Title:

A variational solution of the frictional unloading problem in linear elasticity

A numerical solution is obtained for the frictional unloading problem, of a circular cylindrical rigid punch being removed from a homogeneous, isotropic, incompressible, linear elastic halfspace, when conditions of Coulomb friction act during both loading and unloading. Considering a variational formulation of the elasticity equations, it is shown that at the solution the complementary energy achieves a minimum within the subset of admissable stress states; that is those that obey the equilibrium equations and the boundary conditions on stress. By considering the general traction boundery value problem, the complementary energy is reduced to a quadratic functional of the unknown surface stresses alone, the frictional boundary conditions acting as linear constraints on the minimization. It is then shown that this reduces to a fairly simple analytical form for the stated problem. It is required to find a numerical solution of the frictional unloading problem, but first some known solutions to the loading problems in the frictionless, adhesive, and frictional cases are given, all on incompressible halfspaces. A numerical solution, based on a Finite Element approximation to the above energy principles, is then obtained, not only for the unloading problem, but also for the above loading ones. The solution for the three loading problems is known, the numerical solution comparing favourably, and that for the unloading problem confirms the prediction from a physical argument. During the derivation of the above energy principles, a solution to the general traction boundary value problem is obtained, using the Airy stress function. This is used to calculate the entire stress/displacement state from the numerical solutions. The results obtained for the radial stress are consistent with previously obtained empirical results. A detailed description of the implementation of the numerical method, and the structure of the FORTRAN programme is given, and convergence of the numerical technique proved. Finally a brief attempt at an analytical solution is made, and some results obtained which further confirm the numerical ones.
