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A fast and efficient contact algorithm for fretting problems applied to frettingmodes I, II and III
L. Gallego a, D. Nlias a,, S. Deyber b
a Universit de Lyon, CNRS, INSA-Lyon, LaMCoS, UMR5259, F-69621, Franceb SNECMA, Villaroche, F77556, France
a r t i c l e i n f o
Article history:
Received 5 April 2008Received in revised form 19 May 2009
Accepted 28 July 2009
Available online xxx
Keywords:
Contact mechanics
Fretting
Stick-slip
Conjugate gradient method
Fast Fourier transforms
Numerical method
a b s t r a c t
A computational contact algorithm is presented to solve both the normal and tangential contact prob-
lems that describe fretting contacts between two elastic half-spaces. The coupling between the normal
and tangential contact problems can or not be taken into account. Nevertheless the coupling should
be introduced when materials are dissimilar. Fast and efficient methods are used. The contact solver is
based on a conjugate gradient method and acceleration techniques based on the Fast Fourier transforms
(FFT) are employed. Very good agreements are found with analytical solutions of three fretting exam-
ples representing each fretting mode. However it is shown that these analytical solutions are based on
approximations that can be too strong when materials are dissimilar.
2009 Elsevier B.V. All rights reserved.
1. Introduction
Fretting is a phenomenon that occurs when two contacting sur-
faces are submitted to an oscillating displacement, small and often
tangential. Numerous industrial applications are concerned with
fretting which is amongst the most critical surface damage phe-
nomenon. For instance, fretting occurs in mechanical joints (spline
joints, axial joints, blade-disk dovetail joints, rivets. . .), cables andflex ducts. Fretting is therefore connected with multiple industries.
To study fretting and its damages which are fatigue and wear, one
should solve the state of stress in the contact surface and in the
bulk. This is the aim of the contact mechanics area. Many physical
parameters intervene in a fretting contact. For instance Ambrico
and Begley [1,2] have highlighted the role of macroscopic plas-
tic deformation in fretting fatigue life predictions. Because plastic
deformation produced by fretting occurs over subsurface distancescomparable to the grain size, Goh et al. [3] took into account
the discrete grains and their crystallographic orientation distribu-
tion. However this paper will focus on purely elastic analysis that
remains available in several contacts.
Contact mechanics have been initiated with a famous paper of
Heinrich Hertz [4]. The paper introduced what is now defined as
the solution of a hertzian contact. Hertz considered the contact of
Corresponding author. Tel.: +33 4 72 43 84 90; fax: +33 4 72 43 89 13.E-mail address: [email protected] (D. Nlias).
two elastic bodies submitted to a normal static load. A hypoth-
esis on the surfaces geometry was stated: they are analogous to
semi-ellipsoids and non-conforming. The contact initiates there-
forethrougha single point (spherical or ellipsoidal contact)or along
a line (cylindrical contact). The hertzian theory is based upon the
following hypotheses:
- The contact zone is elliptical;
- The contact is frictionless;
- The elastic half-space body description is used.
The last point is important because it allows using an important
part of theelasticity theorydeveloped in case of elastic half-spaces.
Elastic half-space hypothesis can be used if the next conditions are
fulfilled:
- Thecontact zone issmallin regardof thedimensionof both bodies
in contact. It permits to know that stresses areconcentrated close
to the contact area and are not influenced by distant limit condi-
tions. Dealing with non-conforming surfaces permits to validate
this condition.
- Curvature radii should be much greater than the contact dimen-
sions to validate the previous condition. It implies also that the
surface slope is small and close to a flat plane and it avoids high
pressure peakswhichare not compatible withthe linear elasticity
theory.
0043-1648/$ see front matter 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.wear.2009.07.019
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Many contact problems canbe solved with the help of the Hertz
theory in order to obtain the surface stress distribution and subse-
quent bulk stress and strain states. However when a contact does
not fulfill the Hertz assumptions, the analytical solution may differ
significantly from the real one. Many researchers have proposed
additional analytical solutions for a variety of other elementary
problems, such as the indentation of a half-space by a rigid wedge
or a flatpunch. A remarkable surveyof differentcontactmechanics
solutions is given by Johnson [5]. The elastic half-space theory is
still used for most of these solutions. Most of them are 2D prob-
lems. Some solutions exist also for 3D problems but usually with
a symmetry of revolution. These solutions have been found from
the singular integral equation solution [6,7] or the integral trans-
form (i.e. Fourier transforms) methods [8]. When surfaces cannot
be longer considered as smooth, one may use the solutions given
by Westergaard [9] in the case of sinusoidal surfaces or the statisti-
cal method proposed by Greenwood and Williamson [10] for ideal
rough surfaces with spherical asperity tips. The effect of an elastic
coating [11], inhomogeneous half-space [12] or the elastic-plastic
indentation of a half-space by a conical, a spherical or a pyramidal
indenter [5] have also beenthe subject of manyinvestigations.Ana-
lytical solutions for frictional contact have also been established.
Most of them assume a uniform Coulomb friction coefficient. They
are given for sliding contact by McEwen [13] for cylindrical con-tact, by Hamilton and Goodman [14] for spherical contact and by
Sackfield and Hills [15] for elliptical contact. Another important
problem independently solved by Cattaneo [16] and Mindlin [17]
is the spherical contact submitted to stick-slip behavior. This situ-
ation happens when the tangential load is lower than the product
between the normal load and the friction coefficient. Globally the
contact is sticking, but to respect the Coulomb friction lawover the
entire contact interface, a slipping annulus appears at the periph-
ery of the contact zone. The CattaneoMindlin problem has been
recently generalized for any 2D-geometry by Ciavarella [18,19].
Another example is 2D complete fretting contact solutions that
were obtained through an asymptotic analysis and with an equiv-
alent V-notch plain fatigue specimen by Mugadu and Hills [20].
Numerous analytical solutions exist, however it could be diffi-cult sometimes to find the analytical solution that matches a given
engineering problem. Numerical methods are more convenient.
