an algorithm to solve coupled 2d rolling contact problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2000; 49:1143–1167

An algorithm to solve coupled 2D rolling contact problems

Jos�e A. Gonz�alez and Ram�on Abascal∗;†

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

SUMMARY

This work presents a new approach to the steady-state rolling contact problem for two-dimensional elasticbodies, with and without force transmission. The problem solution is achieved by minimizing a generalfunction representing the equilibrium equation and the contact restrictions. The boundary element method isused to compute the elastic in uence coe�cients of the surface points involved in the contact (equilibriumequations); while the contact conditions are represented with the help of variational inequalities and projectionfunctions. Finally, the minimization problem is solved by the Generalized Newton’s Method with line search.Four classic rolling problems are also solved and commented on. Copyright ? 2000 John Wiley & Sons,Ltd.

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

1. INTRODUCTION

The contact with rolling has occupied an important place in the technical literature relative tomechanical engineering problems. It has been due to the necessity of having design approachesfor mechanical elements of very extended use, as components of rotating machines, bearings,rollers, etc., as well as for wheels and rolling paths, either for cars, transportation and elevationmachines, or trains.The mechanical study of the rolling problems with creep has its beginnings in the works of

Reynolds [1] in 1876, and Carter [2] in 1926, where two-dimensional problems are studied. It canbe also highlighted the pioneer work of Kalker [3], and the papers of Bentall and Johnson [4; 5],and Nowell and Hills [6; 7].The present formulation of rolling problems springs from the works of Kalker [8–10], summa-

rized in his book [11]. In the �eld of the �nite element method (FEM) the work of Singh andPaul [12], Batra [13] could be cited, as well as that of Padovan and Zeid [14], and the prominentworks of Oden and their collaborators in the �eld of the variational inequalities and FEM, mostof which have been collected in the book of Kikuchi and Oden [15].

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

†E-mail: [email protected]

Contract=grant sponsor: DGICYT, Spain (partially); contract=grant number(s): PB 96-1380 and PB 96-1322.

Received 22 October 1999Copyright ? 2000 John Wiley & Sons, Ltd. Revised 2 February 2000

1144 J. A. GONZ �ALEZ AND R. ABASCAL

Although one can �nd a wide number of papers on contact problems related with the boundaryelement method (BEM), among them the works of Anderson [16], Paris and Garrido [17], andAbascal [18–20], there are few relevant works that incorporate rolling, but among these the worksof Wang and Knothe [21], Kong and Wang [22] and Kalker and Van Randen [23] could behighlighted.Mathematical programming techniques (MPT) are very attractive for solving this type of prob-

lems because they allow a more general point of view, and they have very valuable numericaltools. The references using MPT to solve the problems of quasi-static contact are abundant, es-pecially the book of Panagiotopoulos [24], the articles of Klarbring [25; 26], Kwak and Lee [27],Gakwaya et al. [28], De Saxe and Feng [29], the book of Antes and Panagiotopoulos [30], andthe paper of Kong et al. [22].Conry and Seireg [31], Wriggers [32], Alart and Curnier [33], and Sim�o and Laursen [34], are

the precursors of the use of the augmented Lagrangian formulation. This formulation has alsorecently been used in the papers of Wriggers [35; 36], and Str�omberg [37]. Later, Christensen[38; 39] compared the results and convergence obtained solving problems with several mathemat-ical programming methods, enlarging the study of the formulation.The applications of these methods to the case of rolling contact are scarce in literature although

the works of Kalker [11], Braat [40], and those presented by the authors can be cited [41–44].This work presents a new approach to the steady-state rolling contact problem for two-dime-

nsional elastic bodies, with and without force transmission. The problem solution will be achievedby minimizing a general function representing the equilibrium equation and the contact restrictions.The boundary element method will be used to compute the elastic in uence coe�cients of thesurface points involved in the contact; while the contact conditions will be represented with thehelp of variational inequalities and projection functions, the minimization problem �nally beingsolved by using the generalized Newton’s method with line search. Finally, the algorithm presentedis checked by solving several classical rolling problems.

2. THE 2D ROLLING CONTACT PROBLEM

The rolling between two cylinders of parallel axes is the scope of this work. The study is carriedout from a two-dimensional point of view; excluding the displacements in the direction of thecylinder axis and its relative spin from the analysis, assimilating the problem to plane strain.It is assumed that friction between the cylinders obeys the Coulomb law, where normal and

tangential tractions in the slip area are related by a constant friction coe�cient. Adopting thisapproach, each pair of points belonging to each one of the cylinders in rolling (A and B in Figure 1)can be in any one of the following states: separation, adhesion or slip.The mathematical expressions of the contact conditions, using Kalker’s notation [11], can be

classi�ed into two groups: normal or tangential. It is possible to write each condition as a com-plementary relation or as a variational inequality (both being equivalent expressions):

• Normal direction: The Signorini conditions for unilateral contact in the form of a comple-mentarity relation could be expressed as

Complementarity form: �n¿0; pn60; pTn �n=0

Variational inequality: pn ∈R−; �n(pn − p∗n )¿0; ∀p∗n ∈R−(1)

being R− ≡{x∈R : x60}.

Copyright ? 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 49:1143–1167

2D ROLLING CONTACT PROBLEMS 1145

Figure 1. Rolling cylinders: contact zone.

