Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
1/14
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: V7.22.121
HSNV121 - Traction in great deformations
plastics of a bar under loading
thermics
Summary:
This quasi-static thermomechanical test consists in heating a bar of rectangular section uniformly
(3D) or cylindrical (axisymmetric 2D) then to subject it to a traction. One validates the kinematics thus of
great deformations in plasticity (control
STAT_NON_LINE
, key word
deformation: “SIMO_MIEHE”
or “PETIT_REAC”
) for a relation of behavior in great deformations with isotropic work hardening
linear (control
STAT_NON_LINE
, key word
relation
: “VMIS_ISOT_LINE” and
“VMIS_ISOT_TRAC”
) with thermomechanical loading. With modelings hull or plate, them
great deformations in plasticity are accessible thanks to the key word
deformation: “PETIT_REAC”
provided that rotations remain weak.
The bar is modelized by a voluminal element (HEXA20, modeling A) or quadrangular (QUAD4, for
an axisymmetric modeling, modeling B) or by elements of plate or hull (DKT for
modeling C and COQUE_3D for modeling D).
The solution is analytical.
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
2/14
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
1.000 (mm)
1
2
3
4
1.000 (mm)
y
X
Z
1.2
Properties of material
The material obeys a law of behavior in great deformations figure with work hardening
isotropic linear, whose characteristics depend on the temperature.
The traction diagram is given in the plan deformation logarithmic curve - rational stress.
=
=
F
S
F
S
L
L
O
O
.
E
E
T
y
ln (/
)
L L
O
=
=
=
=
°
=
=
=
°
=
=
-
-
0 3
10
1000
20
250000
2500
120
200000
2000
4
1
.
K
T
E
E
T
E
E
y
T
T
MPa
with
C
MPa
MPa
with
C
MPa
MPa
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, initial and current surface. Between the temperatures 20°C and 120°C,
the characteristics are interpolated linearly.
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
3/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
1.3
Boundary conditions and loadings
The bar, initial length
L
O
, locked in the direction
OX
on the face [1,2] is subjected to one
uniform temperature
T
and with a mechanical displacement of traction
U
meca
on the face [3, 4].
sequences of loading are as follows:
1
4
2
3
T
unif
L
O
U
meca
0
120
1
2
0
293.3 mm
1
2
T (S)
T (S)
20
T ()
U
°
C
Temperature of reference:
T
ref.
= 20°C.
Note:
Mechanical displacement is measured starting from the configuration deformed by
thermal loading (T = 1s). To have total displacement, it is thus necessary to add it
thermal displacement obtained at time T = 1s.
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
4/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
2
Reference solution
2.1
Result of the reference solution
For a tensile test according to the direction
X
, the tensor of Kirchhoff
is form:
=
0 0
0 0 0
0 0 0
The tensors gradients of the transformation
F
and
F
and the isochoric tensor of plastic deformations
G
p
are form:
F
F
F
F
F
G
G
=
=
=
=
=
=
=
=
=
=
=
-
-
-
-
F
F
F
J=
FF
F
J F
J
F
F
F
F
J
F
F
F
G
G
G
G
G
yy
yy
yy
yy
/
yy
yy
/
yy
p
p
yy
p
yy
p
p
yy
p
p
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
1 2
and
and
and
det
/
det
det
(
)
/
/
By the law of behavior, one obtains the following relation:
=
- -
-
+
3
2
1
9
2
1
2
K J
K T T
J
J
ref.
(
)
(
) (
)
that is to say
J
T T
J
J
K
T T
ref.
ref.
3
2
3
1 2
3
3
0
-
-
-
+
-
-
=
(
)
(
)
(
)
The stress of Cauchy is written:
J
=
In plastic load for an isotropic work hardening
R
linear, such as:
R p
EE
E E p
T
T
()
=
-
one a:
p
E E
E E
T
T
y
=
-
-
(
)
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
5/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
The integration of the law of flow of the plastic deformation
G
p
give (knowing that
G
p
P
(
)
=
=
0
1
):
G
E
P
p
=
- 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
(
)
/
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
U
L X
U
F
L
U
v
L Y
v
J
F
L
U
v
L Z
X
O
O
y
O
O
Z
O
=
=
-
=
=
-
=
~
~
.
