In this paper, the geometry of borehole forced by two-dimensional stress was studied based on classical rock mechanics. The relation between horizontal principal stress and geometrical parameters of deformed circular borehole was established. The new prediction model of horizontal stress was deduced based on the above quantitative relation. Additionally, the stress computation was realized based on elliptical parameters of borehole. Subsequently, the feasibility and scientificity of the calculation method of horizontal principal stress were further confirmed via an experiment of borehole deformation based on borehole deformation. The feasibility and applicability of the method were verified based on actual drilling data in Xinjiang Province. The results demonstrated that the geometric shape of circular borehole after deformation was elliptical under the action of planar two-dimensional stress, and the horizontal principal stress could be expressed by the elliptic geometric parameters after deformation. The shape structure of the circular borehole under the action of non-uniform horizontal principal stress was elliptical, and the horizontal principal stress calculated using the parameters of the elliptical shape structure was directly proportional to the loading load. Larger the borehole diameter, smaller was the error of the method. The results provide new insights for in-situ measurement and inversion of deep horizontal in-situ stress.

In part I of this study, the great practical importance of the onset of stable crack growth was out-lined, since several researchers have supported that it can serve as a more realistic estimation of the in-situ spalling strength of the rock mass, in comparison with the more frequently used uniaxi-al compressive strength. Moreover, a short review was also provided regarding the empirical techniques that have been proposed in the last six decades for the prediction of the crack initiation stress. The main criticism of each method was also mentioned. Subsequently, a new method was proposed, based on the Trapezoid Rule, an elementary calculus approximation technique. The lat-ter technique possesses a strong and robust physical explanation, easy application, and complete objectiveness. The Trapezoid Rule method was applied to ten rock specimens, eight marbles and two vesicular basalts, that were subjected to uniaxial compressive tests. The results of the newly suggested method were compared to those obtained using the established methods of the existing literature. Ultimately, it displayed exceptionally close results with all other methods for the ten specimens, thus indicating that it can accurately and consistently predict the crack initiation stress for the two rock types.

In part II of this study, a second method for the prediction of the crack initiation stress was sug-gested. The new technique is based on an elementary mathematical calculus theory. Particularly, the one that supports that the points where the second derivative of a function is equal to zero can be considered as possible inflection points of the function. The proposed Second Derivative meth-od fulfilled all the necessary criteria, that were mentioned in part I, so that it can further advance the research field. The method was applied to ten rock specimens, specifically eight marbles and two vesicular basalts, that were subjected to uniaxial compressive tests. The predicted crack initi-ation stresses from the new method were compared with those obtained from the established techniques of the existing literature. The new method had very close results with all other utilized methods for the marbles, thus meaning that the proposed Second Derivative technique can accu-rately and consistently determine the onset of stable crack growth for that rock type. On the con-trary, the new method displayed a poor correlation with the other techniques for the two basalts, hence indicating that further tests need to be conducted in the future for that rock type.