Further consideration of the DMT model for elastic contact

Abstract

One of the early models for the contact of an elastic sphere on a rigid flat, which took account of the action of surface forces, was developed by Derjaguin et al. Making the assumption that the deformed profile of the sphere is Hertzian, they calculated the force of attraction due to surface forces by a “thermodynamic” method. This approach is now shown to be incorrect, and is replaced by a direct “force” method of calculation. The new analysis shows that the adhesional behaviour depends on the sphere radius, elastic modulus and the surface energy of the sphere. For small, fairly rigid spheres with low surface energy, the contact behaviour is the same as that originally predicted by Derjaguin et al., that is, as the sphere and the flat are pulled apart, separation occurs at point contact. However, for larger, more elastic spheres with high surface energy, separation of the sphere and flat occurs at a finite area of contact even though the deformation is still assumed to be Hertzian. The behaviour resembles that predicted by Johnson et al. where a different mode of deformation is assumed. However, in the present work, the instability at a finite contact area is not due to infinite tensile stresses around the contact periphery as in the Johnson—Kendall—Roberts model, but arises from the sharp fall in surface forces as the contact area decreases.