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The propagation of transient waves in a homogeneous, isotropic, linearly elastic half-space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are derived for the interior of the half-space and for all load speeds. Wave front expansions are obtained from the exact solution, in addition to results pertaining to the steady-state displacement field. The limit case of zero load speed is considered, yielding new results for Lamb's point load problem. INTRODUCTIONGround motions excited by moving surface forces arise, for example, from nuclear blasts and from shock waves generated by supersonic aircraft, and they interact with structures causing extensive damage. A mathematical problem of fundamental importance in these applications is that of an elastic half-space whose surface is excited by a normal point load which is suddenly applied and which subsequently moves rectilinearly at a constant speed.In recent years several solutions to this probl em have been given for the surface of the half-space. t Numbers in brackets designate References at end of paper. shown to form the steady-state displacement field when the load speed exceeds both of the body wave speeds. FORMULATION OF THE PROBLEMThe subject half-space problem is depicted in Fig. 1 The constants cd and cs, defined by c! = (X. + 2µ)/v and c; = µ/v, represent the dilatational and equivoluminal body wave speeds, respectively, where X. and µ are the Lam~ constants and v is the material density. The stresses T .. are related to the displacements by lJwhere tensor notation is employed.The boundary conditions at z = 0 take the formwhere o is the Dirac delta function. To represent quiescence at t = 0, the initial conditions appear asFinally, the potentials cf> and ljJ, and the space derivatives of the patentials, are required to vanish at infinity. FORMAL SOL UT IONA solution of the wave equations (1) that satisfies the initial conditions (6) and the boundedness condition at infinity can be computed Soos-n z+i(kx+vy) (1 7a)(1 7b)(1 7c)(1 7d) In view of the symmetry properties u (r,9,z,p) =u (r,-9,z,p) xa xa u (r,9,z,p) = -u (r,-9, z ,p)u (r,9,z,p) =u (r,-9,z,p), za za u. , and hence u., are only inverted for 0 :S 9 <'TT. Since u. has different Ja J Ja forms depending on the speed of the load relative to the body wave speeds, the inversion of each u. is separated into three cases. In particular, S(X)s(X)_ _E...c (mdz-iqr)where ~ is the position vector. uzd is converted into the Laplace transform of a known function by mapping (1 /cd)(mdz-iqr) into t through a contour integration in a complex q-plane. To this end, the singularities of the integrand of U:zd are branch points at -9-for t > twd, where Equation (26) defines one branch of a hyperbola with vertex 2 .!.2 r/p and asymptotes arg q = ±r/z. As shown in Fig. 2 (27) + -by a solid line labeled with qd and qd, this hyperbola is parametrically described by t as t varies from twd ...

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The present paper points out that Kromm’s [1] plane-stress solution, for compressional waves in an infinite elastic plate subjected to radial pressure in a circular hole at its center, has application to still another problem of interest. This is the problem of a stretched elastic plate in which a circular hole is suddenly punched. The plane-stress solution for the tensile circumferential stresses, generated by the unloading mechanism in punching, is given here. This solution is derived independently of Kromm’s work in which a rather special Laplace-transform technique was used. The derivation given here also makes use of the Laplace transform but in a more direct manner, employing the inversion integral and a contour integration. It is also shown that the present inversion technique offers important simplifying features over that used by Selberg [3] in the closely related plane-strain problem. The numerical results presented are of interest in fragmentation studies. It is shown that the dynamic circumferential stress field in the vicinity of the punched hole is quite severe; which would be important to the creation and propagation of radial cracks.

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