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, Key P. MASSIN
:
V7.22.121-B Page:
1/14
Organization (S): EDF/MTI/MN
Handbook 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 (command STAT_NON_LINE, key word deformation: “SIMO_MIEHE”
or “PETIT_REAC”) for a relation of behavior in great deformations with isotropic work hardening
linear (command 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 modelled 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.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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, Key P. MASSIN
:
V7.22.121-B Page:
2/14
1
Problem of reference
1.1 Geometry
y
1.000 (mm)
1
4
2
3
Z
1.000 (mm)
X
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 constraint.
F
F L
=
=
.
S
S
L
O
O
= 0 3
.
=
- 4
-
10
1
K
E
= 1000 MPa
T
y
with T
=
°
20
y
C
E
E
= 250000MPa
E
= 2500
T
MPa
with T
=
°
120 C
E
= 200000MPa
ln (L/L
E
= 2000 MPa
O)
T
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, initial and current surface. Between the temperatures 20°C and 120°C,
the characteristics are interpolated linearly.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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, Key P. MASSIN
:
V7.22.121-B Page:
3/14
1.3
Boundary conditions and loadings
The bar, initial length lo, blocked in direction OX on the face [1,2] is subjected to one
uniform temperature T and with a mechanical displacement of traction umeca on the face [3, 4].
sequences of loading are as follows:
lo
1
4
Tunif
U meca
2
3
T ()
°C
U
120
293.3 mm
20
T (S)
0
1
2
T (S)
0
1
2
Temperature of reference: Tréf = 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.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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, Key P. MASSIN
:
V7.22.121-B Page:
4/14
2
Reference solution
2.1
Result of the reference solution
For a tensile test according to 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
0
0
F =
0 F
2
yy
0
and J=
det F = FF
F = J/F
yy
yy
0
0
Fyy
F
0
0
F = -
J 1/3F
- 1/3
F = J
F = 0 Fyy
0
and det F = 1
-
1/2
F = F
0
0
F
yy
yy
p
G
0
0
G p =
0
G p
p
p
p
1/2
yy
0 and det G = 1
G =
-
(G
yy
)
p
0
0
Gyy
By the law of behavior, one obtains the following relation:
3K
9
K (T - Tref)
1
=
2
(J - 1) -
(J +)
2
2
J
that is to say
2
J 3 - (T - T) J 2
3
- J 1
(+
) - 3 (T - T
ref.
) = 0
3K
ref.
The constraint of Cauchy is written:
J =
In plastic load for an isotropic work hardening R linear, such as:
EE
R (p
T
) =
p
E -
AND
one a:
E - E
p
T
=
(-)
E E
y
T
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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, Key P. MASSIN
:
V7.22.121-B Page:
5/14
The integration of the law of flow of the plastic deformation G p gives (knowing that
G P (p =)
0 = 1):
G P
E p
= - 2
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
)/
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~
U =
X with u~ = (F -)
1 .l
X
L
O
O
v~
J
U =
Y
with v~ =
-
1 L
y
L
F
O
O
v~
U =
Z
Z
lo
2.2
Results of reference
One will adopt like results of reference displacements, the constraint 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 constraint is equal to:
= 1500 Mpa (with T = 120°C)
·
3K = 500.000 MPa µ = 76923 MPa
·
J = 10
. 3
·
= 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
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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, Key P. MASSIN
:
V7.22.121-B Page:
6/14
3 Modeling
With
3.1
Characteristics of modeling
Voluminal modeling:
1 mesh HEXA20
1 mesh QUAD8
Z
5
20
8
17
19
18
7
6
16
13
y
15
1
12
4
1.000 (mm)
9
11
2
10
3
X
Boundary conditions:
N2:
U = U = U
X
y
Z = 0
N9, N13, N14, N5, N17: U X = 0
N1:
U = U
X
Z = 0
N6:
U = U
X
y = 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 106.
3.2
Characteristics of the grid
A number of nodes: 20
A number of meshs: 2
1 HEXA20
1 QUAD8
3.3 Functionalities
tested
Commands
STAT_NON_LINE
COMP_INCR:
DEFORMATION:
“SIMO_MIEHE”
RELATION
:
“VMIS_ISOT_TRAC”
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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, Key P. MASSIN
:
V7.22.121-B Page:
7/14
4
Results of modeling A
4.1 Values
tested
Identification Reference
Aster %
difference
T = 2 Déplacement DX (N8)
303
303.063
0.021
T = 2 Déplacement DY (N8)
110
109.852
0.134
T = 2 Déplacement DZ (N8)
110
109.852
0.134
T = 2 Contraintes 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
Obstruction memory:
8 MW
Time CPU To use:
44.5 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:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, Key P. MASSIN
:
V7.22.121-B Page:
8/14
5 Modeling
B
5.1
Characteristics of modeling
Axisymmetric modeling 2D:
1 mesh QUAD4
1 mesh SEG2
y
4
3
X
1
2
Boundary conditions:
N1:
U y = 0
N2:
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 106.
