Deformation of an Elastic Beam on a Winkler Foundation

We present a simple model for geophysical systems involving sources of deformation, such as magmatic intrusions, subglacial lakes, and the subsurface storage of CO2. We consider the idealized system of a uniform elastic layer overlying a localized region of constant pressure that is surrounded by a Winkler foundation composed of springs. We investigate the effect of source depth and foundation stiffness on the resulting displacement profiles at both the surface and the level of the source. The system is characterized by three key features: the maximum uplift, the maximum subsidence, and the distance to the point of zero displacement. For each of these, we determine asymptotic scaling behavior in the limits of a thin/thick layer and a soft/stiff foundation and form composite curves that allow specific parameter values to be determined from field data. Both two-dimensional and axisymmetric pressure patches are considered, and in the thin-layer limit we derive analytical solutions.