Loading-unloading hysteresis loop of randomly rough adhesive contacts

We investigate the loading and unloading behavior of soft solids in adhesive contact with randomly rough profiles. The roughness is assumed to be described by a self-affine fractal on a limited range of wave vectors. A spectral method is exploited to generate such randomly rough surfaces. The results are statistically averaged, and the calculated contact area and applied load are shown as a function of the penetration, for loading and unloading conditions. We found that the combination of adhesion forces and roughness leads to a hysteresis loading-unloading loop. This shows that energy can be lost simply as a consequence of roughness and van der Waals forces, as in this case a large number of local energy minima exist and the system may be trapped in metastable states. We numerically quantify the hysteretic loss and assess the influence of the surface statistical properties and the energy of adhesion on the hysteresis process.

Figure 1

An elastic half-space in contact with a periodic randomly rough rigid substrate of wavelength $\lambda$.

Figure 2

A schematic representation of elastic displacement field at the interface, as they occur during approaching the substrate to the elastic solid by a quantity $u_{\mathrm{tot}}$. Observe that $u_{\mathrm{tot}}$ is the sum of the mean displacement $u_{\mathrm{m}}$ of the elastic body and substrate penetration $\mathrm{\Delta }$. Also the interfacial displacement and the separation $s$ between the two surfaces are shown.

Figure 3

The deformed shapes of the elastic body at three different separations, $s=92\mu \mathrm{m},s=51\mu \mathrm{m}$, and $s=20\mu \mathrm{m}$, and the rough rigid substrate profile, for (a) loading and (b) unloading. The work of adhesion is $\gamma =0.01\mathrm{J}/{\mathrm{m}}^2$, and the rigid rough substrate profile has a fractal dimension $D_{\mathrm{f}}=1.2$.

Figure 4

The PSD of the rigid substrate profile with fractal dimension $D_{\mathrm{f}}=1.2$, compared to the PSD of the deformed shape of the elastic body at fixed non dimensional penetration $\tilde{\mathrm{\Delta }}=\mathrm{\Delta }/h_{max}=0.4$, obtained for (a) $\gamma =0.01\mathrm{J}/{\mathrm{m}}^2$ and (b) $\gamma =0.04\mathrm{J}/{\mathrm{m}}^2$, for loading (blue curves) and unloading (red curves) conditions.

Figure 5

The PSD of the rigid substrate profile with fractal dimension $D_{\mathrm{f}}=1.2$, compared to the PSD of the deformed shape of the elastic body, obtained for $\gamma =0.04\mathrm{J}/{\mathrm{m}}^2$, (a) for loading and (b) unloading conditions, for different values of the nondimensional penetration $\tilde{\mathrm{\Delta }}=\mathrm{\Delta }/h_{max}$.

Figure 6

The true contact area $A/A_0$ as a function of the dimensionless quantity $m_2^{-1/2}\tilde{\sigma }$ for two different Hurst exponents $H=0.8$ ($m_2=2.7\times {10}^{-3}$), $H=0.9$ ($m_2=1.5\times {10}^{-3}$), and for three different values of energy of adhesion, $\gamma =0.01\mathrm{J}/{\mathrm{m}}^2$ (black curves), $\gamma =0.04\mathrm{J}/{\mathrm{m}}^2$ (blue curves), and $\gamma =0.07\mathrm{J}/{\mathrm{m}}^2$ (red curves). As predicted by the theories, there is a marginal influence of the fractal dimension on the true contact area at small loads under loading conditions. The influence of $D_f$ becomes important during unloading. This leads to the formation of a hysteresis loading-unloading loop, which is strongly affected by the adhesion energy $\gamma .$

Figure 7

The dimensionless load $\tilde{\sigma }=2\sigma /\left(E^{*}{q}_0{h}_{max}\right)$ (a) and the quantity $A/A_0$ (b) as a function of the nondimensional penetration $\tilde{\mathrm{\Delta }}$, for two different Hurst exponents $H=0.8$ ($m_2=2.7\times {10}^{-3}$), $H=0.9$ ($m_2=1.5\times {10}^{-3}$), and for three different values of energy of adhesion, $\gamma =0.01\mathrm{J}/{\mathrm{m}}^2$ (black curves), $\gamma =0.04\mathrm{J}/{\mathrm{m}}^2$ (blue curves), and $\gamma =0.07\mathrm{J}/{\mathrm{m}}^2$ (red curves).

Figure 8

The normalized probability density function (PDF) $p\left(t\right)$ of the local separation $t\left(x\right)$ in the noncontact area $[t\left(x\right)>0]$ for $H=0.7$, and $\gamma =0.04\mathrm{J}/{\mathrm{m}}^2,\tilde{\mathrm{\Delta }}=0.7$ for loading and unloading conditions.

Figure 9

The averaged dimensionless height probability density function $p\left(t\right)$ of 10 rigid rough profiles. $\tilde{h}\left(x\right)=h\left(x\right)/h_{max}$. Calculations are shown for $H=0.8$. Because of the enhanced ergodicity, the trend of the calculated probability density function $p\left(t\right)$ (solid line) follows pretty well a Gaussian random distribution (dashed line).