Splitting failure in brittle rocks containing pre-existingflaws under uniaxial compression

R.H.C. Wong a,*, C.A. Tang b, K.T. Chau a, P. Lin b

a Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, Chinab Centre for Rock Instability Seismicity Research, Northeastern University, Shenyang, China

Received 22 March 2002; accepted 16 April 2002

Abstract

Splitting failure of rock specimen containing pre-existing crack-like flaws under compression is numerically inves-

tigated using Rock Failure Process Analysis (RFPA2D). Crack growth from single, triple and multi-crack-like flaws

contained in numerical specimens are studied. The analysis of parameters, such as angle and length of the flaws,

specimen width and the arrangement of flaw locations, is conducted to examine its influence on the growth and co-

alescence behaviour. Flaw length, flaw location and stress interaction between the nearby flaws are found to be im-

portant factors affecting the behaviour of crack initiation, propagation and coalescence.

� 2002 Published by Elsevier Science Ltd.

Keywords: Crack; Flaw; Splitting; Coalescence; Numerical simulation

1. Introduction

A common mode of failure of rock specimens under uniaxial compression is splitting where the frac-ture surface is approximately parallel to the direction of applied loading. Failure of splitting is generallyobserved in solid cylindrical specimens and has been reported in many different rock types, includinggranitic rocks [1,2], marbles [3,4], and quartzites [5]. Holzhausen and Johnson [6] indicated that there wereseveral possible mechanisms to cause splitting failure. The interaction of growing cracks with a free sur-faces might be one of the possible mechanisms to produce tensile splitting failure. This type of splittingfailure normally involves a sequence of progressive microfracturing. The microcracking results from ahigh tensile stress concentrated at the inhomogeneities, such as crack-like flaws, pore-like flaws or softinclusions, under uniaxial compression. These initiated microcracks will propagate with increased load-ing, and eventually extend to the specimen surface to form a macroscopic splitting failure. The topic of

Engineering Fracture Mechanics 69 (2002) 1853–1871

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*Corresponding author. Tel.: +852-2766-6057; fax: +852-2334-6389.

E-mail address: cerwong@polyu.edu.hk (R.H.C. Wong).

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splitting failure has been widely researched in fields, such as material science, geophysics and rock me-chanics [5–15].In this paper, we analyse the growth of axial cracks from pre-existing crack-like flaws specimen. A

numerical tool, Rock Failure Process Analysis (RFPA2D) code [16,17], has been used to simulate the failureprocess of specimens containing crack-like flaws.Although a number of experimental results of specimens containing pre-existing flaw under uniaxial and

biaxial compression have been published, e.g. [11–15,18–20], there is no existing study investigates sys-tematically on the effect of progressive failure in brittle solids with the changing of the size of specimen, theflaw length, the inclination of flaw, the position of array flaws and multi-flaws with random distribution.Nemat-Nasser and Horii [11] and Horii and Nemat-Nasser [12,13] studied the mechanism of crack co-alescence in specimens made of Columbia resin CR39 under uniaxial, as well as biaxial, compression. Theirspecimens contain a series of flaws which consist of different flaw lengths and orientations. In general, largeflaws control the mechanism of coalescence in a form of axial splitting under uniaxial compression (with noor little crack growth in the small flaws), while under biaxial compression the growth of large cracks arefollowed by growth of smaller cracks and the final failure is a coalescence of the smaller cracks in a form ofshear zone or fault. Ashby and Hallam [14] used both experimental and analytical approaches to study thecrack problem. In their experiments they tested PMMA plate specimens, 10 mm thick, 170 mm high and25–100 mm wide, containing single crack-like flaw with flaw length 2a of 16 mm, inclined at angles w of20�, 30�, 45�, and 60� to the compression axis. They also studied the crack growth in an array of flaws(three flaws with diagonal arrangement). The experimental results showed that cracks initiated from theflaw and interacted with the surfaces of the finite specimen in a way that caused them to grow faster thanthey did in an infinite medium. For the crack study in multi-flaws, they used the observations from theexperimental study of multi-holes [15] and developed a simple analysis based on beam theory to describethis interaction in their work [14]. Wong and Chau [18] and Bobet and Einstein [19,20] conducted researchon crack initiation, propagation and coalescence of flaws in synthetic rock containing two through-goingflaws under uniaxial compression. The size of specimen is fixed with the changing angle of flaw, the po-sition of flaws (from overlapping to non-overlapping) and the frictional coefficient of the flaw. Foroverlapping flaws, linkage of the tensile cracks appears to be leading the coalescence mechanism. For non-overlapping flaws, coalescence through both shear and mixed (tensile and shear) cracks have been ob-served. Comparing to the pervious studies, Ashby and Hallam [14] is a more systematic study on crackgrowth problem. Although the analytical and experimental work of Ashby and Hallam [14] is consideredpromising, much more work is required to further develop reliable constitutive laws for compressivefracture. However, due to the complicated behaviour of brittle rocks under compression, it is difficult to seeat least at present that analytical methods can answer most of the important questions raised by theobservations. The application of RFPA2D code [16,17] shows some promises on genuinely modelling thefailure behaviour of actual rock in a more realistic way. Actually, the RFPA2D code has been used anddemonstrated that it can realistically capture the spatial evolution of damage and shear localization in abrittle solid [21–29].The particular problem studied here is similar to Ashby and Hallam [14] with the changing of the

