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Titrate:
Elements of contacts derived from a hybrid formulation continues Date
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P. MASSIN, H. BEN DHIA, Key Mr. ZARROUG
:
R5.03.52-A Page
: 1/20
Organization (S): EDF-R & D/AMA, ECP
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.52
Elements of contacts derived from a formulation
continuous hybrid
Summary:
This document describes the way in which the elements of rubbing contact are derived from a hybrid formulation
continue problems of contact between solids (2D or 3D) in great transformations and specifies the strategy
of resolution used [bib1], [bib2].
The approach is implemented in Code_Aster. It is usable with modeling STAT_NON_LINE in
assigning to key word CONTACT, under AFFE_CHAR_MECA, key word METHODE=' CONTINUE'.
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Elements of contacts derived from a hybrid formulation continues Date
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:
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Count
matters
1 Introduction ............................................................................................................................................ 3
2 Formulation continues hybrid problem of contact .......................................................................... 3
2.1 Principle of Travaux Virtuels .......................................................................................................... 4
2.2 Pairing and kinematic condition of noninterpenetration ...................................................... 5
2.3 Principle of Action and Réaction .............................................................................................. 6
2.4 Laws of contact ................................................................................................................................. 7
2.5 Quasi-static hybrid formulation ................................................................................................. 9
3 Strategy of resolution ......................................................................................................................... 10
4 Elements of contact ............................................................................................................................ 12
4.3 Tangent matrices of contact ....................................................................................................... 15
4.4 Tangent matrices of friction .................................................................................................. 16
4.5 Expression of the second members of contact-friction ............................................................ 17
4.6 Calculation of the matrices and the second members .............................................................................. 18
5 Implementation .................................................................................................................................... 19
6 Bibliography ........................................................................................................................................ 20
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Elements of contacts derived from a hybrid formulation continues Date
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:
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1 Introduction
There exists, for the processing of the problem of contact-friction, an important “gap” between
discrete formulations and the formulation continues. The implementations in the computer codes are
(often) founded on discrete models such those developed by [bib3], [bib4], [bib5] or, more
recently, [bib6]. The latter are without clear links with the formulation continues. That poses
question of the precision and the nature of the results obtained: towards what does converge one?
The two essential objects of this document are on the one hand the description of the derivation of elements
of contact starting from a hybrid formulation continues of a problem of contact between solids
three-dimensional, deformable, undergoing great transformations, and in addition the detail of
strategy of resolution [bib1], [bib2], [bib7].
Section 2, devoted to the continuous hybrid formulation, comprises five paragraphs. In
paragraph [§2.1], one points out a Lagrangian formalism of Principe of Travaux Virtuels for two
deformable solids which can come into contact. Thanks to an application of pairing and to
Principle of Action and Réaction detailed in the paragraphs [§2.2] and [§2.3],
respectively, one gives a simplified expression of virtual work of the forces of contact.
paragraph [§2.4] is devoted to equivalent writings of the laws of contact and friction which
lend to weak formulations. The continuous hybrid formulation of the problem is specified in
paragraph [§2.5]. One specifies the strategy of resolution in section 3. The latter is founded
on algorithms of fixed point and tangent module. In section 4, one discretizes, by
finite element method, the continuous hybrid formulation suggested in the paragraph [§2.5] and one
evoke the difficulties due to the incompatibility of the discrete models of interfaces. Key words
concerning the implementation of this approach in Code_Aster are given in section 5.
