Crack coalescence in a rock-like material containing two cracks

PII: S0148-9062(97)00303-3

Int. J. Rock Mech. Min. Sci. Vol. 35, No. 2, pp. 147±164, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0148-9062/98 $19.00 + 0.00

Crack Coalescence in a Rock-like Material Containing Two Cracks ROBINA H. C. WONG K. T. CHAU This paper investigates the pattern of crack coalescence and strength of a sandstone-like material containing two parallel inclined frictional cracks under uniaxial compression, with changing values of inclination of preexisting cracks a, bridge angle b (inclination between the inner tips of the two preexisting cracks), and the frictional coecient m on the surfaces of the preexisting cracks. Three main modes of crack coalescence are observed: the shear (S) mode (shear cracking between the two preexisting cracks); the mixed shear/tensile (M) mode (propagation of both wing and shear cracks within the bridge area); and the wing tensile (W) mode (coalescence of wing cracks from the tips of the preexisting cracks). The M-mode and Wmode of crack coalescence can further be divided into two and six types, respectively. Simple regime classi®cations of coalescence in the a±b space are proposed for di€erent values of m (=0.6, 0.7 and 0.9). In general, the Smode mainly occurs when a = b or when b < b*(a, m) = a ÿ ba, with both a and b depending on m; the M-mode dominates when bL>b>b*(a, m) (where bL182.58); and the W-mode is only observed when b>bL. However, more experiments are still required to re®ne the classi®cation. The observed peak strength, in general, increases with m. Our results show that the peak strength predicted by the Ashby and Hallam (1986) model basically agrees with experiments. A minimum occurs at about a = 658 when the peak strength is plotted against a. For a>458, the peak strength is essentially independent of the bridge angle b. # 1998 Elsevier Science Ltd.

INTRODUCTION

The length scale of fractures found in natural rocks ranges from tens of kilometres down to tens of microns. In addition, fractures or joints in rocks are normally of ®nite length or are discontinuous in nature. The mechanism of crack coalescence in the rock bridge area (see the de®nition on Fig. 1) between the preexisting cracks remains one of the most fundamental theoretical problems in rock mechanics to be solved. From the practical point of view, nearly all rock engineering projects involve, to a certain extent, constructions of structures in or on rock masses, which contain both fractures and rock bridges. The crack coalescence in a rock bridge is usually responsible for the failure of many rock structures. For example, in 1991 a large block of rock of about 2000 m3 slid from a steep rock face at the discussed quarry at Shau Kei Wan in Hong Kong. At the time of the incident, blasting had been taking place above the disused quarry for *Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong.

some time in conjunction with the site formation works for the construction of a new housing estate. Interpreted from the investigation report [1], the preexisting parallel joints, dipping toward the rock slope surface, were not fully persistent before the slide occurred; it is likely that crack initiation and growth at the tip of the joints may have been induced by the vibration of blasting. Therefore, it is believed that crack coalescence was the cause of the rock slope failure. Thus, in rock engineering projects the mechanism of crack coalescence of preexisting fractures plays an important role in controlling the stabilities of slopes, foundations and tunnels. However, the mechanisms of crack coalescence in rock masses, containing joints and rock bridge, have not been fully understood. Crack initiation and propagation have been subjects of intensive investigation in rock mechanics, both experimentally and theoretically. The ®rst theoretical study on the growth of preexisting two-dimensional cracks was given by Grith [2, 3]. Irwin [4] further introduced the concept of critical energy release rate and the crack tip stress intensity factor (K). Relating to the ®eld of rock mechanics, a number of experimen-

147

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Fig. 1. Rock bridges in discontinuously-jointed rock.

tal studies have been carried out to investigate the crack initiation, propagation and interaction (e.g. [5± 12]). For a comprehensive literature review on microcrack studies we refer to Kranz [13]. At the same time, many mathematical models were developed to explain and predict the processes of crack growth, interaction and rock failure (e.g. [14±20]). For the discussion on how these shear crack models can be applied to real rocks, we refer to the comparative study by Fredrich et al. [21]. However, most of the previous studies are focused on the mechanisms of crack initiation, propagation and interaction, but relatively few experimental investigations were done to examine the pattern of crack coalescence in the rock bridge area between the preexisting fractures. The pattern of coalescence between two parallel cracks in both modelling materials and natural rocks have been done in direct shear boxes (e.g. [22±25]). However, it is not easy to observe the development of the whole process of coalescence inside the shear box. Recently, Reyes [26] and Reyes and Einstein [27] have performed some uniaxial compressive tests on gypsum samples containing two inclined open cracks. The process of crack coalescence was recorded by a microscope connected to a video camera. As expected, the mechanism of coalescence is controlled by the initial geometric setting of the parallel cracks. To incorporate the e€ect of friction, Shen et al. [28] conducted a series of uniaxial compressive tests on gypsum samples containing both openand closed-fractures. Related numerical simulation of the failure of rock bridge was also done by Shen and coauthors [28±31]. Therefore, Shen et al.'s [28] study seems to be most relevant to the actual process of coalescence in jointed rocks. However, certain aspects of the study by Shen and coauthors [28] can further be improved. Firstly, the gypsum mixture used by Shen et al. [28] is the one by Nelson and Hirschfeld [32]. However, most of the pfactors for their modelling material do not fall within

the range of the p-factors for quartzite and granite, which are the rocks to be modelled (see Ref. [33] or the Appendix A for the de®nition of p-factors). Therefore, a more careful dimensional analysis of the modelling material should be done using the p-theorem in order to simulate the patterns of crack coalescence as close as possible to those observed in real rocks (see [33] or the summary in Appendix A for p-theorem). In addition, we also expect that the pattern of crack coalescence may depend on the types of rocks. Secondly, the e€ect of the inclination of the preexisting cracks a, the bridge angle b (which is the relative angle between the two inclined cracks), and the frictional coecient m of preexisting crack surfaces on the mode of crack coalescence is not fully examined by Shen et al. [28]. In particular, only one frictional coecient (m = 0.7) has been simulated on the crack surfaces, together with the open crack (m = 0), and only 13 combinations of a/b were examined. In addition, no attempt has been made to compare the observed peak strength (the maximum attainable applied stress under a uniaxial compression) and the prediction by existing crack models. Therefore, we attempt here to give a more re®ned study on the pattern of crack coalescence of sandstone-like material and the peak strength. The modelling material to be used here will be analysed by the ptheorem such that it resembles the main characters of sandstone. A total of 87 combinations of a/b and m will be presented in this study, compared to the 26 combinations by Shen et al. [28]. A regime classi®cation for the patterns of coalescence will also be proposed empirically based upon our experimental observations. The peak strength predicted by using the sliding crack model of Ashby and Hallam [17] is compared to our experimental results and the range of its applicability will be discussed. EXPERIMENTAL STUDIES

