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THIS IS A PREPR2VT --- SUBJECT TO CORRECTION

Determination of “the Inelastic Parameters Downloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 -. .-.,.,.....:. .+._...... ufGeologi .Materi-als .Trcm .. . ..

By

H. Reginald Hardy, Jr., Fennsylv~nic S%te University, University Park, pa”

t) Col)YriKht 1967 Amrrirrm Instilute of 11inin~, }Waiiurgid and Prlrolrum Enginrcrs, Inc. ——. IIWH701XJCTION br:adly outlined as follow~:

In spite of the increased research cm the “To study experimentally the mechanical behavior of geologic materials which has been r-whaviorof selected geologic materials and undertaken in recent years, many of the mast analyze these experimental data using the important mechanical properties have nr.)tbeen concept 01 mechanical model~ and the analyti investigated in detail. In particular the mcthotisof viscoclzsticity. In particular t effect of time has received very little attcn- tievelopgovefning equativns far these materi tion from the experimental point of view, cxisedoc suitable laboratory experiments and although its im]mrtance has been suggestsd b; valid at least within a range of stre~s, con many worker%. Many wvrkers have reported that fi,ning])ressure,duration of loading, anti “test “resultsappetiredto be influenced by the temperature consistent with’pr.mcticalo,ppli rate and duraticn of loading, although few have tions..” .. beenwilling to accept the fact jhat such ‘ ;;- ...... ~...... bchavi.oris indicative of an inelastic.material. ~“e .first.phasetifthis.$esearch ~f~grt Lncluded the experimental investigation uf t . .4general study of the mechanical proper- inelastic behavior of.a number of “simple’”. ties of geologic materials”was initiated in @910gic materials using incremental creep 1951 in The Mining Research Section of the experiments, and the associated development Fuels and Mining Practice Division, Mines the necessary experimental facilities and Branch, Canadian Dept. of Energy, Mines and analytical methods. This paper outlines bri Resources, as part of a ftmdaanentalstudy of the experimental and analytical technique bround stress in Canadian Coal Mines.1 The developed, and presents experimental result initial work was restricted to short-period for the initial deformation studies carried tests on typical mine rock under uniaxial on Wombeyan marble. compressive stress, but as the importance of the “time factor” became more abvious, studies THEORY were undertaken to investigate time-dependent behavior; These.ktuditis,which were initiated Papers de”alingwith the behavior of in 195’(,consisted mainly of a comprehensive geologic materials are found to be widely literature search on the subject, and the dispersed throughout the literature. Howeve initial.development of experimental facili- extensive bibliographies dealing with the ties.2?3 This research project was inactive general mechanical properties of these mater during the period 1959-1961 due in part to the als have beck $ublished recently by Griggs - -relocation. .=— of-labo”ratory. . facilities;’- :. -- Hahdin4 and”MitraandWillson.5- In”the more specific field-of time:depe~dent behavior of ~ the fall of.1961 the.project”wbs.reacti-. .geologic.material.s..extensive.bibliora hies vated l~iththe propos_edresearch program being have been.~iven in papers Y Hardy> ! ..—.-. - . -.— -@d ~sra;”{”,~d”Robefisoni=.8.. ~ .8~urre ..*pre~e-ntl,yat ,Thepennsy-l”v&iaS.tai.e-UO) ... . :.:.-.:--” .. .Un~versity P&$k~;,Pa;‘“ “ ‘. ., . :“ . ‘“ ““ .TheVmqJority~of ekpe~iqents’con~ucted. Re?erencei”an”d’illus%?ations-”atend of pape”r..- - investigate the time-dependent behavior of experiment] is found to be geologic materials have been creep experiments carried out under conditions of constant applied load. [The term “creep” wi.llbe defined, in a general sense, throughout this paper 0.6the time-dependent strain observed in a material under a condition of constant str~ss:] Numerous empirical equations have

been 6eveloped to describe this time-dependent Downloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 behavior;“however,it is the author’s opinion that the phenomenological approach providefia: more convenient representation; Here the behavi~r-vft.he material is expressed.in terms of a combination of a numbe”r“ofsimple where e[t] is the time-dependent strain, Un is mechanical models. According to Freudenthal, the mabaitude of the constant stress, t is the “the theory of mechanical models considers time, T is the retardation time defined as combinations of elements which are supposed to behc.vemechanically like the constituent phases of the material but which, apart from the material behavior, have nothing in common with the real material.”9 A number of mechanical 7-=4/’=, ...... [31 models heve been postulated to explain the inelastic deformation of various geologic materials. and El, E2) Nl, and N2 are the mtidelParQ$neter$

The use of mechanical models to describe To investigate the creep behavior of the the deformation behavior of geologic materials test material at various compressive stress haG been discuss~~ by a number of writers in- ievels it was f.mnd convenient co u~e incre- cluding Hard 2~o~10)ll Price,12 Ijnan,13Ito mental loading. ~ this case the specimen and Terada,l{’H~t$15 Attvell,16,1”(s~lu~to. stress was increased in a number of small wicz,18 Tida and Kumazawa,19 Terry,20 increments A:&. After each increme& t the lfiatsushima~land Litwiniszym.22 A review of resulting stresson was maintained co%stant at the Ceneral application of Viscoelasticity, the level for a period of tinw t~, such that including the mechanical model concept, to the tn>> T in which case the factor exP [-t/l’]+Ot behavior of geologic material Under these conditions the strain el[t] deveL- lished recently by Robertson:B ‘a’ been ‘ub- oped during the incremental loading and the peri;d”tn .o? cqnst~t. stress $ollu.wwing,it w~ll ... 21.. . . . ““10 ll”[~t,sushtiaj ‘beofthe form - Studies by Hardy, ~ ...... Pri;e12”aid”othei% Orirock, by.Tek?ry20tincoal and by ROSS23 on.,concrete[a material.in many ways similar to-geologic materials] indicate that deformation of these materials under sus- tained loading may be analyzed satisfactorily in terms of the Burgers model, illustrated in Fig. 1. The theoretical behavior of this model has been investi ated in detail by Frederick and Garcia,2f and more recently by Steppel.25 The governing equation is found to [1,] be ...... 0. ● ......

