solving 2d transient rolling contact problems using the bem and mathematical programming techniques

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 53:843–874 (DOI: 10.1002/nme.315)

Solving 2D transient rolling contact problems using theBEM and mathematical programming techniques

Jos5e A. Gonz5alez and Ram5on Abascal∗;†

Escuela Superior de Ingenieros; Camino de los descubrimientos s=n; E-41092 Sevilla; Spain

SUMMARY

This work presents a new approach to the transient rolling contact of two-dimensional elastic bodies. Asolution will be obtained by minimizing a general B-di=erentiable function representing the equilibriumequations and the contact conditions at each time step. Inertial e=ects are not taken into account andthe boundary element method is used to compute the elastic in>uence coe?cients of the surface pointsinvolved in contact (equilibrium equations). The contact conditions are represented with the help ofvariational inequalities and projection functions. Finally, the minimization problem is solved using theGeneralized Newton’s Method with line search. The results are compared with some example problemsand the in>uence of discretization and integration time step on the results is discussed. Copyright ?2001 John Wiley & Sons, Ltd.

KEY WORDS: transient rolling; contact; mathematical programming; boundary element method

1. INTRODUCTION

The mechanical study of rolling contact problems started in 1875 with Reynold’s investigationof the creep phenomena [1], observed when a rubber cylinder rolling on a steel plate movesforward more than its undeformed rolled perimeter. He attributed this e=ect to the elasticdeformations in the contact area and introduced the idea that the contact region would becomposed of stick and slip zones. It was about 50 years later, in 1926, when Carter publishedhis work [2] presenting the steady-state analytical solution for the two dimensional case ofsimilar cylinders’ materials, and the non-linear relation between longitudinal creep force andcreepage.The Krst semi-analytical solutions of the steady-state case for dissimilar rollers can be

found in the papers of Bentall and Johnson [3; 4] and Nowell and Hills [5; 6]. Concerningthe numerical solution of three-dimensional rolling contact problems, the pioneering work of

∗Correspondence to: Ram5on Abascal, Escuela Superior de Ingenieros, Camino de los descubrimientos s=n, E-41092Sevilla, Spain.

†E-mail: [email protected]

Contract=grant sponsor: Ministerio de Ciencia y Tecnolog5Ma, Spain; contract=grant number: DPI2000-1642

Received 4 September 2000Copyright ? 2001 John Wiley & Sons, Ltd. Revised 19 April 2001

844 J. A. GONZ 5ALEZ AND R. ABASCAL

Kalker [7] should be highlighted. The present formulation of rolling contact springs from thisand other papers by Kalker, [8–10], summarized in his book [11].However, only a few investigations dealing with non-steady contact mechanics phenomena

of elastic bodies in rolling motion can be found, most of which have been published byKalker and are focused on time-domain solutions for two and three-dimensional problems.Historically, the Krst approximation to the transient rolling contact problem was reported

by Kalker for the particular no-slip case [12]; the formulation was extended later to the com-plete adhesion-slip case using an original minimum principle [13]. In both works, a constantHertz normal pressure distribution over time was assumed, and uncoupled normal–tangentialproblems were solved.Later, another view more interested in the wheel–rail response on the frequency domain

for the high-frequency range was proposed by Knothe and Gross-Thebing [14; 15]. With theirapproximation, a linearization of the contact problem around the operation point becomesunavoidable (due to its non-linear character) and very important information such as theharmonic response of the creepages for harmonically varying creep forces, can be obtained.Applicability is restricted to cases where the variables amplitude is small in order to maintainthe linearization hypothesis as valid.Finally, we should mention the work of Nielsen [16], where we can Knd an analytical

approach to 2D unsteady rolling contact of quasi-identical bodies using approximating poly-nomials. Nielsen divides the contact area into one stick zone and one slip zone, avoiding thenon-linear character of the contact boundary conditions and obtains analytical solutions forsome similar material problems.Although many relevant papers concerning contact problems using the Boundary Element

Method (BEM) can be found, among which are the works by Anderson [17], Paris andGarrido [18], and Abascal [19–21], only a small number of them incorporate rolling motion.The works by Wang and Knothe [22], Kong and Wang [23] and Kalker and Van Raden [24],can also be mentioned.Mathematical programming techniques (MPT) are very attractive for solving this type of

problem because they allow a more mathematical point of view, and provide very valuablenumerical tools. References using MPT to solve quasi-static contact problems are numerous,especially the book by Panagiotopoulos [25], the articles by Klarbring [26; 27], Kwak and Lee[28], Gakwaya et al. [29], De Saxe and Feng [30], the book by Antes and Panagiotopoulos[31], and the paper by Kong et al. [23].Conry and Seireg [32], Wriggers [33], Alart and Curnier [34], and Sim5o and Laursen [35],

are the precursors of the use of the Augmented Lagrangian formulation. This formulationhas also recently been used in the papers by Wriggers [36; 37], and by StrQomberg [38].Later, Christensen [39; 40] compared the results and the convergence obtained by solving theproblem with several mathematical programming methods, and enlarged the study of the for-mulation. As before, the applications of these techniques to the case of contact with rolling arescarce in literature, although the work of Kalker [11] using the KOMBI algorithm for transientproblems and those presented by the authors [41–45] for steady-state rolling, can be cited.This work is the natural enhancement of a previous paper on steady rolling [46], presenting

a new approach to the transient rolling contact of two-dimensional elastic bodies under theaction of variable external forces.With the proposed methodology, normal–tangential coupled and uncoupled problems can be

solved. A solution is obtained, after a time integration, by minimizing a general B-di=erentiable

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:843–874

SOLVING 2D TRANSIENT ROLLING CONTACT PROBLEMS 845

function representing the equilibrium equation and the contact restrictions at each time step.The boundary element method is used to compute the elastic in>uence coe?cients of thesurface points involved in contact (equilibrium equations), while the contact conditions arerepresented with the help of variational inequalities and projection functions. The mini-mization problem is then solved using the Generalized Newton’s Method with line search.Finally, the algorithm presented is veriKed by solving several classical non-stationary rollingproblems.

2. THE TRANSIENT ROLLING CONTACT PROBLEM

Transient rolling contact of two cylinders with parallel axes is within the scope of this work.The study is carried out from a two-dimensional point of view, i.e. we assume that axialdisplacements and rotation around normal axes to the contact area (spin) are excluded fromthe analysis, assimilating the problem to plane strain.Transient phenomena appear when the forces applied to the rollers or their rigid body

velocities vary with time. The initial equilibrium conKguration could be either stopped andbraked cylinders under the action of constant loads, or two cylinders in steady-state rolling. Ifexternal excitation (loads and, or, velocities) now vary to a di=erent constant value, a transientregime starts that ends with a new steady-state regime based on a new and di=erent state ofequilibrium.The problem will be purely transient if oscillations produced by the external actions on

the contact area displacements and tractions, have a characteristic wavelength smaller thanthe size of the contact area. In other cases, the problem solution could be obtained using asuccession of enchained stationary problems.Transient rolling is a complex process, highly dependent on the initial situation and the

load history. We suppose in our approximation that the acceleration of particles is not high,so inertial e=ects are not important, see Reference [47], and the elastostatic equations can beused to compute the in>uence coe?cients.For the integration of the non-linear time equations, a Knite di=erence scheme has been

used. To start time integration and simulate transient e=ects, some information has to begiven: initial conditions and load history. The initial state from which the cylinders start theirtransient evolution should be determined by solving either a contact or a steady-state rollingproblem. Load history will be provided in the form of normal and tangential net loads versustime.It is assumed that the friction between the cylinders obeys the Coulomb–Amontons law,

where normal and tangential tractions in the slip area are related by a constant friction coef-Kcient. By adopting this approach, each pair of points belonging to each of the cylinders inrolling (A and B in Figure 1) can experience any of the following contact states: separation,adhesion or slip.

2.1. Contact conditions

The mathematical expressions of the contact conditions, using Kalker’s notation [11], willbe classiKed into two groups: normal or tangential, making it possible to write each contactcondition as a variational inequality expression.



Figure 1. Rolling cylinders. Load history and contact area initial conditions.

