Evolution of the contact between rough viscoelastic solids after decreasing loads: memory erasure and monotonic increase

Abstract

The real area of contact governs, in part, the friction coefficient, yet its time evolution in rough viscoelastic interfaces remains incompletely understood. In experiments of contact between polymethylmethacrylate blocks under decreasing normal loads, Dillavou and Rubinstein have shown that the true contact area exhibits, after unloading, a decreasing phase and long-term memory of the contact state prior to unloading. It is however unclear what modeling ingredients are necessary to reproduce these two features. Here, we investigate these effects using fractional viscoelastic rough contact models. By adapting existing contact theories and numerical simulation methods to fractional viscoelasticity, which induces a wide relaxation spectrum, we reproduce logarithmic aging under constant load, but show that memory of the contact state is erased upon unloading. Indeed, the contact area behaves as if it had always experienced the reduced load, even on short time-scales, contrasting with the response of a standard linear solid. Moreover, none of our results show a decreasing regime of the contact area after unload: we ultimately prove that this is the case for all linear viscoelastic models -- despite capturing logarithmic aging -- leading to the conclusion that additional local internal variables are required to explain both long-term contact memory and contact area reduction after unloading.

Figures (6)

• Figure 1: Illustration of the evolution of the true contact area in a rough contact with fractional viscoelastic materials under constant load. $\tau$ is a characteristic time.
• Figure 2: Discretization of linear viscoelastic model (a) Fractional Zener model with continuum spectrum (b) Generalized Maxwell model with discretized spectrum: an elastic branch with shear modulus ($G_\infty$) in parallel with multiple Maxwell branches composed of a spring ($G_i$) and a dashpot ($\eta_i$).
• Figure 3: Evolution of the true contact area under constant normal load $p_0$ for a fractional viscoelastic behavior with $\nu = 0.2$ and $\nu = 0.8$ ($k = 0.1$). The exponent $\nu$, which controls the breadth of the relaxation spectrum, greatly influences the time scales over which the contact area transitions from its instantaneous value to its long-time asymptote. Smaller $\nu$ values exhibit the characteristic logarithmic aging behavior over more time-scales, with $\nu = 0.2$ staying in the logarithmic regime for over six decades.
• Figure 4: Instantaneous unloading of a single-timescale viscoelastic contact (SLS). (a) shows the evolution of the true contact area when the unloading time $T$ is smaller than the transition time $k + \ln\left(\frac{1-k}{1-\alpha}\right)$: the area keeps evolving after unloading, but does not follow the monotonic curve. When $T$ is larger than the transition time, the area jumps to the asymptotic value for the reduced load $\alpha p_0$, with no evolution thereafter. (b) shows how the transition time depends on the load reduction factor $\alpha$: a larger load reduction has a shorter transition time.
• Figure 5: Evolution of the contact area under ramp unloading using the fractional viscoelastic model with $\nu = 0.3$ and $\nu = 1$ (SLS). The black continuous curve shows the purely elastic behavior, which is followed by the SLS curve shortly after the unloading starts at $t = T = 1$. The curve for $\nu = 0.3$ shows a qualitatively similar behavior, but the unloading does not follow the elastic curve. Instead, the contact area decreases down to the monotonic curve with the reduced normal load $\alpha p_0$, and increases with no memory of the previous contact area, in contrast with \ref{['transition time']}.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3