2
Formulation continues hybrid problem of contact
Two solids I are considered
B (I =,
1 2) deformable, presumedly elastic (for the clearness of
document), in rubbing contact. These two solids occupy in their initial configuration adherence
of two fields 1
and 2
from 3
R and in their current configuration (at the moment T) the adherence of
1
and 2
, respectively. It is supposed that, in their initial configuration, these two solids are
T
T
in a natural state, such as without residual stresses or predeformations. During their
movement, they can come into contact, as indicated on [Figure 2-a]. The border of each
solid
I
B is broken up into parts I
, I
and I
in the initial configuration, of which them
0
G
C
intersections are empty 2 to 2, and out of I
, I
and I
, deformed the preceding ones, in the configuration
0
G
C
current. The description of this partitition is given to [§2.1]. Solid I
B is embedded on I
and
0
subjected to a nominal density of surface forces noted I
G on part I
. In addition, one notes
G
I
F the voluminal field of density of efforts applied to solid I
B (I =,
1 2). Parts of
surfaces
I
, likely to come into contact at the time of the deformation of the two solids, are noted
I
. One supposes the existence of noted regular cards I
describing surfaces I
. These cards are
C
C
defined as follows:
I
I
3
: R
(
éq 2-1
,
I
I
p =,
1
2)
(1 2)
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where
I
is a field limited (of reference) contents in 2
R. Par ailleurs, one indicates by I
transformation of solid I
B, defined by:
I
I
I
: T éq 2-2
I
I
p X
I is noted
N the unit normal with I
external with I
and I is noted
N its opposite in the configuration
p
C
X
current (cf [Figure 2-a]). One indicates by I
U the field of displacements of solid I
B and by I
F it
tensor gradient of deformation, defined by:
Fi =
I
p, T
éq
2-3
p
(I) = ui +Id
p
Appear 2-a: Description of the mechanical problem
2.1
Principle of Travaux Virtuels
By using the notations introduced previously, the local equations of balance, the conditions
initial and the boundary conditions of the problem considered are written in the following form:
2 I
I
I
I
U
I
Div + F
=
in
p
T
2
I I
I
I
N
= G
on
p
G
I
I
U
= 0
on
éq
2.1-1
0
I I
I
I
N
= R
on
p
C
I
U (p 0
,)
I
I
I
I
I
U (p 0
,) = U (p) and
U (p 0
,) = v (p)
in
p. p
0
T
0
where
I
indicate the first tensor of the constraints of Piola Kirchoff, I
is the density on
p
initial configuration, I
U is the field of displacements and I
R is the density of the efforts, due to
possible interactions of contact rubbing between the two solids, unknown factor of the problem. Moreover,
except mention clarifies contrary, exhibitor I takes here, and subsequently, value 1 or 2. Us
let us suppose, inter alia, to concentrate us on the problem of contact, that the density of the forces
voluminal as that of the surface efforts applied to the two solids are null, i.e.,
F I = 0 and gi =.
0
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One indicates by
I
(CA) the space of the fields of displacements kinematically acceptable for
solid I
B. Principe of Travaux Virtuels is written:
For all
I
I
W (CA),
2 I
U
I
-
I
W
. D =
I I
Tr F S
T
I
W
D
I
p
2
I ((p ())
T
éq
2.1-2
- I
I
R p
(, T W
). di
I
C
C
In [éq 2.1-2], the point in fat (.) represent the Euclidean scalar product in
3
R and
T
(*) is
transposed of (*). The trace of a tensor of command 2 is noted Tr and I
S indicates the second tensor of
constraints of Piola-Kirchhoff, related to the first tensor by the following relation:
I
S = (I
F) 1 - I
éq
2.1-3
Without restricting the general information concerning the mechanics of contact (object of this document), one
that the materials constituting the two solids are hyperelastic, i.e. will suppose,
I
I
I
W (I
F)
=
éq
2.1-4
p
I
F
where
I
W is the mass density of local internal energy definite on solid I
B.
2.2
Pairing and kinematic condition of noninterpenetration
To translate nonthe interpenetration, one proceeds as follows:
1) one couples the points of two surfaces of contact: it is pairing,
2) one imposes between the two points of a couple of points paired nonthe penetration according to one
direction given.
The first stage can be modelled while seeking, for any item 1
X T
()
1
= (1
p, T) of the border
1
, the point of 2
who is closest to him. This amounts solving the problems of optimization,
C
C
under constraints, following:
For all 1
1
p (thus 1
1
p = ()
1
,) and any T 0,
C
To find (, T) = (, such as:
1
) 2
2
(, T =
1
)
ArgMin (1
(1 (), T) - 2
(2 ()
2
, T))
2
,
éq 2.2-1
2
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The solution of this problem formally makes it possible to define, at any moment T, an application
of pairing,
A: 1
2
T
éq
2.2-2
(, T)
1
This application defines, also, item 2
p (1
p, T)
2
= ((1
p, T), noted p, paired to 1
p, at the moment
1
T. It defines also item X, paired as in point 1
X.
1
The condition of noninterpenetration between 1
X of 1
and X of 2
is written in the direction of
C
C
1
N = N
-
, the unit normal with 2
, in X, interior to 2
, in the form:
X (1
2 X)
C
T
D =
éq
2.2-3
N
(1
1
X - X) .n 0
A second approach (cf [bib8] for comments) consists in introducing a direction of
acceptable search [bib9]. One of these directions (cf [bib8], [bib2]) is the field speeds
standardized V.
Thereafter one will note X the quantity which represents the scalar product X N
. or the projection of
N
X on N.
One however continues to impose nonthe penetration like previously. The influence, in practice,
choice of the strategy of pairing on the resolution of the problems of contact is shown in
[bib10], [bib8].
Notice 1:
It is pointed out that, under the assumption of small displacements, pairing is made only one
only time. In great transformations, pairing depends on the deformation and introduces
not geometrical linearity of contact.
Notice 2:
Although pairing seems to introduce a dissymmetry of processing between the two
surfaces constituting the interface of contact, the approach remains “democratic”, uninterrupted.
dissymmetry is in fact due to the discretization (by the finite element method) and inspired with
certain authors the main concept/slave [bib11] or of geometrical surfaces/kinematics
[bib1].
