DH CRUDII, Ph.D., 1969 1 of 1. ERRATA P. 220 Write

D.H. CRUDII, Ph.D., 1969 1 of 1.

ERRATA p. 220 write 7:7, (55) as

de/dt - de/dt_. 2. (1 - p.228, L5, insert after (ordinate),

natural logarithm of the

D.318, L8 for --0.66, - 0.25 p. 397, Reply cc the column under log 660 by

-2.82, -4.34, -L.70, -4.'25, -3.78, -3.46

-3.47, -5.15, -3.14 p.399, Replace the column under log 660 by

-2.17, -2.73, -2.20, -2.91, -2.52, -2.93, -3.00 1

A 1,1160AAT01I STUDY OF TIME STRAIN LiERAVIOUR AND ACOUSTIC ...11. I; OF STRESSED ROCK by

DAVID MILNE GRUEN.

Ph.D. thesis, University of London, 1969. ABSTRACT

..ixty seven new experiments on the time strain behaviour of

samples of Pennant sandstone and Carrara marble under constant uniaxial load are reported. Their results, plus a sample of creep

experiments from the literature, have been used to test the various

creep "laws". Rewriting these relationships between strain and duration of the test in terms of strain rates and duration causes the laws to fall into two groups; those proposing that the strain rate declines exponentially with time and those proposing a power law dependence of strain rate on time.

A new method of calculating the parameters of the laws gave

results that suggest the exponential law does not fit the data. An exponential law of transient creep plus a component of steady state creep is also not an adequate description of all the creep data.

As the three main formal theories all predict strain rates in transient creep proportional to the applied stress and inversely proportional to the duration of the experiment, a fourth theory, a

new structural theory which assume: the rate determining process in creep is stress controlled corrosion at the tips of small cracks in the sample, is the only satisfactory explanation of the new data. 3

All four theories predict differin forms of creep after an increment of load butt aLAin, only the structural theory is a satisfactory explanation of seventeen new increment creep experiments. The structural theory is also confirmed -ey eleven new recovery experimonto but the predicted acoustic emission fro alples during creep Limier comprezeion has not been detected because cracks iroi;agating under comprescion emit loss energy then thin they would under tension. The structural theory leads naturally to a plausible theory of static fatigue. It also suggests deforming rock masses can be treated us rigid plastic bodies; this is illustrated by an analysis or thrust faulting in the Canadian Aockiee. Statement on subelliary natter submitted in support of my candidature. Paragraph 33.8 of the relevant University of London thesis regulations requires that I state .fully my own share in any conjoint

work submitted as subsidiary matter in support of my candidature. Two papers (reprints of which are in the pocket at the back of Vol. II of the thesis) have been submitted. They are; Oruden, D.M., Charlesworth, H.A.R.„ 2966„ The Lissiceippian• Jurassic unconformity near Nordegg, Alberta; bull. Canadian Petroleum Geology, 14,266. Crude% D.M., 1968, liethoda of calculating the axes of cylindrical folds; a review, bull. Geol. Soc. Amer., 79,143. I wrote both these papers. Gharlesworthts contribution to the earlier paper was confined to its editing. 5

AccowmUGEEENTS

I am most grateful for the help and guidance of my supervisor, Dr. N.J. Price. This thesis owes much to his criticism. I have also been helped by dicusaions with colleagues in thoiepartmente of Civil Engineering, Geology, lAthematics, Mechanical Engineering, Mining and Physics at Imperial College. In particular, Mr. L Holloway sand Mr. E. Rutter of the Department of Geology have road and criticized Chaptera 1:4 and 9. Er. J. Franklin of the interdepartmental Project on Rock Mechanics at Imperial College provided data on the physical properties of the rook used in this study. Dr. C.J. Talbot, now of the 1:epartment of Geology, University of Dundee, was a kind and attentive host when I visited the Post- graduate ,;chool of Mining in the University of Sheffield. i)uring part of the time that this thesis was written, I was the holder of a Natural Environment Research Council Research .:Audontship. 6

CONTENTS Volume 1

Abstract

Acknowledgements 5

CHAPTER 1: haperimental Methods 1:1 Introduction 12 1:2 Definitions, Units and Symbols 16

1:3 Review of apparatus 25

1:1 Description of deformation apparatus used 29

1:5 Description of rock types used 36 1:6 !ilethod of preparing specimens for creep experiments 41

1:7 Assessment of experimental technieue 41+

Clap22.9. 2: Ar.,15rsiS of axial transient creep data using estimated strain rates

2:1 Two transient creep laws compared 50 2:2 Graphical methods of fitting creep laws 56

2:3 Simple linear regression as a numerical fitting method 60 2:14. Criteria of satisfactory fit 61. 2:5 4easures of goodness of fit 67

2:6 Another straight line for in regression 70 2:7 The two forms of the straight line compared 71 7

2:8 Particular problems in fitting transient creep laws 72 2:9 Criteria for choosinc experiments for analysis 77 2:10 An exponential law of transient creep rejected 81 2:11 An exponential and steady state creep law considered 83

2:12 Lteady state creep removed 90

CHAPTER 3: The analysis of transient creep data using strains ;:l The use of non linear regression 95 3:2 Hardy': method 99 3:3 Hardy's method. modified 102 3:4 The calculation of the initial strain 103 3:3 The remits of Hardy's method 105 3:6 Calculation of initial strain in power laws of transient creep 107 3:7 Interpretation of initial strain 110 3:8 The fit of a power law of transient creep 113 3:9 Discrepancies. between power law fits to the strains and to the strain rates 114. 3:10 :easuros of goodness of fit compared 116 3:11 Conclusions 119 8

cLuor it Z: me further 1.11;v2 of simiaL linear regression

4:1 introduction 130 4:2 A method et tostin6 the identity of sim,ple linear regressions 131 4:3 Experimental design, 1, Used to reduco the cstinr.ted variance in same experiments 134 4.:4 II The effect of previous creep 138

4:5 III, Prediction of duration of experiments 140 4:6 Lateral oree-p e:..periments, 1)44

4:7 ;,cceleratint, er(;ep experiments 149

1,4b Use distribution of residuale 152

CHAPTER 5: Formal theories of creep, Strain hardening and exhaustion theories

5:1 Introduction 155

5:2 Crovan's theory of strain hardening 157

5:3 lIntt's theory oi' strain hardening - 159 5:4. 7zhaustion theories 163 5:5 4att's modification of the exhaustion theorice 169 5:6 Common criticisms of the formal theories of creep 171 5:7 The distribution of the activation energies of elements in the exhaustion theory 172 5:8 Lminor correction to Cottrell's theory 176 9

5:9 uyatt's theory discuesed in detail 178 5:10 increment creep experiments and the exhaustion theory of creep 180

CHAPTER 6: Formal theories of creep, 11 Llacroannlytic (or rheological) theories

6:1 Introduction i62

6:2 The ie1vin Body 185

6:3 The :beaker Body 189

CHAPTER 7: Struotural theories of brittle creep 7:1 Introduction 193 7:2 Schols's theory of brittle creep 195 7:3 Criticism of Schols's theory 200 7:4 Charles' theory of static fatigue 204 7:5 Stress field around cracks under unimcia1 compression 210 7:6 Theories of brittle creep under uniaxial compression 213 7:7 Creep after an increment of load 220

CHAPTER 8: Tests of theories of brittle creep and their results 8:1 Introduction 222 8:2 The prediction of stress dependence of strain rate in transient creep 223 8:3 Results - Pennant sandstone 225 10

8:4 Results — Carrara marble 235 8:5 An assessment of creep thcories 238 6:6 Increment creep tests 240 8:7 Yiecovery tests 24.7 8:8 Permanent deformation in creep 250 8:9 A structural theory of recovery 252

CHAPTER 9: Microscismic emission 9:1 Introduction 256

9:2 The Aysical properties of elastic waves in rock 257

9:3 Review of work on microseismic emission 260

9:4 Experiments on emission in creep under Uninw4P1 compression 263

9:5 Interpretation of these experiments 271

CHAPTER 10: Some consequences of the experimental results for structural geology

10:1 Summary of the results of the thesis 277

10:2 Limitations of the structural theory of creep 283

10:3 The structural theory applied to thrust faulting in the Canadian Rockies 290

References 302 11

Volume 2 List of symbols used. in the Tables 312 Tables 1 - 44 316 Appendix 1: Listings of computer programs 400

Appendix 2: Details of creep experiments 4.24. At rear, reprints of papers In pocket Figures 1 and 2 12

JaAPTLii, 1

1!: < P;Ju1TAL

1.1 tructural geologists attempt to Ascribe and explain the deformation observed in the earthic crust. Soup of them aro more epecific than thic about their iateroot;. Flinn, for inotance, (l9U, p. 385) stated, in an influential paper, It Ic the tack of the etructural geologist to determine, from the study of structures in rocks the stroce system that gave rise to these structures. This van only be done by first determining the strain or deformation the rocks have suffered". lie Flinn sazgested, only the strains and displacements of rocks that took part in a deformation can be obeerved in nature. The stress field can then be determined in two ways; either by its assumed relationship to elements of the tectonic fabric whose analojues have been produced in the laboratory under a known stress field or, where the strain field is known, from the equations connecting stress and strain which are known as the constitutive equationa. The two approaches to discovering the stress field, the study of the p4sical processes that cause (ieformation and the :AuJy or the bulk strain :.)ehaviour, are, of course, inter.lependent* 13

111

'the -ulk strains are, eieely, the OUra of the deformation processes. xperieento in the laboratory on the deformation of rock are, thus, an important part of structural geology. They are, also, of interest to civil and eining engineers who have to design structures on or in rock masses and to geophysicists, who share some of the etruetural eeoloeists' objectives, though differing in the Beans by which they attempt to roach thee. eaaneitive experimental work on rook began by studying deformation under thou implest eeanary conditions rolovant to geology neeely, a uniexial ca:preecive stress. Additional variables can be in eodueed such as a triazial stress field, planes of inhonoeeneity in the rook material, a pore pressure and Lhe effects of heat and fluids until a realistic sisu1ation of natural conditions is built

Up.

The duration of natural deformations, probably cillions of year: in some cases, cannot, of course, ee reproduced in the laboratory. Applying the results of experimente to natural deformations thus, often involves eetraeolatiee over periods at least sie orders of maenituee larger than the duration of the experiments. There is, clearly, a considerable uncertainty in such extrapolations; to minimize it, the time eopendence of the rates of deformation processes should be deterwined as accurately as possible. Observine the strain in a rock specimen with the passing of time 1 .while other =.1oundaryouUtione are hold constant is the sii plest methoa of Jetermirdnut the time train properties reek. :::ach exhAriments are called creep c;:0)riments and the emirical relationships between .Atrain or .,itrain rate) and time derived from them are called creep'law. *,:here aro other types of experiment which can .43 uoed to investigate time ..epon_Leat strains, o.:perimonte in which the str0.-4o or Lhe strain in the specimen is. increaad at a con,ant rates for instance, out tiley all require zome assumpt,ion about the Corm of the ereep law 'before .11.ey can be interpreted. this-thesis -,:onsiee a only era experiments in its ctu4y of time :tependent strain.

aver with ,:he considers:As con,ribations of a 11 number of workers, the Au4y of creep is not very far advanced. !:ost work has been ,one on creep under uniaxial compression with he specimen uner nor ma atl:vapheric conditions. Other work, however, indicAes that teperatare, the re:xonce o fluids in cow ,act -ith tho -o en and .confinind pressure are i ,ox tent. variablee. This thesis reports further experimonts =WI. the simplest eenndary Aonditiona described above. It uttou9tta to uco those results and a reprelA)ntative roap

of results taken from the litoraturo to saedest th4t some 1101,".e ,ed creep laws are unlikely to be s3tizfactory .,jeser4tions of the availabio data. There is, however, at least one reasonable alltien to 4110 constitutive relations. ;be thesis also sLupsts physioal ;recesses .------

I 1;

~., p ' d ' .t t te·! lDu.; to Xl' ttl

e b _ to 00

nd I ale e.t 4 " , 6 " l'

1 . , 01 ,~~ ..• .1

I , 1 ,,2.

'l,S

I l ___ _ 16

1:1 - 1:2 gives details of the phyical properties of the two rock types used. Chapter 1:6 describes the preparation of specimens for creep testing; Chapter 1:7 critically discussea experimental technique.

1:2

Chapter 1:2 considers definitions and units. This thesis is principally concerned with measurements of strain, stress and time. The units of strain need no discussion, As even the tote/ strains in creep experiments on rook under uniaxial compression at room temperatures are small (usually less than 1:), some simplification of terminology can be made. In popular usage, strain is measured by the total fractional deformation. Thus, the strain, E.x, in a. creep experiment after a duration, Ti, is given by (Li - Lo)/Lo where L is the length of the measure at Ti and Lo, the untrained length of the measure at zero time. At small strains* the difference between the total fractional deformation and another common strain measure, natural strain, (dLi/Li), Jaeger (1962, p. is negligible. The extensometere used in most creep experiments measure the displacement of a surface of the specimen. (Chapter 1:4), If the strain field in the specimen is homogeneous, strains can be simply calculatee from displacements (Jaeger, 1962, p. 59). They should 17

1:2 be distinguished from strains measured by bonded strain gauges which measure the change in dimensions of the small part of the surface which they cover. However, for mall strains, the difference between the two measures is negligible. Creep theories (Chapter 5, 6, 7) are generally written in terms of displacements so the total fractional deformation seems the more suitable measure of strain. It has been the practice of British, Comnonwcalth and Berth American mining and civil engineers to measure stresses in pounds or tons weight per square inch. On the Continent, dynes or kilo- grammes weight per square centimetre has been used. This last unit has the advantage of being almost identical to a unit used predomin- antly by geophysicists, the atmosphere; the stress due to the weight of the atmosphere at the earth's surface. The bar is defined 6 as 10 dynes per square centimetre. International standardisation of units on the SI system, (Anon., 1967) is making progress and the system will probably, eventually, be universally accepted. The unit of stress, the newton per square metre (or pascal), is 10-5 bars. Only one paper quoted in the bibliography has used the SI system, however; so to use it in the thesis would only increase confusion. Generally, data taken from the literature remain in their

original units. Most qwetities are, anyway, used only on a. 18

1:2 comparative basis and, in these cases, units are chosen which make the relative sizes of the quantities most clear. A comparative table is provided at the end of this section. Stress is used throughout the thesis in its usual sense* The strese field in a cylindrical specimen loaded in a creep rig through its ends is assumed to be uniform. The principal stress

is Ririe/1 and equal to the load divided by the cross sectional area or the specimen. Assuming deformation takes place at constant volume, then the axial strain is equal to the percentage change in cross sectional area of the specimen. Hence, the error in calculating the stress on a specimen during creep as the load divided by

the oriems1 cross sectional area is less than 1. The sm411 differences between the strain rates in creep experiments on the same rock type at quite widely different loads suggest an error of this magnitude is negligible. Three experiments on creep under uniaxial compression performed by Misra (1962) at elevated temperatures show much larger strains. One experiment on marble at 600°C and 8950 pounds per sq, inch shows a strain of 18:If, during creep, two others on dolomite at 600°C and 5660 pounds per sq. inch and at 700°C and 3775 pounds per sq* inch show strains of 7,;r, and 3.X: during creep. These experiments are discussed in Chapter 4:7 where it is assumed Misra's observations are corrected to strains under constant stress. 19

1:2 The most convenient time unit is the minute, Strain rates are given unless otherwise stated, in microstrains per minute.. Temperature is aeasured in degrees Celsius. The following list of symbols used in the thesis is supplemented at the beginning of Appendix 2 by an explanation of the identifying code of the new creep experiments. An abbreviated for: of this code is used throughout the thesis to refer to individual samples. Prefix A refers to Carrara marble specimens, B to Pennant sandstone specimens. Three following sets of figures (as B 2.3.04) locate the sample in its original block. Other symbols are defined whenever they are used. 20

1:2

LIST OF SYMBOLS USED IN TdESIS A3 are constants used in the exponeatial law of transient

creep, el=A1 ex(-A2ti) al, a2, a3 are estimates of such constants.

Al is the strain rate at zero time. is a strain hardening parameter determining the rate of decrease A2 of the strain rate. A.. is the rate of steady state creep.

• B BI, B2 are constants in the paver law of transient creep, o = Bit 2. bit b2, are estimates of such constants. Bi is the strain rate at unit time. B i 2 s a strain hardening parameter determining the rate of decrease of the strain rate. con is an abbreviation of "the confidence limits on..... Confidence limits are t times the tabAlatea value where t is the value of Student's t with the appropriate degrees of freedom. dw„ is the computed value of the Durbin watson statistic, - u1)2/i4 21

1:2 13 is the strain at time, to after loading has been completed* - t is an estimate of f; / an observation, et 't th e. is the i observation of strain*

ithearithmeticmeanofth.estimatesof.Edwhen(b2A1) is negative it is the estimate of 21*

•e t is an 4.-;stimate of the strain rate at time, t* th 1)th (43.1. is an estimate of the strain rate based on the i and

estiaatesefthestraillt-aUde.ex 1. th is the value of tho estimated strain rate estimated by the

estimates of the parameters of the creep curve* ainf is the estimated strain after infinite duration* log x is the natural logarithm of x. n is the total number of observations of strain during a creep experiment*

R is a variance ratio*

✓ is an estimate of a variance ratio.

IR1 is the ratio of the estimated variance of a simple linear regression to the variance about the mean value of the &pendent variable*

L2 is the ratio of the variance of a fit by the exponential law of transient creep to the variance of a fit by the power law of transient creep to the same experiment. 22

1:2 R3 is the ratio of the valiance of a fit by the exponential law of transient creep with an esti,mat©d steady state creep component removed to the variance of a fit by the power law of transient creep to the same experiment. is the ratio of the variance of a fit by the exponential law of transient creep to the variance of a fit by the exponential law of transient creep with an estimated steady state creep component removed. b = b t 2 R r is the ratio of the value of in a fit of a law, et i , to the observations in a creep experiment to the value of Ri derived from a fit of et = (a1/a2)(1-exp(-a2t)) + a3t by HardY's method (Chapter 3:3). E6 is the ratio of the variance of a power law fit to a transient creep experiment to the variance of a power law fit after the experiment has been modified by the method of Chapter 4:3.

2 is given by Dei eil)2,/n - 3), S is 'the standard deviation associated with the least squares fit'. (see Chapter 3:10).

the time interval from the beginninz of a creep experiment to the i. thstrain measurement. ti, the time interval to the ith strain rate measurement.

U2 the variance is estimated by 23

1:2 2 . u whloh is given by .th u the residual associated with the a. strain rates it is an estimate of U.

w is the weightings it is equal toEwi where wi is the weighting th of the i observation. If there are no strain reversals in the experiment w is equal to (n-2).

I, II, III, IV are parts of a procedure for testing the identity of simple linear regressions (see Chapter 482). ▪indicates that the statistic is significant at the level. •indicates that the statistic is significant at the 1'7, level. The last two symbols are generally omitted for the ratio, ri. 24

1:2 Conversion Table

load %.; of failure Stress,, Bars (tons) load lbs/in'

Pennant Sandstone.

1.5 15 4720 295 2.5 25 7150 4.93 3.5 35 10000 669 4.5 45 12700 665 5.0 50 14300 986 5.5 55 15700 1082 6.5 65 18500 1280 7.5 75 21400 1477 8,5 35 21.2004 1673 Carrara marble 2.5 53 7150 493 3.0 64 8570 592 3.3 70 9440 651 3.6 77 10290 710 3.9 83 11150 770 4.05 86 11600 800

0 5bars = 1 pascal = 1 newton/metro2 25

1:3

Cris (1939) reported the dsin of two single lever creep

testing machines. The more robust of these has been used for a 550 day dur- tion test of a specimen of Solenhofen limestone under

a load of 55,7: of its strength in uniaxial com2ression. The specimen

was loaded through deuble spherical seats between two I beam levers

connected by an Emery hinge. The mechanical advantage of the levers was 100. Deformation was measured between the ends of the levers

by a dial gauge reading to 10-5 inches; this corresponded to

deformations of Er'? inches in the The measurements, however, included bending of the lever beams

and creep in the Eery hinge. The weighted long lever also tended

to act as a seismograph and produce vibration in the specimen. The

irkeortance of these effects was not known.

The more simple creep tester was used for a 4.2 day test on a

single crystal of halite, a 22 day test on a calcite single crystal,

a 1.44. day test on a uniform mudatone and short tests on boric

anhydrite glass and talc schist cylinders. The specimens were placed on one platen of the tester and

loaded through a single ball seating by a yoke from the single lover, Deformation was measured to 10 inches by a MO gauge between a point on the yoke vertically above the centre of the ball seating and the supporting frame of the tester. A furnace could be placed around the specimen to conduct 26

10 experiments at elevated temperaturee, ereep in thu presence or liquids was investigated by surruunding the specimen with a email ,jachot to contain the fluids (Grizgs„ 154o). '.2emperature within, the laboratory wee controlled to -within 0..1°C ana humidity controlled by a calcium chloride drier. Hardy (153) investigated the possibilityof using a hydraulic ram to apply loads to the specimen. Nie early attempts to build an apparatus capable oi maintii 'Jag a load constant to 044Z tor ninety day:: were unsuccessful, Eaedy (1966b) deeuribed a hydraulic apparatu- which incorporated feedback from the load measuring apparatus and produced a load that vas stable to 0.27: over one day, Stability is limited by the precision of the load cell used to monitor load, Oscillations take place in the hydraulic system ftr to five minutes after the load has been changed, They lead to oscillations in the applied load which have to be corrected for in the subsequent analysis by an empirical graphical procedure. Deformation during the early part of the creep experiment is therefore obscured. In Hardy's experiments, strains wore measured by resistance strain gauges toned to the rock. These were estimated to be stable to 3.10- over long periods of time, Hardy (1966a) however reported

only experiments of forty minutes duration on six specimens of 'eombeyan marble. After forty minutes at an initial load, the load 27

1:3

on the specimen was increased and creep observed at the incremented load. Increments were added till the specimen failed. Hardy (1966 b) kept the humidity around the specimen "as low

as possible" and controlled the temperature to 0404°C. Price (1964. a) reported experimente on twelve specimens of Coal l':easure sandstones and mudetoness, all at room temperatures, under uniaxial compression. They were tested in two types of creep rig , a large double lever machine and a series of six smeller triple lever creep machines. In the large machine, load was applied directly through an upper spherical seat to the specimen standing on a lower platen suported on the creep rig frame. The machine levers were pin jointed and this caused a large frictional resistance at the joints - about 2 of the total load4 The minimum load. that could be applied to the specimen was the weight of the levers, 6800 lbs. The smaller rigs wore designed to investigate creep at lower stresses. In these, the levers were supported on tungsten carbide knife edges, this reduced friction to 2 lbs. The minimum load that could be applied was 500 lb. Strain was measured in these experiments by a roller extensometer fixed on the specimen; it had a magnification of 104 Resolution is limited only by the stability of the scale and the viewing telescope.

All the apparatus was set up in a temperature controlled cellar.

1:3 o_ -5 lilt: permitted_ teveL rature range, ) ,Q caulled strains of • There was no heei dity control but the effects of humidity wore minimized by coating the sl:ecimen with bitumen paint. Misra and. Murrell (1965) described 62 uniaxal compression creep exeeriments on eight rock types, igneous, metamorphic and sedimentary, at temperatures up to 75000. Most of these were carried, out on a double lever creep rig with a capacity of twelve tons load. Load is applied by adding weights to a weight pan which exerts an upward pull on the specimen through the lever system and a load reverser. The pull is taken, through a spherical seat, by a spring and jack assembly attached to the creep rig frame, When the ftll, desirea load has been applied, the spring and jack assembly is adjusted to bring the levers approximately horizontal. The load on the assembly is then transferred to the tram; by a thrust bearing. Strains were measured between twe horizontal arms attached to the upper and lower platens. Their approach was measured by a .%:artens type optical extensometor with a magnification of 6,250 or, in later experiments, by a linear variable differential transformer. The furnace for heating the speoimens was mounted inside the load reverser so only the platens and the arms of the strain measuring device needed to be made of a special, creep resistant

steel. The temperature was maintained constant to Vithin 3°C. 29

1:14. The' two creep rigs used in this study, one T45 and one T55 Compression Creep Tester, are commercially available from Samuel eenison 01: Son Ltd., of Leeds. Plans of the two rigs form Figures 1 and 2. (in pocket). They differ in capacity; the T45 can aFply loads of up to five tons while the T55 can apply up to fifteen ton loads. The construction of the T45 is described first as the design of the T55, though similar in principle, is more complex in detail. There are three groups of moving aprts in the T45 attached to a massive frame - an upper load adjuster, a lower load adjuster and a lever system (Figure 1). The position of the upper load adjuster (P (? 18 - 24, symbols refer to the coordinates on the margin of rig. 1) can be adjusted by seven inches relative to the massive frame using the straining hand. wheel (R25.24). Rotation of the handwheel moves a worm gear (R23, not shown on Peg. 3.). The upper load adjuster is attached to the inner screw of the worm through a universal knife edge joint (R21-22). As the inner screw is keyed to part of thy: frame, R23, it cannot rotate. The upper load adjuster is, thus freely euspended from the frame. The lower crosshead of the upper load adjuster contains, in its centre, the lower platen of the rig (R13). 'Then the rig contains a specimen under load, the upper load adjuster is not in contact with the lower load adjuster. The lower 30

1:4- load adjuster (R14. - 19) is attached to the lever system through a universal knife edge joint (R14). The upper crosshead contains the upper platen of the rig in a spherical seat let into the crosshead. There are two levers in the lever system each supported on the frame by knife edges at M16 and EU. Load is applied by a poiseweight (G16) moved on lubricated surfaces on the upper lever by a wire and pulley weight system through the poise propelling handwheel (L-M 15 - 16). For one particular position of the poiseweight close to the knife edges at M16, the lever system is horizontal and perfectly balanced; its weight is borne by the knife edges. The weight of the lower load adjuster is borne by the frame through the screws at D-E15. If the lower load adjuster is then raised by inserting a creep specimen between the platens of the creep rig and raising the upper adjusters the weight of the lower load adjuster is no longer borne by the frames it is applied to the specimen. The weight of the adjuster, about 0.075 tons, is the least load that can be applied to a specimen on the T145 (the T55 has a least load of 0.05 tons). Moving the poiseweight to the left, (towards A) from its equilibrium position depresses the left hand side of the upper lever whose right hand side then exerts an upward pull on the right hand side of the lower lever. The left hand side of the lower lever 31

1:4 exerts a downward pull on the lower load adjuster thus applying load to the specimen. There is a linear relationship between the advance of the poise weight and the load applied to the specimen. The upper lever has been calibrated by the mernfaoturers and the load can be read directly with a precision of 0.02A at five tons on the VO (and with a siAlar precision at 15 tons on the T55). The T55 differs from the T45 mainly in its stronger and more elaborate method of raising the upper load adjuster. The straining handwheel (Li - N6, Fig. 2) operates bevel gears through a chain driven gear reduction (covered by a chain guard). Torque applied to the straining screw by the bevel gears is carried onto the supporting columns of the rig frame through adjustable V shaped sliders (u - V 7 ... 8). Bemuse of this greater complexity, the lugger capacity rig is less adjustable than the smaller rig; the maximum specimen length it can accommodate is four inches. The diameter of the creep specimens is limited by the diameter of the platens. Those supplied are 1.5 inches diameter, tool steel (Hall and. Pickles, "Vampire") hardened to Rockeell C60. further details of both rigs can be obtained from the manufacturer's handbooks. The rigs were erected on anti-vibration rubber blocks and their frames carefully levelled. Particular attention was paid to the levelling of the lower crosshead of the upper load adjuster, this can be separately levelled by adjusting the lengths of the columns 32

1:4 connecting the upper and lower crossheads. The specimen placed in T45 was loaded in the following way.

The poiseweight was moved by the poise propelling handwheel to the zero load position. The straining handwheel was then used to make sufficient room for the specimen between the platens by adjusting the crossheads (specimen size was standardized at one inch diameter and about two inches long in all experiments). The specimen was then inserted, setting it centrally on the lower platen (which is engraved with concentric circles at 0.5 inch spacings). Brass bushes which slide on the columns of the lower load adjuster can then be inserted into the lower crosshead of thd upper load adjuster at B18; the bushes align the upper and lower load adjusters. The poise propelling handwheel was used to move the poiseweight into position to apply a load of 04 ton& As the upper platen was not yet in contact with the specimen, this smell load moved the steelyard to the bottom of its stroke. The lower crosshead of the upper load adjuster was then slowly raised by turning the straining handwheel, taking particular care as the te.per platen aligned itself with the top of the specimen. V]hen the steelyard approached the top of its stroke, the bushes were removed. The specimen then supported a load of 0.1 tons weight. Further load can be applied by moving the poiseweight. If the steelyard approaches the bottom of its stroke before the total load has been applied, it can be raised by the straining hanawheel. 1:4. Strains in the specimen have been measured by linear variable differential transformers supplied by CNS Instruments, Kentish Town, H.V.5. Those "read out extensometers" consist of a transformer on a ceramic former with a moveable ferrite core to each of whose ends a steel rod is attached; these rods move on carefully aligned bearings, To measure the displacement of a point on a surface the hemispherical end of the rod is rested on the surface, usually under its own weight, Displacement of the surface moves the ferrite core within the transformer changing the coupling between primary and secondary windings. This movement changes the output of the transformer yel its constant voltage input. The output is rectified and the resulting DC voltage determined using a voltage comparator method. As supplied, the potentiometer, used to measure the DC voltaree has been calibrated directly in displacement of the core from a reference position, True displacements arc the differences of measured displacements. Thus, apart from routine cheeks, the instrument requires no further calibration. The advantage of a null balance system of measurement such as the read out extensometers employ is that measurements are independent of variations in the supply voltage so that the instrument is stable over long periods of time. Unfortunately, the null balance system is not suitable for continuous recording but proper experimental design can overcome much of this disadvantage (Chapter 443). 34-

-6 Displacements can be read to 10-5 inches and interpolated to 10 inches. The system is linear to 5.10 inches over a range of -2 .5 -1 4.10 inches and to 4, 40.10 inches over its total range of 10 inches. To measure axial displacements, an extensometer was mounted vertically to rest on the upper crosshead of the lower load adjuster, directly above the centres of the two platens. In this position, measurements of displacements are not affected by any rotation of the upper spherically seated platen relative to the crosshead. The extensometer is held in a tufnol holder and retained by nylon screws. The holder is supported by two strips of light metal alloy bolted to the columns of the upper load adjuster. Thus the displacement measured is the relative movement of the centres of the upper crosshead of the low load adjuster and the lower crosshead of the upper load adjuster. This displacement should be identical in creep with the shortening of the specimen. Measurements of the displacements of the lateral margins of the specimens have also been made. Three extensometers were mounted horizontally in brass rings, closed by nylon screws, at aeproleimately 120 degrees to one another. The rings were supported on a brass cylinder fitting closely around the lower platen and hold in position by three brass retaining screws. Thc, extensometer rods are held in light contact with the lateral surface of the specimen by rubber bands. 55

1:1..

The centres of the rods are 29/32 inches from the lower platen surface and thus rest approximately at the mid height of most specimens. This arrangement was designed to pick up any hetero- geneity of deformation rate in a horizontal plane in the specimen. A more elaborate arrangement was used on several occasions in au attempt to distinguish buckling from shear parallel to the platen surfaces. Nine exteneometers were mounted horizontally in three groups of three at 120 degrees to one another. Each group was held by brass screws, one extensometer vertically above the other, in a tuenol column. The columns were screwed to a tufnol cylinder which fits closely around the lower platen and is hold in position on the platen by three brass retaining screws. Again, the extensometer cores were held in light contact with the specimen surface by rubber bands. It was generally Bound that lateral strain rates were too small to be measured reliably. The results are discussed in Chapter /446. The extensometers and the rock specimen are sensitive to changes in the ambient temperature and humidity. Creep experiments were carried, oat in an air conditioned, sound-proofed cellar approached through double doors. Temperature was controlled by thermostats which activate on electric heater in the trunking of the air conditioning when the temperature in the laboratory fell below 18°C. 1:4 - 1:5

Temperature could be maintained to .4. 1°C over a week. Unfortunately the cellar did not have a really adequate damp course. Generally the circulation of air through the air conditioning kept the humidity in the laboratory at a level of about 6Vh. However, heavy rainfall caused the humidity to rise and drought caused tt to fall* Experiments during which the humidity had a range greater than 5, have been rejected from the analysis.

1:5

The creep tests reported in this thesis have been performed on two rock types only, Carrara marble and. Pennant sandstone. A brief description of the rocks and some new determinations of their physical properties follow.

Carrara marble was supplied by the stonemasons, H.T. Clements Ltd., of Rotherhithe. A kerb of the material, 6' x 6" x 3", chosen in the suppliers' yard, was free from any superficial inhomogeneities. It vas a uniform, white, fine grained marble composed solely of calcite. In thin section, the calcite crystals are equigranular, usually about 0,1 mm in diameter, strongly twinned but showing well defined crystal boundaries without granulation« The dimensional orientation of the

calcite graina noted by iamez and Murrell (1964) is absent from this sample as are the small amounts of gra' hitic impurities. Razes and. Uurrell (1964) report a petrofabric analysis of Carrara which showed 1:5 that the fabric was random apart from a possible preferred orientation of twin intersections. No petrofabric work has been done on this sample. Mx.. 3. Iranklin of the Inter Departmental Rock Mechanics Project at Imperial College has suplaied the information below.

6 Voce. No. F- Fi Vn d p.10 3 208/1/8 324 16186 5980 2.705 879 208/1/7 571 /7275 6100 2.706 779, 208/1/4 1524 19017 6260 2.707 737 208/1/2 2343 22692 5890 2.701 748 208/1/6 3657 27291 5690 2.704. 831 208/1/1 4476 29869 6400 2.700 650 208/1/3 5676 34002 6340 2.703 586 208/1/5 5105 31501 6500 2.705 496 208/1/12 7505 38033 6420 2,705 573

Fit F3 are the princiyal compressive stresses in lbs. per sq. inch at failure of a cylindrical specimen about 3.14. inches long, 1.49 inches diameter in the Hoek-Franklin, cell (Hoek and Frnnklin, 1968). Some of the marble specimens do not show discrete, macroscopic failure planes. In these cases failure is defined by a very large increase in dilantancy as the axial stress is increased, 33

1:3 (Brace, Paulding and Scholz, 1966). V is the velocity in metres per second of P waves in the specimen measured by a Cawkell lieterials Tester. d is the dry bulk density in gram., per cubic centimetre of the specimen. p is the effective porosity of the specimen determined by the hydrocarbon resaturation method (Anon., 1960, p. 31). If the marble is pure calcite (density, 2.712 gm, per c.c., Peselnick and. Robie, 1963) then the total porosity of the marble is about 0.24% The value of h is intermediate between the Reuss and. Voigt values (Birch, 1966) and is, hence, consistent with the hypothesis that the material is a pore free, random aggregate of calcite grains. A value of Young's modulus for this sample of Carrara marble can be determined from the slope of a graph of applied stress against the first measured strain in creep experiments on the material (these measurements are listed in Appendix 2). The estimate, 23.2.10 lbs per eq. in., should be compared with values of 70.0.105 (isra and. Jarrell, 1965), 78.5.1051 (birch, 1966) and 138.0.105 (the value On the Voigt theory) and 118.0.105 (the value on the teuss theory). The discrepancy between theoretical and measured values of Young's modulus can be reasonably attributed, to twinning in the calcite crsytals. 39

135 The comparatively low value of Young's modulus for this sample of Carrara marble is also associated with a low uniaxial strength. The mean strength of three specimens deformed in the T55 creep rig was 13,4.00 lbs, per sq. inoi The estimate is in accord with results on the same material from the extrapolation of Franklin's triaxial failure envelope but lower than values determined by Misr, and Murrell (1965, p. 516) and von n (Handin, 1966). Pennant sandstone has been supplied from G.R.C. Quarries Ltd., :ileriston, near Swansea, Glamorgan. It is quarried to provide kerb- stones so the rook used is from the bottom of the quarry away from zones of superficial weathering and close to the area from which the material described by Price (1958, 1964.a) and Misra and Murrell (1965) was taken.

