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Mar 10, 1985 - Micromechanisms of rock failure (axial splitting and shear failure) are examined in light of ... Brittle solids such as rocks, by their nature, contain numer- ...... Let the minimum flaw size be denoted by Cm and the corresponding.
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 90, NO. B4, PAGES 3105-3125, MARCH 10, 1985

Compression-InducedMicrocrack Growth in Brittle Solids' Axial Splitting and Shear Failure H. HORII

AND S. NEMAT-NASSER

Departmentof Civil Engineering,The TechnologicalInstitute, NorthwesternUniversity Evanston, Illinois

Micromechanisms of rock failure (axial splittingand shearfailure) are examinedin light of simple mathematicalmodelsmotivated by microscopicobservations.The elasticity boundary value problem associated with cracksgrowingfrom the tips of a modelflaw is solved.It is shownthat under axial compression,tensioncracks nucleate at the tips of the preexistingmodel flaw, grow with increasing compression, and becomeparallel to the directionof the maximumfar-field compression. When a lateral compressionalso exists,the crack growth is stableand stopsat somefinite crack length. With a small lateral tension,on the other hand, the crack growth becomesunstable after a certain crack length is attained. This is consideredto be the fundamentalmechanismof axial splitting observedin uniaxially compressed rock specimens. To model the mechanismof shearfailure, a row of suitablyorientedmodel flaws is consideredand the elasticity boundary value problem associatedwith the out-of-plane crack growth from the tips of the flaws is solved.It is shownthat for a certain overall orientationof the flaws the growth of the out-of-planecracksmay becomeunstable,leadingto possiblemacroscopicfaulting.On the basis of this model the variations of the "ultimate strength"and the orientation of the overall fault plane with confiningpressureare estimated,and the resultsare comparedwith publishedexperimental data. In addition, the resultsof a set of model experimentson plates of Columbia resin CR39 containing preexistingflaws are reported.These experimentsare specificallydesignedin order to show the effectof confining pressureon the crack growth regime. The experimentsseem to support qualitatively the analytical results.

1.

INTRODUCTION

Brittle solidssuch as rocks, by their nature, contain numerous flaws, cavities,inclusions,and other inhomogeneities.Materials of this kind fail under axial compressionby axial splitting when the confining pressureis zero or very small and by faulting or (macroscopic)shear failure when the confining pressureis moderate but still below the brittle-ductile transition value. The strength, the orientation of the macroscopic failure plane, the dilatancy, and other mechanicalfeaturesare

greatlyaffectedby the confiningpressure.Nucleation,growth, and interaction

of microcracks are considered to be the domi-

nant, controlling micromechanismsof macroscopic failure; see, for example, Paterson [1958, 1978], Griggs and Handin [1960], Brace [1964], Gramberg [1965], Fairhurst and Cook [1966], Mogi [1966], Scholz [1968], Friedman et al. [1970], Hoshino and Koide [1970], Wawersik and Brace [1971], Peng and Johnson[1972], Hallbauer et al. [1973], Olssonand Peng [1976], Tapponnier and Brace [1976], Holzhausen [1978], Holzhausenand Johnson[1979], Wong [1982a, b], and Kranz [1983]. For reviews and references,see Paterson [1978] and Kranz [ 1983]. Models of microcrackinghavebeenpostulatedbasedon the idea that frictional sliding along preexistingcracksresultsin the formation of tension cracks at the tips of the preexisting cracks,and experimentson glassand photoelasticplates have been performed to illustrate this process[Brace and Bornbolakis, 1963; Hoek and Bieniawski, 1965; Bornbolakis,1968]. In addition, model calculations have been made in order to quantify the microfracturingand the associateddilatancy and other macroscopic manifestations[McClintock and Walsh, 1963; Holcornb, 1978; Ingraffea and Heuze, 1980; Dey and Wang, 1981; Kachanov, 1982a, b; Moss and Gupta, 1982; Nernat-Nasserand Horii, 1982]. Copyright 1985by the AmericanGeophysicalUnion. Paper number 4B5089. 0148-0227/85/004B-5089505.00 3105

