Code_Aster
®
Version
6.0
Titrate:
SSNA106 - Hollow roll subjected to a behavior thermoviscoelastic
Date:
19/08/02
Author (S):
PH. BONNIERES, D. NUNEZ
Key
:
V6.01.106-A
Page:
1/6
Manual of Validation
V6.02 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A
Organization (S):
EDF/AMA, CS IF
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
V6.01.106 document
SSNA106 - Subjected hollow roll
with a behavior thermoviscoelastic
Summary:
This case-test makes it possible to in the case of validate the law of LEMAITRE established in Code_Aster behavior
thermoviscoelastic linear. The found results are compared with an analytical solution.
Code_Aster
®
Version
6.0
Titrate:
SSNA106 - Hollow roll subjected to a behavior thermoviscoelastic
Date:
19/08/02
Author (S):
PH. BONNIERES, D. NUNEZ
Key
:
V6.01.106-A
Page:
2/6
Manual of Validation
V6.02 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
R
0
1 m
R
1
2 m
1.2
Properties of materials
Young modulus: E= 1 MPa
Poisson's ratio:
=0.3
Expansion factor:
=0.7
Law of LEMAITRE:
N
m
K
T
G
=
1
1
)
,
,
(
with
1
,
0
1
,
1
1
=
=
=
N
m
K
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 a field of temperature
2
)
,
(
tr
T
R
T
=
R
0
R
1
With
B
C
D
T (R, T)
Code_Aster
®
Version
6.0
Titrate:
SSNA106 - Hollow roll subjected to a behavior thermoviscoelastic
Date:
19/08/02
Author (S):
PH. BONNIERES, D. NUNEZ
Key
:
V6.01.106-A
Page:
3/6
Manual of Validation
V6.02 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/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
3
)
,
(
)
(
*
)
(
*
)
(
)
(
I
I
T
R
T
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
)
(
)
(
)
) (
*
(
The thermoelastic problem are equivalent, while passing by the transform of Laplace is:
(
)
()
=
=
-
=
=
+
-
+
=
+
+
+
+
+
+
+
+
+
+
+
+
+
R
Z
R
R
R
R
R
Dr.
D
p
R
Tr
K
K
I
0
1
'
)
(
)
(
2
3
3
I
I
By eliminating the sign “+”:
(
)
(
)
(
)
(
)
+
+
+
-
+
=
+
+
+
-
+
=
+
+
+
-
+
=
-
+
p
R
K
K
I
p
R
K
K
I
R
p
R
K
K
I
R
Z
R
R
Z
R
Z
R
Z
R
R
2
2
2
)
(
)
(
0
)
(
0
1
'
maybe,
(
) (
)
(
) (
) (
)
+
+
+
+
-
+
=
+
+
+
-
+
-
+
=
=
-
+
p
R
I
K
I
I
K
K
I
K
I
p
R
I
K
K
I
K
I
R
pi
R
I
K
R
R
R
R
R
Z
R
R
2
2
2
)
(
)
(
)
(
0
)
(
1
'
Code_Aster
®
Version
6.0
Titrate:
SSNA106 - Hollow roll subjected to a behavior thermoviscoelastic
Date:
19/08/02
Author (S):
PH. BONNIERES, D. NUNEZ
Key
:
V6.01.106-A
Page:
4/6
Manual of Validation
V6.02 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A
(
) (
) (
)
R
R
K
I
p
R
I
K
I
I
K
K
I
K
I
R
K
I
)
(
)
(
)
(
2
+
=
+
+
+
+
-
+
+
+
According to the equilibrium equation, one has
R
R
R
+
=
'
, one obtains:
(
) (
) (
)
0
'
2
)
'
) (
(
'
)
(
2
=
+
+
+
+
-
+
+
+
+
p
R
I
K
I
R
I
K
K
I
R
K
I
R
K
I
R
R
R
R
R
,
(
)
0
)
(
'
2
2
=
-
+
+
K
I
p
R
R
R
R
,
)
(
'
2
2
I
K
p
R
With
R
R
R
-
+
=
+
what while integrating compared to R gives:
)
(
4
2
2
2
I
K
p
R
R
B
With
R
-
+
+
=
,
boundary conditions
0
)
(
)
(
1
0
=
=
R
R
R
R
give:
)
(
4
)
(
)
(
2
2
1
2
0
2
1
2
0
I
K
p
R
R
B
R
R
I
K
p
With
-
=
+
-
-
=
One thus has by taking again the initial notations:
(
)
-
+
-
=
+
-
+
-
=
-
-
+
-
=
+
+
+
+
+
+
+
+
+
+
+
)
2
(
)
(
)
3
(
)
(
4
)
(
)
(
4
2
2
1
2
0
2
2
1
2
0
2
2
1
2
0
2
2
1
2
0
2
2
1
2
0
R
R
R
I
K
K
I
p
R
R
R
R
R
R
K
I
p
R
R
R
R
R
R
K
I
p
Z
R
Maybe, by taking the opposite transform,
(
)
-
+
+
-
+
-
+
-
+
-
-
-
+
-
=
-
-
-
-
)
1
(
2
)
1
(
0
0
0
3
)
1
(
2
0
0
0
)
1
(
2
2
2
1
2
0
2
2
1
2
0
2
2
1
2
0
2
2
1
2
0
2
2
1
2
0
2
2
1
2
0
Ekt
LT
LT
LT
E
R
R
R
R
R
R
E
K
R
R
R
R
R
R
E
K
R
R
R
R
R
R
E
K
One deduces some
V
and W:
(
)
(
)
(
)
+
-
+
+
-
-
+
-
+
-
-
=
-
-
2
2
2
1
2
0
2
1
2
0
2
2
1
2
0
2
1
2
0
2
1
4
3
4
)
(
1
1
2
1
)
,
(
R
R
R
R
Ekt
R
R
E
R
R
R
R
R
E
R
Ek
T
R
W
Ekt
LT
Code_Aster
®
Version
6.0
Titrate:
SSNA106 - Hollow roll subjected to a behavior thermoviscoelastic
Date:
19/08/02
Author (S):
PH. BONNIERES, D. NUNEZ
Key
:
V6.01.106-A
Page:
5/6
Manual of Validation
V6.02 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A
2.2
Results of reference
Displacement DX on the node 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
6.0
Titrate:
SSNA106 - Hollow roll subjected to a behavior thermoviscoelastic
Date:
19/08/02
Author (S):
PH. BONNIERES, D. NUNEZ
Key
:
V6.01.106-A
Page:
6/6
Manual of Validation
V6.02 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
The problem is modelized in axisymetry
3.2
Characteristics of the mesh
120 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) 0.24
1.110
1.1106
0.05%
5
Summary of the results
The results calculated by Code_Aster are in agreement with the analytical solutions but depend
very strongly of the refinement of the mesh.