Frictional Energy Dissipation in Wavy Surfaces

Lazan [1] describes frictional damping as the rate-independent structural damping caused mainly by slip and micro-impacts occurring at the contact interfaces. For metallic assemblies, frictional damping is the orders of magnitude higher than the material damping, and hence, controls the amplitude of dynamic response of structures to a great extent. Hartog [2], Cattaneo [3], and Mindlin [4] provided the classical solutions to the frictional interactions under sliding and partial-slip (microslip) contacts loaded by a constant normal and monotonically increasing tangential forces. Those solutions listed the static friction and applied tangential forces as the parameters tuning the frictional slip and thus dissipation. In addition, frictional slip and force were found to depend on slip rates and system states [5–7]. In the last four decades, additional parameters such as normal load fluctuations and phase difference between dynamic loads imposed on the contact were shown to affect frictional dissipation significantly. High normal pressures lead to suppression of sliding, whereas low pressures promote frictional slip while decreasing the shear tractions over the interface. Therefore, tuning frictional dissipation is possible by continuous preload-control [8–11]. Imposing a dynamic preload and adjusting the phase difference between the preload and tangential vibrations provides an alternative approach to tuning frictional dissipation [12–15].

The effect of surface geometry on frictional dissipation was first embedded in the response of spherical contacts to cyclic loading by Mindlin et al. [16]. The dissipation was found to scale inversely with contact radius under constant normal load and tangential fluctuation amplitude. More recently, frictional dissipation was investigated extensively in arbitrary 2D contacts [17,18], and rough surfaces [15,19,20]. In those works, surface geometry and compliance were found as major factors affecting dissipation. Utilizing those observations, we aim to employ surface wrinkling in this paper to tune frictional energy dissipation in microslip contacts. Tension-induced wrinkling of elastic thin films was studied by Cerda and Mahadevan [21], and strain incompatibility due to Poisson effect was found as the major wrinkling mechanism. Large mismatch in stiffness of constituents was also a key factor in creating wrinkles on composite surface layers such as soft biological tissues and soft-hard materials interfaces [22–24]. Inspired by the mismatch factor, Chen and Elbanna recently showed that a composite film with periodic arrangement of stiff strips transfer-printed on a compliant substrate wrinkles under tension, and the tension and periodic arrangement control the amplitude of wrinkles (corrugation) [25]. We will investigate a similar configuration as in Ref. [25], and tune frictional dissipation on wrinkled (corrugated) surfaces.

The remainder of the paper is organized as follows. The finite-element model (FEM) of a 2D wavy (wrinkled) surface in contact with a rigid plate will be developed and validated in Sec. 2. The effects of corrugation and wavelength of wrinkled surfaces on frictional dissipation under different boundary conditions are presented in Sec. 3. Section 3 also features a theoretical and numerical wrinkling analysis leading to tunable dissipation via loading and geometric parameters. Conclusions are listed in Sec. 4.

We will estimate the energy dissipation due to microslip in a 2D wavy surface in contact with a rigid flat plate by a finite-element model (Fig. 1). During microslip, part of the contact slips while remaining part sticks. Loading conditions and friction coefficient determine the extent of stick and slip.

where $γ=(1−ν2)/(πE)$⁠, and $λ$⁠, $Δ$⁠, $E$⁠, and $υ$ are the spatial wavelength, peak-to-peak height (Fig. 1), Young’s modulus, and the Poisson’s ratio of the wavy surface, respectively. A 2D finite-element model (FEM) is created in abaqus to obtain frictional dissipation. Six-node modified quadratic plane strain triangle elements (CPE6MH) are used and the mesh is finest nearby the contact and gradually coarsens away from the contact region.

Surface to surface contact formulation with Lagrange multipliers for tangential behavior is used to model the contact case. Several models with different geometry are built for parametric investigation of corrugation effects. The modeling parameters used are tabulated in Table 1. Note that the frictional dissipation is mostly converted into thermal energy. In case of low thermal conductivity, that thermal energy can heat up the contact, and thus, alter the contact geometry and material properties in the vicinity of contact. In this study, we neglect the thermal effects by assuming that the thermal conductivity and heat flow rate at the contact are sufficiently high to inhibit any significant rise in temperature within a loading cycle.

