Response to Dr Greenwood’s Comments on “Extending the Double-Hertz Model to Allow Modeling of an Adhesive Elliptical Contact”

© 2018, The Author(s). An adhesive elliptical contact is normally found in microscale applications that involve cylindrical solids, crossing at an angle between 0° and 90°. Currently, only one model is available to describe the elliptical contact’s surface interaction: the approximate Johnson–Kendall–Roberts (JKR) model which is limited to soft materials. In this paper, a new adhesive elliptical model is developed for a wide range of adhesive contacts by extending the double-Hertz theory, where adhesion is modeled by the difference between two Hertzian pressure distributions. Both Hertzian pressures are assumed to have an equivalent shape of contact areas, the only difference being in size. Assuming that the annular adhesive region is obtained by the area difference between the two Hertzian contact areas, the pull-off force curves can be calculated. In the limiting case of an adhesive circular contact, the results are very close to results from the existing models. However, for an adhesive elliptical contact in the JKR domain, lower pull-off forces are predicted when compared to the JKR values. Unlike the developed model, the shape of the JKR contact area varies throughout contact. Results show, particularly for conditions close to the JKR domain, that it is important to take into account that the adhesive region is the result of the two Hertzian contact areas having a non-equivalent shape

Numerical method for the adhesive normal contact analysis based on a Dugdale approximation

Modeling adhesion between two contacting surfaces plays a vital role in nano-tribology. However, providing analytical models, although desirable, is mostly impossible, in particular for complex geometries. Therefore, much attention has to be paid to numerical modeling of this phenomenon. Based on the adhesive stress description of the Maugis-Dugdale model of adhesion, which is credible over a broad range of engineering applications, an extended Conjugate Gradient Method (CGM) has been developed for adhesive contact problems. To examine the accuracy of the proposed method, the common case of the adhesive contact of a rigid sphere on an elastic half-space is investigated. To further evaluate the accuracy of this method, the adhesive contact of a rigid sphere over a wavy elastic half-space is also studied for different combinations of the amplitude and wavelength. There is good agreement between the analytical solution and the values predicted by the proposed method in the force-approach curves. Moreover, the calculation of pull-off force at a bisinusoidal interface between two surfaces is carried out for various cases to study the effects of different influential parameters including work of adhesion, elastic modulus, radius curvature at a crest, and the wavelength ratio. A curve is fitted on the calculated pull-off force in order to express it as an analytical relation. Similar to the JKR and DMT expressions for the pull-off force of a rigid ball on an elastic half-plane, the fitted curve is not affected by the elastic modulus and is linearly dependent on the radius of curvature and the work of adhesion. In addition, a power law governs the relation between pull-off force and the wavelength ratio. In the end, it is shown that roughness can either increase or decrease the adhesive force at a rough interface depending on the degree of the roughness