size of specimen, the flaw length, the inclination of flaw. In addition, crack growth study in threearray flaws with the changing position of diagonal, vertical and horizontal, and the crack growth studyin multi-flaws with random distribution will also be investigated. The paper will summarise the nu-merical analysis on the growth of wing cracks from pre-existing flaws in specimens subjected to an uni-axial compression. The flaw lies at an angle w to the axial loading direction. The initiation and thegrowth of wing cracks from a single flaw are considered first; then the interaction between the grow-ing wing cracks from three flaws is simulated; finally, the more complex multi-flaws are studied. In allthe cases, plane strain is used and all the specimens are subjected to displacement controlled axial com-pression.

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2. Outline of Rock Failure Process Analysis

Before the numerical simulations are discussed, the essential features of RFPA2D code are summarised inthis section. The RFPA2D is a progressive elastic damage model, it can be used to simulate the deformation,stress distribution and failure induced stress redistribution, fracture initiation and fracture propagation inheterogeneous materials. The consideration of heterogeneity for the elements is achieved by assigning theelements random strength and elastic modulus by assuming a Weibull’s distribution, u (m; a0). Where theparameter m is defined as homogeneity index, which controls the shape of the distribution function relatingto the degree of material heterogeneity; a0 is the parameter related to the mean value of the material pa-rameters for the elements, such as strength and elastic modulus [16].The damage in the model is simulated as the reduction of material elastic properties. In order to quan-

titatively describe the damage accumulation, the code assumes that the Young’s modulus of the damagedmaterial E0 can be estimated by reducing the Young’s modulus of the initially undamaged solid E as

E0 ¼ ð1� DÞE ð1Þ

where D ¼ 1� r=ðEeÞ is the damage parameter, depending on the loading history and determined by thefailure criterion. The Poisson’s ratio m for all the elements is assumed to be constant in this paper.The elastic simulated material, defined by E and m, is discretized using a finite element method to cal-

culate the stress state in each element due to the loading. It is assumed that each element corresponds to amesoscopic scale, i.e. the defect size is small compare with the element size. Although fracture mechanicsplays an important role in the analysis of crack propagation, the assumption of homogeneity adopted infracture mechanics limits its application in heterogeneous materials. Instead of using a fracture mechanicsapproach where fracture propagation is related to a stress intensity factor at the advancing crack tips and iscontrolled by the fracture toughness, a failure approach is adopted in the code, RFPA2D, where micro-fracturing occurs when the stress of an element satisfies a certain strength criterion [16]. In this investi-gation, a Coulomb criterion envelope with a tensile cut-off [30] is used so that the elements may fail either inshear or in tension.The simulation proceeds as follows:

• An external displacement with a constant rate is applied in the vertical direction as shown in Fig. 1 toproduce a compression loading to the sample. No end friction is induced between the specimen and load-ing platen.

• The stress and deformation distribution in the sample is calculated in steps.• The external displacement in vertical direction is then increased gradually.

At steps that the stress in some elements satisfies the strength criterion, these elements start to fail either inshear or in tension based on the failure criterion and become weak elements [16]. The stress and deformationdistribution throughout the sample is then adjusted instantaneously after each element rupture until thewhole sample reaches the equilibrium state. At positions where stress increases due to stress redistribu-tion around a damaged element, the stress may reach the critical value and further ruptures are caused.The process is repeated until no failure elements are present. Then, the next step of external displacement isapplied.