The finite elements analysis is the most developed and most used
method in solid mechanics. However the analysis of real contacts
requires a fine mesh of the volume surrounding thecontactzone to
getaccurate results,leadingto importantcomputingcosts.An alter-
nativeis the useof whatis calledthe semi-analytical method (SAM)
which consists of a numerical summation of elementary solutions
in a contact solver scheme. Bentall and Johnson [21] and Paul and
Hashemi [22] werethe first to introducethese methods.Kalker [23]
gave a mathematical formulation of this method and proposed an
algorithm for its resolution. The interesting point of such a method
is that acceleration techniques can be used. The Multigrid method
[24,25] and the fast Fourier transforms method [2629] have tobe mentioned. Using such acceleration techniques computations
are now fast enough to perform numerical simulation for elastic
rough contact normally loaded. Recent developments in the liter-
ature include the introduction of a thermo-elastic behavior [30],
the effect of coatings [31,32], elastic-plastic behavior and thermo-
elastic-plastic behavior [3339].
This paper deals with a contact model to simulate fretting. Pre-
vious papers have been presented by the authors to compute the
fretting contact problem [4042]. Computations were fast enough
to proceed wear simulations. Indeed wear predictions are usually
obtained through iterative contact computations where the con-
tact geometry is updated to take into account its change due to
wear. Fretting wear predictions have been highlighted under gross
slip regime in a first paper [40] where only the normal contact
Fig. 1. Three different fretting modes depending on the forces and moments trans-
mitted through the contact. Slips are transversal in fretting mode I; slips are radial
in fretting mode II; slips are circumferential in fretting mode III.
problem was needed to solve. The additional tangential contact
problem was later added [41], and fretting wear predictions under
partial slip regime were possible. Simulations were done on clas-
sical contact geometries, but extension to an industrial application
was proved through fretting wear simulations on dovetail joints
between fan blades and turbine engine disk [42]. Whilst previous
papers focused on wearpredictions,the current paperpresents firstin detail thealgorithm developedto solvethe normal andtangential
contact problem by the semi-analytical method. Whereas normal
contact algorithms are frequent in the literature, tangential contact
has been less studied. The second point is the coupling between
normal and tangential effects in stick/slip contact problem. Indeed
most analytical results concerning tangential contacts are given in
the case of uncoupled normal and tangential problems. It is not
longer the case when the elastic properties of the contacting bod-
ies are different, as highlighted in the present study. Finally, since
each component of the force and moment transmitted through the
contact can be considered, the model is able to simulate the three
different fretting modes that have been defined in the literature
[43], as schematically shown in Fig. 1.
2. Contact model
2.1. The basics
The fretting contact model is built from the two classical
equation sets that define the normal and the tangential contact
problems. Let consider two bodies defined by their surface equa-
tions in the Cartesian coordinate frame (Oxyz):
z1 = f1(x,y)z2 = f2(x,y)
. (1)
Each body is assumed to behave as an elastic half-space, the
plane Oxy being the free surface. They are submitted to rigid body
displacements, cf. Fig. 2, resulting in a distribution of contact pres-sure and shear stress at the interface between the two bodies.
Summation of these stresses gives the global forces and moments
transmitted through the contact, cf. Fig. 3. The contact area C isnot known in advance. The normal contact problem, cf. Fig. 4, is
described on a potential contact area G such as C G at timet. Inside the contact area, the gap is nil and the contact pressure
p is positive whereas outside the contact area the pressure is nil
and the gap positive. The definition of gap g includes the initial
body separation h(x,y)=f1(x,y) f2(x,y), the surface elastic deflec-tions uz(x,y) = uz1(x,y) + uz2(x,y) andthe rigid bodydisplacementz = z1 + z2. Thenormal load balance equation is usually added,Pbeing the normal load imposed to the contact. To take into account
the flexion moments Mx and My in the contact, rigid body rotations
are added. These angles x and y are small and are included in the
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Fig. 2. Rigid body displacements of both bodies. Each body is submitted to three
translations and to three rotations.
Fig. 3. Stressesin the contact zone. Contact pressure(p) and shear (qx , qy) aretrans-
mitted between the two bodies. Integration gives the global forces (normal force Pand tangential forces Qx and Qy) and moments (flexion moments Mx and My, and
torsion moment Mz) transmitted through the contact.
Fig. 4. The classical normal contact problem. The initial geometry of body 1 and
body 2 is z1 =f1(x,y) and z2 =f2(x,y), respectively, distance between the surface and
the plane Oxy. A normal force P is transmitted through the contact. Normal rigid
displacements z and normal elastic deflections uz appear.
gap definition. The equation system is set at time t:
pt(x,y) > 0, (x,y) C (a)ht(x,y) + uz (x,y) tz xy + yx = 0, (i, j) C (b)pt(x,y) = 0, (i, j) / C (c)ht(x,y) + utz (x,y) tz xy + yx > 0, (i, j) / C (d)
P
pt(x,y) dS = Pt (e)
P
y pt(x,y) dS = Mtx (f)
P
x pt(x,y) dS = Mty (g)
(2)
When the normal contact is solved the tangential contact can be
solved, cf. Fig. 5. It consists to find the sticking zone ST and theslipping zone SL thatverify C ST SL . Inthe stickingzonethe
slip amplitudes =
sxsy
is nil. Theslip amplitudeis functionof the
surface tangential elastic deflection uT
= ux1uy1
+ux2uy2
andthe tangential rigid body displacement T =
x1y1
+
x2y2
.
The tangential contact tractions (shear) q =
qxqy
verify the
Coulomb friction law. The tangential load balance equation is usu-
ally added. Q=
QxQy
is the tangential load applied to thecontact.
