Characterization of microstructures and fracture toughness in five granitic rocks

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International Journal of Rock Mechanics & Mining Sciences 42 (2005) 450–460 www.elsevier.com/locate/ijrmms

Technical note

Characterization of microstructures and fracture toughness in five granitic rocks M.H.B. Nasseria,, B. Mohantya, P.-Y. F. Robinb a

Lassonde Institute, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada b Department of Geology, University of Toronto, Toronto, ON, Canada Accepted 16 November 2004

1. Introduction Fracture toughness is considered a key material property in characterizing fragmentation behaviour of rocks. In principle, therefore, the same should hold true in all rock breakage mechanisms under dynamic loads as well. These include, crushing and grinding and rock fragmentation process in blasting. However, confirmation of this hypothesis is far from proven. This is partly due to the complex procedures required for determining the fracture toughness even under static loading conditions [1,2], and mostly due to the large number of parameters that must affect fracture toughness of a complex material like rock. A considerable body of experimental data on fracture toughness for a variety of rocks is currently available under these loading conditions [1,3]. However, the complex nature of rock in terms of its inhomogeneity and anisotropy makes any generalization between fracture toughness and its micro-properties extremely difficult. The correlation with the latter has not received the same degree of attention as in other metals and alloys [4], and with strength properties in rock [5,6]. Thus before extending the fracture toughness concepts to highly dynamic processes, it is important to examine its relationship with microstructural properties in greater detail. This is especially relevant as the latter are known to play a dominant role in crack growth and propagation under dynamic loading. The present investigation between fracture toughness and microCorresponding author. Tel.: +1 416 978 5969.

E-mail address: [email protected] (M.H.B. Nasseri).

structure in selected granitic rocks is part of a continuing attempt at seeking such a correlation, initially under static loading conditions, and to be extended in future to dynamic loading conditions. Whittaker et al. [3] have shown that physico-mechanical properties of rocks, such as hardness index and fracture toughness can be correlated with each other and may be useful to predict rock failure. It additionally has been observed by the same authors that fracture toughness values (KIC and KIIC) exhibit an approximately linear behaviour with a number of related parameters. These include hardness index, uniaxial compressive strength, point-load strength, flexure strength, Young’s modulus and acoustic wave velocity, whereas Poisson’s ratio appears to be an exception. Brown and Reddish [7] have shown good relationship between fracture toughness and specific gravity over the range examined. Prikryl [8] has elucidated the strength variation of granitic rocks showing uniform mineralogical composition but diverse grain size distribution and/or fabric. He concluded that the strength anisotropy of individual rock is affected by the shapepreferred orientation of rock-forming minerals. The peak compressive strength of rocks has been observed to decrease inversely with the square root of the grain size by Olsson [5] for marble, by Brace [9] for quartzite, and Fredrich et al. [6] for calcite marble. The latter suggest that the crack length is probably the most critical parameter and the effect of grain size on fracture strength is typically rationalized by noting that in rocks the initial crack size, 2c, scales with the grain size, d. Wong et al. [10] concluded that compressive peak strength decreases with the initial crack density for

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fine-grain marble, but remains constant for coarse-grain marble. It is important to note here that all the mechanical properties of intact rocks are a function of microstructural features and petrofabric characteristics, and correlation of two or more mechanical properties with one another needs to be carried out with appropriate understanding of the microstructural and petrofabric responses to certain type of stress systems. So far the data on the effect of micro-fracture density and microstructures such as micro-fracture length, grain size and grain shape, preferred orientation of minerals in magmatic rocks with low anisotropy, such as granite, on fracture toughness is scanty [11]. Study of such a relationship along with other mentioned parameters will help predict the fracture behaviour of rocks and can lead to an improved understanding of the fracture and fragmentation process in rock.

2. Experimental material Five granites were used to study the relationship between fracture toughness values and their microstructural properties. At this stage of investigation, the selection of the granite samples was based strictly on their availability, and visual and textural differences. No attempt was made to identify them in terms of their lithological origins. The granites are thus described as: Pink Granite 1, Pink Granite 2, Grey Granite 1, Grey Granite 2, and White Granite. Detailed petrographic information for all the granites are given in Table 1.

