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®
Version
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Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
1/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
Organization (S):
EDF/MTI/MN
Manual of Validation
V7.22 booklet:Thermomechanical nonlinear statics of the voluminal structures
Document: V7.22.122
HSNV122 - Thermo plasticity and metallurgy
in great deformations in simple traction
Summary:
One treats the determination of the mechanical evolution of a cylindrical bar subjected to thermal evolutions
and metallurgical known and uniform (the metallurgical transformation is of bainitic type) and with one
mechanical loading of traction.
The relation of behavior is a model of plasticity in great deformations (control
STAT_NON_LINE
, motclé
DEFORMATION: “SIMO_MIEHE”
) with linear isotropic work hardening and plasticity
of transformation (control
STAT_NON_LINE
, motclé
RELATION: “META_EP_PT”
).
The yield stress and the slope of the traction diagram depend on the temperature and the composition
metallurgical. The expansion factor depends on the metallurgical composition.
The bar is modelized by axisymmetric elements.
The mechanical loading applied is a following pressure.
This case test is identical to the case test HSNV101 (modeling B, [V7.22.101]) in the direction where it is about same
material, of the same loading and the same thermal and metallurgical evolutions but in a version
in great deformations.
Code_Aster
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Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
2/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
1
Problem of reference
1.1 Geometry
C
D
H
With
B
has
Z
R
P
Radius: = 0.05 m has
Height: H = 0.2 m
1.2
Properties of material
The material obeys a law of behavior in great deformations with isotropic work hardening
linear and plasticity of transformation. For each metallurgical phase, the slope of work hardening is
data in the plan deformation logarithmic curve - rational stress.
E
=
=
F
S
F
S
L
L
O
O
.
ln (/
)
L L
O
yphase
E
T
phase
L
O
and
L
are, respectively, the initial length and the current length of the useful part of
the test-tube.
S
O
and
S
are, respectively, surfaces initial and current.
Code_Aster
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Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
3/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
C
E
T T
.
T T
H
T T
p
yaust
O
yfbm
O
fbm
aust
O
=
J m
C
W m
C
AP
AP
AP
AP
AP
C
AP
AP
3
1
=
=
=
=
+
-
=
+
-
=
=
-
-
-
-
-
-
-
-
2000000
9999 9
200000 10
400 10
0 5
10
0 3
530 10
0 5
10
15 10
1250 10
5
10
1
1
6
6
6
6
6
6
1
6
6
.
.
.
.
. (
)
.
. (
)
.
.
. (
)
!
!
!
aust
fbm
O
fbm
ref.
F
ref.
B
M
fbm
fbm
H
T T
K
T
K
K
F
Z
=
= -
-
-
=
=
=
=
=
=
-
-
-
-
-
-
235 10
50 10
5
10
2 52 10
0
900
10
2
6
1
6
6
3
1
10
.
.
. (
)
.
.
.
)
(1
C
AP
AP
AP
C
AP
1
!
!
with
C
p
=
heat-storage capacity
=
thermal conductivity
E
=
YOUNG modulus
=
Poisson's ratio
*
aust
=
characteristics relating to the austenitic phase
*
fbm
=
characteristics relating to the phases ferritic, bainitic and
martensitic
=
thermal expansion factor
fbm
ref.
=
deformation of the phases ferritic, bainitic and martensitic
at the temperature of reference, austenite being regarded as
not deformed at this temperature
y
=
yield stress
H
=
EE
E E
T
T
-
K
=
coefficient relating to the plasticity of transformation
F
=
function relating to the plasticity of transformation
The TRC used makes it possible to modelize a metallurgical evolution of bainitic type, on all
structure, of the form:
Z
T
T
T
.
T
fbm
=
=
-
-
<
=
0
1
1
1
1
2
1
1
2
2
2
.
if 60s
if 112 S
if
1.3
Boundary conditions and loadings
·
U
Z
= 0 on face AB (condition of symmetry).
·
traction imposed (following pressure) on the face CD:
()
p T
p T
T
T
p
O
O
=
=
=
360 10 AP
for
for
6 10 AP
60s
6
6
1
1
1
Note:
In great deformations, it is essential to use the following pressure to hold
count current surface and not of initial surface (before deformation).
·
T
T
T
O
=
+ µ
,
µ
= 5°C.s
1
on all the structure.
Code_Aster
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Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
4/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
1.4 Conditions
initial
T
T
O
ref.
=
° =
900
C
2
Reference solution
2.1
Calculation of the reference solution (cf bib [1] and [3])
For a tensile test according to the direction
X
, tensors of Kirchhoff
and of Cauchy
are
form:
=
0 0
0 0 0
0 0 0
and
=
0 0
0 0 0
0 0 0
with
=
J
Variation of volume
J
is given by the resolution of
J
K J
J
HT
HT
3
2
3
2
3
3
0
-
+
- -
=
(
)
where
HT
is the thermal deformation. Celleci applies to an austenitic transformation bainitic:
[
]
HT
aust
aust
ref.
