Geomech. Geophys. Geo-energ. Geo-resour. https://doi.org/10.1007/s40948-018-0080-z
ORIGINAL ARTICLE
Numerical simulation and interpretation of the grain size effect on rock strength Qinglei Yu
. Wancheng Zhu . P. G. Ranjith . Shishi Shao
Received: 13 July 2016 / Accepted: 27 October 2016 Springer International Publishing AG, part of Springer Nature 2018
Abstract Although the grain size effect on rock’s physical–mechanical properties has been widely studied, little attention has been paid to the reasons for this phenomenon. In this paper, an attempt is made to interpret the effect of grain size on uniaxial compressive strength (UCS) from the point of view of mechanics. Based on the microscopic images of rock with different mineral grain size, actual mineral types and grain sizes were identified and characterized using digital image processing techniques and microstructure analysis. By mapping this real rock microstructure into a well-established rock failure process analysis code, numerical specimens were built to model the
grain size effect on the UCS. The results showed that considering only mineral type and grain size, the numerical method can reveal the grain size effect on UCS. Further, an ideal conceptual model is proposed to clearly interpret and discuss the grain size effect on strength. It was found that grain size results in different stress concentrations within rock. Under identical loading conditions, the bigger the grain size is, the stronger the stress concentration is. This different mechanical response can eventually affects the UCS of the rock, resulting in the grain size effect on rock strength. Moreover, fractal geometry is also introduced to quantify rock microstructure with different grain sizes. The results show that there exists a good relationship between the fractal dimension and the grain size, which can describe the grain size effect on rock strength both quantitatively and effectively.
Q. Yu (&) W. Zhu Center for Rock Instability and Seismicity Research, Northeastern University, Shenyang 110819, Liaoning, China e-mail:
[email protected]
Keywords Grain size effect Uniaxial compressive strength Interpretation Fractal dimension Digital image processing
W. Zhu e-mail:
[email protected]
1 Introduction P. G. Ranjith S. Shao Deep Earth Energy Laboratory, Department of Civil Engineering, Monash University, Building 23, Melbourne, VIC 3800, Australia e-mail:
[email protected] S. Shao e-mail:
[email protected]
At the grain-scale level, intact rock consists of assembled mineral particles (grains) with different sizes and the interfaces between the grains (grain boundaries and micro-fractures). Grain size is one of
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the most important micro-structural factors affecting the mechanical behavior of rocks (Suorineni et al. 2009). Therefore, the effect of grain size on rock physical and mechanical properties has been widely studied by several authors. The effect of grain size on physical properties mostly focused on granitic rock. Mineral grain size of rock can influence linear thermal expansion coefficients (de Castro Lima and Paraguassu´ 2004), magnetic properties (Ge and Liu 2014), relative brittleness index (Yilmaz et al. 2009), and even the surface roughness of rock joints (Kabeya and Legge 1997). Moreover, for polyphase rocks the mineral grain size distribution also influences its rheology (Heilbronner and Bruhn 1998). These effects may be related with the properties of mineral and the rock forming process. However, it was also found that the grain size can influence crack initiation stress and propagation in rocks (Hatzor and Palchik 1997; Eberhardt et al. 1999). In general, fracture initiation stress is inversely related to mean grain size. In addition, this effect on fracture initiation stress partially depends on the porosity of rock, as well as rock type. Concerning the effect of grain size on rock strength, it has been long accepted that the strength of brittle rocks decreases with grain size, e.g. linearly with the mean grain size in carbonate rocks (Hugman and Friedman 1979) or inversely with the inverse square root of the grain size for marble (Olsson 1974), quartzite (Brace 1961), and limestone (Fredrich et al. 1991). Moreover, the fatigue strength of rock also decreases almost linearly with mean grain size (Singh 1988). Recently, Wong et al. (1996) studied the relationship between grain size and initial length of microcracks, and further illustrated the influence of grain size on the uniaxial compressive strength (UCS) for Yuen Long marbles. Hatzor et al. (1997) studied the influence of microstructure (i.e. mean grain size and porosity) on the crack initiation stress and ultimate strength of dolomites, and found that both crack initiation and ultimate stresses were relatively insensitive to mean grain size, but the nature of grain arrangement imposes a constraint on the micromechanics of fracture growth and further controls the ultimate strength. Prˇikryl (2001) also demonstrated that the variability of the UCS of low porosity magmatic rocks (e.g. granites) was closely related to changes in the mean grain size of major rock-forming
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minerals. The strength of these rocks increases with the decreasing grain size of rock-forming minerals. More recently, Shao et al. (2014) investigated the effect of cooling rate and constituent grain size on the mechanical behavior of heated Strathbogie granites and observed that the cooling rates have different influences on the strength and elastic modulus of Strathbogie granites with different grain sizes. However, there is little research examining why grain size influences the strength of rock. Brace (1961) linked the relationship between grain size and strength to Griffith’s theory (Griffith 1920). Griffith’s crack length is considered to be proportional to the grain size. By increasing the grain size, grain boundaries, which are widely assumed to be the predominant source of stress-concentrating flaws, increase. Consequently, with increasing crack length, strength decreases. This may be one reasonable illustration. In fact, besides this factor in the aspect of rock microstructure, rock strength should yet depend on mechanical responses of constituents at mesoscale and stress concentration induced by the grain shape and size. In this paper, therefore, another attempt is made to interpret the grain size effect on rock strength from the point of view of mechanics in order to further clarify this phenomenon. Based on microscopic images of thin sections of granites with different grain size compositions, a digital image processing (DIP) techniques based numerical code (i.e. RFPA) is used to build the corresponding numerical specimens with different grain sizes to model the grain size effect. Then a conceptual model that takes grain size into account is proposed to clearly interpret this effect.
