Exact transient results for pure and grazing indentation with friction
Let us consider the elastic half-plane \(y \leq 0\) and suppose that, at the instant \(t=0\), a rigid wedge, of half-angle \(\Theta\) and symmetric with respect to the \(y\)-axis, is applied with its vertex at the origin while the remaining part of the wedge is placed in the half-plane \(y>0\). Then the wedge is pushed with a constant vertical velocity \(V\) into the lower half-plane, but assuming, as opposed to what had been done in the previous literature, that friction arises between punch and half-plane along the region of contact. The problem is solved by using the Laplace transforms of the unknown fields with respect to the time and their Fourier transforms with respect to the \(x\)-variable. The explicit forms of the solutions can be obtained after inversions of the results. In the same way it is possible to study the case in which the die, besides penetrating into the half-plane, moves simultaneously in the direction of the \(x\)-axis. With the same technique, it is possible to write the formal solutions of the problem in which the moving punch is not a wedge but a rigid circle. In all there cases, the most difficult step is that of finding the inverse transforms of the solutions. This proves, once more, that the method of integral transforms is not efficient, exept for simple cases.
scientific article; zbMATH DE number 549822