Transactions on Engineering Sciences vol 14, 1997 WIT Press, ISSN

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1 Simulation of wear with a FE tyre model using a steady state rolling formulation A. Becker, B. Seifert Tyre Research, Tyre Mechanics, Continental AG PO box 169, D Hannover, Germany abecker@conti.de, seifert@conti.de Summary In order to meet market requirements passenger car and truck tyres have to pass abrasion tests which can be simulated with the help of the finite element method effectively. The scope of the present paper is to summarize the governing equations for rolling contact problems and to analyse a 3D tyre problem where the correlation between highway abrasion and factional energy is illustrated. 1 Introduction In tyre industry, all tyres have to pass a lot of different tyre tests before they can be put on the market. One of the most expensive tests is the highway abrasion test, in which specified tyres have to run about to km. The goal of our Finite Element simulation is to reduce the high test costs by a numerical simulation, and to get a better understanding of the mechanical wear processes for a tyre under different load conditions. Based on a continuum-mechanical approach of steady state rolling, a Finite - Element code has been developed by Continental AG. Within a variational setting of a modified penalty formulation for static and kinematic rolling contact problems, only the precise description of the interaction between tyre and road leads to an accurate FE model. As a numerical example highway abrasionresultsof two types of PC tyres (195/60 R 14) will be compared to results derived from the frictional energy of the rolling FE model. The correlation between both will be presented. Furthermore, big differences between the mechanical behaviour of a driven axle and a non driven axle can be explained by means of the presented approach. This is possible by a detailed investigation of the pressure distribution and the slip velocity distribution in the footprint of tyre structures.

2 120 Contact Mechanics HI 2 Rolling kinematics and variational setting In order to describe steady state rolling of tires there are many different approaches, which can be can be distinguished by their kinematical and variational settings. An often used kinematical approach is the so called Arbitrary Eulerian Lagrandian (ALE) formulation where rigid body kinematics are described in an Eulerian and the deformation in an Lagrangian setting. Alternatively effective descriptions can be derived by coordinate systems which are attached to the roling tyre at its center or at a representative material point. The authors refer to the overwhelming literature concerning this subject for instance the work of Padovan [1987], Oden & Lin [1986] and Tallec & Le Rahier[1994] and references therein. On the other hand the underlying variational settings are the foundation for an effective FE implementation and will therefore strongly influence the computationalresults.a formulation which involves acceleration terms is in some way a natural approach for the dynamic tyre rolling problem because it is closely related to Newtons law (e.g. Radovan [1987]). But this direct approach leads to unsymmetric stiffness matrices and futhermore second order derivatives of the shape functions are needed. Both drawbacks can be avoided, if a variational setting based on Hamilton's principle is used. For steady state rolling Hamilton's principle has a remarkable simple structure and leads to an effective finite element implementation. Within the present approach the position of a material point x(t) in the current configuration is a one to one mapping cp of a material point X(t) of the reference configuration at a representative time t It is useful to measure X(t) with respect to a coordinate system which is attached to the center of the tire. Application of the chain rule leads to the following expression for the time derivative of a material point x. x(t) = GRAD^x-X(t) = F-X(t) With respect toreferencecoordinates at an arbitrary chosen time frigid body dynamics are described with the help of an orthogonal tensor R(t) %(') = *(') S which yields the following final expression for the material time derivative x(t) = F-R(t) - = F-Q(0 -X(t) where the skew symmetric tensor Q = R R^ has been introduced. The equation above is the central kinematical relation and itsfirstvariation 8% = SF O(f) -X(t) is the associated kinematical expression in the variational setting of a rolling finite element formulation. In order to get simplified FE matrices there is an alter-

3 Contact Mechanics HI 121 native approach for the variation of the velocity, where the explicit dependence of the deformation is neglected. x(t) = Here the material time derivative and itsfirstvariation are of a very simple structure which simplifies the resultingfiniteelement formulation at the cost of accuracy remarkabely. The following expressions can be derived directly. x = Q *, 8* = Q 8* In order to get a suitable variational structure forfiniteelement implementation the equations of motion are transfered within standard variational calculus in combination with the information of steady state rolling and Gauss' theorem. The resulting variational expression reads = f J which is the continuum mechanical equivalent to Hamilton's principle in analytical mechanics. According to standard conventions in continuum mechanics W, C and 8 E denote the strain energy, therightcauchy Green tensor and the first variation of the Green Lagrange strain tensorrespectively.furthermore p and/ are the associated mass density and the applied external forces. Based on this variational expression straight forward discretization of SE and 8x gives the resultingfiniteelement formulation. 3 Frictional rolling contact In order to achieve a suitable variational setting for the frictional contact problem a perturbed Lagrangean functional is used. See for instance Simo, Wriggers & Taylor [1985] or Wriggers [1996] and references therein for a detailed description. Within this framework the residual terms are G (jc, 8;c) = H (x, 8%) +y^8^ +/,8g, = 0 where the contributions concerning the standard displacement part for the rolling formulation &H (x, Sx) and the normal f^g^ and tangential part ffig^ of the contact forces are separated. The normal contact force / = A, g depends on the geometrically nonlinear gap function g^ and the penalty parameter &. For this normal contribution a standard numerical treatment concerning linearization within the finite element algorithm is performed. Regarding the tangential contact force a regularized Coulomb friction law is used which reads where /, is the contact force vector, g, the nonlinear tangential gap function with penalty parameter \, p, the tangential frictional coefficient, v^ the slip velocity and CO the associated angular velocity. The factor vj/co can be viewed

