Integral properties of solutions to some boundary initial-value problems in linear anisotropic elastodynamics

We study some integral properties of weak solutions to some boundary initial-value problems in the linearized dynamical theory of elasticity. These problems arise during the indentation of an anisotropic half-space by a convex punch having an arbitrary indenting velocity and shape. In contrast to previous studies of the problem, we consider various boundary conditions, e. g., adhesive or frictional, in the case when this non-frictionless boundary initial-contact problem has two orthogonal planes of symmetry, which are both orthogonal to the boundary of half-space. We show that if the non-frictionless boundary initial-contact problem has such a symmetry, then the problem for integral characteristics of the solutions is equivalent to a problem of plane-wave propagation in the same medium. A proof is given that the instantaneous value of the force required to indent the punch during the first supersonic stage of contact is directly proportional to the product of the velocity of indentation and the area of contact at that instant, and is independent of the boundary conditions in the contact region.