In outcrop the rock is a grey, rusty weathering massive sandstone showing large scale channeling. The base of the channels contains pebbles of black mudstone. Kerbs were selected as far as possible free of pebbles. Typical thin sections of the rock show that quartz grains up to lmm. in size form 5& of the rock. The remainder of the framework is made up of about equal proportions of felspar crystals and sandstone and mudstone cleats with a few muscovite flakes. The rock is poorly sorted. The matrix forme about 25;1 of the rock, dominantly composed of 4.0

1:5 clay minerals with a variable cement, locally it may be ferruginous, calcareous or even quartzose. Individual clasts arc angular; the quartz grains show patchy extinction4 Mr. J. Franklin reported the following values for cores drilled parallel to the bedding. Spec. No. F F V d P.106 3 1 P 209/2/31 192 29768 5000 2.678 2120 209/2/24. 421 30643 4950 2.681 2140 209/1/30 747 36522 4960 2.684 1914 209/3/20 1111 34476 5000 2.681 1969 209/1/29 1494 42146 5000 2.684 3096 209/1/25 1762 43242 4960 2.682 1734 209/1/19 2260 45287 4940 2.682 2000 209/2/26 2816 46748 4990 2.682 1635 209/2/23 3716 50181 4.990 2,679 2158 209/1/27 4425 54856 4920 2.685 2420 209/1/21 4789 57851 5000 2.679 24.22 209/2/28 54.02 58222 4980 2,681 1862 209/1/22 6130 64060 5040 2.683 24.38 209/3/32 7260 65697 4900 2.677 1888 Symbols are the same as those in the previous table. 1:5 - 1:6 Price (1958) has shown that there is little difference between the triaxial failure envelope of cores taken parallel and perpendicular to bedding. All the specimens used in this study of creep were cored perpendicular to bedding except one, B2.3.04, cored parallel to bedding. The mean strength of three creep specimens deformed on the T55 rig was 28,600 lbs. per sq. in.. This is consistent with the extrapolation of Franklin's data but higher than values determined by Misra and Murrell (1965, p. 516) or Price (1958). The estimate of Young's modulus perpendicular to bedding is 20.2.1 lbs. per sq. in. consistent with Misra and Murrell's values but considerably lower than those reported by Price (1958).

1:6

A number of techniques have been developed for preparing rock samples for tests of their strength in uniaxial compression (Hoek, 1965, Rowlands, 1967). Those techniques are adequate for preparing specimens from creep tests in =initial compression; the choice of technique depended on the machines available and the form of the rock samples supplied. The rock, as received, is in blocks slightly thicker than the proposed specimen length with their upper and lower surfaces sawn approximately parallel. The upper and lower surfaces of the Pennant sandstone blocks are parallel to bedding but there are no discernable 4.2

1:6 reference planes in the Carrara marble,

The blocks were mounted on the table of a massive' fixed, hand-fed drilling machine and held in a heavy clamp. A number of cores can then be drilled perpendicular to the loser surface of a block with only slight adjustment of the position of the clamp. The position of the cores relative to the blocks was noted in case further work should show any variation of rock properties between the specimene. Once the problems of supplying sufficient water to the diamond drill bit had been solved, the cot's diameters were reproducible to within a thousandth of an inch and core recovery was perfect, The core was used without further preparation of the lateral margins, The ends of the specimens were prepared using a surface grinder with a diamond impregnated wheel. A core was held in a collet in a lathe chuck held on a magnetic chuck whose surface had been ground flat by the surface grinder. The ends of the specimen were then ground parallel to the base of the chuck and hence perpendicular to the axis of the core until all the original end surface had been removed, After grinding, the ends of the specimen still showed some irregularity. This was removed by light manual polishing with Aloxite 600 powder (Carborundum Co, Ltd.). Detailed investigations were made of the dimensions of four prepared specimens, two of Ferment sandstone and two of Carrara marbly.* 4-3

1:6 aeasurements were made by a universal measuring microscope which . -5 is accurate to ;,.10 inches. The instrument consists of a microscope whose height can be adjusted over a table which is adjustable in a plane perpendicular to the axis of movement of the microscope; the microscope focuses on a feeler connected to it which is held vertical by a weak spring. The specimen was mounted in a V block aligned along one of the axes of the table and the coordinates of a grid of points on the ends and the lzteral margins of the specimen were measured. The form of the ends of the specimen can then be most conveniently displayed as contour maps. Deviations from linearity of the contours indicate deviations from flatness in the endo. Differences in orientation and spacing of the contours at each end show lack of parallelism of the ends. The rightness of the ends can be checked from the measurements of the lateral margins of the specimen. The maps suggest that specimens are prepared with ends flat to .4 10 inches but the total error une to lack of parallelism and rightness may amount to as much as a quarter degree or nearly 10-3 inches in a two inch specimen. Small scale irregularities on the ends of specimen can be investigated by a hank Taylor-Hobson "Talysurfn. This uses a -44. diamond stylus to profile asperities of wavelengths down to 10 inches wavelength. The maximum length of a transverse is 3/8 inches. 44.

1:6 — la The irregularities on the ends of the epeoiziens were dominantly holes (of width about equal to their depth), in an otherwise flat surface. Presumably these are caused by the plucking out of individual grains in grinding the epecimen ends; they have a maximum amplitude of 1O' inches. Surfaces of marble specimens are less irregular than those of sandstone specimens. After preparation the specimens were exposed for at least two weeks in the laboratory principally to allow their water content to reach equilibrium with their experimental surroundings. la

A study of the stress distribution in a perfeotly elastic cylinder loaded through platens at its ends showed that the stress distribution may be far from uniform (Balla, 1960). The distribution depends principally on the length to diameter ratios of the specimen, the coefficient of friction between the platens, and the Poisson ratios of the specimen and the platen. Hoek (1965, p. 78) considered that the main factor causing non uniform stress distribution was the frictional restraint of the platens on the specimen ends. Under the worst conditions, however, at least the central third of eho specimen was uniformly stressed if the length diameter ratio was greater than two. Using steel platens on hard rook, the radial deformations of platen and specimen have a 45

1:7 similar magnitude so the constraint is comparatively small. Hardy (1966b) used teflon inserts in an attm2t to reduce the coefficient of friction betseen the platens and the ssecimen. Hoek

(1965, p. 79) criticised the use of deformable inserts because their extrusion induces tensile stresses in the ends of the specimen, possibly splitting them. Further, deformation in creep tests has to be measured by gauges actually attached to the sjccimen itself if inserts are used. This reduces available gauges to optical extensometers like Lamb's (see Price, 1964a) or bonded strain gauges; neither type is entirely satisfactory. Another problem is that, while large deformations are taking place in the inserts, it is difficult to calculate even the load on the specimen. So Hoek's suggested procedure seems preferable.

Sigvaldson (1966) has shown that the mechanical characteristics of the platens of testing machines have an important effect on the results of tests of the compressive strength of cubes of concrete.

With the brass bushes inserted in the lower crosshead of the upper load adjuster and assuming both the load adjusters were originally vertical, :he lower platen of the T45 and T55 is effectively fixed and the upper platen is effectively pinned. The machines thus operate by Sigvaldson's "third philosophy". The spherical seat of the upper platen, however, is not lubricated and is only roughly finished. Friction will ensure that it also is fixed even under

r.;kall loads. 46

1:7

Sigvaldson suggested (19A, p. 10) that "Compression tests should be carried out with both ends of the specimen effectively fixed (this) test is least susceptible to instability, lateral stiffness, specimen misalignment and operator technique.

The spherical seating should tilt freely only during the initial setting up, becoming looked as the load is applied to the specimen".

The creep testing machines used are aesi ned to operate to this specification.

The duration of the creep experiments described in this thesis is measured from the beginning of the adding of further load to the specimen above the 0.1 tons weight necessary to hold the specimen firmly in place (Chapter 1:5). In Chapter 445 the advantages of making the first measurement of strain as soon as possible are discussed. In the experimental system described in Chapter 1:5, the time to the first strain observation is occupied by two processes, the time taken to load the specimen and the time taken to balance tho read out oxtonsonoter. Both of these may be shortened.

With simultaneous operation of the straining hantifheo1 and the poise propelling handwheel, the auration of loading can be reduced to under liolf a ',Amite. So long es the loading rate is relatively constant, the creep rig is not strained. Oscillations of the steelyard can be eemped by reducing the length or its stroke. 4-7

1:7

Delay in measuring the strain can be eliminated by supplementing the read out extensometor with a continuously recording transducer of the type used by Misra and Murrell (1965, p. 513). it can be calibrated at leisure using the read out extonsemeter. Delay can be reduced by setting the read out exteneometer at balance at a strain which previous experiments suggest will be reached early in the experiment.

Thus, using these techniques the time to the first strain observation can be reduced to thirty eeconds. Further reductions require a different system of applying load. Hardy (1966b) has demonstrated the difficulties with transients in hydraulic systems.

Evans (1940) has built a compressed air testing machine capable of aeplying a load of twelve tons in 0.005 seconds and this is suitable for creep experiments of comparatively short duration using Hardy's

(1966a) method of correcting for transients.

The most simple method of chocking the accuracy of the system of strain measurements used (Chapter 1:5), was to compare permanent deformation measured by the extonsometer readings in recovery experiments, (Chapter 8:8), with changes in length determined by the universal measuring microscope. The length of the specimen was determined imeediately before mounting in the creep rig and immediately after removal from it at the end of the completed experiment. The estimates of permanent deformation agreed to within the precision of h.b

1:7 measurement for the two epecimens, 12..1.37), which eere checked* Sp errors due to expansion or movsment of the extensometer or it. holder are probably less than 5.10 inches over five days. Examination of the record of A2.1.10.in creep at 2.5 tons load

(Appendix 2) indiontes that under favourable oonditions, readings may -5 remain constant to within 10 inches over one week. These limits compare reasonably with the results of other experimenters (Chapter 1:3)* It is unlikely that significant improvements in precision can be made vithout considerably more elaborate methods of controlling the ambient environment of the specimen and strain moasurin device. The precision of strain measurements could probably be nearly doubled by doubling the length of the specimen if this were considered to be necessary.

Doubling the length of the specimen end increasing its diameter to, say, 1.5 inches should have little effect on the accuracy with which the specimens can be prepared (Chapter 1:7). Hobbs (1964) has found no significant change in uniaxiel compressive strength of one inch diameter specimens over two inches and up to four inches in length; his data are supported by Grosvenor (1965). Hobbs' data also suggest that a size effect on strength is not important. So it is not necessary to standardize ssecimen lengths closely within the range, two inches to fear inches. ror is it necessary to standardise radii of the specimens closely. 49

1 : 7 (aturements of the parallelism and rightness of specimens have been described in Chapter 1:7. It has been shown that in tho testing machine used small departure from the ideal have no serious effect on results. Departures from flatness of the ends of specimens are more serious. Hoskins pointed out, in the discussion of Hardy (1F)66b), that, even using optimistic estimates of the surface finish

of specimens, irregularities may aeproach 1C of the observed strain at failure of the specimen. It is likely, however, that surface irregularities can undergo considerable permanent deformations without rupturing the eeecimen. Novarthelees, standards of surface finish on specimens used in this thesis aro at least as high as those

of the best GI previous workers, It is not possible to determine the accuracy of creep experiments because there is no means of predicting quantitatively the time stain behaviour of rock. The most useful criterion of experimental method is reproductibility; exi:erimente on the same rock type under the same boundary conditions should not give significantly different results. The reeroductibility of the results of the experiments by the method; suegested above is established in Chapters 4:2 and 4:4. 50

CRAVY211 2

THF ANALYSIS OF AXIAL TRANSIENT CREEP DATA USING ESTIMATIED STRAIN Ref S.

2:1 There is general agreement about a number of features of the creep behaviour of rock (Ramsay, 1967, p. 263), On applying a constant stress to a creep specimen, an elastic strain, E0, takes place instantaneously, Time dependent deformation (creep) then occurs at a rate which decreases with increasing time, If the stress is removed durik; this period, the elastic strain, Bo, is recovered instantaneously; the creep strain is recovered by a time depeniAent process at a decreasing rate. The period of decreasing creep rate is followed by a period in which the creep rate is constant, Creep deformation during this period is permanent and is not recovered

on removing the applied stress, In a final period of creep, the strain rate increases and the specimen eventually fails by static fatigue. The immediately following discussion is confined to the first two stages of creep. The little work that has been done on accelerating creep and recovery is discussed in separate later sections (Chapters 4:7 and 8:b 8:10). 51

2:1

The first modern work on creep is due to Andrade (1910) who investigated the creep of lead, copper and soft alloy wires under tension to strains of about 20%. At suoh large strains, the

str,in measure is important. Andrade found that, after a sufficiently long time, (2/Lt) (dli/dt) was apprrerimntely constants He fitted

L - L = At1/ t o where A is a constant, to the early part of the creep curve and so suggestea a creep law,

L L t = o (1 + Bt143) exp(kt) At comparatively short times or for smaller strains this may be written as

Lt = Lo (2 + Bt ) where B is a constant. Andrade referred to the creep behaviour at short times as beta creep and to the behaviour at long times as viscous flow. Andrade's distinction has been generally accepted though a number of synonyms are used for the two types of creep. Beta creep has been called elastic, primary, transient, logarithmic, Andrade and alpha creep. Viscous creep has been termed pseudoviscous, secondary and steady state creep. This thesis will use transient and steady state as these terms lack any implication of the form of 52

2111 the creep law or its mechanism. Griggs (1939, p. 228) suggested that the total strain in creep, e, vas given by e = A + B log t + Ct The strain referred to is measured by the total fractional deformation, (Lt Lo)/Lo, where Lo is the length of the measure at zero, and Lt is the length at time, t. The parameters, A and Bo were estimated by plotting the oreep curves on semi logarithmic paper. A straight line was drawn through the early values of e. The slope of the line was used to estimate B and the intercept of the line with the strain axis was A. Thus, A is the strain at one time unit (usunrly one day), Griggs. (1)39* figs. 4., 6, 9, 11, 12). The straight line fitted to the earlier data was subtracted from the later data and the differences plotted on ordinary graph paper. A straight line through the origin whose slope was C could then be fitted to the results. Griggs (1940) suggestee that the creep of alabaster at low stresses in the presence of solutions followed the form, B e =At The constants* A and B, were determined from the strain-tine plot on double logarithmic paper. The parameter, A, was the intercept of the straight 1 ne on the strain axis and B the slope of the line; 53

2:1 A is, again, the creep strain at one time unit, Hardy (1956) proposed a creep law derived from the Burger's rheological body. it was e = A + B (1 -exp(-Ct)) + Dt where A was the initial elastio strain, B(1 exp (-Ct)) represented steady state creep. Al B, C, D are constants,. They were determined from a "complete" creep experiment which included both creep under a constant load and recovery under no load. Then the permanent deformation is Dt and the amount of strain recovered under no load is B. The parameter, C, can be determined from the form of the curve shewin: recovery under no load. Unfortunately, the permanent strain observed in the one experiment Hardy had. performed was g. eater than the total creep deformation. Iatsushima (1960), working on the creep of granite in uniaxial compression, suggested a creep law for axial deformation e = Ao + Al exp('.-a,t) 2 exp(-alt) + A B logt + et and e = A + B lost + et for lateral deformation. Price (1964) demonstrated a paradox in a group of three experiments on ;:arkham sandstone. Tie plotted the first experiment on ordinary time-strain coordinates and showed that steady state creep appeared to be the dominant phenomenon after ten days and to continue for another thirty days, its total duration. However, data from all 9+

2:1 three experiments, the longest being 470 days plot close to a straight line on strain - log (time) coordinates (Price, 1964, Fag. 11.b.)

Miara and. Murrell (194 have fitted a power function, e = Ath , to their data. At small fractions of their melting temperature, the creep of the rocks can be described by e = A + B log t. At above half the melting temperature, it is close to e = At1/3. At intermediate temperatures transitional forms, such as, e Ai + A. log t 1 B A3Bt + Alkt are suitable. The tyansition also depends on stress. Constants were estimated by th_ methods described by Griggs (1939), (1940). Le Comte (1963) and Hendren (1968) have fitted e = AtB to their data on the creep of salt, again, using Griggs' (1940) method of calculating the constants. Hardy (1965) has used a numerical technique based on a suggestion by Hartley (1961) to fit the Burgers model to Price's (1964) experiments on Markham sandstone and to a number of his own experiments of much shorter duration. He concluded (p. 11) that the fit was satisfactory. Hardy's method is reviewed in Chapter 3:2, the general problem of the reduction and analysis of transient creep data is discussed in Chapters 2, 3 and 4. This short review shows some of the problems for structural geologists in the interpretation of creep data. Consider, for instance a typical creep experiment on a specimen of Pennant sandstone 55

2:1 under a load equal to half its instantaneous compressive strength (B1.1.31). If the material creeps in the manner suggested by Hardy (1958), then Table 9 shows that after ten years there will be a permanent strain of 50kz in the specimen. If creep follows Misra and Murrell's (1965) proposal, after ten years there will be a creep strain of 0.05;1 and it will take 1028 years to reach a 50;1 strain (Table 10). Creep data, then, are practically useless to the structural geologist unless much closer limits can be placed on the way the data are extrapolated to times longer than the duration of the experiments. At present, it is not possible to limit in any useful way the conditions under which, say, a naturally observed 10;," strain may have taken, place. Nor can anything useful be said about the conditions under which certain distributions of strain through a rock mass may occur. Hardy's conjecture, for instance, is equivalent to suggesting that after a small initial transient strain, further strain hardening is negligible; Misra and Murrell, however, consider that, under atmospheric conditions, strain hardening is always severe. Under s-imilar boundary conditions, entirely different strain distributions would develop in the two types of body. The underlying problem is to determine more exactly the form of the creep curve for rock. This problem is composed of several 56

2:1 - 2:2 others - that is the form of the transient creep curve? When does steady state creep begin to be important? How does the creep rate in each of the stages of creep depend on the ambient environment of the specimen? These problems can best be attacked in the sequenoe in which they arise. First, the form of the transient creep curve can be determined from the earliest part of the creep curve. Steady state creep then begins to be significant when creep can no longer be satisfactorily described by transient creep alone and so on. The next three chapters (Chapters 2, 3, l) discuss the enellysis and reduction of creep data in detail,

2:2

Chapter 2 is concerned with the neelysis of creep data using estimated strain rates* The laws of transient creep can be divided into two main types - an exponential law

czp (1) P"t = A1 (-A2 t) and a power lay • B E = B 1t 2 (2) t E t is the strain rate at time, t, after loading the specimen 2:2

Al = Eo

B1 = El • • where A2 is a constant determining the rate of decrease of E, the rate of strain at zero time. B2 has a similar relation to B1, the rate of strain at unit time. With increas ng duration and hence with increasing strain the rate of deformation given by (1), (2) decreases. A2, B2 may be considered as strain hardening parameters. Equation (1) is more familiar in the form Et = (Al/A2)(1-exp(.4120), the creep law of a Kelvin body or as t = (A1/A2) ex; (1-A3 -1 (mv1.20), the creep law of a general linear rheological body where A is a 3 constant.

Equation (2) can be written as

Et = El + B log t (5) if B, = -1. Equation (5) is then the familiar logarithmic law of transient creep if

Equations (1) and (2) can be compared. Taking natural logarithms of (1) and (') gives log tt = log Al (6) log T.:2t = log B, B2 log t (7) 2:2 Dikrerentiating (6), (7) uith respect to time leads to

d (lob; Et)/dt = -A2 (8)

d (log Et),/dt = B2/t (9)

The value of B2 usually lies between zero and -1. From (8), it can be seen that the rate of change of the logafithm of the strain rate in an exponential law of transient creep is a constant with respect to time. Equation (9) shows that for a power law the rate of change is inversely proportional to time, that is at longer times the rate of change is less rapid thafl at short times. Because the left hand sides of (8) and (9) are monotonic functions of the derivatives of the strain rates the rates of change of the strain rates in the exponential and the power laws of transient creep behave in a similar manner to the right hand sides of (8) and (9). When t is large, the rate of strain hardening in a power law of transient creep will be less than in an exponential law. Vhen t is zero the rate of strain hardening in a power law of transient creep is infinite. At an intermediate value of t the rates of strain hardening are the same. The laws of transient creep, (1), (2), thus present distinctly different descriptions of the behaviour of rook in transient creep and it is unlikely that both laws fit the experimental data equally well. It is indeed, perhaps surprising that both laws are still 59

2:2 in use. But the understandable preference for the quicker, more visual, graphical methods of fitting creep laws to data has led to a lack of quantitative comparisons between the "goodness of fit" of the two laws to data. Graphical methods can also con Use even qualitative comparisons. Consider a group of n observations (xi, yi) displayed on Cartesian coordinates. There are at least three criteria to determine the best fit straight line to the pletted data points. The sum of the distances that the y coordinates of the data points lie from the fitted line may be minimised; the x coordinates may be minimized and the Perpendicular distances from the straight line to the data points may be minimized. aich of these three criteria is most suitable deeends on the structure of the errors involved in the collection of data. Which criterion is used in the graphical fitting of the data depends on the way the data is displayed when the straight line is drawn. Consider the effect on the straight line fitted by graphical methods of increasing the y scale of the Cartesian exec. Discre- pancies in y would then play a dominant role in the fitting process and discrepancies in x would be suppressed. Supposing that strain had been plotted on the y axis against time on the x axis this would be equivalent to suggesting that errors in observing the strain are much larger than errors in observing the time. 60

2:2 - 2:3 Assumptions, than, about the relative size of the scales of the axes are equivalent to assumptions about the structure of errors in collecting the data. A change of scale or oven the decision to include in the strain some of the initial elastic strain will result in a change of the parameters of the line which appears to be "best fit. to the data. Besides the effects of scale on graphical determinations of the creep lays, further npoortainties result from the errors introduced in plotting data and the personal element in determini the best fit straight line* DiffLrent people using the same data may obtain materially different results. Graphical techniques are not reproducible. The solution to the problem of the uncertainties of the graphical techniques, the use of numerical methods, was, until recently comparatively slow and expensive. But advances in the design of electronic digital computers have brought accurate numerical techniques into the cost range of processing this data.

2:3

The first objective of a fitting technique will be to calculate the parameters of the laws of transient creep in a reproducible manner according to a realistic criterion for the fit of the law to 61

2:3 the data. This criterion should be based on a. reasonable model

of the errors involved in the experiment, The quantities measured in creep experiments are strain and. time. The strain measurement will include errors due to instrument drift, =111 chances in ambient temperature and humidity and to vibration and disturbance of the specimen, It is subject to considerably more error than the time measurement which only involves a clock reading when the strain measurement is b-in6 taken and when loading of the si:ecimen finished. A model in which errors in meavuriik time aro negligible compared to errors in measuring strain is thus the most reasonable. Come experimenters have measured the strain at predetermined time intervals. This procedure is equivalent to assuming the time is measured without

error (Acton, 1959, p. 53), The power and exponential laws of transient creep can be transformed, into straight line form. This has been demonstrated in equations (6), (7). Simple linear regression can be used to fit a. straight line to two variables only one of which is subject to error. r,imle linear regression has not been applied to the fitting of crce,,9 data before and there is no discussion of it in the literature on creep and so a short description of the larocess follows. More extended descriptiolle are given by Acton (19!)9) and by Bald (1952, C.hu,ter IC.. in t:isz' books servo as an introduo-

tiun to z-he con:Adorable t it±u.1 litc,rature on thi.: subject.

Li!), then, there aro a grouL of n acervationa of two variables (X, Y) and a pentulatr_d r lationship blAween them

Y = A + (10) True values of the variables and of the parameters of the ritta lino (J1 and B are in upper case; estimates of them obaorva- tion, for instance, are in lower case.

ior any pair of observation,

ZY%:=0-4-11,2:.+11 (11) In this equation u is an estimate of L1 where U ia a random variable mado necessary by thc dei7.arturo of Y frail eaot th dsponthAlce on X The pars,mwt:-ru. is ilnown is tho u.:31,,...rally„ only th absolute value of the ith residua/ is important. th hothor the observation lies ab ve or below the best fit line is, for the moment, of little interest compared 'with tun; far from the lino it lice.

The distance of the line fro.n the observations can be conveniently iric usin the sum of the squares of 'tr.:, residuals*

= b. y (12)

quation (12) avoids diflUultios v‘ith the alzebraie in of the residuals. It iL knozn as a. 'acast squares" criterion. 65 •

511-: indicated summations extend over all n observations. quations (13) and (14) are derived by partial differentiation of (12) with respect to a and b and then applying the conditions for a turning point,

n b E.i = E yi (13) abEx. + (Ex. )2 = xi yi (14) (13) and (14) can be solved. for a and. b

a = (217Y° EKI:x44)

(n rx. ( rx. )2 (15)

b = (ntxiyi - ExItyi)/(nExi2 (573i)2)

Equations (15) and (16) are not symmetrical in xi and yi. Hence, had it been decided that the yi were without error rather than the xi, thevalues of A and B determined would have been different" Equations (15) and (16) show that the first objective of the fittin; technique has been reached though the least squares criterion needs further justification, Equations (15) and (16) can be used to estimate A and B acoording to a model where the error in measuring one variable (the strain) is very much larger than the error in measuring the other (the tine). The next objective is the derivation of tests to determine whether the proposed creep law is a reasonable fit to the data. if the creep lac is a reasonable fit to the data then the residnpls will be unimportant, random noise about the straight line relationship (10) between X and. Y. If the values of the residusls show some dependence on the variables, ,34 y, then the proposed law is not a satisfactory fit to the data, because there still remains in the residnzos a systematic variation which the creep law has not satisfied. There are three ways in which the residuals can depart from randomness. The first of these ways is positive serial correlation. This th is said to occur when there is a tendency for the i residual to be similar to the (i + 1)th residual and the (i - 1)th residual. If the ith resienP1 is positive, the (i. 4. 1)th residual is likely to be positive. If the ith residual is negative, the (i 4. 1)th residual in also likely to be negative. negative serial correlation, the tendency for the ith residual th th to be dissimilar to the (i 4. 1) Said (i - 1) residuals is not important in the analysis of straight line data if there is no a priori suggestion that the data migW be periodic or better fitted by two straight lines. As a test for serial correlation, Durbin and aatson (1951) suggested that du should be calculated 65

2:4

= . )2) / rU, L. 2. (17)

If the residuals are positively serially correlated, dw will tend to be sall. if the residuals are natively serially carrel- ated dv will tend to be large. Durbin and '„atson (1951) tabulate two Groups of oritical values for du aainst n, the number of observations; an upper value of dw, which, if not cxceeed sue.geste that positive serial correlation of thP residuals 1.jilt exist in- the, observations, and a lower value of dw, which, if not exceeded suggests that positive serial correlation exists in the sample. The lower value of dw for a one per cent confidence level ranges from 0.81 for fifteen observations, the minimum number for a satisfactory test, to is47 for eizhty observations* The upper value ranges tram 1.07 to l.,52 over the same interval*

if the line suggested by the estimates of A and L were a good fit to the data, then there would be a large number of observations close to the line and a smaller number of greatc,:r distances from it.

Eormally, if the residuals are randde estimates of the same variable, e, they should be normally distributed with zero mean and constant variance, Non zero moans of the residuals and departures fraa a normal distribution arc the other two ways in which -die residuals can depart from randomness.

The hypothesis that the residuals have a normal distribution. 65

2:4 with zero mean can be tented against the chi-square distribution (Bald, 1952, pp. 740-742). Dixon and Llaesey (1957, pip 222) coament that in the chi-square test, none of the ex2ected flequencies in any of the classes should be less than one and not more than one fifth of the expected frequencies should be leee than five. ause only the variance of the distribution residuele, U, need, be Letiaated, the degrees of freedoa associated with the test are two le:s than the number of cells. iifteon observations, then, is ho minimum number before this test can be applied.'eUrther observations increase the reliability of the test. Wei-square is tabulated by Lindley and Miller (1952, Table 5). If the residuals do have a normal distribution with a mean zero then estimates of the parameters by the least squares criterion will coincide with tee estimates derived by maximum likelihood methods (iald, 1952, e. 204). The estimates of ,1 and. Is are then best in the sense that they are normally distributed with the required parameters as mean values and with the smallest peasible variances (JIald, 1952, p. 927). Aon the least squares criterion has this special significance, it is natural to use it for fitting straight lines to data. The Durbin iiatson statistic and a chi-aquare teat on the residuals can be used to test whether the creep laes are reasonable fits to the data. If both the laws are reasonable fits the next problem 67

2:4 . 2:5 is to determine whether either law is significantly better than the other.

2:5

The variance, u2, is estimated by

u2 = (1/(n-2)) Eui-ri)2 where yfi = a + bxi and so y is the value of Yi estimated by the fitted line.

The variance is a measure of the amount of variation about the fitted line, that is, a measure of the "closeness" of the fitted line to the data. Lecause U is assumed to be normally distributed the hypothesis that two estimated variances are estimates of the same variance can be tested by referring the ratio of the estimates to F tables (Lindley and Miller, 1962, Table 7). If the hypothesis is rejected, then the estimated variances are, each, estimates of different variances, the larger Estimate estimating the larger variance. Then the line having the reeller variance is a better fit to the data. Critical values of F vary with the degrees of freedom that the estimptes of the variance are based on and with the confidence level t which the hypothesis is to be tested. The most common confidence level used in statistics is five per cent (Acton, 1959, pp. 22-23), but, because of the large amounts of data analysed a one i;er cent level is used in this thesis. Confidence limits need careful 68

2:5 interpretation. In this case, if the estimates were estimates of the same variance then a value of F greater than the critical value for a one per cent confidence level would occur once out of a hundred in a long series of tests of this typo. The initial value of F at a one per cent confidence level for fifteen and fifteen degrees of freedom is 3.56. Another use of the variance is to test the hypothesis that the fitted line does not have a slope significantly different from zero, a hypothesis equivalent to suggesting that the data might be as well represented by a constant and that the fitted line has not picked out

any significant variation. For this test, $1 is computed,

= 11:(Y1 • Y.)2/12 (19) where y. = . The significance of 111 is determined by entering tables of the t distribution (LinAley and Miller, 1962, Table 3) with (n — 2) degrees of freedom. Lindley and Miller tabulated t for a two sided risk. For R1, the appropriate risk is one sided, but As the confidence levels for a two sided risk are twice those for a one sided risk, the two per cent level for a two sided risk is equivalent to the one per cent level for a one sided risk. If Ri exceeds the tabulated value at the two per cent confidence level there is less than one chance in a hundred that this value of Ri 69

2:5

can have arisen by chance Pad the hypothesis should be rejected (Hold, 1952# p. 540). The statistic, lii, can also be referred to F tables with one and (n - 2) degrees of freedom, The variance can also be used to calculate confidence limits on the parameters of the creep curve, Th* are (Acton, 1959, p. 23),

t (nu2/(nEaci2 ()

+ t (u25,i7 2/(4: (lExi)) (21) whore t is the critical. value of the t distribution for a two per cent two sided risk with (n - 2) degrees of freedoms The confidence limits mean that, had it been asserted that the true value of the parameter lay with the limits given by (20) and (21), this would be a true statement ninety eight times out of a hundred statements of this type, .!ieparate confidence limits do not clearly indicate the possible range of values of the parameters of the creep law which could conceivably describe the data. To define the straight line that has been fitted* values of both crtep parameters are needed. To define the permitted range of these parameters a joint confidence region is ncoded. If a and b arc plotted on orthogonal axes the joint confide:me region appears as an ellipse with axes inclined, to the axes of the plot (Acton, 1959, p. 28). This ellipse may be approsimated by 70

2:5 - 2:6 either of two parallelograms. This ambiguity arises from the sequence in which the parameters are calculated i.e. whether the slope of the line is calculated and then the intercept with the y axis calculated assuming the slope is known or the other way about. Representing a joint confidence region in this way is complex and it is usual to proceed in another way.

2:6

If the straight line had been described by

Yi = C + D (Xi - X.) (22) then C DX. = A The sum of the squares is then

2 tri 2 = 1E: (y. C - D (x. - x..)) (23)

Applying the conditions for a turning point leads to

E(Yi. C D (xi - xi")) = 0

I: (Yi C D (xi . x.))(x. x.) = 0 Since E(xi » x.) = 0 C =E = y. (24) ..111111•1111101•1110•••• n -D = E c.) (25) E (xi - x.)2 71

2:6 . 2:7 In this form, then, the conditions for a turning point load to two equations each of which may be solved directly for one of the parameters of the fitted straight line. The estimates of the parameters are thus independent (Had, 2.952* p. 534.)* Acton (3.99, p* 28) showed that the joint confidence region for C, D can be revrestmted by an ellipse whose axes are parallel to the axes of the C, D space. Furthermore the semiazos of this ellipse are nearly equal to the confidence limits on Co D* These aro 2 4- t (u in) (26)

D t (0,/ - Ya (27) and the ellipse may be represented without serious error by a reotngle with sides twice the confidence limits, if a pair of parameters be within the confidence limits on C and D then the probability that these parameters could describe the data is better than v.C2.

2:7

the oLthogonal form (Acton, 19i94 p. 26) offers consider. statistical advantag.s, it Gives estimates of parameters which are often not immediately useful. .it can bo seen that A is in both (6) and (7), the strain rate -t one time unit, and C is the strain rate at the moan of the lozarithms of the time in (7) and at the mt:_an of the times in (6). 14hileAcan be directly caapared from 72

2:7 — 2:8 experiment to experiment, C varies vith the duration of each experi— ment and with the form of the creep law. Notice however that C is the most precisely determined of all the strain rates in the fitted data. Paraphrasing Held (1952, p. 536), the variance of the strain rate is oval to the variance multiplied by a. polynominal of the second degree in time (if the creep equation has the form (6), or in the logarithm of the time if the creep equation has the form (7)). The variance of the strain rate takes its minimum value at the mean time y.. Thus the uncertainty about the strain rate increases with the distance between the time at which it was measured and the mean time. The most suitable form of the straight line in the regression analysis varies with circumstances. It is sensible, then, in fitting a straight line to follow the computational scheme set out by Acton, (1959, p. 11) computing the values of C and L (whence A may be found) and providing for both forms of the line.

2:8

Having discussed the general problem of fitting a straight line to a group of data by simple linear regression, some of the problems raised by the form of the transformations, (6), (7), of the transient creep laws to straight line form should now be considered.

The strain rate, 8i,in the time interval (Ti, Ti + 1) may be 7.3

2:8 estimated by

(ei Ti) = (28) Two problems are apparent. The first is the avoidance in (6), (7) of non-computable (that is, negative or sera values of ti) values of log ti. Such values arise when ei is less than or equal to ei. As no creep law proposes a model in which the creep strain decreases with time under compressive load, strain reversals aro considered to be caused by fluctuations in experimental conditions sufficiently large to conceal the increase in strain due to creep in is too large or the time interval. It is not clear whether ei is too small but it can bL supposed that e are ei + 1 i, ei + I estimates of the strain ei at time, t1.