Mz is the torsion moment in the contact, which corresponds to a
small tangential rotation z also considered when computing localslips. The term introduces the variation between time t 1 andt, for example tT = tT
t1T . Note that the tangential contact
problem cannot be written independently of the history. Finally
the equation set is:
qt(x,y) = pt(x,y) st(x,y)/|st(x,y)|, (x,y) SL (a)
utT(x,y) tT ytz
+xtz
= st(x,y) /= 0, (x,y) SL (b)
|qt(x,y)| < pt(x,y), (x,y) ST (c)
utT(x,y) tT(x,y) ytz
+xtz
= 0, (x,y) ST (d)
c
qt(x,y) dS = Qt (e)
c
(xqty (x,y) yqtx(x,y)) dS = Mtz (f)
SL
ST=
C (g)
(3)
Both problems could be solved in terms of unknown elastic
deflections or in terms of constraints, the later being chosen here.
To solve the problem, a discretization is required. Each point of
a regular grid area is associated to a centered rectangular surface
of dimensions a and b along the x and y directions, respectively.
It corresponds also to the distance between two adjacent points
of the grid. The pressure p and tractions q are assumed uniformover each rectangular surface element because the Newman con-
ditions apply on the half-space, i.e. the boundary conditions in
terms of displacements are nil at the infinite, the theory of poten-
tials is used. Results are given by Boussinesq [44] and Cerutti [45]
who used this theory to formulate stresses and strains of an elas-
tic half-space. Love [46] extended this approach to a constant
load on a rectangular patch at the surface of an infinite elas-
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Fig. 5. The simplified tangentialcontact problem alongthe x direction.A tangential
force Qx istransmittedthroughthecontact.Tangentialbody displacementsx , elastic
tangential deflections ux and slips sx appear for one point in the stick area (left) and
one in the slip area (right).
tic half-space, providing relations between surface stresses andelastic deflections at any point of the surface, and between sur-
face stresses and sub-surface stresses. The linear elasticity theory
authorizes thesuperimposition of eachelementarysolution. Finally
the elastic deflections are obtained through the following double
summation:
utxij
=k=1,N
k=1,Ny
Kqxxijkl
(qtxkl
qt1xkl
) +k=1,Nx
l=1,Ny
Kqyxijkl
(qtykl
qt1ykl
)
+k=1,N
l=1,Ny
Kpxijkl
(ptkl
pt1kl
)(a)
utyij
=k=1,N
l=1,Ny
Kqx
yijkl (qt
xkl qt1
xkl) +k=1,N
l=1,Ny
Kqy
yijkl (qt
ykl qt1
ykl)
+k=1,Nl=1,NyK
p
yijkl (pt
klpt1
kl)
(b)
utzij
=k=1,N
l=1,Ny
Kqxzijkl
qtxkl
+k=1,N
l=1,Ny
Kqyzijkl
qtykl
+
k=1,N
l=1,Ny
Kpzijkl
ptkl
(c)
(4)
where Kare the influence coefficients that are function of the elas-
tic constants E1, E2, 1, 2 and the distance between the loaded
point and the point where the deflection is computed. The grid size
is N= Nx Ny. The computation cost of each double summation inEq. (4) is O(N2) , which could become very large if a fine mesh is
used. To reduce computation times, acceleration techniques are
used. These double summations are equivalent to discrete con-
volution products. Liu et al. [29] developed the DC-FFT (Discrete
Convolution and Fast Fourier Transforms) to accelerate compu-
tations and reduce the computation cost to O(NlogN). FFT were
used by several authors, but Liu et al. has clarified the technique
avoiding computational errors that compelled to extend five oreight times the computational grid (i.e. the potential contact area)
towards the contact area. With Lius DC-FFT technique, the com-
putation grid should be extended only two times. The reader could
refer to [29] to obtain more information about the way to obtain
the influence coefficients and how to extent the problem to coated
elastic half-spaces [31]. These formulae highlight a second impor-
tant point which is the coupling between the normal and the
tangential problems that could occur, depending on the material
and geometrical asymmetry of the problem. For example when
an elastic body is pressed against a rigid one, the contact pres-
sure at a given point produces a tangential displacement at any
point of the elastic surface in addition to the normal one. The
degree of coupling is function of the elastic constants. When the
elastic constants are equals, i.e. E1 =E2 and 1=
2, the influencecoefficients Kqxz , Kqyz , Kpx and Kpy are equals to zero. Dundurs andLee [47] established an eponym constant that permits to char-
acterize that degree of coupling: =
12
(121)/G1(122/G2)
(11/G1+(12)/G2)
.
The coupling is nil in the case of bodies with identical elas-
tic properties but also for two incompressible materials (1 =2 = 0.5). The extreme values are 0.5 that are reached whenone body is rigid and the other elastic with a Poisson ratio
nil.
The contact equations that have been presented can be
expressed under a global form through a variational formula-
tion, as introduced by Duvaut and Lions [48] in the mechanical
field. They studied and proved the existence and uniqueness
of the contact problem solution. Kalker and Van Randen [49]
and Kalker [23] resumed this work rewriting the formulae withthe elastic half-space hypotheses. It consists to minimize the
complementary contact energy under constraints, i.e. the con-
tact pressure should remain positive and the tangential traction
is limited by the contact pressure times the Coulomb friction
coefficient.
min =
(p, q) =
C
h + 1
2uz
p dS +
C
WT +
1
2utT ut1T
q dS
, (a)
p 0, (b)q p. (c)
(5)
h*is the non-deformed body separation, i.e. it includes the normal
rigid body displacement added to the initial body separation h. W*
denotes the tangential body displacements (translations and rota-
tion). A matrix formulation of the complementary energy can be
written in the discrete space through scalar products:
min
pT
1
2A
pz p +
1
2A
qxz qx +
1
2A
qyz qy + h
+ qTx
Wx +
1
2A
qxx q +
1
2A
px p ut1x
+qTy
Wy +
1
2A
qyy q +
1
2A
pyp ut1y
,
(a)
pij 0, (b)qij = q2xij + q2yij pij. (c)(6)
In expression (6), p, qx and qy are vectors, of length N and
that contains all values of pij, qxij and qyij. Let set q
= qx
qy. A
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are the matrixes constructed from the influence coefficients, for
example upz = Apz p. The set of equations (6) is a constrained opti-
mization problem. The inequality constraints require employing
the Lagrange multipliers and solving the Karush, Khun and Tucker
optimality conditions. To solve theproblema differentiation is done
with respect to p and q. The Panagiotopoulos [50] method will
be used, i.e. the normal and tangential problems will be solved
separately. When the Dundurs constant is nil, the normal and the
tangential problems can be solved independently, otherwise the
normal and the tangential problems should be solved successively
several times until convergence.
