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:
1/12
Organization (S): EDF/MTI/MN
Handbook 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 (command
STAT_NON_LINE, motclé DEFORMATION: “SIMO_MIEHE”) with linear isotropic work hardening and plasticity
of transformation (command 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 dilation coefficient depends on the metallurgical composition.
The bar is modelled 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.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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:
2/12
1
Problem of reference
1.1 Geometry
Z
Radius: = 0.05 m has
Height: H = 0.2 m
P
C
D
H
R
With
B
has
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 constraint.
F
F
L
=
=
.
S
So lo
E phase
T
phase
y
E
ln (L/lo)
lo and L are, respectively, the initial length and the current length of the useful part of
the test-tube.
So and S are, respectively, surfaces initial and current.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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:
3/12
C
3
-! - 1
1
-! - 1
p = 2000000. J m
C = 9999 9
. W m
C
E
6
aust
= 200000. 10 Pa
= 400. 106Pa + 0.
5 (T - T O
6
y
) 10 Pa
=.
fbm
0
3
= 530. 106Pa + 0.
5 (T - T O
6
y
) 10 Pa
= 15. -
10 6!C-1
H
= 1250. 106Pa - 5. (T - T O
6
fbm
aust
) 10 Pa
- 6! - 1
6
O
6
aust = 23 5
. 1
0 C H
fbm = - 50. 1
0 Pa - 5. (T - T) 10
Pa
ref.
- 3
- 1
fbm = 2 5
. 2 1
0 K F = 0. Pa
ref.
T
= 900! C
K
- 10
-
B = km = 10 Pa 1
F
fbm = 2. (1 - Z fbm)
with
C
=
heat-storage capacity
p
=
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 dilation coefficient
ref.
=
deformation of the phases ferritic, bainitic and martensitic
fbm
at the temperature of reference, austenite being regarded as
not deformed at this temperature
=
yield stress
y
EE
H
=
T
E - AND
K
=
coefficient relating to the plasticity of transformation
F
=
function relating to the plasticity of transformation
The TRC used makes it possible to model a metallurgical evolution of bainitic type, on all
structure, of the form:
0.
if T1
1 =
60s
T -
Z
1
if 1 T
fbm
=
< 2 2 =
112 S
2 -
1
.
1
if T
2
1.3
Boundary conditions and loadings
·
uZ = 0 on face AB (condition of symmetry).
· traction imposed (following pressure) on the face CD:
p T
O
for T
1 p = 6 106
Pa
(
p T)
O
=
6
for T
1 1 =
360 10 Pa
60s
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 O
=
+ µt, µ = 5°C.s1 on all the structure.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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:
4/12
1.4 Conditions
initial
T O
900 C T ref.
=
° =
2
Reference solution
2.1
Calculation of the reference solution (cf bib [1] and [3])
For a tensile test according to direction X, the tensors of Kirchhoff and Cauchy are
form:
0 0
0
0
= 0 0 0 and = 0 0
0 with = J
0 0 0
0 0 0
The variation of volume J is given by the resolution of
2
J 3
HT
- (+
) J 2
3
- J
HT
- 3 = 0
3K
where HT is the thermal deformation. Celleci applies to an austenitic transformation bainitic:
HT
ref.
ref.
ref.
= Zaust aust (T - T) + B
Z [fbm (T - T) + fbm]
Note:
The coefficient K is the module of compression (not to be confused with the coefficients
Kphase 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
the cumulated plastic deformation p is worth
J -
p
y
= Z H + Z H
aust aust
B fbm
with
aust
fbm
y = Zaust y
+ B
Z y
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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:
5/12
The tensors gradients of the transformation F and F and the tensor of plastic deformations G p
are form:
F
0
0
F = 0 F
0 and J= det F
2
yy
= FF
yy
Fyy = J/F
0
0
Fyy
F
0
0
- 1/3
- 1/3
F = J
F
F = J
F = 0 F
0 and det F
yy
=
1
-
F
1/2
yy = F
0
0
Fyy
G p
0
0
G p = 0
G p
p
p
p -
yy
0 and det G =
1 Gyy = (G)/12
p
0
0
Gyy
The law of evolution of the plastic deformation G P is written:
“G p/G p = - 2 “p - 4 K
(1 - Z) “Z
B
B
B
· For
0s T 60s, one has “Zb = 0. There is no plasticity of transformation. One obtains
then:
G p
E p
= - 2
· For
60s T 176s, one has = constant.
