Complex analysis of uniaxial compressive tests of the Mórágy granitic rock formation (Hungary)

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Abstract

Understanding the quality of intact rock is one of the most important parts of any engineering projects in the field of rock mechanics. The expression of correlations between the engineering properties of intact rock has always been the scope of experimental research, driven by the need to depict the actual behaviour of rock and to calculate most accurately the design parameters. To determine the behaviour of intact rock, the value of important mechanical parameters such as Young’s modulus (E), Poisson’s ratio (ν) and the strength of rock (σcd) was calculated. Recently, for modelling the behaviour of intact rock, the crack initiation stress (σci) is another important parameter, together with the strain (σ). The ratio of Young’s modulus and the strength of rock is the modulus ratio (MR), which can be used for calculations. These parameters are extensively used in rock engineering when the deformation of different structural elements of underground storage, caverns, tunnels or mining opening must be computed. The objective of this paper is to investigate the relationship between these parameters for Hungarian granitic rock samples. To achieve this goal, the modulus ratio (MR = Ec) of 50 granitic rocks collected from Bátaapáti radioactive waste repository was examined. Fifty high-precision uniaxial compressive tests were conducted on strong (σc >100 MPa) rock samples, exhibiting the wide range of elastic modulus (E = 57.425–88.937 GPa), uniaxial compressive strength (σc = 133.34–213.04 MPa) and Poisson’s ratio (ν = 0.18–0.32). The observed value (MR = 326–597) and mean value of MR = 439.4 are compared with the results of similar previous researches. Moreover, the statistical analysis for all studied rocks was performed and the relationshipbetween MR and other mechanical parameters such as maximum axial strain (εa,max)for studied rocks was discussed.

1 Introduction

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 (E), should be estimated correctly. There are different empirical relationships between σc and E obtained for limestones, agglomerates, dolomites, chalks, sandstones and basalts [1, 2, 3], among the others.

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 (MR), which is defined as the ratio of tangent modulus of intact rock (E) at 50% of failure strength and its compressive strength (σc).

Figure 1
Figure 1

Hypothetical stress–strain curves [4].

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

The modulus ratio MR = Ec between the modulus of elasticity (E) and uniaxial compressive strength (σc) for intact rock samples varies from 106 to 1,600 [6]. For most rocks, MR is between 250 and 500 with average MR = 400, E = 400 σc.

Palchik [7] examined the MR values for 11 heterogeneous carbonate rocks from different regions of Israel. The investigated dolomites, limestones and chalks had weak to very strong strength with a wide range of elastic modulus. He found that MR is closely related to the maximum axial strain (εa, max) at the uniaxial strength of the rock (σc) and the following relationship was found (Fig. 2):

Figure 2
Figure 2

Relationship between modulus ratio (MR) and maximum axial strain (εa, max) using different carbonate rocks [7].

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

MR=2kεa,max(1+eεa,max)

where k is the conversion coefficient equal to 100 and εa,max is in %. When MR is known, εa, max (%) is obtained from Eq. (1) as

εa,max=kMR0.46k

The expansion of the expression 2/(1+eεa,max)using Taylor’s theorem shows the value of 2/(1+eεa,max)=1 + 0.46 εa, max [8].

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.

2 Laboratory investigations and analyses

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

Figure 3
Figure 3

Main types of rock samples. (a, b) Megacryst-bearing, medium-grained, biotite monzogranites. (c) Medium-grained, biotite monzogranites with elongated monzonitic enclaves. (d) Quartz monzonite.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

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 (d = 50 mm)-sized cylindrical rock samples having the ratio of L/d = 2/1 (here L and d are the length and diameter of a sample, respectively) were prepared (see

Fig. 4). Mechanical properties of granitic rock samples are summarised in Table 1.

Figure 4
Figure 4

A prepared sample in the beginning of the UCS test.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

Table 1

Mechanical properties of investigated Mórágy granitic rock samples.