• Normal direction:The Signorini conditions for unilateral contact in the form of a variational inequality couldbe expressed as

pn ∈�−; �n(pn − p∗n )¿0; ∀p∗

n ∈�− (1)

with �− ≡{y∈�: y60}.• Tangential direction:For tangential direction, fulKllment of the friction law is guaranteed by the followingvariational inequality:

pt ∈C|pn|; st(pt − p∗t )60; ∀p∗

t ∈C|pn| (2)

where Cg is a closed interval of radius g:

Cg ≡{y∈�: |y|6g} (3)

As we can see, there exists a coupling between normal and tangential variables due to thepresence of the friction limit g=|pn| in the tangential direction condition. These inequalitieshave to be fulKlled over the entire contact area for any time .In this way, possible states for each pair of contact points are characterized by a group of

equalities and inequalities:

• StickpA;B

n 60; pAn =pB

n ; pAt =pB

t

�n=0 : st =0(4)

• SlippA;B

n 60; pAn =pB

n ; |pA;Bt |=|pA;B

n |�n=0; sgn(st)=− sgn(pA;B

t )(5)



• Separation

pA;Bn =0; pA;B

t =0; �n¿0 (6)

where is the friction coe?cient; p�n and p�

t are the normal and tangential tractions of thecontact points of the cylinder �=A; B; u�

n; u�t are their respective displacements; �n is the

normal separation in the deformed state, and st is the relative tangential slip velocity. It isimportant to mention that we have considered the tangential vector with opposite directionfor di=erent solids.

2.2. Problem description

Normal and tangential tractions will depend on the Eulerian co-ordinate x (relative to a Kxedaxis situated in the contact area) which is used to allocate each pair of points relative to arigid body position of the cylinders at certain time , i.e. pn(x; ) and pt(x; ).We start the simulation at 0 with two braked or rolling cylinders in contact, equilibrated

with external forces P0, Q0 and presenting initial normal and tangential traction distributionswithin the contact area given by

pn0(x)=pn(x; 0); pt0(x)=pt(x; 0) (7)

obtained after a rolling contact process between the two cylinders (Figure 1).The process to obtain these initial conditions will be di=erent depending on initial velocities.

Initial conditions will be obtained by solving a steady-state rolling problem following theprocess described in Reference [46] if cylinders start from a rolling situation. If cylindersstart braked, we have to solve what is known as a fretting contact problem, quasi-static,incremental and path dependent (due to the presence of friction). Path dependence means thatpn0 and pt0 distributions will depend on how P0 and Q0 loads are applied. In the present work,we have chosen linear (ramp) variation. After the application of these loads, the cylindersare unbraked and rolling starts under the action of external forces P( ) and Q( ). If theseexternal forces remain constant in time, the system will tend to a steady rolling state.The separation of bodies in normal direction, �n, will be

�n(x; )= �n0(x; )− {uAn (x; ) + uB

n (x; )} (8)

where �n0 is the initial distance between the cylinder contact points,

�n0(x; )= �g(x)− �0( ) (9)

�g being the geometric initial separation between two contact points located at x, and �0the sum of two approaching rigid body displacements externally imposed on each cylinder,producing an overlap between them.For the particular case of rolling cylinders, �g can be geometrically obtained as

�g(x)=RA

1−

√1−(

xRA

)2+ RB

1−

√1−(

xRB

)2 (10)

R� being each one of the cylinder radius.



Using a Eulerian description of the particles moving through the contact area, the relativetangential slip velocity between two points of the cylinders A and B, is deKned as

St(x; )=D�t(x; )

D (11)

where D(:)=D is the total time derivative, sometimes called substantial or material derivative(following a particle of Kxed identity relative to Kxed axes) and �t represents the tangentialseparation, given by

�t(x; )= {xA( )− xB( )}+ {uAt (x; ) + uB

t (x; )} (12)

If we expand Equation (11) on its convective and local part (D(:)=D =V@(:)=@x+@(:)=@ ),substitution into the slip velocity expression, yields

St(x; ) = {VA( )− VB( )}+ {VA( )uAt; x(x; ) + VB( )uB

t; x(x; )}+ {uAt; (x; ) + uB

t; (x; )} (13)

where V� is the rigid body’s local speed in the x direction. The third term is the local slip,which vanishes if the >ow is steady, i.e. time independent.Kalker shows in Reference [11] that when there is a mean speed of rolling, Equation (13)

can be approximated by

st(x; )= �0( ) + sgn[V ( )]ut; x(x; ) + |V ( )|−1ut; (x; ) (14)

where ut(x; )= uAt (x; ) + uB

t (x; ) is the tangential displacement distance, sgn[V ( )] is therolling direction,

V ( )= 12{VA( ) + VB( )} (15)

is the mean speed of rolling (positive in x direction), relative slip st(x; ) is the slip velocitydivided by the mean speed absolute value, and

�0( )=V A( )− VB( )

|V ( )| (16)

the creepage, or normalized relative rigid slip.Equation (14) has to be discretized not only in space but also in time, and two expansions

of it can be adopted. To do this, we assume quasi->at contact surfaces, and that the meanrolling velocity V ( ) governs the rotation of the two cylinders, i.e. (V A( )−VB( ))�V�( ).Under these hypotheses, the individual velocity V�( ) of each cylinder can be substituted intothe equations by the mean velocity V ( ).The Krst expansion is obtained from equations (12) and (11), based on the total time

derivative

st(x; )={�0( ) + lim

T →0

ut(x; )− ut(x − V ( )T ; −T )|V ( )|T

}(17)

and presents the advantage that the formulation for each time step only involves tangentialdisplacements at the current and previous time step (see Figure 2), and the drawback is thatwe have to interpolate them between nodes with the shape functions, if V ( )T is not equalto the nodal separation.



Figure 2. Moving particles at current (left) and previous (right) time step.

Equation (17) can be written in the same form as (14),

st(x; ) ={�0( ) + sgn[V ( )] lim

T →0

ut(x; −T )− ut(x − V ( )T ; −T )|V ( )|T

+

+ limT →0

ut(x; )− ut(x; −T )|V ( )|T

}(18)

and calculates the boundary displacement derivatives along the contact area for the last timestep solved ( −T ); this method seems more complicated, but by using this approach onlynodal information will be involved.In the second term of (18) the time derivative can be changed into a space derivative, as

in Equation (14), using an arbitrary distance h:

limT →0

ut(x; −T )− ut(x − V ( )T ; −T )|V ( )|T

= limh→0

ut(x; −T )− ut(x − h; −T )h

(19)

This derivative can now be approximated using a forward Knite di=erence scheme whenparticles moving through the contact area are travelling from right to left, or using a backwardscheme when coming from left to right,

sgn[V ( )] limh→0

ut(x; −T )− ut(x − h; −T )h

≈

ut(xi+1; −T )− ut(xi; −T )xi+1 − xi

if V ( )¡0

ut(xi; −T )− ut(xi−1; −T )xi − xi−1

if V ( )¿0(20)

an explanation of this choice will be addressed later.



A review of the relative slip expressions shows that the rolling velocity always appearsmultiplied by the time increment, revealing the quasi-static character of the problem. Thatmeans the problem can be reformulated as a function of the travelled distance �( ) between 0 and instead of time, making the change of variable

�( )=∫

0|V (�)| d�= 1

2(�A( ) + �B( )) (21)

Considering the relation @(:)=@�= |V ( )|−1@(:)=@ , we can replace |V ( )|T by T� and anew version of Equation (18) dependent only on particle position x, rolling direction sgn[V (�)]and the travelled distance � can be obtained

st(x; �)= �0(�) + sgn[V (�)] ut; x(x; �) + ut; �(x; �) (22)

For all other equations, time variable can be directly replaced by � taken in account ofthe fact that a relation �( ) exists, that makes our new ‘time-variable’ the distance travelled(time is involved only as a parameter) and the problem becomes independent of the rollingvelocity magnitude V ( ).

