Rock-Bit Tooth Friction Analysis
Authors: Cheatham J.B. 
Published: December 2007

Abstract
The influence of friction on the force required for an idealized bit tooth to penetrate a "plastic" rock is analyzed. The rock is assumed to obey the Coulomb-Mohr yield criterion and the tooth is represented by a two-dimensional sharp wedge. At the tooth-rock interface the stress field not only satisfies the yield condition by also is such that the ratio of the shearing stress to the normal stress is equal to the coefficient of friction. The solution relates the force on the tooth to the depth of penetration, the tooth angle, the coefficient of friction between the rock and bit tooth and the angle of internal friction and compressive strength of the rock.
Introduction
Most rocks are brittle at atmospheric pressure but exhibit ductility and increased strength as the confining pressure is increased. In a previous analysis the force required for a bit tooth to penetrate a rigid- plastic rock was computed for two limiting conditions:
a perfectly lubricated tooth-rock interface which should correspond to a lower limit for the penetrating force, and a perfectly rough interface which should provide an upper limit.
Experimental results for rock indentation under confining pressure have confirmed that the analysis generally predicts upper and lower limits for the penetrating force as a function of rock properties and tooth angle. The present analysis extends the previous work to account for the friction between the rock and the tooth.
ANALYSIS
The force required for incipient plastic flow of an idealized model of a rock under a bit tooth is computed subjected to the following assumptions.
BIT TOOTH
sharp two-dimensional wedge (end effects neglected) rigid tooth (elastic deformation of tooth neglected) displacement of tooth is normal to rock surface friction exists between tooth and rock.
ROCK
semi-infinite, isotropic and homogeneous rigid-plastic (elastic deformation of rock neglected) non-work hardening (horizontal stress-strain curve) Coulomb yield condition (linear Mohr envelope) smooth, flat surface
According to the Mohr theory of failure it is assumed that failure occurs when the state of stress is such that the Mohr stress circle becomes tangent to an experimentally determined envelope. The Mohr failure curve is frequently approximated by a straight line and referred to as the Coulomb-Mohr failure criterion. If the rock is under sufficient pressure to be ductile we can use the linear envelope as a yield condition. The Coulomb-Mohr yield criterion is illustrated by Fig. 1 and is expressed by the following equation:
where
= shear stress on radial slip line
= normal stress on radial slip line
= cohesive strength= angle of internal friction
The slip-line field for the plasticity solution of this problem is shown by Fig. 2. The angle depends on the coefficient of friction at the tooth rock interface.
If the solution corresponds to the perfectly lubricated tooth without friction; and if, the solution becomes that for the perfectly rough tooth (see Fig. 1). The two families of slip lines intersect at angles of and each family intersects the direction of maximum compressive stress at angles of. In the triangular region adjacent to the free surface, both families of slip-lines are straight, which indicates a constant-state field in which the mean stress does not vary. Stresses change along the curved slip-lines of the "fan" but are constant along the straight radial lines.
SPEJ
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