Rock sampleνEεciσciεcdσcdεa, maxσcMR

(-)(GPa)(%)(MPa)(%)(MPa)(%)(MPa)(-)
BeR-6_U-100.2474.7760.03050.730.091152.2440.278181.05413.0
BeR-7_U-020.2171.6120.03750.370.095145.150.34174.80409.7
BeR-7_U-040.2574.4470.03759.610.063131.700.33183.39405.9
BeR-8_U-010.2263.3570.06059.840.120165.890.29184.48343.4
BeR-10_U-080.2166.1290.02530.060.04477.300.22137.14482.2
BeR-10_U-180.2372.7940.04864.890.078148.240.2148.39490.6
BeR-10_U-200.2363.7870.03539.280.087133.750.27156.74407.0
BeR-11_U-080.2368.9500.05480.820.104168.940.31204.23337.6
BeR-12_U-020.2279.6600.02934.360.08128.840.18133.34597.4
BK1-1_U-120.2370.1530.03651.740.076131.500.23172.74406.1
BK1-3_U-010.3272.8910.03779.970.053121.050.28184.59394.9
BK1-3_U-030.1969.1640.06571.750.14132.660.22133.62517.6
BK1-3_U-040.1871.8600.04547.930.113112.280.18153.60467.8
BK1-3_U-080.2370.1370.05980.360.147142.790.22172.55406.5
BK1-3_U-120.2557.4250.06667.990.13134.110.27135.14424.9
BK2-1_U-030.2174.2280.05774.610.09131.780.19146.65506.2
BK2-3_U-070.2877.3320.03659.840.068119.120.19143.71538.1
BK2-3_U-150.2280.3650.03548.570.090160.740.24178.41450.5
BK2-3_U-180.273.8190.06980.220.11153.840.23159.16463.8
BK2-4_U-020.276.8200.0688.120.106177.320.26205.62373.6
BK2-4_U-040.2177.7090.04560.070.090130.570.20155.49499.8
BK2-5_U-020.2577.8660.03862.630.070134.140.23166.29468.3
Bkf-1_U-030.2477.6650.05050.460.070120.140.30161.63480.5
Bkf-2_U-030.2260.6020.06576.840.118164.660.39180.93334.9
Bkf-4_U-030.2279.8560.04260.290.083142.380.24179.28445.4
Bkf-5_U-020.2479.8180.03453.900.067135.290.20169.67470.4
Bl-112_U-020.2172.8970.02937.880.093144.190.20164.59442.9
Bp-4_U-050.2576.9920.04169.460.1181.850.24187.69410.2
Bp-4B_U-010.2169.8000.04249.250.109159.850.36184.45378.4
Bp-4B_U-050.2376.2370.03349.370.076148.770.28170.10448.2
Bp-4B_U-130.2777.9240.04974.110.096170.280.25177.91438.0
Bp-4B_U-170.2474.6480.04560.270.083162.610.26181.43411.4
Bp-4B_U-190.2277.1820.05880.430.100160.130.25190.48405.2
Bp-4B_U-230.2474.6830.05380.000.077137.960.24165.23452.0
Bp-5_U-190.2573.5060.03149.730.056121.480.23149.76490.8
Bp-5_U-210.2580.1590.04070.450.064137,000.26171.46467.5
Bx-81_U-030.2265.7820.04553.440.088130.840.29149.28440.7
Bx-82_U-010.2582.9400.04680.510.085162.080.27180.33459.9
Bx-82_U-030.2984.9490.02449.990.044120.6120.2166.87509.1
Bx-83_U-010.2672.8640.03060.560.067150.3210.26169.70429.4
Bx-83_U-030.2578.0720.05790.360.095182.0850.37212.42367.5
Bx-84_U-010.2580.6690.04779.900.073147.60.23178.07453.0
Bx-84_U-030.2781.1440.03969.380.062138.1830.26166.94486.1
Bx-101_U-020.2476.9940.04271.530.058112.50.19142.49540.3
Bx-101_U-040.2679.3000.04860.580.091160.960.23163.19485.9
Bz-921_U-010.2171.5740.05668.790.121164.5730.3192.80371.2
Bz-942_U-010.2373.5110.05373.430.11182.660.28198.58370.2
Bz-1221_U-010.269.5400.04958.250.100165.8360.29213.04326.4
Bz-1311_U-010.388.9370.03575.930.060163.3710.23206.48430.7
Bz-1351_U-010.2567.0530.03450.860.080145.5660.28159.97419.2

