...Presented at the Seventh Annual Meetingof the Tire Society, Akron, Ohio,March 1988.

A THREE-DIMENSIONAL ROLLING CONTACT MODELOF A REINFORCED RUBBER TIRE

L.O. Faria, J.M.Bass, J.T.Oden, and E.B.Becker

The University of Texas at Austin and theComputational Mechanics Company, Inc.

Austin, Texas

AbstractAb"tract. Finite element models of steady rolling and cornering

of a cord-reinforced rubber tire on a rough roadwny are described.!\umerical solutions of severnl test problems nre presented.

1 Introd uctionThe complexity of modern pneumatic tires has limited the use of analyticaltechniques to calculate tire stresses and deformations.

Some simple one-dimensional models have been proposed which accountfor, but do not describe accurately, the behavior of a tire rolling and cor-nering on a rigid foundation. They are reviewed in [1].

The rolling of elastic cylinders has been studied by extending Hertz'stheory of elastic contact. That theory assumes homogenous and isotropicbodies, linear elastic behavior and frictionless surfaces and, therefore, can-not be used to model the tire road interaction.

The structure of a tire is also difficult to treat analytically withoutgross simplification. Its ability to carry loads derives from the stiffening,due to inflation pressure, of anisotropic layers of high modulus flexible cordsembedded in a rubber matrix.

Early analysis considered the tire as a ring or as a membrane. Morerecently finite elements have been used to model tire behavior and someresults are described in the review article of Noor and Tanner [2]: most ofthe studies account for large displacements, anisotropic material behaviorand pressure loading as in [3], [4], [5L but do not include dynamic effects,material nonlinearities and rolling/cornering contact.

In earlier papers (6,7,9,8] we have described variational formulationsof rolling contact problems for general cases of finite elastic deformation.In the present work, we extend these methods to a broad class of three-dimensional rolling contact problems with friction, that allows the analysisof free rolling, cornering, acceleration and braking.

This formulation is applied to the finite element analysis of tires. A lay-ered shell finite element with shear deformation is developed that allows forlarge deflection and rotation. In each layer, orthotropic Hookean materialsor Mooney Rivlin type materials with fiber reinforcement can be used, andthe incompressibility constraint is enforced with Lagrange multipliers.

The contact constraint is enforced with a penalty and the friction term,instead of the usual Coulomb friction, is regularized by a differentiable formthat makes it more suitable for numerical analysis ..

We conclude the study with a presentation and discussion of numericalanalysis with a typical tire geometry.

2 }(inenlatics of Steady State Rolling Con-tact

,\Ve wish to consider the case of a tire-like ~xisymmetric structure, thatrolls on a horizontal surface describing a curv~d path. The analysis can besimplified by assuming a steady motion and thereby removing explicit timedependency from the problem.

We first formulate the kinematics of steady state for the case of straightline rolling. By subtracting the translation, this situation is equivalent to atire spinning about its fixed axis and in contact with a moving foundation.

\Ve decompose the motion of the structure into two successive motions,in which the particle p undergoes first a rigid rotation, and secondly movesto the point with position vector x, the place occupied by the material

1

·.

point p at time t (see Fig. 1).We select as reference coordinates and spatial independent variables y,

places in the rigid spinning structure at an arbitrary time t. The corre-sponding material points p are obtained from

where R is a rotation tensor. Its components, referred to a cartesian coor-dinate frame with orthogonal basis {ei}, e 1 being parallel to the foundationand e3 along the axis of the structure, are given by

[

coswt -sinwt 0][R] = sinwt coswt 0

o 0 1

where w is the spinning angular velocity of the rigid structure about itsa.XIS.

To describe the deformation of the structure we define the map X(y, t)that gives the position of point p at time t as a function of its location aty at time t. ~Ne have

x = X(Y, t)By the chain rule we obtain the velocity v(y, t) as

v(y,t) = [8X(y,t)]8Y 8x(y,t)8y 8t + 8t

(1)

(2)

where ~ represents the velocity field in the rigid spinning cylinder.:In a steady motion w is constant, X(y, t) = x(y) and the expression for

the veloci ty becomes

v(y)

the velocity components are

(3)

_ ...... 00

p - position vector of the particle at t=O.

y - position vector of the particle in the reference configuration

x - position vector of the particle in the deformed configuration.

Figure 1: Straight line rolling - 2d case. Shown is a cylindrical structurewith inner radius Ri and outer radius Ro.

3

aX2 aX2V2= W(Y2- - Yl-)

aYl aY2\Vhen written in terms of the displacement

u=x-y

(4)

(5)

these expressions reduce to the ones in [10] where linear elasticity is con-sidered and the Lagrangian and Eulerian descriptions coincide.