Al where all factors have been defined previously ~2: F+ /+5’+4! &- (’)[ ‘4”- #ya This may be written as 1

.. . . ~ere k.~d-e’”are, re~pe-ctivelyjth-est~ess-aud. “.w~ere “&~ ~s””thkinitan”t~kous elastic”~t”rati’ strain, and El, E2, Nl, and,N2 are the=m~del produced by.the stress increment A~n, ~d en[~] -.-... par&meters.’- ‘“ -“’”- . ‘ is-the time-dependent [creep]-”strain”which-”’. ..- . occurs+at,.$he.const~t-”stres”s

stress compatible with the writer’s experi- mental facilities, A supply of this material in the form of ‘(/8-in.diameter core was kindly made available by M. S. Paterson of Th ‘m’’)=.~[’”+~”~1Australian NatiorialU., Canberra, Au6tralia. I Apparatus and Experimental Technique

.+ +29 ...... [61 Deformation expertients were conducted Downloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 in a test enclosure which was designed to pro .. ,-..... > .,.:.....b A? ,..,.,,..-. .. ..:,... .,,...... ’:: ‘- ...- , vide controlled temperature conditionsof 70 ? and l/10F. Cylindrical test specimens, nominally 3/1+in. in diameter and l-3/~ in. in length were machined from diamond drill core ushig ‘ lathe and an attached tool post grinder [see’ Aez = Aq/Ea. . . . . [71 Ref. 28.] To measure axial strain a set of three foil-type resistance strain gauges were mounted at 120° intervals around the circwn- EXPER~4ENT ferencc of the tect speeb~en. Fig. 2 shows a set of strain-gauged spec~mens ready for Introduction testing.

Relatively few workers have undertaken the T& Gpecirqens‘,leremounted in a loading investi~ation of the inelastic “behaviu~of jig and compressed axially using a specially geologic materials. This has been due in part designed loading system. Complete details of to the many experimental difficulties that must this system have been described in recent ?6 11,29 ~i~ sys.;em, which is be overcome if significant data are to be publications.- > obtained. As a result considerable time has illustrated in Fig. 3, is basically an “on-of been spent by the author in developing a pneumatic-hydraulic servo-luadin~ system whic suitable e~erimcntal facility fur such inves- may be vperated in a variety of test modes. tigations. A description of the design and the incremental ~trcss mode the specimen stre eonstruc~ion oi”this facility has been p’iJ- may be increased in a number of rapidly appli lished,2b Therefore, this aspect of zhe small increments. After each increment the program will be discussed only briefly in this stress may be maintained constant for n peyio paper. of time during which the resulting creep stra is observed ahd recorded. At the end of this ,. . ..Test-Material ~ -- - . - perio$,.thestress Tai be-ihcie?sed.;o the nex level. Using this system it.hasbeen possibl Until recently, engineers and in partic- ttiapply incr~mental’-stressesof up to 1,000 ular mining engineers have been concerned psi in period? of less than four ~econds and. mainly with the beh~vior and properties of the maintain the stress constant to within ? 1/10 specific local geologic materials with which per cent at each level. they were working. The detailed experimental investigation of such materials may be merited Specimen strains were nonitored by the for certain applied studies; however, such three attached strain gauges [designatedas 1 investigations do not appreciably assist in 2/2 and 3/2] and a measurement system which developing our understanding of rock enabled the strains from the three gauges to behavior. The writer favors the selection of recorded separately. Fig. h shows a block a few “simple” materials, approximately diagram of this measurement system. This. equivalent to those encountered in practice and system allowed an estimate to be made of the upon which a concentrated effort would be made strain uniformity in the specimen throughout to the”behavior of these materials. the experiment. It was found that deviation When this is accomplished, the results could be between gauges rarely exceeded 8 per cent. I extended to the geologic materials’encountered general it was estimated that strains could b in specific problems. measured to within t 0.5 vs.” Specimen stress was monitored by a strain-gauge-typeload cel Wombey&n marble was selected as the test [built into the loading.system] and recorded material for..our.initialinvesti ations ,as.:-.. continuously throughout the experiment. Spe, earlier”ktieriments’by Paterson27 had fndicat~d men--s%re=~s-.couldbe-detehin%d to”within i10 psi...... thpt its-deformation behavior.was ?easoaa~ly . . ,. ..-...... -.. ~“reproducible,that it was approximately ‘. .-. . .- ~iso~ropic both~in-its f~bg~c and::-nitp~..- ~-.> ‘epor-brevity,,,ps?..is:Gitten for measureh mechanical”propgti$es;and.that ‘Ii ““microstra rePrtis&ting.a strain of 1o-G. .- .L .: ‘exhibited. ine,last~c;. . behav:~r wit+~n’a:~ge of ...... -,...... DETERMINATION OF THE INELASTIC PARAMETERS OF GEOLOGIC L CREEP EXPERIMENTS SPE-17~ Brf.eflythe experimental procedure was as follows: determine the value of the parameter E2 and enabled the strain-vs-time data to be corrected 1. The test specimen was mounted in the for small fluctuations in the applied load. loading jig. This was placed in position i.a the loading frame and the various strain ma 7* This incremental loading procedure was load transducers were connected into the repeated eight to ten times, at increasing ‘measurement system. stress levels, urrcilthe specimen failed or it was felt desirable tv terminate the experiment.

