R H C Wong Et Al 2001 - Analysis of Crack Coalescence in Rock-like Materials I - Printed

International Journal of Rock Mechanics & Mining Sciences 38 (2001) 909–924

Analysis of crack coalescence in rock-like materials containingthree flawsFPart I: experimental approach

R.H.C. Wonga,*, K.T. Chaua, C.A. Tangb, P. Linb

aDepartment of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, ChinabCentre for Rock Instability and Seismicity Research, Northeastern University, Shenyang, China

Accepted 6 September 2001

Abstract

Fractures in the forms of joints and microcracks are commonly found in natural rocks, and their failure mechanism stronglydepends on the crack coalescence pattern between pre-existing flaws. However, the crack coalescence pattern of rock specimenscontaining three or more flaws has not been studied comprehensively. In this paper, we investigate experimentally crack coalescenceand peak strength of rock-like materials containing three parallel frictional flaws. Three flaws are arranged such that one pair of

flaws lines collinearly and the third flaw forms either a non-overlapping pattern or an overlapping pattern with the first flaw. It isfound that the mechanisms of crack coalescence depend on the flaw arrangement and the frictional coefficient m on the flaw surface.Two ‘‘rules of failure’’ for the specimens containing three flaws are proposed. Rule No. 1: the pair of flaws with a lower value of

coalescence stress will dominate the process of coalescence. Rule No. 2: mixed and tensile modes of coalescence are always the dominantmodes if the coalescence stress of the two pairs of flaws is very close (say within 5%). In addition, it is found that the peak strength ofthe specimens does not depend on the initial crack density but on the actual number of pre-existing flaws involved in the coalescence.

Comparisons of pattern of crack coalescence with the numerical approach are given in Part II of this study, and the two results agreewell. The research reported here provides increased understanding of the fundamental nature of rock failure in uniaxialcompression. r 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction

When a brittle rock is loaded to failure, cracksnucleate and propagate from pre-existing inhomogene-ities, which can be in the form of pores, fractures,inclusions or other defects. Crack initiation andpropagation in solids have been studied since the earlytwenties [1,2]. Particular reference to fractures in rocks,systematic, theoretical and experimental investigationsof crack initiation, propagation and interaction began atabout the middle of the last century and have continuedsince [3–18]. It is recognized that under the compressiveloading, both tensile and shear stress concentrations candevelop at pre-existing inhomogeneities in rock. As thecompression applied to the rock further increases,tensile cracks will be initiated. In the shear sliding crackmodel, this tensile crack is called a wing crack, which

initiates from the tip of pre-existing fracture and growsprogressively parallel to the compression direction. Atthe early stages, when the wing crack is short, thegrowth is dominated by the stress field around the pre-existing fracture from which it grows. As the crackextends, it start to interact with neighbouring micro-cracks, and this interaction ultimately leads to crackcoalescence and final failure of the sample [16].Fracture propagation leading to rock failure is a very

important topic in rock mechanics research. A numberof studies have been done on two-dimensional (2-D)model plates with through going pre-existing fractures[3–29] and some of them have been done on 3-Dspecimens [30–33]. In reality, pre-existing fractures are3-D in nature. The growth mechanisms of a 3-D crackmay be more complicated. Actually, according to theobservations by Germanovich et al. [30–32] andGermanovich and Dyskin [33], unlike in 2-D samples,there are intrinsic limits on the growth of a crack in a3-D model. However, the failure mechanism of rockscontaining 3-D cracks is out of the scope of the present

*Corresponding author. Tel.: +852-2766-6057; fax: +852-2334-

6389.

E-mail address: [email protected] (R.H.C. Wong).

1365-1609/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved.

PII: S 1 3 6 5 - 1 6 0 9 ( 0 1 ) 0 0 0 6 4 - 8

study. In particular, the fundamental mechanisms ofcrack coalescence of 2-D model have not been fullyinvestigated, and the failure process of flawed rocks isstill not fully understood. Thus, the present study isfocused only on the propagation of 2-D cracks. Thefollowing literatures are reviewed and experimentalstudies are investigated on the 2-D models.To study the failure of brittle rocks, Nemat-Nasser

and Horii [12] and Horii and Nemat-Nasser [14,15]investigated the mechanism of crack interactions and thefinal failure pattern in fractured (flawed) plates made ofColumbia resin CR39 under uniaxial as well as biaxialcompression. Their specimens contain a series of flaws1

of different lengths and orientations (e.g. see Figs. 8,9and 17,18 of reference [14]). They showed that flawlength is one of the parameters controlling the failurepattern of the specimens. In general, larger flaws controlthe mechanism of coalescence in the form of axialsplitting under uniaxial compression with little or nocrack growth from the small flaws. Under biaxialcompression, the growth of larger flaws is followed bythe growth of smaller flaws and the final failure is acoalescence of the smaller flaws in a form of shear zoneor fault. Their studies provide fundamental under-standing of macroscopic failure in relation to the crackdistribution. However, the crack growth and its inter-action between two flaws are not fully understood.Reyes [19] and Reyes and Einstein [20] studied thefailure mechanisms of specimens containing two inclinednon-overlapping open flaws. They found that wingcracks and secondary cracks (which initiate after thewing crack) may occur and eventually lead to coales-cence under uniaxial compression. To incorporate theeffect of crack surface friction, Shen et al. [21] conducteda series of uniaxial compressive tests on gypsumspecimens containing both open and closed fractures.