The acceleration a(y, t) can be obtained from the velocity field by asecond application of the chain rule. In the steady case its components are

aX., ax·Qi = -W2(Yl-O' , + a 0' (YlYi - b(iYmYm) (6)

Yl Yl YiIn this expression repeated indices imply summation (i, e,j, m = 1,2)

and b(j is the Kronecker delta

bli = 1 if £ = j, blj = a if e =1= j

the same results can be obtained if the reference system, now denoted byz, rotates with the body with angular velocity u.:. The motion is the mapX(z, t) and the acceleration \'ector is given by

By considering a polar coordinate system centered at the origin we set

For steady motion we have to consider wave like solutions of the form

x (r, B + wt) = X(r, B, t)

. a a a a aUsmg ot = w ao' 00 = -Z2az1 + Zlazz

we can obtain from (7) an expression equivalent to (G),

This analysis will therefore allow us to find solutions described in thetire industry as standing waves and discussed in [8].

We now turn to the three dimensional case shown in Fig. 2. Followingthe same procedure as above, we choose a reference system y in the rigidstructure, spinning and cornering with constant angular velocities wand nrespectively and with the plane normal to the axis inclined with the verticalat a constant angle,. The velocity v(y) and acceleration a(y) are givenby

(8)

(9)

where the velocity field in the rigid spinning and cornering structure hasthe components

= Dcosh)YJ - (w + nsin(-y))Y2

OY2 .ot = (w + nszn(-y ))YlOY3ot = -nCOS(-y)Yl

(10)

Various kinematical quantities can be computed in terrns of the motionx· The deformation gradient F is given by

F = \7X (11)

and the right and left Cauchy Green deformation tensors C and B are givenby

c

5

(12)

Direction of tireheading

• y - reference coordinates

• w - spinning angular velocity

• n - cornering angular velocity

• 'Y - inclination angle •

• V - foundation velocity

• Q' - angle between V and direction of tire heading

Figure 2: Specifications for tire rolling in a circular path.

6

3 A Boundary Value Probleln for RollingContact

'\Fe now establish equations and inequalities for a class of rolling contactproblems. We shall work in the reference configuration and therefore allthe fields will be defined in the reference configuration and function of thereference coordinates.

The boundary value problem for rolling contact consists in finding amotion X that satisfies

3.1 Balance of linear 1110111entU111

DivS + Polll = PoX (13)

\..-here S is the Piola Kirschoff stress tensor, Po the mass density and m thebody force per unit mass, both in the reference configuration, and x is theacceleration vector given by (9).

3.2 Constitutive equationsThe tire structures \ve want to model are complex structures with manydifferent components. It is generally assumed an isotropic behavior for theru bber parts and that the reinforcements (plies, belts) can be modelled astransyersely isotropic.

\Ve consider hyperelastic materials, characterized by a strain energy n"per unit volume in the reference configuration.

For the case of isotropy TV can be expressed as a function of the inyari-ants of the tensor C defined in (12) ,

TV = lV(Ih 12, 13)

I. = trC, tr = trace1 '1 1 2}-{(trC) - -tre2 2

13 = defe

7

(14)

(15)

the stress is given by

s = mvof

In the case of incompressibility. which is the common assumption forrubber like materials, 13 = 1 and W becomes a function of I} and 12 butwe may add to W any multiple of (IJ - 1). Hence (14) is replaced by

(16)

where the pressure p may be regarded as a Lagrange multiplier. The stressis then

ol~'S = - - pF-T (17)of

A typical expression for the strain energy of rubber like materials includethe Neo Hookean

H" = C(II - 3)and the Mooney Rivlin material

(IS)

(19)

where C, CI and C2 are material constants.For a material reinforced by a family of parallel fibers, with initial di-

rection ao it is shown in (11] that IV can be expressed as a function of theim'ariants Ij, 12• 13 defined above and

1~= ao· CaoIs = ao· C2ao

lVt = lV'(IJ, 12, 13, I... Is) (20)

In the case of incompressibility 13 = 1, but again a term -p/2(13 - 1)may be added to tV.

To make clear the relationship between this expression (without com-mitting ourselves to a particular form of the strain energy) and the onesused to predict the elastic constants of cord and rubber plies [12] it is con-venient to write explicitly the constitutive equations in terms of the Cauchystress T = (detF)-ISFT

s

where p is the reaction pressure due to incompressibility l..vj= ~J.!,a is thecurrent direction of the fibers

a FaoII Fao II

and c 0 d denotes tensor product of vectors c and d.The constitutive equations of linear elasticity for an incompressible ma-

terial reinforced by a family of fibers in the direction a can be derivedfrom (21) by deleting all terms of degree two or more in the displacementgradients. That gives

T = 2µTE + p(a· Ea)a 0 a + 2(µL - µT)(a 8 a . E + Ea 0 a) - pI (22)

where E is the infinitesimal strain term with components

µL and µT are the longitudinal and transverse shear moduli and {3 is amaterial constant.