2.3. The model

Classical methods to solve rolling contact problems make use of the half-space approach.This simpliKcation consists of the use of analytical expressions for the half-space to computethe solid equilibrium equations. Due to the numerical capability of the BEM, the proposedmethodology allows the study of curved boundaries without having to use that simpliKcation.In order to compare the results with the available solutions obtained by Kalker (based

on the half-space approach), we will use a model with curved geometry which satisKes thenecessary geometrical requirements of the half-space approach, i.e. the cylinders used in ourexamples will have a potential contact half-length, a, small in comparison with the character-istic dimensions of the domain, radii RA and RB.The selected geometry is represented in Figure 3 where two cylinders are considered; the

larger one has a radius RB=300mm and rotates around a rigid shaft of radius rB=30mm, andthe smaller one has an external radius RA=150 mm and an internal rigid shaft of dimensionrA=15 mm.The boundary element mesh used to solve the examples of problems is also schematically

represented; Figure 3 reveals how nodes inside the contact area are confronted and howthey reside almost in the same geometrical position. According to the small deformationsassumption, the undeformed distance between pairs can be calculated geometrically from themesh, instead of using analytical expressions like (10). Moreover, reKnement of the meshinside the contact area where a stress concentration is expected can also be clearly observed.The cylinders’ external surfaces are discretized by using 19 boundary elements with zero

traction boundary condition, while the internal surfaces are discretized by using 16 elementswith a zero displacement boundary condition in order to represent their being welded to rigidshafts.



Figure 3. Rolling cylinders. Boundary element mesh.

3. PROBLEM FORMULATION

Transient rolling contact is a highly non-linear and path-dependent problem; the source ofthis non-linearity is the boundary conditions used to represent the friction law. Non-linearitymeans that the evolution from di=erent situations to the steady-state general solution is stronglydependent on the starting contact state and the variations in applied external forces.The step-to-step time integration scheme is adopted, forcing the fulKllment of contact con-

straints at all nodes for every discrete time step. Nothing is known about change of variablesbetween time steps, so little incompatibilities vanishing when the time step tends to zero haveto be accepted. In addition, imposing contact conditions only at nodal positions can make in-consistencies appear, due to the approximation of displacement and traction distributions alongthe element using shape functions. These incompatibilities due to the element representationwill only occur in the change of state limits.Historically, the most used algorithms to solve rolling contact problems are denominated

as PANA [31] and KOMBI [11], both based on iterative strategies:

• PANA algorithm was Krst adopted by Kalker [10] for 2D elastic steady-state rolling andlater by Wang [22] and Braat [48] for the viscoelastic case. It is based on uncoupling of theproblem into its normal and tangential parts. First, we suppose known tangential tractionsand obtain normal ones and, after that, normal tractions are known and tangential ones canbe obtained. These two steps are repeated until convergence occurs.

• KOMBI algorithm is introduced by Kalker as a more e=ective alternative to the PANAprocess. This algorithm is based on Duvaut and Lions’ [49] variational theory of contact,



and has been used successfully for 3D transient problems. For this algorithm, the tractionbound has to be estimated and new normal and tangential tractions are obtained by solvinga minimization problem. With new normal pressures, the algorithm can be resubmitted;iteration control on normal tractions is also needed.

Since these ways of iterating could be slow or ine=ective for strongly coupled problems,we propose the use of another more general solution procedure to avoid iteration control onnormal tractions. The suggested algorithm is based on mathematical programming tools andsolves the problem in a combined manner, without separating into their normal and tangentialparts, and with a fully automated iteration control.

3.1. The elastic equations

A geometrical discretization process and subsequent application of the BEM static formulation(see Reference [50]), provide us with the cylinders’ elastostatic equations, in the absence ofinertial forces, when a speciKed distance � was travelled

H�u�(�)=G�p�(�); �=A; B (23)

where the travelled distance dependence appears only in the boundary variables. BEM matricesare not � dependent because when time increases, the geometry remains unchanged withrespect to used axes.Grouping the BEM equations of the two cylinders, and condensing all variables not

associated with the contact areas(u�n(�)

u�t (�)

)=

[S�

nn S�nt

S�tn S�

tt

](p�n(�)

p�t (�)

)+

(g�ne(�)

g�te(�)

)(24)

where u�n, u�

t and p�n, p�t are vectors containing the normal and tangential displacements andtractions on the contact area; S�

rs is a matrix whose ith column represents the contact nodedisplacements in the r-direction due to the application of a unit traction in the s-directionat node i, maintaining the rest of the boundary conditions equal to zero; and the last term(g�ne(�), g�te(�))T, represents the normal and tangential displacements in the contact area dueto externally applied boundary conditions, outside the shaft or the contact area.The external action of a brake on the surface of the cylinder, is an example where the last

term of equation (24) becomes non zero and has to be considered in the analysis.Substituting (24) into the vector form of equation (8) and after a discretization of the

travelled distance parameter on k constant steps (�k = �k−1 +T�), we obtain one of the Krstexpressions of the algorithm for the discrete step �k :

Tkn = f kn + Snnpkn + Sntpkt ; k=1 : : : nts (25)

where

f kn =Tkn0 + gkn; gkn= − (gAne(�k) + gBne(�k))

Snn=−(SAnn + S

Bnn); Snt =−(SA

nt + SBnt)

(26)



nts is the total number of steps, and

pkn= pAn(�k)= pBn(�k); pkt = p

At (�k)= pBt (�k) (27)

are the surface nodal tractions for discrete kth step.Similarly, the tangential displacement di=erence can be obtained in the condensed form as

ukt = g

kt + Stnpkn + Sttpkt (28)

where

gkt =gAte(�k) + gBte(�k)

Stn=SAtn + S

Btn; Stt =SA

tt + SBtt

(29)

Finally, after the resolution of the elastic problem using the BEM and later condensation,we have obtained the linear relation between gap-tangential displacements vector (Tkn ; uk

t ) andnormal–tangential tractions vector in the contact area (pkn; pkt ), expressed as(

Tknukt

)=

(f kn

gkt

)+

[Snn Snt

Stn Stt

](pkn

pkt

)(30)

The equilibrium equation (30) together with the contact conditions expressed by Equations(1) and (2), for all contact point pairs, completes the mathematical description of the prob-lem. However, tangential contact restrictions are expressed by means of relative tangentialdisplacement time derivatives, i.e. slip velocity.To calculate these slip velocities at the contact nodes, two di=erent approximations can be

adopted: the material formulation based on (17), or the convective-local formulation basedon (18).

3.1.1. Material formulation. Tangential relative displacements are discretized at the contactarea with continuous quadratic elements in their usual form,

ut(x; )�ne∑

j=1[Nj(x) utj( )] (31)

where ne is the number of nodes per element, Nj the element shape function associated tonode j (located at co-ordinate xj), and utj( ) is the nodal tangential displacement at instant .These shape functions Nj are the same for each couple of elements in the contact area becausetheir nodes are located at the same x-co-ordinate.When approximation (17) of st is used, we need the tangential relative displacement of

the pair of particles now located at xi co-ordinate, when and where they were T timeago

ut(x − VT ; −T )�ne∑

j=1[Nj(x − VT )utj( −T )] (32)



being necessary to take into account that the values x and (x − VT ) could be located ondi=erent elements.This information can be organized in a vector ut with components

[uk−1t ]i=

n∑j=1[R]ij utj( −T )=

n∑j=1[R(T�)]ij utj(�k−1) (33)

where n is the total number of contact nodes and R can be understood as a rigid body rotationoperator, that situates the position of a node T time ago.At this stage, it is important to note that we calculate the previous particle position con-

sidering a >at contact area. This is not valid for long travelled distances where real geometryshould be considered. Braat studied this e=ect in Reference [48] and observed that the obtainederror is proportional to (a=R).If the equations are organized in vector form, we can write the slip velocity for a step

�k as

skt = ^k +1T�

[ukt − uk−1

t

](34)

where the last equation, corresponding to the Krst node situated at the leading edge of thecontact area, has been modiKed to avoid the need for information out of the contact area (thispoint n, is always supposed to be in the separation state, pnn =ptn =0), writing skt and ^kvectors in the following form:

skt ={st1 (�k); st2 (�k); st3 (�k); : : : ; stn−1 (�k);

1T�[utn(�k)− utn(�k−1)]

}T^k=�0(�k)e�; e�=(1; 1; : : : ; 1; 0)T

(35)

Introducing (33) in matrix form i.e. ukt =Rkuk

t into the slip velocity main expression (34)and considering (30), we Knally obtain the grouped equation for the material approach

skt = ^k +1T�

[#k−1 + gkt + Stnpkn + Sttpkt

](36)

with

#k−1 =−R (gk−1t + Stnpk−1n + Sttpk−1t

)(37)

only depending on known information about the previous time step and the time step size.