3. Microsturctural investigation Micro-crack observational techniques include (1) dye penetration prior to thin section preparation, (2) radiography and X-ray, (3) SEM and (4) TEM, [12]. In recent years, development of computer-aided image analysis programs has greatly facilitated microstructural characterization through analysis of digital images obtained from the thin sections, [8,13,14]. These new techniques are based on direct measurements of crack length, orientation, grain size and shape measurements from thin sections. These methods provide easier handling of larger amount of data collected and therefore provide a more representative assessment of microstructural properties. Microstructural investigation in this study involved examination of thin sections along three orthogonal planes. Mineralogy, microfabric characteristics and mineral size distribution of the five granitic rocks are given in Table 1. Microstructural studies of these five granitic rocks detailed the variation in terms of mineral size distribution, preferred orientation and fabric shape. They also showed that various degrees of

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intergranular and transgranular micro-cracking density and fracture length exist among the five granites (Table 2). The microstructural properties selected for this study are micro-fracture density, micro-fracture length and grain size as observed in thin sections. In our analysis one screen pixel represents 0.94 mm. Thus the shortest resolvable crack and grain size have a cut-off limit of approximately 1 mm. It has been shown that larger micro-cracks are the first to interact mechanically and thus dominate both the fracture process and the transport properties of the rock, [15,16]. Therefore, limiting the observable crack and grain sizes to 1 mm or larger should not be considered a drawback in the present investigation. Correlation of the rock’s microfabric elements with the mechanical properties such as that of fracture toughness requires knowledge of the original orientation of the rock or reference of orthogonal axis of the coordinate system (X, Y and Z axes) to the internal visible fabric of the rock. In the absence of any information regarding natural in situ orientation of the granitic blocks and lack of visible internal rock fabric, thin sections were prepared along three arbitrarily assigned orthogonal XY, YZ and XZ planes. 3.1. Image analysis of micro-cracks A quantitative analysis of micro-cracks was carried out on digital images of petrographic thin sections captured with a video camera mounted on a standard petrographic microscope. The whole process thus consisted of the following stages: image acquisition, image pre-processing, micro-cracks and grain boundary tracing, measurements with automated image analysis programs, INTERCEPT, and data analysis. 3.1.1. Image acquisition This process refers to the capturing of images from the thin sections (3 cm
2 cm) observed through a video camera attached over a standard polarizing petrographic microscope. More than 30 digital images (72
72 dpi each) from each thin section were captured to construct a mosaic. A total of 450 images were captured from three orthogonal thin sections of the five granites. The mosaics for each section were assembled with the graphic program CorelDraw. 3.1.2. Image pre-processing In CorelDraw, two images were drawn, in two separate layers, by tracing the photographic mosaic and cross-checking on the petrographic thin section at the same time. One layer was of the cracks, both intergranular and intragranular, that were visible in thin section. The other layer contained a drawing of all identified grains, each grain being defined by its closed

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Table 1 Mineralogy and microfabric characteristics of granites Granites Pink Granite-1

Microfabric Magmatic heterogranular

Minerals and size variation Quartz abundant 0.1–1.6 mm Plagioclase 0.1–1.4 mm

K-feldspar Moderately abundant 0.1–1.4 mm Micas 0.2–0.8 mm Grey Granite-1

Magmatic heterogranular

Quartz dominates 0.1–2 mm

Plagioclase mostly Microcline 0.1–2.2 mm K-feldspar 0.1–2.2 mm

Grey Granite-2

Magmatic heterogranular

White Granite

Magmatic heterogranular

Magmatic heterogranular

Microdefects and alteration a

Anhedral grains with lobate boundaries Subhedralb to anhedral grains showing Albite, Microcline twinning Subhedral to anhedral grains

Very low microcracking

Platy cleavaged grains

Common cleavage cracks

Anhedral grains with lobate boundaries preferably oriented Subhedral equidimensional grains, Microcline twinning Anhedral grains

Preferred orientation of high density intragranular microcracking in one plain and very little microcracking in other plains Very low microcracking

Micas, 0.5–1 mm

Thin platy grains mostly Biotite

Quartz moderately abundant 0.2–5 mm

Anhedral grains with lobate boundaries

Plagioclase Less abundant 0.2–2.2 mm

Mica Abundant 0.1–1.0 mm

Common subhedral grains with both Albite and Microcline twinning Large subhedral to euhedral grainsc showing intergrowth with Plagioclase Platy and irregular

