Two-dimensional FE model for the simulation of hard, multi-scale

Like in the study of Wriggers and Reinelt [34] real surface roughness is approximated here by a superposition of sine waves with different amplitudes and wavelengths (multi-scale model). Wriggers and Reinelt assumed that the different (length-) scales of surface roughness can be separated when calculating viscoelastic friction. The smaller scale roughness generated viscoelastic coefficient of friction is used as input coefficient of friction at the next larger scale. The proposed technique used scale by scale makes it possible to solve the problem of scale transition (interaction of scales). In [34], an abrasive paper surface being in frictional contact with an elastomer counterpart was modeled through the superposition of three as well as four sine waves (amplitudes of sine waves were 15, 6, 3, and 2 m while the correspond-ing wavelengths were 180, 78, 31, and 21 m). The amplitude per wavelength values of the harmonic components were 0.083, 0.077, 0.097 and 0.095, respectively. Although approxima-tion of surface roughness is similar the viscoelastic solid used is drastically different from that of Pálfi et al. [35]. In [34], a very simple viscoelastic solid characterized by 6 relaxation times was used while, in Pálfi et al. [35], a real viscoelastic solid characterized by 40 relaxation times is studied. Consequently FE simulation reported by Wriggers and Reinelt [34] can be considered as a qualitative analysis for the viscoelastic friction. (For additional details on the importance of number of relaxation times see Appendix C.)

In order to examine the hysteresis portion of the coefficient of friction when smooth rubber block (carbon black filled EPDM rubber of TRW, see Appendix C) slides on a hard, rough surface, two-dimensional FE models were constructed by Pálfi et al. [35]. The micro-scopic surface roughness of the hard counter surface was modeled by two different sine waves (counter surface A and B) having a wavelength of 100 µm (amplitude= 4 µm) and 11.11 µm (amplitude= 1 µm) and by their combination (counter surface A+B) where the counter surface B was superimposed on counter surface A. Consequently the amplitude per wavelength ratio for surface A and B is 0.04 and 0.09, respectively. Fig. 4.14 shows the three different surface

roughness models used in [35]. In order to emphasize the difference among the models they were depicted in a single graph by using different scale in the horizontal and vertical direc-tions. The sliding speed was considered to be 10 mm/s. With the help of surface roughness spectrum analysis of a measured rough surface, one can determine the wavelengths and am-plitudes of its harmonic components. Using the harmonic components of the measured rough surface one by one and their combination one can examine the contribution of each compo-nent of the surface roughness (from micro- to nano-level) to the hysteresis friction. Due to the periodicity in the surface roughness it is sufficient to model a small, repetitive segment of the rubber. Naturally, for such a small model the condition of repetition has to be applied as boundary condition. Due to the repetitive symmetry the length of the 2D plane strain FE mod-el in sliding direction was the wavmod-elength of the corresponding harmonic component of the rough surface. According to this the length of the FE models for counter surface A, B and A+B was 100 µm, 11.11 µm, and 100 µm, respectively. The length of the rigid counter sur-face is larger than the wavelength of the irregularity. Its length was specified to insure contin-uous contact with the rubber during the simulation. It means that the rigid counter surface is composed of identical irregularities having a wavelength of 100 or 11.11 µm, thus many ir-regularities make a contact with the rubber during the simulation. The modeled rubber seg-ment had a height of 500 µm to ensure that the load application zone (upper part in Fig. 4.15), which was defined to remain perfectly horizontal during the simulation, may not be able to affect the contact zone (lower part in Fig. 4.15). In the contact zone between the rubber and the rigid counterpart, the FE mesh consisted of elements having a size of 0.25 µm in x-direction. The element size of 0.25 µm was obtained from sensitivity analysis. The rubber part was considered to obey the large strain linear viscoelastic material behavior and discretized by QUAD80 elements [36], while the rough counter surface was modeled as an analytical rigid surface. The sliding contact between the rubber and the counter surface was modeled as fol-lows. As a first step the rubber segment was squeezed with a pressure of p=1 MPa against the counter surface. During the indentation the relative tangential velocity between contacting bodies was zero. At the end of the loading the uniform acceleration phase of the rubber block began. It took up to sliding speed of 10 mm/s. During the acceleration phase the counter sur-face was fixed in vertical direction and the nodes on the lateral walls of the rubber were forced to move identically. The phase of uniform horizontal motion with constant speed of 10 mm/s began after the acceleration phase had finished. In the course of steps 2 and 3 (accelera-tion and mo(accelera-tion with constant speed) nodes on the lateral walls moved horizontally in the same way (constraint equations have been defined for this purpose), in order to model a finite rubber sample being composed of identical sections, while pressure was applied on the top wall (see Fig. 4.15). Furthermore the same condition has been imposed for the vertical dis-placement of the nodes at both lateral walls. Fig. 4.15 shows the deformed shape of the FE model, according to the boundary conditions discussed above. The real rubber rheology was taken into consideration through 40-term generalized Maxwell-models connected parallel with two-parameter Mooney-Rivlin model (see Appendix C). Like in all the cases presented in this thesis FE models were constructed and solved in MSC.Marc. Since the magnitude of the apparent friction force is sum of the horizontal component of the reaction forces the ap-parent coefficient of friction (µForce) was defined as the ratio of the total horizontal and verti-cal reaction force.

Table 4.1 shows the apparent coefficients of friction at three different temperatures computed by viscoelastic material models with different parameters and zero input real coef-ficient of friction at the contact interface (µinterface=0). The apparent coefficient of friction (COF) computed for counter surface A is the highest at T=25ºC while it is the lowest at -50ºC.

It first rises as the temperature increases, then, having reached a peak value, it begins to fall.

Similar tendency can be observed for counter surface B, however, in case of material model

fitted to loss tangent, the highest COF can be observed at 150ºC. In case of counter surface A+B, the viscoelastic friction for all the temperatures is greater than at counter surfaces A or B separately. Its maximum appeared at T=25ºC while at -50ºC nearly identical hysteresis COF can be computed. At T=25ºC and 150ºC the highest COF is generated at counter surface A+B, followed by counter surface B and then by counter surface A. The higher apparent coef-ficients of friction obtained for surface B can be explained by the higher amplitude per wave-length ratio.

Fig. 4.14. Surface roughness models used in [35]

Fig. 4.15. Deformed shape of the FE model and the applied boundary condi-tions (u – displacement in x-direction; v –

displacement in y-direction) [35]

Table 4.1. Average steady-state apparent coefficient of friction and the average penetration depth at different temperatures in case of 40-term viscoelastic models (µinterface=0) [35]

4.3.6. Three-dimensional FE model for the simulation of hard,