3. Modelling of crack growth from an angled crack-like flaw in a brittle specimen under compression

The numerical specimens for studying the growth of wing cracks from a pre-existing crack-like flaw are170 mm high (i.e. H ¼ 170 mm in Fig. 1). Three parameters are considered in this study for investigating

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the crack growth behaviour: (1) flaw length, (2) flaw angle, and (3) specimen width. Since most brittle solidsare rather heterogeneous materials, the Weibull’s distributions, u1 (5, 200 MPa) and u2 (5, 60 GPa), areused in this paper to represent the strength and Young’s modulus distributions within the specimens, re-spectively. The frictional coefficient of the numerical specimen used in this study is 0.2.

3.1. Wing crack growth from a single flaw with different length

Pre-existing single flaw with length 2a of 10, 20 and 30 mm is numerically simulated to examine theinfluence of flaw length on the growth of wing crack. The specimens are 50 mm wide, with flaw angle winclining 45� to the axial compression direction. The numerical specimens of 170 mm high and 50 mm wideand the corresponding element number is 255� 75 ¼ 19125. The sequence of wing crack growth for thesethree specimens are shown in Fig. 2a. Fig. 2b shows the stress distribution in the specimens obtained in thesame simulations where the light colour shows the maximum stress. Due to the heterogeneity of thespecimens, the crack path is not smooth comparing the experimental results of Fig. 3 of Ashby and Hallam[14]. Actually it is true that the actual crack propagation in real rock materials the crack path is not smooth[31]. The influence of flaw length on the initiation and propagation of wing crack is plotted in Fig. 3 by

Fig. 1. Schematic of numerical specimen.

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Fig. 3. The influence of the varied initial flaw length (10, 20 and 30 mm) on the growth of wing crack under uniaxial compression

showing (RFPA2D simulation).

Fig. 2. (a) Sequences of the growth wing crack under uniaxial compression for specimens containing crack-like flaw at length of 10, 20

and 30 mm (RFPA2D simulation). (b) Stress distribution of the growth of wing crack under axial load for specimens containing crack-

like flaw at length 10, 20 and 30 mm (RFPA2D simulation).

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using the normalized length of the wing cracks (L/a) versus the applied axial stress r. It is shown that wingcrack is easier to nucleate from the larger flaw than from the smaller flaw. The overall crack growth rate(length of the wing crack ‘‘L’’ per unit stress) from the longer flaw is faster (about 2 times) than that fromthe shorter-flaw-specimen. The crack growth rate in flaw length of 2a ¼ 10, 20 and 30 mm are 1.8, 3.6 and10.4 mm/MPa, respectively.Fig. 4 shows the ultimate failure mode of the specimen. For specimen containing shorter flaw (2a ¼ 10

mm), the failure is in splitting mode. For specimen containing longer flaw (2aP 20 mm), it is observed thatseveral cracks initiated in the form of a shear zone at peak stress level. The cracks grew unstably towards tothe tip of flaw and the free surface of specimen (Fig. 4). It is worthwhile to note that this result for thelonger flaw may be different with the prediction from the analytical solution of fracture mechanics in whichthe specimen is considered to be infinite in size and is supposed to be loaded by a remote stress. For ourcase, a finite specimen size is used in our numerical simulations. Comparing to the length of flaw, the size oflarger flaw to the size of specimen is about 0.4–0.6. As observed from Fig. 4, it is found that when the cracksapproached the upper and bottom surfaces of the specimen, the propagation of crack slowed down orsometimes stopped. Based on the above observations from the numerical simulations, two conclusionscan be drawn. First, for the same size of specimen cracks nucleate more easily and grow faster from thetips of longer flaw than that for a shorter flaw. This conclusion can be applied on a specimen contain-ing different length of flaws. For example, Horii and Nemat-Nasser [12] observed that for a specimencontaining different length of flaws where the longer flaw is more conducive to wing crack initiation andpropagation than the shorter flaw. Second, boundary effect is one of the factors to influence the mode offailure and growth of crack. For distance between flaw tip and free boundary of the specimen less thantwo times of flaw length, shear failure may occur if shear stress around the flaw tips is high. This mode offailure is observed when the lateral boundary restraint free and the end boundary is friction free. If theend friction exists under the applied loading of the specimen, splitting failure may occur instead of the shearfailure.