Informati~n about the shear moduli is generally not a\"ailable but, dueto the fact that the cord volume fraction is generally small, the response ofthe cord rubber ply to shear will be essentially that of the isotropic matrix,as implicit in the Gough-Tangorra and Akasaka-Hirano formulas of [12].Therefore, we set µL = µT and lVs = 0 in expression (21).

3.3 Unilateral contact conditionThe contact condition expresses the fact that the motion of the structureis constrained by the foundation. It is given as

X(y)·b~H,yErc

9

(23)

where b is a unit vector normal to the foundation and directed away fromthe origin, H is the distance from the origin to the foundation as shownin Figure 3, and rc is the candidate contact surface of the structure. Letn(y) be the exterior normal at y in rc and denote by cr the stress vector

cr = Sn (24)

The components of cr normal and tangential to the foundation are givenby

UT = Icr - (cr· b)bl

If the unilateral condition (23) holds, then we also have

(25)

(26)

x(y)· b < H -+ an = O,x(y)· b = H -+ Un ~ a for y E rc (27)

Instead of specifying the distance H, it is sometimes convenient to spec-ify a load F acting on the axis at the origin, and normal to the foundation.Or we can fix the axis and move the foundation up with the load. Fromequilibrium we have:

F

3.4 Friction condition

(28)

The friction conditions for rolling contact were discussed in (10] for thecase of linear elasticity. Here we consider the nonlinear case with Coulombfriction although other friction laws could be used.

We assume that the spinning angular velocity w, the cornering velocityD, the inclination I of the rigid structure relative to the foundation andthe velocity vector Vo of the foundation in its plane are given. Then theslip velocity vector s(Y) for points in the contact surface is given by

s(y) = v(y) - Vo

10

H

F

Figure 3: Deformed configuration of the tire showing the distance H be-tween the origin and the foundation, the vertical force F and the torqueT.

11

with v(y) defined in (8). We can work in displacement units by setting

w(y) = s(y)/w

the stick slip conditions are then, on rc

IOOTI < vlOOnl-+ w = 0 , II = coefficient of friction

(29)

IOOTI = vlOOnl-+ there exist a positive number .\ such that (30)

w = -.\(u - (0'" b)b) (31)

3.5 Specified displacelllents and tractionsIf the structure is fixed to a rigid spinning axle then

x(y) = y (32)

for points on the axle.A pressure Po can be specified in part roof the reference surface of the

structure by the condition

(33)

where 110 is the exterior normal to r o.It is also possible to specify forces acting on the structure, like a torque

T (Fig. 3) or a force parallel to the foundation, both acting on the axle,instead of kinematic quantities like w or foundation velocity Vo·

This allows us to consider conditions as free rolling, when no torque isapplied on the axis, or accelerating or braking.

Due to the fac.t that we assume a steady motion it is not possible ingeneral to specify both torque and horizontal force. \Ve choose to specifyT and foundation velocity vo, and the equation to find w is obtained fromequilibrium

r (yxO"hdA.+T=O (34)irewhere (h denotes the component of the vector product in the YJ direction.

12

4 Variational Forn1ulation

The set of equations and inequalities defined in the previous section willnow be given a variational formulation which is the starting point for thefinite element discretization. This variational formulation is also known asthe principle of virtual work on the reference configuration.

Due to the conditions and constraints present in the boundary valueproblem, namely the unilateral conditions (23) and the stick slip condi-tions (30)-(31), the variational formulation consists of an inequality ratherthan an equation. By appropriate regularization it is possible to approxi-mate that inequality by equations depending on a parameter e > 0, whosesolutions approximate the inequality solution arbitrarily close as e goes tozero.

All these expressions and proof of their equivalence can be found in [Sj.Here we just present the final expression of the regularized problem. Itconsists in finding a motion x€, € = (c I, C2, C3), such that

10S(VX,) . VvdF + ire vlan(x)1 { \:V~~~I} .vdA

+ ~ !r (X· b - H)+v· bdAC2 re

+ ~ J (det\?x - 1)adjVvdV = (35)£3 11

J [ox] oy [ ov] oy J- Po - -. - -dv + Pol11 . vdv11 oy ot ay at 11

for all virtual displacements v. :The first term represents the internal virtual work, the second gives the

work done by friction forces with a regularized friction law dependent on£ 1, the brackets meaning that the smaller in absolute value of (~. , I~:I)ischosen.