Presumably in fretting problems, dynamic stick-slip effects
must occur, with a reduction in frictional forces once sliding has
been initiated. It is also well known that thefrictioncoefficient may
change within the contact conjunction depending on how wear
debris remain or not entrapped. In the numerical investigations
presented later in the paper a constant and uniform Coulomb fric-
tion coefficientis assumed for purpose of comparison with analyti-
cal solutions. However the model should hold for non-uniform and
non-constant friction coefficient. This point is left to future work.
2.2. The normal contact problem
To solve a general constrained minimization problem:min
x n((x)), ci(x) 0, i I, a Lagrangian formulation is written:
L(x, ) = (x) i I
ici(x), where i are the Lagrange multipliers.
The differentiation of the Lagrangian formulation and the Karush
Kuhn and Tucker optimality conditions leads to:
xL(x, ) = 0, (7)ci(x) > 0, i = 0, (8)
ci(x) = 0, i 0. (9)
When the constraint ci has a linear form, i.e. aTi
x = bi, the problemis known as a quadratic optimization problem.
The complementary energy is written removing the constantterms that do not depend on p and that will disappear after differ-
entiation:
min (p) =
pT
1
2Apz p +
1
2Aqxz qx +
1
2Aqyz qy + h
+
1
2qTxA
px +
1
2qTyA
py
p
, pij 0, (i, j) P
(10)
Because of thesymmetryproperties of theinfluencecoefficients,
i.e. Kpx = Kqxz and Kpy = Kqyz , it yields:
min
(p) = pT
1
2A
pz p +Aqxz qx +Aqyz qy + h
, pij 0, (i, j) P
(11)
By applying the Lagrangian method to the normal problem
where the constraint is linear:
p
pT 1
2A
pz p +Aqxz qx +Aqyz qy + h
(i,j) p
ijpij
=Apz p +Aqxz qx +Aqyz qy + h = 0,
(12)
pij > 0, ij = 0, (i, j) c (13)
pij = 0, ij 0, (i, j) / c. (14)
The Lagrange multipliers ij are here the gap gij. The normalcontact set of equations is therefore obtained. Several methods can
be used to minimize the quadratic form. Polonsky and Keer [51]
used a contact algorithm based on the conjugate gradient method
(CGM). The choice of the conjugate gradient method is based on its
efficiency in terms of storage: the methodis iterative; and in terms
of rapidity:the rate of convergence is superlinear.For more details,
the reader may refer to the Polonskys paper or alternatively to a
review paper on contact algorithms by Allwood [52].
The Polonskys contact algorithm deals with the normal contact
subjected to a normal load. An extension is given here by taking
into account (i)either a normal load or a normalrigiddisplacement
z, and (ii) the flexion moments transmitted through the contact(i.e. the application point of the normal force is not centered in the
contactarea). c is the contact area,which is not knownin advance.Polonsky minimized the complementary energy on the active set
c, i.e. on the current contact area that evolves duringthe iterativeprocess until convergence to the real contact area c:
min
(p) = pT
1
2A
pz p +Aqxz qx +Aqyz qy + h
, (i, j) c (15)
During the iterative process, the Lagrange multipliers are com-
puted and the complementary conditions are checked. It means
that a newgap is obtained after each iteration, andthe contact con-
ditions are checked to add or remove a point in the current contact
area c. A difficulty is thattheterm h* contains the rigid body dis-placement z which is unknown in the case of an imposed normalload. To linearize the problem, Polonsky avoided inserting the nor-
mal load balance equation into the set of equations to solve andavoided also adding an outer loop to solve the balance equation.
The trick was to force the normal load balance during the iterative
process while the rigid body displacement is approximated at each
iteration. When the rigid normal displacement is imposed the nor-
mal load balance is no longer enforced. To take into account the
flexion moment balance equations, an equivalent method is used.
The normal contact algorithm can be described as follows (Fig. 6):
0. p is initialized. It should verify the normal load and the flexionmoment balance equations if imposed. The entire potential con-
tact area is supposed in contact: c p and ij =0. A variable
Fig. 6. Flow chart of the normal contact algorithm.
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called is set to 0. uqxz and uqy
z are computed with the DC-FFT
technique.
1. upz is computed with the DC-FFT technique. If the normal load
is imposed and/or if the flexion moments are imposed, the rigid
body displacement z and/or the rigid body rotations x and yare approximated.
2. The gap is computed: gij = upzij + hij
, (i, j) P. For the points(i, j) C, gij are the residue of the linear system to solve. They
are also the Lagrange multipliers.3. The conjugate gradient descent direction d is obtained with theresidue and the previous descent direction:
dij gij + G/Golddij, (i, j) C,dij 0, (i, j) C,withG =
(i,j)
C
g2ij .(16)
Then the value ofG is store in Gold and is reinitialized to 1.4. The DC-FFT technique is applied to the descent direction:
rpz = Apz d. (17)
Then the descent step is obtained:
=
(i,j) Cg2ij
(i,j) C
dijrij.
5. The pressure values are updated:
pij pij + dij, (i, j) C. (18)
6. The complementary conditions are checked and enforced:
ifpij < 0, (i, j) C, thenpij 0, C C/(i, j), (19)
ifgij = ij < 0, (i, j) / C, thenpij gij, 0, C C (i, j).
