Analysis of Crack Coalescence in Rock-like Materials

8/7/2019 Analysis of Crack Coalescence in Rock-like Materials

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International Journal of Rock Mechanics & Mining Sciences 38 (2001) 909924

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 strongly

depends on the crack coalescence pattern between pre-existing flaws. However, the crack coalescence pattern of rock specimens

containing three or more flaws has not been studied comprehensively. In this paper, we investigate experimentally crack coalescence

and 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 is

found 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 dominant

modes 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 of

the 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 agree

well. The research reported here provides increased understanding of the fundamental nature of rock failure in uniaxial

compression. r 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction

When a brittle rock is loaded to failure, cracks

nucleate and propagate from pre-existing inhomogene-

ities, which can be in the form of pores, fractures,

inclusions or other defects. Crack initiation and

propagation in solids have been studied since the early

twenties [1,2]. Particular reference to fractures in rocks,

systematic, theoretical and experimental investigations

of crack initiation, propagation and interaction began at

about the middle of the last century and have continued

since [318]. It is recognized that under the compressive

loading, both tensile and shear stress concentrations can

develop at pre-existing inhomogeneities in rock. As the

compression applied to the rock further increases,

tensile cracks will be initiated. In the shear sliding crack

model, this tensile crack is called a wing crack, which

initiates from the tip of pre-existing fracture and grows

progressively parallel to the compression direction. At

the early stages, when the wing crack is short, the

growth is dominated by the stress field around the pre-

existing fracture from which it grows. As the crack

extends, it start to interact with neighbouring micro-

cracks, and this interaction ultimately leads to crack

coalescence and final failure of the sample [16].

Fracture propagation leading to rock failure is a very

important topic in rock mechanics research. A number

of studies have been done on two-dimensional (2-D)

model plates with through going pre-existing fractures

[329] and some of them have been done on 3-D

specimens [3033]. In reality, pre-existing fractures are

3-D in nature. The growth mechanisms of a 3-D crack

may be more complicated. Actually, according to the

observations by Germanovich et al. [3032] and

Germanovich and Dyskin [33], unlike in 2-D samples,

there are intrinsic limits on the growth of a crack in a

3-D model. However, the failure mechanism of rocks

containing 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


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coalescence observed by Bobet and Einstein [26] fall

within the classification of Wong and Chau [23]. Bobet

and Einstein [26] and Bobet [27] investigated the pattern

of crack coalescence under both uniaxial and biaxial

compression. They found that the patterns of crack

coalescence not only depend on the flaw geometry butalso on the stress conditions. Wing cracks initiate at the

flaw tips for uniaxial or low confinement biaxial

conditions, but the location of crack initiation moves

to the middle of the flaw and wing cracks disappear

completely for higher confining stresses. For the relation

between 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 for

wing crack coalescence is normally lower than that for

shear 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 is

exceeded.

Although previous studies provide a general under-

standing of the coalescence pattern between two flaws,

when specimens contain three or more flaws, the crack

interaction between the flaws has not been studiedcomprehensively. This is important because rock con-

tain many flaws. Thus, Wong et al. [28,29] reported very

briefly the results of specimens containing two flaws to

multiple flaws under both uniaxial and biaxial compres-

sion. The number of flaws in the specimens was from 3

to 42. To report the results more comprehensively, we

present in this paper only the results of crack

coalescence and peak stress of rock-like materials

containing three flaws. The research of this paper is of

fundamental importance to understand the mechanism

controlling crack coalescence in the multiple flawed

specimens.

In this study, the flaw angle a; bridge angle b and thefrictional coefficient m are varied under a fixed flaw

length 2c and bridge length 2b, which have been

defined in Fig. 1. Our main interest is to investigate the

dominant factors controlling the failure patterns in

specimens containing three flaws. A further objective of

the present paper is to investigate the failure mechanism

of rock bridges in brittle materials containing multiple

flaws in order to represent fully the failure of intact rock.

The numerical study of the same problem is presented in

Part 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 underground

openings or slopes, and in fracture mechanics involving

multiple flaws. The relevant observations in the first case

are that the collapse of a rock structure containing non-

persistent joints may be preceded by several stages of

crack propagation, interaction and coalescence. Our

investigation should provide the fundamental under-

standing of crack propagation, interaction and coales-

cence in rock under uniaxial compression. With respect

to the contribution to fracture mechanics, the coales-

cence of multiple non-persistent joints is involved in the

fracture of all brittle materials.

