Rock engineering properties are considered to be the most important parameters in the design of groundworks. Two important mechanical parameters, uniaxial compressive strength (σc) and elastic modulus of rock (
Hypothetical stress–strain curves for three different rocks are presented in Fig. 1 by Ramamurthy et al. [4]. Based on the figure, curves OA, OB and OC represent three stress–strain curves with failure occurring at A, B and C, respectively. According to their sample, curves OA and OB have the same modulus but different strengths and strains at failure, whereas the curves OA and OC have the same strength but different modulus and strains at failure. It means, neither strength nor modulus alone could be chosen to represent the overall quality of rock. Therefore, strength and modulus together will give a realistic understanding of the rock’s response to engineering usage. This approach of defining the quality of intact rocks was proposed by Deere and Miller [5] considering the modulus ratio (
The modulus ratio
Palchik [7] examined the
where
The expansion of the expression
The goal of this paper is to check Eq. (1) for Hungarian granitic rocks as well as to study the relationships between characteristic compressive stress level, strain and mechanical properties. These granitic rock samples were investigated previously by Vásárhelyi et al. [8] using multiple failure state triaxial tests.
Laboratory samples originated from research boreholes deepened in carboniferous Mórágy granite formation during the research and construction phases of deep geological repository of low- and intermediate-level radioactive waste. This granite formation is a carboniferous intruded and displaced Variscan granite pluton situated in South-West Hungary. The main rock types are mainly microcline megacryst-bearing, medium-grained, biotite monzogranites and quartz monzonites [9] (see Fig. 3). In spatial viewpoint, the monzogranitic rocks contain generally oval shaped, variably elongated monzonite enclaves (predominantly amphibole–biotite monzonites, diorites and syenites) of various sizes (from a few centimetre to several 100 metres) reflecting the mixing and mingling of two magmas with different composition. Feldspar quartz-rich leucocratic dykes
belonging to the late-stage magmatic evolution and Late Cretaceous trachyte and tephrite dykes cross cut all of the previously described rock types [10]. In general, fractured but fresh rock is common which is sparsely intersected by fault zones with few metre thick clay gauges. Intense clay mineralisation in the fault cores indicates a low-grade hydrothermal alteration.
The samples were tested by using a computer-controlled servo-hydraulic machine in continuous load control mode. The magnitude of loading was settled in kilonewton with 0.01 accuracy, and the rate of loading was 0.6 kN/s. Axial and tangential deformation was measured by strain gauges, which measures the deformation between 1/4 and 3/4 of the sample’s height.
Fifty uniaxial compressive tests were performed in the rock mechanics laboratory at RockStudy Ltd. The NX (
Fig. 4). Mechanical properties of granitic rock samples are summarised in Table 1.
Mechanical properties of investigated Mórágy granitic rock samples.
Rock sample | εci | σci | εcd | σcd | εa, max | σc | |||
---|---|---|---|---|---|---|---|---|---|
(-) | (GPa) | (%) | (MPa) | (%) | (MPa) | (%) | (MPa) | (-) | |
BeR-6_U-10 | 0.24 | 74.776 | 0.030 | 50.73 | 0.091 | 152.244 | 0.278 | 181.05 | 413.0 |
BeR-7_U-02 | 0.21 | 71.612 | 0.037 | 50.37 | 0.095 | 145.15 | 0.34 | 174.80 | 409.7 |
