Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
1/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
Organization (S):
EDF/IMA/MN
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
Document: V7.22.120
HSNV120 - Hyperelastic traction
of a bar under thermal loading
Summary:
This quasi-static thermomechanical test consists in heating a parallelepipedic bar uniformly, it
to subject to an important traction for finally letting it return in a discharged state. One validates thus
kinematics of the great hyperelastic deformations (control
STAT_NON_LINE
, key word
COMP_ELAS
)
for a relation of elastic behavior non-linear (
ELAS_VMIS_LINE
and
ELAS_VMIS_TRAC
) with
thermal loading.
The bar is modelized by a voluminal element (HEXA20, modeling A) or quadrangular (QUAD8,
assumption of the plane stresses, modeling B).
The results obtained by Aster do not differ from the theoretical solution.
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
2/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
1
Problem of reference
1.1 Geometry
1.000 (mm)
1
2
3
4
1.000 (mm)
y
X
Z
1.2
Material properties
The material obeys a law of isotropic nonlinear behavior hyperelastic to work hardening
linear isotropic.
S
y
E
E
E
T
E
=
2.10
5
MPa
E
T
=
2.10
3
MPa
y
=
10
3
MPa
=
0, 3
=
10
4
K
1
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
3/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
1.3
Boundary conditions and loadings
The bar locked in the direction
OX
on the face [1,2] is subjected to a uniform temperature
T
and
a tensile load
F
distributed on the face [3,4]. The sequences of loading are as follows:
1
4
2
3
F
T
unif
0
120
1
2
3
0
1298
1
2
3
T (S)
T (S)
20
T
°
C
()
F
(MPa)
Temperature of reference:
T
ref.
= 20°C.
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
4/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The field of displacement is sought
U
in the form:
(
)
U X y Z
ux
vy
vz
,
=
The gradient of the transformation, the deformation and its mechanical share are then:
(
)
(
)
(
)
(
)
F
E
F F 1
E
E
1
m
=
+
+
+
=
-
=
+
+
+
=
-
=
1
0
0
0
1
0
0
0
1
1
2
2
2
0
0
0
2
2
0
0
0
2
2
0 0
0
0
0 0
U
v
v
U U
v v
v v
T
has
B
B
T
with:
(
)
(
)
has
U U
T
B
v v
T
=
+ -
=
+ -
2
2
2
2
Note:
()
(
)
E
m
eq
B has
B has
B has
= - = -
>
it is supposed that
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
5/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
The relation of behavior is written:
(
)
(
)
(
)
(
)
S
K has
B
G has B
S
S
K has
B
G has B
xx
yy
zz
=
+
+
-
=
=
+
-
-
2
2
3
2
1
3
with:
3
1 2
K
E
=
-
modulate compressibility
To determine
G
by taking account of linear work hardening, one introduces:
·
the modulus of rigidity:
2
1
µ
=
+
E
·
the module of work hardening:
R
E E
E E
T
T
'
=
-
,
The “pseudo variable interns”
p
is worth then:
()
(
)
p
R
B has
R
eq
y
y
=
-
+
=
- -
+
2
3
2
3
µ
µ
µ
µ
E
m
'
'
Finally,
G
is written:
G
R p
B has
y
=
+
-
'
By taking account of the boundary conditions:
S
F
U
S
xx
yy
=
+
=
1
0
(dead load)
(free edge)
The system to be solved is written:
(
)
(
)
(
)
(
)
K has
B
R
B has
R
F
U
K has
B
R
B has
R
y
y
y
y
+
+
+
- -
+
=
+
+
-
+
- -
+
=
2
2
3
2
3
1
2
1
3
2
3
0
µ
µ
µ
µ
'
'
'
'
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
6/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
It is also written:
(
)
(
)
3
2
1
2
1
1 3
3
K has
B
F
U
B has
F
U
R
R
y
+
=
+
-
=
+
+
-
µ
µ
µ
'
'
With
F
fixed, it is thus about a nonlinear system in
U
and
v
, since
has
is quadratic in
U
and
B
quadratic in
v
.
