On the application of finite-element techniques to problems concerning potential distribution and stress analysis in the earth sciences

Summary

The elastic response (i.e., displacement fields, strain and stress distributions) has been determined by structural matrix analysis for a homogeneous rock layer subjected to two-dimensional distributions of vertical displacement applied along a basal boundary. The results obtained compare favorably with biharmonic function solutions obtained bySanford [42]²) and the authors. The analysis was then extended: the effects of different magnitudes of superposed gravitationally-induced stresses, release of a constrained side boundary, and introduction of pronounced heterogeneity were each quantitatively evaluated. Each of these changed conditions effected important alterations in response characteristics, e.g., in configuration of stress trajectories, predicted locations and configurations of initial fracture surfaces, shear stress distributions, and displacement fields.

These results are cited as an example of the utility of finite-element analysis (originally developed for evaluation of aeronautical structural components) for solutions of boundary value problems of a geological/geophysical nature. The breadth of specific physical problems succeptible to evaluation by this method is large and includes (i) behavior of rocks under static/dynamic loading, (ii) seismic response, (iii) heat and fluid flow, and (iv) distribution of potential. Most of the fundamental problems in geophysics, structural geology, geohydrology, geomorphology, glaciology, and engineering geology involve the above categories either directly or otherwise. The finite-element approach to these problems if bound to have enormous significance, for unlike classical mechanics, it can be readily adapted to solution of systems characterized by non-linear, anisotropic, heterogeneous material properties, of any exterior configuration, containing structural discontinuities, and subjected to any viable combinations of load/displacement boundary conditions.