• Tangential direction: For tangential direction, the ful�llment of friction law is guaranteed byComplementarity form: |pt |6�|pn|; �¿0; st =−�pt; st(|pt | − �|pn|)= 0Variational inequality: pt ∈C�|pn|; st(pt − p∗t )60; ∀p∗t ∈C�|pn|

(2)

where Cg is a closed interval of radius g centred in R; Cg≡{z ∈R : |z|6g}.In this way, the possible states in each pair of contact points are characterized by

Stick: pA;Bn 60; pAn =pBn ; pAt =p

Bt ; �n=0; st =0

Slip:

{pA;Bn 60; pAn =p

Bn ; |pA;Bt |= �|pA;Bn |

�n=0; sgn(st)=−sgn(pA;Bt )

Separation: pA;Bn =0; pA;Bt =0; �n¿0

(3)

where � is the friction coe�cient, p�n and p�t are the normal and tangential tractions of the

contact 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.

The separation in the normal direction, �n, will be

�n= �no − (uAn + uBn ) (4)

where �no is the initial separation between the cylinder contact points, �no= �g − �0, being �gthe geometric initial separation between two contact points, and �0 the sum of two rigid bodydisplacements externally imposed on each cylinder, usually as an overlap between them.For the rolling cylinders �g can be approximated by �g= 1

2x2(1=RA + 1=RB), R� being each one

of the cylinder radius, x the Eulerian co-ordinate in the contact area direction which is used toposition each pair of points at each time, relative to a rigid body position of the cylinders.The relative tangential slip velocity between two points of the cylinders A and B, is de�ned as

st = �̇t =d�t(�; �)d�

(5)



where � is the time variable, �= �(x; �) is the cartesian co-ordinate of each point relative to the�xed axes, and �t represents the tangential separation, given by �t =(�A − �B) + (uAt + uBt ).Substitution into the slip velocity expression, Equation (5), yields

st =(VA − VB) + (VAuAt; x + VBuBt; x) + (uAt; � + uBt; �) (6)

where V� is the rigid body speed in the x-direction (coincident with � direction).When the steady-state regime is reached, the time variations disappear and, as Kalker shows

[11], Equation (6) can be approximated by

st = |V | [�0 + sgn(V ) (uAt; x + uBt; x)] (7)

where V = 12

(VA + VB

), is the mean speed of rolling, and �0 =

(VA − VB) =|V |, the creep, or

normalized rigid body relative slip velocity.In this work the boundary displacement derivatives were calculated using a forward �nite di�er-

ences scheme (backward with respect to the true displacements of the surface points). The speedof the tangential slip for a pair of points i, located at co-ordinate xi, under the hypothesis ofsteady-state rolling, can be expressed as

sti = |V |{�0 + sgn(V )

[uAt (xi+1) + u

Bt (xi+1)

hi− uAt (xi) + u

Bt (xi)

hi

]}(8)

where hi denotes the distance between two adjacent points in the rolling direction.Most of the cylinders used for the examples were discretized by using the classical approximation

to contact between cylinders in semi-analytical methods, i.e. the half-space approximation. Themain hypothesis of this approximation is to assume very small contact zone length in comparisonwith the cylinders radius (2a.R�). Under this hypothesis the in uence coe�cients representingthe elastic equations of the cylinders are approximated by those computed using a half-spacecontaining a potential contact zone (Figure 2); and the initial normal separation between twopoints is calculated assuming the real cylinder’s geometry.This hypothesis, not strictly necessary in the present formulation, is a �rst approach to the

problem of stationary rolling from the BEM point of view, and thus permits a comparison of theresults with those existing in the literature based on similar approaches. We should not forget,however, that the proposed methodology allows the study of curved boundaries without additionale�ort, owing to the numerical power of the BEM. To this end, an example of contact rollingbetween two cylinders modelled with their true geometry, not as half-spaces, is included. Also,the problem of the rolling between a tyred cylinder (with a thin and soft external layer) andanother cylinder, will be solved discretizing the thin layer and prescribing the problem boundaryconditions, in contrast with the classical methods that need the analytic expressions of the in uencecoe�cients (when they are available).

3. PROBLEM FORMULATION

The rolling contact is a highly non-linear and path-dependent problem, the source of this non-linearity being the boundary conditions used to represent the friction law. In the steady-stateregime, time dependence disappears and the process is somewhat simpler becoming independentof the solution path.



Figure 2. Half-space approximation: boundary element mesh.

One of the most used iterative schemes to solve the contact problems is denominated PANA[30] that was adopted by Kalker [10] for the case of 2D rolling. This process consists of dividingthe problem into two chained subproblems NORM and TANG, where their names make referenceto the kind of tractions (normal or tangential) which are the unknowns, while the others areassumed to be equal to those of the last iteration. The solution then is obtained by multipleiterations NORM-TANG-NORM: : : until convergence is achieved. Kalker also used an algorithmcalled KOMBI [11], which automatically solves the NORM and TANG problems simultaneously,but with an iteration control that needs some physical knowledge of the problem.Because this iteration method might not be valid for complex problems, we present a more

general solution procedure. Our algorithm will solve the problem in a combined way, withoutseparation in the normal and tangential parts, or a friction limit estimation, and with a fullyautomated iteration control.