It was found that the initial geometric setting of theparallel flaw controls the mechanism of crack coales-cence. The patterns of crack coalescence observed aresimilar to those reported in the study by Reyes andEinstein [20]. The failure of flawed solids may occur intensile and/or shear modes, depending on the geometricrelation between the two pre-existing flaws. Althoughsome of the fundamental mechanisms of crack coales-cence have been established, understanding of thecomplete failure process of flawed rocks and thepatterns of crack coalescence is still incomplete.Based upon the experimental work of Reyes and

Einstein [20] and Shen et al. [21], Wong [22], Wong andChau [23–25] reconsidered the problems of crackcoalescence and the strength between two flaws using arock-like material (made of barite, sand, plaster andwater) under uniaxial compression. Three main factorswere varied to investigate the failure patterns: they areflaw angle ‘a’ (inclination of the flaw), bridge angle ‘b’(angle between two flaws) and frictional coefficient ‘m’ ofthe flaw surface, under the conditions of a fixed flawlength ‘2c’ and a fixed distance between flaws ‘2b’ [23].In general, three main modes of crack coalescence wereobserved as shown in Fig. 1. They are the wing tensilemode (crack coalescence involving the growth of wingcracks along the direction parallel to the compression),the shear mode (links between two flaws along thedirection roughly parallel to the flaw), and the mixedmode (shear/tensile). As illustrated in Fig. 2, Wong andChau [23] proposed a classification of patterns for threedifferent failure modes (tensile, shear and mixed), fordifferent combinations of flaw angle a; bridge angle band frictional coefficient m on flaw surface. Triangles,rhombuses and squares were the data points of the 2-flaw specimens for shear, mixed and wing tensile modes,respectively.Actually, for similar flaw geometry under uniaxial

compression (i.e. similar a; b and b=c ¼ 1:6; the ratio offlaw length and bridge length) the patterns of crack

Tensile Crack

Shear Crack

Tensile Crack Shear Crack

(a) (b) (c)

Flaw

2b

2c

Fig. 1. Three main modes of crack coalescence in 2-flaw specimens, where (a) is wing tensile mode, (b) is shear mode, and (c) is mixed mode (tensile

and shear). The flaw length 2c and the bridge distance between the two flaws 2b are fixed with 2c=12mm and 2b=20mm.

1We will refer to the pre-existing fracture as a ‘flaw’, and the

initiated or propagated fracture as a ‘crack’.

R.H.C. Wong et al. / International Journal of Rock Mechanics & Mining Sciences 38 (2001) 909–924910

coalescence observed by Bobet and Einstein [26] fallwithin the classification of Wong and Chau [23]. Bobetand Einstein [26] and Bobet [27] investigated the patternof crack coalescence under both uniaxial and biaxialcompression. They found that the patterns of crackcoalescence not only depend on the flaw geometry butalso on the stress conditions. Wing cracks initiate at theflaw tips for uniaxial or low confinement biaxialconditions, but the location of crack initiation movesto the middle of the flaw and wing cracks disappearcompletely for higher confining stresses. For the relationbetween the strength and the pattern of crack coales-cence of specimens, Wong [22] and Wong and Chau [24]found that the compressive strength of the specimen forwing crack coalescence is normally lower than that forshear crack coalescence. Furthermore, Wong and Chau[25] found that the strength of cracked solids does not

depend linearly on the number of pre-existing flaws(density) once a threshold value of flaw density isexceeded.Although previous studies provide a general under-

standing of the coalescence pattern between two flaws,when specimens contain three or more flaws, the crackinteraction between the flaws has not been studiedcomprehensively. This is important because rock con-tain many flaws. Thus, Wong et al. [28,29] reported verybriefly the results of specimens containing two flaws tomultiple flaws under both uniaxial and biaxial compres-sion. The number of flaws in the specimens was from 3to 42. To report the results more comprehensively, wepresent in this paper only the results of crackcoalescence and peak stress of rock-like materialscontaining three flaws. The research of this paper is offundamental importance to understand the mechanismcontrolling crack coalescence in the multiple flawedspecimens.In this study, the flaw angle a; bridge angle b and the

frictional coefficient m are varied under a fixed flawlength ‘2c’ and bridge length ‘2b’, which have beendefined in Fig. 1. Our main interest is to investigate thedominant factors controlling the failure patterns inspecimens containing three flaws. A further objective ofthe present paper is to investigate the failure mechanismof rock bridges in brittle materials containing multipleflaws in order to represent fully the failure of intact rock.The numerical study of the same problem is presented inPart II [34].There are two general areas where a study of this type

could prove useful: in problems of stability of rock incivil engineering, such as the excavated undergroundopenings or slopes, and in fracture mechanics involvingmultiple flaws. The relevant observations in the first caseare that the collapse of a rock structure containing non-persistent joints may be preceded by several stages ofcrack propagation, interaction and coalescence. Ourinvestigation should provide the fundamental under-standing of crack propagation, interaction and coales-cence in rock under uniaxial compression. With respectto the contribution to fracture mechanics, the coales-cence of multiple non-persistent joints is involved in thefracture of all brittle materials.