2. Experimental studies

In order to have a good comparison between our

present study and the previous study, the mixture of the

modelling material is the same as that used by of Wong

and Chau [23], which is a mixture of barite, sand, plaster

and water with a mass ratio of 2 : 4 : 1 : 1.5. The average

values of unit weight, uniaxial compressive strength,

tensile strength and frictional coefficient of the model-

ling material are gm 17:68 kN/m3, scm 2:09 MPa,

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 0:6 and (b) is the

classification for m 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) 909924 911


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stm 0:35 MPa and mm 0:62; respectively. The aver-age value of the tangent Youngs modulus (Em) at 50%

of peak strength is 0.33 GPa and the Poissons ratio (nm)

is 0.19. The fracture toughness KIC of the modelling

material is 0.0443 MPam1/2. The p factors of dimen-

sional analysis of this artificial material have been found

comparable to the physical ranges of the p factors forsandstone; therefore, Wong and Chau [23] concluded

that the material is appropriate as a sandstone-like

modelling material. The overall dimensions of specimens

containing three flaws are 60mm wide 120 mm

long25 mm thick. To simplify the present analysis,

the bridge length 2b (distance between two flaws) and

the flaw length 2c are fixed at 20 and 12 mm,

respectively.

Flaws were created by inserting steel shims into three

slots in the mould template and removing them during

curing (Fig. 3). Different degrees of the roughness of the

flaw surface are created by applying different numbers

of punch marks to the smooth steel shims (Fig. 4). Thefrictional coefficients on the flaw surfaces are measured

by the titling test on specimens with a through going

flaw. The mean frictional coefficient on flaw surfaces

simulated by inserting plain steel shim is 0.6, while

that simulated by steel shim with punched-indentations

is 0.7.

Two different flaw angles a were used to investigate

the effect of flaw geometry on the pattern of crack

coalescence. The chosen flaw angles are 451 and 651,

where 651 is the preferred orientation for the frictional

flaws (m 0:620:7) of the specimen to slide under

uniaxial compression [23]. The layout of specimenscontaining three flaws is shown in Fig. 5. For sake of

later 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 (between

flaws and ) varies from 751 to 1201 with increments

of 151. Thus, there are two different bridge angles

between the three flaws. As is illustrated in Fig. 2, the

flaw settings of b 451 result in a shear coalescence

pattern, and the other settings of b 751 to 1201 result

in mixed and tensile coalescence modes. Therefore, we

can investigate whether coalescence occurs along therock bridge of b1 (i.e. shear crack coalescence), along

that rock bridge of b2 (i.e. tensile and mixed crack

coalescence), 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 be

examined.

To obtain reliable results in the experiment, the

sample preparation procedures were under well control.

The modelling materials were weighed by using the

electronic weighting balance to a 70.01. Each mixing

procedure was under time control where for the mixture

of barium sulphate and sand it was 4 min. Then cold

water was added evenly and mixed until all particles had

been wetted (4 min). Finally, plaster was added and

mixed evenly until the mixture became a churn-like

paste (7 min). The mixture was then poured into a

mould under vibration (4 min).

To prevent the boundary condition of specimen

affecting the results of experiments, the positions of

three pre-existing flaws were designed as far away as

possible from the side boundaries of specimen. Other-

wise, local failure may be observed instead of crack

coalescence between the pre-existing flaws during the

testing.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.

R.H.C. Wong et al. / International Journal of Rock Mechanics & Mining Sciences 38 (2001) 909924912


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In this study, two specimens with the same parameters

have been cast and tested. If the pattern of crack

coalescence for both specimens is the same, the mean

value of the peak strength is taken. If only one fails by

crack coalescence and the other fails but without crack

coalescence, one more specimen with the same para-

meters was prepared and tested. If both specimens failed

with no crack coalescence, two more specimens with the

same parameters were prepared and tested. If, again, no

crack coalescence was observed for these additional

specimens, the mean peak strength of these specimens is

recorded for comparison purposes only.

