Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
1/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.51
Contact discrete friction in 2D and 3D
Summary:
Discrete modelings of the contact with friction in 2D and 3D, areas of slip being
respectively 1D and 2D, are proposed starting from a mixed variational formulation stresses
displacements. The conditions of contact and friction are treated with the nodes of surfaces of contact of
solids implied with taking into account of great displacements. The law of friction of Coulomb is treated
by operators STAT_NON_LINE and DYNA_NON_LINE after definition of the conditions of contact and of
friction under the key word CONTACT of AFFE_CHAR_MECA.
In 2D as in 3D, various methods are usable for the modeling of the problem:
·
taking into account of the contact and friction using multipliers of Lagrange,
·
taking into account of the contact using multipliers of Lagrange and friction using
penalization,
·
taking into account of the contact and friction using penalization.
The subjacent algorithms are inspired by the active stresses [bib2] [bib9] and by the predictor-correctors
usually used in plasticity.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
2/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Count
matters
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
3/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
1 Introduction
Discrete modelings of the contact with friction in 2D and 3D, areas of slip being
respectively 1D and 2D, are proposed starting from a mixed variational formulation stresses
displacements. The conditions of contact and friction are treated with the nodes of surfaces of
contact of the solids implied with taking into account of great displacements. The law of friction of
Coulomb is treated by operators STAT_NON_LINE and DYNA_NON_LINE after definition of
conditions of contact and friction under the key word CONTACT of AFFE_CHAR_MECA.
In 2D as in 3D, various methods are usable for the modeling of the problem:
·
taking into account of the contact and friction using multipliers of Lagrange,
·
taking into account of the contact using multipliers of Lagrange and friction using
penalization,
·
taking into account of the contact and friction using penalization.
The subjacent algorithms couple the method of the active stresses [bib2] to determine them
areas of contact and an algorithm of resolution of Newton inspired of the methods of the type
predictor-corrector, usually used in plasticity, for friction, in order to determine them
areas of slip [bib1] [bib7] [bib10].
In 2D as in 3D, the user has a whole of methods panachant dualisation and
regularization by penalization:
·
dualisation of the conditions of contact and friction,
·
dualisation of the conditions of contact and regularization of the conditions of friction,
·
regularization of the conditions of contact and friction.
The compared interests of these methods are well-known: the dualisation introduces news
unknown factors but it provides an exact solution; the regularization provides only one approximation of
solution dependant on a parameter chosen by the user but it does not introduce a news
unknown factors.
The document begin with a general presentation from the laws from friction. One presents then
discretization of these laws like their linearization for their integration within the method of
Newton. One details the algorithms then allowing to solve these problems. End of the document
draft of the use practices these methods within Code_Aster and of their post front processing
to approach the conclusions.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
4/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
2
Formulation of the model
2.1
The criterion of Coulomb
Are 2 solids being able to come into rubbing contact:
N
T
Are
N
the outgoing normal on the surface of contact,
N
U.
=
N
U
displacement following this normal,
G
existing initial play between 2 solids,
N
N
.
.
=
N
the normal force exerted by one of surfaces
on the other and
N
N
N
T
-
=.
shearing.
U
1
U
2
U
1
U
2
U
1
U
2
U
1
U
2
T
1
T
2
N
1
Solid 1
Solid 2
U
1
U
2
U
1
U
2
U
1
U
2
U
1
U
2
U
1
U
2
U
1
U
2
T
1
T
2
Solid 1
Solid 2
More precisely, for two solids (1) and (2) in contact: the area of contact is either specific, or
linear is surface. The force of shearing then has as a direction in the area of contact one
vector
T
located in the tangent plan
)
,
(
2
1
T
T
indicated on the figure above. One defines:
T
N
N
N
N
R
R
=
-
=
1
1
1
1
1
1
)
).
.
((
)
.
(
,
T
R
=
the force of shearing exerted by the solid (2) on the solid (1) per unit of area of contact.
Let us write the system of equations and inequations having to be checked by these sizes:
(
)
(
)
=
-
=
-
=
-
=
-
+
-
=
+
0
0
)
(
.
0
0
)
.
.
(
0
.
.
.
1
2
2
2
1
1
1
2
1
2
2
1
1
µ
µ
N
T
T
N
T
N
N
G
G
R
T
U
U
U
N
U
N
U
N
U
U
N
U
N
U
&
&
&
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
5/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
where
1
N
is the outgoing normal with the solid (1) and
2
N
the outgoing normal with the solid (2) opposed to
1
N
.
first set of equations and inequations corresponds to the management of the contact; it will not be detailed and
we return to [bib9]. The second corresponds to the description of friction obeying the criterion of
Coulomb. It utilizes several fields and binds between them: normal pressure, shearing and
tangent displacement. It can be included/understood as follows:
·
If
N
T
µ
<
,
-
0
=
and
0
=
T
u&
·
If
N
T
µ
=
,
-
0
>
and
R
U
=
T
&
One can give following graphic interpretations:
T
1
T
2
N
T
1
T
2
member
slipping
member
slipping
member
slipping
N
2
T
1
T
In the space of the stresses, the effort of rubbing contact can be only inside the cone of
Coulomb: if it is strictly inside, the contact is adherent; if it is on the surface of the cone, it
contact is slipping. One can thus give another representation of this criterion for a situation of
contact known:
R
member
slipping
slipping
N
µ
N
µ
-
T
u&
Friction induces the concept of threshold; we will see now how to formulate in manner
general of other laws of friction by using this concept.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
6/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
2.2
General formulation for various criteria
The selected criteria of friction are form:
()
0
R
G
where
()
R
G
is a convex function. The field of nonslip is defined by the interior of the convex one.
Two criteria of friction of the form
()
0
R
G
are particularly used:
·
the criterion of Tresca where:
0
)
(
-
=
K
G
R
R
and
K
=constante
One notes
C
the convex disc
C
of radius
K
centered at the origin defined by:
{
}
K
C
=
R
R
.
The condition of nonslip is then defined by the membership of
R
inside
C
.
In the event of slip, for
R
located on the border of
C
, direction of slip
T
of
u&
is given by the normal to the criterion in
R
, as indicated below:
T
1
T
2
R
T
C
-
K
+k
T
1
T
2
R
C
-
K
+k
·
the criterion of Coulomb where:
0
)
,
(
)
,
;
(
-
=
N
N
K
G
µ
µ
R
R
and
N
K
µ
=
The value of
K
depends on
0
).
.
(
=
N
N
N
the normal component of the force exerted by
one of surfaces on the other and of
µ
, the coefficient of friction of Coulomb. In the event of
slip, for
R
located on the border of
G
who is a cone, the direction of slip
T
of
u&
is not given by the normal to the criterion in
R
, but by the normal with the convex disc
C
of radius
N
K
µ
=
.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
7/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
T
1
T
2
N
T
T
1
T
2
N
T
N
µ
=
R
Determination of the direction of slip for
the criterion of Coulomb in the reference mark of the vectors
(
)
2
1
, T
T
·
the criterion of Mohr-Coulomb where:
0
)
,
,
(
)
,
,
,
(
-
=
C
K
R
C
G
N
N
µ
µ
R
and
N
C
K
µ
+
=
It is particularly used to characterize the behavior of interfaces the géomatériaux one (clays
in particular).
C
is the cohesion of material and
µ
the coefficient of friction (
=
tan µ
, where
is the angle
of friction). Again
0
).
.