(29) el- 4":(e i 1 4.e1.)/2

t. = (T (30)

It is then natural to weight (e.„ t.) twice as it is based on two estimates(e.l i), (ei 1, Ti 1). A check on this conven- tion will be discussed when regression on strains rather than strain rates is described. The second problem arises from the difficulty of assigning an appropriate time, ti, to the strain rate, el.. As an illustration, the creep strain in the time interval Ti, Ti can be derived when 7'

2:6 a logarithmic creep law oi:erates, thu logarithmic law

thus (T + - T) = 2 (log Ti * - log T4) where is the strain rate in the interval (Ti' Ti + /)

8i = (131/ti) by the logarithmic orc.ep law, when t.a. is the appropriate time.

Ti *

(.51)

7quations for a power law of transient creep, oi - B t/32, and for an exi,onentiza law of trensiLnt creev, 11.1 exp (-k2t), can be similarly derived. They are (32) and (33) B) ti (Ti (13,, 1)Itiftl%*2 1)(ti ti) (32)

ti = (1/A2) log (A9(t + - ti)(ex(-A2ti) ex(-At 1))) (33)

2quationa (32) and (33) show that the value of ti depends on the parameters of the transient creep curve. If the value of t. is critical. to the fitting procedure an iteration routine to generat and modify successive estimates of the paramritere of the laws would be necessary. 715

2:8 Under any creep law Ti lies between ti and (ti ti 1)/2, as every transient creep law predicts a strain rate that is always decreasing. To investigate the dependence of the creep parameters, on the technique of choosing i i a sample of creep experiments was chosen from the recent literature and the creep laws fitted to them in two rays, the first using

Ti = t. (34)

4- and the second, T. = t. t. .1. 1 (35) 2

The sample covered a wide range of conditions and rock types. The fits of both the exponential and the power law of creep were determined and tabulated in Tables I and 2. Table 1 shows that for the exponential law it mAkes little difference which technique is used to estimate the parameters of the creep law. Each estimate of each parameter is well rithin the standard error of the estimate of the parameter by the other technique. Tests of hypotheses would give the same results by either method. Seven of the experiments have fits -ehich allow definite positive serial correlation by either fitting.,technique. (Table 1, oolemu dm)* Remember that statistics significant at the 1:: level are followed by two apostrophes. One experiment shows no evidence of serial correlation (PR 1(3)). 76

2:8

Table 2 shows less- agreement between estirmtes of the parameters of the creep law. :However, there is no evidence aL the level to suggest that any pair of estimates by the two methods are significantly different, Eon° of the experiments show evidence of positive serial correlation (Table 2, column dw). All the fits in both tables have significant slopes (columns r1). On the evidence of this sample of creep data, the technique of choosing, t. is not critical either to the estimation of parameters or to decisions about hypotheses if ti is chosen within the limits

Ti, (Ti Ti

In this thesis, ti is defined by (36), ti = (TeTi + 1)1, (36)

1iriting Ti = Ti Dt, (36) gives (37)

ti e (1 4. DOI) i . 2 (37) Hence,

t. = Ti + Dt .;:(Dt)/Ti (38)

Equation (38) holds if Dt is less than T. From (38), t. lies between T. and (T T + 1)/2 and is a convenient form for coputation, Having outlined these attempts at solving the: particular problems raised by the transformation of the creep laws to straight line form, the calculation of the parameters of the best fit straight linos 77

2:8 - 2:9 can follow. A computer program 1, listed in Appendix 1, has been written in Fortran IV for the IBr. 7094 computer. It has been used to fit the power and exponential transient creep laws to a. large sample of creep experiments and to some of the new experiments which form part of the work of this theeie. The parameters of the fits are tabulated in Tables 3 and 4, respectively.

2:9

Ideally all the creep experiments so far performed, on rock in uniaxial compression would have been included in the analysis. This would have led to considerable problems of data handling. It should also be recognised that all the experiments do not provide equally severe tests of whether the pre:20130d laws of transient creep (1), (2) are reasonable explanations of the experimental data. Nor are ail the experiments equally suitable for testing whether, if both forma are reasonable, ono is a better fit than the other. It is not surprising that some creep experiments are better than others for testing these hypotheses. Some ezperiments, those of Price (1964)0 for instance, were not designed to inieetigate transient creep at all but to investigate steady state creep, but they should not be excluded for this reason as they uay still provide useful information. Rough indices of the usefulnees of a creep experiment in the 78

2:9 proposed teats can be developed in the folloAme way. If it is supposed that the creep rates predicted by the two laws (1), (2) of transient creep at t = 1 are the same, then from (6) and (7)

log Ai - A2t = log B1 + B2 log t (39)

and as t = 1 .

log Al - A2 = log J (4.0) The difierenoe, B, of the logarithm of the strain rates predicted by the two laws can be written, usin, (39) and (40), as

D = B, 106- t 4. A2 (t - 1) (41) where B2 is negative, generally having a value about (-1), A2 is positive with a value considerably less than 1. The absolute value of A2 is generally considerably less than the absolute value of Be The parameter, D is a measure of the difference in values of strain rates predicted by the two transient creep laws. Large absolute values of D indicate a large difference in the predicted values of the strain rates and hence the possibility of severe tests. Consider the variation of the absolute value of D with time. D is zero at t = 1, The modulus of D reaches a maximum at = At, it is zero again 2 4 at -B2 log t = A2 (t - 1). It then increases as t increases. In this region of increasing D, it can be made as large as is necessary 79

2:9 to distinguish between the creep laws by designing experiments in which t took sufficiently large values. This possibility will be taken up again in Chapter 4.:5 when experimental design is considered. Notice that the units in which t has been measured have not entered the discussion so far nor has the meaning of the assumption that the predicted strain rates are equal at t = 1, an assumption which affects the choice of units for t.

Intuitively, it can be seen that as the strain rate in a transient creep experiment is largest at the beginning of the experiment a large number of measurenonts of strain which would be estimates of significantly different strains can be made then in a comparatively short time. These measurements will have great weight in the estimation of the parameters of the creep law so that the strain rates estimated from these parameters should be similar for the same group of data in the early parts of the experiment. It is convenient to suggest that when t = ti the strain rates estimated by either law will be approximately oval* This provides units for t. The ratio, to/ti, might be a useful index of D and hence of the usefulness of a particular experiment in testing the hypotheses about the creep lass. The content of the sample of creep experiments chosen is not then random, It is biased towards those experiments with high values of tn/ti* This ratio and the value of to is tabulated in 80

2:9 Table 3. Some almost classic creep experiments with relatively low values of the ratio have been included in the erelysis for comparison. Two experiments have been chosen from Griggs (1939); one, on Solohofen limestone is still the loneest duration compression creep experiment which has been run on rook* the other* on a single crystal of halite, has been chosen because Griggs considered it deviated from the logarithmic creep lay. The three experiments on Markham sandstone taken from Price (1964.) have been included to look at what is a possible specimen to specimen variation in the same rook type at the same load. Together they present a detailed and important picture of creep in the rock over 1+70 days. A large number or experiments have been taken from Misra (1962). Misra's work is the only comprehensive investigation of the effect of temperature on the creep of rock in uniaxial compression. The experiments analysed are at the extremes of the temperature ranges he covered for each rock type. Three of Uisra's experiments were carried through to the failure of the rock in creep, these will be considered separately (Chapter 4.:7). Dare for Griggs (1939) experiments was taken from his figures 4. and 9. Data on Price's work was taken from Hardy (1966) pp. 55-58 and from Price (1964) Fig. lla. Asra (1962) listed numerical values of strain and time for his experiments and so eliminated any 81

2:9 2:10 uncertainties involved in determining thee values from graphs.

Data determined from graphs are listea in Appendix 2 together with

numerical values of strain and time for all new experiments.

2:10

The values of the creep parameters in Tables 3, 4. and their

significance will be discuased later (Chapter 8). A necessary preliminary is to establish that the propoeed creep laws are a

satisfactory desoription of the data.

In Table 4 twenty six of the forty eight fits of the exponential creep law Should be rejected since they show significant positive serial correlation (column dw). -3tatistics significant at the 1 level are followed by two apostrophes. in contrast, 'Yable 3 shows

that there are no grounds for rejecting any of the power law fits en the basis of positive serial correlation. Aside from whether the fits are satisfactory it is also possible to test whether the power law is a significantly better fit to the data than an exponential lay. The null hypothesis that the variances estimated by the two fits arc not significantly different can be tested by the ratio of the tabulated. teetc, Li, of the significance of the slopes. This is possible since the means of the strain rate data are approximately the same in each case. Notice that all the elopes of the fitted lines are significant. R. is the ratio of the 82

2:10 tabulated values of R1. In all but seven cases the fit of the power law is significantly better at the 15.;- level than the fit of the exponential law. A further case is significant at the 524 level. There are no cases where the exponential law is a better fit than the power law. Of the six cases where there is no reason to prefer the power law, five show no significant positive serial correlation in the fit of the exponential law to them. So the two criteria recommend reasonably consistent courses of action. To illustrate these results, the data from four typical creep experiments has been plotted, first as the logarithms of the strain rates against time in diagrams 2:1, 2:2 when, on the exponential law, the plots should be linear. The same data has also been plotted as the logarithms of the strain rates against the logarithms of the time (diagrams 2:3, 2:4); the plots should then be linear on the power laws. Inspection shows that only the power law plots (diagrams 2:3, 2:4) seem to be linear. One possible explanation of the significance tests in Tables 3, 4 is that the exponential transient creep law fits experimental data only in so far as it resembles a power law of transient creep. For small ratios of the time of the initial reading to the time of the final reading the exponential lay will be close to the power law of transient creep, at larger ratios tizc difference between 83

2:10 - 2:11 the laws is more pronounced. Thus low values of tr/tl should lead to non significant values of R, and the Durbin Watson statistic for the exponential creep law fits. The four experiments from Price (1964) are a good example of this.

2:11

It can be argued in defence of the exponential law of transient creep that the experiments analysed contain a large steady state creep component. Instead of attempting to fit a creep lay of the form, tt = al exp (-a2t), a creep law,

tt = al exp (-a2t) + a3 (42) should be fitted. This might be closer to a power lay at large times than an exponential law alone and the tabulated data will result.

The product, a2tn is tabulated in Table 4.. If al is the total amount of transient creep in a Kelvin body, al exp(-a2tn) is the total amount of transient creep remaining at the end of an experiment and exp(-a2tn) is the amount remaining as a proportion of the total amount. Tables (Comae, 1950) show exp(-2.3) = 0.1, exp(-4.6) = 0.01, exp(-6.9) = 0.001 whence it seems that in a large number of experiments transient creep is nearly exhausted and steady state creep might be expected to be the dominant phenomenon. Diagrams 2:1 — 2:2

Typical plots of log (strain rate) against time. Legend Vertical axis (abscissa) apixopriate time of strain observation in minutes. Horizontal axis (ordinate) natural logarithm of strain rate in miorostrains per minute.

Diagram 2:1 Open triangles, Anhydrite, 15°C, 4.720 psi, (Misra, 1962). Filled triangles, ,B21310.4

Diagram 2:2 Open squares, ills1:3.5, filled squares A2:2:40.

Comment It the exponential law were obeyed, data from each experiment would fall on a straight line. Some early observations have been cm itted from the plot to clarify it.

dia. 2 :1

A A A A A A

A A A A • • A A A A A

2 loo '200 • '500 '1000 • '1500 '2000 CO A I% Diagrams 2:3 ... 2:4

Typical plots of log (strain rate) against log (time)

Legend Vertical axis, (abscissa) natural logarithm of the appropriate time of the strain observation in minutes. Horizontal axis, (ordinate) natural logarithm of strain rate in microstrains per minute.

Diagram 20, Open triangles, Anhydrite, 15°C 4720 psi (Misra, 1962). Filled triangles, 152:3:04.

Diagram 2$4, Open squares, B1:1:35, filled squares, A2:2:40.

Comment If the power law were obeyed, data from each experiment would fall on a straight line. -6

• dia. 2:3

•• AAA A -4

A A A A A

. • A A A • • A A A

2 •

A .6. • • • • 6. A

... A 0 A A

I , I I I 2 ' Ti '2 I4 5 6 7 8 9

2:11 - 2:12 The work of Price and Griggs in particular is characterized

by experiments of long durational and low values of to/t1. Price's

work showed that a number of very long experiments are satisfactorily

fitted by an exponential creep law. Misra's results showed that comparatively short experiments are not well fitted by the exponential

law. This brings into question the steady state creep hypothesis.

2:12

One way of testing the steady state creep hypothesis which is

closely related to the previous work is described below.

Price (1964) has suggested a method of determining the steady

state creep rate in that part of a creep experiment in which steady

state creep is believed to dominate. If the data is plotted on

strain time coordinates the steady state creep rate is the slope of

a line fitted to the creep curve. At least five data points will be needed for a reasonably

accurate estimate of the steady state strain rate. If, n r e. = m n-5 4- 5

Ti m 5 n 5 the slope b of the best fit line by a least squares criterion is given by (Acton, 1959, p. 14), 91

2:12 .

1:(T. - T )(e. b m 1 5E:(T1 Tm )2

Having estimated the steady state creep rate by (43), strains due to steady state creep can be subtracted from the total strain and the fit of the new data to a transient creep law can be investigated by the methods already described. Table 5 tabulates the fit of an exponential law of transient creep to the data. This method has two main disadvantages. The first is that five data points must be sacrificed to estimating the steady state strain rate. If they are assumed to be due to steady state strain they cannot be used to estimate the transient strain law. The sacrifice of five points will reduce the possibility of distinguishing between hyl2otheses about the form of the remaining transient creep curves. The tests of these hypotheses can be constructed from the ratio of the estimates of the variance of the residuals in the three types of fit. This ratio can then be referred to F tables (Lindley and filler, 1962, Table 7) with (n - 2) and (n' 2) degrees of freedom, (nt - 2) is the weighting (w in Table 5) of the modified data, (n - 2) is w in Table 3. When (n' - 2) is less than five, these tests are unreliable and the results have not been tabulated. Nor is there ) 2

2:12 any point in fitting creep curves to less than four data points.

This has resulted in a reduction in the number of comparisons that can be made. The statistic R contains the test of the hypothesis 3 that there is no significant difference in the estimates of the variance of the ,oeer law fit ane the steady state fit. sixteen experiments are significantly better fitted by a power law of

t_ansient crece at the 1 level than by an exponential law when the steady state creep has been removed. One experiment (FRI (1)) ie significantly better fitted by the exponential law. Five of the modified exponential fite should be rejected since they show definite positive serial correlation. Only seven experiments show eignifi- cantly better fits at the 1:: level to an exponential creep law when steady state creep has been removed. The ratios of the variances for this test are tabulated as

Table sugsests then that some experiment* arc not satisfactorily explained by the exponential plus steady c tate law, (40 and that in cases where a comparison gives a definite anneer, exeeriments are better explained by a eceeer law of transient creep. Table e contains the results of reducing the steady state strain rate in the five exp riments shoeing significant positive serial correlation, The original fit end the fit with an exponential alone are included for coepaxison.

In all five cases, the fie, with the highest steady state strain rate, that ie the fit based on the stead/ state strain rate calculated 93

e:12

from the final five strain readings shows the least variance. For

the experiments on the gramodiorite, the olivine, and the Solenhofen limestone there is an increase in the variance as the strain rate is

decreased. This increase seems to level off in the anhydrite

experiments,

The Durbin eatson statistic shows a reduction with decreasing.

steady state strain rate in the fits to the Solenhofen limestone experiments. All the fits still show significant positive serial

correlation.

Two observations should be made on Table 6; firstly the

significant positive serial correlation in five of the experiments is unlikely to be due to systematic overestimation of the steady state

strain rate.

Secondly that the variance of the resifThels is lowest for the fits with the highest steady state strain rates.

The first point suggests that the five experiments are not satisfactoril;, fitted by an exponential plus a steady state creep and the second would be a natural consequence of it.

In an attempt to confirm this conclusion, that there are some creep experiments which cannot be fitted reasonably by an exponential plus a steady state creep law, the sample of experiments can be analysed in a different manner, using not the strain rates but the

strains. As all previous workers on creep in rocks have fitted 94 creep laws to the measured strains rather than to calculated strain rates, the analysis that follows will also provide a check on the fitting techniques that used strain rates. 95

CHAPTER 3

TiL AALLYSIS Of Ti ANSI, CR2EP DATA u-ia,.; SLIAINS

3:1

It might be thought that the differences between succesAve strain observations could be used in a calculation of the parameters of the creep curve because strain rates have been successfully used in Chapter 2. Suppose successive observations of strain, (ei, Ti),

(ei + T + 1) had been made. Integration of the power law of (b + 1) creep Et = bitb2 , leads to Et - E0 = (b1/(b2 + 1)) t 2 when, as is generally the case, V2 is greater than mirms one (Chapter 3:6) Then, (b + 1) b + 1) ei + 1 - ei = (b2 + 1))(T + 1 - Ti. 2 ) 1) = (b1/(b2 + 1)) (b2 ((1 + (Dt/T))b + 12 1)

by writing T 4. 1 = T + Dt. Because, (1 + (Dt/Ti)72 4' 1) = 1 + b + 1)(Dt/T1) (b2 + 1) b2

(Dt/Ti)2 na Using, as an approximation, the first two terms of the series, 96

3:1 ei 4. 1 - °i = (b1/(b2 + 1)) Ti(b2 1) (Dt/Ti) b, (ei 4. 1 • ei)/Dt = biTi

Thus, using the strain differences is equivalent to using the

strain rates. The final form is an underestimate of the strain rate in the interval (T Ti 4. 1). As b2 is generally close to minus one, the approximation is gross unless (ht/T1) is less than 0.01, which is, of course, a most restrictive condition on the design of the experiment. This problem is equivalent to that of choosing an appropriate value of ti in Chapter 2:8. If, instead., the. first reading were subtracted from succeea3ng readings, then (b 2 ei - el ( T = (b1/(b2 1))(Ti 2 1

e. - e = (b 4. 1)) T (b2 1) (1 (b, + 1) 1 1 1 2 i 1/4114i) ) Itit is tempting to suggest (T1/Ti)(b2 1) approaches zero as ta,. becomes large. Indeed, it does, but very slowly. Taking (b 2 + 1) 0.1 0.1 as 0.1, (0.1) = 0.7951 (0.01) = 0.63, (0.001)041 = 0.50. Thus, the approximation is useless over practical values of (Ti/T). If an estimate of the instantaneous strain (perhaps from ultra- sonic pulse velocity measurements on the specimen) is used as the first reading, the results might generally be better than those 97

3:1 obtained using the first reading of a creep experiment but they woul,L still be far from, satisfactory. If the creep strain recorded in an experiment is 10% of the instantaneous strain, a typical error of 17; in estimating the instantaneous strain leads to an unacceptable 1V, error in estimating the creep strain. Analogous results can be derived, for an exponential law of creep (Chapter 3:4). The only method so far proposed of calculating the parameters of a creep curve directly from the strains uses non linear regression (Hardy, 1965). So far in this thesis regression has been applied only to functions where the value of Y is a linear function of the value of X, Y = A 4, BE (1) If (1) does not satisfactorily describe the relationship between X and. Y, further fUnctions can be introduced. For instance (2) might be tried

Y = A 4. DX + CZ 4. DV ( 2 ) where Z = f (x) 1 V = r 2(x) Linear regression can be extended to the further variables (Held, 1952, pp. 658-649). Unfortunately, some of the forms of the transient creep laws c)3

3:1 discussed here fall into a further class where the parameters Al B.

C, D which should be estimated do not vary linearly with Ye 'whereas linear regression can be used to estimate parameters in (3), Y = A 112C C exl) (X) relationships such as Y = A BX C exp (DX) (4.) when the parameters are estimated bj the conventional "least squares" technique, leqd to non linear partial differential equations which are very difficult to solve. There is a general method of performing non linear regression if a set of initial estimates of the parameters can be found that are sufficiently close to the true values of the parameters. Then the non linear equations can be rewritten in a linear form in terms

of the difference between the initial estimates of the parameters and their true values. Unfortunately, this method does not lead to quantitative statistics of goodness of fit. Existing statistics are based only on mologies with linear regression. But non linear regression does avoid the difficulties associated with the over estimation of rates of steady state creep and it allows the use of

all the data for the estimation of all the parameters. Hartley (1961) has provided a computational scheme for non linear regressions. He described a procedure for fitting a function i9

- 3:2 Y = f to a resi.onse, y, to various inputs, The paramuters, (01 CI) were estimated from wand tha. inlJuts. In fitting creep laws, Eardy (l96) identified the rcsonsc, y, as the strain in the eNperiment and the inputs (xi x. as functions ofth e 4) duration of the experiment when this strain uas measured. '2ho ixoblcm, then, in Hartley's tsrainology (19,11, p. 27)), is to deta•mineasetof-suchCI that

n '2(C) = " f(711; c)) hal is a Pirimum. Surviation is over all the inputs and the n strain measurements. The analysis is analogous to simple linear regression but Hartley showed how convergence to au absolute minimum could be euaranteed under certain conditions. The detail of the method is complex and will not be discussed here.

3:2

The 1,xincipal practical difficulty of Hartley's method is the estimitioneaninitialsetofi;..ilardy (l965) suggested a method of obtaining the initial estimates of the parameters in a transient creep law ox' the form (5) :!;.t = ( 1,012)(1 exi) (-A„,t)) A3t (5) 100

3:2 The data were plotted manually on Caltesian strain and time axes and the time, to, at which steady state creep begins to dominate the form of the creep curve v:ere marked off. The slope of the curve at times longer than to gave a3, the initial estimate of A. Choosing three other points with coordinates (e7/8 7/8 to),

(e1/8 t 1/8 to), (°1/2 t 1/2 t0) allowed the calculation (Hardy, (1965, Appendix 2)).

I 3 = 1, 2* 3

(al/a2) = f21/(2f3 fi f2)

= ti/log ((a/a21/((a/a2) fi)), 3r. a2 = 12-2 1/3

Hardy's method is elaborate and still does not guarantee convergence of the iteration procedure, though convergence did occur in all the cases in which he applied, the method. A condition of convergence to an absolute minimum is that there should be no other maxima, minimn or saddle points in the region between the initial estimates and the true values of the parameters. Hartley (1961, p. 274.) commented, "when the surface may have numerous local maxima and/or minima and/or saddle points it is necessary to search the parameter space at a wide grid in an 3:2 attempt to locate a point in the region of convergence". The convergence of one experiment, Eil(1), ehich was fitted by Hardy (19S5, Table 1) has been investigated in detail. The choice of al, a2, a3 is not unrestricted, they must be positive* It is not reasonable to cheese a3 so that it is greater than (en/tn) or to choose (al/a2) so that it is greater than en. supposing the choice of initial constants ie further restricted by en, - (a /a ) = at 1 2 n (6) only al and a2 need be chosen. The best estimate of al, a2, a3 are aperoached by an iteration procedure and. Hardy (1965, Appendix 4) gave a computer program to perform these calculations. The final results (A1/A2, A2, A3) of the iteration process for various

values of a1, a2 are given in Table 7. Hardy's iteration procedure stops after four iterations or when the new estimates of the parameters produce a fit less than one per cent better than the previous estimates. Exact agreement between the final estimates for varying initial values of al, a2 cannot therefore be expected. Table 7 shows that the agreement is to better than 1 part in 1000 for all the parameters. Better agreement could presumably be reached by further iterations. The results are also in good agreements with Hardy's (1965) estimates though he did not determine a3 sufficiently accurately for more than

a rough comparison to be made. 102

3:3

Notice in Table 7 that the region of convergence is large and that no other mivima were found in the area outside this large region. These two observations suggest that mimme. determined without a preliminary survey may be treated with reasonable confidence.

I have derived a simpler method of making initial estimates of the parameters of (5) based on the results of the large number of fits of (5); tabulated in Hardy (1966, Table 2), based on data in Hardy (1966, Tables 7 - 39). These tables have been used to compile Table 8 in which the experiments are designated by Hardy's code. inspection of Table 8 shows that A2tn is always greater than 6. Thus at least 99.75 per cent (exp (-6) = 0.00248) of the total transient creep strain due to the load has taken place by the time the experiments were terminated.. The total transient creep strain can therefore be approximated by (111/A2). It varied from forty to eighty per cent of the total recorded creep strain, en, and exoeptione ally, (PRe2(3))„ reached 99 per cent of the total strain. The remainder of the total creep strain is duo to steady state oreep. As estimates of the parameters of (5), it is simple to define

(a3/a2) = 0.75 on

a = 0.01 t 2 n

a3 = 0.25 en it 103

3:3 3:4 I have modified Hardy's computer program (Hardy, 1965, Appendix 4) to generate al, a2, a3 from the experimental data. Comparison with Hardy's method for FR1(1) shows that the estimates of the parameters derived by both methods are in excellent agreement. This agreement is a natural consequence of the large region of convergence FR1(1) shows in Table 7.

314

The measured strain at time, t, in a transient creep experiment is the sum of et, the creep strain, and en initial elastic displacement, eo. The estimation of eo presents a problem when working with strains (it is, of course, no problem when working with strain rates). Hardy (1965, pp. 39-43) implicitly suggested that the problem could be solved by transferring the origin of the strain time coordinates to the first strain time reading. This is equivalent to defining new variables, et, t such that

et = et e1 and

t. = t t1 Assuming that creep takes the form (5) and that in the early stages of creep when Hardy's suggestion will lead to most error, steady state creep makes a negligible contribution to the total creep 104.

3:4. strain, (5) written in terms of the new variables, will be (7) in terms of the old variables

et - el = (al/a2)(1 exp (-a2(t ti))

= (al/a2)(1 exp (-act) exp(a 2t1)) (7) If a2ti is less than 0.15, exp (a2t1) can be written with less than one per cent error as (8),

exp (a2t1) a (1 ta2t1) (8) Using (8), (7) gives (9)

(et - el) = (al/a2)(1-exP(-a2t) a2tIexP(-a2t1)) (9) Because et = (al/a2)(1 exp (-a2t)), the difference between

(5) in the new variables and the old variables reduces to the assumption (10)

el a alt1 exP (-a2t1) Again, in the old variables, el = aiti if a2t1 is less than 0.15. Thus the percentage error in et caused by the transfer of the origin is

(100 ela2yet) (10) 105

3:5

Errors caused by Hardy's suggestion are thus minimized if the first reading is taken shortly after the loading of the specimen is complete and the experiments have a long duration.

An estimate of co has already been made by the authors in the creep experiments of Mara (1962), Griggs (1939) and Price (1964)* Only the new creep experiments of Tables 3, 4, 5 have been modified. All of the experiments in Tables 3, 4, 5 have had (5) fitted to them by the modification of Hardy's method. Convergence occurred in every case except Misrule experiment on dolomite* This experiment will be considered in more detail in the section of the thesis tint discusses tertiary creep (Chapter 4:7). Table 9 contains the best estimates of the parameters and the value of the Durbin Watson statistic for each fit. Unfortunately the Durbin Watson statistic has no quantitative significance in non linear regression. If it had, it would seem that a further group of experiments shoeing significant positive serial correlation had been added to those of Table 5. This rise might have been attributed to the fact that Hardy's method uses all the data, not excluding the last five readings in an experiment. The larger amount of data gives more precise answers to hypotheses about the distribution of residuals.

Another qualitative indication of satisfactory explanation can 106

3:5 be derived from the estimates of steady state strain rates (column a3). There is reason to suspect that the estimates of steady state strain rate is Table 5 may be systematically high. If the experiments can be satisfactorily explained by an exponential plus a steady state creep law, then, the estimates of the steady state creep rate in Table 9 should be either systematically lower than those of Table 5, or about the same value. A power law of transient creep would predict a less rapid rate of strain hardening in transient creep than an exponential. If the data followed a power law of transient creep, the discrepancy which the steady state creep rate would attempt to accommodate would increase with time. The steady state creep rate may eventually exceed the estimates of Table 5. Comparison of Tables 5 and 9 shows that estimates of the steady state strain rates by Hardy's method are consistently higher than those tabulated in Table 5, which indicates that some of the data is unlikely to be reasonably explained by an exponential law of transient creep plus steady state creep. The data of Table 9 thus qualitatively confirm those of Table 5 and again auggest that there are some experiments which are not satisfactorily explained by a creep law of the form (5). Comparing the transient creep laws it was found that besides providing a satiafactory explanation for n31 the experiments in the 107

3:5 3:6 sample, the power law of transient creep tended to fit the data better than the exponential law and better than an exponential law plus a steady state creep component (Tables 3, 4, 5). This coincidence was to be expected when the regressions were linear. When regression is non linear, goodness of fit should be investigated separately and the remainder of Chapter 3 is given over to this investigation.

3:6

The results of Hardy's method can be compared with the fits of a power law to strain data if the initial strain, Eo„ can be estimated. Errors of the type discussed in equations 7 ... 10 can be avoided in the following way. As B2 Et = B1t

Et B de :: Bit 2dt

B2 4. 1t Et = (0082 1)) t Jo (u)

If B2 is greater than .61 the right hand side of (11) can be evaluated directly, (12)

108

B, + 1 E = (Bi/(B + 1))t (12)

if B2 is less than (-1), (11) becomes (13) .0, + 1) —(L, + 1) - so = (13//(B2 + 1))( (lit 4 ) (1/b " )) (13)

Tiw right hand side of (13) is indeterminate. If, however, the lower limit of integration are changed from zero to one time unit, +1) + 1))(1 t ) (14)

El. is the strain at 1 time unit, (14) is formally reminiscent or (5). If 132 = .1 tile right bona aide of (11) is again indeterminate. /t dt/t has been dicussed by Hardy (1914, pp. 357-.3(51) because

the integration cannot be performed by the usual rule for integrating rowers of t. His solution is to define a special function log t such that log t = / dt/t. All the algebraic properties of Y3./f t

natural logarithms can be derived from this definition of the logarithm of t. ;4ith the new limit of integration (11) can be evaluated as (15),

Et "' = /1 log t (is) Hardy (1914, p. 361) also showed that log t tends to infinity more slowly than any positive power of t. There is thus a transition

109

3:6 between creep laws of the typo (12), in which the creep strain tends to infinity more quickly than of the type (15), through the logarithmic creep law, (15), to a creep law (14) where the strain tends to a constant value, f1 + (B1/(B2 + 1)), as t tends to infinity. The transition is through creep laws with larger rates of strain hardening. An estimate of can be derived from (12) (or an estimate of

1 from (14) or (15) depending on the value of B2) since the estimates of Et, t are observations and estimates of B1, B2 have already been derived by strain rate fits to the data, Table 3. Each observation (ei„ ti) provides by substitution in (12) a separate estimate ea

of Eo. If a power law is a satisfactory fit to the data then the estimates of E will be normally distributed about ems,the o arithmetic mean of the estimates, ea.

i =n B + 1 n eom 2 (16) == 1 - (B2/02 1)) ti The estimated variance of the estimated mean a2 is given by

eom)2 (n - 1) s2 = oi (17) and confidence limits on the estimate are

e OM + to = con e0 (18) where t is the critical value of Students t distribution at 10;:- no

3:6 - 3:7 confidence for a two sided risk with (n 2) degreus of freedom.

The computer program, 2, listed in Appendix 1, was written to perform the calculations of (16),, (17) and (18). It requires a slight modification for the cases when B„, is less than (-1). There are no cases in the large sample of experiments where B2 is

exactly -1. eom is tabulated in Table 10. When (b + 1) is negative the tabulated value is the mean estimated strain at 1 time unit.

3:7

The initial strain, E0, should be interpreted carefully. The most reasonable physical interpretation is that the initial strain

is an elastic strain. As load is imposed, on the specimen elastic displacements result from surface forces (Jaeger, 1962, pp. 115-121). These displacements travel through the specimen as dilational and shear elastic waves with the velocity of sound in the specimen (Jaeger, 1962, pp. 131-138). The displacements, then, are practically instantaneous and are revovered at the same speed when the load is removed. All other displacements are time dependent and, in creep experiments, should be predicted by the time laws of creep. Experimentally determined, values of the instantaneous strain in a rock experiment arc difficult to find. Evans (1958) reported

a series of experiments using a compressed air loading machine 111

5:7 capable of applying a lead uf up to 12 tons in 0.005 seconds. Under stresses which were smol fractions of the coa:xessive strength of the material, Evans founa that the time dependent strain varied from 40,7, of the total strain "at slow loadinz rates" in a sandstone to 15;7 in a granite. These results need interpretation. First the time dependent strain Evans measured was wholly recoverable.

'Arens had placed his specimens in a state of ease by subjecting them to cycles of stress over the ranee of the later experiments. Cc ho stressing continued till the stress strain curve was revoatable. Four or five voles was usually suffieient. Evans calculated the instantaneous elastic strain from what he termed the short range integral strain curve. If, after some time at a constant load, the load on a specimen is slightly reduced the resulting deformation, Evans suggested, would be free of time dependent strain and the instantaneous elastic modulus could be measured. A series of such measurements at different loads form together a strain curve, the short range integral strain curve. Walsh (1365) has put forward a theory to explain the apparent increase in Llastic modulus over short stress ranges. He assumed rock was an isotropic elastic body with randomly orientated narrow cracks, Under load these cracks close and sliding on the cracks contributes to the total deformation. On reduction of load sliding is oliposea by frictional stresses between the cram surfaces. 112

3:7 VIalsh showed (p. 406) that this theory gave plausible estimates of the coefficient of friction between cracks in a granite under ueievial stress. while Walsh's work reenires modification to take account of the stress dependence of the coefficient of friction (Ldrrell 1965), it may offer an explanation of part of Evans' results without invoking any time dependent mechanism. Evans results however show independently of the short range integral strain curve that some time dependent mechanism affected the stress strain curves. For instance, the sandstone showed, Evans (1958, Fig. 1), 16• less strain when loaded, in 0.1 seconds. So whtle there is no need to accept that as much as 4.V.) of the total deformation of the sandstone "at slow loading rates" was time dependent there is good evidence for time dependent strains of 16:1 of the elastic deformation for a sandstone, 9 for concrete and about Ti!, for a granite. Larger tine dependent strains can be expected if the material is not in a "state of ease" for then irrecoverable strain will contribute to the total. Recovery experiments described, in this thesis (Chapter 8:9) on Pennant sandstone at room temperature in uniaxial compression show irrecoverable strains up to 20 per cent of the recoverable strain. A similar figure is reported by Hardy (1959, fig. 27) for Steep Rock Lake iron ore, A reasonable upper limit on the time dependent strain within 113

- 3:8 the first minute of a creep experiment might be 40,,Z of the initial strain. This limit is intended to provide a guide for estimates of the amount of initial strain. A lower limit on the time dependent strain is zero* Thus there is a large range of possible values of E0.

3:8

Equation (11) may be transformed to a straight line form by taking logarithms. Then, log( Et 0) - log (DIA 2 + 1)) + (i2 + 1) log t (19)

A computer program, listed in Appendix 1 has been written to determine by simple linear regression the slope and intercept with the log (creep strain) axis of (19). It uses em in Table 10 as an estimate of Bo and (ei,ti) as estimates of(Et, t is assumed to be without error. The results are tabulated in Table 11. is necessary. If B2 is less than (-1), a modification of program 3 Equation (14) can be written as (20) (B2 + 1) El - 01/02 + 1)) - Et = -(B1/(B2 + 1))t (20) Then on taking logarithms of (20),

log (E1-(B1/(B2+1))-Et) = log (-B1/(B2+1))+(B2+1)log t (21) 114

3:8 - 3:9 and simple linear regression can be used to estimate the parameters of (21), providing an estimate of the strain at infinite time, lus•can be made.

