Transition from Static to Dynamic Friction in an Array of Frictional Disks

The nature of an instability that controls the transition from static to dynamical friction is studied in the context of an array of frictional disks that are pressed from above on a substrate. In this case the forces are all explicit and Newtonian dynamics can be employed without any phenomenological assumptions. We show that an oscillatory instability that had been discovered recently is responsible for the transition, allowing individual disks to spontaneously reach the Coulomb limit and slide with dynamic friction. The transparency of the model allows a full understanding of the phenomenon, including the speeds of the waves that travel from the trailing to the leading edge and vice versa.

Figure 1

The model consists of $N$ identical disks (here and in the simulations below, $N=10$) which interact via Hertz and Mindlin forces between themselves and the substrate below. A constant force $F_y$ is applied to press them against the substrate, and an external force $F_x$ is applied to the first disk, increasing it quasistatically until a pair of complex eigenvalues gets born, signaling an oscillatory instability. From that point on, the Newtonian dynamics takes the system from static to dynamical friction.

Figure 2

Two examples of a bifurcation of a pair of imaginary parts of eigenvalues upon increasing the external force $F_x$. A pair of real eigenvalues coalesces when the pair of imaginary eigenvalues bifurcates. The first bifurcation results in an oscillatory instability, but the force $F_x$ is not sufficient to trigger a transition to dynamical friction. The second does.

Figure 3

Dynamical response of $M(t)$ at various values of $F_x$. The two lowest values are within the range of existence of an oscillatory instability (see Fig. 2), but $F_x$ is not large enough to trigger a transition to dynamical friction. For $F_x=0.062$, the system is stable and $M(t)$ remains minute. The largest value of $F_x=0.083$ is after the birth of the new pair of complex eigenvalues, and now the oscillatory instability triggers the transition to dynamical scaling.

Figure 4

The actual trajectory in the $x\text{−}y$ plane (for $t\approx 17500$) that is induced by the instability. Here we show the fourth grain ($i=4$), but this is representative for all grains.

Figure 5

Upper panel: The wave of breaching of the Coulomb law, starting with the first disk and ending with the last. Every realization is noisy, but averaging on 25 realizations yields this plot which supports a wave velocity $V_1(F_x)$ that depends on the external force $F_x$. Lower panel: The wave of actual sliding, starting with the last disk and ending with the first. This backward running wave has a velocity $V_2$ that does not depend on the external force $F_x$.

Figure 6

Projection of the shear-wave eigenfunction of the Jacobian matrix on the degrees of freedom $q_{\ell }$, $x$ modes for $\ell =1–10$, $y$ modes for $\ell =11–20$, and angular modes for $\ell =21–30$. This mode is predominantly a $y$ shear wave that is responsible for initiating the sliding motion. The wavelength of this up-down shear wave is approximately 2.

Figure 7

The transition from static to dynamical friction as measured by the dynamics of the center of mass. Upper panel: The motion of the $x$ component of the center of mass for $F_x=0.087$. The upper inset shows the $y$ component of the center of mass that oscillates without growth. The lower inset shows the transition itself from static to dynamics. Lower panel: The dynamics of the $x$ component of the center of mass for different values of $F_x$. The inset shows how the velocity of the center of mass depends on $F_x$.