A New Method for the Determination 

of Three-Dimensional Deformation 

Anisotropy in Rock Specimens

GEOLOGICAL SURVEY BULLETIN 1 2 5 8 E

A New Method for the Determination 

of Three-Dimensional Deformation 

Anisotropy in Rock Specimens

By RICHARD A. FARROW

CONTRIBUTIONS TO ENGINEERING GEOLOGY

GEOLOGICAL SURVEY BULLETIN 1258-E

A new laboratory technique for measuring 

orthotropic deformation anisotropy in single 

rock specimens and comparison of results with 

structural features in the source rock mass

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1973
UNITED STATES DEPARTMENT OF THE INTERIOR 

ROGERS C. B. MORTON, Secretary

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

Library of Congress catalog-card No. 72-600297

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.G. 20402 - Price 25 cents (paper cover)

Stock Number 2401-00269
CONTENTS

Page

Abstract................................................................................................................ El

Introduction........................................................................................................ . 1

Ellipsoid of deformation............ ........................ .............................. 2

Test procedure and data reduction.................................................................... 3

Features of the granite and results of the test............----.--.---..-.-.-.-.....-...-.--.- 6

Conclusions............................................................................................................. 11

References cited..................................................................................................... 12

ILLUSTRATIONS

Page

FIGURE 1. Diagram showing dimensions of the specimen tested for this

study................................................................................................ E3

2. Equal-area lower hemisphere projection of successive changes 

of position of the principal axes of the strain ellipsoid for 

the first loading cycle................................................................. 4

3. Equal-area lower hemisphere projection of successive changes 

of position of the principal axes of the strain ellipsoid for 

the second and third loading cycle........................................... 5

4. Graphs showing microstrain per pound per square inch as a

function of hydrostatic pressure............................................... 7

5. Equal-area lower hemisphere projection of succcessive positions of the principal axes of the strain ellipsoid for the 

first loading cycle.......................... ............................................ 8

6. Equal-area lower hemisphere projection of successive positions of the principal axes of the strain ellipsoid for the 

second and third loading cycles................................................ 9

7. Graphs showing net principal strain magnitudes as a function of hydrostatic pressure. . . . 10

in

CONTRIBUTIONS TO ENGINEERING GEOLOGY

A NEW METHOD FOR THE DETERMINATION

OF THREE-DIMENSIONAL DEFORMATION

ANISOTROPY IN ROCK SPECIMENS

By RICHARD A. FARROW

ABSTRACT

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- -4- e yy- -f- czz-.

These three strains, the principal strains, are designated ei, e z, 

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.

NORTH

FIGURE 1. Dimensions of the specimen tested for this study. Under hydrostatic stress, strain-gage rosettes on the three orthogonal faces measured 

unequal strains. These data were resolved into three ellipses and then were 

combined mathematically to form an ellipsoid of strain which rotated and 

changed shape under changing applied pressure.

Strain measurements were made in three orthogonal planes by 

three-component rectangular strain-gage rosettes bonded with 

epoxy to the surface of the specimen. The sample was coated with 

a clear semirigid epoxy compound to protect the electrical connections and to prevent entry of hydraulic fluid. Pressure was
E4 CONTRIBUTIONS TO ENGINEERING GEOLOGY

applied with a manual pump in increments as shown in figures 2-7 

and was held while strain readings were recorded from a digital 

strain indicator. The whole procedure was carried out as rapidly 

as possible to minimize any effects of creep.

FIGURE 2. Equal-area lower hemisphere projection of successive changes of 

position of the principal axes of the strain ellipsoid for the first loading 

cycle (loading-step numbers 1-18). Aei, Ae2 , and Acs are maximum, intermediate, and minimum compressive-strain changes. Also shown are poles 

of foliation (F), quarry floor (Q), three sets of prominent joints (Ji), and 

two sets of moderate joints (Ji>).

