Frictional dynamics of viscoelastic solids driven on a rough surface

We study the effect of viscoelastic dynamics on the frictional properties of a (mean-field) spring-block system pulled on a rough surface by an external drive. When the drive moves at constant velocity $V$, two dynamical regimes are observed: at fast driving, above a critical threshold $V^c$, the system slides at the drive velocity and displays a friction force with velocity weakening. Below $V^c$ the steady sliding becomes unstable and a stick-slip regime sets in. In the slide-hold-slide driving protocol, a peak of the friction force appears after the hold time and its amplitude increases with the hold duration. These observations are consistent with the frictional force encoded phenomenologically in the rate-and-state equations. Our model gives a microscopical basis for such macroscopic description.

Figure 1

Sketch of the viscoelastic interface model. (a) The interface consists in blocks (labeled $i,i+1,\cdots$) located at the positions $h_i,h_{i+1},\cdots$ (empty squares) and bound together via a combination of springs ($k_1,k_2$) and dashpots (${\eta }_u$). Driving is performed via springs $k_0$ linked to the position $w=Vt$. (b, c) The asperities that provide the contact (highlighted with ellipses) hinder the sliding of the block on its substrate, with a random force $f_i^{\text{dis}}\left(h_i\right)$ acting as a microscopic static friction force. (c) The solid sled on the substrate and the asperities under stress are changed.

Figure 2

(a) Mean-field model of Eqs. (1) and (2). (b) Equivalent model if the solution is stationary.

Figure 3

Evolution of the stress $\sigma$ over time for three velocities. At slow driving, $V=0.001$ (large amplitude), $V=5$ (small amplitude), the stress oscillates periodically, with an amplitude that we denote $\mathrm{\Delta }\sigma \left(V\right)$. At faster driving, $V=9.8$ (lower curve, almost always constant), the stress reaches a stationary value after a very short transient. The precise choice of initial condition only impacts the transient regimes. We used $k_0=0.01$, $k_1=0.1$, $k_2=0.9$, and $\overline{z}=0.1$.

Figure 4

Friction force against driving velocity, for different values of $k_0$ ($k_0=0.3,0.1,0.01,0.001,0.0001$, from left to right); and constant $k_1=0.1$, $k_2=0.9$, $\overline{z}=0.1$. In the slow driving regime (on the left), motion occurs mostly through the abrupt slips during which stress falls from the stress before slip, ${\sigma }_B$ (up), to the stress after slip, ${\sigma }_A$ (down). In the faster driving regime, the dynamics is stationary and friction decreases with velocity. Crosses indicate the values of $V$ for which the numerical integration has actually been performed.

Figure 5

Phase diagram in the $k_0\text{−}V$ plane, for different sets of $(k_1,k_2)$ values: from bottom to top we used $(k_1=0.0,k_2=1.0),(0.1,0.9),(0.5,0.5),(0.9,0.1)$. In the lower-left region, the system behaves in a nonstationary way; i.e., we have stick-slip motion. For $k_0\ge k_2$, the system is never nonstationary, even at quasistatic driving, as predicted in Ref. [35]. The bold dashed line has slope one, indicating a scaling of the critical velocity as $V^c\sim k_0^{-1}$ for small $k_0$'s. The thin dashed vertical lines indicate the asymptotic behavior $V^c\to 0$ when $k_0\to k_2^{-}$.

Figure 6

Velocity weakening in the steady-slip regime ($V>V^c$), far from the critical point. We observe that $\sigma$ decreases when the velocity increases. For $V\to \infty$, the decay in the friction force toward the limiting large-velocity value ${\sigma }_{V=\infty }$ goes as $\sim V^{-1}$. Inset: plot in the log-log coordinates. The dashed line gives the pure power law with exponent $-1$. We used the same color code as in previous figures (curves that go to larger values are the lower $k_0$'s).

Figure 7

Slide-hold-slide experiment. Left: Evolution of the stress $\sigma$ over time under this protocol, around the time when driving is resumed ($t=0.015$ here). The driving velocity is $V=1000$. Right: For many different hold times $\mathrm{\Delta }t$, we record the maximal deviation ${\left(\mathrm{\Delta }\sigma \right)}_{\text{max}}$ from stationary stress (blue points). This deviation corresponds to the gap between static and kinetic friction coefficient. In our model we observe a linear growth of the gap with $\mathrm{\Delta }t$ while in experiments a logarithmic increase is measured. The dashed line gives the purely exponential saturation: $\mathrm{\Delta }{\sigma }_{\text{max}}=\mathrm{\Delta }{\sigma }_{\infty }(1-e^{-\mathrm{\Delta }t/\tau })$, with $\tau =2$ and $\mathrm{\Delta }{\sigma }_{\infty }\equiv \mathrm{\Delta }{\sigma }_{\text{max}}(\mathrm{\Delta }t=\infty )$.

Figure 8

Friction force as a function of velocity for the same set of values as described in the caption of Fig. 4, using the RS formalism. The form of the steady-sliding part of the curves is introduced by hand. We used a value $D_c=0.06$ to achieve the best fitting. The RS formalism yields the form of ${\sigma }_{\text{max}}$ and ${\sigma }_{\text{min}}$ for $V<V^c$.

Figure 9

Sketch of the one-dimensional viscoelastic interface model. The interface itself (bold black line) consists in blocks set at discrete lattice sites $i,i+1,\cdots$ along the $x$ axis (in higher dimensions, $x\in {\mathbb{R}}^d$). The blocks can evolve along the $z$ axis: they are identified by their locations $h_i,h_{i+1},\cdots$ (empty squares) and bound together via a combination of springs ($k_1,k_2$) and dashpots (${\eta }_u$). The viscoelastic interaction introduces an internal degree of freedom ${\varphi }_i$, represented by full squares (blue). Driving is performed via springs $k_0$ linked to the position $w=Vt$ (thin purple lines). The disorder force $f_i^{\text{dis}}$ (red) for the site $i$ derives from a disordered energy potential $E_i^{\text{dis}}\left(z\right)$ (gray).

Figure 10

Convergence of the stress to its limit value $\sigma \left(V\right)$, as the binning $\varepsilon$ decreases. We selected velocities $V$ close to but larger than $V^c$ for the five values of $k_0$ studied (from top to bottom: decreasing $k_0$ and increasing $V$). Convergence is reached for the three largest values of $k_0$.