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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
8/22 4. DEVELOPED PROGRAM
Material parameter obtainment:
- Young’s modulus determination: linear regression
ÍNDICE
Pequeño ajuste
9/22 4. DEVELOPED PROGRAM
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
10/22 4. DEVELOPED PROGRAM
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
10/22 4. DEVELOPED PROGRAM
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
11/22 4. DEVELOPED PROGRAM
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
12/22 4. DEVELOPED PROGRAM
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%
13/22 4. DEVELOPED PROGRAM
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
13/22 4. DEVELOPED PROGRAM
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
14/22 5. ANALYSIS OF RESULTS
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
15/22 5. ANALYSIS OF RESULTS
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
15/22 5. ANALYSIS OF RESULTS
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
15/22 5. ANALYSIS OF RESULTS
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
15/22 5. ANALYSIS OF RESULTS
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
15/22 5. ANALYSIS OF RESULTS
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
16/22 5. ANALYSIS OF RESULTS
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
17/22 5. ANALYSIS OF RESULTS
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
18/22 5. ANALYSIS OF RESULTS
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
18/22 5. ANALYSIS OF RESULTS
)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
19/22 5. ANALYSIS OF RESULTS
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
19/22 5. ANALYSIS OF RESULTS
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
20/22 5. ANALYSIS OF RESULTS
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