2. Experimental studies

In order to have a good comparison between ourpresent study and the previous study, the mixture of themodelling material is the same as that used by of Wongand Chau [23], which is a mixture of barite, sand, plasterand water with a mass ratio of 2 : 4 : 1 : 1.5. The averagevalues of unit weight, uniaxial compressive strength,tensile strength and frictional coefficient of the model-ling material are gm ¼ 17:68 kN/m3, scm ¼ 2:09MPa,

Fig. 2. Classifications of coalescence of a 2-flaw specimen with

different combinations of flaw angle a; bridge angle b and frictionalcoefficient m: (a) is the classification for m ¼ 0:6 and (b) is the

classification for m ¼ 0:7: Triangles, rhombuses and squares were thedata points of the 2-flaw specimens for shear, mixed and wing tensile

modes, respectively. The S-regime is the regime in which the shear

mode of crack coalescence is expected to occur. The M-regime is the

regime in which the mixed shear/tensile mode of crack coalescence is

likely to occur, and the W-regime is the regime in which wing crack

failures are expected (after Wong and Chau [23]).

R.H.C. Wong et al. / International Journal of Rock Mechanics & Mining Sciences 38 (2001) 909–924 911

stm ¼ 0:35MPa and mm ¼ 0:62; respectively. The aver-age value of the tangent Young’s modulus (Em) at 50%of peak strength is 0.33GPa and the Poisson’s ratio (nm)is 0.19. The fracture toughness KIC of the modellingmaterial is 0.0443MPam1/2. The p factors of dimen-sional analysis of this artificial material have been foundcomparable to the physical ranges of the p factors forsandstone; therefore, Wong and Chau [23] concludedthat the material is appropriate as a sandstone-likemodelling material. The overall dimensions of specimenscontaining three flaws are 60mm wide� 120mmlong� 25mm thick. To simplify the present analysis,the bridge length 2b (distance between two flaws) andthe flaw length 2c are fixed at 20 and 12mm,respectively.Flaws were created by inserting steel shims into three

slots in the mould template and removing them duringcuring (Fig. 3). Different degrees of the roughness of theflaw surface are created by applying different numbersof punch marks to the smooth steel shims (Fig. 4). Thefrictional coefficients on the flaw surfaces are measuredby the titling test on specimens with a through goingflaw. The mean frictional coefficient on flaw surfacessimulated by inserting plain steel shim is 0.6, whilethat simulated by steel shim with punched-indentationsis 0.7.Two different flaw angles ‘a’ were used to investigate

the effect of flaw geometry on the pattern of crackcoalescence. The chosen flaw angles are 451 and 651,where 651 is the preferred orientation for the frictionalflaws (m ¼ 0:620:7) of the specimen to slide underuniaxial compression [23]. The layout of specimenscontaining three flaws is shown in Fig. 5. For sake oflater discussions, the three flaws are labelled as , and, respectively.As shown in Fig. 5, there are two bridge angles b1 and

b2 for the three-flaw model. In the experiments, b1(between flaws and ) is fixed at 451, and b2 (betweenflaws and ) varies from 751 to 1201 with increments

of 151. Thus, there are two different bridge anglesbetween the three flaws. As is illustrated in Fig. 2, theflaw settings of b ¼ 451 result in a shear coalescencepattern, and the other settings of b ¼ 751 to 1201 resultin mixed and tensile coalescence modes. Therefore, wecan investigate whether coalescence occurs along therock bridge of b1 (i.e. shear crack coalescence), alongthat rock bridge of b2 (i.e. tensile and mixed crackcoalescence), or along the rock bridge of both b1 and b2:Then the possible relevance of the coalescence in the 2-flaw-specimens to the 3-flaw-specimens can also beexamined.To obtain reliable results in the experiment, the

sample preparation procedures were under well control.The modelling materials were weighed by using theelectronic weighting balance to a 70.01. Each mixingprocedure was under time control where for the mixtureof barium sulphate and sand it was 4min. Then coldwater was added evenly and mixed until all particles hadbeen wetted (4min). Finally, plaster was added andmixed evenly until the mixture became a churn-likepaste (7min). The mixture was then poured into amould under vibration (4min).To prevent the boundary condition of specimen

affecting the results of experiments, the positions ofthree pre-existing flaws were designed as far away aspossible from the side boundaries of specimen. Other-wise, local failure may be observed instead of crackcoalescence between the pre-existing flaws during thetesting.

Fig. 3. Flaws are created in the specimen by inserting stainless steel

shims into three slots in the mould template.

Fig. 4. The stainless steel shims with different roughness used in

creating the flaws in the modelling material. The top and lower shims

give a frictional coefficient of 0.6 and 0.7 on the surfaces of the flaws,

respectively.