2.3
Principle of Action and Réaction
By using the procedure of pairing, described in the paragraph [§2.2], Principe of Action and of
Reaction (PAR) is written in the following local form:
1
R (1
p, T) 1
2
D
+ R p T D
éq
2.3-1
C
(1,) 2 = 0
C
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By taking account of the principle of the action and the reaction [éq 2.3-1] and the assumption made on
pairing, one writes the low balance of the efforts as follows:
(1 2
W, W) (CA) 1 × (CA) 2
2
2 I
2
U
I
-
I
W
.
D
=
I I
R F S
I
W
D
I
p
2
I ((p ())
i=1
(T)
i=1
éq
2.3-2
- R (p, T). [W]
D
C
C
In [éq 2.3-2], one used the following notation:
[] * (1p) = () 1
* (1
p) ()
-
1
2
*
p
éq
2.3-3
Moreover, one posed
1
= and
1
R = R. One announces, finally, that the density of efforts 2
R is prolonged by
C
C
zero at the points of 2
without opposite on 1
.
C
C
2.4
Laws of contact
One breaks up the density of effort of contact R into a normal part which indicates the pressure
normal of contact and another tangential
r. Ainsi, the effort of contact R is written:
R = N + R
éq
2.4-1
where N is the unit normal, defined in the paragraph [§2.2].
The laws of Signorini are written in the following form:
,
0
D,
0
D
= 0
éq
2.4-2
N
N
where D is the directed distance, defined by [éq 2.2-3].
N
By introducing the function characteristic of
-
R, noted:
(
if
X), 1 X 0
=
éq
2.4-3
,
0 if X > 0
and the multiplier (known as of increased contact [bib3]), noted G, defined by:
N
G = - D
éq 2.4-4
N
N
N
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where is a strictly positive reality (noted
N
COEF_REGU_CONT in the fochiers of command), the laws
of contact [éq 2.4-2] are written as follows:
- (G G
éq
2.4-5
N)
= 0
N
For the phenomena of friction, one uses the isotropic laws of Coulomb which are written like
follows:
R
µ
(p, T)
(p, T)
If R
< µ
=
(p, T)
(p, T)
then
v (p, T) 0
éq 2.4-6
If R
= µ
=
-
(p, T)
(p, T)
then
0; v (p, T)
R (p, T)
where µ is the coefficient of friction of Coulomb and v are tangent relative speed. One defines
this speed v, in a given point of the surface of contact, by:
v (
1
1
1
1
p,
2 p,
p, T) = (Id - N N)
(T)
(T)
-
= (Id - N N) v (1
p, T) éq
2.4-7
T
T
One notes thereafter
X the projection of X on the tangent level on the surface of contact, defined by
X = (Id - N N) X
, where the symbol indicates the tensorial product.
An equivalent formulation of the laws [éq 2.4-6] is as follows [bib12]:
R =
µ
éq
2.4-8
-
G =
éq
2.4-9
0, ()
()
0
1
G = +
v
éq 2.4-10
In these quantities, definite on, is a strictly positive parameter, is one
C
semi-multiplier (vectorial) of friction, G is the semi-multiplier (vectorial) of friction
increased and
is projection on the ball unit.
(0 1
,)
The laws of friction are supplemented by the equation (of exclusion type) following:
D = 0 or
G
éq
2.4-11
N
(1 - () N) = 0
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2.5
Quasi-static hybrid formulation
In the continuation of the document, one adopts the assumption of quasi-staticity of the movement. They are neglected
terms of inertia. With a stage of loading given (K), corresponding to the fictitious moment T, one
K
suppose known the fields at the moment T
noted I
U, and
, and the new ones are sought
K 1
-
K 1
-
K 1
-
K 1
-
fields at the moment T.
K
While using [éq 2.3-2], [éq 2.4-5], [éq 2.4-9] and [éq 2.4-11], one derives the weak, hybrid formulation (with
three fields: U = (1
U, 2
U),
and
) and quasi-statistics of the problem of contact, following:
To find (u1, u2
1
2
,
;
K
K
K
K) (CA) × (CA) × H × H
(
w1, w2
,) (CA) 1 × (CA) 2 × H × H
2 Tr F S
W
I
(I I ((I D
G
G
W
D
p
)) - (nk) nk []
C
nk
C
i=1
éq
2.5-1
-
µ (G
G. W
D
0
nk) K
0,1 (K) [
]
()
=
C
K
C
- 1 {- G G * D 0 éq
2.5-2
K
(nk) nk}
=
C
C
N
- µ (G
nk)
K {-
G
. D
K
0,1 (
K
)}
()
C
C
éq
2.5-3
+
(1 - (G D
,
0
nk)
=
C
K
C
G = - D
éq 2.5-4
nk
K
N
nk
G = + Id - N N X
éq
2.5-5
K
K
K
(
K
) [K
K
]
where =
is a plus coefficient (is noted
K
COEF_REGU_FROT in the files of
T
K
order Code_Aster) and (K
*) is the increment of (*), at the moment T. It is noted that a diagram of Euler has
K
summer used to discretize in time the field speeds.
