EFFECT OF SPECIMEN SIZE ON CONFINED COMPRESSION …-L44 Elastic Modul-is Models for Indiana Limestone 65 45 Elastic Modulus for Indiana Limestone (a3 = 2 ksi) 65 46 Elastic Modulus

-. I - ,

AD-751 548

EFFECT OF SPECIMEN SIZE ON CONFINED

COMPRESSION TESTING OF ROCK CORES

Peter Jay Huck

IIT Research Institute

Prepared for:

Advanced Research Projects Agency

August 1972

DISTRIBUTED BY:

National Technical Information ServiceU. S. DEPARTMENT OF COMMERCE5285 Port Royal Road, Springfield Va. 22151

L.Woo

S• ..• IIT RESEARCH INSTITUTES• i' .• r-•Technology Center

•i" '• •Chicago, Illinois 60616

i T • •IITRI Project No. D6059

S• Final Technical. Report

4,EFFECT OF SPECIMEN SIZE.,. ON CONFINEDCOMPRESSION TESTING OF ROCK CORES

S~by

Peter Jay Huck "

SMonitored by '"D

-Bureau of MinesSU. S. Department of the Inte-rior NOV 10• 1972

Washington, D. C.

S•August l 1972

The views and conclusions contained in this Aocument areI those of the author and should not be intex-preted as necessarily

representing the official policies, either expressed or implied!T of the Advanced Research Projects Agency, Bureau of Mines, orS~the U. S. Government.

NATO "' L©,, TECHNICALIS

I , _ ---- •,,

\lIT ESERCH NSTT.T

r i

-7;

Unclassified 3200.8 (AtL 1 to Encl 1)'

security Clasifienttia Mar 7, 66DOC UMENT CONTRO L DATA R D

(.. .. .cf.,oioc.I, n of (if['. body of , If-.. .. . l -- d . . - .-.. ..d. -l.I . ..d)Of4IGINA TINQ c r"lV Ty (Corp.,it.i aItho-) &2. vO r ' SECURI TY ZI..ASSIPICA TIOlJ

lIT Research Institute, Chicago, Ill. Unclassified

3 REPOkr TITLE.

Effect of Specimen Size on Confined Compression Testingof Rock Cores

A. O1SCRIPTIV. NO rEs ('-pco oPe.-a rl • ,td inclusive dat v)

"Final Technical Report7 AU THO'I(5) (-l18,t n iaer. roddlo Initao., last tnnae)

Peter J. Huck I;

6 r~r F 0074- AL 7o. =TTANO VAES 17b. 140. Or REFS_pril 1972 _,._,____ .. 1 733

.80 CONTRACT on cR/NT No go. ONIGIIAIOR'S kLPORT NUMBCRIS)SITNH0210009h. PROJECT 1.0

ARPA Order No. 1579________C.9. OTHER PO,i ho(NI (Any other Iumbors det may be h SSIgIIeod

Amendment 2 OV'O€°)d°Proga _d__11I0. W TFIUV TIOU N STATEPArtJT

Approved for public release, distribution unlimited

11 SJPP.EMEINTAfI N OTlE - fi2. SPONSORING MILITARY ACT'VITY

Director, Advanced ResearchProjects Agency

13. ADS-qAC1-

An experimental study was conducted to determine theinfluence of specimen size on the mechanical response ofrock. Specimens of Charcoal Black granite (Cold Spring,Minnesota) and Indiana Limestone (Bedford, Indiana) rangingin size from 2 -in. dia. to 36-in. dia. (3 2 -in. dia. forthe granite) were tested in triaxial compression. Testdata included axial and circumferential strain at up to30 locations on the largest specimens, and axial and radialstresses. Data for loading, unloading and reloading conditionswere collected. The loading data were fit to models describingbulk modulus and shear modulus, from which other moduli weredetermined. The reduction in strength over the size rangeand confining pressures employed ranged from 20 to 50 percent.

Iriro FORM 4 A7DD I, toV : ie 7.37 Unclassified

(g) lasslzc:!1 o

uTnclassiffed 3200.8 (Att 1 to En! 1) i"S"cur:t ClisificIion Mar 7, 66 L--

LitSK A I LINK 8i L. 14K CKEY WORDS

Large Triaxial Tests OLL

Scale EffectCharcoal Black Granite LIndiana Limestbne

':1

'7 1

Unclassified(•) Security Classification w.

IEFFECT OF SPECIMEN SIZE ON CONFINED -

COMPRESSION TESTING OF LARGE ROCK CORES

by

Peter Jay Huck1 1 lIT Research InstituteTechnology Center

T Chicago, Illinois 60616

ARPA Order Number 1579Program Code Number IFIOName of Contractor lIT Research InstituteEffective Date of Contract 29 Dec 1970Contract Expiration Date 29 Jan 1972Amount of Contract $75,791Contract Number H0210009Principal Investigator and Madan M. Singh

Phone Number 312/225-9630 Ext. 4784Project Engineer and Peter J. Huck

Phone Number 312/225-9630 Ext. 4735Short Title of Work Triaxial Tests on Large Rock Cores

"Sponsored by

"Advanced Research Projects Agency

ARPA Order No. 1579

The views and conclusions corntained in this document are thoseT Yof the author and should nor be interpreted as necessarily

representing the official policies, either expressed or implied,of the Advanced Research Projects Agency or the U. S. Government.

I"Approved for public release; distribution unlimited"

lIT RESEARCH INSTITUTE

FOREWARD

This is the final report on lIT Research Institute

((IITRI) Project No. D6059, entitled, "Effect of Specimen size

on Confined Compression Testing of Rock Cores" covering the

work period 29 December 1970 to 29 December 1971. This program

T !was performed under Contract No. H0210009 with the Bureau of

Mines of the U. S. Department of the Interior, with Mr. Egons R.Podnieks and later Dr. Syd Peng of the Twin Cities Mining Research

Center acting as technical monitors. The program was sponsored

by the Advanced Research Projects Agency of the U. S. Departmentof Defense under ARPA order no. 1579, Amendment 2.

""- The project was conducted under the direct supervision

of Dr. Madan M. Singh, who served as program manage Mr. Peter

J. Huck was project engineer. Other IITRI staff members contri-

buting to the overall research effort included Drs. R. H. Cornish

and A. Semnelink, and Messrs. L. A. Finlayson, P. A. Hettich,

E. J. Smith, J. Vosatka and A. Wawryszyn.

Respectfully submitted,

lIT RESEARCH INTITUTE

Madan M. Singh, ManagerSoil and Rock Mechanics

• " APPROVED:

R. H. CornishDirector of ResearchMechanics of Materials Division 4

MMS/ps


ST

TABLE OF CONTENTS

Section

1 INTRODUCTION 2

2 PREVIOUS STUDIES 3

2.1 Effect of Size 3

2.2 Effect of Confinement 6

3 EXPERIMENTAL APPARATUS 7

3,1 Small Test Cells 9

3.2 48-Inch lest Cell 9

3.3 Specimen Preparation 13

3.4 Instrumentation and Data Reduction 14

_ 4 EXPERIMENTAL PROGRAM 17

4.1 Non-Destructive Tests 17

4.1.1 Results of Non-Destructive Tests 20

Charcoal Black Granite

4.1.2 Resultj of Non-Destructive Tests 27"Indiana Limestone

"4.2 Classification Tests 32

4.2.1 Petrographic Analyses 35

4.2.2 Tensile and Compressive Tests 37

4.3 Description of Constitutive Equations 37

4.4 Triaxial Compression 40

4.5 Poisson's Ratio and Young's Modulus 43

4.6 Charcoal Black Granite 44

4.6.1 Hydrostatic Behavior 44

4.6.2 Triaxial Behavior 47

4.6.3 Young's Modulus and Poisson's Ratio 55

4.7 Indiana Limestone 58

4.7.1 Hydrostatic Behavior 58

4.7.2 Triaxial Behavior 58

4.7.3 Young's Modulus and Poisson's Ratio 60


iiL

!

m TABLE OF CONTENTS (Cont'd.)

Section Page

4.8 Strength Properties 644.8.1 Charcoal Black Granite 644.8.2 Indiana Limestone 72

5 CONCLUS IONS 74

5.1 Hydrostatic Behavior 745.2 Triaxial Behavior 74

S5.3 Effect of Specimen Size on Scrength 755.4 Effect of Scale on Type of Failure 775.5 Recommendations 77

REFERENCES 78

LAI

hI

L

U lIT RESEARCH INSTITUTE

ILLUSTRATIONS1 81IFigure Pag

1i Typical "'riaxial Cell Schematic 8

2 Schemat'. of System for12 in. 10T- axial Specimens

3 Schematic of 48 in. I. D. Triaxial Cell Layout 11

4A 32 in. Grai.ite Specimen Being Loaded into Large 12Triaxial ChamberI 3 • 32 in. Granite Specimen after Testing at Confining 12Pressure of 2000 psi

Spec.:"eno Orientation and Array of Measurement 15

6: '.-c il ' V'elocity S-stem for Small Cores 18

6b Acc -•.c Velocity System for Large Cores 18 1

7 Time of Arrival Measurements for a Single Specimen 20at Several Frequencies

8 requency of Occurrence of Shore Scleroscope 22Readings for Charcoal Black Granite

9 Frequency of Occurrence of Schmid& Hammer 23Readings for Charcoal Black Granite

10 Frequency o-. Occurrence of Sonic Velocity 25Determinations in Charcoal Black Granite

"11 Shore Scleroscope Readings for Indiana Limestone 29

12 Frequency of Occurrence of Schmidt Hammer Readings 30for Indiana Limestone

13 Sonic Velocities for Indiana Limestone 31

14 Acoustic Velocities in 36 in. Dia Indiana 33"Limes tone

7- 15 Idealized Hydrostatic Behavio- of Granular 39Materials

16 Variation of Shear Modulus with Mean Stress 42

17 Mean S, :ess vs. Volumetric Strain (2" Granite) 4518 Mean Stress vs. Volumetric Strain (4" Granite) 45

19 Mean Stress vs. Volumetric Strain (12" Granite) 46

20 Mean Stress vs. Volumetric Strain (32" Granite) 46

21 Bulk Modulus for 2" Granite 48

: 22 Bulk Modulus for 4" Granite 48


iv

ILLUSTRATIONS (Cont' d)

"Figure &a-e



25 Mean Stress ve. Volumetric Strain - Granite 50Models

26 Bulk Modulus vs. Mean Stress - Granite Models 50

27 Shear Modulus vs. Mean Stress - 2 in. Granite 51




31 Elascic Modulus for 2" Granite 54

32 Elastic Modulus for 4" Granite 54

33 Elastic Modulus vs. Mean Stress - 12" Granite 55

34 Elastic Modulus vs. Mean Stress - 32" Granite 55

35 Poisson's Ratio for 2 in. Granite 56

36 Poisson's Ratio for 4 in. Granite 56

37 Poisson's Ratio for 12" • Granite 5738 Poisson's Ratio for 32" • Granite 57

- 39 Mean Stress vs. Volumetric Strain for all 59Limes tone

40 Bulk Modulus for All Limestone 59

41 Shear Stress vs. Shaar Strain for Limestone 61

42 Shear Modulus vs. Shear Strain for All 62L. Limestone

43 Shear Modulus vs. Shear Stress for All Limestone 63 A

-L44 Elastic Modul-is Models for Indiana Limestone 65

45 Elastic Modulus for Indiana Limestone (a3 = 2 ksi) 65

46 Elastic Modulus for Indiana Limestone (a3 = 4 ksi) 66

47 Elastic Modulus for Indiana Limestone (03 = 6 ksi) 66"48 Elastic Modulus for Limestone (03=8 ksi) 67

49 Elastic Modulus for Limestone (a3 = 10 ksi) 67


v

ILLUSTRATIONS (Con t' d)

IFigure Page

50 Poisson's Ratio Models for Indiana Limestone 68

-51 Poisson's Ratio for Indiana Limestone (a = 2 ksi) 6851 Poisson's Ratio for Indiana Limestone (03 = 2 ksi) 6852 Poisson's Ratio for Indiana Limestone (a3 = 4 ksi) 68 1

53 Poissons Ratio for Indiana Limestone (03 = 6 ksi) 6954 Poisson's Ratio for Indiana Limestone (a3 = 8 ksi) 69

55 Poisson's Ratio for Indiana Limestone (a3 = 10 ksi) 69

56A Failure Stresses for Charcoal Black Granite 70

T 56B Failure Stress for Indiana Limestone 70

X

.11


vi

ABSTRACT i

An experimental study was conducted to determine the

influence of specimen size on the mechanical response of rock.

Specimens of Charcoal Black granite (Cold Spring, Minnesota)

and Indiana Limestone (Bedford, Indiana) ranging in size from

2-in. dia. to 36-in. dia. (32-in. dia. for the granite) were

tested in triagial compression. Test data included axial andk. circumferential strain at up to 30 locations on the largest

specimens, and axial and radial stresses. Data for loading,

unloading and reloading conditions were collected. The loading

data were fit to models describing bulk modulus and shear modulus,

from which other moduli were determined. The reduction in

strength over the size range and confining pressures employed

ranged from 20 to 50 percent.

SRAT;


1

1.0 INTRODUCTION

One of the problem areas in mining and underground

construction is that little is known about the mass behavior

of rcck. It is recognized that the rock mass behaves differentlyI: from small specimens that can be tested in the laboratory, but

the effect of scale is not well understood, particularly undergeneralized stress conditions. Since large scale tests aredifficult and expensive even when they are possible, investi-

gation into the influence of specimen size will help to esta-

blish a relationship between the behavior of small specimens

in routine laboratory tests and the mass behavior of rock in

the field. This program was intended to study the scale effect

in triaxial tests over a range of specimen sizes.

Two rock types, Charcoal Black Granite (Cold Spring,

Minnesota) and Indiana limestone (Bedford, Indiana) werestudied during this program. A previous program considered

the granite, as well as another rock type, so that additional

data on large granite specimens was available. Four sizes of

specimens, 2-in. dia, 4-in. dia, 12-in. dia, and 32-in. dia

(36-in. die. for the limestone) were tested in triaxial com-

pression. All triaxial specimens were strain gaged to allow

the determination of elastic moduli. The strain data were

fit to models representing hydrostatic and triaxial behavior

to identify any change in mechanical properties with size. The

failure data were correlated to quantify the reduction in

strength wikh increasing size.

