Code_Aster
®
Version
3
Titrate:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
1/10
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
V6.04.121 document
SSNV121 - Rotation and traction hyper-rubber band
of a bar
Summary:
This test of quasi-static mechanics consists in making turn of 90° a parallelepipedic bar, with
to subject to an important traction for finally letting it return in a discharged state. One validates thus
kinematics of the great deformations hyper-rubber bands (control
STAT_NON_LINE
[U4.32.01], key word
COMP_ELAS
), and thus in particular great rotations, for a relation of elastic behavior linear.
The bar is modelized by a voluminal element (HEXA8, modeling A) or plan (QUAD4, assumption
plane deformations, modeling B).
Results obtained by
Code_Aster
do not differ from the theoretical solution.
Code_Aster
®
Version
3
Titrate:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
2/10
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 St - 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 in the new configuration:
1
2
3
4
1 '
2 '
4 '
Overall rotation (0 < T < 1 S)
2 '
4 '
1 '
3
T
Traction (1 S < T < 2 S)
Code_Aster
®
Version
3
Titrate:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
3/10
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 and one
lengthening of a factor
in the direction
y
.
(
)
(
)
U X Y Z
X
Y
X
Y
,
=
- -
+
-
1
0
The gradient of the transformation and the deformation of Green-Lagrange are then:
(
)
F
E
=
-
+
=
=
+
0
1 0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
2
2
E
E
with
The relation of behavior leads then to a tensor of Lagrangian stresses diagonal:
(
)
(
) (
)
(
) (
)
S
E
E
S
S
E
E
xx
yy
zz
=
-
+
-
=
=
+
-
1
1
1 2
1
1 2
The boundary condition of the equilibrium equation then enables us to determine the value of
lengthening
:
()
(
)
(
)
(
) (
) (
) (
)
T
S
E
T
yx
xx
=
=
+
-
+
-
+
+
=
FS
1
1
1
1 2
1
2
2
The stress of Cauchy is given by:
(
)
=
=
=
+
=
+
1
1
1
Det F F S F
T
xx
zz
yy
yy
xx
S
S
Lastly, the force exerted on the faces:
·
[2,4]:
[]
[]
F
y
yy
yy
O
S
S
=
=
2 4
2 4
,
,
·
[4,3]:
[]
(
)
[]
F
X
xx
xx
O
S
S
=
=
+
4 3
4 3
1
,
,
·
[1,2,3,4]:
[
]
(
)
[
]
F
Z
zz
zz
O
S
S
=
=
+
1 2 3 4
1 2 3 4
1
,
,
where
[]
S
O
initial surfaces of the faces represent.
Code_Aster
®
Version
3
Titrate:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
4/10
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 stress of Cauchy and the force
exerted on the faces [2,4] and [4,3].
At time T = 2 S:
One seeks
T
such as lengthening
=
0 1
.
==>
T
= 31 096.154 MPa.
The stress of Cauchy is then:
xx
zz
yy
=
=
=
11 013.986 MPa
31 096.154 MPa
The exerted forces are:
[]
[]
[
]
F
F
F
X
O
y
O
Z
O
S
NR
S
NR
S
NR
=
×
=
×
=
×
12.115.385
31 096.154
12 115.385
.