The discussion of our experimental studies is divided into four sections. The ®rst section discusses the physical properties of a sandstone-like modelling material, the second section on the technique in preparing the cracked specimens, the third section on the testing procedure in loading the cracked specimens, and the fourth section on the general experimental observation. Modelling material and its physical properties Full scale testing on a rock mass containing a speci®ed number of cracks with predetermined orientations is either prohibitively expensive or impossible. One common recourse to the problem is to conduct experiments under conditions that are attainable, but at the same time the patterns of coalescence involved in the prototype have to be preserved in the model experiments. In order to achieve such a goal, dimensional analysis has to be used in designing the size and prop-

WONG and CHAU: CRACK COALESCENCE

erties of the specimens [33]. Thus, all dimensional parameters involved in the experiments have to be scaled according to the p-theorem, which is discussed brie¯y in the Appendix A. In addition, two additional characteristics of the material are considered here to ensure the suitability of the modelling material: (1) brittle character of the modelling material and (2) dilatancy of the material under uniaxial compression. Although dilatancy has not been properly discussed in most of the previous studies, it is one of the most important properties for the rock-like materials [34, 35]. The most comprehensive review on how to select a modelling material for rocks is probably by Stimpson [34]. Although there are a number of modelling materials that can be considered as rock-like (e.g. [32, 36, 37]), the present modelling material is, however, motivated by the material used by Barton [38] and Bandis et al. [39]. It consists of barite, uniformlygraded sands, plaster and water. The barite powder of speci®c gravity 4.5 is used as a dense ®ller for increasing the relative density of the modelling material. The sand is used to provide the frictional behaviour of the modelling material, and it is a mixture of 40% ®ne (90±150 mm) and 60% coarse sands (150±300 mm). ``Glastone 2000 Dental stone'' is used as the cementing plaster, and its setting time ranges from 7 to 9 min. The uniaxial compressive strength for this plaster at 2 h is 9 MPa. After testing several trial mixtures, we ®x the mixture ratio barite/sand/plaster/water by weight as 2/4/1/1.15. The samples are oven-dried at 1058C for 5 days in order to achieve a brittle character. The uniaxial compressive strength (scm) of the model material is measured on cylindrical specimens with 56 mm in diameter (Dm) and 112 mm in length (Lm). The indirect tensile strength (stm) of the material is determined by the Brazilian test using solid discs 56 mm in diameter (Dm) and 28 mm in thickness. The frictional coecient mm of the model material is measured by tilt test on blocks of specimens with size 60 mm  120 mm  25 mm. The testing procedure of uniaxial compressive test and the Brazilian test complies with the ASTM designation D2938-86 [40] and ASTM designation C496-71 [41], respectively. Four transducers are used to measure the horizontal and

149

vertical displacements. Two of the transducers are set to touch the mid-point of the cylindrical specimen along a diametral line, while the other two transducers are set to touch the top of the platen along another diametral line, which is perpendicular to the previous one. The transducers and loading record are connected to a KYOWA UCAM-5B data logger which is further linked to an IBM PC for data recording. The mean values of unit weight, uniaxial compressive strength, tensile strength and frictional coecient mm of the model material are gm=17.68 kN/m3, scm=2.09 MPa, stm=0.35 MPa and mm=0.62, respectively. The mean value of the tangent Young's modulus (Em) at 50% of peak strength is 329 MPa and the Poisson's ratio (nm) is 0.19. The modulus/strength ratio Em/scm is 158. According to Fig. 2 of Stimpson [34], the modulus ratio (Em/scm) of 158 closely resembles that of sandstone and shale. Therefore, the mechanical properties of sandstone and shale should be compared with those of the modelling material by using p-factors as indices [33] to decide whether the modelling material is classi®ed as a ``sandstone-like'' or ``shale-like'' material. The mechanical properties and fracture toughness KIC of sandstone and shale are listed in Table 1. The mechanical properties of sandstone and shale are taken from Farmer [42, 43], the fracture toughness KIC of sandstone and shale are taken from Atkinson and Meredith [44]. (The measurement procedure for KIC of the modelling material will be discussed in more detail at the end of this section). Using the dimensional analysis, six p  dimensionless products or p-factors (gL/E, KIC/(sc L), sc/st, E/sc, n and m) of the modelling material are obtained (see the Appendix A). Except for two p-factors, it is found that most of the p-factors fall within the ranges for sandstone and shale. In particular, the p-factor ``n'' of the modelling material does not fallpwithin the range for shale and the p-factor  ``KIC/(s L)'' of the modelling material does not fall within the range for sandstone (see Table 1). However, the di€erence of p-factor ``n'' between the modelling material and shale pis  about 46%, but the di€erence of p-factor ``KIC/(s L)'' between the modelling material and sandstone is only 5%. Therefore, it is concluded that the modelling material is more sandstone-like

Table 1. The comparisons of the prototype rock and modelling material. All of the mechanical properties for sandstone and shale are from Farmer [42, 43], except that KIC is from Atkinson and Meredith [44] Parameters or p-factors of material sc (MPa) st (MPa) E (GPa) p KIC (MPa m) g (kN/m3) L (m) gL/E p KIC/(sc L) sc/st E/sc n m

Sandstone

Shale

Modelling material

20±170 4±25 3±35 0.22±2.66 23 0.53 6  10ÿ7 0.015±0.183 5±7.4 150±206 0.02±0.20 0.47±0.67

10±100 2±10 2.5±15 0.61±1.34 22 0.24 6  10ÿ7 0.125±0.274 5±10 150±250 0.02±0.13 0.49±0.62

