Significance of Elastic Coupling for Stresses and Leakage in Frictional Contacts

We study how the commonly neglected coupling of normal and in-plane elastic response affects tribological properties when Hertzian or randomly rough indenters slide past an elastic body. Compressibility-induced coupling is found to substantially increase maximum tensile stresses, which cause materials to fail, and to decrease friction such that Amontons’ law is violated macroscopically even when it holds microscopically. Confinement-induced coupling increases friction and enlarges domains of high tension. Moreover, both types of coupling affect the gap topography and thereby leakage. Thus, coupling can be much more than a minor perturbation of a mechanical contact.

Figure 1

(a) Contact setup: a rigid indenter sliding along the $x$ axis at constant velocity $v_0$. Vector components in the $x$, $y$, and $z$ direction are called longitudinal, transverse, and normal, respectively. (b) Cross sections of a compressible (left) and confined (right) body loaded by sinusoidal surface stresses, whose extrema are indicated by red arrows. Stripes and shapes represent coupling-induced longitudinal (top) and normal (bottom) displacements, respectively.

Figure 2

Panels (a),(b),(d) and (c),(e) relate to Hertzian indenters of radius $R$ and randomly rough indenters, respectively. (a) Contact pressure $p_{\mathrm{z}}$ (top row) normalized to the maximum regular Hertzian contact pressure $p_{\mathrm{H}}$, relative longitudinal $v_{\mathrm{l}}^{\mathrm{rel}}$ (second row), transverse ${\dot{u}}_{\mathrm{t}}$ (third row) velocity, and the maximum principal surface stress ${\sigma }_{\mathrm{I}}^{\max }$ (bottom row) for the reference ($\nu =0.49$, $h\to \infty$, left column), the semi-infinite compressible (middle column), and the confined nearly incompressible (right column) elastomer. The applied normal load is $F_z=0.01E^{*}{R}^2$ yielding a Hertzian contact radius of $r_{\mathrm{H}}\approx 0.2R$ (gray circles). Coupling-induced relative changes in (b) contact area $\mathrm{\Delta }{A}_{\mathrm{c}}/A_{\mathrm{c}}$ and (d) friction coefficient $\mathrm{\Delta }\mu /{\mu }_{\mathrm{c}}$ as functions of dimensionless normal load $F_z/E^{*}{R}^2$. (c) Rough contact cross section, and (e) normalized relative change of the friction coefficient vs the dimensionless normal pressure $ph/E^{*}{\lambda }_{\mathrm{l}}$ for confinement-induced coupling. Moreover, $\overline{g}$ and ${\lambda }_{\mathrm{l}}$ are the rough surface’s root-mean-square gradient and long-wavelength cutoff, respectively.

Figure 3

Top row: maximum tensile stress normalized to $E^{*}$ for a (a),(b) compressible and (c),(d) confined elastomer. Panels (a),(c) were deduced from static simulations and lateral stress added in postprocessing, whereas (b),(d) are based on full sliding simulations. Bottom row: (e)–(h) leakage current density for different fluid flow and sliding directions for the confined layer at a relative contact area of approximately 19%. Each panel was produced using the same randomly rough indenter and, for (e)–(h), the same fluid-pressure difference. Results are normalized to the maximum value in frictionless conditions. (i) 3D contact around the critical constriction for the square roughness profile against either (j),(l) a semi-infinite, $\nu =0.25$ elastomer or (k),(m) a confined, $\nu =0.49$ elastomer. Blue color indicates the frictionless contact area, while orange marks the case with friction. The purple region is the overlap of the two. Constrictions open in sliding direction (j),(k) but close in the transverse direction (l),(m).