2. The over-all facility was activated an, Downloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 a general check-out of all ~ystems conducted. ANALYSIS OF EXPERIMENTAL DATA A warm-up period:of at’least ’12hours-was. .. . !.. , ,. . . .,,.. .*..-.. .- ~ allowed, which was followed-by a second check- Stress-Strain Curve~ “’- out and calibration of all systems. Utiliz@ the strain-vs-time and the “3. During the initial pha.scof the loadin, stress-vs-time data from the “recordercharts, pro~ram the three axial specimen strain Ew,uges the indivf.dualstress-strain data for the three and the load cell were connected intv the 12- strain gauges [1~%, 2/2, and 3/2] ‘de~ obtained chariaelY-T recorder system. The recorder From these data the average stress-strain data chart m.ttirwas started and allowed to run for and the per cent strain deviatiun &Yv which approximately 15 min to eotablish reliable indicates the degrc? of nonuniform loading, zero values for the strain and stre~s channels were co~~uted. Fiu. > shows these data from and to check on the over-all stability and the experiment on specfmen WO-7. The incii- noise level 0S the measurement system. vidual stress-strain curves shown in Fig, 5A include the time-dependent struins experienceci 4. Tne load controller was then set to during sustained loading at the various ievels. its auto-m~de and the compressive stress on thf In most experiments the stress-strain curves $pecimen increased through a number vf incre- for the three gauges were found to be in good ments [AoD.s1,200psi] up to a stress of ap- agreement. Fig. 5B sh[>wsthe variation ~~fAe7. proximately ‘(,000psi [Initial experiments con. with strc. level. The deviation is initially ductecion specimens OS Wombeyan marble indi- high, falling rapidly to a low value and re- cated that belcw a stress of approximately maining there until the material starts exhi- ‘(,000psi this material efiibited negligible biting appreciable time-’dcpendentstrain. At inelastic oehavior.] The stress was maintaine~ this point the deviation begins to increase constant at each of these levels fvr approxi- rapidly with increasing stress level. Fig. 5C mately 15 min. before it was raised to the nex illustrates the average”stress-strRin curve. level.“ Quring this phase of the experiment th, “Here the contribution of the elastic and “the recorder.channelsmonitoring the outPut.,Qf the tirnezdepen’dentgtiai?s kre5hOWn “5epar&t~lY.. . - three specimen strain’gauges’were“constantly the vertical sectioxs of-the .curve indicating- ch~_ck’edt-overif-ythat-the ;pe?tien was hein”g the”total time-de~knde’ntstrain observed-during loaded uniformly.. . . each s,ustai.nealoading period, and the d%shed section indicating the total strain curve, 5. At the ‘O.OO psi level the load cell including the t.ixe-dependentstrains. signal Was shifted to the 2-ch~e~ !ifast!~ Y-T recorier system and the signal from one of Instantaneous Elastic Strain the axial strain ~auges was shifted to the Y-smis input of the Y-XT recorder. A timing To ccmpute the value of”the parameter E2 marker system in each of these recorders, whicl in the Burgers model it is necessary to deter- was interlocked to a master marker system, was mine the magnitude of *he instantaneous elastic activated insuring that the data on all three strain [&E] resulting from a rapidQ applied recorders were correlated in the. incremental load. In i,nitialexperiments an attempt was made to estimate the value of AeE 6. A series of incremental creep.experi- directly from the strain-vs-time.curves. This , ments was then initiated. These consisted of method was found satisfactory for stress levels rapidly raising the specimen stress by a small where the material was effectively elastic in increment [)+00< Aan <750 psi] ~d recording behavior. Above this level however> the strain the resulting transient strain-vs-time data fo vs-time curves were fotu ‘ tc be complicated by a period of approximately 1+0min. After this superimposed inelastic effects and by small t@ the stress.was again .rapidly-~ecre%s?aby fluet;atioqs.$nzth.e ~pplied load. It was found approx.tiatelythe .same.amount.to the next .leve necessary therefore, to deve”lop’amore refined” and the next.transient strain-vs~time curve .. method of analysi”sfor use in the.inelastic ...-...... ’ . Recorded:-,“tiringthe incremental lotiirig””“. Fe-gion. .- -. -period.and -the-period”ofapprox~ately constan .. ,...- ---- . - stress which followed it,’the specimen stress. !J&ls;@”alysik;i:v-olved-’the‘use~ofboth the .was~.also--recorded.The-sedata w&re~u5ed..tci strain-vs-$ime md. stress-vs-time”data.““.Figi.~ ... . ,...... ,,, . . . - . !- . L .-.- ;,...... JE-170”7.— ,, H. REGINALE HARDY. JR. under theoe conditions the maximum error in illustrates sections of the strain-vs-time &E introduced by neglecting the inelastlc and stress-vs-ttie recordings, Section AB contribution to the instantu~ous strain wil [similarlyA’B’] represents the end of the be less than 1,5 per cent.l”)ll previous experiment conducted at a sustained stress of Lc[l]. At time t ~ O the stress is Time-Dependent Strain rapidly increased and eventually becomes stable at a new level LC[2]. However, due to the Two difficulties”were encountered”ini- typical stress fluctuations [B’D’,E’.F’G1,which tially in analyzing the tirhe-dependentstrai

normally accompany an Incremental stress change data. First it was not possible to determinDownloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 and which are reflected in the strain-vs-time by inspection the time at which this strai~ curve, it is Got possible to’estimate the” - ‘ was initiated. Secbcdly’the ’time~dependcnt instantaneous elastic strain ACE associated with strains observed were relatively small,‘ran the stress increment [,LC (2.)-LC(I)] by a from 5-200 vs in 40 min. and it was therefor simple inspection of the curve, Instead the . necessary to correct ,thgoriginal experimen follvwing procedure was used: strain-vs-t.tmccurves for the small fluctu- ations in the applied load. The procedure 1. The final equilibrium level sf LC[2], developed to overcome these difficulties wil at point G’, was determined by inspection of bs outlined briefly. the over-all stress-vs-ttie curve. The difficulty of accurately determinin 2. The line G’H’ was drawn through G’ the point e[t] = O, t = O on the strain-vs- parallel to the tixe axis, cutting the stre6s- curve was overcome once the procedtic descri time curve at point D’; the chart displacement in the last sectiun for accurately determini As~ cvrrcsponding to the incremental stress deE was developed. On the strain-vs-ttie cu [Lc (l)-LC(+ shown in Fig. 6 the Point D represents both limit of the instantaneous elastic strain an i. The corresponding time l’Lwas deter- the point e[t] z 0, t - 0 on the tLne-depend mined, in relation to the timing markers Ml and strain-vs-time curve. M2 . The procedure developed for correcting 40 At the corresponding time on the strain- strain-vs-time data far fluctuations in the vs-ttie curve a vertical line KL was drawn applied load is briefly as follows: which cut the curve at the point D. 1. Figs. “{Aand “(Bshow diagrammatica 5. The chart displacement [QdE] corre- &n enlarged ‘~iewof the inelastic section of spcmding to the instantaneous clastic strain the strain-vs-time curve ~d the .strcss.[l@d [&E],is represented by.the distance .KO as. .. .-vs-time.curve,.sh.~’,.mearlier in.Fig. f,. A .. .: ‘shown;----- ...... ,-...... ‘vertical grid* is first sup”erimpo”seaon’both &rvek uiin$ “-the-ti%e”nia~keisfor ‘refe”re’n .6,,’TIIcclastic modulus [E2] associated wi’th. the instantaneous elastic strain was determined 2. The chart displacements [dl, d2, . from Eq. 7: dn] corresponding to the observed strain at times tl) t2, etc,, are tabulated along with Ea -ZV&+’+%~~ chart displacements [Asl~ fW2~ ..*, ~nl corresponding to the deviation of the applie where stress from the control point LC[2] l%r the time.

