background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
1/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
Organization (S):
EDF-R & D/AMA















Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.506



SSNV506 - Elastoplastic indentation of a block
by an elastic spherical indentor




Summary:

This test relates to the modeling of the indentation of an elastic sphere on a half-plane with the behavior
elastoplastic. The objective is to test the functionalities related to the contact on an example comprising one
non-linearity material.
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
2/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
1
Problem of reference
1.1 Geometry
Radius of the sphere
R = 500 mm
Imposed displacement
100 mm

1.2
Properties of material
Two different modelings to represent the rigid sphere:
Material rigidification: E=2,1E9 Mpa and
= 0,3
Rigidification by conditions kinematics
Block: Steel, law of perfect elastoplastic behavior.
Modulate Young
E=210000 MPa
Poisson's ratio
= 0,3
Modulate work hardening
And = 0
Yield stress
y
= 50 MPa

1.3
Boundary conditions and loadings
The deformations are axisymmetric and the block forming the plan is supposed to be embedded on its basis.
An imposed displacement is applied:
·
Loading of 0 with ­ 100 mm on the higher part of the sphere in the models A and D
·
Loading of 0 with ­ 100 mm on the surface of contact of the sphere in the models B and C
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
3/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The results of reference result from the book quoted below [bib1].
p
m
= 3
0
with p
m
the contact pressure (page 171).
Rjohnson = p
m
has = 3
0
has if A is the surface of contact.
However in perfect plasticity
= 0,368 A ²/R according to the analysis of Richmond (page 200)
Finally, one obtains:
Rjohnson = 3
R
0
/0,368
Rjohnson: Normal reaction of contact of the solid mass on the sphere
R: Radius of the sphere
: Displacement of the node of the solid mass
0
: Yield stress of the solid mass
This result is valid under the following assumptions:
axisymmetric problem,
perfectly plastic material (coefficient 0,368 results from this assumption)
small deformations
rigid sphere.

2.2
Results of reference
The results of reference are obtained starting from the preceding formula. It is valid for
complete model in 3D.
Note:
In our study, Rjohnson depends only on displacement, one can write the relation under
the following form thanks to the facts of the case: Rjohnson = 640 270
with Rjohnson in
newton and
in millimetre. is directly connected to the moment of calculation.
The value of the normal resultant of contact coming from ASTER is given on a district of
1 radian of opening in axisymmetric 2D and on a district of
/2 for the model 3D (by symmetry, it
is enough to modelize the quarter of the problem).
Thus, the values of reference are:
in axisymmetric 2D: Rref = Rjohnson/2
= 101902,1
in 3D
: Rref = Rjohnson/4 = 160067,5

2.3
Uncertainties on the solution
Analytical solution.

2.4 Reference
bibliographical
[1]
“Contact Mechanics” - K.L. JOHNSON - Cambridge University Close - chapter 6 p. 153-201
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
4/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
3 Modeling
With
3.1
Characteristics of modeling
The symmetry of revolution of the problem allows an axisymmetric modeling: The sphere and the block
are represented respectively by a half disc and the cut of half of the block, with a grid with
axisymmetric elements 2D.
A contact of the node-mesh type is defined between the two structures.
A loading in imposed displacement is applied to the higher part of the sphere rigidified by
a high Young modulus.
Boundary condition:
·
symmetry of revolution: the nodes located on the axis Y (group of nodes “LB” and “LS”) are
locked according to direction X (DX = 0),
·
embedding of the base: the nodes of group “PLANX” are locked according to
directions X and Y (DX = DY = 0),
·
the rigid movements of body are removed by imposing a connection following there enters it
node E pertaining to the sphere and the node D pertaining to the solid mass.
Loadings:
An imposed displacement is applied to the higher part of the sphere (group of nodes
“NDPL”) according to the direction Y: Loading of 0 with ­ 100. mm
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
5/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
3.2
Characteristics of the mesh
A number of nodes: 916
A number of meshs and type: 625 QUAD4 and 289 SEG2
3.3 Functionalities
tested
Controls Key word
factor Key word
AFFE_MODELE
AFFE
MODELING = “AXIS”
DEFI_MATERIAU ECRO_LINE
SY
D_SIGM_EPSI
AFFE_CHAR_MECA CONTACT
METHOD = “FORCED”
STAT_NON_LINE COMP_ELAS
COMP_INCR
NEWTON
RELATION = “ELAS”
RELATION = “VMIS_ISOT_LINE”
STAMP = “TANGENT”
REAC_ITER = 1