(
<
=
N
N
N
so that the contact remains maintained.
2.3
Formulation by differential inclusions
One notes
V
the whole of displacements kinematically acceptable of the problem. The relation enters
the speed of relative slip
u&
and the shear stress
R
translated the two possible states of
system: not slip or relative slip following the normal direction to the convex disc
C
.
For the three criteria presented, the function
)
(R
u&
and its reciprocal
)
(U
R &
both belong
with the under-differentials of two combined pseudopotentials, so that one can write:
)
(R
U
C
&
and
)
(
*
U
R
&
C
.
The appearance of included differential comes from the not-differentiable character of the laws from
contact-friction. Indeed,
C
indicate the indicating function of the convex disc
C
of radius
K
,
centered at the origin, previously definite. It is such as:
+
=
if not
if
0
)
(
C
C
R
R
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
8/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
)
(R
C
is then the under-differential of
C
in
R
. It merges with the normal external with
C
in
R
.
U
U
&
&
K
C
=
)
(
*
, where
K
is the threshold of friction resistance, is combined of Fenchel of
indicating function
C
.
)
(
*
u&
C
is positively homogeneous degree 1. This function is interpreted
like the density of power dissipated in the slip. Using the concepts of under-differential,
one can establish the following relations for
u&
and
R
associated:
()
(
)
()
()
()
()
.
.
;
,
.
;
,
0
)
.(
0
)
(
)
R
(
*
*
*
*
U
R
U
R
U
v
U
v
U
v
R
U
U
R
R
R
R
U
R
U
&
&
&
&
&
&
&
&
&
&
&
&
K
V
V
C
C
C
C
C
C
C
C
=
=
+
-
-
-
=
Note:
1) The two combined pseudopotentials presented are nondifferentiable.
2) Once known the normal reaction for the criterion of Coulomb, one is reduced locally to
a criterion of friction of Tresca whose threshold is worth
N
K
µ
=
.
3) The adopted local criteria having a circular form one deduces from it that
)
(R
U
C
&
imply
that there exists
positive reality such as
R
U
=
&
.
4) The formulation of the problem of speed suggests an incremental numerical resolution of
problem of friction. The resolution of the problem of balance will thus be presented under
incremental form.
2.4
Resolution of the problem of balance.
One considers two solids of total volume
whose surface of contact is
C
. To simplify, one
will suppose the existence of a differentiable deformation energy to characterize the answer of
two solids separated with external stresses. In fact, one can show that the results given
hereafter are independent of this assumption. One notes
V
the whole of the fields of displacement
kinematically acceptable, constrained by the respect of the conditions of contact and friction on
the interface.
The balance of the two solids in the absence of friction is written:
To find
U
field of displacement kinematically acceptable such as:
[
]
{
}
.
),
(
)
(
)
(
)
(
)
(
))
(
(
min
arg
V
W
W
W
V
-
-
-
=
v
v
v
U
U
v
v
U
v
In elasticity,
=
D
))
(
(
)
(
v
v
is the deformation energy. The function
)
(v
W
represent it
work of the external forces. A condition necessary and which becomes sufficient if
is strictly
convex so that this balance is checked is that:
()
0
L
U
U
U
=
-
=
-
ext.
D
DW
D
)
(
)
(
where
D
is the operator derived Gâteaux and
ext.
L
is the linear form associated the external forces.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
9/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
With the introduction of friction, the problem must be tackled in incremental form. One is
conduit [bib3] [bib4] with the problem of minimization following on the unit
V
fields
kinematically acceptable constrained by the respect of the conditions of contact and friction on
the interface:
U
known, to find
V
U
such as:
[
]
)
(
)
(
))
(
(
min
arg
*
v
U
v
v
U
U
U
v
+
-
+
+
=
+
W
T
C
V
.
U
U
+
is thus solution of:
+
-
+
+
)
(
))
(
(
min
v
U
v
v
U
v
W
D
K
D
C
T
V
C
.
where
T
v
is the tangential component of the increment of relative displacement of solid 2 compared to
solid 1 along the surface of contact, with the conventions adopted with [§2.1].
Using the relations
T
T
C
K v
v
=
)
(
*
and
T
T
C
v
R
v
.
)
(
*
if
C
R
one deduces from it that
U
U
+
is
solution of the problem of following MinMax, on space
V
fields kinematically acceptable:
)
,
(
R
v
U
R
v
+
J
Max
Min
V
where:
)
(
)
)
(
.
(
))
(
(
)
,
(
v
U
R
v
R
v
U
R
v
U
+
-
-
+
+
=
+
W
D
D
J
C
C
T
C
The presence of the indicating function in this expression indicates that shearing
R
on
surface contact
C
belongs to the convex disc of friction
C
.
2.5 Formulation
variational
If
is convex, the problem of
MinMax
to solve in an equivalent way in the form is put:
To find
V
U
and
C
R
, together of independent variables such as:
(
)
0
,
+
R
U
U
J
This amounts solving the system of equations to following balance:
,
0
)
(
,
0
))
(
-
=
-
+
+
R
U
R
v
L
v
R
U
U
(
C
C
C
C
T
ext.
C
T
D
D
D
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
10/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
or in an equivalent way:
(
)
(
)
.
.
.
),
(
,
0
))
(
(
on
1
C
C
T
ext.
frot
D
=
=
-
+
+
T
N
R
R
U
v
L
L
U
U
As in the preceding section,
ext.
L
is the linear form associated the external forces. The form
linear
frot
L
is associated the forces of shearing exerted by solid 2 on the surface of
contact of solid 1. It will be also noted that the variational formulation makes it possible to find not
only equilibrium equations of the system but also membership of
T
U
with under
differential of
C
.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
11/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
3
Contact rubbing 2D and 3D in Code_Aster
We saw the formulation previously continues problem of rubbing contact. We go
now to examine how it is expressed in discrete form.
3.1
Formulation of the problem
With each pitch of time N, one seeks to check the total balance of the structure:
frot
cont
ext.
N
int
L
L
L
U
L
-
-
=
)
(
where:
·
int
L
is the operator of calculation of the internal forces
·
ext.
L
is the vector of the external forces
·
cont
L
is the vector of the forces of contact
·
frot
L
is the vector of the forces of friction
Moreover, the field of displacement
N
U
is subjected to a whole of conditions equality and inequality which
include/understand connection by connection:
·
nc
N
nc
D
U
With
)
(
·
C
N
C
D
U
With
=
)
(
·
0
U
U
With
=
-
-
)
(
1
N
N
sg
·
frot
N
N
G
L
U
U
With
=
-
-
)
(
1
where
0
where
·
L
: together possible connections of contact (active and nonactive)
·
NC
: together nodes of potential surfaces of contact which are not in contact
(nonactive connections)
·
C
: together nodes indeed in contact (active connections)
·
SG
: together adherent nodes of contact
·
G
: together slipping nodes of contact
·
G
SG
C
=
,
=
NC
C I
,
NC
C
L
=
·
C
With
is the matrix of the nodes in contact
·
sg
With
is the matrix of the nodes in adherent contact
·
G
With
is the matrix of the nodes in slipping contact
Because of incremental nature of the resolution of balance, one can rewrite these equations and
inequations in the form:
I
frot
I
cont
ext.