3.1. The elastic equations

The discretization process and the application of the BEM, provides us with the elastic equationsof each cylinder

H�u�=G�p�; �=A;B (9)

Grouping the BEM equations of the two cylinders, and condensing all the variables not associatedwith the contact areas (

u�nu�t

)=

[S�nn S�ntS�tn S�tt

](p�np�t

)+

(g�neg�te

)(10)

where u�n; u�t and p

�n; p

�t are the vectors that contain the normal and tangential displacements and

tractions on the contact area, g�ne; g�te are the normal and tangential displacements associated with

the external boundary conditions, S�rs is a matrix whose columns represent the contact nodes



displacements in the r-direction due to the application of a unit traction in the s-direction at node i,maintaining the rest of the boundary conditions and tractions equal to zero.Substituting Equation (10) in the vectorial version of the normal separation Equation (4), we

obtain one of the departure expressions of the algorithm

Tn= Tn0 − (gAne + gBne)− (SAnn + SBnn)pn − (SAnt + SBnt)pt (11)

that in compressed form could be written as

Tn= dn + Snnpn + Sntpt (12)

where

dn = Tno + gn; gn = −(gAne + gBne)Snn = −(SAnn + SBnn); Snt = −(SAnt + SBnt)

(13)

with

pn= pAn = pBn ; pt = pAt = p

Bt (14)

The matrix form of the slip velocity of each pair of points is obtained applying the T operator tothe tangential displacements vector, providing us with the �nite di�erence approach of its tangentialderivative. If Equations (8) are organized in matrix form, we can write

st = ^+ T(uAt + u

Bt

)(15)

where the constant |V | has been included in the st vector (it is important to keep in mind that ifV is negative the st sign will not change), and the last equation has been modi�ed to obtain apositive-de�nite matrix T (the last point of the contact area, point n, is supposed that it is alwaysin separation, pnn =ptn =0), writing the st and ^ vectors in the form

st =

st1 =|V |st2 =|V |st3 =|V |· · ·

stn−1 =|V |(uAtn + u

Btn

)

; ^=

�0�0�0· · ·�00

(16)



and the matrix T as

T=sgn(V )

− 1h1

1h1

0 · · · 0 0

0 − 1h2

1h2

· · · 0 0

0 0 − 1h3

· · · 0 0

· · · · · · · · · · · · · · · · · ·

0 0 0 · · · − 1hn−1

1hn−1

0 0 0 · · · 0 1

(17)

Substituting the expression of the tangential displacements in Equation (15) �nds that

st = ^+ T(gAte + g

Bte

)+ T

(SAtn + S

Btn

)pn + T

(SAtt + S

Btt

)pt (18)

or in condensed form

st = ^∗ + Btnpn + Bttpt (19)

where

^∗ = ^+ gt ; gt = T(gAte + g

Bte

)Btn = T

(SAtn + S

Btn

); Btt = T

(SAtt + S

Btt

) (20)

In this way, after the resolution of the elastic problem using BEM and later condensation, thelinear relationship between the normal separation and the slip velocity can be obtained, (Tn; st), asthe normal and tangential tractions in the contact area (pn; pt):(

Tnst

)=(dn^∗)+[Snn SntBtn Btt

](pnpt

)(21)

This equation together with the contact conditions expressed by Equations (1) and (2), for allthe pairs of contact points, complete the description of the problem.

3.2. The contact constraints

Alart and Curnier [33] developed a mixed penalty–duality formulation of the frictional contactproblem inspired by the augmented Lagrangian method to treat its multivalued aspects. Theyderived an unsymmetrical operator using a quasi-augmented Lagrangian formulation (to palliatethe absence of a genuine one) showed its properties, and established a necessary and su�cientcondition on the friction coe�cient that guarantees the uniqueness for the solution of curved,discrete, small slip contacts, and the convergence of Newton’s method.Inspired by that work, Christensen [38] proposes a variation of it, giving place to a system

of B-di�erentiable‡ equations, that allows him to use the convergence theorems and the powerful

‡ The notion of B-di�erentiability was introduced by Robinson [47], its concept being related to the non-linearity of thedirectional derivative. In the cited reference, Robinson studied and derived many properties of B-di�erentiable functions.See also the work of Pang [45].



Newton’s algorithm developed by Pang [45] for this kind of equations, formulated �rst for thecontact problem by Klarbring [46].In this work we compute the discretised elastic equations of rollers using the boundary element

method, and following the cited works we formulate the contact conditions in a quasi-Lagrangianway, reducing the solution of the problem to obtain the zero of a vectorial function.

3.2.1. Normal problem If the tangential tractions are assumed to be well known (pt = �pt), theproblem could be considered exclusively in function of the normal variables, obtaining the denom-inated Normal problem.The possible states for each pair of points according to the normal direction are two: contact

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

• Contact:�ni =0; pni60⇒pni + r�ni60 (22)

• Separation:�ni¿0; pni =0⇒pni + r�ni¿0 (23)

where r¿0 is a penalty parameter.

These constraints can be summarised in the vectorial equation:

pn= min(0; pn + rTn) (24)

an equality that will only be veri�ed if each pair of points ful�ls the contact conditions of one ofits possible states.Another form of writing this condition is

−min(−pn; rTn)= 0 (25)

which is equivalent to imposing the condition that the normal tractions are always of compression(−pni¿0), and that the overlapping between cylinders does not exist (�ni¿0), and that at leastone of the two variables is zero.The Normal problem can be formulated by combining the normal part of the elastic equations

(21) with the contact conditions de�ned by Equation (24) or Equation (25):

�n(�n; pn)=[Tn − (dn + Snt �pt)− Snnpnpn −min(0; pn + rTn)

]= 0 (26)

or

�n(�n; pn)=[Tn − (dn + Snt �pt)− Snnpn

−min(−pn; rTn)]= 0 (27)

and its solution is obtained by computing the zero of a non-linear vectorial function.

3.2.2. Tangential problem The formulation of the tangential part of the problem is dependent onnormal tractions. Let us suppose that the distribution of normal tractions is known (pn= �pn). Toformulate the contact constraints it is convenient to have previously introduced two de�nitions:the Euclidean distance between a point and an interval, and the projection of a variable over aninterval.