B
fbm
ref.
fbm
ref.
Z
T T
Z
T T
=
-
+
-
+
(
)
(
)
Note:
The coefficient
K
is the module of compression (not to be confused with the coefficients
K
phase
relating to the law of plasticity of transformation)
In plastic load, for an isotropic work hardening
R
linear, such as:
R
Z
H
Z H
p
aust aust
B
fbm
=
+
(
)
cumulated plastic deformation
p
is worth
p
J
Z
H
Z H
y
aust aust
B
fbm
=
-
+
with
y
aust
y
aust
B
yfbm
Z
Z
=
+
Code_Aster
®
Version
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Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
5/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
The tensors gradients of the transformation
F
and
F
and the tensor of plastic deformations
G
p
are form:
F
F
F
F
F
G
G
=
=
=
=
=
=
=
=
=
-
=
and
and
and
F
F
F
J=
FF
F
J F
J
F
F
F
F
J
F
F
F
G
G
G
G
yy
yy
yy
yy
-/
yy
yy
-/
yy
p
p
yy
p
yy
p
p
yy
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
2
1 3
1 3
1 2
det
/
det
det
/
p
p
G
=
-
(
)
/
1 2
The law of evolution of the plastic deformation
G
P
is written:
“/
“
(
)
“
G
G
p
K
Z
Z
p
p
B
B
B
= -
-
-
2
4
1
·
For
0
60
S
S
T
, one has
“Z
B
=
0
. There is no plasticity of transformation. One obtains
then:
G
E
p
p
=
-
2
·
For
60
176
S
S
T
, one has
=
constant
.
To integrate the law of evolution of the plastic deformation, it should be supposed that the stress of
Kirchhoff
vary very little, it estàdire that the variation of volume
J
is very small. Under this
assumption, one obtains
(
)
G
E
E
p
p
K Z
Z
B
B
B
=
-
-
-
2
4
2
2
/
The component
F
gradient of the transformation is given by the resolution of:
F
G F
G
p
p
3
3 2
1
0
-
-
=
µ
(
)
/
Lastly, the field of displacement
U
(in the initial configuration) is form
U
X + Y + Z
=
U
U
U
X
y
Z
. The components are given by:
(
)
(
)
(
)
U
F
X
U
J F
Y
U
J F
Z
X
y
Z
=
-
=
-
=
-
1
1
1
/
/
Code_Aster
®
Version
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Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
6/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
2.2 Notice
In the case test HSNV101 (modeling B), the coefficients of material were selected of such
manner not to have conventional plasticity
“p
=
0
during the metallurgical transformation which takes place
between moments 60 and 122s. Indeed if one writes the criterion of chargedécharge in this interval of
time, one obtains
F
p
= -
2750 - 250
with
=
MPa
360
who cancels himself only for only one value of the cumulated plastic deformation
p
.
For the law of behavior written in great deformations, the criterion of chargedécharge is written
between these two moments
F
J T
p
=
-
()
2750 - 250
with
=
MPa
360
In this case, as long as the variable
J
remain lower than the value obtained at time
T
= 60s, one will have
“p
=
0
. However the value of
J
is a function only of the value of the thermal deformation (
stress
is constant and the coefficient
K
is independent of the metallurgical phases and
temperature).
In this interval of time, thermal deformation
HT
is given by the following equation:
HT
T
T
=
-
-
-
-
-
8173 10
11807 10
2 90763 10
7 2
4
3
.
.
.
One traces cidessous the thermal deformation as well as the variation of volume
J
, solution of
the equation of the 3
ème
degree, according to time.
Time (S)
Thermal deformation
EDF
Electricity
from France
Mechanical department and Digital Models
Evolution of the thermal deformation enters moments 60 and 112s
agraf 08/11/1999 (c) EDF/DER 1992-1999
- 7.0
- 6.8
- 6.6
- 6.4
- 6.2
- 6.0
X
- 3
10
60
70
80
90
100
110
Thermal deformation according to time
Code_Aster
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Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
7/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
Time (S)
J
EDF
Electricity
from France
Mechanical department and Digital Models
Variation of J enters moments 60 and 112s
agraf 08/11/1999 (c) EDF/DER 1992-1999
9.790
9.795
9.800
9.805
9.810
9.815
9.820
9.825
X
- 1
10
60
70
80
90
100
110
Variation of volume
J
according to time
It is noted that the variable
J
decrease and increases same manner as the deformation
thermics. In this case, to know the moment from which the variable
J
is higher than the value
obtained at time 60s, it is enough to know the moment for which the thermal deformation is identical
with that obtained at time
T
= 60s. One finds by the resolution of the equation cidessus
T
= 84.46s.