Since about 1970, the use of high-resolutionscanningelectron microscopeshas produceda considerableamount of information on the sourcesof microcracksand their growth in responseto applied loads [Brace et al., 1972; $prunt and Brace, 1974; Dengler, 1976; Tapponnier and Brace, 1976; Kranz, 1979; Wong, 1982b]. It has been observedthat isolated

preexisting,Griffith-type cracks are seldom seen to be the major source of microcracking,and many sources(or stress concentrators),other than the preexistingcracks, have been identified. In Westerly granite, for example, Tapponnier and Brace [1976] report that sets of grain boundary, low-aspect ratio cavities, as well as suitably oriented interfaces of two different minerals(biotite often being involved),produce most of the microcracks. For marble, Olsson and Peng [1976] observe, with the aid of optical microscopy,cracks forming at the intersectionsof inclined (relative to maximum axial compression)lamellae (or slip bands [Cottrell, 1953]) and grain boundaries.More recently,Wong [1982b] has made a detailed microscopicobservationof the fracture processin Westerly granite at high pressuresand temperatures.Wong confirms many early observations. In addition,he reportsfor prefailure loadingsthe presenceof microcracksat high angles(15ø-45ø) relative to the axial compression,which can slip by overcomingthe frictionalresistancealongtheir surfaces. It is essentialthat the developmentof analytic models for microcrackingbe guided by these physical observations.A model of this kind is consideredin section 2, and its relation to the microscopicallyobserveddeformationprocesses is discussed.It is concludedthat although preexisting,frictional microcracks are seldom observed to be the source of micro-

cracking,the shearcrack model proposedby Brace and Bombolakis [1963] still presentsa reasonableidealization, if it is properlyinterpretedas a "microflaw"associated,for example, with a set of grain boundary cavities,a soft inclusion,a cleavage,a slip band, or evenwith a high-anglefrictional crack.A microflaw of this kind may have frictional resistance,cohesive shear resistance(due to its in-plane plastic deformation), or

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HORII AND NEMAT-NASSER:AXIAL SPLITTING AND SHEARFAILURE

Fig. la

Fig. lb

Fig. lc

Fig. 1. Sketch of stress-induced microcracks(a) at a biotite grain boundary, (b) at the intersectionof an inclined

lamellaand a grainboundary,and (c) at preexisting grainboundarycavities.Figuresla and lc are in Westerlygranite [from Tapponnier andBrace,1976],andFigurelb is in Tennessee marble[from OlssonandPeng,1976].

both, dependingon which actual mechanismit may be approximating. In the presentwork we use this model flaw as a basicsourcefor microcrackingand assumethat it represents adequately the effectsof actual flaws. In section2 a mathematicalmodel of curvedcrack growth from the tips of a model "microflaw" is defined as a twodimensional,elasticityboundaryvalue problem.The exactfor-

mulation of this boundary value problem is given, and numerical resultsfor curvedcrack growth are obtained.(The problem of noncoplanarcrack growth under far-field tension (but not compression)has been extensivelydealt with in the literature; see,for example,Banichuk[1970], Lo [1978], Pala-

a macroscopic shearfractureplane.This localized,highmicrocrack densityzone materializeswhen the magnitudeof the appliedaxial compression is closeto the ultimatestrengthof the specimen.(It should be pointed out, however,that some experimentalobservationsseem to suggestthe initiation of localizeddeformations at ratherlow stresses; see,for example, $oga et al. [1978].) Most of the microcracks are in the axial

direction; see Friedmanet al. [1970], Wawersikand Brace [1971], Dunnet al. [1973], Hallbaueret al. [1973], Olssonand Peng [1976], Tapponnierand Brace [1976], and Wong [1982b] for discussions and references. In section 3 we consider a mathematical model for simulat-

niswamyand Knauss [1978], Wu [1978], Cotterell and Rice [1980], Nemat-Nasser [1980], Hayashi and Nemat-Nasser