2.09 0.35 0.33 0.04 17.683 0.012 6  10ÿ7 0.193 5.97 158 0.19 0.62

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WONG and CHAU: CRACK COALESCENCE

than shale-like. In simulating sandstone, the geometric scale factor o (=gmEp/gpEm where Ep and gp are the Young's modulus and density of prototype rock, respectively) for the modelling material is 44.2, while the stress scale factor w (=ogp/gm) for the modelling material is 57.6. These two parameters can be used to scale up the size of crack and strength of the modelling specimens to those of the prototype rock (i.e. sandstone for this case). Therefore, the uniaxial compression strength sc=2.09 MPa, the tensile strength st=0.35 MPa and Young's modulus E = 0.33 GPa of the modelling material give a prototype sandstone with sc=120.4 MPa, st=20.2 MPa and E = 19 GPa. To check the brittleness and dilatancy of the material, typical stress±strain curves of the modelling material have been scaled up to the prototype rock and are plotted in Figs 2 and 3. The solid lines are the curves for the modelling material, while the dotted lines are those for sandstone adopted from Fig. 4.13 of Farmer [43]. As shown in Fig. 2, the axial stress drops rapidly as the peak stress is passed, such a sudden drop in stress is a good indicator of the brittle behaviour of the material. As shown in Fig. 3, the compressive volumetric strain reduces near the peak strength, and as the peak stress is passed volumetric dilatancy occurs. Thus, the dilatant character of the modelling material is evident. Therefore, it is concluded that our modelling material can be classi®ed as a ``sandstonelike'' material and can be used to study the mechanisms of crack coalescence of a fractured rock mass of sandstone.

To end this section, we present here the mode I fracture toughness (KIC) of the modelling material, which will be used in the calculation of peak strength by using the Ashby±Hallam model (to be considered in later sections). The measurement of KIC was done on a simply-supported round bar with a single-edge notch of depth C at mid-span, and the loading is applied through a three-point bending using a BEMEK loading machine [45, 46]. The test specimen is normally referred to as SENRB (single-edge notched round bar). The span-to-diameter ratio (S/Dm) must be ®xed at 3.33 and depth-of-notch-to-diameter ratio (C/Dm) must be less than 0.6. The width of the notch is 1 mm, and the specimen is 44 mm in diameter (Dm) and 190 mm in length (Lm). The span is ®xed at 146.5 mm (i.e S/Dm=3.33 as required). Four specimens are prepared with di€erent values of C which are 2, 3, 4 and 5 mm respectively, giving C/Dm<0.6 as required. The calculation for the fracture toughness KIC of SENRB is given by [46]: KIC ˆ

10:627‰C=Dm ‡ 19:646…C=Dm †5:5 Š0:5 P , …1 ÿ C=Dm †0:25 D1:5 m

…1†

where P is the point load acting on the mid-span of the specimen at failure. The testing procedure is done according to the manual of the BEMEK advanced rock tester. A load cell of capacity of 5 kN is attached on the loading frame to record P accurately during the testing. The mean value of the fracture toughness of p the modelling material is KIC=0.0443 MPa m.

Fig. 2. Axial stress versus the axial and lateral strains for both the modelling material and sandstone. The typical stress± strain curves of the modelling material have been scaled up to the prototype rock.

WONG and CHAU: CRACK COALESCENCE

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Fig. 3. Axial stress versus the volumetric strain for both the modelling material and sandstone. The typical stress±strain curves of the modelling material have been scaled up to the prototype rock.

Preparation of cracked samples The dimensions of our specimens are 60 mm  120 mm  25 mm. A large guillotine similar to the one shown in Fig. 1 of Barton [38] was manufactured. Two short and sharp knives are pressed simultaneously to the specimens using hydraulic pressure; however, such an impact-fracturing technique is found very dicult to control the exact length of the cracks. Consequently, we have decided to use preinstalled stainless steel sheets, with their removal during curing. In particular, the preexisting cracks are created in the specimens by inserting two thin steel sheets (0.3 mm in thickness) into two slots in the template. The positions and orientations of the slots are predetermined to give a di€erent combination of the inclination of the cracks (a) and rock bridge angle (b), which is the relative inclination between the cracks as shown in Fig. 4. As mentioned in the Introduction, a varies from 35 to 758 with 108 increments, while b varies from 45 to 1208 with 158 increments. The thin sheets are 12 mm in length (2c) and the distance between two cracks is kept constant at 20 mm (2b) (see Fig. 4 for the de®nition of b and c). Using the geometric scale factor, it is straightforward to show that a specimen with cracks of 12 mm in length with 20 mm in bridge length actually simulates a prototype rock with 0.53 m joints with 0.88 m bridge distance. To obtain a crack surface with closed contact, the steel sheets were removed carefully after about 3±5 min of curing, then further expansion of the plaster mixture during curing closed the slots, and thereby created preexisting cracks with closed surfaces. This procedure is

quite similar to that used by Shen et al. [28]. Although our steel sheet seems relatively thick compared to those used by Reyes and Einstein [27] (0.25 mm) and

Fig. 4. A solid containing two neighbouring sliding cracks of length 2c with wing cracks of length l subject to uniaxial compression. The bridge distance between the two preexisting cracks is 2b. The crack inclination a and rock bridge angle b are de®ned together with the location of inner and outer crack tips.

152

WONG and CHAU: CRACK COALESCENCE Table 2. The experimental and theoretical results of the peak strength of the cracked specimens p p p b (8) Normalized peak stress s1 …pc†/ Normalized peak stress s1 …pc†/ Normalized peak stress s1 …pc†/ KIC (m = 0.6, f = 318) KIC (m = 0.7, f = 358) KIC (m = 0.9, f = 418)

a (8)

experimental 35 45 55 65 75 35 45 55 65 75 35 45 55 65 75 35 45 55 65 75 35 45 55 65 75 35 45 55 65 75

45 45 45 45 45 60 60 60 60 60 75 75 75 75 75 90 90 90 90 90 105 105 105 105 105 120 120 120 120 120

6.68 5.17 4.74 4.41 4.93 5.35 5.19 4.68 4.47 5.01 5.27* 4.95 4.56 4.48 5.01 5.35 4.94 4.55 4.62 5.00 6.34 5.81 4.68 4.68 5.05 5.02* 5.07 4.67 4.53 4.37*

theoretical 6.60 5.19 4.62 4.54 4.91 6.18 5.03 4.54 4.48 4.79 5.95 4.93 4.48 4.42 4.71 5.88 4.89 4.46 4.40 4.68 5.95 4.93 4.48 4.42 4.71 6.18 5.03 4.55 4.48 4.79

experimental 6.45 5.84 4.79 4.47 4.93 6.48 5.55 4.82 4.59 5.07 6.31 5.67 4.84 4.61 5.07 6.22 4.85* 4.75 4.59 5.05 6.84 5.69 4.86 4.71 5.07 6.43 5.37 4.70 4.58 4.57*

theoretical 7.37 5.53 4.80 4.63 4.95 6.78 5.32 4.70 4.55 4.83 6.47 5.19 4.62 4.49 4.74 6.73 5.14 4.59 4.47 4.71 6.47 5.19 4.62 4.49 4.74 6.78 5.32 4.70 4.55 4.83

experimental ÿ 5.85 5.63 4.80 5.15 ÿ 5.62* 5.59 4.88 5.17 ÿ 5.60* 5.57 4.89 5.19 ÿ 5.61 5.31 4.94 5.16 ÿ 6.20 5.63 4.98 5.19 ÿ 5.45* 5.43 4.86 5.13

theoretical ÿ 6.39 5.19 4.83 5.03 ÿ 6.01 5.03 4.72 4.89 ÿ 5.80 4.93 4.64 4.80 ÿ 5.73 4.89 4.62 4.77 ÿ 5.80 4.93 4.64 4.80 ÿ 6.01 5.03 4.72 4.89