The uniaxial compression tests of the specimens were

performed in a Wykeham Farrance WF-5562s loading

machine. This is a load control machine available in our

rock mechanics laboratory. The average loading rate is

about 0.002 kN/s, and it takes about 2530 min to load

one specimen to failure. Two LVDTs were installed in

the front and behind the specimen for measuring the

vertical deformation of the specimen. Only three of

specimens fail suddenly after peak applied stress;

otherwise, the recorded displacement rate is rather

steady up to peak applied stress and even after.

Therefore, there should be no appreciable differencebetween the displacement and load control in our

particular case. Thus, the loading process can be

considered as displacement-controlled approximately.

The setting of the apparatus is shown in Fig. 6 where a

load cell of 5 kN is placed below the lower loading

platen to measure the applied load. To reduce the

friction 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 the

specimen failed, which is identified by the drop in the

applied load. All the loading and displacement records

are transferred to and stored in an IBM PC through a

KYOWA UCAM-5B Data Logger.In all the reported experiments, no local failure was

observed, thus no sample boundaries affect to the results

of our experiment.

3. The coalescence of cracks

3.1. Comparisons of the patterns of crack coalescence

between specimens containing two and three flaws

Wong and Chau [23] concluded that there are three

modes of coalescence in the bridge area, wing tensile,shear and mixed (tensile and shear), for specimens

containing two flaws. To compare the patterns of crack

coalescence between specimens containing two and three

flaws, Figs. 7 and 8 report all the failure patterns

for various values of flaw angle (a 451 and 651),

bridge angle (b 451; 7511201) and frictional co-efficient (m 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 crack

coalescence) are the same as those used in Fig. 6 of

Wong and Chau [23] which is also given in Fig. 9 here.

For the S-type coalescence, crack links between the tip

of two flaws along the direction roughly parallel to the

flaw. For the MI-type coalescence, the growing wing

cracks, which initiated from the two tips of the flaws, are

coalesced by a shear crack in the middle of bridge area.

For the MII-type of coalescence, a growing wing crack

is coalesced by a shear crack that appeared at the other

tip of a flaw. The WI-type of coalescence is a simple

coalescence between two wing cracks. The WII-type of

coalescence is resulted as a growing wing crack coalesces

with the other flaw. The WIII-type of coalescence is a

1

2

Outer Flaw Tip

Outer Flaw Tip

Inner Flaw Tip

Inner Flaw Tip

1

3

2

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. The

bridge angle ofb1 is fixed at 451, while b2 vary from 751 to 1201. The

length of flaw 2c is fixed at 12mm. The bridge distance between the

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

shown.



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growing wing crack joining the outer tip of the other

flaw. From comparisons of Figs. 7 and 8, for the same

a=b it is observed that the type of failure patterns in the3-flaw specimens are the same as those for the 2-flaw

specimens. Coalescence in the 3-flaw specimens can

again be identified as either shear S, wing tensile WI or

mixed (tensile and shear) MI and MII depending on the

values ofa; 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 was

not observed in all specimens with three flaws (Fig. 8).

For 3-flaw specimens with bridge angles ofb1; b2 451;1201, coalescence occurs only for b1 451 but not for

b2 1201 (Fig. 8).

The classification given in Fig. 2 of this study suggests

that the appearance of these modes of coalescence

depends on the values of a; b and m: The patterns ofcrack coalescence for 3-flaw specimens in the a bcspace for m 0:6 and 0.7 were superimposed onto theregime classification given in Fig. 2; and the results are

plotted on Fig. 10. The triangles, rhombuses and

squares in circles are used to denote the data points

for shear, mixed and wing tensile modes observed in the

3-flaw specimens, respectively. Except for one specimen

with a=bc=m 451=1051=0:6 (see Fig. 10a), it is foundthat all of the experimental results for 3-flaw specimens

fall within the same regimes classification of 2-flaw

specimens.

3.2. General observation for 3-flaw specimens

Experimental observations (see Fig. 8) show that

crack coalescence occurred in 14 out of the 16 geometric

settings. There are three possible scenarios in the process

of crack growth. (1) In about 27% of the specimens,

tensile cracks (wing cracks) initiate first at the tips of the

two flaws (either the flaw , or ) followed by wing

crack initiation from a third flaw at a later stage.

