Onset of sliding across scales: How the contact topography impacts frictional strength

When two solids start rubbing together, frictional sliding initiates in the wake of slip fronts propagating along their surfaces in contact. This macroscopic rupture dynamics can be successfully mapped on the elastodynamics of a moving shear crack. However, this analogy breaks down during the nucleation process, which develops at the scale of surface asperities where microcontacts form. Recent atomistic simulations revealed how a characteristic junction size selects if the failure of microcontact junctions either arises by brittle fracture or by ductile yielding. This work aims at bridging these two complementary descriptions of the onset of frictional slip existing at different scales. We first present how the microcontact failure observed in atomistic simulations can be conveniently “coarse grained” using an equivalent cohesive law. Taking advantage of a scalable parallel implementation of the cohesive element method, we study how the different failure mechanisms of the microcontact asperities interplay with the nucleation and propagation of macroscopic slip fronts along the interface. Notably, large simulations reveal how the failure mechanism prevailing in the rupture of the microcontacts (brittle versus ductile) significantly impacts the nucleation of frictional sliding and, thereby, the interface frictional strength. This work paves the way for a unified description of frictional interfaces connecting the recent advances independently made at the micro- and macroscopic scales.

Figure 1

Geometry of the problem. The inset presents the schematic shear stress ${\sigma }_{xy}$ profile predicted by LEFM at a distance $r$ from a macroscopic rupture front. A nonlinear region (I) exists at the immediate vicinity of the tip, followed by a linearly elastic region (II), where ${\sigma }_{xy}$ is dominated by the square root singularity. Further away from the tip (III), nonsingular contributions dominate the profile of ${\sigma }_{xy}$, which converges toward the far-field stress conditions. At the onset of sliding, the microcontacts within the nonlinear region (I) can either break by brittle fracture of their apexes or by plastic yielding [28, 30, 31]. Our work aims at describing how these different failure mechanisms occurring at the scale of asperity contact, i.e., “hidden” within (I), impact the onset of sliding and the frictional strength.

Figure 2

The ratio of the process zone size to the length of the junction mediates the work required to initiate sliding. (a) Normalized external work required to initiate sliding along a single uniform junction as function of the ratio between the process zone size $l_{pz}$ and the resisting junction size $(l-w_0)$ for different types of interface properties and geometries. Panels (b) and (c) present the evolution of energies during the onset of sliding, which occurs respectively at $t=92t^{*}$ and $t=35t^{*}$. The two events share the same elastic properties and $G_c=4G_c^{\mathrm{ref}}$, but their respective interface cohesive laws lead to $l_{pz}/(l-w_0)=3.5·{10}^{-2}$ and $l_{pz}/(l-w_0)=3.5$. The dashed lines in panels (b) and (c) present the buildup of elastic strain energy in the absence of interfacial slip discussed in the Appendix.

Figure 3

Evolution of the frictional strength in the presence of microcontact junctions for two representative interfaces differentiated by their respective characteristic junction size ($d_A^{*}<w<d_B^{*}$). (a) The gray circles recall the data discussed previously in Fig. 2. The blue circles correspond to homogeneous (single-junction) interfaces. The red circles are associated to multiasperity interfaces with a heterogeneous microstructure but the same average fracture energy $G_c^{\mathrm{ref}}$. [(b), (c)] Enlargements of the vicinity of the critical nucleus $(x=w_0)$ revealing the origin of the frictional strength difference between interfaces $A$ and $B$ in the presence of microcontacts. Colors depict the shear stress profile existing before the onset of sliding while an artificial vertical displacement $(u_y(x,y)=u_x(x,y))$ is applied to help visualizing the slip profile along the interface (200 times magnification). The evolution of junctions strength is depicted with a gradation from black (${\tau }^{\mathrm{str}}={\tau }_c$) to white (${\tau }^{\mathrm{str}}=0$). The sketches located in the top right of each plot associate the failure of the coarse-grained multicontacts interfaces $A$ and $B$ to the corresponding failure mechanism of surface asperities discussed in Fig. 1.

Figure 4

At a macroscopic distance from the interface, the evolutions of the stress fields observed during the dynamic failure of the heterogeneous interfaces $A$ (top) and $B$ (bottom) comply with LEFM predictions for an interface fracture energy corresponding to the average value $G_c^{\mathrm{ref}}$. On the left panels, shear stress at the vicinity of the propagating slip front is mapped using the same color scale. To mimic the experimental measurements, the white lines highlight the position along which the components of the Cauchy stress tensor are presented on the right panels in red. The stress fields predicted by LEFM at the vicinity of a shear crack are plotted in blue for a fracture energy equal to $G_c^{\mathrm{ref}}$. Note that the mismatch visible in the simulation profiles of ${\sigma }_{xy}$ is caused by the shear wave traveling ahead of an accelerating shear crack which is not included in LEFM solutions of Eq. (A18) [43, 44].

Figure 5

Elastodynamic solution in the absence of interfacial slip. The dynamic fields are mediated by the vertical propagation of a shear wave front characterized by $\mathrm{\Delta }\tau =\mu /c_s{\dot{\mathrm{\Delta }}}_x$. $t^{*}=h/c_s$ is the time needed by the front to travel between the top and bottom surfaces and $n\in \mathbb{N}$ is the total number of reflections observed at the top boundary.

Figure 6

Shear stress profiles before the onset of sliding for the two different failure mechanisms selected by the size of the process zone. In the left plot [$l_{pz}\ll (l-w_0)$], the stress concentrates at the very edge of the junction and the subsequent rupture corresponds to the sharp crack-like event studied in Fig. 2. In the right plot [$l_{pz}>(l-w_0)$], the stress is uniform over the junction and leads to the ductile failure presented in Fig. 2.

Figure 7

External force required to trigger sliding as function of the process zone size for the simulations reported in the Figs. 2 and 3 of the paper. In the large process zone limit, the data follow $F_{\mathrm{ext}}^{\mathrm{str}}$, whose evolution predicted by Eq. (4) is depicted by the black and gray solid lines for the two studied geometries, respectively $w_0/l=0.05$ and $w_0/l=0.1$. With shorter process zone sizes, the external force saturates at $F_{\mathrm{ext}}^{\mathrm{lefm}}$, the value predicted from brittle fracture. The blue and red dots present the values observed for respectively the homogeneous and heterogeneous large-scale simulations. Please refer to the presentation in the main text for more information about the different setups.

Figure 8

From left to right: Typical force vs slip profile observed during the shearing of two interacting asperities in molecular dynamics simulations (see Ref. [32] for a detailed presentation of the method and setup). Such behavior can be conveniently described by the exponential cohesive law given in Eq. (A14) and derived from a Rose-Ferrante-Smith [58] type of universal binding potential. The exponential cohesive law allows for saving the cost of describing the fine details of asperity contact and, in return, the coarse-grained junctions can embed the microcontact failure behavior into the macroscopic response of frictional systems. More notably, these coarse-grained junctions also reproduce the essential observations of the molecular dynamics simulations: the brittle-to-ductile transition in the failure of microcontact junctions controlled by an identical characteristic length scale (see the discussion in the main text). Examples of a very brittle cohesive law in dark blue (${\tau }_c^b$; ${\delta }_c^b$) and a more ductile one in cyan (${\tau }_c^d=0.1{\tau }_c^b$; ${\delta }_c^d=10{\delta }_c^b$) having the same fracture energy.