BeR-7_U-04 | 0.25 | 74.447 | 0.037 | 59.61 | 0.063 | 131.70 | 0.33 | 183.39 | 405.9 |
BeR-8_U-01 | 0.22 | 63.357 | 0.060 | 59.84 | 0.120 | 165.89 | 0.29 | 184.48 | 343.4 |
BeR-10_U-08 | 0.21 | 66.129 | 0.025 | 30.06 | 0.044 | 77.30 | 0.22 | 137.14 | 482.2 |
BeR-10_U-18 | 0.23 | 72.794 | 0.048 | 64.89 | 0.078 | 148.24 | 0.2 | 148.39 | 490.6 |
BeR-10_U-20 | 0.23 | 63.787 | 0.035 | 39.28 | 0.087 | 133.75 | 0.27 | 156.74 | 407.0 |
BeR-11_U-08 | 0.23 | 68.950 | 0.054 | 80.82 | 0.104 | 168.94 | 0.31 | 204.23 | 337.6 |
BeR-12_U-02 | 0.22 | 79.660 | 0.029 | 34.36 | 0.08 | 128.84 | 0.18 | 133.34 | 597.4 |
BK1-1_U-12 | 0.23 | 70.153 | 0.036 | 51.74 | 0.076 | 131.50 | 0.23 | 172.74 | 406.1 |
BK1-3_U-01 | 0.32 | 72.891 | 0.037 | 79.97 | 0.053 | 121.05 | 0.28 | 184.59 | 394.9 |
BK1-3_U-03 | 0.19 | 69.164 | 0.065 | 71.75 | 0.14 | 132.66 | 0.22 | 133.62 | 517.6 |
BK1-3_U-04 | 0.18 | 71.860 | 0.045 | 47.93 | 0.113 | 112.28 | 0.18 | 153.60 | 467.8 |
BK1-3_U-08 | 0.23 | 70.137 | 0.059 | 80.36 | 0.147 | 142.79 | 0.22 | 172.55 | 406.5 |
BK1-3_U-12 | 0.25 | 57.425 | 0.066 | 67.99 | 0.13 | 134.11 | 0.27 | 135.14 | 424.9 |
BK2-1_U-03 | 0.21 | 74.228 | 0.057 | 74.61 | 0.09 | 131.78 | 0.19 | 146.65 | 506.2 |
BK2-3_U-07 | 0.28 | 77.332 | 0.036 | 59.84 | 0.068 | 119.12 | 0.19 | 143.71 | 538.1 |
BK2-3_U-15 | 0.22 | 80.365 | 0.035 | 48.57 | 0.090 | 160.74 | 0.24 | 178.41 | 450.5 |
BK2-3_U-18 | 0.2 | 73.819 | 0.069 | 80.22 | 0.11 | 153.84 | 0.23 | 159.16 | 463.8 |
BK2-4_U-02 | 0.2 | 76.820 | 0.06 | 88.12 | 0.106 | 177.32 | 0.26 | 205.62 | 373.6 |
BK2-4_U-04 | 0.21 | 77.709 | 0.045 | 60.07 | 0.090 | 130.57 | 0.20 | 155.49 | 499.8 |
BK2-5_U-02 | 0.25 | 77.866 | 0.038 | 62.63 | 0.070 | 134.14 | 0.23 | 166.29 | 468.3 |
Bkf-1_U-03 | 0.24 | 77.665 | 0.050 | 50.46 | 0.070 | 120.14 | 0.30 | 161.63 | 480.5 |
Bkf-2_U-03 | 0.22 | 60.602 | 0.065 | 76.84 | 0.118 | 164.66 | 0.39 | 180.93 | 334.9 |
Bkf-4_U-03 | 0.22 | 79.856 | 0.042 | 60.29 | 0.083 | 142.38 | 0.24 | 179.28 | 445.4 |
Bkf-5_U-02 | 0.24 | 79.818 | 0.034 | 53.90 | 0.067 | 135.29 | 0.20 | 169.67 | 470.4 |
Bl-112_U-02 | 0.21 | 72.897 | 0.029 | 37.88 | 0.093 | 144.19 | 0.20 | 164.59 | 442.9 |
Bp-4_U-05 | 0.25 | 76.992 | 0.041 | 69.46 | 0.1 | 181.85 | 0.24 | 187.69 | 410.2 |
Bp-4B_U-01 | 0.21 | 69.800 | 0.042 | 49.25 | 0.109 | 159.85 | 0.36 | 184.45 | 378.4 |
Bp-4B_U-05 | 0.23 | 76.237 | 0.033 | 49.37 | 0.076 | 148.77 | 0.28 | 170.10 | 448.2 |
Bp-4B_U-13 | 0.27 | 77.924 | 0.049 | 74.11 | 0.096 | 170.28 | 0.25 | 177.91 | 438.0 |
Bp-4B_U-17 | 0.24 | 74.648 | 0.045 | 60.27 | 0.083 | 162.61 | 0.26 | 181.43 | 411.4 |
Bp-4B_U-19 | 0.22 | 77.182 | 0.058 | 80.43 | 0.100 | 160.13 | 0.25 | 190.48 | 405.2 |
Bp-4B_U-23 | 0.24 | 74.683 | 0.053 | 80.00 | 0.077 | 137.96 | 0.24 | 165.23 | 452.0 |
Bp-5_U-19 | 0.25 | 73.506 | 0.031 | 49.73 | 0.056 | 121.48 | 0.23 | 149.76 | 490.8 |
Bp-5_U-21 | 0.25 | 80.159 | 0.040 | 70.45 | 0.064 | 137,00 | 0.26 | 171.46 | 467.5 |
Bx-81_U-03 | 0.22 | 65.782 | 0.045 | 53.44 | 0.088 | 130.84 | 0.29 | 149.28 | 440.7 |
Bx-82_U-01 | 0.25 | 82.940 | 0.046 | 80.51 | 0.085 | 162.08 | 0.27 | 180.33 | 459.9 |
Bx-82_U-03 | 0.29 | 84.949 | 0.024 | 49.99 | 0.044 | 120.612 | 0.2 | 166.87 | 509.1 |
Bx-83_U-01 | 0.26 | 72.864 | 0.030 | 60.56 | 0.067 | 150.321 | 0.26 | 169.70 | 429.4 |
Bx-83_U-03 | 0.25 | 78.072 | 0.057 | 90.36 | 0.095 | 182.085 | 0.37 | 212.42 | 367.5 |
Bx-84_U-01 | 0.25 | 80.669 | 0.047 | 79.90 | 0.073 | 147.6 | 0.23 | 178.07 | 453.0 |
Bx-84_U-03 | 0.27 | 81.144 | 0.039 | 69.38 | 0.062 | 138.183 | 0.26 | 166.94 | 486.1 |
Bx-101_U-02 | 0.24 | 76.994 | 0.042 | 71.53 | 0.058 | 112.5 | 0.19 | 142.49 | 540.3 |
Bx-101_U-04 | 0.26 | 79.300 | 0.048 | 60.58 | 0.091 | 160.96 | 0.23 | 163.19 | 485.9 |