UCS, uniaxial compressive strength

Table 1 summarizes the value of elastic modulus (E), crack damage stress (σcd), uniaxial compressive strength (σc), Poisson’s ratio (ν), crack initiation stress (σci), axial failure strain (εa, max), maximum volumetric strain (εcd), crack initiation strain (εci) and MR for each of the studied 50 samples.

The values of elastic modulus (E) and Poisson’s ratio (ν) were calculated by using linear regressions along linear portions of stress–axial strain curves and radial strain–axial strain curves, respectively. The values of crack initiation stress (σci) and crack damage stress (σcd) were calculated based on the following methods:

2.1 Onset dilatancy method

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.

Figure 5
Figure 5

Axial stress–volumetric strain curve with the threshold of crack initiation and crack damage and failure stress for Hungarian granitic sample (uniaxial compression case).

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

2.2 ‒ Crack volumetric strain method

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,

Figure 6
Figure 6

Crack volumetric strain method for crack initiation threshold determination for Hungarian granitic rock sample (uniaxial compression case).

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

εV=2ε1+εa
εVcr=εVεVel
εVel=12vE(σ1+2σ3)

εa and εl are the axial and lateral strain; σ1 and σ3 are the axial and confining stress and E and ν are the Young’s modulus and Poisson’s ratio, respectively.

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.

2.3 Change of Poisson’s ratio method

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).

Figure 7
Figure 7

Poisson’s ratio method for crack initiation threshold determination for Hungarian granitic rock sample (uniaxial compression case).

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

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 MR in each of 50 studied granitic rock samples is between 326.4 and 597.4 with the mean of 439.4. The range of MR obtained by Deere [15] is between 250 and 700 with the mean of 420 for limestone and dolomites. The range of MR obtained by Palchik [7] is between 60.9 and 1011.4 with the mean value of 380.5 for carbonated rock samples. The mean value of MR in this study is similar to the mean value of MR obtained by Deere [15] and Palchik [7]. Fig. 8 shows the value of MR for all studied samples in this study. As shown in Fig. 5, the range of MR =326.4–597.4, observed in this study, is less than the range of MR obtained by Deere [15] and Palchik [7].

Figure 8
Figure 8

Observed values of modulus ratio (MR) in each of 50 examined rock samples.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

The ranges of the elastic modulus (E), Poisson’s ratio (ν), crack damage stress (σcd) and uniaxial compressive strength (σc), axial failure strain (εa, max) and maximum volumetric strain (εcd), crack initiation stress (σci) and crack initiation strain (εci) for the studied 50 samples are presented as follows:

57.425GPa<E<88.937GPa0.18<v<0.3230MPa<σci<90MPa77MPa<σcd<182MPa133.34MPa<σc<213.04MPa0.02<εci<0.060.18<εa,max<0.190.04<εcd<0.14

The ranges of εa,maxεcdandσcdσcand σciσcratios are 1.49–5.28 and 0.5–0.9 and 0.2–0.5, respectively. These values are different from the values of σcdσc=0.51.0andεa,maxεcd=1.516.91 obtained by Palchik [7]. They are also different from the values of σcdσc=0.710.84obtained by Brace et al. [11], Bieniawski [16], Martin [17], Pettitt et al. [18], Eberhardt et al. [19], Heo et al. [20] and Katz and Reches [21] for granites, sandstones and quartzites. The range of σciσcfor most rocks falls in the range of 0.3–0.5.