Fig. 4. Ultimate failure modes for specimens containing pre-existing crack-like flaw with the length of 10, 20 and 30 mm (RFPA2D

simulation).

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Fig. 5. Stress distribution of the growth of wing crack uniaxial compression for specimens containing crack-like flaw at angle of 25�,45� and 60� (RFPA2D simulation).

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3.2. Wing crack growth from a single flaw with different angles

Fig. 5 shows the numerical results of the stress distributions of the growth of wing cracks from the pre-existing flaw at angles w of 25�, 45� and 60� under gradually increased axial compression. Light colourshows the maximum stress. The specimens are 50 mm wide with flaw length 2a of 20 mm. Fig. 6 shows theinfluence of the angles w of the pre-existing flaw on the crack initiation and propagation behaviour. It isfound that for flaws with orientation near 60� to the axial loading direction wing crack propagation is easierto occur. The crack initiated from the flaw inclining at 25� is the most difficult to propagate. According tothe analytical analysis of Ashby and Hallam [14], the most favourable flaw angle for crack nucleation isequal to w ¼ 1=2 tan�1ð1=lÞ, where l is the frictional coefficient of flaw surface. Ashby and Hallam [14]inserted two brass shims into the slot to control the friction between flaw surface. The frictional coefficientof l in their study is about 0.25 (by back calculation from the experiments). Comparing to the experimentalstudy of Wong and Chau [18], the frictional coefficients l are in the range of 0.6–0.9 (the calculated flawangle of w is at about 24–29.5�). Their experiments showed that the most favourable flaw angles w for cracknucleation were in the range of 25–30� from axial load. Therefore, it seems that the favourable flaw anglefor crack nucleation depends on the frictional coefficient l of flaw surface. In this study, the numericalsimulation of the l of the specimen is about 0.2 with one element width of the opening of the flaw (about0.67 mm). According to the crack nucleation easier at the angle of 60�, the frictional coefficient between theflaw may be very low or even zero (open crack). A further experimental study should be carried out toclarify what is the favourable angle for crack nucleation in the low frictional coefficient of the flaw surface.Fig. 7 shows the ultimate failure modes of three specimens. It is found that only the specimen con-

sistingof flaw inclined at 25� fails in splitting mode. The other two failed in a combined mode ofsplitting and shear ultimately. The reason for this failure is more or less the same as the discussion inSection 3.1 where the distance between the flaw tip and the side boundary is less than two times of flawlength.

3.3. Wing crack growth from a single flaw with different specimen width

Figs. 8 and 9 show the numerical result of the influence of the specimen width (25, 50, and 100 mm) onthe growth of the wing crack. The specimens contain a flaw inclined at 45� with the length 2a of 20 mm. Thenumerical simulations shown in Fig. 8 reveal that the wing cracks grow more easier in narrow specimens,

Fig. 6. The influence of the varied initial flaw angle (25�, 45� and 60� from axial load) on wing crack growth in specimens under uniaxial

compression (RFPA2D simulation).

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and the specimens are split into two thin columns where the width is about half of flaw length. It indicatesthat a strong interaction between the growing crack and the side boundary of the specimen exists. FromFig. 8, it can be seen that when the narrow specimen was loaded, an outward bending of the ligamentsappeared on either side of the growing wing cracks. This phenomenon was also observed in the experimentof Ashby and Hallam [14] and explained by them with a beam model. Ashby and Hallam [14] studied thecrack growth behaviour of inclined flaws in silicone rubber plates (in which large elastic deflections arepossible) with detailed measurements of the shape-change of the plates during loading. Their observationsand measurements indicated that the speed of crack propagation would increase if additional bendingdisplacements appear in the plate. They reported that when a narrow specimen is loaded, the growth ofwing cracks splits the specimen into vertical columns. These narrow columns would bend outward andcause a larger sliding displacement in the finite plate. This extra displacement increases the stress intensity inthe crack tips and makes it to grow further. The final failure of the narrow columns is caused by buckling.However, if the dimensions, H (height) and W (width), of the specimen are much larger than the length ofthe pre-existing flaw, a slight confinement may exist at the tip area of the wing cracks. This may cause thewing crack growth in a stable manner as shown in Fig. 8 for specimen of 100 mm wide. Crack growthbehaviour shown in Fig. 9 also confirms this observation.Fig. 10 shows the failure modes of the specimens with different width (25, 50 and 100 mm). For narrow

specimen (W ¼ 25 mm), buckling occurs when the cracks are close to the two ends of the specimen.Buckling is due to a strong interaction between the cracks and the free boundary of the specimen. For thespecimen with a width of 50 mm, this interaction reduces. The failure occurs by inducing shear cracksinstead of the buckling tensile crack near the flaw tip. In specimen with a width of 100 mm, however, thefailure occurs by the unstable en echelon fractures, which results in a manner more complicated than simplesplitting.From the above observations, it can be concluded that the growth rate of wing cracks in wide specimen is

much slower than that in narrow specimen. Furthermore, for the narrow specimen, if the splitting column isthinner than the half-length of flaw, strong interaction between the cracks and the free boundary of the