Superscripts indicate references listed at the end of thereport


2

- --. •'-- - 'i •.i i • -.-- .•',,- •=• ••, •.• -• .. ....... -

2.0 PREVIOUS STUDIES

2.1 Effect'of SizeA number of investigators have studied the strength

of pillars and coal cubes as a function of size. Their resultsare usually expressed as a power function in one of the

following or similar forms:a•Va Rb i

c •P. aB (2)• c

t cc _ ba h (3)

in which

i.•.= failure stress

V = volume

R = the pillar width-height ratio

a = cube dimension

b, h = pillar width and height respectively.• 2

Salaman and Munro summarized reported values for the exponentsa, b, a and f in formulas of the form of eq. (1) and (3) bymaking use of the relationships between ( a, S) and (a,b)implied by the dimensional terms V,R,b and h. This summaryis given as:

. a b

Salam n and -0.66 0.46 -0.067 0.59Munro F-0.16 +0.14 +0.0.48 +0.14

Greenwald -0.183 0.50 -0.1-11 0.72et a13

Steart 4

Holland and -1.00 0.50 -0.167 0.83Gaddy t

-4l RSERC INTIUT

Salaman and Munro that the variation in the exponents

may be a function of specimen size, and that there may exist acritical size above which the effect of increasing volame is Jnegligible.

Investigators concerned with cube strength have typically

fotiud values of1 in Eq. (2) on the order Af -0.5.

For rocks, the types of relationships developed are5slightly different. According to the Weibull theory

im v

where a = tensile or compressive strength of the rock fxom a

standard laboratory test,

am equivalent strength of the rock mass,

V vm = volume of the rock mass,

v = volume of the test sample, and

a = constant (with values near 10 for rocks).

6The relation established by Protodyakonov was oi the type

} : m

where s = spacing between major discontinuities in the rock

mass, e.g., joints, beds.

a = dimension of the test specimen, usually diameter

for compression tests, and= mass fract coefficient (in compression: 1-2 for

igneous rocks, 1-3 for competent sedimentary rocks,

3-10 for weak rocks; in tension: approximatelydouble these values).

7 .8 9Grobbelaar based on the work of Epstein8, Bieniawski , and

others found that the formulae relating the modal strength of


4

the weakest elements and its standard deviation, based on the

weakest link theory, are:

= - a (2logN)0.- 1/2 log(logN) + log(4 ) (2logN)-0"5

-0 5and r (N)= x r (12logN);•s s

or as(N) = (No) (logho/logN)0.5

Where N = number of flaws in the large cubic specimen,

No = number of flaws in the small cubic specimen or

unit cube,CN =modal strength of the weakest link in a sample

containing N elements,=average strength of a unit cube of material

containing Nu elements,as =standard deviation of the modal strength of

samples containing No elements,

and ys(N) = standard deviation of the modal strength

of the weakest element in a sample containing

N elements.

These formulae are based on the "weakest link theory",

which can be analyzed mathematically if it is assumed that

the frequency of occurrence of events is a continuous function

(e.g. Weibull 5 or normal distribution). The "links" in this

case are the macroflaws (or cracks) in the bulk material; not

the microflaws.

Glucklich and Cohen 10,11 have indicated that effects

other than statistical exist, since the total stored elastic

energy increases with specimen volume. The energy released

at onset of fracture is related to initiate fracture; in other

words, this reduces the rock strength. This phenomenon has

been recently discussed by Baecher1 2 .

14


5

2.2 Effect of Confinement

There have been numerous studies investigating theSeffects of various aspects of confinement on rocks. It is not

internded to review all of these completely in this section.lMost of the pertinent work has been briefly discussed bySwanson1 3 . The earliest experimental work was performed by

Adams 1 4 and von Karmen 1 5 . However, significant headway was16not made until the initiation of work by Griggs and his t

coworkers 1 7 ' 1 8 . Since then, of course, a number of researchers

have conducted various types of studies under pressure several19of which were presented at a symposium on rock deformation .

Baidyuk2 0 has summarized some of the Russian and American work.

Research in this area is still very active 13,21,22 All ofthis work has been performed with small rock specimens, a fewinches in diameter. As a result considerable light has been

shed on the behavior of the rock matrix and the criteria offailure. Refinements to the Griffith hypothesis have been

proposed 23,24,25 and appear to explain the rock fractureprocess under confinement fairly well. The extrapolation of

these theories to larger rock masses is of doubtful valueand hence large scale field testing has to be resorted to 2 6 .

The U. S. Bureau of Mines has undertaken a rather comprehensiveprogram to collect field data with the intention of correlating

it into a hypothesis 2 7 The contributions of Hoek 2 8

Bieniawski9, Wawersik2 9 Cook and Houpert 3 1 to the mechanismof brittle failure in rock deserve to be noted even though thestudies were not conducted under a confined state of stress.


6

In order to conduct triaxial tests for the range of

specimen sizes used on this project, four triaxial cells were

set up as shown below:

Chamber Specimen Maximum MaximumI. D. (in.) Diameter (in.) Chamber Pressure Axial Load

(ksi) lb. x10 6

4.0 1.95 30 0.375

6.5 3.65 30 0.990

14.7 12 20 3.40

48.3 32 & 36 20 axial 36.510 confining

In a standard triaxial cell the axial load is supplied by an

external loading machine. However, in order to achieve the

large end, loads required for the tests in this program, these

chambers were separated into two regions by sliding pistons.

The general configuration is shown in Fig. 1. One region

contained the specimen, and was pressurized to the desired

confining pressure. Axial load was transmitted to the rock

by the sliding piston. The maximum axial stress in the rock

depends upon the ratio of the rock and piston areas and the

difference in the confining pressure and the axial chamber

pressure. The specimen stresses are given by the following

equations:

a2 = o 3 = P 3

ALaO (PI-P3 -

1 )Ar

•i y + Laa1 3

where

A= piston areaP

Ar = specimen area


7

• •i q - r- f lrflg r •-wiui nstt • ..... sUUCSW ..... - rw_••4 urro -r

SChLamber Closure with

SInstrumentation Lead-

Out

Confining Pressure;¢:•.... .. Region

Test Specimen

S~Sliding Piston with

SLoading Blocks

•- • Axial Pressure Region

FIG. I TYPICAL TR!AXIAL CELL SCHEMATIC

8z

'1t•-•, •_ •,r•I

P axial chamber pressureP3 confining chamber pressure

Au = deviator stress

01,02 and a3= principal stresses.

Reference to Figure I and the above equations confirms that

if P1 = P3, Aa = 0 and the specimen is under hydrostatic

stress (al = a2 = 03). For P3 = 0, the specimen is unconfined,with 3 0 and 01 =AO = P1 /Ar"

3.1 Small Test Cells

The three smaller test cells are incorporated intoa testing system with centralized controls, instrumentation,

and pumping systems. A schematic of this system is shownin Figure 2. The tests conducted in this system consisted of

initial hydrostatic loading up to the desired confining pres-

sure, followed by triaxial compression at constant 03 abovethat pressure. At least one load-unload-reload cycle wasobserved for each test. Provision was made for pressure

cross-connections between the confining pressure and axialpressure chamber volumes to insure hydrostatic conditionsduring the hydrostatic test phases.

3.2 48-Inch Test Cell

This test cell is shown in Figures 3 and 4. The basic

unit is a 48" I.D. by 86" working length chamber having i

20,000 psi design working pressure. The chamber walls are

built up from rings 12" long which are held in place by a

3/4" thick liner on the inside diameter. The entire axialload is carried by a flexible reaction frame which was built

up froria steel strap. The 140 ton weight of chamber issubstantially less than the weight that would have beenrequired by conventional chamber design. This is thelargest chamber currently available at lIT Research Institute,and is capable of applying axial loads of 36 million pounds


9

CeIF-P41P

;4 t-4r- (5YLFH4

r44

CN (L) Ft clq

H4

040

C1.4

P-4

100

4 _ _ _ _ _ _ _'

(U U)

W > 1

V) <

3.000

coj

-*24

44 0

ci c

kd

Fig. 32 IN. GRANITE SPECIMEN BEINGLOADED INTO LARGE TRIAXIAL CHAMBER

Fig 3 32 IN. GRANITE SPECIMEN AFTERTESTING AT CONFINING PRESSUREOF 2000 PSI

"12

to rock specimens as large as 3 1/2 ft. in diameter.

As can be seen in the schematic, the pumping and control

systems for this unit are simpler than for the small chambers.A separate pump was used at each end of the chamber. Accumulators

were not used because the chamber volume itself is large in com-

parison with other available pressure chambers.

The operation of this chamber is similar to that of the

smaller chambers, except that the turn-around time between testsis on the order of a week instead of an hour. The tests in this

chamber differed slightly from those in the smaller test cells.

In order to maintain seal integrity, a positive pressure differen-

tial of at least 200 psi was maintained across the sliding piston

during the "hydrostatic" portions of the triaxial tescs, and the

axial pressure was not allowed to drop below 400 psi at the bottom

of the load-unload-reload cycle. Had these precautions not been

taken, there would have been danger of upsetting the piston seal,

thus aborting the remainder of the test.

303 Specimen Preparation

Specimens of both rock types (Indiana limestone and Char-

coal Black granite) were delivered to IITRI in the form of coresranging from 12 in. to 36 in. in diameter, Extra rock was included

to permit coring of the 2-in, and 4-in, diao cores, These smaller

cores were cut parallel to the axis of the larger cores so as not

to introduce complications because of anisotropy. All cores were

cut with a 1:2 dia to length ratio with the exception of thelargest cores, which were all 60 in. long. Thus the aspect ratio

of the Charcoal Black granite cores was 32/60 = 1:1.87 and that

of the Indiana Limestone was 36/160 = 1:1.67.

End preparation of the 2-in. and 4-in. dia cores con-sisted of facing and grinding on a lathe until the ends wereplane and parallel to 0.0001 inches. The larger cores were

capped in a specimen cage to permit handling. The cappingmaterial used was a steel-filled epoxy. Figure 4A and 4B

each show an assembled 32-in. specimen in its cage. The cage

tie rods were designed with end fittings that would accept

13

tensile load only. Since the maximum tensile load that could beapplied to the cage by these tie rods corresponded to 30 psi

compressive stress in the rock specimen, the effect of the cageon the rock was negligible.

An array of foil strain gages were mounted on each speci-

men as shown in Figure 5. The number used ranged from threerosettes on the 2-in. cores to thirty rosettes on the 32-and 36-in. cores. These were two-element reosettes with 1/4 in. gagelength placed with the direction of rolling parallel to the

specimen axis. The gage placement procedure included the follow-

ing steps:

"* grind the rock surface"* apply two thin coats of gage cement"* visual inspection for voids in the cement base"* affix gage and solder leads"* apply two coats of gage coat for water proofing* check gage for continuity and response (soft

eraser) and replace if necessary.

The instrumented cores were waterproofed with latex ce-

ment over the foil gages, and at least two coats of latex paintwith a thickening agent over the entire rock surface. Water-

proofing is very irportant in a triaxial test since both theloading conditions and the character of the rock can be changedby intrusion of fluid into the rock voids. In those tests where

the specimens could not be loaded to failure on the second loadcycle, the specimen was recovered intact and stripped of paintfor visual inspection. In several cases, traces of oil werefound under the paint, but typically the specimens were completelydry. In no case was there enough oil to do more than dampen a

very small area on the specimen surface.

3.4 Instrumentation and Data Reduction

The instrumentation on this program included foil strain

gages, and pressure gages. All data were recorded using anautomatic digital data acquisition system. The pressures

were read using Hiese bourdon gages and inserted


14

[I6

2 1

4 .'

Fig. 5 SPECIMEN ORIENTATION AND ARRAY

I

S15 1

Ii

manually onto the data system output as they were read.

Due to the bulk of data involved, the data were reducedand plotted using an 1108 Univac computer. Two programs were

employed in the reduction of each triaxial test. The first pro-gram plotted the raw strain data to show the behavior of the in-

dividual foil gages. On occasion, the response of individualstrain gages differed radically from the majority of the gages.The cause of this behavior was then evaluated as either a key-punching error and corrected, or as actual failure in the opera-tion of the gage, and the data discarded. Reasons for impropergage operation include actual gage failure, or a flaw in thespecimen near the gage location. After the editing operation,the data were reduced by a program that solved the bridge equa-tion for each gage location and printed out the individual strainsand the strain averaged over the specimen. These data were evalu-ated for any consistent variation in strain distribution, such asmight be caused by barreling of the specimen. If any variationactually occurred, its magnitude was too small to detect under

the normal gage to gage variation. In addition to printed out-put, the following plots were produced for each test:

Shear strain vs. deviator stress

Volumetric strain vs. mean stressiCircumferential and axial strain vs. axial stress

Elastic, shear and bulk moduli vs. mean stress.

Data from each test was then assembled by hand to show trends

from test to test.


16

•!i77 7- 71 F: Mr P_• •

4.0 EXPERIMENTAL PROGRAM

The experimental work consisted of non-destructivetests and unconfined compression tests on 2-in, die. cores, and

the major series of triaxial compression tests.

4.1 Non-Destructive Tests

A series of non-destructive tests were conducted on allcores. These tests provide information on the varibilitybetween specimens as well as an indication of the uniformityof individual specimens. These tests include:

* Schmidt Hammer- * Shore Sclerscopeo Dilatational Wave Velocity

In general these measurements were made on the cores at thelocations of the foil strain gages. The Schmidt hammer re-bound test was found to be strongly affected by the size of

Sthe small cores. The hammer is in contact with the rock fora relatively long time during impact, and the rebound heig.,,.is influenced by the method being used to hold the small cores.After experimenting with several clamping and bedding arrange-ments, the most consistent technique found was to measure theSchmidt hammer rebound on the ends of the specimen rather thanthe sides. The specimen being tested was placed on a largeblock of the parent rock to reduce the amplitude of the re-flected wave. This technique was used for all 2-in. dia. and4-in. dia. specimens for the Schmidt hammer tests. Largercores were tested on the cylinderical surface at gage location.