2 Numerical simulations of grain size effect The numerical models for grain size effect were set up based on the experimental data using digital image processing (DIP) techniques. Then numerical simulations under uniaxial compressive conditions were carried out using RFPA code and the grain size effect is discussed. 2.1 Experimental data In the experiments, Strathbogie granite was used to fabric specimens, which was collected from the region
Geomech. Geophys. Geo-energ. Geo-resour.
surrounding the Strathbogie batholith, which is a large composite body covering more than 1500 km2 in area 150 km north-northeast of Melbourne in central Victoria. Strathbogie granite is a high-level, discordant, granitoidlate Devonian intrusion in south-eastern Australia. Three categories of Strathbogie granite [i.e. fine-grained granite (FG), medium-grained granite (MG) and coarse-grained granite (CG)] were selected to prepare cylindrical specimens with a diameter of 22.5 mm and a length-to-diameter ratio of approximately 2.0, as shown in Fig. 1. A series of uniaxial compressive tests were carried out to examine the changes of properties under different cooling rate conditions and considering the grain size effect. The experimental results have been published (Shao et al. 2014), in which the effect of grain size on the mechanical properties was also discussed. In this paper, the aim is to interpret the effect of grain size on strength from the point of view of mechanics. The fundamental geo-mechanical properties of the three types of Strathbogie granite simulated in this study are summarized in Table 1. Before the tests, the grain sizes of FG, CG and MG granites were determined using microscopic imaging. Some microscopic images of thin sections of the three types of granites are shown in Fig. 2. According to the length scale of the images indicated by the white line, the imaging area is about 0.7352 9 0.9834 mm2 (height 9 width). From the microscopic images (Fig. 2), we observe that the microstructural fabric of granite is composed of mineral grains, voids, crystal defects and grain boundaries.
2.2 Numerical model In this section, the microscopic images of FG granite and CG granite are processed and analyzed to identify the shape of grains, obtain the morphological information and further statistic the grain size distribution. Next, two images with the actual size of 0.7352 9 0.9834 are assumed to be the representative volume element (RVE) in the FG granite and CG granite respectively and the morphological information is mapped into RFPA code. The homogenization method (Liu et al. 2004) is used to build the corresponding numerical specimens with different grain size distributions. 2.2.1 Image analysis Image analysis provides a powerful tool to simplify the microstructure, identify the mineral composition and obtain the morphological information. From Fig. 2a, c, it is observed that each mineral grain occupies a certain area with similar color properties. Therefore, the segmentation method based on region growing with a seed and a threshold is developed to automatically scan the microscopic images and identify the mineral grains. This segmentation method first selects a pixel as the seed point, then examines neighboring pixels of the initial seed point and determines whether the pixel neighbors should be added to the region according to the threshold. The process is iterated and the region grows until the boundaries with distinct color properties. The pixels in the region are assigned the same label and assumed to be a mineral grain. In addition, DIP techniques, i.e. median filter and noise removal, are also used, combined with the segmentation method, and a small
Fig. 1 Close views of typical a FG, b MG and c CG Strathbogie granite specimens
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Geomech. Geophys. Geo-energ. Geo-resour. Table 1 Physical–mechanical properties of the three types of Strathbogie granite Fine-grained (FG)
Medium-grained (MG)
Coarse-grained (CG)
Rock type
Micro-granite
Grano-diorite
Granite
Bulk density, kg/m3
1805.7
2699.7
2703.4
Compressive strength, MPa Elastic modulus, GPa
215.97 8.97
154.97 8.57
112.97 8.87
Porosity, %
2.159
0.914
1.156
Fig. 2 Microscopic images of thin sections. a FG granite, b MG granite and c CG granite
part of the work is finished manually due to the complex color properties of the heterogeneous microstructure. Figure 3 shows the results for the mineral grain identification. Compared with the original images (Fig. 2a, c), the mineral information in the resulting images is consistent with that in the microscopic images, ignoring a few microcracks within mineral grains. During image processing for mineral identification, the size of each mineral grain is also recorded by counting the number of pixels. Therefore, the equivalent diameters of mineral grains are calculated Fig. 3 Morphological information on mineral grains of a FG granite and b CG granite identified by DIP techniques
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by assuming that the mineral grains are round and their own areas are equal to the corresponding regions with real shapes. By statistic analysis, the grain size distribution of FG granite and CG granite are determined, as shown in Fig. 4. The results of grain size calculations revealed that the grain sizes (i.e. the equivalent diameters) of FG granite are distributed mainly from 0.12 mm to 30 lm, with a few grains larger than 0.15 mm, while the grain sizes of CG granite are distributed mainly from 0.2 mm to 90 lm, with a few larger grains over 0.3 mm. The average grain sizes are 63.4 lm 4 for the FG granite and
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30
Number of grains
Fig. 4 Grain size distributions of a FG granite and b CG granite
25 20 15 10 5 0
0
50
100
150
200
Equivalent diameter of grains /μm
(a) Number of grains
8 7 6 5 4 3 2 1 0
0
100
200
300
400
