## Constraint on Stress Tensor from Slip on a Single Fault Plane

##### Abstract

Given a fault plane and its slip vector, the stress tensor which caused the displacement is sought. Two constraints are considered: first, a geometrical constraint that the shear stress applied to the fault plane is parallel to the slip [Wallace, 1951; Bott, 1959]; second, a frictional constraint that the shear to normal stress ratio equals tan?0 [Coulomb, 1776]. This is done in two steps. In a first step, the stress tensors that satisfy the geometrical constraint are sought. For tensors belonging to the vectorial space of solutions, shear and normal stress magnitudes become a function of the principal stress orientation s1, s2, s3 and are mapped, extending a study by McKenzie [1969]. The relationship between Mohr's [1882] (?nt) plane and these maps is described. In a second step, it is investigated which among these tensors also satisfy the frictional constraint. Within this more restricted vectorial space, there is a relationship between principal stress magnitudes, represented ?= (?1-?2)/( ?1 – ?3) and s = (?1 – ?3)/ ?1, and the principal stress orientation s1, s2, s3. Both the range of s and the spatial distribution of s1, s2, s3 are more restricted than when the geometrical constraint alone is considered. As when the geometrical constraint is solely considered [McKenzie, 1969], the principal stress orientations s1, s2, s3 may lie significantly away from and up to right angle to the P, B, T axis. However, this can happen only in two cases: (1) either the effective stress difference, s, has reached a high value, which is unlikely to happen if enough pre-existing fractures are available to release the stress, or (2) ?2 becomes close to either ?1 or ?3 and therefore barely distinguishable from it; in that case the delocalisation of the principal stress orientations is best described by a tendency for s2 to exchange role with either s1 or s3. When the stress difference remains small and ?2 reasonably away from ?1 and ?3, sf, s2, s3 approach positions that we define as the Pf, B, Tf axis and that are obtained from the P, B, T axis by a rotation of angle ?0/2 around B and towards the slip vector. This explains why the P, B, T axis gives reasonable estimates of the principal stress orientations [Scheidegger, 1964] despite objections [McKenzie, 1969]. However, whenever the fault plane can be distinguished from the auxiliary plane, Pf, B, Tf should give a better estimate [Raleigh et al., 1972]. In an area where many fault planes are available and a uniform tensor is assumed, the scatter in the plane orientations contains information about both the relative position of ?2, represented by ?, and the relative stress difference, s: the higher s or the closer ? to either 0 or 1, the more scatter. This information could then be extracted by inverse methods. Because a friction law would constrain these inverse methods more tightly, it may show the necessity of non-uniform tensor to explain scattered fault planes.