The third term represents the relaxation of the unilateral constraint(23), and the fourth the relaxation of the incompressibility constraint.

The first term on the left side of the equation gives the virtual work ofinertia forces and the last the virtual work of exterior forces.

13

If instead of specifying the distance H we specify the load F, equation(35) is modified in a similar way as presented by (13].

We denote by U the displacement associated with the load F and assumethat in the underformed state the fonndation touches the structure at adistance R from the origin as is shown in Fig. 4. The condition of non-penetration will be

U ~ R - X(y) . b, for y in r c

and the variational problem will become:Find a motion Xe and a number U such that equation (35) holds with

the third term replaced by

for all virtual displacements v and numbers V.

5 Finite Elelnent ModelsTwo types of isoparametric finite elements were developed to numericallysolve the discrete form of(35): they are the three dimensional 20 node brickelement and a shear deformable 9 node shell element, both sho\\'n in Fig.5.

For the brick element the geometry and motion over each element areinterpolated from its nodal values by serendipity functions. Constitutivelaws of Hookean type or isotropic rubber models like the neo-Hookean (18)or Mooney Rivlin (19) can be employed.

The penalty term enforcing incompressibility is underintegrated, bywhich we mean that a Gauss quadrature rule of 2x2x2 is used, and thecontact and friction terms are integrated with Gauss or Simpson surfaceintegration rules.

The shell element is based upon the degenerated concept originally in-troduced in [14] and later generalized to nonlinear material and geometricbehavior. In [15], Hughes gives a detailed description of the element, usingan updated Lagrangian formulation.

14

R

Figure 4: Undeformed and deformed geometry of the structure \vith normalload F and associated displacement U.

15

TJ

Reference Surface

Pseudo-normal

Figure 5: Elements used: (a) 20 node brick; (b) 9 node shell element.

16

The degenerated concept consists in specializing the three dimensionalcontinuum equations to the form of a shell, by enforcing kinematic andstatic conditions.

First it is assumed that the motion of points in the continuum varieslinearly across the thickness and can be given by the motion of a referencesurface and of a fiber normal to it.

Secondly, the assumption of fiber inextensibility is compensated by en-forcing a state of zero normal stress in the fiber direction.

The element has 5 degrees of freedom at each node corresponding tothree displacements and two rotations of the fiber. This allows for sheardeformations to be taken into account and to model moderately thick shellsand layered shells weak in shear.

The thickness of the shell is diyided into layers for the purpose of fiberintegration and different constitutive laws can be employed in each layer.

Finally, the three dimensional brick and the shell element can be em-ployed simultaneously to model different parts of a structure.

6 NUlnerical Results

The finite element approximation of (35) leads to a system of nonlinearequations that are solved by an iterative method. In the examples shown be-low, Newton Raphson's corrections were used, but more elaborate schemeslike the Riks or Crisfield methods [7] can be used, especially in cases wherebifurcations or limit points are expected, at high angular velocities.

In [6] numerical results obtained with the 20 node brick modeling of arubber cylinder were presented. Here we use the 9 node shell element tomodel an idealized tire geometry, made of Hookean rna terial and in contactwi th a rigid foundation.

The figures show the initial configuration of the tire and two stages ofits deformation obtained by inflation and changing the distance betweenthe axis and the foundation.

These are preliminary results, designed to test the element in largedisplacements and rotations, before implementing in full the formulationdescribed in sections 2 and 3.

17

AcknowledgementThe support of this effort by NASA Langley Research Center under

contract No. NAS 1-18137 is gratefully acknowledged.

IS

Figure 6: Undeformed configuration, showing mesh H = 21.

10

Figure 7: Undeformed configuration, showing cross-section.

20

.'

I.I,

L _

-- - ... - -- ---- ...- --lI

Figure 8: Deformed configuration, H = 2-1.5.

21

Figure 9: Deformed configuration, cross section in the 1"21"3 plane.

/Iv

.--,-- I --.-.-

-

Figure 10: Deformed configuration, H = 24.5.

23

"

Figure 11: Deformed configuration, 3d view of exterior surface, H = 24.5.

24

..

/r::~-- r\ --z~""/'--,v .......... ~/~h,. ~ '~ "'-:'\/- ""'-. .... -'

// "~ ,/ 7,\ I -

'I i- :;>- -\0 " /

..."

f ,rl :1

..;::

::: '\n ~

? ~f\I :.~~c~~~~. ~ ~~;:.......