Equivalent diameter of grains /μm
(b) 138.92 lm for the CG granite. In fact, pores with different sizes exist in the two types of granite (Shao et al. 2014). However, the purpose of the present study is to interpret the grain size effect on the compressive strength and the porosities of the two types of granite are thus not taken into account in the numerical modeling. 2.2.2 Model set-up Homogenization theory was used to build numerical specimens with different grain sizes. In homogenization theory, it is usually assumed that a composite material is locally formed by the spatial repetition of very small microstructures, i.e. microscopic cells, when compared with the overall macroscopic dimensions of the structures of interest (Seo et al. 2002; Sanchez-Palencia 1980). The small microscopic cell in homogenization theory should have the same average mechanical response as the actual heterogeneous material, i.e. the RVE of the heterogeneous material (Liu et al. 2004). In the present study, since
the microscopic images of FG granite and CG granite contain abundant microstructural information, they are assumed to be the REV of FG granite and CG granite. Two vertical sections of cylindrical specimens with a ratio of about 2 between the height and the diameter were then constructed on the basis of the RVE. Using mineral composition analysis, the mineral grains identified by DIP techniques were classified basically into three types on the basis of different colors, i.e. quartz, feldspar and mica, as shown in Fig. 5. By counting the pixels occupied by each mineral, the proportions of the three minerals (i.e. quartz, feldspar and mica) were determined respectively to be 24.3, 40.1 and 12.2% for FG granite, and 23.7, 54.4 and 10.3% for CG granite. Other areas represented in pink are assumed to be the interfacial transition zones (ITZs), and their proportions are 23.4% for FG granite and 11.7% for CG granite. In order to model the physical–mechanical behavior of Strathbogie granites with different grain sizes, two macroscale granite specimens were constructed
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Geomech. Geophys. Geo-energ. Geo-resour. Fig. 5 Mineral composition analysis for RVE: (a) FG granite and (b) CG granite
(a) by the spatial repetition of a representative volume element (RVE) composed of rock-forming minerals according to homogenisation theory. The size of the RVE is equal to that of the microscopic images, i.e. 0.7352 mm 9 0.9834 mm (width 9 height). Before the construction, however, the RVE (Fig. 5) represented by the image needs to be transformed into a vectorized one, which can be accepted by numerical methods. In this regard, a simple method was developed to complete the transformation, in which the pixel within the image is considered as a square element in RFPA. Therefore, the coordinates of the four corners can be calculated by the ratio between the image size and the physical size. The detailed illustration of the vectorized transformation is derived from Yu et al. (2014). Figure 6 shows macroscale numerical specimens of FG granite and CG granite in terms of Young’s modulus, and the size is 2.95 mm 9 5.8816 mm. This means that each specimen consists of 3 9 8=24 RVEs. To consider more details of rock microstructure, the domain of the specimens is divided into 350 9 700 = 245,000 elements. In the numerical specimens, the gray level indicates the magnitude of the Young’s modulus of the elements. The higher the brightness of one element, the bigger the Young’s modulus is. The mechanical properties of the three minerals are listed in Table 2 and the mineral grains are considered as homogeneous material. The determination of their mechanical properties is by reference to Liu et al. (2004). Moreover, some microcracks within mineral grains, as shown in Fig. 2, cannot be fully considered in simulation due to computational limitations. Therefore, the UCS of these single crystals is reduced proportionally for comparison with the experimental results. After the numerical specimens of granite are constructed, the uniaxial compressive tests are simulated using RFPA. The external displacement at the rate of 0.0001 mm/step is applied on the top end step-
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Mica
Mica
Feldspar
Feldspar
Quartz
(b)
Quartz
by-step until the specimens lose their load-bearing capability and its initial value is 0.005 mm. 2.3 The constitutive law of element In RFPA code, an elastic damage constitutive law is proposed and incorporated into a finite element code for simulating the failure process in rock (Zhu and Tang 2004; Tang 1997). The damage constitutive relation of an element under uniaxial compressive and tensile stress is illustrated in Fig. 7. The element is initially considered elastic, with elastic properties defined by Young’s modulus and Poisson’s ratio. Correspondingly, the stress–strain relation of an element is considered linear elastic until the given damage threshold is attained, and is then modified by softening. The maximum tensile stress criterion and the Mohr–Coulomb criterion are applied to check the stress state of an element in order to capture the principal modes of damage and fracture of rock. The tensile stress criterion is considered preferentially. When the tensile stress in an element reaches its uniaxial tensile strength, ft0, r3 \ ft0
ð1Þ
The damage variable D of the element under uniaxial tension can be expressed as: 8 e [ et0 ; > e : 1 e etu : where r3 is the minor principal stress, e is the tensile strain, and k is the residual tensile strength coefficient, which is given as ftr = kft0. The parameter et0 is the strain at the elastic limit, strain. etu is the ultimate tensile strain of the element, describing the state at which the element would be completely damaged. The ultimate tensile strain is defined as etu = get0, where g is the ultimate strain coefficient.