4 122 Contact Mechanics HI as a regularized slip velocity with tanh(#) as an arbitrary chosen smoothing function (Zheng [1995]). 4 Frictional energy within the rolling formulation In general frictional energy is the product of the tangential forces within the contact area (i.e. the footprint area of the tyre) times the regularized slip velocities. W FRIC = f\ > t\\footprint CO It is obvious from the equation above, that the numerical errors of the frictional contact algorithm and of the steady state rolling formulation have to be within prescribed tolerances in order to get reliable results. As mentioned above the tangential forces in the footprint area are a nonlinear function of the slip velocities whereas the slip velocity in the footprint area itself is a result of three dimensional finite element solution of the rolling tyre. 5 The tyre 5.1 The tyre components v\ = x = Q -xl. + vl. - vl if \footpnnt * axis \ground 1 Tread 2 Cap ply 3 Belt ply 4 Ply 5 Filler 6 Liner 7 Sidewall 8 Flange cushion 9 Apex 10 Bead Figure 1: The tyre construction The tyre is a complex structure with a lot of different components (Fig. 1). Every component (either a cord reinforcement or a rubber compound) has a specific task. Here, these components are explained only, which have direct influence on the wear behaviour. The tread is the direct connector between the vehicle and the ground, his

5 Contact Mechanics III 123 compound material properties have to achieve a long lifetime of the tyre add his damping properties have to guarantee good traction and braking results. The cap ply is a textile cord directed into circumferential direction. This reinforcement has big influence on the dynamic contour, which has direct correlation to the high speed durability. The two steel belt layered package (with same positive and negative angle against the circumferenrtial direction) is the most important component in the tyre to achieve the necessary stiffness. To get the final shape of the tyre, all components are heated during the production process within a mold. 5.2 Parameter variations Tyres have to fulfill a lot of very different tyre properties - handling, comfort, durability, wear, braking, hydroplaning, rolling resistance etc. - as well as possible. To achieve these often contradictious goals, tyre tests with different tyre specifications are carried through. In this investigation the shape of the mold and all compounds are not changed. The variations are made with following constructive parameters: cap ply belt width belt angle belt construction belt cord distance piecewise single or double layered changed in lateral direction variied in a small range changing the construction of the cord itself variied by some millimeters A greater test plan is carried through (Walloch [1995]), combining different variations of above parameters. Out of this test plan, two variants with small changes of the parameters but with significant differences in the tyre test results are selected for further simulative investigations. 6 The tyre wear tests At Continental, there exist different labor tests to describe the characteristics of wear and abrasion. One testing machine determines the wear resistance capacity of the different compounds. Hereby, homogeneous rubber samples are used. Other in- door test equipments determine the irregular wear sensitivity of the tyre itself under certain global conditions (slip angle, camber angle, wheel load, velocity). To get a clear response on the global wear image, four identical tyres are mounted on a car and the driver has to cover a long distance with fixed conditions. In this case, the route is a km drive on highways with a velocity of approx. 160 km/h. This very expensive abrasion test delivers the,,weight loss" of the tread rubber, the abrasion depth, and the wear image. The results are always separated between the driven and the non-driven axle. For above mentioned two variants, very different abrasion images are received (fig. 2). The variant I shows strong center wear especially at the driven axle. Variant II shows even wear, whereby the lifetime is increased by factor of 2!

6 Contact Mechanics III Driven axle Variant I Eslimated Life: km Variant II Estimated Life: km =r 200 % o CX PR Non driven axle Estimated Life: km Estimated Life: km =170% 43 10r 7 Wear simulation 7.1 Tyre model Figure 2: Global wear results of highway abrasion tests In fig. 3, the discretisized cross-section with all components is presented. The axisymmetric model has approx NDOF, whereby the 3D model has NDOF (fig. 3). A hybrid element formulation is used and a composite material model describes the cord rubber layers. In the present example, the ground is assumed as a rigid plane surface. Figure 3: Discretisized cross-section