3 Effect of mechanical properties on MR value

3.1 Relationship between MR, σc and E for all granitic rock samples

The relationship between uniaxial compressive strength (σc), MR and E is shown in Fig. 9. It illustrates that how uniaxial compressive strength influences MR and E for all studied rock samples.

Figure 9
Figure 9

Influence of uniaxial compressive strength (σc) on elastic modulus (E) and the value of MR for all studied samples.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

As it is clear, the elastic modulus is related to σc, with R2 = 0.06 very small. It also demonstrates that increase in the value of σc from 133 to 213 MPa does not influence E value. It can be seen from Fig. 9 that MR is related to σc, with R2 = 0.61. These values, however, are different from the values found by Palchik [7] for carbonated rocks. In his studies, the elastic modulus is partly related to uniaxial compressive strength with R2 = 0.55 and increase in the value of σc does not influence MR value (R2= 0.021 is very small).

3.2 Relationship between MR value and different strain and stress of the rock

The calculated values are compared with the international published relationships.

3.2.1 Relationship between MR and maximum axial strain (ea, max) for all studied samples

The observed and analytical (Eq. 1) relationships between εa, max and MR for all rock samples exhibiting εa, max < 1% are plotted in Fig. 10. It is clear that the calculated Diederichs Eq. (1) and observed values of MR for studied rock samples are similar. Fig. 11 presents the relative and root-mean-square errors between the calculated Diederichs Eq. (1) and observed MR at εa, max < 1%. As it is clear, the relative error (ζ , %) for studied samples is between 0.28% and 25% and root-mean-square error is (χ = 50). Comparing the values with the results obtained by Palchik [7] for carbonated rock samples, the relative error is between 0.08% and 10.8% and the root-mean-square error is 43.6.

Figure 10
Figure 10

Observed and analytical (Eq. 1) relationship between εa, max and MR.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

Figure 11
Figure 11

Relative (ζ , %) and root-mean-square (χ) errors between calculated (Eq. 1) and observed MR.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

The relative (ζ , %) and root-mean-square (χ) errors between the observed and calculated parameter Π have been calculated as:

ζ(m)=2|obs(j)cal(j)|obs(j)+cal(j)×100
χ(m)=j=1n[obs(j)cal(j)]2n1

where Πobs(j) is the observed value of parameter in the jth sample, here is MR, Πcal(j) is the calculated value of parameter in the jth sample, j = 1, 2,...,n, is the number of tested samples, here is 50.

3.2.2 Relationship between MR and maximum volumetric strain ecd for all studied samples

The observed values between MR and εcd are plotted in Fig. 12. As it is clear, these parameters are partially related (R2 = 0.2) for studied rock samples. Palchik [7] however, found a good relationship (R2 = 0.85) between these two parameters for carbonated rock samples.

Figure 12
Figure 12

Relationship between MRandεcd(%).

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

3.2.3 Relationship between MR and crack damage stress scd for all studied samples

Fig. 13 shows the relationship between MR and crack damage stress (σcd) for all studied rock samples. As it can be seen, there is a relationship (R2 = 0.41) between these two parameters.

Figure 13
Figure 13

Relationship between MRandσcd.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

3.2.4 Relationship between MR andσcdσcfor all studied samples

The relationship between MR and σcdσcis presented in Fig. 14. As it can be seen, there is practically no relationship (R2 = 0.0009) between them.

Figure 14
Figure 14

Relationship between MRandσcdσc.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

3.2.5 Relationship between MR and εa,maxεcdfor all studied samples

As shown in Fig. 15, there is practically no correlation between these two values.

Figure 15
Figure 15

Relationship between MRandεa,maxεcd.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

3.2.6 Relationship between σcdσcandεa,maxεcdfor all studied samples

The relationship between σcdσcandεa,maxεcdis presented in Fig. 16. As it can be seen, there is a relationship (R2 = 0.4). Palchik [22], however, found the relationship (R2 = 0.7) for carbonated rock samples.