For each pressure increment, and for the cumulative pressure 

changes, ellipsoids of strain change and total strain were denned 

using algorithms similar to those given by Obert and Duvall 

(1967). The method consists of finding the orientation of subsidiary principal strain axes and the magnitude of shear strain, y, for 

each of three orthogonal planes. For each plane, two measure-
DETERMINATION OF DEFORMATION ANISOTROPY E5

ments of normal strain were averaged from gages on adjacent 

faces. This average was used for normal strain, c, in the computations which lead to the three principal strain-magnitudes solutions, cj, c2, and c3, which, together with their direction cosines, 

define the deformation ellipsoid.

FIGURE 3. Equal-area lower hemisphere projection of successive changes of 

position of the principal axes of the strain ellipsoid for second (loadingstep numbers 21-25; dots) and third (loading-step numbers 31 and 32; triangles) loading cycles. Aei, Ae2, and Aes are maximum, intermediate, and 

minimum compressive strain changes. Also shown are poles of foliation 

(F), quarry floor (Q), three sets of prominent joints (Ji), and two sets 

of moderate joints (Ja).

The orientation and magnitude of the principal axes of the 

strain ellipsoid can be shown graphically by several techniques. A 

variation of the methods described by Ramsay (1967, p. 134) is 

used here. Two diagrams define the changing strain tensor during
E6 CONTRIBUTIONS TO ENGINEERING GEOLOGY

deformation: (1) a lower hemisphere equal-area diagram onto 

which the axes of the ellipsoid are projected and which specifies 

the attitude of the strain ellipsoid at any given hydrostatic pressure, and (2) a diagram on which the principal strain magnitudes 

are plotted as a function of hydrostatic stress.

The simplifying assumption of elastic theory, that the deformation ellipsoid exhibits a constant and linear change at all stress 

levels, was not applicable to the Mount Airy "granite." In strainrelief overcoring, which has been performed at the Mount Airy 

quarry, the main interest is the behavior of the deformation ellipsoid for stress changes from 0 to the maximum relieved stress. For 

long-term monitoring of stress changes, the main interest is the 

behavior of the deformation ellipsoid at various confining stresses 

for incremental stress changes.

FEATURES OF THE "GRANITE" AND RESULTS OF THE TEST

The petrology of the Mount Airy "granite" has been discussed 

in detail by Dietrich (1961). He described the rock as a leucogranodiorite consisting of "approximately 55 percent oligoclase, 

20 percent quartz, 20 percent microcline, and 5 percent other constituents, chiefly biotite, epidote, and muscovite. * * * as exposed 

at the quarry, [it] is a foliated to nonfoliated holocrystalline, 

hypidiomorphic, medium-grained phanerite * * *. It has the overall appearance of being light gray with rather evenly disseminated 

black spots and (or) black streaks."

An unpublished geologic sketch map of the quarry made by F. T. 

Lee in 1970 shows the axial trace of an anticline which trends 

N. 70° E. and which is defined by sheeting joints parallel to the 

slope of the quarry floor. The specimen tested for this report 

shows neither foliation nor megascopic fractures. A detailed analysis of microfractures, similar to that made by Norman (1967) 

in granitic rocks near Atlanta, Ga., has not yet been made by the 

present author. Such microfractures could partly account for the 

measured deformation anisotropy of the rock (Norman, 1967).

Tests on the oriented specimen of Mount Airy "granite" demonstrated the dependence of both the orientation and the magnitude 

of the principal deformation axes upon the amount of stress. The 

orientation of the changes of the principal deformation axes relative to north, east, and vertical for incremental hydrostatic stress 

changes is shown in the equal-area projections (figs. 2, 3). The 

average magnitudes of these changes for the three loading cycles 

are shown in figure 4, and the orientations of principal strain axes 

are shown in figures 5 and 6. The net principal strain magnitudes 

as a function of hydrostatic stress are presented in figure 7. The 

calculations were made using the computer program described by 

Nichols, Abel, and Lee (1968).
DETERMINATION OF DEFORMATION ANISOTROPY E7

The results for the intervals between 700 and 900 psi (pounds 

per square inch) hydrostatic pressure on the first loading cycle 

were so anomalous that a deformation ellipsoid could not be constructed.

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FIGURE 4. Graphs showing microstrain per pound per square inch as a 

function of hydrostatic pressure.