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3
Strategy of resolution
The problem describes by the formulation [éq 2.5-1] with [éq 2.5-5] is strongly nonlinear. Indeed, in addition to
not “traditional” linearity due to the framework of the great deformations (and that possibly due to
non-linear behaviors of materials), several levels of non-linearities inherent in
phenomena of rubbing contact, can be distinguished:
·
Non-linearity due to the ignorance of the effective surface of contact.
It is solved by an iterative strategy which can be brought closer to the method of
active constraints implemented in Code_Aster (cf [bib13] (tallies linear), [bib14]
(tallies non-linear)).
·
The geometrical non-linearity of contact defined notices of them 1 of the paragraph [§2.2]).
It is solved by a fixed algorithm of point on the geometry.
·
Non-linearity related to the threshold of friction (dependant on the field, unknown).
This non-linearity is also solved by the method of the fixed point.
Other non-linearities are treated by an algorithm of the type modulates tangent (or Newton
generalized). The general outline of the algorithm is thus the following:
I. Boucle on the stages of time:
fields U, and known at the preceding stage
II. Loop on the geometry:
N = Nd, = D
and
T = D (new pairing)
III. Loop on the thresholds for friction:
= S (only for the terms of friction)
IV. Loop on the surface of the contacts with a method of the type forced active:
= D
V. Boucle of tangent module:
generalized linearization
VI. End of the loop of tangent module
VII. End of the loop of the method of the active constraints
VIII. End of the loop on the thresholds
IX. End of the loop on nongeometrical linearity
X. Fin of the loop on the stages of time.
Notice 3:
By defect in Code_Aster the initial statute of the nodes slaves is not contacting. In addition
is initialized to zero. That amounts starting by solving the problem without contact with
S
first iteration, then then with contact but without friction, then then to activate it
friction with fixed contact pressures, i.e. to solve a problem of Tresca.
S
Notice 4:
The test of stop of the geometrical loops and threshold is based on the relative variations of
fields of relative displacements. Three key words: ITER_FROT_MAX, ITER_CONT_MAX and
ITER_GEOM_MAX make it possible the user to control the course of calculation.
Notice 5:
The loop of geometry is necessary especially for the problems with surfaces of contact
lefts. In practice only one correction often sufficient and is advised
(ITER_GEOM_MAX=1) if there are few geometrical modifications and kinematics of surfaces of
contact.
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The problem solved by the algorithm of the tangent module is a nonlinear problem of elasticity (it
could be of elastoplasticity in great deformations or others) with a term of friction of
Tresca and a reactualized pairing. It is written:
D
To find (1
U, 2
U,) (CA) 1 × (CA) 2 × H × H;
(
w1, w2 *
,) (CA) 1 × (CA) 2 × H × H
G1 (1
U, 1
W) -
G w1 D
D
N
()
C
N
D
D
éq
3-1
-
µ
G. 1
W D
0
D
S
0 1 ()
()
()
=
C
,
D
D
G2 (2
U, 2
W) +
G W 2D
D
N
N (
()
C
D
D
D
éq
3-2
+
µ
G. 2
W
D
0
D
S
0,1 ()
(
()
()
=
C
D
D
D
- 1 (- D G) * D = 0 éq 3-3
C
Nd
N
- µ
D
S (-
G
. D
0 1 ()
()
+
C
,
D
éq
3-4
(1 -)
D = 0
C
D
where
1
G (..) and 2
G (..) virtual work of the internal efforts to solids 1 indicates
2
B and B
respectively and where G
G
and
are such as:
D
N
D
G = - X
éq 3-5
N
N []
D
Nd
G = +
[U]
éq 3-6
D
D
The reference to time “T” is omitted to reduce the writing. Moreover, in the integrals on
K
surface contact one replaced D by
D to simplify the notations.
C
Notice 6:
One solves the problem with iteration I by using the thresholds of frictions µ obtained using
S
contact pressures of the iteration i-1. With convergence, the criterion of stop on the threshold is defined
S
compared to the relative variations enters and.
S
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4
Elements of contact
One describes in this section the space discretization of the problem, defined by the equations [éq 3-1] in
[éq 3-6], within the framework of the finite element method.