However, no matter which wing crack initiates first,

crack coalescence occurs only between two flaws (flaws

and ) at failure. (2) In about 60% of the specimens,

wing cracks initiate from only two flaws (either between

flaws and or flaws and ), with no wing cracknucleating from the third flaw during the whole loading

process and the final coalescence also does not involve

the 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 eye

observation. It is observed that crack initiates first at

either inner tip or outer tip of the flaws, followed by

crack growth at the other tip of the same flaw (see the

definitions of inner and outer tips in Fig. 5). In general,

the growth of cracks at the outer tips is faster than that

observed at the inner tips. The growth rate of each inner

crack is not the same. When an inner crack grows

rapidly, the other inner tip of flaw normally grows much

slower and even seems to stop growing. This is because

of the higher stress concentration around the growing

inner crack tip and causing the crack to grow further.

With a nearby propagating inner crack, a high stress

concentration at the neighbouring inner crack tip will be

affected. A further discussion of stress distribution

within the bridge area will be presented in Part II of

this study [34]. The types of cracking in the bridge area

between the three flaws can be wing tensile, shear, or a

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



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mix of these. Furthermore, it is observed that if the

growth rate of the inner cracks is the same, no

coalescence occurs even when the applied stress drops.

In general, when an inner crack coalesces with the

neighbouring inner crack, the applied stress will

decrease. The test is stopped until the axial stress drops

to 70% of the peak stress.

Fig. 8 illustrates the very important feature that crack

coalescence occurs only between two flaws either

between flaws and or between and , and never

between flaws and . What makes the flaw to

coalesce with the flaw but why not the or the

reverse order? Why crack coalescence does not occur

between 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 0:6 and 0.7. The

notations 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.



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To demonstrate the above rules, the observed data ofthe 3-flaw specimens are summarised in Table 1. The

peak strength results (or coalescence stress, in general,

crack coalescence was observed at about the peak

strength of specimen) for the 2-flaw specimens are

tabulated 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 the

parameter set a=b1; b2=m 451=451; 751=0:6; thereare two possible angles of coalescence bc 451 (coales-

cence between flaw and ) or bc 751 (coalescence

between flaw and ). The coalescence stress for 2-flaw

specimen with a=bc 451=451 is 1.67 MPa, comparedwith 1.59 MPa for specimen with a=bc 451=751 (thiscoalescence stress is smaller than 1.67 MPa). Fig. 11

shows the peak strength and crack coalescence of 2-flaw

specimens of b 451 and 751 and 3-flaw specimen of

b1=b2 451=751 with the same m (0.6) and the same atogether. The mode of coalescence for the 3-flaw

specimens is clearly the same as that for the 2-flaw

specimen with a=bc 451=751 (between flaws andin Fig. 11a). Therefore, Rule 1 applies in this case.

That is, a bc value that corresponds to the smaller

coalescence stress seems to prevail in the process of

crack 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-flaw

specimen with the parameter set a=b1;b2=m 65=45; 75=0:6; the coalescence stress for 2-flaw speci-men with a=bc 651=451 is 1.42 MPa, for specimenwith a=bc 651=751 is 1.45MPa. If Rule 1 is theonly rule for coalescence, the angle of coalescence bc for

data set a=b1;b2=m 65=45; 75=0:6 should be 451(coalescence between flaws and ) instead of 751

(coalescence between flaws and ). However, the

coalescence in 3-flaw specimen is between flaws and

(Table 1 and Fig. 11b). Consequently, Rule 2 is

formulated for crack coalescence for 3-flaw models as:

when the coalescence stress of the two pairs of flaw is

very close (say within 5%), mixed and tensile modes of

crack coalescence always dominate. All data that

comply 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, a

conformity of 86%. If b2 equals to 751 or 901, the

conformity is 100%. Since these rules of crack coales-

cence are rather preliminary based on limited tests,

Shear

Crack

S

Shear

Crack

WingCrack

M I

Shear

Crack

WingCrack

M II

Wing

Crack

W I

WingCrack

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]).



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numerical simulation studies were conducted and will be

presented in Part II of this study [34].

Up to this stage, it cannot be explained why crack

coalescence does not occur between flaws and . The

following section attempts to address this question by

comparison with Nemat-Nasser and Horii [12].