7 Contact Mechanics III 7.2 Calculation steps The whole calculation is carried through in several steps: A)Axisymmetric calculation B) 3D Static deflection C) 3D Rolling calculation Prestrains and temperature effects Mold width > rim width Internal pressure Vertical wheel load Velocity Frictional behaviour Slip and camber angle 7.3 Static results Figure 4: Measured and calculated footprint of variant n After every step, it is useful to check the accuracy of the results versus tyre test results. So inaccuracies of the original input can be discovered, that means geometrical or material deviations. One important intermediate check is the comparison of the calculated versus the measured static contact area (called footprint", see fig. 4). A more round footprint is theresultfor the variant I and a more rectangular footprint for the variant II. For tyres a pointwise anti-symmetric distribution of the footprint shape and the contact stresses is given due to the special construction. 7.4 Rolling results Under the conditions of internal pressure 2.1 bar, wheel load 4240 N, velocity 80 km/h, rim width 6J* 14, prestrain of cap ply cords, the comparison of the frictional energy for the non driven axle is carried through (Becker [1996]). The frictional coefficient is variied in order to investigate the wear behaviour on different grounds. For both variants the result versus the frictional coefficient is

8 726 Contact Mechanics HI compared infig.5. The average quotient between variant I and II is the same as the calculated lifetime factor after the highway abrasion test. R 14, Non driven axle Price oral coefficient [-] Figure 5: Comparison of total frictional energy for both variants After the,jree rolling" calculations, a driving moment is added to the structure. This driving moment is necessary to compensate all resistances of the car and of the tires. The characteristic difference between the non driven and the driven state is clearly seen in the dynamic footprint pressure distribution (fig. 6). The resulting contact force of this pressure distribution for the driven tyre is approximately 100 times higher as for the non driven one.,free rolling", M* Rolling with driving moment, M* =150 Nm Figure 6: Dynamic pressure distribution (non driven and driven tire) With the above mentioned rolling friction law, a relationship between the dynamic contact pressures and the slip velocities is given. The slip velocities under the driving moment are high in the leading edge area, very small in the footprint center and very high in the trailing edge area (fig. 7a). The product of tangendal forces and tangential slip velocities lead to the frictional energy distribution (fig. 7b). Here, wear occurs mainly in the trailing area and with a smaller contribution in the leading area. Furthermore, it can be

9 Contact Mechanics III 727 seen that this tyre will have center wear because of the peak in the middle of the trailing area. Figure 7a: Slip velocities Figure 7b: Frictional energy distribution of variant I 7.5 Wear image * Variant I: FE-smJateon v=60 km/h MA=150 Nm ^Variant I: highway abrasion test A * Variant II: FE-SunUabon v=60 knvh MA=150 Nm * ^Variant II: rttfway abrasion test ffi " Z5 S ^ I Figure 8: Wear image measured and calculated The integration of the frictional energy distribution in circumferential direction (fig. 7b) leads to the wear image: this wear image is presented in lateral direction and can be compared to the abrasion depth of the highway abrasion test. For the driven axle, both variants show very fine correlation for both, the depth itself and the distribution in lateral direction. 7.6 Wear while cornering When the car is driven with high velocity through any curve, this cornering situation delivers the biggest contribution to tyre abrasion. For any FE calculation, this is simulated with a specified camber angle and a big slip angle. In this situation, the side force can reach the same value as the vertical wheel load. Under this driving manoeuvre, the whole load is concentrated on one shoulder only (fig. 9).

10 128 Contact Mechanics III Figure 9: Dynamic pressure distribution (velocity 80 km/h, slip angle 8 ) 8. Summary In the present article a finite element algorithm for steady state rolling contact problems has been introduced which allows to analyse rolling tyres efficiently. The presented numerical results are in very good overall agreement with road tests of car tyres and give a remarkable good prediction of tyre wear. 9. References A. Becker [1996], Wear simulation withfiniteelements, Continental AG, Internal report B96/4.1/17. J.T. Oden & T.L. Lin [1986], On the general rolling contact problem for finite deformations of a viscoelastic cylinder, Comp. Methods Appl. Mech. Eng.,Vol. 57, pp J. Padovan [1987], Finite element analysis of steady and transiently moving/ rolling nonlinear viscoelastic structure -1. Theory, Comp. & Structures, No. 2, pp J.C. Simo, P. Wriggers & R.L. Taylor [1985], A perturbed lagrangian formulation for thefiniteelement solution of contact problems, Comp. Methods Appl. Mech. Eng.,Vol. 50, pp P.Le Tallec & C. Rahier [1994],Numerical models of steady state rolling for non-linear viscoelastic structures infinitedeformations, IJNME, Vol.37, F. Walloch [1995], Irregular wear - Final Report, Continental AG, Internal report AN 95/4.2/31. P. Wriggers [1996], Finite element algorithms for contact problems, Archives of Computational Methods in Engineering, Vol.2, pp D. Zheng [1995], Frictional rolling contact element, Continental AG, Internal report AN