De�nition 1. The Euclidean distance DCg from a point z ∈R to a closed and centred intervalCg ≡ [−g; g], g∈R+; R+ ≡ {x∈R : x¿0}, is de�ned as

DCg(z)= max(z − g; 0)−min(z + g; 0) (28)

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

PCg(z)= z − sgn(z)DCg(z) (29)

If these functions are applied componentwise to vectors z∈Rn, their de�nition and representationwill be analogous:

DCg(z)∈Rn; [DCg

(z)]i=DCgi(zi); PCg (z)∈Rn; [PCg (z)]i=PCgi (zi) (30)

Using the previous de�nitions 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�| �pni |⇒ |pti − rsti |6�| �pni | (31)

• Slip:sti 6=0; pti sti¡0; |pti |= �| �pni |⇒ |pti − rsti |¿�| �pni | (32)

• Separation:sti 6=0; �pni =0; pti =0⇒C�| �pni |= ∅ (33)

It is easy to show that ful�lling these inequalities can be guaranteed by satisfying

pti =PC�| �pni |(pti − rsti) (34)

where r¿0, is a penalty parameter, having for convenience the same value as that of the parameterused for the Normal problem.The Tangential problem could be written by organising in a system of equations the tangential

part of Equation (21) and the contact conditions (34):

�t(st ; pt)=

[st − (^∗ + Btn �pn)− Bttptpt − PC

�|�pn|(pt − rst)

]= 0 (35)

where, as in the normal problem, the equilibrium is imposed by the �rst equation, and the contactconditions (applied point by point) are established by the second one.

3.2.3. Coupled problem The solution of the coupled normal–tangential problem is more com-plicated than each one separately, because the normal tractions, and thus the friction limit, areunknowns. To formulate the problem we have used the direct combination of Equations (26) or(27) and (35), substituting in the Tangential problem the �xed friction limit �|�pn| by anotherone, a function of normal tractions. The alternatives for this limit are: −� min(0; pn + rTn) and



−� min(0; pn), and it is always possible to project onto the valid region of the friction cone. Thereare then two functions that can de�ne the problem

�(�n; st ; pn; pt)=

Tn − dn − Snnpn − Sntptst − ^∗ − Btnpn − Bttptpn −min(0; pn + rTn)

pt − PC−� min(0; pn+rTn)(pt − rst)

=0 (36)

or

�(�n; st ; pn; pt)=

Tn − dn − Snnpn − Sntptst − ^∗ − Btnpn − Bttpt

−min(−pn; rTn)pt − PC−� min(0; pn)

(pt − rst)

=0 (37)

If it is desired to �x the global external loads applied to the roller as an input parameter,characterizing the problem, the obtained formulation should be modi�ed to include these loadvalues.The resultant of the normal and tangential loads (P, Q), applied on the discretized contact

surface can be calculated by integrating over the contact zone

P= −ec∑i=1

{ne∑j=1

[(∫�iNj d�

)pnj

]}; Q=

ec∑i=1

{ne∑j=1

[(∫�iNj d�

)ptj

]}(38)

where ec is the number of elements, ne the number of nodes by element and Nj the element shapefunction associated with node j. These expressions can also be written in vector form as

P=−fTpn; Q= fTpt (39)

and easily included in the general formulation.In Equation (39), two new equalities have been formulated, being necessary to consider as

unknowns two variables: the creep �0, and the overlap �0. The initial normal separation and thecreep vector can be written as

Tn0 = Tg − �0e; ^= �0e� (40)

where

eT = (1; 1; : : : ; 1; 1); eT� =(1; 1; : : : ; 1; 0) (41)

then, the system of equations is completed, and if we refer to the unknowns vector z:

zT =(TTn ; sTt ; pTn ; pTt ; �0; �0

)(42)



the coupled functions describing the problem can be written in the following forms:

�(z)=

Tn − Tg + �0e − gn − Snnpn − Sntptst − �0e� − gt − Btnpn − Bttpt

pn −min(0; pn + rTn)pt − PC−� min(0; pn+rTn)

(pt − rst)P + fTpn

Q − fTpt

=0 (43)

and

�(z)=

Tn − Tg + �0e− gn − Snnpn − Sntptst − �0e� − gt − Btnpn − Bttpt

−min(−pn; rTn)pt − PC−� min(0; pn )(pt − rst)

P + fTpnQ − fTpt

=0 (44)

both being B-di�erentiable functions, because of the non-linearity of the contact conditions, whichimplicates non-linear directional derivatives. This is the reason why the functions are divided intotwo parts:

�(z)=�LD(z) +�NLD(z) (45)

where �NLD(z) is the part of the function which has a non-linear directional derivative in somepoints:

�NLD(z)=

0

0

−min(0; pn + rTn)−P−� min(0;pn+rTn)(pt − rst)

0

0

(46)

and �LD(z) is the rest, having linear directional derivative at all its points.In a similar way, �NLD(z) can be obtained by grouping the non-linear terms of �(z):

�NLD(z)=

0

0

−min(−pn; rTn)−PC−� min(0; pn)

(pt − rst)0

0

(47)

�ND(z) being the other part.



4. SOLUTION PROCEDURE

To solve the problem the generalized Newton’s method with line search (GNMls) has been used.GNMls is an extension of the Newton’s method for B-di�erentiable functions formulated by Pang[45]. A similar algorithm has been also used in Reference [38] to solve the usual contact problem,not rolling, discretizing the equilibrium equations by the �nite element method, and showing thatthis method is well suited for solving this kind of problems.