A 1 B 1)) (22)

If strain hardening is rapid, there may be some observations of

exceed e strain which inf.' this leads to non computable values of the left hand side of (21). Non computable values occurred in

A2.1.10. Rather than further modify the computational procedure to deal with one experiment, A2.1,10 has been omitted from Table 11. A direct comparison between the parameters estimated from the strain rates and from those estimated from the strains can be made by the values of (b2 + 1) in Tables 10, 11. Those in Table 10 are estimated from the strain rates, those in Table 11 are estimated from the strains. The agreement is reasonable, the estimates by one method generally falling well within the, confidence limits of the estimates by the other method.

3:9

One cause of discrepancies in the values of the parameters estimated by the different methods is the effect of the estimate of the initial strain, o0. Ten of the fits to the data in Table 11 show significant positive serial correlation. This may also be 115 due to the estimate of the initial strain and, in Tables 12 - 21, the effect of varying eo in each of the tun experiments is investigated. Only three of the ten experiments show positive serial correla• tion which is significant at the 5):: level for all the plausible values of the initial strain. Uisra's experiments on anhydrite at 600 degrees and at 8265 (Table 16) and 10370 psi (Table 17) while significantly positively serially correlated at the 55 level do not show significant positive serial correlation at the 2.5)L level, The experiment on Pennant sandstone (Table l4) is not signifioant at the l level. Exact agreement between the estimates of the parameters calculated by both methods cannot be expected because the methods. are based on slightly different models of the creep process. It has already been remarked that the fitting procedure for strain rates smooths away possible negative strain rates. On the other hand, strain reversals present no special problem in the calculation of the parameters using the measured strains. This difference cannot be the only one between the models for none of the three experiments which show significant positive serial correlation at the five per cent level contain strain reversals. Another possible cause of discrepancies is that errors in the strain measurements are cumulative. Suppose that an event outside the usual range of laboratory phenomena caused an abnormally large 116

30 3:10 strain in a short time interval. This event would add an abnormal increment to all succeeding strains but it would affect only one strain rate measurement, that for the time interval in which it occurred. The strain rate erocedure will thus be more robust against abnormal events than the strain curve.

The estimation of parameters using strains was however only incidentally aimed at juftifying the assumptions made in the calculation of the parameters using strain rates. The primary purpose was to calculate measures of goodness of fit.

3:10

Three measures have been calculated for all the fits tabulated 2 in Tables 12 - 21. Two of these, the estimated variance, u and

l' have already been used. The third measure, S is one used by Hardy (1965, Table 1) "the standard duration associated with the least squares fit". The parameter et, is the value of E t estimated from the estimated parameters of the strain fit. (b2 +1) et - ec, = (b1/(b2 1)) t

(n 3)52 = E(at )2 - et The joint behaviour of these three parameters divides the experiments of Tables 12 - 21 into two types. 117

3:10 2 In the first type, Tables 12 - 1 as o varies u doz,n not reach a minimum, it growc smaller as the slope of the straight line approaches zero. stay reach a minimum and R1 may reach a maximum. In the second type, Table.,' 17 - 21, u2 does reach a minimum and this uuually coincides with a maximum value of Ri. S2 may take a minimum value but this is not generally related to turning points of either of the other two statistics, The right hand side of (19) can be written

log (Et - E0) = log E0 + /og ((Et/B0) - 1) (23)

qb is nearly equal to Eo, that is, when the creep deformation is negligible compared with the initial deformation, E0, (23) can be written using Fierce (1956, 639), log IT0 + log ((Et-E0)/IT.0) = log Lo + (Et/E0) -2 (24)

Then (19) becomes,

Et/E0 = -log Bo + 2 + log(L)+1)) + (B2+1) log t (25)

If was close to (-1), then, by (15)

= B1 log t (is)

Equation (15) can always be satisfied by choosing

B1 = E0 (B2 + 1) (26) 118

3:10

Thus, as the estimated slope of the fitted line in (19) becomes closer to zero due to varying oo, the condition (15) is approached and the parameters are always satisfied by (26). The variance decreases aa o and (b2 1) decrease. If the data of an experiment are already approximately fitted by (15), then the variance will decrease continuously as eo is decreased. Tables 12 - 16 are examples of this effect. If (15) is not a good fit to the data with the initial estimate of e o the variance will rise to a maximum as eo is decreased and then begin to fall, Table 19. When (15) is not a. good fit then the variance may take a minimum value, Tables 17 - 21. Lstimatea of the variance of the fitted straight line to the data are thus sensitive to the estimate of the initial strain in a complex manner. Tables 12, 14., 17, /8, 19 show that the minimum estimate of the variance may occur in a fit that shows positive serial correlation. In Tables 12 - le, a minimum, if it occurs, is at an implausible estimate of eo. Thus, a minimum variance criterion for the estimate of e is not reasonable. Criteria for classing co based on either a maximum value of `1.P or a minimum value 2 of S can be rejected on similar grounds. One way towards an estimate of goodness of fit is to regard the estimates of e as a priori estimates. Then R o I would probably provide the most robust measure of goodness of fit as it should be 119

3:10 3:11 less sensitive than u2 and S2 to small changes in eo. The ratio of the values of El from Table 9 and Table 11 is tabulated as Rb in Table 11. F tables with (n - 2) dezrees of freedom will provide a qualitative guide to the significance of R. At the one per cent level there are twenty experiments better fitted by a power law, equation (11), than by a creep law of the for, (5). Two experiments, .liera's on Pennant sandstone at 3500 and. B.1.1.30 show significantly better fits to an exponential law,

The qualitative evidence of the strain fits on the evidence of is in broad agreement with Table 5. This is illustrated by diagrams 3:1, 3:2, 3:3, 3:4. Observations from four typical creep experiments have been plotted with the fits of the power law and by Hardy's method on strain - 10 (time) coordinates. TAagrams 3:1, 3:2, 3:3, 3:4

Comparisons of Hardy and power law Mato strain data.

Legend Vertical ax (abscissa), time of observation from the commencement of loading, in minutes, logarithmic scale. Horizontal axis (ordinate), strain in micro.. strains after the first observation. Filled circles - observations Open circles - power law fit to observations Open squares - Hardy fit to observations Diagram 3:1 B1.1.31 Dia6ram 3:2 - B2.3.04 Diagram 3:3 - Anhydrite, 15°C, 4.720 psi, from (Uisra, 1962) Liagram 3:4 B1.I.35

Comment Comparison of the slopes of the power law and Hardy fits shows a characteristic pattern. The slopes are identical at 3 points along their length. The steeper elope of the Hardy fit after the third point is due to the increasing importance of steady state creep with increasing time in the Hardy law. Chapter 2:9 predicted the rest of the pattern from the discussion of the difference between the exponential and power laws. There are two points

where the slopes of the two curves should be the same. The slope of the exponential fit first increases and then decreases relative to the power law curve. dia. 30 • 0 0

100

m 0 B• 0 •

50 0

•o

i 20 0 I 10 !!! 0 • 0 10 100 1000

100 dia.3:2

0 • • • • 0 o o o 50 0 •8i i Y i 20 o 0 0 o ip o • 10 • 0 o o

0 10 100 1000 dia.3:3

dia .3:4

0 •0

I I I o o i o o eo

o •0 . a • • o o° °

. , 10 100 1000 126

: 11

let eeems likely that the tit of power law (11) to the data is generally better than a fit of the creep law (5). It is unlikely that any more exact statement than this can be made if the strains are the experimental data fitted. Thi situation should bo contrasted with the test, R3, tabulated in Table 5 which is exact. The general effect of fitting the strains in an experiment instead of the strain rates is, if the strain at zero time has to be estimated, as if another parameter has been added to the two which need to be estimated. An experiment from gixra (1962), on anhydrite at 600C and

!4.4-0 psi is a good example of this. The choice of eo used by

Usra and :airrell (1965) led to a plot (eisra and i.furrell, 1965, o and this thesis, diagram 5:5 a) which suggested that the data needed to be represented by tee power law terms,

Et = 94V.32 410015 where t is in seconds. Tables 10 and 11 showed that with a smaller instantaneous strain, co = —260, the data can be adequately represented by

-t = 716t 0.17 where t is in minutes. The plot, aiagram 5:5, b snows that the 127

3:11 s/o2e of the initial part of the curve is reduced. ',fith o = -780, o diagram 3:5, e, the slope is further reduced and the observations appear to be even clost,r to a single straight line.

Diagram 3:5 demonstrates that !Lisra and lairrell's (196, p. 518) conjecture that the creep law pay contain, under certain conditions, a number of power law terms is based on their inadequate method of fitting the creep laws.

Table 22 shows the effect of varying the estimate of eo in fits to 10 exveriments by Hardy's method (7 of which have boon used in Tables 12 - 21). Table 22 shows for Hardy's method something that has already been seen in Tables 12 - 21 for the power law fits to the strains, both the st:f.tistics of fit and the estimates of the parameters of the creep law are sensitive to variations in tht, estimate of eci.

So, unless oo is known a priori, methods of fitting creep laws to experiments using the measured strains lead to le.:s exact tests of hypotheses about the fits and to less exact estimates of the parameters of the fits than methods using the estimlted strain rates. Diagram 3t5

on the fit of a power law. The effect of the choice of eo send (a) eQ = 0 (b)e = -280 (c)e 0 = -780

Vortical axis, (abscissa), time in minutes Horizontal axis, (ordinate), strain in microstrains Solid triangles are observations

Comment The experiment is from Misra (1962), on anhydrite at 600°C and 5/40 psi. cia. 3: 5 10000

A— A---"ArArka, •

,A-Ar— A ______,L...---A A-----1-----A A ..-----A. • A.------'AL A • • ...... ---• ------4r...... „---- • • 1000

1000 100 £10 100 130

CHAPTER /4..

SOME FURTHER USES OF SIMPLE LINEAR REGRESSIONS

4j1

In chapters 2 and 3 methods of fitting creep laws to data were described and discussed. In chapter 2:3, simple linear regression was introduced as an adequate method of calculating creep laws. This method leads in Chapter 2:4 to criteria for satisfactory fit and in Chapter 215 to means of measuring goodness of fit which are of sufficient power to suggest the rejection of an exponential law of transient creep (Chapter 2:10),

In Chapter 3, the fitting of creep laws to strains was discussed and it was shown to be less convenient than fitting creep laws to strain rates. The difficulty of interpreting criteria of fit obtained using the strains was a further disadvantage of this method.

The rest of this thesis will therefore be concerned with the interpretation of the results of power law fits to the estimated strain rates and there will be no further detailed consideration of the possibility of exponential Iris of transient creep. Nor will data based directly on measured strains be used. 131

4:1 - 4:2 Chapter/. is concerned with developing some further uses of

the simple linear regressions. They have applications in the

reduction of data, experimental design and in the investigation of the onset of tertiary creep. Regressions are used hero first

to investigate the reproduotibility of the creep experiments.

Auy analysis of creep data beyond the fitting of creep laws

to individual creep el9eriments assumes that if an experiment were repeated on the same rock type under the same experimental conditions an identical creep curve (within the limits of experimental error) would be produced. This hypothosi,:. cannot be invectigated without

a hypothesis about the form of the creep curve. Assuming a power law of transient creep, the technique set out by Held (1952, pp . 571 584) can be used.

A prelude to the annlysis is the calculation for ouch experiment of the quantities listed below:

LMN STRAIN = E8 w I/

t W a. i ZAN 7.1bLE = E VI

SSDY =Dti 132

4:2

SSDX = 5(6i - 602

SPDXY = 1:01. 8,)(ti - to)

SSDYX = - 4e$)2 whereti ." is the value of i estimated from the calculated pare- meters of the creep curve and xi is the weighting of individual observations,

Eli

Computer program 1 calculated there quantities and listed them as data for comparison tests. The results for each experiment arc collected in Appendix 2. Testing the identity of several regressions takes place by four separate tests. The identity of the variances of each of the rogreesions is first tested. The variance is estimated by 8SDYX/(w - 2)0 Held (1952, p. 579) ointed out that Bartlett's test is an exact means of testing identity but it le, simpler to use Hartley's ..Aximum F ratio test (Acton, 1959, p. 89). If it cannot be reasonably assumed that the variances are estimates of the same variance, then, Raid implied, the analysis should beebaiidoned. When only two regressions are being compared though it is possible to continue, (Held, 1952, p. 573), but further tests are only approximate. 133

4:2

The estimated variances, u', of the experiments on Pennant sandstone and of Price's (1964) experiments on Markham sandstone are tabulated in Table 23. The statistic, I, the maximum ratio of the variances can be referred to Acton (1959, p. 259, Table 15). It shows that there is no reason for supposing that the variances of experiments performed at the same load are significantly different. The next test is of the parallelism of the regressions. The statistic, II, should be referred to F tables with the tabulated degree of freedom. If it is not significant there is no reason to suppose the regressions are not parallel. Then the third teat of the linearity of the regression on the means of the individual regressions can be applied. This is only possible when there are more than two individual regressions. Thus the statistic, III,has only been calculated for the tests on Markham sandstone. It should be referred to F tables with the tabulated degrees of freedom. Because the statistic, III, is not significant, the identity of the regressions can be tested. The statistic, IV, should be referred to F tables, again with the tabulated degrees of freedom. Hald (1952, pp. 579 - 5)has set out the calculations involved in a form suitable for a computer program. This is listed, program 4, in Appendix 1. It generates the statistics, II, III and IV with their appropriate detTees of freedom from an input of data for comparison tests. The theory of the analysis will not be discussed here. 134.

4:2 — 4:3 Instead, refer again to Table 23. The statistic, IV, shows that for all but one group of experiments, there is no reason to doubt the identity of experiments performed at the same load. As the experiments at the same load can be reasonably assumed to be identical, regressions based on all the data at a particular load can be calculated. A computer program, 5, listed in Appendix 1 has been written as a modification of program 1 to perform the calculations. The results are listed in Table We. They show that the confidence limits on the creep parameters have been narrowed slightly and the significance of the slope is slightly increased (compare the values of r 1 to those tabulated in Table 3). The main advantage of a regression based on all the data at a particular load (what will be referred to as a pooled regression) is that it forme a convenient method of reducing the data for further analysis of, say, the stress or temperature dependence of creep rate,

4:3

Some further new experiments excluded from the analysis of Chapters 2 and 3 by their comparitively low values of tit will now be considered. The parameters of power law fits to them are displayed in Table 25. full details or the observations in the experiments are given in Appendix 2. Included in Table 25, are estimates of the variances of the experiments, u,2 135

4.:3 Comparison of values of u2 froa Table 25 with those of Table 23 show that a number of the estimates of variance in Table 25 are significantly higher than those in Table 23. The difference is attributed to poor experimental design in some of the experiments. Design of experiments will be discussed briefly here and some suggestions to improve design will be made. Held (1952, po 536) pointed out that "the values of the independent variable should (therefore) be chosen as far apart as possible in order to determine the regression line with the least possible uncertainty". Supposing the duration of an experiment to be given by considerations of convenience, then Bales criterion i...t1,?lies that the values of the logarithms of the times of observations should be °quell,/ spaced. If C is a constant,

C = log (ti 4. I) - log (ti)

so that C log (ti and

exp (nC) = (1)

Equation (1) shows that the optimum number, n, of observations that can be made in an experiment of duration, tn 1 depends on the time, t1, at which the first observation is made and the value of the constant* C. 136

4:3 If logarithmic creep is taking place, the difference, Do, between successive observations of strain, e e. 1 takes the form (2)

= bi(log ti 4 106 ti) C'1.1 (2) so that Do = bl log (ti 1/ti)

By comparing equations (1) and (2),

De/b, = C ( 3)

Thi value of the constant C can be determined if a value of De is chosen. Because it is advantageous to maximize the number of observations during the experiment, Hald (1952, p. 536), De should be chosen as the least strain difference which can be reliably determined. The size of De can be estimated by investigating the precision of readings of strain taken when the strain rate is negligible. The range of five readings of strain taken within five minutes of one another two days after the beginning of a creep experiment is rarely more than two micro inches. This range is not dependent on the time interval between the observations providing the environment round the specimen is constant (and, of course, the rock has not deformed in the time interval), A reasonable estimate of De is thus five micro inches. 137

Valusz of log b1 for Pennant sandstone and Carrara marble have been tabulated in Table 3. The values of b 2 tabulated show that these experiments asproximate logarithmic creep (that is, b2 = -1). A value of b1 of 5 miorostrains per minute (log b1 = 1.62) is a reasonable approximation over a wide variety of stresses to the behaviour of the Carrara marble and Pennant sandstone. Thus, C for a two inch long specimen in equation 3 is about Taking

t/ to be two minutes, optimum values of time at which observations should be taken can be generated from equation (1). They are 2, 3.3, 5.4., 9.0, 14.8, 24.4, 40.2, 66.2, 109.2, 180.0, 2964,3, 489.4, 806.8, 1330,3, 36164, 59620, 9329.6 and. so on. The times listed above are, the times at which strain rates should be used. Such a series can be approximated by taking one observation a strain in each interval between the listed times. .kdditional observations may only add to the estimated variance though they are, of course, a chock on the stability of the experimental environment. All the experiments with estimated variances over 1.00 in Table 25 have been examined using the series above. Only the first observation in any of intervals has been retained in the data. The results of a. power law regression on tho modified dats are tabulated in Table 26. The details of the modifications are included immediately behind the details of the experiment in Appendix 2, R6 is the ratio of the 138

— 404 estimated variance of the unmodified experiment to the variance of the modified experiment. In each case the estimated variance of the modified experiment is less than that of the unmodified case, in five cases the reduction is significant at the 5 level and in three of these cases the reduction In significant at the 4; level. Confidence limits on the parameters of the regressions are also generally tightened. Values of the estimated variances in Table 26 are now within the range of those in Table 23. The methods outlined in Chapter 441 can be used to test the hypothesis that the modified and unmodified experiments are not significantly different. The results are tabulated in Table 27 where the modified experiments are identified by a postcript A.

Table 27 shows that there is no reason to suspect any significant difference between the experiments (apart, of course, from the differences in the estimated variances).

4:4

An inspection of Tables 3 and 25 shows that three specimens have been used for more than one creep experiment. Specimens A2.2.43, A2.2.42 and 1.1.30 have been used for two experiments and

A2.24,41 has been used in three experimented Price (1960) has shewn in experiments on Chislet siltstono and South Kirby mudstone that previous creep has no apparent effect on the creep of a specimen 139

4:4 providing the specimen has sufficient time to recover from the previous load and no internal stresses have been released by that load.

The effect of previous creep has been examined in Table 28. The latter experiments arc those on A2.2.43 at 3 tons load, A2.2.41

at 3.6 and 3,9 tons load and. B1.1.30 at 4.5 tons load. B2.3.04 was loaded to 2 tons for 3 days, allowed to recover under 0.1 tons load for 212 days then again loaded to 2 tons. The tabulated tests on the creep of this specimen are tests of the identity of the two creep experiments under 2 tons load. The hypothesis that previous creep has no effect can be tested by testing whether later experiments on the specimen can be reasonably assumed to be identical with other experiments at the sane load. The methods of Chapter 4:1 can be used for this test. Table 28 shows that only the experiments at 3 tons load on Carrara marble show any significance difference and then only at the 5,g; level, they are not significant at the 2i% level. Table 28 also contains tests of the identity of the new experiments on Pennant sandstone at loads of 3.5 and 5.5 tons from Tables

3 and 25. As might be expected from Table 23, the experiments at 5.5 tons show no significant evidence of any difference in behaviour. Nowever, the experiments at 3.5 tons show significant differences at the 5 level though again they are not significantly different at the 22;I: level. 110

4:4 415 Table 28 has shewn the compatibility of the new experiments introduced in Table 25 with those of Table 3. Though these experiments add little to the precision with which the creep behaviour of

Carrara marble and Pennant sandstone has been defined by the experiments in Table 3, they confirm the overall pattern of behaviour. Their titles in Table 25 show that many of the creep experiments were preludes to increment oreep experiments in which creep was observed after a load increment when the specimen had been creeping for some time. Increment experiments will be discussed in Chapter 8.

4:5

Continuing this discussion of experimental design, consider transient creep according to a pOwer law when b2 is not close to

(-1). (b, + 1) Then, as ei = (b1/(b2 + 1)) ti

/t) log (ei = (b2 + 1) log t + 1 (4) Because, C. + 1 = ei 4. De

(Serei) = (b2 + 1) log (ti liti) (5)

Equation (5) can be compared to equation (2). Predictions of optimum observation times for an experiment where b2 is less than

(-1) based on (2) would generally be conservative and less 124

4:5 observations than were possible would be made. Notice that the

number of observations which can be made in a fixed time can be

expected to increase exponentially with (b, 1). Since much of the strain in the experiment will still be in

the very early stages of the creep curve, ei will not vary strongly with time. Generally the factor, ei, will compensate for any tendency of Do to increase with the duration of the experiment and withtheintervalbetweenreadings.Treatinge.as approximately constant shows the similarity in form between (5) and (2). The only use to which the transient creep eeperiments have been put so far is to test the fit of power and exponential laws of transient creep to the data. It has been shewn in Chapter 2:4. that at least sixteen strain observations (that is, fifteen measurements of strain rates) are necessary before either of the tests can be made. Using t1 as 2 minutes and b as 5 microstrains per minute substitution in equation (2) shows that the duration of the experiment should be about 3 days.

Inspection of equation (2) shows that if t is reduced the duration of the experiment can be reduced proportionally. Thus if t1 is halved the duration of the experiment may be halved. Possible methods of loading the specimen quickly and of monitoring the early behaviour of the specimen have been discussed in Chapter 1. Greater reductions in the Duration of the experiment can be made by reducing 14-2

..5

De, the least reliably perceptible strain. .question (1) shoves that halving De reduces the duration of the experiment from about

three days to three hours. ;lays of reducing De have also been discussed in Chapter 1.

It will be shown in Chapter d that the strain rate in transient

creep can be considered to be approximately proportional to the load

on the specimen. Doubling the load doubles the strain rate in the

specimen. Equation (2) shows that the duration of the experiment

thus increases exponentially with decreasing load. This relation-

ship leads to a loeer limit on bl for practical transient creep

With b equal to 4-microstrains per minute, 16 strain 1 observations will take about 15 days. If b1 is 3 microstrains per minute, 1 year will be needed and 2000 years will be needed if b 1 is 2 microstrains per minute.

The values of strain rates at particular times are not immediately obvious on inspecting creep data. It may be useful

to discuss the limits of measurement in terms of strains. Since 16 strain observations are necessary in a creep experiment, say, at

intervals of five microinches, a transient creep experiment should

cover a displacement of at least 75 microinohes (or 37.5 microstrains

in a 2 inch long specimen). Supeosing that the experiment was not

intended to test hypotheses about the shape of the creep curve but

simply to estimate the parameters of the curve than, say, only half 14.3

4:5 the number of observations would be needed. Inspection of the details in Appemdix 2 of the experiment on A2.1.10 at 2.5 tons shows that this experiment is below the limit at which reasonable results can be expected from creep tests using the present apparqtus. Less than ten microstrains deformation can be observed in 1 week, this, despite the fact that the specimen is loaded to over half its failure load. There would be little point in conducting experi-

ments at lower stresses on this material. It has been suggested Coates and Parsons, 1966, Parsons and

Hedley, 1966) that the creep properties of a rock may be used in another way as a measn of classifying the rock. Parsons and Medley (1966, p. 330) suggest the use of the strain rate at about 200 minutes after the experiment began. If a classification can be based on only one creep parameter, a strain rate, then it should be based on the strain rate at the arithmetic mean of the logarithms of the times at which the observations were made. At this point the variance of the strain rate estimated from the power law fit is least and the strain rate is most accurately determined. Confidence limits on the strain rate are then given by equation (26), (Chapter 2:6). 2 C ± t (u /N)M (26), (2:6) Because the variance is a function only of the apparatus in a well 144

14.5 — 4:6 designed experiment and t is almost constant t large valuos of n

the width of confidence limits is inversely proortional to the

square root of no the number of observation. Equation (26) (2:6)

allows the prediction of confidence limits, in advance so that tis 11,1mber of observations necessary to determine a strain rate to a given precision can be calculated.

From this number the duration of the experiment can be

determined given an approximate value of the strain rate at 1 minute.

Thus the detail of a classification of creep properties can be increased with the square root of tie number of observations in the creep experiments.

4:6

The augention that a certain minimum amount of strain is

necessary in a creep experiment if the parameters of the creep curve are to be determined can be applied to reduce the large amount of

data on lateral creep collected during the new creep experiments that have been described.

A technique for the measurement of lateral creep has been described in Chapter 1 and the analysis of the measurements will now be aiscussed.

A program, computer program 6, listed i ependix 1, has been written to analyse the creep data in the same way as the axis.]. data 145

4:6 (Chapter 2:8 and 2:10). It analyses any three records separately and then creates a fourth record, ei(4), by taking the arithmetic mean of the three previous records.

ei(1) + ei(2) + ei(3) = 3 ai(4).

Generally, the ith observations in each of the lateral records are contemporaneous. When an observation is missing in one record, th the (i - 1) observation is used again. Program 6 was designed to analyse the records of any of the groups of three transducers mounted in the same plane. In fact, little systematic uovement has been observed« In Table 29 the number of lateral strain observations in each experiment is tabulated as n and with it, the resulting number of strain measurements in each record after reduction by strain reversals. The analysis is terminated if the number of strain measurements is less than three, the program prints out "insufficient data". If deformation takes place at constant volume then lateral creep rates can be predicted in terms of the axial creep rate. The volume, v, of the specimen radius, r length L, is given by 14 2 v = 3. 2 r L Taking logarithms and differentiating leads to

(dV/V) = (2d11/0 (WL) 146

and at constant volume -dr/r = d4/11

Supposing that the specimen is 2 inches long and 1 inch in diameter, the lateral creep rate will be one eighth the axial creep rate.

It has been shewn in Chapter I.:5 that at least 37.5 micro inches deformation are required to estimate the parameters of the creep

curve with the present apparatus. So, at constant volume, 300

micro inches (150 microstrains) exipl deformation at least would take place in a specimen showing adequate lateral deformation. On this hypothesis, no estimates of lateral creep parameters can be made

from any of the experiments on Carrara marble or from experiments

at loads below six tons on Pennant sandstone.

Inspection of Table 29 confirms this hypothesis. There are only six of the experiments out of a total of 45 outside the limits suggested which have sufficient data to allow further analysis. So discussion of lateral creep will be confined to the five experiments

on Pennant sandstone at above six tons for which records were made.

The parameters of power law fits to the experiments have been caloulated using computer program, 6. They are collected in Table 30. The data from which these parameters have been derived has been

placed in Appendix 2. Generally the experiments are satisfactorily described by a I47

4:6 power law of creep. There are exceptions, however. One experiment, B1.1.17, E(1), shows significant serial correlation even at the 1 level. Another, B1,1.26 E(3), does not have a significant value of r . 1 In two cases, B1.1.20 E(l) and B1,1.37 E(1) there was insufficient data to determine the values of the parameters. Two hypotheses about the experiments can be tested by the method outlined in Chapter 4:26 The first is that the arithmetic mean of the lateral experiments shows the same rate of strain hardening as the axial creep data. This can be examined by testing the parallelism of the experiments, L(4), with the axial creep data. The results are tabulated in the first part of Table 31. One experiment, 131.1.20 E(4), shows a significant departure from parallelism. This is associated with an anomalously low variance in the lateral experiment. Three experiments give creep parameters which might be considered the same as those for axial creep. Since the predicted strain rate in lateral creep should be half that in axial creep. the identity of axial creep rates with lateral creep rates suggests that the volume of the specimen is increasing in creep, eilatancy under load has been reported by Brace, Paulding and Scholz (1966) in constant stress rate tests on a variety of crystalline rocks. Dilitanoy begins to be apparent when the stress on the specimen reaches half the failure stress and was attributed to the opening of axial cracks in the specimen. 348

Because the specimens of Pennant sandstone are under at least og of their failure load, dilation in creep may be attributed to the same process (Chapter 7:6 contains further discussion).

One experiment, B1.1.17 E(4), may not be dilatant. The secone hypothesis tested was that the experiments E(1), E(2), E(3) have identical creep parameters. In two cases, B1.1,20 and 31.1.37, one experiment does not show sufficient creep to allow its parameters to be determined. Insufficient data has been taken as evidence that the experiment is not identical with those experiments having sufficient data. The remaining experiments may be compared however and the results of these comparisons are in the second part of Table 31. Two groups of experiments, those on 331.1.17 and 131.1.26, show significant differences in estimated variances. This, in itself, suggests that the groups of experiments do not have identical parameters. The suggestion of identity in the tests IL, IV for these experiments need not be given any weight.

The hypothesis of parallelism for the experiments on B1.1.36 can be rejected at the 1% level. There only remains then the two groups of data which contain only two experiments, the two experiments in each group can at least be considered to be identical, Thus none of the five groups of experiments contain three experiments with creep parameters which might reasonably be thought to be identical. Either the specimens themselves or the applied 4:6 - 447

stress field is heterogeneous. :!ore information could be gained

by continuing the experiments to failure by static fatigue and

attempting to relate the geometry of the failure surface to the lateral creep behaviour.

4:7

All the experiments examined so far can be satisfactorily explained by a power law of transient cree, and none require the addition of steady state creep. Various criteria have been proposed for steady state creep to become a significant process; the moat restrictive is due to Misra and Murrell (1965, p. 533). They suggest that "at sufficiently high temperatures (above half the absolute melting, 0.5 Tn. of the rock) and at sufficiently high stresses, rocks will exhibit steady state creep."

Misra and Murrell (1965) consider that two of the experiments they report show complete recovery from strain hardening. One of them, on anhydrite at room temperature, has too few readings to be

Finalysed in dete31; the other, on dolomite at 600°0, (0,54 Tm) and 5660 psi has been included in the group of experiments from Misra

(1962) discussed in Chapter 2. There is no reason to believe it shows steady state creep.

Misra (1962) also performed two experiments on rock specimens that showed static fatigue above 0.5 1,1. One on marble at 600°C 150

4:7 (0.54. T) and 8,950 psi failed after twenty one days; the other, on dolomite at 700°C (0.6 Tm) and 3,775 psi failed after 1070 minutes. Both might be expected to show the successive stages of transient, steady state and accelerating creep. As the duration of a creep experiment increases it should be

expected that the fit of a power law of creep becomes increasingly

poor, because, steady state creep and then accelerating creep

become an increasingly larger proportion of the total creep deformation.

The results of a power law fit and details of measurements are

collected in Appendix 2. The experiment on dolomite shows a

decreasing strain rate to an inflexion point between 300 and 350

minutes and an increasing strain rate beyond this point. The

experiment on marble has two inflexions betueen 6,000 and 5,200

minutes and between 9,000 and 13,000 minutes.

If the later observations are removed from these experiments

the estimated variance should be reduced. In Table 32 the parameters of the power law fits are tabulated against these reduced

durations of the experiments. The hypothesis that the variance of

the complete experiment is significantly greater that the variance of that part of the experiment before the inflexion of the strain

rates can be tested by referring the ratio of the variances to F

tables. Both ratios are significant at the 5r. level. This

suggests that a variance ratio test is an adequate means of testing 151

4:7 whether accelerating (or tertiary) creep is taking place in a specimen. Notice that the warning of impendinG failure is quite long, 3 to 5 times the duration of the experiment up to the point of inflexion. Both experiments are adequately described, by a power law of transient creep when tertiary creep has been removed. However, there might still be reason to suppose steady state creep was taking place if the variance of the fit of the power law was particularly large or further truncation of the data reduced it. Table 32 showed that further truncation of the two experiments does not decrease the estimated variance, The variances of the truncated experiments are of the order of others of Misra's experiments at similar temperatures and stresses. Various hypotheses about the variances of Misra's experiments fail. It might be supeosed that experiments on the same rock type at the same temperature would have similar variances. However, experiments on olivine at 750°C show significantly higher variances at lower stresses. Those on anhydrite at 600C are not similar , the experiment at 8265 psi has an anomalously high variance. None of these experiments show significant decreases in estimated variance when their duration is reduced. The truncated dolomite experiment at 700C has the lowest variance of any of Miera's high temperature experiments, the truncat d marble experiment has a 152

4:7 — 4:8 significantly lower variance than one of the experiments on

olivine and one on anhydrite and it is not significantly higher

than two further anhydrite experiments and one further olivine experiment.

There are no grounds, then, for suggesting that the creep

behaviour of the two static fatigue experiments up to the beginning of accelerating creep cannot be adequately described by a power law

of transient creep. Similarly the experiment on dolomite at 600°C and 5660 psi has a variance typical of others at this temperature.

Reduction of the duration from 15,000 minutes to 2,000 minutes gives

no significant reduction in variance. Steady state creep does not seem to be distinguishable in these experiments.

4.:8

Further information about the variance of the fits of power laws to these creep experiments can be gathered by investigating the distribution of the size of the residuals. In Chapter 2:4e a chi square test on the residuals was suggested for significant departures from a normal distribution with zero mean. It was not necessary to use this test on the residuals to reject an exponential law of transient creep (Chapter 2:10), the evidence of positive serial correlation is sufficient.

The chi square teat is not as convenient to program as the 1:43

4:8

test for serial correlation of residuals. The size and number

of the coils that should be created depends both on the total number

of observations in the experiment and the variance of the fit. So

a sub routine has not been added to computer program 1. Instead

the residuals listed by the program have been taken from Appendix 2

and the arithmetic done by hand. The results of chi square tests on all the experiments in Table 3 with 21 or more strain obeervations form Table 23.

The sizes of residuals should be symmetrically disposed about

their mean (zero) and if they are normally distributed, half 6f the total number aeould be within ± 0.6745 u of the mean (refer to

Lindley and :.filler' 1962, Table 2). This divides the size distribu-

tion into four cells each having an expected content of a quarter of

the total number of observations, The sum of the squares of the differences between the observed and expected values divided by the

expected value can then be tested. for significance against chi square tables with two degrees of freedom. If the tabulated value of chi

square is exceeded then thellOpothesis of a normal size distribution with zero mean can be rejected...

Only Misra's experiment on anhydrite at 9560 psi is significant

at the 15 level (Table 33,.column Chi-square). Detailed examination

of this experiment shows an anomalously large variance due to large fluctuations in the strain rate measured over cemparatively short 1.54.

• 4:8 periods of time. No creep lay: predicts these and they are not

present in others of Misra's experiments. These changes maybe

due. to changes in ambient condition the experiment or to

inhomogeneities in the creep specimen.

Thus while the general picture.ie clear, that is, that the residuals can generally be assumed to be normally distributed with

zero mean, details, such as the effect of steady state and accelerating

creep on the residuals must wait till experimental evidence becomes

available.

