Experimental Determination of the Mechanical Properties and Deformation Constants of Mórágy Granitic Rock Formation (Hungary)

Rock engineering properties are considered to be the most important parameters in the design of ground works. Two important mechanical parameters, uniaxial compressive strength (σ_c) and elastic modulus of rock (E) should be estimated correctly. There are different empirical relations between σ_c and E obtained for limestones, agglomerates, dolomites, chalks, sandstones, and basalts (Vásárhelyi 2005; Palchik 2007; Ocak 2008; Vásárhelyi and Davarpanah 2018; Asszonyi et al. 2016), among the others.

Hypothetical stress–strain curves for three different types of rock are presented in Fig. 1. by Ramamurthy et al. (2017). 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 response to engineering usage. This approaching of defining the quality of intact rocks was proposed by Deere and Miller (1966) considering the modulus ratio (M_R), which is defined as the ratio of tangent modulus of intact rock (E) at 50% of failure strength to its compressive strength (σ_c).

The modulus ratio (M_R = E/σ_c) between the modulus of elasticity (E) and uniaxial compressive strength (σ_c) for intact rock samples varies from 106 to 1600 (Palmström and Singh 2001). For most rocks M_R is between 250 and 500 with average M_R = 400, E = 400 σ_c.

Palchik (2011) examined the M_R values for 11 heterogeneous carbonate rocks from different regions of Israel. The investigated dolomites, limestones and chalks had weak to very strong strength with wide range of elastic modulus. He found that M_R is closely related to the maximum axial strain (ε_a,max) at the uniaxial strength of the rock (σ_c) and the following relationship was found (see Fig. 2):where k is a conversion coefficient equal to 100, and ε_a,max is in  %. When M_R is known, ε_a,max (%) is obtained from Eq. (1) as:

Since the expansion of the expression 2/(1 + \(e^{{\varepsilon_{a.\hbox{max} } }}\)) using Taylor’s theorem shows the value of 2/(1 + \(e^{{\varepsilon_{a.\hbox{max} } }}\)) = 1 + 0.46 ε_a,max (Palchik 2013).

The goal of this paper is to check Eq. (1) for Hungarian granitic rock samples and study relations between characteristic compressive stress level, strain and mechanical properties, as well. Additionally, the validity of recently proposed mathematical model by Palchik (2019) for granitic rock samples was investigated. These granitic rock samples were investigated previously by Vásárhelyi et al. (2013) using multiple failure state triaxial tests (Figs. 3, 4).

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 (Buda 1985). In spatial viewpoint the monzogranitic rocks contain generally ovalshaped, variably elongated monzonite enclaves (predominantly amphibole-biotite monzonites, diorites, and syenites) of various size (from a few cm to several hundred 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 tephryte dykes crosscut all of the previously described rock types (Király and Koroknai 2004). In general fractured but fresh rock is the common which is sparsely intersected by fault zones with few meter thick clay gauges. Intense clay mineralization 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 kN with 0.01 accuracy, rate of loading was 0.6 kN/s. Axial and tangential deformation were measured by strain gauges, that measures the deformation between ¼ and ¾ of the sample’s height.

Fifty uniaxial compressive tests were performed in the rock mechanics laboratory at RockStudy Ltd. The D = 50 mm sized cylindrical rock samples having the ratio L/D = 2/1 (here L and D is the length and diameter of a sample, respectively) were prepared. Mechanical properties of granitic rock samples are summarized in Table 1.

Table 1 shows the value of Modulus of elasticity (E), Modulus of rigidity (G) and Bulk modulus (K), uniaxial compressive strength (σ_c), Poisson’s ratio (ν), Axial failure strain (ε_amax), and M_R for each of the studied granitic samples.

The values of Tangent Young’s modulus (\(E_{tan} )\) is defined as the slope of a line tangent to the stress–strain curve at a fixed percentage (50%) of the ultimate strength, the value of Average Young’s modulus \(\left( E \right)\) is defined as the slope of the straight-line part of the stress–strain curve, the value of Secant Young’s modulus \(\left( {E_{sec} } \right)\) is defined as as the slope of the line from the origin (usually point (0; 0)) to some fixed percentage of ultimate strength, usually 50% and Poisson’s ratio (\(\nu\)) were calculated by using linear regressions along linear portions of stress–axial strain curves and radial strain–axial strain curves, respectively The International Society for Rock Mechanics suggests three standard methods for its determination (Ulusay and Hudson 2007). The value of \(G_{tan}\), \(G_{sec}\) and \(K_{tan}\), \(K_{sec}\) are calculated based on the following equations.

