Rock/Bit-Tooth Interaction for Conical Bit Teeth
Authors: Miller T.W. , Cheatham J.B. 
Published: December 2007

Abstract
Insert-type bit teeth are axially symmetric and therefore cannot be properly described by a two-dimensional, plane-strain analysis of a wedge-shaped tooth. Since few exact plasticity solutions exist for axially symmetric problems, an approximate method is developed and applied to the interaction of a conical bit tooth with rock. Numerical results of the approximate method compare favorably with the known exact solutions. Alter making allowance for the experimental result of little or no lip formation around an indentation, numerical results using the approximate method correlate well with experimental measurements for the indentation of conical teeth into marble and sandstone.

Introduction
Previous studies of the interaction of bit teeth with rock have been concerned primarily with two-dimensional wedge-shaped teeth. This provides a good approximation to the shape of most bit teeth, and the two-dimensional analysis simplifies the problem. However, some bits have inserts or so-called "buttons", which cannot be adequately represented by a wedge. Also, "end effects" exist for the finite-length wedges, whereas cones, flat-circular cylinders and hemispheres, while axially symmetric, are truly three-dimensional. These axially symmetric punches pose a more difficult problem analytically; hence one of the objectives of this paper is to provide an approximate method for analysis of this class of problems. problems. The analysis of problems concerned with the indentation of a rigid/plastic half space by an axially symmetric indenter is more difficult than the analogous problem in plane strain due to the radial expansion that occurs with increased distance from the axis of symmetry. The axially symmetric solutions are also of interest because they give insight into other practical problems, such as analysis of Rockwell-type hardness tests and end effect corrections for plane-strain solutions.

Discussion of Exact Solutions
Ishlinski has solved the problem of the indentation of a plastic half space by a perfectly smooth, flat punch by tedious hand calculations, whereas Shield and Cox et al. have rigorously solved the perfectly smooth, flat-punch problem for metals and soils, respectively, using finite-difference techniques requiring a digital computer. Mroz has developed a graphics technique for the above problem for metals, but it, too, is a long, tedious procedure. Lockett has solved problems for the indentation of metals by perfectly smooth, conical indenters with half-cone angles between 52.5 degrees and 90 degrees, also using finite-difference equations. Berezancev has presented values of average pressure on cones with hall-cone angles of 15 degrees and 30 degrees for indentation of both metals and soils.
Eason and Shield have given the solution for the indentation of metals by a perfectly rough, flat indenter, again using finite-difference equations. Problems for perfectly rough, flat indenters Problems for perfectly rough, flat indenters penetrating soils and perfectly rough, conical penetrating soils and perfectly rough, conical indenters penetrating soils and metals have not yet been solved to the authors' knowledge.
Spencer has presented an approximate solution for an annular, flat punch using perturbation methods. This method does not seem applicable to conical indenters, however, since it requires a fixed punch radius that is large with respect to other dimensional quantities of the problem. If the approximate method presented here were applied to the annular punch problem, it would give results corresponding to Spencer's first-order solution.
It is the purpose of the present paper to compare the results of the approximate method of finding pressure distributions on a conical indenter with pressure distributions on a conical indenter with some of the exact results discussed above; and then to use other results of the approximate method to analyze experimental data. The numerical results used to analyze experimental data are calculated using a modified slip-line field for a formation.