comparative study of analytical expressions for the …...one stage model ramberg-osgood up to 0.2%...

Real, E., Arrayago, I., Mirambell, E. and Westeel, R.

Comparative study of analytical expressions for the modelling of

stainless steel behaviour

Fourth International Experts Seminar Ascot, UK 6-7 December 2012

2/22 1. INTRODUCTION

• Stainless steel: nonlinear stress-strain behaviour described by analytical material models. •Different material models based on Ramberg-Osgood expression: •Material models use parameters (E0, s0.2, n,…) fitting experimental tests.

•Parameter obtainment: - Experimental data - Standards - tables - expressions

Different values: for different material models Standards: tables and analytical expressions for different stainless steel grades

n

2.00

002.0E

s

ss

Research significance:

obtain the main parameters for each material model

compare different material models, determine best approach

The aim of this paper is

1.- to present a program which provides (from the experimental data): the mechanical properties the nonlinear coefficients which better fit different material models

2.- to analyze the differences between current material models to: determine the most appropriate approach suggest some expressions for the material parameters

3/22 1. INTRODUCTION

4/22 2. EXISTING MATERIAL MODELS

Material Models Definition Parameters

One stage model Ramberg-Osgood up to 0.2% proof stress E0, s0.2, n

Two stage models

Mirambell -Real up to ultimate strain E0, s0.2, n, su, *

pu, m

Rasmussen up to ultimate strain E0, s0.2, pu

Gardner up to 1% strain E0, s0.2, n0-0.2, s1.0,

*p1.0, n0.2-1.0

Three stage models

Quach up to ultimate strain E0, s0.2, n0-0.2

Hradil et al. up to ultimate strain E0, s0.2, n0-0.2, s1.0,

*p1.0, n0.2-1.0, su, *

pu, n1.0-u

5/22 2. EXISTING MATERIAL MODELS

EN 1993-1-4, Annex C

01.0

2.0ln

20lnn

s

su

2.05.31ms

s

200 GPa for austenitic and austenitic-ferritic excluding 1.4539, 1.4529, 1.4547 (195 GPa)

220 GPa for ferritic

E0

σ 0.2 and σu

n

Table 2.1

or Table 4.1

m

for

for

2.0

m

2.0u

2.0u

2.0

2.0

n

2.00

E

002.0E

ss

ss

ss

s

ss

2.0ss

2.0ss

6/22 3. TEST DATA

42 stainless steel stress-strain curves

Austenitic grades:

1.4301 (12 coupons) 1.4541 (5 coupons)

1.4435 (5 coupons) 1.4307 (2 coupons)

Ferritic grades:

1.4003 (3 coupons) 1.4509 (6 coupons)

1.4016 (6 coupons) 1.4521 (3 coupons)

All coupons were annealed and tested in rolling direction

Stress

Strain

7/22 4. DEVELOPED PROGRAM

Material properties:

E0, σ0.2, σu, εu, etc.

Optimized nonlinear parameters n, m

Office Excel sheet with automatic processes (VBA module)

From any experimental stress-strain curve

through

ÍNDICE

Representative value of E0 : importance of the selected set of points

Initial experimental data dispersion

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

0

5

10

15

20

25

0,0E+00 5,0E-05 1,0E-04 1,5E-04

s (MPa)

Importance and effect of the considered last point

0

100

200

300

0,0E+00 1,0E-03 2,0E-03 3,0E-03

s (MPa)

0

100

200

300

0,0E+00 1,0E-03 2,0E-03 3,0E-03

s (MPa)

0

100

200

300

0,0E+00 1,0E-03 2,0E-03 3,0E-03

s (MPa)

Nonlinear branch

Considered first point


Material parameter obtainment:

- Young’s modulus determination: linear regression

ÍNDICE

Pequeño ajuste


Material parameter obtainment:

- Young’s modulus determination

- Simple calculation of the proof stresses

From E0 σ0.01 , σ0.2 , σ1.0 …

E0

E0

σ0.2


Material parameter obtainment

Nonlinear parameter optimization:

Least square adjustment: minimizing the error between curves

Error definition

250

252

254

256

258

260

262

264

266

268

270

0.0018 0.00185 0.0019 0.00195 0.002Strain (mm/mm)

Str

ess (

MP

a)

Analytical

modelExperimental

e(common)

P(σ)

