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Titrate:
SSNV506 - Elastoplastic Indentation of a half-plane by a indentor Date
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Author (S):
P. MASSIN, Key Mr. KHAM
:
V6.04.506-B Page:
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Organization (S): EDF-R & D/AMA
Handbook 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.

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SSNV506 - Elastoplastic Indentation of a half-plane by a indentor Date
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:
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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
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2
Reference solution

2.1
Method of calculation used for the reference solution

The results of reference result from the book quoted below [bib1].
pm = 3 0 with pm the contact pressure (page 171).
Rjohnson = pm has = 3 0 A 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 model 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 Press - chapter 6 p. 153-201
Handbook of Validation
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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
blocked according to direction X (DX = 0),
·
embedding of the base: the nodes of group “PLANX” are blocked 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
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3.2
Characteristics of the grid

A number of nodes: 916
A number of meshs and type: 625 QUAD4 and 289 SEG2

3.3 Functionalities
tested

Commands Key word
factor Word-key

AFFE_MODELE
AFFE
MODELING = “AXIS”

DEFI_MATERIAU ECRO_LINE
SY

D_SIGM_EPSI
AFFE_CHAR_MECA CONTACT
METHOD = “FORCED”

STAT_NON_LINE COMP_ELAS
RELATION = “ELAS”

COMP_INCR
RELATION = “VMIS_ISOT_LINE”
NEWTON
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 step of final time:



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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 frame located on the axis Y (group of nodes “LB”)
are blocked according to direction X (DX = 0).

All nodes belonging to the sphere (group of nodes
“MAT1”) are blocked according to direction X (DX = 0).
·
Embedding of the base: the nodes of “PLANX” are blocked 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
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5.2
Characteristics of the grid

A number of nodes: 458
A number of meshs and type: 419 QUAD4 and 171 SEG2.

5.3 Functionalities
tested

Commands Key word
factor Word-key

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
RELATION = “ELAS”

COMP_INCR
RELATION = “VMIS_ISOT_LINE”
NEWTON
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 modelling only the surface of contact of the sphere
rigidified by conditions kinematics.

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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 blocked according to direction X (DX = 0),

nodes located in the plan (O, X, y) (group of nodes
“SBXY”) are blocked according to direction Z (DZ = 0),

the nodes of the sphere (group of nodes “SPHSUP”) are
blocked according to directions X and Z (DX = DZ = 0)
·
Embedding of the base: the nodes of group “BASE” (plane Y=0.) are blocked
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 grid

A number of nodes: 6852
A number of meshs and type: 5326 HEXA8, 387 PENTA6 and 183 QUAD4.
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7.3 Functionalities
tested

Commands Key word
factor Word-key

AFFE_MODELE
AFFE
MODELISATION = “3D”

DEFI_MATERIAU ELAS


ECRO_LINE
SY

D_SIGM_EPSI
AFFE_CHAR_MECA
CONTACT
METHOD = “FORCED”

STAT_NON_LINE COMP_ELAS
RELATION = “ELAS”

COMP_INCR
RELATION = “VMIS_ISOT_LINE”
NEWTON
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 grid in 3D makes lose it
exact character of the axisymmetric case. Moreover, for savings of time of calculation and space
memory, the grid 3D is refined less than that in 2D.
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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 blocked according to direction X
(DX = 0),

nodes located in the plan (O, X, y) (groups of nodes
“SBXY” and “SSXY”) are blocked according to direction Z
(DZ = 0).
·
Embedding of the base: the nodes of “BASE” (plane Y=0.) are blocked 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
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9.2
Characteristics of the grid

A number of nodes: 6993
A number of meshs and type: 5544 HEXA8, 407 PENTA6 and 191 QUAD4

9.3 Functionalities
tested

Commands Key word
factor
Key word

AFFE_MODELE
AFFE
MODELISATION = “3D”

DEFI_MATERIAU ELAS


ECRO_LINE
SY

D_SIGM_EPSI
AFFE_CHAR_MECA CONTACT
METHOD = “FORCED”

STAT_NON_LINE COMP_ELAS
RELATION = “ELAS”

COMP_INCR
RELATION = “VMIS_ISOT_LINE”
NEWTON
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. Mais calculation is even more
tiresome because a quarter of the sphere is with a grid.
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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
blocked according to direction X (DX = 0),
·
embedding of the base: the nodes of group “PLANX” are blocked 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
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11.2 Characteristics of the grid

A number of nodes: 688
A number of meshs and type: 625 QUAD4 and 241 SEG2.

11.3 Functionalities
tested

Commands Key word
factor Word-key

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
RELATION = “ELAS”

COMP_INCR
RELATION = “VMIS_ISOT_LINE”
NEWTON
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 method CONTRAINTE.

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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 blocked according to direction X (DX = 0),

nodes located in the plan (O, X, y) (group of nodes
“SBXY”) are blocked according to direction Z (DZ = 0),

the nodes of the sphere (group of nodes “SPHSUP”) are
blocked according to directions X and Z (DX = DZ = 0)
·
Embedding of the base: the nodes of group “BASE” (plane Y=0.) are blocked
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 grid

A number of nodes: 2236
A number of meshs and type: 1638 HEXA8, 126 PENTA6, 725 QUAD4, 27 TRIA3 and 26 SEG2.
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SSNV506 - Elastoplastic Indentation of a half-plane by a indentor Date
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13.3 Functionalities
tested

Commands Key word
factor Word-key

AFFE_MODELE
AFFE
MODELISATION = “3D”

DEFI_MATERIAU ELAS


ECRO_LINE
SY

D_SIGM_EPSI
AFFE_CHAR_MECA
CONTACT
METHOD = “CONTINUES”

STAT_NON_LINE COMP_ELAS
RELATION = “ELAS”

COMP_INCR
RELATION = “VMIS_ISOT_LINE”
NEWTON
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 grid in 3D makes lose it
exact character of the axisymmetric case. Moreover, for savings of time of calculation and space
memory, the grid 3D is refined less than that in 2D.


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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 grid 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 grid is not sufficiently
refined, the reaction given by Aster moves away appreciably from that of reference.

To model only the sphere by its surface of contact rigidified by conditions kinematics
a saving of time allows.

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Outline document