Quartz abundant 0.1–3 mm

Anhedral grains

Plagioclase abundant 0.1–4 mm K-feldspar abundant 0.1–7 mm

Subhedral to anhedral grains

Micas 0.2–1.0 mm

Irregular grains

Quartz abundant 0.1–2.5 mm

K-feldspar 0.1–3 mm

Anhedral grains with sutured recystalised grain boundary, Euhedral to suhedral grains with both Albite and Microcline twinning Subhedral grains

Micas 0.01–5 mm

Platy grains

K-feldspar 0.2–2.2 mm

Pink Granite-2

Microstructures

Plagioclase 0.1–3 mm

Anhedral grains

Very low density of intragranular microcracking No microcracking

Low microcracking with alteration, sericitized and chloritised Microcracking along the cleavage plane High density of intragranular microcracking with multi grain cracks, undulatory extinction No microcracking no sericitization

Low microcracking

Common cleavage cracks Very high density of intragranular and microcracking in all the three orthogonal sections with preferred orientation of cracks, some healed cracks observed Low microcracking with numerous filled veins and alterations High intragranular and intergranular microcraking usually of conjugate type highly altered Little microcracking Absence of mirocracks in one of the planes and long intergranular microcracks crosscutting various minerals Very little microcracking but moderately altered Low microcracking, intergrowth with plagioclase, altered with numerous filled and healed cracks Well developed cleavage fractures

a

Anhedral grain: A crystalline solid without well formed faces. Subhedral grain: A crystalline solid having imperfectly developed faces. c Euhedral grain: A crystalline solid with well formed faces. b

sectional boundary and each such closed contour filled with a colour corresponding to the mineral identity of the grain. Fig. 1 shows examples of the resulting images, with the ‘grain layer’ and the ‘crack layer’ superimposed. These two layers are then each converted into a bitmap image in TIFF format. In order to be analysed by the

image analysis programs, the size of the final image has to be less than 2 Mb. 3.1.3. Digital Image Analysis of the cracks The TIFF image of the ‘crack layer’ was analysed by the ‘Intercept method’ [14] using the program INTERCEPT.

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Table 2 Fracture density and average fracture length and fracture toughness values in the granites Rock types

Pink Granite -1 XY-Plane YZ-Plane XZ-Plane Grey Granite-1 XY-Plane YZ-Plane XZ-Plane Grey Granite –2 XY-Plane YZ-Plane XZ-Plane White Granite XY-Plane YZ-Plane XZ-Plane Pink Granite-2 XY-Plane YZ-Plane XZ-Plane

Average fracture density Daver (cm/cm2)

KIC Average (MPa .m1/2 fracture length, mm & & Std.Dev.) Std. Dev.

Average P-wave velocity (m/s)

1.15 1.31 0.67

1.04

0.5570.22

2.3270.07

4580

60

2.80 0.52 0.82

1.38

0.6370.29

2.0470.13

4115

1.08

38

1.31 2.70 2.58

2.1

1.0070.62

1.6570.15

3305

1.2170.83

1.4870.057

4030

1.0

41

0.45 1.30 2.14

1.29

34

27

1.88

48

8.00 3.32 4.06

5.12

1.3070.68

1.2570.175

3765

Quartz

Feldspar

Fracture density (cm/cm2)

Average grain % size (mm)

Average grain % size (mm)

0.72

23

0.59

51

0.90

25

0.87

1.24

35

1.0

1.05

The intercept method calculates parameters such as boundary lengths, densities, orientation distribution, etc., by scanning the image along sets of parallel lines, successively oriented over a 1801 span, and counting the number of boundaries intercepted by these lines. The parameters output by the program for all faces analysed are given in the 2nd–5th data columns of Table 2. The average fracture density, in cm/cm2, is calculated by simple arithmetic averaging of the densities obtained along the three orthogonal planes. 3.1.4. Digital Image analysis of grain sizes Grain sizes were measured and analysed from TIFF images of grains and grain boundaries by the method outlined in [14]. In the present study, the only output used is grain size distribution, shown in Fig. 2 for the five granites studied.

