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

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1 On the Computation of Elastic Elastic Rolling Contact using Adaptive Finite Element Techniques B. Zastrau^, U. Nackenhorst*,J. Jarewski^ ^Institute of Mechanics and Informatics, Technical University Dresden, D Dresden, Germany urz. tu-dresden. de ^Institute of Mechanics, University of the Federal Armed Forces Hamburg, D Hamburg, Germany de Abstract The computation of the stress and strain of a rolling wheel on the rail requires the thorough analysis of the contact condition including slip. Prom the physical point of view it is important to use an appropriate description of the surface roughness and to take account of the rolling condition because this differs significantly from static contact. As afirstapproach a two dimensional model of a wheel running with constant velocity over an infinite elastic strip is investigated. Because of the necessity to use a dense mesh in the contact area and in order to avoid a transient computation it is advantageous to use an arbitrary Lagrangian Eulerian description (ALE). This paper contains a description of the ALE approach for the problem of the elastic-elastic contact of a wheel rolling with constant velocity over a rail of infinite length. The roughness of the surface is used in the formulation of a physically motivated penalty formulation and is exploited for the normal contact condition. The approach for tangential contact of rough surfaces according to Willners model is stated for completeness. A concept of an error estimator on the basis of the surface traction error as a criteria for the refinement of the mesh in the contact area is presented. As a numerical example the solution for a roller is presented in order to demonstrate the practicability of the introduced approach.

2 130 Contact Mechanics HI 1 Introduction The analysis of the contact behavior of wheels is of great interest because of the significance of the contact condition, e.g. for sliding of vehicles and for the prediction of wear. Since static contact and dynamic contact, especially rolling contact differ a lot, special emphasis is laid on the treatment of rolling contact. The main goal of the herewith published preliminary results of this research is the computation of stresses between the wheel and the elastic rail in typical railway applications. Although the real rolling on a rail is a transient process, in a first approach only the steady state motion of rolling of a wheel on an infinite elastic strip is solved. This allows the direct application of the ALE-method. The computation of contact problems is highly nonlinear. There exist only very few analytical solutions for two-dimensional problems like Hertzian contact and steady state rolling of homogeneous cylinders (Bufler [1]) against each other. Due to the three-dimensional character according to the shape of the wheel and the rail a numerical solution has to be obtained. The computation of contact is often done by application of the penalty method with a numerical suitable penalty parameter. In order to treat realistic surfaces it is natural to apply Willners approach [4] exploiting statistical information about the roughness of a surface thus leading to a nonlinear constitutive description for normal and tangential contact. Both laws will be stated here and the computation of the contact pressure is done by this rule. Since the main concern is to use a correct rolling contact condition for elastic-elastic contact, this will be described in more detail and the numerical results presented here are confined to sticking of wheel and rail. Nevertheless it is advantageous to introduce a local slip velocity and a rolling contact penalty parameter to compute the tangential forces. Some basics for error measures of surface traction errors and adaptive meshing are discussed. The presentation of numerical results will be postponed until the effectiveness of the proposed method can be proved, e.g. in comparison to the Z^-method. 2 Basic Equations of Motion For a pair of two contacting bodies Hamiltons Principle reads as 2 / ( (^ -6W' + I 0B' -rjdv + I T' T] da) J \ T _ -I J J ' *i ^~* B' d(b') da dt = 0. (2.1)

3 Contact Mechanics HI 131 In this expression 6T= f Q (B) is the variation of the kinetic energy and dt, dt dv (2.2) 6W = I P -GRADr? dv (B) (2.3) is the variation of the internal energy expressed in terms of the second Piola Kirchhoff stress tensor P. B^ and T* are the body forces and the surface loads acting at body / (/ = 1,2) respectively, while 77 is a test function (virtual displacement). The second term in equation (2.1) describes the virtual work of the contact forces, TN is the normal contact stress, 5g is the variation of the gap, TT is the vector of the tangent contact stresses and 8s is the variation of the slip vector. The integration is performed over the contact zone dc(b). For an efficient numerical solution of the rolling contact problem an Arbitrary Lagrangian Eulerian (ALE) description of the contacting bodies is chosen according to Nackenhorst [2]. Basic idea of this approach is the decomposition into a the rigid body motion and the deformation, which allows to compute a steady state rolling motion without integration with respect to time. Fig. 2.1 gives a scheme of the mathematical idea of the decomposition. The decomposed total motion \& is introduced as the rigid body motion % composed with the deformation c/>, which is measured with respect to the moving reference configuration %. initial configuration deformed configuration reference configuration Figure 2.1: ALE-decomposition into rigid body motion and deformation. Within this kinematical frame the material derivative is defined as d(---) dt dt +GRAB( (2.4)