~
~
~
with
with
1
1
2.2
Results of reference
One will adopt like results of reference displacements, the stress of Cauchy
and
cumulated plastic deformation
p
.
At time T = 2 S (
T
= 100°C, traction
U
)
One seeks total displacement (thermal + mechanical) such as the stress
that is to say equalizes with:
= 1500 Mpa (with
T
= 120°C)
·
3K
= 500.000 MPa
µ
= 76923 MPa
·
J
= 103
.
·
= 1453 MPa
·
p
= 0,2475
·
G
p
= 0,609
·
F
= 1,289
·
F
= 1,303
·
~
U
= 303 mm
·
~
v
= 110 mm
2.3
Uncertainty on the solution
The solution is analytical. With the round-off errors near, one can consider it exact.
2.4 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
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
6/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
3 Modeling
With
3.1
Characteristics of modeling
Voluminal modeling:
1 mesh HEXA20
1 mesh QUAD8
1
2
3
4
1.000 (mm)
X
y
Z
5
6
7
8
10
11
9
12
15
16
19
20
17
18
13
Boundary conditions:
N2:
N1:
N6:
U
U
U
X
y
Z
=
=
= 0
U
U
X
Z
=
= 0
U
U
X
y
=
= 0
N9, N13, N14, N5, N17:
U
X
= 0
Charge: Traction on the face [3 4 8 7 11 16 19 15] + assignment of the same temperature on all them
nodes.
The total number of increments is 21 (1 increment between T = 0s and 1s, 20 increments between T = 1s and
2s)
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: 20
A number of meshs: 2
1 HEXA20
1 QUAD8
3.3 Functionalities
tested
Controls
STAT_NON_LINE
COMP_INCR:
DEFORMATION:
“SIMO_MIEHE”
RELATION
:
“VMIS_ISOT_TRAC”
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
7/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
4
Results of modeling A
4.1 Values
tested
Identification Reference
Aster %
difference
T = 2 Displacement DX (N8)
303
303.063
0.021
T = 2 Displacement DY (N8)
110
109.852
0.134
T = 2 Displacement DZ (N8)
110
109.852
0.134
T = 2 Stresses SIGXX (PG1)
1453
1458.51
0.379
T = 2 Variable
p
VARI (PG1)
0.2475
0.2504
1.182
4.2 Parameters
of execution
Version: NEW 5.04.14
Machine: CLASTER
Overall dimension memory:
8 MW
Time CPU To use:
44.5 seconds
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
8/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
5 Modeling
B
5.1
Characteristics of modeling
Axisymmetric modeling 2D:
1 mesh QUAD4
1 mesh SEG2
4
1
2
3
y
X
Boundary conditions:
N1:
N2:
U
y
= 0
U
y
= 0
Loading:
Traction on the face [3 4] (mesh SEG2) + assignment of the same temperature on all the nodes
The total number of increments is 21 (1 increment between T = 0s and 1s, 20 increments between T = 1s
and 2s)
Convergence is carried out if the residue resi_glob_rela is lower or equal to 10
6
.
5.2
Characteristics of the mesh
A number of nodes: 4
A number of meshs: 2
1 QUAD4
1 SEG2
5.3 Functionalities
tested
Controls
STAT_NON_LINE
COMP_INCR:
DEFORMATION:
“SIMO_MIEHE”
RELATION
:
“VMIS_ISOT_LINE”
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
9/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
6
Results of modeling B
6.1 Values
tested
Identification Reference
Aster %
difference
T = 2 Displacement DX (N3)
110
109.85
0.134
T = 2 Displacement DY (N3)
303
303.06
0.021
T = 2 Stresses SIGYY (PG1)
1453
1458.5
0.379
T = 2 Variable
p
VARI (PG1)
0.2475
0.2504
1.182
6.2 Parameters
of execution
Version: 5.04.14
Machine: CLASTER
Overall dimension memory:
8 MW
Time CPU To use:
32.7 seconds
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
10/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
7 Modeling
C
7.1
Characteristics of modeling
Modeling plates DKT thickness 1000 mm:
1 mesh QUAD4, 2 meshs TRIA3
1 mesh SEG2
1
2
3
4
y
X
6
5
Boundary conditions:
N2:
N1:
U
U
U
X
y
Z
X
y
Z
=
=
=
=
=
=
0
0
0
0
0
0
U
U
X
Z
=
=
0
0
Loading:
Traction on the face [3 4] (mesh SEG2) + assignment of the same temperature on all the nodes
The total number of increments is 21 (1 increment between T = 0s and 1s, 20 increments between T = 1s
and 2s)
Convergence is carried out if the residue resi_glob_rela is lower or equal to 10
6
.
