Surface Displacements due to a Steadily Moving Point Force

In this paper we shall investigate the normal surface displacement of the isotropic elastic half-space z > 0, due to a normal point force, F, which moves with constant speed, V, in the xdirection over the surface z = 0. This problem defines the Green's function for a class of elastodynamic contact problems for the half-space in which a frictionless rigid indenter moves at constant speed over the surface. We assume that the force has been moving for a long time, so that the stress and displacement fields have achieved a steady state when viewed in a frame of reference that moves with the force. The more challenging transient problem in which the half-space is initially quiescent and the force moves for a finite time was investigated by Payton (1964), who gave results for the special case where Poisson's ratio u = 1⁄4. A more convenient solution to the steady-state problem for general Poisson's ratio was given by Churilov (1977), who also extended the argument to give simple solutions to some elastodynamic contact problems (Churilov, 1978). The steady-state problem is self-similar and we can therefore use equilibrium and dimensional arguments to demonstrate that the stresses must decay with R-2 and the displacements with R -l , where R is the distance from the moving force (see, for example, Willis, 1967; Barber and Sturla, 1992). The problem therefore reduces to the determination of the q%dependence of the normal surface displacements in a cylindrical polar coordinate system (r, th, z) moving with the force. Churilov's solution is based on his observation that the governing elastodynamic equations in the moving frame of reference have the same form as the elastostatic equations for a fictitious anisotropic material. The problem is thereby reduced to that of the point force acting on a half-space of this fictitious material, for which Churilov uses the method of Sveklo (1964), which is in turn based on a technique due to Smirnov and Sobolev (see for example Eringen and Suhubi, 1975, Section 8.8). However, it can be shown that Churilov's fictitious anisotropic material has physically reasonable (i.e., positive definite) elastic constants if and only if the speed V in the real problem is less than the shear wave speed c~. In the present paper, we shall demonstrate that the SmirnovSobolev technique can be applied directly to the elastodynamic problem, without invoking the analogy to anisotropic elastostatics. This will permit us to obtain simple closed-form expressions for the normal surface displacements over the entire speed range and also points up some interesting relationships between the