(20)
7. If neededthe balance equations are enforced.The followingcon-vergence criterion is tested:
=
(i,j) Ppij poldij(i,j) Ppij
. (21)
The pressure values are memorized: pij pold ij.In step 1, the approximation of the rigid body displacements is
based on the gap definition in the contact zone: gij = upzij + uqx
zij+
uqyzij
+ hij z yij x +xij y = 0. By summation over the contactarea the three following equations arise:
(i,j) cup
zij+ uqx
zij+ uqy
zij+ hij z yij x xij y = 0, (22)
(i,j) c
(upzij
+ uqxzij
+ uqyzij
+ hij z yij x xij y)/xij = 0, (23)
(i,j) c
(upzij
+ uqxzij
+ uqyzij
+ hij z yij x xij y)/yij = 0. (24)
If the three equations are written on the current contact zone
C
, an estimation of the unknown rigid body displacements canbedone. It consists therefore to solve the matrix system:
C1
Cxi
Cyj
C
1/xi
C
1
Cyj/xi
C
1/yj C
xi/yj C
1
zyx
=
Cup
zij+ uqx
zij+ uqy
zij+ hij
C
(upzij
+ uqxzij
+ uqyzij
+ hij)/xi
C
(upzij
+uqx
zij
+uqy
zij
+hij)/yj
. (25)
Part or totality of the system is solved depending on what body
displacements are needed. In step 7 the balance equations are
enforced. The contact pressures are corrected by a linear func-
tion: pij pold ij(a + b xi + c yj). Constant a is sought if the normalload balance is enforced, b and c if the flexion moment balance is
enforced. Finally it consists to solve part or totality of the following
matrix system:
CpijS CpijxiS CpijyjS
CpijxiS
Cpijxi2S
CpijxiyjS
CpijyjS
CpijxiyjS
Cpijyj
2S
abc
= P
My = PxpMx = Pyp
.
(26)
The role of variable is to reinitialize the conjugate gradientdirection when nil. It occurs at step 5 when the complementary
conditions are checked. When the gap of a point that was not in
the current contact zone at the previous iteration becomes nega-
tive, then this point is added in the current contact area and the
conjugate gradient method is reinitialized for the next iteration.
2.3. The tangential contact problem
The tangential contact problem will be solved with an equiva-
lent method. Theform to minimize from a differentiationofq is thefollowing:
min
(q) = qTx
1
2A
qxx qx +Apx p +Aqyx qy +Wx
+qTy
1
2A
qyy qy +Apyp +Aqxy qx +Wy
,qij pij, (i, j) c
(27)
The constraint which is not linear in this case can also be written
as:q2x +q2y
2P p2 0.
Therefore applying the Lagrangian method leads to:
min
(q) = qTx
1
2Aqxx qx +Apx p +Aqyx qy +Wx
+qTy 1
2
Aqy
y qy+
Apyp
+A
qxy qx
+Wy+ij
q2x + q2y
2P
p
2
Aqx
x Aqy
x
Aqxy Aqy
y
qx
qy
+
Apx p +WxA
pyp +Wy
=
ij/pij . .. ij/pij
I 0
0 I
qxqy
,
(28)
q2
xij+ q2
xij< pij, ij = 0, (i, j) ST, (29)
q2xij
+ q2xij
= pij, ij 0, (i, j) SL. (30)This setof equationsis equivalent to Eq.(3), where the Lagrange
multipliers are now the slip amplitudes. The problem is non-linear
because W* contains unknowns which are the tangential bodydisplacements x, y, z and the Lagrange multipliers thatare unknowns. The methodology presented for the normal contact
problem will be applied here for the tangential contact problem.
The problem can be solved taking into account (i) the tangential
loads Qx and Qy or the tangential rigid displacements x and y, and(ii) the torsion moment transmitted in the contact Mz. The trick to
linearizethe equation systemis to force thetangentialload balance
andthe torsion moment balance using an estimation of the respec-
tive rigid body displacements. To linearize the system respectively
to the Lagrange multipliers that are unknown, an estimation of
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Fig. 7. Flow chart of the tangential contact algorithm.
them will be done at each iteration and used in the next one to get
a linear matrix system:A
qxx A
qyx
Aqx
y Aqy
y
+
ij/pij . .. ij/pij
I 00 I
qxqy
=
A
px p +Wx
Apyp +Wy
.
Finally the tangential contact algorithm can be describedas follows
(Fig. 7):
0. q is initialized. It should verify the tangential loadand the torsion
momentbalance equations if imposed. Theentire contact area isfirst assumed sticking: ST C, SL = ,and ij =0. A variablecalled is set to 0. upx and u
py are computed with the DC-FFT
technique.
1. uqx
x , uqy
x , uqx
y and uqy
y are computed with the DC-FFT technique.
If the tangential load is imposed and/or if the torsion moment is
imposed, the rigid body displacement x, y and/or the rigidbody rotation z are approximated.
2. The slip is computed:
sij = utTij T, (i, j) C (31)
3. The Lagrange multipliers are obtained from the norm of the slip
in the current slip zone:
ij= sij sgn(sij qij), (i, j) SL (32)
4. The complementary conditions on the Lagrange multipliers are
checked:
ifij < 0, (i, j) / SL, then 0, ST ST (i, j). (33)
The calculus of the error that quantifies the non-colinearity
between the shear and slip vectors is done in the current slip
zone:
sl = (i,j) Psij + ij(qij/pij)(i,j) Pij
. (34)
5. The conjugate gradient descent direction d is obtained with theresidue and the previous descent direction:
dij sij + G/Golddij, (i, j) ST,
dij
sij + ijqij
pij
+ G/Golddij, (i, j) SL,
with G =
(i,j) C
sij + ij
qijpij
sij + ijqij
pij
.
(35)
Then the value ofG is stored in Gold and is reinitialized to 1.6. The DC-FFT technique is applied to the descent direction:
r =
rxry
=
Aqx
x Aqy
x
Aqx
y Aqy
y
dxdy
,
then
rij rij + ijdij/
pij
. (36)
Then the descent step is obtained:
=
(i,j) C(sij + ij(qij/pij)) (sij + ij(qij/pij))
(i,j) Cdij rij
. (37)
7. The shear values are updated:
qij qij + dij, (i, j) C. (38)8. The complementary conditions on shears are checked and
enforced:
ifqij pij, (i, j) ST,
thenpij 0, SL SL (i, j), = 0, qij pij(qij/qij). (39)
9. If neededthe balance equationsare enforced. Thefollowing con-
vergence criterion is tested:
=
(i,j) C
(qxij qxoldij)2 + (qxij qxoldij)2
(i,j) C
q2
xij+ q2
yij
. (40)
The shear values are then memorized: qij qold ij.In step 1 the rigid body displacement estimation is based on the
slip definition in the contact zone:
sij +ij
pij
qxijqyij
=
upxij
upyij
+
uqxxij
uqxyij
+
uqyxij
uqyyij
xy
yij zxij z
+ ij
pij
qxijqyij
= 0.
