| The principal stress trajectories (PST) obtainable from the inversion of data on various
types of coeval natural stress indicators provide important information on the real
stress fields of the Earth’s crust. In particular, once the PST field has been obtained,
the stress field can be calculated from equilibrium conditions without regard to
the crust rheological properties. In the case under consideration the equilibrium
conditions form a closed hyperbolic system of differential equations on unknown
magnitudes of the principal stresses. The PST concept is significantly complicated in
presence of faults, natural interfaces, and other surfaces of discontinuity where
according to laboratory experiments and drilling results the stress trajectories are usually
refracted. Unfortunately, the phenomenon of the PST refraction is poorly understood
theoretically and often ignored in mathematical modelling which leads to unjustified
conclusions.
The full investigation of the phenomenon has been carried out for discontinuity surface D
for which we distinguish "+" and "-" sides. Inclinations of axes of the extreme principal
stresses T1 and T3 (T1 >T3, tension is positive) from both sides are different but should be
potentially compatible which means existence of stress magnitudes ensuring equilibrium. To
check the compatibility of axes at any point, they should be orthogonally projected in a
special way onto the plane tangent to the surface. This yields the so-called shear
sectors S+ and S- containing all the possible shear stress direction vectors p+
and p-, |p+| = |p-|=1, at a local point. The T1 and T3 axes from both sides are
potentially compatible if and only if the intersection of S- and S+ is nonempty.
Indeed, if the intersection is empty the necessary condition of equilibrium, p-=p+,
cannot be fulfilled. Let now the S- sector with the central angle φ- be given at any
point on the surface D. Then possible orientations of the T1 and T3 axes at this
point from the "+" side can be presented by the stress orientation sphere which
is divided into 3 pairs of areas: tension, compression and compression-tension.
Tension areas can not contain poles of the T3 axes whereas poles of the T1 axes are
prohibited for the compression areas. The areas of compression-tension, which are
spherical digons with angle φ-, may contain the T1 or T3 pole but not both poles
simultaneously. In some particular cases, when the vector p- has a unique direction
(φ-=0), the compression-tension digons disappear and the stress orientation sphere
degenerates into a "beach ball" diagram which is characteristic for the earthquake focal
mechanism solutions. If the potentially compatible pairs of the T1 and T3 axes
are given, then the jumps of the stress values across the surface D can be easily
calculated.
The results obtained have a wide range of applications. For example, no trihedron of the
principal stress axes can be associated with a slip along already existing fault. In fact, 6
values are to be determined. These values define orientation of the potentially compatible
stress axes on both sides of the fault. Existing restrictions are too weak to ensure that these
values can be determined locally. |