I I ;:~~- ~~.., :=:::;;;=- .-._- ~~

Z,2;:Z:, ! ~ ;:_.~~II ' m] :•• 3: ?'

I ... ---- ;::"",,.,c-~;:: I., ..... ~.._-~:f:. I ..1"';-.-- "'--"-"1;;~;~; :~~-;.:~; I

I, I : ~--- ;:;~-" nl~~~~ I~~miU ..~

.--:i~:i !-;,--.: I . .... ---- ~~~: g~m~...-.-... -...-. =I -.".-

Lc . ... ....,. ..- ...... "' ....... ... _4_ ~ r;~-"~-"'." ~:.=: ..-.....\Ji · ...-...... -- ...... f1f~~· .....,....-· ..-...... : ~:~~:.:.· ....;.:-r- 8V, I I ~ ...- .~~-::c

\ , I '1 < I. -1/1 II \, \.

-' -~ / \.

L. ~,~ ,t!/-, ~~: .....,,"""'i7

~~ ~:: .....>//"(J/

/c......

I \:S)J/

Figure 12: Deformed configuration, H = 24.5. The shaded region representscontact with the foundation.

Figure 13: Deformed configuration, H = 2G.9.

2G

Figure 14: Deformed cross-section, H = 2G.9.

')-_I

References

[1] Frank, F. and Hoffenberth, W., "Mechanics of the Pneumatic Tire,"Rubber Chemistry and Technology, Vol. 40, pp. 271-322, 1967.

(2] Noor, A.K, and Tanner, J.A., "Advances and Trends in the Develop-ment of Computational Models for Tires," Computers and Structures,Vol. 20, pp. 517-533, 1985.

(3] Rothert, H. and Nguyen, B., "Comparative Study of the Incorporationof Composite Material for Tire Computation," in Composite Structures2, ed. Marshall, I.H., Applied Science Publishers, 1983.

[4] Ridha, R.A., "Computation of Stresses Strains and Deformation ofTires," Rubber Chemistry and Technology, Vol. 53, pp. 849-902, 1980.

[5] Kennedy, R.H., Patel, H.P. and McKim, M.S., "Radial Truck InflationAnalysis," Rubber Chemistry and Techn,?logy, Vol. 54, pp. 751-766,1981.

[6] Bass, J.?vL, "Three-dimensional Finite Deformation, Rolling Contactof a Hyperelastic Cylinder: Formulation of the Problem and Compu-tational Results," Computer.! and Structures, Vol. 26(6), pp. 991-1004,1987.

[7] Oden, J.T., Becker, E.B. Lin, T.1., Demkowicz, L., "Formulationand Finite Element Analysis of a General Class of Rolling ContactProblems \Vith Finite Elastic Deformation," MAFELAP83 Confer-ence, BruneI University, United Kingdom, 1983.

,(8] Oden, J.T., Bass, J.M., and Lin, T.L., UA Finite Element Analysis

of the General Rolling Contact Problem for a Viscoelastic RubberCylinder," Tire Science and Technology,(to appear), 1987.

[9] Oden, J.T. and Lin, T.L., liOn the General Rolling Contact Problemfor Finite Deformation of a Viscoelastic Cylinder," Computer Methodsin Applied Mechanics and Engineering, Vol. 57, pp. 297-367, 1986.

[10] Johnson, K1., Contact Mechanics, Cambridge, Cambridge, 1985.

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•

[11] Spencer, A.J.M., Continu.u.m Theory of the Mechanic~ of Fiber Rein-forced Compo~ite~, CISM Courses and Lectures, Springer, Wien, NewYork, 1984.

[12] Walter, J.D. and Patel, H.P., "Approximate Expressions for the ElasticConstants of Cord Rubber Laminates," Rubber Chemi~try and Tech-nology, Vol. 52, pp. 710-724, 1979.

[13] Duvaut, G., "Problemes de contact entre corps solides deformables,"in Application" of M ethod~ of Functional A nalysi~ to ProbleT'f1-3in M e-chanic~, ed. Germain, P., Lecture Notes in l\'lathematics, No. 503,Springer, Berlin, 1976.

[14] Ahmad, S., Irons, B.M. and Zienkiewicz, O.C., "Analysis of Thickand Thin Shell Structures by Cun'ed Finite Elements," InternationalJournal for Numerical Methods in Engineering, Vol. 2, pp. 419-451,1970.

[15] Hughes, T.J.R. and Liu, \V.I~., "1\onlinear Finit.e Element Analysisof Shells: Part 1: Three Dimensional Shells," Computer Methods inApplied Mechanics and Engineering, Vol. 26, pp. 331-362, 1981.

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