An alternative to the Maugis model of adhesion between elastic spheres

The problem of how contact between elastic spheres is modified by the surface forces which act between them is becoming of great interest with the possibility of studying such contacts experimentally in the atomic force microscope and in the surface force apparatus. Experiments are frequently interpreted with the help of the JKR adhesion theory, in which the action of the surface forces is represented by the change in surface energy involved in uniting or separating the two surfaces. This has two drawbacks: it is strictly valid only when the Tabor parameter mu = sigma(o)[R/(E*(2) Delta gamma)](1/3) is large, though in fact it appears that the condition mu greater than or equal to 1 is sufficient, and it gives no help with the attractive forces before contact takes place, which may cause 'jumping on'. However, it has one enormous virtue: the results are described by simple algebraic equations, which all can manipulate as required. In contrast, full numerical analyses, such as the pioneering solution by Muller, Derjaguin and Toporov and a more recent solution by Greenwood, give only the results the authors choose to pass on. These establish when the JKR solution is valid and indicate the ways in which it is incomplete; but stop there. For this reason, an analytical solution by Maugis using a simplified force-separation law, that the adhesive stress has a fixed value whenever the separation is less than a critical distance, has proved most valuable. However, it too has a drawback, though a minor one: the shape of the gap involves elliptical integrals, which is inconvenient when the extension to visco-elastic spheres is considered. Here we present an alternative to the Maugis theory, based on the combination of two Hertzian pressure distributions. The results are very close to those of the Maugis model, but the gap shape now involves only elementary functions.