On the static and dynamic equilibrium of concentrated loads in linear acoustics and linear elasticity

Abstract

The displacement field in an unbounded linear elastic fluid subjected to a time-dependent point force is obtained by using integral transform techniques. Differentiation of the displacement field yields the pressure field. It is shown that the pressure on the surface of a spherical ball B r of radius r centered at the point where the force is applied is statically equivalent in the limit as r→0 to only one-third of the force. The remaining two-thirds are carried by the inertia terms. It is also shown, by an independent reasoning, that a point force cannot be carried in static equilibrium by a linear elastic fluid. The displacement field corresponding to an unbounded isotropic linear-elastic solid subjected to a time-dependent point force (the Stokes solution) is also obtained by using integral transform techniques. As is well-known, the tractions of the Stokes solution on the surface of a spherical ball B r are statically equivalent in the limit as r→0 to the force itself; consequently, the inertia terms do not contribute to the dynamic equilibrium of B r. The contrast between the response of a fluid and that of an isotropic solid under the action of a point force is discussed.