The values of crack initiation stress (σ_ci) and crack damage stress (σ_cd) were calculated based on the following methods:

The relationship between uniaxial compressive strength (σ_c), M_R and E is shown in Fig. 10.

As it is clear, the elastic modulus is related to (σ_c), (with R² = 0.06 very small). It also demonstrates that increase in the value of σ_c from 133 to 213 MPa doesn’t influence E value. It can be seen from Fig. 10 that M_R is related to (σ_c), (with R² = 0.61). These values, however, are different from the values found by Palchik (2011) 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 σ_c doesn’t influence \({\text{M}}_{\text{R}}\) value (R² = 0.021 is very small).

The observed and analytical (Eq. 1) relationships between ε_{a max} and M_R for all rock samples exhibiting ε_{a max} < 1% is plotted in Fig. 11. It is clear that the calculated (Eq. 1) and observed values of M_R for studied rock samples are similar. Figure 12 presents relative and root-mean square errors between the calculated (Eq. 1) and observed M_R at ε_{a max} < 1%. As it is clear, the relative error \(\left( {\varsigma ,\% } \right)\) for studied samples is between 0.28 and 25% and root-mean square error is (χ = 50). Compare the values with the results obtained by Palchik (2011) for carbonate rock samples, the relative error is between 0.08 and 10.8% and the root-mean square error is 43.6. The relative \(\left( {\varsigma , \% } \right)\) and root-mean-square (χ) errors between the observed and calculated parameter \(\varPi\) have been calculated as:where \(\varPi_{obs\left( j \right)}\) is the observed value of parameter in jth sample, here is M_R, \(\varPi_{cal\left( j \right)}\) is the calculated value of parameter in jth sample, j = 1, 2,…, n, is the number of tested samples, here is 50.

Correlation between Tangant Young’s modulus \(\left( {E_{tan} } \right)\) and Secant Young’s modulus, Tangant Modulus of rigidity (\(G_{tan}\)) and Secant Modulus of rigidity (\(G_{sec}\)), Tangant Bulk modulus (\(K_{tan}\)) and Secant Bulk Modulus (\(K_{sec}\)), Tangant Poisson’s ratio (\(\vartheta_{tan}\)) and Secant Poisson’s ratio (\(\vartheta_{sec}\)), Tangant Modulus ratio (\(M_{Rtan}\)) and Secant Modulus ratio (\(M_{Rsec}\)) were investigated (Fig. 13).

According to Fig. 13a, we can see good linear correlation between tan and secant Young’s modulus \(,\) tan and secant modulus of rigidity (Fig. 13b), tan and secant modulus ratio (Fig. 13e), tan Young’s modulus and rigidity modulus (Fig. 13f), secant Young’s modulus and rigidity modulus (Fig. 13g).

Palchik (2019) proposed a simple stress–strain model for very strong (\(\sigma_{c} > 100\;{\text{MPa}}\)) carbonate rocks based on Haldane’s distribution function (Haldane 1919).

where \(\sigma_{a}\) is a current axial stress (MPa), \(\varepsilon_{a}\) is a current axial strain (%) at the axial stress \(\sigma_{a}\), \(\sigma_{c}\) is a uniaxial compressive strength (MPa), parameter (s) is a constant involved in a canonical form of Haldane’s function. For very strong limestones and dolomites (\(\sigma_{c} > 100\;{\text{MPa}}\)) the value of s is between 0.5 and 1, and exponent \(\theta\) is defined as follows

And can be presented in the following form by using Taylor’s series expansion:when s = 1.1:

The estimated and observed stress–strain relationships for four granitic rock samples are presented in Fig. 14. The relevant modelling parameters used in Fig. 14. are followings:

According to our calculations, the proposed model by Palchik (2019) for very strong limestones and dolomites is in good agreement with the elastic region of stress–strain relationships observed in this study for very strong granitic rock samples. However, there is a significant difference in non-linear part of stress–strain relationships between the observed and estimated values.