To integrate the law of evolution of the plastic deformation, it should be supposed that the constraint of
Kirchhoff varies very little, it estàdire that the variation of volume J is very small. Under this
assumption, one obtains
2
p
- 2 p - 4
K (Z - Z
B
B
B/2)
G = E
E
The component F of the gradient of the transformation is given by the resolution of:
1
F 3 -
F -
= 0
µ G p
(G p 3 2
)/
Lastly, the field of displacement U (in the initial configuration) is form
U = U X +u Y + U Z
X
y
Z
. The components are given by:
U = (F -)
1 X
X
uy = (J/F -)
1 Y
uz = (J/F -)
1 Z
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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:
6/12
2.2 Notice
In the case test HSNV101 (modeling B), the coefficients of material were selected of such
manner not to have traditional plasticity “p = 0 during the metallurgical transformation which takes place
between moments 60 and 122s. En effet if one writes the criterion of chargedécharge in this interval of
time, one obtains
F = - 2750 p
- 250 with =
360 MPa
who cancels 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) - 2750 p
- 250 with =
360 MPa
In this case, as long as the variable J remains 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 (
constraint is constant and the coefficient K is independent of the metallurgical phases and of
temperature).
In this interval of time, thermal deformation HT is given by the following equation:
HT =
- 7
T 2
-
- 4
T -
-
8173 10
11807 10
2 90763 10 3
.
.
.
One traces cidessous the thermal deformation as well as the variation of volume J, solution of
the equation of the 3rd degree, according to time.
EDF
Department Mécanique and Modèles Numériques
Electricity
Evolution of the thermal deformation enters moments 60 and 112s
from France
X
- 3
10
- 6.0
- 6.2
- 6.4
- 6.6
Thermal deformation
- 6.8
- 7.0
60
70
80
90
100
110
Time (S)
agraf 08/11/1999 (c) EDF/DER 1992-1999
Thermal deformation according to time
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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:
7/12
EDF
Department Mécanique and Modèles Numériques
Electricity
Variation of J enters moments 60 and 112s
from France
X
- 1
10
9.825
9.820
9.815
J 9.810
9.805
9.800
9.795
9.790
60
70
80
90
100
110
Time (S)
agraf 08/11/1999 (c) EDF/DER 1992-1999
Variation of volume J according to time
It is noted that the variable J decreases 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 constraint of Kirchhoff which is not constant on
the interval of time ranging between 60 and 176s. Ceci will impact the calculation of ux displacement 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 constraint of Cauchy, the Boolean indicator of plasticity and plastic deformation
cumulated p. Les various moments of calculations are T = 47, 48, 60, 83, 84, 85 and 176s. Pour the calculation of
displacement, the initial length of the bar in the direction of loading is of 0.2m.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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:
8/12
In all the cases, one has
·
3K = 500.000 MPa (module of compression) µ = 76923.077 MPa
At time T = 47s, one has Zb = 0, T =
°
665 C, = 282 MPa
HT = -
-
55225
.
10 3
J = 0 983855
.
= 277 45
. MPa
y = 282 5. MPa
p = 0
= 0
p
G = 1
F = 10012
.
U = -
-
8 4347
.
10 4
m
At time T = 48s, one has Zb = 0, T =
°
660 C, = 288 MPa
HT = -
-
5 6
. 4 1
0 3 J = 0 983508
.
= 28325
. MPa
- 3
y = 280. MPa
p = 1327
.
10
= 1
p
G = 0 997
.
F = 100256
.
U = -
-
5 9639
.
10 4
m
At time T = 60s, one has Zb = 0, T =
°
600 C, = 360 MPa
HT = -
-
7 0
. 5 1
0 3 J = 0 979337
.
= 352 56
. MPa
- 2
y = 250. MPa
p = 3 7295
.
10
= 1
p
G = 0 9281
.
F = 103959
.
U =
-
6 47595
.
10 3
m
At time T = 83s, one has Zb = 0 442138
.
, T =
°
485 C, = 360 MPa
HT = -
-
7
07867
.
10 3
J =.
0 979249
= 352.53 MPa
- 2
y = 249 978
.
MPa
p =.
3
7295 10
= 0
p
-
G = 0 8841277
.
F =.
106514
U =.
