Code_Aster ®
Version
6.0
Titrate:
SSNA106 - Hollow Cylindre subjected to a behavior thermoviscoelastic Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.106-A Page:
1/6

Organization (S): EDF/AMA, CS IF
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
V6.01.106 document

SSNA106 - Subjected hollow Cylindre
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.
Handbook of Validation
V6.02 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
6.0
Titrate:
SSNA106 - Hollow Cylindre subjected to a behavior thermoviscoelastic Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.106-A Page:
2/6

1
Problem of reference

1.1 Geometry
D
C

T (R, T)



With
B

R0

R

1

R0
1 m
R1
2 m

1.2
Properties of materials

Young modulus: E= 1 MPa
Poisson's ratio: =0.3
Dilation coefficient: =0.7

Law of LEMAITRE:
N


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

K 1
K
m
m



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 a field of temperature
2
T (R, T) = tr
Handbook of Validation
V6.02 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
6.0
Titrate:
SSNA106 - Hollow Cylindre subjected to a behavior thermoviscoelastic Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.106-A 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 + T
(R, T) I
3
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

The thermoelastic problem are equivalent, while passing by the transform of Laplace is:


R
2
+
= +
(I + +
+
K
)
- +
+
K Tr
(
) I
I
3 +
3

p


+
D
1
+
'=
R


R
= (+ - +r)

Dr.
R
+

0
Z =
(+
R) = +
R
By eliminating the sign “+”:


1
'


0
R +
(R -) =

R


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


p

R 2
2

R
R (I + K) - K (

(
)


R +
+ Z) +

= I + K R - K (R +
+ Z) +

p
p
maybe,


1
'
(
) 0
R +
R -

=

R


K
R 2


(
)

Z =
R +
-

I
pi

(I + K) K
R 2

2

I K K
I K
R

R (I + K) -
(
(
)




R +
)
(+)
+

= I + K R -
(R +) (+)
+

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

Code_Aster ®
Version
6.0
Titrate:
SSNA106 - Hollow Cylindre subjected to a behavior thermoviscoelastic Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.106-A Page:
4/6


(I + K) K
+
2
(I + K
)
+ R (I + K


)


-
(
I
K
R
+ +
= (I + K


)

R
) (
)


R
I
I
p



According to the equilibrium equation, one has = R

'
+, one obtains:
R
R

(I + K)
2
K
I + K
R

(I + K) '



,
R +r (I + K) (R
'
R +
)


R
-
(2 R + R 'R) (
)
+
= 0



I
I
p

R

(
2

2

,
R + R
'
R) +
= 0
p (I - K)


2
R
2 +

R '= A +
what while integrating compared to R gives:
R
R
p (K - I)

2
With
B
R
= +
+


,
R
2
2
R
4 p (K - I)
boundary conditions (R)
give:
R
=
(R)
R
= 0
0
1

With = -
(2
2
R + R)
2 p (K - I) 0
1

2 2
R
R
0 1
B = 4P (K - I)
One thus has by taking again the initial notations:

2 2

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

4 p (I - K)
R


2 2

R R
+
=
(2
R
R
R

0 +
2
1 - 3 2 + 0 1)

+
4 p (I - +
K)
2
R

+
2
2
+

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


+
p (I - +
+
K) I
2
Maybe, by taking the opposite transform,



2
2
R R


-
1
(
LT
E) 2
2
2
0 1
R
R
R
0
0

-
0 + 1 -
-



2


2k
R




2
2




R R
- LT
2
2
2
0 1

=
0
1
(- E


) R
R
R
0 + 1 - 3
+ 2
0
2k
R







2
2
-
R
R
LT
2
2
2
0 +
0
0
1
(- E) (
Ekt
R
R
R
E
0 + 1 - 2
)

1
-


+
1
(-
)
2


K
R


One deduces some and W:
V
1 - 2


2 2
LT
R R

2
2
2 2
-
Ekt
(R
R)
3Ekt
R R
2
2
0 1
-
- 0 +



(
W R, T) =
R
(1 - E) R R
1 E
R

0 + 1 -
2
+ (-
)


1
+
0 1 + 2
2

Ek



R


4

(41 - 2)

R

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

Code_Aster ®
Version
6.0
Titrate:
SSNA106 - Hollow Cylindre subjected to a behavior thermoviscoelastic Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.106-A Page:
5/6

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
V6.02 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
6.0
Titrate:
SSNA106 - Hollow Cylindre subjected to a behavior thermoviscoelastic Date:
19/08/02
Author (S):
PH. Key BONNIERES, D. NUNEZ
:
V6.01.106-A Page:
6/6

3 Modeling
With

3.1
Characteristics of modeling

The problem is modelled in axisymetry

3.2
Characteristics of the grid

120 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) 0.24
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 grid.

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

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