Ae#x##f# dg ...... [8] 3. The observed strain [eob] at tl) t2 .... tn are calculated from the relation

and ,,. ‘e& =’434.:.. ..-: ..[,0]

A~%l$%$a ...... [9] and the stress deviation at tl, t2, .... tn calculated from the relationship [email protected] .k~ are :&he stress-and strain-.s.ensi-...:. tivities of he two recording systems. -- .d-~k;$;dk::-::.:.’:-.’- . . [1 ----- .. .‘. -...... - The incremental’s~resstikwere applied “,~e has shown that the following gri

where AC$Cwill be positive for stresses greater than the control stress, and all other factors have been defined previously. em(t)= #f? ]-+fi&/&#l [ 1 4. TLe strain deviations [&c] caused by the stress fluctuations Ao_cwill be associated almost entirely with the series elastic element in the material. This is the same element #-(?t ...... [l4l responsible fur the instantaneous elastic Downloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 strain discussed earlier. The strain deviation where may therefore be calculated,.simply fr”om. “..: -, .”.’““: ,. . A=:A ~“/E; 8;’”;’:“.’.”’;””.’: :’[15] ,A”+”=. ”dq++--, , ., , . [~] “ #sdn%4f/E,, . . ..o. [161

where E2 is the modulu~ of the elastic element determined earlier from Eq. ‘(. &=q/#& ...... o.. [l-/l 5. The corrected strain [ec] at tl, t2, .... tn is then obte.inedfrom the relationship The purpose of the analysis is to find the values of A, B and C which make the theoretical expression [Eq. I.]+]fit the experimental data as closely as possible. Oficethese values a?e ecfi~ti -Aec, .e. .. [13: determined, the model para!!etersEl, illand N2 may be calculated from =q. ?.5through l-(.

6. Finally the values of ec are plotted Since Eq. 14 is nonlinear the least square against the corresponding times tl, t2, .... values of A, B and C cannot be found directly. ‘n) as shown in Fig, 7C, t,uobtain the Instead initial approximations ~, B. and Co corrected creep curve. must first be determinetiand these carrected by cm iterative procedure to give the best values. Determination of Model Parameters from Creep Data Determination of Initial Pammters :’ . Once the corrected creep data were “~btained A nu~ber uf methods were.i.nve,stigatedfor .the”next.... “s&p-waS to”~aly~e-these data-in- “dkteiininintithe’ini.tia$”p”atiatie$ers.30 }~~ts+er- terms of-the Burgers model~ -:Inearlier szudies -the most practical-method involved-the use of - by the writer and others this analysis waz data’from three specific points on the correcte “normally~a?r’iedout by graphical co-nk”tticti(m.“ creip curv”e; Briefl~ t“he:”meth~diS .as f~lldws: When this form of fialysis was investigated fn more detail, however, it was found that it 1. Fig. U shows a typical corrected cree~ was too sub~ective and that the results obtaine“d curve OPA. A point P is selected on the curve, for the model parameters were not consistent. at a location just outside where the line AB To overcome this the writer established a becomes tangent to the curve. The corresponciir program, in the summer of 1964, to in~stigate time T associated with this point is noted. various meams for analyzing the data. 2, Three uther points, Pl, P2 and P3, are In the procedure finally developed,31 a set located on the curve with time coordinates T/8, of initial [rough] parameters was determined 7T/8 and T/2, respectively. The strain coor- from the correcte+ creep-curve, aqd,then),using dinates [el, e2,-and e5] of.these points are. . . these parameters and”the Gauss-Newton”method, determined. the least squares values of the model parameters we~e obtained. Initially the Gauss-Newton 3. The slope [tan CZ]of the Line AB, procedure was applied manually, but as this was wh%ch is tangent to the linear portion of the fo~d to be too time,conswing, a program for curve, is determined, This slope is equal to this analysis W?S written for the IBM-1620 Co, one of .the.requiredpa?~eters~ .- . computer;” ~e-over-aQ-&alysis procedure will- : : ” ;-’:-:---- :“: -””-- ; -., --:: “’: be”outlincclhere briefly, but no atttimp twillbe 4-. The parametei,Ao is”csQculated from made’to discuss-in”detail either.the”Gauss.. L ‘ ‘-the-relationship ‘.-. ..-....- .-” .‘--””:“- ‘---’” Newton met’hod-o:,.thecomputer program.” ....-”...... : ...... ,..,. ..-.,.. ...7P-- .,------.—:.. ,..-.-. --.-,.-’. ‘“..,..’ ... - ,, ...... -.:,,...., .,.,.. ..— .~..,- . . - Eq;-.. -6“rnay.,be... wriitte~-+n,>he.-..- form. .-,‘:) ‘.“’-. : . ,: . ““’:.“.;:- ,:,.-. :.”’,.. :. .;:-:<,. ... .-, -..,.~’ .? ...... :.:.....W.. ;., . :.:,,.-...... ’ ... :~=.-..-.,... . ., — .

SPE-1707 .---H. Rl?CTNAT,ll—.-. ..——HARTIY.-- -, -.,..TR.

RESULTS a 24 General 4* = ‘- “-A 2?; !*” . . [,8] The experimental and o.aalyticaltechni .. described earlier @ this paper have,been and applied to the investigation of the ihelast behavior of a number of carbqnate rqcks and ,qze ##==cezfn,d191 potash. In particular selected results obtDownloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 i during the initial inves,tigatiunof the beh ...... -, ... ” ‘“ .. ... ‘,,. ,’of Vombeyan marble are presented. Thes& il where em and tm ark the coordinates of’the trate both the precision and sta~ility of th Points ,PljP2y Md P3j”and Co is the parqmeter -experimental facilities, and thesuitability determined in step 3. the analysis technique.