Further statistical analysis of the data is poetponed till after the discussion of theories of transient creep in Chapters 52 6 and 7. 155

CHAPTER 5 FORMAL THEORIES OF ma) - I STRAIN HARDENING AND EXHAUSTION THEORIES

5:1

The time laws of creep are widely aL.plieable. They seem to apply to metals and to non-metals, to single crystals, poly-crystals and to amorphous substances. Consequently the theoretical explanation of these laws should "depend only on general properties common to all or most solids and not on special mechanisms e.g. involving dislocations, which belong to some only". (Cottrell, 1955, p.198). Cottrell suggested that theories of this:Aype should be called formal theories and that those whose aim was to explain the particular creep behaviour of some material were another typo of theory, "structural" theories. Structural theories attempt to explain creep in terms of the physical processes involved in the deforming materials. They thus correspond with what farofalo (1965) has termed micro-mechanical theories of creep. Recent reviews (Pratt, 1967, Garofalo, 1965) indicate that while considerable progress has been made since 1953 in structural theories of creep, there has been little advance in formal theories. 156

5:1 Cottrell's original distinction therefore seems both real and useful that it is followed in Chapters 5, 6 and 7. A preliminary discussion of creep in terms of the "elements of deformation" of the formal theories of creep in Chapters 5 and 6 leads to an attempt to determine the physical nature of these elements in rock in Chapter 7. The principles of formal oreep theories are outlined in Chapter 5:2. Mott's development, of these principles is reviewed in

5:3. There, I have extended Mott's theory to describe creep after an increment of load. Formal exhaustion theories of creep are introduced in 5:4. end Wyatt's theory in treated separately from Cottrell's in 5:5. Criticism of the formal theories of creep seems to have been rather perfunctory (5:6). A short discussion of Boltzmann's Law of Energy Distribution is a necessary preliminary to more detailed criticism. There is an ambiguity in Cottrell's description of eleaents of deformation and I show that the elements must exist separately from a source of energy for them (5:7). This analysis exposes a small error by Cottrell (5:8) which has tended to obscure a larger blunder by Wyatt. I have corrected these and eyatt's theory is then identical with Cottrell'e (5:9). Correction of Cottrell's short derivation of the exhaustion theory leads to a new extension of the theory to creep after an increment of load (5:10). 157

5:2

Cottrell (1953, p. 199) suggested that formal theories of creep were based on three main ideas: 1) Every creep specimen has a yield stress, defined as that stress a% which flow can take place without the help of local fluctuations in the stress field. During creep, the applied stress is smaller than the yield stress and the creep rate is limited by the frequency of fluctuations in the stress field. 2) At the beginning of creep, the yiela stress is equal to the applied stress. Hence the initial creep rate is indefinitely high, even at very low temperatures. 3) As the creep strain increasOs the yield stress increases, that is, strain hardening takes place and the yield stress rises progressively above the applied stress. The increasingly large fluctuations which are needed to bridge the gap between the yield stress and the applied stress do not occur as frequently and the rate of floe slows down. The rate of flow will decrease until it is equal to the rate of recovery from strain hardening. Steady state creep is then observed. These ideas may be put more formally. Consider a. crystal with a yield strength, Fy, under a stress F. Deformation will occur if there is a local fluctuatio# of the stress to Fy. The energy needed for such a fluctuation in a volume V is V (i g)2/2G 153

5:2 where G is the appropriate elastic modulus, the shear modulus if Fy is the yield strength in shear (Cottrell, 1953, p. 12). The probability, p, of such an event is

p_= C exp(- V (F F)/2GICT)/V (1) where C is a constant, K is Boltzmann's constant and T is the absolute temperature. The creep rate, 6, may then be written

4 . v C exp (-V (F:y F)2/2GKT)(V (2) v is the strain due to one fluctuation. Orowan (1947) suggested that the difference between the yield strength, Fy, and the applied stress, F, could be written

F F = he (3)

His suggestion has been generalized (Atott and Nabarro, 1948) to

(F = he (4) where h is a coefficient of strain hardening. If M = 0.3, the strain, et, at time, t, is by intssration of (2)

e t = A log (1 4. Bt) ( ) where A = 2GKT/Vh and B = vC/2GKT. When Bt is large compared to one (5) may be written

(6) et = A log B + A log t 159

5:2 - 5:3 Differentiation of (6) with respect to time results in the characteristic power law of transient creep

A / t (7)

F was originally thought to be the strength of the bond between two adjacent molecules of the material. However, calculation showed that this assumption led to negligible creep rates at room temperature in materials that were known to deform under these conditions. Orowan (194.7) therefore su_gested that there were weak regions in the specimen with yield stresses considerably less than b" (or, alternatively, there were areas of stress concentration where the stress was considerably higher than F). Even in this form the theory required some extension to describe these weak regions (or deformation elements) more exactly.

5:3

Mott (4953) attempted an elaboration based on the assumption that in metals the elements of deformation were dislocations in the structure of individual crystals. his theories will not apply to amorphous or glassy materials. The motion of the dislocations under the applied stress is held up against obstacles. Supposing that the energy required to overcome one of these obstacles is U,then 160

5:3

enNbDexp (-U/Kt) (8) where n is the frequency of vibration ef the dislocation loop (or

Frank-Read Source), N, number of obstacles for unit volume, b,

Burger's vector of the dislocation, 0, area swept out by the dislocation before it reaches the next obstacle. Assuming that the dislocations were held up at particles of precipitate from the pure metal, Mott suggested 2 10 = L , n = 10 c.p.s., U = B(1 - F/F )3/2 (9)

where L is the separation of the precipitate particles and B = 2 0.2 b L F (Mott and Nabarro, 1948). Mott then assumed that strain hardening took place such that

(Fy - F) = he • then

U = Bhe/Fy (lo)

Strain hardening, he suggested could be caused by the back stress of dislocations piled up at obstacles on the Frank Read source that generated them. Alternative1y, strain hardening could result from the movement of even a few dislocations since piled up groups of dislocations produce larger internal stresses than the same number dispersed randomly. There is no necd,then, to generate new dislocations from Frank-Read sources. Substituting (10) in (8) gave 161

5:3

8 = H exp (- Jo3l2) (n) where J = B(bify)3/2/1U and H=nNb L2.

=hen tis large, integration of (11) gives

et = J-2/3(1og 4t)2/3 (12) -2/ where M = .

In pure metals, there are no precipitate particles. Cottrell (1952) has suggested that dislocation may then be held up at the intersection of the dislocation surface with that of another dislocation. These intersections are called jogs. Then Mott (1953) suggested

U =,B (1- 1=o/10y), A = L2, a = 1012 c.p.s. (13)

B is the energy of a jog, L is the spacing of the intersections

Then as U = Bhe/Fy ('4) t =nNbL2 exp (-Bhe/VCT) by substituting (14) in (8). Integrating

et = log (1 + Pt) (15) where Q = KTIVBh, P = N b 12 11/Q.

Equation (15) can be recognised as similar in form. to (5). It is more general than tha first form of Mott's theory, as it applies to pure crystals as well as to crystals containing foreign particles. ;43 Wm extended the zecona form GI' ,4ott'a theory to incre2ent or,_5 tot. In these, the ffpociwne ir as;:mmed te oree,,- to a

4train e at a stress ie before the 1ii tress ie raised in a o o short time to FJ.* The strain e at a time t after this event is given by

. • 2. b t (16)

EAV 1:-(6r coWl) oxP (17) whre 1/ KTP 1 = n b '1

k) (18) Then • et Q1 log (I)1t vihore k = eXJ? ( ) • (19) OAS by difVerentiating (lb)

t - /(t (1)) Tho von' of thu strain raU curve in a conventional creep exerimLnt at F1 is by dif.vt_rentiating (1!))

tt = *1.1;1/(1 t) GO) or when F1t i& large), com4?ared to once

= Qi/t (21 163

5:3 - 5:4 Comparing (21) and (19) shows that k/P1 is, in Wyatt's (1953) terminology for increment tests, the repeat time. 4att (1953) suggested that increment tests, plotted on et - log t axes can be displaced parallel to the log t axis to bring them parallel to the parent curve. This is equivalent to constructing a new variable, T,

T = t + k/Qi

Plotting (19) as 6 = Q /T on log 8 - log T axes should give a slope parallel to the parent curve. If (FI 10) is small, Q1 = Q0 and the parent and increment curves should be identical. Even if Q, is not equal to 0 the increment curve should be identical to a conventional creep curve at Fr Increment creep tests can thus lead to tests of Mott's theories.

5:4

Another important group of theories based on ()rowan's ideas are the exhaustion theories (Garofalo, 196, pp. 159 - 163). In exhaustion theories the creep specimen is considered as an aggregate of a large number of elements of deformation. These elements have the following properties in transient creep. 1) Each has a characteristic yield stress greater than the applied stress. The element cannot flow till the applied stress has been augmented by an amount, the activation stress, Fe 5 :4 which is the difference between the aeplied stress and the yield stress of the element. 2) Augmentation of the stress occurs' by local fluctuations. The activation energy of these elements, U(Fa) is assumed to be the same function of E for all the elements. Cottrell (1953, a p. 200), on the hypothesis that the elements were dislocation jogs, suggested

U(Fa) = Aloa = 13(1 - .F/y . (22)

This suggestion also leads to simplification in the analysis. ;4) When an clement is activated, a displacement takes place which contributes an increment v to the strain. For simplicity, assume that v is the same for all elements and that after a displacement, the activation stress of the element becomes so large that the element does not move again. 4) A further simplification in the theory is to assume that the displacements are stochastic events in the sense of Mogi (1962, . 158). This would imply that triggering (Orowan, 1947) is absent. With these assumptions now let N(Fa, t) dFa be the number of elements with activation stresses between Fa and Fa + dFa at a time t after the start of creep. If n is the frequency of the fluctuations then the chance that an element with activation energy

Fa will be displaced ("jump") in the time interval, dt, is

165

.524. U at N(Fat t)aFe, exp a dt (23)

The creep rate, 64., is then

00 t = v/ p t N(Fa t) f a Ci,4) 0 Since each element is displaced only once the number of elements available for deformation at each activation stress decreases at a rate

d(N(Fattl/dt = -N(Fa t) f (25)

On integration from zero time to t (25) gives

,N(Fa t) = N(Fal0) exp (-f(i)t) (26)

where N (F a 0) is the aistribution of the elements at the start of crook) (t = 0). Substituting (26) into (24.) loo = v N(F,O) exp (-f (flt) f (c) dFa (27)

Cottrell (l953, p. 201) suf.gezted that the distribution 1!(Fc4,0) might be considered uldform in the interval zero to Jim and zero outside this interval. Theo N = N (s,0), whore N is a constant. substitutins (22) in (23), (28) can bu derived

f(Fa) = n exp (-AiaAT) (28) i66

Differentiation of (28) with renpect to F a gives ug;a) = n (-A/K) exp (9A/KT) (-WA) df(Fa) = f(Fa) dFa (29) albstituting (29) into (27) gives

f(0) 8t = (vIILT/0/ exp (-f(Fa)t) df(Fa) (30) f(F)

f(F ) t = (vNAT/At)(exp(-f(Ea)t) m (31) f(o)

Mira and. Murrell (1965, p. 526) have discussed the evaluation of

(31). At small values of nt the first two terms of the exponential series are adequate they slkgest. Then exp(-f(0)t) 2. 1 - nt.

exp (-f(Fm)t) = 1 - nt exp (-AFWKT). (31) can then be written

et = vliNATa (1 exp (.4Fm/KT))/A (32)

At longer times, they suggest exp (-f(0)t) = 0, exp (-f(Fm)t) = 1

These conditions represent a situation in which all the lowest activation energy elements have been exhausted while those with much higher activation energies are unaffected. Then (31) can be evaluated

et = vNET/At (p. 526 (13)) (33) 167

b:4. Integration of (33) gives

et o1 = (vNET/A) log t (34) el is the strain at one time unit. :,equation (30 difrers from the form given by Cottrell (19530 p. 201, (5.42)). Suppose, however, only that exp(-f(Fa)t) = 1. Then the resulting form of the creep law should be applicable at shorter times than (34) and (33). (31) becomes

tt = (vNxT/At)(1 exp (-nt)) (35)

et de = (vMKT/A) (1/0dt- (exp(-nt)dt) (36) b where b is the shortest time at which (35) applies. To carry out the integration, it is convenient first to write nt = y, then

t (1/t)dt = jy ( 1/y) dy, = nb A bt. and_ then -at = x -x (exp(-nt)(t)dt;// (exgx)/x)dx -b4 -bt -x (exP(7) gexp(x),/x) dx (37) -00 )//e. 163

(37) can be recognised. as the difference of two exponential integrals (Jahnks, and Fade 1945, pp‘1-5). Carrying out the integration in (37)

et eb = (vAKTVA)(log nt - log b' Ei(-4D9 Ei(-nt)) ( 8)

If b' is small, (-b') = 0.5772 + log b' (Jahnke Emde, 194.5, p. 2). equation (38) becomes - vi(-nt)) et - (vNKT/A)(log nt + «5772 (39)

If it is assumed that nt is greater than 5, say, then (39) can be written

t = (vNKT/k)(log nt + 0.5772)

Fi. (-5) = 0.001148. Equation (40), then, is a precise form of Cottrell's expression (p. 201 (5.42)) and correots Cottrell's oversight.

Cottrell gave (1953, p. 202) a simpler method of deriving (so). He remarked that when f(F11) is small exp(-f(Fdt) is practically a step function of Fa, being nearly zero or nearly one and changing sharply- from one value to the other centre about a value f(P)t = 1. The exhaustion process is then equivalent to the advance of this step function across the distribution curve N(11a) towards the higher values of Fa The position of the step after a time t is given by = nt exp (-11Ft/KT) — 5:5

't 4T/A) 106 r.t. The crep strain at tine t is et

t - (v T/A) log nt (2) 5:5

'oyatt's derivation (1953) eV a creep equation by an exhaustion also al.peare to be simpler Vlan Cottrell's (1953) original derivation. Vqatt considered x elements exchanging energy and suggested that "the number, dx, which will have energies between A and A dA given by Boltzme.nn's lcv",

dx = G exp (-A/KT) da (43) where C i3 a constant. 0 Integrating (43), x = C (exg-A/g19) KT 010

At any instant the number of elements with energy A or greater i8 por idx x exp (—AM) A If there are then N(8,t) element at time t vita activation energies between E and E dr_ (behaving as eluants exchanging energy) the number of elements which will "jump" in the next time interval, is, 170

5:5 from OW

d N(E,t) = -N(E,t) exp (-FAT) rat (45)

Integration of (45) from zero time to t gives

N(E,t) = N(E,o) exp (-nt exp (-2/KT)) (46)

Consider then the value of E at which n(4,t) is some fraction, p, of its initial value

N(E,t)/11(13,0) = p = exp (-nt exp (-E/KT)) (47)

Differentiation of (47) gives

dE /dt = KT It (48)

(46) is independent of P so the thermal front remains constant in shape as it advances towards the higher activation energy elements. The creep strain rate is then, Wyatt suggested, the product of the elements eliminated per second and the contribution to the macroscopic strain of each element, vt

8t v (E,0) dE/dt

v N(,0) KT/t (49)

et - el = v N (BO) KT log t (50) by integration of (49), ",yatt's expression (50) differed from that due to Cottrell (1955, p. 201, (5.42)) by the absence of n, the frequency of 1/A and of the constant, 0.5772 (compare (40)). 171

5:6

Teefore a detailed discussion of Cottroll's and Vyatt's theories, some of the criticisms of them both should be noted. Criticism has centred on the values of n derived from the theories. It has already been noted that, depending on the mechanism of deformation, 0 Mott (1953) has suggested that n is about 10 - 1012 cycles per second.. Davis and Thompson (1950) using an earlier form of the exhaustion theory baseu on precipitate particles rather than jogs as obstacles determined n to be about one cycle per second. ilott suggested (Cottrell, 1953, p. 204) that they had determined the frequency of jumps of the dislocation loops which had set into action avalanches of further jumps. Wyatt (1953) determined n for pure copper by a group of 18 22 increment experiments. He found n to vary from 10 to 10 with increasing strain. This discrepancy has been attributed (by Wyatt) to experimental error. Another criticism (Garofalo, 1950, p. 162) was that the linear increase of transient creep rate with temperature predicted by the theories is at variance with experimental evidence. Misra and iIurrell„ however, reported (1965 , p. 527) that the prediction "is in accordance with our observations on rocks and those of Hyatt (1953) on pure metals". Aisra and Murrell (1965) pointed out that there is no natural 172

5:6 5:7 explanation of Andrade and steady state creep by an ettension of the exhaustion theory. Cottrell (1953, pp. 206-210) has attempted such an extension by suggesting that an element may jump more than once at higher temperatures and stresses than those at which logarithm:10 creep is experienced* Removal of the exhaustion restriction leads at sufficiently long times to a steady state where rates of exhaustion and creation of element are equals. At intermediate tines the creep curves approximate to Andrade creep.

5:7

The short aiscussion given in 5:6 suggests that the theories are reasonably robust to criticism apparently because criticism has been perfunctory. So, consider the first steps in both theories based on the Boltzmann Law of Energy Distributions Joos (1951, p. 580) stated this law as

f(i) = N exp (-U(i)/kT)/ Exp(-0i)/KT) DTi (51) where Ef(i)DTi = N, the total number of systems. DTi is the volume of a cell containing systems of energy U(i) at a density f(i) which is thus a measure of the probability of finding a system of energy, U(i), per unit time per unit volume. Cottrell (1953, p. 12) has suggested that the denominator of the right hand side of (51) is a factor determined by the frequency 175

5:7 of attempts to establish fluctuations in the system and "by the entropy changes accompanying these fluctuations". In normal usage, probabilities range in value from zero to one. However a particular thermodynamic probability of an event has been defined (Doles, 1951, p. 572) as the number of different ways in which this event could happen. The thermodynamic probability of an event is thus, usually, a large positive number. The distribution of the energies of the systems can be examined by comparing f(i) under conditions where the denominator is constant. Then, it can be seen from (51) that the probability of finding a system of energy U(i) declines exponentially with the energy of the system. It is not clear from either tyatt or Cottrell's work whether they identity the elements of deformation with systems following Boltzmann's Law and exchanging energy or whether there exists both elements of deformation and an external source which provide energy in a manner governed by Boltzmann's Law. The first possibility would give a simpler physical description of the creep process. Taking up this possibility, the number, N(U (Fa),0), of elements with activation energies between 11(fd and U(Fa) dU(Fa) at zero time is

N(U(Fa),0) = N(0) exp (-0(Fa)/ET) (52) 174.

5:7

N(0) is the number of elements with activation energies between 0 and dU(Y M) Equation (52) describes how activation energies are distributed among the elements of deformation. Cottrell sug6ested (1953, p.200)

0(Fd = A a (22)

From (22), (53) can be written by analogy with (52)

N(Fal0) = N(0) exp (.AF/KT) (53) and then by substitution from (23)

N(Fa,0) = N(0) f(Fa)fn (54.)

Equation (54) shows that the distribution of activation stresses cannot be considered uniform in the interval 0 to P the maximum activation stress. Cottrell's original assumption of uniformity was an analytic convenience. I now consider the consequence of (54). Substituting (54) in (24.)

00 = (v/:) N(0) f (Fa) exp(-f(Fa)t)f(Fa)dif (55) gr* By substituting (29) into (55) F (-vErN(0)/n1 f( m) f(Fa) exp(-f(Fa)t)df(%) f(0) f(Fj = (tiKTN(0)/nA)(exp(-f(Fdt)(f(ct-1)/t") f(0) 175

5:7 If n. is greater than 40 cycles per second, exp(-nt) = 0, for all practical values of t, (exp (-40) = 4.22 10-18). Then (56) gives (57),

tt = (vKTN(0)4A)(exp (-f(Fm)t)(f(Fm) t 1)44) (57)

if the strain rate is to take a finite value, exp(-f(Fi )t) should be non zero. Thus exp (AF,m)/KT) should be of the same (or larger order) as nt. In typical creep experiments it has a range of about 104, thus if n is greater than 40, exp(-A.Fro/KT) 6 must be less than 2.5 x 10 . Over the early part of the creep curve exp(-nt(exp(-ARWE)) will be close to one, then (57) takes the form

8 = (vKTN (0)/At2) (58) Equation (58) does not show the usually ocserved form of the transient creep law. Compare (33) and (7) to (58). The disparity arises from the initial assumption that the elements taking part in the deformation process exchange energy among themselves. I have shown that the energy which activates the deformation elements should be assumed to come from another source, possibly from the small fluctuations in temperature within the creep specimen (Orovan, l9.7). The new enllysis (52) - (58), also shows that the form of the creep curve is dependent on the energy distribution of the elements 176

- 5:8 of deformation. If there are more elements at higher energies strain hardening will be less than a logarithmic creep law. In the example just discussed where the number of elements decreases exponentially with the activation energy strain hardening is more severe than by a logarithmic creep law.

5:8

I now consider my remarks on the evaluation of (56). They apply with equal force to the evaluation of (31). If n is greater than 40, (31) becomes (59)

8t = (vKTWAt)(exp (-nt exp (-APIKT)) (59) In the early part of the creep curve exp(-nt(exp(-AFIAT))

will be close to one, then from (59)

at = vKTN/At (60)

(60) is identical with (33) and (7) in form. If AFIll is very large then (60) may represent the whole of the transient creep curve. Equation (59) can be written

et de = (VITN/A) (exp(-f(F t)/t)dt (61) eb b m

An integral of the same form as (61) ha,: been evaluated in detail 177

5:8 in (37), (61) can then be written

et = (v4TIVA)(Ei (-f(Bdt) 0.772) (62) as b is very small. if f( is small, (62) becomes

t = (vKTN/A) log (nt exp(-.eiiKT)) (63)

Equation (63) should be compared.to (40) and to Cottrell (1953, p. 201 (5.42)). Notice the factor exp (-AFJKT) which accounts for the discrepancy between Davis and Thoepsons' (1951) observed value of n and that suggested by Mott (1953). Considering new the derivation of the creep law by Cottrell's step function approximation, define an element with zero activation energy as certain to jump in a time dt. As there can be no probability higher than certainty, the thermodynamic probability of this event can range only from n to zero by (23). The advance of the step function can be measured b the value F0 of the activation stress of those elements which are certain to have jumped in time, t. Thus

n.= nt exp (-AFIKT)

= KT log t A' --(vliKT/A) log t (64) et - Equations (41) and (42) should be compared with (64) and the absence of n from (64) noted. 178

- 5:9 Equation (64) is consistent in form with (60), (33) and (7). gy short derivation of the exhaustion theory result is now in good agreement with the longer derivation I corrected.

5:9

So far, only Cottrell's work has been PxAmined in detail and it has been shown that the various forms of the creep law that are possible are due to various assumptions about the distribution of the elements of deformation. Turning now to Viyatt's theory, notice that the final form (50) takes account only of elements with activation energy E. It is clear from (43), however, that the specimen contains other deformation elements which will contribute to the deformation. Equation (50) is therefore incomplete. The strain, de, in time, dt, after a time t due to elements of energy E is given by (45) as

de = -N(E,t) exp (-E/KT) n dt Because

N(E,t) = N(E-10) exp (-nt exp (-E/KT) The strain rate due to all the elements is

= y( N(E 00) exp ,nt exp(-E/KT)) exp (-E+ KT 179

5:9 f(E ) = vntT N(E,O) exp (-nt f (E)) df (E) (65) f(0) f(Em) = (vkT N(E,0)4)(exp(-nt f (E)) (66) f(o) as f(Em) = exp (-Em/KT), Supposing E = AF, dE = AC2

Then N(E,O), the number of elements between E and E dE is written more exactly as

N(E 0) dE = N(F, 0) dP

N(E,o) = N(F.0) = N A A using (27) and (28). Then (66) can be written

f(E ) t = (vETN/At)(exp (-nt f(E)) m (67) f(0) (67) is identical to (31). Thus, I have shown Wyatt's and Cottrell's theories lead to identical results. Dyatt's simplification is only apparent and the observed dependence of the creep law on the distribution of the elements of deformation in Cottrell's work applies to Wyatt's work. 180

5:10

I have used Cottrell's step approximation as a convenient basis to extend both his and eyatt's theories to increment creep experiments.

Suppose that after a strain eo in time to at stress Fo the stress is increased to El. After a time t at F the position of the step is given by (64).

1 = t exp (-A2 Ft/KT)

The strain in time, t is et = vN - (Ft px is the position of the step after t o. The strain increment Fi (=Fi Fo) displaces the step, Fi, towards the origin.

et = (vBET/A1) log t VA (Fx - Fi) from (64)

Negative values of et have no physical significance but no strain is recorded till after the elapse Of a eaiting time, tw, defined by (vNRT/A1) log tw = vN (Fx - 11)

et = (vNKTIA1) (log t log tw) (65)

At times greater than t the strain rate curve takes the form

t = MT/A1 t

So if 11 the measurements in a creep experiment before tw are disregarded, the subsequent strain rate curve should be parallel to the parent curve on log u - log t coordinates (and. identical to it if Fl = Fde 161

5:10

Wyatt (1953) made an intuitive sug,;estion of the form of increment creep curves resulting from exhaustion creep processes which is equivalent to

et = (vNKT/A0)log (t - tr) (66) tr, he defined, as a repeat time. (66) can be written

et = (vNKT(A0) log t (1 - tilt)*

As tr was generally small compared to t, log (1 - tr/t) = -tr/t

Thus

et = (vNKT/A0) log t trit (67)

Writing tit = log to and Ao = A, the two forms (67) and (65) are equivalent for small stress increments and repeat times, but under other conditions, Wyatt's theory is misleading. I have shown that increment creep experiments are a possible way of distinguishing between ?ott's strain hardening theories, predicting curves of the form (20)and (21), and the exhaustion creep theories giving creep curves of the form (65). This will be used in Chapter 8.

1a2

CHAPTER 6

FORIAL THEORD3 OF CREEP II — nacroanalytic (or rheological thevnies.

6:1

In Chapter 6, I give an account of the rheological theories of creep based on their governing differential equations, an approach which is new to the geological literature. A simple new description of the 'Becker" body forms 6:3. Garofalo (1965, p. 199) has grouped together a number of formal theories of creep as macroanalytic theories. Macroanalytic theories attempt to derive a constitutive equation for the specimen. then this is solved for the appropriate boundary conditions a constitutive equation relates stress to strain and time. The general

constitutive equation in one dimension has the form m na dF 2F 1.01' + Al at + A2 d — + el,e0+ Am -4-Ex at2 dt d2e =e B1 sk4 + B2 2 414-4412 dne (1) dt dt dtn where A, B are constants, F is the stress on the specimen and e is the strain. 183

6:1 The ideal bodies described in the classical mechanics of continua art special cases of the general equation. For instance

A F o (2) is the constitutive equation of a perfectly elastic body in one dimension«

AoF = B1 (de/dt) (3) is the constitutive equation of a perfectly viscous body. Notice that (2) and (3) are equilibrium equations. An idealised creep experiment would apply a constant axial load to the creep specimen without effects due to the contact between the platens and the specimen or due to the momentum of the loading device. As an analogue of a creep experiment on a perfectly elastic body, consider a mass, M, suspended by a string so that it just rests on a perfectly elastic spring. At time, t = 0, the string is out. Then the motion of the mass, M, is analogous to the approach of the platens in a comprer.sion creep experiment. If the mass IA is displaced down the positive x axis,

I&• =Mg-Lx (4) where x = 0 when t = 0 and I is the elastic modulus of the spring. The solution to equations of the fora (4) is well known (Starling l81. e:1 and Woodall, 1957, p. 15). The solution shows that the mass describes simple harmonic motion about an equilibrium position, L. If the mass had rested on a perfectly viscous element its equation of motion would be (5).

M 'x = Mg R (5) where R is the viscosity of the element. The solution to (5) is the sum of two parts, the solution to the homogeneous equation, *I (R/M) = 0, and a particular solution, x = Mg/R,

x = (g2g/R2)(1 exp (-Rt/U)) Mgt/r (6)

Handin (1966, p. 289) has tabulated values for the pseudo viscosity of rock up to 500°C and below 5kb confining pressure; they are at le4st 101'5 poises, Thus the first term of the right hand side of (6) is negligible for rock and (6) may be written

x = Mgt/R (7)

If the mass, M, rested on a perfectly elastic spring and a perfectly viscous element, the resulting motion would be that of a Kelvin body. Kelvin, himself, suggested (Reiner, 1960, p. 113) that the Kelvin body was a structural model of a perfectly elastic, vesicular solid with its pores filled with a viscous liquid. This does not mean that the constitutive equation of a Kelvin body is an exact solution (with the appropriate bounanry conditions) of the 16)

6:1 - 6:2 behaviour of a perfectly elastic vnicular solid. If such a solution could be obtained it might ahow a different behaviour from that ahem by the Kelvin body. Kelvin's su4Gestion has only the force of an analog.

6:2

The motion of a Kelvin body in creep in uniaxial compression can be written a 'A = ms - RA (8)

The solution of (8) is composed f the solution to the homo- gent:our; equation*

't + (Wm) I + (4/2) x = 0 (9)

and a particular solution of (8). equation (9) in the well known equation of damped harmonic motion (Starling nrd Aoodall* 1958, pp. 21. - 22). It has the solution,

x = A exp (-b+(b2 + k2)4) A2 exp(-b + t(b2 ")2) (10)

. are constants depending on the boundary conditions,

b = ♦ LP,1)"'4 A particular solution be (8) is

x = Mg/L

Aiz; complete solution of (8) is 186

6 t 2 2 x = Al exT)- (-b * t (b + 3c2):) 2 -I- A2 exp (-b + t (b + k')2) Mg/L Analysis of (11) can be divided into three, depending • on the relative values of b and k (Gaskell, R.* 1958, p. 77). Typical ,o 16 11 values of R and L for rock below OL) 0 are 10 and 10 0 Generally* then, b is greater than k. The two cases, b is less than k and b is equal to k* are not important. If b is less than k,

x= Ai exp (-bt) sin (ikt + A2) (12) where i = (-1)2 The sine term indicates that the motion is oscillating and the factor* exp (-bt), decreases the amplitude of successive oscillations. When b = k*

x = (A1 A2t)(cxp (-bt)) (13) rquation (13) represents the most rapid decay of the original disturbance; motion is described as being critically damped. 4hen b is greater than k and x= 0 when t = 0 and i = 0 when t = 0, Ai in (11) is given by,

Al = (b :43/1, + (14/1)(b2 - )s/2(b- k')2 04)

A2 = (b Mg/L + (Mg/L)(b2 k2)2)/2 (b k2)2 Mg/L (15)

137

Substituting (11+) and (15) in (21),

(M44)(10.0 +(b2 + k) ) exp (-b + t(b2 k2)2) -(b -(1)2 + k2)) exp (-b t(b2 k2);),/2(b2-sk2)) (16)

If b is very much greater than k (16) can be simplified to (17)

x = (Lt /L)(1 exp(-b + t (b2

= (U,/L)(1 - exp (-WO) (17)

Ecuation (17) is identical to the relations derived by Murrell and nisra (1962) and Hardy (1965) by postulating the ovilibrium

= Lx + RA

So, :siren b is very much greater thqn k, the acceleration of the platens is negligible in the subsequent analysis. 31uation (17) can be interpreted in terms of damped harmonic motion. If the origin of the x coordinate system in displaced by x0(=(Mg/L)) in the direction of motion so that y = - x0), then (8) can be rewritten as

4. AY = 0 (18)

y = (x - x0) = -(174./WeAP (-Lt/r) (19) Equation (18) is again (9), the equation of damped harmonic motion+ The effect of the laeight, Mg, is equivalent to a displacement, 138

6:2 x = (Mg/L), of the system from the equilibrium position under the weight, Mg. When b is very much greater than k, equilibrium is approached very slowly and nearly attained at very large times.

Equation (18) permits the derivation of a creep law for an increment expeiment. The increment can be represented in the idealized creep experiment by a second weight, Ml, resting on the spring and dashpot until a time, ti, after the string supporting M was cut, the string supporting A1 is cut. Then, from (16),

(M + Mi) + nSr Ly = !Lg (20)

Since the acceleration of the platens is negligible, (20) can be written, for b very much greater than k,

(Fit ) (L/M)y=g and.

y = Ai exp (-Lt/R) Mig/L (21)

Ance y = w(Mg/L) exp (-Ltl/R) when t = 0, from (19),

Ai = -(g/L)(Mi exp (-Lt1/11))

Equation (21) can then be written

y = (M1g/L)(1 exp (-Lt/R)) -(Mg/L) exp (-L (t1 + t)/R) (22) 189

6:2 - 6:3 Equation (22) uhows that the motion is the sum of the motions due to the loads at zero and tl. In other words, Boltzmann's superposition principle (Ramsay, 1967, p. 273) holds when the acceleration of the platens is negligible. The displacement, et, since t1 will be

et = y + (Mg/L) exp (-Lti/R)

et = ((Mig/L) + (ag/L) oxp t1/R))

(1 - exp (-Lt/R)) (23) when t1 is very large

of = (M1 /L)(1 exp (-Lt/fl)) (24)

.question (2z) is identical with the result derived by Hardy (1965) by postulating that the specimen was in equilibrium when was added. The effect, then, of departures from equilibrium can be seem by comparing (23) and (24). The strain rate increases without changing the rate of strain hardnning.

6:3

If the increment had been (-MO (the equivalent of removing the first weight) the revovery curve would, be, from (23),

o = (Ug/L)(1 - exp (-Lti/R))

(1 exp (-Lt/R)) (25) 190

6:3 Earlier in this thesis it was shown tint the exponential law of transient creep is a less satisfactory fit to the data than a power law. Locker formed this opinion in 1925 (Crowan, 1967, p. 199). He investigated the iroblem of what distribution of viscosities of the viscous pores in a Kelvin body would load to a logarithmic creep curve. Differentiation of (17) with respect to time gives

1 = (Mg/R) exp (-14/r) (26) Equation (26) can be rewritten in tyrms of the inverse viscosity, 1 'relaxance', r for analytical convenience

= Mg r exp (-14'0 (27) and Assume that N(ra) dra elements have relazancos between ra and r2. Then ra dra and that all the relaiovires lie between r1 the displacement due to all the relaxances in a time interval Dt is Dx

r2 Dx = Dt ° N(ra)dra Mg ra exp (—Lrat) (28) ri

If the relaxances have a distribution

!(r3) dra = dra (C/i'3) (29) where C is a constant of the material; in the Limit of Dx 191

6:3

r2 .mg exp (-L r t) ri a r, = (C Og/Lt)(exp (-L ar t))xi! (30)

°Nation (29) can be interpreted by Kelvin's structural model. If the viscosity of the pores is controlled by the rate at which fluid can escape from them, sme 1 pores will have higher viscosities and lower relaxances than large pores. (29) would imply that there are more small pores than large pores, a reasonable assumption. The evaluation of funotions of the form (30) has been exhaust- ively discussed in Chapter 5:4. If Lr2 is very large and Lri is very sma/1, (30) represents a logarithmic creep law. If only hri is very smol, then the strain hardening is more rapid than a logarithmic law. Physically, this would represent a sample with some very small pores. If Lr2 is very large then strain hardening is less rapid than a logarithmic law and this represents a sample with some very large pores. The form of the creep curve ter an increment creep test for what might be called a Becker body can be derived in the same way as the conventional creep curve. Differentiating (23) with respect to time

8 = (4/R)(1 exp(-Lt/R) exp (-L(t WA)) (31) 192

6:3 Then the strain rate in a Becker body is by comparison with (30)

r ro = 2 CI g'exp (-rLt) ` GM exp (..rL(t + t )) 8 1 1 r1

t = (Chlig/Lt) (exp (-Lt) ) ' (32) 4-1 r2 + (CMg/L(t + t1))(exp + tl)) r1 (32) takes its simplest form when Lri is very small and Lr2 is very large, then

t = (C Mig/Lt) + (CMg/L (t + ti)) (53)

Equation (33) can be recognised as the sum of the motion due to the increments at zero and tl. If the motion due to the stress increment at zero is subtracted from (33), there remains

bt = (C idig/Lt) (34)

Equation (34) will be parallel to the parent curve on log Es - log t t coordinates. The recovery curve where Hig = ..Mg can be written directly from (33)

at = (Gtig/1)(1/t l/(t ti)) (35)

Equation (34) is a possible measn of distinguishing the Becker body from specimens behaving according to Mott's and Cottrell's theories. 193

CHAPTER 7

STRUCTURAL THEORIES OF BRITTLE CRFEP

7;1

Many of the transient creep experiments considered in this thesis were conducted under conditions where the specimen is brittle, that is, it fractures at qm/111 strains with loss of cohesion between the fracture surfaces. It is well known, for instance, that nearly all rooks fracture under uniaxial compression at room temperatures. Brittleness has certain implications. Pratt (1967) has pointed out that for a material to be able to undergo a general deformation 'there must be a sufficient number of independent slip systems, distributed homogeneously and able to interpenetrate, with enough mobile dislocations on them to accommodate the applied strain rate'. So at least one of these conditions is not fulfilled for most rocks at room temperatures. For the rocks to be capable of a general deformation the component minerals should be deformable. Observed slip systems for rock forming minerals have recently been compiled by Handin (1966) and Watchman (1967). Data on calcite have been added by 194.