*Specimens show partial-surface-contact along the preexisting cracks.

by Shen et al. [28] (0.01 mm), it is not dicult to show that our cracks are actually in closed contact. For example, when the specimens are put in front of a light source, no light ray is observed through the crack surfaces. There are, however, about eight specimens which allow some light to come through on the crack lines partially (see footnote of Table 2). Thus, we conclude that the cracks in these specimens are only in partial contact, and, as will be shown later, their peak strength is lower than the expected value. Further evidence that the crack surfaces are actually in closed contact will be presented in the previous section when the experimental results are discussed. Three di€erent frictional coecients (m = 0.6, 0.7 and 0.9) on the crack surfaces are simulated by changing the roughness of the stainless steel sheets. In particular, Fig. 5(a) shows the three types of steel sheets used: (1) plain steel sheet [m = 0.6 on crack surfaces and labelled with A in Fig. 5(a)], (2) steel sheet with punching-indentations at 4 mm in spacing [m = 0.7 on crack surfaces and labelled with B in Fig. 5(a)], (3) steel sheet with punching-indentations at 2 mm in spacing [m = 0.9 on crack surfaces and labelled with C in Fig. 5(a)]. Actually the crack surfaces are created by pulling out of the inserted steel sheets; and therefore the ®nal roughness of the crack surfaces is in the form of parallel wrinkles, as shown in Fig. 5(b). The left broken specimen has m = 0.7 and the one on the right

has m = 0.9. The sharper the grooves on the crack surface, the higher is the frictional coecient m. The frictional coecients on the crack surfaces are measured by the tilting test method on specimens with throughgoing cracks, and the corresponding mean friction angles f are 31, 35 and 418, respectively (note that f = tanÿ1m). Since the friction angle for m = 0.9 is greater than the lowest predetermined crack inclination a = 358 (i.e. f = 418>358), frictional sliding is not expected to occur in these specimens under uniaxial compression. Consequently, for m = 0.9 the inclination of preexisting cracks a varies only from 45 to 758 (as shown in Table 2). Therefore, a total of 84 combinations of a, b and m are tested initially. Testing procedure The uniaxial compression tests of the cracked specimens are performed in a Wykeham Farrance WF55623 loading machine. To measure the applied load accurately, a load cell of 5 kN is placed below the lower loading platen. To reduce the friction between the specimen and the loading platens, two pieces of polythene sheet are inserted. All specimens are loaded until either the preexisting fractures coalesce or the specimen fails. The loading rate is ®xed at 0.002 kN/s, and it takes about 20±25 min to complete the loading test. Crack coalescence occurs in all the specimens except the following sets of a/b/m: 658/1208/0.6, 758/

WONG and CHAU: CRACK COALESCENCE

153

the ®lm is used to study the failure mechanism when it is replayed in slow motion. Our interest here is on the overall failure pattern in the bridge area but not on the propagation of the hairline microcracks; therefore, no microscope is installed for microcrack observation. General experimental observations According to the loading record, applied load drops slightly at about 50% of the peak load in most of the specimens. This probably corresponds to the onset of sliding along the preexisting cracks, although such sliding is not evident in our video recording. The crack initiation normally occurs as a sudden event in most of our observations. Sixty percent of the crack initiations appear ®rst at the inner tips of the preexisting cracks, then crack growth follows at the outer tips of the preexisting cracks (see the de®nitions of inner and outer tips in Fig. 4). About 20% of crack initiations occur in the reverse order, that is cracking at the outer tips is followed by those at the inner tips. For the ®nal 20% of specimens, crack initiation seems to appear simultaneously at both the inner and outer tips of the preexisting cracks. The sequence of cracking at either the outer or inner tips seems to be independent of both a and b. In general, the growth of cracks at the outer tips is faster than that observed at the inner crack tips. Most parts of the process of crack coalescence between the growing cracks at the inner crack tips are normally slow enough to be captured by either the naked eye or by video camera. The types of cracking in the bridge area can appear as either tensile, shear, or a mix of both. The shear secondary crack in the bridge area is de®ned as a crack parallel to the preexisting crack surfaces [47]. A detailed discussion on the pattern or mode of crack coalescence will be deferred to the sections after we present the Ashby and Hallam [17] model in the next section.

SLIDING CRACK MODEL

Fig. 5. Photo (a) showing stainless steel sheets with di€erent roughness used in creating the sliding cracks in the modelling material. The top, middle and lower sheets give a frictional coecient m of 0.6, 0.7 and 0.9 on the surfaces of the sliding cracks, respectively. Photo (b) showing the ®nal surface roughness of the cracks for m = 0.7 (left) and m = 0.9 (right). Several parallel wrinkles appear on the crack surface after the pulled-out steel sheet. The sharper the grooves on the crack surface, the higher is the frictional coecient m.

458/0.7 and 758/608/0.9. The experimental results are listed in Table 2. The entire process of crack initiation and propagation is recorded by a video camera, and

Although experimental studies seem to suggest that microcracking is mainly tensile but not shear in nature (e.g. [8]) and some authors also argued that the sliding crack model cannot account for all of the experimental observations (e.g. [48, 49]), the ``sliding crack model'' consisting of a sliding shear crack and two tensile wing cracks remains one of the most popular models to describe the inelastic dilatancy of rocks. A myriad number of sliding crack models have been proposed in predicting the brittle failure of rock specimens in compression (e.g. [16, 17, 50±53]). In particular, the sliding crack model assumes a frictional crack of inclination a and length 2c subject to nonhydrostatic compressive loading as shown in Fig. 4. When the resolved local shear traction along the inclined crack surfaces exceeds the frictional resistance, sliding along the main or preexisting shear crack occurs and causes wing cracks to nucleate and grow (see Fig. 4).

