Code_Aster
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Version
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Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
1/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
Document: V6.04.112
SSNV112 - Hollow roll into incompressible
(great deformations)
Summary:
This test makes it possible to validate the quasi-incompressible elements in great deformations, in statics for one
three-dimensional, axisymmetric or two-dimensional problem (plane deformations). A cylinder is considered
hollow subjected to an internal radial displacement. The material has a Poisson's ratio equal to 0.4999 and one
use the quasi-incompressible elements (modeling
INCO
) with the deformations of SIMO_MIEHE.
Four modelings are carried out for this problem. Modelings A and B make it possible to test
quasi-incompressible modeling 3D (
3d_INCO
), on the one hand with HEXA20 (A) and on the other hand with
TETRA10 (B). Modelings C and D are studies 2D being based on mixed mesh QUAD8 and
TRIA6. Modeling C is the study in plane deformations (
D_PLAN_INCO
), modeling D is a study
axisymmetric (
AXIS_INCO
).
This test is similar to the test SSLV130, which tests the quasi-incompressible elements under the assumption of
small deformations.
The numerical results are satisfactory for all modelings.
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Titrate:
SSNV112 - Hollow roll into incompressible
Date:
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Key
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V6.04.112-A
Page:
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V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
1
Problem of reference
1.1 Geometry
Z
X
R
y
P
With
B
45°
C
D
E
F
Internal radius
= 0.1 m has
External radius
B = 0.2 m
Co-ordinates of the points:
WITH B
E
F
C
D
X 0.1.0.2.0.1 * cos (45) 0.2 * cos (45) 0.1 * cos (22.5) 0.2 * cos (22.5)
y
0 0.0.1 * sin (45)
0.1 * sin (45)
0.1 * sin (22.5) 0.1 * sin (22.5)
Z
0 0
0
0
0
0
1.2
Properties of material
E = 2.10
5
MPa
= 0.4999
1.3
Boundary conditions and loadings
Radial displacement
m
U
5
0
10
.
6
-
=
(expansion)
Ur
Code_Aster
®
Version
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Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
3/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
2
Reference solution
2.1
Method of calculation
For the studied problem, displacement
U
is radial and thus of the form
0]
0,
[U,
=
U
.
One deduces the general form from it from the tensor of the deformations in great deformations:
+
+
=
=
1
0
0
0
)
1
(
0
0
0
)
'
1
(
2
2
R
U
U
T
FF
B
as well as the form of the tensor of the stresses, which is written simply if one takes into account does it
that
1
det
=
=
F
J
for an incompressible problem:
D
D
p
B
I
µ
+
-
=
, that is to say:
=
=
=
+
+
-
+
-
+
-
=
-
+
+
+
-
+
-
=
-
+
-
+
+
-
=
0
3
2
)
1
(
)
'
1
(
3
1
)
1
(
)
'
1
(
3
1
)
1
(
)
'
1
(
2
3
1
2
3
1
2
3
2
2
3
1
2
3
1
2
3
2
Z
rz
R
zz
rr
R
U
U
p
R
U
U
p
R
U
U
p
µ
µ
µ
The writing of the equilibrium equations leads to the checking of only one equation:
0
'
=
-
+
R
rr
rr
who allows to determine the pressure
p
knowing the field of radial displacement
U
:
(
)
()
+
-
+
+
-
+
-
+
=
R
R
U
R
U
R
U
R
U
U
U
p
R
U 2
2
2
3
2
3
4
1
'
1
'
1
''
)
'
1
(
'
µ
2.2
Particularization of the solution
The condition of incompressibility is written
1
det
=
F
with
+
+
=
1
0
0
0
1
0
0
0
'
1
R
U
U
F
. Displacement
U
thus check the following differential equation:
0
'
'
=
+
+
U
U
U
Ru
éq 2.2-1
The imposed loading is as follows
0
U
U
=
in
R = has
.