Geomech. Geophys. Geo-energ. Geo-resour. Fig. 6 Numerical model for a FG granite and b CG granite
3×0.9834=2.9502mm 0.9834mm
8×0.7352mm=5.8816mm
0.7352mm Numerical REV for FG granite built by RFFA based on DIP techniques
(a) 3×0.9834=2.9502mm 0.9834mm
8×0.7352mm=5.8816mm
0.7352mm Numerical REV for CG granite built by RFFA based on DIP techniques
(b) When an element is under compressive and shear stress, the Mohr–Coulomb criterion, expressed as
follows, is chosen to define the second damage threshold:
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Geomech. Geophys. Geo-energ. Geo-resour. Table 2 Mechanical and thermal parameters of three mineral grains
Parameter
Quartz
Feldspar
Mica
ITZ
Young’s modulus/GPa
95.6
69.7
88.1
35
Poisson’s ratio
0.079
0.301
0.248
0.25
Uniaxial compressive strength/MPa
700
600
300
150
Frictional angle
60
40
30
30
Ratio of compressive to tensile strength
15
12
10
10
Fig. 7 Elastic damage constitutive law of element subjected to uniaxial stress. ft0 and ftr are uniaxial tensile strength and residual uniaxial tensile strength of the element, respectively, and fc0 and fcr are uniaxial compressive strength and residual corresponding strength of the element, respectively
r1
1 þ sin u r3 fc0 1 sin u
ð3Þ
In this case, the expression for the damage variable D can be described as: ( 0 e\ec0 ; kec0 D¼ ð4Þ e ec0 : 1 e where r1 is the major principal stress, and u is the internal friction angle of the mesoscopic element. e is the compressive strain, ec0 is the compressive strain at the elastic limit, k is the residual strength coefficient, and fcr/fc0 = ftr/ft0 = k is assumed to be true when the element is under either uniaxial compression or tension. After the D value is determined, the elastic modulus of damaged material is defined as follows: E ¼ ð1 DÞE0 ;
ð5Þ
where E and E0 are the elastic moduli of the damaged and the undamaged material, respectively. Accordingly, the elastic modulus of the damaged element at
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different stress or strain levels can be calculated. Therefore, the progressive failure process of a brittle material subjected to gradually increasing static loading can be simulated. To avoid the problems caused by zero elastic modulus, when D = 1.0 in Eq. (5), the elastic modulus of a damaged element is specified as a small number, such as 1.0 9 10-5 MPa. In addition, the failure (or damage) in every element is assumed to be the source of an acoustic event because the failed element must release its elastic energy stored during the deformation. Therefore, by recording the number of damaged elements and the associated amount of energy release, RFPA is capable of simulating acoustic emission (AE) activities, including the AE event rate, magnitude and location. 2.4 Simulated results 2.4.1 The effect of grain size on UCS Figure 8 shows the numerically obtained stress–strain curves of FG granite and CG granite, and the
Geomech. Geophys. Geo-energ. Geo-resour.
180
FG granite CG granite
120 90
60 30
(a)
0 0
0.0005
0.001
0.0015
0.002
0.0025
Strain 250
FG granite CG granite
St ress /MPa
200
2.4.2 The effect of grain size on failure process
150 100 50
0
(b)
0
0.005
0.01
0.015
0.02
0.025
0.03
Strain
Fig. 8 Stress–strain curves a numerical and b experimental
comparison with the experimental results. It is observed that the magnitudes of the peak stress from the numerical tests are 151.90 MPa for FG granite and 121.25 MPa for CG granite. There exists a difference of 30.65 MPa between the two types of granite, which is clearly caused by the different grain size of minerals. However, the experimental results for the two types of granite are 217.26 and 118.69 MPa, and the difference is 98.57 MPa. One of the reasons that the grain size effect in the simulations is weaker than that in the experiments may possibly result from the interfacial transition zones. The proportion of the interfacial transition zones in the numerical specimen of FG granite is double that in the numerical specimen of CG granite, and the influence of the interfacial transition zones may be enlarged during the simulation. Furthermore, the effect of grain size can be modeled by the RFPA code, although the difference of the UCS between the numerical specimen of FG granite and CG granite is smaller than that obtained experimentally. By comparing the numerical stress–
During the simulation of uniaxial compressive testing, the number of damaged elements and the associated amount of energy release can be automatically recorded by RFPA to simulate the acoustic emission (AE). Figure 9 shows the comparison of the AEs between the FG and the CG granite at each loading level. Due to the brittle failure of the specimens, the peak AE counts for the FG and the CG granite are 40373 and 33192, respectively. To depict more information on AE variation, the peak AE counts are ignored in Fig. 9, as shown in the numbers. For the FG granite the damage initiated at the loading displacement of 0.009 mm, while for the CG granite, the damage initiated at the loading displacement of 0.0075 mm. Since the mechanical properties of each constituent input into the numerical specimens
33192
200
40373
FG
160
CG
AE count s
St ress /MPa
150
strain curve of CG granite with the experimental result, the UCS obtained numerically approaches that obtained experimentally, which further calibrates the mechanical parameter input into the numerical model to some extent. However, it should be noted that the elastic modulus obtained numerically is quite different from that obtained experimentally. The numerically obtained elastic moduli for the FG granite and CG granite are 72.53 and 70.98 GPa, which is in a normal range for granite. However, the experimental results are 8.972 and 8.87 GPa (Shao et al. 2014), which may be abnormal for granite. It could be related with the measuring equipment of deformation.
120 80 40 0
0
0.005
0.01
0.015
Loading displacement /mm Fig. 9 Comparison of simulated AE counts between the FG and the CG granite during failure process
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were identical, it is concluded that the stress of some local elements within the CG granite is slightly higher than that within the FG granite at the same loading level, which is definitely caused by the stress concentration. With the displacement increasing, the stress of these elements within the CG granite first reaches their strength and causes the elements to be damaged (e.g. the loading displacement equals 0.0075). That is to say, the large mineral grains result in a strong stress concentration, and such a stress state eventually affects the UCS of rock. Figures 10 and 11 depict the failure process of the FG and the CG granite in terms of elastic modulus. The FG granite failed at the loading step of 78 (the loading displacement was 0.0127 mm) and the CG granite failed at the loading step of 56 (the loading displacement was 0.0104 mm). The numbers (e.g. 4 in Step78-4) behind the dash means that the individual plots are induced at the same loading level. Therefore, the symbols in Figs. 8 and 9 also indicate the brittle failure of the numerical specimens. From the figures, the failure processes of the two types of granite are similar. The microcracks mostly initiate at the boundaries parallel to the loading direction between mineral grains and scatter within the specimen at the pre-peak stage. After the peak, the microcracks become clearly localized, continue to propagate and coalesce, and finally form macroscopic shear bands. Both inter-
Fig. 10 Failure process of FG granite
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granular and trans-granular cracks occur during the failure process, but inter-granular cracks dominate specimen failure.
