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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

ARTICLE IN PRESSGModel

WEA-99257; No.of Pages15

Wear xxx (2009) xxxxxx

Contents lists available at ScienceDirect

Wear

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / w e a r

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
http://dx.doi.org/10.1016/j.wear.2009.07.019http://www.sciencedirect.com/science/journal/00431648http://www.elsevier.com/locate/wearmailto:[email protected]://dx.doi.org/10.1016/j.wear.2009.07.019http://dx.doi.org/10.1016/j.wear.2009.07.019mailto:[email protected]://www.elsevier.com/locate/wearhttp://www.sciencedirect.com/science/journal/00431648http://dx.doi.org/10.1016/j.wear.2009.07.019


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2 L. Gallego et al. / Wearxxx (2009) xxxxxx

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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L. Gallego et al. / Wearxxx (2009) xxxxxx 3

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


(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


, (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


7/15






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)


8/15






(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


9/15






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)


10/15






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.


11/15






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.


12/15






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.


13/15






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).


14/15






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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