In this study, two specimens with the same parametershave been cast and tested. If the pattern of crackcoalescence for both specimens is the same, the meanvalue of the peak strength is taken. If only one fails bycrack coalescence and the other fails but without crackcoalescence, one more specimen with the same para-meters was prepared and tested. If both specimens failedwith no crack coalescence, two more specimens with thesame parameters were prepared and tested. If, again, nocrack coalescence was observed for these additionalspecimens, the mean peak strength of these specimens isrecorded for comparison purposes only.The uniaxial compression tests of the specimens were

performed in a Wykeham Farrance WF-5562s loadingmachine. This is a load control machine available in ourrock mechanics laboratory. The average loading rate isabout 0.002 kN/s, and it takes about 25–30min to load

one specimen to failure. Two LVDTs were installed inthe front and behind the specimen for measuring thevertical deformation of the specimen. Only three ofspecimens fail suddenly after peak applied stress;otherwise, the recorded displacement rate is rathersteady up to peak applied stress and even after.Therefore, there should be no appreciable differencebetween the displacement and load control in ourparticular case. Thus, the loading process can beconsidered as displacement-controlled approximately.The setting of the apparatus is shown in Fig. 6 where aload cell of 5 kN is placed below the lower loadingplaten to measure the applied load. To reduce thefriction between the specimen and the loading platens,two pieces of polythene sheet were inserted. All speci-mens were loaded until either the flaws coalesced or thespecimen failed, which is identified by the drop in theapplied load. All the loading and displacement recordsare transferred to and stored in an IBM PC through aKYOWA UCAM-5B Data Logger.In all the reported experiments, no local failure was

observed, thus no sample boundaries affect to the resultsof our experiment.

3. The coalescence of cracks

3.1. Comparisons of the patterns of crack coalescencebetween specimens containing two and three flaws

Wong and Chau [23] concluded that there are threemodes of coalescence in the bridge area, wing tensile,shear and mixed (tensile and shear), for specimenscontaining two flaws. To compare the patterns of crackcoalescence between specimens containing two and threeflaws, Figs. 7 and 8 report all the failure patternsfor various values of flaw angle (a ¼ 451 and 651),bridge angle (b ¼ 451; 751–1201) and frictional co-efficient (m ¼ 0:6 and 0.7) for 2- and 3-flaw-specimens,respectively.The notations S (shear mode crack coalescence), MI,

MII (mixed shear/tensile mode crack coalescence), WI,WII, WIII and WII/III (wing tensile mode crackcoalescence) are the same as those used in Fig. 6 ofWong and Chau [23] which is also given in Fig. 9 here.For the S-type coalescence, crack links between the tipof two flaws along the direction roughly parallel to theflaw. For the MI-type coalescence, the growing wingcracks, which initiated from the two tips of the flaws, arecoalesced by a shear crack in the middle of bridge area.For the MII-type of coalescence, a growing wing crackis coalesced by a shear crack that appeared at the othertip of a flaw. The WI-type of coalescence is a simplecoalescence between two wing cracks. The WII-type ofcoalescence is resulted as a growing wing crack coalesceswith the other flaw. The WIII-type of coalescence is a

β1

β2

α

Outer Flaw Tip

Outer Flaw Tip

Inner Flaw Tip

Inner Flaw Tip

1

32

Fig. 5. The layout of specimens containing three flaws. The inclina-

tions of the pre-existing flaws a used in this study are 451 and 651. Thebridge angle of b1 is fixed at 451, while b2 vary from 751 to 1201. Thelength of flaw 2c is fixed at 12mm. The bridge distance between the

two flaws 2b is fixed at 20mm. The inner and outer flaw tips are also

shown.


growing wing crack joining the outer tip of the otherflaw. From comparisons of Figs. 7 and 8, for the samea=b it is observed that the type of failure patterns in the3-flaw specimens are the same as those for the 2-flawspecimens. Coalescence in the 3-flaw specimens canagain be identified as either shear S, wing tensile WI ormixed (tensile and shear) MI and MII depending on thevalues of a; m and the coalescence angle bc; defined as thebridge angle along which the crack coalescence occurs(i.e. either b1 or b2). Unlike the studies on 2-flaw models(Fig. 7), coalescence with a bridge angle b of 1201 wasnot observed in all specimens with three flaws (Fig. 8).For 3-flaw specimens with bridge angles of b1; b2 ¼ 451;1201, coalescence occurs only for b1 ¼ 451 but not forb2 ¼ 1201 (Fig. 8).The classification given in Fig. 2 of this study suggests

that the appearance of these modes of coalescencedepends on the values of a; b and m: The patterns ofcrack coalescence for 3-flaw specimens in the a� bcspace for m ¼ 0:6 and 0.7 were superimposed onto theregime classification given in Fig. 2; and the results areplotted on Fig. 10. The triangles, rhombuses andsquares in circles are used to denote the data pointsfor shear, mixed and wing tensile modes observed in the3-flaw specimens, respectively. Except for one specimenwith a=bc=m ¼ 451=1051=0:6 (see Fig. 10a), it is foundthat all of the experimental results for 3-flaw specimensfall within the same regimes classification of 2-flawspecimens.