3 Interpretation of grain size effect using a conceptual model 3.1 The conceptual model To clearly interpret the grain size effect on UCS, an ideal conceptual model which can take the grain size into account is proposed to explain why the grain size can affect the strength from the point of view of mechanics. The conceptual model is also constructed according to homogenization theory. Therefore, a conceptual RVE composed of the main rock-forming minerals is first defined artificially, as shown in Fig. 12a. Similar to the Strathbogie granite, the conceptual RVE is assumed to be composed of three minerals. In Fig. 12a, quartz grains are represented by light gray, feldspar grains by gray and mica grains by black. The three minerals have identical sizes within the RVE but different sizes within the conceptual specimens. The proportions of the three minerals (i.e. quartz, feldspar and mica) in the RVE are 32, 48 and 20%. Due to the difficulty in determining the mechanical properties of
Geomech. Geophys. Geo-energ. Geo-resour.
Fig. 11 Failure process of CG granite Fig. 12 The conceptual model using a conceptual RVE a The conceptual RVE and b a conceptual specimen with the equivalent diameter of grains d = 1.128 mm, 2 9 4 RVEs
10mm
Mica
20mm
Quartz
Feldspar
(a) ITZs between different mineral grains, the ITZs are not taken into account in the conceptual model and the mineral grains directly contact each other. The conceptual specimen is shown in Fig. 12b. It is composed of 2 9 4 RVEs and its size is 10 mm 9 20 mm (width 9 height). Therefore, the equivalent diameter (d) of grains within this specimen
(b) is 1.128 mm. For investigating the grain size effect on compressive strength, 7 conceptual specimens with and different grain sizes were constructed in total in the same way and the specimens are of the identical size. Therefore, the equivalent diameters of grains are 0.752 mm (3 9 6 RVEs), 0.564 mm (4 9 8 RVEs), 0.376 mm (6 9 12 RVEs), 0.282 mm
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3.2 Simulated results
260
Un iax ial co mp ressive stren gth /MPa
(8 9 16 RVEs), 0.188 mm (12 9 24 RVEs) and 0.094 mm (24 9 48 RVEs), respectively. For convenient illustration, the symbols CS1, CS2 to CS7 are used to indicate the 7 specimens, respectively. The whole domain of the specimen is divided into 240 9 480 elements for numerical calculation. The uniaxial compressive tests are numerically carried out using RFPA code in the following section.
250 240 230 220 210
y = -50.08x + 254.0 R² = 0.978
200 190 180
The stress–strain curves of the conceptual specimens are shown in Fig. 13. From the curves, it is clear that the specimens represent brittle failure. According to the construction of the conceptual specimen, the proportions of the three minerals are constant for the 7 conceptual specimens. Therefore, the elastic moduli of the specimens are constant, although the specimens have different grain sizes. It further reveals that the elastic modulus of rock is the average effect of the elastic moduli of rock-forming constituents. However, the UCS of the conceptual specimen is quite different from the elastic modulus. Figure 14 shows the variation of UCS with the equivalent grain size. The UCS of the conceptual specimens approximately linearly decreases with the equivalent grain size, similar to previous studies (Hugman and Friedman 1979; Olsson 1974; Brace 1961; Fredrich et al. 1991; Singh 1988; Wong et al. 1996). In nature, the failure of rock is determined by the stress state and the strength of one point within a rock sample. In this simulation, the mechanical properties of the three minerals input into the conceptual model are identical.