4
Results of modeling A
4.1 Values
tested
Identification Displacement
(mm)
Reference
Aster
% difference
Reaction (NR)
20
­ 2.03804E+06
­ 2.06806E+06
1.473
Reaction (NR)
40
­ 4.07608E+06
­ 4.04698E+06
­ 0.714
Reaction (NR)
60
­ 6.11412E+06
­ 5.82730E+06
­ 4.691
Reaction (NR)
80
­ 8.15217E+06
­ 7.66632E+06
­ 5.960
Reaction (NR)
100
­ 1.01902E+07
­ 9.11899E+06
­ 10.512

4.2 Remarks
The most important error is for the last result. It remains acceptable nevertheless.
We illustrated the deformation of the solid mass to the pitch of final time:
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
6/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
5 Modeling
B
5.1
Characteristics of modeling
The symmetry of revolution of the problem allows an axisymmetric modeling: The block is represented
by the cut of its half and the sphere is represented by its surface potentially in contact, they
are with a grid with axisymmetric elements 2D.
A contact of the node-mesh type is defined between the two structures.
A loading in imposed displacement is applied to all the meshs representing the sphere,
rigidified by conditions kinematics.
Boundary condition:
·
Conditions of symmetry:
nodes of the jig located on the axis Y (group of nodes “LB”)
are locked according to direction X (DX = 0).
All nodes belonging to the sphere (group of nodes
“MAT1”) are locked according to direction X (DX = 0).
·
Embedding of the base: the nodes of “PLANX” are locked according to directions X
and Y (DX = DY = 0).
·
The rigid movements of body are removed by imposing a rigid connection, following y,
between the node E pertaining to the sphere and the node D pertaining to the solid mass.
Loadings:
An imposed displacement is applied to the part representing the sphere (group of node “MAT1”)
according to the direction Y: Loading of 0 with ­ 100. mm
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
7/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
5.2
Characteristics of the mesh
A number of nodes: 458
A number of meshs and type: 419 QUAD4 and 171 SEG2.
5.3 Functionalities
tested
Controls Key word
factor Key word
AFFE_MODELE
AFFE
MODELING = “AXIS”
DEFI_MATERIAU ELAS
ECRO_LINE
SY
D_SIGM_EPSI
AFFE_CHAR_MECA CONTACT
METHOD = “FORCED”
STAT_NON_LINE COMP_ELAS
COMP_INCR
NEWTON
RELATION = “ELAS”
RELATION = “VMIS_ISOT_LINE”
STAMP = “TANGENT”
REAC_ITER = 1


6
Results of modeling B
6.1 Values
tested
Identification Displacement
(mm)
Reference
Aster
% difference
Reaction (NR)
D = - 20 mm
­ 2.06771E+06
­ 2.31620E+06
1.456
Reaction (NR)
D = - 40 mm
­ 4.04742E+06
­ 4.23518E+06
­ 0.703
Reaction (NR)
D = - 60 mm
­ 5.82779E+06
­ 6.07847E+06
­ 4.683
Reaction (NR)
D = - 80 mm
­ 7.66673E+06
­ 7.91027E+06
­ 5.955
Reaction (NR)
D = ­ 100 mm
­ 9.11942E+06
­ 9.79599E+06
­ 10.508