3.2. General observation for 3-flaw specimens

Experimental observations (see Fig. 8) show thatcrack coalescence occurred in 14 out of the 16 geometric

settings. There are three possible scenarios in the processof crack growth. (1) In about 27% of the specimens,tensile cracks (wing cracks) initiate first at the tips of thetwo flaws (either the flaw , or ) followed by wingcrack initiation from a third flaw at a later stage.However, no matter which wing crack initiates first,crack coalescence occurs only between two flaws (flawsand ) at failure. (2) In about 60% of the specimens,

wing cracks initiate from only two flaws (either betweenflaws and or flaws and ), with no wing cracknucleating from the third flaw during the whole loadingprocess and the final coalescence also does not involvethe third crack. (3) In the remaining 13% of specimens,wing cracks initiate from all three flaws at the same time,but no crack coalescence is observed at failure.The process of coalescence between the growing wing

cracks is normally slow enough to be captured by eyeobservation. It is observed that crack initiates first ateither inner tip or outer tip of the flaws, followed bycrack growth at the other tip of the same flaw (see thedefinitions of inner and outer tips in Fig. 5). In general,the growth of cracks at the outer tips is faster than thatobserved at the inner tips. The growth rate of each innercrack is not the same. When an inner crack growsrapidly, the other inner tip of flaw normally grows muchslower and even seems to stop growing. This is becauseof the higher stress concentration around the growinginner crack tip and causing the crack to grow further.With a nearby propagating inner crack, a high stressconcentration at the neighbouring inner crack tip will beaffected. A further discussion of stress distributionwithin the bridge area will be presented in Part II ofthis study [34]. The types of cracking in the bridge areabetween the three flaws can be wing tensile, shear, or a

Fig. 6. The layout of the loading system with the displacement recording system.


mix of these. Furthermore, it is observed that if thegrowth rate of the inner cracks is the same, nocoalescence occurs even when the applied stress drops.In general, when an inner crack coalesces with theneighbouring inner crack, the applied stress willdecrease. The test is stopped until the axial stress dropsto 70% of the peak stress.

Fig. 8 illustrates the very important feature that crackcoalescence occurs only between two flaws eitherbetween flaws and or between and , and neverbetween flaws and . What makes the flaw tocoalesce with the flaw but why not the or thereverse order? Why crack coalescence does not occurbetween flaws and under uniaxial compression?

Fig. 7. The mode of crack coalescence for specimens containing two flaws. The angles b represent the bridge angle with m ¼ 0:6 and 0.7. Thenotations S (shear mode crack coalescence), MI, MII (mixed mode crack coalescence), WI, WII, WIII and WII/III (wing tensile mode crack

coalescence) are the same as those proposed in Fig. 6 of Wong and Chau [23] or Fig. 9 of this paper.


What is the dominant factor that controls the failurepatterns in the multiple flawed specimens? The followingsection attempts to address these questions by formulat-ing two rules of coalescence for the 3-flaw models.

3.3. Rules of coalescence for solids containing 3 frictionalflaws

By comparing the experimental mode of failureobservations on 3-flaw specimens with those for 2-flaw

specimens, the following rules of coalescence areformulated:

Rule 1. Crack coalescence always occurs betweenthat pair of flaws for which the coalescence stress issmaller.

Rule 2. Mixed and tensile modes of crack coalescenceare always the dominant modes when the values of thecoalescence stress between the pairs of flaws are veryclose (say within 5%).

Fig. 8. The mode of crack coalescence for specimens containing three flaws with 16 combinations of a; (b1; b2) and m: The bridge angles b1 shows theangle between flaws and , b2 shows the angle between flaws and . The notations S, MI, MII and WI are the same as those in Fig. 9.


To demonstrate the above rules, the observed data ofthe 3-flaw specimens are summarised in Table 1. Thepeak strength results (or coalescence stress, in general,crack coalescence was observed at about the peakstrength of specimen) for the 2-flaw specimens aretabulated in Table 1 together with the results for the 3-flaw specimens with the same a; m and bc (coalescenceangle). For example, for the 3-flaw specimens with theparameter set a=ðb1;b2Þ=m ¼ 451=ð451; 751Þ=0:6; thereare two possible angles of coalescence bc ¼ 451 (coales-cence between flaw and ) or bc ¼ 751 (coalescencebetween flaw and ). The coalescence stress for 2-flawspecimen with a=bc ¼ 451=451 is 1.67MPa, comparedwith 1.59MPa for specimen with a=bc ¼ 451=751 (thiscoalescence stress is smaller than 1.67MPa). Fig. 11shows the peak strength and crack coalescence of 2-flawspecimens of b ¼ 451 and 751 and 3-flaw specimen ofb1=b2 ¼ 451=751 with the same m (0.6) and the same atogether. The mode of coalescence for the 3-flawspecimens is clearly the same as that for the 2-flawspecimen with a=bc ¼ 451=751 (between flaws andin Fig. 11a). Therefore, ‘‘Rule 1’’ applies in this case.That is, a bc value that corresponds to the smallercoalescence stress seems to prevail in the process ofcrack coalescence. In the lower part of Table 1, all data

that comply with ‘‘Rule 1’’ are marked with super-script ‘‘1’’.However, some data in Table 1 do not comply with

Rule 1. For example (see Table 1), for the 3-flawspecimen with the parameter set a=ðb1; b2Þ=m ¼65=ð45; 75Þ=0:6; the coalescence stress for 2-flaw speci-men with a=bc ¼ 651=451 is 1.42MPa, for specimenwith a=bc ¼ 651=751 is 1.45MPa. If ‘‘Rule 1’’ is theonly rule for coalescence, the angle of coalescence bc fordata set a=ðb1;b2Þ=m ¼ 65=ð45; 75Þ=0:6 should be 451(coalescence between flaws and ) instead of 751(coalescence between flaws and ). However, thecoalescence in 3-flaw specimen is between flaws and(Table 1 and Fig. 11b). Consequently, ‘‘Rule 2’’ isformulated for crack coalescence for 3-flaw models as:‘‘when the coalescence stress of the two pairs of flaw isvery close (say within 5%), mixed and tensile modes ofcrack coalescence always dominate’’. All data thatcomply with ‘‘Rule 2’’ are indicated by the superscript‘‘2’’ in Table 1.Table 1 shows that 12 of the 14 coalescence sets of 3-

flaw data conform to these rules of coalescence, aconformity of 86%. If b2 equals to 751 or 901, theconformity is 100%. Since these rules of crack coales-cence are rather preliminary based on limited tests,