300 CS7 CS6 CS5 CS4 CS3 CS2 CS1
250
St ress /MPa
200 150 100
50 0 0
0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004
Strain
Fig. 13 Stress–strain curves of the conceptual specimens with different grain sizes
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0
0.2
0.4
0.6
0.8
1
1.2
Equivalent diameter of grains /mm Fig. 14 The effect of grain size on the uniaxial compressive strength using the conceptual model
Therefore, it must be the difference of grain sizes that affects the internal stress distribution, resulting in differences of UCS. In the next section, the stress distribution within the conceptual specimens with different grain sizes will be analyzed to explain why. 3.3 Interpretation 3.3.1 The stress distribution Figure 15 shows the maximum and the minimum principal stress distributions on the middle horizontal section of the conceptual specimens with different grain sizes under the applied external displacement of 0.005 mm. Due to the redundancy of plotting all data together, only three specimens’ results are given in the figure (i.e. CS1, CS3 and CS6). From Fig. 15a, it can be observed that the maximum principal stress is not uniform, but fluctuates within a certain range due to the differences of mechanical properties between the minerals. Generally, the difference between the maximum and the minimum of the maximum principal stress increases with the grain size. In fact, the maximum of the maximum principal stress increases with the grain size, but this is not readily apparent. However, Fig. 15b shows that there are many tensile stress zones (below zero) on the middle horizontal section despite the specimens being under compressive conditions. Moreover, the minimum of the minimum principal stress generally decreases with
Geomech. Geophys. Geo-energ. Geo-resour. 24
St ress /Mpa
23 22
21 20
19 CS1
18
(a)
0
2
CS3
4
CS6
6
8
10
Position /mm
Maximum principal st ress /Mpa
23.8 23.7
23.6 23.5 23.4
23.3 23.2
3
0
(a)
-1
-2 CS1
CS3
CS6
-3 0
2
4
6
8
10
Position /mm
Fig. 15 The stress distribution on the middle horizontal section within the conceptual specimens a the maximum principal stress and b the minimum principal stress
the grain size, which means that the tensile stress on the middle horizontal section increases with grain size. In this case, the stress state for some points within the specimens represents the compressive stress on one direction and the tensile stress on the perpendicular direction, which is adverse. In addition, both the compressive stress and the tensile stress increase with grain size. Therefore, under the same loading conditions, the specimen with larger grain size fails more easily with increasing external displacement. Eventually, the experimentally-obtained UCS of the specimen composed of large mineral grains is lower than of that composed of small mineral grains. Figure 16 shows more clearly the effect of grain size on the stress within the conceptual specimens. The maximum of the maximum principal stress (i.e. compressive stress) on the middle horizontal section of the conceptual specimens increases with grain size (Fig. 16a), while the minimum of the minimum principal stress decreases with the grain size (i.e. tensile stress increases, see Fig. 16b).
Minimum principal st ress /Mpa
St ress /Mpa
0.6
0.8
1
1.2
0
0
(b)
0.4
Equivalent diameter of grains /mm
2 1
0.2
-0.5 -1 -1.5 -2
-2.5
(b)
0
0.2
0.4
0.6
0.8
1
1.2
Equivalent diameter of grains /mm
Fig. 16 a Maximum value of the maximum principal stress and b the minimum value of the minimum principal stress on the middle horizontal section versus the equivalent diameter of grains
Based on the above analysis, it is clear that the grain size of rock-forming minerals can affect the stress concentration within rock. Under the same loading conditions, the bigger the mineral grain size, the more adverse the stress state within the rock. This will cause a specimen with a relatively large grain size to fail at a lower loading level, and finally affects the UCS of rock. This analysis can interpret the effect of grain size on the UCS of rock (Hugman and Friedman 1979; Olsson 1974; Brace 1961; Fredrich et al. 1991; Singh 1988; Wong et al. 1996), as well as on fracture initiation stress (Hatzor and Palchik 1997).
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3.3.2 Fractal dimension
Nr ¼
X
nr ði; jÞ
ð7Þ
i;j
nr ði; jÞ ¼ l k þ 1
ð6Þ
where the subscript r denotes the result using the scale r. Considering contributions from all blocks, Nr is counted for different values of r as
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The fractal dimension can then be estimated from the least squares linear fit of log (Nr) versus log(1/r). Figure 17 plots the fractal dimension of the conceptual specimens with different grain sizes. The best-fit straight line shows that the fractal dimension generally decreases with grain size. The smaller the grain size, the bigger the fractal dimension. This means that rock with small mineral grain sizes has a relatively complex microstructure. The UCS of the conceptual specimens has a positive relationship with the fractal dimension of rock microstructure (Fig. 18). In general, the UCS increases with the fractal dimension of rock microstructure. For the Strathbogie granite studied experimentally by Shao et al. (2014), as shown in Figs. 1 and 2, the relationship between the fractal dimension of rock microstructure and the UCS is shown in Fig. 19. The figure shows that the UCS of the FG granite and the CG granite is positively proportional with the respective fractal dimension of the microstructure. However, the case for MG granite is unusual. The UCS of the MG granite is relatively lower but the fractal dimension of its microstructure is high. Although this type of Strathbogie granite can be defined as medium-grained according to the mean grain size (Shao et al. 2014), the composition of the MG granite is more complex, as shown by comparing the microscopic images of three types of granite microstructure (Fig. 2). The sizes of mineral grains within the MG granite are distributed in a much wider 3
Fract al d imension
Fractal geometry provides a mathematical model for many complex structures found in rock engineering (Kruhl 2013), such as rock joint roughness (Li and Huang 2015; Bagde et al. 2002), anisotropy (Gerik and Kruhl 2009), fracture pattern (Babadagli and Develi 2003), crystal-distribution pattern (Peternell and Kruhl 2009), grain size (Billi and Storti 2004) and spatial distribution of rock acoustic emissions (Kusunose et al. 1991; Xie et al. 2011; Zhang et al. 2015). In our study, fractal geometry was introduced to delineate the microstructure of rocks with different mineral grain sizes. Figure 2 reveals that the color of every type of rockforming mineral grain is distributed in a specific range, and, in general, differ from that of other minerals. For example, quartz usually presents as light gray, feldspar presents as gray and mica presents as dark gray or even black. According to Table 2, the descending order of the strength of minerals is quartz, feldspar and mica. That is to say, there exists a positive relationship between the textures of the microscopic images (Fig. 2) and the strength of the minerals. On this premise, the fractal dimensions of the microscopic images and the conceptual specimens are estimated to relate with the effect of grain size. The fractal dimension of rock microstructure in terms of images was estimated using the so-called differential box-counting method (DBC), proposed by Sarkar and Chaudhuri (1994). A brief description is as follows. Consider an image of size M 9 M as a threedimensional (3-D) spatial surface with two coordinates (x, y) denoting pixel positions on the image plane, and the third coordinate (z) denoting pixel gray level. In the DBC method, the x, y plane is partitioned into non-overlapping grids of size s 9 s, where M/ 2 C s [ 1 and s is an integer. Then let an estimate of r = s/M. On each grid, there is a column of boxes of size s 9 s9s’, where s’ is the height of each box, G/ s’ = M/s and G is the total number of gray levels. Assign numbers 1, 2,…to the boxes. Let the minimum and maximum gray level of the image in the (i, j)th grid fall in box number k and l, respectively. The boxes covering this grid are counted in the number as
2.8
2.6 y = -0.488x + 3.043 R² = 0.969
2.4 2.2
2 0
0.2
0.4
0.6
0.8
1
1.2
Equivalent diameter of grains /mm Fig. 17 The relationship between fractal dimension and the equivalent diameter of grains for the conceptual specimens
Uniaxial compressive st rengt h /MPa
Geomech. Geophys. Geo-energ. Geo-resour.