3.1.2. Convective-local formulation. Another possibility is to apply the convective-localapproximation of the slip velocity, (18), by a pointwise application of the scheme givenby (20) to the vector form of tangential displacements found in (28).



To apply this scheme, we deKne a new operator T depending on the rolling velocity sign.If V ( )¡0, then

T=−

− 1h1

1h1

0 · · · 0 0 0

0 − 1h2

1h2

· · · 0 0 0

0 0 − 1h3

· · · 0 0 0

· · · · · · · · · · · · · · · · · · · · ·

0 0 0 · · · − 1hn−2

1hn−2

0

0 0 0 · · · 0 − 1hn−1

1hn−1

0 0 0 · · · 0 − 1hn−1

1hn−1

(38)

and if V ( )¿0, then

T=

1h1

− 1h1

0 : : : 0 0 0

1h1

− 1h1

0 : : : 0 0 0

01h2

− 1h2

: : : 0 0 0

: : : : : : : : : : : : : : : : : : : : :

0 0 0 : : : − 1hn−3

0 0

0 0 0 : : :1

hn−2− 1

hn−20

0 0 0 : : : 01

hn−1− 1

hn−1

(39)

being

hi=(xi+1 − xi) (40)

Using this operator, the tangential displacement derivatives at the contact area could beexpressed by

u�t; x(�)= h

�te(�) + [B

�tn B�

tt ]

(p�n(�)

p�t (�)

)(41)

where

h�te(�)=Tg

�te(�); B�

tn=TS�tn; B�

tt =TS�tt (42)



B�rs being a matrix whose ith column represents the r-direction x displacement derivativedistribution at the contact nodes due to the application of a unit traction in s-direction at nodei, maintaining the rest of the boundary tractions as equal to zero and free nodal displacements,and �=A; B.Then, the x-direction di=erentiation of the boundary tangential displacements of k step, can

be easily obtained as

ukt; x= h

ktx + Btnpkn + Bttpkt (43)

with the following deKnitions:

hktx =h

Ate(�k) + hB

te(�k)

Btn=BAtn + B

Btn; Btt =BA

tt + BBtt

(44)

Finally the slip can be expressed by

skt = ˆk + [hk−1

tx + Btnpk−1n + Bttpk−1t ] +1T�[uk

t − uk−1t ] (45)

where the local time derivation appears grouped into the last term, and slip velocity andcreepage vectors di=er from the previous deKnition (35) because only nodal information is nownecessary. The problem of this formulation is that space and time derivatives have di=erentapproximation errors. The error of the space derivative is proportional to the distance betweennodes, while the time derivative is proportional to the increment of the distance travelledT�= |V |T . These errors will be equal if T�=Le=2, being Le the element length in thecontact area.This leads to a new organization of the non-dimensional slip velocity and creepage in a

complete vector form

skt ={st1 (�k); st2 (�k); st3 (�k); : : : ; stn−1 (�k); stn(�k)}Tˆk=�0(�k)e; e=(1; 1; : : : ; 1; 1)T

(46)

Equation (45) can be expanded considering (28), obtaining an expression very similarto (36),

skt = ˆk +

1T�[Vk−1 + gkt + Stnpkn + Sttpkt ] (47)

with a vector

Vk−1 = − [gk−1t −T�hk−1tx ]− [Stn −T�Btn]pk−1n − [Stt −T�Btt]pk−1t (48)

that groups past information and that can be easily obtained for each time step.

3.2. The contact constraints

Alart and Curnier [34] developed a mixed penalty-duality formulation of the frictional contactproblem inspired by the Augmented Lagrangian method, to treat its multivalued aspects. Theyderived a non-symmetrical operator from quasi-augmented Lagrangian to palliate the absenceof a genuine one, exhibiting its properties and establishing a necessary and su?cient condition



of the friction coe?cient in order that the solution of curved, discrete, small slip contacts beunique, and guarantee the convergence of Newton’s method.Inspired by that work, Christensen [39] proposes a variation of the same formulation,

obtaining a system of B-di=erentiable equations, that allows him to use the convergencetheorems and the powerful Newton’s algorithm developed by Pang [51], Krst formulated forthe contact problem by Klarbring [52].In this work, we compute the discretized elastic equations of the rollers using the Boundary

Element Method, and following the cited works we formulate the contact conditions in a quasi-Lagrangian way, by reducing the solution of the problem to obtain the zero of a vectorialfunction.

3.2.1. Normal problem. If the tangential tractions at step k are assumed to be well known(pkt = Xpkt ), the problem could be considered exclusively as a function of the normal variables,obtaining the denominated normal problem.There are two possible states for each pair of points according to the normal direction:

contact and separation. This means that each couple should satisfy the following conditions:

• Contact:

�ni =0; pni60 ⇒ pni + r�ni60 (49)

• Separation:

�ni¿0; pni =0 ⇒ pni + r�ni¿0 (50)

r ∈�+ being a dimensional penalty parameter.These constraints can be summarized in the vectorial equation

pn=min(0; pn + rTn) (51)

equality, that will only be veriKed if each pair of points fulKl the contact conditions.This condition can also be written in a more compact form by means of the expression

−min(−pn; rTn)=0 (52)

which is equivalent to imposing the condition that the normal tractions are always incompression (−pn¿0), that cylinders overlapping do not exist (Tn¿0), and that at least one ofthe two variables is zero.The normal problem can be formulated by combining the normal part of the elastic

equations (30) with the contact conditions deKned by Equation (52) for certain distancetravelled �k ,

�kn(Tkn ; pkn)=

[Tkn − f kn − Snt Xpkt − Snnpkn

−min(−pkn; rTkn )

]= 0 (53)

Its solution is obtained by computing the zero of a non-linear vectorial function. It isimportant to remark that the normal part only depends on current time step, and the problembecomes path dependent only when friction is considered.



3.2.2. Tangential problem. The formulation of the tangential part of the problem is dependenton normal tractions. Let us suppose that the distribution of normal tractions is known (pkn= Xpkn)at the current step. To formulate the contact constraints, it is convenient to have previouslyintroduced two deKnitions: the euclidean distance between a point and an interval, and theprojection of a variable over an interval.

De+nition 1. The euclidean distance DCg from a point z ∈� to a closed and centred intervalCg ≡ [−g; g]; g∈�+, is deKned as

DCg(z)= max(z − g; 0)−min(z + g; 0) (54)

De+nition 2. The projection function PCgof a point z ∈� on the closed and centred intervalCg ≡ [−g; g]; g∈�+, is deKned as

PCg(z)= z − sgn(z)DCg(z) (55)

If these functions are applied componentwise to vectors z∈�n, their deKnition andrepresentation will be analogous

DCg(z)∈�n; [DCg(z)]i=DCgi(zi); PCg(z)∈�n; [PCg(z)]i=PCgi

(zi) (56)

Using the previous deKnitions and assuming a constant friction coe?cient , the equationsrelated with the possible states of each contact point i for the tangential case will be:

• Adhesion:

sti =0; |pti |6| Xpni| ⇒ |pti − rsti |6| Xpni

| (57)

• Slip:

sti �=0; pti sti¡0; |pti |=| Xpni| ⇒ |pti − rsti |¿| Xpni

| (58)

• Separation:

sti �=0; Xpni=0; pti =0 ⇒ C| Xpni

|= ∅ (59)

It is easy to show that the fulKllment of these inequalities can be guaranteed by satisfying

pti =PC| Xpni |(pti − rsti) (60)

where r ∈�+ is a dimensional penalty parameter, having for convenience the same value thatthe parameter used for the normal problem.The tangential problem could be written by organizing in a system of equations, the tan-

gential part of the equilibrium equation given by (36) or (47) and the contact conditions (60),particularized to the present step k.It is possible to use the material approach, with a consequent equation

�kt (s

kt ; p

kt )=

skt − ^k − 1

T� [#k−1 + gkt + Stn Xpkn + Sttpkt ]

pkt − PC| Xpkn|(pkt − rskt )

= 0 (61)



or a convective-local approximation with a very similar form given by

�kt (s

kt ; p

kt )=

skt − ˆk − 1

T� [Vk−1 + gkt + Stn Xpkn + Sttpkt ]

pkt − PC| Xpkn|(pkt − rskt )

= 0 (62)

where, as in the normal problem, equilibrium is imposed with the Krst equation, while thecontact conditions (applied point by point) are established by the second one.