By summation over the entire contact area, one obtains the three
following equations:
(i,j) cup
xij+ uqx
xij+ uqy
xij x yij z +
ijpij
qxij = 0, (41)
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(i,j) c
upyij
+ uqxyij
+ uqyyij
y +xij z +ij
pijqyij = 0, (42)
(i,j) c
xi
up
xij+ uqx
xij+ uqy
xij x yij z +
ijpij
qxij
yj
upyij
+ uqxyij
+ uqyyij
y +xij z +ij
pijqyij
= 0.
(43)
An estimation of the rigid body displacements that are unknowncan now be done. It consists to solve the following matrix system:
C1 0
Cyj
0
C1
Cxi
Cyj
C
xi
C(x2
i+y2
j)
xy
z
=
C
upxij
+ uqxxij
+ uqyxij
+ ijpij
qxijC
upyij
+ uqxyij
+ uqyyij
+ ijpij
qyij
Cyj
upxij
+ uqxxij
+ uqyxij
+ ijpij
qxij+xi
upyij
+ uqxyij
+ uqyyij
+ ijpij
qyij
. (44)
Part or totality of the system is solved depending on what body
displacements are needed. In step 9 the balance equations are
enforced. The contact shears are corrected in the current stick
zone ST with a linear function: qxij qxij + a c yj and qyij qyij + b + c xi. Constants a and b are sought if the tangential loadbalance is enforced, c if the torsion moment balance is enforced.
Finally it consists to solve part or totality of the matrix system:
ST1 0
STyj
0
ST
1
ST
xi
ST
yj
ST
xi
ST
(x2i
+y2j
)
ab
c
= Qx/S CqxijQy/S Cqyij
Mz/S +
C(yj qxij xi qyij)
. (45)
The role of variable is to reinitialize the conjugate gradient direc-tion when nil. It occurs at steps 5 and 8 when the complementary
conditions are checked. When a point progresses from stick to slip
conditionor conversely, the conjugate gradient method is reinitial-
ized for the next iteration.
3. Fretting mode I
This first fretting mode, which is also the most studied starting
from Cattaneo [16] and Mindlin [17], corresponds to transversal
slips. The problem is an example of fretting mode I. Two elastic
bodies are brought into contact with a normal load P. A force tan-gential to the contact surface is then applied. A tangential force
along the x direction is considered here. Accordingly to Coulomb
this tangential load is limited by the normal load times the friction
coefficient, i.e. Qx
-
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Table 1
Three configurations to simulate fretting mode I, the difference being the number of influence coefficients that are considered.
Case I.1 I.2 I.3
Elastic properties
E1 (GPa) 200 200 200
1 0.3 0.3 0.06
E2 (GPa) 200 200 200
2 0.3 0.3 0.42
Coupling 0 0 0.1938
Influence coefficients Kqx
x
K
qy
x Kpx
Kqxy
Kqyy
Kpy
Kqxz
Kqyz
Kpz
CPU time/increment (s) 12.4 78.6 298.2
Contact radius a (mm) Analytical 0.3011
Numerical +0.75% +0.75% (0.55 +0.75)%
Stick radius c (mm) Analytical 0.1398
Numerical 0.61% +10.62% (79.0 +52.6)%
Hertz pressure p0 (MPa) Analytical 2106.491
Numerical +0.001% +0.001% +1.461%
Min shear in the stick area q0 (MPa) Analytical 112.874Numerical 0.002% 0.485% 43.76%
Rigid body displacement z (mm) Analytical 9.0665 103Numerical 0.006% 0.006% 0.061%
Rigid body displacement x (mm) Analytical 0.8637 103Numerical +0.012% 0.116% +19.6%
ForcaseI.1 only theinfluence coefficientslinkingthe localtractionand theelasticdeflectionin thesamedirectionare used.Sincethetangential load isalongx, thecoefficients
in the y direction are omitted. For case I.2 all influence coefficients are used except the coefficients linking the normal and the tangential problems because their values are
nil when the materials are identical. For case I.3, materials are dissimilar, but the equivalent Young modulus is equal to previous cases, therefore all influence coefficients
should be considered. Computation times are also indicated. Comparisons between analytical and numerical results are done.
that the CattaneoMindlin solution is an approximation because
slips and elastic deflections along the y direction are neglected
(or assumed nil). To illustrate this point, the angle between the
x direction and the slip direction obtained in case I.2 is plotted in
Fig. 9.
Pressure, shears and slips are plotted in Figs. 10 and 11. Obvi-
ously the analytical solutions and the numerical ones found in
simulation I.1 are found equivalent. Simulation I.2 exhibits some
differences in terms of shear distributions that are however quite
close to the (approximated) analytical solution. Large differences
appear in caseI.3 comparedto theanalyticalsolutionwith an equiv-
alent Young modulus, outlining the fact that the CattaneoMindlin
solution should be no more considered as valid when the Dundurs
constant is not nil. A helpful mean to illustrate the main differences
Fig.9. Anglebetweenthex axisand the slipdirections obtained withsimulationI.2.
It highlights that the CattaneoMindlin assumption, which is that the y component
of the slips and shears are nil, is a quite strong assumption.
between these three configurations is to plot thestreamlinesof the
shear vectors, see Fig. 12.
The fretting loop can be obtained through the rigid body dis-
placement, cf. Fig. 13. Case I.3 gives a different fretting loop. It can
be observed that points A and E of the loading path are not super-
imposed on the fretting loop, outlining a slip ratcheting behavior. In
addition after one unloading-loading the tangential displacement
is lower.
