A pseudo three-dimensional finite element formulation for elastostatic problems and its geophysical applications

Abstract

A special class of three-dimensional elastostatic problems sometimes encountered in geology, geophysics, and engineering involves structures which can be reasonably represented by two-dimensional bodies but for which two-dimensional load functions are not suitable approximations, for example, ocean tidal loading of continental margins and localized loading of linear geological features such as faults and dipping strata. A pseudo three-dimensional finite element formulation capable of treating these problems has been developed which involves a system of two-dimensional finite element equations with sinusoidally distributed loads along the third direction. Solutions to the system of equations yield three components of displacement at each nodal point. By means of Fourier synthesis a variety of arbitrarily shaped load functions can be constructed. The present technique does not require the exceedingly large computer storage capacity usually necessary for solving general three-dimensional elastostatic problems. The present finite element formulation has been applied to the problem of tidal gravity response due to ocean tidal loading on continental margin structures. Two different continental margin models were adopted, one having a uniform mantle and the other having a layered laterally heterogeneous mantle. Applied sinusoidally distributed loads include spatial wavelengths between 50 and 1000 km. The results indicate that for the surface vertical displacement and the load-induced gravity variation the influence of the transition between oceanic and continental crustal and upper mantle structures is most pronounced for loads located over the transition zone and is significant only within a limited region surrounding the transition zone. For any given location and load wavelength the values of the vertical surface displacement resulting from loads acting on the transition region, as expected, lie between the corresponding values resulting from loads acting on a purely oceanic crustal and upper mantle structure and on a purely continental crustal and upper mantle structure.