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This kind of sliding crack model has been well analysed by many authors, however, the discussion given in this section follows the analysis by Ashby and Hallam [17] and is similar to that given by Wong et al. [54]. It is because Ashby-Hallam's [17] crack model is one of the most well-analysed models, and, more importantly, it is not mathematically complicated and thus is easy to use. As remarked by Fredrich et al. [21] and by Wong et al. [54], the peak strength of cracked rock can be estimated using Ashby-Hallam's [17] model and expressed in terms of the crack length (2c), fracture toughness (KIC), frictional coecient (m) on the surfaces of the preexisting cracks, the inclination of preexisting crack (a) and the number of microcrack counts per unit area (Eo). In particular, Ashby and Hallam [17] derived the following approximate expression for the mode I stress intensity factor KI at the tip of the wing cracks, which nucleate from a preexisting inclined crack of length 2c when the solid is subject to uniaxial compression s1 (as shown in Fig. 4) KI …sin 2c ÿ m ‡ m cos 2c† p ˆ s1 pc …1 ‡ L†3=2   1  0:23L ‡ p , 3…1 ‡ L†1=2

…2†

where c is the angle measured from s1-direction to the direction along the shear crack surface (i.e. c = 908 ÿ a), L = l/c the normalized length of the wing cracks (l is the length of the wing crack as shown in Fig. 4), and m the frictional coecient along the main shear crack. The orientation of the shear crack for which the nucleation of the wing crack is most favourable is given by 2c = tanÿ1(1/m) [17]. If peak strength is to be predicted, crack interaction and coalescence must be incorporated into the analysis. Therefore, Ashby and Hallam [17] derived the following KI due to crack interactions using beam theory KI p ˆ f2Eo …L ‡ cos c†=pg1=2 , s1 pc

…3†

where Eo is the initial crack density (Eo=c2Na, Na is the number of cracks per unit area). Combining Equations (2) and (3) gives the following total stress intensity factor KI for the wing cracks with crack interaction KI ‰sin 2c ÿ m ‡ m cos 2cŠ p ˆ s1 pc …1 ‡ L†3=2

 r 1 2Eo …L ‡ cos c†  0:23L ‡ p : ‡ p 3…1 ‡ L†1=2 

…4† In this study, the peak strength for rock failure is obtained as the applied stress s1 corresponding to a growth of the wing cracks of about the distance between the rock bridge distance 2b (i.e. lmax=2b sin b,

where lmax is the maximum possible value for l). Therefore, the peak uniaxial compressive strength smax 1 can then be calculated by setting L = Lcr=lmax/c, into Equation (4) KI=KIC, and s1=smax 1  KIC ‰sin 2c ÿ m ‡ m cos 2cŠ ˆ p smax 1 pc …1 ‡ Lcr †3=2   1  0:23Lcr ‡ 3…1 ‡ Lcr †1=2  1=2 ÿ1 2Eo …Lcr ‡ cos c† ‡ …5† p Predictions p of the normalized peak strength (smax …pc†/KIC) by using Equation (5) are also listed 1 in Table 2 for all of our specimens. The detailed comparison between the prediction and experimental results will be given in the last section (to follow). MODE OF CRACK COALESCENCE

When crack coalescence occurs, two main types of cracking can be identi®ed in the rock bridge area: wing cracks, which are tensile in nature; and secondary cracks, which are mainly shear in nature and are normally parallel to the preexisting shear cracks. Based upon these two types of cracks and their combinations, Fig. 6 summarizes nine di€erent patterns of crack coalescence observed in our experiments. In particular, there are three main modes of crack coalescence: (1) shear crack coalescence [S-mode in Fig. 6(a)], (2) mixed shear/tensile crack coalescence [M-mode in Fig. 6(b and c)] and (3) wing tensile crack coalescence [W-mode in Fig. 6(d±i)]. S-mode crack coalescence In the shear crack coalescence, wing crack nucleation at both the inner and outer tips of the preexisting cracks will normally occur ®rst. But before the wing cracks can propagate further, secondary shear cracks will nucleate from both kinks at the inner tips. The propagation of these secondary cracks will lead to a shear crack coalescence in the bridge area while the outer wing cracks continue to extend to the edges of the specimen, as shown in Fig. 6(a). This kind of coalescence is mainly induced by a high shear stress concentration in the bridge area; indeed, Shen et al. [28] demonstrated such a high shear stress concentration occurs after the wing cracks have propagated for a certain distance for the case of a/b = 458/458 using the displacement discontinuity method DDM (see Fig. 15 of [28]). Our observations suggest that whenever the two preexisting main cracks are in alignment the shear interaction between the preexisting cracks becomes dominant. M-mode crack coalescence The mixed crack coalescence can further be divided into two types: MI and MII as shown in Figs 6(b and

WONG and CHAU: CRACK COALESCENCE

155

Fig. 6. Nine di€erent patterns of crack coalescence observed in our experiments. The letters S, M and W indicate the shear, mixed shear/tensile, and wing tensile modes of coalescence, respectively. The triangular, rhombic, and square representations for the coalescence will be used again in Figs 9±11 for regime classi®cation.

c). For the MI-type coalescence, wing cracks normally initiate from the inner tips. However, as the wing cracks grow further, anti-symmetric coalescence in the bridge area will suddenly occur with the appearance of a secondary shear crack joining the tips of the two inner wing cracks. The orientation of this secondary shear crack is roughly parallel to the preexisting cracks. It is, however, unclear whether this secondary crack nucleates from the centre of the bridge area or it propagates from one wing crack to the other. For the MII-type coalescence shown in Fig. 6(c), the lower

wing crack will ®rst appear and propagate through most parts of the bridge area, then it is followed by the sudden appearance of a shear crack joining the upper preexisting crack and the tip of the wing crack. Again, it is unclear whether it nucleates from the tip of the upper preexisting crack or from the wing crack of the lower preexisting crack. This pattern is probably caused by a slight di€erence in the frictional coecient between the upper and lower crack surfaces. Although the pattern of our M-mode coalescence di€ers slightly from that observed by Shen et al. [28], they have

