Summary: The Mohr-Coulomb (M-C) failure criteria is by far the most widely used model for granular materials (including rocks and soils) in many geotechnical industries due to its simplicity. Its parameters are conventionally obtained by performing a series of pressurized triaxial compression tests (TCTs). The peak stress from each confining pressure is plotted at a single point as the maximum vs. the minimum principal stresses. The M-C parameters are obtained by the best linear fitting through the failure envelope formed by the peak stress points of interest. The TCT is time-consuming, costly, and destructive. Is there an experiment that can provide the failure envelope with one single test, even at the atmospheric pressure? This seems impossible, at least has never been rigorously reported. In this work, we present our finding of such a test. The paper first provides the experimental setup and the theoretical solution for a hemispherical scratch test of a polycrystalline diamond cutter (PDC) at low rate of cut (ROC) and low depth of cut (DOC); the theoretical characteristic response of the test is identified and validated. Next, the formula for the rock-cutter interfacial coefficient of friction (COF) is derived, which provides a theoretical guide for the friction measurement for shaped cutters. Then, the nominal average drag and thrust stresses are evaluated, which are observed experimentally and then proved theoretically to behave in a linear relation, analogous to the M-C envelope from the TCTs. We also provide a first method that can self-consistently evaluate the M-C parameters for the crushed zone, which plays a very important role in rock cutting. By comparing the M-C results for three rocks at different loading conditions, good agreements under different scratch conditions are made against the M-C parameters for several rocks from the TCTs.

Rotary steerable drilling technology is fundamentally aimed at achieving precise wellbore trajectory control. As a representative directional tool, a dynamic push-the-bit RSS generates steering force during rotary drilling through the interaction between its extendable steering pads and the borehole wall, and it is distinguished from static push-the-bit RSS by the rotational friction that develops at the pad–wall interface. To further clarify the influence of friction on the resultant steering force and the build-up rate, this study develops a steering-force optimization model that explicitly incorporates tangential friction, validates the model, and then conducts numerical simulations to examine how PDC bit design parameters and formation properties affect the build-up rate. The results indicate that the friction-aware optimization model can achieve a higher build-up rate. Quantitatively, relative to the friction-free allocation model that is commonly used as the baseline in push-the-bit BUR prediction, the friction-aware formulation increases the final lateral displacement from approximately 28.4 to 30.6 mm in the analytical comparison (+7.7%) and from approximately 24.3 to 26.9 mm in the full-scale finite-element comparison (+10.7%) over the same steering-force action time. In soft formations with a low internal friction angle, a bit design combining a moderate gauge-protection dimension, an appropriate inner cone angle, and a large crown radius can effectively enhance lateral cutting and steering-force transmission, thereby improving build capability and trajectory stability. These findings provide a theoretical basis for improving build-rate efficiency in push-the-bit rotary steerable drilling systems.