Shear Crack

S

Shear Crack

Wing Crack

M I

Shear Crack

Wing Crack

M II

Wing Crack

W I

Wing Crack

W II

Wing Crack

W III

Fig. 9. Six different patterns of crack coalescence observed in the 2-flaw specimens. The notations S, M and W indicate the shear, mixed (shear/

tensile) and wing tensile mode crack coalescence, respectively (after Wong and Chau [23]).


numerical simulation studies were conducted and will bepresented in Part II of this study [34].Up to this stage, it cannot be explained why crack

coalescence does not occur between flaws and . Thefollowing section attempts to address this question bycomparison with Nemat-Nasser and Horii [12].

3.4. Comparison with Nemat-Nasser and Horii (1982)

It is instructive to compare the observations of thisstudy to those by Nemat-Nasser and Horii [12], whoused Columbia resin CR39 as the modelling material.The specimens were 6mm thick, flaw lengths about12mm, flaw widths or openings about 0.4mm, and eachcrack was lined with two 0.2mm thick brass shims inorder to reduce friction between the two flaw faces. Theflaw distance (bridge length) was 12mm and a was 451.The specimens contained two rows of two parallelcollinear flaws with b1 of 451 and b2 of 901 (estimated bydirect measurements on Figs. 17(a–c) and 18(a–b) of

Nemat-Nasser and Horii [12]). In order to give a cleardiscussion and illustration, Fig. 12 reproduces theexperimental observation given in Fig. 18(b) of Nemat-Nasser and Horii [12]. Three of the flaws are namedsimilar to the 3-flaw specimens of , and . Underuniaxial compression, wing cracks initiate and propa-gate (the solid line) from the tips of the flaws. The wingcracks from the lower row flaw tips (e.g. flaw )propagate upward to the upper one (e.g. flaw ), andthose wing cracks from the upper row flaw tips growdownward to the lower one. However, the specimensfailed by axial splitting rather than localized coalescencefailure. In contrast, coalescence failures were formed inthe specimens for this study (see Fig. 8) under uniaxialcompression for the same values of a and b: Thisdiscrepancy between the present study and that byNemat-Nasser and Horii [12] may have resulted from:(i) their material and the one used in this study areconducive to different modes of failure even thoughboth are brittle; and (ii) their frictional coefficient m

Fig. 10. (a,b) Modes of crack coalescence for specimens containing three flaws superimposed onto the classifications given in Fig. 2. Symbols ,

and indicate the shear, mixed shear/tensile and wing tensile modes of coalescence observed in the 3-flaw specimens, respectively.


(which is actually not given) may be very smallcomparing to that of the present study. As illustratedin Wong and Chau [23], deviation of the orientation ofwing cracks from the line of flaw decreases with increaseof m: To further illustrate the second possible reason,Fig. 8 from Wong and Chau [23] is redrawn in Fig. 12(the small figure at the left lower corner) together withthe reproduction of Nemat-Nasser and Horii [12]. Asshown in the figure, if a higher value of m had been usedon the surfaces of the flaws, the path for the growth ofwing cracks would have been more likely to follow apath linking the flaw tips (between flaws and ,indicated by the dotted line in Fig. 12). Therefore, it isclear from the small figure of Fig. 12 that specimens witha higher value of the frictional coefficient m on the flawsurfaces are more conducive to wing crack coalescencecompared to cases of small m values (as in theexperiment of Nemat-Nasser & Horii [12]). If zero mvalue has been used on the surfaces of the flaws, the pathfor the growth of wing cracks would have been morelikely to follow a path linking the flaw tips betweenand (indicated by the dotted line in Fig. 12). However,

a higher value of m had been used in our study, thereforeno crack coalescence is observed between flaws andin our 3-flaws study. As reviewed from Fig. 8, for thosecracks initiated from flaws and , the growth of innertip of flaw propagates towards flaw , while thegrowth of outer tip of flaw grows towards the edge ofspecimen under uniaxial compression. For the same flawarrangement under a biaxial compression [28], second-ary crack can initiate at the outer tip of flaw ,propagate towards flaw and coalesce. In this case,failure involves three flaws.