Fields I
are supposed to be approached by fields I
H polygonal and thus them
surfaces of contact by
. The border is made up of NR elements:
C
CH
CH
C
Nc
CH
= Uej
J 1
=
Following approximate spaces are introduced:
·
(CA) I: approximate space of () I
CA of dimension Mr. One indicates by I J
NR, for J =, 1 m
H
I
I
basic functions. Thus, the approximate field of displacements, noted I
U, is written it like
H
follows:
I
m
I
U =
NR,
U
éq 4-1
H
I I J
J
j=1
where I
U are the components of the vector of displacements on the basis. Each I
U has two
J
J
or three components according to whether the problem is 2D or 3D in the Cartesian base of
reference (E, E, E). E thereafter will be noted
N the vector which is written:
1
2
3
D
= E N
. , E N
. , E N
.
who is not anything else that vector N expressed in the base
D
(1 D 2D 3) T
D
D
(E, E, E).
1
2
3
·
H: approximate space of H of dimension NR. J is noted
basic functions. Thus,
H
C
density of normal effort of contact approached, noted breaks up in the form
H
following:
Nc
=
H
J
J éq 4-2
j=1
J being components of
on the basis.
H
·
H: approximate space of H, dimension NR. J is noted
basic functions. Thus it
H
m
semi-multiplier of approximate friction is written:
H
Nm
=
éq 4-3
H
J
J
j=1
where are to them the components of on the basis.
J
H
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Discretization of the problem, defined by the equations [éq 3-1], [éq 3-6], the system gives not
linear according to:
To find 1
U, 2
U,
and
such as,
H
H
H
H
for all 1
W, 2
W,
and
with 1 J m 1
, K m 1
, L NR
1
and
m
:
J
K
L
m
1
2
C
Nm
G1 (1
1
,
1 J
U, W NR
G
1
W .n NR, 1 D
H
J
) -
H
D
N (
J
) J
D
()
D
CH
éq
4-4
-
µ
G. 1
W
NR, 1 D
0
D
S
0,1 (hd
) (
)
J
J
()
()
=
D
CH
G2 (2
2
2, K
U, W NR
G
2
W .n N2, A D
H
K
) +
H
D
N (
K
) K
D
(()
D
C
D
H
éq
4-5
+
µ
G.
2
W
N2, A D 0
D
S
0,1 (hd
) (
)
K
K
(()
()
=
D
C
D
H
- 1 (-
H
G D 0
éq 4-6
H
D
N)
L
=
D
C
L
H N
- µ
D
S (-
G
. D
H
0 1 (H
,
)
m
()
+
D
C
m
H
éq
4-7
(1 -. D 0
D)
m
=
C
H
m
H
where:
H
G = - X N
.
éq 4-8
N
H
N [H]
D
D
H
G = +
U
éq 4-9
H
[]
D
H D
That is to say still to find 1
U, 2
U,
and
for all 1 i1 m 1
, i2 m 1
, p NR 1
and Q NR,
i1
i2
p
Q
1
2
C
m
such as for all 1
W, 2
W,
and
with 1 J m 1
, K m 1
, L NR
1
and
m
:
J
K
L
m
1
2
C
Nm
1
1
G
m
1
,
1 I
U NR 1 X, 1
W NR, 1 X
i1
()
J
J
()
I =11
Nc
1
2
p
m
m
1
,
1 i1
2
2, i2
-
-
X NR
-
X NR
With
1
,
1 J
.n
W .n NR
D
C
D p
N i1
()
i2
(D ()
D (
J
D)
()
H
p=1
I =11
i2=1
Nm
m1
m2
Q
1
,
1 i1
2
2, i2
-
µ
+
U NR
-
U NR
With
1
,
1 J
.
NR
D = 0
W
C
D
S
0,1
Q
i1 () i2
(D ()
(J)
()
()
D
H
q=1
I =11
i2=
1
D
éq 4-4bis
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2
2
G
m
2
2, I
U N2 X, 2
W N2, X
i2
()
K
K
()
i2=1
Nc
1
2
p
m
m
1
,
1 i1
2
2, i2
+
-
X NR () -
X NR
With
2
2, K
.n
W .n NR
With D
C
D
p
N i1
i2
(D ()
D (
K
D)
(D ()
H
p=1
I =11
i2=1
Nm
m1
m2
Q
1
,
1 i1
2
2, i2
+
µ
+
U NR () -
U NR
With
2
2, K
. W
NR
D = 0
With
C
D
S
0,1
Q
i1
i2
(D ()
(K)
(
D ()
()
D
H
q=1
I =11
i2=
1
D
éq 4-5bis
- 1
Nc
1
2
p
m
m
1
,
1 i1
1
,
1 i2
1
(-)
+
X NR () -
X NR
With ()
L
.n D = 0 éq
4-6bis
C
D p
D
N
i1
i2
(D)
D L
H N
p=1
I =11
i2=1
Nm
Nm
m1
m2
- µ
D
S
Q
Q
1
,
1 i1
2
,
1 i2
-
+
U NR
-
U NR
With
m
. D
C
Q
0,1
Q
i1 () i2
(D ()
()
m
H
q=1
q=1
I =11
i2=
1
D
Nm
+
(1 -
. D
0
D)
Q
m
=
C
Q
m
H
q=1
éq 4-7bis
The not-differentiable quantity is projection on the ball unit, defined by:
X if X (0,)
1
éq 4-10
0,1 (X)
()
= X
if not
X
The “generalized” differential of this application is as follows:
X
if X (0,)
1
1
.
