Code_Aster
®
Version
8.1
Titrate:
SSNA104 - Hollow roll subjected to a pressure, linear viscoelasticity
Date:
02/11/05
Author (S):
PH. BONNIERES, S. LECLERCQ, L. SALMONA
Key
:
V6.01.104-B
Page:
1/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
Organization (S):
EDF-R & D/AMA, EDF-R & D/MMC, CS IF
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
Document: V6.01.104
SSNA104 - Hollow roll subjected to a pressure,
linear viscoelasticity
Summary:
This case-test makes it possible to validate the laws of LEMAITRE and LEMA_SEUIL established in Code_Aster in the case
of linear viscoelastic behavior. The found results are compared with an analytical solution.
Code_Aster
®
Version
8.1
Titrate:
SSNA104 - Hollow roll subjected to a pressure, linear viscoelasticity
Date:
02/11/05
Author (S):
PH. BONNIERES, S. LECLERCQ, L. SALMONA
Key
:
V6.01.104-B
Page:
2/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
1
Problem of reference
1.1 Geometry
Dimensions of the cylinder:
R
0
1 m
R
1
2 m
Appear 1.1-a: Cuts hollow roll and loading
1.2
Properties of materials
Young modulus: E= 1 MPa
Poisson's ratio:
=0.3
Law of LEMAITRE:
N
m
K
T
G
=
1
1
)
,
,
(
with
1
,
0
1
,
1
1
=
=
=
N
m
K
Law LEMA_SEUIL:
(
)
1
,
2
3
3
2
,
,
G
with
=
=
=
With
With
T
on all the mesh
10
10
-
=
S
Being given the value of the various parameters materials, the two laws are absolutely identical and
can thus be compared with the same analytical solution.
1.3
Boundary conditions and loading
Boundary conditions:
The cylinder is locked out of DY on the sides [AB] and [CD].
Loading:
The cylinder is subjected to an internal pressure on [DA] P
0
=1.E-3 MPa
Code_Aster
®
Version
8.1
Titrate:
SSNA104 - Hollow roll subjected to a pressure, linear viscoelasticity
Date:
02/11/05
Author (S):
PH. BONNIERES, S. LECLERCQ, L. SALMONA
Key
:
V6.01.104-B
Page:
3/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
2
Reference solutions
2.1
Method of calculation used for the reference solutions
The whole of this demonstration can be read with more details in the document [bib1].
In the case of a linear viscoelastic isotropic material, one can describe the behavior with the course
time using two functions
)
(T
I
and
)
(T
K
so that strains and stresses
can be written:
(
)
(
)
3
)
(
*
)
(
*
)
(
)
(
I
D
T
Tr
D
K
D
T
D
K
I
T
-
+
=
where
3
I
indicate the matrix identity of S/N 3
and * the product of convolution:
-
=
T
D
G
T
F
T
G
F
0
)
(
)
(
)
) (
*
(
One finds
kt
E
T
K
kt
E
T
I
2
1
)
(
,
1
)
(
+
=
+
=
The pressure P is imposed
0
at the moment t=0, the internal pressure is worth
0
P
)
(
)
(
T
H
T
p
=
where
-
<
-
=
0
1
0
0
)
(
T
if
T
if
T
H
with in this case
0
=
One uses the transform of Laplace Carson
-
+
=
=
0
)
(
))
(
(
)
(
dt
E
T
F
N
T
F
L
N
F
NT
From where
0
P
=
+
p
The solution of the elastic problem are equivalent is:
+
-
=
+
+
Z
R
R
R
R
0
0
0
1
0
0
0
1
2
2
1
2
2
1
where
2
0
2
1
2
0
0
R
R
R
P
-
=
One determines
+
Z
by the condition on
+
Z
data by the boundary conditions:
+
+
+
+
+
+
+
+
+
-
=
+
-
+
=
=
K
I
K
K
I
Z
Z
Z
Z
2
)
2
(
)
(
0
From where
(
)
+
-
+
=
+
Ek
p
p
Z
1
2
1
.