ing microscopiceventswhich may be instrumentalin the inceptionof macroscopicshearfailure. The model consistsof a [1981], and Karihaloo [1982] for a review). Crack paths, row of "microflaws"in an infinitelyextended,linearlyelastic, crackextensionlengths,and their relationto the appliedover- two-dimensionalsolid under far-field lateral, as well as axial, all loadsare established, usingan incrementalapproach;this compression. (From the mathematicalpoint of view,thisrepis a considerableimprovementon our previouscalculations resents a ratherdifficultproblemin two-dimensional elasticity, which were basedon a straightcrack path [Nemat-Nasserand whichhasnot beendealtwith before.In thispaper,we outline Horii, 1982]. It is shown that in the presenceof some (very a powerfultechniquefor solvingthis and othersimilarprobsmall) far-field lateral tension the growth processof cracks lemsinvolvinginteractingflawsor inhomogeneities. We note emanating from the tips of a model flaw in an extendedsolid that problemsof this kind alsooccurin many geologicalforbecomesunstableafter a certain length is attained, with the mations;see,for example,$egall and Pollard [1980], Pollard cracks growing spontaneouslyin the direction of maximum et al. [1982], and Deng and Zhang [1984].) It is shownthat axial compression. The phenomenonof axial splittingis dis- because of interaction,the growthregimeof cracksemanating cussed in terms of this model. from the tips of the flawsmay suddenlybecomeunstableat a Another aspectof brittle failure relatesto the distributionof certainaxial load, with cracksgrowingspontaneously in the microcrackswithin the sampleand to the evolutionof this directionof maximum axial compression.For this mathematdistributionas the peak axial load is approached. Here, a ical modelwe definethe smallestaxial stressat instabilityas greatdealhasbeenlearnedby the useof opticalmicroscopy, the ultimate strengthand the associatedoverall orientation of acousticemission,scanningelectronmicroscopy, and other the row of flaws as the fault orientation. In this manner, the methods;see,for example,$cholz[1968], Hoshinoand Koide variation of the ultimate strengthand the orientation of the [1970],Hallbaueret al. [1973],Liu andLit•anos [1976],So•la overall fault plane with confiningpressureare estimated,and et al. [1976],LocknerandByerlee[1977],and Won,t[1982b]. the resultsare comparedwith somepublishedexperimental

It isreported thatastheaxialloadisincreased, thegrowthof data. microcracks accelerates, leadingto theformation of regions of In addition, we report the resultsof a set of relevant "model highcrackdensityin a narrowzonewhicheventually becomes experiments"which are specificallydesignedto test the effect

HORII AND NEMAT-NASSER.' AXIAL SPLITTINGAND SHEARFAILURE

of the confining pressure on the crack growth regime. The models consist of thin plates of Columbia resin CR39, containing preexisting,straight, thin flaws of different sizes.It is shown that, in the absenceof lateral confinement, cracks emanating from the tips of large and compliant flaws grow under axial compressionin the axial direction, leading to axial splitting. On the other hand, the presenceof confining pressure arrestscrack growth of this kind. However, at a certain stage of loading, cracks emanatingfrom the tips of setsof smaller, less compliant flaws suddenly grow in an unstable manner, forming a localizedcrackedzone. 2.

AXIAL SPLITTING

Under axial compression,microcracks may nucleate at boundaries of suitably oriented, preexistingmicroflaws and grow in the direction of the axial load. A flaw may be a preexisting, soft inhomogeneity, such as biotite in granite (Figure la) or a slip band or lamella in a plastically deformed grain of marble (Figure lb), or it may be a connectedset of low-aspectratio grain boundary cavities (Figure lc) [Olsson and Penq, 1976; Tapponnierand Brace, 1976; Wonq, 1982b]. From the mechanicspoint of view, any material or geometric discontinuity may serve as a stressconcentrator and may hencepromote crack nucleationand crack growth. For example, Brace and Bombolakis[1963] and Hoek and Bieniawski [1965] have shown that a suitably oriented, Griffith-type, preexisting crack in a glass plate can grow out of its own plane when the plate is compressedaxially. In this case,tension cracks are nucleated at the tips of the preexistingcrack becauseof the relative sliding of the faces of the preexisting crack. Nemat-Nasser and Horii [1982], on the other hand, use a thin (6 mm) plate of Columbia resin CR39, containing a small slit (instead of a crack) of about 0.4 mm width, fitted with two 0.2-mm-thick brass sheets, and they show that a similar crack nucleation and growth is observed under axial compression.Typical examplesare shown in Figures 2 and 3. (In fact, a similar crack nucleation and growth regime results if an inclusionis embeddedand glued to the plate [see NematNasser, 1983,p. 186, Figures 3 and 4].)

the flaw resembles

3107

a crack with frictional

resistance.