5. The parameter B. is calculated from the Instantaneous Elastic Modulus [E2~ relationship It waG observed that %he?e was a renar similarity between the stress-strain curves all specimens tested. AU curves exhibit t ~’ m : (=m/3) 001 ● [2”1 sane initial curvature followed by a linear a region from about h,000 to 8,OOO psi. l?hi where followed by a region of increasing curvatur which appears to be associated with the ini ation of time-dependent behavior. The stre tm strain-curves for two specimens [wo-6 and WO which were defomned following exac~ly the s loading program, were found to be vzry near ~ = k-$’~.j:-- ● ● [211 identical. The elastic modulus E2, associa with the instanta~eous elastic response of material, was found to vary with stress lev and all other factors have been defined for these two specimens as shown in Fig. ~. previously. a stress level of 200-500 psi the modulus i the order OC 3 x 106 psi.. With incrcfising IL2astSquares Analysis stress the value rises to approximately 9 x . psi [at.a stress of.8,000 psil,..remainingn ..‘I’he-s’tandard.meth.odof function”fiti,ing.by...constantat ,thisvalue,up to.failure. . “~he.teg~ique bf-l@as$;s~ua@632_c&ot. be - ,,-:’-- -.: -:...” ::.”” “.:::< ‘“. “:” applied wh’~n-’thefunction “c”ontainsnonlinear Results of Incremental Creep “Exper&nts .... terms [e.g., e,=t/B]6 . . . . method “of least squaresinstead’analysis‘he”~ss”N%Onwa~ used. “’ A total of 26 incremental creep e~erim In this method am iterative procedure is was carried out on Specimens WO-5, -6 S@ - employed in which Parameters Ao, B. and ‘C. Table 1 summarizes the experimental details are used as initial estimates of A, B, and C the results for each test. The experimenta and the analysis carried out to give values data were found to fit the incremental form Al, Bl, a~d Cl, which are better estimates of the Burgers model [Eq. k] extremely well. A, B, and C. The values Al) B1 and Cl them- particular, Figs. 10 and 11 illustrate the selves are then used as the new initial esti- of the time-dependent portion of the Mrgers mates of the parsanetersand the analysis carried model [Eq. 6 or 14] to the observed crzep out again. This procedure is repeated k times behavior. Fig, 10 shows the creep data for UUtil the values Of the Par-eterS Ak, I?kand Specimen WO-5 at a stress level where the Ck obtained provide.a satisfactory fitto the - inelastic behavior is,relative,ly-slight,th experimental data. total creep strain after sustained loading approximately 40 minutes being of the order A Fortran program using the Gauss-Newton 10 MS. Fig, 11 illustrates the behavior of methoi was written for the IBM-1620computer, sane specimen at a stress level near the fa This allowed efficienthandlingof the large stress. Here the total creep strain was of v.ol~e..ofdata and permitted as many itetiations o~g.i @ [email protected][email protected] as necessa~.to-be car~iedout. The details period of approximately 70.rninutes.In-both of this Fotirm.progrm are-published else-...- -cases the aaalysis provided a theoretical c wher”e%?”j~~” - “-”” ““ “ ; “.‘-”“ ‘=”~:”; ‘which’,f~ttedthe”data extr6mely-’~eLl”~he-” ...... — ...... :....-..,;. -standard deviation being 0.37%,ndjl.b ~s, ,. . . . ,...... ~‘“-.”.“”.’ ..’ -“-. .. ..” respectively. --- “.. --;-, - ._ z ..-,-. .,— .-=.-. ,, . ..- ...... ,. ... , ...... ,.._...: .------.,:...... :. .:--,...... :.-.:~...... - ...,------...... $...... -.. ,’...... ”; ‘L ,-. ,.-. . DETERMINATION OF THE INELASTIC PARAMETERS OF’GEOLOGTG 112 MATERLA.LSFROM INCREME ~ CREEP EXPERIMENTS SPE-170

functional relationship appeared to be approxi- +Although the experimental data were found mately linear, with the magnitude of the to fit the incr,’mentalcreep fo?zmof the parameters decreasing with increasing stress Burgers model extremely well, initial experi- level. h this same region the parameter E2 ments indicated that +.hemodel ~areneterfi was found to be relatively independent of stres~ [El, NI, Ez, N2 andT] aswell as the final Below 7)700 psi no data Mere available for the 6teady-state strain rate C were functionsof parsxneterfiEl> Nl? ~d N2 as the inelastic, the ~ial stress. “Morb iefiriedexperiments strains associated with stresses.belowthis which were conducted on SpecimefikWO-5 to WO-7 point were too”ame.11to,be dgtected; hoyever,. verified this fact. Figs. 12 aid 13 show the i= this region E2 was”found to decrease with Downloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 variation o? these factors.as a function of decreasing stress. This initial region-of low axial stress [CPzz]using the,combl,nedresults modulus is often associated with ‘theclosure of for SpechnensWO-6and wO-7, u cracks and pores in the’rnaterial;hu

To investigate the stress dependency of th Recently another series of experiments wa6 Burger model parameters in more detail, the conducted on specimens of Wombeyan marble to experimental results were analyzed using the verify further the initial results presented in IBM-STUFF Program [IBM-1620 Universal Function this paper, md to investigate the effect of Fitting progrsm] ad fitted by least squares to longer loading periods on the incremental creep a straight-line relationship* [X = ~ + Ale, behavior. An initial series of fiimilarexperi- vhere X Is the particular factor, o is the axia ments wac also conducted on specimens of Ik2bo compressive stress, and Au and Al are the and Missisque marble and on potash. The resultl intercept and the slope of.the theoretical of these experiments will be publishetiin the curve]. These theoretical curves arc shown on near future. Figs. U and 13, and Table 2 lists the coeffici ents A and A associated with the stress An expanded program of investigation of dependency of the various factors. Where an the inelastic behavior of geologic materials isolated inconsistent data point appeared, it is being planned at the Pennsylvania State U. was rejected from the STUFF analysis. by the a~thor.” Tnis program will include the investigation of a wide variety of rock types DISCUSSION under conditions of both uniaxi~ and triaxial stress, A newly acquired closed-loop loatiing . . . The results of the experiments conducted system capable of being computer programmed on specimens of Wombeyan marble indicate that will be used in these studies. the facility ~d the exper@ntal and analfiica . techniques developedhave proved,~u be Ii30St . ‘“.The”results-of”:the;presentyttidi~st~em-” satisfactory.- Copsisteritresultswere obtaine{ s-elves-provide.n-oimmedia~e-solut{bn-,totiY. --- from’-corrected”cree~’’~u$vkswhtise-m-&irnum’-str6i of the numerous mining engineering problems. was less.than 5 ~s.. The.reproducibility of”.. Howeve”r,tneydo”’provide further basic infor- the’experiments 1s verified bythe excellent mation on the mechanical behavior of the” agreement in the results from the experiments c material itself. Until recently, geological &ecti”ens WO-6and W0-~. materialc have usually been considered to be elastic for the purposes of mathematical With few exceptions the creep data for analysi6. This simplification has not been Wombeyan marble was found to fit the increments possible or desirable in the analysis of many creep form of the Burgers model extremely well, recent problems and instead the material has The typical quality of the fit is exemplified t been considered to be inelastic. the results shown in Figs. 10 and Il. The low standard deviation between the experimental emi Examples of the tne of problems and the theoretical curves quoted in Table 1 provides approach being used in their solution are given further evidence ”of.this fact, - - . in two papers published’recently by the Polish Academy of Science18 and Litwiniszym.22 These The model parsunetersE amd N2 were ~, NI, are concerned with the solution of mining Btres found to be functions of t e applied stress in problems when the associated rock is represent the region above approximately 7,700 psi. The by various mechanical models. An extension of *Attempts were made to fit the expertiental. this work-has recently .beencairied Out by - -1.man13-atthe”Ui ofMinnesota, He hbsbeen --- data”t5 high@ order poljrnomiali”but.inmoat”‘- cases the ii@rovement ffi.thequality of’the” cone’e”%edwith th”e&nad.ysisof-sub-idenc,e’based fit was relatively-sm~l.-:’Uhtil:@rther creep on the’assumption.that-rock behaves:as.an .“.... inelastic material. ‘. - -- ‘ ~ studies are conducted.at.,stres~.es.in.,the~...a,. .—...... ~..,:,T.“--- ...... ------..7=-. -.--.—---.,,. region “whereinelastic behavior appears tti.- : commence it-is..felt%hat-a”straight-line.fi.t. — - At present koriside~abledlffic~%y”is” E&ei’i&ice&.Ln ahalyiing.miiiihg’problems”be’caus is adequate, : 7 ~....7’ ‘- .- ““: ‘:~;:.- : -. .