Pmodel(σ)

P(σ’) nearest to P(σ)

Analytical model Experimental curve

e(considered)

Ai

2

0.1

ki

2

kimAki

2

01.0

)()(·minCe

s

ssssError definition:

Error definition

250

252

254

256

258

260

262

264

266

268

270

0.0018 0.00185 0.0019 0.00195 0.002Strain (mm/mm)

Str

ess (

MP

a)

Analytical

modelExperimental


Material parameter obtainment

Nonlinear parameter optimization:

e(common)

P(σ)

Pmodel(σ)

P(σ’) nearest to P(σ)

e(considered)

Analytical model Experimental curve

Ai

2

0.1

ki

2

kimAki

2

01.0

)()(·minCe

s

ssssError definition:

Least square adjustment: minimizing the error between curves


Program output:

1.4509 - Test: material parameters

s0.01 245 MPa

E0 206 880 MPa s0.05 303 MPa

s0.1 320 MPa

u 17.6% s0.2 331 MPa

s1.0 352 MPa

s10 452 MPa

Initial stress-strain data:

0

100

200

300

400

0.000 0.002 0.004 0.006

Stre

ss (

MP

a)

Strain (mm/mm)

Ferritic 1.4509: experimental data


Program output:

1.4509 - Modelling: nonlinear parameters

Ramberg-Osgood Mirambell-Real Rasmussen Gardner Hradil et al.

n 14.39 n 14.62 n 14.43 n0-0.2 14.36 n0-0.2 14.38 m 1.75 m 1.73 n0.2-1.0 1.45 n0.2-1.0 1.32 n1.0-u 4.43

E0.2 10 891 E0.2 10 729 E0.2 10 864 E0.2 10 914 E0.2 10 894 E1.0 2 126

Optimized for

strains up to 0.2%

Optimized for

strains up to 5%

Optimized for

strains up to 1%


Program output:

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02

s (MPa)

Material model comparison Ferritic grade 1.4509

Test (corrected)

Ramberg-Osgood

Mirambell-Real

Rasmussen

EN 1993-1-4

Gardner

Three-stage


Program output:

325.0

350.0

375.0

400.0

425.0

0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02

s (

MPa

)

Material model comparison Ferritic grade 1.4509

Test (corrected)

Ramberg-Osgood

Mirambell-Real

Rasmussen

EN 1993-1-4

Gardner

Three-stage

14/22 5. ANALYSIS OF RESULTS

Considering Rasmussen model : needs only 3 parameters ε=f(σ) expression is similar to EN 1993-1-4, Annex C Good agreement to experimental data for strains up to 1%

Nonlinear parameters from different material models are very similar


Nonlinear parameters fitted from Rasmussen material model are considered

Analysis: - accuracy of the classical expression for n - applicability of σ0.2/σu expressions to ferritic stainless steels - accuracy of the original expression for m

Considering Rasmussen model: needs only 3 parameters ε=f(σ) expression is similar to EN 1993-1-4, Annex C Good agreement to experimental data for strains up to 1%

Nonlinear parameters from different material models are very similar


0

50

100

150

200

250

300

350

400

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Str

ess

(M

Pa)

Strain (mm/mm)

Nonlinear parameter n definition

Experimental


0

50

100

150

200

250

300

350

400

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Str

ess

(M

Pa)

Strain (mm/mm)

Nonlinear parameter n definition classical expression

Experimental

σ0,2

σ0,01

01.0

2.0ln

20lnn

s

s


0

50

100

150

200

250

300

350

400

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Str

ess

(M

Pa)

Strain (mm/mm)

Nonlinear parameter n definition classical expression

Experimental

n=10 (original expression)

01.0

2.0ln

20lnn

s

s


0

50

100

150

200

250

300

350

400

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Str

ess

(M

Pa)

Strain (mm/mm)

Nonlinear parameter n definition new proposal

Experimental

σ0,05

σ0,2

05.0

2.0ln

4lnn

s

s


0

50

100

150

200

250

300

350

400

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Str

ess

(M

Pa)

Strain (mm/mm)

Nonlinear parameter n definition new proposal

Experimental

n=10.7 (new proposal)