4.1. Generalized Newton’s method

The GNMls algorithm for solving the non-linear equation G(z)= 0, G(z) being a B-di�erentiablefunction, could be resumed by the following steps:

Step 0: Let z0 be an arbitrary initial vector, and let s, �, � and � be given scalars s¿0, �∈ (0; 1),�∈ (0; 1=2), and �¿0 but small.Step 1: In general, given zk with G(zk) 6= 0, obtain a direction Wkz solving the equation

G(zk) + BG(zk)Wkz = 0 (48)

where BG(zk) means B-derivative.Step 2: Let �k = �m

ks, obtain mk as the �rst non-negative integer for which

(zk)−(zk + �mks Wkz )¿− ��mks′(zk ; Wkz ) (49)

where

(zk)= 12G(z

k)TG(zk); ′(zk ; Wkz )=−G(zk)TG(zk) (50)

in which (zk) is the merit function and ′(zk ; dkz ) its directional derivative (Armijo-type linesearch, see Reference [48]).Then, taking into account (50), Equation (49) can be expressed also as

(zk + �mks Wkz )6(1− 2��m

ks)(zk) (51)

Step 4: Set the new solution vector

zk+1 = zk + �kWkz (52)

Step 5: Terminate if (zk+1)6�, otherwise return to Step 1 and iterate until convergence isreached.An expanded version of the algorithm, a study of its convergence, and some details related with

the use of this algorithm to solve complementarity problems and variational inequalities, could befound in the paper by Pang [45].Often the solution of Step 1, the generalized Newton equation, could involve a big computational

e�ort because of the non-linearity of BG(zk)Wkz . In order to reduce this e�ort some previousresearchers solved accurately contact problems avoiding the points where the non-linearity couldhappen or neglecting the non-linear part of the derivative in these points.In rolling problems, as in contact ones, the points to avoid are those where the inequalities

formulated during the iterative process of solution change to equalities. These points, or lines, arethe frontiers between zones where the functions �(zk) or �(zk) have di�erent slopes and then,



their derivatives do not have the same value for all the directions. During the iterative process it isvery unusual to have a trying solution zk over one of these points, this is because the convergenceis not usually a�ected if those points are avoided, or some part of their derivative neglected.The non-linear part of the generalized Newton’s equations, is linearized and computed as

BG(zk)Wkz =(OGLD + @GNLD)Wkz (53)

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

4.2. First option, function �(z)

In the case of the function �(z) the linear part of the derivative is

O�LD =

I 0 −Snn −Snt e 0

0 I −Btn −Btt 0 −e�0 0 I 0 0 0

0 0 0 I 0 0

0 0 fT 0 0 0

0 0 0 −fT 0 0

(54)

The pseudo-Jacobian, @�NLD is obtained by a di�erent way depending on the state of eachpoint i:

1. Contact restriction non-linear part: (�NLD(z))3 =−min(0; pn + rTn)• Case 1: pni + r�ni¿0

[(@�NLD(z))3]i= [0; 0; 0; 0; 0; 0] (55)

• Case 2: pni + r�ni =0 (modi�ed):[(@�NLD(z))3]i= [0; 0; 0; 0; 0; 0] (56)

• Case 3: pni + r�ni¡0[(@�NLD(z))3]i= [−r; 0;−1; 0; 0; 0] (57)

where ‘modi�ed’ denotes the use of a simpli�ed linear version of the directional derivative,instead of its true non-linear expression.

2. Friction law restriction: (�NLD(z))4 =−PC−� min(0; pn+rTn)(pt − rst)

• Case 1: pni + r�ni¿0[(@�NLD(z))4]i= [0; 0; 0; 0; 0; 0] (58)

• Case 2: pni + r�ni¡0 and |pti − rsti |¡−�(pni + r�ni)[(@�NLD(z))4]i= [0; r; 0;−1; 0; 0] (59)



• Case 3: pni + r�ni¡0 and |pti − rsti |=−�(pni + r�ni) (modi�ed):[(@�NLD(z))4]i= [0; r; 0;−1; 0; 0] (60)

• Case 4: pni + r�ni¡0 and |pti − rsti |¿−�(pni + r�ni)[(@�NLD(z))4]i= [r� sgn(pti − rsti); 0; � sgn(pti − rsti); 0; 0; 0] (61)

• Case 5: pni + r�ni =0 and |pti − rsti |¿0 (modi�ed):[(@�NLD(z))4]i= [0; 0; 0; 0; 0; 0] (62)

• Case 6: pni + r�ni =0 and |pti − rsti |=0 (modi�ed):[(@�NLD(z))4]i= [0; 0; 0; 0; 0; 0] (63)

4.3. Second option, function �(z)

In the case of the function �(z) the linear part of the derivative is

O�LD =

I 0 −Snn −Snt e 0

0 I −Btn −Btt 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

(64)

The pseudo-Jacobian, @�NLD is obtained also in di�erent forms depending on what is the stateof each point i.

1. Contact restriction non-linear part: (�NLD(z))3 =−min(−pn; rTn)• Case 1: (−pni)¡r�ni

[(@�NLD(z))3]i= [0; 0; 1; 0; 0; 0] (65)

• Case 2: (−pni)= r�ni (modi�ed):[(@�NLD(z))3]i= [0; 0; 1; 0; 0; 0] (66)

• Case 3: (−pni)¿r�ni[(@�NLD(z))3]i= [−r; 0; 0; 0; 0; 0] (67)

2. Friction law restriction: (�NLD(z))4 =−PC−� min(0; pn)(pt − rst)

• Case 1: pni¿0[(@�NLD(z))4]i= [0; 0; 0; 0; 0; 0] (68)

• Case 2: pni¡0 and |pti − rsti |¡−�pni[(@�NLD(z))4]i= [0; r; 0;−1; 0; 0] (69)



• Case 3: pni¡0 and |pti − rsti |=−�pni (modi�ed):