4 132 Contact Mechanics HI where % describes therigidbody velocity of a spinning wheel as Thus, for example the material velocity reads as <9y X = -^r + n x %. (2.5) y = ^ + GRAD<f>.%. (2.6) Using this, the variation of the kinetic energy can be rewritten as, ' X) (^ + GRAD?7 %) dl/. (2.7) Some final basic transformations lead to the following expression y 6T df = - y ( y <X^ T? - y GRAD7? x) dl/) d^, (2.8) ti ti (B) which provides symmetric matrices for the subsequent finite element approximation. It has been found to be advantageous to introduce a coordinate system following the axis of the wheel, which is moving with constant velocity. Thus it is assumed that a wheel is rotating around a fixed axis, whereas the road or rail is moving under it horizontally. Within this frame the rigid body motion is described for the wheel as X = ^xx (2.9) with the angular velocity ffc and for the road or rail according to X = % (2.10) The advantage of the ALE description is combined with higher numerical effort for the formulation of the sticking condition. From equation (2.6) obviously follows that the sticking condition for rolling contact (2), i.e. the vanishing relative motion of sticking particles, is not derived automatically from the identical displacement because the displacement gradient of the contacting points is not necessarily the same. Thus an additional penalty term a <;2 do, (2.12) is added to the left hand side of equation (2.1) to impose the sticking condition of rolling surfaces.

5 Contact Mechanics HI 3 Constitutive Laws of Contact Interfaces According to Willner [4] a statistical description of the surface roughness is used to derive constitutive laws of contacting surfaces. For each asperity Hertzian contact is assumed in normal direction and the hypothesis of maximum shear in tangential direction is used. By use of a probabilistic method constitutive laws of the contact interface are as follows 7 0 oo oo 7 0 d( (3.1) (3.2) Herein 7, % and ( are the gap, curvature and tip height of an asperity scaled by the statistically derived root mean square values of the contacting surfaces respectively, s is the slip vector, N is the normal contact force of one asperity with the gap 7 and the tip curvature %, T is the tangential force of an asperity with the gap 7 and the slip s and P is the probability that a tip with gap 7 and curvature K, is in contact. These laws of contact interface behavior are illustrated in fig. 3.1 for a typical steel to steel contact "S ts3 13 So normalized 3 4 gap y Figure 3.1: Pressure-gap function (left) and contact shear stress as a function of pressure and slip (Willner [4]). 4 Finite Element Approach The description of the rolling contact problem by the equations (2.1) in addition with the constitutive interface laws (3.1), (3.2) and nonlinear material behavior is a highly nonlinear problem. Thus, for the numerical solution an

6 134 Contact Mechanics HI incremental approach is necessary wherein the equations are linearized consistently. After linearization a finite element approximation of the problem is performed by common techniques, which lead to quasi static incremental equilibrium equations for the steady state rolling process, * + <fc+ t- t. (4.1) In this equation *K is the tangential stiffness matrix, W is the inertia matrix due to the motion, *Kc is the contact stiffness matrix, *Ks is the contact stiffness generated from the linearization of the ALE-sticking conditions, *~^*fg is the matrix of the applied body and surface loads, * is the matrix of the inertia forces, % is the matrix of the contact forces, *f«is the matrix representing the non-equilibrium of the sticking conditions and % is the matrix of the internal force representing the actual stress state. 4.1 Error Estimation and Adaptivity The reliability of finite element results is of great interest especially in applications where no or only few experimental verifications are possible. Error estimation and adaptive mesh refinement methods are developed since more than one decade. In present the adaptive technique has become a common tool in commercial finite element codes, most of them are based on the well accepted Zienkiewicz/Zhu (7?) method [6]. As shown in [3] for critical boundary value problems such as contact analysis this method is not very suitable because the boundary conditions are not emphasized explicitly. Another drawback of commercial FE-codes in general originates from the fact that the common mesh refinement algorithms are not able to deal with arbitrary element types. Especially in three-dimensional computations often these algorithms support thetrahedrical elements only. For alternative adaptivefiniteelement techniques with special emphasis to contact analysis it is referred to [5] and the literature cited there. Because of the aforementioned reasons it is advantageous to follow the concept suggested by Nackenhorst [3]. Basic idea of this approximation is to apply mesh refinement only to those parts of the surface of special interest, namely the contact region. For this an approximation error in the surface tractions is given by \e\c= r, -Te da, (4.2) where TC is a continuous approximation of the surface tractions through equations (3.1) and (3.2) and TC is the representation of the surface tractions by commonfiniteelement solutions.