The velocity of the dilatational wave was measured atall gage locations on all cores. The system used for the

small cores is shown in Figure 6A. The input signal is a tsingle cycle square wave. This is amplified and used to drivea 1 MHz piezoelectric transmitting transducer. The signaltransmitted through the rock is teceived by a similar transducerand displayed on a dual beam oscilloscope through an internalvariable delay line. This signal is compared with the input


17

Function Sync. Dual Beam

Generator Oscilloscope

Amplifier

Input Signal

PiezoelectricS.. - Transducers

Fig. 6A ACOUSTIC VELOCITY SYSTEM FOR SMALL CORES

Sn

Function Dual Beam

Generator Oscilloscope

Amplifier

Mechanical Delay Line 72.4 jsec Preamp

Fig. 6B ACOUSTIC VELOCITY SYSTEM FOR LARGE CORES

18

4

signal, and the variable delay line adjusted to achieve coin-

cidence. The elapsed time is then read from the delay line.

The system delay is easily measured by removing the rock speci-

men and placing the transducers face to face. Figure 6B shows

the system used for the 32-in. and 36-in. dia cores. A

mechanical delay line was introduced to allow the use of rapidsweep rates, and a preamp boosted the received signal. The

operating frequency ranged from 300 kHz for the large coresto I MHz for the small specimens. A check was made to deter- 3mine if the frequency response of the rock influenced the

apparent time of arrival by making measurements on a single2-in. dia specimen at frequencies between 10 kHz and 2 MHz.

The results are plotted in Figure 7, and indicate no variation

in time of arrival across the entire frequency range. Theamplitude of the received signal is, of course, strongl.ydependent on input frequency.

4.1.1 Results of Non-destructive Tests-Charcoal Black Granite 3

Of the three rapid non-destructive tests used to

characterize the properties of the individual granite cores,the sonic velocity determinations were perhaps most useful.

The Schmidt hammer test was found to be size dependent, theresults of tests on the 2-in. dia cores being somewhat question-

able. Various limitations of the Shore scleroscope test

apparatus make this test inappropriate for determining grossproperties in granite, unless test procedures are used that

cannot fit into the category of "rapid non-destructive" test.For best results, the scleroscope should be firmly held

vertically over the horizontal specimen surface, as conditionsof plumbness and perpendicularityare critical. This test

measures hardness over a very small area on the surface of the

specimen, the rebound being affected by the type of individualgrain that is impacted, and the proximity of the grain boundary.

Recommended procedures for good results would involve cutting

and polishing a small specimen that can be tested in a

19

C/)2

rz

00 r4

(L) u00 U2)4 i

En

z

CJ 4-1

L/)- *N H

0 erF4 LjrI 0 U1) Z4

0 Lr c)-4~

o *0 -t~r ý$4 C)

144 .0%14.4 C14 (

U1) 'r-4

4Jr

cci 0v-4 O U)

Q) ý0

40 0

0 4) 4

r44

20

mounted scleroscope apparatus. The striker should be targetedon individual grains, and a number of replicate tests conducted

to determine the hardness of the individual constitutent

minerals in the rock. This procedure was not possible in thisprogram since the individual cores could not be sectioned

prior to triaxial testing. The scleroscope was hand held

t vertically by reference to an attached level bubble, and the

test cores rolled over the floor to make the desired test

point horizontal. Figure 8 is a histogram showing the scleroscopereadings for the granite. The readings taken on 12 in. dia

cores are shaded. This size was the only one having twodistinct peaks at scleroscope readings of 50 and 95. The

2 in., 4 in. and 32 in. dia. specimens displayed only the upper

peak at readings of approximately 60 to 90. Note, also that

nearly all readings above 100 occured on the 12-in. dia speci-

mens.

The Schmidt hammer tests proved more interesting,

giving the first evidence that the 12-in, dia granite hadproperties significantly different from the other sizes, It

should be remembered that the 32-in. dia cores and the 12-in,

dia cores were quarried on different occasions, although from the

same quarry. The 2-in. and 4-in. dia cores were drilled from

a piece of 32-in. dia stock during the conduct of this researchprogram, and should display properties similar to the 32-in,

cores. Figure 9 shows histograms of granite specimens. Thedistributions of results on 12-in. and 32-in. dia cores had

standard deviations near 2 1/2 Schmidt hammer units, but the2

means of the two distributions differ by approximately two

standard deviations, that is, 5.1 units. The 2-in. and 4-in.

cores gave readings that were influenced by their small size.Various clamping and bedding arrangements were tried in aneffort to eliminate rebound of the cores and to reduce the

experimental scatter within each size. The most consistentresults were obtained by testing the ends of the cores, which


21

II

80-Open - total for all rock sizesShaded - contribution of 12 in.

dia. cores

Li o

0

S40-

Z41420 I

131

2 30 4b 5 610 710 80 90o 10 1 120

Shore Scleroscope Reading

7I

kI

Fig. 8 FREQUENCY OF OCCURRENCE OF SHORE SCLEROSCOPEREADINGS FOR CHARCOAL BLACK GRANITE

22

• .• • °• ,_ ,•,. • , .• ... . • •-. ,• . • ,,._ _ ••.••...-- ,.. .•, -- •,•, _ . ... . . .-- .•-L • .•. .. ... .. .... • .•7 --4

I

12 In. p Granite10 Mean = 34.9

5 Std. Dev. =4.14

30 40 50 60 i

10 4 In. o Granite

J Mean = 47.0 !w 5o ~St!. Dev. = 3.77

I I Il I30 40 50 60

I0 i

W 0 F 12 In.c1)GraniteF Mean = 54 -6

r_ _5 Std. Dev. 2.65 I

30 4060

10i32 In. ýp GraniteMean = 49.5

Std. Dev. =2.34

S~i

30 40 1 50 6'0Schmidt Hammer Reading

Fig. 9 FREQUENCY OF OCCURRENCE OF SCHMIDT HAMMERREADINGS FOR CHARCOAL BLACK GRANITE

23

were placed on heavy cubes of granite. This increased the

path length of the stress wave in the core, and reduced the

amplitude of the reflected wave at the risk of introducing

variations due to anisotropy. The problem was not completely

solved, since both 2-in. and 4-in. dia cores displayed stan-

dard deviations of approximately 4 units, and depressed mean

values, especially in the case of the 2-in. dia specimens. 4

The sonic velocity determinations confirmed both of

the above tentative conclusions, that the 12-in. cores wereslightly more competent, and that the variations in the 2-in.,4-in. and 32-in. dia. Schmidt hammer readings were caused bythe small specimen size. The first three histograms it:.Figure 10 show the sonic velocities measured in 2-in., 4-in.

and 32-in. dia cores. The measured velocities spread rather

uniformly, across the range of 4.3 to 5.1 km/sec. The

accuracy of the measuring system on replicate individual readings

was found to be about 2%, so that the observed variation isinherent in the specimens. The last histogram in Figure 10shows all the granite sonic data, with the normal distribution

curve for the 12-in. cores shown separately. Again, the 12-in.dia cores are significantly more competent than the othergranite cores, with sonic velocities greater by about 20%.

A detailed summary of the mean sonic velocities andstandard deviations for each granite core is presented in

Table I. This shows that much of the scatter in Figure 10results from variation from core to core. The 4-in. and 32-in.

dia cores consistently have low standard deviations for individualcores, so that the normal distribution curve shown in Figure 10with standard deviation = 0.118 provides a good visualization

of the greatest possible measuring system errors.

The anisotropy of the Charcoal Black granite was

evaluated by checking the variation of sonic velocity on threedifferent diameters of the 2-in., 4-in., and 12-in. dia cores.


24

IIII£~

10[ 2 In. p Granite

L- Mean = 4.676S. Dev. = 0.235

I I I -I

4.00 4.50 5.00 5.50 6.00

10[- 4 In. T Granite

"5- Mean = 4.609

"4 Std. Dev. = 0.244

4.00 4.50 5.00 5.50 6.00~44

10- |32 In. p Granite

| Mean = 4.701ýZ 5 -Std. Dev. = 0.233

4.00 4.50 5.00 5.50 6.00

2,4 and 32 12 in. p

10 in. T Granite Granite

5I Iek

4.00 4.50 5.00 5.50 6.00

• Sonic Velocity - km/sec

Fig. 10 FREQUENCY OF OCCURRENCE OF SONIC VELOCITY

DETERMINATIONS IN CHARCOAL BLACK GRANITE

25

TABLE I

SONIC VELOCITY FOR INDIVIDUAL CHARCOAL BLACK GRANITE CORES(km/sec)

2" dia #6 #7 #8 #9 #10 all 2"

mean 4.83 4.79 4.53 4.40 4.75 4.676std. dev - - 0.235

4" dia #11 #12 #13 #18 #19 all 4"mean 4.60 4.77 4.78 4.37 4.36 A.609std. dev 0.17 0.19 0.18 0.18 0.16 0.244

12" dia #45 #46 #47 #48 #49 all 12"mean 5.55 5.39 5.48 5.27 5.35 5.409std. dev 0.064 0.055 0.095 0.064 0.041 0.118

"32" dia #2 #3 #4 all 32"

mean 4.69 4.97 4,45 4.701std. dev 0.094 0.085 0.096 0.233 t

UT RESEARCH INSTITUTE

26

In all cases, the variation between mean velocities on differentdiameters was less than the standard deviation of the individualmeans. This indicates that any consistent anisotropy present

S~was smaller than the core-to-core variations. The 12-in. diacores were also checked to determine if a consistent variation

existed along the length of the cores. In this case also, the

core-to-core variation was large in comparison to any consistent

anistropy. The 32-in. dia cores were not checked because the

appropriate directions were not marked on the individual cores

at the quarry. The data is summarized in Table 2. a

i 4,1'.2 Results of Nondestructive Tests-Indiana LimestoneThe comments regarding the Shore scleroscope and

Schmidt hammer tests on granite apply to a lessor degree on the

limestone. The limestone does not have the large grain size

of the granite, so the scleroscope should show a single peakvalue, and the reduced sonic velocity should result in more

consistent rebound hammer tests on the small cores.

The results of all Shore scleroscope readings on

limestone are plotted in Figure 11. There was little variation

related to core size in these tests. The tests scattered

about a mean value of 13.95 with a standard deviation of 4.56

scleroscope units.

The Schmidt hammer results are plotted for the individual

specimen sizes in Figure 12. The 2-in and 4-in. dia cores

were tested parallel to the axis in the same manner as the

small granite cores, and display little shift in the mean values.

The sonic velocity determinations are plotted for each

size and for all sizes together in Figure 13. Again, there is

no significant variation between the different sized specimens.

The anisotropy of the Indiana limestone was investi-gated by evaluating the three 36-in. dia cores. The 45

individual acoustic velocity tests on these cores had a mean


27

TABLE 2

ANISOTROPY OF CHARCOAL BLACK GRANITE(km/sec)

3Radial Variation Axial Variation

col. 1 col. 2 col. 3 row 1 row 3 row 5

2" diamean 4.70 4.69 4.64std. dev 0.327 0.203 0.194

4" diamean 4.61 4.59 4.63std.dev 0.379 0.139 0.155

12" diamean 5.42 5.42 5.40 5.39 5.44 5.42std. dev 0.135 0.113 0.110 0.110 0.145 0.086

':• iIT RESEARCH INSTITUTE

ii• 28

;A

200

mean = 13.95

5 std.dev. = 4.56S 150-

4

50

00 20 30 40

Shore Scleroscope Reading

S~Fig. II SHORE SCLEROSCOPE READINGS FOR INDIANA LIMESTONE

t Jt

29

30- -)

_ Al)32 in. dia. Limestone 4 in. dia. Limestone

Smean = 24.2 mean = 28.05.4Z 20 std.dev. = 2.48 w 20 std. dev. = 3.55u $4

• 0 0

20 30 40 20 30 40

Schmidt Hammer Reading • Schmidt Hammer Reading

60 60.-12 in. dia. Limestone

0 mean =28.2S50 50 -

I std.dev. = 3.93Uf

a) 40I0- 36 in. dia. Limestone !p 4 40

o mean = 29.8Uo 0 std.dev. = 4.10

o 30 30

S54

W o)

Q)20 -20SO 0

r

10 10 u

20 30 40 20 30 40 1

Schmidt Hammer Reading Schmidt Hammer Reading

Fig. 12 FREQUENClY OF OCCURRENCE OF SCHIIIDT HAMMER READINGSFOR INDIANA LIMESTONE

30

-T- --- --. ,-.-- -.

2 in. dia.limestone5

10-.Cl)

5 4 in. dia. limestone

SI tI III

0 36 in. dia. limes tone

S 15-Sall limestone

10

5-

Sonic Velocity - (km/sec)1

Fig.13 SONIC VELOCITIES FOR INDIANA LIMESTONE

31

MIT

value of 4.021 km/sec, with 95% confidence levels at +0.017

km/sec and a standard deviation of 0.058 km/sec. 'ihis value of

4.021 +0.017 km/sec is taken as the acoustic velocity in

Indiana limestone. These 45 readings may be separated into 3 3

groups of 15 readings, each group spaced at 60-deg intervals

around the axis of the core, or as 5 groups of 9 readings,

each group spaced at 10 in. intervals down the length of the

core. In this fashion the variation in acoustic velocity can

be considered as a function of direction in the horizontal

plane, or as a function of the vertical depth in the bedding

plane. This nonemclature, and the variation of ac-ustic

velocities as a function of direction and depth are indicated

in Figure 14. Note that in all cases, the 95% confidence

levels for the individual subgroups include the mean value

of 4.021 km/sec computed for the entire 45 reading group.

Looking at the variation of acoustic velocity, it

appears that the maximum velocity would be measured along a

diameter between 2-5 and 3-6, along a quarry bearing of N-80*-W,

and the lowest velocity would lie along a bearing of N-10°-E.