Fig. 7. Ultimate failure modes for specimens containing crack-like flaw at angle of 25�, 45� and 60� (RFPA2D simulation).

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Fig. 8. Stress distribution in crack growth under uniaxial compression for specimens with width 25, 50 and 100 mm (RFPA2D sim-

ulation).

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specimen occurs and this interaction causes buckling failure. Similar phenomenon can also be observed inthe multi-flaw-specimens.

4. Modelling of crack growth from an array of crack-like flaws in a brittle specimen under compression

4.1. Wing crack growth from three array flaws

According to the previous experimental studies on the specimens containing multi-flaws [11,13,14], wingcracks grow from each flaw stably when the length of wing cracks is short. However, when the wing cracksbecome longer, they start to interact with each other in a way that will slow down the growth of crack, oreven stop until a suddenly coalescence occurring between cracks.In order to study the influence of geometric setting of crack-like flaws and the interaction between flaws,

specimens of 170 mm high and 100 mm wide containing three array flaws with the same length 2a of 20 mmwere numerically tested. The corresponding element number of this specimen is 255� 150 ¼ 38250. Thearrangement of the array flaws is in diagonal (D Model), vertical (V Model) and horizontal (H Model)directions as shown in Fig. 11. Fig. 11 shows the stress distribution of the three model specimens beforecrack initiation. Light colour shows the maximum stress concentration.

Fig. 10. Ultimate failure modes for specimens with width 25, 50 and 100 mm (RFPA2D simulation).

Fig. 9. The influence of the specimen width (25, 50 and 100 mm) on the growth of wing crack under uniaxial compression (RFPA2D

simulation).

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Fig. 11. Stress field showing the interaction between flaws (RFPA2D simulation).

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Fig. 12 shows the numerical results of crack growth in the associated specimens. It can be seen that crackgrowth is easier in D Model specimen than in V Model and H Model specimens. The data of the crack

Fig. 12. Stress distribution of crack growth under uniaxial compression for specimen containing three crack-like flaws (D Model, V

Model and H Model, RFPA2D simulation).

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growth in D Model specimen is plotted in Fig. 13 whereas the definitions of the flaws I, II and III have beengiven in Fig. 11. For comparison, the data for an isolated flaw in the specimen with the same size (Fig. 9 forW ¼ 100) is also plotted in Fig 13. It can be seen that although the stress level for crack nucleation in eachflaw are more or less the same, the crack growth from flaws I, II and III extend rapidly at a load well belowthat for an isolated crack. As referring to Figs. 8 and 11, the free boundary away from the tips of pre-existing flaws of I and III of Fig. 11 is less than that of an isolated flaw (Fig. 8). Thus, the tips of flaw nearthe free boundary may cause the crack to grow more easily. For the middle flaw (II) of Fig. 11, crackgrowth still more easily than the isolated flaw although the boundary condition around the tips of flaw isthe same. This numerical result agrees well with the experimental observation made by Ashby and Hallam[14]. They concluded that the wing cracks grow more easily when other cracks are nearby. Although thenumerical simulation for DModel specimen shows that cracks may grow more easily when another crack isnearby, the scenario is not always the case. To accentuate this point, a comparison between three simu-lations of D Model, V Model and H Model are studied.From the results shown in Figs. 11 and 12, there are two interesting observations. First, as seen from the

