPa)
YS/UTS
809.0
937.3
874.0
0.9256
920
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
900
Log true stress
920
σy ≈ YS (MPa)
880
860
880
860
840
840
820
820
0.006 0.008 0.01
0.02
0.04
Log true strain
Logarithmic least square fits
Yield point not fixed
0.06
True values
N (Y -U)
N (Y - (F+U)/2)
N (Y -F)
N (Y - (Y+F)/2)
0.006 0.008 0.01
0.02
0.04
Log true strain
Logarithmic least square fits
Yield point fixed
0.06
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University of Cantabria
Page 24 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
122
0.06488
0.9703
0.04866
0.9357
81
0.05361
0.9633
0.04214
0.9378
51
0.04055
0.9728
0.03521
0.9629
23
0.02806
0.9929
0.02884
0.9925
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Microalloyed Steel E690 (3)
1000
True stress
800
600
400
200
0
0.00
0.01
0.02
0.03
True strain
0.04
0.05
0.06
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
840.0
953.2
905.0
0.9282
940
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
920
Log true stress
Log true stress
940
900
880
920
900
880
860
860
840
840
0.006
0.008 0.01
0.02
Log true strain
Logarithmic least square fits
Yield point not fixed
0.04
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
0.006
0.008 0.01
0.02
Log true strain
Logarithmic least square fits
Yield point fixed
0.04
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University of Cantabria
Page 26 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
67
0.05133
0.9833
0.04797
0.9797
56
0.04385
0.9830
0.04227
0.9818
49
0.03691
0.9924
0.03763
0.9920
38
0.03451
0.9889
0.03715
0.9842
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Microalloyed Steel E690 (4)
1000
True stress
800
600
400
200
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
True strain
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
924.0
1051.8
994.0
0.9296
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
Log true stress
Log true stress
Stress-strain curve
1000
980
960
940
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y- (Y+F)/2)
1000
980
960
940
920
920
0.006
0.008 0.01
0.02
Log true strain
Logarithmic least square fits
Yield point not fixed
0.04
0.006
0.008 0.01
0.02
Log true strain
Logarithmic least square fits
Yield point fixed
0.04
Report/SINTAP/UC/07
University of Cantabria
Page 28 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
156
0.05953
0.9774
0.04821
0.9568
108
0.05185
0.9741
0.04334
0.9584
63
0.03891
0.9826
0.03581
0.9787
32
0.02939
0.9936
0.03089
0.9919
Report/SINTAP/UC/07
University of Cantabria
Page 29 of 26
Normalised Steel 4135A (1)
1000
True stress
800
600
400
200
0
0.00
0.01
0.02
0.03
0.04
True strain
0.05
0.06
0.07
900
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
782.0
962.0
905.0
0.8641
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
Log true stress
Log true stress
Stress-strain curve
800
900
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
800
0.006 0.008 0.01
0.02
Log true strain
Logarithmic least square fits
Yield point not fixed
0.04
0.06
0.006 0.008 0.01
0.02
0.04
Log true strain
Logarithmic least square fits
Yield point fixed
0.06
Report/SINTAP/UC/07
University of Cantabria
Page 30 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
107
0.09232
0.9939
0.08710
0.9920
67
0.08394
0.9916
0.08200
0.9913
42
0.07221
0.9953
0.07613
0.9933
21
0.06676
0.9893
0.07821
0.9678
Report/SINTAP/UC/07
University of Cantabria
Page 31 of 26
Normalised Steel 4135A (2)
1200
True stress
1000
800
600
400
200
0
0.00
0.01
0.02
0.03
0.04
0.05
True strain
0.06
0.07
0.08
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
690.0
1123.1
1042.0
0.6622
1000
900
800
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
700
0.006 0.008 0.01
0.02
Log true strain
0.04
Logarithmic least square fits
Yield point not fixed
0.06
Log true stress
Log true stress
1000
900
800
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
700
0.006 0.008 0.01
0.02
Log true strain
0.04
Logarithmic least square fits
Yield point fixed
0.06
Report/SINTAP/UC/07
University of Cantabria
Page 32 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
95
0.18575
0.9901
0.22388
0.9641
66
0.20777
0.9931
0.24229
0.9761
31
0.24804
0.9938
0.27957
0.9832
12
