Adhesive Contact of Model Randomly Rough Rubber Surfaces

Abstract

We study experimentally and theoretically the equilibrium adhesive contact between a smooth glass lens and a rough rubber surface textured with spherical microasperities with controlled height and spatial distributions. Measurements of the real contact area A versus load P are performed under compression by imaging the light transmitted at the microcontacts. A(P) is found to be non-linear and to strongly depend on the standard deviation of the asperity height distribution. Experimental results are discussed in the light of a discrete version of Fuller and Tabor’s (FT) original model (Proc R Soc A 345:327, 1975), which allows to take into account the elastic coupling arising from both microasperities interactions and curvature of the glass lens. Our experimental data on microcontact size distributions are well captured by our discrete extended model. We show that the elastic coupling arising from the lens curvature has a significant contribution to the A(P) relationship. Our discrete model also clearly shows that the adhesion-induced effect on A remains significant even for vanishingly small pull-off forces. Last, at the local asperity length scale, our measurements show that the pressure dependence of the microcontacts density can be simply described by the original FT model.

Appendix 1: FT model

In FT’s model, the summits of the spherical asperities have a Gaussian height distribution given by $\phi (z)=\frac{1}{\sigma \sqrt{2\pi }}\exp (-z^{2}/2\sigma ^{2}).$ We consider here non-dimensional heights $\zeta \equiv z/\sigma$ so that the height distribution becomes $\phi (z)=\frac{1}{ \sqrt{2\pi }}\exp (-\zeta ^{2}/2).$ We follow the normalization suggested by Greenwood [20] and define the normalized load ${\overline{P}},$ indentation depth ${\overline{\delta }}$ and contact radius ${\overline{a}}$ as

$${\overline{P}}=\frac{P}{Rw};\quad {\overline{\delta }}=\beta \frac{\delta }{R};\quad {\overline{a}}= \beta \frac{a}{R}$$

(5)

with

$$\beta \equiv \left( \frac{E^{*}\,R}{w}\right) ^{1/3}.$$

(6)

Using this normalization, JKR equations now write

$${\overline{\delta }}={\overline{a}}^{2}-\sqrt{2\pi {\overline{a}}};\quad {\overline{P}}= 4/3 {\overline{a}}^{3} - \sqrt{8\pi {\overline{a}}^{3}}$$

(7)

during loading. Assuming that the asperities do not jump into contact, the average load can be expressed as a function of the normalized mean plane separation $h \equiv d/\sigma$ as

$${\overline{P}}(h)=\frac{N Z}{\sqrt{2\pi }} \int _{0}^{\infty } f({\overline{\delta }}) \exp \left[ -(h+Z{\overline{\delta }})^{2}/2\right] {\text {d}}{\overline{\delta }},$$

(8)

where $Z \equiv (Rw^{2}/E^{*2}\sigma ^{3})^{1/3}=4/3\pi ^{2/3}\theta _{\text {FT}}$ is FT’s adhesion parameter without the numerical factor $(3/4)/\pi ^{2/3}.$ As detailed in Appendix 2 of [20], the above expression can be turned to an explicit integral which can readily be evaluated numerically. Similarly, one can write the microcontact density $\eta$ as

$$\eta (h)=\eta _{0} \int _h^{\infty } \phi (\zeta ){\text {d}}\zeta =-\eta _{0}/2 [{\text {erf}}(h/\sqrt{(}2) )-1],$$

(9)

where $\eta _{0}$ is the microasperity density.