An exact solution for the first stage of frictional collision of a die and an elastic half-space

The first stage of frictional collision of a die and an isotropic linear elastic half-space is studied. The die is a blunt convex rigid body having an arbitrary shape with two orthogonal planes of symmetry, which are both orthogonal to the boundary of the half-space. The problem is investigated using the technique of integral characteristics of solutions to boundary-initial value problems introduced by Borodich. We consider non-frictionless boundary-initial contact problems, for example, adhesive or frictional. Expressions are obtained for the relations between time, depth of indentation, and velocity of the body. In particular cases, when the body is an elliptic paraboloid, a blunt four-sided pyramid or an elliptic cone, some expressions have simple algebraic forms. A proof is given that the expressions are independent of the boundary conditions in the contact region.