Another assumption we make in our model is that the contact loads are small and localized to the peaks of the corrugated profile, and hence do not alter the corrugation. Larger contact loads, however, could result in stress fields diffusing into the soft substrate and modify the corrugation. Hence, our study and results are applicable only to the small contact loads and deformations.

The FEM results of normal contact for different geometry and loading parameters are compared with the analytical expressions in Eqs. (4) and (5). The average errors in contact half-length and pressures are within 5% and 16%, respectively, for all the FE models with the parameters listed in Table 1.

where $p0$ is the maximum normal contact pressure. Figure 2 shows the comparison between the tangential tractions predicted by the analytical expressions in Eq. (7) and the FEM for a specific case (λ = 1.8 mm, $Δ$ = 0.06 mm, constant loading condition) under three different loading conditions: (a) $Q/fP=0.2$⁠, (b) $Q/fP=0.3$⁠, and (c) $Q/fP=0.4$⁠. The tractions correlate well within the stick region, and the maximum mismatch happens at the boundary of the stick–slip zone. The maximum mismatch does not exceed 20%.

“Frictional dissipation” parameter in abaqus is used to trace the mechanical losses due to friction in the FEM. We compared this parameter against the integral definition of energy dissipation in Eq. (3) for the same case as described in Fig. 2 of $Q/fP=0.3$ for a quarter loading period. Contact pressure and slip displacements used in the integral formulation are retrieved from the FEM, and numerical integration over the contact patch is performed. As shown in Fig. 3, the maximum error between the integral formulation and readily available “frictional dissipation” parameter is about 14%. The mismatch is mostly due to the mismatch in shear tractions at the edge of the stick–slip boundary, and error in contact half-length predictions. Similar error prevails for all of the loading conditions, and thus, does not introduce any superficiality to the results. Therefore, we will report the frictional dissipation obtained directly from the FEM.

First, we consider the case with the constant normal displacement at the center of the rigid plate. Then, we apply a varying tangential load as described in Eq. (2) while keeping the normal displacement constant. Figure 4 shows the influence of corrugation amplitude, $Δ$ on the frictional energy dissipation. Here, we choose the wavelength as 1.8 mm, and $Δ$ varies an order of magnitude (from 0.06 mm to 0.6 mm). Energy dissipation shown here is normalized to the smallest value obtained at Q/fP = 0.3. As evident from the figure, less corrugated surfaces dissipate significantly higher energy. For a given normal displacement, flatter surfaces distribute the normal load over larger contact areas and the contact pressure across those areas are less concentrated. Both of those observations lead to increase in contact slip and the extent of slip regions, and thus, facilitate higher frictional dissipation. Figure 4 also shows the trends for different tangential loads. As expected, higher tangential loads lead to higher dissipation thanks to increase in shear tractions, slip, and the extent of slip zones. Flatness of the contact patches is not only controlled by the corrugation amplitude but also by the wavelength. Next, we show the effects of both parameters on the frictional dissipation as a contour plot.

To obtain the contour plot shown in Fig. 5, we vary the wavelengths from 0.9 to 3.6 mm, and the corrugation from 0.06 to 0.6 mm, while keeping the tangential load as Q/fP = 0.3. We follow the same normalization in Fig. 4 and find out that the energy dissipation varies approximately an order of magnitude. The contours show that larger wavelengths and smaller corrugations yield higher energy dissipation. This is mainly because flatter surface profiles lead to more frictional slip. In contrast, when the wavelength is small, the contact tip is sharper, and that leads to more concentrated contact pressures and less slip. This trend is expected to prevail for other loading cases with varying contrast in contours. As evident in Fig. 4, the maximum energy dissipation for $Q/fP$  = 0.4 is nearly twice as the maximum for $Q/fP$  = 0.2. Those differences will impact the contrast in the contour plots.

where $G1$⁠, $G2$ and $ν1$⁠, $ν2$ are the shear moduli and Poisson’s ratios for sphere and flat surfaces, respectively. For a given $Q/fP$ and constant $f$⁠, the energy dissipation given in Eq. (8) scales with $(P02/a)$⁠. Substituting Hertzian contact relations for a fixed normal penetration, $a∼R1/2$⁠, and $P0∼R1/2$⁠; the energy dissipation can be related to the sphere radius as $ΔW∼(P02/a)∼R1/2$⁠. Therefore, flatter surfaces are expected to dissipate more energy than corrugated surfaces under constant normal penetration in both 2D and 3D contact configurations.

