A new method for the determination of three-dimensional deformation anisotropy in rock specimens

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A new laboratory technique for determination of anisotropic deformation in rock involves the application of hydrostatic stress to a single instrumented prism of rock. Measured strains are used to calculate a deformation ellipsoid. Determinations on Mount Airy granite agree well with published data. Directions of anisotropy in the test sample and structural features within the source rock mass appear to be related. INTRODUCTION Calculations of in situ stresses in rocks are made using measurements of length changes of linear elements that are taken when a mass of rock containing the linear elements is freed from the surrounding rock. Whether the linear element is quartz d-spacing (Friedman, 1967), borehole diameter (Merrill and Peterson, 1961), or a strain gage, bonded either to the rock itself (Leeman, 1964) or to an intermediate carrier (Nichols and others, 1968), the deformation characteristics of the rock must be determined in order to calculate stress in the rock. Simplifying assumptions, such as that of linear elastic isotropy of the host rock, may yield stress determinations that are seriously in error (Becker and Hooker, 1967). Strains measured on the rock surface (or in the borehole) theoretically can be restored, under properly chosen triaxial stress conditions in the laboratory, to their original measure. A more practical approach was described by Becker (1968), who developed equations defining the relationships among stress, strain, and deformation in a thick-walled cylinder of rock subjected to triaxial loading. Among the limitations to his approach, as he pointed out, is the necessity for a geometric alinement between the cylinder and a plane of elastic symmetry. To obtain this required geometric alinement of the El E2 CONTRIBUTIONS TO ENGINEERING GEOLOGY « borehole, the elastic symmetry must be determined by laboratory testing (Panek, 1966). In April 1969 the U.S. Geological Survey began a stress-field investigation at the quarry of the North Carolina Granite Corp. at Mount Airy, N.C. Fieldwork was done in cooperation with U.S. Bureau of Mines personnel who had previously worked in the area (Hooker and Johnson, 1969). Studies of long-term stresschange measurements and stress-relief overcoring are continuing. The present report concerns a new deformation test performed in the laboratory on a sample taken at the time of overcoring. ELLIPSOID OF DEFORMATION To simplify sampling and testing programs and to define the symmetry-of-deformation properties of rocks, a procedure wasdevised to measure anisotropic deformation under hydrostatic compression. The test is based on the assumption that when subjected to hydrostatic stress a circle of anisotropic material will deform into an ellipse and a sphere will deform into an ellipsoid. Anisotropy of this type is characterized by three mutually perpendicular axes. Along each of these axes the stiffness, or compliance, coefficients have different values. Materials possessing directional elastic properties with this particular symmetry are called orthotropic. Douglass and Voight (1969) have described an ellipsoid of Young's modulus for some North American granites which indicates that the assumption of orthotropic behavior may be justified for most granitic rocks. One method of portraying the ellipsoid of strain (Timoshenko and Goodier, 1951, p. 224) is to define it as the locus of_tips of all radial vectors, r, whose magnitude is given by /?=^/V|e|, where k is a constant and e is the normal strain in the direction of r. The ends of these vectors will all lie on the surface described by ±k2 = exX2 -f eyy2 + ezZ* + yyzyZ -f yfZXZ + yryXy, where x, y, and z form a right-handed Cartesian coordinate system, and yxy is the shear strain in the xy plane. If k=l, and all strains are positive but not equal, the surface is an ellipsoid. If the principal axes of the ellipsoid are rotated to aline with the x, y, and z axes, the equation simplifies to ±k= exx-4e yy-fczz-. These three strains, the principal strains, are designated ei, ez, and £3 for the maximum, intermediate, and minimum principal strains respectively. Throughout this report compressive strains are reckoned positive. DETERMINATION OF DEFORMATION ANISOTROPY E3 TEST PROCEDURE AND DATA REDUCTION An oriented specimen of Mount Airy granite of Stuckey and Conrad (1958) was deformed under hydrostatic pressure in a triaxial cell. The dimensions of the sample are shown in figure 1.