Two-dimensional single asperity contact model for the simulation of

Here the asperity-asperity interaction is simulated qualitatively using a two-dimensional plane strain FE model (see Fig. 4.5) in which both the stationary and the moving asperities are modeled by cylinders. Notwithstanding that the radius of cylinders is greater than the radius of curvature of typical asperities it is believed that the FE simulations per-formed by Soós and Goda [28] are able to capture the essential characteristics of the asperity-asperity interaction. For the sake of clarity it must also be noted that such a single asperity-asperity contact model does not take into consideration the interaction between the asperities located on the same surface. The interfacial adhesion is neglected and the material behavior is mod-eled by using the simplest description of rubber rheology i.e. the small strain linear viscoelas-tic Standard-solid model (see Appendix A) with glassy modulus of 7.5 MPa and relaxed mod-ulus of 1.5 MPa. Contrary to the broad relaxation spectrum of real rubber-like materials the material model used has a single relaxation time ( 5.5810^3 s) only. These material model parameters result in practically zero loss modulus (E”) and loss factor (tan ) at excitation (angular) frequencies lower than 0.1 rad/s or higher than 10⁵ rad/s. The peak value of loss modulus and loss factor curves appears at an excitation (angular) frequency of about 100 rad/s, which corresponds to sliding speed of v8mm/s ( f v 2R, where f denotes the

excitation cyclic frequency in [Hz]). The simulations have been performed under approxi-mately constant mean normal force as well as constant overlap (s =0.0378 mm = const). In the former case, the actual magnitude of overlap was determined by the fact that the mean normal force (F_y) to be kept approximately constant during the simulations independently of sliding speed. To compute the mean normal force reaction forces in y-direction were averaged over a cycle of contact.

Fig. 4.5. Two dimensional (plane strain) FE model. Asperities of contacting rough surfaces are replaced with cylinders having radius of R=0.25 mm and the steel asperity is modeled as a

perfectly rigid body with velocity varied between 0.1 mm/s and 100 mm/s. u_x and u_y denote displacement in x- and y-direction. [28]

During the mechanical interaction, work is done as the steel cylinder moves forward. This work is required to deform the rubber in front of the hard asperity. At the same time elastic energy is recovered from the rear. Since rubber has viscoelastic material behavior it exhibits hysteresis and thus one portion of the work done is lost. If the interfacial adhesion and the friction at steel/rubber interface caused by physical processes other than adhesion are neglect-ed (the input coefficient of friction was set to 0), hysteresis is the only source of frictional work. If the rubber were ideally elastic the elastic energy recovered would be identical the work done to deform rubber and no energy would be lost. In this case, the frictional work would be zero. Fig. 4.6 shows the normal force F_y (reaction force in y-direction) and tangen-tial force F_x (reaction force in x-direction) as a function of the position of steel asperity. At smaller sliding speeds rubber is less stiff (excitation frequency dependent material behavior of rubber) thus to obtain approximately the same mean contact normal force a larger overlap is required. By increasing the sliding speed the elastic modulus of rubber increases and the over-lap - to obtain approximately the same mean force – decreases due to the increasing excitation frequency. In Fig. 4.6a, each curve has the same tendency, that is, the absolute value of the force increases as the steel asperity moves toward the center of rubber one, reaches a maxi-mum and then falls to zero. As it can be seen, both at small and high sliding speeds the curves, as a good approximation, are symmetrical. The reason for this is that the rubber be-haves elastically at very low (rubbery region) and very high excitation frequencies (glassy region). At sliding speeds between the rubbery and the glassy region (for example at 2.5mm/s) the curves of normal force are asymmetrical. The force reaches its maximum before the centers of asperities would be in the same horizontal position (x=0). When rubber behaves elastically the tangential force curves (see Fig. 4.6b) resemble a sine wave. In these cases, the average value of tangential force, over a cycle of contact, is zero. In the viscoelastic region (for example at v=2.5mm/s) force curves lose their sine wave form. Therefore, the average

tangential force and hereby the apparent friction force will be zero no longer. Here the appar-ent friction force is defined as the mean value of the tangappar-ential force over a cycle of contact.

(a)

(b)

Fig. 4.6. (a) Normal force F_y and (b) tangential force F_x as a function of horizontal position of steel asperity at different sliding speeds (approximately constant mean normal force) [28]

Figs. 4.7a-b show similar tendencies for sliding speed dependency of the average apparent friction force and the apparent coefficient of friction. Amount of energy loss and hence mag-nitude of the apparent friction force and apparent coefficient of friction are, as anticipated, very small at low (v0.1mm/s) and high (v10³ mm/s) sliding speeds due to the quasi-elastic material behavior. Between these speed domains the rubber used shows rate-dependent material behavior (viscoelastic region). As it can be seen in Fig. 4.7a the energy loss and the apparent friction force are much greater in case of constant overlap than in case of approxi-mately constant mean normal force. At the same time, the difference in apparent coefficient of friction is somewhat smaller. The cause of this is that the apparent coefficient of friction is defined, in conventional sense, as  F_x/F_y. In case of constant overlap, both the tangential and the normal force increase with increasing sliding speed while, in case of approximately constant mean normal force, the increasing sliding speed has an influence on the magnitude of tangential force only. Since the rubber becomes stiffer with increasing sliding speed, the

mag-nitude of overlap and the size of contact area have to vary in case of approximately constant mean normal force.

(a) (b)

Fig. 4.7. Variation of (a) the average apparent friction force and (b) the apparent coefficient of friction as a function of sliding speed [28]

4.3.2. Two-dimensional single asperity contact model for the