0.31911
0.9971
0.33914
0.9943
Report/SINTAP/UC/07
University of Cantabria
Page 33 of 26
Quenched Steel 4135B
2000
True stress
1600
1200
800
400
0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
True strain
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
1245.0
2039.3
1981.9
0.6282
2000
2000
1800
1800
1600
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
1400
1200
0.008
0.01
0.02
Log true strain
Logarithmic least square fits
Yield point not fixed
Log true stress
Log true stress
Stress-strain curve
1600
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
1400
1200
0.008
0.01
0.02
Log true strain
Logarithmic least square fits
Yield point fixed
Report/SINTAP/UC/07
University of Cantabria
Page 34 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
54
0.39377
0.9890
0.44835
0.9767
33
0.47407
0.9970
0.50530
0.9941
20
0.52602
0.9992
0.54161
0.9986
10
0.56462
0.9998
0.56748
0.9998
Report/SINTAP/UC/07
University of Cantabria
Page 35 of 26
Austenitic Steel
1000
True stress
800
600
400
200
0
0.00
0.10
0.20
0.30
True strain
0.40
0.50
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
236.7
968.0
611.8
0.3869
1000
1000
600
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
800
Log true stress
Log true stress
800
400
600
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
400
200
200
0.01
0.1
Log true strain
Logarithmic least square fits
Yield point not fixed
0.01
Log true strain
0.1
Logarithmic least square fits
Yield point fixed
Report/SINTAP/UC/07
University of Cantabria
Page 36 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
19
0.27312
0.9753
0.23153
0.9599
17
0.24982
0.9768
0.21521
0.9639
14
0.21247
0.9809
0.18973
0.9731
11
0.16724
0.9975
0.16269
0.9970
Report/SINTAP/UC/07
University of Cantabria
Page 37 of 26
Aged Stainless Steel
1000
True stress
800
600
400
200
0
0.00
0.05
0.10
0.15
True strain
0.20
0.25
0.30
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
335.0
934.9
712.0
0.4705
1000
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
800
Log true stress
Log true stress
800
1000
600
400
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
600
400
0.01
Log true strain
0.1
Logarithmic least square fits
Yield point not fixed
0.01
Log true strain
0.1
Logarithmic least square fits
Yield point fixed
Report/SINTAP/UC/07
University of Cantabria
Page 38 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
65
0.24793
0.9953
0.23898
0.9945
41
0.22953
0.9975
0.22840
0.9975
25
0.21337
0.9993
0.22022
0.9986
14
0.21991
0.9984
0.22967
0.9970
Report/SINTAP/UC/07
University of Cantabria
Page 39 of 26
Stainless Weld Steel
1000
True stress
800
600
400
200
0
0.00
0.05
0.10
0.15
0.20
0.25
True strain
0.30
0.35
0.40
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
454.0
987.4
684.0
0.6637
1000
800
1000
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
900
Log true stress
Log true stress
900
700
600
500
800
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
700
600
500
0.01
Log true strain
0.1
Logarithmic least square fits
Yield point not fixed
0.01
0.1
Log true strain
Logarithmic least square fits
Yield point fixed
Report/SINTAP/UC/07
University of Cantabria
Page 40 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
95
0.19908
0.9516
0.14552
0.9136
63
0.16607
0.9544
0.12891
0.9277
37
0.12807
0.9668
0.10887
0.9542
17
0.08970
0.9926
0.08765
0.9923
Report/SINTAP/UC/07
University of Cantabria
Page 41 of 26
Aluminium
250
True stress
200
150
100
50
0
0.00
0.02
0.04
0.06
0.08
True strain
0.10
0.12
0.14
Stress-strain curve
σy ≈ YS (MPa)
σu (MPa)
UTS (MPa)
YS/UTS
82.6
257.2
224.0
0.3687
200
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
100
90
Log true stress
Log true stress
200
True values
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
100
90
80
80
0.01
Log true strain
Logarithmic least square fits
Yield point not fixed
0.1
0.01
Log true strain
Logarithmic least square fits
Yield point fixed
0.1
Report/SINTAP/UC/07
University of Cantabria
Page 42 of 26
Number
of points
σy - σu
σy -
σ + σu
2
σy - σ
σy -
σy + σ
2
Yield point not fixed
Yield point fixed
N
r
N
r
127
0.29646
0.9881
0.32501
0.9827
58
0.35820
0.9985
0.35863
0.9985
31
0.37875
0.9997
0.36629
0.9990
17
0.36302
0.9993
