Modeling Elastic Stresses in Long Ridges with the Displacement Discontinuity Method

Elastic stresses at ridges formed by (a) erosion and (b) volcanism are calculated here by the method of particular integrals with a displacement-discontinuity boundary element method. The two geologic scenarios require treating the far-field stresses in the analyses differently. For a continuous body, the elastic governing equation requires that the second partial derivatives of the vertical and horizontal far-field normal stresses with respect to horizontal and vertical directions be defined throughout the solution domain. This constrains admissible solutions for continuous bodies in both two- and three-dimensional analyses. For example, if the horizontal far-field normal stress varies linearly with depth over any depth interval in a continuous body, then it must be continuous and vary linearly over the entire elevation range of the solution domain. This far-field stress distribution permits the description of a ridge formed by erosion of the surrounding material. A discontinuous or a piecewise linear distribution of horizontal far-field stresses does not yield admissible solutions for a continuous body but can apply to a ridge constructed by volcanism. Stresses induced by topography in volcanic ridges will inhibit dikes from propagating to the surface, favor the formation of near-surface magma chambers, and promote slope instability. The analyses demonstrate that topography and the extant tectonic stresses will not by themselves fully determine the stresses in a ridge; the geologic history is also important.