I
I
N
N
int
L
L
L
U
U
U
L
-
-
=
+
+
-
-
)
(
1
1
subjected to:
·
C
I
I
N
N
C
D
U
U
U
With
=
+
+
-
-
)
(
1
1
that is to say
1
-
=
I
C
I
C
D
U
With
with
1
2
1
1
1
1
)
(
-
-
-
-
-
-
-
=
-
=
+
-
=
I
N
C
I
C
I
N
C
C
I
N
N
C
C
I
C
U
With
D
U
With
D
U
U
With
D
D
·
0
U
U
With
=
+
-
)
(
1
I
I
N
sg
that is to say
1
-
=
I
sg
I
sg
D
U
With
·
I
frot
I
I
N
G
L
U
U
With
=
+
-
)
(
1
where
0
We have the discretized formulation of a problem of rubbing contact. We will see them
various manners of taking into account the whole of conditions (or stresses) equality and inequality
who relate to the field of displacements: dualisation or regularization.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
12/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
3.2
Dualisation of the conditions of contact and friction
To take into account the stresses relating to the field of displacements, it is possible of
dualiser i.e to utilize them in balance through multipliers of Lagrange (like
that is done for the boundary conditions kinematics in the code). 3 assemblies are introduced of
multipliers of Lagrange:
·
C
µ
relating to the conditions of contact
·
sg
µ
relating to the conditions of adherence
·
G
µ
relating to the conditions of slip
Balance is written then in the following form:
ext.
I
G
G
I
sg
sg
I
C
C
I
I
N
N
int
T
T
T
L
µ
With
µ
With
µ
With
U
U
U
L
=
+
+
+
+
+
-
-
)
(
1
1
subjected to:
·
1
-
=
I
C
I
C
D
U
With
·
1
-
=
I
sg
I
sg
D
U
With
·
I
G
I
I
N
G
µ
U
U
With
=
+
-
)
(
1
where
0
and
I
C
I
G
I
G
K
µ
µ
µ
=
=
This system allows the following interpretation of the multipliers of Lagrange:
·
I
C
C
T
µ
With
is the whole of the nodal forces of contact
·
I
sg
sg
T
µ
With
is the whole of the nodal forces of adherence
·
I
G
G
T
µ
With
is the whole of the nodal forces of slip
Note:
1) In the expression of balance, the condition of contact became an equality. Indeed,
this equation is written for the nodes really in contact (for the active connections).
It is a logic which takes as a starting point the the method of the active stresses established in
code for the processing of the unilateral contact [bib9]. It is nevertheless imperative to check
condition:
0
µ
>
I
C
2) Indeed: For a connection the operator
C
With
associate the fields displacements
1
U
and
2
U
the sum relative displacements
2
1
N
U
N
U
.
.
2
1
+
compared to the normals with
surface contact is the scalar
(
)
1
2
1
.n
U
U
-
. The operator
T
C
With
associate the scalar
C
µ
nodal forces
1
N
C
µ
and
2
N
C
µ
applying to the solids (1) and (2) respectively. These
nodal forces are equivalent to external forces
N
C
µ
-
applying to the solids
(1) and (2) respectively, which amounts transferring the term
I
C
C
T
µ
With
equation
of balance from left to right. Except for a surface scalar term
N
C
µ
-
is equivalent to
N
N
what implies the positivity of
C
µ
in the event of contact.
3) It is also checked that for the nonactive connections one has well:
1
-
I
nc
I
nc
D
U
With
.
4) Matrices
G
sg
C
nc
With
With
With
With
,
,
,
vary during reiterated. We will clarify these
variations more in detail with [§4].
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
13/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
3.3
Dualisation of the conditions of contact, regularization of the conditions
of friction
It is possible of dualiser the conditions of contact and to regularize the conditions of friction. One
understands by there the fact of facilitating the processing of friction by removing of its graph the infinite slope
in 0 i.e.:
T
u&
R
E
T
N
µ
-
N
µ
This graph calls for several observations:
·
the concept of adherence with properly spoken disappeared, all the nodes slip. It is defined
nevertheless by:
the node
I
is “adherent” if, being given
T
T
E U
R
&
=
,
N
µ
<
R
· more the slope
T
E
is strong, more the regularized graph approaches the graph not regularized
· in fact of regularization of the conditions of friction, it is rather about regularization of
conditions of adherence
Taking into account the preceding remarks, one rewrites balance in the form:
ext.
I
G
G
I
I
N
sg
sg
T
I
C
C
I
I
N
N
int
T
T
T
E
L
µ
With
U
U
With
With
µ
With
U
U
U
L
=
+
+
+
+
+
+
-
-
-
)
(
)
(
1
1
1
subjected to:
·
1
-
=
I
C
I
C
D
U
With
·
I
C
I
I
N
sg
T
E
µ
U
U
With
µ
<
+
-
)
(
1
·
I
G
I
I
N
G
µ
U
U
With
=
+
-
)
(
1
where
0
and
I
C
I
G
I
G
K
µ
µ
µ
=
=
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
14/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
3.4
Regularization of the conditions of contact and friction
With same logic that in the preceding paragraph, one defines the graph of the law of contact
regularized:
NR
E
N
D
N
And one deduces the form from it from balance:
ext.
I
G
G
I
I
N
sg
sg
T
I
C
I
C
C
NR
I
I
N
N
int
T
T
T
E
E
L
µ
With
U
U
With
With
D
U
With
With
U
U
U
L
=
+
+
+
-
+
+
+
-
-
-
-
)
(
)
(
)
(
1
1
1
1
subjected to:
·
)
(
)
(
1
1
-
-
-
<
+
I
C
I
C
NR
I
I
N
sg
T
E
E
D
U
With
U
U
With
µ
·
I
G
I
I
N
G
µ
U
U
With
=
+
-
)
(
1
where
0
and
)
(
1
-
-
=
=
I
C
I
C
NR
I
G
I
G
E
K
D
U
With
µ
µ
Note:
The term
)
(
1
-
-
I
C
I
C
C
NR
T
E
D
U
With
With
is calculated only for the active connections. The use of
the operator left positive
[]
=
+
if not
0
0
if
X
X
X
a compact writing of the law of contact allows
regularized for all the possible connections in the form
[
]
+
-
-
1
I
C
I
C
C
NR
T
E
D
U
With
With
. It would be
possible to use it to write the law of regularized adherence. We preferred to preserve
partition between the various states of the connections because this presentation is closer to
numerical integration in the code.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
15/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
4 Resolution
algorithmic
4.1
Linearization of the various terms
It is the method of Newton which is used in Code_Aster for the resolution of the problems not
linear [bib8]. It is in fact necessary to linearize the various terms appearing in
the expression of balance.
4.1.1 Forces
interns
The linearization of the operator of forces intern compared to
I
U
conduit with:
I
I
N
I
N
N
int
I
I
int
I
N
N
int
I
I
N
N
int
I
N
N
U
K
U
U
L
U
U
L
U
U
L
U
U
U
L
U
U
+
+
+
+
+
+
-
-
+
-
-
-
-
-
-
.
)
(
.
)
(
)
(
1
1
1
1
1
1
1
1
I
N
K
is the coherent tangent matrix which includes non-linearities behavioral and geometrical.
4.1.2 Forces of contact
In the absence of regularization of the forces of contact, one does not carry out any linearization. In
contrary case, one with the linearization:
1
1
1
1
1
-
-
-
-
-
-
I
C
I
C
NR
I
I
C
I
C
NR
I
C
I
C
I
C
C
T
T
T
T
E
E
D
With
U
With
With
µ
With
µ
With
Note:
·
During the linearization of
I
C
T
C
µ
With
the index i-1 in the matrix of contact appeared
1
-
I
C
With
. In
effect, at the time of the determination of
I
U
, only the state of contact rubbing with the iteration i-1 is known.