3.2.3. Coupled problem. The solution of the coupled normal–tangential problem is much morecomplicated than solving each one separately, because normal tractions and thus the frictionlimit, are now unknowns. To formulate the problem we have used the direct combinationof Equations (53) with (61) or (62), by substituting into the tangential problem the Kxedfriction limit | Xpkn| at step k by another function of the unknown normal variables at the samestep. Alternatives for this limit are: −min(0; pkn + rTkn ) and −min(0; pkn), both of whichmake projections onto the valid region of the friction cone when a solution is possible to beachieved. If the Krst alternative is selected to obtain the normal pressure bound, there will betwo functions that can deKne the problem; one based on the material approximation

�k(Tkn ; skt ; pkn; pkt )=

Tkn − f kn − Sntpkt − Snnpkn

skt − ^k − 1T� [#

k−1 + gkt + Stnpkn + Sttpkt ]

−min(−pkn; rTkn )pkt − PC− min(0; pkn+rTkn )

(pkt − rskt )

=0 (63)

or another using the Convective-Local deKnition

�k(Tkn ; skt ; pkn; pkt )=

Tkn − f kn − Sntpkt − Snnpkn

skt − ˆk − 1T� [V



(pkt − rskt )

=0 (64)

If it is desired to introduce the global external loads applied to the roller as an inputparameter, by characterizing the problem, it is useful to modify the obtained formulationincluding these load values.The sum of normal and tangential loads (P(�), Q(�)) applied on the discretized contact

surface when the travelled distance is � can be calculated by integration over the contact area,

P(�k) =−ec∑i=1

{ne∑

j=1

[(∫YiNj dY

)pnj(�k)

]}(65)

Q(�k) =ec∑i=1

{ne∑

j=1

[(∫YiNj dY

)ptj(�k)

]}(66)



where ec is the number of elements, ne the number of nodes per element and Nj the elementshape function associated to node j. These expressions can also be written in vector form as

P(�k)= − fTpkn; Q(�k)= fTpkt (67)

and easily included in the general formulation. Observe also that external loads are calculatedby using normal and tangential tractions in the contact area without referring them to theglobal axes, and that this approach will be valid only for small contact areas.As from (67) two new equations have been formulated, it is necessary to consider as

unknowns other two variables which vary with distance: the creep �0(�), and the overlap�0(�). The initial normal separation and the creep vector can be expressed as

Tn0 = Tg − �0(�)e; ^= �0(�)e�; ˆ = �0(�)e (68)

Now the unknowns vector at the time step k, zk , is deKned as

zk =(Tkn ; skt ; pkn; pkt ; �0(�k); �0(�k)) (69)

and the coupled functions describing the problem using the material formulation can bewritten as

�k(zk)=

Tkn − Tg + �0(�k)e − gkn − Sntpkt − Snnpkn

skt − �0(�k)e� − 1T� [#



(pkt − rskt )

P(�k) + fTpkn

Q(�k)− fTpkt

=0 (70)

while if the convective-local formulation form is used, it will be

�k(zk)=

Tkn − Tg + �0(�k)e − gkn − Sntpkt − Snnpkn

skt − �0(�k)e − 1T� [V



(pkt − rskt )

P(�k) + fTpkn

Q(�k)− fTpkt

=0 (71)

both being B-di=erentiable functions, because of the non F-di=erentiable contact conditions,that involve non-linear directional derivatives. This is why it is convenient to operate withthe functions divided into two parts

�k(zk)=�kLD(z

k) +�kNLD(z

k) (72)



where �kNLD(zk) is the part presenting a non-linear directional derivative for some

points

�kNLD(z

k)=

0

0

−min(−pkn; rTkn )−PC− min(0; p k

n+rT kn )(pkt − rskt )

0

0

(73)

and �kLD(zk) is the rest, presenting a normal linear directional derivative for all its points.

In a similar way, �kNLD(zk) can be obtained by grouping the non-linear terms of �(zk),

�kLD(zk) being the linear part.The concept of B-di=erentiability is related with the non-linearity of the directional deriva-

tive. Only if a function F(z) is F-di=erentiable, its derivative at z in an arbitrary direction dis linear and satisKes the equation F′(z; d)=OF(z)d, where OF(z) is the Jacobian matrix.

4. SOLUTION PROCEDURE

To solve the problem, the generalized Newton’s Method with line search (GNMls) hasbeen used. GNMls is an extension of Newton’s Method for B-di=erentiable functions for-mulated by Pang [51]. This algorithm has also been used to solve static contact problems[39] (discretizing equilibrium equations with Knite element method) and rolling stationaryproblems [45] (with BEM), showing that the method is well suited for solving these kind ofproblems.

4.1. Generalized Newton’s method

Application of GNMls algorithm to solve our non-linear equations G(zk)= 0, G(zk) being aB-di=erentiable function, can be summarized in the following steps:

• Step 0. Start integration on the distance travelled (k index) with �k−1 = �0 and obtain z0

solving a quasi-static or steady rolling contact problem of the model loaded with P(�0)and Q(�0).

• Step 1. Use zk0 = zk−1 as the initial vector, P(�k) and Q(�k) as external loads. Let s, +, ,

and - be given scalars s¿0, +∈ (0; 1), ,∈ (0; 1=2), and -¿0 but small.• Step 2. Start iterations (j index) to Knd a solution for Kxed step k:• Step 3. Given zkj with G(zkj ) �= 0, obtain a direction dk

j solving the equation

G(zkj ) + BG(zkj)dkj = 0 (74)

where BG(zkj ) means B-derivative.



• Step 4. Let �j=+mjs obtain mj as the Krst non-negative integer for which

Z(zkj )−Z(zkj + +mjs dkj )¿−,+mjsZ′(zkj ; d

kj ) (75)

with

Z(zkj ) =12G(z

kj )TG(zkj ) (76)

Z′(zkj ; dkj ) = BZ(zkj )d

kj =G(z

kj )TBG(zkj )d

kj =−G(zkj )TG(zkj ) (77)

where Z(zkj ) is the merit function and Z′(zkj ; dkj ) its directional derivative.

• Step 5. Then, considering Equations (76) and (77), Equation (75) can be expressedalso as

Z(zkj + +mjsdkj )6(1− 2,+mjs)Z(zkj ) (78)

• Step 6. Set the new solution vector

zkj+1 = zkj + �j dk

j (79)

• Step 7. Terminate if Z(zkj+1)6-, otherwise return to Step 2 with zkj = zkj+1 and iterate untilconvergence is reached

• Step 8. Having found solution zk , increase the distance �k to �k+1, make k= k + 1 andreturn to Step 1 until the steady state regime is reached, i.e. zn � zn+1.

Often the solution of Step 3, the generalized Newton equation, could involve a big com-putational e=ort because of the non-linearity of BG(zkj )dk

j . In order to reduce this e=ort,some previous researchers solved contact problems accurately by avoiding the points wherethe non-linearity could occur, or neglecting the non-linear part of the derivative in thesepoints.In rolling problems, as in contact ones, the points to avoid are those where the in-

equalities formulated during the iterative solution process change to equalities. On thesepoints, or lines, the slopes of functions �k(zkj ), �

k(zkj ) are discontinuous, then their deriva-tives do not have the same value for all directions. Since it is unusual to have a trialsolution zkj exactly over one of these points during the iterative process, the convergenceis not usually a=ected if those points are avoided, or if some part of their derivative isneglected.The non-linear part of the generalized Newton equation is linearized and computed as

BG(zkj )dkj =(OGLD + @GNLD)dk

j (80)

where OGLD is the Jacobian of the LD part, and @GNLD is a pseudo-Jacobian of the NLDpart, computed by avoiding the non-linear points.



4.2. Material formulation, function �k(zk)

In the case of minimizing the function �k(zk) for Kxed step k, the linear part of the deriv-ative is

O�LD =

I 0 −Snn −Snt e 0

0 I − 1T�Stn − 1

T�Stt 0 −e�

0 0 0 0 0 0

0 0 0 I 0 0

0 0 fT 0 0 0

0 0 0 −fT 0 0

(81)

presenting no k dependence when the integration step size is maintained to be constant.The pseudo-Jacobian, @�NLD is obtained in a di=erent way depending on the state of each

point i at iteration j of time step k.

