An upper bound for viscoelastic pull-off of a sphere with a Maugis-Dugdale model

ABSTRACT

We develop an extension of the Maugis-Dugdale solution for viscoelastic spheres. We show that we can define two characteristic Tabor parameters, a larger one ${\mu }_R$ corresponding to relaxed modulus $E_R$ and a smaller one ${\mu }_I$ for instantaneous modulus $E_I$ of the material. Only if both are very large (corresponding to the JKR regime, ${\mu }_R>{\mu }_I>5$) the pull-off load increase due to viscoelastic effect is possibly very large at large pulling speeds, as given by existing solutions and approximately equal to the ratio $E_I/E_R$, and otherwise the amplification at very high speeds is much reduced and we give a very simple upper bound of the pull-off load as a function of the relaxed Tabor parameter, independently on the exact form of the viscoelastic linear modulus. An example detailed calculation is given for standard material and constant velocity of load reduction. A dependence on preload is found.

KEYWORDS:

• Viscoelasticity
• peeling
• JKR solution
• DMT-JKR models
• Dugdale-Maugis model

Acknowledgements

MC acknowledges support from the Italian Ministry of Education, University and Research (MIUR) under the program “Departments of Excellence” (L.232/2016). Also, proff. H.Gao from Singapore University and Qunyang Li from Tsingua University for having inspired this model with their fruitful discussions.

Notes

¹ For example, for standard material $\hat{C}\left(\frac{\hat{c}-\hat{a}}{\hat{V}}\right)=1-\left(1-\frac{E_R^{\ast }}{E_I^{\ast }}\right)exp\left(-\frac{\hat{c}-\hat{a}}{3\hat{V}}\right)$.

² If the load is not slow there is a small change since the curve on loading for high speeds of loading tends to be Hertzian, as the work of adhesion is reduced instead of increased as it is on unloading, as it could be obtained with the equation

32 $\begin{array}{rl}& \frac{w}{C\left(\frac{c-a}{3V}\right)E_R^{\ast }}={\sigma }_c\frac{a^2}{\pi R}\left[\sqrt{m^2-1}+\left(m^2-2\right){tan}^{-1}\sqrt{m^2-1}\right]\\ & +\frac{4}{\pi E_R^{\ast }}{\sigma }_c^{2}a\left[\sqrt{m^2-1}{tan}^{-1}\sqrt{m^2-1}-m+1\right]\end{array}$32