Ideas and methods for local recovery of tectonic stresses from fault-slip data: a critical
Authors: Mukhamediev Sh. A. 
Published: May 2019

In the research aimed at determining tectonic stresses from fault-slip data (the seismological data on the focal mechanisms of earthquakes, geological data on slickensides, etc.), in the past few decades, it has become a predominant practice to use the approach that we refer to as the method of the local kinematic reconstruction (MLKR) of stresses and paleostresses. In the MLKR, ignoring the equilibrium conditions, the authors assign a studied block (macrovolume x) a certain symmetric tensor T which they call without explanation a stress tensor and which is, in their opinion, the only cause of the observed slips. In the MLKR, the principal axes and the ratio of the differences of the principal values of tensor T (the so-called reduced tensor TR) are reconstructed locally, without taking into account the interaction of x with the contacting blocks, i.e., in such a manner as if macrovolume x were isolated. Tensor TR is determined based on the analysis of N events (N ≥ 4) that occurred in x over the time span Δt using only the data on the slip directions and on the orientation of the slickensided planes. This approach ignores the rate of change of the stresses, previous deformation history, and mechanical properties of the block, as well as the ratio of Δt to the stress relaxation time in the block.

In this review, the key ideas of MLKR are discussed and it is shown that the underlying concept of this method is fundamentally fallacious and can lead to results that are arbitrarily inconsistent with reality since under a change in the ignored factors, tensor TR can become almost arbitrary with the same set of slips. According to the mechanics of deformable solids (MDS), uniform stresses in a quasi-statically deformed macrovolume x are genetically related to the self-equilibrated surface forces acting on x and are completely independent of deformations. In contrast, the “stresses” in MLKR are genetically caused by strains and not related to surface forces. As a result, MLKR misses the possibility to balance x, i.e., to satisfy the inviolable conservation laws of momentum and angular momentum. Besides, the TR object that is reconstructed in MLKR is not objective: frame indifferent. In the attempts to achieve the desired objective, followers of the MLKR have to implicitly return to the representations that have been rejected as early as in Cauchy’s works: they do not separate the universal laws of dynamics from the mechanical properties of a particular medium. Specifically, they postulate some a priori subjective interrelations between the elements of the sought tensor TR and the slip directions, thus formulating the constitutive relations of the medium which differ from author-to-author but are attributed a meaning of the universal laws. The information about TR in the MLKR is derived from these relations rather than from the laws of mechanics. Due to this, the notions of stresses and constitutive laws in the MLKR fundamentally differ from the respective notions in the MDS. The followers of the MLKR constantly neglect the fact that the observed slip pattern not only reflects the sought stresses but also the other factors – at least, the mechanical properties of a particular medium, which should also be reconstructed from the observations rather than postulated speculatively. In the Appendix to the review, by the example of a perfectly plastic medium, we recapitulate our previously suggested scheme in which the problem of reconstructing the field of equilibrium stresses and the problem of reconstructing the constitutive relations (in this case, it is the form of the plastic potential of the medium) are separated and solved sequentially. In media that are not perfectly dissipative, separating these problems is problematic. Together, these problems constitute an absolutely new problem that has no analogs in the MDS and waits for its solution from ambitious and competent researchers.