[(@�NLD(z))4]i= [0; r; 0;−1; 0; 0] (70)

• Case 4: pni¡0 and |pti − rsti |¿−�pni[(@�NLD(z))4]i= [0; 0; � sgn(pti − rsti); 0; 0; 0] (71)

• Case 5: pni =0 and |pti − rsti |¿0 (modi�ed):

[(@�NLD(z))4]i= [0; 0; 0; 0; 0; 0] (72)

• Case 6: pni =0 and |pti − rsti |=0 (modi�ed):

[(@�NLD(z))4]i= [0; 0; 0; 0; 0; 0] (73)

4.4. Starting vector

Pang’s algorithm has been used to solve each one of the non-linear problems. Assuming someknown values of P and Q, the iteration is started from an initial solution z0 that usually implicatesan overlap between the two cylinders. This can be easily obtained choosing a negative initialseparation. For the examples presented here we always started the iteration using the vector

z0T= (−TTg ; 0; 0; 0; 0; 0) (74)

and in all the cases the iteration process converged to a good solution rapidly, the results beingalmost equal when using �(z) or �(z) (e.g. three or four �gures equal).

4.5. Alternative formulation

An alternative way to solve the problem is to consider the resultant of the forces on the contactzone, P and Q, as unknowns, and the overlap �0 and the creep �0 as known values, then

�zT = (TTn ; sTt ; pTn ; pTt ; P; Q) (75)

The Jacobian matrices of the linear parts of the functions ��( �z) or ��( �z), for the new vector ofunknowns are

O ��LD =

I 0 −Snn −Snt 0 0

0 I −Btn −Btt 0 0

0 0 I 0 0 0

0 0 0 I 0 0

0 0 fT 0 1 0

0 0 0 −fT 0 1

(76)



and

O ��LD =

I 0 −Snn −Snt 0 0

0 I −Btn −Btt 0 0

0 0 0 0 0 0

0 0 0 I 0 0

0 0 fT 0 1 0

0 0 0 −fT 0 1

(77)

the derivatives of its non-linear parts being the same. In this other case the trial solution used forthe �rst iteration was the same as that used previously (74).

5. RESULTS

Three classical application examples in growing order of complexity are presented. The �rst ex-amples have been solved with models based on the half-space approach because the obtainedresults will be compared with those obtained by authors who have solved these same problems byother techniques, but using this approach. Only in the last one the model used represents the realgeometry of cylinders.In the �rst example, the case of rolling between cylinders of the same material and radius

(similar) is solved. In the second one the case of two cylinders of di�erent material (dissimilar) isexposed. For the third example the results for the case of rolling of a cylinder covered with a �nelayer of elastic material (tyred), will be presented. A last example compares the results obtainedfor the same problem when using the real geometry or the half-space approach.Each cylinder has been discretized, having a potential contact area of small dimensions compared

with the characteristic dimensions of the domain, the mesh being re�ned on the contact zone torepresent accurately the stress concentrations in its surroundings. In Figure 2 the boundary elementmesh used is represented in a schematic form, where the contact points of the two contact surfacesare in the same geometrical position.Discontinuous quadratic elements (DQE) have been used to discretize the contact boundaries

and their displacements and tractions. DQE uses quadratic shape functions to approximate the dis-placements and tractions inside the element on the boundary, but the displacements between twocontiguous elements can be discontinuous. Those elements represent more accurately a displace-ment and traction distribution when there is a change in the contact state from one element to thenext.The other boundaries have been discretized with continuous quadratic elements (CQE), except

at the corners, where partial discontinuous elements are used (only at one of its extremes, corre-sponding with the corner). Most of the elements used are of the same size and they are evenlydistributed over the di�erent contact surfaces.The contact conditions of each node are checked using the cylinders’ geometry and the dis-

placements and tractions corresponding to this point, accepting the small incompatibilities that canbe derived by a change of the contact status along an element. This choice has been checkedwidely by the authors for steady-state rolling problems and shows no additional complications asthe friction coe�cient is increased. Man and Aliabadi [49] reported some problems with quadratic



Figure 3. Tangential tractions for the rolling ofsimilar cylinders.

Figure 4. Dissimilar cylinders: two slip zones in thesame direction.

elements, but they used continuous elements, solved a path-dependent problem (not steady state)checking the contact status of each node by a decision-tree including the status of the other elementnodes, and they used a trial-and-error iterative method to solve the problem.To �nish, we will show the excellent behaviour of the quadratic discontinuous elements in this

kind of problems, and the power of the algorithm presented, solving the rolling of two dissimilarcylinders. With this proposal for each of the three load ratios, Q=P=0:075; 0:004, and −0:083,�ve di�erent friction coe�cients, �=0:1; 0:2; 0:4; and 0.8, were solved.After some numerical experiments, the parameters used in Pang’s algorithm for all the cases

were: s=1, �=0:9, �=0:1, �=10−15; and for the penalty parameter r=0:1.

5.1. Similar problems

The simplest rolling contact problem consists of two identical rotating cylinders. The results willbe compared with the corresponding ones obtained using the Carter solution [2]. This problem hasalso been solved by means of semi-analytic methods by Bentall and Johnson [4] and Kalker [8].The modelled cylinders have a radius of R=100 mm. The friction coe�cient is �=0:1. And the

two cases presented are characterized by the values of the resultant of the normal and tangentialforces applied on the contact area, the initial overlapping �0 and the creep �0 being unknowns.The �rst case is such that Q=�P=0:51, while for the second Q=�P=0:84.The used boundary element mesh is shown in Figure 2, where 21 discontinuous quadratic

elements are used to represent the contact area displacements and tractions.In Figure 3, the tangential tractions obtained are represented and compared with the solution

of Carter. The normalization factors are �|pn0|, friction coe�cient times the absolute value of themaximum normal traction for the equivalent Hertz problem; and a, half the length of the contactarea. The agreement between the two results is excellent.