Code_Aster
®
Version
7.2
Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
4/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
The solution in displacement is thus:
=
=
+
+
+
-
=
0
)
2
(
0
0
2
Z
R
U
U
has
U
U
R
R
U
The tensor of the deformations thus has as an expression:
=
=
=
=
+
+
=
+
+
=
0
1
)
2
(
)
2
(
2
0
0
2
0
0
2
2
Z
Z
R
zz
rr
B
B
B
B
R
has
U
U
R
B
has
U
U
R
R
B
And the stresses are worth:
=
=
=
+
+
+
-
+
+
-
+
-
=
-
+
+
+
+
+
-
+
-
=
-
+
+
-
+
+
+
-
=
0
3
2
)
2
(
3
1
)
2
(
3
1
3
1
)
2
(
3
2
)
2
(
3
1
3
1
)
2
(
3
1
)
2
(
3
2
2
0
0
2
0
0
2
2
2
0
0
2
0
0
2
2
2
0
0
2
0
0
2
2
Z
Z
R
zz
rr
R
has
U
U
R
has
U
U
R
R
p
R
has
U
U
R
has
U
U
R
R
p
R
has
U
U
R
has
U
U
R
R
p
µ
µ
µ
with
p
obtained by integration of [éq 2.2-1] which is worth:
(
)
[
]
C
R
has
U
U
Log
R
Log
R
has
U
U
has
U
U
R
has
U
U
p
+
+
+
+
-
+
+
+
-
+
=
2
0
0
2
1
2
0
0
0
0
2
0
0
)
2
(
)
(
)
2
(
3
)
2
(
2
6
)
2
(
µ
where
C
is a constant
One obtains finally the following numerical values:
.
20
9326
.
19
006
.
40
9566
.
99
.
0
9955
.
59
10
00067
.
3
2
.
0
10
.
6
1
.
0
5
5
=
=
=
=
=
-
=
=
=
=
=
-
-
zz
zz
rr
rr
R
R
U
R
U
R
in
in
Code_Aster
®
Version
7.2
Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
5/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
The passage in the Cartesian system is done using the following relations:
(
)
xx
rr
R
yy
rr
R
xy
rr
R
=
+
-
=
+
+
=
-
-
-
cos
sin
sin cos
sin
cos
sin cos
sin cos
sin cos
cos
sin
2
2
2
2
2
2
2
2
2
2.3
Sizes and results of reference
One compares with the values of reference:
·
displacements (U, v) at points A and F,
·
stresses (
xx
,
yy
,
zz
,
xy
) at points A and F,
·
stresses of Von Mises and Tresca as well as the eigenvalues of the tensor of
stresses at point A.
Code_Aster
®
Version
7.2
Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
6/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
3 Modeling
With
3.1
Characteristics of modeling
Mesh with incompressible elements 3D of type HEXA20 only
X
y
With
B
F
E
45°
Face with imposed pressure
Face locked in dx
Locked face
normally
Along axis Z:
·
total thickness E = 0.01
·
2 layers of elements
Limiting conditions:
DDL_IMPO =
GROUP_NO = ' FACSUP' DZ =
0.
GROUP_NO = ' FACINF' DZ =
0.
faces AEFD (z=0 and Z = 0.01)
GROUP_NO = ' FACEAB' DX =
0.
face AB
FACE_IMPO = GROUP_MA = ' FACEEF'
DNOR =
0.
face EFF
GROUP_MA = ' FACEAE' DNOR =
- 6.10
- 5
face AE
3.2
Characteristics of the mesh
A number of nodes: 1501 nodes
A number of meshs: 240 HEXA20
Face with imposed radial displacement
Code_Aster
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Version
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Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
7/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
3.3 Functionalities
tested
Controls
AFFE_MODELE MODELING “3D_INCO”
GROUP_MA
DEFI_MATERIAU ELAS
AFFE_CHAR_MECA DDL_IMPO
GROUP_NO
FACE_IMPO
GROUP_MA
PRES_REP
GROUP_MA
STAT_NON_LINE COMP_INCR
RELATION ELAS
DEFORMATION SIMO_MIEHE
NEWTON
REAC_ITER
1
3.4
Sizes tested and results
Displacements and the stresses are evaluated at points A and F. the components of the field
EQUI_NOEU_SIGM are tested at point A only.
