Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
1/10
Organization (S): EDF/IMA/MN
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.104
SSNV104 - Contact of two phères
Summary:
Calculation consists in crushing two quarters of spheres one on the other to test the algorithm of unilateral contact
in statics. The solution is compared with the analytical result of Hertz. This functionality re-enters within the framework
mechanics of the structures having a nonlinear behavior.
Three axisymmetric modelings suggested differ by the type of finite elements used (modeling
AXIS, meshs TRIA3, QUAD4 and QUAD8).
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
2/10
1
Problem of reference
1.1 Geometry
Two half-spheres: axisymmetric model
With
B
Z2
Z1
I
G
H
radius R = 50 mm
G'
y
C
D
X
Taking into account the differences in grid, positions of the nodes H and I, indicated here in manner
approximate, are different in 3 modelings (one will note nodes ha, IA, HB, IB, HC, IC).
1.2
Material properties
E = 20.000. MPa
= 0.3
1.3
Boundary conditions and loadings
DX blocked on the axis AC
DY imposed ­ 2 mm for the nodes of AB
+2 mm for the CD nodes
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
3/10
2
Reference solution
2.1
Method of calculation used for the reference solution
Analytical method for calculation of yy in G (and G'): it is the solution of a problem of Hertz
- E 1
2h

=
yy
1-2 R
where H is imposed crushing (H = 2 - (- 2) = 4 mm).
2.2
Results of reference
yy at the point G (analytical solution).
Displacements in three points of the edge (of which G).
xx, yy and zz in a mesh being pressed on AG.
2.3 Reference
bibliographical
[1]
G. DUMONT
: “Method of the active constraints applied to the unilateral contact”
Note HI-75/93/016.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
4/10
3 Modeling
With
3.1
Characteristics of modeling
3.2
Characteristics of the grid
Nodes: 110 nodes.
Meshs: 170 TRIA3 and 53 SEG2.
3.3 Functionalities
tested
AFFE_CHAR_MECA
LIAISON_UNIL_NO
STAT_NON_LINE
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
5/10
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
Tolerance
yy mesh M1 node N110 (G)
­ 2798.3
­ 2636.5
5.8%
7%
yy mesh M8 node N110 (G)
­ 2798.3
­ 2976.7
6.4%
7%
DX node N110 (G)
0.
1.14 10­17
10­17 (ABS)
10­13
DY node N110 (G)
0.
­ 2.63 10­4
10­14 (ABS)
10­13
DX node N100 (ha)
­ 1.23616 10­1
DY node N100 (ha)
­ 1.43310 10­1
DX node N92 (IA)
­ 1.76462 10­1
DY node N92 (IA)
­ 6.94980 10­1
4.2 Remarks
The node G belongs to 2 meshs (M1 and M8): one can make the average of both yy corresponding
to compare with the theoretical value, from where an error of 0.3%.
4.3 Parameters
of execution
Version: 3.02.24
Machine: CRAY C90
Obstruction memory:
8 MW
Time CPU To use:
5 seconds
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
6/10
5 Modeling
B
5.1
Characteristics of modeling
5.2
Characteristics of the grid
Nodes: 410.
Meshs: 30 TRIA3, 324 QUAD4, 93 SEG2.
5.3 Functionalities
tested
AFFE_CHAR_MECA
LIAISON_UNIL_NO
STAT_NON_LINE
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
7/10
6
Results of modeling B
6.1 Values
tested
Identification
Reference
Aster
% difference
Tolerance
yy mesh M31 node N291 (G)
­ 2798.3
­ 2477.
11.5%
12%
DX node N291 (G)
0
­ 2. 10­17
10­17 (ABS)
10­13
DY node N291 (G)
0
­ 5.8. 10­17
10­17 (ABS)
10­13
DX node N287 (HB)
­ 1.22920 10­1
DY node N287 (HB)
­ 1.62911 10­1
DX node N285 (IB)
­ 1.89036 10­1
DY node N285 (IB)
­ 6.29666 10­1
6.2 Parameters
of execution
Version: 3.02.24
Machine: CRAY C90
Obstruction memory:
8 MW
Time CPU To use:
6 seconds
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
8/10
7 Modeling
C
7.1
Characteristics of modeling
7.2
Characteristics of the grid
Nodes: 660.
Meshs: 176 QUAD8 and 80 SEG3.
7.3 Functionalities
tested
Commands
Keys
AFFE_CHAR_MECA
LIAISON_UNIL_NO
[U4.25.01]
STAT_NON_LINE
[U4.32.01]
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
9/10
8
Results of modeling C
8.1 Values
tested
Identification
Reference
Aster
% difference
Tolerance
yy mesh M1 node N660 (G)
­ 2 798.3
­ 2419.8
13.5%
14%
DX node N660 (G)
0
­ 3.4 10­17
10­17 (ABS)
10­13
DY node N660 (G)
0
­ 3.4 10­14
10­14 (ABS)
10­13
DX node N644 (HC)
­ 1.10481 10­1
DY node N644 (HC)
­ 1.11082 10­1
DX node N606 (IC)
­ 2.10057 10­1
DY node N606 (IC)
­ 7.37809 10­1
8.2 Parameters
of execution
Version: 3.02.24
Machine: CRAY C90
Obstruction memory:
8 MW
Time CPU To use:
6 seconds
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SSNV104 Contact of two spheres
Date:
04/02/98
Author (S):
I. VAUTIER
Key:
V6.04.104-A Page:
10/10
9
Summary of the results
The comparison of each of 3 modelings to the analytical reference (limited to the point G) is
satisfactory for modeling A, less for modelings B and C.
The comparison of 3 modelings between them shows that:
· the deformations are superposable,
· the stress fields have marked enough local differences, in particular in
zones Z1 and Z2. The phase of independent validation should make it possible to include/understand
why.
In the command files, one ensures oneself of the not-degradation of the results by testing them
displacements in two nodes of the edge, H and I. occupying Ces nodes of the positions slightly
different according to modelings', one should not seek to compare the values of the cases A, B, C.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HI-75/96/044 - Ind A