éq
4-11
X
0,1 (X)
()
X =
X X
I -
X D
2 X if not
X
One writes it in the following generic form:
X (0,) (X) X
= [K
1
(X)] X
éq 4-12
with
Id
if X (0,)
1
[K (X)]
= 1
.
X X
éq
4-13
Id -
2 if not
X
X
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:
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Notice 7:
One can generalize [éq 4-13] as follows [bib15] [bib16]:
Id
if X (0,)
1
[K (X)]
= 1
.
X X
éq
4-14
Id -
2 if not
X
X
where 0 1.
The use of the algorithm of the tangent module for the resolution of the definite nonlinear problem
by the equations [éq 4-4] to [éq 4-9], place gives, with each iteration N + 1, the resolution of one
linear system of the form:
N
+ N
+ N
NT
NT
n+
U 1 N
L
U
U
H
1
N
N
C
0
n+1
L éq
4-15
H
= N2
N
N
n+1
N
0
F
L
H
3
where
N
indicate the matrix of tangent rigidity traditional of the two solids to state N. Quantities
N 1
+
N 1
+
N 1
U,
+
and
correspond to the increments of the vectors components of the fields
H
H
H
approached displacements, field of density of approached normal effort and field
semi-multiplier of friction approached with the iteration N + 1. In addition, [N
], [N
and
C
U]
[N]
[N], [naked] [N
and
F]
the assembled matrices of contact indicate and
matrices assembled of
friction. Second members of the system [éq 4-15], noted N
N
N
L, L
and
L are calculated of
1
2
3
traditional manner by holding account of the contribution of the terms of contact and friction.
Their expression is given in this paragraph after that of the tangent matrices.
4.3
Tangent matrices of contact
The elementary contributions to the tangent matrices of unilateral contact are given below,
where, for clearness, the basic functions of spaces (CA) 1
CA were renumbered of 1 with
H
(
and
) 2h
~
m + m and noted I
NR with for convention:
1
2
~
For 1 I m
I
NR ()
,
1 I
= NR ()
1
~
For m I m + m
I
NR ()
2, I 1
m
= - NR
(eA ()
D
)
1
1
2
· For []
if the problem is 3D (the version 2D deduces some immediately while not holding
count component on E) one has then for I =,
1 1
, J m
+ m
3
C
1
2
[E
~
] *
= -
I
E
NR N D éq
4.3-1
, U
D
() J ()
I
J
E
D
where the exhibitor E, added with the application of pairing, returns to the fact that into discrete, one
D
work with several cards and that pairing must hold account of it.
Handbook of Référence
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:
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· For [U
] (of penalization type) with 1 < I, J m + m, one a:
1
2
[E
~
~
NR NR E E
N N D
éq
4.3-2
U]
=
I
W
, U
N
D
() J ()
I
J
T
E
D
D
· For [C] with 1 I, J NR:
C
[
1
(
)
E
C]
-
D
I
J
=
éq
4.3-3
,
() ()
I
J
-
D
E
N
4.4
Tangent matrices of friction
Before detailing the form of the matrices of friction, one indicates per T and T the two vectors
1
2
covariants of the tangent plan (in a point) and by ()
* the tangential part of the vector (*).
One indicates by [P] the operator of projection on the level defined by normal N:
[] = [Id - N]
which one associates the matrix P in the Cartesian base of reference defined by:
1 - 2
N - N N - N N
1
1 2
1 3
P = [
N N 1 N
N N
éq
4.4-1
1 2
3]
2
= -
-
-
1 2
2
2 3
- N N - N N 1 - 2
N
1 3
2 3
3
where (1
N, N2, n3) are the components of normal vector N.
[N
K]
While noting
following quantity (cf [éq 4-13]):
[N
K] = [K (hn
Gd)]
with
hn
N
Gd = H +
[N
U
H] D
While noting:
I
= I
(T, T
1
2) = (
I
I
1
2)
for 1 I NR
m
éq
4.4-2
I
= I
T
for 1 I NR
m
One has as follows:
NR
NR
NR
NR
m
m
m
m
Q
=
T
1
T
Q
Q
+
Q
=
Q
1
1
2q
2
Q Q =
Q
Q
q=1
q=1
q=1
2q q=1
who allows to have the values of the semi-multipliers in the local Cartesian base associated
surface contact.
