Code_Aster ®
Version
3
Titrate:
SSNV122 Rotation and following traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.122-A Page:
1/8
Organization (S): EDF/IMA/MN
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
V6.04.122 document
SSNV122 - Rotation and following traction
hyper-rubber band of a bar
Summary:
This test of quasi-static mechanics consists in making turn of 90° a parallelepipedic bar and to
to subject to an important traction by means of following forces. One validates the kinematics of large thus
deformations hyper-rubber bands (command STAT_NON_LINE [U4.32.01], key word COMP_ELAS), and thus in
private individual great rotations, for a relation of elastic behavior linear, as well as the catch in
count following forces (command STAT_NON_LINE [U4.32.01] key word TYPE_CHARGE:“SUIV”).
The bar is modelled by a voluminal element (HEXA8, modeling A).
The results obtained by Code_Aster do not differ from the theoretical solution.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
3
Titrate:
SSNV122 Rotation and following traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.122-A Page:
2/8
1
Problem of reference
1.1 Geometry
y
1.000 (mm)
1
2
3
4
X
1.000 (mm)
1.2
Material properties
Behavior hyper-rubber band of SAINT VENANT - KIRCHHOFF:
E
E
E = 200.000. MPa
S
= (
tr
+
1 +) (1 -
2)
(E) 1
E
1 +
= 0 3
.
1.3
Boundary conditions and loadings
The loading is applied in two times: first of all, an overall rotation of the structure,
followed by a traction exerted by following forces.
Overall rotation (0 < T < 1 S)
Traction (1 S < T < 2 S)
T = ­ p NR
NR: normal
external with
face [2, 4].
2 '
4 '
1
2
2 '
4 '
3
4
1 '
1 '
3
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
3
Titrate:
SSNV122 Rotation and following traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.122-A Page:
3/8
2
Reference solution
2.1
Method of calculation used for the reference solution
It is about a plane problem. One can seek the solution in the form of a rigid rotation followed of one
dilation of a factor has in a direction and B in the other:
X
- Y
(
B - Y)
- X - bY
rotation

traction




Y
X
X has
that is to say U = AX
- Y







Z
Z
Z
0

The gradient of the transformation and the deformation of Green-Lagrange are then:

2
0
- B

0
E
0

has -
0
1


X

ex =
F
=
has
0
0 E


= 0 ey
0
2
where
2
B - 1

0 0


1

0
0


0
E y =
2
The relation of behavior leads to a tensor of Lagrangian constraints diagonal (with
and µ coefficients of Lamé):
S
= + 2µ
+
xx
(
) E
E
X
y
E
= 1+ 1 - 2
S
= E
+ + 2µ
yy
X
(
)
(
) (
)
E y
where
E
µ = (21+)
S
= E
+ E
zz
X
y
One deduces the tensor from it from the constraints of Cauchy, him so diagonal:

B
has
=
=
= 1
X
S y
y
S X
Z
S Z
has
B
B has
Finally the boundary conditions are written:
= 0 (edge libr)
E
= -
X

(traction)
y
p
One can moreover calculate the efforts exerted on the faces:
[1, 3]
F = - B S
y
y
O [1, 3]
[3, 4]
F = 0
X
[
- ab S
on the lower side of the face
Z