Identification Reference
Aster
% difference
With
U
0. 6.6703
10
- 21
-
v
6. 10
5
6.0046 10
- 5
0.077
xx
99.9566 99.3400
0.617
yy
59.9955 60.9543
1.598
zz
19.9326 19.2770
- 3.289
xy
0. - 1.1617
-
VMIS 138.5226 138.6161
0.067
TRESCA 159.9521 160.0601
0.068
PRIN_1 - 59.9955 - 60.8372
1.403
PRIN_2 19.9326 19.2770
- 3.289
PRIN_3 99.9566 99.2229
- 0.734
VMIS_SG
138.5226 1.8.6161 - 0.067
Identification Reference
Aster
% difference
F
U
2.1218 10
5
- 2.1219
10
- 5
0.007
v
+2.1218 10
5
2.1219
10
- 5
0.007
xx
20.003 20.029
0.129
yy
20.003 19.980
- 0.115
zz
20.003 20.001
- 0.011
xy
20.003 20.026
0.111
3.5 Remarks
One obtains very good results since for all the examined sizes, the difference between
solution obtained with the code and the analytical solution is lower than 0.1% for displacements and
lower than 1.6% for the stresses.
Code_Aster
®
Version
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Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
8/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
4 Modeling
B
4.1
Characteristics of modeling
Mesh with incompressible elements 3D of type TETRA10 only
With
B
C
D
E
F
y
X
45°
Normally locked face
Face locked out of Dy
Face with imposed pressure
AB is on axis OX (contrary to modeling A).
The mesh was obtained with GMSH for a density of 0,01.
Limiting conditions:
DDL_IMPO =
GROUP_NO = ' FACSUP' DZ =
0.
GROUP_NO
= ' FACINF'
DZ
=
0.
faces AEFD (z=0 and Z = 0.01)
GROUP_NO
= ' FACEAB'
DY
=
0.
face AB
FACE_IMPO = GROUP_MA = ' FACEEF' DNOR =
0.
face EFF
= GROUP_MA = ' FACEAE' DNOR =
- 6.10
- 5
face AE
4.2
Characteristics of the mesh
A number of nodes: 2064
A number of meshs: 1121 TETRA10
Face with radial displacement imposed Face locked out of Dy
Code_Aster
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Version
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Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
9/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
4.3 Functionalities
tested
Controls
AFFE_MODELE MODELING “3D_INCO”
GROUP_MA
DEFI_MATERIAU ELAS
AFFE_CHAR_MECA DDL_IMPO
GROUP_NO
FACE_IMPO
GROUP_MA
STAT_NON_LINE COMP_INCR
RELATION
ELAS
DEFORMATION
SIMO_MIEHE
NEWTON
REAC_ITER
1
4.4
Sizes tested and results
One notes the results obtained for the points A and F.
Identification Reference
Aster
% difference
With
U
6. 10
5
6.009
10
- 5
0.158
v
0. 2.65
10
23
-
xx
59.9955 - 60.90
1.512
yy
99.9566
98.63
- 1.323
zz
19.9326
19.39
- 2.707
xy
0. - 2.765
-
VMIS 138.5226
TRESCA 159.9521
PRIN_1 - 59.9955
PRIN_2 19.9326
PRIN_3 99.9566
VMIS_SG
138.5226
Identification Reference
Aster
% difference
F
U
2.1218 10
5
2.1198
10
- 5
0.096
v
2.1218 10
5
2.1198
10
- 5
0.096
xx
20.003
19.94
- 0.302
yy
20.003 19.90
- 0.496
zz
20.003 20.025
0.110
xy
20.003 - 19.90
- 0.535
4.5 Remarks
The results obtained are completely correct since the stresses are obtained with one
precision lower than 3% even 0.5% at the point F. the variation is a little more important here than for
HEXA20, but can be explained by the fact why the loading is imposed here in manner a little less
specify since displacement U at point A, is defined only with one accuracy of 0.158% against 0.077%
(evening factor 2, that one finds on the stresses).
Code_Aster
®
Version
7.2
Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
10/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
5 Modeling
C
5.1
Characteristics of modeling
Mesh with incompressible elements 2D of type QUAD8 and TRIA6
X
y
With
B
F
E
45°
Face with imposed pressure
Face locked in dx
Normally locked face
Limiting conditions:
DDL_IMPO =
GROUP_NO = ' GRNM11'
DX =
0.
side AB
FACE_IMPO = GROUP_MA = ' GRMA12'
DNOR =
0.
dimensioned
EFF
= GROUP_MA = ' GRMA13'
DNOR =
- 6
. 10
- 5
face AE
Name of the nodes:
WITH = N2, B = N361, C = N121, D = N584, E = N155, F = N503
5.2
Characteristics of the mesh
A number of nodes: 591
A number of meshs: 200 TRIA6, 50 QUAD8.