1
15441 10 2 m
At time T = 84s, one has Zb = 0 461361
.
, T =
°
480 C, = 360 MPa
HT = -.
-
7 06031 10 3
J =.
0 979305
= 352.55 MPa
- 2
y = 249 977
.
MPa
p =.
3 7296 10
= 1
p
-
G = 0 8828104
.
F =.
106593
U =.
1
17051 10 2
At time T = 85s, one has Zb = 0 480584
.
, T =
°
475 C, = 360 MPa
HT = -
-
7
04032
.
10 3
J =.
0 979367
= 352.57 MPa
- 2
y = 249 976
.
MPa
p =.
3 73044 10
= 1
p
-
G = 0 8815276
.
F =.
106671
U =.
118644 10 2
At time T = 176s, one has Zb = 1, T =
°
20 C, = 360 MPa
HT = -
-
1068
.
10 2
J = 0 968132
.
= 348527
.
MPa
- 2
y = 90. MPa
p = 5 9432
.
10
= 1
p
G = 0 82814
.
F = 110053
.
U =
-
17743
.
10 2
m
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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
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 Note 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
y
C
D
N13
N11
N12
N9
N10
N7
N6
N8
N1
N3
N4
N2
N5
With
B
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 106.
3.2
Characteristics of the grid
A number of nodes: 13
A number of meshs and types: 2 meshs QUAD8, 6 meshs SEG3
3.3 Functionalities
tested
Commands
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
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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:
10/12
4
Results of modeling
4.1 Values
tested
Identification
Reference
Aster
% difference
T = 47 Déplacement DY (N13)
8.4347 104 m
8.4303 104 m
0.052
T = 47 Variable p VARI (M1, PG1)
0.
0.
0.
T = 47 VARI (M1, PG1)
0
0
0
T = 47 Contrainte SIGYY (M1, PG1)
282. 106 Pa
282. 106 Pa
1.47 107
T = 48 Déplacement DY (N13)
5.9639 104 m
5.9755 104 m
0.194
T = 48 Variable p VARI (M1, PG1)
1.3260 103
1.3263 103
0.024
T = 48 VARI (M1, PG1)
1
1
0.
T = 48 Contrainte SIGYY (M1, PG1)
288. 106 Pa
288. 106 Pa
6.80 107
T = 60 Déplacement DY (N13)
6.476 103 m
6.4553 103 m
0.319
T = 60 Variable p VARI (M1, PG1)
3.7295 102
3.7294 103
0.002
T = 60 VARI (M1, PG1)
1
1
0
T = 60 Contrainte SIGYY (M1, PG1)
360. 106 Pa
360. 106 Pa
3.36 106
T = 83 Déplacement DY (N13)
1.1544 102 m
1.1449 102 m
0.826
T = 83 Variable p VARI (M1, PG1)
3.7295 102
3.7294 102
0.002
T = 83 VARI (M1, PG1)
0
0
0
T = 83 Contrainte SIGYY (M1, PG1)
360. 106 Pa
360. 106 Pa
2.05 106
T = 84 Déplacement DY (N13)
1.1705 102 m
1.1607 102 m
0.833
T = 84 Variable p VARI (M1, PG1)
3.7296 102
3.7294 102
0.005
T = 84 VARI (M1, PG1)
1
1
0
T = 84 Contrainte SIGYY (M1, PG1)
360. 106 Pa
360. 106 Pa
7.51 106
T = 85 Déplacement DY (N13)
1.1864 102 m
1.1762 102 m
0.859
T = 85 Variable p VARI (M1, PG1)
3.7304 102
3.7305 102
0.002
T = 85 VARI (M1, PG1)
1
1
0
T = 85 Contrainte SIGYY (M1, PG1)
360. 106 Pa
360. 106 Pa
7.64 106
T = 176 Déplacement DY (N13)
1.7743 102 m
1.7615 102 m
0.719
T = 176 Variable p VARI (M1, PG1)
5.943 102
5.9219 102
0.354
T = 176 VARI (M1, PG1)
1
1
0
T = 176 Contrainte SIGYY (M1, PG1)
360. 106 Pa
359.93 106 Pa
0.003
4.2 Parameters
of execution
Version: 5.02.10
Machine: claster
Obstruction memory:
128 Mo
Time CPU To use:
146.5 seconds
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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
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.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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
Intentionally white left page.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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