Code_Aster
®
Version
8.1
Titrate:
SSNA104 - Hollow roll subjected to a pressure, linear viscoelasticity
Date:
02/11/05
Author (S):
PH. BONNIERES, S. LECLERCQ, L. SALMONA
Key
:
V6.01.104-B
Page:
4/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
One finds by the transform of opposite Laplace
(
)
(
)
Eht
Z
E
T
-
-
-
=
2
1
1
)
(
, in the same way in
applying the transform of Laplace reverses on
R
and
, one finds
(
)
(
)
-
-
+
-
=
-
+
Eht
E
R
R
R
R
2
1
1
0
0
0
1
0
0
0
1
2
2
1
2
2
1
One deduces some:
(
)
(
)
-
-
-
-
-
-
=
-
-
-
Ekt
Ekt
Ekt
V
E
K
R
R
E
K
R
R
E
K
2
1
0
0
0
3
2
1
2
3
0
0
0
3
2
1
2
3
2
2
1
2
2
1
&
and while integrating with
0
)
0
(
=
V
;
(
)
(
)
-
-
-
-
-
-
-
=
-
-
-
Ekt
Ekt
Ekt
V
E
E
T
R
R
K
E
E
T
R
R
K
E
E
1
2
1
0
0
0
3
2
1
2
3
0
0
0
3
2
1
2
3
2
2
1
2
2
1
.
One deduces radial displacement from it
(
)
(
)
(
)
+
-
-
-
+
+
=
-
T
R
R
K
E
R
R
E
R
T
R
W
Ekt
2
2
1
2
2
1
2
3
2
1
3
2
2
1
1
1
)
,
(
2.2
Results of reference
Displacement DX on the node B and stresses SIXX, SIYY and SIZZ out of B
2.3
Uncertainty on the solution
0%: analytical solution
2.4 References
bibliographical
[1]
PH. BONNIERES: Two analytical solutions of axisymmetric problems in
linear viscoelasticity and with unilateral contact, Note HI-71/8301
Code_Aster
®
Version
8.1
Titrate:
SSNA104 - Hollow roll subjected to a pressure, linear viscoelasticity
Date:
02/11/05
Author (S):
PH. BONNIERES, S. LECLERCQ, L. SALMONA
Key
:
V6.01.104-B
Page:
5/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
3 Modeling
With
3.1
Characteristics of modeling
The problem is modelized in axisymetry.
3.2
Characteristics of the mesh
1000 meshs QUAD4
3.3
Functionalities tested
Controls
DEFI_MATERIAU LEMAITRE
STAT_NON_LINE COMP_INCR
LEMAITRE
4
Results of modeling A
4.1 Values
tested
Identification Moments Reference
Aster Variation
%
DX (B) 0.9
2.14498
E3
2.14493496E03
0.002%
SIXX (B)
0.9
0.0
4.8168 E6
4.8168 E6
SIYY (B)
0.9
2.7912 E4
2.759 E4
1.5%
SIZZ (B) 0.9 6.66
E4
6.635 E4
0.5%
Code_Aster
®
Version
8.1
Titrate:
SSNA104 - Hollow roll subjected to a pressure, linear viscoelasticity
Date:
02/11/05
Author (S):
PH. BONNIERES, S. LECLERCQ, L. SALMONA
Key
:
V6.01.104-B
Page:
6/6
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A
5 Modeling
B
5.1
Characteristics of modeling
The problem is modelized in axisymetry
5.2
Characteristics of the mesh
1000 meshs QUAD4
5.3
Functionalities tested
Controls
DEFI_MATERIAU LEMA_SEUIL
STAT_NON_LINE COMP_INCR
LEMA_SEUIL
6
Results of modeling B
6.1 Values
tested
Identification Moments Reference
Aster Variation
%
DX (B) 0.9
2.14498
E3
2.14493496E03
0.002%
SIXX (B)
0.9
0.0
4.81687 E6
4.8168 E6
SIYY (B)
0.9
2.7912 E4
2.759 E4
1.5%
SIZZ (B) 0.9 6.66
E4
6.635 E4
0.5%
7
Summary of the results
The results calculated by Code_Aster are in agreement with the analytical solutions but
very strongly depend on the refinement of the mesh.