4. Peak strength of flawed specimens

Table 1 shows that the peak strength for specimenswith the same a; b and m are basically the same,regardless of whether they contain two or three flaws.In other words, peak strength appears not to decreaseproportionally with the initial flaw density.These observations not only appear in 3-flaw specimens,but also were observed in the modelling specimens

Table 1

A comparison of the experimental peak strength for specimens containing two flaws and three flaws

Two flaws

a (1) b1 (1) m ¼ 0:6 m ¼ 0:7

bc (1) Peak strength (MPa) bc (1) Peak strength (MPa)

45 45 45 1.67 45 1.88

75 75 1.59 75 1.80

90 90 1.59 90 1.57

105 105 1.88 105 1.84

120 120 1.64 120 1.73

65 45 45 1.42 45 1.44

75 75 1.45 75 1.49

90 90 1.49 90 1.48

105 105 1.51 105 1.52

120 Nob 1.46 120 1.48

Three flaws

a (1) b1; b2 m ¼ 0:6 m ¼ 0:7

bc (1) Peak strength (MPa) bc (1) Peak strength (MPa)

45 45, 75 751 1.59 751 1.73

45, 90 901 1.57 901 1.73

45, 105 105a 1.60 1051 1.61

45, 120 Nob 1.69 45a 1.70

65 45, 75 752 1.43 752 1.58

45, 90 902 1.42 902 1.52

45, 105 451 1.51 451 1.62

45, 120 451 1.51 Nob 1.58

aThese results do not comply with the ‘‘rule of failure’’.bNo crack coalescence occurs at failure.

751 Data comply with Rule 1.

902 Data comply with Rule 2.


containing multiple flaws (18 and 42 with the specimensize of 400mm� 400mm� 25mm [24]). This is alsoprecisely what was observed for the Hong Kong graniteby Wong and Chau [25], and Yuen Long marble byWong et al. [35] that peak strength does not decreasewith initial flaw density once a threshold value of flawdensity is exceeded. We speculate that the peak strengthfor a specimen is not proportional to the number offlaws, and the following hypothesis is thus proposed.The peak strength for flawed specimens does not dependon the total number of pre-existing flaws, but only on thenumber of flaws actually involved in the formation of thefailure pattern. (Note, the above conclusion is fromspecimens with fixed flaw spacing of 20mm [24], andfrom Yuen Long marble with varying flaw spacing from53 to 106 mm [35]).To examine this hypothesis, the model by Ashby and

Hallam [16] is employed. Ashby and Hallam [16] derived

the following total stress intensity factor KI for thegrowth of wing cracks:

KI

s1ffiffiffiffiffipc

p ¼ðsin 2c� mþ m cos 2cÞ

ð1þ LÞ3=20:23Lþ

1ffiffiffi3

pð1þ LÞ1=2

" #

þ2e0ðLþ cos cÞ

p

� �1=2; ð1Þ

where s1 is the uniaxial compression, c is the anglemeasured from the s1-direction to the direction alongthe flaw surface (i.e. c ¼ 901� a), 2c is the length of thepre-existing flaw, L ¼ c=c is the normalized length of thewing cracks (c is the length of the growing wing crack),m is the frictional coefficient along the shear or frictionalflaw, and the flaw density e0 is defined as Nc2=A (N is thenumber of flaw per area A). Although strictly speaking(1) is for the case of multiple initial flaws, it was foundthat it can also be applied to the specimen containing

Fig. 11. (a,b) The mode of crack coalescence and the peak stress of specimens containing two flaws and three flaws with the same a; b and mpresented here for discussion on the two rules of coalescence for solids containing three flaws.


two flaws (see Table 2 of [23] which is also given inTable 2 of present study). Therefore, the peak uni-axial compressive strength smax1 of a flawed solid can beestimated as Wong and Chau [23]:

smax1

¼KICffiffiffiffiffipc

p ½sin 2c� mþ m cos 2c�

ð1þ LcrÞ3=2

(

� 0:23Lcr þ1ffiffiffi

3p

ð1þ LcrÞ1=2

" #

þ2e0ðLcr þ cos cÞ

p

� �1=2)�1

; ð2Þ

where KIC is the fracture toughness (0.0443MPaOm forour modelling material), Lcr ¼ cmax=cðcmax ¼ 2b sin b isthe maximum possible value for the length of the

coalesced wing cracks, and 2b is the distance betweenthe two flaws). In this study, the initial flaw density ofspecimens containing three flaws is e0 ¼ 0:015(e0 ¼ Nc2=A note that N ¼ 3; A ¼ 0:06m� 0.12m andc ¼ 0:006m). Predictions of the normalized peakstrength (ðsmax1 OðpcÞ=KICÞ) by using Eq. (2) and theexperimental observations are tabulated in Table 3. Todemonstrate the above hypothesis, the peak strength ofthe specimens containing 18 and 42 flaws [24] aretabulated in Table 3 together with the results for the 3-flaw specimens with the same flaw length 2c, bridgelength 2b, a; b and m: For the specimen containing 18flaws, 3 rows of 6 collinear flaws are placed at the centralregion. For the specimen containing 42 flaws, the flawarrangement is 9 rows of 4 collinear flaws at the centralregion and 2 rows of 3 parallel flaws at the upper andlower ends of specimen. The density of the multipleflawed specimens for 18 and 42 flaws are e0 ¼ 0:016 and0.038, respectively (Table 3). It is found that if initialflaw density e0 is used in the calculation, the prediction ismuch lower than the experimental observations. Thehypothesis was then tested by using the number of flawsinvolved in the formation of the failure pattern incalculating flaw density ef : For the 3-flaw specimen, ef ¼0:01 is used, because the observations presented in thisstudy show that final crack coalescence involves onlytwo flaws but not three. For 18- and 42-flaws specimens,the number of flaws involved in the failure pattern are 15and 5. Thus the values of the adjusted crack density areef ¼ 0:0135 and 0.0045, respectively. It is found that thepredicted peak strength based on flaw density ef agreeswell with the experiments, as shown in Table 3.

