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
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
Organization (S):
EDF/IMA/MN
Manual 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 (control
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 (control
STAT_NON_LINE
[U4.32.01] key word
TYPE_CHARGE:“SUIV”
).
The bar is modelized by a voluminal element (HEXA8, modeling A).
Results obtained by
Code_Aster
do not differ from the theoretical solution.
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
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
1
Problem of reference
1.1 Geometry
1
2
3
4
1.000 (mm)
1.000 (mm)
y
X
1.2
Material properties
Behavior hyper-rubber band of COMING SAINT - KIRCHHOFF:
(
) (
) ()
S
E 1
E
=
+
-
+ +
=
E
E
E
1
1 2
1
200 000
0 3
tr
.
MPa
=.
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.
1
2
3
4
1 '
2 '
4 '
Overall rotation (0 < T < 1 S)
2 '
4 '
1 '
3
Traction (1 S < T < 2 S)
T
=
p
NR
NR
: normal
external with
face [2, 4].
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
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
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
Z
Y
X
Z
B
Y
X has
Z
U
X
bY
AX
Y
-
-
=
-
-
-
rotation
traction
that is to say
0
The gradient of the transformation and the deformation of Green-Lagrange are then:
F
E
=
-
=
=
-
=
-
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
2
1
2
2
2
B
has
E
E
E
has
E
B
X
y
X
y
where
The relation of behavior leads to a tensor of Lagrangian stresses diagonal (with
µ
and
coefficients of Lamé):
(
)
(
)
(
) (
)
(
)
S
E
E
S
E
E
E
E
S
E
E
xx
X
y
yy
X
y
zz
X
y
=
+
+
=
+
+
= + -
=
+
=
+
µ
µ
µ
2
2
1
1 2
2 1
where
One deduces the tensor from it from the stresses of Cauchy, him so diagonal:
X
y
y
X
Z
Z
B
S has
has
B S
B S has
=
=
=
1
Finally the boundary conditions are written:
(
)
(
)
X
y
p
=
= -
0
free edge
traction
One can moreover calculate the efforts exerted on the faces:
[]
[]
[]
[
]
[
]
[
]
1, 3
3, 4
1, 2, 3, 4
on the lower side of the face
on the higher side of the face
1, 3
1, 2, 3,4
1, 2, 3,4
F
F
F
y
y
O
X
Z
Z
O
Z
O
B S
B S has
B S has
=
=
=
-
-
0
where
[]
S
O
initial surfaces of the faces represent.
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
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
2.2
Results of reference
One adopts like results of reference displacements, the deformations of Grenn-Lagrange, them
stresses 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
E
E
X
y
=
=
= -
0.9539
0.105
0.045
The stresses of Cauchy are worth:
X
y
Z
=
=
=
0
26610.3 MPa
6597.6 MPa
Lastly, the exerted forces are:
[]
(
)
F
F
F
X
y
O
Z
S
NR
NR
=
= -
= -
0
25384
6.9228 10
lower side
9
1 3
,
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
Eric LORENTZ “a nonlinear relation of behavior hyperelastic” Notes intern
EDF/DER HI-74/95/011/0
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
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
Intentionally white left page.
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
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
Voluminal modeling:
1 mesh HEXA 8
1 mesh QUAD4
1
2
3
4
1.000 (mm)
X
y
Z
5
6
7
8
·
rigid phase of rotation 0
T
1 S
[]
[]
[]
[]
3,7
1,5
2,6
4,8
=
=
DX
DY
DZ
DX
T
DY
T
DZ
DZ
DZ
0
0
0
1000
2
1000 1
2
0
0
0
=
= -
= -
-
=
=
=
sin
cos
·
phase of traction: 1s
T
2s
-
boundary conditions (
TYPE_CHARGE: “DIDI”
)
[]
[]
[]
[]
3,7
1,5
2,6
4,8
=
=
DX
DY
DZ
DY
DZ
DZ
DZ
0
0
0
0
0
0
0
=
=
=
=
=
-
loading: pressure (negative) on the face [2, 4, 8, 6]
(
PRES_REP
): net [2, 4, 8, 6] (QUAD4):
NEAR
= 26610.3 (T1).
3.2
Characteristics of the mesh
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
EXCIT
EXCIT
DEFORMATION: “GREEN”
TYPE_CHARGE: “DIDI”
TYPE_CHARGE: “SUIV”
[U4.32.01]
CALC_NO
OPTION: “FORC_NODA”
GEOMETRY: “DEFORMED”
[U4.61.03]
CALC_ELEM
OPTION: “EPSG_ELNO_DEPL”
[U4.61.02]
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
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
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
Stresses SIXX (PG1)
0
8. 10
10
Stresses SIYY (PG1)
26610.3
26610.3
0
Stresses SIZZ (PG1)
6597.6
6597.6
0
Stresses SIXY (PG1)
0
10
26
Stresses SIXZ (PG1)
0
10
11
Stresses SIYZ (PG1)
0
10
10
Deformation EPXX (PG1)
0.105
0.105
0
Deformation EPYY (PG1)
0.045
0.045
0
Deformation EPZZ (PG1)
0
10
16
Deformation EPXY (PG1)
0
10
14
Deformation EPXZ (PG1)
0
10
14
Deformation EPYZ (PG1)
0
10
16
Nodal reaction DX (NO3)
0
10
3
Nodal reaction DY (NO3)
6.3462 10
9
6.3461 10
9
0.001
Nodal reaction DZ (NO3)
1.7307 10
9
1.7307 10
9
0.004
4.2 Remarks
Calculation of the nodal force:
The force applied
F
on a face described by a linear mesh is distributed by:
1/4
1/4
1/4
1/4
F
node
=
1
4
F
4.3 Parameters
of execution
Version: 3.05.32
Machine: CRAY C90
Overall dimension memory:
8 MW
Time CPU To use:
33.59 seconds
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
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
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 pitch
time on the other.