4. Fretting mode II
When slips are radial, as for indentation, the fretting is defined
as mode II. This mode can be obtained when an elastic sphere is
pressed against a flat of different elastic properties. In the fric-
tional case, the Hertz theory is no longer validor should be used
cautiously. The coupling between the normal and the tangential
problems produces a radial shear stress and slip distributions. An
outer slip annulus appears between radii c and a, c being the radiusof the inner circular stick area and a the contact radius. The ana-
lytical solution is due to Spence [54] who proposed a closed form
solution of the Hertz contact problem with finite friction when the
indentation is processed progressively. This solution is based on
an approximation which is that, if the pressure distribution has
an effect on the tangential problem, the new tangential solution
is not used to modify the pressure distribution. In other words, it
is equivalent to solve successively the normal and the tangential
problem only one time. The Spence solution links the stick radius
to the friction coefficient, the Dundurs constant and the complete
elliptic integral of the first kind K:
a
2clna + ca c =
K ca , (46)
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Fig. 10. Dimensionless analytical and numerical contact pressure and shears. Results are plotted along the x in (a) and (b), and along the y axis in (c) and (d). Results are
given for the loading point A in (a) and (c) and for the loading point B in (b) and (d). When qy is not indicated, its value is nil.
Fig. 11. Analytical and numerical slips. Results are plotted along the x and y directions, top and bottom, respectively. Results correspond to the total accumulated slips
between the loading points O and A. When sy is not indicated, its value is nil.
Fig. 12. Streamlines of the shear vectors in the contact area for the three cases. (a) Simulation I.1, (b) simulation I.2 and (c) simulation I.3.
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Fig. 13. Fretting loop. Analytical result and results of simulations I.1 and I.2 are
superimposed. Results of I.3 are shifted along the direction where the tangential
load is first applied.
Fig. 14. Ratiobetween the contact radius a andthe stick radius c versus the normal
load Pfor several friction coefficients for fretting mode II.
where
K
c
a
= K
1
c
a
2.
The previous sphere with radius R=10 mm is brought into con-
tact on a flat. The material properties are E1 =200 GPa, 1 =0.06,E2 =200 GPa, 2 =0.42, i.e. the Dundurs constant is equal to=0.1938. The normal load P=400N, is applied gradually every40N. Computations are done for a range of friction coefficient from
0.01 to 0.25. The Spence lawimplies that the ratio c/a remains con-
stant whenthe normal is increasing, which is notfound numerically
at the beginningof the loading, see Fig. 14. It can be concluded that
the dimensionless stick radius c/a reaches a constant value, as pre-
Fig.15. Comparisonofthe analyticalsolution ofSpenceand resultsobtainednumer-
ically. Theratiobetweenthe contact radius a andthe stick radius c is functionof the
friction coefficient and the Dundurs constant .
dicted by Spence, only after a transient period in which the stick
radius decreases from a to c (as given by Spence).
The numerical results are compared to the analytical solution
of Spence in Fig. 15. The main differences could be observed for
extreme values. When the friction coefficient increases, both thestick radius and the maximum contact pressure increases, see
Fig. 16(a) and (b), respectively. However the maximum pressure
tends to stabilizewithhighfrictioncoefficients dueto a smaller slip
area. Note that when the stick radius is important, the shear distri-
bution becomes numerically wavy, due to an insufficient number of
load steps. A solution obtained after only one increment is plotted
in Fig. 17 for a friction coefficient =0.1, and compared to the oneobtained with 10 increments. The difference suggests that several
loading increments are required to reach the analytical solution
given by Spence.
The shear results are given in Fig. 18 for three different fric-
tion coefficients at each loading increment. As discussed above the
shear distribution has a wavy shape at high friction coefficient,
i.e. =0.25, see Fig. 18(c). This numerical artifact, which disap-pears when using more load increments, perturbs the stick radius
result. In addition it shouldbe noted that, whereas theSpence solu-
tion assumes a Hertz pressure distribution, the maximum contact
pressure found numerically when coupling normal and tangential
problems reaches 1.04 times the Hertz pressure due to high shear
stress that perturbs the normal problem. It explains why in Fig. 15
the numerical solution diverges from the analytical one for high
frictioncoefficient, the numerical solution being probablythe exact
one. At low coefficient of friction (=0.01) the Spence law is accu-rate because the normal problem becomes hardly affected by the
tangential solution. The difference between the numerical and the
analytical solutions is here attributed to the mesh whichis not fine
enough to catch accurately a very small stick disk. Finally for all
Fig. 16. (a) Dimensionless contact pressure and shear obtained for various friction coefficients and at the final loading increment. When the friction coefficient becomes
high, more increments are required to avoid waviness in the numerical solution. (b) The maximum pressure increases with the coefficient of friction.
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Fig. 17. Shear obtained for a friction coefficient =0.1 with a load path separated
in only oneor tenincrements.The difference suggests that a quantity of increments
is necessary to obtain the permanent contact evolution described by Spence.
simulations carried out in mode II shear and slip vectors are radial,
as shown in Fig. 19 for a friction coefficient =0.1.
5. Fretting mode III
When circumferential slips are involved, the fretting mode is
called mode III. It happens for example when a sphere pressed to a
flat surface is submitted to an additional torsion moment. An outer
slip annulus appears, the circular stick area being limited by the
radius c. Whenelastic properties are similar, normal and tangential
problems areuncoupled and a close form solutionexists. Ithas been
given by Lubkin [55]. The corresponding shear expression is:
q =3P
2a2
1 r
2
a2
1/2, c r a (47)
q=
3P
2a21
r2
a2
1/2
2 +
k2D (k) Fk, K (k) Ek, ,r c (48)
k =
1
c/a2 = 1 k2, k = c/a, and
= arcsin
1
k
k2 r/a21
r/a
2 . (49)
F and E are the elliptical integrals of the first and second kindrespectively, of modulus k and amplitude . D is thecompleteellip-tical integral of modulus k given by D(k)
=(K(k)
E(k))/k2, with
K and E the complete elliptical integrals of the first and secondkind respectively. The rigid body angle is linked to the stick limit
Fig. 19. Streamlines of the shear vectors on the contact area for fretting mode II.