5. Conclusions

In this study, experimental results on the mechanismof crack coalescence and on the peak strength of rock-like materials containing three flaws under uniaxialcompression loading were presented. The specimensused in this study are made of a sandstone-like materialand contain three parallel frictional flaws. Variousvalues of inclination of these flaw angles a; the bridgeangle b and the frictional coefficient m were used in ourparametric studies. For specimens containing threeflaws, it was found that:

* Crack coalescence occurs between only two flaws (notthree).

* The mechanisms of crack coalescence depend on thecoalescence stress of the pair of flaws. The lowervalue of coalescence stress between the pair of flawswill dominate the process of coalescence.

* Mixed and wing tensile modes of coalescence aremore likely to occur than shear mode, if the

Wing Crack µ = 0.9 0.7 0.6 0.0

Pre-existing Flaw

From Fig.8 of Wong and Chau [23]

1 in

Possible path for crack growth if µ is higher

Possible path for crack growth if µ is zero

β2

β1

3

2

1

Fig. 12. Shows the effect of m on the path of wing crack propagation.The specimen with two parallel rows of collinear pre-existing cracks is

adopted from Fig. 18 of Nemat-Nasser and Horii [12]. For a higher m;the possible path for crack growth may occur between flaws and .

For a lower mE0; the possible path for crack growth may appearbetween flaws and . The small figure showing the effect of m on thepath of wing crack from Fig. 8 of Wong and Chau [23] is redrawn here

for comparison.


Table 2

The experimental and theoretical results of the peak strength of specimens containing two flaws (Wong and Chau [23])

a (1) b (1) Normalized peak stress s1OðpcÞ=KICðm ¼ 0:6Þ Normalized peak stress s1OðpcÞ=KIC ðm ¼ 0:7Þ

Experimental Theoretical Experimental Theoretical

45 45 5.17 5.19 5.84 5.53

45 75 4.95 4.93 5.67 5.19

45 90 4.94 4.89 4.85a 5.14

45 105 5.81 4.93 5.69 5.19

45 120 5.07 5.03 5.37 5.32

65 45 4.41 4.54 4.47 4.63

65 75 4.48 4.42 4.61 4.49

65 90 4.62 4.40 4.59 4.47

65 105 4.68 4.42 4.71 4.49

65 120 4.53 4.48 4.58 4.55

aSpecimens show partial-surface-contact along the pre-existing flaws.

Table 3

Experimental and theoretical results for the peak strength of specimens containing three flaws and multiple flaws

3-flaw specimen

a (1) b1; b2 (1) Coalescence

angle bc (1)Normalized peak strength (s1OðpcÞ=KIC) (m ¼ 0:6Þ

Experimental Theoretical

Three

flaws e0 ¼ 0:015Two flaws

ef ¼ 0:01

45 45, 75 75 4.92 4.19 4.93

45, 90 90 4.85 4.16 4.89

45, 105 105 4.96 4.19 4.93

45, 120 No 5.22 F F65 45, 75 75 4.44 3.80 4.42

45, 90 90 4.40 3.78 4.40

45, 105 45 4.67 3.96 4.54

45, 120 45 4.69 3.96 4.54

a (1) b1; b2 (1) bc (1) Normalized peak strength ðs1OðpcÞ=KICÞ (m ¼ 0:7Þ

Experimental Theoretical

Three flaws

e0 ¼ 0:015Two flaws

ef ¼ 0:01

45 45, 75 75 5.36 4.38 5.19

45, 90 90 5.36 4.34 5.14

45, 105 105 5.00 4.38 5.18

45, 120 45 5.27 4.71 5.53

65 45, 75 75 4.91 3.86 4.49

45, 90 90 4.72 3.84 4.46

45, 105 45 5.02 4.03 4.63

45, 120 No 4.91 F F

Multiple flawed specimen (Wong and Chau [24])

a (1) Initial flaws Coalescence

angle bc (1)Flaws involved in failure pattern Normalized peak strength

ðs1OðpcÞ=KICÞ (m ¼ 0:6Þ

Number e0 Number ef Experimental Theoretical

e0 ef

45 18 0.016 75 15 0.0135 4.39 4.06 4.38

45 42 0.038 75 5 0.0045 6.72 2.84 6.63


coalescence stress between the pair of flaws is veryclose (within 5% of each other).

* The frictional coefficient m of flaw surface can affectthe pattern of coalescence of the cracked solids.

* The uniaxial peak strength for cracked specimensdoes not depend on the total number of flaws butonly on the number of flaws actually involved in theformation of the shear zone of the failure pattern.

Our observation provides a better understanding onthe failure behaviour of crack coalescence between threeflaws. In addition, to further examine the reason ofcrack coalescence occurring between only two flaws, acomprehensive numerical study will be presented inPart II of the paper [34]. In particular, we focus on thestress distribution within the bridge area.

Acknowledgements

The study was supported by the Research Project No.A-PA42 of the Hong Kong Polytechnic University toRHCW. The laboratory assistance by C.Y. Chim isappreciated.

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