- -- I II III

DEPARTMENT OF MECHANICAL ENGINEERING

UNIVERSITY OF UTAH ,SALT LAKE CITY. UTAH 84111 '•'

t~.

Springfield, Ve 22151

Sponsoredl by Advanced Research Projects AgencyARPA Order No. 1579, Amend. No. 2Program Code No.--TF10

UTEC ME 72-027

FRACTURE MECHANICS APPLICATIONS TO ROCK

W. S. BrownS. R. SwansonW. E. Mason

FInAL REPORT

Contract No. H0210002

Submitted ToU.S. Bureau of Mines

Twin Cities Mining Research CenterTwin Cities, Minnesota

University of UtahCollege of EngineeringSalt Lake City, Utah

February 1972

The views and conclusions contained in this document arethose of the authors and should not be interpreted asnecessarily representing the official policies, eitherexpressed or implied, of the Advanced Research ProjectsAgency or the U. S. Government.

UnclassifiedSeruritv Classifica(ion

DOCUMENT CONTROL DATA - R & P(Security Classification of title. bodr of ahttract and indexing annotario- mu.. t he entered when 1av ,.,erall report is etatsilied) ,

I ORIGINATING ACTIVITY (Cirporate author) 120. REPORT SqCUR•TY CLASSIFICATION

College of Engineering UnclassifiedUniversity of Utah ,b. GROUPSalt Lake City, Utah 84112 A

3. REPORT TITLE

FRACTURE MECHANICS APPLICATIONS TO ROCK

4. DESCRIPTIVE NOTES (•ype of report and inclusive dates)

Final Report 21 December 1970 to 21 December 1971S. AUTHORISI fFirst name, middle initial, last name)

Wayne S. BrownStephen R. SwansonWilliam E. Mason

6. REPORT DATE 7". TOTAL NO. OF P.GES 7b. NO. OF REFS

February 1972 92 4311. CONTRACT OR GRANT NO ga. ORIGINATOR'S REPORT NUMBER(S)

H0210002 ME 72027b.PROJECT NO. UTEC -

ARPA Order No. 1579, Amend. No. 2,.Program Code No. 1 F10 ,b. OTHER REPORT NO(IS (Any other numbers that may be asjI.-,ed

this report)

d.

10. DISTRIBUTION STA ." '-Q

Unlimited

31 SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

"ARPA

13. AeBSIRACTIResults of a study to investigate the influence of non-uniform stress fields

on the failure strength of rocks are presented. Three rock types, Westerly granite,Nugget sandstone, and Tennessee marble, were tested to failure in .'nconfined tension,torsion, and bending tests, and in triaxial compression and extension tests.Specimens were prepared with notches of various sharpness to vary the intensity ofthe stress gradients in the specimens. Analyses were performed to determine how themagnitude of the stress varied across the specimen. Two significant experimentalobservations were made. First, the fracture stress v;Ya: insensitive to the notchconfiguration for all rocks tested. Second, local stresses existed in the specimensnear the crack tips signif;cantly above the stress required to fracture the rockin a uniform stress fiPid. A preliminary correlation of the increase in strengthexhibited by a specimen in a non-uniform stress field is presented based on acritically stressed distance that varies with the maximum stress.'

DD .ON`O`1473 14_7_3OSeV.u rv 6a..mfic'.ion

ARPA ORDER NUMBER CONTRACT NUMBER

1579 H0210002

PROGRAM CODE NUMBER PRINCIPAL INVESTIGATOR

IFlO Wayne S. Brown(801) 581-7064

NAME OF CONTRACTORPROJECT SCIENTIST

U. S. Bureau of Mines

Mr. Syd Peng(612) -125-4582EFFECTIVE DATE OF CONTRACT

21 December 1970 SHORT TITLE OF WORK

Fracture Mechanics ApplicationsCONTRACT EXPIRATION DATE to Rock

21 December 1971

AMOUNT OF CONTRACT

$ 55,364.).

ID

APR2!?7 197?

FRACTURE MECHANICS APPLICATIONS TO ROCK

TABLE OF CONTENTS

Page

ABSTRACT i...... ........................

ACKNOWLEDGMENTS . i.........................

INTRODUCTION . . ................. .. ... ... 1Previous Work ....... ......................... .. 1

II. EXPERIMENTAL METHODS ........ ...................... 3Rock Types .......... .......................... 3Specimen Preparation ........ ..................... 4Test Methods .......... ......................... 5

III. RESU!TS .......... ........................... 6STens onTests .. . .. . . . .. .. . . . . . . . . . . . .. 7Ten on Tss...............................7Extension ............ ......................... 7Torsion ......... ............................ 8Beam Bending ........ ..... ...................... 8Notched Compression Tests. ................... 9

IV. ANALYSIS OF RESULTS. .................................. 9Finite Element Analysis ...... .................. .. 10Results of Analysis ........ ...................... 11Fracture Stresses in Test Specimens ..... .............. 11Critically Stressed Region ....... ................. 12Sharp Crack Fracture Mechanics ....... ............... 13Multiaxial Stress Effects ..... ................... .. 16

V. DISCUSSION ........ ........................... .. 17VI. SUMMARY AND CONCLUSIONS ........ .................... 20

REFERENCES .......... ............................. 21TABLES ............ ............................... 24FIGURES .............. ............................ 32

APPENDICES ............ ............................ 73

ABSTRACT

Results of a study to investigate the influence of non-uniform stress

fields on the failure strength of rocks are presented. Three rock types,

Westerly granite, Nugget sandstone, and Tennessee marble, were tested to

failure in unconfined tension, torsion, and bending tests, and in triaxial

compression and extension tests. Specimens were prepared with notches of

various sharpness to vary the intensity of the stress gradients in the

specimens. Analyses were performed to determine how the magnitude of the

stress varied across the specimen. Two significant experimental observations

were made. First, the fracture stress was insensitive to the notch configura-

tion for all rocks tested. Second, local stresses existed in the specimens

near the crack tips significantly above the stress required to fracture the

rock in a uniform stress field. A preliminary correlation of the increase

in strength exhibited by a specimen in a non-uniform strcss field is present-

ed based on a critically stressed distance that varies with the maximum stress.

ACKNOWLEDGMENTS

This research was supported by the Advanced Research Projects Agency

of the Department of Defense and was monitored by Bureau of Mines underContract No. H0210002.

Mr. Lynn Jensen and Mr. Stan Duncan were extremely helpful in conduct-

ing the experiments and were assisted by Mr. Jay Rogerson and Mr. Wayd Weber.

ii

FRACTURE MECHANICS APPLICATIONS TO ROCK

I. INTRODUCTION

Most previous laboratory investigations of rock fracture have considerej

only conditions of uniform stress. It has been suggested that the degree ofthe non-uniformity of stress may have some bearing on fracture in rock and

this has been demonstrated to some extent in rock and other materials. This

could be of considerable practical importance as many rock mechanics problems

involve non-uniform stresses. A particular example would be in rock cutting

where the stress is localized in the vicinity of the cutting tool. Thus,rather than being uniform the stress is highly concentrated. The effect of

the non-uniformity of stresses on rock fracture has been recognized by Cook (1)

for this problem. Stress gradients occur in other engineering problems where

loads are localized or non-uniform.

It appears that improvements in the design of cutting systems will be

facilitated by a detailed understanding of the stress and strain fields in

the region of rock involved. This approach has been pursued by Cheatham et al (2,3)

and Pariseau (4) among others for bit penetration problems. However, thisknowledge of the stress field must be accompanied by a corresponding knowledge

of the rock fracture properties under the appropriate conditions of high stress

gradients.

In the present research program, the objective is to obtain a detailed

and quantitative understanding of rock fracture under conditions involving

gradients of stress. To accomplish this a series of laboratory tests and

theoretical analyses have been carried out. The results and their interpreta-

tion are presented in this report.

Previous Work

Although of fundamental importance in many rock mechanics problems, the

effects of stress gradients on fracture have been little studied. It has

long been appreciated that certain indirect tests for determining the censile

strength of rock, such as bending tests, gave strength values higher than

those measured in direct tension. Jaeger and Cook (5) present a discussion

of the experimental work on the subject, and present an analysis of the effect

of stress gradients based on critically stressed volume. The size effect onfracture strength has been interpreted by some investigators to be a result

of stress gradient effects ( 6,7). However it appears that little direct

work on the effect of stress gradients on fracture in rock has been published.

Stress gradients have been recognized as a variable in materials other

than rock. A fundamental theoretical approach is due to Weibull (3) who

considered the statistical effect of spaci .... size on strength. Fracture is

considered to result from internal defects, and the statistical defect sizeand distribution throughout the material introduces a size effect. Stress

gradients enter by virtue of the volume of material at a given stress level.