5.2. Dissimilar problems

If the cylinders rolling are of di�erent material, the problems become more complicated becausethere can appear several adhesion-slip transition points. Dundur’s constant

�=

[(1− 2�A)(1 + �A)]EA

− [(1− 2�B)(1 + �B)]EB

(1− �2A)EA

+(1− �2B)EB

(78)

quanti�es the similarity between the properties of both materials, and together with the frictioncoe�cient, the loads applied Q=�P, and the radii of the cylinders, are the parameters that governthe distribution of the slip and adhesion zones.Previous works have shown the existence of four di�erent regimes for steady-state rolling prob-

lems. The �rst regime implicates one slip zone, is the case of similar problems previously analysedand shown in Figure 3. During the second and third regimes two slip zones, in equal or in oppositedirections, respectively, are present. For the last regime, the fourth, three slip zones appear; twoof these zones are located at the beginning and at the end of the contact zones, both slipping inequal direction; while the third zone, located between them, is slipping in the opposite direction.For all the problems, the constant parameters were: an equivalent radius 1=R=1=RA + 1=RB, of

100 mm for the two cylinders, a friction coe�cient �=0:1, and a Dundur’s constant �=0:288(corresponding approximately to aluminium and steel). Di�erent regimes are obtained by the ap-plication of di�erent forces, the values of the load parameter Q=�P considered were 0.75, −0:83and 0.04. The boundary element mesh used coincides with the one utilized to solve the case ofsimilar cylinders.These parameters were the same as those used by Bentall and Johnson [4] and by Nowell

and Hills [6] in order to compare our results with theirs. The cited works share the followingassumptions: the contact zone is approximated by two half-spaces, the normal tractions and thelength of the contact zone have the Hertzian values, and the tangential tractions have no e�ecton normal displacements. The di�erences between them are in the approximation of the tangentialtractions, one by triangular distributions and the other by Chebyshev polynomials.The tangential tractions and the slip velocities obtained with the proposed algorithm have been

plotted in Figures 4–6, and compared with the corresponding results of Nowell and Hill presentedin the cited work. The normalization factor for the slip is R=(a�|V |).For the second regime Q=�P=0:75, Figure 4, the agreement between the tangential tractions

is excellent, the situation of the slip and adhesion zones is the same, but the slip velocity di�erssomewhat.To verify if the di�erences in the slip velocity are caused by the �nite di�erences scheme used,

Equation (8), the tangential derivatives involved, Equation (7), were also computed by using thehypersingular boundary element method (HBEM).HBEM basically results from the di�erentiation of the singular integral representation of the

displacements, the starting point for usual BEM, with respect to the collocation point components,and then a usual discretization of the boundary displacements and tractions. In general, the strainsand stresses (displacement derivatives) computed from a known solution of displacements andtractions on the boundary by the HBEM, are more accurate than the corresponding ones obtaineddirectly from this boundary result.



Figure 5. Dissimilar cylinders: two slip zonesin opposite directions.

Figure 6. Dissimilar cylinders: three slip zones.

The HBEM formulation of the displacement derivatives starts from the following integral rep-resentation:

0= cikjh uj; h + lim�→0

{ujbijk�

}+ lim�→0

{∫(�−e�)

[Vikjuj −Wikjpj] d�}

(79)

where � is the boundary, cikjh and bijk are coe�cients that depend upon the local geometry ofthe boundary and upon the shape used for e�, and Vikj and Wikj are the hypersingular nuclei. Thedetails of the HBEM method and the expressions of nuclei could be obtained in the paper ofGuiggiani [50]. Substituting the displacements and tractions obtained for the problem of Figure 4in the HBEM approximation, we obtain a more accurate slip velocity. This new curve, denotedas ‘Hyp. Deriv.’ in Figures, is closer to the curve obtained using the �nite di�erence schemeproposed, showing that the accuracy of this scheme is good enough for the discretization used.The small incompatibilities with the law of friction shown by the hypersingular derivative, are notimportant enough to lead us to opine that their correction would produce slip velocities similarto the Nowell and Hills velocities, therefore we accept our results as the correct solution tothe problem. An example of the third regime is shown in Figure 5. Now the load parameter isQ=�P=−0:83, the results obtained being quite similar to those corresponding to Nowell and Hills,and showing again small di�erences in the slip velocity. The hypersingular derivatives are, as forthe second regime, close to the �nite di�erences values.The last regime, the fourth, with three slip zones, is obtained if a load parameter of Q=�P=0:04

is prescribed. In Figure 6 the results for this case are plotted. Again, the values of tangentialtractions are quite similar, but there are small di�erences in slip velocities. The behaviour ofHBEM results is also similar.

5.3. Dissimilar problems with variable friction coe�cients

The object of the following examples is to show the good behaviour of the results obtained usingthe methodology presented in solving problems with di�erent friction coe�cients, for which otherauthors, [48], reported some problems if the models were discretized with quadratic elements.



Figure 7. Tangential tractions versus friction coe�-cient: two slip zones in the same direction.

Figure 8. Tangential tractions versus friction coe�-cient: two slip zones in opposite directions.