The radial behavior is reached numerically.
Fig. 20. Dimensionless analytical and numerical absolute shear traction. Results
are plotted along the radial axis x. Results are similar for cases III.1 and III.2. When
materials are dissimilar (case III.3), the analytical solution is no longer valid.
radius c:
z =3P
4Ga2k2D(k). (50)
The radius c is related to the moment:
Mz =Pa
4
32
4+ kk2
6K(k) + (4k2 3)D
3kK(k) arcsin(k)
3k2
K(k)
2
0
arcsin (k sin )
(1 k2 sin2 )3/2 d D(k)/2
0
arcsin (k sin )
(1 k2 sin2 )1/2 d
.
(51)
Fig. 18. Evolution of the dimensionless shear and pressure for (a) =0.01, (b) =0.1 and (c) =0.25. When friction is important, more increments are needed to avoid
waviness in the results.
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Table 2
Three configurations are analyzed for simulating fretting mode III.
Case III.1 III.2 III.3
Elastic properties
E1 (GPa) 200 200 200
1 0.3 0.3 0.06
E2 (GPa) 200 200 200
2 0.3 0.3 0.42
Coupling 0 0 0.1938
Influence coefficients Kqx
x
Kqyx K
px
Kqx
y
Kqyy
Kpy
Kqxz
Kqyz
Kpz
CPU time/increment (s) 108.9 211.2 124.4
Contact radius a (mm) Analytical 0.3011
Numerical +0.75% +0.75% 0.55%
Stick radius c (mm) Analytical 0.2088
Numerical +1.232% +1.232% 59.70%
Hertz pressure p0 (MPa) Analytical 2106.491
Numerical +0.001% +0.001% +3.209%
Min shear in the stick area q0 (MPa) Analytical 0Numerical 0% 0% 0%
Rigid body displacement z (mm) Analytical 9.0665 103Numerical 0.006% 0.006% 0.160%
Rigid body displacement z (deg.) Analytical 0.0831
Numerical 7.471% 0.043% 78.31%
Thedifference is thenumber of influencecoefficients thatare takeninto account. Forcase III.1 onlythe influencecoefficientslinking thelocal tractionand theelasticdeflection
in the same direction are used. For case III.2 every coefficients are used except the coefficients linking the normal and tangential problems because there values are nil when
materials are similar. For case III.3, materials are dissimilar, but the equivalent Young modulus is equal to the previous cases, then all influence coefficients should be taken
into account. Computation times are given. Comparisons between analytical and numerical results are done.
Fig. 21. Ratio qr/q is used to define the computational errors induced after numerical simulation for case III.1 (a) and case III.2 (b). Indeed shear obtained should be
circumferential and its radial component nil. Therefore to properly simulate fretting mode III, all the influence coefficients used in case III.2 are necessary.
Fig. 22. Streamlines of the shear vectors on the contact area for the three cases. (a) Case III.1, (b) case III.2 and (c) case III.3. A non-negligible radial component appears when
materials are dissimilar (case III.3).
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Please cite this article in press as: L. Gallego, et al., A fast and efficient contact algorithm for fretting problems applied to fretting modes I, II and
III, Wear (2009), doi:10.1016/j.wear.2009.07.019
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Three cases are simulated equivalently to fretting mode I simu-
lations. A first step consists to apply the normal load P=400N, and
in the second step a torsion moment Mz =5 Nm is imposed. Forcase III.3, it is executed in 10 increments per step because of cou-
pling that induces a non-linear behavior. The contact configuration
remains a sphere with radius R=10 mm brought into contact on a
flat.
Computation time increases when considering coupling. The
shear distribution and the stick limit radius c found numerically
when the bodies have identical elastic properties (cases III.1 and
III.2) are in very good agreement with the analytical solution, as
shown in Fig. 20.
The only difference between the numerical results obtained for
configurations III.1 and III.2 is a radial component of shears that
appears in case III.1 but not in III.2, as shown in Fig. 21. Obviously
for symmetry reasons the correct solution should be exempted of
radial shear. This is attributed to the fact that the problem is solved
in a Cartesian coordinate frame. It can be concluded that, when
using a Cartesian coordinate frame for solving a torsional contact
problem (i.e. with a radial symmetry) the crossed influence coeffi-
cients should not be omitted. When dissimilar materials are used
it canbe noticed that the symmetry of revolution is still preserved:
shear is nilat the originand the contact area remains circular. How-
ever analytical solutions are inappropriate. Simulation III.3 is verysingular compared to the two other cases; the stick radius is found
to be 60% lower than the Lubkin solution [55], as shown in Table 2
andalsoin Fig. 20. This difference canbe explainedby observingthe
streamlines of the shear vectors, Fig. 22, where a significant radial
component is found in addition to the circumferential one.
6. Conclusion
A semi-analytical contact code has been presented to model
fretting contact. The conjugate gradient method is adapted to solve
the coupled normal and tangential problem while minimizing the
complementary energy. Computation times are contained thanks
to the use of the DC-FFT technique to compute the convolution
products between the surface stresses and the influence coeffi-
cients. In the present algorithm each component of the applied
force and moment vectors are considered, which means that the
center of pressure and shear is not centered in the contact area.
Indeed normal contact with flexion moments and tangential con-
tact with torsion moment can be computed. The efficiency of the
method was investigatedthroughthree examples correspondingto
the fretting modes I (tangential displacement), II (normal displace-
ment) and III (torsional displacement) as defined by Mohrbacher.
It has been found for mode I that, when the contacting materials
have similar elastic properties, the numerical solution slightly dif-
fers from the analytical one that neglects slip in the perpendicular
direction to the tangential force. On the other hand when the con-
tacting bodies behave elastically differently it has been shown for
modes I, II (for high friction coefficients) and III that the numericalsolutions with full coupling do not match the analytical ones.
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