Brittle Fracture Theory Predicts the Equation of Motion of Frictional Rupture Fronts

We study rupture fronts propagating along the interface separating two bodies at the onset of frictional motion via high-temporal-resolution measurements of the real contact area and strain fields. The strain measurements provide the energy flux and dissipation at the rupture tips. We show that the classical equation of motion for brittle shear cracks, derived by balancing these quantities, well describes the velocity evolution of frictional ruptures. Our results demonstrate the extensive applicability of the dynamic brittle fracture theory to friction.

Figure 1

The experimental system and definition of the dynamic stress drop. (a) Two identical PMMA blocks are used in a stick slip friction experiment. The elastic medium is considered to be 2D with a quasi-1D frictional interface, as the 5.5 mm width of the blocks is smaller than any other system dimensions. Full 2D tensorial strains are measured every $1\mu \mathrm{s}$ at 16–19 locations by miniature Rosette strain gauges mounted $\sim 3.5\mathrm{mm}$ above the interface (blue squares). Slip events are nucleated by applying minute out of plane perturbations at $x\approx 0$ (green arrows). (b) The evolution of the real area of contact, $A(x,t)$, for a typical rupture front propagating along dry interface (see the text). Nucleating at $x\approx 0$, the front rapidly accelerates towards the Rayleigh wave speed, $C_R\approx 1237\mathrm{m}/\mathrm{s}$, leaving in its wake a reduced area of contact. (c) Shear stresses ${\sigma }_{xy}$ plotted relative to the rupture tip position $x_{\mathrm{tip}}=105\mathrm{mm}$. Green line: Rupture event in (b), propagating with the local rupture velocity $C_f\approx 0.95C_R$. Blue line: A slow rupture propagating at $C_f\approx 0.1C_R$. ${\sigma }_{xy}^0$ is the initial stress level, prior to the rupture arrival, while ${\sigma }_{xy}^r$ denotes the residual stress measured at times corresponding to $x-x_{\mathrm{tip}}=-40\mathrm{mm}$. Their difference $\mathrm{\Delta }{\sigma }_{xy}={\sigma }_{xy}^0-{\sigma }_x^r$ defines the dynamic stress drop. Smaller $\mathrm{\Delta }{\sigma }_{xy}$ accompany slower ruptures.

Figure 2

Determining rupture velocity scaling. Colors represent the analysis of three rupture events that include the examples in Fig. 1. $l$: Rupture length. (a) Solid dots are measured $\mathrm{\Delta }{\sigma }_{xy}$. $\mathrm{\Delta }{\sigma }_{xy}$ near $x=0$ are extrapolated to $\mathrm{\Delta }{\sigma }_{xy}=0$ at $x=0$ [21]. (b) Measured rupture velocity profiles $C_f(l)$. (c) The profiles of $\mathrm{\Delta }{\sigma }_{xy}$ yield static energy release rates $G_{\mathrm{II}}^S(l,\mathrm{\Delta }{\sigma }_{xy})$ (see Ref. [21] for details). Dashed line: The measured fracture energy profile $\mathrm{\Gamma }$. Slow rupture fronts (blue line) are associated with $G_{\mathrm{II}}^S\approx \mathrm{\Gamma }$, while ruptures rapidly accelerate towards $C_R$ (green line) once $G_{\mathrm{II}}^S\gg \mathrm{\Gamma }$. (d) $C_f(l)$ profiles collapse to a single functional form $C_f(G_{\mathrm{II}}^S/\mathrm{\Gamma })$, given by (black line) the classical equation of motion for shear cracks, Eq. (2).

Figure 3

Generality of the equation of motion. $C_f(l)$ profiles, in experiments with dry (a) and boundary-lubricated interfaces (b), having significantly different values of $\mathrm{\Gamma }$ and spanning the whole sub-Rayleigh regime. All rupture events follow the LEFM prediction (black curves) given by energy balance [Eq. (2)] for brittle cracks (right).

Figure 4

Comparison to LEFM predictions for rupture velocities approaching $C_R$. (Left) Maximal material velocities ${\dot{u}}_x$ measured at points ($l$, $y=3.5\mathrm{mm}$) for the ruptures presented in Fig. 3. (Right) Collapse of these measurements is obtained, as described by LEFM, when ${\dot{u}}_x$ amplitudes are plotted with respect to $G_{\mathrm{II}}^S/\mathrm{\Gamma }$ and scaled by $\sqrt{\mathrm{\Gamma }}$. Note that ${\dot{u}}_x$ amplitudes span 2 orders of magnitude and increase significantly as $C_f\to C_R$. The solid line designates the values predicted by LEFM. The values of ${\dot{u}}_x$ provide a more detailed comparison with LEFM than the equation of motion for $C_f\to C_R$ as occurs for $G_{\mathrm{II}}^S\gg \mathrm{\Gamma }$.