Nucleation and propagation of solitary Schallamach waves

We isolate single Schallamach waves—detachment fronts that mediate inhomogeneous sliding between an elastomer and a hard surface—to study their creation and dynamics. Based on measurements of surface displacement using high-speed in situ imaging, we establish a Burgers vector for the waves. The crystal dislocation analogs of nucleation stress, defect pinning, and configurational force are demonstrated. It is shown that many experimentally observed features can be quantitatively described using a conventional model of a dislocation line in an elastic medium. We also highlight the evolution of nucleation features, such as surface wrinkles, with consequences for interface delamination.

Figure 1

Schematic of the experimental setup, with reference coordinates. Images on the right show the contact regions for the cylindrical (top) and spherical (bottom) lens geometries. Scale bar corresponds to 200 $\mu \text{m}$.

Figure 2

Four frames from a high-speed sequence showing the nucleation of Schallamach waves (top row) with a schematic side view (bottom row). (a) Circular contact region before application of tangential force. (b) Change in shape of contact region and accompanying buckling instability that initiates wave nucleation. (c) The surfaces readhere ahead of the lens (point B). (d) A single Schallamach wave pulse travels through the contact region. $v_s=20\mathrm{mm}/\mathrm{s}$, spherical lens.

Figure 3

Wrinkles accompanying the nucleated wave. (Left) Initial pattern, with wrinkle wavelength ${\lambda }_0=18\mu \text{m}$. (Right) Increase in wrinkle wavelength to ${\lambda }_1=40\mu \text{m}$ due to continued application of tangential stress.

Figure 4

Schallamach wavefront and its propagation features. (Top row) Sequence of images showing propagation of a solitary Schallamach wave. The full length of the contact region is about $20\times$ the length shown in the images. (Bottom) Three-dimensional intensity plot, derived from the images, showing various features on the elastomer surface. A, initial adhesive contact between the surfaces; B, front of Schallamach wave; C, extent of trapped air pocket comprising the wave; D, wrinkles on the surface; and E, incomplete readhesion after wave passage. Wave velocity $v_w\simeq 110\mathrm{mm}/\mathrm{s}$. $v_s=2.5\mathrm{mm}/\mathrm{s}$, cylinder lens.

Figure 5

Propagation of single Schallamach wave pulses in an adhesive contact. The waves retain their profile over long distances. The length of the contact region is 2.5 cm. Scale bar corresponds to $250\mu \text{m}$; wave velocity $v_w\simeq 380\mathrm{mm}/\mathrm{s}$. $v_s=10\mathrm{mm}/\mathrm{s}$, cylinder lens

Figure 6

Pinning of a single Schallamach wave by static dirt particles. The wavefront approaches a single dirt particle A (top row), and is bent by it (point B, middle row). The wave then regains its original shape, while leaving behind an air pocket C around the dirt particle.

Figure 7

Burgers vector and group velocity of a Schallamach wave pulse. (a) Mean surface displacement due to a single Schallamach wave, for various values of $v_s$; the magnitude of the jump denotes the $\left|\mathbf{b}\right|$ of the wave. (b) Space-time diagram showing local velocity $|{\mathbf{v}}_p|$ for points on surface, $v_s=2.5\mathrm{mm}/\mathrm{s}$. AB and CD denote the front and rear of the wave pulse. Cylinder lens.

Figure 8

Normal $\left(F_N\right)$ and tangential $\left(F_T\right)$ forces, and generation frequency ($n$) for Schallamach waves. (a) Time variation of $F_T$ (blue or dark gray) and $F_N$ (green or light gray) for $v_s=1\mathrm{mm}/\mathrm{s}$. Each oscillation of $F_T$ represents the propagation of a single wave. Critical force for nucleation $F_C$ is marked by point ${\mathrm{A}}_1$. (b) Dependence of $n$ on $v_s$. The critical velocity for Schallamach wave formation is $v_c=150\mu \mathrm{m}/\mathrm{s}$. Cylinder lens.

Figure 9

Geometry of equivalent line during propagation of Schallamach wave. $2a$ is the width of the contact region. $\mathbf{t}$ denotes tangent vector along the discontinuity line, which has inclination $\theta$. $\mathbf{b}$ is the Burgers vector. Cylinder lens.