resolution. Moreover, they should have relatively stable color characteristics, similar to the conceptual specimens or the FG and the CG granite.
260 250 y = 100.17x - 51.327 R² = 0.9631
240 230
4 Conclusions
220 210 200 190 180 2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
Fractal dimension Fig. 18 The relationship between uniaxial compressive strength and the fractal dimension for the conceptual specimens 3 UCS FD
200
2.9
150
2.8
100
2.7
50
2.6
0
Fractal dimension
U niaxial compressive st rengt h /MPa
250
2.5
FG
MG
CG
Fig. 19 The UCS and fractal dimension of the Strathbogie granite microstructure
range than in the other two types of granite. This includes the size of both the small grains within the FG granite and the large grains within the CG granite. Therefore, the grain size composition is quite different from that of the other two types of granite, which results in the fractal dimension being highly estimated. In fact, the grain size range of the studied rock materials should be considered as an important factor in the investigation of the grain size effect on rock strength, and mentioning the grain size range makes it more meaningful (Tavallali and Vervoort 2010). When the effect of grain size is looked at a relatively wide range, other parameters such as porosity can play a role. It is noteworthy that, when fractal geometry is used to quantify rock microstructure in terms of microscopic images, the images must have an identical
The aim of this paper is to interpret the grain size effect on the UCS of rock from the point of view of mechanics. In order to model this effect for heterogeneous rock, integrated digital image processing techniques, mineral composition analysis, homogeneity theory and numerical modeling methods were used. Digital image processing techniques and mineral composition analysis were used to distinguish the mineral grain type and morphological information from the microscopic images of two categories of Strathbogie granite. Next, homogenisation theory was used to build numerical specimens with different grain sizes and RFPA code was used to carry out numerical tests under uniaxial compression. The simulated results showed that the UCS of specimens with large grain size was lower than that of specimens with small grain size and under the same loading conditions, the specimen with large grain size initiates damage earlier than that with small grain size. Considering only mineral type and grain size, the numerical method can also reveal the grain size effect on UCS which is basically consistent with the experimental results. To clearly interpret this effect, a serial of ideal conceptual specimens with different grain size was built in the same way for uniaxial compressive tests. The stress analysis revealed that grain size can influence stress distribution and concentration within rock. With increasing grain size, the maximum of the maximum principal stress increases and the minimum of the minimum principal stress decreases (i.e. tensile stress increases). The effect of grain size on the minimum principal stress is stronger than the maximum principal stress. This stress state can affect the rock failure process and eventually affect rock strength. Fractal geometry provides a tool to quantify rock microstructure in terms of images at the mineral grain scale. On the premise that there exists a positive relationship between the gray level of rock microstructure images and the strength of rock-forming minerals, the fractal dimensions of the conceptual specimens and Strathbogie granite were estimated using a
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differential box-counting method. For the conceptual specimens, the fractal dimension relates well with the grain size and UCS. For Strathbogie granite, the fractal dimensions of FG and CG granite microstructures relate well with the UCS, with the exception of MG granite with a wide range of grain size. In cases where the mineral grain size cannot actually be measured, the fractal dimension could become a relative index to describe rock microstructure. However, the rock microstructure image must have identical resolution and relatively stable color characteristics, similar to the FG and CG granite. When studying the grain size effect, the specimens are usually cored from the same place for comparison. The rock-forming process and mineral composition percentages should be similar. Therefore, it can be concluded that the difference of stress induced by grain size is one of the most important factors in the grain size effect from the aspect of mechanical properties. Acknowledgements The present work is partially funded by the National Science Foundation of China (Grant Nos. 51574060 and 51525402). This support is gratefully acknowledged.