7:1 Santhanam and Gupta (1966). Calcite and quartz are the two minerals that have been most intensively studied and in neither mineral vas there appreciable dislocation mobility at room temperature. In general deformation by crystal twinning or slip is insensitive to confining pressure. Only calcite, marble and hPlite have been reported to show stress-strain curves insensitive to confining pressure (Paterson, 1967). Deformation in other rocks can be mIpposed to be cataclastic. The increasing ductility of rocks with increasing confining pressure is due to the inhibiting of fracture propagation, (Pratt, 1967). Darrell (1965) has shown that the brittle ductile transition observed in rocks occurred when the stress required to propagate a creek rose to the stress required to overcome sliding friction on the crack. Other features of tha stress-strain curves of rocks are.11so adequately explained on the assumption that the rock is a perfectly elastic body containing an array of pre-existing cracks (Brace and t'ialsh, 19,6). This chapter therefore develops a theory of ores. in brittle materials based on the assumptions 1) Dislocation motion is either negligible or results in the dislocations being held up at obstacles with the possible nucleation of cracks. 2) The material contains pre-existing cracks. 194.

7:1 Santhanam and Gupta (1968). Calcite and quarts are the two minerals that have been most intensively studied and in neither mineral was there appreciable dislocation mobility at room temperature.

In general deformation by crystal twinning or slip is insensitive to confining pressure. Only calcite, marble and halite have been reported to show stress-strain curves insensitive to confining pressure (Paterson, 1967). Deformation in other rocks can be supposed to be cataclastic. The increasing ductility of rocks with increasing confining pressure is due to the inhibiting of fracture propagation, (Pratt, 1967). Murrell (1965) has shown that the brittle ductile transition observed in rocks occurred when the stress required to propagate a crack rose to the stress required to overcome sliding friction on the crack. Other features of th stress-strain curves of rooks are:.also adequately explained on the assumption that the rock is a perfectly elastic body containing an array of pre-existing cracks (]race and Walsh, 19A).

This chapter therefore develops a theory of creep in brittle materials based on the assumptions 1) Dislocation motion is either negligible or results in the dislocations being held up at obstacles with the possible nucleation of cracks. 2) The material contains pre-existing cracks, 195

7:1 7:2 These assumptions do not exclude dislocation models of the brittle creep. Misra and Murrell (1965) considered that both exhaustion and. strain hardening theories could be applied to their experiments. In both these theories the elements of deformation are dislocations in the structure of individual crystals. Notice, however, that both theories envisage the eventual immobilization of the dislocations at obstacles (jogs in the slip planes seem to be the simplest type of obstacle). In discussing creep in terms of the behaviour of cracks rather than crystal dislocations, one type of elastic dislocation is being preferred only to another because cracks, can, anyway, be represented by arrays of crystal dislocations, Cottrell (1963). The change is one of scale, not of kind.

7:2

It seems that only one theory, due to Scholz (19Aa), has been developed specifically to describe creep in brittle rock. It is reviewed briefly below where it is shewn to be unsatisfactory. Scholz (1968a) suggested that a creep specimen could be considered as a large number of small homogeneous regtens (elements) which undergo static fatigue according to (1),

t = (1/a) exp ((EIKT)b(F (1) za F )) 196

7:2 where a, b are oonstants. E is the activation energy of the corrosion reaction that leads to static fatigue. ais the is the stress instantaneous failure stress of the element and Fa on the element causing failure at tr, the mean fracture time. Two further assumptions were necessary; that as each element fails it contributes an amount, v, to the eeeel strain and that each region acts independently and only fails once. Scholz derived the theory for volumetric strains but he commented (p. 3299), "v can also be considered to be the increment of axial or lateral strain", If P(Fddt is the transitional probability of fracture, that is the probability that an element at a stress, Fa, will fail in the time interval, dt, tr is defined by (2)

f p(y-)dt = 1 (2) o

Thus P(Fa) = litr (3)

Substituting (3) in (1) gives

P(Fa) = a exp ( (WKT) b(Fm Fa)) (e)

If 10a'0 are the number of elements under stress, Fa at time, t, then the probability of one of these elements failing in the subsequent time interval, dt, is

f(c) = N(Fet) P(Fa) (5) 197

d(N(Fatt))/dt = 11(Fet) F(Fa) (6)

The axial creep rate is then

•F tt = v m N(Fet) F(Fa) dFa (7)

The resemblance in terminology with Cottrell's derivation of the transient creep law due to an exhaustion process is intentional, It emphasizes Scholz's comment, (p. 3299) "the derivation of the creep law given above is formally identical to the exhaustion theory". Integration of (6) from zero to time, t, gives

N(Fatt) = N(Fal0) exp (-P(Fa)t) (8)

Differentiation of (4) leads to

u(P(Fa)) = -b P( a)dF021. (9)

Assuming that the initial distribution, N(Fa,0) is uniform in the interval zero to FM and zero outside it, then N(a00) = N. Equation (7) can then be written

= vN a t 2 P(F ) exp • (-F(a)t) dFa

= (vii/bf ex? (-P(Fdt d(P(F)) (10) 138

7:2 The integration of (10) has already been discussed in the description of exhaustion theories of creep, Chapter 5:4, It leads to

t = vN/bt

Scholz's contribution was based on the assumption (1). This is composed of two statements.

tf = c exp (b(Fm - F)) (12)

t = d exp (/O) (13)

Equation (12) described the static fatigue of the elements at constant temperature and (13) their static fatigue at constant stress, echo's suggested that (12) could be verified by experiments on the static fatigue of homogeneous specimen* of silicates such glass. Scholz, then, has assume that a creep specimen is composed of a number of elements of the same dimensions and with identical physical and chemical prpperties (that is, they all obey the same law of static fatigue). The stress distribution in each element is assumed to be uniform and the elements are each stressed to different stresses in the range from zero to the instantaneous compressive strength of an clement. Under compression of the specimen, tensile stresses are assumed to be absent. 199

7t2 There are immediate difficulties with these assumptions. One of these is the definition of the instantaneous compressive strength of an element. Fracture of bodies under compression is invariably attributed to tensile stresses at cracks and other stress concentrations within the body, Scholz was clear,(1968a, p. 3298) however, that there are no tensile stresses within the specimen. So it is difficult to envisage that any fracture coubd ocour. Notice, also, that the stress distribution within the specimen is specialized. If the stress distribution within the elements is uniform, then their boundaries will be free of shearing stresses, for instance. Scholz has not discussed what arrangement of the elements would produce this *tress distribution. However, if the elements are to have perfectly smooth margins to eliminate shearing stresses then the specimen may not cohere. Even is the above theoretical difficulties can be resolved, the application of Scholz's theory would still be restricted to that well class of rocks composed of identical isotropic elements.

A new theory is developed below. It is based on recent work relatin,_ to the stress strain curves of brittle rock, Beck can then be treated as a perfectly elastic substance containing an array of pre-existing cracks (Brace and Walsh, 1966). To provide background for this theory, it will be necessary to review the data on static fatigue of silicates, the principal rock forming material. 200

7:2 - 7:3 Scholz (1968a) has drawn attention to the extensive experimental studies of the static fatigue of various types of glass reported by Schmitz and Metcalfe (1966), Mould and Southwick (1959) and

Glathart and Preston (1946). Other studies are reported by

Charles (1958) and Gurney and Pearson (1946).

7:3

Charles (1958) and Gurney and Pearson (1946) demonstrated that static fatigue in glass was negligible in a vacuum. It has also shown that vacua reduce the effects of static fatigue on basalt

(Krokosky and Husak, 1968), on ceramics, (Baker and Preston, 1946), on sintered alumina, (Pearson, 1956) and on fused silica rods (Le

Roux, 1965, Hammond and Revitz, 1963). Charles (1958), Sohoening

(1960), and Gurney and Pearson (1946) demonstrated that static fatigue of glass was accelerated by high concentrations of water vapour. Le Roux (1965) demonstrated the same effect of water vapour in the fatigue of fused silica, Gurney and Pearson (1946) showed that the presence of carbon dioxide in the surrounaing environment accelerated fatigue. These studies show that the fatigue of a wide range of brittle materials is dependent on the ambient environment. The common hypothesis of these experimenters was that static fatigue is due to stress aided corrosion at the tips of micro cracks gal

7:3 in the specimen. This causes the cracks to lengthen. After a period of time at a sustained stress a crack reaches a critical length, Jaeger (1962, p. 85), and propagates unstably, causing fracture 60 the specimen. The Griffith criterion for the initiation of crach propagation does not predict the behaviour of a propagating crack. Wells and Post (1958) have shown that a propagating crack under eeinvial tension in a direction normal to its direction of propagation will extend its own plane to a surface boundary. This result has been confirmed experimentally by Brace and Bombolakis (1963) and by Hoek (1965) for cracks in glass sheet.

Since all the experiments on static fatigue wore performed on specimens under bending or unioTiel tension, the above model of the process was adequate to describe the results.

Hoek (1965) has confirmed empirically the Griffith criterion for fracture initiation from open cracks in glass plates in uniaxial compression. In a modified form to allow for friction between the crack surfaces, it also applies to closed cracks. The behaviour of the propagating crack in compression is much more complex than in tension* Brace and Bombalakis (1963) reported experiments on open cracks in glass plates under uniaxial compression. At the critical stress, branch fractures propagated from the ends of the cracks and "became 202

7:3 nearly parallel with the direction of compression ... when this direction was attained further crack growth stopped apparently because of the decrease of tensile stress concentration at the tip of the crack considerable increase in compressive stress is required to cause additional growth of these cracks" Brace, Paulding and Scholz (1966) refer to this sequence of events as "crack hardening". Scholz's theory can also be criticised for the form of (12). Taking logarithms of (12),

log tf = log c + b(Fal - Fa) 410

So, from (14), a plot of the logarithm of the time to failure of the fatigue specimen against the applied stress should be linear. The three main groups of data that Soholz quoted, Charles (1959), Mould and Southwick (1959) and Glathart and Preston (1946), were collected for the purpose of determining the functional relationship between Fa and tr. All these authors displayed the data on F log t plots, To connect data collected under similar environ- a f mentaL conditions, they draw best fit curves, not straight lines, through the data. The curves are generally concave upwards. Glathart and Preston (1946, p. 189) explicitly reject (12) - "Baker (Baker and Preston, 1946) adopted the rather natural method of plotting (Fa against log tf,) and obtained very definitely curved 203

7:3 • lines the curvature being more obvious because of his longer range or timeintervals". They report that the data are adequately explained by (15)

(15) log tt = -a 4. b/Fa Mould and Southwick (1959) consider four proposed static fatigue laws to explain their data and that of Glathart and Freston (1946). Besides (15), they tried (16), (17) and (18).

log tt = a -(b/Fa) log Fa (16)

which was sue vested by Stuart and Anderson (1953),

log tt = -a + (/Fe. ) (17)

due to Elliott (1958) and

log tf = -a -b log Fa (18)

where a, b are positive conztants (though not the same constants in each equation). Equations (15) - (18) are predicted by various models of the corrosion process at the crack tip. Equation (18) was the only static fatigue law "in complete agreement with the data obtained in the study" (Mould and Southwick,

1959: P. 591). Charles (1958) reported that his data was well fitted by (18). Unfortunately the full experimental data have not been published 204

7:3 - 7:4. by any of the authors and the graphical representations are too

small to describe the data accurately. Charles (1958) conducted tests on groups of specimens at the same pre set stress. He then selected the mode of the logarithm of the time to failure and plotted it against the logarithm of the stress. A best fit straight line could have been fitted to this group of data by simple linear regression using the logarithm of the stress as the independent variable. It is doubtful whether the stress in the other two groups of experiments was sufficiently closely controlled to allow it to be treated as an independent variable. Notice, also, that "averages" of the times to failure of the groups of specimens were plotted. Lecause the averages were unidentified, it is probable they are arithmetic averages of the times to failure. The form, (18), would require the arithmetic averages of the logarithms of the times to failure be plotted against the logarithm of the stress. The fit of various functions to the static fatigue data, thus, is still a matter of opinion but the weight of evidence seems to favour (18) over (12). Charles* (1958) theory of static fatigue might form a more satisfactory basis for a theory of brittle creep than that of Scholz.

7:4.

Charles (1958) considered a highly elliptical hole of major a3rts, 205

7:4 I,* in a flat glass plate subject to an average tensile stress, Sy, in a direction perpendicular to the major axis of the crack. He suggested

L = r((si) ♦ k (19) x where tx is the velocity of the crack in the x direction.

Sx is the tensile stress at the tip of the crack tangential to the crack surface, k is the corrosion rate at zero stress. Suppose

f(S ) = e(S IS ) (20) x cr n where n is a positive constant, c the maximum velocity of the crack and S or is the tensile strength of the atomic bonds at the •crack tip. As

SiSy = 2(11/r)2 (21)

Scr/Sy = 2 (Lcrir)4 (22) where r is the curvature at the crack tip, Lcris the length at which the critical stress, Sorg for rupture of bonds at the crack tip occurs, Equations (21), (22) are derived from the theory of stress concentrations around holes in perfectly elastic bodies (Jaeger, 1962, p. 85). Substituting (20), (21) in (19) gave (23)

206

7:4 • L = c (L/L r1/2 k X or. ) (24) Charles suggested that if static fatigue is to take place, stress activated corrosion must occur at a much greater rate than stress free corrosion. Then the crack would grow with constant curvature, r, until it reaches its critical length, Ice If stress free corrosion were as important as stress activated corrosion the crack growth would occur with increasing radius of curvature and the stress concentration might be seriously reduced. The corrosion rate at sere stress, k, can therefore be neglected in (23) in comparison with the stress dependent corrosion rate. The temperature dependence of crack growth can be introduced by the assumption that corrosion is a rate process with an activation energy, A. The experimentally determined activation energy of the o process below 1!A C is close to that for the diffusion of sodium atoms in glass. Charles suggested that the sodium atoms catalyse the hydrolysis of the oxygen silicon bond in glass by creating free hydroxyl ions. Equation (23) can be written • Lx = B(L/LCr )11/2 ems . (-A/KT) (22) Integrating (24) with respect to time gave (25),

or d1(1/1or)'-n/2 y B exp /K0dt 2J7

7:4 (n 2)/2 (214cr/(n-2))(Ocr/L0) -1) = B exp (-A/KT) (25) then n and tfare large (25) can be written, (n.2)/2 (2 Ler/B (n2)) cxp (A/10)(Lor/1,0) (26)

Taking logarithms of (26),

log tr = (n/2) log Lcr * log D (27) where (n-2)/2 D = o (w2) (n-2) ex, (A/KT) (2) can be rewritten as (28)

L = rS' /452 or oar (28) Eubstituting (28) into (27)

log tr = -n log S. w log D' where -A/2 D' = (r5o2r /4)

Equation (29) gave the static fatigue law (18). The parameter, n, can be determ.nederom the slope of a log - log 5 plot, Charles (1958) reported a value about tf 16. The growth of sub critical cracks under tension has now been directly observed in glass microscope slides ('iederhorn, 1967) and in sapphire (i'rioderhorn, 1968). 208

7:4. It was found that the growth of a crack can be divided into two stages, a stage where crack motion is relatively slow and a stage of catastrophic motion initiated when the crack is long enough to satisfy the Griffith criterion for crack initiation. In the first region the dependence of crack growth on applied and relative humidity can be divided into three further stages. At the lowest velocities, the crack velocity, L, is exponentially dependent on the applied force, P. L = A exp (BP). A, B are constants. A is dependent on the relative humidity. Under a tension of 0.8 Kg, • L 2.11.10+6H2 + 1.02.10-5 H where His the relative humidity. ,ieuerhorn sug ested that crack propagation in this region (crack velocities below 10*.2cm. per second at 100Z humidity) was due to corrosive attack by water vapour at the crack tip. At higher velocities, the crack velocity is nearly independent of the applied force, a condition Uederhorn attributed to the limiting of crack velocity by the rate of water transport to the crack tip. At even higher velocities the crack velocity is again exponentially dependent on the applied force but it is independent of the relative humidity. 2.j9

7:4 The rate of crack growth in glas- immersed in ui.stilled water is slightly higher than for a crack in a saturated atmosphere. Stress dependence has the same form. The time dependence of static fatigue is thus controlled by crack growth in the first region. The time taken to transverse the other two regions is negligible by comparison. Uederhorn's data on the stress dependence of crack growth in glass in the first region can be replotted on double logarithmic coordinates, log L — log P, which is equivalent to assuming a stress dependence of the form, L = CPA. Taking logarithms, • log L = log C n log P The data seem to be adequately explained by this relationship. • Indeed, compared to the exponential relationship, L = A exp (B?), the scatter or the data is reduced. A more detailed analysis cannot be justified since numerical values of L and P were not given and P has a small range. The parameter, n, can be determined approximately from the slope of the fitted line and has a value about 19. The agreement between Liederhorn's data and that of Charles (198) for the value of n for glass is reasonable, particularly as they were working on different glasses. So the main assumption of Charles' theory is plausible. 210

7:5

The extension of Charles' theory to the growth ofcracks under uniaxial compression involves some problems with the stresses at the crack margins. Presumably, the effect of tensile stress at the crack tips at the atomic level is to stretch the bonds between the atoms allowing easier passage to diffusing sodium ions and hence increasing corrosion rates. Compressive stress at the crack tips will inhibit corrosion and the stress dependent corrosion rate for compressive stresses may be less than the corrosion rate at zero stress. Under these conditions, crack growth may result in the elimination of the stress concentration.

Jaeger (1962) has reviewed the general problem of stress concentrations round holes in a perfectly elastic medium. Consider an elliptical hale of major axis, a, and minor axis, b, in an infinite perfectly elastic plane. The solution (Jaeger, 1962, pp. 198 - 199) for the stresses used elliptical coordinates, x = c cosh z cos v, w = c sinh z sin v, then a = c cosh z„ b = c sinh z and the hole is bounded by z = so. The plane is subject to stress, PI, at infinity inclined to the x axis (which lies along the major axis of the ellipse) at an angle y. The tangential stress, Sx, at the boundary of the hole is, by Jaeger (1962, p. 199, (29)), 211

7:5 2 2 2 S = P (2ab (a - b x 1 ) cos 2y - (a b) cos 2(y . v)) (30) 2 2 2 a b - (a --b ) cos 2-tt tensile stresses are negative.

The stress at the crack tip is given by (30) with v = 0. For a flat crack, a is much greater than b. Then, (30) can be written as (31),

Sx = (P/b)(a-(b + a) cos 4) (31)

Equation (31) shows that if PI is tensile, the stress at the crack tip is tensile except where y is very close to zero; that is, when the major axis of the crack is nearly parallel to the applied tension.

If P is compressive, S is compressive except when y is very close 1 x to zero, The rteTimum value of the tensile stress at the crack tip is P (1 + 2a/b) when F1 is tensile (y = 90 degrees) and F1 when P2 1 is compressive = 0).

Then P is compressive and y = 0, notice that the tensile stress at the crack tip is independent of the form of the crack, Unless

Fi approaches the tensile strength of the atomic bonds at the crack tip the crack cannot propagate catastrophically.

Hock (1965, p. 16) pointed out that while the maximma tensile stress tangential to the crack surface or flat cracks occurred near the crack tip, it did not occur at the crack tip. He simplified

(50) by assuming v is small and b is smAll compared to a, Hoek 212

7:5 (1965, Appendix I). By difierentiating the resulting expression with respect to v, Hoek was able to show that the 1112.Ximum tensile stress, -t' near the crack tip is given by = p . 2 1 ( &In y sin y) (32)

zo = b/2a

Shen PI is compressive the negative sign in (32) is a.eropriate; is compressive t will then always be negative (tensile) when Pi except when sin y = 1 or 0, then St is indeterminate. Notice that as the positive sign in (32) should be used when Pi is tensile, St is always tensile and considerably larger than its value when Pi is compressive.

Hoek's approximation leads to errors -ohm y is close to zero or 90 degrees, This can be seen by comparing (32) pith (31), (which is exact) or from the predicted positions of Sty These are given by (33),

Vt = -b (tan y a sec y), (Hoek, 1965, Appendix 1). The errors arise because some products of b and trignometric functions or y removed by the simplification of (30) are not negli6ible when the trignometric functions take extrcme values, Amore elaborate analysis than Reek's is required to determine the exact situation, it will not be attempted here. Instead, 213

7:5 -.7:6 notice that symmetry considerations suggest that the maximum tensile stress is at the crack tip when the orack major axis is parallel or perpendicular to the principal stress and that (33) suggests that in other positions the maximum tensile stress is at some distanoe from the crack tip. The situation is more complex when the crack is closed. Hoek (1965, p. 24) used the same approximations as he made in the case of open cracks to show that on AcClintock and 'L'ialshts hypothesis of the behaviour of closed cracks,

s = P sin y (cos y m sin y) (3k) -t o 1 where m is the coefficient of friction on the crack surface. The stress', is tensile for values of cos y greater than m sin y. t Taking is to be equal to one, closed cracks inclined at more than 45 degrees to will net then grow in uniaxial compression.

7:6

now use this discussion of stress distribution around cracks and Chariest theory to explain brittle creep in uniaxial compression.

Suppose that a sub-critical crack in uniaxial compression ext nds in its own plane by stress corrosion aue to the tensile stress near the crack tips and that when it reaches a critical length, it propagates in the manner described by brace and sombolakis (1:;63). 214

7:6 Then, the principal contribution to creep strain comes from strains about propagating cracks. Once those cracks have propagated, they are stable or "crack har uoned". This last assumption can, sq ain, be recognised as characteristic of exhaustion theories of creep. Oack (146) has shown that results for stresses around flat cracks in two dimensions can be exscnded naturally to three dimensions, to flat cracks with a circular plan. Those cracks have been termed "penny" shaped cracks. The maximum tensile stress on the crack margin lies in the plane of the minimum and maximum principalAl stresses. it differs only by a constant frim the value predicted by equation (50) for cracks in 2 uimensions. Thers seem to be no recorded observations of the behaviour of sub-critical cracks in uniaxial compressions Perhaps the most likely assumption is that they grog by corrosion in the same direction as critical cracks propagate. Suppose, however, there are M (L,y) aL cracks in the creep specimen with lengths at zero time between Lo and Lo 4- dL at angles to the ppincipal stress between y and y + dy. If each crack caused a strain increment, v, on propagating, the total strain, de, due to these cracks is Id (Lmy) vaL. The time, tr, for a crack length Lo to grow to its critical length Ler is given from (25) by 10/2 -((n-2)/2) tf, = exp (A/KT) Lor (2/B(n-2))L -((n-W2) (35) =TELo 215

7:6 defining F. Similarly the time (tt dt) for a crack of length (Lo + dL) to grow tc Lor is given by

-((n-2)/2) (t+,-dt) = E (L + dL) (36)

Azbtracting (56) from (35),

-((n -2)/2) "((n"'2)/2) dt = FL" (1 - (1 + 4/1,0)) (37)

IV dt and dL are :moll :and n is large, (37) can be written, -n/2L = ((n-2)/2) EL0 dL (38)

Then the strain rate at tf due to the propagating cranks is do/dti, = M(La) v dL/dt,

de/dtt = (2/E(n-2)) Lo M (Loy) v (39)

It would be reasonable to expect more short cracks than long ones. Thus L (L,y) is unlikely to be independent of Lo. Unfortunately there is no direct .ay to determine the distribution of crack lengths, Gilvarry (1961) suggested the basis of an indirect method. He considered the size distribution of the fragments in the single fracture of an infinitely extensive brittle body due to the propagation of internal flaws. He divided internal flaws into three types, 7:6 depending on the number of flaws against which they terminate. There are volume, facial and edge flaws terminating against zero, one and two flaws. Further classes were excluded because of the fragments at fracture were four sided. He found

g = exp (-(x/k) (x/.)2 (x/i)3) where g is the volume (or weight) passing a mesh size, x; k, j, are the average spacings of edge, facial and volume flaws. If x is small then this may be written

g = 1 - exp (-(x/k)) and if x is very SMail

G = (x/k) (40) so that the smallest mesh size fragments are controlled by edge flaws. The weight, dg, in a size interval, dx, is given by differentiating (40),

dg/dx = 1/k (41)

As dg = NOL3 where N is the number of fragments in size interval, the number of fragments with sverage size, L, is given by

N = k" L-3 (42)

Equation (4.2) has been confirms experimentally by a number of workers (Gilvarry and Bergstrom, 1961). In eartioular, Hamilton and Knight (1958) report the exponent of L to be about -2.75 for

217

7:6 Pennant sandstone. angle fracture has been defined by Calvarry (1961) as "fracture by an external stress system which is instantly and permanently removed when the first one or fow Griffith cracks begin to propagate. subsequent flaws are activated by stress waves preduccd by propagation of prior ones ...." A-sume, then, that the smallest size fragments are bounded by flaws close to their original lengths so that the lengths distribution of the flaws can be written,

(y)L-.1a = M (L,y) (43)

Substituting (43) in (39) gave (44), (n-2m)/2 de/dtf = (2/(n-2)E) Lo M(y)v (44)

-(2/(11..2)) As Lo = (tf/E) from (35), (44) can be written

-2(m.1)/(n..2), - (n-2m)/(n-2) de/dtf = (2/(11..2)) E t • M(y)v (45)

(2m-2) 11,-2m m-1) deldtf = ((B ep(-A/KT)) (2/(11..2)0 jor 1/(n-2) M(y)v (4.6)

From (32)

S = S 2 (L (sin2y - sin y) (117) or y• 226

usinz OhArlos. terminology for crack:, inclinQa at y to the ,rincipal

oovressivo stress ::. :ubutitutie6 (4.1) into (4i) aivos (48) (42-2) r1.2m de/dtr = ((I cx,p (.1.A/Y,T)) (2/(n)t) n(—l) 1/(n-0 C: siWy — sinLy)/ . r.'-' M(y)v (46) Y or )

The creep rate of the whole specimen is the 51113 of the values of (48) over all the appropriate value of y. :Equations (34 and (47) are inaccurate when y is near zero cr ninety degrees. Cracks at very hieh or very /ow angles to tht: co;zpressivo stress will make

- only a all contribution to the total strain as the tensile stre:sos at their tips arc comaratively small. Thqre is, then, probably no serious error in evaluating the ewe only between the omits of, say, eighty five and five degrees. it is not possible to prcodict the creo:) rate fro:a (48) booause there are considerable uncertainties in the values ot A, Bo v and (y). li,wevar, as (48) )redicts the time, temperature and stress del)endence of transient creep rate in th oimon, the theory can still be tortcu. It can also be erten ed to describe the creep behaviour of the pocinen in inercent tests.

7:7

A crack langth, Lo, at zero will have leni;tho after 2i

7:7 From (35),

-(n-2)/2 -(n-2)/2 El(L o

(n-2)/2 ..2/(n-2) Ll Lo(1 " t1Lo /E1) (49) rAmilarly a crack leneth, (Lo + dL), will have lonGth, LI after

(n-2/2 -2(n-2) Li = (Lo + t, (Lo dL) /E1) (50)

Me difference, (Li - 11), is approviltely (1,1,-2)/2 ...2,/(n-2)

Equation (51) is an inereasinzli coarse approximation as t1 increases+ Lecause there were 2 (L o y) dL cracks between Lo and Lo + dL„ the crack density after time tl, Li is given by (n-2)/2 -2/(n-2) g(Le y) dL = M1 (1-t1140 /El) dL (2)

Suppose that after a duration tip at Zyl, the stress is raised to Sy2, Then from (34) the creep rate at a time, t, after this event can be written,

(st- 1(2 de/dt = (2/(n-21.71dLi v dL (53) 22.,)

7:7 Fslatien (53) can be written in ter; of thn original length of the cracks at zero time using (4.9) and (52),

(n-2)/2 -2/(n-2) ae/dt = (2,/(n-2)E2) (L(1-t, Lo /T) (11-2m)/2 (n-2)/2 -2/(n..2) 1(1,0$y)/(1-t1 ()

Thc creep rate at same time, t (greater than t1)• under stress, can be written from (39)

n1.24/2 de/dtx = (2/(n-2)131) L0 M(Ley)v

whore tx = E from (35).

ubstituting in (5.4.) gives -(n-2m+2) (u-2) do/dt = do/dte E2/E1. (l-t//tx) (55)

ince tx = t1 (Ei/E) t, (55) gives the increment curve in terms of the parent curve and is thus a suitable for for ,pvidicting the increment curve,

Notice that when tx is large compared to tl (55) can be written approvimAtely as 2a

7:7

-(n-2m)/(n-2) deidtx = B1 ((q1-32) t_ + t 1 x ) (56) where R is the strain rate on the parent curve at 1 time unit.

(56) is a form clove to that suEzested by the formal theory of strain hardening in Chapter 5:3. Chapter 7:6 and 7:7 have outlined a nevi structural theory of brittle creep, It is the only theory based on the hypothesis that creep is due to stress aided corrosion at the tips of small cracks with the creep specimen and the only structural theory in accord with the recent experimental work on the stress strain curves of brittle rook. 222

CHAPTER 8

VESTS OF THEORIES OF BRITTLE CRP Alit) Miit RESULTS

8:1

Chapters '!), 6, 7 have outlined the theoretical background of transient creep under uniaxial compression. In Chapter 8, means of testing the various theories of creep will be described and the results discussed. The forms of transient creep predicted by exhaustion and strain hardening theories, by the theory based on the Becker body and by the 'structural' theory of brittle creep are collected below with references to their derivation. It has not been possible both to remain close to the original nomenclature of the theories and to avoid confusion of symbols. The symbols in the equations will be defined, fully in the text. They are, for the structural theory, (2m-2) n-2ta 81; = ((B exp (-A/KT)) (2/(n-2)t) 1/(n-2) 2n(m-1) sin2y - sin y)/3 ow Chapter 7:6, (48), 223

3:2 • to: the straiu hlrdenik; theory,

= (1 1-1t ), uhater 5:3, (J), for the exhaustion theory,

t t = v :XT/At, Chapte 5:44 (33), fur the i-,ecker theory,

= C Mg / Lt, Chapter 6:3, (30)

Chapter 8 first disousses the new experimental evidence from this thesis on the stress and time dependence of strain rate in transient creep. Then, more elaborate forms of creep experiments, increment and recovery tests, are discussed.

Consider, then, the stress dependonco of strain rate, g thD structural theory of creep. From 7:4, (48)1

PI (1) j where C' is a constant, 0 is th co:apressivo stross on Y the f*ecimen. th,1 strain hardenik; theory, cause

gt = yl/(1 t) from 3:31 (20),

At = / (2) 224.

8:2 2 where Fit is large compared to one and Fi = nNbli Ai. From Chapter 50, Q1 = KTFy/Bh, where K is Boltzmann's constant, T is the absolute temperature and F is the yield stress of a deformation element at time, t. The parameter, h, is the coefficient of strain hardening and B is the activation energy of an element of deformation. Misra and Murrell (1965,. p. 572) have pointed out that, as the instantaneous strain is dominantly elastic under the conditions of their creep experiments, h is approximately equal to Young's modulus, E. Thus, Fy - F = Bet, where et is the strain in creep. Further, since the creep strain is negligible compared to the elastic strain, they suggest, Fy = F, where F is the applied stress. There are, however, reasons to suppose that strains in creep may be up to 4.01;',; of the elastic strains (refer to Chapter 3:7)* A great deal of the creep strain will take place in the first minute of deformation so that the creep strain, et, will vary by a comparatively small amount over the interval covered by strain observations. Thus, F = 1.4F to a good approximation and, hence, by substitution in (2),

If C F rr where G is a constant. In the exhaustion theory, AYa = B(1 . 1/Fy), from Chapter 5:4 (22), where the symbols have the same interpretation as in the strain 225

8:2 - 8:3 hardening theory. As 0 +F=F,A= 40. Providing F is la a smoi (or it does not vary greatly during creep, compare the strain hardening theory), A = B/F and hence 'I' = G

fIf by substitution in 5:4. (33). C is a constant. The Becker body leads, by writing Mg = F to simplify terminology, to

= C (5) where WI"' is a constant. Thus, three of the theories predict that the strain rate in transient creep at a given time is proportional to the load on the specimen. In the fourth, the structural theory, the strain rate is proportional to the load raised to a power.

823

Consider the data on the creep of Pennant sandstone at room temperature in Table 3. Table 3 has shown that creep behaviour can be expressed by two parameters ol, the strain rate at one minute and a strain hardening parameter, b2. If the strain hardening parameters are stress dependent, then the stress dependence of the strain rates will vary with the duration at which these strain rates have been measured. 226

813 The data of Table 3 have been reduced so that creep at any particular load is reiresented by a regression based on all the experiments at that load. The results of this process have been displayed in Table 2/e. To the mate of Table 24., should be added regressions based on one experiment (BI.1.19) at 2.5 tons load and one experiment (8I.1.30) at 4.5 tone load. These regressions can now be tested for parallelism by the method of Chapter 4:2. The results of this test (numbered 10 are in Table 34. and show no significant departure from parallelism (Column II). There is, thus, no reason to believe that the strain hardening parameters are stress dependent. The weighted mean, b2., of the strain hardening parameters is also calculated by computer program, le. It has been placed in Table 344 The weighted mean is the arithmetic mean of the strain hardening parameters when each is weighted by the sum of the squares of the residuals in the experiments it represents. The test of parallelism does not take account of any possible ordering of the strain hardening parameters by load. As a further test, the values of the logarithms of the strain hardening parameters, log b2, have been regressed against the logarithms of their loads (the independent variable). The results, numbered 1R, are in Table 35. The data have been set out in Table 36. The slope of the regression is far from being significant so it is unlikely that 227

8:3

the values of b for Pennant 2 sandstone are stress dependent. The values of b 2 are plotted against stress in diagram 8:2. The values of the logarithms of the strain rate in each experiment at one minute, one hour and at a time, m, are given in Table 36. The time, m, is the weighted mean of the means of the logarithms of the estimated times of the strain rates in the experiments. The strain rate, tt, at any time, t, is, of course, given by log tkt = log bl b2 log t. in diagram 8:1 the strain rates

at time, in, are plotted against the stress on the specimen as a percentage of the failure stress. The values from Table 36 have been regressed against the logarithms of the loads as the independent variable. The para-

meters of the regressions are tabulated in Table 35 as 2R, (log 61 1), 3R, (log%0), and 411t, (log tin). Reference to Table 35 shows that the regressions, 2R, 3R and

4R do not differ significantly from parallelism. Hence, the stress dependence of the strain rates can be estimated from any of the regressions. The regression, 4R, should be preferred since the strain rates used in it are closest to the strain rates at the mean time and, hence, are most precisely determined (refer to Chapter 2:5). Diagram 8:1

Dependence of strain rates on stress.

Legend Vertical axis, (abscissa), stress as a percentage

of the short term failure stress of the specimen.