Figure 16
Figure 16

Relationship between σcdσcandεa,maxεcd.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

3.2.7 Relationship between MR and crack initiation stress (σci)

The relationship between MR and crack initiation stress (σci) is presented in Fig. 17. As it can be seen, there is practically no relationship between them (R2 = 0.08).

Figure 17
Figure 17

Relationship between M R and crack initiation stress (σci ).

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

3.2.8 Relationship between MR and crack initiation strain (εci)

Fig. 18 shows the relationship between MR and crack initiation strain (εci). As it is shown, there is a relationship (R2 = 0.13).

Figure 18
Figure 18

Relationship between MR and crack initiation strain (εci).

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

3.2.9 Relationship between MRandσciσcd

Fig. 19 presents the relationship between MRandσciσcd.As it is clear, there is practically no relationship (R2 = 0.03).

Figure 19
Figure 19

Relationship between MRandσciσcd.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

3.2.10 Relationship between MRandεciεcd

Fig. 20 shows the relationship between MRandεciεcd.As it can be seen, there is practically no relationship (R2 = 0.06).

Figure 20
Figure 20

Relationship between MRandεciεcd.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

3.2.11 Relationship between σciσcdandεcdεci

Fig. 21 shows the relationship between σciσcdandεcdεci.As it can be seen, there is a relationship (R2 = 0.54).

Figure 21
Figure 21

Relationship between σciσcdandεcdεci.

Citation: Studia Geotechnica et Mechanica 41, 1; 10.2478/sgem-2019-0010

4 Results and discussions

The laboratory compressive tests, statistical analysis and empirical and analytical relationships have been used to estimate the values of MR = Ec and its relationship with other mechanical parameters for granitic rocks. Studied rock samples exhibited the wide range of mechanical properties (57.425 GPa < E < 88.937 GPa, 0.18 < ν < 0.32, 77.3 MPa < σcd < 212.42 MPa, 133.34 MPa < σc < 213.04 MPa, 0.18 < εa max < 0.19, 0.04 < εcd < 0.14). From the results of this study, the following main conclusions are made:

  1. The mean value of MR mean = 439 for all granitic rock samples observed in this study and the mean value of MR mean = 420 obtained by Deere [15] for limestone and dolomite and the mean value of MR mean = 380.5 obtained by Palchik [7] for carbonated rock samples are similar. However, the range of MR = 326.42–597.42 obtained in this study is narrower than the range of MR = 250–700 obtained by Deere [15] and the range of MR = 60–1,600 obtained by Palchik [7].
  2. The observation confirms that there is no general empirical correlation (with reliable R2) between elastic modulus (E) and uniaxial compressive strength (σc), MR and maximum volumetric strain (εcd), MR and crack damage stress σcd.
  3. The analytical l relationship (Eq. 1) between εa max and MR offered by Palchik [7] or carbonated rock samples was investigated for granitic rock samples in this study. It is observed that this relationship can also be used for granitic rocks. The relative error (ζ , %) for studied samples is between 0.2% and 24.5% and root-mean-square error is (χ = 50) . Comparing the values with the result obtained by Palchik [7] for carbonated rock samples, the relative error is between 0.08% and 10.8% and the root-mean-square error is 43.6.
  4. The observed correlation between MR and εcd for studied granitic rock sample is R2 = 0.2 . Palchik [7], however, found a good relationship ( R2 = 0.85 ) between these two parameters for carbonated rock samples.
  5. It is established that there is a correlation between σciσcdand εcdεciwithR2=0.54.
    1. Based on the obtained results, there is practically no relationship between MR and (εa,maxεcd),(σcdσc),(σci),(σciσcd),εcdεci;however, there is a relationship between MR and (εcd),(σcd) and (εci).

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].

Acknowledgements

This paper has been published with the permission of Public Limited Company for Radioactive Waste Management (PURAM). The project presented in this article is supported by National Research, Development and Innovation Office – NKFIH 124366 and NKFIH 124508 and the Hungarian-French Scientific Research Grant (No. 2018-2.1.13-TÉT-FR-2018-00012).