PF.-17O7 H. REGINALD HARDY. JR.

little is known of the mechanical behavior Eng. Exp. Station, U. of Utah, Salt Lake of the rock itself, It iB only through a large C:ty, Utah [1962]. number of experimental investigations, such as 6. Hardy, H. R,, Jr.: “New Rock Mechanics the present one, that.this behavior will-be Progrsm Underway by Canadian Government” understood and the parameters nece6sary for the Mining Engineering [1965]~, 62. successful analysis of the various problems 7. Murrell.. S. A. F. and Mi~ra. A. K,; “Ti obtained. De~endentStrain or ‘Creep”ln Rocks and SimilarNoIi-MetallicMaterials”, Bull.; ACKNOWLEJX2MENTS Inst. MiningMet. [1962]g71.;No. =53. Downloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 8, Robert60n, E. Ci$ ~ The author wishes to express his gratitude State.of Stress in the.E&rth’s Crust; .to the merab,e,rs”of his graduate:committee.,,R. .@erican @sivier pub;.co~.,Inc. 11964] -n. .. -@ic_urel,S).Fre,derick”,-W.D. .T@wryand C. W. . . .. 101. ---- . . - . . -. - ., Smith~for their constant interest and sugges- “ “9, Fre&ien%nalj”’~,M,: The Inelastic Behavi tions”duririgthis pro~ect.” The frequenCdis- of Engineering McteTials and Structures, cussions with Frederick, the autho$’B director, John Wiley amd Sons, Inc., New York [19 and with C. T. Holland [formerly Head of the - 10, Hardy, H. R., Jr.: “Inelastic Behavior Mining Dept. at V.P.I.] have assisted greatly Geologic Materials, Part I - Experimenta in a successful completion of this work. The and Analytical Techniques, antiInitial author is particularly indebted to D. H. Pletta, Studies on Wozbeyan M@rble”, Internal Head of tileEngineering Mechanics Ikpt. for his Report 114P65/155P,Fuels and Mining Pra cncouragement, both academic and personal, tice Div., Mines Branchj Ccin.Dept. of during this project. Energy, Mines and ReGource6, Ottowa [196 Il. Hardy, H. R., Jr.: “Inelastic Behavior, The author would like to thank A. Ignatieff, of Geologic Materials”) PhD Thesis, Engr Chief of the.Fuels and Mining Practice Division. Mechanics Dept., Virginia Polytechnic of the Mines Branch, Canadian Department of Institute, Blacksburg, Va. [June, 1965]. Energy, Mines and Resources, for his assistsrace 12. Price, N. J.: “A Study of the Time-Stra in establishing this research program. k. M. Behavior of Coal-MeaGure Rocks”) Int. Jo Gray, Head of the Physics section, Fuels and Rock Mech. Minine Sci. [1964]~, 2“(7. Mining Practice Div., formed the link between 13. =, H. F.; “A Viscoelastic Analysis o the Mine~ Branch and V. P. I. during this Mine Subsidence ‘inHorizontally’Lam’inat research and his constagt assistance and encour- Strata”) PhD Thesis, School of Mineral a agement are greatly appreciated. Pi Okulich, Metallurgical Eng., U. of Minncs-ta,. “formei’lyattached to the Physics section and” Minneapolis [1964], presently Research Aid,’MinirigDept.,”Zhe “14. Ito, I..and ’Terada,N.;” “A Phcwlugical Pennsylvania’StateU.”was responsible for the ‘:Study on the DynamicaYBehaviotiof Rocks Pr?PFr:tioGi?f.tti$.~ajority ‘of,:hetest”.. .“‘“..’;-:”Jour.”Min. Metall, Inst. Japan [1962] @ ‘specimensand-the preparationyof.the-final-- : --- -723.-[In-J@paese]. ‘- - ; ---- -: . “The Development of Appara .di’swings. . .. and. .graphs...... 15. Hunt,’Co A.: .’ -”” -’-:. ‘to Determine’”tfieTime~De@6ndetitPro@%ies The author”would also like to e~ress his of Rock”Salt and Their Measurement on’ thanks to M. S. Paterson of the Australian -SeveralSpecimens”, Internal Report FMP National U., Canberra, Australia, for supplying 60MIN, Fuels and Mining Practice Div., the samples of Wombeyan marble used in the Mines Branch, Cdn, Rpt. of Energy, Mine experiments. and Resources, Ottawa [1962]. 16. Attwell, P. B.: “Composite Model to Sim REFERENms late Porous Rock”, En&ineering [1962] 19 57)+●