second column of Fig. 11, a high stress concentration (light colour) occurs at flaw tips, and stress inter-action occurs in between the nearby flaw area. The numerical simulation for D Model specimen shows thatcracks may initiate more easily when the growth direction of wing cracks is running away from the in-teraction stress field (see the first and second column of Fig. 12). However, for the V Model, zone of stressinteraction between the nearby flaws locates vertically around the tips of flaws and this is also the locationfor tensile crack initiation and propagation. With this stress interaction, a higher stress is required for crackinitiation (about 20 MPa). Second, for the H Model, numerical simulations show that crack growth fromthe central flaw is much slower than the cracks from the two side flaws. It is because the growth of tensilecracks in the central part is suppressed by the stress interaction from the two side flaws. Fig. 14 shows moreclearly that the crack growth behaviour is strongly affected by stress interaction between flaws, crackgrowth rate in V Model and H Model is slower than that of an isolated flaw. Therefore, it can be concludedthat stress interaction is one of the important factors that determines the initiation and propagation ofcrack from the tips of pre-existing flaw, and stress interaction depends strongly on the locations of theinitial flaws. The conclusion from Ashby and Hallam [14], that cracks grow more easily when another crackis nearby is not always plausible. Similar phenomena will be shown again in the following subsectionsregarding crack growth in specimens containing multi-crack-like flaws.

Fig. 13. Crack growth under uniaxial compression in specimen containing three pre-existing crack-like flaws arranged in D Model.

The definition of flaw I, II and III refers to Fig. 11, the data of isolated crack is obtained from Fig. 9, W ¼ 100 mm (RFPA2D sim-

ulation).

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Fig. 15 shows the ultimate failure patterns for the three model specimens. Although the crack propa-gation path in the D Model is longer, the coalescence stresses of the three different models are more or lessthe same (also seen from Fig. 14).

4.2. Wing crack growth from random distributed multi-flaws

The complexity of crack initiation, propagation, interaction and coalescence can be demonstrated moreclearly in the following numerical simulation on specimen containing a number of randomly distributedcrack-like flaws with different lengths and angles. Fig. 16a shows the sequence of progressive failure andFig. 16b shows the stress distribution of a multi-flaw-specimen. Light colour shows the maximum stressconcentration. From these figures, there are four interesting observations.

First––crack nucleation. Tensile cracks nucleated first from a 45� large flaw, which located near the freeboundary at the right side of the specimen (see Fig. 16a, stage b). The distance between the tip of flaw andthe free surface is less than a half of the flaw length. However, the longer flaw located at the centre of thespecimen suffers from no crack nucleation at this stress level. Thus, confinement around the flaw tips is

Fig. 14. Influence of flaw arrangement (D Model, V Model and H Model) on the growth of crack under uniaxial compression. The

normalized crack length is taken by the mean value from three flaws I, II and III. The crack growth of the isolated crack is plotted from

comparison (RFPA2D simulation).

Fig. 15. The failure modes of the specimens containing three crack-like flaws at different arrangement: D Model; V Model; and H

Model.

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Fig. 16. (a) Sequences of crack growth under uniaxial compression for specimen containing multi-flaws (RFPA2D simulation).

(b) Stress distribution of growth of cracks under uniaxial compression for specimen containing multi-flaws (RFPA2D simulation).

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clearly one of the factors that influences the crack nucleation. After stage b, tensile cracks initiated fromthose flaws located along the diagonal (more or less) of the specimen (see Fig. 16a, stages c–d). All theseflaws consist of different lengths (long and short) and different angles (large and small) from the axial loaddirection. In general, cracks initiate from the larger flaws first (see Fig. 16a, stage c), then initiate from theshorter flaw (Fig. 16a, stage d). Therefore, it can be confirmed that for a multi-flaws-specimen, flaw lengthand the location of flaw are the important factors in determining the nucleation of crack.

Second––crack propagation. It is observed that although cracks initiated from the flaw near the freesurface of the specimen (see Fig. 16a, stage b), cracks propagate to the direction of the axial compressiononly, but do not grow towards the free boundary (Fig. 16a, stages c–e). The final failure of the specimen iscaused by lateral buckling (Fig. 16a, stages f and i). The same phenomenon was also observed in the ex-periment by Nemat-Nasser and Horii [11]. They produced a pre-existing flaw close to a free boundary(about half of flaw length distance) with different boundary profiles: dome shape, concave shape andstraight line shape. Cracks occurred at the tip of the flaw, propagating along the direction of axial force androughly parallel to the free boundary. This observation confirmed our discussion in Section 3.3 that if thesplitting column is thinner than the half-length of flaw, strong interaction between the cracks and the freeboundary of the specimen occurs and causes buckling failure.

Third––unstable crack growth. It is observed that after the peak stress, cracks still grow continuouslyeven if the stress drops to the half of the maximum load (Fig. 16a, stages f–j).

