The Sliding and Rolling of a Cylinder at the Nano-scale

The behavior of a nano-scale cylindrical body (e.g. a fiber), lying on a substrate and acted upon by a combination of normal and tangential forces, is the subject of this investigation. As the scale decreases to the nano level, adhesion becomes an important issue in this contact problem. Thus this investigation treats the two-dimensional plane strain elastic deformation of both the cylinder and the substrate during a rolling/sliding motion, including the effect of adhesion using the Maugis model. For the initiation of sliding, the Mindlin approach is used, whereas for rolling, the Carter approach is utilized. Each case is modified for nano-scale effects by including the effect of adhesion on the contact area and by using the adhesion theory of friction for the friction stress. Analytical results are given for the normal and tangential loading problems, including the initiation of sliding and rolling in terms of dimensionless quantities representing adhesion, cylinder size, and applied forces. Figure 1. Contact of a cylinder with a half-space under normal and tangential loading. INTRODUCTION Adhesion of cylindrical bodies on a substrate is encountered in nano-wires, carbon nano-tubes and nano-fibers, and in different fields such as microbiology, microelectronics, and MEMS/NEMS devices. Determination of the forces necessary to roll or slide a cylindrical body on the substrate are important quantities to know in these applications. In some cases it is important to manipulate these single fibers to form a structure whereas in other instances the sliding and rolling motions are important in contamination removal processes. In this paper the adhesion of contacting cylinders, or equivalently a cylinder in contact with a half space, at the nanoscale is considered. If the cylinder is subjected to a combined tangential and normal loading it may remain at rest, roll, slide or undergo a complex motion depending on the magnitudes and the application points of the loading. The elastic behavior of the cylindrical body and half-space with adhesion are investigated using the plane strain theory of elasticity. A similar problem was treated in the thesis by Sari [1]. Numerous studies have been conducted on the adherence of spherical bodies. Bradley [2] found the pull-off force required to separate two rigid spherical bodies, of radii R1 and R2, to be 2 F wR where R = R1R2/( R1+ R2) is the equivalent radius of curvature, w is the work of adhesion w= 1+ 212, with the surface energies of the contacting bodies 1 and 2, and the interface energy of the two surfaces 12 . Johnson, Kendall and Roberts (JKR) presented a theory on the adherence of deformable elastic bodies [3]. In the JKR approximation, the adhesion outside the contact region is assumed to be zero, and the contact area is larger than the Hertz contact area. The pull-off force was found to be F = (3/2) wR. Derjaguin, Muller and Toporov (DMT) presented another theory where the adhesion force is considered outside the contact area, but the form of the contact stress distribution is assumed to be unaffected [4]. The same pull-off force as the Bradley relation was found. It should be noted that F is independent of the elastic properties of the materials, for both JKR and DMT theories. Body 2 y