Stresses Around Boreholes in Bilinear Elastic Rock
Authors: Haimson B.C. , Tharp T.M. 
Published: December 2007

Abstract
It is commonly accepted that the radial and tangential stresses around boreholes arc independent of the rock elastic properties when the assumptions of linearity, homogeneity, and isotropy are made. Although rock is never perfectly linear, the stress/strain relationship can often be linearized as a first approximation, which vastly simplifies the analytical approach. However, the slope of the linear relationship in compression (Ec) is almost always higher than that in tension (Et), and this bilinear behavior can and should be incorporated in the analytical approach to any problems involving mixed stresses at a point - e.g., stresses around boreholes and cavities, stresses along the vertical diameter in a Brazilian test, stresses in uniaxially loaded rings, stresses in bent beams. The problem of a circular hole under internal pressure and hydrostatic loading at infinity has been worked out. The resulting stresses differ considerably from those obtained using the common assumption of linearity. In particular, when no external loading exists, the particular, when no external loading exists, the tangential stress (sigma theta) at a borehole wall is expected to equal the internal pressure (pi) in the borehole (sigma theta = pi). However, the bilinear character of rock yields the expression
The internal pressurization of hollow cylinders was suggested as a possible technique for determining tensile strength. The findings reported here dispute the suitability of the method since the tensile and compressive Young's moduli must be known in order to calculate 0. Laboratory testing shows that generally the internal pressure, required to initiate rupture around a borehole is higher than the uniaxial tensile strength of the rock, in accord with the results of this paper.

Introduction
Linear elasticity is generally assumed in both field and laboratory situations involving stress and displacement in rock. That assumption allows the direct application of a considerable body of theoretical stress solutions. Unfortunately, those solutions are only as good as the underlying assumptions, which are sometimes of questionable validity in rock. The typical nonlinearity of the stress/strain curve in rock has long been recognized but is usuals ignored. The stress/strain relationship can often be linearized as a first approximation, which vastly simplifies the analytical solution. However, the slope of the line in compression (Ec) is almost always higher than that in tension (Et). The ratio Et/Ec can in fact vary between 1:1 in very tight rocks, to 1:2 in some limestones, to 1:20 in weak sandstones, to 0 in no-tension soils (see Table 1). Hence, with respect to its complete deformation spectrum, rock stress/strain relationship can at best be simplified into a bilinear curve with the point of intersection at zero stress (Fig. 1). The point of intersection at zero stress (Fig. 1). The assumption of bilinearity can and should be incorporated in the analytical approach to any problems that involve mixed stresses at a point. problems that involve mixed stresses at a point. The bilinear assumption has been employed by a number of investigators to represent this behavior. Burshtein and Fairhurst have derived bilinear stress formulas for rectangular beams in flexure, and Adler has done the same for beams of circular and more general cross-sections. These efforts cover only a few of the many cases in which both tensile and compressive stresses exist. The emphasis of the present paper is on bilinear stress equations for thick-walled cylinders.
TABLE 1 -- YOUNG'S MODULI IN TENSION AND COMPRESSION
Et Ec Rock Type (10(6) psi) (10(6) psi) Et/Ec Westerly granite 2.5 10.5 0.24 Austin limestone 1.7 2.3 0.74 Carthage limestone 5.1 9.2 0.55 Indiana limestone 1.6 3.9 0.41 Georgian marble 3.4 6.1 0.56 Tennessee marble 7.7 11.1 0.69 Russian marble 1.3 3.0 0.43 Star Mine quartzite 11.0 11.0 1.00 Arizona sandstone 1.7 6.6 0.26 Berea sandstone .6 3.4 0.18 Millsap sandstone 0.1 1.9 0.05 Tennessee sandstone 0.2 2.4 0.08 Russian sandstone 1.7 8.3 0.21
SPEJ
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