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

It is instructive to compare the observations of this

study to those by Nemat-Nasser and Horii [12], who

used Columbia resin CR39 as the modelling material.

The specimens were 6 mm thick, flaw lengths about

12 mm, flaw widths or openings about 0.4 mm, and each

crack was lined with two 0.2 mm thick brass shims in

order to reduce friction between the two flaw faces. The

flaw distance (bridge length) was 12 mm and a was 451.

The specimens contained two rows of two parallel

collinear flaws with b1 of 451 and b2 of 901 (estimated by

direct measurements on Figs. 17(ac) and 18(ab) of

Nemat-Nasser and Horii [12]). In order to give a clear

discussion and illustration, Fig. 12 reproduces the

experimental observation given in Fig. 18(b) of Nemat-

Nasser and Horii [12]. Three of the flaws are named

similar to the 3-flaw specimens of , and . Under

uniaxial compression, wing cracks initiate and propa-

gate (the solid line) from the tips of the flaws. The wing

cracks from the lower row flaw tips (e.g. flaw )

propagate upward to the upper one (e.g. flaw ), and

those wing cracks from the upper row flaw tips grow

downward to the lower one. However, the specimens

failed by axial splitting rather than localized coalescence

failure. In contrast, coalescence failures were formed in

the specimens for this study (see Fig. 8) under uniaxial

compression for the same values of a and b: Thisdiscrepancy between the present study and that by

Nemat-Nasser and Horii [12] may have resulted from:

(i) their material and the one used in this study are

conducive to different modes of failure even though

both 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.



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(which is actually not given) may be very small

comparing to that of the present study. As illustrated

in Wong and Chau [23], deviation of the orientation of

wing cracks from the line of flaw decreases with increase

of 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 with

the reproduction of Nemat-Nasser and Horii [12]. As

shown in the figure, if a higher value ofm had been used

on the surfaces of the flaws, the path for the growth of

wing cracks would have been more likely to follow a

path linking the flaw tips (between flaws and ,

indicated by the dotted line in Fig. 12). Therefore, it is

clear from the small figure of Fig. 12 that specimens with

a higher value of the frictional coefficient m on the flaw

surfaces are more conducive to wing crack coalescence

compared to cases of small m values (as in the

experiment of Nemat-Nasser & Horii [12]). If zero m

value has been used on the surfaces of the flaws, the path

for the growth of wing cracks would have been more

likely to follow a path linking the flaw tips between

and (indicated by the dotted line in Fig. 12). However,

a higher value ofm had been used in our study, therefore

no crack coalescence is observed between flaws and

in our 3-flaws study. As reviewed from Fig. 8, for those

cracks initiated from flaws and , the growth of inner

tip of flaw propagates towards flaw , while the

growth of outer tip of flaw grows towards the edge of

specimen under uniaxial compression. For the same flaw

arrangement 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 specimens

with 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 decrease

proportionally 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 0:6 m 0:7

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

45 45 45 1.67 45 1.8875 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 0:6 m 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.



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containing multiple flaws (18 and 42 with the specimen

size of 400 mm 400 mm 25 mm [24]). This is also

precisely what was observed for the Hong Kong granite

by Wong and Chau [25], and Yuen Long marble by

Wong et al. [35] that peak strength does not decrease

with initial flaw density once a threshold value of flaw

density is exceeded. We speculate that the peak strength

for a specimen is not proportional to the number of

flaws, and the following hypothesis is thus proposed.

The peak strength for flawed specimens does not depend

on the total number of pre-existing flaws, but only on the

number of flaws actually involved in the formation of the

failure pattern. (Note, the above conclusion is from

specimens with fixed flaw spacing of 20 mm [24], and

from Yuen Long marble with varying flaw spacing from

53 to 106mm [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 the

growth of wing cracks:

KI

s1

ffiffiffiffiffipc

psin 2c m m cos 2c

1 L3=20:23L

1

ffiffiffi3

p1 L1=2

" #

2e0L cos cp

1=2; 1

where s1 is the uniaxial compression, c is the angle

measured from the s1-direction to the direction along

the flaw surface (i.e. c 901 a), 2c is the length of the

pre-existing flaw, L 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 frictional

flaw, and the flaw density e0 is defined as Nc2=A (N is the

number of flaw per area A). Although strictly speaking

(1) is for the case of multiple initial flaws, it was found

that 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.