The curves of magnitudes of principal strain changes and principal strains (figs. 4, 7) were projected through the interval 700 

to 900 psi. The curves for subsequent loading cycles suggest that 

no large errors in magnitude were introduced by using this procedure. During the 700- to 900-psi pressure interval, the trend of 

clockwise rotation of the e2 and c3 axes ceased. The attitude of 

e3 remained nearly fixed thereafter, and rotation of the e l and 

c2 axes began. This behavior continued during the second and third 

loading cycles, perhaps as a result of permanent closure of one or 

more sets of microfractures favorably alined with the c3 direction. 

Factors not yet thoroughly evaluated are the effects of specimen 

geometry, the cumulative effects of creep in the sample and in
E8 CONTRIBUTIONS TO ENGINEERING GEOLOGY

the strain gage adhesive, and strain resulting from temperature 

changes as the rock was deformed.

FIGURE 5. Equal-area lower hemisphere projection of successive positions 

of the principal axes of the strain ellipsoid for the first loading cycle, ei, 

£2, and es are maximum, intermediate, and minimum compressive strains. 

Also shown are poles of foliation (F), quarry floor (Q), three sets of 

prominent joints (Jj), and two sets of moderate joints (Jo).

The directions of high modulus that I calculated agree very well 

with previously published values. The average direction of maximum modulus in the horizontal plane is N. 51° E. over a stress 

range from 0 to approximately 4,000 psi for specimens from the 

Mount Airy quarry, as determined by Hooker and Johnson (1969, 

p. 14). The solution of the directions is based on the assumption 

that one of the principal modulus directions is vertical and that 

the other two lie in the horizontal plane. The information in the
DETERMINATION OF DEFORMATION ANISOTROPY E9

present report shows that the direction of greatest strain, corresponding to the direction of minimum modulus, is nearly vertical 

for pressures up to about 3,000 psi. It is recognized that the two 

remaining principal modulus directions are forced to be horizontal 

by an ellipsoidal solution. Nonetheless, the average attitude of 

minimum strain is N. 42° E., 10° NE; and that of intermediate 

strain varies from N. 50° W. to N. 80° W., with the plunge 

increasing from nearly horizontal to nearly vertical as pressure 

increases.

FIGURE 6. Equal-area lower hemisphere projection of successive positions 

of the principal axes of the strain ellipsoid for the second and third loading cycles, ei, e2 , and £3 are maximum, intermediate, and minimum compressive strains. Loading-step numbers 21-25 (dots) are for second loading 

cycle; loading-step numbers 31 and 32 (triangles) are for the third. Also 

shown are poles of foliation (F), quarry floor (Q), three sets of prominent 

joints (Ji), and two sets of moderate joints (J2).
E10 CONTRIBUTIONS TO ENGINEERING GEOLOGY

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DETERMINATION OF DEFORMATION ANISOTROPY Ell

Directions of deformation axes appear to be related to attitudes 

of geologic features mapped in the quarry by F. T. Lee (unpub. 

data, 1970). The specimen tested for the present report was taken 

from a site in the quarry where an indistinct foliation trends 

N. 35° E. and dips 70° NW. and where three sets of joints, somewhat enlarged and emphasized by blasting, occur. The attitudes 

of these joints are N. 25° E., vertical; N. 80° W., 8° NE. (a sheeting joint) ; and east-west, vertical. Lee described the first two as 

major (prominent) and the third as intermediate (moderate). At a 

nearby site, about 125 feet northeast of the overcoring location, 

the foliation strikes N. 12° E. and dips 60° NW. At this site a 

prominent joint strikes N. 35° W. and is vertical. A moderate joint 

strikes N. 60° E. and dips 60° SE. The quarry floor at the sampling site slopes N. 30° W. at about 12°. These data are plotted on 

the equal-area diagrams (figs. 2, 3, 5, 6). Under hydrostatic loading, maximum deformation of the specimen should occur in the 

direction normal to planes parallel to open fractures. In order to 

examine the relationship between the poles (normals) of the 

planes of joints and foliation and the principal strain axes, the 

poles and axes are plotted together. After the first-cycle instability, previously noted, the direction of least strain change, A£3 , 

remains fairly constant for all pressures. The axes Aci and Ac2 

rotate counterclockwise as pressure increases; however, they 

remain roughly in the plane which contains poles of two prominent joints, one moderate joint, the quarry floor, and the foliation. 

