Frictional Sliding without Geometrical Reflection Symmetry

The dynamics of frictional interfaces plays an important role in many physical systems spanning a broad range of scales. It is well known that frictional interfaces separating two dissimilar materials couple interfacial slip and normal stress variations, a coupling that has major implications on their stability, failure mechanism, and rupture directionality. In contrast, it is traditionally assumed that interfaces separating identical materials do not feature such a coupling because of symmetry considerations. We show, combining theory and experiments, that interfaces that separate bodies made of macroscopically identical materials but lack geometrical reflection symmetry generically feature such a coupling. We discuss two applications of this novel feature. First, we show that it accounts for a distinct, and previously unexplained, experimentally observed weakening effect in frictional cracks. Second, we demonstrate that it can destabilize frictional sliding, which is otherwise stable. The emerging framework is expected to find applications in a broad range of systems.

Popular Summary

Frictional motion—arising from, for example, tectonic plates sliding over one another along an earthquake fault—is of enormous scientific, technological, and economic importance. Physical factors that weaken frictional interfaces are particularly important because they lead to unstable sliding accompanied by undesired bursty releases of energy, such as squeaking door hinges. Here, we report on a new generic weakening effect associated solely with the size and shape of the sliding bodies.

Traditionally, frictional resistance to sliding is believed to depend on the pressing force that brings the sliding bodies together and on the frictional interaction between the bodies along the contact interface. When the frictional resistance decreases dynamically (e.g., when the sliding velocity increases), unstable frictional sliding occurs. The frictional resistance can also be effectively reduced if the sliding bodies are made of different materials. We theoretically and experimentally show that, in fact, a similar effect exists for sliding bodies made of the same material but of different size or shape (i.e., when the system is not geometrically symmetric). Since no system is perfectly symmetric, this effect generically exists in a broad range of man-made and natural frictional systems and has major implications on their dynamical behaviors.

We expect that our findings will inform various tribological and geophysical investigations.

Figure 1

Examples of physical systems featuring frictional interfaces separating bodies made of identical materials without geometrical reflection symmetry. (a) A thin block sliding over a thicker block. (b) A block of finite height $H$ sliding atop a semi-infinite bulk. Sliding occurs in the $x$ direction. (c) An idealized schematic geometry of tectonic subduction motion.

Figure 2

Experimental results. (a) Snapshots of the spatial profile of the contact area $A$ of rupture fronts in the “thin-on-thick” setup [see Fig. 1 and Ref. [49] for additional details]. These fronts propagate to the right at velocities $c$ indicated in the legend of panel (b) ($0.900c_R<c<0.993c_R$, see Ref. [57]), where $x=0$ corresponds to the tip of each rupture front. The contact area is normalized by its value $A_0$ before the passage of the front. (b) The slip-velocity profiles corresponding to the snapshots in panel (a) (see Ref. [57] for details). (c) $\mathrm{\Delta }A/A_{\infty }$, where $\mathrm{\Delta }A$ is the magnitude of the real contact area undershoot and $A_{\infty }$ is the asymptotic value (see inset) vs the maximal slip velocity $v_m$ [see panel (b)] for both the thin-on-thick setup (red symbols) and the geometrically symmetric “thin-on-thin” setup (blue symbols). Different symbols correspond to different experiments, and their size roughly corresponds to the measurement error. The red line is the best linear fit for the red symbols. (Inset) The contact area profile for $c=0.993c_R$ in the thin-on-thick setup [red line, already appearing in panel (a)] and in the geometrically symmetric thin-on-thin setup (blue line).

Figure 3

Analytical results. (a) The effective shear modulus ${\mu }^{\mathrm{eff}}$ of the thicker block, in units of $\mu$, vs the dimensionless wave number $q=kW$, cf. Eq. (7). (Inset) The variation of the effective Poisson’s ratio ${\nu }^{\mathrm{eff}}(q)$ with $q=kW$. In both the main panel and the inset we used $\nu =0.33$, which is relevant for PMMA [49, 70]. (b) The response function $G_y$, quantifying the effective bimaterial contrast according to ${\mu }^{\mathrm{eff}}(q)$ and ${\nu }^{\mathrm{eff}}(q)$ (for the thicker block; the thinner one is represented by plane-stress conditions), corresponding to selected values of $q=kW$. The corresponding values of the elastic moduli ${\mu }^{\mathrm{eff}}(q)>\mu$ and ${\nu }^{\mathrm{eff}}(q)>\nu$ are marked in panel (a) and its inset using the same color code. (c) $\mathrm{\Delta }{\sigma }_{yy}$ given in Eq. (8), normalized by the experimentally applied normal stress ${\sigma }_0=4.5\mathrm{MPa}$, vs the slip velocity $v$, where the propagation velocity was set to $c=c_R\simeq 1237\mathrm{m}/\mathrm{s}$. The gray dashed line is the red line in Fig. 2.

Figure 4

Linear stability: Simplified analysis. Imaginary (a) and real (b) parts of solutions to the linear stability spectrum in Eq. (10). Here, $\mathrm{Im}(c)>0$ implies an instability. Note that only one solution branch is discussed (other solution branches exist as well, but they are not discussed here). The solid lines show numerical solutions to Eq. (10), and the dashed lines show the approximate analytic solutions obtained by a linear expansion around $c=c_R$. The parameters used are $f=0.9$ and $\beta =0.3$, where $\gamma \equiv \mu /\mathbf{(}{\sigma }_0{c}_s{f}^{\prime }(v)\mathbf{)}$ is varied according to the legend. (c) The instability threshold ${\chi }_c$ (i.e., for $\chi \equiv \gamma f<{\chi }_c$, sliding is stable for all $k$) vs $\beta \equiv c_s/c_d$. The open symbols show direct numerical results, and the solid line is the prediction in Eq. (13).

Figure 5

Linear stability: Generalized analysis. $\mathrm{Im}[c/c_s]$ (i.e., the rate of exponential growth or decay of perturbations; $\mathrm{Im}[c]>0$ corresponds to instability) vs $kH$ for a broad range of physical parameters. In all panels, the parameters are the same as in Fig. 4 with $\gamma =3$. (a) The dependence of $\mathrm{Im}[c(kH)/c_s]$ on $\eta$ for $\mathrm{\Delta }=1$. The curve $\eta =\infty$ identifies with the blue curve of Fig. 4. The case $\eta =1$ corresponds to a symmetric system and is thus stable for all $k$. (b) The dependence of $\mathrm{Im}[c(kH)/c_s]$ on $\mathrm{\Delta }$ for $\xi =1$ and $\eta =\infty$. (c) The dependence of $\mathrm{Im}[c(kH)/c_s]$ on $\xi$ for $\mathrm{\Delta }=0.5$ and $\eta =\infty$.