13/16

two flaws (see Table 2 of [23] which is also given in

Table 2 of present study). Therefore, the peak uni-

axial compressive strength smax1 of a flawed solid can be

estimated as Wong and Chau [23]:

smax1

KICffiffiffiffiffipc

p sin 2c m m cos 2c1 Lcr

3=2

(

0:23Lcr 1ffiffiffi

3p

1 Lcr1=2

" #

2e0Lcr cos c

p

1=2)1; 2

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

coalesced wing cracks, and 2b is the distance between

the two flaws). In this study, the initial flaw density of

specimens containing three flaws is e0 0:015(e0 Nc

2=A note that N 3; A 0:06 m 0.12 m andc 0:006 m). Predictions of the normalized peakstrength (smax1 Opc=KIC) by using Eq. (2) and the

experimental observations are tabulated in Table 3. Todemonstrate the above hypothesis, the peak strength of

the specimens containing 18 and 42 flaws [24] are

tabulated in Table 3 together with the results for the 3-

flaw specimens with the same flaw length 2c, bridge

length 2b, a; b and m: For the specimen containing 18flaws, 3 rows of 6 collinear flaws are placed at the central

region. For the specimen containing 42 flaws, the flaw

arrangement is 9 rows of 4 collinear flaws at the central

region and 2 rows of 3 parallel flaws at the upper and

lower ends of specimen. The density of the multiple

flawed specimens for 18 and 42 flaws are e0 0:016 and0.038, respectively (Table 3). It is found that if initial

flaw density e0 is used in the calculation, the prediction ismuch lower than the experimental observations. The

hypothesis was then tested by using the number of flaws

involved in the formation of the failure pattern in

calculating flaw density ef: For the 3-flaw specimen, ef

0:01 is used, because the observations presented in thisstudy show that final crack coalescence involves only

two flaws but not three. For 18- and 42-flaws specimens,

the number of flaws involved in the failure pattern are 15

and 5. Thus the values of the adjusted crack density are

ef 0:0135 and 0.0045, respectively. It is found that thepredicted peak strength based on flaw density ef agrees

well with the experiments, as shown in Table 3.

5. Conclusions

In this study, experimental results on the mechanism

of crack coalescence and on the peak strength of rock-

like materials containing three flaws under uniaxial

compression loading were presented. The specimens

used in this study are made of a sandstone-like material

and contain three parallel frictional flaws. Various

values of inclination of these flaw angles a; the bridge

angle b and the frictional coefficient m were used in ourparametric studies. For specimens containing three

flaws, it was found that:

* Crack coalescence occurs between only two flaws (not

three).* The mechanisms of crack coalescence depend on the

coalescence stress of the pair of flaws. The lower

value of coalescence stress between the pair of flaws

will dominate the process of coalescence.* Mixed and wing tensile modes of coalescence are

more likely to occur than shear mode, if the

Wing Crack

= 0.90.7

0.6

0.0

Pre-existing

Flaw

From Fig.8 of Wong

and Chau [23]1 in

Possible path for crack

growth if is higher

Possible path for

crack growth ifis zero

2

1

3

2

1

Fig. 12. Shows the effect ofm 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 ofm on the

path of wing crack from Fig. 8 of Wong and Chau [23] is redrawn here

for comparison.



14/16


15/16

coalescence stress between the pair of flaws is very

close (within 5% of each other).* The frictional coefficient m of flaw surface can affect

the pattern of coalescence of the cracked solids.* The uniaxial peak strength for cracked specimens

does not depend on the total number of flaws but

only on the number of flaws actually involved in theformation of the shear zone of the failure pattern.

Our observation provides a better understanding on

the failure behaviour of crack coalescence between three

flaws. In addition, to further examine the reason of

crack coalescence occurring between only two flaws, a

comprehensive numerical study will be presented in

Part II of the paper [34]. In particular, we focus on the

stress distribution within the bridge area.

Acknowledgements

The study was supported by the Research Project No.

A-PA42 of the Hong Kong Polytechnic University to

RHCW. The laboratory assistance by C.Y. Chim is

appreciated.

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Analysis of Crack Coalescence in Rock-like Materials

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