0.34943
0.9983
Report/SINTAP/UC/07
University of Cantabria
Internal Use Only
0.6
1.75
0.5
1.50
0.4
1.25
N/N
N
Page 43 of 26
0.3
1.00
0.75
0.2
0.50
0.1
N (Y - (F+U)/2) / N (Y - U)
N (Y - (F+U)/2)
0.25
0
0.1
0.2
0.3
N (Y - U)
0.4
0
0.5
0.6
1.75
0.5
1.50
0.4
1.25
N/N
N
0
0.3
0.2
0.1
0.2
0.3
N (Y - U)
0.6
1.00
0.50
N (Y - F)
N (Y - F) / N (Y - U)
0
0.25
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0
0.6
1.75
0.5
1.50
0.4
1.25
N/N
N
0.5
0.75
0.1
0.3
0.2
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0.6
1.00
0.75
0.1
0.50
N (Y - (Y+F)/2)
N (Y - (Y+F)/2) / N (Y - U)
0
0.25
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0.6
1.75
0.6
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
0.5
1.50
1.25
N/N
0.4
N
0.4
0.3
1.00
0.2
0.75
0.1
0.50
N (Y - (F+U)/2) / N (Y - U)
N (Y - F) / N (Y - U)
N (Y - (Y+F)/2) / N (Y - U)
0.25
0
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0.6
Comparisons between different N values as a function of the last point considered
Yield point not fixed
Report/SINTAP/UC/07
University of Cantabria
Internal Use Only
0.6
1.75
0.5
1.50
0.4
1.25
N/N
N
Page 44 of 26
0.3
0.2
1.00
0.75
0.1
0.50
N (Y - (F+U)/2)
N (Y - (F+U)/2) / N (Y - U)
0
0.25
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0
0.6
1.75
0.5
1.50
0.4
1.25
N/N
N
0
0.3
0.2
0.1
0.2
0.3
N (Y - U)
0.6
1.00
0.50
N (Y - F)
N (Y - F) / N (Y - U)
0
0.25
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0
0.6
1.75
0.5
1.50
0.4
1.25
N/N
N
0.5
0.75
0.1
0.3
0.2
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0.6
1.00
0.75
0.1
0.50
N (Y - (Y+F)/2)
N (Y - (Y+F)/2) / N (Y - U)
0
0.25
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0.6
1.75
0.6
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
0.5
1.50
1.25
N/N
0.4
N
0.4
0.3
1.00
0.2
0.75
0.1
0.50
N (Y - (F+U)/2) / N (Y - U)
N (Y - F) / N (Y - U)
N (Y - (Y+F)/2) / N (Y - U)
0.25
0
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0
0.1
0.2
0.3
N (Y - U)
0.4
0.5
0.6
Comparisons between different N values as a function of the last point considered
Yield point fixed
Report/SINTAP/UC/07
University of Cantabria
Internal Use Only
Page 45 of 26
0.6
1.3
N yield point fixed / N yield point not fixed
N (Y - U)
N yield point fixed
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
N yield point not fixed
0.5
0.8
0
N yield point fixed / N yield point not fixed
0.5
N yield point fixed
0.9
0.1
0.2
0.3
0.4
N yield point not fixed
0.5
0.6
1.3
N (Y - (F+U)/2)
0.4
0.3
0.2
0.1
0
N (Y - (F+U)/2) / N (Y - (F+U)/2)
1.2
1.1
1
0.9
0.8
0.7
0
0.1
0.2
0.3
0.4
N yield point not fixed
0.5
0.6
0
0.6
0.1
0.2
0.3
0.4
N yield point not fixed
0.5
0.6
1.3
N yield point fixed / N yield point not fixed
N (Y - F)
0.5
N yield point fixed
1
0.6
0.6
0.4
0.3
0.2
0.1
0
N (Y - F) / N (Y - F)
1.2
1.1
1
0.9
0.8
0.7
0
0.1
0.2
0.3
0.4
N yield point not fixed
0.5
0.6
0
0.6
0.1
0.2
0.3
0.4
N yield point not fixed
0.5
0.6
1.3
N yield point fixed / N yield point not fixed
N (Y - (Y+F)/2)
0.5
N yield point fixed
1.1
0.7
0
0.4
0.3
0.2
0.1
0
N (Y - (Y+F)/2) / N (Y - (Y+F)/2)
1.2
1.1
1
0.9
0.8
0.7
0
0.1
0.2
0.3
0.4
N yield point not fixed
0.5
0.6
0
0.1
0.2
0.3
0.4
N yield point not fixed
0.5
0.6
1.3
N yield point fixed / N yield point not fixed
0.6
N (Y - U)
N (Y - (F+U)/2)
N (Y - F)
N (Y - (Y+F)/2)
0.5
N yield point fixed
N (Y - U) / N (Y - U)
1.2
0.4
0.3
0.2
0.1
1.2
1.1
1
0.9
N (Y - U) / N (Y - U)
N (Y - (F+U)/2) / N (Y - (F+U)/2)
N (Y - F) / N (Y - F)
N (Y - (Y+F)/2) / N (Y - (Y+F)/2)
0.8
0.7
0
0
0.1
0.2
0.3
0.4
N yield point not fixed
0.5
0.6
0
0.1
0.2
0.3
0.4
N yield point not fixed
Comparisons between different N values
as a function of whether or not the yield point is fixed
0.5
0.6
Report/SINTAP/UC/07
University of Cantabria
Internal Use Only
Page 46 of 26
N (Y - U) not fixed
N (Y - (F+U)/2) not fixed
N (Y - F) not fixed
N (Y - (Y+F)/2) not fixed
N (Y - U) fixed
N (Y - (F+U)/2) fixed
N (Y - F) fixed
N (Y - (Y+F)/2) fixed
0.6
0.5
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Y/T
All different definitions of N versus Yield/Tensile Ratio
0.16
0.14
0.12
0.10
N
N
0.4
0.08
0.06
0.04
0.02
0.00
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
0.96
Y/T
Detail of the previous figure
0.98
1.00
0.9
1