Love's problem

SUMMARY Explicit expressions for the displacements generated in a non-gravitating, homogeneous, semi- infinite half-space by uniform surface pressure applied over a rectangular region are presented. These complement expressions for the associated stress field given by Love in 1929. The problem of the surface loading of a Cartesian elastic half-space is associated with the name of Boussinesq (1885) who showed that the components of displacement and stress at any point within the half-space can be expressed in terms of the various spatial derivatives of elastic potential functions. Many people have contributed to this area of elasticity theory by producing explicit expressions for the deformations produced in response to a surface pressure distribution of specific shape and form. For example, Boussinesq (1885) provided the solution for a point load, while Lamb (1902) and Terazawa (1916), using Fourier-Bessel transforms, addressed the case of uniform pressure applied within a circular boundary. Love (1929) revisited the problem of the circular or disc load using Boussinesq's potential method. He also introduced a new class of loading problem—that of uniform pressure applied within a rectangular region of the surface. We refer to this as Love's problem. Love (1929) presented only a partial solution to this problem, in that he provided expressions for the stress field within the half-space, but not for the displacement field. Presumably this omission reflects the technical motivation of his study—that of the safety of foundations. In many cases of practical interest, the surface load is not laterally uniform. Multiple disc loads often are used to approximate laterally varying surface loads; however, a flat surface is much easier to tile with rectangles. By solving Love's problem, we provide a convenient basis for modelling arbitrary surface loads applied to an elastic half-space. Crustal motion geodesists have become increasingly interested in elastic loading signals manifest as seasonal fluctuations in the position time-series produced at continuous GPS stations and by related space geodetic techniques (Heki 2001; Mangiarotti et al. 2001; Dong et al. 2002; Elosegui et al. 2003). These surface loads are associated with changes in atmospheric and seafloor pressure, and shifting masses of snow, ice, and surface and subsurface water. Many problems associated with localized loads have a spatial scale which is very small compared with the radius of the Earth, allowing the problem to be analysed within the framework of a Cartesian half-space. In most parts of the world if these loads are applied for time periods of ∼1 yr of less, then viscous effects may be neglected and we may assume an elastic structure for the half-space. In some cases it may be appropriate to assume a uniform elastic half-space, at least as a first approximation. Of course, well established numerical techniques exist for analysing loading problems with more complex Earth models that incorporate the Earth's curvature and radial variation in elastic structure (Farrell 1972; Elosegui et al. 2003). Nevertheless, the solution to Love's problem may be of interest because it may provide a reasonable approximation in some contexts, or because it may be used to test computer codes based on more sophisticated geoelastic models.