The maximum variation is on the order of 1-1/2 percent of the

mean value. Considering the variation of acoustic velocity

with depth, there may exist a trend of increasing velocity

with depth. The slight reversal of this trend at row no. 5may be a real effect caused by the presence of the bedding plane

in the formation. In any case, the variation is on the order

of 1% or less. The acoustic velocities for all subgroups lienear or within the 95% confidence levels for the entire group

of measurements.

The results of all nondestructive tests on both rocktypes are summarized in Table 3.

4.2 Classification Tests

In addition to the nondestructive tests to evaluatethe specimen-to-specimen variability, petrographic analysis,


32

1 6

_ I

Specimen Orientation

and Array of MeasurementLocations

3

0

5 -.05I .- '- •

o

0>. 1-4 2-5 3-6

Direction of Measurement (column to column)

S4.05

A. 4.021 + 0.017r4.oo

.. i.oý 35.95

1 2 3 4 5Depth of Measurement (row number)

Fig. 14 ACOUSTIC VELOCITIES IN 36-IN. DIA INDIANALIMESTONE

33

33.

S RTABLE 3

SUMMARY OF NONDESTRUCTIVE TEST RESULTSSHOWqING MEANS AND STANDARD DEVIATIONS

Shore Schmidt Sonicsclerescope Hammer Velocity (ks/sec)

mean a mean a mean G

2 in. dia 63.8 - 34.9 4.14 4.68 0.24granite

4 in. dia 83.1 - 47.0 3.77 4.61 0.24granite

12 in. dia 75.4 - 54.6 2.65 5.41 0.11granite

32 in. dia 74.7 - 49.5 2.34 4.70 0.23 J

granite _ _ _ __________1

2 in..dia 12.7 3.48 24.2 2.48 3.99 0.10limestone

4 in. dia 14.6 4.53 28.0 3.55 4.00 0.06limestcne I V

12 in. dia 14.4 4.40 28.2 3.93 4.00 0.06limestone ___

36 in. dia 14.4 4.99 29.8 4.10 4.02 0.06limestoneI

liT RESEARCH INSTITUTE

34

tensile and unconfined compression tests were conducted to

define the rock types.

4.2.1 Petrographic Analyses

Petrographic analyses for the two rock types were con-

ducted under the direction of Dr. A. Howland of Northwestern

University in Evanston, Illinois. These reports are reproducedbelow.

Charcoal Black Granite

Macroscopic

Gray medium-grained, equigranular; grain size generally

1-3 mm; light gray feldspar, black hornblende, and dark

brown to black biotite recognizable.

Microscopic

Testure: porphyritic: rectangular crystals ofplagioclase 2.5-3.5 mm. long in asubhedral to anhedral interlockingaggregate of plagioclase, alkalifeldspar, biotite, hornblende, andquartz with grain size largely inthe 0.3 - 1.0 mm. range.

Mineralogy

Plagioclase ca. 50%

Euhedral to subhedral crystals in two distinctsize ranges, giving a porphyritic texture. Finepolsynthetic twinning and strong oscillatoryzoning ranging from about An25 in the cores toAn1 4 in the outer zones; thin untwinned albiticrims. Many inclusions of biotite, hornblende,and magnetite. Slightly clouded with finesericitic alteration,

Alkali feldspar ca. 15%

Subhedral to anhedral grains, fine perthiticintergrowths, grid microcline twinning visiblein some grains.

Biotite ca. 15%

Euhedral to subhpAval plates, brown, many darkhaloes around zircon inclusion.

Quartz ca. 10%

Strong tendency to interstitial occurrence,lIT RESEARCH INSTITUTE

35

filling in between rectangular feldspar grains.

Hornblende ca. 7%

Pale green to yellow green pleochroism, cor-plexly twinned and intergrown with magnetite,some contain cores of relict pyroxene (includedin hornblende percentage).

i Magnetite ca. 3%

s Separate grains and intergrowths in hornblende.

Apatite

Many needles and stout prisms included inother minerals.

: ~Zircon

: ~Inclusions in biotite.

I ~This rock would classify as a granodiorite,

Indiana Limestone

Macroscopic

Fine-grained porous light gray limestone; abundant oolites

and fossil fragments visible with the hand lens.

Microscopic

Clastic limestone composed of oolites 0.1 - 0.15 mm. indiameter, fossil fragments ranging up to 1.5 mm. in length

and about 0.1 mm. thick generally alined parallel to the

bedding, angular to subangular polycrystalline calcite

fragments, and calcite cement.

Oolites show both concentric and radiating structures,

the former much more abundant. Some have overgrowths of

calcite. They form about 35% of the rock.

Fossil fragments with either fibrous or granular internal

structure are about 30% by volume. Grains of polycrystal-

line calcite and clear single plates of calcity cement

are about 20% by volume.

Pore space is estimated as about 15% of the thin section,

but some interstitial material may have been lost in the

grinding of the section.


-I36

4.2.2 Tensile and Compressive Tests

Tensile and unconfined compression tests were conducted

on 2-in. dia specimens in accordance with ASTM specifications.

The unconfined compression test specimens were instrumentated,

and the data included in the body of the analysis as triaxial

tests having a3 =0.

rock type tensile compressive

strength strength(psi) (psi)

granite 1340 18700limestone 290 3970

4.3 Description of Constitutive Equations

In order to conveniently describe and analyze therather large amount of data generated during this program, thedata for the individual tests were fit to descriptive models.

The variation in the model parameters is then analyzed for therange of specimen sizes and confining pressures employed in

the study. The models that were chosen are described in greaterdetail in a previous report, as well as the underlyi-g theory.

Briefly, the constitutive equations for an isotropic Anear

elastic material are written in terms of the volumetric responseand the distortional response through the use of the deviatoric

stress and strain tensors, defined by:

F•. 6k deviator strain (1)iLj ij 3 K Sk ij

oiji -. 1 k deviator stress (2)ij 13 - kk 6ij

in which

a.. = stress tensor-ij

i = strain tensor

6.. = Kronecker delta

1 liT RESEARCH INSTITUTE

37

The linear elastic isotropic constitutive equation given by:

ij= ij 'kk + 2 pij (3)

in which -A and p. are Lame's constants, are thereby transformed

into

SoM. = . ek + 2G(E' (4)

in which G and k are respectively the shear and bulk moduli,

given by:

k= 0kk (5) H38_kkj

G ~(6) 9

ii

Nonlinear behavior is incorporated by using incremental

constitutive equations and defining G and k as functions ofone or more of the stress invariants. Inelastic behavior

may be incorporated by defining separate load and unloading 2

moduli.

A nonlinear hysteretic model used by Seaman andWhitman32 o study the behavior of sand appears to be suitablefor representing the hydrostatic behavior of rock. Thestress-strain curve for this model is shown in Figure 15.For virgin loading

n

am A v (7)

and for unloading and reloading

am A2 (6v n (8)

in which

i is mean stressi mlIT RESEARCH INSTITUTE

38

n~1JC/2 m=AA -v, n

mm A 2( v vi

FI

Volumetric Strain, F

FIG. 15 IDEALIZED HYDROSTATIC BEHfNIOR OF GRANULAR MATERIALS

(after Seaman and Whitman

39

e is volumetric strain

Ev is residual volumetric strainA,, A2 and n are material properties

This model represents both the nonlinear and the inelastic

behavior of the rock observed under hydrostatic loading.

Application of the mathematical model in a computer code would

perhaps be most convenient in an incremental form using tangentvalues of bulk modulus, kto This modulus is a function of

mean stress and for a given stress represents the slope of the

curve at that stress level.

For loading

dm do n Al/n Sn-lYn (9)kt m = A m(9

1 4m

and, for unloading and re.loading

do (10)nl~kt= m n A/n 2 nl (0)d-2-mv

4Note that in this model the modulus for unload-reload is

larger than the modulus for virgin loading at any given

stress level. Only loading models were generated in this work.

4.4 Triaxial Compression

A triaxial compression test normally consists of two

phases. First, a hydrostatic confining stress, Oc, is applied

to the specimen so that the principal stresses are all equal

to ac. Then, two principal stresses, a2 and a3, are kept

constant at ac while the third principal stress, 01, iscA

increased. For these conditions one of the components of thedeviator stress is

all a -1/3 (aI + 2o3) =2/3 (oi-o (11)


40

If the material is isotropic two principal strains, 62and e3' are also equal, meaning that the corresponding compo-

nent of deviator strain is

Ei =61i " 1/3 (el + 2 3) 2/3 (1 -63) (12)

Thus, for triaxial conditions the shear modulus may be

* •determined from

all a - U3 (13)

S11 2(sl - P3)

* o

At low stress levels shear modulus increases relatively

rapidly with mean stress; however, at higher stresses it

remains relatively constant. Torsional wave velocities were

measured in cylinders of different granites exposed to hydro-2 2static pressure between 1 kg/cm and 4,000 kg/cm by Birchand Bancroft• 3 From the wave velocities the modulus of rigidity,

which is identical to the shear modulus, was determined and is

shown as a function of mean stress in Figure 16. Also shown

in Figure 16 are values of shear moduli for specimens of Char-

coal Black granite determined from triaxial tests performedduring this study and a previous study. The moduli determined

by Birch and Bancroft and those for the Charcoal Black graniteare in good agreement and increase approximately with the 1/10power of the mean stress.

Values of shear modulus shown in Figure16 are tangent

values, Gt, which refer to the :lope of the deviator stress

value. For nonlinear behavior the tangent shear modulus is

taken as a function of mean stress and may be calculated from

Gt = 1/2 d(°l - 03) (14)d(eI -'I3T

The functional relationship with mean stress may be written

as

Gt = c 0.1 (15)


41

t-4H 0

11 CU 0)

4JW)4-4 t-4

CC/3

4-4 -r-4

t4 -r4 u/

r-A ý40)

0) Q) Or 4

4.)4

cu 040,

o'-410 -

41 C

in which c is a constant.

4.5 Poisson's Ratio and Young's Modulus

Two constants are necessary to describe the stress-

strain behavior of an isotropic elastic material. One set of

constants, k and G, which separate volumetric and deviatoric

behavior have been described previously. Another pair of

constants which are commonly used are Poisson's ratio, v, and

Young's modulus, E. The constants E and v are derived directlyfrom a triaxial test in which the lateral stresses, a2 and 03,are equal to zero. For this special test condition.

a 1, and (16)

v=- s3 (17)

For more general test conditionsE 9kG (18

3k + G ,and (18)

3k- 2G0 (19)Sv = 2(3k+G)

SThe use of tangent values of bulk and shear moduli, k andt

Gt, in equations (18) and (19) will result in tangent values

for Poisson's ratio, vt, and Young's modulus, Et.

For hydrostatic loading the deviatoric stresses and 3

strains are zero and G is undetermined. Therefore, E and v

are also indeterminate from a hydrostat. For a triaxial test

with oa equal to zero, E and v may be computed from (16) and

(17). For a triaxial test with 03 unequal to zero but

constant, tangent values of E and V may be computed from

Et d and (32)

d• 3 (33)


43

!P

4.6 Charcoal Black Granite

4.6.1 Hydrostatic Behavior

The hydrostatic stress-strain data for the CharcoalBlack granite was fit to the nonlinear hysteretic model described

by Seaman and Whitman. The model parameters were deter-

mined by plotting log mean stress as a function rf log volu-metric strain. Recall that this model is given by

S=A~n

The parameters A and n were computed for each size specimen,grouping the five tests at each size together. The values forA and n were used to compute tha bulk modulus by using

the transform 2

) kt =_._m =n (n-1) /n= -. nAý mf~)f

v

where kt is the tangent bulk modulus and the other quantities are

as given above. The hydrostatic stres'9-strain behavior for the

four specimen sizes is shown in Figs.17 to 20, together withthe models fit to each specimen size. The data from the 2-ino

dia specimens fit the model closely at stress above about 2 ksi.

This deviation is expected because the model predicts zerostiffness at zero pressure. The data from the 4-in. dia

specimens appeared to be slightly stiffer than the 2-in.specimens, but the same model parameters provided a reasonablefit to the data. The 12-in. dia specimens were considerablystiffer than the smaller cores, and displayed less strain-

hardening. The 31-in. dia specimens were similar in responseto the 2-and 4-in, dia cores. The model for the two smaller

sizes is also indicated with the mcdel for the 32-in. dincores in Figure20 for comparison. The difference in the twomodels is not large, and reference to Figurelg will show thatthe 4-in. dia data would fit either model equally well. The

HT RESEARCH INSTITUTE

44

kI

Confining Pressbre10 .n

o 2 ksi m = Apv9 c , 4 ksi A = 2.77 x 10

8 A 6 ksi,,i / 0 8 ksi /

7 o '- " 71 s

2 4 6 8 10 12 14 16 1$ 20 22 24 26 28VOtdumetric Strain (in/in x 10-)

v•,> 17 MEAN STRESS VOLUMETRIC STRAIN (2" GRANITE)

10[ Confining Pressureo 2 ksi

8A 6 ksi V0< 8 ksi

0 7"-ý4CL)

5V

3n 8VO0& A =2.77 x108

U) 13n = 1.73r- -2

I 0 L0

2 4 6 8 10 12 14 16 18 20 22 24 26 28

Volumetric Strain ev (in/in x 10-4)

Fig. 18 MEAN STRESS vs. VOLUMETRIC STRAIN (4" GRANITE)

45

iA

LI) = As

A = 4.66 x 108 n = 1.30 K>

6---

x• 5 V5 Confining Pressure

0 2 ksi4 0 4ksi

0A 6 ksi

3to] 8 ksi(2 12" G T0 ksEi

2 4 6 8 10 12 14 16 18 20 22

U-4

VoIlumetric Strain c (i-n/in x 10-4v

Fig. 19 MEAN STRESS vs. VOLUMETRIC STRAIN(12"1 GRANITE)

n Confining Pressure6- o 2 ksi

= A7xn× [ 4 ks i Cm AF 8.U - A 6 ks i A =4.00x10

"4n = 1.73

3_

4J 2 _ A 2.77xl0[

1- 2 in. model

2 4 6 8 10 12 14 16 1 20Volumetric Strain Fv (in/in x 10

Fig. 20 MEAN STRESS vs. VOLUMETRIC STRAIN(32" GRANITE)

46

deviation of the 12-in. cores from the other three sizes may

be explained by the fact that the 2- and 4- in. dia specimensS~were cored from a waste piece of 32-in. dia core. The 12-in.

dia specimens were probably obtained from a different part

of the quarry, and were found to display slightly different

properties in the non-destructive tests. The fact of greater

stiffness and less strain-hardening may indicate a relative

lack of.macroflaws in these specimens.

