Code_Aster ®
Version
8.1
Titrate:
SSNA104 - Hollow Cylindre 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

Organization (S): EDF-R & D/AMA, EDF-R & D/MMC, CS IF
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
Document: V6.01.104

SSNA104 - Hollow Cylindre 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.
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA104 - Hollow Cylindre 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

1
Problem of reference

1.1 Geometry

Dimensions of the cylinder:
R0
1 m
R1
2 m
Appear 1.1-a: Coupe of the hollow roll and loading

1.2
Properties of materials

Young modulus: E= 1 MPa
Poisson's ratio: =0.3

Law of LEMAITRE:
N


1
1
1
G (, T) =
with
=,
1
=,
0 N = 1

K 1
K
m
m



Law LEMA_SEUIL:



G (, T)
2
3
= A
with A =
, = 1




on all the grid
3
2
10
S 10
=


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 blocked out of DY on the sides [AB] and [CD].

Loading:
The cylinder is subjected to an internal pressure on [DA] P0 =1.E-3 MPa
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA104 - Hollow Cylindre 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

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 I (T) and K (T) so that strains and stresses
can be written:


D (T)
D (Tr ((T))
(T) = (I + K) *
- K *
I
3

D

D
where I indicates the matrix identity of row 3
3
T
and * the product of convolution: (F * G T
) () = F T (-) G () D
0
1

1
I T is found
() =
+ kt, K T () =
+ kt
E
E
2

One imposes the P0 pressure on the moment t=0, the internal pressure is worth
0 if T - < 0
p (T) = H (T) P where H (T) =
with in this case = 0
0

1 if T - 0

One uses the transform of Laplace Carson
+
F (N) = L (F (T)) = N
-
F (T) E NT dt
0
From where +
p = P
0
The solution of the elastic problem are equivalent is:


R 2

1 - 1
0
0

2
R




2

2
+

P R
=

R
0
1+ 1
0 where
0 0
=




2
R



2
2
R - R

1
0
+

0
0
Z





One determines +
by the condition on +
data by the boundary conditions:
Z
Z

+
+
+
= 0 = (I + K +
+
)
- K (
2 + +
+
) = I +
+
- 2K
Z
Z
Z
Z

+

(2 -) 1p
From where = 1
.
Z

+



p + Ek
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA104 - Hollow Cylindre 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

-
One finds by the transform of opposite Laplace
(T) = (1 - (1 -
2) Eht
E
Z
), in the same way in
applying the transform of Laplace reverses on and, one finds
R



R 2

1 - 1
0
0


2
R




2

+
=

R
0
1+ 1
0




2
R




0
0
(1 - (1 -
2) - Eht
E
)






One deduces some:
3 1 -
2
R 2


- Ekt
K
E
- 1
0
0




2
2
3
R




2

3
1

& =
- 2
R
- Ekt
0
K
E
- 1
0

V



2
2
3
R





0
0
- K (1 -
2) - Ekt
E
)






and while integrating with ()
0 = 0;
V
3 1 - 2
R 2


- Ekt

E
- K 1 T
0
0




2


2
3rd
R




2


=
3 1 - 2
R

- Ekt
0

E
- K 1 T
0
.
V



2


2
3rd
R




(1 - 2)

0
0
-
(1 - - Ekt
E
)

E



One deduces radial displacement from it
1
R 2
1 - 2

2
3
W (R, T) =
1
Ekt
R
R (1+)
+
(3 (1 - 2) - E)

+

2

K 1 T
2


E
R
2
2 R

2.2
Results of reference

Displacement DX on the node B and constraints 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
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA104 - Hollow Cylindre 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

3 Modeling
With

3.1
Characteristics of modeling

The problem is modelled in axisymetry.

3.2
Characteristics of the grid

1000 meshs QUAD4

3.3
Functionalities tested

Commands


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%

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA104 - Hollow Cylindre 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

5 Modeling
B

5.1
Characteristics of modeling

The problem is modelled in axisymetry

5.2
Characteristics of the grid

1000 meshs QUAD4

5.3
Functionalities tested

Commands


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 grid.
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/05/005/A

Outline document