A classic result of the Weibull theory is that the strength of a material

under uniform stress varies with volume as

a1 ou = V2

02 0u (l1

where and m are material parameters. The details of

this have been presented by Jaeger and Cook (5). The use of a Weibull-type statistical theory has been often employed in considerations of brittle

fracture. It has been criticized however on the grounds that it is a "weakest-

link" type model that is based on catastropic propagation of a crack once frac-

ture has been initiated. It has been established that in many materials con-

sidered brittle in some sense, a coalesence of defects is required before final

fracture is produced ( 9). On this basis Hasofer (1O) has developed a "parallel-

link" statistical model to describe brittle fracture in steel. One of thefeatures of the results is a much less pronounced dependence on si:e as com-

pared to the Weibull model. The effect of stress gradient on fracture in

ceramics has been investigated by Weiss, Chait, and Sessler (11). The apparent

stress-strain behavior of their ceramics were linear to the point of fracture

and thus would be expected to be sensitive to stress concentrations. The usual

maximum str,..s criteria would predict

Snet Kt = 0max (2)

2

where K is the stress concentration factor. Instead, they found that the

ceramics approximately followed the relationship

Cr ;Kt(3)'net = 0max

In this work the weak dependence on stress concentration was explain-

ed on the basis of stress gradients by means of the Weibull theory.An alterna-

tive interpretation was also suggested that considers a distribution of defects

at a given spacing in the material. The ratio of the defect spacing to a

length characterizing the stress gradient introduces a size (and stress gra-

dient) effect into the interpretation of fracture. Their model predicts that

an inhomogeneous material, as characterized by a large defect spacing, will

be insensitive to stress concentrations. This is supported by the findings

of Wright and Byrne (12) who introduced various notches into concrete speci-

mens and found a negligible effect on the net section fracture stress in ten-

sion.

An experimental study of the effect of stress gradients on stresses in

brittle plastics has been presented by Durelli and Parks (13). A strong

dependence of fracture on stress gradients was found. Their data were corre-

lated by considering the failure stress to be a linear function of the loga-

rithm of the volume of material stressed above 95% of the tensile strength.

Size and stress gradient effects have been long noted in metal fatigue

(14), and are ;Ynlained by statistical flaw distributions as well as size

parameters such as the width of a plastic slip-band. Stress gradients have

been considered in spall fracture (15,16); however, this problem is complicated

by wavepropagation and strain rate effects.

In the following an experimental program to investigate fracture in

Westerly granite, Nugget sandstone, and Tennessee marble will be described.

Fracture results will be given for these rocks under a variety of stress con-

ditions, principally involving specimens with stress risers so as to introduce

stress gradients. These results will be discussed in detail.

II. EXPERIMENTAL METHODS

Rock Types

3

Three rock types were tested in this program; these are Westerly granite,

Nugget sandstone, and Tennessee marble. All three rock types have been tested

previously in this laboratory and elsewhere (17-25).

Specimen PreparationThe tests carried out in this program were tension, extension, and tor-

sion of cylindrical specimens, and bending and compression of prismatic (rec-tangular) specimens. These tests were carried out on both smooth specimens,

and with the exception of the bending tests, on specimens with various types

of notches to serve as stress-risers. The notch configurations are shown in

Figure 1. The notches can be seen to have tip radii of 1/8, 1/32, and 0.0015

inches.

The cylindrical specimens were cored from large blocks and surface

ground on the lateral surface and ends to a uniformity of within + 0.0003

inch. The rectangular specimens were cut fronm the same blocks, and the com-

pression specimens were ground on all sides.

The 1/32 and 1/8 radii notches were produced in the specimens by

grinding wheels with semicircular ends. The smallest notch was produced ty

first establishing a V-shaped notch by grinding , and then furthershaping the tip by cutting with a 0.003 inch diameter diamond impregnated

wire, thus giving a controlled notch tip radius of 0.0015 inch. The notches

formed by these techniques are relatively smooth and reproducible.

The nutches for the tension and extension specimens were placed in the

circumferential direction as shown in Figure 1. The notches for the torsion

specimens were machined both in circumferential and longitudinal directions,

as shown in Figures 1 and 2. The smooth (unnotched) extension specimens

were 3/4 inch diameter, the notched extension specimens were I inch diameter,

while both notched and unnotched tension and torsion spLcimens were 1 inch

diameter. All of the notches were placed 1/8 inch deep, so that the net

section of the circumferentially notched 1 inch diameter specimens was 3/4

inch in diameter. In addition, tension specimens with a 2 inch diameter

were also tested, both unnotched and with 1/8 inch deep circumferential

notches. All specimens had an L/D ratio of 2 or more.

4

In addition to the above specimens, a number of tension tests werecarried out on a separate block of Nugget sandstone using a slightly dif-

ferent notch configuration shown in Figure 3. These notches were cut tovarious depths with a 0.030 thick cut-off saw blade. The notch tips werethen sharpened with 0.003 and 0.008 inch diamond impregnated wire. Com-

pression tests were run on the specimen shown in Figure 4. These edge-notched rectangular bars were used for both unconfined and confinedcompression tests. The specimens used for the bend tests are shown in Figure 5.

In the tests under confining pressure the specimens were jacketed with

laboratory Tygon tubing. Since the analysis of the notched specimens underpressure assumes that hydrostatic pressure exists around the surface of thenotch as well as on the lateral surface of the specimen, care was taken to

ensure that this condition prevailed. This was accomplished by filling thenotch with an RTV rubber (Dow-Corning 732) that was soft enough to transmit

hydrostatic pressure, Trouble was experienced initially with the Tygonjacket being cut by the sharp edge of the notch flank as pressure compressed the

RTV rubber excessively. This was solveu by stiffening the notch filler

slightly by placing a neoorene rubber O-rina around the notch before filling theremainder of the void with the RTV rubber.

Test MethodsThe servo-controlled, electro-hydraulic triaxial testing system used

for previous tests (26,27 ) was emplcyed for both the tension, compression,and extension testing. The apparatus was adapted to tensile testing by con-

structing theapparatus shown in Figure 6. A proving ring type load cell wasdesigned and fabricated to provide the sensitivity for accurately measuring

the small loads encountered in testing brittle materials in tension. Straingages mounted on the ring provided the stress signal and calibration showed

the instrument was both linear and reproducible. The specimen was bondedtn metal end tabs using a filled epoxy cement. It was found that this jointwas stronger than the rock and no bond failures occurred in these tests.

The extension test apparatus is shown in Figure 7. This test was con-ducted by applying confining pressure to the specimen with sufficient end

load applied to maintain a condition of hydrostatic pressure. The end loadwas then -educed and since the closure piston diameter is larger than the

5

specimen diameter the axial stress in the specimen is reduced. This is

equivalent to superposing hydrostatic pressure on a tensile test. Some

initial difficulty was experienced in the extension test due tG fracture

of the extension piston. This was solved by using a high toughness

maraging steel piston. The axial load in the extension test was measured

with a bonded strain gage load cell built into the closure plug and located

inside the pressure vessel as shown in rigure 7. Pressure was measured

with a manganin coil, and all of the data were recorded on a pen-type Offner

recorder.

Alignment of the specimen in both tension and extension testing is acritical problem since bending must be avoided. In both tests the align-

ment problems were minimized by very carefully grinding the ends of the

specimens to ensure parallelism. Universal joints were used in the tensionlinkage to prevent bending loads being Induced by the test machine. The

same effect was accomplished in the extension tests by allowing slack in

the connecting linkage.

The torsion tests were carried out without confining pressure usinga standard laboratory Tinius Olsen torsion testing machine. A special

thin-walled steel tube torque cell of appropriate size was designed, con-

structed, and calibrated for these tests. Strain-gages were used as the

output sensors and data were recorded on a Houston Instruments Co. X-Y

recorder.

Bending tests were carried out on rectangular specimens 1/2 inch thick

by 1 inch wide by approximately 6 1/2 nches long. A four point loadapparatus was used as shown in Figure 8. A 10,000 pound Instron Tester

was used for these tests.

The compression tests were performed on edge notched rectangular speci-mens 1/2 inch thick by 1 inch wide by 2 1/2 inches long. These tests were

run both unconfined and with confining pressure.

III RESULTS

The result. from the tension, extension, torsion, and bending tests aredescribed in this section. The results presented refer in most cases to the

average maximum stresses in the specimen, and are based on the minimum cross-

sectional area of the notched section for the circumferentially notched

6

specimens. A consideration of the actual state of stress in the specimensis deferred until the next section. The sign convention employed is that

compression is considered positive.