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    Király E. Koroknai B. (2004). The magmatic and metamorphic evolution of the north-eastern part of the Mórágy Block. Annual Report of Geological Institute of Hungary from 2003 pp. 299-318.

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    Brace W.F. Paulding B.W. Scholz C. (1966). Dilatancy in the fracture of crystalline rocks. Journal of Geophysical Research 71 3939-3953.

    • Crossref
    • Export Citation
  • [12]

    Martin C.D. Chandler N.A. (1994). The progressive fracture of Lac du Bonnet granite. International Journal of Rock Mechanics and Mining Sciences 31 643-659.

    • Crossref
    • Export Citation
  • [13]

    Diederichs M.S. (2007). The 2003 Canadian Geotechnical Colloquium: mechanistic interpretation and practical application of damage and spalling prediction criteria for deep tunnelling. Canadian Geotechnical Journal 44 1082-1116.

    • Crossref
    • Export Citation
  • [14]

    Cieslik J. (2014). Onset of crack initiation in uniaxial and triaxial compression tests of dolomite samples. Studia Geotechnica et Mechanica 1 23-27.

  • [15]

    Deere D.U. (1968). Geological considerations. In: Rock mechanics in engineering practice Edited by K.G. Stagg O.C. Zienkiewicz. pp. 1-20.

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    Bieniawski Z.T. (1967). Mechanism of brittle fracture of rock. International Journal of Rock Mechanics and Mining Sciences 4 395-430.

    • Crossref
    • Export Citation
  • [17]

    Martin C.D. (1993). Strength of massive Lac du Bonnet granite around underground openings PhD thesis Department of Civil and Geological Engineering University of Manitoba Winnipeg.

  • [18]

    Pettitt W.S. Young R.P. Marsden J.R. (1998). Investigating the mechanics of microcrack damage induced under true-triaxial unloading. In: Eurock 98 Society of Petroleum Engineering pp. SPE 47319.

  • [19]

    Eberhardt E. Stead D. Stimpson B. (1999). Quantifying progressive pre-peak brittle fracture damage in rock during uniaxial compression. International Journal of Rock Mechanics and Mining Sciences 36 361-380.

    • Crossref
    • Export Citation
  • [20]

    Heo J.S. Cho H.K. Lee C.I. (2001). Measurement of acoustic emission and source location considering anisotropy of rock under triaxial compression. In: Rock mechanics a challenge for society Edited by P. Sarkka P. Eloranta . Swets and Zeitlinger Lisse Espoo pp. 91–96.

  • [21]

    Katz O. Reches Z. (2004). Microfracturing damage and failure of brittle granites. The Journal of Geophysical Research 109(B1).

  • [22]

    Palchik V. (2013). Is there link between the type of the volumetric strain curve and elastic constants porosity stress and strain characteristics? Rock Mechanics and Rock Engineering 46 315-326.

    • Crossref
    • Export Citation
  • [23]

    Asszonyi Cs. Fülöp T. Ván P. (2015). Distinguished rheological models for solids in the framework of a thermodynamical internal variable theory. Continuum Mechanics and Thermodynamics 27(6) 971-986.

    • Crossref
    • Export Citation
  • [24]

    Vásárhelyi B. Kovács D. (2017). Empirical methods of calculating the mechanical parameters of the rock mass. Periodica Polytechnica Civil Engineering 61(1) 39-50.

  • [25]

    Asszonyi Cs. Csatár A. Fülöp T. (2016). Elastic thermal expansion plastic and rheological processes – theory and experiment. Periodica Polytechnica Civil Engineering 60(4) 591-601.