1. Brown, A,: “Ground Stress Investigations 17 ● Attwell, P. B,: “Response of Rocks to H in Canadian Coal Mines”, Trans.,AIME [1958] Velocity Impact”, Trans., Inst. Min. 211, 879$... Metall. [,1~61171,= 2:’ =dy, H. R:, Jr.: “Tti’e~EependentDefor-- 18~ Salu~towicz~ A..“y “Rocks’as-a”Vi6coelast mation and Failure of Geologic Materials”, Niidium”,Polish Academy of Science, Colorado School ofMhes Quarterly [1959] Mchivium Gornictwa [1958]j, 141. [m 54,134. Polishj— 3....~rdy, H. R., .Jrp:. “Design of Instrumen- .!?*- Iida, K. and Kumazawa, M.: “Viscoelast tation for the Measuremen~ of Time-Dependent . Prope&ies of Rocks’’,”Jour.Earth Scienc ‘-‘Strain-in-StressedRock-Specimens”, Trans., -: - --.--NagoyaUniversity [:~95-~In: ISA [1962]I-,”147. - : - - ““English] ?..Griggs, D. T. .&d %r&n, J,: Memoir.79,” 20..Terry, N. B.:{ Vhe Elabtic~Propertieso ‘GSA Rock Ikfofiation.Symposi~m [19601: Coal;”Part 6,.SomeMeasurements of Inte~ .“ ‘5~f~~r~~:~:~-~”~”-Wi~S-0~,”.Je ~*:--’’~Biblt~-:~-- ““=”;;-,‘“Diunj5ing-a.fid.Sorne.”Consideratiotis”of ViSco s “og~phy”of’Geom~:ch~icsl’,:Bull. 119,_Ut@--,>“ -:-ElasticBehavior’’,’MRE.Report do. 2080j . -... .,.... 1.+ ------...... ,=, ... “L~--Britisli’NatiohallCoal Board [-1957]s.: DETERMINATION OF THE INELASTIC PARAMETERS OF GEOLOGIC

21. Matsuchima, S.: “OrIthe Flow and Fracture Geologic Materials, Part 11 - Review and of Igneous Rocks”, Bull. No. 36, Disaster Analysis of Data from the Literature on Prevention Research Inst., Kyoto il.,Kyoto Creep in Rock Under Uniaxial Compression”, Japan [1960]. [InEnglish] Internal Report FMP 66/51 P, Fuels and 22. Litwiniszym, J.: “The Effect of Time on Mining Practice Div., Mines Branch, Cdn. Deformation and Stress of Rock Bodies”, Dept. of Energy, Mines and Resources, Polish Academyof Science, &chivium Ottawa [1966]. # Gornictwa-=Hutnictwa”[19551 ~j””4790 [~ Polish]...... ’ “ : ‘. ... Downloaded from http://onepetro.org/SPEDRM/proceedings-pdf/67DRM/All-67DRM/SPE-1707-MS/2087759/spe-1707-ms.pdf by guest on 02 October 2021 “23● Roes, A. D.: .Elasticit~ Shrinkage”of Concrete; Mechanical—— ProP- erties of Non-Metallic Brittle Materials, ... Butterworths Scientific Publications, nndon-[19581.1:-: . ‘: “... 24. Frederick, D,andC%rciaJ M..A.$- ‘Ine ~ Theoretical Behavior of an Inelastic Material Model”, Bull., Eng. Exp. Station Serie~ No. 85, Virginia polytechnic Institute, Blacksburg, Va. [1953]. 25. Steppel, S.: “The Differential Equations Representing Several Simple Theological Models and Their Solutions for Various Conditions”, Internal Report FMP 64/87P, Fuels and Mining Practice Div., Mines Branch, Cdn. Ikpt. of Energyj Mines and Resources, Ottawa [1964]. 26. Hardy, J. R., Jr.: “Design’and Construc- tion of a Facility for Research on the Inelastic Behavior of Geologic Materials”, Mines Branch Research Report R 165, Dept. of Energy, Mines and Resources, Ottawa [1965]. 27. Paterson,. M. S.: “Experimental Deformation and Faulting in Wombeyan Marble”,.Bull., GSA [19581Q; 465.- 28, Hardy, H. R.~ Ok~ichj ~. J and Kapeller, F“.. ““Pre~arationof Srnall”CYlindrical . .Test S~ec-tiens-ofGeologic M&teriala”. -“. . . . inte”~al:Report:.m166/~8,p,:~ek. .w.~jv. .. MiningYractice Div.,~Mln@s Br.%~ch~cd?. .. .,..” . . .. Dspt-.of”Energy, Mines and Resources, .- . . Ottawa [1966]: “ 1“ 29* Hardy, H. R., Jr.: “A Loading System for the Investigation of the Inelastic Prop- erties of Geologic Materials: Testing Techniques for Rock Mechanics”, ASTM Special Technical Paper, STP402 [1966]. 30. Steppel, S.: “MathematicalProcedures fo~ Fitting the Burger Theological Model to Experimental Data”) Internal Report FMP 65 30 P, Fuels and Mining Practice Div., Mine Branch, Cdn. I%pt. of Mines and Technical Surveys,.Ottawa [1965].-. : ‘. .. . . 3i: Hardy, H. R.j Jr., Steppel, W. and Toews~ N. A.: “The Analysie of ‘incrementalCreeI Data Using the Burgers Mechanical Model”, wnort in Pre-oaration[1965].

32 ● Il&ing, V: E.-: Statie~ical-Ad~ustmentof -Data,John Wiley and.Sons,.Inc.~ New..Y= TTW31’0: ‘ .-” -. ::

3; ● .iIa-itiey,. H..o.: “The.Modified Gauss-Newt”c Method for Fitting of Non-Linear Regressi( Functions by I.east-Squ&?es’’~--Technornetric~ [196113. NO.-2, 269. “ “-

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S~mrnaryof’DetailsandExperlmenta3 Results from Incremental ,... Creep .Expcriments on Wombeyan hlarble Specimens ,, $