·
The term
1
1
-
-
I
C
I
C
NR
T
E
With
With
contribute a new share to the tangent matrix of the problem,
while the term
1
1
-
-
I
C
I
C
NR
T
E
D
With
contribute a new share to the second member.
4.1.3 Forces of slip
Taking into account the definition of the forces of slip, they can be expressed:
)
(
)
(
)
(
)
(
1
1
1
1
I
I
N
G
I
I
N
G
I
G
I
I
N
G
I
I
N
G
I
C
I
G
K
U
U
With
U
U
With
U
U
With
U
U
With
µ
µ
µ
+
+
=
+
+
=
-
-
-
-
They check the conditions indeed:
I
G
I
I
N
G
µ
U
U
With
=
+
-
)
(
1
where
0
and
I
C
I
G
I
G
K
µ
µ
µ
=
=
and reveal the two unknown factors
I
C
µ
and
I
U
.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
16/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
In practice, however, it is considered that the knowledge of the threshold of slip is acquired with
the preceding iteration, which amounts being brought back to a criterion of Tresca for each iteration. With
convergence the threshold is obviously fixed: there are not thus more differences between the thresholds with the course
iterations. Another approach, used in [R5.03.52], amounts solving, for each state of
contact, the solution with friction and thus to consider that the threshold of slip is a fixed point. Our
approach takes as a starting point this type of method while being less constraining and thus less expensive in
time CPU but can be less robust.
I
G
µ
is thus approximated by:
)
(
)
(
)
(
)
(
1
1
1
1
1
1
1
1
1
1
I
I
N
I
G
I
I
N
I
G
I
G
I
I
N
I
G
I
I
N
I
G
I
C
I
G
K
U
U
With
U
U
With
U
U
With
U
U
With
µ
µ
µ
+
+
+
+
-
-
-
-
-
-
-
-
-
-
Linearization of
I
G
µ
compared to
I
U
in the expression given previously led to
following formulation:
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
.
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
I
N
I
G
I
I
G
I
N
I
G
I
N
I
G
I
N
I
G
I
G
I
N
I
G
I
I
G
I
G
I
N
I
G
I
N
I
G
I
G
I
G
K
K
K
U
With
U
With
U
With
U
With
U
With
U
With
U
With
U
With
U
With
µ
who is still written:
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
-
+
I
N
I
G
I
I
G
I
G
I
N
I
N
I
G
I
N
I
G
I
G
I
N
I
G
I
I
G
I
G
I
G
I
N
I
G
I
I
G
I
G
I
N
I
N
I
G
I
N
I
G
I
G
I
N
I
G
I
I
G
I
G
I
N
I
G
I
N
I
G
I
G
I
G
T
T
T
T
K
K
K
K
K
U
With
U
With
With
U
U
With
U
With
U
With
U
With
µ
U
With
U
With
With
U
U
With
U
With
U
With
U
With
U
With
U
With
µ
Note:
·
Terms
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I
N
I
G
I
G
I
G
I
N
I
N
I
G
I
N
I
G
I
G
I
N
I
G
I
G
I
G
T
T
K
K
U
With
With
With
U
U
With
U
With
U
With
With
news brings
contributions to the tangent matrix of the problem.
·
In 2D,
I
G
µ
is linearized in
1
1
1
1
1
1
-
-
-
-
-
-
=
I
G
I
N
I
G
I
N
I
G
I
G
K
µ
U
With
U
With
who is independent of
I
U
, there is not thus
not new contributions to the tangent matrix.
4.1.4 Forces
of adherence
In the absence of regularization of the forces of adherence, one does not carry out any linearization. In
contrary case, one with the linearization:
I
I
sg
I
sg
T
I
N
I
sg
I
sg
T
I
G
I
sg
I
G
sg
T
T
T
T
E
E
U
With
With
U
With
With
µ
With
µ
With
1
1
1
1
1
1
-
-
-
-
-
-
+
4.1.5 Notice
Matrices of contact
C
With
, of slip
G
With
and of nonslip
sg
With
are brought to be
modified during iterations of Newton if contacts change statute or if one
geometrical reactualization takes place: they are thus subscripted
I
C
With
,
I
G
With
and
I
sg
With
. In the contrary case
I
C
With
,
I
G
With
and
I
sg
With
do not vary during not reiterated.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
17/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
5 Resolution
The resolution strictly speaking is approached in this part. Will be presented the total system
solved in the framework of the method of Newton and the algorithm of processing of the rubbing contact
(under-iteration). The initial state with the iteration i=1 for all these methods corresponds to the resolution of one
problem without contact nor friction. If there is indeed no detected contact, this initial state
corresponds to the solution of the problem in the linear elastic case for example, if not it is modified
as of this iteration in order to take into account the connections which would violate the conditions of contact
unilateral.
5.1.1 Dualisation of the conditions of contact and friction in 2D
Method of Newton:
·
With the iteration
I
, at the total level, resolution of the system (one voluntarily does not make
to appear the conditions of Dirichlet):
1
1
1
1
1
)
(
-
-
-
-
-
-
-
-
-
=
I
G
T
I
G
I
sg
T
I
sg
I
C
T
I
C
I
N
int
ext.
I
I
N
µ
With
µ
With
µ
With
U
L
L
U
K
Algorithm of rubbing contact:
·
Determination of the connections in contact
- Initial State:
L
NC
N
=
0
,
=
0
N
C
,
=
0
N
SG
,
0
0
/
=
N
G
. All points of surface
potential of contact are lack of contact.
- If i=1 first elastic design without taking into account of the contact. That is to say then
{
}
0
t.q.
connections
1
0
1
1
<
-
=
=
U
With
D
D
nc
nc
nc
N
C
. If
=
1
N
C
the solution without contact is
valid. If
1
N
C
then
1
1
N
N
C
SG
=
,
0
1
/
=
N
G
and resolution of the system of equations Ci
above with the new conditions for iteration 1.
- If geometrical reactualization
{
}
0
t.q.
connections
<
=
I
nc
I
N
C
D
,
I
N
I
N
I
N
C
G
G
=
- 1
and
I
N
I
N
I
N
G
C
SG
-
=
.
- If not
1
-
=
I
N
I
N
C
C
,
1
-
=
I
N
I
N
SG
SG
,
1
-
=
I
N
I
N
G
G
·
Resolution of the system
1
1
1
1
1
1
)
(
- +
- +
-
-
-
+
- +
=
-
-
=
+
I
sg
C
I
I
sg
C
I
G
T
I
G
I
N
int
ext.
I
sg
C
T
I
sg
C
I
I
N
D
U
With
µ
With
U
L
L
µ
With
U
K
One carries out for that a resolution per blocks:
)
)
(
(
)
(
1
1
1
1
1
1
1
1
1
1
1
I
sg
C
T
I
sg
C
I
G
T
I
G
I
N
int
ext.
I
N
I
I
G
T
I
G
I
N
int
ext.
I
sg
C
T
I
sg
C
I
N
I
sg
C
+
- +
-
-
-
-
-
-
-
+
- +
-
- +
-
-
-
=
-
-
=
µ
With
µ
With
U
L
L
K
U
µ
With
U
L
L
µ
With
K
With
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
18/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
·
Checking of the state of the sliding joints
- If for a connection
J
0
)
(
.