The tangent bulk modulus as predicted by the mo6els

and the observed values are shown in Figures 21 to 24. The

model predictions fit the data well except at low stress

levels. There is somewhat more scatter in this form of data A

presentation, as may be expected from the fact that the values rep-

resent differentiation of experimental data over short inter-

vals. Finally, the derived models are compared in Figures 25

and 26. It appears that there is little effect of specimen

size on the hydrostatic behavior of Charcoal Black granite.

The greater stiffness of the 12-in, dia cores is probably

caused by slightly different properties in the rock. This is

confirmed by the non-destructive tests, which would not be ex-

pected to be influenced by specimen size.

4.6,2 Triaxial Behavior

The triaxial behavior of the charcoal black granite

is represented by a model used by Birch and Bancroft33 S

Gt =ca

in which

Gt = tangent shear modulus

= mean stressm

c = model parameter

The triaxial behavior for all sizes of the granite

was represented by a single model, the data being shown in

Figures 27 to 30. By following the data for an individual


47

121Confining Pressure

-- 10 o 2 ksix × 4 ksi

8 A 6 ksi0 8 ksi 03"• ~ ~6 -- V 10 ks i -

9~ 10

C Mean Stress m(pi x 10

Fig,. 21 BULK MODUAISt FOR 2" GRANITE

2 Confining Pressureo 2 ksi

10 -_ 4 ksi"A 6 ksi

3" ( -- ¢ 8ksi

. V I0 ksi

6 a00

04

2

1 2 3 4 5 6 73 8 9 10Mean Stress o (psi x 10)

Fig.22 BULK MODULUS FOR 4" GRANITE

48

S-" '- - - - -' " , i • -" .

I

Confining Pressure14 o 2 ksi

" 12 -- 0 4 ksi

o A 6 ksi10 70 8ksi

S66 4

r- 4

0 2

1• 2 3 4 5 6 7 8 9 103

Mean Stress • (psi x 103)m

Fig. 23 BULK MODULUS FOR 12'" GRANITE

'KConfining PressureS140 2 ksi

--4 12 [] 4 ksi

x A 6 ksi A

• 10 0 8 ksi 0

,- V iOksi

00

8

S 6

" 40

-td 2I-I

1 2 3 4 5 6 7 8 9 103

Mean Stress om (psi x 10 )

Fig. 24 BULK MODULUS FOR 32" GRANITE

49 :1

MIT

10i

12" Dia.

S~9-

32" Dia.

Cc) m n0 In .I 0

X, and 4" Dia.

F 4-

3ore Dia.. n A2712 in. 1.73 2.77x10 a4 in. 1.73 2.77x0

12 in. 1.30 4:66xi08S i32 in. 1.73 4.006xI0

"2 4 6 8 10 12 14 16 18 20 22 24-4Volumetric Strain E (in/in x 10-)v

Fig. 25 MEAN STRESS vs. VOLUMETRIC STRAIN -

GRANITE MODELS

0 10

12 in.Dlij

r4 in and 4 in Dia

0 2 in. Dia.

1 2 3 4 5 6 7 8 9 1013Mean Stress o (psi x 10)m

Fig. 26 BULK MODULUS vs. MEAN STRESS - GRANITEMODELS

50

410

1 0 Confining PressureI-- 0 ksi

ME-in S4 ksi 1.895 x 10 )

100

. . -. O 2 s

z ~

61* 0.

Sc=1.645 x 106 G c j0.

3 1

Mean Stress m (psi x 103)

Fig. 27 SHEAR MODULUS vs. MEAN STRESS -2 in. GRANITE

"t 5 0.1

X•x--•'895 x 106 0 C•

c "q 3 0 0) cr 45xlO0

:-• - (psi 3 103)iSMean Stress m(pix1

- Fig. 28 SHEAR MODULUS vs. MEAN STRESS -4 in. GRANITE

51

S10 Confining Pressure0 2?ksi 8 8ksi [Ii4 ks i 7 10 ksi 66

00

100

3- 1.845 m

~ 31.653 10

illii 21111 It ii

v4r41.895 10

35552

test, it can be seen that the shear modulus does increase with

mean stress at intermediate stress levels. As yielding begins,

the shear modulus decreases rapidly, deviating from the model.Inspection of Figure 27, in which the effect shows most clearly,

reveals that the point at which yielding occurs is higher for

the tests at higher confining stress, exactly as would be

expected.

4.6.3 Young's Modulus and Poisson's Ratio

The values of shear and hulk moduli derived from their

respective models may be employed to predict values for otherelastic moduli. This was done for Poisson's ratio and the elastijc

modulus, using the relationships given in Eq. (18) and (19) on

page 43.kG 3k-2G

3k+G and V2(3k+G)

in which E = Young's modulus

v = Poisson's ratio

k = 1Lilk modulus

G = shear modulus

The predicted values for the elastic modulus are shown in

Figures3l to 34, together with the observed values. The

agreement is good at low and intermediate stress levels. Athigher stresses, yielding begins, reducing the observed

stiffness.

Predicted and observed values for Poisson's ratio are

displayed in FigureL 35 to 38, Inspection reveals that the

trends predicted by the model are present, but the valuespredicted are lower in some cases than the observed values.

The model is fallacious at low stress levels, where the low

predicted bulk moduli act significantly. As yielding cccurs,

Poisson's ratio increases due to dilational effects, resultingin values greater than 0.5. Yielding occurs at higher stress

levels on the tests at higher confining pressures. The


53

16

~14

11:10 0O 4cn 0 0

r v Confining Pressure A v6 0 0 zero

020s

0004 ksiA 6 ksi

.rj 8 ks i

4' i10 12 1416 132022 242628 30

31Fig. 31 ELASTIC MODULUS FOR 2" GRANITE

-40

0A12~~0 00 Q 1 a ý

A)A

0 2ý !zs0 4 ksi.`

85 80 ksli in" l4css(p0 s

0202 14 16 18202426283

Mea Stes (pi s1

154

f 14 - l

12-- 0 0 oA12 O

0•0 Confining Pres~c~re

08 0 2~ ksio 7 ksi

A 6 ksi

6 8 iýsi

4 -- 7 10 ksi

(2-cu

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Mean Stress m (psixl03)

Fig. 33 ELASTIC MODULUS vs. MEAN STRESS - 12" GRANITE

16-

S14-

r-4

x 12 --

S10 Confining Pressure

o[ 4 ksiS6

" A 6 ksi

o 4 - 8 ksi

rV 10 ksi.( 2cu

2 4 6 8 10 12 14 16 18 20

Mean Stress om (psixl03)

Fig. 3" ELASTIC MODULUS vs. MEAN STRESS - 32" GRANITE

55

kv

•OA V

S. 0.5 0 0

0 0 0"'j0.4 0o 0

S 10 0 R0 0.3- _ 0 A

Continuing Pressureo Zero 4 ksi ,, 8 psi

S0 2 ksi 11 6 ksi '1.0 ksi

2 4 6 1 10 12 14 lb 18 20 22 24 26 28 303

Mean Stress (psi x 103)• m[ F, 35 POISSON'S RATIO FOR 2 in. GRANITE

Confining Pressure o

O 2 ksiO 4 ksi 00

0.5 A 6 ks O 08 "' 0 •

c 0.4 & Bksi'• v7 10 ks i 0 o 0a q vS0.3 A v-

019 oAO© F v v V ,, ,

° 0.2 0c0•0 v

, 0 . i0

2 4 6 8 10 12 1" 136 18 20 22 24 26 28 30

Mean Stress u (psi x 10 3

Fig. 36 POISSON's RATIO FOR 4 in. GRANITE

56

i0.5 0 Confining Presqure0.5- o 2 ksi04o 4 ksi

A 6ksiS0

.3 - 0 0-- O , 0 8 ksiV 10 ksi

w :•0.2 0 13 A•- .A&

0 00

2 4 6 8 10 12 14 16 18 20 22 24 26 28

Mean Stress u (psi x 103m>

Fig. 37 POISSONS RATIO FOR 12" 0 GRANITE

Confining Pressure

0

I St2 ks(i

•, 0.• - 6 ksi •'

2 4 6 810 16 8 ksi

- 440' -03 - oO E3O0oD a (3O 0 CO 0• A

0"2

[2 4 6 8 10 12 14 16 18 20

Mean Stress o -(psi x10

i•Fig. 38. POISSOdS RATIO FOR 32" 1 GRANITE

.m57

unconfined tests shown in Fig. 35 begin yielding at a meanstress of 2 or 3 ksi (axial stress of 6 or 9 ksi), while thetest at a3 = 10 ksi did not yield until it reached a mean

stress of 23 ksi (, = 49 ksi). The larger specimens behavedin a more or less similar manner, with little influence of

specimen size. A

4.7 Indiana Limestone

4.7.1 Hydrostatic Behavior

The limestone typically displayed a strain-softeningbehavior under hydrostatic loading. The mean stress vs. Jvolumetric strain data were fit to the nonlinear hysteretic modelwith no significant variations attributable to specimen size.The data from most tests scatter about the curve given by theparameters below.

m v

A = 0.490xi0 6

n = 0.75

Note that for n less than one, the model is concave toward thestrain axis. The data for all tests are shown in Figure 39.The model for bulk modulus derived for these parameters is I

shown in Figure 40 together with the observed data. Thebulk modulus is given by

kt = n An-I . (n-l)/nm

in which A and n have the values given above. This model pre-dicts a very high modulus at low stress levels, which is not

the case in the real material. The actual data appear to haveslight maxima in the area of 2 ksi, and then decay along thecurve given by the model,

4.7.2 Triaxial Behavior

The behavior of the Indiana limestone did not correspondto the one-tenth mean stress model. The triaxial data for all


58

k _2

Iq

10 ~i9 A 0

8|

nn

A~A.0 0

A 0 0.9 0 10 S=e0i7en Dia

60 o 0 00

5 , 0

2- 0 00"1A 12 in.

00 0 0,3i~

0 00

1 2 3 4 5 6 7

"lumetric Strai ' (10 in./in.)

Fig. 39 MEAN STRESS vs. VOLJUME'TRIC STRAIN FORALL LIMESTONE

5n -1k = 0:An n Specimen Dia.e. 4 tm 0 _9 in.

0 4 A 6• z• 2 in.A 12 Ain.

36 in.S2 - 0O O oCl 0"

2 i0 0 0,

1 2 3 4 5 6 7 8 9 10 11 12 133

Mean Stress ) (psi x 10

Fig. 40 BULK MODULUS FOR ALL LIMESTONE

59

Indiana limestone tests were plotted in the form:

S= f

A =deviator stress G1 - 03

A• = shear strain 6, - s3

This is shown in Figure 41, and it can be seen that the dataare bounded by the line AG = 3.55 x I0 6 () 0' 3 At very low

and very high stress levels the measured strains are larger

than would be predicted from the model. The shear modulus,

Gt, for all tests is shown in Figure 42 as a function of shearstrain. The actual moduli are nearly constant at shear strains

less than 500 microstrains. As the shear strain increases, the

shear modulus drop uniformly until yielding occurs. Above

the yeild point the modulus drops rapidly. This behavior is

indicated by the solid symbols on Figure 42, which are the

trajectory followed by test 38, a 12-in. dia specimen at 6000

psi confining stress. The model generated by Figure 41 can be

transformed to yield the tangent shear modulus. This is indi-

cated on Figure 42 ° It can be seen that the model represents

the shear modulus behavior of intermediate strain levels.

The model AG = B (A 5)r can also be transformed to

express shear modulus as a function of deviator stress. Thistransform, togetber with the measured data are shown inFigure 43 The data follow the model at intermediate stresses,but roll off at low stress levels and above yield.

4.7.3 Young's Modulus and Poisson's Ratio

In order to consider the elastL: modulus and Poissonrsratio, the models for shear modulus and bulk modulus are com-

bined to produce E and v values for comparison with the ob-

served quantities. In the case of the shear modulus model,

the deviation from the model is extreme at low deviator stresses.

The shear moduli represented by the curved line in Figure 43would better account for this roll-off. The models produced


60

}6

B 3.55 x 106

r.0 rO. 30

10

cn

;>

*161

_ :=B2))

o -/ -

I I

i oi->n G rB( )r-1

0 r= 0.30 ,SA0S B = 3.55x40

0 0

0o

"0 0 Ao0

0 A'pecimen Dia.O

0 2 in. 0 0 o0 0 4 in. 'Ia 12 in. A

0 36 in. 0 oA indicates typical trajectory as

^ displayed by test 38 0IZr (refer to section 14.7.2 in text) 0

0

OCI r-,4-4

Shear Strain .,- (in./in.)

Fig. 42 SHEAR MO7'JLUS vs. SHEAR STRAIN FOR ALL LIMESTONE

62

Specimen Dia._ o 2 in .

So4in.

A 12 in.

0 36 in.

A00

0 9000 G

1.0

I r-lA&

Gj rB (AUr 0.30 0

x 0.1 0 0A A

A

A 0

63

Shear Stres (psi xf 10

Fig 43 SHEAR MDLSv.SERTESFOR ALLLMETN

•" B • ,' .. 63

for elastic moduli are shown in Figure 44 as a function of

mean stress. The dashed lines indicate the change in the models

that result from the use of the curved line in Figure 43

The data for each level of confining stress are compared with

these mclels in Figures 44 through 49. As in the case of the

shear mocel, which established an upper limit for the shear

moduli, the observed elastic moduli are also bounded by themodel. The material is reasonably well represented at inter-mediate stress levels, but is softer than indicated by the

model at both low and high stresses.