1. Contact restriction non-linear part:(�NLD(zkj )

)3=−min(−pk

n; rTkn )

• Case 1. −pkni¡rTk

ni

[(@�NLD(zkj ))3]i=[0; 0; 1; 0; 0; 0] (82)

• Case 2. −pkni= rTk

ni, (modiKed)

[(@�NLD(zkj ))3]i=[0; 0; 1; 0; 0; 0] (83)

• Case 3. −pkni¿rTk

ni

[(@�NLD(zkj ))3]i=[−r; 0; 0; 0; 0; 0] (84)

where ‘(modiKed)’ denotes the use of a simpliKed linear version of the directionalderivative, instead of its true non-linear expression.

2. Friction law restriction: (�NLD(zkj ))4 =−PC− min(0;pkn+rTkn )(pk

t − rskt )

• Case 1. pkni+ rTk

ni¿0

[(@�NLD(zkj ))4]i=[0; 0; 0; 0; 0; 0] (85)


ni¡0 and |pk

ti − rskti |¡−(pkni+ rTk

ni)

[(@�NLD(zkj ))4]i=[0; r; 0;−1; 0; 0] (86)


ni¡0 and |pk

ti − rskti |=−(pkni+ rTk

ni), (modiKed)

[(@�NLD(zkj ))4]i=[0; r; 0;−1; 0; 0] (87)




ni¡0 and |pk

ti − rskti |¿−(pkni+ rTk

ni)

[(@�NLD(zkj ))4]i=[r sgn(pkti − rskti ); 0; sgn(p

kti − rskti ); 0; 0; 0] (88)


ni=0 and |pk

ti − rskti |¿0, (modiKed)

[(@�NLD(zkj ))4]i=[0; 0; 0; 0; 0; 0] (89)


ni=0 and |pk

ti − rskti |=0, (modiKed)

[(@�NLD(zkj ))4]i=[0; 0; 0; 0; 0; 0] (90)

4.3. Convective-local formulation, function �k(zk)

Using function �k(zk), the linear part of the derivative ��LD is the same as O�LD, onlyreplacing e� with e.The pseudo-Jacobian @�NLD is obtained in a similar way, also depending on the state of

each point i.

4.4. Starting solution for contact problems

The contact boundary conditions of the cylinders when they are initially braked have to beobtained by solving a quasi-static contact process, and it is important to note that the sub-stitution of VA=VB=0 in the convective-local formulation (13) yields the same formulationproposed by Christensen in Reference [39] to solve the contact problem. That means we canalso solve this initial contact process with a small code modiKcation and using a su?cientnumber of time steps to model energy dissipation by friction correctly. The results of thiscontact problem are later organized in vector form

z0 = (Tn; 0; pn; pt ; �0; 0)T (91)

to start time integration with a feasible solution for the numerical scheme proposed forrolling.Pang’s algorithm has been used to solve each of the non-linear problems for subsequent

time steps, starting to iterate with the solution obtained for the last time step, i.e. zk0 = zk−1.

Owing to the small di=erences between the correlative solutions, when the time step is smallenough, the iteration process usually converges rapidly to a good solution.

5. IMPLEMENTATION CONSIDERATIONS

Two di=erent element types have been implemented for numerical experiments, quadraticcontinuous (QCE) and quadratic discontinuous (QDE). The discontinuous elements use twodi=erent set of nodes for geometrical and variables approximation, the latter being completelyinside the element and producing a non-zero shape function value in the element edges. Bothtypes proved their good behaviour in Reference [46] for 2D steady-state rolling problems, butthis will not exactly be the case for transient ones.



Due to the necessity of a C 0 continuous tangential displacement approximation all alongthe contact area for the material formulation (17), only QCE will be used. If QDE ele-ments are used in combination with this approach, the convergence is not guaranteed. That isbecause tangential displacements are discontinuous between elements and, depending on therelation between element lengths, time step size and rolling velocity, these displacements onthe element’s interfaces are used or not by the solution algorithm.When an approximation based on the convective-local formulation (18) is used, derivatives

of tangential displacements at nodal positions have to be obtained. This deKned nodal deriva-tive condition is automatically satisKed by QDEs with the shape functions and their use isdirect. However, the quadratic interpolation of the slip velocity makes small incompatibilitiesappear in the adhesion-slip transitions. This e=ect is not important for steady-state problemsbecause incompatibility is Kxed in time and the problem is not incremental, but for transientproblems, incompatibility travels all along the contact area causing divergence of the GNMlsalgorithm.The problem of adhesion-slip incompatibilities can be completely removed by adopting

a linear interpolation of tangential displacements between nodes. That yields a piecewiseconstant approach for the last term of Equation (18) like (20), an expression that can beused with QDE and QCE because it is uniquely deKned at node positions taking into accountwhere the particle comes from. It is also important to note that using (20), material andconvective-local formulations are the same when used with QCE in regular meshes and anintegration step equal to node to node distance.It was pointed out in Section 2, that the mesh has to be reKned on the contact area to

represent the stress concentration accurately on its surroundings, but we also have to takeinto account the transient character of the problem in the meshing criteria. If we call thecharacteristic time variation of the magnitudes c, and the required element length Le, torepresent result variations at the contact area correctly, Le has to verify that Le�V c. Afterthe element size decision, an integration time step has to be chosen. Numerical experimentswith material and convective-local formulations reveal that good results are obtained whenthe time step is maintained inside the interval (Le=86T�6Le=2).

6. RESULTS

The examples of problems presented are classiKed into two groups, similar and dissimilar,taking into account the similarity of the cylinders’ materials. Similar problems are easierto solve because under the half-space assumption, there exists uncoupling between normaland tangential contact tractions, i.e. Snt and Stn matrices of equations ((26) and (29)) be-come equal to zero. Logically, no cross in>uence between normal and tangential variablesimproves the GNMls convergence. As a general rule, it is observed that dissimilar prob-lems need double the number of iterations to obtain a feasible solution needed by simi-lar ones. After some numerical experiments, the optimal parameters for Pang’s algorithmwere: s=1, +=0:9, ,=0:1, -=10−15, and the penalty parameter r=0:1 N=mm, for all theexamples.We will restrict ourselves to the cases where loads are applied through rigid shafts and non-

zero external boundary conditions, such as the action of an external brake in the boundaryare not considered, i.e. in Equation (24) g�ne= g�te=0. The possibility of varying step size



in the incremental algorithm has not been implemented, and it is maintained to be constantduring the whole integration process.In all the following examples, the in>uence coe?cients representing the equilibrium equation

were computed by BEM. The cylinder’s geometry is approached by quadratic isoparametricelements.Two examples are compared with available results from Kalker (based on the half-space

approach and Hertz normal pressure assumption) found in References [13; 53].

6.1. Similar problem

The Krst transient rolling contact problem consists of two rotating cylinders of identicalmaterial. This problem does not present a normal–tangential coupling under the half-spacehypothesis, and Hertz normal pressure solution is the Krst-order approximation for normaltractions.The friction coe?cient between surfaces is =0:1 and two cases are considered, both

starting from a static contact situation but with di=erent applied forces.

6.1.1. Constant external loads. Let us suppose Krst that the two cylinders are initially brakedwith applied external loads Q=P=0:75. The initial tractions of this problem are knownanalytically: the Hertz distribution for normal tractions, and the Cattaneo [54] distributionfor tangential tractions. When the brake is released, transient phenomena start, and end withanother analytical solution obtained by Carter [2] for similar cylinders in steady state rollingmotion.The boundary element mesh used is that which is represented in Section 2.3 with 40 QCEs

evenly distributed in the contact area. The time integration step is the half-length of a contactelement T�=Le=2.In Figure 4, the BEM tangential tractions are represented and compared with the solution

obtained by Kalker for di=erent distances travelled. Agreement between the two results isvery good but with some small di=erences.To complete the problem description, we have also represented the slip velocity. For this

variable, the comparison is not possible because there are no available results from otherauthors.The normalization factors used in Figure 4 are: |pnH|, the friction coe?cient times the ab-

solute value of the maximum normal traction for the equivalent Hertz problem; and bH =2aH,the length of the contact area.