Horizontal axis, (ordinate), strain rate in microstrains per minute at time, m (Chapter 8:3)

Open circles, Carrara marble

Filled circles, Pennant sandstone - 6- dia. a:.1

O 4

O O O

O • • 2 • • • • • •

0 '10 '20 '30 '40 50 '60 80 90 '100 Diagram 8:2

Dependence of strain hardening parameters, b2, on stress.

Legend Vertical axis (abscissa), stress as a percentage of the short term failure strength of the specimen. Horizontal axis (ordinate), value of b2 from Table 36.

Open circles, Carrara marble Filled circles, Pennant sandstone. O8 09 0S. 07. OE, OZ. 01. °Ott 06. 0L1 1

0 o• • • • 0 • • • • 0 0

Z:41 'EV 232

8:3 The variance of a strain rate estimated by a regression can be estimated (Held 1952, 18.3,J2 and Chapter 2:5). ':hen the strain rate is estimated at a time close to the mean time, the variance is given by u2/w, without serious error. The variance of the strain rate ehou/d be the same as that estimated from the regression, 4R, if the residuals in 4R arise solely from errors in estimating the strain rates. Thus, the power law dependence of strain rates on load can be tested by referring the ratio of the two estimated variances to F tables, A weighted estimate of the variance of the strain rates in regression, 4.R can be derived from Tables 23 and 24 and from the data on B1.1.19 and. B1.1.30 in Appendix 2. It is 0.0094. The annlysis of variance conducted here has been tabulated by Bald (1952, p. 537, Table 18..5), The variance of the individual strain rates given in Table 36 can be identified with the "within sots" variance of Hald. The variance estimated from the regression, 4R, can be identified with the "about the regression line" variance; it has six degrees of freedom, Clearly, the variance of the strain rate is significantly less than the variance estimated from the regression, 4R. So the stress dependence of the strain rates cannot adequately described by a power law.

Examination of equation (7) shows that, when m is close to one, 233

8:3

small changes in m will cause large changes in the value of the stress dependence; the time dependence is, however, much less sensitive. Changes in the value of m with the stress are not implausible in the structural theory. They lead, however, to a more complex model. Further, at least two experiments at different stresses are required to calculate a value of m and at least three to test a power law hypothesis. So, if m changes rapidly with stress, neither n nor m can be usefhlly determined from the creep experiments. To test the hypothesie that departures from a power law depend nee of strain rates on stress are due to changed in the exponent of the power law with stress, the data of Table 36 has been split into two groups. One group, contains experiments at 3.5 tons load and below. The other group, B, contains experiments at loads of 4.5 tons and above. The results of performing regressions similar to 2R, 3R and 4.1Z are tabulated as 2R A, 3RA, 4M, 2RB, 3RB and 44 B in Table 35. The fits of a power law to 3RA, 4RA, 3BB and 4R.B. all show significantly reduced variances. In particular, the variances estimated by 31A and 4RA are compatible with a power law dependence of strain rate on stress. Votice that the confidence limits on 4.RA are narrow (Table 35); the appropriate value of t, however, is 31.82 (Lindley and Miller,

1962, Table 3). Thus b should lie between 0.39 and 0.77 and 2 234.

8:3 there is only one chance in a hundred that it is greater than 0.77. Clearly those experiments at loads of 3,5 tons and less cannot reasonably be considered to have strain rates that follow a power law dependence on stress with an exponent of one. iecause the strain hardening parameters of the sandstone experiments are not siGnificantly different, the weighted mean, of the values of b2 for the experiments at loads of 3.5 tons and less can be used to write, from the structural theory,

(n 24/(n - 2) = 0.930 and from their stress dependence

2n(m - 1)/(n . 2) = 0.58

These equations can be uniquely solved for n and m as the root, n = 22 can always be discarded. solution gave n = 8.3, m = 1.22; these values are in the ranges suggested in Chapter 7;6. Assuming that no which measures the increase in corrosion rate caused by stretching the mineral lattice, is a constant of the mineral. and is not stress dependent, values of m can be calculated for higher loads. They are listed below. Load in tons a 2a(m-1 )/ ( n-2) 44,5 0.97 . 0.079 5.0 1.24. 0.63 6.5 1.44 1.16 7.5 1.06 046 8. 1.22 0.58 235

8:3 8:4. A.s m is not a constant, thu value of 2n(m-1)/(n-2) does not represent the exponent of the power law dependence of strain rate on stress. A rough value of this exponent comes from 311B and 11.RB; it is about 1.2 (Table 35, Column .1)2). So at about a third of the

failure strength, the exponent approximately doubles. Evans (1958, p. 182) reported a similar phenomenon in creep experiments on concrete.

6:4

Data on the stress dependence of strain rate in creep in

Carrara marble shows a complication that is not present in the data on Pennant sauustone. The experiments themselves have been slimmqrized in Tables 3, 25 and 26. Table 35 contains the parameters of the regressions on the data from experiments at 3 tons, 3.6 tons and 3.9 tons load "pooled" by the methods described in Chapters 4:2 and 483. These regressions and the data from single experiments g at 2.5, 3.3 and 4..05 tons load have been used to compute l' t60 and um for Table 36. The strain rates at time, m, are plotted against the stress as a percentage of the short term failure stress in diagram 8:1.

Following the same procedure as with the Pennant sandstone data, the strain hardening parameters of the Carrara marble experiments 6

8:4 arc compared, by testing the parallelism of the regressions. The results, RP, are in Table 344 they indicate that the hypothesis of parallelism can be rejected at the 1;:) level. However, persisting with the analysis of the d ta from Yable 36, the regressions 511, 6, 7R and 81 were performed.

The regression, 5R, of the logarithms of the strain hardening parameters against the logarithms ce the loads has a value of r1 which is significant at the 5,T. level, log b2 showing a significant decrease with increasing load. Values of b2 are plotted against stress in diagram 8:2.

From 8R, the regression of the values of log :;m against the logarithm of the loads,

2 n(m - 1)/(n - 2) = 4.79 oa) and from 2P,

(n 2m)/(n - 2) = 1.123 (9) Solution of (8) and. (9) gives a negative value of n which is clearly physically absurd. Another problem raised by the limitations for physical reasons of the value of n is the time dependence of the strain rate in the experiment on A2.1.10 at 2.5 tons load. From this,

(n 2m)/(n - 2) = 2.11 (10)

Inspection of equation (10) shows that if m = 0 and n = 10 the value 237

3i4 of the right hand side of (10) is 1.25. The lowest reasonable values of b, for A2.1.10 is about 1.8. Thus, either n must be about 1 with m zero or in must be negative. Consider the possibility that m is negative. This implies that the number of cracks increases with the length of the crack. At loads above three tons, where shorter cracks will be mqking their contribution to creep, there is no necdAw suppose that m is negative and the number of cracks can, then, be supposed to decrease with their length. Thus, the crack length distribution has a maximum grouped about those cracks which propagate early in transient creep experiments at about 3.0 tons load,. If the experiments on Carrara marble at 2.5 tons and 3.0 tons load are omitted from the analysis the regressions can, now, reasonably be supposed to be parallel (refer to 2a, Table 34). The reduced body of data can, again, be examined by regressing logarithms of the strain rates against the logarithms of the loads. The results 62A, 7r A, 8EA are shown in Table 35.

From 8RA, 2n(m 1)/(n — 2) = 2.36 (11) From 2 FA, (n 2m)/(n — 2) = 0.976 (12)

Equations (11) and (12) lead to estimates of n as 98.5 and

In as 2,16. The estimate of n is large. However, little confidence can be placed in the value of b 2. from Table 34, lower values of b 238

8:4. . 8:5 would lead to considerably lower estimates of the value of n. The variances estimated from the regressions, 7RA and 8RA, show that deviations from a power law fit cannot reasonably be due to uncertainty about the strain rates in the experiments. Again, variations in the value of are a possible cause of this discrepancy.

8:5 Chapters 2, 30 4 have been concerned with the form of the time dependence of strain rates in the creep experiments. There, it has been shown that a power law dependence of strain rate on time is a reasonable description of all the experiments. In Chapters

5 and 6, the three main formal theories of creep have been outlined; in these, the value of the strain hardening parameter, b2, is restricted to minus km. This is not, in general* an adequate description of the time dependence of the creep rate.

The average value of b2 for the Pennant sandstone data is — U.924 (from 1P, Table 34.). None of the experiments have values of b2 which are exactly minus one (Tables 3, 00. The experiments at 6.5 tons have a best estimate of b2 which is considerably larger than minus one; the possibility that the true value of b2 is minus one can be rejected at the 1;‘; confidence level. The data on the creep of Carrara marble (Table 3, 35) show that b2 is stress dependent (SR, Table 35). The values ofb2 for the 239

8:5

(:xp - riments at low stresses (2.5 and j tone load) are significantly

less than minus one at the confidence level.

The creep data thus show distinctly different patterns of behaviour for Pennant sandstone and Carrara marble. 1:ith Pennant

sandstonethevalueofb,4 is just Greater than minus one and the stress dependence of the strain rates is linear to a crude approxima-

tion. Carrara marbl_ shows a stress dependent b2 and the strain rates are roughly proportional to the square of the stress.

The structural theory attributes the difference in behaviour to differing corrosion reactions in silicates and carbonates resulting in different values of n and to differing crack length distributions.

The, length distribution of cracks in the tcio rook types can be derived from the calculated values of an.

The most renounced difference betueen the trio distributions is the relative deficiency of the marble in len4 and short cracks; the crack len,,th distribution has a maximum (Chapter 8:4). Brace

(19644 p. 153) suggested that the maximum crack longth in a rock s ple was a function of the grain size. An extension of this hypothesis would suggest that the crack length distribution was a function of the grain size distribution. Then, the clustering of the size distribution of the cracks about a broad maximum would appear a consequence of the equigranular texture of the marble described in Chapter 1:5. 24.0

8:6

The forms of creep after an increment of load has been derived from the various theories of creep in Chapters 5:3, 5:10, 6:3 and 7:7. They are collected below: Structural theory

D.2m+21/(1...2) 8t = (de/dtx)(E201)(1-t At + E1t/R2)) (7:7, 55) Strain hardenin; theory,

8t Q1/(t 401) (5:3, 19) Exhaustion theory,

tt = v NKT/Alt (5:10, 65) when t is greater than tw; the strain rate is zero where t is less than t w Becker body theory,

tt = (Cg/L)(Mlit + M/(t + t1)) (6:3, 33)

The equations given above are for experiments in which there has been a single increment of load. Additional load increments complicate the theories and their effect will not be discussed here. There seem to be no published accounts of single increment creep experiments on rock in the literature. The analysis that follows is based on seventeen neat increment experiments; six on 241

8:6

Carrara marble and eleven on Pennant sandstones

Experimental procedure was to raise the load after the specimen bad crept for some time in a conventional creep test. The load was raised by the poise propelling hand wheel (Figures 1 and 2). If the steelyard approached the bottom of its stroke before the desired increment load had been added, the steelyard was raised by the straining handwheel. The steelyard was finally adjusted to its position before the increment was added by the straining handwheel. Duration of the increment experiment was measured from the beginning of the loading procedure. The observations during the increment experiments are collected in Appendix 2. An outline of them, from a power law fit to the experiments forms Table 37. Because features of the creep behaviour in increment tests can be predicted from the creep behaviour at lower load (the 'parent' curve) and the various theories of creep, the experiments can be used to test the four creep theories. These tests will be described and assessed, theory by theory, beginning with the structural theory. By re-arranging equation (7:7, 55) the strain rate at a time, te on the parent curve can be derived from the observed strain rate on the daughter curve.

(n-2m+2)((n-2) (de/dtx) = bt(Ei/E2)(1-ti/(ti t/E2)) (14) 242

8:6

From (7:a, 47) and (7:6, 35)

/2 = (Sy2/Sy1) (15) where5y2 is the stress after the increment of load, and Syl is the original stress, The stress dependence of the crack growth rate, n, has already been determined in Chapter 6:3 and 8:4. The time, tx, is given by tx = t1 + (E1/E2) t.

Computer program, 1, generates 8t and t in the normal way from an input of the strain and time observations of the increment experiment* The program can be simply modified, if t1, (E /E2) nd (n-2ni + 2)/(n-2) are included in the input, to generate(de/dtx) and tx from et and t using (14)* These new estimated strain rates and their times, (de/dt om,can be regressed against one another in x t the same way as the strain rates and times, at, t, were* The resulting regression should be, on the structural theory of creep, identical with the regression from the parent curve. This may be tested by the methods set out in Chapter 4:2. The parameters of the regression ana the results of the tests of identity are set out in Table 41 (Columns III, IV). None of the experiments in Table 41 show significant departures from identity at the 1;:, level. So, the structural theory, even in its present approximate form can be regarded as offering a satisfactory prediction of the behaviour of rock in increment tests. 245

0:6 It follows from (50, 19), that on the strain hardening theory, such that, when it should be possible to find a constant, (k/P1), added to the time, t, at which each observation of the strain rate, tt$ is made, the strain hardening parameter of the resulting regression should be the same as that of the parent curve. The two regressions are, log 6t = log Q0 — log t for the parent curve and log tt = log Q1 — log (t 4.14/V1) for the daughter curve. In these, Q0, Qi, are the strain rates at one time unit in normal creep experiments at loads, Ito, 11. Computer program, 1, was modified to calculate an estimate of (k/P 1) and the resulting creed curve. Identity can, of course, be tested by the methods of Chapter 4:2. The parageters of the regression on the modified data, the value of the constant, tr (= k/Pi) and the values, TT, IV, of the statistics from the test of identity are tabulated in Table 38.

Since k = exp (e0 %1), (5:3, 18) and eo = Q log (1 Pot1) from (5:3, 15) or eo = o log (P ot]) for large values of t1, k = exp (log (Pot1)• Qa/Q1) thus 244

8:6 2 As Fo = AL A/Q0 (5:3, 15) 1 Q/ = Q1/Q0, tr = (‘I t1/Qo)cQ3. Fo/F1 As Qoand Qi are proportional to Fo and Fi, tr = (Yit2/R0)

Thus tr should be less than t, and can be calculated from Fo and t1. Four experiments in Table 38 show significant departures from identity at the l level. So the strain hardening theory cannot be considered a reasonable explanation of the behaviour of rook in increment tests. The exhaustion theory predicts "waiting periou.s" at the beginning of increment experiments. In these periods, no strain takes place. “aiting periods should be obvious in an inspection of the observations of strain. They do seem to occur in five experiments. In the other experiments, the waiting period, if it occurs, must be less than the duration to the second observation of strain in the incre— ment experiments. Estimates of the waiting period in each experiment, t $ are listed in Table 39. Creep motion after the end of the waiting period is given by (5:10, 65). This equation shows that the rate of strain hardening in the daughter curve when measurements made,during the waiting period are excluded should be the name as the rate of strain hardening in the parent curve, and that the strain rate in the daughter curve 245

3:6 is increased by Fi/Fo compared to the strain rate in the parent curve. Computer program, 1, can be adapted to adjust the daughter curve so that its identity with the parent can be tested by the methods of Chapter 4:2. The parameters of the regressions on the adjusted daughter curves and the statistics, II, IV, of the tests of identity can be found in Table 39. yhen possible waiting periods are removed from experiments on A2.2.43 and 81.1.06, there is insufficient data for fitting a straight line. One of the fifteen remaining experiments does not pass the test of identity at the 4] level. Further tests could have been carried out if there were a method of predicting the value of the waiting period, tw. The data, themselves, prevent the choice of tw at random, so the test of parallelism does have some value. Notice, however, that values of tw from the five experiments with obvious waiting periods are not consistent with estimates from the other experiments. So, explanations of increment experiments by the exhaustion theory are also suspect and the exhaustion theory cannot be considered a reasonable explanation of all the experimental data. The motion of a :eeeker body in an increment creep experiment is given by (6:3, 33). The motion can be considered as due to two causes, each represented by a term on the right hand side of 246

3:6 (6:3, 33). If the motion due to the 'memory' of the parent curve, (CMg/L)(1/(t1 + t)), is subtracted from the total motion, the remainder should show a rate of strain hardening identical with the

parent curve and a rate of strain equal to MI/M of the strain rate in the parent curve.

The calculations have been carried out by modifying program, 1. The parameters of the regressions on the modified data and the results of the test of the identity form Table 40. One experiment, (B1.1.13), does not pass the test of identity at the 1 level. In six experiments, the creep rate estimated from the parent curve and

due to the "memory" of previous creep exceeds the creep rate measured

in the increment experiment. Thus the strain rate due to the

increment of load is negative and the parameters of the regression

are not computable. Together with the experiment on B1.1.13, the negative strain rates suggest that the Becker body theory is not a

satisfactory explanation of the increment experiments.

In semmary, then, the analysis of the increment experiments

confirms the conclusions drawn from the analysis of the time and

stress dependence of the strain rates in creep experiments. It demonstrates that the three principal formal theories of creep are

not reasonable explanations of the data; the structural theory

is the only theory considered that is satisfactory. 247

8:7

One of the advantages of a structural theory is that it predicts proCesses which it may be possible to observe and measure.

The two final groups of experiments described in this thesis describe attempts to observe two effects predicted by thestructural theory; permanent deformation during transient creep and the elastic waves emitted by propagating cracks. The structural theory suggests that transient creep is caused by the propagation of microcracks within the specimen when they grow to their critical lengths by stress aided corrosion. The cracks do not heal when the original stress is removed; much of the deformation occurring in transient creep must thds be reckoned permanent. However, the accepted description of creep behaviour (Chapter 2:1) regarded transient creep deformation as recoverable when the stress is removed. So experiments on recovery test the structural theory of creep. Few recovery experiments have been performed. Hardy (1958) reported one experiment on Steep Rock Lake iron ore and Price (1964a) reported eight on a nodular muddy limestone from Warsop Colliery. Eleven new recovery experiments, all on Pennant sandstone, have been performed. Experimental procedure was similar to that in increment experiments (Chapter 8:6). After the specimen had been creeping under load for some time, the load was lowered by moving 448

8:7 the poise propelling handwheel (see Figures 1 and 2). If the steelyard approached the top of its stroke before the load on the specimen had been lowered to 0,1 ton, the steelyard as lowered by the straining handwheel. The steelyard was finally adjusted to its position before the load was lowered by using the straining handwheel. Again, the duration of the experiments was measured from the commencement of unloading.

In two experiments, B1.1.18 and B1.1.25, the data on creep under load has not been used in the aeelysis of the recovery experiments. The form of those experiments under load, both of which had lasted for a month, was strongly affected by a decline in the humidity of the laboratory of about 10 over this period. The recovery experiments lasted only two days and can probably be considered to have taken place under constant ambient conditions, All the recovery experiments have been fitted to a power law of creep. The results form Table 42. However, examination of the strain observations (listed in Appendix 2) shows an unexpected feature. All the experiments except those on B1.1.56 and B2.5,04, show recovery ceased after about a day and the specimens again began to contract. This contraction will be referred to as abnormal recovery, The experiments on B1.1,190 131.1.20 and 11.1.26 reached lengths shorter than those at the beginning of recovery. There is, thus, insufficient data to fit power laws of creep to the experiments 249

8:7 on 131.1.20 and B1.1.26. The experiments on B1.1.36 and B2.3.04 show no significant recovery after one day. Part of the appearance of abnormal recovery can be explained by reference to (6:3, 35), on the assumption that the apecimen is behaving as a Becker body.

8 = (CIg/L)(1/t 1/(t t1)) (6:32 35)

The larger of the two terms on the right hand side of (6:3, 35) describes the normal recovery of the specimen with the load removed. The seeder term is the 'memory' of creep under load and is an abnormal recovery; t1 is the duration of creep under load. As t becomes of the order ofti, the rate of abnormal recovery approaches the rate of normal recovery so the total rate of recovery becomes very &eau.

The rate of normal recovery can be estimated by adding the rate of abnormal recovery to the total recovery rate observed. The rate of abnormal recovery can be estimated from creep under load. The rate of normal recovery should, on the eeoker model, be identical with the creep rate in the specimen under load. Program 1 has been modified to add the rate of abnormal recovery to the data used to calculate the regressions for Table 42 and then to calculate new regressions using the modified data. The identity of these new regressions with the parent curves can be tested by the methods of 250

8:7 - 8:8 Chapter 4:2. The parameters of the new regressions and the statistics, II, IV, of the tests of identity are in Table 44. sour experiments are excluded from Table 24; two, B1.1.18

and B1.1.25, because the data on the parent curve is not sufficiently reliable anti the other two, 81.1.20 and B1.1.26, because there is insufficient data on them in Table 42. Inspection of Table 44 shows that only one experiment shows

significant departure from identity with the parent curve. However, the two experiments in which abnormal recovery was very pronounced, 81.1.20 and B1.1.26, can be tentatively added to this total so that three of the nine experiments cannot be considered as adequately explained by the Becker model.

8:8

In all the recovery experiments, rates of recovery at the end of the experiment were either negligible or abnormal. So, it can

be reasonably supposed that emax r , the largest observed strain in recovery, is a good estimate of the final value of the strain* The difference between this final value and the initial length of the specimen is placed in Table 43 under the heatiing (ed. - opap.„ r)*

nf is the final strain observation in the parent curve, In every experiment, the specimen, after unloading and being allowed to recover is shorter than its original length by about 10;:, 6:8 of the total strain under - load. In experiments at low loads, the percentage is higher; 16.7%, for instance, in B1.1.29. The observed contraction is about 4. to 5 times the size of the total contraction observed in creep under load and 10 to 25 times the maximum expansion observed in recovery. One explanation of the data of Table 43 is that nearly all the strain that takes place in creep under load is permanent strain. The strain is permanent only, of course, in the sense that it is not recovered in the duration of the recovery experiments. Important amounts of permanent strain musts .then, have taken place before the first observation of strain in creep under load. The total permanent strain before this first measurement can be estimated by subtracting the. observed strain in creep under load from the total permanent strain. The reult is listed under (elf - ein Table 43. Chapter 3:6 described, a method of estimating an. initial strain, , from creep under load by integrating the power law of creep. This estimate can also be used to calculate the strain before the first observation, it is tabulated, as (elf - e om) in Table 43. Comparison of (e - c ) and (e - e ) shows that the two lf lf of quantities are of the same order of magnitude. A.closer agreement could not be expected. Qien the strain hardening parameter is close to one, the estimate of eom is very sensitive to its value. tihen the estimate of b2 is very close to one, then 00 may too small - 8:8 — 8:9 the experiment on B1.1.36 is an example. Vhen b is not so close, edm may be too large. Thus, the evidence of Table 43 is at least consistent with the suggestion that most of the strain that takes place in transient creep in .Pennant sandstone is permanent (in the previously Mined sense of this word). The Leckevbody theory of recovery failed to predict the large amount of permanent strain observed after creep under load. The general failure of rheological theories to predict the permanent strain of rocks under load had already been described by Evans (Limns, ;i.11., in the discussion of Terry and Morgans, 1958) thirty years ago. In what folloes, an attempt is made to outline a structural theory of recovery, d:9

Suppose that, as in Chapter 7:6, permanent deformation is due to the growth of cracks by stress aided corrosion at their tips and their subsequent propagation (allowing sliding on their surfaces). Suppose, also, that most of the cracks are stable in their new positions after the load has been removed. Cracks and voids which arc permanently closed by the application of . a load are, of course, included in this total of permanent deformation. 2)

8:9

.'ealsh (1965) has pointed out that movement on cracks whose

surfaces are in contact is opposed by frictional ferces between

the surfaces of the cracks. No motion takes place till sliding forces are sufficient to overcome friction. The frictional forces

between the crack surfaces will be reduced with time as individual

asperities on the surfaces fail by a corrosion process. Thus,

cracks whose motion is held up by frictional forces between the

crack surfaces can be expected to move eventually; normal recovery

can be attributed to this motion.

The effect of friction is unimportant if thl.e, stress that brought

the crack surfaces into frictional contact is much above the load. during recovery. Providing no permanent deformation has taken piece,

removal of load from a crack closed at a relatively high stress will

result, simply* in the crack opening, again. As the load during recovery is only 0.1 ton, friction is only important on cracks that closed at low loads. All the specimens were stressed during the

parent experiment to relatively high loads so it might be expected

that rates of recovery would be independent of load during creep. The strain rate in the recovery experiments after one hour has been

calculated by the method used to compile Table 56. in Table 442

the strain rates from the estimated normal recovery rates do not

appear to show any strong stress dependence; nor do the total 2,54

8:9 recovery rates in Table 42 show a consistent pattern of stress dependence. Price (1964a) has suggested that abnormal recovery is due to residual stresses in the specimen. If creep under load involves the interaction of some propagated cracks, then internal stresses will remain when the specimen is unloaded. Abnormal recovery can be considered as taking place by the same physical process as normal recovery under a different stress field. Abnormal recovery might be considered as a measure of the amount of interaction between cracks in the specimen. On a structural hypothesis of recovery, then, the amount of recovery would be smell and roughly equal to the amount of creep under low loads (in this case, loads of about 0.5 ton). The total normal recovery in the experiments on Pennant sandstone has been estimated by (e r elr) in Table .3.3 It is in reasonable agreement with the structural hypothesis. The data from the recovery, experiments thus support the structural theory of transient creep. They show that permanent deformation does take place during transient creep and, that the amount is approximately equal to estimates of the total amount of creep deformation. Recovery is A less marked process than previously thought, Neither the total amount of recovery nor the form of 255

8:9 recovery creep curves appear to be stress dependent. Recovery is consistent with the hypothesis that it is due to the sliding of cracks temporarily held up by frictional forces between their surfaces. 256

CHAPTER 9

411.1;;ROSEISlaU LAI6 Si QN

9:1

The velocity of propagation of critical cracks in rocks under

ImiAxinl tension has been shown to be extremely fast and to approach a theoretical limit of AL of the velocity of elastic shear waves (Dienawskj., 1967, p. 40). Deformation caused by the propagation of these cracks is thus sufficiently fast to emit elastic waves and set up oscillation throughout the volume affected by the redistribution of the stress field round the crack. Detection and analysis of the movements on the surface of specimens caused by these elastic waves would give information about the events occurring in the interior of the specimen in the same way that earthquakes offer information about the earths interior. The events causing the movement are called mioroseisms, by analogy with earthquakes, and the data recorded through sensing instruments, microseismic emission. This chapter on the possible applications of microseismic emission to the study of brittle ert,,v, first reviews the properties of elastic waves in rock and then considers recent studies of 257

9:1 — 9:2 microseismic emission. Some new experiments are described and. these lead to suggestions for further studies.

9:2

Analysis of the equations of motion of an infinite elastic medium shows that waves can be propagated in the medium with two different velocities (Kolsky, 1953, p. 13). Dilational waves, also called longitudinal, primary or P waves travel with velocity ((L + ziwa)i, where L, L are Lame's constants and R is the density of the medium. Distortional waves also called transverse, secondary, shear or S waves travel with velocity (1002. If the medium has a boundary then a third type of wave motion, aayleigh waves, is possible. The displacements due to waves of this type die away exponentially with distance from the free surface. The rate of propagation of these waves is slightly less than that of S waves. It can be shown that the attenuation of the waves with dqpth in the medium is proportional to their frequency (Kols4y, 1953, p. 21). If the medium shows any anisotopy of elastic properties a further type of wave motion, Love waves, is possible. The behaviour of elastic waves is si —tar to that of more familiar forms of wave motion. At boundaries of the medium, waves are both reflected and refracted. It has been shown (RolsIcy, 1953, p. 24.) that when either a P or an S wave impinges on a boundary a 4,8

9:2 wave of each type is reflected and a wave of each type is refracted.

The partition of energy between these waves depends on the nature of the original wave, its angle of incidence on the boundary and the relative elastic constants of the two media. The complexity of the behaviour of an elastic wave at a single interface suggests that it would be very difficult to predict its behaviour in a medium with several boundaries. Exact solutions have enty been obtained for simple cases such as the propagation of a wave down the axis of an infinitely long cylindrical bar

(Kolsicy, 1953, p. 42). It is not generally possible then to deduce the motions of the source of an elastic wave from observations of the motions of the boundaries of the medium containing the source, even when the source is perfectly elastic. If the medium is anisotropic or an aggregate of anisotropic particles as most rocks are the problem is even more complex. Real solids are never perfectly elastic and some mechanical energy is always lost in propagating a uisturbance through them. This loss occurs by a number of mechanisms collectively called internal friction, There are at least two important kinds of internal friction (Gordon and Davis, 1968); that duo to static hysteresis and that due to velocity gradients set up in the material and associated with 259

9:2 a viscous response to deformation. A characteristic of static hysteresis is that the hysteresis curve is independent of the rate of traverse of the curve. This characteristic leads to internal friction which is independent of the frequency of the wave. The internal friction of dry rock is generally independent of frequency and amplitude, (Bradley and Fort, 1966) so Gordon and Davis (1968) have suggested that the dominant process of internal friction may be static hysteresis by sliding on networks of cracks in the specimen. Static hysteresis by sliding on individual cracks has been discussed by Walsh (1966) and Orowan (1967). Orowan (1967, p. 201) has however pointed out that the "Becker" body (Chapter 6:3) also has frequency independent internal friction. Thus viscous deformation cannot be ruled out provided that the viscosity of the elements is distributed in the correct way. interstitial fluias can cause preferential attenuation of the higher frequency and higher amplitude components (Gordon and Davis, 1968, p. 3925). Notice also that frequency and amplitude dependence of internal friction are customarily determined by forced resonance methods in which pick-up and specimen resonances are controlled. A microseism will excite resonances in the specimen pickup system which result in some frequencies in the emission being preferentially amplified. This effect and the effect of interstitial fluids 260

9:2 - 91) further complicates the interpretation of boundary motion in terms of source events.

9:3

Malone (1965, p. 75) distinguished two approaches to the experimental study of microseisms. The first used statistical treatment of emission records to investigate empirical relationships between emission an specimen deformation. The other approach attempted

to analyse discrete emission events. Only Scholz (1968b) has used this second approach in studies of microseisms from rook. The first detailed laboratory investigations of microseismio emission from rock are due to Obert and Duvall (194.5). They performed 17 experiments on a variety of rock types out into 4" x 2c 2e rectangular blocks with a li" diameter hole drilled between the centres of the two largest faces. The hole accommodated an 8" long, 1:4" diameter Rochelle salt piezo-electric microphone. Output of this microphone was amplified and recorded on a paper oscillograph. The microphone was held in contact -with the rook by its own weight. Lead on the specimen was applied by a hydraulic press fitted with a special valve. The load was applied in increments and after each increment the valve was closed and the hydraulic compressor shut off

to prevent machine vibration obscuring emission from the specimen. :rainsion vas recorded for four minutes before the load. was again 26].

9:3 raised. Load increments were applied until the specimen failed. Obert and Duvall found that at loads between 1/8 and i of the crushing strength of the specimen (depending on its lithology) high rates of microseismic emission were produced and nearly always associated with visible cracks in the specimen. The cracks originated from the margin of the central hole and occurred in a vertical plane passing through the centre of the bole. As the load was further incremented the rate of microseismic emission fell reaching a minimum between i and of the crushing strength. The emission rate then either rose steadily till failure occurred or remained constant till close to the failure load when a rapid rise in emission rate occurred. Emission at low loads was much reduced when a different specimen eometry was used. With anvil shaped apecimens the load at which emission was first recorded was increased and there was little visible cracking before failure. There was still some tendency however for emission to be at a minimum at about 7Q of the crushing strength. Similar patterns of emission rate Uth increasing load have been observed in experiments on cylindrical rock specimens where the load has been increased at a constant rate (Goodman, 1963, Knill, Franklin and Malone, 1967). Mogi (1962) has shown that emission during creep in bending 262

9:3 decreased with time before increasing again as time dependent failure is approached. This observation has been confirmed by Brown (1965) for simple tension tests. Emission has also been recorded during stress decreases. Goodman (1965) showed that this emission decreased when the load on the specimen was cycled at below 8O) of the crushing strength of the material. After about 12 cycles no emission is observed on decreasing the load. Most of the emission is recovered if the specimen is left for some days under no stress. Abort and. Duvall also reported emission while the stress was being decreased. No emission took place, however, under constant load after the load has been decreased. Mogi (1962) has investigated the amplitude and interval distribution of microseisms during creep in bending. He has shown that the amplitude distribution followed the same relationship as the amplitude distribution of foreshooks and aftershocks of shallow earthquakes. He also showed that the intervals between shocks were randomly distributed. Scholz (1968o) reported measurements of emission from cylindrical specimens of various rock types under compression at a constant axial strain rate. He found emission even at very small percentages of the failure stress. This emission however quickly died down. At stresses of about half the fracture stress emission began "to 263

9:3 - build up once more and steadily increases until just before fracture when a very rapid acceleration of activity occurs". Scholz showed that the rate of emission was proportional to the dilatancy in the specimen. Because he attributed dilatancy to the opening of microfractures in the specimen, he concluded emission was connected with miorofracturing. Scholz (1968o, p. ]428) reviewed Mogi's and brown's work and concluded that "the microfracturing behaviour of rock in uniaxial tension is also very similar to the behaviour in compression. The observed difference between the propagation of cracks in tension and oompression therefore seems to have very little effect on microfracturing behaviour.

9:4

This brief review indicates that there has been no work on microseismic emission under constant compressive load since the pioneer observations of Obert and Duvall (1945). It is doubtfUl whether the specimens used in these tests were under even an approximation to a uniaxial stress field. The stress distribution round a circular hole in a perfectly elastic plate due to a uniaxial load at infinity gives a qualitative indication of the stress distribution in the specimen. It may be shown (see Jaeger, 1962, pp. 167-180 264

9:4 that the stress tangential to the holo boundary was tensile where it was perpendicular to the applied load and equal in magnitude to the stress at infinity. Where the tangential stress is parallel to the applied load it is compressive and three times the magnitude of the stress at infinity. Hoek (1965, pp. 126-159) showed phetoelastically that the analytic solution was a reasonable approximation to the stress distribution in a five inch square, 1/8" thick plate with a diameter central hole. The disturbance in the uniaxial stress field was no approximately, at distances 1 hole diameter away from the hole perpendicular to the applied load.

Unfortunately Hoek's results cannot be simply extended to the specimen configuration used by ()bort and Duvall because the lateral margin of the specimen is less than a hole diameter away from the hole margin. Moreover the introduction of the free surface reduces the area perpendicular to the applied load and probably increases the tangential compressive stress parallel to the load compared to the analytical solution. Presumably then the tangential stress perpendicular to the load also increases in magnitude. The hole in the specimen thus acts as a stress concentration of at least the magnitude predicted by the analytic solution and disturbs the supposed uniaxial stress field over the whole of the specimen. Some new investigations of microseismic emission in creep under 265

a stress field which is a closer approximation to a uniaxial compressive field are described below. Much of the apparatus and technique used has already been described by !thin, Franklin and eione (1967) and Malone (1965) and need not be described in detail here. A Pennant sandstone specimen was prepared as if for a creep test and placed in the Denison T55 compression creep tester. An Endevco 2150 transducer was attached to the specimen using a steel check piece, vaseline and a rubber band. Cable from the transducer was carefully taped down and connected to an Environmental Equipments

LeLed 'CVA-2 charge amplifier. The emission record eas stored on magnetic tape using a Ferrograph 631 tape recorder. The characteristics of this model are similar to those of the Tandberg 62 tape recorder used by Neill, Franklin and Malone (1967). Because the experiments were only exploratory the initial analysis of the recorded signals was considerably simplified compared to the analysis of Hall, Franklin and Malone (1967). If the signal level under constant compressive load. was greater than the level recorded under no load then this was regarded as evidence of microseismic emission from the speoimen in creep in compression. There are at least two complications in the interpretation of a change of emission rate under load. Malone (1965, p. 80) has shewn 266

9:4. that increasing load on the specimen could reduce by up to 40;t: the

response of the specimen to lateral excitation. The reduction was

due to restraint on the ends of the specimen, Increasing load may

also improve the contact between the specimen and the platens of the testing machine. This will raise the extraneous noise level

since transmission through the testing machine will be improved, This factor may be evaluated by recording after the specimen has been under load for sufficient time to allow any microseismic emission

to die away. Recordings were played back into a hacal SA 535 Universal. Counter- Timer set in the frequency mode. In this mode the Counter Timer counts all pulses of greater than 140mV peak to peak amplitude.