1
1
1
<
+
-
-
-
I
I
N
I
G
T
I
G
U
U
With
µ
,
}
{J
G
G
I
N
I
N
-
=
and
}
{J
SG
SG
I
N
I
N
+
=
- If not
)
(
)
(
1
1
I
I
N
I
G
I
I
N
I
G
I
C
I
G
U
U
With
U
U
With
µ
µ
µ
+
+
=
-
-
- So at least a connection changed state, return to the resolution of the system of equations
for same iteration I, but with the connections not slipping identified above.
·
Checking of the state of the adherent connections
- If for a connection
J
I
C
I
sg
µ
µ
µ
,
I
sg
I
sg
I
C
I
G
µ
µ
µ
µ
µ
=
,
}
{J
SG
SG
I
N
I
N
-
=
and
}
{J
G
G
I
N
I
N
+
=
·
Checking of the state of the connections of contact
- If the connection
J
supposed not activates active, is violated, i.e that whose play is it
more negative, is added with the whole of the active connections,
}
{J
C
C
I
N
I
N
+
=
and
}
{J
G
G
I
N
I
N
+
=
, return to the resolution of the system of equations for the same iteration
I
,
but with the connections not slipping identified above.
- If for a connection
J
0
<
I
C
µ
,
}
{J
C
C
I
N
I
N
-
=
}
{J
SG
SG
I
N
I
N
-
=
}
{J
G
G
I
N
I
N
-
=
(in
function of the type of the connection)
·
Update
I
U
,
I
C
µ
,
I
G
µ
,
I
sg
µ
,
I
sg
C
+
With
and
I
G
With
.
5.1.2 Dualisation of the conditions of contact and regularization of friction in 2D and
3D
For this modeling, the algorithm is the same one for the 2D and the 3D.
Method of Newton:
·
With the iteration
I
, at the total level, resolution of the system:
1
1
1
1
1
)
(
)
(
-
-
-
-
-
-
-
-
-
=
+
I
N
I
N
F
I
G
T
I
G
I
C
T
I
C
I
N
int
ext.
I
I
N
F
I
N
U
K
µ
With
µ
With
U
L
L
U
K
K
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
19/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Note:
·
The matrix
I
N
F
K
contains the contributions of terms of slip and of adherence is:
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
=
I
N
I
G
I
G
I
G
I
N
I
N
I
G
I
N
I
G
I
G
I
N
I
G
I
G
I
G
I
sg
I
sg
T
I
N
F
T
T
T
K
K
E
U
With
With
With
U
U
With
U
With
U
With
With
With
With
K
·
In the preceding expression, a contribution is preceded by the sign -. The command of the terms
of this contribution is the same one as that of the other terms. The effect of this contribution
is particularly destabilizing for the total behavior of the tangent matrix with
system, more particularly when one is far from balance and thus with the beginning from
resolution with each new pitch of time. One thus decides to only take it into account
partially by affecting it of a coefficient
]
1
,
0
[
. One advises to use an initial value
from 0.5 for this coefficient and to decrease it if convergence is not obtained. In the case
where
0
=
convergence always seems to be obtained but is particularly slow.
When one is close to the solution, it is on the other hand very useful to have a value of it
coefficient equalizes to 1 in order to accelerate convergence. That is done automatically
in the code when residue RESI_GLOB_RELA is lower than
3
10
-
.
·
The matrix
I
N
F
K
contains the contributions of the new second members of adherence,
that is to say:
1
1
-
-
=
I
sg
I
sg
T
I
N
F
T
E
With
With
K
Algorithm of rubbing contact:
·
Determination of the connections in contact
- Initial State:
L
NC
N
=
0
,
=
0
N
C
,
=
0
N
SG
,
0
0
/
=
N
G
. All points of surface
potential of contact are lack of contact.
- If i=1 first elastic design without taking into account of the contact. That is to say then
{
}
0
t.q.
connections
1
0
1
1
<
-
=
=
U
With
D
D
nc
nc
nc
N
C
,
1
1
N
N
C
G
=
,
0
1
/
=
N
SG
. If
=
1
N
C
solution without contact is valid. If
1
N
C
then
1
1
N
N
C
SG
=
,
0
1
/
=
N
G
and resolution of
system with the new conditions for iteration 1.
- If geometrical reactualization
{
}
0
t.q.
connections
<
=
I
N
I
N
C
D
,
I
N
I
N
I
N
C
G
G
=
- 1
and
I
N
I
N
I
N
G
C
SG
-
=
.
- If not
1
-
=
I
N
I
N
C
C
,
1
-
=
I
N
I
N
SG
SG
,
1
-
=
I
N
I
N
G
G
·
Resolution of the system
1
1
1
1
1
1
1
)
(
)
(
-
-
-
-
-
-
-
=
-
-
-
=
+
+
I
C
I
I
C
I
N
I
N
F
I
G
T
I
G
I
N
int
ext.
I
C
T
I
C
I
I
N
F
I
N
D
U
With
U
K
µ
With
U
L
L
µ
With
U
K
K
One carries out like previously a resolution per blocks.
·
For all the connections,
)
(
)
(
1
1
1
1
I
I
N
I
G
I
I
N
I
G
I
C
I
G
U
U
With
U
U
With
µ
µ
µ
+
+
=
-
-
-
-
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
20/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
·
Checking of the state of the connections of contact
- If the connection
J
supposed not activates active, is violated, i.e that whose play is it
more negative, is added with the whole of the active connections
}
{J
C
C
I
N
I
N
+
=
and
}
{J
G
G
I
N
I
N
+
=
, return to the resolution of the system of equations for same iteration I,
but with the connections not slipping identified above.
- If for a connection J
0
<
I
C
µ
,
}
{J
C
C
I
N
I
N
-
=
}
{J
G
G
I
N
I
N
-
=
·
Checking of the state of the sliding joints and “adherent”
- If for a connection
J
I
G
I
C
I
I
N
I
G
T
E
µ
µ
U
U
With
=
<
+
-
-
µ
)
(
1
1
, then
)
(
1
I
I
N
I
sg
T
I
sg
E
U
U
With
µ
+
=
-
,
}
{J
SG
SG
I
N
I
N
+
=
and
}
{J
G
G
I
N
I
N
-
=
.
·
Calculation of the tangent matrices
I
N
F
K
and
I
N
F
K
(if RESI_GLOB_RELA
<1.E-3
,
.
1
=
)
·
Update
I
U
,
I
C
µ
,
I
G
µ
,
I
sg
µ
,
I
C
With
,
I
sg
With
and
I
G
With
.
5.1.3 Regularization of the conditions of contact and friction in 2D and 3D
For this modeling, the algorithm is the same one for the 2D and the 3D.
Method of Newton:
·
With the iteration
I
, at the total level, resolution of the system:
1
1
1
1
1
1
)
(
)
(
-
-
-
-
-
-
+
-
-
-
=
+
I
C
T
I
C
NR
I
N
I
N
F
I
G
T
I
G
I
N
int
ext.