Comparison between estimated [Eqs. (10) and (14)], and observed axial stresses for studied rock samples shows a very good linear correlation between these stresses, and thus, Eq. 10 can predict stresses well (Fig. 15).

The laboratory compressive tests, statistical analysis, and empirical and analytical relations have been used to estimate the values of M_R = E/σ_c and its relationship with other mechanical parameters for granitic rocks. Studied rock samples exhibiting the wide range of mechanical properties (57.425 GPa < E < 88.937 GPa, 0.18 < ν < 0.32, 77.3 MPa < σ_cd < 212.42 MPa, 133.34 MPa < σ_c < 213.04 MPa, 0.18 < ε_{a max} < 0.19, 0.04 < ε_cd < 0.14). The ranges of \(\frac{{\varepsilon_{a max} }}{{\varepsilon_{cd} }}\) and \(\frac{{\sigma_{cd} }}{{\sigma_{c} }}\) and \(\frac{{\sigma_{ci} }}{{\sigma_{c} }}\) ratios are 1.49–5.28 and 0.5–0.9 and 0.2–0.5, respectively. These values are different from the values of \(\frac{{\sigma_{cd} }}{{\sigma_{c} }}\) = 0.5–1.0 and \(\frac{{\varepsilon_{a max} }}{{\varepsilon_{cd} }} =\) 1.51–6.91 obtained by Palchik (2011). They are also different from the values of \(\frac{{\sigma_{cd} }}{{\sigma_{c} }}\) = 0.71–0.84 obtained by Brace et al. (1966), Bieniawski (1967), Martin (1993), Pettitt et al. (1998), Eberhardt et al. (1999), Heo et al. (2001), Katz and Reches (2004) for granites, sandstones and quartzite. The range of \(\frac{{\sigma_{ci} }}{{\sigma_{c} }}\) for most rocks falls in the range of 0.3 to 0.5. Moreover, the relative and root-mean square errors between the calculated (Eq. 1) and observed M_R at ε_{a max} < 1% were calculated. The relative error \(\left( {\varsigma , \% } \right)\) for studied samples is between 0.28 and 25% and root-mean square error is (χ = 50), while Palchik (2011) calculated the relative error is between 0.08 and 10.8% and the root-mean square error is 43.6. Furthermore, the relationships between deformation constants were investigated. Based on our analyses, there is correlation between tan and secant Young’s modulus (R² = 0.91), tan and secant modulus of rigidity (R² = 0.90), tan and secant modulus ratio (R² = 0.95), tan Young’s modulus and rigidity modulus (R² = 0.96), secant Young’s modulus and rigidity modulus (R² = 0.95), Tangant Modulus ratio (\(M_{Rtan}\)) and Secant Modulus ratio (\(M_{Rsec}\)) with the value of (R² = 0.95). The estimated [Eqs. (10) and (14)] and observed stress–strain relations were plotted for all studied granitic rock samples. The values of the root-mean-square errors (χ), was also calculated for samples (based on Eq. (9)). For sample(a), this value is 2.3 MPa and 1.91 MPa, for sample(b), this value is 18.71 MPa and 21.39 MPa, for sample(c), this value is 7.32 MPa and 4.97 MPa, for sample(d), this value is 34.7 MPa and 21.87 MPa. Palchik (2019) observed a very good linear correlation (0.983 < \(R^{2}\) < 0.99) between estimated and observed axial stresses for very strong (\(\sigma_{c} > 100 \;{\text{Mpa}}\)) carbonate rocks, his results are in good agreement with our obtain results (0.997 < R² < 0.999) for granitic rock samples.

From the results of this study, the following main conclusions are made:

Notably, for a more precise and fundamental description of the mechanical behaviour of rock, one should apply nonequilibrium continuum thermodynamics along the lines of Asszonyi et al. (2015) and beyond. These relationships can be used for determining the mechanical parameters of the rock mass, as well (Vásárhelyi and Kovács 2017).