    • Crossref
    • Export Citation
  • [26]

    Vásárhelyi B. Davarpanah M. (2018). Influence of water content on the mechanical parameters of the intact rock and rock mass. Periodica Polytechnica Civil Engineering 62(4) 1060-1066.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Vásárhelyi B. (2005). Statistical analysis of the influence of water content on the strength of the Miocene limestone. Rock Mechanics and Rock Engineering 38 69-76.

    • Crossref
    • Export Citation
  • [2]

    Palchik V. (2007). Use of stress-strain model based on Haldane’s distribution function for prediction of elastic modulus. International Journal of Rock Mechanics and Mining Sciences 44(4) 514-524.

    • Crossref
    • Export Citation
  • [3]

    Ocak I. (2008). Estimating the modulus of elasticity of the rock material from compressive strength and unit weight. Journal of the Southern African Institute of Mining and Metallurgy 108(10) 621-629.

  • [4]

    Ramamurthy T. Madhavi Latha G. Sitharam T.G. (2017). Modulus ratio and joint factor concepts to predict rock mass response. Rock Mechanics and Rock Engineering 50 535-366.

  • [5]

    Deere D. Miller R. (1966). Engineering classification and index properties for intact rock. Techn. Report. No. AFWL-TR-65-116 Air Force

  • [6]

    Palmström A. Singh R. (2001). The deformation modulus of rock masses - comparisons between in situ tests and indirect estimates. Tunnelling and Underground Space Technology 16 115-131.

    • Crossref
    • Export Citation
  • [7]

    Palchik V. (2011). On the ratios between elastic modulus and uniaxial compressive strength of heterogeneous carbonate rocks. Rock Mechanics and Rock Engineering 44 121-128.

    • Crossref
    • Export Citation
  • [8]

    Vásárhelyi B. Kovács L. Kovács B. (2013). Determining the failure envelope of intact granitic rocks from Bátaapáti. GeoScience Engineering 2(4) 93-101.

  • [9]

    Buda Gy. (1985). Formation of Variscan collisional granitoids. (in Hungarian) Candidate thesis Eötvös University Budapest Hungary.

  • [10]

    Király E. Koroknai B. (2004). The magmatic and metamorphic evolution of the north-eastern part of the Mórágy Block. Annual Report of Geological Institute of Hungary from 2003 pp. 299-318.

  • [11]

    Brace W.F. Paulding B.W. Scholz C. (1966). Dilatancy in the fracture of crystalline rocks. Journal of Geophysical Research 71 3939-3953.

    • Crossref
    • Export Citation
  • [12]

    Martin C.D. Chandler N.A. (1994). The progressive fracture of Lac du Bonnet granite. International Journal of Rock Mechanics and Mining Sciences 31 643-659.

    • Crossref
    • Export Citation
  • [13]

    Diederichs M.S. (2007). The 2003 Canadian Geotechnical Colloquium: mechanistic interpretation and practical application of damage and spalling prediction criteria for deep tunnelling. Canadian Geotechnical Journal 44 1082-1116.

    • Crossref
    • Export Citation
  • [14]

    Cieslik J. (2014). Onset of crack initiation in uniaxial and triaxial compression tests of dolomite samples. Studia Geotechnica et Mechanica 1 23-27.

  • [15]

    Deere D.U. (1968). Geological considerations. In: Rock mechanics in engineering practice Edited by K.G. Stagg O.C. Zienkiewicz. pp. 1-20.

  • [16]

    Bieniawski Z.T. (1967). Mechanism of brittle fracture of rock. International Journal of Rock Mechanics and Mining Sciences 4 395-430.

    • Crossref
    • Export Citation
  • [17]

    Martin C.D. (1993). Strength of massive Lac du Bonnet granite around underground openings PhD thesis Department of Civil and Geological Engineering University of Manitoba Winnipeg.

  • [18]

    Pettitt W.S. Young R.P. Marsden J.R. (1998). Investigating the mechanics of microcrack damage induced under true-triaxial unloading. In: Eurock 98 Society of Petroleum Engineering pp. SPE 47319.