In like fashion, Poisson's ratio models are shown in

Figure r50 and the data at each confining stress in Figures 51

through 55 These data are less consistent, and about all

that can be said is that both the models and the data increase

with mean stress.

In discussing the elastic modulus and Poisson's ratio,it should be remembered that the models for these properties

were derived using the bulk modulus model, which was fit to

the hydrostatic data only. The models for G, E, and v can be

applied uniquely only to the triaxial data.

4.8 Strength Properties

The strength of a rock is typically displayed by

plotting the locus of failure points as a function of the

loading conditions. The failure data for Charcoal Black granite

and Indiana limestone are shown on Figures 56 A and 56 B as

functions of mean stress and J2"

4.8.1 Charcoal Black Granite

Sufficient data now exists to produce a failure curve

for the small granite specimens. The 2-in. and 4-in. dia cores

scatter about the curve indicated in Figure 56A , with no indi-

cation of any influence due to size in this range. These cores

all failed catastrophically, with no indication of slippage

along joints or weakness planes prior to failure.lIT RESEARCH INSTITUTE

64

_________

Confining Stresses (3) as noted

7 2 ksi 4 ksi 6 ksi 8 ksi 10 ksi

'' 6

A 5 5

4

2

1 2 3 4 5 6 738 9 10 11 1 31

S• 3 -

!Mean Stre-ss (psi x 103)•'

SFi,".44 ELASTIC MODULUS MODELS FOR INDIANA LIMESTONE

C oSpecimen Dia.

6 - 0 2 in.0 4 in.

U,9S5 -€12 in.i ~~ ~ -"<36 in.

23 ksi3~i~

'-4

•0

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mean Stress %; (psi x 10 )

Fig. 45 ELASTIC MODULUS FOR INDIANA LIMESTONE (c 3-2 ksi)

65

7

0

- 5 Specimen Dia.

S4o 0 2 in.S"1 0 4 in.

LL3- A 12 in.

0 < 36 in.,-4 2

S 2 0

-z'o o°-A

2 4 5 6 7 8 9 10 I[ 12 13 14CMI S: ' oss (psi x 10

F Vg 46 HI,ASTIC MOIU)[1,11S FOR INDIANA UIME'STONE (' 4 ksi)

-4[3

Specimen Dia.

S5 0 2 in.'40 4 in.

CL 4 A l 2 in.

-4 3

0o• 2 A

. 0

W AA

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mean Stress om (psi x 103 )

Fig. 47 ELASTIC MODULUS FOR INDIANA LIME-STONE (03 = 6 ksi)

66

7 Specimen Dia.

-0 02 in.X 6 O4in.

A 12 in.• 5

4

:3-5 3 -

o 2-- 0zoa

0

2 3 4 5 0 7 8 9 10 11 12 13 143

Mctil Stress (psi x 10 )

II !'ig. 48 K.AS°IC MOD _S FOR IME.STONE ('°3 = 8 ksi)

,: t) •

5 - Specimen Ilia. 00 2 iln. 0

S-- in. S

o 0

2 0000 0

r-4

Po

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mean Stress o (psi x 103)

Fig. 49 ELASTIC MODULUS FOR LIMESTONE (o3=10 ksi)

67

%4

0.5Confining stresse

0.4 s noted (ksi)0 o"-,4• 03--

.Y 0.2 /0 S0.1I- ksi 4 ksi 6 ksi 8 k i 10 ýsi

1 2 3 4 5 6 7 8 9 10 11 12 13 14Mean Stress , m (psi x 10 )

Fig. 50 POISSON'S RATIO MODELS FOR INDIANA LIMESTONE

0.5 2 -2ksiSpecimen Lia.

0 0.34 00 0 2 in.

o 0.3 0 2 4 in.

Q _ 12in.

w <> 36iin; 0.2 -

0

.C-4

1 2 3 4 5 6 73 8 9 10 11 12 13 14Mean Stress •m (psi x 10 )

Fig. 51 POISSON'S RATIO FOR INDIANA LIMESTONE ( 23 " 4 ksi)

0.56 .. • •Specimen Dia.0 0.4

'• O 2 in.ca 3 CP 4 in.

r4 0. C 12in.

"O 9A

0 36 in.

1 2 3 4 5 6 7 8 9 10 11 12 13 14Mean Stress -) (psi x 10 3

Fig. 52 POISSON'S RATIO FOR INDIANA LIMESTONE (c,3 =4 ksi)

!6

0.5

S0.4- Specimen Die.

o 0 2 in.u 0.3-- o 4 in. 0

A 12 in.S-= ~0.2 --

w2 0.1-

0 -A I I --j1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mean Stress o (psi x 103) S~m

Fig. 53 POISSON'S RATIO FOR INDIANA LIMESTONE (o3 = 6 ksi)

0.4 Specimen Dia. 0 0 0 000.4'A

o 0 2 in. 00 S•0.3k; 3 4 in. 0A • O

2 70012 in0.2 L-

o~~ 0 2 in

00 AA Q

.t122 n

AA01 0

1 2 3 4 5 6 7 8 910 1112 13 143

Mean Stress ur (psi x 10)

Fig.54 POISSON'S RATIO FOR INDIANA LIMESTONE ( 3 8 ksi)

0.5Specimen Dia.

04 0 2 in.~ .3r4 n A

-W 4in. /*

S0.2/

.~0.1 00

1 2 345 6 7 8 91011 12 1314

Mean Stress 0 m (psi x 10)

Fig.55 POISSON'S RATIO FOR INDIANA LIMESTONE (03 =10 ksi)

69

Sii

40-n4 failure for2 in. and

- 4 in. dia.cores

30-

Specimen Dia.

2 2in.

20 4 4in.

V 32 in.

~10 V 32 in.(Huck & Singh, 1971)

typical stress trajectory V

10 20 30 40

Mean Stress m (ksi)

Fig. 56A FAILURE STRESSES FOR CHARCOAL BLACKGRANITE

12 in. dia.

10 Specimen Dia.0 2 in.8 - 4 in. /i

•~1 46in.[• 4 in. dia. ~

2 in. dia.•' 2,.1>"

2 4 6 8 10 12 14 16

Mean Stress o m (ksi)

Fig. 56B FAILURE STRESS FOR INDIANA LIMESTONE

70

-.," --

No failure curve is defined for the 32-in. dia granite

as yet. These cores fail by a series of slippages, displayingability to carry increased load long after failure begins. In

several cases, the 32-in. dia cores survived two complete load-

unload cycles, and were found to be fractured only upon removalfrom the test cell. The highest stress state reached by these

cores is shown on Figure 56A, but failure is initiated at some

point lower on the stress trajectories. An evaluation of the

scale effect can be made by comparing the maximum axial stressattained in the 32-in. dia cores with the axial stress obtained i

by the intersection of the stress trajectories with the failure

curve for small specimens. The following table shows these results.K- Table 4 SCALE EFFECT ON FAILURE OF GRANITE i

Failure data for Failure curve Ratio32-in.dia small cores 32-in. _ small cores

(stress in ksi) (stress in ksi)

3 a a 01 am 013 m m 1 M

2 11.9 31.6 15 41 0.79 0.77

4 12.5 29.4 21.5 56.5 0.58 0.526 15.0 33.0 126 60 0.57 0.55

It can be seen that the 32-in. dia specimens failed at one-half z

to three-quarters of the strength attained by the 2-in. and

4-in. dia cores. The stress trajectory for the 2 ksi confining

pressure test is nearly parallel to the failure curve, and

one o. the weaker 2-in. dia specimens failed at nearly the

same point as the 32-in. dia core. Tests were condacted on

32 in. dia Charcoal Black granite specimens on a previousprogram yielding somewhat lower strengths. These data are

indicated by open symbols of Figure 56A, and show the amount

of scatter observed in the large specimens. None of the 12-in.

dia granite cores failed at stress levels possible in that

test cell.


71

4.8.2 Indiana Limestone

The strength of the smell Indiana limestone cores ran

counter to the expected trend of reduced strength with increased

size. Inspection of Figure56B reveals an apparent loss of

cohesive strength in the 2 in. and 4 in. dia cores, as compared

with both the 12 in. dia triaxial tests and the 2 in. die

unconfined compression test. This may be caused by distress (disturbance)

in the limestone maLtix during the hydrostatic loading phaseof the tri3xial tests. Such an effect would be greater for

tests run at higher confining stresses, and smaller for tests

run on larger cores, the latter due to arching action in thedistressed limestone. That is, particles in the outer layers

of the core move into point-to-point contact, so that the

distressed region displays no change in frictional strength,

only slight volume change and greatly reduced cohesion. Inthis configuration, a portion of the hydrostatic stress may

be carried around the interior of a large core by frictional

arching in the outer distressed region, resulting in somewhat

reduced stresses and les- cohesion loss in the interior.

Whether this hypothesis is correct is not known at this time,

An alternative cause for this behavior would be the intrusion

of liquid and pressure into the pore volume. This is not felt.

to be the cause, because the trend became noticeable early

in the test series, and extreme precautions against liquid

intrusion were taken. All cores were given two coats of latex

paint by dipping before any triaxial tests were conducted.

After the first two triaxial tests on the small rocks were

conducted, the remaining eight 2 in. and 4 in. die cores were

given additional coats of latex, so that a heavy layer of

watE-proof paint was built up on the cores. Secondly, had

pressure intrudeu into the rock, the state of stress wculd .. avebaen radically changed, together with the apparent mechanicalresponse of the rock. However, all sizes of limestone displayed

the same moduli and were modeled by the same laws. Hence it

lIT R.-SEARCH INSTITUTE

72

seems that the only influence was an actual loss of cohesivestrength, probably due to distress in the limestone matrix

near the surface of the specimens.

The 36 in. dia limestone cores failed at stress levels

ranging from 85 to 21 percent of the 12 in. dia core strengths,

the greater strength reduction occuring at higher stress levels,

as was ilso the case for the charcoal black granite.

117 RESEARCH INSTITUTE

73

5.0 CONCLUSIONS

A total of 36 triaxial tests were conducted on Charcoal

Black granite and Indiana limestone specimen, ranging from

"2 in. dia to 36 in. dia. Because of the bulk of data collected,

the stress-strain data are presented in terms of descriptive

models.

5.1 Hydrostatic Behavior

The hydrostatic model for both rock types is the

following, previously used by Seaman and Whitman. 2

m= A n

from which

t (n-l)/n

in whicham =mean stress

Svolumetric strain 4

kt = tangent bulk modulus

A and n model parameters

5.2 1. _axial Behavior

The granite was fit to a shear model used by Birch

Sand Bancroft. 33

G c o 0at m

in which

Gt = shear modulus

c a model parameter


74

At ý--hQ

5.0 CONCLUSIONS

A total of 36 triaxial tests were conducted on Charcoal

Black granite and Indiana limestone specimens ranging from

2 in. dia to 36 in. dia. Because of the bulk of data collected,

the strer -strain data are presented in terms of descriptive A

models.5.1 Hydrostatic Behavior

The hydrostatic model for both rock types is the

following, previously used by Seaman and Whitman2

nam = A v

from which n-1 (n.-l)/nk n Ckt nm

in which.

a mean -tress

volumetric strain

kt = tangent bulk modulus

A and n model parameters

5.2 Triaxial Behavior

The granite was fit to a shear model used by Birch

and Bancroft.Gt c a

S~m

in which

Gt = shear modulus

c = a model parameter

lIT RESEARCH INSTITUTE74

The limestone behavior was found to be bounded by thefollowing model:

Gt r BI/in which

L AO deviator stress

B and r model parameters

The parameters for all models are listed in Table 5.

The models were used to compute values for elasticmodulus and Poisson's ratio. The agreement with the observeddata was good for elastic modulus at stress states below

yielding, and only fair for Poisson's ratio.

The only major influence of specimen size on the stress-

strain behavior of either rock type occured in the 12-ino dia

granite cores. These cores also displayed significantly

higher sonic velocity and Schmidt hammer readings, and thedifference in model parameters is considered to be the result

of slightly different rock properties rather that, the influence

of scale.

5.3 Effect of Specimen Size on Strength

Sufficient data is not yet available to preciselydescribe the scale effect on strength, or to evaluate the

statistical hypotheses describing scale effect. The 2-in. and

4-in. dia Charcoal Black granite cores appear to share acommon failure criteria, while the 32-in. dia granite specimensfailed at stress levels only t/2 to ''4 of the strengths

achieved by the small cores.

The 36-in. dia Indiana limestone cores failed at

stress levels of approximately 3/4 those attained by the 12-

in. limestone.

[IT RESEARCH INSTITUTE

75

TABLE 5

HYDROSTATIC MODEL PARAMETERS

A n model

2 in. andl 2.774 in. dia 817 t

granite ~ 1 ~ 17 s

12 in. dia 4.66granite v 7 13 lnl~ (n-l)/n

32 in. dia 4.00granite x 0 1.73A

all sizes 0.490limestone x 16 0.75

TRIAXIAL MODEL PARAMETERS

_____________ Parameters -Model

2 in. and 4in.dia granite =c l.89x106 Gt = a

612 in. dia c 1 .895xl0 for granitegranite AI32 in. dia c = .735x106

granite________(r)/

all sizes r = 0.30 t r '(olimes tone B=3.55yx104 fo lietn

IIT RESEARCH INSTITUTE

76

5.4 Effect of Scale on Type of Failure

There was a drastic change in failure with increasing

specimen size for both rock types. The 32-in. dia granite and

36-in. dia limestone specimens failed very gradually and

displayed increasing load carrying ability long after slippage

was initiated. The failure of the small cores was very rapid,

with all strength lost immediately upon initiation of failure.

If initiation of failure is considered rather than maximum

stress attained, the influence of specimen size on strength

would be even greater than that presented above.

5.5 Recommendations

Additional data on these and similar rock types shouldbe accumulated. The gradual failure and extended load carrying

ability of the larger specimens appears to be similar tofailures observed in rock masses. It seems likely that models for

large specimens may be developed that will be more effective

in predicting mass behavior.