156

WONG and CHAU: CRACK COALESCENCE

demonstrated by using DDM that such mixed mode coalescence is indeed possible. In particular, for this particular preexisting geometric setting of cracks, high tensile stress is ®rst expected in the bridge area. Then, it is followed by a high shear stress concentration in the ligament area if the tensile stress is released by the propagation of a tensile fracture in the bridge area (see Fig. 16 of [28]). This sequence of stress evolution also matches with what we have observed. W-mode crack coalescence The wing crack coalescence can be divided into three main types WI, WII and WIII, and their combinations WI/II, WI/III, WII/III, or a total of six types of coalescence. As shown in Fig. 6(d), the WI-type coalescence is simply the coalescence between two wing cracks. When the lower tip of the upper crack is o€set further from the upper tip of the lower crack (say br 1058), the WII-type coalescence may result as the wing crack from the lower main crack nucleates, propagates through the bridge area, and joins the upper main crack, as shown in Fig. 6(e). In the WIII-type coalescence shown in Fig. 6(f), a wing crack usually nucleates from only the inner tip of the upper preexisting crack, propagates through the bridge area, and joins the outer tip of the lower preexisting crack. This type of coalescence occurs when both b and m are of larger values (e.g. m = 0.9 and b = 1208). For the WI/ II-type coalescence, it is clear by comparing Figs 6(d, e and g) that it is a combination of the WI and WII patterns. Similar observation also applies to the WI/III and WII/III cases. For the coalescence shown in Fig. 6(g), the whole process happens so fast that the sequence of crack propagation is uncertain. For the WI/III case, wing cracks ®rst appear at the inner and outer tips of the two main cracks, then coalescence occurs between two inner wing cracks and, at the same time, a downward propagating outer wing crack collides with the inner tip of the lower crack, as shown in Fig. 6(h). Finally, for the WII/III coalescence shown in

Fig. 6(i) wing cracks nucleate from both the inner crack tips, then the lower wing crack coalesces with the outer crack tip of the upper preexisting crack while the upper wing crack collides with the surface of the lower preexisting crack. To end this section, we should note that the coalescence patterns of MII, WII, WIII, WI/II, WI/III and WII/III are all antisymmetric, and all of these coalescence patterns have another appearance which is the up-side-down image of the original pattern. E€ect of m on the mode of crack coalescence In this section, we compare the observed patterns of crack coalescence between our study and those by Shen et al. [28]. Figure 7 plots the observed coalescence for a/b = 458/1208 with changing values of m (=0.6, 0.7 and 0.9) together with the observation by Shen et al. [28] for a/b = 458/1208 and m = 0.7. By comparing Figs 7(b and d), it is evident that our observation agrees with failure mode observed by Shen et al. [28] when the frictional coecient on the crack surface is the same. However, Figs 7(a±c) show that the mode of coalescence depends not only on a and b but also on the frictional coecient m on the crack surfaces. To further investigate why WI-type shifts to WIIand WIII-types when the frictional coecient increases, we plot the path of the wing cracks, which emanate from a shear crack with inclination a = 458, as a function of m in Fig. 8, together with the numerical simulation by DDM done by Shen et al. [28] for m = 0 and 0.7. For m = 0.7, the path of the wing crack propagation agrees quite well with the prediction by Shen et al. [28]. In general, both experimental observations and numerical predictions suggest that deviation of the direction of wing cracks from the line of preexisting shear crack decreases with the increase of m. If we compare the inclinations of the line of coalescence in the rock bridge area for the cases m = 0.6, 0.7 and 0.9 shown in Figs 7(a±c), respectively, it is clear

Fig. 7. Pattern of coalescence versus the frictional coecient (m = 0.6, 0.7 and 0.9) for a = 458 and b = 1208 together with the observed pattern by Shen et al. [28] for m = 0.7.

WONG and CHAU: CRACK COALESCENCE

157

that the angle of deviation of the wing crack from the existing main cracks decreases with m. This observation agrees with our conclusion obtained from Fig. 8, and, therefore, it provides an explanation for the shift of modes from WI to WIII as m increases. REGIME CLASSIFICATION OF CRACK COALESCENCE

Fig. 8. The relative orientation of the wing cracks versus the frictional coecient m on the preexisting crack surfaces together with the numerical simulation by Shen et al. [28] using displacement discontinuity method (DDM).

As discussed in the previous section, there are main modes of coalescence, namely: the shear mode, the mixed shear/tensile mode and the wing tensile mode. From their appearance, nine di€erent patterns of crack coalescence can be classi®ed as shown in Fig. 6. It is natural to ask: Is there any correlation between a, b and m and these patterns of coalescence? This section is set forth to answer this question. Figures 9±11 plot the regimes of crack coalescence for di€erent modes in the a±b space for m = 0.6, 0.7 and 0.9 respectively. The plots are for a from 30 to 808 and for b from 40 to 1258. The triangles, rhombuses and squares at each grid point in these ®gures indicate a crack coalescence of S-, M- and W-modes, respectively, and the notations used here are the same as those given in Fig. 6. The dotted region on the left with an upward strip parallel to a = b is called the Sregime, indicating that shear mode of crack coalescence is expected to occur within this parameter space; the central M-regime indicates that mixed shear/tensile mode of crack coalescence occurs in this regime; and, ®nally, the dotted rectangular region on the right is

Fig. 9. The regime classi®cation in the a±b space for m = 0.6. The dotted S-regime with a strip parallel to a = b on the left is for the shear mode coalescence, the M-regime in the centre is for the mixed mode coalescence, and the dotted W-regime on the right is for the wing tensile mode coalescence. The symbol of `?' is for sample failure without crack coalescence.

158

WONG and CHAU: CRACK COALESCENCE

Fig. 10. The regime classi®cation in the a±b space for m = 0.7. Other captions are the same as those for Fig. 9.

called W-regime, indicating that wing crack failures are expected. Because only a ®nite number of points in the a±b space have been chosen in our experiments, the regime boundaries cannot be identi®ed with very high con®dence. Nevertheless, these ®gures provide the

®rst regime classi®cations for crack coalescence of this kind, and can be used as a benchmark-solution for further studies. As indicated by Figs 9±11, S-mode failure is expected if a = b or if b < b*(a, m) for a given value

Fig. 11. The regime classi®cation in the a±b space for m = 0.9. Other captions are the same as those for Fig. 9.