Poles of two moderate joints lie in the direction of greatest modulus and at N (horizontal). Possibly, microfractures parallel to 

the moderate joints closed permanently during the first loading 

cycle. The nonlinear stress-strain curves (fig. 7) show that the 

rock stiffens under increasing load, thus tending to support the 

assumed presence of microfractures.

CONCLUSIONS

The directional deformation characteristics of rocks can be 

determined from single rock specimens that are properly strain 

gaged and subjected to hydrostatic pressure. The immediate value 

of the method is to permit more precise stress calculations from 

strain measurements, such as those obtained with the solid-inclusion borehole probe of Nichols, Abel, and Lee (1968). Knowledge 

of the direction and magnitude variation of the deformation 

anisotropy of the host rock will remove the necessity of simplifying, perhaps erroneously, assumptions of isotropic elastic properties of the host rock. Additionally, this knowledge may permit 

the deciphering of a record of discrete episodes of tectonic loading 

which have impressed permanent changes on the rock fabric.
E12 CONTRIBUTIONS TO ENGINEERING GEOLOGY

REFERENCES CITED

Becker, R. M., 1968, An anisotropic elastic solution for testing stress relief

cores: U.S. Bur. Mines Rept. Inv. 7143, 15 p. 

Becker, R. M., and Hooker, V. E., 1967, Some anisotropic considerations in

rock stress determinations: U.S. Bur. Mines Rept. Inv. 6965, 23 p. 

Dietrich, R. V., 1961, Petrology of the Mount Airy "granite": Virginia

Polytech. Inst. Bull., Eng. Expt. Sta. Ser. 144, v. 54, no. 6, 63 p. 

Douglass, P. M., and Voight, B., 1969, Anisotropy of granites a reflection of

microscopic fabric: Geotechnique, v. 19, no. 3, p. 376-398. 

Friedman, Melvin, 1967, Measurement of the state of residual elastic strain in

quartzose rocks by means of X-ray diffractometry: Norelco Reporter,

v. 14, p. 7-9, 35. 

Hooker, V. E., and Johnson, C. F., 1969, Near-surface horizontal stresses

including the effects of rock anisotropy: U.S. Bur. Mines Rept. Inv. 7224,

29 p. 

Leeman, E. R., 1964, Borehole rock stress measuring instruments, pt. 2 of

The measurement of stress in rock: South African Inst. Mining and

Metall. Jour., v. 65, p. 82-114. 

Merrill, R. H., and Peterson, J. R., 1961, Deformation of a borehole in rock:

U.S. Bur. Mines Rept. Inv. 5881, 32 p. 

Nichols, T. C., Jr., Abel, J. F., Jr., and Lee, F. T., 1968, A solid-inclusion

borehole probe to determine three-dimensional stress changes at a point

in a rock mass: U.S. Geol. Survey Bull. 1258-C, 28 p. 

Norman, C. E., 1967, Microfractures in brittle rocks their relationship to

larger scale structural features and existing ground stresses: Ohio State

Univ. Ph. D. dissert., 169 p. 

Obert, Leonard, and Duvall, W. L, 1967, Rock mechanics and the design of

structures in rock: New York, John Wiley & Sons, Inc., 650 p. 

Panek, L. A., 1966, Calculation of the average ground stress components from

measurements of the diametral deformation of a drill hole: U.S. Bur.

Mines Rept. Inv. 6732, 41 p. 

Ramsay, J. G., 1967, Folding and fracturing of rocks: New York, McGrawHill Book Co., 568 p. 

Stuckey, J. L., and Conrad, S. G., 1958, Explanatory text for geologic map of

North Carolina: North Carolina Dept. Conserv. and Devel., Mineral

Resources Bull. 71, p. 22-23. 

Timoshenko, S., and Goodier, J. N., 1951, Theory of elasticity [2d ed.]: New

York, McGraw-Hill Book Co., 506 p.

U. S. GOVERNMENT PIUNTING OFFICE : 1973 O - 489-653