Tension Tests

A summary of the fracture stress. for the direct tension tests is

given in Table 1. The results are given for both the 1 inch and 2 inchdiameter specimens, smooth and with three notch configurations, and for

the three rock types. A comparison of the smooth an6 notched specimennet-section fracture stresses shows that the strength of all three rocksare very insensitive to the presence of the notches. This appears to bea fundamental result, which will be discussed in more detail in the nextsection. The tension results are seen to exhibit considerably more

scatter than compression tests of these rocks. Although this iiay be aninherent material property, it undoubtedly also reflects the experimvntaldifficulties in direct tension tests. The tensile strength of the Tennessee

marble unnotched specimen is in good agreement with values reported byWawersik (25). Examples of the fracture surfaces are shown in Figure 9.

The results of preliminary tension tests run on Nugget sandstone with

a slightly different notch configuration are given in Table 2. Thesenotches were formed by sharpening the end of a 0.030 wide straight notch,as shown in Figure 3. The results are interesting in that several differentnotch depths were employed. It should be noted that the Nugget sandstoneused for these tests was obtained from a different location than that used in

all the other studies, and had a somewhat higher tensile strength.

ExtensionThe fracture stresses for the extension tests (i.e. tension under con-

fining pressure) are given in Table 3. The stresses for unnotched exten-

sion specimens of Westerly granite are shown in FigurelO along with previousresults by Mogi (22). It can be seen that the data are in good agreement,although in somewhat different ranges of confining pressure. Similar plots-For Nugget sandstone and Tennessee marble are shown in Figures 11 and 12. Thenet-section stress difference for the notched extension tests is shown inFigures 13-15. The effect of the notches is similar to that seen in the

tension tests, and will be analyzed further in the next section. The fractures

7

in all specimens appeared to be tension fractures, with the surfaces more or less

normal to the specimen axis. Typical examples are shown in Figure 16.

Torsion

Unconfined torsion tests were run on unnotched cylindrical specimens,

and specimens with either circumferential or longitudinal notches of the

configuraticns : ;-wn pr~viously. A summary of the resu.lts is given in Table

4. The nominal shear stress which is included has been ca'cu ed from the

usual formulaTc (4)

where T is the torqje, c is the distance from the neutral axis to the outer

surface, anc J is the polar moment of inertia of the cross section. For the

circumferentially notched specimens, the net section diameter has been used.

It can be seen that the failure torque is only moderately affected by thenotch; thus the nominal shear stress is actually increased. Photographs

of typical fractured specimens are shown in Figure 17. The spiral fractures

that result from tension fracture can be readily observed in both the smooth

and notched specimens.Beam Bending

Simple rectangular beam specimens were loaded in four point bending. Theresults are given in Table 5 O.hich shows the fai'ure load, calculated bending

moment, and nominal bending stress calculated from the formula

Mc (5)ab = I

where M is the moment, c is the distance from the neutral axis to the oLter sur-

face, and I is the moment of inertia for the cross section. The bending tests

were much more reproducible than the direct tension results, and exhlibited only

nominal scatter. The specimens app dred to break randomly in the central part

of the beam. As would be expected, the bending stress is considerably higher

than the direct tensile strength for each rock.

Sections were cut in the longitudinal directicn from, the central part

of some of the beams after fracture occurred. These sections were polished

and examined with a scanning electron microscope. Typical results are shown

in Figures 18-20. Significant defects were observed, primarily associated

with grain boundaries. However a comparison of t•e rock located on the tension

anO ccmpression sides, and also on the beam neutral axis shows no significant

difference.

8

Notched Compression TestsSeveral tests were carried out on rectangular (prismatic) compression

specimens with a side notch as shown in Figure 4 . These tests were pri-marily qualitative and were motivated by an attempt to induce a t ear orfaulting type fracture, as opposed to the tension fractures characteristicof all the other tests in this program. A typical result is shown in

Figure 'i for both unconfined and confined tests. The cracks in the rockcoecimens appear, however, to be tension cracks. It is not clear at this

time if a variation of this specimen configuration may besuccessful in pro-ducing the desired result.

IV. AMALYSIS OF RESULTSThe experimental results presented in the previous section show a

complex picture of rock behavior under conditions of stress gradients. Inthis section the interpretation of these results will be studied.

Fundamental to an understanding of fracture under conditions of stressgradient is a knowledge of the stresses in the test specimens. In generala stress analysis is needed since test specimens with stress gradients arebasically statically indeterminate. For specimens with reasonably simpleconfigurations, the stress analysis per se is not an extremely difficult

task, particularly as numerical techniques are rather generally available.One of the more successful nf these, the finite-element method, was used

extensively in this program and will be described subsequently.lhe question of the material behavior description to be used in the

analysis is P re difficult. It -is well known that rock is inelastic tosome degree depending on the rock type and state of stress, and the authorshave previously been involved in the development of constitutive equations

(i7) for rock inelastic behavior. This inelastic behavior is evidenced evenin uniaxial tension. The tension stress-strain curves shown in Figures 22

through 25 illustrate this. These curves, measured by means of strain gagedspecimens (18) show two features that are particularly interesting.First, the rock shows bulking (volume expansion) similar to that seen in com-pression, as evidenced by the lateral strain measurements. Second, hysteresis

and permanent set are seen on unloading, particularly in the lateral strain

measurements. Similar observations have been made by Wawersik (25). These

observations undoubtedly reflect micro-cracking and/or grain boundary sliding.I 9

In spite of these evidences of inelasticity, there is justificatiopfor using linear elastic theory in the test specimen stress analysis. First,

the non-linearity of the axial stress-strain curves is not large. Even

though one may intuitively believe that stresses are considerably relieved

by inelastic effects in the immediate vicinity of stress concentrations,

this evidence is not readily available as it is not manifested in overall

stress-strain curves. Second, even though it may not be possible to cal-

culate realistic stresses in the presence of say, a sharp crack, it has been

shown in some cases to be of great utility to use linear elastic solutions.

This is the cvse, for example, in linear elastic fracture mechanics where

the use of energy release rate or stress intensity factor can in many cases be

based on elastic stress solutions. Finally the relative ease of using

linear elastic theory both for the test specimens and in applications is a

significant factor. Or the basis of these remarks, the specimen stress

analysis is being carried out using both linear elastic and nonlinear-

inelastic theory. Linear solutions have been carried out and are presented

below and used in the analysis of results. The inclusion of inelasticity

in the stress analyses is currently under study and will be reported in

fiture work.

Finite Element Analysis

In order to carry out the numerical stress analysis of the cylindrical

notched specimens, an axisymmetric finite element computer program was used.

This program which was developed specifically for use in this research program,

utilizes quadrilateral elements which are made up of four separate triangular

elements. Within each triangle the displacements are assumed to vary linearly.

Results of analysis of several check cases have shown the program to be very

efficient as well as accurate. As previously mentioned, only linear elastic

analyses have been carried out for the test specimen configurations to date,

however work is currently proceeding on modifications to the finite element

program which will allow for nonlinear and inelastic behavior of the rock

material.

Finite element analyses have been performed for all six of the cylindrical

notched bar configurations. Advantage was taken of the symmetry conditions

which allow only one quarter of a bar section to be considered. The finite

10

7-l

element mesh used to represent the one inch diameter bar with a 1/8 inchradius notch is shown in Figure 26. Similar meshes were used to representthe other test specimen geometries with appropriate refinements made in

the cases of the notches with smaller radii.

Results of Analysis

The results obtained for the notched cylindrical bars are shown in

Figures 27-32. With this information the stress in both the tension and

extension test specimens can be determined. This is accomplished by multi-plying the stress difference in the test specimens by the appropriate stress

ratio from these figures, and then superposing the hydrostatic pressure forthe extension specimens. This superposition procedure of course depends onthe linearity of the stress analysis. The stress concentration factors

obtained from the finite element solutions are presented in Table 6.

As a check on the accuracy of the results a comparison was made with

the classic Neuber solution (29). This solution is for a similar configura-

tion with the exception that the notch is assumed to be hyperbolic and

"deep" so that the section away from the notch is of infinite diameter. Thusthe Neuber solution is appropriate for the specimens only in the immediate

vicinity of the notch tip. However, if errors were to exist in the finite

element results, they undoubtedly would occur where the stress gradient ishighest, i.e., in the vicinity of the notch. A comparison of the results

for the one inch diameter bars is given in Figures 27-29. These resultsshow the finite element and Neuber solutions to be in very good agreement

near the tips of the notches. Thus, confidence can be placed in the numeri-

cal results.