Next, we study the case with constant normal load applied at the center of the rigid plate. Similar to the case with the displacement boundary condition, we generate the normalized energy dissipation versus corrugation and the contour plots as shown in Figs. 6 and 7. Energy dissipation is normalized to the smallest dissipation value obtained for Q/fP = 0.3. In Fig. 6, the wavelength is fixed at 1.8 mm, and corrugation amplitude is varied from 0.06 to 0.6 mm. For the contour plot shown in Fig. 7, both the wavelength and corrugation are varied as in Fig. 5. In contrast with the displacement controlled case, both Figs. 6 and 7 show that more corrugated surfaces dissipate more energy. For a given normal load, more corrugated surfaces lead to smaller contact radii and slip zones, but higher contact pressures and shear tractions over the slip zones. Those competing factors nearly cancel out each other, and thus, variation in corrugation results in only slight change in energy dissipation. This competition and weak-dependence on corrugation can be seen in 3D contacts too. As evident from the analytical expression given in Eq. (8), the energy dissipation in spherical contacts is inversely proportional to the contact area under a constant normal load. Therefore, more corrugation leading to smaller contact areas is expected to result in higher frictional dissipation. Mathematically, we can express the contact area in terms of applied load and sphere radius, and substitute in Eq. (8), and show that $ΔW∼R−1/3$ for the constant loading condition. In other words, flatter surfaces will dissipate less energy. However, the energy dissipation depends weakly on the sphere radius when compared to the displacement-controlled case. This is also reflected in the contrast of the contours presented in Figs. 5 and 7. Displacement-controlled case leads to an order of magnitude variation in dissipation in contrast to twofold change in the load-controlled case. We also observe in Fig. 6 that the energy dissipation increases significantly with tangential load. As described before, this is due to substantial increase in shear tractions and the size of the slip zone.

The results presented above suggest that the frictional dissipation can be tuned by surface wrinkling. Surface wrinkling can be induced by various methods discussed in Sec. 1. Here, we study a wrinkling model consisting of stiff strips of length L and height t periodically embedded in a long compliant substrate with height h [28]. As discussed in Ref. [28], the high contrast in elastic properties of the substrate and embedded strips result in significant shifts of the neutral axis off the center of the substrate. Therefore, any distant concentric loading as shown in Fig. 8 leads to spatially varying moments and bending, and thus, wrinkling. The spacing between periodic strips, s and one strip length determine the half wavelength of the periodic-wrinkled profile due to the distant loading imposed by the uniaxial tension p shown in Fig. 8. In Secs. 3.3.1 and 3.3.2, we will first solve this problem numerically and analytically, and relate the wrinkled profile to the distant loading and spacing of the strips. Then, we will utilize the results of Secs. 3.1 and 3.2 to investigate the effect of surface wrinkling on frictional dissipation.

A finite-element model of the geometry shown in Fig. 8 is created in Mathematica (second-order triangular elements). Two-dimensional plane strain formulation is used, and both the substrate and strips are assumed as linear isotropic elastic materials. Note that in Ref. [28], the substrate was assumed as hyperelastic, and thus, finding analytical solution to the problem was difficult especially at large deformations. Despite the large mismatch between the elastic properties of the substrate and strips, they are assumed perfectly bonded. The geometry and material properties used are summarized in Table 2.

In the finite-element simulations, the spacing between the strips, s and the distant uniaxial tensile strain, $εa=p/Esubs$ will be varied, and the distance between the peak and the valley of the upper-edge profile will be reported as the corrugation amplitude, i.e., $Δ$ assuming a sinusoidal profile. The wavelength of that profile will be $λ=2(s+L)$⁠. A representative contour plot of the axial strain is shown in Fig. 9 within the composite strip–substrate domain of one wavelength (shown at the top of Fig. 9). This contour is obtained from the FE model with $s$  = 0.5 mm and $εa$  = 0.08. Axial strains vanish near the stiff strips, and reaches maximum on the edge of the substrate away from the strips. Also, the strain values exceed the applied tension at those edges because of bending-induced tension. Together with this bending-induced excessive tension, alternating periodic arrangements of the stiff strips on different edges of the substrates (top/bottom) result in periodic corrugation. Figure 10(a) shows the wrinkled profile at the top edge of the substrate (y = 0.2 mm in the undeformed configuration) as obtained from the same FE simulations in comparison with the sinusoidal curve that has the same wavelength and corrugation amplitude. Wrinkled profile follows the analytical curve reasonably well throughout one wavelength, and thus, can be assumed sinusoidal. Next, we analyze the spatial variation of axial strains along the neutral axis shown in Fig. 10(b). The axial strain is drawn along the neutral axis, $N0$ shown at the top diagram of Fig. 10(b), of the middle part of the composite. Figure 10(b) reveals the distribution of axial strains along one period of the composite. One can obtain the strains in the whole composite domain by mirroring the strains periodically.

