Abstract
If an incompressible elastic layer bonded to a rigid plane is placed close to a second rigid plane, adhesive interactions between the surfaces can cause elastic instabilities. These lead to spatially non-uniform traction and gap distributions which exhibit a regular pattern with a characteristic wavenumber. However, real surfaces are never completely plane. In this paper, we consider the influence of surface roughness on the instability, with particular reference to the force-displacement relation. With random roughness profiles, the traction distributions are always spatially irregular, so the onset of instability is more difficult to define. One approach is to monitor the amplitude of the power spectrum of the distribution near the characteristic wavenumber. Since surface roughness generally reduces the mean adhesive traction, we might expect it to exert a stabilizing effect. Numerical results confirm this for moderate to large RMS amplitudes, but show that low RMS roughness can actually trigger the instability in ranges where the uniform layer would be stable. The resulting traction-displacement relation is then found to be approximately linear with a slope close to that at the point where the uniform solution loses stability.
Highlights
If two bodies with plane surfaces are placed close together, they may experience attractive [e.g., van der Waals’] forces, or forces involving both attractive and repulsive ranges (Jones, 1924; Maugis, 2013)
Since the attractive forces must eventually decay with increasing separation, they have the character of a “negative spring,” which can trigger an elastic instability
If the bodies are incompressible [Poisson’s ratio ν = 0.5], or if a body comprising a thin elastic layer bonded to a rigid plane surface is attracted to another rigid plane surface, the instability may result in a nonuniform [typically periodic] pattern of alternating regions of contact and separation
Summary
If two bodies with plane surfaces are placed close together, they may experience attractive [e.g., van der Waals’] forces, or forces involving both attractive and repulsive ranges (Jones, 1924; Maugis, 2013). If the bodies are incompressible [Poisson’s ratio ν = 0.5], or if a body comprising a thin elastic layer bonded to a rigid plane surface is attracted to another rigid plane surface, the instability may result in a nonuniform [typically periodic] pattern of alternating regions of contact and separation. We can define the mean interface energy per unit area as If both surfaces are plane [i.e., smooth], the state u(ξ , η) = 0, g(ξ , η) = g, σ (ξ , η) = σ (g) is clearly an equilibrium state, but it will be unstable to small sinusoidal perturbations of dimensionless wavenumber ζ if there exists any ζ such that. The critical wavenumber is defined by the maximum of the curve in Figure 2, from which we deduce that the uniform solution will be unstable if and only if −σ ′(g) > E/0.482h
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