References Babadagli T, Develi K (2003) Fractal characteristics of rocks fractured under tension. Theor Appl Fract Mech 39(1):73–88 Bagde MN, Raina AK, Chakraborty AK, Jethwa JL (2002) Rock mass characterization by fractal dimension. Eng Geol 63(1–2):141–155 Billi A, Storti F (2004) Fractal distribution of particle size in carbonate cataclastic rocks from the core of a regional strike-slip fault zone. Tectonophysics 384(1–4):115–128 Brace WF (1961) Dependence of fracture strength of rocks on grain size. In: Proceedings of the 4th symposium rock mechanics. University Park, USA, March 1961, pp 99–103 de Castro Lima JJ, Paraguassu´ A (2004) Linear thermal expansion of granitic rocks: influence of apparent porosity, grain size and quartz content. Bull Eng Geol Environ 63(3):215–220 Eberhardt E, Stimpson B, Stead D (1999) Effects of grain size on the initiation and propagation thresholds of stress-induced brittle fractures. Rock Mech Rock Eng 32(2):81–99 Fredrich JT, Evans B, Wong TF (1991) Effect of grain size on brittle and semibrittle strength: implications for micromechanical modelling of failure in compression : Fredrich, J T; Evans, B; Wong, T F J Geophys ResV95, NB7, July 1990, P10907–10920. Int J Rock Mech Min Sci Geomech Abstr 28(2–3):A136
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Ge K, Liu Q (2014) Effects of the grain size distribution on magnetic properties of magnetite: constraints from micromagnetic modeling. Chin Sci Bull 59(34):4763–4773 Gerik A, Kruhl JH (2009) Towards automated pattern quantification: time-efficient assessment of anisotropy of 2D patterns with AMOCADO. Comput Geosci 35(6):1087–1097 Griffith AA (1920) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A Math Phys Sci 221(587):163–198 Hatzor YH, Palchik V (1997) The influence of grain size and porosity on crack initiation stress and critical flaw length in dolomites. Int J Rock Mech Min Sci 34(5):805–816 Hatzor YH, Zur A, Mimran Y (1997) Microstructure effects on microcracking and brittle failure of dolomites. Tectonophysics 281(3–4):141–161 Heilbronner R, Bruhn D (1998) The influence of three-dimensional grain size distributions on the rheology of polyphase rocks. J Struct Geol 20(6):695–705 Hugman RHH, Friedman M (1979) Effects of texture and composition on mechanical behavior of experimentally deformed carbonate rocks. Am Assoc Pet Geol Bull 63(9):1478–1489 Kabeya KK, Legge TFH (1997) Relationship between grain size and some surface roughness parameters of rock joints. Int J Rock Mech Min Sci 34(3–4):146.e115–146.e141 Kruhl JH (2013) Fractal-geometry techniques in the quantification of complex rock structures: a special view on scaling regimes, inhomogeneity and anisotropy. J Struct Geol 46:2–21 Kusunose K, Lei X, Nishizawa O, Satoh T (1991) Effect of grain size on fractal structure of acoustic emission hypocenter distribution in granitic rock. Phys Earth Planet Inter 67(1–2):194–199 Li Y, Huang R (2015) Relationship between joint roughness coefficient and fractal dimension of rock fracture surfaces. Int J Rock Mech Min Sci 75:15–22 Liu HY, Roquete M, Kou SQ, Lindqvist PA (2004) Characterization of rock heterogeneity and numerical verification. Eng Geol 72(1–2):89–119 Olsson WA (1974) Grain size dependence of yield stress in marble. J Geophys Res 79(32):4859–4862 Peternell M, Kruhl JH (2009) Automation of pattern recognition and fractal-geometry-based pattern quantification, exemplified by mineral-phase distribution patterns in igneous rocks. Comput Geosci 35:1415–1426 Prˇikryl R (2001) Some microstructural aspects of strength variation in rocks. Int J Rock Mech Min Sci 38(5):671–682 Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Springer, Paris, pp 45–128 Sarkar N, Chaudhuri BB (1994) An efficient differential boxcounting approach to compute fractal dimension of image. IEEE Trans Syst Man Cybern 24:115–120 Seo YS, Jeong GC, Kim JS, Ichikawa Y (2002) Microscopic observation and contact stress analysis of granite under compression. Eng Geol 63(3–4):259–275 Shao S, Wasantha PLP, Ranjith PG, Chen BK (2014) Effect of cooling rate on the mechanical behavior of heated Strathbogie granite with different grain sizes. Int J Rock Mech Min Sci 70:381–387
Geomech. Geophys. Geo-energ. Geo-resour. Singh SK (1988) Relationship among fatigue strength, mean grain size and compressive strength of a rock. Rock Mech Rock Eng 21(4):271–276 Suorineni FT, Chinnasane DR, Kaiser PK (2009) A procedure for determining rock-type specific Hoek–Brown brittle parameters. Rock Mech Rock Eng 42(6):849–881 Tang CA (1997) Numerical simulation of progressive rock failure and associated seismicity. Int J Rock Mech Min Sci 34(2):249–261 Tavallali A, Vervoort A (2010) Failure of layered sandstone under Brazilian test conditions: effect of micro-scale parameters on macro-scale behaviour. Rock Mech Rock Eng 43(5):641–653 Wong RHC, Chau KT, Wang P (1996) Microcracking and grain size effect in Yuen Long marbles. Int J Rock Mech Min Sci Geomech Abstr 33(5):479–485
Xie HP, Liu JF, Ju Y, Li J, Xie LZ (2011) Fractal property of spatial distribution of acoustic emissions during the failure process of bedded rock salt. Int J Rock Mech Min Sci 48(8):1344–1351 Yilmaz Gunes N, Karaca Z, Goktan RM, Akal C (2009) Relative brittleness characterization of some selected granitic building stones: influence of mineral grain size. Constr Build Mater 23(1):370–375 Yu Q, Zhu W, Tang C, Yang T (2014) Impact of rock microstructures on failure processes—numerical study based on DIP technique. Geomech Eng 7(4):375–401 Zhang R, Dai F, Gao MZ, Xu NW, Zhang CP (2015) Fractal analysis of acoustic emission during uniaxial and triaxial loading of rock. Int J Rock Mech Min Sci 79:241–249 Zhu WC, Tang CA (2004) Micromechanical model for simulating the fracture process of rock. Rock Mech Rock Eng 37(1):25–56
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