Experiment Czi( .2) o-f .69- s(4) E,(5) N’, ! E2 Number (1) ~ ps 10b psi 109 psl-sec 106 psi 109 A J& psi-see I 5-(1) 7026 7672 646 543 2.5 72.8 .0003 0.35 253,9 18.5 11.0 23.080 5- (2) 767?. 8181 509 ‘ 1095 3:3 81.6 .0014 0.70 154.7 12.6 11.2 5708 5- (3) 8181 8685 504 1223 4.0’ .?2. 1 .0033 0.43 ,25. , 2.8 ; 10.8 262?. 5 -(4) 8685 8964 2.79 1320 $ 1.3 290.7 .0011 0. 3? 2.23.0 64.8 ,.. , 11.5 8068 5- (5) 8964 9720 756 “2400 9.4 39.9 .0044 0.86 80.6 3,2 11.2 2218 5- (6) 9720 10, 140 420 : 2700 , 4.9 190. z .0024 0.39 86.1 16.4 ., 10.6 4272 5- (7) 10.140 10,880 740 12,000 25.4 121.6 .0055 1.54 29. Z 3.6 ‘ .10.7 1993 !. 5- (8) 10,880 11, 557 677 2200 ‘ , 4.9 16.9 .0058 0.88 137.3 .3.3 11.9 1978 ;,,5-(10) 10, 880 I& 005 L125 2610, ; 27.7 ; 905.9 .0032 0.80 40.6 36.8 .’ 11.3 3677 ‘,, ,. ,, ,, ,, 5 -’(11) 12,005 1.2.453 448 : 4680’ (Ii 88.7 583.9 .0121 1..79 5.1 3.0 1027 I “ 9.9 4 ; WO-6 6- [1) 7487 8101 r 2386 . II 11.4 I 81.2 .0047 0.74 53.9 4.4 8.6 1719 ,, ,: ,,,’ 6- (2) 8101 653 219? 15,6 45,3 .0057 1.42 41.8 1.9 . 9.1 1540 ; 8754 ,, ‘., ,,, 6 - (3) 8754 9380 626 2064 19.6 108. Z .0074 1.19 31.9 3.5 8.6 IZ63 .,’. J,;. ‘-=;Vt 6- {4) 9380 10,010 630 “ 2184’ 21.5 84. ? .0067 1.97 .?9. 3 2.5 8.4. 1505 ,: ,1, ,! ., I,* 6 - (5) 10,010 LO. 628 6i8 2300 35.4 117.4 .0125 1.70 17.5 2.1 8.9 854 ,, ‘~. , .;, ,: 6- (6) 10, 628 11,256 628 2294 56.4 118.2 .0136 4.11 11.1 1.3 8.9 830 :1,! 6 -:(7) 11,256 11,892 636 2303 :’ ,78.”2 180.9 . 0Z34 3.20 8.1 1.5 ‘.,. 9.2 509 ,s 6- i8j 11,892 I 2, 544 652 440 1170:9 158.3 ~ 4.86 3.8 0,6 8.8 70 W;O-7’ 7 - (1) 7468 8080 612 2440 13.8 42.9 .0030 1.08 44.5 1.9 9.7” 2700 ,’” 7- (z) 8095 8740 645 23z0 15.2 .23.7 .0049 1..28 4.3.4 1.0 10.2 1772 ,; 7- (3) 8759 9370 611 J-2449 15.9 117,2 .0049 0.97 38.3 4.5 9.7 1898 ,, 1505 !,. 7 -“(4) 9386 9994 608 ; 2313 19.9 63.3 .0066 1.44 30.6 1.9 9.6 ,, 7 -’(5) 10.031 0,658 627 2391 24.0 i 22.1 .0070 2.12 26.2 3.2 8.9 1516 ,,,, 7- (6) 10,640 1,283 643 2394, 37.4 ‘ 105.1 .01Z4 2.89 17.2 1.8 ,.” 10.0 909 ,, 7-(7) L1,212 1,851 639 238Z 48.9 237.8 .0138 2.14 13.1 3.1 861 , :,, 40 7- (8) 11,912 2, %2 650 333 II 20.9 1 27.9 ! .318Z 2.07 31.0 0.9

.“, , L ,, ;. ‘, .! ,, ,) ,, ,; TABLE 2

i’.

rv.. !4080. St _— ~.. . ‘Parameter Cttlt. x“,(z) AO Tarjan. eOf x (1) “- ., Esrtmafe(+) )., l ,! i. ,.. , , ,,, ,$. =1 10b ~s] ,,; . 17.31 I 32.5 -1.035 x IO-Z 11.50 ,.. IG9 p,t. s.zc. 2.25 6.04 -} .678 X 10-4 1.14? ,, N1 ,., , iub ~s, 9.27 ‘?.41 1.770 x 10-5 0.310 ,,, ~,’,, ~2 ,, ,. 1: ,, X2 lo~ p.l-, e< 5572 -0.4219 9.Z5 x 104 *’, . I T’z’s :,, .” ,,, . .2.67 ~“lo-~ J. 57 x 10-6 s.86 x 10-6 ... ,,, ,,+.: -211.6 3.13X 10-2 1?.?4 — (1) f:,esc. :,m!s a’!+v ‘t~,~ll iac!brs except Al and the variance. ,, (~ ?h,s rcprcstm,, ti;e mean (.Iuc .d the particular par.amrter over theslre~s : E~

N ~“ —

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... ,...... ,. ...,..!-...... , ...... -: . . ,, .. .-: . *m. .:....,...... ,, ...... b.,. ,.. ~~~ ~igOl’”- Burg-er M;cha~ical Model

...... ,. -’. --, .,.. .< -. -:-& -. . .- ..-: ...... ,. . . .. -...... ,. . .,.

A: Group of Test Specimens w~th Attached Strain Gages. . ------.

..—. ,. ., ..

. . ., ------Pi-l QGii5s ——. —. PtWMAT; C OFI -1 r HYDRAULIC

POWER SUPPLY I n I 1 l— .-INPUT +R PROORAH

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.> .,,-, # ,. .,. . ..+’. ,-. “. 1..~‘ “FEEDMCK TRANs!31JGkRs ; 1. .“ . . I ““

I

Fig, 3 - Sitnplific+d Block Diagram of Loading System for Rock Deformation Experiments

1“ ? I I I 9TOR?N 1 .! PRESbURE}:RA~’DUCERS L-– __––-–-,-...... ,. ... .- - -.. . .’ .- ,.- ...+ “...... ,., .-. ..”...... ‘. .-. . .,.. ... ----- ,...... : .,...... -. . ,...... ----- ...... “MANUAL ---., -61 ..,...... “..’ ,-. - R’F:ADOUT . “SLOW: 12:CHANNEL SYSTEM ...... — ,- ,= -- - - I- -1’ ~ MAIN Y-T RECORDER SYSTEM SWITCH , SWITCH I ~ [J. B.-I) --- .-

......

Y-XT

RExh3ER Y-T - SYSTEM .LI?JE” . .. \ h ...... -— ----- — — .. .- .,

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, ‘. .:

A

400 -

0 0[!1 0121 013) EaCH INCREMENT EO”a L8 POOO PS, “o e 000 e000 !2000 “kAXIAL STRESS-PSI AXIAL STRESS- PSI

Fig. 5

A. STRAIN VERSUS Yl~E CHART ,

..-. . t +-Lw “ I

0. LOAD VERSUS TIME CHART

A. STRAIN VERSUS TIME CHARI r- . “1’. M2. .m3 I ~, “~- ,1 d H’-o LC(2 C. CORREC7ED CREEP CURVE c’ 1 F’ l---~I . ...=.— ‘;:...-.-:7-:‘---B: _:TRE~S ‘VERSUS-TIMECHA!Y‘“ ./ - --- &-+- &Loll) “r-- Ml , I M2 &J3 ..’ 1 . ..._- ‘A. ‘L TIME ““”’”” TIME .

.

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Fig. 8 - Method Used to Determ,ne the F,g.9- Variat,on of E2 with Strees Level Initial Parameters

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