Fracture Stresses in Test Specimens

An apparent result giveinfa the previous section is that the notched

rock specimen net section fracture stresses were relatively insensitive to

the presence of the notches. On the other hand, this means that the maximum

stress predicted at the notch tip is sensitive to the stress concentration

factor or perhaps the stress gradient. This can clearly be seen in Figure

33 where the notch tip stresses are plotted vs. the stress concentration

factor. A similar effect can be seen in the bending tests, as the maximum

tensile stress in the bending specimens is on the order of 60 to 100 % higher

than in direct tension.

11

The differences between the three rocks can be seen more clearly in

Figures 34-36 where the net section stress is plotted vs. the stress con-

centratlon factor on log-log coordinates. The data have been fitted with a

least squares straight line; the slopes of the lines for the three rocks

are apparently different.

Critically Stressed Region

One of the methods used in the literature to understand fracture under

conditions of stress gradient is to postulate that fracture cannot propagateuntil the stress reaches a critical value over a finite-sized region. The

region may be a critical volume or perhaps a critical linear distance. The

size of the region may depend on the distribution and magnitude of thestress in the region. This latter problem is often approached by means of

Weibull statistics ( 8,30), involving the distribution of flaws in

the material. As discussed in the introduction, the use of Weibull statistics

has often been criticized on the basis that it postulates a "weak-link in

series" fracture mechanism, while for certain materials there is a somewhat

separate process of the initiation of micro-cracks and the final catastrophic

propagation. This has also been approached statistically as well as with more

strictly physical concepts such as stress over dislocation-related lengths.In view of the evidences of micro-cracking in rock specimens prior to final

fracture for both tension and compression conditions, it seems plausible toinvestigate the concept of a critically stressed length. This will be

approached first by considering only the maximum tension stress in the un-

confined tension and bending tests. Subsequently, considerations of the

triaxial stress fields will be made.

Tentatively the critically stressed region will be defined by two para-

meters; the size of the region in which the tensile stress exceeds the direct

tensile strength of the rock, and the maximum stress in the region. These

two parameters have bren plotted, one vs. the other, in Figures 37-39 on

log-log coordinates. Included in these plots are the notched 1 and 2 inch

diameter specimen tension results and the bending test results. The

plots in Figures 37,78 , and 39 are apparently quite scattered and the impli-

cations are not fully evident. To consider this further, it is useful to

consider what the limit could be as the stress concentrations are increased.

This will be discussed in the following.

12

[I Sharp Crack Fracture MechanicsAs the radius of curvature of the notch tip is decreased, the stress

distribution approaches that for a sharp crack. In this case the stress

distribution is well known ( 31) and the axial stress on the notch mid-

plane in the immediate vicinity of the notch can be written simply as

.K1a .. (6)

where r is the distance from the crack (notch) tip and K1 is a constant thatdepends on the load and geometry and is termed the stress intensity factor.As explained in texts on linear elastic fracture mechanics (9) the valje of K1

at the instant when the crdck propagates is regarded as a material property, andif the material is under conditions of plane strain is termed Klc, thecritical stress intensity factor for the material. The stress intensity factor

K1 is related to the geometry and loads by a stress analysis, either analy-tical or numerical (32,33). For example, the stress intensity factor for a

sharp crack of length 2c in an infinite plate is given by (32).

K1 = ao - (7)

where ao is the applied stress at the plate boundary (away from the crack).The stress analysis for a circumferentially cracked cylinder is available

(34) and has been also compared to numerical solutions performed by theauthor (33). The analytical solution t;kes the form

I = Onet f(k.)4/F (8)

where anet is the net section axial stress, d is the specimen diameter atthe net section, D is the outside specimen diameter, and f(b) is a numerical

factor. Thus Klc can be easily calculated from tests of cracked cylinderspulled in tension.

The 0.0015 notch tip radius specimens were designed to simulate sharp

crack conditions; this represented the sharpest tip that could be reproducibly

13

produced using the methods described earlier. Whether or not this tip

radius is sufficiently small is open to question and should be investigated

further. Taking the position that further sharpening of the notch tip

would not affect the apparent strength, K1c values can be calculated fromthe previous equation. These are shown in Table 7.

The K values can be related to the fracture energy for the mAterialic

by the use of the well knuwn Irwin formula (35)

(1- 2) fE = Gc = 2yF (9)

where YF is the fracture energy, which includes but is usually much largerthan the surface energy specified in the original Griffith fracture criterion.

Using the average Klc values, YF has been calculated and is shown in Table 8.

Also shown ate values presented in the literature (25,36-40). The com-parison with the previous value of YF obtained for Tennessee marble byWawersik (25) is very close, thus lending some confidence in these results.

It should be cautioned that although the present values look reasonable,

the extremely close agreement is probably fortuitous.

It is interesting to examine the tests on Nugget sandstone (termed

Nugget sandstone II) presented ip Table 2 in terms of Klc factors. This

sandstone camd from a different block than that used in the rest of the

program, and had slightly higher tensile properties although apparently the

same compressive strength. A number of tests of notched specimens were run

on three ditferent notch configurations and four different notch depths. Alt~o:,!O'

somewhat scattered, the results appear to be insensitive to the notch tipradius. Also, if Klc were a material constant independent of notch size,

the net section stress for specimens with different notch depths would fol-

low the prediction of equation 8. A comparison of the net section stress

in the specimens with that predicted on the basis of equation 8 and an

average value of Klc equal to 280 psi v'iiT is shown in Figure 40. Although

again the scatter is large, the results could be interpreted to be following

the predicted variation.

14

" r m •• m

The concepts of Klc and fracture energy discussed above interact withthe concept of a critically stressed distance in the following way. Considerthe critically stressed distance vs. maximum stress plots shown previouslyin Figures 37-39. As the maximum stress is raised, the distance8 over which the streý.s must be raised above the tensile strength decreases.

This distance does not approach zero as the stress gets very large, however,

but instead approaches a minimum limit. This can be seen by consideringa sharp crack stressed just below the critical stress. As can be seen fromequation 6 , the predicted stress is infinite at the crack tip, however, theregion a can be obtained from the stress distribution around the crack. Asa first approximation this distance can be obtained from equation 6 byconsidering r to be equal to 6 when the stress is equal to the tensile

strength. Thus

min = 1 Ki-)2 (10)

and this establishes a minimum value for 6. This expression is approximateas equation 6 only holds for small r. It is interesting to note that a

sinilar expression is commonly used in fracture mechanics for estimitingyield zone size, with GTS replaced by the yield strength (9). These mini-mum values of have been added to Figures 37-39.

15

Multiaxial Stress Effects

It is possible to generalize the concept of a critically stressed

region discussed above to include multiaxial stress effects. To do this

it is necessary to consider a function of the full stress tensor as defining

the boundary of the critically stressed region, instead of merely the

maximum principal stress. A number of possibilities exist for defining

this function of the stresses, such as the classical Coloumb-Mohr criteria

and others discussed by Jaeger and Cook (5). Other possibilities exist,

as for example those discussed by Mogi (28) and Wawersik (2). Rather

arbitrarily a uniform stress criterion was selected that is based on the

octahedral shear stress and octahedral normal stress. To establish the

specific form of the critericil to be used, the fracture values in the

unnotched tension and extension tests were employed. The plots could be

fitted with a straight line as

Westerly granite: F = /JA2, - 0.310 Jl - 1.75 ksi (11)

Nugget sandstone: F = VJ 2 , - 0.346 J1 - 0.80 ksi (12)

Tennessee marble: F = VJ-2, - 0.33 J, - 1.05 ksi (13)

where fracture is assumed to take place when F=O. The stress invariants are

related to the octahedral stresses and are given by

J = 1/2 Sij S i (14)

l ykk (15)

where Sij is the deviatoric stress and the summation convention is employed.

It has been established (21,28) that the above criteria will not pr.dict

failure in both triaxial compression and extension experiments. In -he

present case however only extension type fractures are involved and thus

the above limitation may not be serious.

A material distance 6 was then defined for each specimen tested with

nonuniform stresses, including the notched tension and extension one inch

diameter specimens, the unnotched torsion, and the bending test specimens.

The distance 6 was defined as the radial distance in the cylindrical speci-

mens over which the stresses at the midplane at fracture exceeded the value

F=O that defines fracture for conditions of uniform stress. Similarly 6

16

v a3 defined in the direction of the beam thickness. Thus the width of the

beams and the circumference of the cylinders was ignored.