05.0

2.0ln

4lnn

s

s


56789

101112131415

No

nli

ne

ar

pa

ram

ete

r n

Optimized

Originalexpression

Proposal

5

8

10

13

15

18

20

23

25

No

nli

ne

ar

pa

ram

ete

r n

Distribution of nonlinear parameter n A

ust

en

itic

s Fe

rrit

ics


0.25

0.50

0.75

1.00

σ0.2/σu

0.25

0.50

0.75

1.00

1.25

1.50

1.75

σ0.2/σu

Experimental

Original expression foraustenitics

Original expression forall alloys

Distribution of σ0.2/σu parameter A

ust

en

itic

s Fe

rrit

ics

)5n(0375.01

E1852.0

E1852.0

0

2.0

0

2.0

u

2.0 s

s

s

s

for austenitic

and duplex

for all alloys


y = 144.67x + 0.4587

0.60

0.65

0.70

0.75

0.80

0.0014 0.0015 0.0016 0.0017 0.0018 0.0019 0.0020

σ0.2/σu

σ0.2/E0

Experimental ferritic

Distribution of σ0.2/σu parameter for Ferritic Stainless Steels

)5n·(0375.01

E1852.0

0

2.0

u

2.0

s

s

s

0

2.0

u

2.0

E14546.0

s

s

s

Original expression Proposal linear expression


)5n·(0375.01

E1852.0

0

2.0

u

2.0

s

s

s

0

2.0

u

2.0

E14546.0

s

s

s

Original expression Proposal linear expression

0.25

0.50

0.75

1.00

1.25

1.50

1.75

σ0.2/σu Experimental

Original expressionfor austenitics

Original expressionfor all alloys

Proposal

Distribution of σ0.2/σu parameter for Ferritic Stainless Steels


Nonlinear parameter m

u

2.05.31ms

s

Original expression:

0

50

100

150

200

250

300

350

400

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Stre

ss (

MP

a)

Strain (mm/mm)

m=3.5 (original expression)

m=1.71 (experimental value)

Experimental


Nonlinear parameter m

New proposals:

u

2.03.21ms

s

u

2.01ms

s

Austenitics Ferritics

u

2.05.31ms

s

Original expression:

0

50

100

150

200

250

300

350

400

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Stre

ss (

MP

a)

Strain (mm/mm)

m=3.5 (original expression)

m=1.71 (experimental value)

Experimental


1.0

1.5

2.0

2.5

3.0

3.5

4.0

No

nli

ne

ar

pa

ram

ete

r m

1.50

1.75

2.00

2.25

2.50

2.75

3.00

No

nli

ne

ar

pa

ram

ete

r m

Optimized

Original expression

Proposal

Au

ste

nit

ics

Ferr

itic

s Distribution of nonlinear parameter m

21/22 6. CONCLUSIONS

• Parameters proposed in EN 1993-1-4 are not accurate enough. • Optimized nonlinear parameters: similar results for all analyzed material models. new expressions for determining n and m are proposed. • New linear approximation for σ0.2 /σu for ferritic stainless steels is also proposed.

05.0

2.0ln

4lnn

s

s

u

2.03.21ms

s

u

2.01ms

sfor austenitics for ferritics

0

2.0

0

2.0

u

2.0

E14546.0

E1852.0

s

s

s

sfor austenitics

for ferritics

New proposals obtained from a limited number of test data.

22/22 7. FUTURE WORK

Further research in order to: adjust the new proposals extend their applicability to cold-formed stainless steel

Real, E., Arrayago, I., Mirambell, E. and Westeel, R.

Fourth International Experts Seminar Ascot, UK 6-7 December 2012

Comparative study of analytical expressions for the modelling of

stainless steel behavior

01.0

2.0ln

20lnn

s

s

05.0

2.0ln

4lnn

s

s

u

2.05.31ms

s

u

2.03.21ms

s

u

2.01ms

s

0

2.0

0

2.0

u

2.0

E14546.0

E1852.0

s

s

s

s

EN 1993-1-4, Annex C Proposal

for austenitic and duplex

stainless steels

)5n(0375.01

E1852.0

E1852.0

0

2.0

0

2.0

u

2.0 s

s

s

s

for all alloys

for austenitics

for ferritics

for austenitics for ferritics

for

for

2.0

m

2.0u

2.0u

2.0

2.0

n

2.00

E

002.0E

ss

ss

ss

s

ss

2.0ss

2.0ss

comparative study of analytical expressions for the …...one stage model ramberg-osgood up to 0.2%...

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