5.2
Characteristics of the grid
A number of nodes: 4
A number of meshs: 2
1 QUAD4
1 SEG2
5.3 Functionalities
tested
Commands
STAT_NON_LINE
COMP_INCR:
DEFORMATION:
“SIMO_MIEHE”
RELATION
:
“VMIS_ISOT_LINE”
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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, Key P. MASSIN
:
V7.22.121-B Page:
9/14
6
Results of modeling B
6.1 Values
tested
Identification Reference
Aster %
difference
T = 2 Déplacement DX (N3)
110
109.85
0.134
T = 2 Déplacement DY (N3)
303
303.06
0.021
T = 2 Contraintes 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
Obstruction memory:
8 MW
Time CPU To use:
32.7 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:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, Key P. MASSIN
:
V7.22.121-B Page:
10/14
7 Modeling
C
7.1
Characteristics of modeling
Modeling plates DKT thickness 1000 mm:
1 mesh QUAD4, 2 meshs TRIA3
1 mesh SEG2
y
6
1
4
X
2
5
3
Boundary conditions:
N2:
U = 0 U = 0 U = 0 = 0 = 0 = 0
X
y
Z
X
y
Z
N1:
U = 0 U = 0
X
Z
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 106.
7.2
Characteristics of the grid
A number of nodes: 8
A number of meshs: 4
1 QUAD4
2 TRIA3
1 SEG2
7.3 Functionalities
tested
Commands
STAT_NON_LINE
COMP_INCR:
DEFORMATION:
“PETIT_REAC”
RELATION
:
“VMIS_ISOT_TRAC”
AFFE_CARA_ELEM
COQUE:
EPAIS
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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, Key P. MASSIN
:
V7.22.121-B Page:
11/14
8
Results of modeling C
8.1 Values
tested
Identification Reference
Aster %
difference
T = 2 Déplacement DX (N3)
110
108.81
1.076
T = 2 Déplacement 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
Obstruction memory:
8 MW
Time CPU To use:
23.67 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:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, Key P. MASSIN
:
V7.22.121-B Page:
12/14
9 Modeling
D
9.1
Characteristics of modeling
Modeling coques_3d thickness 1000 mm: 1 mesh QUAD9, 2 meshs TRIA7
1 mesh SEG3
y
12
8
14
1
4
17
15
5
11 9
7
16
X
2
10
6 13
3
Boundary conditions:
N2:
U = 0 U = 0 U = 0 = 0 = 0 = 0
X
y
Z
X
y
Z
N5:
U = 0 U = 0
X
Z
N1:
U = 0 U = 0
X
Z
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 106.
9.2
Characteristics of the grid
A number of nodes: 17
A number of meshs: 4
1 QUAD9
2 TRIA7
1 SEG3
9.3 Functionalities
tested
Commands
STAT_NON_LINE
COMP_INCR:
DEFORMATION:
“PETIT_REAC”
RELATION
:
“VMIS_ISOT_TRAC”
AFFE_CARA_ELEM
COQUE:
EPAIS
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
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, Key P. MASSIN
:
V7.22.121-B Page:
13/14
10 Results of modeling D
10.1 Values
tested
Identification Reference
Aster %
difference
T = 2 Déplacement DX (N3)
110
108.38
1.476
T = 2 Déplacement DY (N3)
303
303.4
0.132
T = 2 Contrainte 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
Obstruction memory:
8 MW
Time CPU To use:
57.88 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:
HSNV121 - Traction in great plastic deformations of a bar
Date:
16/11/01
Author (S):
V. CANO, E. LORENTZ, Key P. MASSIN
:
V7.22.121-B Page:
14/14
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 constraint and 1.2% on the variable
of work hardening. For elements of plate and hull the use of deformation:
“PETIT_REAC” gives satisfactory results with percentages of error of 3% on the effort
or the constraint and lower than 0.7% on the variable of work hardening.
Handbook of Validation
V7.22 booklet: Thermomechanical nonlinear statics of the voluminal structures
HI-75/01/010/A
Outline document