,
,
,
4 3
2 4
1 2 3 4
At time T = 3 S:
The bar returned in its initial state:
=
=
=
0
0
0
F
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:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
5/10
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:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
6/10
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
Boundary conditions:
0
1
1
2
3
R T
()
T
(3,7): DX = 0
DY = 0
(1,5): DX = - 1.000 R (T)
DY = - 1.000 R (T)
(2,6): DX = - 2.000 R (T)
(4,8): DX = - 1.000 R (T)
Loading: Traction on the face [2,4,8,6]
net [2,4,8,6] (QUAD4): FY = 31 096.154 F (T) MPa
0
1
1
2
3
F T
()
T
3.2
Characteristics of the mesh
A number of nodes: 8
A number of meshs: 2
1 HEXA8
1 QUAD4
3.3 Functionalities
tested
STAT_NON_LINE
COMP_ELAS
DEFORMATION: “GREEN”
CALC_NO
OPTION: “FORC_NODA”
GEOMETRY: “DEFORMED”
Code_Aster
®
Version
3
Titrate:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
7/10
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
Identification
Reference
Aster
% difference
T = 2 Displacement DX (NO2)
100
100
0
T = 2 Displacement DY (NO4)
1100
1100
0
T = 2 Stresses SIGXX (PG1)
11013.986
11013.986
0
T = 2 Stresses SIGYY (PG1)
31096.154
31096.154
0
T = 2 Stresses SIGZZ (PG1)
11013.986
11013.986
0
T = 2 Stresses SIGXY (PG1)
0
10
9
/
T = 2 Stresses SIGXZ (PG1)
0
10
10
/
T = 2 Stresses SIGYZ (PG1)
0
10
10
/
T = 3 Displacement DX10 (NO2)
0
10
11
/
T = 3 Displacement DY (NO4)
0
10
12
/
T = 3 Stresses SIGXX (PG1)
0
10
9
/
T = 3 Stresses SIGYY (PG1)
0
10
10
/
T = 3 Stresses SIGZZ (PG1)
0
10
9
/
T = 3 Stresses SIGXY (PG1)
0
10
11
/
T = 3 Stresses SIGXZ (PG1)
0
10
10
/
T = 3 Stresses SIGYZ (PG1)
0
10
11
/
T = 2 nodal Force DX (NO8)
3.0289 10
9
3.0288 10
9
0.002%
T = 2 nodal Force DY (NO8)
7.774 10
9
7.774 10
9
0
T = 2 nodal Force DZ (NO8)
3.0289 10
9
3.0288 10
9
0.002%
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: 3.03.30
Machine: CRAY C90
Overall dimension memory:
8 MW
Time CPU To use:
62.3 seconds
Code_Aster
®
Version
3
Titrate:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
8/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
5 Modeling
B
5.1
Characteristics of modeling
Modeling 2D plane deformations
1
2
3
4
y
X
Boundary conditions:
0
1
1
2
3
R T
()
T
3:
DX = 0
DY = 0
1:
DX = - 1.000 R (T)
DY = - 1.000 R (T)
2:
DX = - 2.000 R (T)
4:
DX = - 1.000 R (T)
Loading:
Traction on the face [2,4]
net [2,4]: FY = 31 096.154 F (T) MPa
0
1
1
2
3
F T
()
T
5.2
Characteristics of the mesh
A number of nodes: 4
A number of meshs: 2
1 QUAD4
1 SEG2
5.3 Functionalities
tested
STAT_NON_LINE
COMP_ELAS
DEFORMATION: “GREEN”
CALC_NO
OPTION: “FORC_NODA”
GEOMETRY: “DEFORMED”
Code_Aster
®
Version
3
Titrate:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
9/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
6
Results of modeling B
6.1 Values
tested
Identification
Reference
Aster
% difference
T = 2 Displacement DX (NO2)
100
100
0
T = 2 Displacement DY (NO4)
1100
1100
0
T = 2 Stresses SIGXX (PG1)
11013.986
11013.986
0
T = 2 Stresses SIGYY (PG1)
31096.154
31096.154
0
T = 2 Stresses SIGZZ (PG1)
11013.986
11013.986
0
T = 2 Stresses SIGXY (PG1)
0
10
- 10
/
T = 3 Displacement DX (NO2)
0
10
12
/
T = 3 Displacement DY (NO4)
0
10
12
/
T = 3 Stresses SIGXX (PG1)
0
10
10
/
T = 3 Stresses SIGYY (PG1)
0
10
10
/
T = 3 Stresses SIGZZ (PG1)
0
10
10
/
T = 3 Stresses SIGXY (PG1)
0
10
10
/
T = 2 nodal Force DX (NO4)
6.0577 10
6
6.0577 10
6
0
T = 2 nodal Force DY (NO4)
15.5481 10
6
15.5481 10
6
0
6.2 Remarks
Calculation of the nodal force:
The force applied
F
on a face described by a linear mesh is distributed by:
1/2
1/2
F
node
=
1
2
F
6.3 Parameters
of execution
Version: NEW 3.03.30
Machine: CRAY C90
Overall dimension memory:
8 MW
Time CPU To use:
48.3 seconds
Code_Aster
®
Version
3
Titrate:
SSNV121 Rotation and traction hyper-rubber band of a bar
Date:
23/07/99
Author (S):
E. LORENTZ
Key:
V6.04.121-A
Page:
10/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A
7
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 law of behavior, the errors do not cumulate a pitch of
time on the other.