6.1.2. Varying external loads. The previous example is now revisited but by applying di=er-ent loads. Let us suppose that we start with the contacting cylinders not clamped but stoppedwith a net tangential force equal to zero. The load Q=P=0 is then linearly increased un-til it reaches 0:75 when the distance travelled is �=aH =8=3, keeping the normal load Pconstant. In addition, a friction coe?cient of 0:1, time step T�=Le=2 and the same dis-cretization of the previous example, are used.The tangential tractions obtained are represented in Figure 5 together with the dimension-

less relative slip velocity. The Krst curve corresponds to the initial time step (�=aH =1=30).Logically, the Knal steady state coincides with that obtained in the previous example becausethe Knal applied forces are the same.



Figure 4. From Cattaneo to Carter, Q=P=0:75. Similar cylinders.

6.2. Dissimilar problem

If the rolling cylinders have di=erent properties (steel–aluminium for example), the problembecomes more complicated because several adhesion-slip transition points can appear. Thecombined elastic constant + (called Dundurs’ constant)

+=[(1− 21A)(1 + 1A)]=EA − [(1− 21B)(1 + 1B)]=EB

(1− 12A)=EA + (1− 12B)=EB(92)

quantiKes the similarity between the properties of both materials, and together with the frictioncoe?cient, applied loads Q=P, and the radii of the cylinders, are the parameters governingthe distribution of the slip and adhesion zones.

6.2.1. Constant external loads. Let us now consider the case of two dissimilar cylindersstopped and in contact without applied net tangential force that start to roll with Q=P keptto zero. The Hertz analytical solution is now not strictly valid due to the presence of friction,but there is an approach by Goodman [55] based on neglecting of the in>uence of tangentialtractions on normal pressure, which is going to be used as an initial condition to start timeintegration.



Figure 5. Ramp load Q=P from 0 to 0:75. Similar cylinders.

This example was proposed by Kalker in Reference [13] to check his formulation forsolving dissimilar roller problems, assuming a constant Hertz normal pressure and free rolling(zero net tangential load).To solve the problem, we keep the same mesh, changing the material properties to obtain

a value of Dundurs’ constant +=1 and a friction coe?cient =0:3. A constant value of theload parameter Q=P=0 is considered and, for comparison to be made with literature results,normal–tangential interaction is deactivated making Snt =Stn=0 in the formulation. The timestep used was T�=Le=4.The results obtained for free rolling of dissimilar cylinders with normal Hertz tractions, are

compared in Figure 6 with Bentall solution [3], the transient e=ects disappear after rolling thecontact area 1.8 times.The results presented next in Figure 7 are obtained for the same problem but now consid-

ering the coupling e=ects between normal and tangential tractions. The starting solution hasno analytical expression and has to be obtained numerically using the scheme proposed inSection 4.4; 40 load steps have been used to compute the contact tractions due to the linearapplication of the normal load.It can be observed that a peak is obtained as a maximum in the normal tractions, and that

it disappears through the trailing edge when cylinders have rolled the contact area 0:6 times.



Figure 6. From Goodman to Bentall Q=P=0. Material formulation.Dissimilar cylinders. Uncoupled case.

The steady state solution is very close to that obtained by neglecting coupling e=ects, butsome di=erences are observed during the transient history.

6.2.2. Varying external loads. The last example consists of two dissimilar rotating cylindersin steady-state free rolling motion. Free rolling means that there is no net tangential loadapplied to the rollers, while being maintained in contact with a constant normal load XP.The transient phenomena studied appear due to the sudden action of a harmonic variation

on the normal load given by P= XP=1 + 0:2 sin(�3=aH); where aH is half the length of thecontact area obtained with load XP in the equivalent Hertz problem.The same model has been used, but discretization of the contact area has been decreased

to 21 QCEs in order to prove their good behaviour with a lower number of elements as well.The selected formulation is the convective-local and the used material constants are, Dundurs’constant value +=0:288 combined with a friction coe?cient =0:1. The time step used wasT�=0:637Le=2.The results for the Krst load cycle are presented in Figure 8 for di=erent travelled distances.

The initial solution used for this example corresponds to the steady-state rolling contact prob-lem obtained with constant normal load, P= XP=1. This initial problem was solved by Nowelland Hills [5] but without considering the full coupling of tractions.



Figure 7. From Goodman to Bentall Q=P=0. Material formulation.Dissimilar cylinders. Coupled case.

It can be observed that the contact area length varies in phase with the normal load andthat two-one-two adhesion zones appear in an alternating sequence.This Krst cycle is completely transient, giving place to a periodic quasi-steady state com-

posed by the last instant represented in Figure 8, followed by those represented in Figure 9.There are two main di=erences with the constant case produced by the harmonic normal load:bigger slip zones and higher slip velocities; both will increase roller wear.

6.3. Norm–tang vs. coupled iteration

The coupled problem, Equations (70) and (71), can also be solved for each time stepusing norm–tang (N–T) iteration, i.e. a chained solution of normal and tangential problems,obtaining exactly the same results.We have used the GNMls method to solve the two individual problems for each N–T

iteration, Equation (53) for the normal part, and Equation (61) or (62) for the tangentialpart, until convergence was reached. The number of N–T iterations to obtain the solution ishighly dependent on the tolerance Kxed for the zero of the objective function, i.e. -. It hasbeen observed that for the most di?cult problems (coupled and dissimilar ones), each N–Titeration decreases the objective function in one order of magnitude, making values -¡10−15

possible.



Figure 8. Harmonic normal force on two dissimilar rotating cylinders.Convective-local formulation. Transient period.

Figure 9. Harmonic normal force on two dissimilar rotating cylinders.Convective-local formulation. Periodic steps.

The N–T algorithm, in comparison with the coupled scheme, increases the number ofiterations needed to obtain the solution. However, the overall computing time is not drasticallya=ected because the size of the system to solve is halved.



For the particular case of the Goodman to Bentall dissimilar problem, the N–T algorithmalmost triples the computing time compared with the Coupled scheme. Statistics for thisparticular problem reveal an average of 6 N–T iterations to obtain the solution for one timestep. Inside one of these iterations, the GNMls algorithm employs 3 and 12 iterations to solvethe Normal and Tangential parts, respectively, compared with the 6 GNMls iterations neededby the Coupled scheme to achieve the same solution.

7. CONCLUSIONS

The transient two-dimensional rolling contact problem between cylinders can be solved usinga new numerical technology based on the use of boundary element method and mathematicalprogramming techniques.In this new technology BEM has been used to compute the constant in>uence coe?cients

based on true cylinder geometry, not on the half-space approximation, while MPT was usedto model the contact conditions and to solve the highly non-linear problem associated withtransient rolling at each time step.The algorithm and proposed formulations prove to be very robust and e?cient for 2D,

similar and dissimilar transient problems. It simpliKes the procedure to obtain the solutionof a transient rolling problem, and minimizes physical knowledge of the particular problemnecessary to control the convergence of the non-linear iteration process, being an e=ectivealternative to other algorithms.New solutions for old and new problems have been computed using this new algorithm,

showing its versatility and accuracy for all the cases.In conclusion, the proposed methodology is a new numerical tool that permits a general

and accurate solution of transient rolling problems.

ACKNOWLEDGEMENT

This work was funded by Spain’s Ministerio de Ciencia y Tecnolog5ia, in the framework of ProjectDPI2000-1642.

REFERENCES

1. Reynolds O. On rolling friction. Philosophical Transactions of the Royal Society of London 1876; 1:155–174.2. Carter FW. On the action of a locomotive driving wheel. Proceedings of the Royal Society of London, Series

A 1926; 112:151–157.3. Bentall RH, Johnson KL. Slip in the rolling contact of two dissimilar elastic rollers. International Journal of

Mechanical Sciences 1967; 9:389–404.4. Bentall RH, Johnson KL. An elastic strip in plane rolling contact. International Journal of Mechanical Sciences1968; 10:637–663.

5. Nowell D, Hills DA. Tractive rolling of dissimilar elastic cylinders. International Journal of MechanicalSciences 1987; 30(6):427–439.

6. Nowell D, Hills DA. Tractive rolling of tyred cylinders. International Journal of Mechanical Sciences 1987;30(12):945–957.