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:
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By using the notations established previously, one deduces the following elementary contributions
with the tangent matrices of friction:
· For [N
B] we have for 1 < I NR and for 1 < J m + m:
m
1
2
[1
~
B]
iT
J
N
= -
() µ
()
()
éq
4.4-3
,
K P
NR
D
U
I
J
D
S
[]
E
· For the elementary contribution in the tangent matrix of friction [N
B us
U]
let us have for 1 < I, J m + m:
1
2
[
~
~
Bne
= -
()
()
(P
)
K P
éq 4.4-4
U]
µ NR I NR J T N D
,
W U
D
S
[]
I
J
E
·
[nF]
For
and this whatever 1 < I, J NR we have:
m
D () µs iT
= -
([
N
) Id - K]
[
J
()
F
éq
4.4-5
U]
D
E
I, J + (1 -
() () M
D () I J
D
E
M whose components are [M] =
T T. is metric current base covariante.
4.5
Expression of the second members of contact-friction
One clarifies here the second members of the system [éq 4-15], noted N
N
N
L, L
and
L by not holding account
1
2
3
that contribution of the terms of contact and friction. One has as follows:
· For N
L:
1
Nc
m +
1 m2
~
N
L =
N
p
-
N
J
E ~ I
X .n NR N NR D
wi
C
D p
()
N (J
D)
() D ()
H
p=1
j=1
éq 4.5-1
Nm
m +
1 m2
~
+
T
N
µ
~
P
+ P
N J
I
U NR NR D
0 1
C
D
S
,
Q ()
Q
J () ()
()
H
q=1
j=1
· For N
L:
2
Nc
m +
1
1 m2
~
N
N
p
N
J
I
L
1
(
)
()
X .n NR
()
() D
=
éq 4.5-2
D
p
D
N
J
D
I
-
+
CH
N
p=1
j=
1
Handbook of Référence
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Elements of contacts derived from a hybrid formulation continues Date
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:
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· For N
L:
3
Nm
Nm
m +
µ
1 m2
~
N
D
S
iT
Q
N
Q
N
N
J
L
P
U NR
D
=
-
+
Q
0,1
Q
J
I
CH
()
()
q=1
q=1
j=
1
éq 4.5-3
Nm
-
(1 -
D
D)
iT
Q
N
C
Q
H
q=1
In these expressions, one will note that subscripted quantities N correspond to the values of the fields
discrete currents 1
U, 2
U,
and
with iteration N of the step of time to which calculation is carried out. Only
i1
i2
p
Q
N
U
has a a little particular significance, since it corresponds to the increment of displacement since
J
the beginning of the step of time considered, i.e. to the increment of displacement enters iteration 1 and
iteration N.
4.6
Calculation of the matrices and the second members
The effective calculation of the solutions of the systems [éq 4-4] with [éq 4-7] requires:
~
· the definition of the basic functions I
I
I
NR
,
and.
· the numerical formula of integration used for the terms of contact and friction.
The defining choice of basic functions of spaces of approximation and the choice of a formula
of numerical integration (joined to the total formulation of the problem of contact) an element defines of
contact. However, this choice is not obvious a priori. Indeed, formulation being of the mixed type, it
is appropriate to pay attention to considerations of compatibility between the elements of surface of
target or main contact and those of the contacting surface of contact or slave. In Code_Aster one has
fact the choice of taking the same functions of form for the contact that for the finite elements on
which surfaces of contact rest. The discretizations of the fields of densities of efforts are
operated in discrete spaces corresponding to the traces of the fields of displacements of the solid
known as slave on the surface of contact. This choice, within the model frameworks, is at least supported
mathematically [bib17] [bib18] [bib19].
With the configuration of the elements of the surfaces of contact suggested with [Figure 4.6-a] the choice of
surface slave and that of surface Master rests on the stated considerations of compatibility
previously. Indeed, if one makes like choice of surface slave, that which is with a grid more
finely then the product of the functions of form 1 2
. intervening in integrations of rigidities
and of the second members remains polynomial for this element. A contrario, if one chooses like surface
slave the surface with a grid the product of the functions of form 1 2
. its character loses
polynomial there exists compact of nonnull measurement on which the product 1 2
. zero are worth - what
makes illicit integrations of the contributions to the rigidity and the second member of the contact.
convergence of the algorithm is not then assured any more.
From a practical point of view, one advises with the user whose calculation would not converge to invert it
main role of surfaces and slave, to check that the surface slave which it chooses is discretized with
less as far as surface Master and to use the same type of element on both sides of surface
of contact.
