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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)-.

Report/SINTAP/UC/07

University of Cantabria Page 2 of 26

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

University of Cantabria Page 9 of 26

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

Report/SINTAP/UC/07

University of Cantabria Page 11 of 26

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

University of Cantabria Page 19 of 26

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

University of Cantabria Page 21 of 26

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

University of Cantabria Page 23 of 26

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

University of Cantabria Page 25 of 26

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

Report/SINTAP/UC/07

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

Report/SINTAP/UC/07

University of Cantabria Page 27 of 26

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

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