Tectonic deformations result from a condition of internal stress caused, in turn, by primary and secondary forces. In the geological literature, a great deal of discussion is based on a direct connection between forces and deformation, completely by-passing the concept of stress. This paper is a contribution in the intermediate field of stress relations. It presents the complete solutions of certain stress systems caused by various forms of boundary forces. Furthermore, the location and attitude of the fault surfaces likely to be associated with them is determined.
The basic concept of stress is briefly reviewed and some of the fundamental differences between the force-vector and the stress-tensor are pointed out. The fallacy of applying the familiar methods of vector-addition of forces to problems in stress is demonstrated.
For certain systems of external boundary forces acting on a portion of the earth's crust, the internal stress distribution can be calculated by means of the familiar equations of elasticity. Appropriate calculation methods for two-dimensional cases are shown and the basic equations applicable to a series of important boundary conditions are derived. The examples here presented include: (1) superposed horizontal compression with constant lateral and vertical gradients; (2) horizontal compression with exponential attenuation; and (3) sinusoidal vertical and shearing forces acting on the bottom of a block. The latter equations provide solutions for differential vertical uplift and for the important case of drag exerted on the bottom of the crust by convection currents in the substratum. Diagrams show configuration of the stress trajectories and distribution of the maximum shearing stress for the resulting stress systems.
A parallel series of diagrams shows the disposition between the relatively stable and unstable segments of the blocks and the probable attitude of the fault surfaces likely to be associated with the individual stress systems. The construction of the fault surfaces is based on the original stress distributions alone, the influence of local stress alterations due to the occurrence of fracture being disregarded; The full effect of this inter-action is not known, due to the extreme complexity of the problem. The fault patterns shown are strictly applicable only to the initial stages of fracture, but may also represent fair approximations during the more advanced stages, since the original stress remains the dominating influence and stress-alterations due to faulting diminish rapidly with distance.