O 1, 2, 3,4
1, 2, 3, 4]
[
]
F =
Z
Z ab So


[1, 2, 3,4] on the higher side of the face
where So [] represent initial surfaces of the faces.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
3
Titrate:
SSNV122 Rotation and following traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.122-A Page:
4/8
2.2
Results of reference
One adopts like results of reference displacements, the deformations of Grenn-Lagrange, them
constraints of Cauchy and forces exerted on the faces [1, 3], [3, 4] and [1, 2, 3, 4] in end of
loading (T = 2 S).
One seeks p such as dilation has = 1,1
==> p = ­ 26610.3 MPa.
Dilation B and displacements are then:
B = 0.9539 E = 0.105 E = -
X
y
0.045
The constraints of Cauchy are worth:
= 0
= 26610.3 MPa
=
X
y
Z 6597.6 MPa
Lastly, the exerted forces are:
F
=
X
0
F
= -
y
25384 So [1,] NR
3
F
= -
9
Z
6.9228 10
NR
(lower side)
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
Eric LORENTZ “Une nonlinear relation of behavior hyperelastic” internal Note
EDF/DER HI-74/95/011/0
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
3
Titrate:
SSNV122 Rotation and following traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.122-A Page:
5/8
Intentionally white left page.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
3
Titrate:
SSNV122 Rotation and following traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.122-A Page:
6/8
3 Modeling
With
3.1
Characteristics of modeling
Voluminal modeling:
1 mesh HEXA 8
1 mesh QUAD4
Z
5
6
y
7
8
1
2
1.000 (mm)
3
4
X
· rigid phase of rotation 0 T1 S
[3,7] DX = 0
DY = 0
DZ = 0
[
T

T
1,5] DX = 1000
-
sin
DY
1000 1 cos
DZ
0

2
= -
-


2
=
[2,6]
DZ = 0
[4,8]
DZ = 0
· phase of traction: 1s T 2s
-
boundary conditions (TYPE_CHARGE: “DIDI”)
[3,7] DX = 0 DY = 0 DZ = 0
[1,5]
DY = 0 DZ = 0
[2,6]
DZ = 0
[4,8]
DZ = 0
-
loading: pressure (negative) on the face [2, 4, 8, 6]
(PRES_REP): net [2, 4, 8, 6] (QUAD4): PRES = ­ 26610.3 (T1).
3.2
Characteristics of the grid
A number of nodes: 8
A number of meshs: 2
1 HEXA8
1 QUAD4
3.3 Functionalities
tested
Order
Keys
STAT_NON_LINE
COMP_ELAS
DEFORMATION: “GREEN”
[U4.32.01]
EXCIT
TYPE_CHARGE: “DIDI”
EXCIT
TYPE_CHARGE: “SUIV”
CALC_NO
OPTION: “FORC_NODA”
GEOMETRIE: “DEFORMEE” [U4.61.03]
CALC_ELEM
OPTION: “EPSG_ELNO_DEPL”
[U4.61.02]
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
3
Titrate:
SSNV122 Rotation and following traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.122-A Page:
7/8
4
Results of modeling A
4.1 Values
tested
The values are tested at the end of the loading (T = 2s)
Identification
Reference
Aster
% difference
Displacement DX (NO2)
­ 1953.94
­ 1953.92
0
Displacement DY (NO2)
100.
100.
0
Constraints SIXX (PG1)
0
8. 10­10
Constraints SIYY (PG1)
26610.3
26610.3
0
Constraints SIZZ (PG1)
6597.6
6597.6
0
Constraints SIXY (PG1)
0
10­26
Constraints SIXZ (PG1)
0
10­11
Constraints SIYZ (PG1)
0
10­10
Deformation EPXX (PG1)
0.105
0.105
0
Deformation EPYY (PG1)
­ 0.045
­ 0.045
0
Deformation EPZZ (PG1)
0
10­16
Deformation EPXY (PG1)
0
10­14
Deformation EPXZ (PG1)
0
10­14
Deformation EPYZ (PG1)
0
10­16
Nodal reaction DX (NO3)
0
10­3
Nodal reaction DY (NO3)
­ 6.3462 109
­ 6.3461 109
­ 0.001
Nodal reaction DZ (NO3)
­ 1.7307 109
­ 1.7307 109
0.004
4.2 Remarks
Calculation of the nodal force:
The force applied F to a face described by a linear mesh is distributed by:
1/4
1/4
F
= 1
node
F
4
1/4
1/4
4.3 Parameters
of execution
Version: 3.05.32
Machine: CRAY C90
Obstruction memory:
8 MW
Time CPU To use:
33.59 seconds
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
3
Titrate:
SSNV122 Rotation and following traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.122-A Page:
8/8
5
Summary of the results
It appears at the end of this test that the numerical solution coincides remarkably with the solution
analytical. It will be noticed however that the strong not linearity due to great rotations requires
a relatively fine discretization in time, without being penalizing on the precision since,
contrary to an incremental relation of behavior, the errors do not cumulate a step
time on the other.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A