6.2 Remarks
The results are almost identical to those of modeling A.
One notices a calculating time reduced by modelizing only the surface of contact of the sphere
rigidified by conditions kinematics.
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
8/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
7 Modeling
C
7.1
Characteristics of modeling
The symmetry of the problem makes it possible to represent in 3D only one quarter of the model: the sphere and the block
are represented respectively by the surface of contact of the sphere and a quarter of cylinder, with a grid
with solid elements 3D CUB8.
A contact node-mesh is defined between the sphere and the block.
A loading in imposed displacement is applied to all the surface of the sphere rigidified by
conditions kinematics.
Boundary condition:
·
Conditions of symmetry:
nodes located in plan (O, y, Z) (group of nodes
“SBYZ”) are locked according to direction X (DX = 0),
nodes located in the plan (O, X, y) (group of nodes
“SBXY”) are locked according to direction Z (DZ = 0),
the nodes of the sphere (group of nodes “SPHSUP”) are
locked according to directions X and Z (DX = DZ = 0)
·
Embedding of the base: the nodes of the group “BASES” (plane Y=0.) are locked
according to directions X, Y, and Z (DX = DY = DZ = 0).
·
The rigid movements of body are removed by imposing a connection following there enters it
node E pertaining to the sphere and the node S pertaining to the solid mass.
Loadings:
An imposed displacement is applied to all surface representing the sphere (group of nodes
“SPHSUP”) according to the direction Y: Loading of 0 with ­ 100. mm
7.2
Characteristics of the mesh
A number of nodes: 6852
A number of meshs and type: 5326 HEXA8, 387 PENTA6 and 183 QUAD4.
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
9/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
7.3 Functionalities
tested
Controls Key word
factor Key word
AFFE_MODELE
AFFE
MODELING = “3D”
DEFI_MATERIAU ELAS
ECRO_LINE
SY
D_SIGM_EPSI
AFFE_CHAR_MECA
CONTACT
METHOD = “FORCED”
STAT_NON_LINE COMP_ELAS
COMP_INCR
NEWTON
RELATION = “ELAS”
RELATION = “VMIS_ISOT_LINE”
STAMP = “TANGENT”
REAC_ITER = 1


8
Results of modeling C
8.1 Values
tested
Identification Reference Displacements
Aster
% difference
Reaction (NR)
D = ­ 20 mm
­ 3.201351E+06
­ 3.986829E+06
24.536
Reaction (NR)
D = ­ 40 mm
­ 6.402702E+06
­ 7.608190E+06
18.828
Reaction (NR)
D = ­ 60 mm
­ 9.604053E+06
­ 1.107936E+07
15.361
Reaction (NR)
D = ­ 80 mm
­ 1.280540E+07
­ 1.355198E+07
5.830
Reaction (NR)
D = ­ 100 mm
­ 1.600675E+07
­ 1.643281E+07
2.662

8.2 Remarks
The results are less precise than those resulting from modelings 2D. The mesh in 3D makes lose it
exact character of the axisymmetric case. Moreover, for savings of time of calculation and space
memory, the mesh 3D is refined less than that in 2D.
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
10/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
9 Modeling
D
9.1
Characteristics of modeling
The symmetry of the problem makes it possible to represent in 3D only one quarter of the model: The sphere and the block
are represented respectively by a quarter of sphere and a quarter of cylinder, with a grid with
solid elements 3D CUB8.
A contact node-mesh is defined between the sphere and the block.
A loading in imposed displacement is applied to the higher part of the sphere rigidified by
a high Young modulus.
Boundary condition:
·
Conditions of symmetry:
nodes located in plan (O, y, Z) (groups of nodes
“SBYZ” and “SSYZ”) are locked according to direction X
(DX = 0),
nodes located in the plan (O, X, y) (groups of nodes
“SBXY” and “SSXY”) are locked according to direction Z
(DZ = 0).
·
Embedding of the base: the nodes of “BASE” (plane Y=0.) are locked according to
directions X, Y, and Z (DX = DY = DZ = 0).
·
The rigid movements of body are removed by imposing a connection following there enters it
node E pertaining to the sphere and the node S pertaining to the solid mass.
Loadings:
An imposed displacement is applied to the higher part of the sphere (group of nodes
“CHIMPO”) according to the direction Y: Loading of 0 with ­ 100. mm
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
11/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
9.2
Characteristics of the mesh
A number of nodes: 6993
A number of meshs and type: 5544 HEXA8, 407 PENTA6 and 191 QUAD4