5.3 Functionalities
tested
Controls
AFFE_MODELE MODELING
“D_PLAN_INCO”
GROUP_MA
DEFI_MATERIAU ELAS
AFFE_CHAR_MECA DDL_IMPO
GROUP_NO
FACE_IMPO
GROUP_MA
STAT_NON_LINE COMP_INCR
RELATION ELAS
DEFORMATION
SIMO_MIEHE
NEWTON
REAC_ITER
1
Face with imposed radial displacement
Code_Aster
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Version
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Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
11/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
5.4
Sizes tested and results
One notes the result obtained for the points A and F.
Identification Reference
Aster
% difference
With
U
0. - 5.93
10
- 21
-
v
6. 10
5
6.0046 10
- 5
0.077
xx
99.9566 99.7104 - 0.246
yy
59.9955 - 61.0467 1.752
zz
19.9326 19.5237 - 2.052
xy
0. 1.9020
-
VMIS 138.5226 19.1945 0.485
TRESCA 159.9521 160.7273
0.485
PRIN_1 - 59.9955 - 61.0318
1.727
PRIN_2 19.9326 19.5237 - 2.052
PRIN_3 99.9566 99.6955 - 0.261
VMIS_SG
138.5226 139.1945
0.485
Identification Reference
Aster
% difference
F
U
2.1218 10
5
- 2.1212
10
- 5
- 0.029
v
+2.1218 10
5
2.1212
10
- 5
- 0.029
xx
20.003 20.0456 0.213
yy
20.003 19.9883 - 0.073
zz
20.003 20.0048 0.009
xy
20.003 20.0252 0.111
5.5 Remarks
As for modeling 3D, the results obtained are completely satisfactory.
Code_Aster
®
Version
7.2
Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
12/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
6 Modeling
D
6.1
Characteristics of modeling
Incompressible elements axi (TRIA6 + QUAD8)
Center cylinder
y
0.01m
E
C
With
F
D
B
Face with imposed pressure
Node locked out of Dy
X
Limiting conditions:
DDL_IMPO =
GROUP_NO = ' FACSUP'
DY =
0.
y=0.1
GROUP_NO
= ' FACINF'
DY =
0.
y=0
FACE_IMPO = GROUP_MA = ' FACEAE'
DX = 6. 10
- 5
face AE
6.2
Characteristics of the mesh
A number of nodes: 175.
A number of meshs and types: 20 QUAD8, 40 TRIA6.
6.3 Functionalities
tested
Controls
AFFE_MODELE MODELING
“AXIS_INCO”
GROUP_MA
DEFI_MATERIAU ELAS
AFFE_CHAR_MECA DDL_IMPO
GROUP_NO
FACE_IMPO
GROUP_MA
STAT_NON_LINE COMP_INCR
RELATION
ELAS
NEWTON
REAC_ITER
1
DEFORMATION
SIMO_MIEHE
Face with imposed displacement
Code_Aster
®
Version
7.2
Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
13/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
6.4
Sizes tested and results
One notes the results obtained for A and F.
Identification Reference
Aster
% difference
With
U
6. 10
5
6.0000 10
- 5
0.00
v
0. 5.71 10
21
-
xx
59.9955 - 59.8619
- 0.223
yy
19.9326 19.9708
0.192
zz
99.9566 99.9171
- 0.039
xy
0. - 3.03
10
- 7
-
VMIS 138.5226 138.3727
- 0.108
TRESCA 159.9521 159.7790
- 0.108
PRIN_1 - 59.9955 - 59.8619
- 0.223
PRIN_2 19.9326 19.9708
0.192
PRIN_3 99.9566 99.9171
- 0.0039
VMIS_SG
138.5226 138.3727
- 0.108
Identification Reference
Aster
% difference
F
U
3.0007 10
5
3.0011
10
- 5
0.038
v
0. 4.90
10
- 22
xx
0. 2.59
10
- 2
-
yy
20. 19.9975 - 0.013
zz
40.006 39.9965
- 0.024
xy
0. - 4.87
10
- 3
-
6.5 Remarks
The precision obtained is very good since all the stresses are obtained with a precision
lower than 0.5%.
Code_Aster
®
Version
7.2
Titrate:
SSNV112 - Hollow roll into incompressible
Date:
23/09/03
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.04.112-A
Page:
14/14
Manual of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A
7
Summary of the results
With a Poisson's ratio
very near to 0.5, the results of the solution are found
analytical incompressible in great deformations, with a completely correct precision.