Refraction of the principal stress trajectories and the stress jumps on faults and contact surfaces: Part 1. Non-constrained regular trajectories

Rock masses contain ubiquitous multiscale heterogeneities, which (or whose boundaries) serve as the surfaces of discontinuity for some characteristics of the stress state, e.g., for the orientation of principal stress axes. Revealing the regularities that control these discontinuities is a key to understanding the processes taking place at the boundaries of the heterogeneities and for designing the correct procedures for reconstructing and theoretical modeling of tectonic stresses. In the present study, the local laws describing the refraction of the axes of extreme principal stresses T ₁ (maximal tension in the deviatoric sense) and T ₃ (maximal compression) of the Cauchy stress tensor at the transition over the elementary area n of discontinuity whose orientation is specified by the unit normal n are derived. It is assumed that on the area n of discontinuity, frictional contact takes place. No hypotheses are made on the constitutive equations, and a priori constraints are not posed on the orientation on the stress axes. Two domains, which adjoin area n on the opposite sides and are conventionally marked + and ?, are distinguished. In the case of the two-dimensional (2D) stress state, any principal stress axis on passing from domain ? to domain + remains in the same quadrant of the plane as the continuation of this axis in domain +. The sign and size of the refraction angle depend on the sign and amplitude of the jump of the normal stress, which is tangential to the surface of discontinuity. In the three-dimensional (3D) case, the refraction of axes T ₁ and T ₃ should be analyzed simultaneously. For each side, + and ?, the projections of the T ₁ and T ₃ axes on the generally oriented plane n form the shear sectors S ⁺ and S ^?, which are determined unambiguously and to whose angular domains the possible directions p ⁺ and p ^? of the shear stress vectors belong. In order for the extreme stress axes T ₁ ⁺ ,T ₃ ⁺ and T ₁ ^? , T ₃ ^? to be statically compatible on the generally oriented plane n, it is required that sectors S ⁺ and S ^? had a nonempty intersection. The direction vectors p ⁺ and p ^? are determined uniquely if, besides axes T ₁ ^? , T ₃ ^? and T ₁ ⁺ , T ₃ ⁺ , also the ratios of differential stresses R ⁺ and R ^? (0 ≤ R ^± ≤ 1) are known. This is equivalent to specifying the reduced stress tensors T _R ⁺ and T _R ^? The necessary condition for tensors T _R ⁺ and T _R ^? being statically compatible on plane n is the equality p ⁺ = p ^?. In this paper, simple methods are suggested for solving the inverse problem of constructing the set of the orientations of the extreme stress axes from the known direction p of the shear stress vector on plane n and from the data on the shear sector. Based on these methods and using the necessary conditions of local equilibrium on plane n formulated above, all the possible orientations of axes T ₁ ⁺ , T ₃ ⁺ are determined if the projections of axes T ₁ ^? , T ₃ ^? axes on side — are given. The angle between the projections of axes T ₁ ⁺ , T ₁ ^? and/or T ₃ ⁺ , T ₃ ^? on the plane can attain 90°. Besides the general case, also the particular cases of the contact between the degenerate stress states and the special position of plane n relative to the principal stress axes are thoroughly examined. Generalization of the obtained results makes it possible to plot the local diagram of the orientations of axes T ₁ ⁺ , T ₃ ⁺ for a given sector S ^?. This diagram is a so-called stress orientation sphere, which is subdivided into three pairs of areas (compression, tension, and compression-extension). The tension and compression zones cannot contain the poles of T ₃ ⁺ and T ₁ ⁺ axes, respectively. The compression-extension zones can contain the poles of either T ₁ ⁺ or T ₃ ⁺ axis but not both poles simultaneously. In the particular case when the shear stress vector has a unique direction p ^? on side ?, the areas of compression-extension disappear and the diagram is reduced to a beach-ball plot, which visualizes the focal mechanism solution of an earthquake. If area n is a generally oriented plane and if the orientation of the pairs of the statically compatible axes T ₁ ^? , T ₃ ^? and T ₁ ⁺ , T ₃ ⁺ is specified, then, the stress values on side + are uniquely determined from the known stress values on side ?. From the value of differential stress ratio R ^?, one can calculate the value of R ⁺, and using the values of the principal stresses on side ?, determine the total stress tensor T ⁺ on side +. The obtained results are supported by the laboratory experiments and drilling data. In particular, these results disclose the drawbacks of some established notions and methods in which the possible refraction of the stress axes is unreasonably ignored or taken into account improperly. For example, it is generally misleading to associate the slip on the preexisting fault with the orientation of any particular trihedron of the principal stress axes. The reconstruction should address the potentially statically compatible principal stress axes, which are differently oriented on opposite sides of the fault plane. The fact that, based on the orientation of the intraplate principal stresses at the base of the lithosphere, one cannot make a conclusion on the active or passive influence of the mantle flows on the lithospheric plate motion is another example. The present relationships linking the stress values on the opposite sides of the fault plane on which the orientations of the principal stress axes are known demonstrate the incorrectness of the existing methods, in which the reduced stress tensors within the material domains are reconstructed without allowance for the dynamic interaction of these domains with their neighbors. In addition, using the obtained results, one can generalize the notion of the zone of dynamical control of a fault onto the case of the existence of discontinuities in this region and analyze the stress transfer across the system of the faults.