9.3 Functionalities
tested
Controls Key word
factor
Key word
AFFE_MODELE
AFFE
MODELING = “3D”
DEFI_MATERIAU ELAS
ECRO_LINE
SY
D_SIGM_EPSI
AFFE_CHAR_MECA CONTACT
METHOD = “FORCED”
STAT_NON_LINE COMP_ELAS
COMP_INCR
NEWTON
RELATION = “ELAS”
RELATION = “VMIS_ISOT_LINE”
STAMP = “TANGENT”
REAC_ITER = 1


10 Results of modeling D
10.1 Values
tested
Identification Reference Displacements
Aster
% difference
Reaction (NR)
D = ­ 20 mm
­ 3.201351E+06
­ 3.963968E+06
23.822
Reaction (NR)
D = ­ 40 mm
­ 6.402702E+06
­ 7.653342E+06
19.533
Reaction (NR)
D = ­ 60 mm
­ 9.604053E+06
­ 1.111985E+07
15.783
Reaction (NR)
D = ­ 80 mm
­ 1.280540E+07
­ 1.337793E+07
4.471
Reaction (NR)
D = ­ 100 mm
­ 1.600675E+07
­ 1.628419E+07
1.733

10.2 Remarks
The results are almost identical to those of modeling C. But calculation is even more
tiresome because a quarter of the sphere is with a grid.
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
12/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
11 Modeling
E
11.1 Characteristics of modeling
The symmetry of revolution of the problem allows an axisymmetric modeling: The block is represented
by the cut of its half and the sphere is represented by its surface potentially in contact, they
are with a grid with axisymmetric elements 2D.
A contact of the node-mesh type is defined between the two structures.
A loading in imposed displacement is applied to the higher part of the sphere rigidified by
a high Young modulus.
Boundary condition:
·
symmetry of revolution: the nodes located on the axis Y (group of nodes “LB” and “LS”) are
locked according to direction X (DX = 0),
·
embedding of the base: the nodes of group “PLANX” are locked according to
directions X and Y (DX = DY = 0),
·
the rigid movements of body are removed by imposing a connection following there enters it
node E pertaining to the sphere and the node D pertaining to the solid mass.
Loadings:
An imposed displacement is applied to the higher part of the sphere (group of nodes
“NDPL”) according to the direction Y: Loading of 0 with ­ 100. mm
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
13/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
11.2 Characteristics of the mesh
A number of nodes: 688
A number of meshs and type: 625 QUAD4 and 241 SEG2.
11.3 Functionalities
tested
Controls Key word
factor Key word
AFFE_MODELE
AFFE
MODELING = “AXIS”
DEFI_MATERIAU ELAS
ECRO_LINE
SY
D_SIGM_EPSI
AFFE_CHAR_MECA CONTACT
METHOD = “CONTINUES”
STAT_NON_LINE COMP_ELAS
COMP_INCR
NEWTON
RELATION = “ELAS”
RELATION = “VMIS_ISOT_LINE”
STAMP = “TANGENT”
REAC_ITER = 1


12 Results of modeling E
12.1 Values
tested
Identification Displacement
(mm)
Reference
Aster
% difference
Reaction (NR)
20
­ 2.03804E+06
­ 2.09057E+06
2.577
Reaction (NR)
40
­ 4.07608E+06
­ 4.09426E+06
0.446
Reaction (NR)
60
­ 6.11412E+06
­ 5.84817E+06
­ 4.350
Reaction (NR)
80
­ 8.15217E+06
­ 7.68357E+06
­ 5.748
Reaction (NR)
100
­ 1.01902E+07
­ 9.13216E+06
­ 10.383