The small scale effect with respect to elastic properties

suggests that effort should be diverted from strain gage

instrumentation to obtaining additional failure data. The

strain gages should probably be replaced with deflection gages

operating over the length and diameter of the specimens, at

least in the larger sizes. This will give an indication of the

gross rock behavior, including slippage.

The sonic velocity mapping was found to be effective

in predicting variations in basic properties, the system

used being both rapid and accurate. The use of this test

should be increased in future programs. The amount of Shore

scleroscope testing should probably be reduced sharply, and

not used as an indicator of core to core variability.


77

L_ L •

VI

REFERENCES

1. Huck, P. J. and Singh, M. M., "Triaxial Tests on Large Rocki " Specimens" Final Report, contract DASA-01-69-C-0134 forDept. of Defense, Advanced Research Projects Agency, July 1971.

2. Salamen, M. D. G., and Munro, A6 Ho,"A Study of the Strengthof Coal Pillars" J. of the S. Africa Inst. of Mining and

Metallurgy, Sept. 1967.

3. Greenwald, H. P., Howath, H. C., and Hartman, l.,"Experimentson the Strength of Small Pillars of Coal in the PittsburgBed" U, S. Bureau of Mines Tech. Paper 605, 1939.

4. Steart, F. A.,"Strength and Stability of Pillars in Coal Mines"J. Chem, Metall. Min, Soc. S. A, vol 54, 1954.

5. Weibull, Wo,"A Statistical Theory of Strength of Materials",Ingretensk Akad, Handl., nll, 1939 p. 5-45.

6. Protodyakonov, M. M.,"Methods for Evaluating of Cracks andStrength of Rocks in Depth' Fourth Intl. Confo Rock Mech.and Strata Control, Columbia U., New York, N. Y,, 1964,Addendum.

7. Grobbelaar, C., "A Theory for the Strength of Pillars",Pillarco, Pretoria, S. Africa, June 1970, 103p.

8. Epstein, B., "Statistical Aspects of Fracture Problems",Jour. Appl. Phys. v 19, Feb. 1948.

9. Bieniawski, Z. T., '"echanism of Brittle Fracture of Rock",D.Sc (Eng.) Thesi, U. of Pretoria, S. Africa, 1968.

10. Glucklich, J., Cohen, L. J., "Size as a Factor in theBrittle-Ductile Transition of Some Materials.' Int. Jour.ofFract. Mech. Dec. 1967, p. 278-289.

11, Glucklich, J.. Cohen, L, Jo, "Strain-Energy and Size Effectsin a Brittle M~aterial", ASTM Materials Research and Standards,v 8, n1J, Oct. 1968, p. 17-22.

12. Baecher, C. B.,"The Size Effect in Brittle Fracture'' M. S.Thesis, Mass, Inst. Tech. June 1970, 190p.

13. Swanson, S. R., "Development of Constitutive Equations forRocks", Ph.D. Thesis, Univ. of Utah, Dec. 1969, 140p.

14. Adams, F. D. and Nicholson, Jo T.p "An Experimental Investi-ation into the Flow of Marble", Phil. Trans, Roy, Soc.London), Ser. A, v 196, 1901, p. 363-401,


78

REFERENCES (Cont' d)

15. von Karman, T., "Festigkeitversuche unter Allseitigem DrucW,Zeitschr. des Vereins deut. lng,, v 60, 1911, p. 1749-1757.

16. Griggs, D. T., "Deformation of Rocks Under High ConfiningPressures", J. Geol., v. 44, 1936, p. 541-577.

17. Griggs, D. T., and MillerP, W. B., "Deformation of Yule 4Marble: Part 1--Compression and Extension Experiments onDry Yule Marble at 10,000 Atmospheres Confining Pressure,Room Temperature" Geol. Soc. Am, Bul, v. E-' 1951, p. 853-862.

18t Handin, J. W. and Griggs, D., Deformation of Yule Marble:Part 11--Predicted FabriL Changes", Geol. Soc. Am., v. 62,1951, p. 863-886.

19M Griggs, D. and Handin, J.,"Rock Deformacion", Geol. Soc.Am., Mem. 79, March 1, 1970, 3 8 2 p.

20. Baidyuk, B. V., 'lMechanical Properties of Rocks at High

Temperatures and Pressures,' Consultants Bureau, PlenumPub. Corp. New York, 1967.

21. Morgenstern, N. R, and Tamuly Phukan, A. L., "Non-LinearStress-Strain Relations for a Homogeneous Sandstone",Intl. Jour, Rock Mech. Mng. Sci., v. 6, 1969, p. 127-142.

22, LaMori. P. N., "'Static Determination of the Equation ofState of Cedar City Tonaiite", DASA Rept. No, 01-69-C0053, May 1970, 78p.

23. McClintock, F. A,, Walsh, J. B.o"Friction on GriffithCracks in Rock under Pressure", Proc. 4th Nat. Cong. Appl.Mech., Berkeley, 1962, p. 1015-1022.

24. Brace, W. F., "Brittle Fracture of Rocks" Proc. Intl. Conf.on State of Stress in the Earth's Crust, American Els-vierPubl. Co., New York, 1964, p. 111-174.

25. Murrell, S. A. F., "The Effect of Triaxial Stress Systemson the Strength of Rocks at Atmospheric Temperatures",Geophys. Jour,, v. 10, 1965, p. 231-282.

26. Pfefferle, W. and Smith, C. R., "Phase I Flatjack Tests",Air Force Rept. No. SAMSOTR-70-381, Oct. 6, 1970, 4 2 p.

27. Obert, L., Personal communication.

28. Hoek, E., "Brittle Failure of Rock", Chapter 4 in RockMechanics in Engineering Practice (ed. Stagg and Zienkiewicz),J. Wiley, London, 1968, p. 99-124.


79

REFERENCES (Cont'd)

29. Wawersik, W. R., "Detailed Analysis of Rock Failure inLaboratory Compression Tests", Ph.D. Thesis, Universityof Minnesota, 1968.

30. Cook, N. G. W., "The Failure of Rock", Int. Jour. Rock Mech.Mng. Sci., v. 2, 1965, p. 389-403.

31. Houpert, Ro• "La r~sistance a la rupture des granites",Revue de l'industrie min~rale, May 15, 1968, p. 21-23.

32. Seaman, L., and Whitman, R. V., "Stress Propagation in Soils",Final Report -Part IV, Stanford Research Institute, MenloPark, California, for Defense Atomic Support Agency,DASA 1266-4, June 1964.

33. Birch, F., and Bancroft, D., "The Effect of Pressure on theRigidity of Rocks-I", Jour. Geol. v. 46, Jan-Dec 1938.

~A

A -3


80

APPENDIX

PLOTTED "EST DATA

This appendix presents the stress-strain data for the

individual triaxial tests. The plots included are deviator stress

as a function of shear strain and mean stress as a function ofS~ volumetric strain. The tests are arranged by rock type, core

diameter and confining stress in that order. Pertinent test

data is specified on each plot.


A

g'

00

mC:'

C.3

,:)tI%

C)~

~-10.00 6.00 10.00 20.00 30.00 40.00 150 .0 0SHEAR STRAIN MICRQ-STF<FUNS(X1Oý-*ý2)

A-i

S~o

C0

Lo

0 *1TEST NUMBER 14

0 DIAMETER 2 INCONFIrMING PRESSURE 0 PSI

e-4 CHARCOAL BLACK GRANITE

WD

ZIl: +

+

I +-

I4 + '

+

01 +

"44-4++4-

C2+oII I •"

0;-10.00 0.00 10.00 20.00 30.00 40.00 50.00

VOLUMETRIC STRAIN MICRO-STRAINS(X1O**2)

A-2

"QI"ICD.

TEST NUMB-R 15

SDIAMETER 2 INC CONFINING PRESSURE 0 PSIc;- CHARCOAL BLACK GRANITE

c'+

+ ++'-.4a: + '

•+

÷ Ii

r I I I P I I•-10.00 0.00 10.00 2•0.O) "30.09 4;0.00 50.00

SHEAR STRAIN MICRO-STRAINS(X1O.*2) "

A-3

:31

01i

TEST NUMBER 15DIAMETER 2 IN

¶1 CONFINING PRESSURE 0 PSIC~l L-HARCOAL BLACK GRANITE

{ x" C;-

+ +

+ ++z+

ci: ++w~++z, -+ I

+++++

In_ ++0 -;-

++ I

-I'- i

"-10.00 0.00 10.00 2O.00 .'49.09 40.00 50.00•-1VOLUMETRIC STRAIN MICRO-STRAINS(XIO**2) 0

A-4

cnu2 CD

CD CONFINING PRESSURE 0 PSI• • •- CHARCOAL BLACK GRANITE

I-=

ci-

C14-

cm

o Ln

°

CHARCAL BLATRAINMITE -TRNSXO,2

6-5

CD1

(/)

0D

CIc;a

10.00 0.01.0 2.0 300 00 00

SHA0TANMCO-TAN(l*2

A-

C3

0

Co

L.-

TEST NUMBER 17C DIAMETER 2 IN

CONFINING PRESSURE 0 PSI,- CHARCOAL BLACK GRANITE

03

Ml

bbL I 2.l-

0

-10.00 0.ao 10.00 20 .on 30.00 40.00 50.00VOLUMETRIC STRAIN MICRO-STRAINS(X1O+.2)

A-6

I

+ +,=;- +

c-J

03

I + + ++i

IJJ ICc

CD

"I I

S-H+ T E S T N U M B E R 6;-.DIAMETER 2 INCONFINING PRESSURE 2000 PSI

SCHARCOAL BLACK GRANITE 23980 ,c-r

-ý 10.00 O.0o 10.00 20.00 30.00 410.00 50.00SHEAR STRAIN MICRO-STRAINS(XCAIT2)

A-7

CI)

o

U,!

Vd: +

Z ' + 4++

Wd +

tJ -+- itME E :

C+ ONIIG RSSR 00 S

Ld +

4++

I-2

o #+ 4-

TEST NUMBER 6DIAMETER 2 IN

+CHARCOAL BLACK G3RANITE 29980

-10O.00 0.00 10.0O 20.00 30.00 4-o.00 50.00VOLUMETRIC STRAIN MICRO-STRAINS(XIO;*2)

A-8

C,-

co

°4 1

CD +

o- +

++

0,C

++CDo ++

t-- +H+

+

TEST NUMBER ?DIAMETER 2 INCONFINING PRESSURE 4000 PSICHARCOAL BLACK GRANITE

-20.00 0.0 2.00 40.00 60 .00 80.00 !00 .00SHEAR STRAIN MICRO-STRAINS(XIO**2)

A-9

0

-. II

,w,; -4

c~,! "I'

0+

0+

+

V +

--. +

•-5* c 5 - + I+Lb +

SZ +(Y C - + IC~w; 4-

W'-4 +

+

z+li: +

+

+oD +

-. 4 +

++

+

0N +

4++ TEST NUMBER 7DIAMETER 2 IN

"444+ CONFINING PRESSURE 4000 PSI4++ CHARCOAL BLFCK GRANITE

.4f+

2.0 40.00 60.00 80.00 100

-20.00 0.90 20.Oa ,100 ao0 8~a o.00

VOLUMETRIC STRAIN MICRO-STRRINS(XIO**0)

A-1O

- __..._.. ._ --__-.- .. _ - ,_, __=_, -. --L i

c'_

C311+

CD-

i +0

t Cc

>+

0.

("0

- +TEST NUMBER 8

DIRMETER 2 INCONFINING PRESSURE 6000 PSICHRRCORL BLRCK GRRNITE

Co

-20.00 0.00 20.00 40.00 60.00 80.00 100.00SHEAR STRAIN MICRO-STRAINS(X1O**2)

AI

01

++

at+

+

WA; +

+

4+

Z4 4+

++

* ~TEST NUMBER* 11AIRETER 2 INj~j.CONFINING PRESSURE 6000 PSI

CC *+ CHRRCOAL BLACK GRANITEI

-20.00 0.00 20.00 40.00 60.00 80.00 ido.coVOLUMETRIC STRAIN MICRO-STRAINS(X1O**2)

A- 12

i! o

to:-I

I +In

C31C31

t o~0 C.4

++0

.1.1 ++

++0.0 1.00- + +++

+

C2 +++

S+S+ DIRNMETER 2 IN

++ CONFINING PRESSURE 10000 PSI

+ CHRRCORL BLACK GRRNITE

-20 .012.00 24.00 36.02 48.00 60.00SHEAR STRAIN MICRO-STRAINS(X1*0 2)

SA-13

cot0II30+

o

0 ++

0+

C)iC3i

I +S2 +

S• +

V)

S • +

C)

++++

? 4'-Lo +H Ht t

++ TEST NUMBER 9+ DIAMETER 2 IN

++ CONFINING PRESSURE 10000 PSI+I ST IN CHARCORL BLACK GRRNTTE01 + +

12.00 0.00 12.00 24.00 36.00 48.00 60.00VOLUMETRIC STRAIN MICRG-STRINS(XO*A2)-4

A-14 I

caa

C44

¾' Wi

ca+ca

I-2

cc6-4

Wa2co+

* TEST NUMBER 19R0'MTE

++ CONFINING PRESSURE 2000 PSI

-50.00 0.00 50.00 100.00 150.00 200.00 250.00

SHEAR STRAI1N MICRO-STRAINS(XIO*.1)

A-15

I+

* ÷

++

++

+ 2

- ~TEST WUIStER 19B |+ D+IhITER 4 IN

COIDWNIN6 PRqESSUIRE f2000 PST+ CHACR 1LFECK SRMITE

-1.0 0.00 11.00 .0.1.00 18 .O0 4C-.00 G55,10SHEAR STRAIN MICRO-STRRINSCXIO**2)

I ~A--17.

caK~:-+a

•- +

.4.