WONG and CHAU: CRACK COALESCENCE

of a and m. In order to ensure whether the S-mode coalescence dominates when a = b, we have tested three additional specimens with a/b = 608/608 for the three di€erent frictional coecients (m = 0.6, 0.7 and 0.9). As expected, S-mode coalescence prevails in all of these tests. Since we expect that a slight deviation of either a or b from the line a = b would not change the mode of coalescence drastically, therefore, a strip instead of a line is drawn on the a-b space. The actual width of this strip should be found by further experimental studies. Based upon our experimental results, we propose the following relation for b*: b*…a, m† ˆ a ÿ ba,

…6†

where a and b are functions of m and their values are tabulated in Table 3 for m = 0.6, 0.7 and 0.9. The unit of a in Equation (6) is in degrees. The M-regime is bounded by b* < b < bL, where bL is about 82.58. The estimate for bL given here is taken as the average of 908 and 758; more experiments should be carried out if a more accurate value is required. Finally the W-regime is bounded by b>bL, and the boundary of this regime is found independent of a and m (comparing Figs 9±11). Note that the regime boundary b = b* shifts to the left when m increases from 0.6 to 0.7 or 0.9; therefore, a lower value of m is more conducive to the S-mode coalescence instead of the M-mode for intermediate bridge angle (e.g. b = 608). Conversely, a higher value of m favours M-mode coalescence for intermediate bridge angle (e.g. b = 608).

159

Table 3. The values of a and b for various values of m m

a (8)

b

0.6 0.7 0.9

86.25 71.25 80.00

0.375 0.375 0.500

EFFECT OF a, b AND m ON PEAK STRENGTH

Another main purpose of this study is to examine how the strength of rock is a€ected by the geometric setting of the two preexisting cracks and the frictional coecient on these crack surfaces. The prediction by using the Ashby-Hallam model [17], which is summarized brie¯y in the previous section, is also presented here for comparison. Table 2 tabulates 84 observed and predicted peak strengths of our specimens containing di€erent combinations of a, b and m. Except for two sets of data (a/ b = 358/458 and 458/1058 in m = 0.6), it is clear from the table that the peak strength of the cracked specimens increases with the frictional coecient m on the preexisting crack surfaces. It is the further evidence to show that most of our man-made preexisting crack surfaces are indeed in closed contact. The variation of the normalized peak strength with a and b is not apparent from the table. Therefore, pFigs 12±14 plot the normalized peak strength (s1 …pc†/KIC) versus the preexisting crack angle a for m = 0.6, 0.7 and 0.9 respectively. The dotted lines are the theoretical prediction using the Ashby-Hallam model, while the solid lines with symbols are the observed values. In general,

Fig. 12. The normalized peak strengths versus the crack angle a for m = 0.6 and for various values of b, together with the theoretical predictions by using the Ashby-Hallam model [17] (



).

160

WONG and CHAU: CRACK COALESCENCE

Fig. 13. The normalized peak strengths versus the crack angle a for m = 0.7 and for various values of b, together with the theoretical predictions by using the Ashby-Hallam model [17] (



).

our experimental data show that the peak strength of the specimen drops with a when a < a*, achieves a minimum at a = a*, then increases slightly with a when a>a*. Based upon our experimental data, a*

equals approximately 658, depending on the exact value of m. In fact, this observation can be veri®ed analytically using the Ashby-Hallam model [17]. To see this, we ®rst notice that the minimum peak

Fig. 14. The normalized peak strengths versus the crack angle a for m = 0.9 and for various values of b, together with the theoretical predictions by using the Ashby-Hallam model [17] (



).

WONG and CHAU: CRACK COALESCENCE

strength should correspond to the theoretical prediction with the crack inclination being equal to the most favourable orientation af. However, the most favourable orientation of the shear crack in turn depends on the frictional coecient on the sliding shear crack. In particular, Ashby and Hallam [17] showed analytically that af equals 908 ÿ cf, where cf=12tanÿ1(1/m). For m = 0.6, 0.7 and 0.9, we have the most favourable crack inclination being af=60.58, 62.58 and 66.08, respectively; these agree quite well with our observation of about 658. Furthermore, the Ashby-Hallam model should not be applied without modi®cation if the cracking in the bridge area involves a shear mode of failure, such as those shown in Figs 6(a±c). Modi®cation to the theory is also required for the WII-type of coalescence because the failure is not caused by direct coalescence of wing cracks. However, how this modi®cation is to be made requires more detailed analysis in the future. The variations of the peak strength versus the bridge angle b are plotted on Fig. 15 for the case of m = 0.7. For a = 35 and 458, there is one common feature in these curves. That is, there is a distinct minimum at b = 908. In fact, we can show analytically that the minimum peak strength for a constant a appears at b = 908 by substituting L = b sinb/c into Equation (4), then setting dKI/db = 0. For a>508, both theory and experiment show that the peak strength is basically independent of the bridge angle b. Plots similar to those given in Fig. 15 were also drawn for m = 0.6 and 0.9. Since the conclusion from these plots is basically the same as that obtained from Fig. 15, these ®gures are not reported here.

161

For a = 458 (Fig. 15), there seems no well-de®ned pattern in the b-dependency of the peak strength. To verify the correctness of our observation, Fig. 16 plots the peak strength observed in this study together with that by Shen et al. [28] versus the bridge angle b for a = 458 and m = 0.7. The agreement between the two studies is surprisingly good, considering the di€erences in the method of fabrication and loading procedure used. CONCLUSION

We have presented in this paper an experimental study on the coalescence mechanism in and peak strength of a sandstone-like material containing two inclined frictional parallel cracks subject to uniaxial compression. The observed peak strengths are also compared with the predictions by the sliding crack model of Ashby and Hallam [17]. The modelling material is classi®ed as a sandstonelike material, based upon a dimensional analysis using p-factors. In addition, both brittleness and dilatancy of this modelling material are ensured by choosing a correct proportions of barite, sand, plaster and water (2:4:1:1.15 by weight). The two inclined cracks of 12 mm in length are created by inserting stainless steel sheets of various roughnesses into two slots with predetermined orientations and spacings. Three di€erent frictional coecients (m = 0.6, 0.7 and 0.9) are simulated on the crack surfaces. A total of 87 specimens, with di€erent combination of crack inclination a, rock bridge angle b and frictional coecient on the preexisting crack surfaces m, have been cast and tested. Three

Fig. 15. The normalized peak strengths versus the bridge angle b for m = 0.7 and for various values of a, together with the theoretical predictions by using the Ashby-Hallam model [17] (



).