12.2 Remarks
The results are slightly better than those of modeling A.
One notices a calculating time 5 times higher than the latter, using the FORCED method.
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
14/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
13 Modeling
F
13.1 Characteristics of modeling
The symmetry of the problem makes it possible to represent in 3D only one quarter of the model: the sphere and the block
are represented respectively by the surface of contact of the sphere and a quarter of cylinder, with a grid
with solid elements 3D CUB8.
A contact node-mesh is defined between the sphere and the block.
A loading in imposed displacement is applied to all the surface of the sphere rigidified by
conditions kinematics.
Boundary condition:
·
Conditions of symmetry:
nodes located in plan (O, y, Z) (group of nodes
“SBYZ”) are locked according to direction X (DX = 0),
nodes located in the plan (O, X, y) (group of nodes
“SBXY”) are locked according to direction Z (DZ = 0),
the nodes of the sphere (group of nodes “SPHSUP”) are
locked according to directions X and Z (DX = DZ = 0)
·
Embedding of the base: the nodes of the group “BASES” (plane Y=0.) are locked
according to directions X, Y, and Z (DX = DY = DZ = 0).
·
The rigid movements of body are removed by imposing a connection following there enters it
node E pertaining to the sphere and the node S pertaining to the solid mass.
Loadings:
An imposed displacement is applied to all surface representing the sphere (group of nodes
“SPHSUP”) according to the direction Y: Loading of 0 with ­ 100. mm
13.2 Characteristics of the mesh
A number of nodes: 2236
A number of meshs and type: 1638 HEXA8, 126 PENTA6, 725 QUAD4, 27 TRIA3 and 26 SEG2.
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
15/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
13.3 Functionalities
tested
Controls Key word
factor Key word
AFFE_MODELE
AFFE
MODELING = “3D”
DEFI_MATERIAU ELAS
ECRO_LINE
SY
D_SIGM_EPSI
AFFE_CHAR_MECA
CONTACT
METHOD = “CONTINUES”
STAT_NON_LINE COMP_ELAS
COMP_INCR
NEWTON
RELATION = “ELAS”
RELATION = “VMIS_ISOT_LINE”
STAMP = “TANGENT”
REAC_ITER = 1


14 Results of modeling F
14.1 Values
tested
Identification Reference Displacements
Aster
% difference
Reaction (NR)
D = ­ 20 mm
­ 3.201351E+06
­ 3.986829E+06
24.536
Reaction (NR)
D = ­ 40 mm
­ 6.402702E+06
­ 7.608190E+06
18.828
Reaction (NR)
D = ­ 60 mm
­ 9.604053E+06
­ 1.107936E+07
15.361
Reaction (NR)
D = ­ 80 mm
­ 1.280540E+07
­ 1.355198E+07
5.830
Reaction (NR)
D = ­ 100 mm
­ 1.600675E+07
­ 1.643281E+07
2.662

14.2 Remarks
The results are less precise than those resulting from modelings 2D. The mesh in 3D makes lose it
exact character of the axisymmetric case. Moreover, for savings of time of calculation and space
memory, the mesh 3D is refined less than that in 2D.
background image
Code_Aster
®
Version
8.2
Titrate:
SSNV506 - Elastoplastic indentation of a half-plane by a indentor
Date
:
15/02/06
Author (S):
P. MASSIN, Mr. KHAM
Key
:
V6.04.506-B
Page:
16/16
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
15 Summary of the results
The results obtained are good. However, a more important variation enters the reference and the results
3D exists. It is possible to fill it by refining even more the mesh but it should be paid in
place memory and in calculating times.
The size of the elements is very important. If they are too large, one can see appearing on the curve
reaction according to the displacement of the “waves” (loss of linearity of this curve). Each
“vague” corresponds to the setting in contact of an element. Moreover, if the mesh is not sufficiently
refined, the reaction given by Aster moves away appreciably from that of reference.
To modelize only the sphere by its surface of contact rigidified by conditions kinematics
a saving of time allows.