2.3
Uncertainty on the solution
The solution is analytical. Two errors are made on this solution. The first gate on
calculation of the bainitic proportion of phase created. Preliminary metallurgical calculation does not restore
exactly the equation of [§1.2] giving
Z
fbm
according to time, this is why results of
reference presented cidessous is calculated with the bainitic proportion of phase calculated by
Code_Aster.
The second error is the assumption made on the stress of Kirchhoff
who is not constant on
the interval of time ranging between 60 and 176s. This will impact the calculation of displacement
U
X
and of
plastic deformation
G
P
.
2.4
Results of reference
One will adopt like results of reference displacement in the direction of the loading of
traction, the stress of Cauchy
, the Boolean indicator of plasticity
and plastic deformation
cumulated
p
. The various moments of calculations are
T
= 47, 48, 60, 83, 84, 85 and 176s. For the calculation of
displacement, the initial length of the bar in the direction of loading is of 0.2m.
Code_Aster
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Version
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Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
8/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
In all the cases, one has
·
3K
= 500.000 MPa (module of compression)
µ
= 76923.077 MPa
At time
T
= 47s, one has
Z
B
=
0
,
T
C
=
°
665
,
=
282
MPa
HT
y
p
J
p
G
F
U
= -
=
=
=
=
=
=
= -
-
-
55225 10
0 983855
277 45
282 5
0
0
1
10012
8 4347 10
3
4
.
.
.
.
.
.
=
MPa
MPa
m
At time
T
= 48s, one has
Z
B
=
0
,
T
C
=
°
660
,
=
288
MPa
HT
y
p
J
p
G
F
U
= -
=
=
=
=
=
=
= -
-
-
-
5 64 10
0 983508
283 25
280
1327 10
1
0 997
100256
5 9639 10
3
3
4
.
.
.
.
.
.
.
.
=
MPa
MPa
m
At time
T
= 60s, one has
Z
B
=
0
,
T
C
=
°
600
,
=
360
MPa
HT
y
p
J
p
G
F
U
= -
=
=
=
=
=
=
=
-
-
-
7 05 10
0 979337
352 56
250
3 7295 10
1
0 9281
103959
6 47595 10
3
2
3
.
.
.
.
.
.
.
.
=
MPa
MPa
m
At time
T
= 83s, one has
Z
B
=
0 442138
.
,
T
C
=
°
485
,
=
360
MPa
HT
y
-
p
-
J
.
.
p
.
G
F
.
U =.
= -
=
=
-
7 07867 10
0 979249
352 53
249 978
3 7295 10
0
0 8841277
106514
115441 10
3
2
2
.
.
.
=
=
=
=
=
MPa
MPa
m
At time
T
= 84s, one has
Z
B
=
0 461361
.
,
T
C
=
°
480
,
=
360
MPa
HT
y
-
p
-
.
J
.
.
p
.
G
F
.
U
.
= -
=
=
-
7 06031 10
0 979305
352 55
249 977
3 7296 10
1
0 8828104
106593
117051 10
3
2
2
=
=
=
=
=
=
MPa
MPa
.
.
At time
T
= 85s, one has
Z
B
=
0 480584
.
,
T
C
=
°
475
,
=
360
MPa
HT
y
-
p
-
J
.
.
p
.
G
F
.
U
.
= -
=
=
-
7 04032 10
0 979367
352 57
249 976
3 73044 10
1
0 8815276
106671
118644 10
3
2
2
.
.
.
=
=
=
=
=
=
MPa
MPa
At time
T
= 176s, one has
Z
B
=
1
,
T
C
= °
20
,
=
360
MPa
HT
y
p
J
p
G
F
U
= -
=
=
=
=
=
=
=
-
-
-
1068 10
0 968132
348 527
90
5 9432 10
1
0 82814
110053
17743 10
2
2
2
.
.
.
.
.
.
.
.
=
MPa
MPa
m
Code_Aster
®
Version
5.0
Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
9/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
2.5 References
bibliographical
One will be able to refer to:
[1]
V. CANO, E. LORENTZ: Introduction into Code_Aster of a model of behavior in
great deformations elastoplastic with isotropic work hardening internal Note EDF DER
HI-74/98/006/0
[2]
A. Mr. DONORE, F. WAECKEL: Influence structure transformations in the laws of
behavior elastoplastic Notes HI74/93/024
[3]
F. WAECKEL, V. CANO: Law of behavior great deformations élasto (visco) plastic
with metallurgical transformations [R4.04.03]
3 Modeling
3.1
Characteristics of modeling
N1
N2
N3
N4
N5
N6
N7
N8
N9
N10
N11
N12
N13
With
B
C
D
y
WITH = N4, B = N5, C = N13, D = N12.