If the amplification of the tape recording is decreased until the measured pulse count is of the order of 10 only a few pulses above this critical amplitude are being received. The count under no load can be directly compared to the count under load if the amplifi-

cation of the signal is unaltered. The ratio of these counts is the ratio of the number of pulses per second exceeding 140mV peak to peak amplitude.

The simple analysis described is adequate for its immediate purpose and requires no calibration.

Two experiments were performed on Pennant sandstone. specimens. Procedure was to load the specimen to 0.1 ton. Then, switch on the 267

9:4. charge amplifier and tape recorder to allow them to warm up for twenty minutes. 5 minutes of emission was then recorded. viith the tape recorder still running the load was raised to eight tons and fifteen minutes of emission was recorded, The end of loading activity was marked on the tape by the cessation of the characteristic noise of the loading gear train, this could be checked by noting the reading on the tape position indicator of the Ferrograph as loading ended. End of loading could thus be fixed to at least the nearest

15 seconds.

An insignificant decrease in emission under 8 tons load compared to emission at 0.1 tons resulted from both specimens. Neither emission record showed any time dependence over the fifteen minute interval at eight tons load. Emission level was monitored at 0.1 toms load during the second experiment and averaged about 8 mV at the output of the charge amplifier at full gain. This low level compares favourably with the levels of extraneous noire recorded by Malone (19;:,5, p. 50). The emission appeared to be fairly uniformly distributed across that portion (below oitc) of the spectrum of audio frequencies where the transducer has a linear frequency response. A visual analysis on an oscilloscope revealed nu obvious peaks. 'ant; scion at both loads was played back through a variable filter into the Counter Timer.

No significant change occurred with bottom cut operating at 50, 250,

500 or 1000 c.p.s. 268

It can be concluded that if microseiems occur in creep in compression in Pennant sandstone, the accelerations at the margins

of the specimen due to the events are smeller than those due to events under constantly inoreasine; compressive loads.

Another series of experiments were designed to repeat Obert and Duvall's work, to establish to significance of the holes in their specimens and to further check experimental technique.

Rectangular blocks of Carrara marble were prepared with dimensions, 3" x x by a rock saw. The square faces were ground flat and parallel by mounting the blocks in clamps on the table of a surface grinder and finished by hand with fine carborundum powder. Half inch diameter holes were drilled, between the centres of oeposite 3" x faces in three specimens numbered 5, 6 and 7. The accelerometer was mounted at the centre. of another 3" x 1" face on specimen 5 coupled to the rock with vasoline and held, in position by an elastic band. Since the specimen face was sawn flat no cheek piece was necessary to maintain the mounting position. The specimen was then placed on its square ends in the fifteen ton creep testing machine. The recording system was set up as before,

The experiment on specimen proceeded as in Obcrt and Duvall's work. mission was recorded for four minutes under 0.1 tons load. 269

9:4 The load was then incremented in one ton steps pausing for four minutes after each step till the specimen failed under 5 tons load. Ratios of recorded noise at 1, 2, 3, 4. tons to noise at 0.1 ton load are given in the accompanying table. They show a significant increase of noise over the noise level at 0.1 ton load. Experiments were conducted in a similar manner on specimens

6 and 7 recording emission at 0.1, 1, 2, 3, 4., and 4L- tons load. An increase in emission level under load is again apparent though due to the more complex loading pattern there is no simple relationship between load and emission level.

Eussion RATIOS Specimen 1 4.11 No. 2 3 4 tons load

5 1.313 1.398 1,90 1.538 6 1.054 1.061 1.023 1.096 0.961 1.043 7 1.023 1.004 1.129 1.273 1.168 0.981

Three specimens similar to 5, 6 and 7 but without the central hole were tested in a similar way. Emission was recorded for 4. mAriltes at 0.1, 6, 7, 8 tons load. Specimen 2 failed as it was being loaded to nine tons. Emission ratios are listed below. There is no significant increase ofemission level at higher loads. 270

EIISSION RATIOS

Specimen 6 7 8 tons load ao.

2 0.966 1.046 1.067 3 0.962 0.943 0.964 4 0.757 0.820 1.371

Specimens 2 and 3 were than drilled to leave a in diameter osntral hole and again tested at 0.1, 1. 2. 3. 4 and 4i tons load. The emission ratios show increases in the amount of omission under load.

ElISSION RATIOS

Specimen 1 2 3 4 4-l- tons load No.

2 1.233 .4187 1.213 1.498 1.589 1.457 3 1.093 1.067 1.161 0.972 1.079 1.023

This series of experiments show that significant emission ratios are correlated with holes in the specimens. Unlike the experiments of Obert and Duvall there was no visible cracking around the central hole in any specimen before failure. But then the diameter hole is probably less of a stress concentration than the 1? diameter holes in Obert and Duvall's specimens. iotico however that cracking 272.

9:4 - 9:5 occurred in their specimens where the tensile stress tangential to

the hole was a maximum. It is reasonable to associate the local failure of the specimen under tensile stress with microseismic

emission as Obert and Duvall have associated the cracking with

microseismic omission. Failure to observe microseismic emission under uniaxial compression can then be attributed to the smaller amounts of energy radiated from fractures propagating under uniaxial compression.

9:5

The problems of the energy balance for propagating fractures have been reviewed by Bienawski (1967). It can be shewn that

dW/do = dWe/dc + dVik/do weere dWe is the stored elastic strain energy transformed into surface energy, dWs, and kinetic energy dWk, as the crack grows a length do. Bienawski also showed experimentally that the terminal volosity of a propagating crack in norite plates under uniaxial tension is not reached till the crack is 25 times its original "critical" length. dW k thus rises rapidly in early stages of crack growth and hence de rice as is a constant of the material. e s Cracks growing under tension in the rock specimen are stopped only when they enter a zone of lower stress in the heterogeneous 272

:5 stress field of the specimen (Scholz, 1968c, p. 1429). Cracks growing, under compression tend to "harden" at less than twice their original length. So propagating cracks under tension will generally grow further and hence faster than cracks in compression and will thus release considerably more strain energy. This statement is not in disagreement with Scholz's remark (1968c, p. 1428) that the behaviour of the two types of cracks is similar - Scholz seemed to refer to their qualitative behaviour, rather than their quantitative behaviour. There remains the problem of the microseismic emission observed under dynamic compressive loading (Goodman, 1963: fill, Franklin and :alone, 1967, Scholz, 1968b, c). Scholz's work should be discussed separately as the emission he observed was considerably be_; once the audio range of frequencies in which the other experimenters worked. There is no evidence in the other two groups of work that emission originated from within the specimen being deformed. Analysis of this statement can be improved by considering another hypothesis, that sudden changes in the strain rate applied to the specimen cause transients in the loading frame which themselves produce en masse vibration of the specimen loading frame system. Changes in the strain rate could be caused by non uniform elasticity in the specimen. Typically such non uniformity could be caused by microfracturing 273

9:5 within the specimen.

Malone (1965, p. 80) reported evidence of en masse movement of the system used by Franklin and galone (1967) under lateral excitation. Such movement would result in large levels of emission at resonant frequencies or the system. Unfortunately

Goodman (1963) did not investigate the frequency spectrum of the microseismic emiseion heerecorded. Xnill, Franklin and Malone

(1967, p. 107) report only a partial analysis sufficient to show that the amplitude of the emission varies with frequency.

A complete analysis of the frequency spectrum of microseisms emitted from rock in the audio frequency range under uniaxial tension is reported by Chugh, Hardy and. Stefanko (1968). They show that peaks of emission in the frequency spectrum correspond to resonances observed in the specimen-loading system under external excitation.

It seems possible then that the microseismic emission detected in the. audio frequency range from experiments under varying load is more a reflection of the characteristics of the specimen-loading frame system than of any microseisms that may occur,

Scholz (1968c) has extended the frequency spectrum of the detection apparatus. He found that most microseismic emission occurred in the range 100At to 1 Mc. He estimated that by increasing 274

9:5 the frequency response of the detection system he increased the

sensitivity of the system by "two or three orders of magnitude". Many of the difficulties associated with analysis in the audio frequency range could thus be avoided.

Scholz (196814 pp. 1451 - 1452) used the arrival times of a large microseismio event at six transducers placed on a specimen

to calculate the location of the event and (for the first time) establish that the source of the event was within the specimen.

It is reasonable then to prefer the slightly different description Scholz (1968b) presented of the dependence of emission rate on stress to those of Goodman (1963) and Franklin and Malone (l67).

Events recorded at less than 90;',: of the failure stress appeared to be randomly distributed in the• specimen and to occur at random intervals. Above this stress the events group around the eventual failure plane and occur at non random intervals. Triggering of one event by another is suLgested by a non random distribution of intervals, (Scholz, 1968b, pp. 1451 - 1452, 1968e).

Scholz (1968d) has investigated the magnitude distribution of microseismic emission and has shown that it has the same form as the magnitude distribution of the fore shocks and aftershocks of earthquakes. This is described by the Actenberg Richter law

lot;N=a-rbE 275

9:5 where N is the number of ecents of magnitude M on the Gutenberg Richter scale and a and b are constants. Mogi (1962) showed that b varied with rock type for emission ender creep tension. Scholz showed that b is only slightly dependent on lithology in compression but depends strongly on stress. At qmpll fractions of the failure stress large numbers of small events are observed, as the stress increases, b decreases indicating that larger numbers of larger events occur. By using an empirical relationship between the amplitude of the emission and the energy of the microseism, Scholz showed the large microseisms were emitted from cracks which propagated over large areas. Ho concluded that the area of propagation of a crack was an increasing function of the applied stress.

Scholz's work suggests several ways in which microseismic emission can be used to investigate creep in compression. First emission can be distinguished from extraneous noise by concentrating on frequencies well above the audio range. Second, that the emission can be used to locate the source of microseisms and thirdly, that emission may give some information about the microfracturing which is believed to cause microseisms. Unfortunately, the location of microseismic sources requires a multichannel tape recorder capable of recording well above the audio range. Such instruments are extremely expensive. Convenient 276

ana/ysi of the magnitude distribution of microseisms also requires elaborate apparatus. It seems, therefore, that bc:fore emission can be used to help interpret the processes occurring in creep in compression an increase in investment of "two or three orders of magnitude" will be necessary.

This investigation has shown that microseismic emission during creep under tension can be detected with simple apparatus. ;Itich more complex aparatus will be necessary, however, to detect and analyse the lower energy emission from propagating cracks during creep in compression. 277

CHAPTER 10

SOME CONSEQUENCES OF THE EXPERIMENTAL RESULTS FOR STRUCTURAL GEOLOGY

10:1

The results of the experiments described in this thesis are summarised again in Chapter 10:1. Having reiterated in 10:1 that the new structural theory is the only reasonable interpretation of the data, 10:2 explores the limitations of the theory at high strains and at high temperatures. At high temperatures, diffusion aided by the presence of solvents eliminates cracks. At strains less than the short term failure strain, a critical crack density is reached at which individwo cracks begin to interact on propagation, leading to failure by static fatigue. The strain before failure is often insignificant compared with subsequent deformation along the failure surfaces. In that case, 10:3 shows that deformation is appropriately analysed not by elasticity theory but by the theory of plastic rigid bodies; this is illustrated by an analysis of thrust faulting in the Canadian Rockies. Chapter 2:1 showed that the laws of transient creep fall into two groups; one proposing that the strain rate declines exponentially 278

10:1 with time, the other showing a power law dependence of strain rate on time. Simple linear regression was used to calculate the parameters of the transient creep curves (Chapter 2:4). This led, naturally, to simple tests of the fit of the proposed laws to the data (2:5). These tests decisively reject the exponential law of transient creep as an explanation of the data (2:11). They also suggest that the exponential law with added steady state creep is not a reasonable explanation of some experiments (2:12). The only previous attempt to use numerical methods to calculate the parameters of the creep curve used the measured strains directly. This led, however, only to qualitative tests of fit because parameters enter the regression equations non linearly (3:1). This and other methods employing the strains are unsatisfactory for another reason; they attempt to estimate three parameters (the two parameters of the creep curve and the instantaneous strain) from one fitted line (3:10) and so cannot lead to any quantitative statistics of fit.

Varying the estimates of the instantaneous strain in an experiment led to significant changes in the estimates of the creep parameters (3:9). In particular, if the creep sttain is less than

10, of the instantaneous strain and the specimen follows a power law of creep, the data can always be apparently satisfied by a logarithmic creep law (3:10). 279

10:1

Creep strains can however be calculated by integrating with

respect to time the creep laws written for the strain rates and

substituting the values of the creep parameters (3:6). Previous

workers on creep have tended to underestimate creep strains when

creep is approximately logarithmic. Estimates of creep strain

based on estimates of creep parameters derived from the strain rates are much closer to the values suggested by Evans'(1940) experiments with very fast loading machines (3:7)*

Integrating the creep laws demonstrated that when the strain hardening parameter, b2, is less than minus one, creep strain tended to a finite limit, (el - bl/(b2 + 1)). It can also be seen that

the frequent criticism of the logarithmic creep law because it led to a singularity at zero time is trivial. The time to apply load

to a specimen tan never be zero and the creep strain between any two times greater than zero can be simply determined (3:6).

A number of hypotheses cannot be tested without the prior knowledge that the power law of transient creep is the only reasonable explanation of the transient creep data so far proposed. In

Chapter 4:2, the reproducibility of creep experiments at the same stress on different samples of Pennant sandstone was examined; a technique of testing whether several regressions can reasonably be considered to be identical demonstrated that it is, after all, meaningful to talk about the creep behaviour of Pennant sandstone 280

10:1 at a certain stress. The same technique was used again to show that lateral creep of Pennant sandstone specimens at 65;:, of their failure stress and above showed the same rate of strain hardening as the axial creep curves in these experiments. In some oases, the lateral creep curves could reasonably be considered to be identical to the axial curves, indicating dilatanoy in creep attributable to the opening of axial cracks in the specimen (4:6).

There is no reason to suppose that steady state creep is taking place in any of the experiments examined in Chapter 2. Two further experiments from Misra (1962) which showed static fatigue at temperatures over half their melting temperature have been examined in detail as these are conditions under which steady state creep is likely to take place. When accelerating creep had been removed, the variances of the experiments were not significantly high so, again, there seemed no evidence for steady state creep (447).

writing the power law of transient creep as

2 et = bt b then the three for“al theories of transient creep (the strain hardening, exhaustion and Becker body theories) suggelt in Chapter 5. ond 6, that bl, the strain rate at one minute* is proportional to the applied stress and b2, the strain hardening parameter, is minus one. 281

1U:1 The sixty two new experiments on Pennant sandstone and Carrara marble show distinctly different forms of stress and time dependence of their creep rates. For Pennant sandstone, b is generally 2 slightly greater than minus one; bl, below 357; of the failure stress, follows a power law dependence on stress with an exponent (0.58) significantly different from one, at higher stresses the exponent is larger (Chapter 8:3 and diagram 841) No creep deformation was detected in the Carrara marble specimens below half their short term failure strength. Above this value, the strain hardening parameter, b2, is stress dependent, varying from —2011 at low stresses to more than minus one at above two thirds of the short term strength. The exponent of the power law of the stress dependence averages over two (8:4). Thus the formal theories of creep do not explain the experimental data from either of the two rock types. The only structural theory of brittle creep (Scholz, 1968a) is based on an unrealistic model, of the stress distribution in a creep specimen (7:2). A new theory of brittle creep is therefore developed (7:6) based on the hypothesis that creep is due to stress aided corrosion at the tips of pre existing cracks in the perfectly elastic specimen. Differences in creep behaviour in the two rock types are due to different corrosion reactions and to different length distributions of the cracks (8:5). Increment tests are another approach to testing creep theories. 282

10:1 The forms of creep after an increment of load have been derived for each of the formal theories (Chapters 5 and 6) and the structural theory (7:7) in terms only of the load increment and the form of the parent curve. The theories can then be tested by calculating the parent curve from the daughter curve and comparing the calculated curve with the measured parent curve. Again the new structural theory provided the only satisfactory explanation of the data (8:6). A series of recovery experiments on Pennant sandstone confirmed the structural theory's prediction that much of the strain in transient creep is irrecoverable (8:8). The recovery that did occur could have been caused by delayed return movement on cracks activated at very low stresses during the parent experiment (8:9). Another attempt to confirm the structural theory by detecting acoustic emission from cracks propagating during creep in compression was unsuccessful. Emission had previously only been detected in creep in tension; this proved easy to repeat, showing that, as might be expected on theoretical grounds, cracks propagating under tension emit more energy than under compression. Emission during creep in compression might be detected by apparatus sensitive to frequencies about a million hertz. When the new structural theory of creep describes creep behaviour, there are major differences from the model presented at the beginning of Chapter 2. Notice, in particular, that there is no evidence for 283

10:1 - 10:2 a period of creep at a constant strain rate intervening between the initial stage of transient creep and the final stage of accelerating creep. Further, much of the deformation during transient creep is irrecoverable.

10:2

The conditions under which the new structural theory derided in Chapter 7 are expected to apply to rock behaviour can be outlined by considering the breakdown of the theory at large strains and under high temperatures. Suppose that, on propagation, a crack intersects a second crack. The new intersecting array may be unstable depending on the length and orientation of the cracks; further growth and propagation could take place. As the array grows in size, the energy released at each propagation increases and with it, the possibility that the array will intersect other cracks as it propagates* leading again to array growth. Intersection can then result in increasingly large arrays of cracks which grow by an ever accelerating process until they extend through the specimen and it finally fails. In other words, intersection may result in accelerating creep. The probability of an intersection occurring depends on the density of cracks in the specimen. A high density leads to a high probability of intersection. At low densities, even if one intersection 284

1012 does occur, the chance of it leading, on further growth and propels.- tion, to a subsequent intersection is slight. Griggs (1940, pi, 1018) showed that there was a critical strain before the onset of accelerating creep and it is natural to suppose that this represents a critical crack density. As stress and the duration of stress increase, pre-existing cracks grow and propagate, increasing their surface area and the crack density. Specimens having similar initial crack length distributions will share a critical crack density which can be reached by increasing either the applied stress or the duration of the applied stress. These considerations lead to a theory of static fatigue. Suppose a creep specimen breaks after one minute at a stress, s; the increase in the initial crack density is (b2 + 1) b1 h /(b2 + 1) where h is the proportion of the total time to failure occupied by transient creep; h is about 0.3 (Chapter 4:7). If a specimen of the same rock type breaks after tf at a stress sf, then, equating the increases in crack density (b, + 1) b /b = t ls lf f (b s 1) /sP 2 f tf log s - log sf = ((b2 + 1)/p) log t f (1) 2d5

10:2 where the strain rate depends on the stress to the power, p. Equation (1) is sieiler to the relationship derived by Charles (1958). for failure due to a single crack (7:4, 29). It implies that the logarithm of the time to failure decreases with the increase of the logarithm of the applied stress; a result in reasonable agreement with Griggs (1940). It b9 is less than minus one, integration of the power law led to a form, (3:6„ 14).

which implies that creep strain asymptotically approaches a finite 1 2 value, E1 b /(b 1). In Carrara marble, b2 approached minus one as the stress increased; then the finite strain is not much greater than b1/(b2 1), a value less than the creep strain after one minute at higher stresses. Clearly then the finite strain represents a crack density less than the critical one. Thus accelerating creep cannot begin and the specimen will not fail under the applied, stress; there are insufficient sources of weakness. This is a structural interpretation of Price's (1960 conclusion that there existed a long term strength for some rocks. For Carrara marble it will be between 64;:, and 7O of the short term strength. Notice that this interpretation also supports Price's conclusion that strain hardening below the long term strength is more severe 286

10:2 than above it. The theory of static fatigue outlined above is them in reasonable agreement with experimental evidence. It implies that the total strain in a creep specimen is unlikely to exceed the failure strain in a short term test. Confining pressure on the creep specimen considerably reduces the length of propaeation of the crack. Hoek (1965, p. 59) has shewn this for pre-existing open cracks in plate glass. The effect is due to higher frictional stresses between the crack's surfaces (7.urrell, 1965) causing more rapid energy dissipation. As the probability of an intersection increases with the crack area but the displacement around it, only with the crack length, larger strains occur under confining pressure before the onset of accelerating creep. As an example consider a crack which propagates for a distance, 214, when the ratio of the principal stresses is 0.025. Then the chance of an intersection is preportional, if propagation takes place from both crack ends, to 8L2; displacement is proportional to 4.I. If the ratio of the principal stresses is increased to 0.05 at propagae tient Hoek (1965, p. 59) observed that the length of propagation was approximately L. So the chance or an intersection is reduced fourfold but the displacement is only halved at the higher confining presure. Four times as many propaeations will occur at the higher confining pressure before an intersection takes place and 287

10:2 these propagations will produce twice the total displacement before an intersection at the lower confining pressure. Increasing confining pressure at a constant load also dramatically reduces the tensile stresses at crack tips, (Hoek, 1965, Chapter 9) and this should result in a considerable reduction of the creep rate. Robertson (1960) reported a hundredfold reduction in the creep rate of Solenhofen limestone with an increase of confining pressure from one to two kilebars. Further iu roases had little effect on the creep rate, however; Briggs (1940) found an increase in the creep rate of specimens of alabaster (in contact with its own saturated solution) with increasing confining pressure. As an explanation, Goranson (1940) drew attention to the increased mobility of the solute. This may also play a part in the breakdown, with increasing temperature, of Charles' (1953) model of stress aided corrosion. Charles (1953, p. 1559) commented "At temperatures below 150°C, the delayed failure results are consistent with the conclusion that sodium ion diffusion at a reacting interface controls the delayed phenomena ..e It seems clear that the corrosion process resulting in extensive silica network breakdown assumes importance at the higher temperatures and acts wits sufficient rapidity to suppress stress orientated corrosion except at very high stress values". Above 1500C sodium ions are sufficiently mobile, then, to allow the hydrolysis, of the glass to proceed independent of their 288

10:2 concontration,t;oranson (1940) has considered how the activity of a solute is affected by stress. He showed (p. 1026) that at a stressed face activity is increased by a compressive stress and reduced by a tensile stress. Diffusion of solute away from a face under compressive stress may leave the bulk of the solution supersaturated and deposition will occur at zones under tension. This, then, is a mechanism for the removal of cracks from a specimen. At transitional temperatures small cracks will remain, explaining the observed dependence of static fatigue on stress, only at high stresses. Concentrations of tensile stress will be removed most quickly where they are largest, resulting in preferential removal of long cracks, simply interpreted as an increase for the specimen in the value of m in the relationship, 7:6 (43), from the structural theory,

(Y) L-m = 414,Y) 7:6 (4.3) As m is increased, the rate of strain hardening, proportional to t-(n-2:)/(*.2), (7:6, 48) decreases. The exponent of the power law dependence of strain rate on stress (at constant n and m), 2n(m 1)/(n - 2) increases. Thus, the transition from the structural theory of creep to creep at higher temperatures is accompanied by a decrease in the rate of strain hardening and an increase in the stress dependence of the strain rate. Taking n to be about 8 for 289

10:2 silicates (Chapter 6:3) and m to have increased to 2 gave Andrade creep,

(n 20/(n — 2) = 2/3 and an exponent of 2.67 for the power law dependence of strain rate on stress; Misrals experiments on peridotite at 750C showed this type of behaviour (Misra lturrell, 1965, p* 530). Goranson (190, p. 1026) pointed out that the mechanism causing the transition from the structural theory can be considered as a diffusion process along the grain boundaries from zones of high compressive stress to less stressed zones with the solution acting as the medium of transport. But activation energies for diffUsion can also be dramatically reduced by phase changes and the chemical reactions that take place in metamorphism (Orman, 1967, p. 216). Clearly then, the upper limit of the applicability of the structural theory is the lower limit of regional metamorphism.

Turner and Verhoogen (1960) placed the lower limit of regional metamorphism (the zeolite facies) at temperatures of two to three hundred degrees and confining pressures of two to three kilobara corresponding to depths of burial of about ten kilometres. Above this limit, complete recovery from strain hardening may be possible. steady state creep may also tame place at lower temperatures and pressures if the rock is in contact with a solution in which the mineral phases are appreciably soluble. 290

.L0:2 - 1U:3 The limit suggested is appreciably lower than that found from laboratory studies on polycrystalline monomineralic aggregates (17Tatt, 19e7). V,teady state creep is not possible below hall the melting temperature of the more covalent, lower symmetry minerals because it is only above this temperature that dislocations become mobile. The aggregates differ, however, from natural rooks in the absence of natural colutions and other minerals with which they might react. The conditions under which the structural theory will apply to rock deformation are thus, roughly, those of Holland and Lambert's (1969, p. 201) first regime, "an essentially non-metamorphic regime below the level of diagenesis." Rocks deformed only in this regime may esaily be recognised. eccrystallisation of mineral phases is absent. Large, apparently homogeneous deformations of strain markers such as ooids and reduction spots, does not occur, So the structural theory is some way towards answering the question posed at the beginning of Chapter 2 about the conditions under which lq% strains take place. Unhealed fractures can be expected at grain boundaries and across grains.

10:3

Having defined the conditions under which the new structural theory can be expected to apply to rocks, its implications for 291

10:3 deformation under this regime can now be considered. Particular attention will be paid to an example - the dynamic analysis of thrust faulting in the Canadian Rockies. The obvious applications of a structural theory of brittle creep to shallow earthquake studies, both in the mechanism of slow growth and slip on a fault under constant boundary stresses and in-the mechanism of the earthquakes themselves should not obscure its implications for structural geology. The dynamic analysis would be a different kind of problem, more difficult to formulate and less quantitative than, say applying the pattern of diselacements round a dislocation in a perfectly elastic medium to the pattern of die:placements observed round a natural fault, as Walsh (1968) had done. It might be thought that because the creep strains are small in this regime the deformation is negligible. However, Holland and Lambert (1969, pp. 203-304) suggested that the superficial folds of the Jura and the larger scale folding of the Pre Alps and Helvetides took place under this regime; so did the deformation of the sedimentary cover of the Appalachians and much of the Basin and Range province of the western United States. It is easy to add to these examples. In Zritain, for instance, there is the well documented deformation in Bristol-Somerset coalfield (lellaway and -:;e1ch, 1948). The best examen le is probably the Canadian Rockies (Shaw, 1963). 292

10:3 There a total shortening of at least sixty miles can be established on the Mississippian limestones across their eighty mile width of outcrop. There are reasons to believe the shortening may be as much as 150 miles. Extensive exploration for oil means that the amount of factual, three dimensional information that can be applied to structural interpretation may be unequalled elsewhere in the world (Shaw, 1963, p. 254)

The Canadian Rockies are the eastern (or frontal) ranges of part of the North American Cordillera; they face the Great Plains.

Ten to fifteen thousand feet of Mesozoic elastics and Palaeozoic limestone and shales rest on a pre Cambrian basement whose upper surface now dips at about two degrees tb the south west (Bally,

Gordy, Stewart, 1966). The sedimentary column thickens in this direction and the rocks pass into deeper water facies. Reoonstrustion of the facies pattern in the sediments showed that the movement direction of the thrust slices is approximately perpendicular to the trend of the mountain belt. As this direction lies in the symmetry plane of major structures in the belt, deformation aepears to approximate plane strain with the sedimentary cover sliding north eastwards over the basement towards the stable area of the

Plains.

The high symmetry of the deformation suggests that a fairly detailed dynautic analysis may be possible. But it is not intended 293

103

to present one here, only to illustrate hoe the structural theory

of creep may contribute to a solution.

According to the structural theory of creep, the stresses in

a rock mass undergoing a deformation in the first regime,can be

calculated from the equations of equilibrium for a perfectly elastic

body, if due account is taken of stress concentrations within the

body. Time dependent effects can be removed from the constitutive equations.

The stress distribution in a perfectly plastic block under

gravity being pushed across a flat, rough plane in plane strain has

been analysed by Hefner (1951). It has been usea by Charlesworth (1959) in an attempt to account for the principal feature of the

Rocky :ountains deformation, the imbricate thrust faults which rise very gradually out of the zone of decoilement at the base of the

block and then steepen more rapidly towards the free surface of ths block. The style of these faults is shown in sections by Shaw

(/963), Bally et al. (1966) and O'Brien (1960).

llafner (1951, p. 383) suggested that failure occurred in the rock mass by the increase of the applied stress field. Around the area where failure takes place there is a considerable stress reduction while further away the stress field remains substantially unchanged. The fault provides local relief for the most intensely 294

10:3

stressed portion of the body. Further build up of the stress

field is presumed to go on until renewed faulting takes place

at some distance from the first fracture where the stress relief

was negligible. The new fracture can be predicted using the old

stress field. Hefner sueeested that thrust faults would develop at thirty degrees to the trajectories of maximum principal stress

(the-faults have the characteristic concave upwards form).

Hefner's figure 6 (critical parts are reproduced as diagram 10:1a) showed that failure probably takes place first in the top

left hand corner of the block. If the effect of the failure is

local, faulting resumes at other localities on the same fault

trajectory outside the range of local stress relief. There seems no reason why this process should stop until the block is actually

n,slit along this same trajectory. Then the stress the upper fragment

of the block can maintain is simply the sliding friction on the fault

plane. The stress in the regten of the fault will not again approach

the values causing initial failure in the rock so the situation

which Hefner illustrated in Fig. 6b is improbable. At x 404,

y= 0, the rock mass has failed, at x = -12, y = 0, the already

fractured rock mass is supporting stresses three times those at its initial failure. If the parameter o (Fig. 6b) is reduced to bring

the stress 's to a more likely value, the boundary between the stable Diagram 10:1 The form of thrust faults in blocks of various homogeneous materials, sliding to the right over a rough surface a)perfectly elastic b)plastic body obeying von Misis yield criterion

c) rigid plastic body obeying the Coloumb Naylor yield

criterion

LOOM Thrust faults are indicated by solid lines with superimposed arrows indicating direction of movement, Other solid lines (10;1a) are the trajectories of the larger principal stress. The

dotted line (10:1a) is the boundary between the unfaulted area (to the right) and the faulted area when c = a. Dashed lines (10:10 indicate the attitude of bedding. The upper surfaces of the blocks = 0) are stress

free. d i a • 10:1

X = 712 x .: -4

c 2)7

/00 and unstable fields approaches the horizontal (Hefner, 1951, p* 568) and the principal stress trajectories then lose their curvature

and with it, much of the original close resemblance to the structural cross sections of the Rockies. Jaeger (1962, p. 178) commented

"In practice, when failure takes place at a point there will be a redistribution of the stress system before further fracture occurs and it is this redistributed stress system which determines the direction of the next failure. A complete theory must begin with a criterion of failure and equation of flow and use an extension of the methods developed for the perfectly plastic solid." The structural theory of creep suggests that the deformation that can take place before the formation of discrete failure planes is insignificant compared to that taking place after failure. This is, of course, one of the characteristics of a perfectly plastic body (Hill, 1950, p. 14). Cruden (1966) suggested that the Rockies might be treated as a perfectly plastic body slipping across a perfectly rough plane towards a rigid area. Providing the rock mass obeys the von :Uses yield criterion, much of Nye's (1951, 1957) analysis of glacier flow can be directly taken over. liye (1951) analysed a steady state equilibrium of the glacier in which material was supplied to the flow from upstream and removed from the top of the glacier by erosion. The analogy with the evolution of the Rockies is supported 29d

10:3 by the western provenance of the elastics deposited in front of the toes of the rising thrust slices and by the eastward decrease in the age of the uppermost elastics (Shaw, 1963). As the structural theory of creep explains, faults are not necessarily caused by increases in the stresses at the boundaries of the problem; they can arise by static fatigue at constant stress. It is this latter mechanism that causes faults under a steady state equilibrium. Ode (1960) pointod out that geological faults can be interpreted as velocity discontinuities in plastic flow. ielcmity discontinuities can only exist along characteristic surfaces of the velocity eauations. (Ode, 1960, p. 311). So solution of a problem in plastic flow is primarily a matter of determining the geometry of the slip surfaces.

The solution to Nyc's particular problem is well known. Gravity has no effect on the form of the slip surfaces which follow the form of those in the lower half of the Prandt1 plate problem

(Bill, l95)r p. 228). They are a family of cycloias tangent to the base of the block and rising slowly eastwards to make an angle of forty-five degrees with the free surface of the block, illustrated in diagram 1O:lb and Nye (1951, Fig. 5b). Unfortunately rock specimens in short term laboratory tests do not follow the von 11.sest yield criterion. The stresses at yield 299

10:3 show considerable dependence on confining pressure (Murrell, 1905). At confining pressures below those at the brittle ductile transition, (of the first kind, Murrell, 1965) yield in short term tests is accompanied by a drop of the etredses to those at sliding friction on the fault surface.

The structural theory of creep has drawn attention to the possibility that failure in geologic deformations could be by static fatigue. Laboratory experiments on static fatigue terminate in failures that are as violent releases of energy as those in short term tests. The violence is not only due to the release of energy stored in the rook specimen but also to the instability of dead weight loading systems under even slight falls of the stress that the specimen will bear. Cook (1967, p. 398) has shown that a very stiff hydraulic testing machine is necessary if the complete stress strain curve for a rock specimen is to be determined. The stress drop observed on failure reflects the characteristics of the testing machine. Simply because static fatigue takes place at lower stresses than failure in short term tests the stress should be less.

At very long times, the stress drop will be very small; the shear stress at static fatigue should be close to the sliding friction on the fault surface (unless this, also, is time dependent). Thus 300

lu:3 the stress field in a rock mass undergoing a geological deformation will only locally exceed the values necesFary to cause sliding on fault surfaces.uch local excesses will, however, be sufficiently largo to cause earthquakes.

Jerk on the sliding friction of rock surfaces showed it

approximately followed the Coulomb lavier law (Lyerlee, 1967). This

lava can be taken as a yield criterion but unfortunately the criterion

leacis to much more complex equations for the velocities and hence for the slip line fields (Ode, 1960).

One simple problem has however been solved. If the rock mass

within the thrust slice is rigid, as the structural theory of creep

would indicate, then the equilibrium equations need only be satisfied

on the boundaries of the slice. This deformation style has been

called "plus flow" (Nye, 1951, p. 560). Jaeger (1962, p. 182) than indicted that the form of the slip lines was a (1o6arithmio) spiral.

The geometry of the flow illustrated in diagram 10:1c shows that the

spiral is tangential to the lower surface of the bock and makes

the amaze of friction (about thirty degrees) with the upper free surface.

There are a number of interesting geometric consequences of

this type of flow. In particular, a marker horizon immediately

above the fault surface will parallel the fault surface only 'when movement along tle total length of the fault has taken place, 301

10:3 There will also be a deoroase in lip of strata within the block away from the emergent fault toe so strata above the thrust should lie in a shallow asymmetric syncline, Both those features arc well

shown in :haw's section (1963, Fie. 3), particularly in the Bighorn

thrust sheet. in conclusion, then, the structural theory of creep leads

naturally to the suggeLtion that, „here it is followed, deforming rock masses can be treated as 1,lastic bodies undergoing plug flow.

The consequences of this are not alvays easy to work out but in at least one case, thrust faulting in the i;ansdian Rockies, the results are in eood agreement with field observAions. 3-.2)2

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