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. . 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  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).
The modulus ratio MR = E/σc between the modulus of elasticity (E) and uniaxial compressive strength (σc) for intact rock samples varies from 106 to 1,600 . For most rocks, MR is between 250 and 500 with average MR = 400, E = 400 σc.
Palchik  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):
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
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.  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  (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 . 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
Mechanical properties of investigated Mórágy granitic rock samples.
|Rock sample||ν||E||εci||σci||εcd||σcd||εa, max||σc||MR|
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, , 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.
2.2 ‒ Crack volumetric strain method
Martin and Chandler  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 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  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 , 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  is between 250 and 700 with the mean of 420 for limestone and dolomites. The range of MR obtained by Palchik  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  and Palchik . 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  and Palchik .
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:
The ranges of
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.
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  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  for carbonated rock samples, the relative error is between 0.08% and 10.8% and the root-mean-square error is 43.6.
The relative (ζ , %) and root-mean-square (χ) errors between the observed and calculated parameter Π have been calculated as:
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  however, found a good relationship (R2 = 0.85) between these two parameters for carbonated rock samples.
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.
3.2.4 Relationship between MR and
for all studied samples
The relationship between MR and
3.2.5 Relationship between MR and
for all studied samples
As shown in Fig. 15, there is practically no correlation between these two values.
3.2.6 Relationship between
for all studied samples
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).
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).
3.2.9 Relationship between
Fig. 19 presents the relationship between
3.2.10 Relationship between
Fig. 20 shows the relationship between
3.2.11 Relationship between
Fig. 21 shows the relationship between
4 Results and discussions
The laboratory compressive tests, statistical analysis and empirical and analytical relationships have been used to estimate the values of MR = E/σc 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:
- – 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  for limestone and dolomite and the mean value of MR mean = 380.5 obtained by Palchik  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  and the range of MR = 60–1,600 obtained by Palchik .
- – 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.
- – The analytical l relationship (Eq. 1) between εa max and MR offered by Palchik  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  for carbonated rock samples, the relative error is between 0.08% and 10.8% and the root-mean-square error is 43.6.
- – The observed correlation between MR and εcd for studied granitic rock sample is R2 = 0.2 . Palchik , however, found a good relationship ( R2 = 0.85 ) between these two parameters for carbonated rock samples.
- – It is established that there is a correlation between
- ‒ Based on the obtained results, there is practically no relationship between MR and
however, there is a relationship between MR and (εcd),(σcd) and (εci).
- ‒ Based on the obtained results, there is practically no relationship between MR and
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].
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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