Figure 11 shows the corrugation amplitudes obtained from the FEM and analytical expression of Eq. (13) for different strip spacing and applied strain. The corrugation amplitude increases linearly with the applied strain. Also spacing between the strips influences the corrugation amplitude to a great extent. Influence of applied strain and spacing is well captured by Eq. (13). Yang et al. studied the wrinkling of a PDMS substrate with stiff silicon ribbons by FEA [28]. For small strains (2.5% elongation in their study), they reported corrugations of 13.5, 11, and 8.5 μm for s/L ratios of 150/500, 250/400, and 350/300 μm, respectively. Using the same material properties and geometry of Ref. [28] in Eq. (13), we predict the corrugation amplitudes of 12.9, 10.9, and 8.7 μm for the same s/L ratios. Therefore, we will use Eq. (13) to predict the corrugation amplitudes and tune frictional damping accordingly.

After establishing the relationship between the wrinkled profile (wavelength and corrugation), and the applied strain and the strip spacing, we transform the contour plots presented in Figs. 5 and 7 to the applied strain–strip spacing domain (Fig. 12: (a) for displacement and (b) for force boundary conditions). The transformation from the wavelength-corrugation to the applied strain–strip spacing domain is achieved by the corrugation expression given in Eq. (13), and the approximate geometric relation of $λ=2(s+L)$⁠. Applied strains in those contour plots are limited to 0.5 to ensure relevance in practice.

The displacement boundary condition case presented in Fig. 12(a) shows that increasing the strip spacing and decreasing the applied strain will lead to flatter surface profiles, i.e., less corrugation, and thus, higher frictional dissipations. At a fixed strip spacing of 1.5 mm, the frictional dissipation can be reduced approximately an order of magnitude by increasing the applied strain from 0 to 0.5. Note that Eq. (13) includes the combined effect of the applied strain and strip spacing, and hence, one-to-one transformation between the wavelength-corrugation, and the applied strain–strip spacing domains is not possible. This results in differences between Figs. 12(a), 12(b), 5, and 7.

For the normal load boundary condition case shown in Fig. 12(b), larger strip spacing and larger applied strain leads to higher energy dissipation. In this case, the energy dissipation does not vary as much as in the displacement boundary condition case for the reasons explained above (see Fig. 7).

The effect of geometry of a 2D sinusoidal surface, specifically the corrugation amplitude and wavelength on the contact frictional energy dissipation is studied by numerical and analytical techniques. The FEM results are first validated with known analytical expressions, and then, the frictional dissipation is linked to the corrugation and wavelength of the surface profile. Finally, tension-induced surface wrinkling of a composite surface consisting of a compliant substrate and stiff strips attached periodically is studied. The parameters influencing the wrinkled profile (applied tension and strip spacing) are then used to tune the frictional dissipation. Key findings of the paper can be listed as follows:

• For a 2D sinusoidal surface in contact with rigid plate, large wavelength, and small corrugation will lead to high frictional dissipation under constant normal displacement boundary condition. This is due to the flatter surfaces are expected to dissipate more energy than the corrugated surfaces under constant normal penetration.

• This behavior is reversed and only slight change in energy dissipation is observed under constant normal force boundary condition because the competing factors that cause the change in frictional dissipation are nearly canceled out each other.

• Wrinkling of a composite surface in the form of stiff strips embedded periodically in a compliant substrate can be controlled by tension. The spatial distribution of the stiff strips and applied tension can be used to tune the corrugation of the wrinkled profile and thus frictional dissipation.

This work was supported by the National Science Foundation under Grant No. NSF-CMMI-1462870.