The cylinders used in this case are the same as used previously for the problem of dis-similar cylinders, their size, mechanical properties and discretization being the same. For eachload ratio, Q=P=0:075, −0:083, and 0.004, �ve di�erent friction coe�cients will be solved,�=0:1; 0:2; 0:3; 0:4 and 0:8, but for each case the number of slip zones and their directions didnot change.The tangential tractions are shown in Figures 7–9, all the curves being smooth and without

ripples. Normal tractions are not plotted, but their values and shape are similar to the correspondingone of the Hertz problem, but not exactly the same. This has also been reported by other authors,as for example by Nowell et al. [51], for fretting problems. The coupling between the normal andtangential tractions causes di�erences in the normal tractions obtained for the same problem butwith di�erent friction coe�cients. This is the reason for which the tangential tractions obtainedon the slip zones for di�erent friction coe�cients and similar location, sometimes do not exactlymatch.

5.4. Tyred cylinder

In the next two examples we suppose that one of the cylinders used in the previous problemsis covered with a �ne layer (a tyre) of a much less rigid elastic material than the material ofthe cylinders. The thickness of the layer is b, similar to the size of the contact area, but muchsmaller than the radius of the cylinders (b.R). This problem was solved by Nowell and Hills [7]using semi-analytic methods, based on a previous work of Bentall and Johnson [5]. To be able tocompare our results with the results of these authors, we will use their hypotheses assuming thatthe two cylinders are rigid bodies, and one of them of radius R rotates on a layer of elastic materialattached peripherally to the other (Figure 10). From the BEM standpoint, it is only necessary todiscretize the layer boundary, using a su�ciently large number of elements (Figure 11), and toassume that the normal load is applied by means of an overlap �0 between the cylinders.The results for the �rst tyre problem are presented in Figure 12. They correspond to a friction

coe�cient �=0:1, a relationship b=a=0:1, a Poisson’s ratio �=0:5 (incompressible material), andrelations between the tangential and normal resultant loads, Q=�P=− 0:74; 0:41 and 0:05.



Figure 9. Tangential tractions versus friction coe�-cient: three slip zones.

Figure 10. Tyred cylinder.

Figure 11. Model and discretization used for the tyred cylinder.

The results obtained are quite similar to those of the cited reference, and the adhesion or slipzones, localized in the same place. All the presented curves in Figure 12 have been centred tofacilitate their comparison, since for the present problem the centre of the contact area su�ers ashift due to the tangential load. Nowell and Hills carried out a similar process with their results.The last case involves a somewhat thicker tyre, b=a=0:1, a Poisson’s ratio �=0:3, and the same

friction coe�cient. The contact zone is now discretized in 31 quadratic discontinuous elements.As can be seen from Figure 13, the results obtained are quite consistent with those reported byNowell and Hills, except for a small di�erence in the traction peak appearing on the left of thecontact zone.

5.5. Half-space versus cylinder approximation

As a �rst check of how accurate the half-space approximation is, we have solved a dissimilarproblem using the cylinder’s in uence coe�cients instead of those corresponding to thehalf-space, which was used to solve previous examples. The formulation and procedures usedwere the same, the only exception being discretization. Now we used the model and mesh ofFigure 14, that is constituted by two cylinders of radii 150 and 300 mm, rotating around two rigidshafts of radii 15 and 30 mm, respectively. The friction coe�cient adopted was �=0:2, Dundur’sconstant �=0:288, and load parameter Q=�P=0:375.



Figure 12. Tangential tractions on the tyre(b=a = 0:1).

Figure 13. Tyred cylinder (b=a = 0:5).

Figure 14. Dissimilar cylinders discretizations. Figure 15. Comparison between half-space andcylinder approaches.

For this rolling problem, the contact zone was discretized with 21 discontinuous quadraticelements, the x-co-ordinates used for nodes of two cylinders being the same as in the half-spacemodel. For the rest of the discretization we have used continuous quadratic elements.The contact zone tangential tractions and slip velocities obtained using the half-space and cylin-

ders in uence coe�cients, are plotted in Figure 15. For this problem, the results are nearly identical,



showing the validity of the half-space approach and the power of BEM to analyse rolling contactproblems.

6. CONCLUSIONS

A new methodology for the solution of the steady-state two-dimensional rolling problem betweencylinders, based on mathematical programming techniques and on the boundary element method,has been proposed.In this methodology, the in uence coe�cients of the elastic equations have been obtained by

means of BEM, and the highly non-linear problem associated with rolling contact with frictionhas been solved using the generalized Newton’s method with line search.The convergence of the mathematical programming problem has been improved in compari-

son with other algorithms experienced by the authors, based on complementarity relations or onquadratic mathematical programming methods.Several examples of classical rolling problems have been solved. The tangential tractions com-

puted in all the cases were close to the tractions obtained by other authors, but the slip velocitiesdi�er somewhat. A checking process on these slip velocities, based on the Hypersingular boundaryequation method, showed that they are accurate and valid solutions for the rolling contact problem,therefore we accept the obtained values as the correct solution of their corresponding problems.The algorithm has been shown to be very e�cient. It simpli�es the procedure to obtain the

solution of a rolling problem, and minimizes the physical knowledge of the particular problemnecessary for controlling the convergence of the non-linear iteration process, and is an e�ectivealternative to the actual algorithms.The presented techniques could be extended to solve other more complex contact rolling prob-

lems, but it would be necessary to develop new strategies to take into account the di�erent aspectsnot yet included (i.e. in the 3D case: unknown slip direction, spin movement, etc.).In conclusion, the new methodology presented in this paper simpli�es the solution of rolling

contact problems, more so if we think of the e�ort necessary to solve the same problem by meansof trial and error or semi-analytic methods.

ACKNOWLEDGEMENTS

This work was partially funded by Spain’s DGICYT in the framework of Projects PB 96-1380 and PB96-1322.

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an algorithm to solve coupled 2d rolling contact problems

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