4. Fracture toughness

crack is defined as the fracture toughness. This parameter can be used to predict the nature of fracture onset depending upon the crack size and its sharpness (leading to the stress concentration effects), the stress level applied and the material property. The fracture toughness values for the selected rocks were determined by the [2] recommended method for a three-point bending configuration. It consists of applying a threepoint load on a cylindrical rock sample with a prescribed chevron notch cut perpendicular to the axis of the sample. The sample dimensions in these tests were 30 mm in diameter and 120 mm in length. A MTS loading frame was used in conjunction with the Teststar SX software to load the samples. The fracture toughness value is calculated on the basis of the following parameters: sample diameter, depth of notch, support span and load at failure. In this geometry, KIC in (MPa. m1/2) is given by K IC ¼ Amin F max =D1:5 ,

(1)

where The material property associated with the ability to carry loads or resist deformation in the presence of a

Amin ¼ ½1:835 þ 7:15ao =D þ 9:85ðao =DÞ2 S=D,

(2)

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Fig. 1. Minerals and microcracks traced for (A) Pink Granite-1, XY Plane, (B) Grey Granite-1, YZ plane, (C) Grey Granite-2 (D) Pink Granite-2 and (E) White Granite, XZ plane.

where the failure load, Fmax, is in kN, the specimen diameter, D is in cm, ao is the chevron tip distance from specimen surface (0.15 D), S is the distance between the two supports in the 3-point-bending beam with the Chevron notch and the factor Amin is dimensionless. The KIC determined based on average values of 5 to 7 tests and the standard deviation for the rock types are

given in Table 2. It is to be noted that the fracture characteristics were measured along three orthogonal planes and expressed as arithmetic mean fracture density. The fracture toughness values on these rocks were measured along several arbitrary planes (but not necessarily coincident with the set planes for microstructure measurements) with the assumption that it would reflect the effect of the general crack size and

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30 20

200 10 0 0.00

(A)

0.04

0.08 0.12 0.16 Grain Size, cm

25 300

20 15

200

10 100

0 0.20

0 0.00

5 0 0.05

(B)

0.10 0.15 Grain Size, cm

0.20

15 200 10 100

5

0 0.00

(C)

Cumulative Number of Grains

20

300

Number of Grains Per Size Class

Cumulative Number of Grains

150

0.10

0.15

100

50

0 0.00

0 0.05

1000

0.20

0.05

0.10

0.15

(E)

Grain Size, cm

0.20 0.25 Grain Size, cm

0.30

0.35

Number of Grains Per Size Class

400

30

400

455

Number of Grains Per Size Class

40

Cumulative Number of Grains

Cumulative Number of Grains

50 600

Number of Grains Per Size Class

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0 0.40

50 300

200

30

20 100

Number of Grains Per Size Class

Cumulative Number of Grains

40

10

0

0 0.0

(D)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Grain Size, cm

Fig. 2. Histograms showing the grain size distribution for all the minerals, (A) Pink Granite-1, (B) Grey Granite-1, (C) Grey Granite-2, (D) White Granite and (E) Pink Granite-2. (The curve is a cumulative curve showing the number of grains that are greater than the size in the abscissa).

density. The respective P-wave velocities in the specimen samples shown in Table 2 are given only as a reference. The seismic velocities were measured in block samples by ultrasonic techniques along several representative

directions to yield the average P-wave velocity in the rock type. No attempt has been made, however, to correlate these values with either fracture characteristics or fracture toughness at this time.