162

WONG and CHAU: CRACK COALESCENCE

Fig. 16. The normalized peak strengths versus the bridge angle b for m = 0.7 and a = 458 together with the results by Shen et al. [28] (



).

main modes of crack coalescence are observed: the shear mode, the mixed shear/tensile mode and the wing tensile mode. More careful examination of the coalescence patterns leads to nine di€erent types of coalescence under these three main modes. A regime classi®cation for the coalescence modes in the a±b parameter space is proposed for m = 0.6, 0.7 and 0.9. In general, shear mode coalescence occurs when b < b*(a, m) = a ÿ ba, where both a and b depend on m (see Table 3 for their values); mixed mode coalescence occurs when b* < b < bL (where bL182.58); and wing tensile mode coalescence occurs when b>bL. Nevertheless, more experimental and theoretical studies are recommended to give a better regime classi®cation of coalescence. The peak strength prediction by Ashby and Hallam [17] compares well with experimental results. Both experiments and theory show that the peak strength increases with m on the preexisting crack surfaces. A minimum is observed at about a = 658 when the peak strength is plotted against the crack inclination a, compared to the theoretical minima at 60.5, 62.5 and 668 for m equal to 0.6, 0.7 and 0.9, respectively. When a>458, both experiments and theory show that the peak strength is more or less independent of the bridge angle b. For a < 458, there seems no wellde®ned b-dependency of our observed peak strengths. However, for a = 458 the b-dependency of our observed peak strengths agrees surprisingly well with the results obtained by Shen et al. [28]. Nevertheless, further modi®cation of the Ashby-Hallam model [17] should be done to give a better prediction of peak

strength, especially when cracking in the bridge involves shear mode of failure. AcknowledgementsÐThe study was supported by the Sta€ Development Program of the Hong Kong Polytechnic University to RHCW. We are grateful to Dr Nick Barton for suggesting this research topic and providing invaluable comments throughout the study. The laboratory assistance by Mr C. W. Leung is also appreciated.

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APPENDIX A For the sake of completeness, we review in this Appendix the key elements of the dimensional analysis or the so-called p-theorem, which is basically a consequence of the so-called ``principle of dimensional homogeneity'' or the principle that only quantities of the same dimension can be added, subtracted or equalized. The p-theorem is a consequence of the supposition that if a physical equation is complete (i.e. all variables of the problem are included), the equation must be true irrespective of changes in measurement units. According to Martins [55], the ®rst application of the dimensional analysis was probably ®rst recognized by Francois Daviet de Foncenex in 1761. The same concept was also evidently used by

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WONG and CHAU: CRACK COALESCENCE

Poisson in 1883, Legrende in 1823 and Rayleigh in 1877 before Fourier published his principle of dimensional homogeneity in 1980, which basically states that all terms of a physical equation must have the same dimension or that every correct physical equation is dimensionally homogeneous [55]. The practical application of the dimensional analysis was not widely recognized until the celebrated ptheorem was published by Buckingham in 1914 [56, 57]. Discussions on the usefulness of p-theorem can be found in Refs. [33, 58]; and a number of useful textbooks on dimensional analysis are also available [59±66].

Table 4. Dimensional matrix of the present problem st M L T

1 ÿ1 ÿ2

sc 1 ÿ1 ÿ2

g 1 ÿ2 ÿ2

L 0 1 0

E 1 ÿ1 ÿ2

KIC 1 ÿ1/2 ÿ2

n

m

0 0 0

0 0 0

According to the p-theorem, if a functional relationship for a physical phenomenon involves n variables (such as sc, st, E, g, L, KIC, n and m in our problem) as F…Q1 , Q2 , . . . , Qi , . . . , Qn † ˆ 0,

…A:1†

then it is always reducible to the following form: f …p1 , p2 , . . . , pi , . . . , pnÿr † ˆ 0,

…A:2†

where r is the number of independent fundamental units needed in specifying the units of all n variables Qi (i = 1, 2, . . ., n) (such as sc, st, E, g, L, KIC, n and m) and pi (i = 1, . . ., n ÿ r) is the so-called pterms or p-factor. These dimensionless p-factors can be formulated as a product of the variables Qj (j = 1, 2, . . ., n) pi ˆ Qk11  Qk22  . . .  Qkmn ,

…A:3†

for i = 1, 2, . . ., n ÿ r, where ki (i = 1, 2, . . ., n) is the unknown exponents to be determined. Say, for example, in our problem, all variables given in Table 1 can be expressed in terms of mass M, length L and time T, which are summarized in Table 4. Let the dimension of any variable Qj be ‰Qj Š ˆ M aj Lbj T cj ,

…A:4†

where j = 1, . . ., n and the square bracket indicates the dimension of the quantity inside. Then, by substituting Equation (A.4) into Equation (A.3)) each p-factor can be expressed as ‰pi Š ˆ M a1 k1 ‡...‡an kn Lb1 k1 ‡...‡bn kn T c1 k1 ‡...‡cn kn :

…A:5†

Thus, the dimensionless requirement for the p-factor requires the powers of M, L and T to vanish, and this leads to the following system of three equations: b1 k1 ‡ b2 k2 ‡ . . . ‡ bn kn ˆ 0, a1 k1 ‡ a2 k2 ‡ . . . ‡ an kn ˆ 0, …A:6† c1 k1 ‡ c2 k2 ‡ . . . ‡ cn kn ˆ 0:

Whenever n is larger than the rank r of the matrix of the coecients ai, bi, and ci, we have more unknowns k than equations. Then, we have in®nite solutions for exponents ki (i = 1, 2, . . . , n) that satis®es the dimensionless requirement; or in other words, we can have in®nite choices for the p-factors, but not all of them are independent. As discussed by Obert and Duval [33], the number of independent pfactors or dimensionless products can be obtained by studying the rank of the matrix formed by any three di€erent columns in Table 4. For our present case, the rank is two, therefore, the expression Equation (A.1) can be written in terms of six independent p-factors (i.e. 8±2 = 6), and f …gL=E, sc =st , E=sc , KIC =…sL1=2 †, , m† ˆ 0:

…A:7†

And, of course, this is only one of the many possible forms. This is how we came up with the p-factors given in Table 1. Dimensional analysis can be applied to experimental studies by reducing the number of tests and to theoretical studies by reducing the speci®c relationship between variables to ®xed power laws. The main application of the p-theorem remains in experimental investigations, as the number of experiments can be reduced drastically and the conditions of model testing can also be made under more attainable conditions. Two experiments (say a prototype and a model experiment) are ``physically similar'' if the p-factors of them are identical (i.e. the same phenomenon is expected to occur) [56, 57]. Therefore, p-theorem provides a useful framework for designing ecient and meaningful experiments. But, of course, the dimensional analysis may also lead to erroneous conclusion if any important variable of the problem is not identi®ed properly. Accepted for publication 9 July 1997