I
I
N
F
I
N
E
D
With
U
K
µ
With
U
L
L
U
K
K
Note:
·
The matrix
I
N
F
K
contains the contributions of terms of contact, slip and
of adherence is:
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
+
=
I
N
I
G
I
G
I
G
I
N
I
N
I
G
I
N
I
G
I
G
I
N
I
G
I
G
I
G
I
sg
I
sg
T
I
C
I
C
NR
I
N
F
T
T
T
T
K
K
E
E
U
With
With
With
U
U
With
U
With
U
With
With
With
With
With
With
K
·
The matrix
I
N
F
K
contains the contributions of the new second members of adherence,
that is to say:
1
1
-
-
=
I
sg
I
sg
T
I
N
F
T
E
With
With
K
·
The term
1
1
-
-
-
I
C
T
I
C
NR
E
D
With
contains the contributions of the new second members of
contact.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
21/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Algorithm of rubbing contact:
·
Determination of the connections in contact
- Initial State:
L
NC
N
=
0
,
=
0
N
C
,
=
0
N
SG
,
0
0
/
=
N
G
. All points of surface
potential of contact are lack of contact.
- If i=1 first elastic design without taking into account of the contact. That is to say then
{
}
0
t.q.
connections
1
0
1
1
<
-
=
=
U
With
D
D
nc
nc
nc
N
C
,
1
1
N
N
C
G
=
,
0
1
/
=
N
SG
and
)
(
0
1
1
1
C
C
NR
C
E
D
U
With
µ
-
=
.
- If geometrical reactualization
{
}
0
t.q.
connections
<
=
I
nc
I
N
C
D
,
I
N
I
N
C
G
=
,
0
/
=
I
N
SG
- If not is
{
}
0
t.q.
connections
<
=
I
nc
I
N
C
D
,
)
(
1
-
-
=
I
C
I
I
C
NR
I
C
E
D
U
With
µ
,
I
N
I
N
C
G
=
,
0
/
=
I
N
SG
·
For all the connections,
)
(
)
(
)
(
1
1
1
1
1
I
I
N
I
G
I
I
N
I
G
I
C
I
I
C
NR
I
G
E
U
U
With
U
U
With
D
U
With
µ
µ
+
+
-
=
-
-
-
-
-
·
Checking of the state of the sliding joints and “adherent”
- If for a connection
J
)
(
)
(
1
1
1
-
-
-
-
<
+
I
C
I
I
C
NR
I
I
N
I
G
T
E
E
D
U
With
U
U
With
µ
, then
)
(
1
I
I
N
I
sg
T
I
sg
E
U
U
With
µ
+
=
-
,
}
{J
SG
SG
I
N
I
N
+
=
and
}
{J
G
G
I
N
I
N
-
=
·
Calculation of the tangent matrices
I
N
F
K
and
I
N
F
K
(if RESI_GLOB_RELA
<1.E-3
,
1
=
)
·
Update
I
U
,
I
C
µ
,
I
G
µ
,
I
sg
µ
,
I
C
With
,
I
sg
With
and
I
G
With
.
5.1.4 Dualisation of the conditions of contact and friction in 3D
For the resolution of this problem, one defines the statute of the connections using regularization: initially
all the adherent connections are treated by regularization with a term of penalization
T
E
determined by the code. One increases the value then repeatedly of
T
E
by
T
T
E
E
10
=
and one
remove the connections not checking the condition
I
C
I
I
N
I
G
T
E
µ
U
U
With
µ
<
+
-
-
)
(
1
1
. When the process
is stabilized, the adherent connections and the sliding joints are treated by multipliers of
Lagrange and the penalization does not appear.
Method of Newton:
·
With the iteration
I
, at the total level, resolution of the system (one voluntarily does not make
to appear the conditions of Dirichlet):
1
1
1
1
1
)
(
)
(
-
-
-
-
-
-
-
-
-
=
+
I
G
T
I
G
I
sg
T
I
sg
I
C
T
I
C
I
N
int
ext.
I
I
N
F
I
N
µ
With
µ
With
µ
With
U
L
L
U
K
K
Note:
·
The matrix
I
N
F
K
contains the contributions of terms of slip is:
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
=
I
N
I
G
I
G
I
G
I
N
I
N
I
G
I
N
I
G
I
G
I
N
I
G
I
G
I
G
I
N
F
T
T
K
K
U
With
With
With
U
U
With
U
With
U
With
With
K
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
22/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Algorithm of rubbing contact:
·
Determination of the connections in contact
- Initial State:
L
NC
N
=
0
,
=
0
N
C
,
=
0
N
SG
,
0
0
/
=
N
G
. All points of surface
potential of contact are lack of contact.
- If
I
=1,
{
}
0
t.q.
connections
1
0
1
1
<
-
=
=
U
With
D
D
nc
nc
nc
N
C
,
1
1
N
N
C
SG
=
,
0
1
/
=
N
G
,
=
T
E
(Max {diagonal terms of
1
N
K
})
25
.
0
- If geometrical reactualization,
{
}
0
t.q.
connections
<
=
I
nc
I
N
C
D
,
I
N
I
N
I
N
C
G
G
=
- 1
and
I
N
I
N
I
N
G
C
SG
-
=
.
- If not
1
-
=
I
N
I
N
C
C
,
1
-
=
I
N
I
N
SG
SG
,
1
-
=
I
N
I
N
G
G
·
Resolution of the system
1
1
1
1
1
1
)
(
- +
- +
-
-
-
+
- +
=
-
-
=
+
I
sg
C
I
I
sg
C
I
G
T
I
G
I
N
int
ext.
I
sg
C
T
I
sg
C
I
I
N
D
U
With
µ
With
U
L
L
µ
With
U
K
One carries out for that a resolution per blocks like previously.
·
Checking of the state of the connections of contact
- If the connection
J
supposed not activates active, is violated, i.e that whose play is it
more negative, is added with the whole of the active connections,
}
{J
C
C
I
N
I
N
+
=
and
}
{J
G
G
I
N
I
N
+
=
, return to the resolution of the system of equations for the same iteration
I
,
but with the connections not slipping identified above.
- If for a connection
J
0
<
I
C
µ
,
}
{J
C
C
I
N
I
N
-
=
}
{J
SG
SG
I
N
I
N
-
=
}
{J
G
G
I
N
I
N
-
=
(in
function of the type of the connection)
·
Checking of the state of the adherent connections
- If for a connection
J
I
C
I
sg
µ
µ
µ
,
I
sg
I
sg
I
C
I
G
µ
µ
µ
µ
µ
=
,
}
{J
SG
SG
I
N
I
N
-
=
and
}
{J
G
G
I
N
I
N
+
=
·
Checking of the state of the sliding joints
- If for a connection
J
I
C
I
I
N
I
G
T
E
µ
U
U
With
µ
<
+
-
-
)
(
1
1
,
}
{J
G
G
I
N
I
N
-
=
and
}
{J
SG
SG
I
N
I
N
+
=
- If not
)
(
)
(
1
1
1
1
I
I
N
I
G
I
I
N
I
G
I
C
I
G
U
U
With
U
U
With
µ
µ
µ
+
+
=
-
-
-
-
- So at least a connection changed state, i.e one detected an adherent connection among
slipping supposed connections, then
T
T
E
E
10
=
and return to the resolution of the system
equations for same iteration I, but with the connections not slipping identified
above.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
23/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
·
Calculation of the tangent matrices
I
N
F
K
(if RESI_GLOB_RELA
<1.E-3
,
1
=
).
·
Update
I
U
,
I
C
µ
,
I
G
µ
,
I
sg
µ
,
I
sg
C
+
With
and
I
G
With
.