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5. Results The petrofabric of the granitic samples is illustrated in Fig. 1A–E, along certain specific planes. They show a wide range of grain size and mineral distribution. More interestingly, there appears to be no consistency between mineral composition and the presence of intergranular or intragranular micro-cracks. However, in general, contrary to expectation, micro-cracks appear to be more numerous in quartz grains than feldspar. The same has also been observed in Westerly granite. This has been attributed to the relatively large compressibility of quartz [13]. The grain size distribution for the five rock types is shown in Figs. 2A–E. The curve in these figures is a cumulative curve showing the number of grains that are greater that the size in the abscissas. The mineral grain size of Pink Granite-1 varies from 0.1 to 1.8 mm (Fig. 2A) with the average quartz grain size of 0.72 mm and the average feldspar grain size of 0.6 mm, with feldspar being the dominant mineral (51%) followed by quartz (23%). The micro-cracks in the specimen rock are of intragranular type and are confined within quartz and feldspar grains, having a maximum length of 1 mm and average length of 0.55 mm, (Table 2). Grey Granite-1 is characterized by mineral grain size ranging from 0.25 to 2 mm (Fig. 2B), with an average quartz grain size of 0.9 mm, which makes up 25% of this rock. The average feldspar grain size for the rock is 0.87 mm and is the dominant mineral (60%). The microcracks are of intragranular type and are found in both the quartz and feldspar grains. The preferably oriented micro-cracks have an average length of 0.63 mm with maximum length of 1.4 mm cutting through the larger quartz grains. The Grey Granite-2 shows mineral grain size distribution of 0.1 to 2.5 mm (Fig. 2C) with characteristic euhedral (a crystalline solid with well formed faces) feldspars of equidimensional type with an average grain size of 1.1 mm. This can be compared to anhedral (a crystalline solid without well formed faces) quartz grains, which have an average size of 1.24 mm. Unlike the two previous rocks, the grain–grain boundary contact type in this rock is mostly of straight line contact type, mainly due to euhedral nature of the feldspar grains. The Feldspars (38%) are slightly more abundant than quartz minerals (35%). The average micro-crack length in this rock is of the order of 1.0 mm, which is almost twice as great as the two earlier mentioned rocks. The micro-cracks are of intergranular in nature, and some of them are greater than 3 mm which is larger than some of its grain size (Fig. 1C). The White Granite, unlike all the other four rocks, is characterized by extreme variation in grain sizes (Fig. 2E). As is evident from Fig. 1E, quite a large number of smaller recrystalized quartz grains

(0.01–0.20 mm) are found to be concentrated between much larger quartzofeldspatic grains (1.0–3.5 mm) suggestive of a shear zone. The quartz content is about 34% and that of feldspar is 41%. Though the average micro-crack length in White Granite is 1.21 mm, this rock is characterized by extremely long micro-cracks (5 mm long), which is easily observable on the polished surface of the rock specimens. These long itergranular micro-cracks run mostly perpendicular to the long axis of quartz grains. The Pink Granite-2 is found to be much coarser on the average compared to the other granites, with its mineral grain sizes varying from 0.1 to 7.0 mm (Figs. 1D and 2D). Some of the long feldspar grains show recrystalized or healed micro-cracks. The average feldspar grains size in this rock is about 1.88 mm and the average quartz grains are 1.05 mm. It is made up of 48% feldspar and 27% of quartz. This rock also contains the highest average micro-crack length of 1.3 mm and they are of intergranular type but mostly found in quartz grains. The maximum micro-crack length is in the order of 4.5 mm. It shows an average micro-fracture length of 1.3 mm with a high value of standard deviation of 0.68, presumably due to its nonuniform micro-fracture length and much higher number of micro-fractures (50%) above 1 mm up to a value of 4.5 mm. Fig. 3 shows the variation of micro-fracture length versus the number of micro-fractures for the five granites under study for certain specific planes. These figures clearly show the respective variations in microfracture characteristics for each rock type. The relatively large variation in both the Grey Granite and the Pink Granite-2 compared to the others is also to be noted. For the planes mapped, the Grey Granite-2 may be considered to be somewhat different than the others, as the micro-cracks in this case appear to have an approximate bimodal distribution. The above results along with the corresponding fracture toughness values are summarized in Table 2. It should be noted that the latter were measured with no particular reference to axes, in contrast to the microstructural measurements such as fracture density and grain size distribution. However, in view of the number of tests performed, the fracture toughness values shown in Table 2 must be considered average for each rock type. Therefore, any comparison between fracture toughness and the microstructural characteristics at this stage can only be done on the basis of their respective averaged values. The average fracture length for each rock type is obtained by direct measurements on the mosaic and by dividing the total length by the total number of the cracks. The fracture toughness KIC values vary from a low value of 1.25 MPa m1/2 for the Pink Granite-2 to 2.32 MPa. m1/2 for the Pink Granite-1.

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(A)

(B)

(C)

(D)

457

(E) Fig. 3. Histograms showing variation of microfracture length versus the number of micro-fractures for (A) Pink Granite-1, XZ plane, (B) Grey Granite-1, XZ plane, (C) Grey Granite-2, XZ plane, (D) Pink Granite-2, YZ plane and (E) White Granite, XY plane.