7 Contact Mechanics HI Numerical Example In order to demonstrate the practicability of the proposed method an elastic roller on an elastic strip has been analyzed. The roller is turning with constant velocity under free rolling conditions. The surface has been assumed to be rough in the sense of Willner and in the contact zone no sliding shall occur. The adopted discretization is according to the ALE-approach dense in the contact area only and coarse in the distant parts of both bodies. The chosen mesh is shown in fig. 5.1 for a deformed state with vertical movement of the axis of 0.04 R- The angular velocity is such small that the exploitation of the rolling contact condition is sufficient while dynamical effects are negligeable. 5R Figure 5.1: Deformed mesh of a rolling elasic wheel on an elastic rail. The plot of the overall deformed configuration shows no difference to Hertzian contact, although in the contact area itself two significant deviations are found. One difference is the development of a smooth boundary of the contact area because of the adopted model of rough normal contact. The appropriate normal stress distribution is shown in fig. 5.2 in a normalized form, since a quantitative comparison with the Hertzian contact is meaningless because of the rigid core of the wheel and the small thickness of the rail. This rigid support leads to development of a broader zone of high pressure at approximately 90% of the maximum value of the static Hertzian problem. And because of the magnitude of indention the width of the contact zone is greater than the value a as being computed according to Hertz. The contact shear stresses are very small thus having little influence on the shear distribution in the inside of the wheel.

8 136 Contact Mechanics III normalized coordinate -J d. 1.0 Figure 5.2: Normalized contact pressure and shear stress distribution of a rolling wheel. maximum shear Tmax Anax 0,09 Q.Q9 Figure 5.3: Maximum shear stress distribution of a rolling wheel on an elastic strip.

9 Contact Mechanics HI 137 O.l-i -O.H Figure 5.4: Development of the normalized shear stress distribution of a rolling wheel due to variation of the ratio of Youngs moduli. Fig. 5.3 presents the distribution of the maximum shear stress inside both bodies. The maximum strain occurs as known from the circular disk inside. Due to the formerly discussed boundary conditions its value of 0.45 Pmax is approximately 30% higher than the value of the corresponding Hertzian solution. The occurrence and distribution of contact shear stresses of rolling cylinders is according to Bufler [1] dependent on the material parameters of the contacting bodies. For cylinders with identical Youngs moduli and Poissons ratio no contact shear stresses will occur. The underlying assumption of Bufler is, that the contact zone is small with respect to the total measures of the body, thus leading to an equivalent stiffness of the contacting cylinders. Both the rigid inner cylinder of the wheel and the finite thickness of the rail lead to different stiffness of the contacting bodies and thus to a shear stress distribution even in the case of identical material parameters. The variation of the Youngs modulus of the contacting bodies changes the maximum shear stress significantly and finally leads to a change of the direction of the tangential forces in the contact area. Nevertheless the maximum value remains small in comparison to the case of elastic-rigid contact. 6 Conclusions Centrifugal forces do not influence the contact behavior significantly at realistic speed, but the rolling contact conditions have to be taken into account

10 138 Contact Mechanics HI accurately. The computation of the slip condition is of central importance since the tangential forces directly correspond to the slip velocity. Further investigation will be necessary for the development of constitutive laws for friction. Especially the transition from sticking to sliding has to be investigated for the model of a rough surface. As known from Nackenhorst in this context the exploitation of adaptive remeshing as theoretically discussed in chapter 4 will be necessary. Acknowledgment This work is supported by the DFG (Deutsche Forschungsgemeinschaft) under contract Z141/5-1, which is gratefully acknowledged. Key-Words elastic-elastic rolling contact, adaptive finite element method, constitutive contact laws, arbitrary Lagrangian Eulerian (ALE) description References [I] Bufler, H. Zur Theorie der rollenden Reibung, Ingenieur-Archiv, 1959, 27, [2] Nackenhorst, U. On the finite element analysis of steady state rolling contact, in: Contact Mechanics I (eds M.H. Aliabadi & C. A. Brebbia), pp , Proceedings of the 1st International Conference on Computational Methods in Contact Mechanics, CMP, Southampton, [3] Nackenhorst, U. An adaptivefiniteelement method to analyze contact problems, in: Contact Mechanics II (eds M.H. Aliabadi & C. Alessandri), pp , Proceedings of the 2nd International Conference on Computational Methods in Contact Mechanics, CMP, Southampton, [4] Willner, K. & Gaul, L. A penalty approach for contact description by FEM based on interface physics, in: Contact Mechanics II (eds M.H. Aliabadi & C. Alessandri), pp , Proceedings of the 2nd International Conference on Computational Methods in Contact Mechanics, CMP, Southampton, [5] Wriggers, P. Finite element algorithms for contact problems, Archives of Computational Methods in Engineering, 1995, 2,4, [6] Zienkiewicz, O.C. & Zhu, J.Z. A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Num. Meth. Engg., 1987, 24,