In either

case, the qualitative results turn out to be the same' under axial compression,tension cracks are nucleatedat the tips of the flaw at about a 70ø angle with respect to the flaw direction. These cracks grow in a stable manner, curving toward the direction of axial compression. The two-dimensional,elasticityboundary value problem associated with the idealized

microflaw

PP' and with the corre-

sponding curved cracks PQ and P'Q' (Figure 4) satisfiesthe following conditions'

uy+ =uy

rxy+ =rxy = -rc+ttay

onPP'

and

ao=rro=O

onPQandP'Q'

whereuyis the displacement in the y direction,r,,yis the shear stressand (7yis the normalstresson PP', aois the hoopstress, rr0 is the shear stressin polar coordinateson the extended crack PQ, the superscriptplus denotes the value of the involved quantity on the positiveface (definedby the positive y direction) of the flaw, and the superscriptminus is the value on the other side. The exact formulation of this boundary value problem is presentedin Appendix A. On the basisof this formulationthe curvedcrack path is calculatedincrementally, and the correspondingcrack length is obtained as a function of the applied load. Before discussingtheseresults,it may be of some interest to examine the relative influence of plastic glide and frictional sliding of the preexistingflaw, PP', on the crack growth process.To this end we have usedthe straight out-of-plane crack profile and, following the method of solution given by NematNasser and Horii [1982-1,have obtained the normalized axial

compression, la•l(r•c)•/2/Kc,as a functionof the normalized

crack length, l/c, for a2/a• = -0.08 and different values of re and tt' Kc is the mode I fracture toughnessof the matrix material, a• is the axial stress,•72is the lateral stress(tension is regarded postive), and c is half the length of the preexisting mathematical flaw. Typical resultsare presentedin Figure 5. It is seen that a flaw which can only glide plastically with no friction (rc -• 0, tt = 0) producesessentiallythe same qualitative result as a flaw which can undergo frictional sliding with 2.1. A Model for "Flaws" no plastic resistance(tt -• 0, rc -0). Indeed, as pointed out at From the above commentsand based on physical observathe beginning of this section,the presenceof any kind of flaw, tions reported in the literature, as discussedin section 1, it is even a "rigid" inclusion, would result in a similar out-of-plane clear that there are numerous complex mechanisms which crack growth under axial compression. may be responsiblefor microcrackingin an actual rock speciFigure 6 shows the crack profiles for rc --0, calculated inmen. To produce an analytically manageable model of a crementally, and comparesthesewith experimentally observed microflaw, a drastic idealizationis necessary.The microscopic onesas well as with thosecalculatedapproximately by Nematobservationsseem to suggestthat one of the major mechaNasser and Horii [1982]. It is seen that the present solution nisms of microcrack nucleation and growth is the shear deformethod yields results that compare better with experimental mation due to glide (plasticflow), or slip along the lamellae,or observations. possiblydue to the frictional slidingalong the interfacecavities or cleavagecracks.The simplestreasonablemodel which representssuch mechanismsis a line discontinuity,a "micro- 2.2. Effect of Confinementon Crack Growth flaw," PP' in Figure 4, which can glide in its own plane, The most significantresult which emergedfrom the analyti-

havinga glide or shearresistance of magnitude where re is the cohesiveyield stress,tt is an average friction

cal calculations

is the effect

of lateral

stress on the crack

growth process.Figure 7 shows typical axial load versusexcoefficient, and ayis the (variable)compressive forcetransmit- tended curved crack length relations (incrementallyobtained ted acrossthe flaw; ay is a functionof x. In general,re may using the formulation of Appendix A) for Zc= 0, indicated depend on the strain history of the flaw material, but for lateral tensions(a2/a• < 0), and lateral compressions (a2/a• > simplicity it may be taken to be a constant; this corresponds 0). As is seen,the presenceof slight lateral tension renders the to perfect plasticity. When tt