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