The correlation between average fracture densities with its corresponding fracture toughness value is shown in Fig. 4. Except for the White Granite, the correlation appears to be good, as fracture toughness is seen to systematically decrease with increasing average fracture density. However, the correlation between fracture toughness and the average micro-fracture length appears to be much better, with no anomalous results in all the five rock types (Fig. 5). The reason for this has been analysed further by examining the dimensional characteristics of micro-fractures in the various rock types. Figs. 6 and 7 illustrate two such cases for the Grey Granite-1 and the White Granite. These show that the Grey Granite-1 is characterized by higher fracture density than the White granite along certain specific planes (e.g. XY-plane for the former and XZ-plane for the latter). However, in terms of average

Fig. 4. Variation of fracture toughness with average micro-fracture density for the five granites.

fracture length, the White Granite shows an average fracture length, which is nearly twice that of the Grey Granite-1. Fig. 8 shows some of the anomalously long transgranular micro-cracks along its XZ plane. A

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Fig. 5. Variation of fracture toughness with average micro-fracture length for the five granites.

Fig. 8. Anomalously long transgranular micro-cracks in White Granite cutting across large and recrystalized smaller quartz grains along XZ plane, Q ¼ quartz, F ¼ feldspar.

Fig. 6. Characteristics of micro-cracks in thin section for Grey Granite -1 along XY plane.

similar analysis shows that the average fracture length in Pink Granite-2 is much higher than the rest of the rock types. The correlation between fracture toughness and microstructure has been shown in a modified plot in Fig. 9. The fracture toughness is plotted against the inverse of the square root of the average grain size (d) and average fracture length (FL) for all the rock types studied. Both show excellent correlation, especially with KIC against average fracture length that yields a correlation coefficient of 0.97. This essentially mirrors the correlation shown in Fig. 5 for KIC against average fracture length.

6. Discussion

Fig. 7. Characteristics of micro-cracks in thin section for White Granite along XZ plane.

It is assumed that the existing correlations between strength and microstructural properties should also be reflected in dealing with fracture toughness [6,17,19]. The excellent agreement shown in this investigation between fracture toughness (Mode 1) and inverse square-root of grain size is in accord with those obtained by previous investigators dealing with strength properties [5]. The corresponding porosity-strength correlations [18] have not been attempted here, mainly due to the extremely low porosity values exhibited by these granitic samples. The latter fact also makes the grain size correlation more obvious as claimed in [18], which states that in low porosity rocks, crack initiation stress is extremely sensitive to mean grain size and initial flaw

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Fig. 9. The correlation between fracture toughness and microstructures, the inverse of the square root of the average grain size and average microfracture length for the five granites.

length is shown to approach the mean grain size value. It is possible to theoretically estimate the critical flaw size from the measured values of fracture toughness and the applied stress for comparison with observed microfracture dimensions. This constitutes part of the continuing investigation in the present study, although such attempts in the past have shown wide discrepancies between calculated and observed values in some cases [15]. It should also be noted that the micro-fracture dimensions studied here consist of millimetre or submillimetre scale. In contrast, it has been shown, at least in some alloys, that fracture toughness decreases with decrease in grain size [4]. Finally, the presence of a fracture process zone is an important attribute of macro-crack growth that is characteristic of the fracture toughness measurement technique employed in this study. Delineation of this zone and the study of its effect on fracture propagation properties also constitute essential aspects of a continuing investigation.

fracture toughness in the rock types tested. The correlation has been shown to be particularly good between fracture toughness and average micro-fracture length (or the inverse of the square root of average fracture length). The combination of high microfracture density and length is found to be the single most significant parameter in lowering fracture toughness value. The anomalous behaviour of the White Granite in the correlation curve can be explained based on its significantly higher micro-crack length than the other granites under investigation. It shows that reliance on only one or the other of these two parameters can lead to erroneous conclusions on the role of microfracture characteristics. Quantitative evaluation of mineral type, grain size, and shape along specified planes, measurement of fracture toughness exactly along the same planes, and tracking the resulting fracture path vis-a`-vis micro-cracks are expected to shed further light on the relationship between physical properties and mechanical properties such as fracture toughness of rocks.

7. Conclusions The results of studies on micro-fracture density, micro-fracture length and fracture toughness show a good correlation among these parameters, and confirms the belief that micro-fracture density and micro-fracture length are major contributors to the measured value of

Acknowledgements The authors wish to acknowledge the contributions of Dr. U. Prasad in the measurement of fracture toughness and through many helpful discussions. They are also

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grateful to the Science and Engineering Research Council of Canada and the Ontario development challenge fund for their financial support.

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