Code_Aster ®
Version
6.0
Titrate:
SSNA105 - Hollow Cylindre subjected to a pressure, linear viscoelasticity
Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.105-A Page:
1/6

Organization (S): EDF/AMA, CS IF
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
V6.01.105 document

SSNA105 - Hollow Cylindre subjected to a pressure,
linear viscoelasticity, contact

Summary:

This case-test makes it possible to in the case of validate the law of LEMAITRE established in Code_Aster behavior
viscoelastic linear. The found results are compared with an analytical solution.
This test takes again same modeling as the case-test SSNA104A to which one adds a cylinder (pellet) and one
draft the contact.
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
6.0
Titrate:
SSNA105 - Hollow Cylindre subjected to a pressure, linear viscoelasticity
Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.105-A Page:
2/6

1
Problem of reference

1.1 Geometry

The diagram is not on the scale, the difference between the two cylinders was amplified for the best
visibility.



R1
0.82
R2
0.92
R3
1.
R4
2.

1.2
Properties of materials

The pellet is made up of an elastic material, the sheath consists of a viscoelastic material.
The elastic data coincide for two 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


Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
6.0
Titrate:
SSNA105 - Hollow Cylindre subjected to a pressure, linear viscoelasticity
Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.105-A Page:
3/6

1.3
Boundary conditions and loading

Boundary conditions:
The cylinder is blocked out of DY on the sides [AP, LP], [AG, BG] and [CP, PD] [CG, PG].

Loading:
The cylinder is subjected to an internal pressure on [DP, AP], this pressure is calculated of such kind
that at the moment t=0, the sheath has the same behavior as the cylinder modelled in the test ssna104a.

R
E R
R
3
(22 - 21)

- 1
if - 1 T 0
p (T)
R
R


1
=


2
2 21 (1 -)

[
WITH B (R R C D Ge Ekt HT
K
if
T
3 - 2 +
(+ - +) +]
0 < 5

with
2
R
R
E
3
P R
2 -
2
With =
1
, B =
,
0 3
C =
with P
2 2
R
R 1
(+)
2
2
R - R
0 =1.E-3 MPa, pressure of the test ssna104a.
1 (1 -)
2
4
3

1
2
(1 - 2) 2

2
3 R
2
P R

2
R
D =
(+) r4 + 3
1
, G = -
,
4
H = K
, K =
0 2
1 - 2 + 1
2
(1 -
2)


E
2
2
2

R
2
2nd
2
2 R
R
R
R
2 -
3

3
1
2

One treats the contact between the two cylinders.
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
6.0
Titrate:
SSNA105 - Hollow Cylindre subjected to a pressure, linear viscoelasticity
Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.105-A Page:
4/6

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].

Phase without contact

One wants to find the value of p (T) to apply to the internal wall of the pellet for which the contact
1
place has.
For the pellet, one finds:

2
R

1 - 2
0
0

2
R




2

2


R
p (T) R
0
1
2
0
=
+
where
1
1
=




2
R



2
2
R - R


2
1

0
0

2




(1+)
R 2
W
=
1 - + 2
2
=.
E


R 2
R


(21+)
The condition of being written contact: (
W R) - (
W R) = 0, one has R
R
R

3 - 2 = 2
(1 -)
3
2
E
R

E
From where
3
=
- 1





R
(
2 1 -) 2
2
R
E R - R
3
(2 2
2
1)
p lim =
- 1
.
1




2
R
R
-
2
2
1
1 (
) 2

Phase with contact

It is wanted that as from the moment t=0, the sheath has same behavior as in the test ssna104a.
When there is contact, one a:
W (R) = W (R) + R - R,
P
2
G
3
3
2
thus by recovering the value of displacements in the test ssna104, one must obtain:
p R 3
1
1 2
3
0 3

R 24
-

2
W (R
4
)
1
3 1 2
.
P
2
= r3 - r2 +
Ekt
R
2
2
(+)
-


+
2
(- (-) E)


+ K
T
2

R
2
2
4 - r3


E
r3

r3


Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
6.0
Titrate:
SSNA105 - Hollow Cylindre subjected to a pressure, linear viscoelasticity
Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.105-A Page:
5/6

The stress field of the pellet is given by

R 2

R 2

1 - 2
1
- 1
2
0
- 2
0
0





2
R
R






2
2



R

R
0
1
2
1
2
0
=


1
+
-
2
0
+







2
R
R








0
0
Z




2
p R
2
p R
with
1 1
=
and
0 1
=
.
1
2
2
R - R
0
2
2
R - R
2
1
2
1
1+

Like




, one finds: Z = 2 (
-
1
0).
Z =
Z -
((2 - + Z =
1
0)
) 0
E
E

1+

1+
R 2
R 2
W
One thus has =
- (+ + =
1 - 2




1 -
0 +
2
1
-
1
2
0
2 =
R
Z)
(
) (
)
E
E
E
R
R
R


1+

2
R
W (R)
R 2 1
1 2
, one finds a little more p (T) given by the formula
P
2
=
2 (-
)

1 -
-
+ 1
0
2

E





1


r2
high.

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
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
6.0
Titrate:
SSNA105 - Hollow Cylindre subjected to a pressure, linear viscoelasticity
Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.105-A Page:
6/6

3 Modeling
With

3.1
Characteristics of modeling

The problem is modelled in axisymetry

3.2
Characteristics of the grid

750 meshs QUAD4

3.3
Functionalities tested

Commands
DEFI_MATERIAU
ELAS
LEMAITRE
AFFE_CHAR_MECA
CONTACT
STAT_NON_LINE
COMP_INCR
LEMAITRE
COMP_ELAS
ELAS

4
Results of modeling A

4.1 Values
tested

Identification Moments
Reference
Aster
Variation (%)
DX (B)
0.9
2.14 E3
2.14 E3
­ 0.953
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

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 grid.

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Outline document