The work done in producing tangenti

The work done in producing tangential displacements is stored as elastic strain energy in the solids and is therefore recoverable under suitable circum- stances. Also, the tangential displacement is not necessarily constant throughout the contact area and it is possible to generate conditions in which some regions are in sliding contact whilst others are stuck. For example, Mindlin [l] treats the case in which two spheres are pressed together with a constant normal force P and then subjected to a tangential force F. If F > pP sliding occurs throughout the contact region, but for lower values of F a central circular region remains stuck whilst sliding or microslip occurs in the surrounding annulus.
During an impact, conditions are considerably more complex than this, since the contact area is continually changing. When it is increasing, new regions of contact will be laid down free of tangential stress, but changes in the applied tangential force will involve a redistribution of stress. In a more recent paper, Mindlin and Deresiewicz [2] considered a wide range of problems involving varying normal and tangential forces and emphasised that the response of the system to small changes depends upon the whole previous loading history. For this reason, they restricted their attention to cases in which the initial state was reached by applying first the normal force and then the tangential force.
In the impact problem, the tangential loading history is not known a priori since it depends on the interaction between the compliance of the
contact and. the motion of the spheres. This difficulty can be overcome in a numerical solution by advancing through the period of the impact in small discrete time increments. Starting from known values of normal and tangen- tial velocity, the displacements in a time increment can be found and these define the boundary conditions of the instantaneous contact problem. The changes in velocity components during the time increment are then found from the contact forces by considerations of momentum.
In a previous publication [ 31 the method of Mindlin and Deresiewicz was used to solve each of the incremental contact problems in such a proce- dure. However, with this method the state after the nth step is described
as the sum of n irreducible components. In other words, the previous history of the system has to be continuously available at each step and this places
a limit on the number of time increments which can conveniently be used.
The previous history of the system only influences the instantaneous behaviour in so far as it determines the locked-in tangential displacements
in those regions of the contact area which are stuck. Hence, if the distribu- tion of tangential displacement (and/or traction) can be approximated by a suitable series of functions, the amount of information carried through the procedure will be independent of the number of time increments used. This is the method used in this paper. The results from the two methods have been found to agree closely in cases to which both can be applied.