5.1.5 Convergence
The convergence of the algorithm with multipliers of Lagrange for the contact without friction in
a finished number of iterations was shown in [bib2]. For the problems with friction, of
results of convergence with unicity of the solution to the discretized problem are established in
[bib6] for low values of the coefficient of friction of Coulomb. The results are established in
using an algorithm of point fixes associated with a method of multipliers of Lagrange. For
each problem of solved contact, one studies the problem of associated friction. Once the aforementioned
solved, one solves a new problem of contact and so on. These methods are however
different from those presented here and one cannot thus have results of convergence
theoretical for these last.
The condition of fastening of the points which come in contact is particularly important for
to ensure the convergence of the method with multipliers of Lagrange. Indeed when a point
returns to the contact during reiterated its tangential displacement remains free. A condition of not
slip would be far too constraining. The algorithm would oscillate then between two states with or without
contact in examples of the type of that presented in [V6.04.105]. The point which is attached is
thus regarded as free from the point of view of the slip. One can then calculate the normal reaction
as well as the tangential reaction by using the assumption of slip and one estimated of the increment
of displacement of initial slip.
The use of the penalization alone makes it possible to avoid these oscillations while making it possible to slacken them
stresses on the preceding system. Coupled with a method of multipliers of Lagrange, one
find pathology announced above.
5.1.6 Notice
The method of usual search linear RECH_LINEAIRE of STAT_NON_LINE is not usable
in this case. Indeed, a correction takes place already on the field of displacement, which is
supposed to be optimal for the realization of the conditions of contact. One thus risks if it is corrected
field during iterations of Newton not to have compatibility between displacements more and
reactions, which makes the method particularly unstable and involves an absence of convergence.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
24/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
6
Compatibility with the boundary conditions of Dirichlet
In the case of the methods with multipliers of Lagrange, one can observe incompatibilities
with the fact of imposing boundary conditions of the Dirichlet type. Indeed, it is necessary that physically it
problem has a direction. One cannot deal with problem of contact in the direction of axis Z if all
the points have a null displacement according to Z. As we will see it, to deal with such a led problem
with a singularity of the matrices of the type
T
C
C
With
K
With
1
-
with the processing of the boundary conditions of
Dirichlet by double lagrange of Code_Aster.
6.1
Writing of the boundary conditions
While taking as a starting point the the reference material [R5.03.01] of STAT_NON_LINE, the dualisation of
boundary conditions of Dirichlet
)
(T
D
U
DRUNK
=
conduit with the system of equations following to solve:
1
1
)
(
-
-
-
=
-
=
+
I
N
D
I
I
N
int
ext.
T
I
DRUNK
U
U
B
U
L
L
B
U
C
One notes then
K
the matrix of rigidity of the system such as:
=
0
B
B
C
K
T
This matrix has a reverse of the form:
=
-
G
F
F
E
K
T
1
such as:
0
EB
=
T
.
One checks thus that for each boundary condition I one with the property
0
EB
=
T
I
.
6.2
Return to the problem of contact
The matrix
T
C
C
With
K
With
1
-
can be also written
T
C
C
EA
With
since vectors of connection
C
With
do not make
to intervene that degrees of freedom of displacement.
·
It from of results whereas if a vector from connection J of the matrix
C
With
is a linear combination
boundary conditions of the Dirichlet type it checks the following property:
0
EA
=
T
J
C
.
stamp
T
C
C
EA
With
is then singular because it has a column of zeros. In
practical, without particular processing, one finishes in the code on a message of stop of the type
STOP ON MATRIX OF CONTACT-FROTTEMENT SINGULIERE. The detection of these
singular columns was implemented in the code in order to eliminate from the relations from
contact-friction this type of relations and to avoid the stop previously described.
·
It from of results whereas if a vector from connection J of the matrix
C
With
a combination contains
linear of the boundary conditions of the Dirichlet type and is written
J
C
I
I
J
C
With
B
With
+
=
, it checks
following property:
T
J
C
T
J
C
With
E
EA
=
. One can then have a matrix
T
C
C
EA
With
singular
because it has two identical lines. This detection is not for the moment not available
in the code and one finishes in the code on a message of stop of the stop type on matrix of
contact-friction singular.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
25/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
6.3
Illustration on a simple example
The two preceding situations can meet for the same example of study. That is to say one
surface being able to slip into the xOy plan. It is supposed locked in direction X. If direction of
blocking corresponds to the one of the main directions of slip which the user can give in
the command file one is found in case 1 of [§7.2]. If the direction of blocking is tilted
compared to the main directions of slip then one finds oneself in case 2 of [§7.2]. One
of the two directions of slip is of too much to characterize the physical system.
6.4 Notice
This problem of compatibility between contact-friction and the boundary conditions does not appear
with the regularized methods insofar as one adds rigidity with total rigidity and that
one does not make elimination as in the calculation of the lagranges.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
26/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
7
Implementation in Code_Aster
The call to the routines for the contact with friction in Code_Aster takes place at the same place in
STAT_NON_LINE
that for the unilateral contact without friction. The phase of prediction to the pitch of
time N is based on the increment of load between the pitches of time n-1 and N.
7.1 Algorithms
These developments are accessible under control STAT_NON_LINE. They are activated by
key word CONTACT of the control AFFE_CHAR_MECA with which one defines the areas of contact
possible. The whole of the parameters of the model is provided in CONTACT under the key word
FRICTION:
FROTTEMENT= “WITHOUT”
“COULOMB”
The various algorithms are chosen by the key word METHOD according to logic:
“LAGRANGIAN” METHODE=
Dualisation contact friction 2D and 3D
“PENALIZATION”
if indicated E_T, dualisation contact, regularization friction
if
E_T
and
E_N
informed, regularization contact and friction
One introduces also the COULOMB key words for the value of the coefficient of friction of Coulomb, and
COEF_MATR_FROT the coefficient of taking into account of the negative component of the tangent matrix
of friction ranging between 0 and 1. The user can thus define the loading in the following way
:
CHA =AFFE_CHAR_MECA (MODELE= MO,
CONTACT= _F (GROUP_MA_1 = ISOL1, GROUP_MA_2 = ISOL2
If the method used is “LAGRANGIAN”, no other indication is necessary in 2D and it is necessary
to provide COEF_MATR_FROT
in 3D. In the case of the penalized methods it is necessary to give the value of
coefficient of penalization E_T, E_N and the COEF_MATR_FROT in all the cases.
The loading thus defined is then used in STAT_NON_LINE:
RESU = STAT_NON_LINE (MODEL = MO, CHAM_MATER = CHMAT,
EXCIT = _F (LOAD = CHA),
NEWTON=_F (REAC_ITER=1),
SOLVEUR = (METHOD = “LDLT”) etc…);
It is noticed that one recomputes the tangent matrix with all the iterations of Newton
(NEWTON=_F (REAC_ITER=1)).
Note:
With
GCPC
like method of resolution, one can make only contact without friction
for the moment. One does not recommend however to do it because the performances in
term of time calculation are not good with this method. One recommends the employment of
MULT_FRONT with a renumerotation of the “MONGREL” type.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
27/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
7.2
Geometrical reactualization
The geometrical evolution is one of non-linearities constitutive of the rubbing contact. Within
Code_Aster, its taking into account is done by a method of point fixes (for more precise details,
to defer to [bib9]). One points out the use of the key word:
REAC_GEOM= “WITHOUT” pitch of reactualization
“AUTOMATIC” reactualization managed by the code
“Control” N reactualizations at the beginning of pitch of time, then afterwards
each convergence until the n-1ème. N is defined by:
NB_REAC_GEOM=n
7.3 Post
processing
Different post processing is possible. For example, the calculation of the efforts of contact can be
carried out in control POST_RELEVE_T by calculating the resultant of the nodal forces on
group meshs representing one of surfaces of contact.
