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

Rock engineering properties are considered to be the most important parameters in the design of groundworks. Two important mechanical parameters, uniaxial compressive strength (σ_c) and elastic modulus of rock (E), should be estimated correctly. There are different empirical relationships between σ_c 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 (M_R), which is defined as the ratio of tangent modulus of intact rock (E) at 50% of failure strength and its compressive strength (σ_c).

Figure 1

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 1,600 [6]. For most rocks, M_R is between 250 and 500 with average M_R = 400, E = 400 σ_c.

Palchik [7] 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 a 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 (Fig. 2):

Figure 2

where k is the 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

The expansion of the expression 2/(1+eεa,max)${2}/{\left( 1+{{e}^{{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{a,max}}}}} \right)}\;$using Taylor’s theorem shows the value of 2/(1+eεa,max)=${2}/{\left( 1+{{e}^{{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{a,max}}}}} \right)}\;=$1 + 0.46 ε_{a, 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.

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

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

UCS, uniaxial compressive strength

Table 1 summarizes the value of elastic modulus (E), crack damage stress (σ_cd), uniaxial compressive strength (σ_c), Poisson’s ratio (ν), crack initiation stress (σ_ci), axial failure strain (ε_{a, max}), maximum volumetric strain (ε_cd), crack initiation strain (ε_ci) and M_R 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 (σ_ci) and crack damage stress (σ_cd) 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

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 ε_Vcr is calculated as a difference between the elastic volumetric strain ε_Vel and volumetric strain ε_V determined in the test,

Figure 6

(3)εV=2ε1+εa$${{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{V}}}=2{{\varepsilon }_{1}}+{{\varepsilon }_{\text{a}}}$$(4)εVcr=εV−εVel$${{\varepsilon }_{\text{Vcr}}}={{\varepsilon }_{\text{V}}}-{{\varepsilon }_{\text{Vel}}}$$(5)εVel=1−2vE(σ1+2σ3)$${{\varepsilon }_{\text{Vel}}}=\frac{1-2v}{E}\left( {{\sigma }_{1}}+2{{\sigma }_{3}} \right)$$

ε_a and ε_l 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

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

Figure 8

The ranges of the elastic modulus (E), Poisson’s ratio (ν), crack damage stress (σ_cd) and uniaxial compressive strength (σ_c), axial failure strain (ε_{a, max}) and maximum volumetric strain (ε_cd), crack initiation stress (σ_ci) and crack initiation strain (ε_ci) 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$$\begin{align}& 57.425\,\text{GPa}<E<88.937\,\text{GPa} \\ & \text{0}\text{.18}<v<0.32 \\ & 30\,\text{MPa}<{{\sigma }_{\text{ci}}}<90\,\text{MPa} \\ & \text{77}\,\text{MPa}<{{\sigma }_{\text{cd}}}<182\,\text{MPa} \\ & \text{133}\text{.34}\,\text{MPa}<{{\sigma }_{c}}<213.04\,\text{MPa} \\ & \text{0}\text{.02}<{{\varepsilon }_{\text{ci}}}<0.06 \\ & 0.18<{{\varepsilon }_{\text{a,max}}}<0.19 \\ & 0.04<{{\varepsilon }_{\text{cd}}}<0.14 \\ \end{align}$$

The ranges of εa,maxεcdandσcdσc$\frac{{{\varepsilon }_{\text{a,max}}}}{{{\varepsilon }_{\text{cd}}}}\text{and}\,\,\,\frac{{{\sigma }_{\text{cd}}}}{{{\sigma }_{\text{c}}}}$and σciσc$\frac{{{\sigma }_{\text{ci}}}}{{{\sigma }_{\text{c}}}}$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 σcdσc=0.5−1.0andεa,maxεcd=1.51−$\frac{{{\sigma }_{\text{cd}}}}{{{\sigma }_{\text{c}}}}=0.5-1.0\,\text{and}\,\frac{{{\varepsilon }_{\text{a,}\,\text{max}}}}{{{\varepsilon }_{\text{cd}}}}=1.51-$6.91 obtained by Palchik [7]. They are also different from the values of σcdσc=0.71−0.84$\frac{{{\sigma }_{\text{cd}}}}{{{\sigma }_{c}}}=0.71-0.84$obtained by Brace et al. [11], Bieniawski [16], Martin [17], Pettitt et al. [18], Eberhardt et al. [19], Heo et al. [20] and Katz and Reches [21] for granites, sandstones and quartzites. The range of σciσc$\frac{{{\sigma }_{\text{ci}}}}{{{\sigma }_{\text{c}}}}$for most rocks falls in the range of 0.3–0.5.

The laboratory compressive tests, statistical analysis and empirical and analytical relationships 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 exhibited 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_amax < 0.19, 0.04 < ε_cd < 0.14). 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 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].