A method based on the radon transform for three-dimensional elastodynamic problems of moving loads

An integral transform procedure is developed to obtain fundamental elastodynamic three-dimensional (3D) solutions for loads moving steadily over the surface of a half-space. These solutions are exact, and results are presented over the entire speed range (i.e., for subsonic, transonic and supersonic source speeds). Especially, the results obtained here for the tangential loads (one of these loads is along the direction of motion and the other is orthogonal to the direction of motion) are quite new in the literature. The solution technique is based on the use of the Radon transform and elements of distribution theory. It also fully exploits as auxiliary solutions the ones for the corresponding plane-strain and anti-plane shear problems (the latter problems are 2D and uncoupled from each other). In particular, it should be noticed that the plane-strain problem here is completely analogous to the original 3D problem, not only with respect to the field equations but also with respect to the boundary conditions. This makes the present technique more advantageous than other techniques, which require first the determination of a fictitious auxiliary plane-strain problem through the solution of an integral equation. Our approach becomes particularly simple when there is no angular dependence in the boundary conditions (i.e., when axially symmetric problems regarding their boundary conditions are considered). On the contrary, no such constraint needs to be fulfilled as regards the material response (and, therefore, the governing equations of the problem) and/or also possible loss of axisymmetry due to the generation of shock (Mach-type) waves in the medium. Therefore, the solution technique can easily deal with general 3D problems having a largely arbitrary radial dependence in the boundary conditions and involving: (i) material anisotropy in static and dynamic situations, and (ii) asymmetry caused by changes in the nature of governing PDEs due to the existence of different velocity regimes (super-Rayleigh, transonic, supersonic) in dynamic situations.