7. Kalker JJ. On the rolling contact of two elastic bodies in presence of dry friction. PhD Thesis, Delft TechnicalUniversity, Netherlands, 1967.

8. Kalker JJ. A minimum principle for the law of dry friction, with applications to elastic cylinders in rollingcontact. Part 1: Fundamentals. application to steady rolling. Journal of Applied Mechanics 1971; 38:875–880.

9. Kalker JJ. The mechanics of the contact between deformable bodies. In Proceedings of IUTAM Symposium,Pater AD, Kalker JJ (eds). Delft University Press: Delft, 1975.



10. Kalker JJ. Contact mechanics algorithms. Communications in Applied Numerical Methods 1988; 4(20):25–32.11. Kalker JJ. Three-Dimensional Elastic Bodies in Rolling Contact. Kluwer Academic Press: Dordrecht, 1990.12. Kalker JJ. Transient phenomena in two elastic cylinders rolling over each other with dry friction. Journal of

Applied Mechanics 1970; 37:677–688.13. Kalker JJ. A minimun principle for the law of dry friction, with applications to elastic cylinders in rolling

contact. Part 2: application to nonsteadily rolling elastic cylinders. Journal of Applied Mechanics 1971; 38:881–887.

14. Knothe K, Gross-Thebing A. Derivation of frecuency dependent creep coe?cients based on an elastic half-spacemodel. Vehicle System Dynamics 1986; 15:133–153.

15. Gross-Thebing A. Frequency dependent creep coe?cients for three-dimensional rolling contact problems. VehicleSystem Dynamics 1989; 18:359–374.

16. Nielsen JB. New developments in the theory of wheel=rail contact mechanics. PhD Thesis, IMM Lyngby, 1998.17. Anderson T. The boundary element method applied to two-dimensional contact problems with friction. In

Boundary Element Methods, Brebbia CA (ed.). Springer: Berlin, 1982.18. Paris F, Garrido JA. An incremental procedure for friction contact problems with the boundary element method.

Engineering Analysis with Boundary Elements 1989; 6:202–213.19. Abascal R, Dom5inguez J. Dynamic response of two-dimensional >exible foundations allowed to uplift.

Computers and Geomechanics 1990; 9:113–129.20. Abascal R. 2D Transient dynamic friction contact problems. I numerical analysis. Engineering Analysis with

Boundary Elements 1995; 16:227–233.21. Abascal R. 2D Transient dynamic friction contact problems. II applications to soil–structure interaction problems.

Engineering Analysis with Boundary Elements 1995; 16:227–233.22. Wang G, Knothe K. Stress analysis for rolling contact between two viscoelastic cylinders. Transactions of

ASME 1993; 60:310–317.23. Kong XA, Wang Q. A boundary element approach for rolling contact of viscoelastic bodies with friction.

Computers and Structures 1995; 54(3):405–413.24. Kalker JJ, Van Raden Y. A minimum principle for the frictionless elastic contact with application to

non-hertzian half-space contact problems. Journal of Engineering Mathematics 1972; 6(2):193–206.25. Panagiotopoulos PD. Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy

Functions. BirkhQauser: Basel, 1985.26. Klarbring A. A mathematical programming approach to three-dimensional contact problems with friction.

Computer Methods in Applied Mechanics and Engineering 1986; 58:175–200.27. Klarbring A. Contact problems with friction by linear complentarity. In Unilateral Problems in Structural

Analysis II, CSIM Courses and Lectures, delPiero G, Maceri F (eds). Springer: New York, 1987; 197–219.28. Kwak BN, Lee SS. A complementarity problem formulation for two dimensional friction contact problems.

Computers and Structures 1987; 28(4):469–480.29. Gakwaya A, Lambert D, Cardou A. A boundary element and mathematical programming approach for frictional

contact problems. Computers and Structures 1990; 42(3):341–353.30. De Saxe G, Feng ZW. New inequality and functional for contact with friction: the implicit standard approach.

Mechanical Structures and Machines 301–325; 19:1991.31. Antes H, Panagiotopoulos PD. The Boundary Integral Approach to Static and Dynamic Contact Problems:

Equality and Inequality Methods. BirkhQauser: Basel, 1992.32. Conry TF, Seireg A. A mathematical programming method for design of elastic bodies in contact. Journal of

Applied Mechanics, ASME 1971; 38:387–392.33. Wriggers P, Sim5o JC, Taylor RL. Penalty and augmented lagragian formulations for contact problems.

Proceedings of NUMETA’85 Conference, Swansea, U.K., 1985; 97–106.34. Alart P, Curnier A. A mixed formulation for frictional contact problems prone to newton like solution methods.

Computers in Applied Mechanics and Engineering 1991; 32:353–475.35. Sim5o JC, Laursen TA. An augmented lagragian treatment of contact problems involving friction. Computers

and Structures 1992; 42:97–116.36. Zavarise G, Wriggers P, Schre>er BA. On augmented lagragian algorithms for thermomechanical contact

problems with friction. International Journal for Numerical Methods in Engineering 1995; 38(2929–2949).37. Wriggers P. Finite element algoritms for contact problems. Archives of Computational Methods in Engineering

1995; 2:1–49.38. StrQomberg N. An augmented lagragian method for fretting problems. European Journal of Mechanics A=Solids

1997; 16(4):573–593.39. Christensen PW. Algorithms for frictional contact problems based on mathematical programming. Master’s

Thesis, Thesis No. 628, Division of Mechanics, Department of Mechanical Engineering, LinkQoping University,Sweden, 1997.

40. Christensen PW, Klarbring A, Pang JS, StrQomberg N. Formulation and comparison of algorithms for frictionalcontact problems. International Journal for Numerical Methods in Engineering 1998; 42:145–173.



41. Gonz5alez JA, Abascal R. Using the boundary element method to solve rolling contact problems. EngineeringAnalysis in Boundary Elements 1998; 21:392–395.

42. Abascal R, Gonz5alez JA. Solving rolling contact problems using BE and mathematical programming algorithms.In Advances in Computational Structural Mechanics, Topping BHV (ed.). Civil-Comp. Press, Edinburgh, U.K.,1998; 9–18.

43. Gonz5alez JA, Abascal R. Solving rolling contact problems using boundary element method and mathematicalprogramming algorithms. In Boundary Element Techniques, Aliabadi MH (ed.). Queen Mary and WestKeldCollege University, London, U.K., 1999; 115–128.

44. Gonz5alez JA, Abascal R. Solving 2D rolling problems using the norm-tang iteration and mathematicalprogramming. Computers and Structures 2000; 78(1–3):149–160.

45. Gonz5alez JA, Abascal R. Solving rolling contact problems using boundary element method and mathematicalprogramming algorithms. Computational Models in Engineering Science 2000; 1(3):141–150.

46. Gonz5alez JA, Abascal R. An algorithm to solve coupled 2D rolling contact problems. International Journal forNumerical Methods in Engineering 2000; 49:1143–1167.

47. Wang G, Knothe K. The in>uence of inertia forces on steady-state rolling contact. Acta Mechanica 1989;79:221–232.

48. Braat GFM. Theory and experiments on layered, viscoelastic cylinders in rolling contact. PhD Thesis, DelftTechnical University, Netherlands, 1993.

49. Duvaut G, Lions JL. Les Inquations en Mecanique et en Physique. Dunod: Paris, 1972.50. Brebbia CA, Dominguez J. Boundary Elements: An Introductory Course (2nd edn). Computational Mechanics

Publications: Southampton, 1992.51. Pang JS. Newton’s method for b-di=erentiable equations. Mathematics of Operations Research 1990;

15(2):311–341.52. Klarbring A. Mathematical programming in contact problems. In Computational Methods in Contact Mechanics,

Aliabadi MH, Brebbia CA (eds), chapter 7. Computational Mechanics Publications: Southampton, 1993.53. Johnson K.L. Contact Mechanics. Cambridge University Press: Cambridge, 1985.54. Cattaneo C. Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Rendiconti dell’ Accademia

Nazionale dei Lincei 1938; 27:342–348.55. Goodman LE. Contact stress analysis of normally loaded rough spheres. Transactions of ASME, Series E,

Journal of Applied Mechanics 1962; 29:515–522.


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