One draws the attention to the structure of data VALE_CONT which is produced for each calculation
implying rubbing contact. It is printed as follows in the form of table:
MATABLE=POST_RELEVE_T (ACTION=_F (INTITULE=' INFOS FROTTMNT',
GROUP_NO=' ESCLAVE',
RESULTAT=U,
INST=10.,
TOUT_CMP=' OUI',
NOM_CHAM=' VALE_CONT',
OPERATION=' EXTRACTION',),);
IMPR_TABLE (TABLE=MATABLE);
The information printed in each node slave is as follows:
·
CONT: indicator of rubbing contact
- 0: no the contact
- 1: slipping contact
- 2: adherent contact
·
PLAY: value of the play
·
RN: normal reaction of contact normalizes
·
RNX: component according to DX of the normal reaction of contact
·
RNY: component according to DY of the normal reaction of contact
·
RNZ: component according to DZ of the normal reaction of contact
·
GLIX: component according to T
1
tangential slip (local reference mark)
·
GLIY: component according to T
2
tangential slip (local reference mark)
·
GLI: normalizes tangential slip
·
RTAX: component according to DX of the tangential force of adherence
·
RTAY: component according to DY of the tangential force of adherence
·
RTAZ: component according to DZ of the tangential force of adherence
·
RTGX: component according to DX of the tangential force of slip
·
RTGY: component according to DY of the tangential force of slip
·
RTGZ: component according to DZ of the tangential force of slip
·
X-ray: component according to DX of the force of rubbing contact (RNX+RTAX+RTGX)
·
RY: component according to DY of the force of rubbing contact (RNY+RTAY+RTGY)
·
RZ: component according to DZ of the force of rubbing contact (RNZ+RTAZ+RTGZ)
·
R: force of rubbing contact normalizes
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
28/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Moreover, it is possible to trace this information for a post graphic processing:
IMPR_RESU (MODELE=MO,
RESU=_F (FORMAT=' CASTEM' or “GMSH” or “IDEAS”,
RESULTAT=U,
NOM_CHAM=' VALE_CONT',
NOM_CMP= (“CONT”, “RNX”, “RNY”, “RNZ”, etc.),);
One presents below a cylinder in a boring with interaction of rubbing contact. One traced
normal forces of contact, tangential forces of adherence and the indicator of contact.
7.4 Precautions
of use
These precautions of use are about the same ones as those stated in [R5.03.50]. One them
recall here:
·
to check that the normals on the surfaces of contact are outgoing (to be wary in particular if one has
used operators of symmetrization in the maillor gibi),
·
attention with the contact friction in quadratic 3D if the meshs of edge are QUAD8
(to avoid using HEXA20 to net volume): to use HEXA27 preferably, or
many PENTA15 whose TRIA6 sides are the meshs of contact,
·
to remove, by boundary conditions of Dirichlet adapted, the movements of body
rigid; it is not necessary that the structure holds only by the contact or friction. In others
terms, that wants to say that a calculation made in elasticity with control STAT_NON_LINE without
to treat the contact must pass,
·
in the event of structure “held” only by the contact, one can add a spring of weak
rigidity to maintain it,
·
not to use method of search linear, a STAT_NON_LINE incompatible with
processing of the conditions of contact-friction
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
29/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
8 Conclusion
Discrete modelings of contact-friction with slip surfaces 1D and 2D were
established in Code_Aster. These modelings usable with
STAT_NON_LINE
and
DYNA_NON_LINE
are accessible under
AFFE_CHAR_MECA
by the operator of contact with friction
in great displacements
CONTACT
.
Contrary to [bib5] modelings suggested are not pressed on dedicated finite elements.
They are based on the mesh of surfaces coming in contact and make it possible to retranscribe node with
node conditions of contact friction between surfaces after discretization of the formulation
variational corresponding. The method extends then without difficulty of small displacements to the case
great displacements. Indeed, the absence of use of finite elements, between surfaces being able
to come in contact, avoids the great distortion of the latter, in the case of great displacements. One
can then use is conditions of direct connections nodes to nodes for mesh
initially compatible, that is to say conditions of connections nodes to nodes balanced according to one
approach by projection of the master-slave type for incompatible mesh. The different ones
conditions of connection are developed in documentation [R5.03.50] available on the processing
contact without friction in great displacements.
In the case of slip surfaces 1D one could only develop a using algorithm
multipliers of Lagrange. The finished convergence of this type of algorithm is proven for the contact
unilateral without friction [bib2] and in the case with friction for low values of the coefficient of
friction of Coulomb [bib6]. In the case of slip surfaces 2D, the rubbing contact is treated
either by dualisation or by regularization with various mixings.
One always advises the use of the dualisation on the contact and friction for the 2D:
method does not utilize of new tangent matrices and it does not present difficulties
major in term of use except compatibility with the conditions of Dirichlet, cf [§6]. In 3D,
one always advises to use the dualisation on the contact and friction; nevertheless it can afterwards
blow being interesting to test the method with dualisation of the contact and penalization of friction:
the problems of compatibility with the conditions of Dirichlet are then less and times of
calculation can be reduced. One insists on the other hand on the very strong dependence of the result with
value of the terms of penalization. For systematic studies, one can nevertheless test
validity of the solution regularized compared to the solution with dualisation, used like reference
on a standard study.
Code_Aster
®
Version 7.1
Titrate:
Contact discrete friction in 2D and 3D
Date:
08/10/03
Author (S):
NR. TARDIEU
,
P. MASSIN
Key
:
R5.03.51-C
Page:
30/30
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
9 Bibliography
[1]
BEN DHIA H., MASSIN P., TARDIEU NR., ZARROUG Mr.: Various algorithms for
problems of contact friction 2D and 3D in great displacements, GIENS 2001.
[2]
DUMONT G.: Algorithm of active stresses and unilateral contact without friction, Review
European of the Finite elements, Flight 4, n°1, 1995
[3]
DUVAUT G., LIONS J.L. : Inequations in mechanics and physics, Dunod, Paris, 1972.
[4]
EKELAND I., TEMAM R.: Convex analysis and variational problems, Bordered, 1974.
[5]
G. JACQUART: A finite element of contact with friction, Reference material of
Code_Aster [R5.03.41].
[6]
LICHT C., PRATT E., RAOUS Mr.: Remarks one has numerical method for unilateral contact
including friction, International Series off Numerical Mathematics, vol. 101, 1991, pp. 129-144.
[7]
MASSIN P., BEN DHIA H.: 2D and 3D algorithms for frictional problems with small
displacements, Proceedings off the European Congress one Computational Methods in Applied
Sciences and Engineering, ECCOMAS 2000, Barcelona, September 11-14, 2000.
[8]
TARDIEU NR.: Quasi Static Nonlinear algorithm, Reference material of
Code_Aster [R5.03.01].
[9]
TARDIEU NR.: Unilateral contact by conditions kinematics, Documentation of
Reference of Code_Aster [R5.03.50].
[10]
ZHONG Z.: Finite Element Procedures for Problems Contact-Impact. Oxford University
Near, p.146-148, 1993.