Nevertheless, one can choose to fix
U
(thus
has
) and to solve a linear system in
F
and
B
(of which one
deduced
p
and
v
):
·
(
)
has
U U
T
=
+ -
2
2
·
1
1
6
3
1 3
1
1
2
2
3
+
-
=
+
+ +
=
+
U F
K B
K has
R
U F
B
has
R
y
µ
µ
µ
µ
'
'
·
(
)
p
B has
R
y
=
- -
+
2
3
µ
µ
'
·
(
)
v
B has T
=
+
+
-
1 2
1
It then remains to express the stress of Cauchy:
()
=
1
Det F F S F
T
That is to say here:
(
)
xx
xx
yy
zz
U
v
S
=
+
+
=
=
1
1
0
2
As for the force exerted on the face [3,4], because of assumption of died loads, she is written
simply:
[]
F
F
F
X
O
O
y
Z
F S
S
=
=
=
where
: initial surface of face 3,4
0
0
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
7/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
2.2
Results of reference
One will adopt like results of reference displacements, the stress of Cauchy and the force
exerted on the face [3,4] (in 3D only):
At time T = 2 S (
T
= 100°C, traction
F
)
In fact, one seeks
F
such as lengthening:
U
=
0 1
,
·
K
= 166.666 MPa
µ
= 76.923 MPa
R'
= 2.020 MPa
·
has
= 0.095
·
0.90909
10
47 500
104.76
153.85 10
128.85 10
MPa
6
3
3
.
F
B
F
B
F
B
-
=
+
=
=
= -
1 298
0 046
·
p
=
8.91 10
2
·
v
= -
3.70 10
2
·
xx
xy
X
yy
xz
y
zz
yz
Z
=
=
=
=
=
=
=
=
=
1 399.66 MPa
0
1.298 10 NR
0
0
0
0
0
0
9
F
F
F
At time T = 3 S
(
)
T
F
=
=
0
0
,
The bar returned in its initial state:
U
=
=
=
0
0
0
p
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]
E. LORENTZ: A nonlinear relation of behavior hyperelastic - internal Note EDF
DER HI-74/95/011/0
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
8/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
3 Modeling
With
3.1
Characteristics of modeling
Voluminal modeling:
1 mesh HEXA20
1 mesh QUAD8
1
2
3
4
1.000 (mm)
X
y
Z
5
6
7
8
10
11
9
12
15
16
19
20
17
18
13
Boundary conditions:
N2:
N1:
N6:
U
U
U
X
y
Z
=
=
=
0
U
U
X
Z
=
=
0
U
U
X
y
=
=
0
N9, N13, N14, N5, N17:
U
X
=
0
Charge: Traction on the face [3 4 8 7 11 16 19 15]
3.2
Characteristics of the mesh
A number of nodes: 20
A number of meshs: 2
1 HEXA20
1 QUAD8
3.3 Functionalities
tested
Controls
Keys
STAT_NON_LINE
COMP_ELAS:
DEFORMATION:
“GREEN”
[U4.32.01]
RELATION:
“ELAS_VMIS_LINE”
“ELAS_VMIS_TRAC”
EXCIT:
CHARGE:
THERMICS
CALC_NO
OPTION:
“FORC_NODA”
[U4.61.03]
GEOMETRY:
“DEFORMED”
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
9/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
T = 2 Displacement DX (N8)
100
100.
0.
T = 2 Displacement DY (N8)
37
37.005
0.013
T = 2 Displacement DZ (N8)
37
37.005
0.013
T = 2 Stresses SIGXX (PG1)
1399.66
1399.67
0.001
T = 2 Stresses SIGYY (PG1)
11013.986
10
10
/
T = 2 Stresses SIGZZ (PG1)
0
10
10
/
T = 2 Stresses SIGXY (PG1)
0
10
12
/
T = 2 Stresses SIGXZ (PG1)
0
10
12
/
T = 2 Stresses SIGYZ (PG1)
0
10
11
/
T = 2 Variable
p
VARI (PG1)
8.9110
2
8.91 10
2
0.