Be reproduced 4.6-a: numerical Problème of integration on surfaces of contact
Handbook of Référence
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:
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In addition, even by using linear basic functions, the exact calculation of the terms of the matrices
and of the second members resulting from the contact is practically delicate. The quantities which we would have with
to integrate are, from pairing, the characteristic function and projection, nonregular
(cf [Figure 4.6-a]). One can however give of it a “approximation” by methods of integration
numerical. Various techniques of integration were implemented:
· integration at the tops,
· integration at the points of Gauss,
· integration of the simpson type.
One advises with the user nonsatisfied with these results passing from the one to the other of these methods
in the order given above, the integration of the simpson type being richest but also more
expensive.
5 Implementation
This formulation is usable with method “METHODE”: “CONTINUE” under the key word
“AFFE_CHAR_MECA”. The parameters of the method and are specified by the key words
N
“COEF_REGU_CONT” and “COEF_REGU_FROT”, respectively.
Behind key word “INTEGRATION” one can use either word “GAUSS” or word “NOEUD”.
axisymmetric modelings are taken into account by another key word “MODEL_AXIS”. This last
word “OUI” or word “NON is followed”.
Notice 8:
Choices “GAUSS” or “NOEUD” must be always specified even with a modeling
axisymmetric since key word “MODEL_AXIS” does nothing but specify modeling. The word
key “NOEUD” is recommended for left surfaces of contact (not-plane).
Let us note that in the implemented formulation, the multiplier of contact and the semi-multiplier
of friction are fields defined on surface slave. For that, a modeling
“SURF_DVP_2D” in the case 2D or “SURF_DVP_3D” in the case 3D, must be added in
version 6 of Code_Aster. It will be noted that in version 7, it is not necessary any more to make this addition and that
surfaces of contact do not need more to be specified on the level of the assignment of the model.
Handbook of Référence
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:
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6 Bibliography
[1]
BEN DHIA H., VAUTIER I.: “A formulation to treat the contact friction in 3D in
Code_Aster ", Report/ratio EDF HI-75/99/007/A, 1999
[2]
BEN DHIA H., VAUTIER I., ZARROUG Mr.: “Problems of contact rubbing into large
transformations: the continuous one with discrete ", Revue Européenne of Eléments Finis, vol. 9,
pp. 243-261, 2000
[3]
ALART P., CURNIER A.: “A mixed formulation for frictional contact problems preaches to newton
like solution methods ", Comp. Meth. Appl. Mech. Engng., vol. 9, pp. 353-375, 1991
[4]
HALLQUIST OJ: “Nike 2D: In vectorized implicit, f.e code…,” Report UCID-19677, flight.
University off California, 1986
[5]
LAURSEN T.A., SIMO J.C. : “A continuum-based finite element formulation for the implicit
solution off multibodies, broad deformation frictional contact problems, “Int. J. Numer. Meth.
Engrg, vol. 36, pp. 3451-3485, 1993
[6]
CHRISTENSEN P.W. KLARBRING A., PANG J.S., STROMBERG NR.: “And Formulation
comparison off algorithms for frictional contact problems, “Int. J. Numer. Meth. Engng. vol. 42,
pp. 145-173, 1998
[7]
BEN DHIA H., ZARROUG Mr.: “Hybrid frictional contact particles-in elements,” Revue
European of Eléments Finis, n° 9, pp. 417-430, 2002
[8]
BEN DHIA H., ZARROUG Mr.: “Mixed frictional local contact elements and “volumic” contact
interfaces ", ECCM'01 CD-ROM, Cracow, Poland, June 26-29, 2001
[9]
BEN DHIA H., DURVILLE D.: “Pun: Year implicit method based one enriched kinematical
thin punt model for sheet metal forming simulation ", J. off Materials Processing Technology,
vol. 50, pp. 70-80, 1995
[10]
BEN DHIA H., ZARROUG Mr.: “Hybrid Elements of rubbing contact”, 5th Colloque
national in calculation of the structures, 15-18 Mai, vol. I, pp. 253-260, 2001
[11]
HALLQUIST OJ GOUDREAU G.L., BENSON D.J.: “Sliding interfaces with contact-impact in
broad-scale langrangian computations ", Comput. Methods. Appl. Mech. Engrg., flight 51,
pp. 107-137, 1985
[12]
BEN DHIA H.: “Modelling and numerical approach off contact and dry friction in simulation off
sheet metal forming ", in WCCM2, pp. 779-782, 1990
[13]
DUMONT G.: “Algorithm of active constraints and unilateral contact without friction”, Revue
European of the finite elements, vol. 4, n° 1, pp. 55-73, 1995
[14]
VAUTIER I.
: “Unilateral Contact by conditions kinematics”, Rapport EDF,
HI-75/97/033/A, 1997
[15]
MASSIN P.: “Algorithms of optimization for the modeling of the contact with friction in
2D and 3D ", [R5.03.51], HI-75/01/001/A, 2001
[16]
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Handbook of Référence
R5.03: Nonlinear mechanics
HT-66/04/002/A
Outline document