The oblique compression of two elastic spheres

Abstract A fundamental problem in the behaviour of the packing of spheres is that of the oblique compression of just two spheres. Here, the solution of this problem is obtained for the case of two identical homogeneous isotropic elastic spheres, since much use can then be made of the existing symmetry. In particular, the normal and shear components of traction on the contact area can be treated separately. Considerations of the normal force show that the contact area is circular and, furthermore, that this part of the solution is precisely that of normal Hertzian contact. To obtain that part of the solution corresponding to shear, two criteria are used. The first is that of no slip between the spheres, and the second is that the energy flux across the contact area must obey the appropriate symmetries of the problem. These symmetries are sufficient to make the solution unique. This solution differs greatly from that obtained when the spheres are first compressed normally and then sheared. In particular, it is shown that if slip does occur, then it will be in the form of sliding; whereas in the latter case, slip occurs only within a circular annulus.