  • [19]

    Eberhardt E. Stead D. Stimpson B. (1999). Quantifying progressive pre-peak brittle fracture damage in rock during uniaxial compression. International Journal of Rock Mechanics and Mining Sciences 36 361-380.

    • Crossref
    • Export Citation
  • [20]

    Heo J.S. Cho H.K. Lee C.I. (2001). Measurement of acoustic emission and source location considering anisotropy of rock under triaxial compression. In: Rock mechanics a challenge for society Edited by P. Sarkka P. Eloranta . Swets and Zeitlinger Lisse Espoo pp. 91–96.

  • [21]

    Katz O. Reches Z. (2004). Microfracturing damage and failure of brittle granites. The Journal of Geophysical Research 109(B1).

  • [22]

    Palchik V. (2013). Is there link between the type of the volumetric strain curve and elastic constants porosity stress and strain characteristics? Rock Mechanics and Rock Engineering 46 315-326.

    • Crossref
    • Export Citation
  • [23]

    Asszonyi Cs. Fülöp T. Ván P. (2015). Distinguished rheological models for solids in the framework of a thermodynamical internal variable theory. Continuum Mechanics and Thermodynamics 27(6) 971-986.

    • Crossref
    • Export Citation
  • [24]

    Vásárhelyi B. Kovács D. (2017). Empirical methods of calculating the mechanical parameters of the rock mass. Periodica Polytechnica Civil Engineering 61(1) 39-50.

  • [25]

    Asszonyi Cs. Csatár A. Fülöp T. (2016). Elastic thermal expansion plastic and rheological processes – theory and experiment. Periodica Polytechnica Civil Engineering 60(4) 591-601.

    • Crossref
    • Export Citation
  • [26]

    Vásárhelyi B. Davarpanah M. (2018). Influence of water content on the mechanical parameters of the intact rock and rock mass. Periodica Polytechnica Civil Engineering 62(4) 1060-1066.

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Journal information
Impact Factor


CiteScore 2018: 1.03

SCImago Journal Rank (SJR) 2018: 0.213
Source Normalized Impact per Paper (SNIP) 2018: 1.106

Figures
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    Hypothetical stress–strain curves [4].

  • View in gallery

    Relationship between modulus ratio (MR) and maximum axial strain (εa, max) using different carbonate rocks [7].

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    Main types of rock samples. (a, b) Megacryst-bearing, medium-grained, biotite monzogranites. (c) Medium-grained, biotite monzogranites with elongated monzonitic enclaves. (d) Quartz monzonite.

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    A prepared sample in the beginning of the UCS test.

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    Axial stress–volumetric strain curve with the threshold of crack initiation and crack damage and failure stress for Hungarian granitic sample (uniaxial compression case).

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    Crack volumetric strain method for crack initiation threshold determination for Hungarian granitic rock sample (uniaxial compression case).

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    Poisson’s ratio method for crack initiation threshold determination for Hungarian granitic rock sample (uniaxial compression case).

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    Observed values of modulus ratio (MR) in each of 50 examined rock samples.

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    Influence of uniaxial compressive strength (σc) on elastic modulus (E) and the value of MR for all studied samples.

  • View in gallery

    Observed and analytical (Eq. 1) relationship between εa, max and MR.

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    Relative (ζ , %) and root-mean-square (χ) errors between calculated (Eq. 1) and observed MR.

  • View in gallery

    Relationship between MRandεcd(%).

  • View in gallery

    Relationship between MRandσcd.

  • View in gallery

    Relationship between MRandσcdσc.

  • View in gallery

    Relationship between MRandεa,maxεcd.

  • View in gallery

    Relationship between σcdσcandεa,maxεcd.

  • View in gallery

    Relationship between M R and crack initiation stress (σci ).

  • View in gallery

    Relationship between MR and crack initiation strain (εci).

  • View in gallery

    Relationship between MRandσciσcd.

  • View in gallery

    Relationship between MRandεciεcd.

  • View in gallery

    Relationship between σciσcdandεcdεci.

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