Fourth––stress distribution. The initially uniform stress distribution at the initial stage (Fig. 16b, stage a)changes to a highly non-uniform pattern (Fig. 16b, stages e–f) with both crack initiation and propagation.High stress concentration (the light colour) occurs at the lower part of the specimen when the stress levelreaches the peak value. That is why more cracks initiated from these flaws but not from the flaws located atthe upper part of specimen. When cracks propagate to the end surface of the specimen, stresses are totallyreleased and concentrate only along the undamaged areas (Fig. 16b, stage j). This scenario is difficult to beobserved in the conventional experiment.Fig. 17 shows the stress–strain curve of the multi-flaws-specimen. In Fig. 17, the solid curve shows the

axial stress plotted against the axial strain for the specimen shown in Fig. 16, and the dotted curve showsthe corresponding curve of the cumulative crack length that is calculated by summing the steps of crackpropagation. The points a–j marked on the stress–strain curve correspond to the Fig. 16 at different stagesa–j, respectively. From Fig. 17 it is known that the first tension crack nucleates at the point marked b,followed by the first crack coalescence at point d. After that, the stress–strain curve shows that the overallelastic modulus reduces due to the continuous nucleation of tensile cracks. According to the curve for thecumulative crack length, every big increase of the cumulative length of crack propagation results in an

Fig. 17. Stress–strain curve for specimen containing multi-flaws with crack accumulative value. The points a–j corresponding to the

Fig. 16 at different stages of a–j (RFPA2D simulation).

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associated considerable stress drop. The biggest stress drop indicates the axial splitting of the specimen(stages i–j).

5. Conclusions

Using the RFPA2D, the growth of brittle cracks from pre-existing crack-like flaws under uniaxialcompression has been numerically studied. A constant external displacement rate is applied in the verticaldirection to produce a compressive loading to the sample. The boundary condition of the ends of specimenis without lateral restraint that is no end friction is induced at the end of loaded specimen. Numerical resultssuggest that under uniaxial compression, axial splitting is the dominant failure mode. Based on theparametric analyses, the following conclusions have been obtained:

(1) The influence of flaw length, flaw angle and specimen width on crack nucleation, propagation and fail-ure mode.(i) For flaw length: independent of the specimen size, cracks are easier to nucleate from larger flaws

than from smaller flaws. For the same size (width and length) of specimen, in general the growthrate of longer flaw in the specimen is about twice of that specimen containing the shorter flaw.

(ii) For flaw angle w: for flaws with orientation of 60� to the axial loading direction, crack grows mosteasily.

(iii) For specimen width: wing cracks grow more easily in narrow specimens and split the specimen intocolumns.

(2) Boundary effect on crack growth and failure mode.(i) For wing cracks approaching to the upper and bottom boundary, the growth of wing crack slows

down.(ii) For distance between the tip of flaw and free boundary lesser than two times of flaw length, shear

failure may occur if shear stress around the flaw tips is high enough.(iii) For distance between the tip of flaw and free boundary smaller than half-length of flaw, strong

interaction between the cracks and the free boundary occurs and causes buckling failure.(3) Crack nucleation and growth in multi-flaw-specimens.

(i) Flaw length and the location of flaw are the important factors to determine the nucleation of crack.In general, the longer flaw and the closer the flaw to the free boundary of specimen, the easier forcrack nucleation.

(ii) Stress interaction is the most important factor affecting the initiation and propagation of crackfrom the tips of flaw. For flaws arranged in a diagonal line of the specimen, cracks generally growmore easily; but for flaws arranged in vertical or horizontal direction, the growth of middle flawwill be suppressed by the stress interaction from the nearby flaws.

(iii) The numerical simulation of RFPA2D can show the failure process very clearly including thegrowth of wing cracks and the change of stress field at the stage of pre- and post-failure withthe growth of wing cracks, which is difficult to observe in the conventional experiment.

Acknowledgements

The work presented here was made possible by the financial support from the RGC project PolyU5050/99E of the Hong Kong Polytechnic University to RHC, Wong and was partially supported by the ChineseNational Key Fundamental Research ‘‘973 Programme’’ (G19980407000) and the National Natural Sci-ence Foundation (no. 49974009) to CA, Tang.

1870 R.H.C. Wong et al. / Engineering Fracture Mechanics 69 (2002) 1853–1871

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