A general mathematical model for the collision between a free-fall hammer of a pile-driver and an elastic pile

In this paper we consider a general mathematical model for the collision between the free-fall hammer of a pile-driver and an elastic pile whose ends are furnished with a bearing. When the free-fall hammer collides with the pile, the displacement of a cross-sectional area of the pile is the weak solution of an initial-boundary value problem involving a linear wave equation with memory boundary conditions. We generalize this problem into a nonlinear one with more general boundary conditions. Then we obtain the unique solvability and the regularity of the weak solution of this nonlinear problem. The unique solvability is shortly discussed in regard to the Galerkin method. The regularity result is obtained by a combination of a fixed-point technique and an energy method, and the convenience of this procedure is also pointed out.