0a

U)w +

w. +

:1 E

a- + 1• 8

CP

• • +

6- a

VOUMpI + ETNEBE 3 5 i

•- • OLUMTRICSTRAIN MICRO-STRRINS(XIO..2)

mA-- 8

-i + - - -- ~~

C)

-l -vo

0

|- 1 +++

oz +

N + "+ ++

LrH

C3 + 74-

, Of

0,

O+

A -19i"-

a;TSTNMBR i

CD

A-19

IIo

C3

01

0÷ + +

SI+

+

+

c +

+

4

+ jONIIG RSSR 00 S

++

Ii +

Lu

VOLUMTRIC TRAINNICRESTNUBRINXO*2+ A-2

A02

Co

0 1'+

S + +

, + +

+

0)

+x + +

c))

w+

Sj+

°l +

>01

0 +

6 - + + TEST NUMBER 11

,÷+ DIAMETER 4 IN- CONFINING PRESSURE 6000 PSI

0 ++ CHARCOAL BLACK GRANITE

SIII i1

-18.00 0.00 18.00 36.00 54.00 72.00 90.00SHEAR STRAIN MICRO-STRRINS(XIO**2)

A-21

IO

UO

>13C3-

C3- II

C?

o ~+-

++

C' + +

CO

c 09 + +

0r1

0+

+ +

STRINMIROSTAIS(1O.2

A 2 + 2C: + + CRCB L RNT°+ +

"DAEE I I N I

SVOLMETIC TRAN MCHRCO-STRLAINGRAITE*

c;-

°3

+

* )

Cl)

j

(f) +

ii ° I +:4yj -o

6 ++CM 4

TEST NUMBER 12DIAMETER 4 INCONFINING PRESSURE 9000 PSICHARCOAL BLACK GRANITE

0,I I IICl +

0-12.50 0.00 12'.50 25.00 37.50 50.00 62.50C;lEAR STRAIN MICRO-STRAINS(XIO+*2)

A-23

o

d -S~+

CD

wO .- + +.

o

--4-

++

. -; +F E T N M B R1

C31.C64 A

S-F DIAMETER 4 INCONFINING PRESSURE 8000 PSI

o ++ +++ CHARCOAL BLACK GRANITE

CD

LLI

-L12.50 0.00 12.50 25.00 37.50 50.Q0 52.50VOLUMETRIC STRAIN MICRO-STRAINS(XIO**2)

,C,:

A-24

i-i

4+ ETNUBR 1

Co

cý1- +

++

ca +

++

c1-

-4

0' +

+

cr-

>+

CO

-+

++

F-- , +

S+o.o0,'TEST NUMBER 13

SI4+DIAMETER 4 IN

t . CONFINING PRESSURE 10000 PSI

-12.50 .00 12.5C 25.00 37.50 50.00 R2N50

SHEAR STRAI N MI CRO-STRRINS(X0 2)

S~A-25 j

oI +

CD

C) ++I3

+

-4 +

Z0 I

" ci+

+ +

r+4

+") +0")

++fir + *

Z I +_-1-

.- -0+ TEST NUMBER 13DIAMETER 4 IN

+4 CONFINING PRESSURE 10000 PSICHARCOAL BLACK GRANITEo_° + ++

6-'12.50 0.00 12.50 25.00 3? .50 50.00 62.50VOLUMETRIC STRAIN MICRO-STRAINS(XIO**2)

A-26

I

C3

-4-

I +

!+

014

x/c• +

ol÷

++

[ ++ ++

T S N- 46+- +

+4lt+(I)

40.0 0.0 .0 soE STNU BRAI S 'lD +l

+H+ I+EER1+I

i ~ ~~~SHEAR STRAIN MICRO-TFISXOI

AA-27

"' II

CD}

0

c.-

Lo

cý-F

C)•

S+=+

01 +

LoJ

•" +4+

~c)

(f)

CE ++

11o + ++

-+ ++ TEST NUMBIER 46+i i-tI DIAMETER 12 IN

+ CONFINING PRESSURE 2000 PSI

S++ CHARCOAL BLACK GRANITEC)+

I I I I I I

0-40.00 0.00 '10.00 80.00 120.00 160.00 200.00

VOLUMETRIC STRAIN MICRO-STRAINS(XIO.*1)

A-2 8

!~o/

04

+

W+ + +

II '÷+

01+ +

++

(I)

Co + +

i~~~u CH3OLBRC RNT

+

00A-2

+ TEST NUMBER ItcnDIAMETER 12 IN

+ +CONFINING PRESSURE 4000 PSI

CHARCOAL BLACK GRANITE

'!5.0 0.05.0 100.00 150.00 200.00 250.00

SHEAR STRAIN MICRO-STRAINS(X1O+*l)

A-29

CIi

C3

C)I

~c "Co

;--

F-4

0+ +

+

++

+ +-

S+S+ +

Ld +-4 ++

+ ++

1)- 4- +

z+ +

+ ++So -w- ++ +

c o + TEST NUMBER 49SDIAMETER 12 IN

4-- CONFINING PRESSURE 4000 PSI

0 + CHARCOAL BLACK GRANITE I-50.00 0.00 50.00 100.00 150.00 200.00 250.00

VOLUMETRIC STRAIN MICRO-STRAINS(X!O**I)

A- 30414- -''-l - -4.

Al

0A

-coa

+

E *..401CD) 14

u++

W+

CID

CD_ +0m TETNMBR 4

+ TS CFNLISPESUER 4 008S

+ CHARCORL BLRCK GRANITE

Cý3.0 0.00 33.00 70.00 105.00 140.00 1?5.0OSHEAR STRAIN MICRO-STRAINSCXlO~*1)

A- 31

oi-

C2j

Co

4-4

CD +

++LI +

+

WC,)w 4.

)D

z

C2CD

TEST NUMIBER 48DIRMETER 12 IN

++ CONFINING PRESSURE 6000 PSICHARCOAL BLAICK GRANITE

C2

o ÷-4-

035.00 0.00 35.00 ?0.00 105.00 140.00 175.00VOLUMETRIC STRAIN MICRO-STRAINS MO..

A- 32

-I "-"- ' -- " - "-- -

7-7ý -ý---71-77-'_7"7N

+ 7o.A

a

i I I 13+

+4

cm:

C .4+

a 4

C

TEST NWABR 4DIA*MTE 12 IN

*CONFINING PRESSURE 3000 PSIa ___ CHARCOAL BLAICK SMAITE

-50.00 0.00 50.00 100.00 150.00 200.00 250.00SHEAR STRAIN MICRO-STRAINS(X1O**1)

A- 33

WWAIo I1,7-7

+

cm

C43

a*ma

U)a

LU +

ca

ca

0;TET UH 45DIANETER It IN

+ CONFININS PRESSURE 5000 PT'~+ CHARCOAL 9LACK SRAITE

oa +

-*50.00 0.00 50.00 100.00 150.00 200.00 250.00VOLUMETRIC STRAIN MICRO-STRAINSCX1O..1)

A- 34

° I

•- +5

cý-

o

>

310I

0OFIINgPESUE 010 S

II II

1-4+

011

>!

DIMEE 12 IN

A-35

tI

I."

-1 +

CD-

I ÷ +

Ld

'--4

U~4.

I+* +

Co

+ + 4

:±iI÷ + +

0 -- TEST NUMBER '7I DIAMETER 12 INI +CONFINING PRESSURE 10000 PSI

f+ CHARCOAL BLACK GRANITE

•III I I-. 00 0.00 40.00 980.00 120.00 160.00 200.00

VOLUMETRIC STRAIN MICRO-STRAINS(X1O**1)

A-36

A

1 001

c- ++

+

0~ +

M-l ++0) +

W; +

++

S• +0. +

U) cd +

++ +

S+

-+ TEST NUMBER +

+ DIAMETER 32 IN>CONFINING PRESSURE 2000 PSI

0 * CHARCOAL BLACK GRANITE

-1.O 001.0 50.0

* SHEAR STRAIN MICRO-STRAINS(XIO.*2)

S~A-37

-l --+

CD

t-4~

M- +

+

0I +

4-8

0- +oj +

+

CD"

Io ++0")j +

(Od +

+CD +

(1) +

++

+

� +-: +

+

Co +0 +

U- +

++$

+

.4+ TEST NUMBER 4DIAMETER 32 INCONFININ6 PRESSURE 2000 PSI

+4+- CHARCOAL BLACK GRANITECD +*1

-o10.00 0.00 10.00 20.00 30.00 40.00 50.00VOLUMETRIC STRAIN MItCRO-STRAINS(X1O*.2)

A-38

01

c6-cv)

0

VoIC)

c•-

C)C;I

II +

r-4

0..+T-4

0 - +

•c + TEST NUMBER 3DIAMETER 32 INCONFINING PRESSURE 4000 PSI

CDCHARCOAL BLACK GRANITE• I I I i

10.00 0.00 10.00 20.00 30.00 40.00 50.00

SHEAR STRAIN MICRO-STRINS(XGR **2)A

A-39

04

C~o+

Ct)

CD_ +CW. +

+

U--) ++id +

,0 +

3.s

0 4+

"•o *-

+

z I I I I+

+ +I41-4

TEST NUMBER 3

DIAMETER 32 INS O+CONFINING PRESSURE 4000 PSI

CHARCOAL BLACK GRANITECD +01000 0.00 10.00 20.00 30.00 40.00 50.00VOLUMETRIC STRAIN MICRO-STRAINS(X1O**t2)

A-40

C3D

ov

r--t,-

'LI C:

coo

S++

C2 ~TEST NUMBER 2A 4DIAMETER 32 INCONFINING PRESSURE 6000 PSI

CHARCOAL BLACK GRANITE

6-25.00 0.00 25.00 50.00 75.00 100.00 125.00SHEAR STRAIN MICRO-STRAINS(X1O..1)V

A-41

o-,-4

,-4

:;ý

c) +

U) c

zc

Cr12

th + i

.- + .

+?

-- + TEST NUMBER 2A

DIRMETER 32 IN+ CONFINING PRESSURE 8000 PSI

CHARCORL BLRCK GRRNITE

° 2'

-; 25.00 0.00 25.00 50.00 75.00 100.00 125.00

VOLUMETRIC STRAIN MICRO-STRAINS(XIOD),I

A-42

II

o3

Or)

C,,

itio+++

01

c~j

+Co

+ +

Cc

>TEST NUMBER 2+

DIAMETER 32• IN+ .• CONFINING PRESSURE 6000 PSI++ CHARCOAL BLACK GRRNITE

0.

0 I IIS-10.00 0.00 10.00 20.00 30.00 40.00 50.00

SH01AR STRATN MICR+-TTRRINS(XIO 20)LA-4

DIMEE 32 IN-

I

S tC3

C)

4 i-o +

U) ++

C3 + +1

Z•s- +

C" ++U))LI) w+

+ +R ENCJLL. + 41

Ld+j

coo +

C~l +

++

CDAro +

TEST NUMBER 20

+DIAMETER 32 INCONFINING PRESSURE 6000 PSI

++ CHARCORL BLACK GRANITE

-]0.00 O.00 10.00 20.00 30.00 40.00 50.00VOLUMETRIC STRAIN MICRO-STRAINS(XIO..2)

A-44

ICCA

1 4

ca-

4f

+*

,,I+4

'V +4.

0I--C3

44+-

I 4"TEST NUMBER 2 0

4* CONFINING PRESSURE 0 PSIINDIANA LIMESTONE FAILURE 4175

050.00 0.00 50.00 100.00 150.00 200.00 250.00

SHEAR STRRIN MICRO-STRAINS(XIO**1)

A-45

AA0

¶J.-

i0

I i

$0CD

&P

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CONFINING PRESSURE 0 PSIS* INDIRNR LIMESTONE FRILURE 41?5

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VOLUMETRIC STRAIN MICRO-STRAINS(XlO.,,)

A-463

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A-47

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i0 7 TEST NUMBER 22: ~DIAMIETER 2 INi CONFINING PRESSURE 0 PSIS•1. INDIANA LIMESTONE FAIILURE 3330

5-S .00 0.00 50.00 100.00 150.00 200.00 250.00

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TEST NUMBER 28"" DIJIF1METER + 2 IN

CONFINING PRESSURE 4000 PSIcs

INDIANA LIMESTONE FFIILURE 3GB6

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• "3 TEST NUMBER 280 IRiNETER 2- IN

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A-54+

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TEST NUMBER 29OIBMETER 2 INCONFINING PRESSURE 6000 PSI

INDIBNA LIME STONE

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SHEAR STRAIN MICRO-STRAINS(XlO.I)

A-55

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DIAMETER 2 IN+ CONFINING PRESSURE 6000 PSI

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VOLUMETRIC STRAIN MICRO-STRAINS(X1O**I)

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"DIAMETER 4 IN

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CONFINING PRESSURE 9000 PSIINDIANR LINE 3TONE +

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A-77

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DIAMETER 12 INCONFINING PRESSURE 10000 PSIINDIANA LIME STONE

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SHEAR STRAIN MICRO-STRAINS(X1O+*2)

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A-80

M- I

C- TEST PRESSUR E 200 P

INDIRM•-LimESToNE FAILUJRE 6335

10.00 0.00 10.10 9-.00 3.00 40.00 50.00• SHEAR STRAIN MICRO-STRRINS(XIO**2)

A-81

3 4+W4W

TETNM+aDIEE 6ICOFNN 4.SM 20 S

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SCONFINING PRESSURE 4000 PSIS[INOIRNA LIME STONE

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A-83

( . ..5) ...... + .. • • • • ... _ • •,= • • •. ,.*• . • - . ... . ... ._. • . .• : -• ... . .- r - "'" -'•• • .'•"F•.' ±'•• •

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I .. . I' I i"0-300.0 0.00 30.00 60.00 90.00 120.00 150.00

SHEAR STRAIN MICRO-STRAINS(X10+*1)

A-85

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TEST NUMBER 41DIAMETER 36 INICONFINING PRESSURE 6000 PSIEND IAN9 LIME STONE

-30.00 0.00 30.00 60.00 30.00 120.00 150.00

VOLUMETRIC STRAIN MICRO-STRAINS(XIO**I)

A-86

EFFECT OF SPECIMEN SIZE ON CONFINED COMPRESSION …-L44 Elastic Modul-is Models for Indiana Limestone 65 45 Elastic Modulus for Indiana Limestone (a3 = 2 ksi) 65 46 Elastic Modulus

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