A variation ir 6 was observed similar to that seen previously for

uniaxial stress. In an attempt to correlate this variation, the values of

6 were plotted versus the peak value of F at the notch tip or specimen

outer surface. These plots are shown in log-log coordinates in Figures 41-43.Although the plots exhibit a great deal of scatter, general trends can

be observed.

The plots of Figures 41-43 are in general quite sensitive to the

experimentally determined net section stresses. Changes on the order of 25%

or less in the experimental values would make all the points fall in on a

straight line, with the exception of the unconfined tension and torsion data.

This may well stem criom the original choice of the uniform stress criterion

for F. Further worK on fracture in mixed biaxial tension--compression

is needed to answer this question. Systematic difference may be pre-

sent in the data, as the tension ard torsion results are somewhat dif-

ferent.It should be pointed o-.t that Figures 41-43, although scattered,

encompass an extremely wide range of experimeintal variables. The stresses

range from triaxial tension to compression, and exceed the allowable stress

for uniform stres; conditions by large factors in some cases. Thus, it is

significant that the empirical criterion illustrated in Figures 41-43 can

successfully correlate the fracture strength.

V. DISCUSSION

One of the fundamental results observed in the experimental data is

the insensitivity of net section fracture stress to stress concentrations.

This insensitivity has been observed in other materials such as concrete (12),

cast iron (14), ceramics (11), and to a lesser extent in plastics (13, 39),

steel and aluminum (41). It is widely believed to be d& to micro-structural

defects and inhomogeneity. As a result of this insensitivity of fracture

to stress concentration, linear elastic theory predicts st.-esses in test

specimens under conditions of stress gradient that are greatly in excess of

17

the fracture stress under uniform stress conditions. Undoubtedly part ofthis effect is due to the assumption of linear elasticity in the stress

calculation, and this assumption will be investigated in future work.

However, unlike compression results, tension stress-strain curves of rocks

and many of the other materials mentioned above are not greatly nonlinear.

Thus, macroscopically observable st.,ess-strain behavior will no4, furnish a full

explanation of the high predicted stresses under nonuniform str!ss conditions.

This observation of increased allowable stress under conditions of non-uniformstress is commonly observed, as for example in the familiar result that the

bending strength of brittle materials is higher than the direct tensile

strength.This increase of apparent allowable strength under conditions of

non-uniform stress has practical implications. For example, rock cutting

inevitably involves concentrated loading. According to the test results

described ezrlier, a rational design of a cutting tool based on rock

fractu relingth must be based on both exe magnitude of the stress and the

distance, or perhaps volume, of the stressed area.A preliminary correlation of the experimental data has been suggested

in this report through the concept of a critically stressed distance that

varies as a function of the maximum stress. An extremely large range of

variables and test conditions has been brought together through this

relationship. The plots of this relationship shown in Figures 41-43 are

seen to be quite scattered: however changes in the experimental data on the

order of 25%, or about 50% for the torsion data, would establish a smooth

relationship. Thus, a useful, if ad hoc, relationship seems to have been

achieved. Systematic differences in the test results mnay exist however, and

it is hoped that future work may provide a more fundamental explanation of

non-uniform stress fracture.

A possible method of connecting "sharp crack" fracture mechanics theory

to fracture under conditions of lower stress gradient has been suggested

through the critically stressed distance approach. The critically stressed

region was seen to dccrease with increasing stress gradient, but is postulated

to reach a minimum that could be approximated by

•min "I's (10)

18

The stress intensity factors (K lc) could be obtained by testing cracked

specimens. Values were calculated based on the assumption that the 0.0015radius notched specimens could be treated as sharply cracked, and good

agreement was reached with fracture energy values available in the litera-

ture, particularly with the data of Wawersik (25). However, the possibility

exists that a sharper notch tip may give different values.

Weiss et al (11, 41-43) have studied the effect of material inhomogeneities

on fracture, and rave observed in other materials some of tie features noted

here for rocks. They have developed a rathepr simple model based on a

uniform distribution of flaws as illustrated in Figure 44. Simplifying

assumptions lead to an equation for fracture of notched specimens as

'Ts yF + 4x (6anet Kt r (16)

where Kt is the cxterior notch stress concentration factor, r is the notch

radius and x is the distance from the notch tip to the inte'nal flaw. If

the flaw spacing is b, an average value of x is b/2 which gives

n Ts • (17)•net K Ktr

which immediately shows that for b=O (homogeneous material) the notch has

its full effect so that

n Ts (18)Onet -Kt

while for finite b/r (inhomogeneous material) a lower notch efect is seen.

It seems likely that a refinement of this approach may furnish an explanation

for the experimental data.

Both critically stressed distances and critically stressed volumes

have been employed in the literature in stress-gradient fracture, as well

as total stressed volume as in the Weibull theory. The difFerence betvieen

these approaches is subtle and it is difficult to justify a choice strictly

on the brsis of the present data. The critical distance concept has been

often employed in metal fatigue, primarily on the basis that a stressed

distance is associated with crack growth. On the other hand, the Weibull

approach is based on a "weak link in series" concept, and thus may appear

19

to be more suitable for brittle materials. However, apparent evidence of

microcr ck growth in tension well before final rupture were reported here

and elsewhere. This clearly is an area for future work.

VI. SUMMARY AND CONCLUSIONS

A test program has been carried out on fracture under conditions of

stress gradient on Westerly granite, Nugget sandstone, and Tennessee marble.

A wide variety of laboratory test methods were used including tension,

extension, torsian, beam bending, and compression. Stress gradients were

introduced into the tension and extension test specimens by means of

notches. The results were analyzed by means of finite element solutions.

The notched specimen net section fracture stress was found to be

insensitive to the notch configuration for all rocks. However, the local

stresses in the specimens were much higher than that required to cause

fracture in uniformly stressed specimens. This result is typical of many

materials, and is evidenced by bending strengths being higher than direct

tensile strength in many brittle materials. This apparent icrease in

ctrength is undoubtedly important in rock me, hanics problems involving

stress gradients.

A preliminary correlation of the increase in strength exhibited by a

specimen in a non-uniform stress field is presented based on a critically

stressed distance that varies as a function of the maximum stress was

presenteO. Although the data were scattered, and systematic differences

may also exist, this concept correlates a very large range of variables

with reasonablF accuracy. The minimum critically stressed distance was

dppl'oximated by the use of fracture mechanics concepts. This correlation

is presented as a preliminary result and will be refined as a part of a

research effort currently in progress.

20

Igloom

REFERENCE'.

1. Cook, N. G. W., "Analysis of Hard-Rock Cuttability for Mactidnes,"Rapid Excavation - Problems and Progress, The American Institute ofMining, Metallurgical, and Petroleum Engineers, Inc., New York (1970).

2. Cheatham, J. B., and Gnirk, P. F., "The Mechanics of Rock FractureAssociated with Drilling and Depth," Proc. Eight Sym. Rock Mech.:Failure and Breakaqe of Rock, Port City Press, INc., Baltimore, 410-429 (1967).

3. Cheatham, J. B., Paslay, P. R., and Fulcher, C. W. G., "Analysis ofthe Plastic Flow of Rock Under a Lubricated Punch," J. Appl. Mech.,35, 87-94 (1968).

4. Pariseau, W. G., "Wedge Indentation of Anisotropic Geologic Media,"Proc. Twelfth Sym. Rock Mech., 529-546 (1970).

5. Jaeger, J. C., and Cook, N. G. W., Fundamentals of Rock Mechanics,

Methuen and Co. Ltd., London (1969).

6. Brown, E. T., "Strength-size Effects in Rock Material," MineralResources Research Center, Prog. Report 24, University of Minnesota,138-150 (1971).

7. Hodgson, K., and Cook. N. G. W., "The Effects of Size and StressGradient on the St-ength of Rock," Proc. Second Cong. Int. Soc. RockMech., 2, 3-5 (1967).

8. Weibull, W., "A Statistical Theory of the Strength of Materials,"

Ingeniors Vet'ýnskaps Akademien, Handlingar, No. 151 (1935).

9. Tetelman, A. S., and McEvily, A. J. Jr., Fracture of Structural Mate-rials, John Wiley and Sons, Inc., New York (1967•).

10. Hasofer, A. M., "A Statistical Theory of Brittle Fracture of Steel,"The International Journal of Fracture Mechanics, 4, No. 4, 439-451 (1968).

11. Weiss, V., Chait, R., and Sessler, J. G., "Fracture of Ceramics," Pro-ceedings of the First International Conf. on Fracture, Sendai, Japans,3, 1307-1320 (1965).