Bz-921_U-01 | 0.21 | 71.574 | 0.056 | 68.79 | 0.121 | 164.573 | 0.3 | 192.80 | 371.2 |
Bz-942_U-01 | 0.23 | 73.511 | 0.053 | 73.43 | 0.11 | 182.66 | 0.28 | 198.58 | 370.2 |
Bz-1221_U-01 | 0.2 | 69.540 | 0.049 | 58.25 | 0.100 | 165.836 | 0.29 | 213.04 | 326.4 |
Bz-1311_U-01 | 0.3 | 88.937 | 0.035 | 75.93 | 0.060 | 163.371 | 0.23 | 206.48 | 430.7 |
Bz-1351_U-01 | 0.25 | 67.053 | 0.034 | 50.86 | 0.080 | 145.566 | 0.28 | 159.97 | 419.2 |
UCS, uniaxial compressive strength
Table 1 summarizes the value of elastic modulus (
The values of elastic modulus (
In this method, [11], crack initiation threshold is visible on the axial–volumetric strain curve (Fig. 5) when it diverges from the straight line. In practice, small deviation of the stress–volumetric strain curve from the straight line can make some difficulties to define one point determining the threshold of crack initiation.
Martin and Chandler [12] proposed that crack initiation could be determined using a plot of crack volumetric strain versus axial strain (Fig. 6). Crack volumetric strain εVcr is calculated as a difference between the elastic volumetric strain εVel and volumetric strain εV determined in the test,
εa and εl are the axial and lateral strain; σ1 and σ3 are the axial and confining stress and
Crack volumetric strain is calculated on the basis of these two elastic constants and is strongly sensitive to its value. This is probably why this method does not give objective values.
Diederichs [13] proposed a method of crack initiation threshold identification based on the change of Poisson’s ratio. The onset of crack initiation can be identified by the analysis of the relationship of Poisson’s ratio, evaluated locally, to the log of the axial stress (Fig. 7).
However, in this paper, the results obtained from the first method were used for further analysis. The reason is
that, based on the findings by Cieslik [14], this method gives more precise results for granitic rock samples.
Table 1 also summarizes that the value of
The ranges of the elastic modulus (
The ranges of
The relationship between uniaxial compressive strength (σc),
As it is clear, the elastic modulus is related to σc, with
The calculated values are compared with the international published relationships.
The observed and analytical (Eq. 1) relationships between εa, max and
The relative (ζ , %) and root-mean-square (χ) errors between the observed and calculated parameter Π have been calculated as:
where Πobs(
The observed values between
Fig. 13 shows the relationship between
The relationship between
As shown in Fig. 15, there is practically no correlation between these two values.
The relationship between
The relationship between
Fig. 18 shows the relationship between
Fig. 19 presents the relationship between
Fig. 20 shows the relationship between
Fig. 21 shows the relationship between
The laboratory compressive tests, statistical analysis and empirical and analytical relationships have been used to estimate the values of
The mean value of
The observation confirms that there is no general empirical correlation (with reliable
The analytical
The observed correlation between
It is established that there is a correlation between Based on the obtained results, there is practically no relationship between
Notably, for a more precise and fundamental description of the mechanical behaviour of rock, one should apply non-equilibrium continuum thermodynamics along the lines of Asszonyi et al. [23, 25] and beyond. These relationships can be used for determining the mechanical parameters of the rock mass, as well [24, 26].