Charge: the total number of increments is of 102 (4 increments of 0 with 46s, 2 increments of 46 with 48s,
6 increments of 48 with 60s, 26 of 60 with 112s, 4 of 112 with 116s and 60 increments until 176s).
convergence is carried out if the residue (resi_glob_rela) is lower or equal to 10
6
.
3.2
Characteristics of the mesh
A number of nodes: 13
A number of meshs and types: 2 meshs QUAD8, 6 meshs SEG3
3.3 Functionalities
tested
Controls
Keys
DEFI_MATERIAU
META_MECA_FO
[U4.23.01]
META_PT
AFFE_CHAR_MECA
PRES_REP
[U4.25.01]
STAT_NON_LINE
EXCIT
TYPE_CHARGE
SUIV
[U4.32.01]
COMP_INCR
RELATION
META_EP_PT
DEFORMATION
SIMO_MIEHE
Code_Aster
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Version
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Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
10/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
4
Results of modeling
4.1 Values
tested
Identification
Reference
Aster
% difference
T =
47 Displacement DY (N13)
8.4347 10
4
m
8.4303 10
4
m
0.052
T =
47 Variable
p
VARI (M1, PG1)
0.
0.
0.
T =
47
VARI (M1, PG1)
0
0
0
T =
47 Stress SIGYY (M1, PG1)
282. 10
6
AP
282. 10
6
AP
1.47 10
7
T =
48 Displacement DY (N13)
5.9639 10
4
m
5.9755 10
4
m
0.194
T =
48 Variable
p
VARI (M1, PG1)
1.3260 10
3
1.3263 10
3
0.024
T =
48
VARI (M1, PG1)
1
1
0.
T =
48 Stress SIGYY (M1, PG1)
288. 10
6
AP
288. 10
6
AP
6.80 10
7
T =
60 Displacement DY (N13)
6.476 10
3
m
6.4553 10
3
m
0.319
T =
60 Variable
p
VARI (M1, PG1)
3.7295 10
2
3.7294 10
3
0.002
T =
60
VARI (M1, PG1)
1
1
0
T =
60 Stress SIGYY (M1, PG1)
360. 10
6
AP
360. 10
6
AP
3.36 10
6
T =
83 Displacement DY (N13)
1.1544 10
2
m
1.1449 10
2
m
0.826
T =
83 Variable
p
VARI (M1, PG1)
3.7295 10
2
3.7294 10
2
0.002
T =
83
VARI (M1, PG1)
0
0
0
T =
83 Stress SIGYY (M1, PG1)
360. 10
6
AP
360. 10
6
AP
2.05 10
6
T =
84 Displacement DY (N13)
1.1705 10
2
m
1.1607 10
2
m
0.833
T =
84 Variable
p
VARI (M1, PG1)
3.7296 10
2
3.7294 10
2
0.005
T =
84
VARI (M1, PG1)
1
1
0
T =
84 Stress SIGYY (M1, PG1)
360. 10
6
AP
360. 10
6
AP
7.51 10
6
T =
85 Displacement DY (N13)
1.1864 10
2
m
1.1762 10
2
m
0.859
T =
85 Variable
p
VARI (M1, PG1)
3.7304 10
2
3.7305 10
2
0.002
T =
85
VARI (M1, PG1)
1
1
0
T =
85 Stress SIGYY (M1, PG1)
360. 10
6
AP
360. 10
6
AP
7.64 10
6
T =
176 Displacement DY (N13)
1.7743 10
2
m
1.7615 10
2
m
0.719
T =
176 Variable
p
VARI (M1, PG1)
5.943 10
2
5.9219 10
2
0.354
T =
176
VARI (M1, PG1)
1
1
0
T =
176 Stress SIGYY (M1, PG1)
360. 10
6
AP
359.93 10
6
AP
0.003
4.2 Parameters
of execution
Version: 5.02.10
Machine: claster
Overall dimension memory:
128 Mo
Time CPU To use:
146.5 seconds
Code_Aster
®
Version
5.0
Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
11/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
5
Summary of the results
The results found with Code_Aster are very satisfactory with percentages of error
lower than 0.9%, knowing that the analytical solution of reference makes the dead end on certain aspects
what precisely takes into account the solution of Code_Aster. This can explain the differences
observed.
Code_Aster
®
Version
5.0
Titrate:
HSNV122 - Thermo plasticity and metallurgy in great deformations
Date:
18/02/00
Author (S):
F. WAECKEL, V. CANO
Key:
V7.22.122-A
Page:
12/12
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
Intentionally white left page.