7.2
Characteristics of the mesh
A number of nodes: 8
A number of meshs: 4
1 QUAD4
2 TRIA3
1 SEG2
7.3 Functionalities
tested
Controls
STAT_NON_LINE
COMP_INCR:
DEFORMATION:
“PETIT_REAC”
RELATION
:
“VMIS_ISOT_TRAC”
AFFE_CARA_ELEM
HULL:
THICK
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
11/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
8
Results of modeling C
8.1 Values
tested
Identification Reference
Aster %
difference
T = 2 Displacement DX (N3)
110
108.81
1.076
T = 2 Displacement DY (N3)
303
303.4
0.132
T = 2 Effort NXX (PG1)
1453 E+03
1497.4 E+03
3.059
T = 2 Variable
p
VARI (PG1)
0.2475
0.246
- 0.591
8.2 Parameters
of execution
Version: 5.04.14
Machine: CLASTER
Overall dimension memory:
8 MW
Time CPU To use:
23.67 seconds
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
12/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
9 Modeling
D
9.1
Characteristics of modeling
Modeling coques_3d thickness 1000 mm: 1 mesh QUAD9, 2 meshs TRIA7
1 mesh SEG3
1
2
3
4
y
X
5
10
6
12
8
14
7
13
9
11
17
16
15
Boundary conditions:
N2:
N5:
N1:
U
U
U
X
y
Z
X
y
Z
=
=
=
=
=
=
0
0
0
0
0
0
U
U
X
Z
=
=
0
0
U
U
X
Z
=
=
0
0
Loading:
Traction on the face [3 4] (mesh SEG3) + assignment of the same temperature on all the nodes
The total number of increments is 21 (1 increment between T = 0s and 1s, 20 increments between T = 1s
and 2s)
Convergence is carried out if the residue resi_glob_rela is lower or equal to 10
6
.
9.2
Characteristics of the mesh
A number of nodes: 17
A number of meshs: 4
1 QUAD9
2 TRIA7
1 SEG3
9.3 Functionalities
tested
Controls
STAT_NON_LINE
COMP_INCR:
DEFORMATION:
“PETIT_REAC”
RELATION
:
“VMIS_ISOT_TRAC”
AFFE_CARA_ELEM
HULL:
THICK
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
13/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
10 Results of modeling D
10.1 Values
tested
Identification Reference
Aster %
difference
T = 2 Displacement DX (N3)
110
108.38
1.476
T = 2 Displacement DY (N3)
303
303.4
0.132
T = 2 Stress SIXX (PG1)
1453
1496.6
3.001
T = 2 Variable
p
VARI (PG1)
0.2475
0.2458
0.680
10.2 Parameters
of execution
Version: 5.04.14
Machine: CLASTER
Overall dimension memory:
8 MW
Time CPU To use:
57.88 seconds
Code_Aster
®
Version
5.0
Titrate:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, P. MASSIN
Key
:
V7.22.121-B
Page:
14/14
Manual of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
11 Summary of the results
Results found with Code_Aster and
deformation: “SIMO_MIEHE”
are very satisfactory
with percentages of error lower than 0.4% on the stress and 1.2% on the variable
of work hardening. For elements of plate and hull the use of
deformation:
“PETIT_REAC”
give satisfactory results with percentages of error of 3% on the effort
or the stress and lower than 0.7% on the variable of work hardening.