T = 3 Displacement DX (N8)
0
10
13
/
T = 3 Displacement DY (N8)
0
10
13
/
T = 3 Displacement DZ (N8)
0
10
14
/
T = 3 Stresses SIGXX (PG1)
0
10
10
/
T = 3 Stresses SIGYY (PG1)
0
10
11
/
T = 3 Stresses SIGZZ (PG1)
0
10
11
/
T = 3 Stresses SIGXY (PG1)
0
10
11
/
T = 3 Stresses SIGXZ (PG1)
0
10
11
/
T = 3 Stresses SIGYZ (PG1)
0
10
11
/
T = 3 Variable
p
VARI (PG1)
0
0
/
T = 2 nodal Force DX (N8)
1.081710
8
1.0817 10
8
0.003
T = 2 nodal Force DY (N8)
0
10
5
/
T = 2 nodal Force DZ (N8)
0
10
6
/
4.2 Remarks
Calculation of the nodal force:
The force applied to the face [3,4],
F
X
, is distributed between the various nodes according to weighting
following:
·
nodes nodes:
/
1 12
F
X
·
nodes mediums:
4 12
/
F
X
1/12
4/12
1/12
1/12
1/12
4/12
4/12
4/12
4.3 Parameters
of execution
Version: NEW 3.03.15
Machine: CRAY C90
Overall dimension memory:
8 MW
Time CPU To use:
47.2 seconds
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
10/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
5 Modeling
B
5.1
Characteristics of modeling
Modeling 2D forced plane:
1 mesh QUAD8
1 mesh SEG3
1
2
3
4
y
X
5
8
7
6
Boundary conditions:
N2:
N1:
N5:
U
X
=
0
U
X
=
0
U
X
=
0
U
y
=
0
Loading:
Traction on the face [3 4 7] (mesh SEG3)
5.2
Characteristics of the mesh
A number of nodes: 8
A number of meshs: 2
1 QUAD8
1 SEG3
5.3 Functionalities
tested
Controls
Keys
STAT_NON_LINE
COMP_ELAS:
DEFORMATION:
“GREEN”
[U4.32.01]
RELATION:
“ELAS_VMIS_LINE”
“ELAS_VMIS_TRAC”
EXCIT:
CHARGE:
THERMICS
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
11/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
6
Results of modeling B
6.1 Values
tested
Identification
Reference
Aster
% difference
T = 2 Displacement DX (N4)
100
100
0
T = 2 Displacement DY (N4)
37
37.004
0.013
T = 2 Stresses SIGXX (PG1)
1399.66
1399.67
0.001
T = 2 Stresses SIGYY (PG1)
0
10
12
/
T = 2 Stresses SIGXY (PG1)
0
10
12
/
T = 2 Variable
p
VARI (PG1)
8.9110
2
8.91 10
2
0
T = 3 Displacement DX (N4)
0
10
14
/
T = 3 Displacement DY (N4)
0
10
13
/
T = 3 Stresses SIGXX (PG1)
0
10
10
/
T = 3 Stresses SIGYY (PG1)
0
10
10
/
T = 3 Stresses SIGXY (PG1)
0
10
10
/
T = 3 Variable
p
VARI (PG1)
0
0
/
6.2 Parameters
of execution
Version: 3.03.13
Machine: CRAY C90
Overall dimension memory:
8 MW
Time CPU To use:
123,8 seconds
Code_Aster
®
Version
3
Titrate:
HSNV120 - Hyperelastic traction of a bar
Date:
21/05/96
Author (S):
E. LORENTZ
Key:
V7.22.120-A
Page:
12/12
Manual of Validation
V7.22 booklet: Thermomechanical statics nonlinear of the voluminal structures
HI-75/96/030/A
7
Summary of the results
The numerical and analytical results coincide remarkably. One can however be astonished by
execution time manifestly longer for modeling in plane stresses (123,8 S) that for
the 3D (47,2 S). The difference is explained by a discretization in time much finer for
plane stresses, related to problems of convergence (the algorithm of resolution of the equation
nonlinear scalar in
p
is still rudimentary).