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SSNV112 - Hollow Cylindre into incompressible


Date:
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:
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Organization (S): EDF-R & D/AMA

Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
Document: V6.04.112

SSNV112 - Hollow Cylindre 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 grids 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.
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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1
Problem of reference

1.1 Geometry
y
R
E
F

C
45°
D
P
Ur
With
B
Z
X
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 = 2.105 MPa
= 0.4999

1.3
Boundary conditions and loadings

Radial displacement U
5 m
0
10
.
6
-
=
(expansion)
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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2
Reference solution

2.1
Method of calculation

For the studied problem, displacement U is radial and thus of the form
U =
0]

0,

[U,

.
One deduces the general form from it from the tensor of the deformations in great deformations:

2

1
(+ u')
0
0


B =
T
FF =
U
0
1
(+) 2 0

R


0
0
1


as well as the form of the tensor of the constraints, which is written simply if one takes into account does it
that J = det F = 1 for an incompressible problem:
D
= - pId + µ B, is:

2
2
1
U 2 1

= - p +
rr
µ 1 (+ u') - 1 (+) -
3
3



R
3

1
2
2
U 2 1

= - p +

µ- 1 (+ u') + 1 (+) -
3
3



R
3


1
2
1
U 2 2
= - p + µ - 1
(+ u') - 1
(+) +
zz
3
3


R
3


=
=
= 0
R
rz
Z

The writing of the equilibrium equations leads to the checking of only one equation:
-
'+ rr
= 0
rr

R
who allows to determine the pressure p knowing the field of radial displacement U:


U 2

1
4
2
U
u'
U (1+ u') 2
(+)
R

p'= µ 1 (+ u' U
) '-
'
1+
-
+
-

3
3


2



R
R
R
R
R




2.2
Particularization of the solution
1+ u'
0
0

U

The condition of incompressibility is written det F = 1 with F = 0
1+
0. Displacement U

R

0
0
1
thus check the following differential equation:
ru'+u + u' U = 0








éq 2.2-1
The imposed loading is as follows U = U 0 in R = A.
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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:
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The solution in displacement is thus:


U = - R + 2
R + U (U + 2a)
R
0
0


U = U = 0

Z

The tensor of the deformations thus has as an expression:

2
R
B =
rr

2
R + U (U + 2a)
0
0


2

R + U (U + 2a)
B
=
0
0


2
R

B = 1
zz

B
= B
= B = 0
R
Z
Z


And the constraints are worth:



2
2

2
R
1 R + U (U +

2a)
= - p +
0
0
1
rr
µ
-
-

3 2
R + U (U + 2a) 3
2
R
3


0
0



2
2

1
R
2 R + U (U + 2a)
0
0
1
= - p +

µ-
+
-

3 2
R + U (U + 2a)
3
2
R
3


0
0




2
2

1
R
1 R + U (U + 2a)
0
0
2
= - p +
zz
µ-
-
+

3 2
R + U (U + 2a) 3
2
R
3

0
0



=
= = 0
R
Z
Z


with p obtained by integration of [éq 2.2-1] which is worth:

U U
(
+ 2a

)
2
(
+ 2)
p = µ 0
0
-
0
0
-
() + 1
(
+ 2) + 2
0
0
+

6r

2
(U U has
3U U
(
+ 2a) + R 2
0
0
) Log R Log [U U has R] C
2



where C is a constant

One obtains finally the following numerical values:

in R = 0.1
U = 6. 10 5
-
in R = 0.2
= 3.00067 10 5
-
R
ur
= 59
- .9955
= 0.
rr
rr


= 99.9566


= 40.006

= 19.9326
= 20.
zz
zz
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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The passage in the Cartesian system is done using the following relations:

= cos2 +
2
xx
rr
sin - 2 sin cos
R

= sin2 +
2
yy
rr
cos + 2 sin cos
R

= sin cos -
2
2
xy
rr
sin cos - 2
R (cos - sin)

2.3
Sizes and results of reference

One compares with the values of reference:

·
displacements (U, v) at points A and F,
·
the constraints (xx, yy, zz, xy) at points A and F,
·
constraints of Von Mises and Tresca as well as the eigenvalues of the tensor of
constraints at point A.
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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3 Modeling
With

3.1
Characteristics of modeling

Grid with incompressible elements 3D of type HEXA20 only


y
B
Face blocked in dx
F
With
Normally blocked face
E
Face with imposed pressure
Face with imposed radial displacement
45°
X


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 grid

A number of nodes: 1501 nodes
A number of meshs: 240 HEXA20
Handbook of Validation
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HT-66/03/008/A

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3.3 Functionalities
tested

Commands



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 constraints are evaluated at points A and F. Les 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. 10­5
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 10­5 - 2.1219
10-5
0.007

v
+2.1218 10­5 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 constraints.
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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4 Modeling
B

4.1
Characteristics of modeling

Grid with incompressible elements 3D of type TETRA10 only


F
Normally blocked face
y
D
E
C

45°
With
B
X
Face with imposed pressure
Face blocked out of Dy
Face with radial displacement imposed Face blocked out of Dy


AB is on axis OX (contrary to modeling A).
The grid 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 grid

A number of nodes: 2064
A number of meshs: 1121 TETRA10
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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4.3 Functionalities
tested

Commands



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. 10­5 6.009
10-5 0.158

v
0. 2.65
10­23
-

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 10­5 2.1198
10-5
­ 0.096

v
2.1218 10­5 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 constraints 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 constraints).
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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5 Modeling
C

5.1
Characteristics of modeling

Grid with incompressible elements 2D of type QUAD8 and TRIA6

y
B
Face blocked in dx
F
With
Normally blocked face
E
Face with imposed pressure
Face with imposed radial displacement
45°
X


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 grid

A number of nodes: 591
A number of meshs: 200 TRIA6, 50 QUAD8.

5.3 Functionalities
tested

Commands



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

Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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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. 10­5
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 10­5 - 2.1212
10-5
- 0.029

v
+2.1218 10­5 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.
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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6 Modeling
D

6.1
Characteristics of modeling

Incompressible elements axi (TRIA6 + QUAD8)

Center cylinder
Node blocked out of Dy
y
E
F
0.01m
C
D
With
B
X
Face with imposed pressure
Face with imposed displacement


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 grid

A number of nodes: 175.
A number of meshs and types: 20 QUAD8, 40 TRIA6.

6.3 Functionalities
tested

Commands



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

Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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6.4
Sizes tested and results

One notes the results obtained for A and F.

Identification Reference
Aster
% difference
With
U
6. 10­5
6.0000 10-5 0.00

v 0. 5.71 10­21
-

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 10­5 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 constraints are obtained with a precision
lower than 0.5%.
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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SSNV112 - Hollow Cylindre into incompressible


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7
Summary of the results

With a Poisson's ratio very close to 0.5, one finds the results of the solution
analytical incompressible in great deformations, with a completely correct precision.

Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
HT-66/03/008/A

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