12. Wright, W., and Byrne, J. G., "Stress Concentrations in Concrete," Nature,203, 1374-1375 (1964).

13. Durelli, A. J., and Parks, V., "Relationship of Size and Stress Gradientto Brittle Fracture Stress," Proc. Fourth U. S. National Nat. Cong, Appi.Mech., 2, 931-938 (1962).

14. Ouvinall, R. C., Stress, Strain and Strength, McGraw Hill Book Co., New14. York, p. 231 (1967).

21

15. Gilman, J. J., and Tuler, F. R., "Dynamic Fracture by Spallation inMetals," Int. Journal of Fracture Mech., 6, No. 2, 169-182 (1970).

16. Peck, J. C., Berkowitz, H. M., and Cohen, L. J., "The Relationshipof the Critical Stress Versus Impulse Theory of Spall Fracture to theStress Gradient Theory," Int. Journal of Fracture Mech., 5, No. 4,297-303 (1969).

17. Brown, W. S., and Swanson, S. R., "Constitutive Equations for WesterlyGranite and Cedar City Torgalite for a Variety of Loading Conditions,"Final Report, DASA-2473, University of Utah, March 70.

18. Brown, W. S., and Swanson, S. R., "Stress-Strain and Fracture Propertiesof Nugget Sandstone," Final Report, AFWL-7R-71-54, University of Utah,March 71.

19. Swanson, S. R., and Brown, W. S., "The Influence of State of Stress onStress-Strain Behavior of Rocks," ASME Paper No. 71-Met-M (1971).

20. Brown, W. S., Swanson, S. R., and Wawersik, W. R., "Influence ofDynamic Loading, Biaxial Loading, and Pre-Fracturing on the Stress-Strain and Fracture Characteristics of Rocks," Final Report, DASA-2713,University of Utah, March 71.

21. Brown, W. S., Swanson, S. R., and Wawersik, W. R., "Further Studies ofDynamic and Biaxial Loading of Rock," Final Report, UTEC ME 72-015,University of Utah, in publication (19727.

22. Wawersik, W. R., and Brown, W. S., "Creep Fracture in Rock in UniaxialCompression," Final Report, UTEC ME 71-242, University of Utah, December1971.

23. Brace, W. F., Paulding, B. W., and Scholz, C., "Dilatancy in the Fractureof Crystalline Rocks," J. Geophys. Res., 71, 3939-3953 (1966).

24. Birch, F., "The Velocity of Compressional Waves in Rocks to 10 Kilobars,Part l," J. Geophys. Res., 65, 1083-1102 (1960).

25. Wawersik, W. R., "Detailed Analysis of the Rock Fracture in LaboratoryCompression Tests," Ph.D. Thesis, University of Minnesota (1968).

26. Brown, W. S., and Swanson, S. R., "Influence of Load Path and Stress onFailure Strength and Stress-Strain Properties of Rock," Technical ReportAFWL-TR-70-53, University of Utah, April 70.

27. Swanson, S. R., and Brown, W. S., "An Observation of Loading Path Inde-pendence of Fracture in Rock," Int. J. Rock klech., Min. Sci., 8, 277-281 (1971).

28. Mogi, K., "Effect of Intermediate Principal Stress on Rock Failure," J.Geophys, Res., 72, 5117-5137 (1967).

22

29. Neuber, H., "Theory of Netch Stresses: Principles of Exact StressCalculation," J. W. Edwards, Publisher, Inc., Ann Arbor, Mich., (1946).

30. Weibull, W., "The Phenomenon of Rupter in Solids," Ingeniors VetenskapsAkademien, Handlinger, No. 153 (1939).

31. Williams, M. L., "On the Stress Distributions at the Base of a StationaryCrack," J. of Applied Mechanics, 109-114, (1957).

32. Paris, P. C., and Sih, G. C., "Stress Analysis of Cracks," FractureToughness Testing and Its Applications, ASTM STP No. 381 (1965).

33. Swanson, S. R., "Finite Element Solutions for a Cracked Two LayeredElastic Cylinder," Engineering Fracture Mechanics, 3, 283-289 (1971).

34. Williams, J. G., and Constable, I., "Some Observations on Klc Determina-tion for Round Bars in Tension and Bending," Int. J. of Fracture Mech.,5, No. 2 (1969).

35. Irwin, G. R., "Crack-Toughness Testing of Strain-Rate Sensitive Materials,"Trans. ASME, J. of Eng. for Power, 86, 444-450 (1964).

36. Brace, W. F., and Walsh, J. B., "Some Direct Measurements of the SurfaceEnerg of Quartz and Orthoclase," The American Mineralogist, 47, 1111-1122(1962).

37. Santhanam, A. T., and Gupta, Y. P., "Cleavage Surface Energy of Calcite,"Rock Mech., Min. Sci., 5, 253-259 (1968).

38. Bieniawski, Z. T., "Fracture Dynamics of Rock," The Int. J. of FractureMech., 4, 415-430 (1968).

39. Perkins, T. K., and Bartlett, L. E., "Surface Energies of RocksMeasured During Cleavage," Soc. Pet. Eng. Jour., 3, 307-313, (1963).

40. Perkins, T. K., and Krech, W. W., "Effect of Cleavage Rate and StressLevel on Apparent Surface Energies of Rocks," Soc. Pet. Eng. Jour.,6, 308-314 (1966).

41. Weiss, V., Schaeffer, G., and Fehling, J., "Effect of Section Sizeon Notch Strength," ASME J. Basic Eng., 88, 675-681 (1966).

42. Weiss, V., "Application of Weibull's Statistical Theory of Fractureto Sheet Specimens," ASME paper No. 62-WA-270 (1962).

43. Weiss, V., and Yukawa, S., "Critical Appraisal of Fracture Mechanics,"Fracture Toughness Testing and Its Applications, ASTM STP No. 381 (1965).

23

Table 1. Tension Test Results

Net

Specimen No. Notch Section Stress-Psi Nominal Dia.-in.

6 Smooth 1260101 Smooth 1190 3/4

115 Smooth 1935 1116 Smooth 1745 134. 1/8 1470 2

2 1/8 1256 1

99 1/32 1090 1

1-2 1/32 1370 2

4 .0015 1196 1

2-2 .0015 1340 2

7 Smooth 518 112 Smooth 455 113 Smooth 700 1

14 Smooth 706 1t 102 Smooth 830 3/4

11-2 Smooth 732 2

117 Smooth 733 1

6-2 1/8 576 2

8 1/8 527 1

88 1/32 635 1

4-2 1/32 553 2

110 .0015 535 1

ill .0015 442 1

5-2 .0015 538 2

Marble

89 Smooth 885 3/4

90 Smooth 735 3/4

91 Smooth 928 3/4

12-2 Smooth 1260 2

118 T-17 1013 1

9-2 1/8 1260 2

100 1/8 1310 1

7-2 1/32 1110 2

95 1/32 1140 1

96 .0015 1240 1

8-2 .0015 1060 2

24

II

Table 2. Tension Tests for Nugget Sandstone, Block 2

Diameter atSpecimen Nominal Notch Notch Net Section Outside Notched

No. O.D. Radius Depth Stress-psi Diameter Sectionin. in. in. D-in. d-in.

0-47 1 Smooth Smooth 1310

0-58 1 Smooth Smooth 1110

0-59 1 Smooth Smooth 1270

0-60 1 Smooth Smooth 1120

0-9 1 Smooth Smooth 1070

0-10 1 Smooth Smooth 1500

0-11 1 Smooth Smooth 1260

0-61 1 .0015 1/16 695 0.972 0.850

0-62 1 .0015 1/16 666 0.960 0.826

0-63 1 .005* 1/8 691 0.940 0.723

0-64 1 .005* 1/4 697 0.950 0.452

0-65 1 .0015 1/4 757 0.970 0.475

0-66 1 .0015 1/8 564 0.920 0.682

0-67 1 .0015 3/16 592 0.971 0.586

0-68 1 .0015 3/16 556 0.906 0.535

0-69 1 .004 1/4 726 0.975 0.497

0-70 1 .004 1/8 622 0.973 0.720

25

Table 3. Extension Test Results (All 1")

Westerly Granite Nugget Sandstone Tennessee Marble

Notch Confining Spec. Net Section Spec. Net Section Spec. Net SectionType Pressure- ksl No. Stress-psi No. Stress-psi No. Stress-psi

Smooth 5 113 7760 114 759052 6145 50 5603 112 6610

10 84 11800 98 12300 109 1170051 10250

15 11 1331520 85 22900 97 22900 106 2330030 86 30800 94 31800 105 31800

1/8 5 8 6120 13 620016 6990

10 38 12500 14 10900 60 1180020 41 18900 7 18400 59 2160030 39 28200 43 31100 58 30000

53 26800 44 2600045 28800

1/32 5 28 5170 27 6390I 10 18 10500 55 11300 64 1060022 11000

20 77 18500 80 19400 63 2030024 18100

30 78 21700 81 26000 62 2260025 25300

22.2 21 24100

0.0015 5 15 6100 9 474010 48 11500 ?5 10900 68 11100

54 13680 75 969032 11200

20 10 16800 36 20500 104 2150033 17400 67 9240

30 76 21400 37 29300 66 2260034 22200

26

ILO 00.CV

S- 4J U C%J w~r U) 0.0

xC? Oý MC%J

4LJ 0M-c~'

(I) U 3 3 ) 0 C

.5 000 00 I ~l4A 4.) I-- C3 e CM OD a..-

4J C) , Z -

w) 4.) -z t C 'C~j rIcn

U) M

4.) ul )W 0 rI- nfl7% C) l

.0ý LOu 0 C'JO

c4. gli C) L Ol~. a

o)) LfO LO t

4.)

V)

CJ C'.J LO

0 con (y C 0

4JU

a'3 4-

0L.4- 0 u4-U)_ 0 S-C r_

o E W.0 0-

27

Table 5. Bending Test Results

Rock Type Spec. Specimen Max. Bending BendingNo. Dimensions Load Moment Stress

b=in. h=in. lbs in-lb psi

TennesseeMarble B-1 1.003 0.497 104 78.0 1890

B-2 1.010 0.538 128 96.0 1970

B-3 1.005 0.510 117 87.7 2014

WesterlyGranite B-4 1.050 0.564 286 214.5 3853

B-5 1.058 0.508 222 166.5 3659

B-6 1.047 0.493 197 147.7 3484

B-7 1.050 0.497 210 157.5 3644

NuggetSandstone B-8 1.025 0.535 121 91.8 1877

8-9 1.018 0.520 108 81.4 1775

B-10 1.027 0.522 127 95.1 2040

B-li 1.026 0.753 240 180.0 1856

28

Table 6. Stress Concentration Factors Obtained

from the Finite Element Analyses of the Notched Cylindrical Bars

Notch Radius Kt, Axial Stress Concentration Factor(inch)

1" Diameter Bar 2" Diameter Bar

1/8 2.06 2.52

1/32 3.58 4.40

0.0015 14.66 18.03

29

.a

Table 7. KIc Values Calculated from

0.0015 Tip Radius Specimen Tests

K (K*Rock Specimen No. lc Zn IcTS

psi Fi-nch

Westerly granite 4 545 .0118

Westerly granite 2-2 769 .0235

Nugget sandstone 110 244 .0155

Nugget sandstone 111 202 .0106

Nugget sandstone 5-2 309 .0249

Tennesse marble 96 565 .0320

Tennesse marble 8-2 608 .0371

30

06 ~ ' CU CLC S

to 9 'U. L 0 ' m 41fm S- fa S10 -

r- 0.0. r.0-4 GOCV

w ~ ~ ~ ' S 'US. S- 6 0 c par - r - r - r -

ri 4 - 1.- 2: 41:3 V *... ~ %

(9 -'US S-S $- % % - -

r- 222 CV)~ (n) ýn M'Ua c 0 m

4J 4J +1 r_. r_ r_ C- r_ C_ C CM1 go'JIJ% -he- 'U w m m m

C C4- to tn (A

4J 4C 4C 0 0 0 41 'U 'U 'U C C C C C

0 41 di ) - u- S- $-. 41 r- C *- *e ** *.* *q *-

to (A 0 4141410 w 0 US$.- 41 3 u m.N u - -Z -x)d b> . 001of1 333 x 4xto0 c W141L. S- S- L. - S- S- ..-

Ul-

S-

0 Cý 4-'C a Dr .0 L)C CMC

j%; 1.0 0) C%;f C;' ; 8 ;C 61 ;

4-) Incc r 0

_ 4-'

'U +j CrD 0- c+ 0 WnIn- CO I4-)0 00a *' .0 .0 +) 4- 0 -o In C 0 4-' 41In I

m- w4. to -0u~C 4-) (n E541414-1 'U "a 'U 'U -C U 'U C I 1 ES'U D C_ U1 r- CV) V) (0 0 E j.-Jr

'U CD 41 (D 41 V) to Le) Cr- .J>))/ 41 41 - 4-i 1) 041 g '.-.l 4)

In n to r- to v 0 C n (m~ (n 'US. 4. (Al W 0 N Q 4J N 4) C M ( " 0 c 'US- c(D ( D a 4-' o .,4-4- -' +) 0 S.. .0 4)1 .-r- 41 W 't

4-) t~l C- S. $-- X~ US. f'- N 4A 0 0 CL U)4-) 4-) 0 --In cm C- c U 'Ur 4-'' M S- -s 0 S- - In S- M "

(D = o) 4U - ~ S- = 0 Sa- 0 0 00 tv WU 41

31

1 /8 in. 1/ in. 1/q in.

Notch Raji us Notch Radius Notch Paraius

= 0.0015 in. 1/32 in. = 1/8 in.

Figure 1. Notched Rock Specimens

32

SI

1/8 radiusnotch

FIGURE 2. Torsion specimen with longitudinal notch.

1Notch depth and,_..notch tip radius

I •---1 variable.SNotch width =0.030

2½

FIGURE 3. Notched specimens used in Nugget sandstoneBlock II tension tests.

33

t2

1

9. ,

2

FIGURE 4. Notched compression specimen.

FIGURE 5. Bend test specimen.

34

• • .... . . ... .

urfvjl irn T ioa-'i cell,

Connector bolt

Metal end tab

Rock s-ýezi-jen

U joint

1u--e 6. Schemitic of Tensirr Tet pý rtu

3 5

Loading pist~on

Wedge an.' 0-rinq seal

Pressure vessel

Steel end-tab

Ty'gor tubinr jacket

RTV fiflld notch

Rýck srecivnen

Sýteel end-tafr bondei torock specirlen

Strain-a~iqed load cell

Cone and seal

base plup

[Vpctrical leads

Figure 7. Schewmatic C exttensionr apparatus

36

Steel cylinder

Rock beam specimen

Steel bars

FIGURE 8. Beam bending test apparatus.

37i _____________________

V &At4t A.

Westerly Granite No. 2, 1/8 Notch

Nugget Sandstone No. 1, 1/8 Tennessee Marble No. 89,Notch Unnotched

FIGURE 9. Fracture surface of tension specimens.

38

V

50

0 Univ. of Utah

40 0 Mogi (Ref. 28)

U 0230 !a

20 002

10'4-

0"U 8

0 10 20 30 40 50

Confining Pressure, a1 (KSI)

Figure 10. Tension and Extension Test Results for

Westerly Granite (Unnotched)

39

*

0M 30

v0' 20

00

~ 0

In

CU

0 10 20 30

Confining Pressure, a, (KSI)

Figure 11. Tension and Extension Test Results for

Nugget Sandstone (Unnotched)

40

, 30

U)0V)

20

10

20 103

U•

Confining Pressure, a, (KSI)

Figure 12. Tension and Extension Test Results

for Tennessee Mlarble (Unnotched)

£4.A.

49-

Q)4-3

c.4JC; 0

5-

00

cli C) 4- 4-)a

00~U .y C t

004 0

5--

CDU In 0

01 4 CDU) 4

m~ 00 a

O4'8 OaJ! SSJI LeM U)43S

4-jO42

CV)

4)

4,

LO 0(Nj 4 A

Inr-

4-)

4.)4,

U)

j4-)

.C *r 0

F U) 0

LL

00~0 m C3 a

4-U)

.C 00

DO LO A

Ck Q -0

ooKN -a)aqjpScZJLPU~:a a

43 U

p - - - ---- -- - -- -- -

Ud)

En

I.1

13 04 1-

13

d)

-he4-) 0

(A 4-)

U4 4.)

.4-) 0 -4

00

C)f

0 -.0

C U -

000 d0

0o 0 0

44

Tennessee Marble No. 59, 1/8 Notch

'Oes r 8dU11ae COpY*

Westerly Granite No. 51, Nugget Sandstone No. 45,Unnotched 1/8 Notch

FIGURE 16. Fracture surface of extension specimens.

45

*Ind1I 0

R P6po u,

~~~46

tot

0

LO E

CL

CL) CL

C 0

4A4

C))

* U)

E

C6

A, S nLD( -, - LA (/

47 '

"00.

4,)

44.

4)0

4-)

0

03

I-

4)

4)

C41r U_

Eu

4.)

484

LA0

100

c'Ji

4-~~0Ln

0

0 0

0

4J

C

'U

L.0

490

& ý -4 W W WUi

Rero70duied fbl'romlOPY

[iNl?'%: IjP

-Q4-

* 4-)

(1)

.4-)

4J)

0.0

L-o

500

affimbz

-800c3 C1

I -6004J

0w

S' -400I--

.-

x

-200

0i I I I I I

-2 0 2 4 6

Strain (x 104) in/in ExtensionFIGURE 22. Principal stress-strain curves for

tension test of Westerly granite.(From Reference l1)

51

-1400

A D

-1200 B C

-1000

-800

-600 Specimen Lateral AxialC Number Stra i n Stra i n

No. 59 A D

-400 No. 60 B C

-200

0 . I I I I

-2.5 0 2.5 5.0 7.5 10.0

Strain (x 10-) in/in Extension

Figure 23. Principal stress-strain curve for tension test of Nuggetsahdstone specimens. (From Reference 18)

52

LLJ 0

4-J

Ln C)

*u- Wa

riw -i

C:) LO Cl

w CO

LO ~

1 ~ 4-)4-Dt -A

C)6

LA~S E .

0

Cii

C) CD 0 CD C

f~~i L4-t n n ~

[Sd Isa'Als IP,34

53o

I4(A

C%j -I

Ch

U;'

4-D

C) t

4,

4-

0~0C

C%j

+ 0n

0o c

C) Cm co 1EG) uG

Li) OUL CA0 )

CO)

'ULI) C) C

LIP)

!sd + Lo)

54-

~zt

1.0

0.9

0.7

0.6

0.5

* 0.4

0.3

0.2

0.1

0

0.1 0.2 0.3 0.4 0.5

Fijure 26. Hinite-elerent Grid tor )/8 in. Radius Ncthe.,

Test Srecimen.

55

jl"Ji~lfjlii l r...... ,

2.5

2.0 NeuberFinite-element:

0-0 z

AOr1.5

.5

0 0

0 0.1 0.2 0.3 p /I

R

-icure 27. Conpat.ison of Finite-elemeflt Res'I ts with

Neuber Sol':t'on in Vicinity 0 in. Radius

56

4.0

Neuber

Finite-element:3.0

0O- OZ

A- or

a/GNOM 13 - ceo

2.0

I!

1.0

0 0or 1 or2 'o7 _•p : 1/32"

Figure 28. Con.Darison of "-inite 21ement Rcs',ilts with

Neuber Solution in ?'icinit,, o 1/32 in. Radius

Nrotch

757

15

- Neuber

10)

"CFnom

0 •",P =.0015"

0 0.1 0.2 0.3

R

Neuber-o iS" r

2

Onom

IR

FIGURE 29. Comparison of finite-element results with Neubersolution at mid-plane for 1 in. diameter specimenwith 0.0015 in. radius notch.

'• 58

o <-

oo

uo~C 4J)

o•o'U

E,"4

0 r*- to U

Sa..ý

00 -60.)0UE

4,J ,-C

P'UI

0

0

t4Y

-- o

-~ 5E

LLJ

•j5 9

H~0 0 00D

0 000 "J

4ECA

o ~:i

0r

W -4

0 E00

Winr-~ a)

o 4- (0

4 -)

0 Ui C14

CD Nb b 0

LflU

00

20

0

10- 0

anom 0

0

5 00

S000 0 0 0 0)•-0 .. p, I .0015"

0 0.2 0.4 0.6 0.8

R

43 Q] -a8

2 2

nom

1

l " -•--p = .00]W,0.2 0.4 0.6 0.8

R

FIGURE 32. Elastic finite-elerent rid-rlane stresses for 2 in.diameter specimen with 0.0,c15 in. radius notch.

61

F 0 Westerly granite

& Nugget sandstone

24 03 Tennessee marble

20 2009B

0

* 16

0)

S12

8 AA

00

0 1 4 8 12 15 20

Kt Axial Stress Concentration Factor

FIGURE 33. Notch tip tensi e stress in tension test specimens.

62

_'M u g ",

0 lin.

S2 in.

3.3C1

C-. 3. 2

a-3.1 0

% 3.0I"-

0.0 .2 .4 . 1.2

Log1 oKt

FIGURE 34. Effect of stress concentration factor on net-section stressin notched tension tests of Westerly granite.

1 in.3.0 I-I2 i n.

";'•c_2.8 - 00 [

El

S2.7 -* 04.) 2.6

L

o 2. 1.2 1.a_ 0., .2- . •

Log 1 oKt

FIGURE 35. Effect of stress concentration factor on net-section stressin notched tension tests of Nugget sandstone.

63

41

o 0CC

4~)

0co

o

a,

4-)

rCA

0w

00

tn o

4J 00.-,

4-) r-

ow

C)U) -

Cý Cý C0co m *4 a,

LS" LeAO i )O

6.).

0 2 in. fiireter notckhed tensile

o 1 in. dianeter notched tensile

0 .ea,' BendingH K 25.2 rn= K()

I min aTS

4.8

4.4 E

b Ia 4.0

0

3.6El

3.2

2.8

-. 4 -. 8 -1.2 -1.6 -2.0 Log 10 6-in.

Figure 37. Critically stressed distance for non-uniformstresses in Westerly granite.

65------

0 2 in. dianeter notched tensile0 1 in. diameter notched tensile

SBeam Bending

4.8 m6min : (TS

4.4

4 .0 -

00

-j 3.6-

ETSmI

oI

3.2 0

oI

I

2.8 I I-. 4 -. 8 -1.2 -1.6 -2.0 Log 10 6-in.

Figure 38. Critically stressed distance for non-uniformstresses in Nugget sandstone.

66

0 2 in. diameter nntced tensileo 1 in. dianeter netche.' tensilp

0 Beam Bending

5.2 1 KIC 2

5.2m _ an TV 7(-,)

4.8

4.4-

Sx

EI

000 4.0-

0

3.6 00

3.2

2.8 I I I I

-. 4 -. 8 -1.2 -1.6 -2.0 Loq1 iO .

Figure 39. Critically stressed distance for non-uniform stressesin Tennessee marble.

6!7

* 800

0

S* 7000

4.)Wn 600

0 0

0 0 Curve calculated from.KIC 280 csivT-

4Q)

500

Notch tiD radius = 0.0015 in.400

0 .05 .10 .15 .20 .25

Notch Depth, In.

FIGURE 40. Effect of notch depth on net section stress for Nuggetsarndstone (Block II).

58

0 Notched extensionA Notched tension, I in.- Notched tension, 2 in.

o Beam Bending2.4 Torsion 0

~0

2.0

0,

1.6-ciI

.5-- I

InI

x 1.2_

E 0o I

.8 /

00'I

.403/f0 Q

- E

-. 4 -. 8 . -1.4 -1.6 -1.8

Loql 0 6-in.

FIGURE 41. Critically stressed distanct for multiaxial stressgradients in Westerly granite.

6 r4

0 Notched extensien3.2 A Notched tension, 1 in.

0 Notched tension, 2 in.Q Beam Bending0 Torsion

2.8

2.4 0 0

2.0

'%0

1.6

'0i O•x /000

LA. 1.2 _0 0

•0

00o- 0.8 '1 A

I

0.4 10, I

' i I00'

-0.4 'I, E

S-.8 p i I I I I I I I I I I I

-. 4 -. 6 -. 8 -1.0 -1.2 -1.4 -1.6 -1.8

Loglo 6 -in.

FIGURE 42. Critically stressed distance for multiaxial stressgradients in Nugget sandstone.

70

0Notched extensionANotched tension, 1 in.

2.8 • Notched tension, 2 in.

o Beam bending0 Torsion

2.4 0 /0

0 /

2.0 /

/I

1.6 /

"0"

~1.2

.8

04

0

-.4

x0

.o L

".4 /

.8O * i i I I I I I

-. 4 -. 6 -. 8 -1.0 -1.2 -1.4 -1.6 -1.8

Log 1 0 6-in.

FIGURE 43. Critically stressed distance for multiaxial

stress gradients in Tennessee marble.

71

S0 0 0

J b2

0 2 02:

FIGURE 44. Simplified flaw model for inhomogeneous material,after Weiss, et al.

72

90

0CI 0 0 0

li 0 4J 4J 4.1P) I L - 0 0 0U. 4J 4) 40 w 0

4- 4

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