## 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 (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}).

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):

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 _{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.

## 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

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.

Mechanical properties of investigated Mórágy granitic rock samples.

Rock sample | ν | E | ε_{ci} | σ_{ci} | ε_{cd} | σ_{cd} | ε_{a, max} | σ_{c} | M_{R} |
---|---|---|---|---|---|---|---|---|---|

(-) | (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 (*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.

### 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,

ε_{a} and ε_{l} are the axial and lateral strain; σ_{1} and σ_{3} 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).

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].

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:

The ranges of

## 3 Effect of mechanical properties on *M*R value

### 3.1 Relationship between *M*_{R}, σ_{c} and *E* for all granitic rock samples

The relationship between uniaxial compressive strength (σ_{c}), *M*_{R} and *E* is shown in Fig. 9. It illustrates that how uniaxial compressive strength influences *M*_{R} and *E* for all studied rock samples.

As it is clear, the elastic modulus is related to σ_{c}, with *R*^{2} = 0.06 very small. It also demonstrates that increase in the value of σ_{c} from 133 to 213 MPa does not influence *E* value. It can be seen from Fig. 9 that *M*_{R} is related to σ_{c}, with *R*^{2} = 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 *R*^{2} = 0.55 and increase in the value of σ_{c} does not influence *M*_{R} value (*R*^{2}= 0.021 is very small).

### 3.2 Relationship between *M*_{R} value and different strain and stress of the rock

The calculated values are compared with the international published relationships.

### 3.2.1 Relationship between *M*_{R} and maximum axial strain (*e*_{a, max}) for all studied samples

The observed and analytical (Eq. 1) relationships between ε_{a, max} and *M*_{R} for all rock samples exhibiting ε_{a, max} < 1% are plotted in Fig. 10. It is clear that the calculated Diederichs Eq. (1) and observed values of *M*_{R} for studied rock samples are similar. Fig. 11 presents the relative and root-mean-square errors between the calculated Diederichs Eq. (1) and observed *M*_{R} at ε_{a, 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.

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

where Π_{obs}(* _{j}*) is the observed value of parameter in the

*j*th sample, here is

*M*

_{R}, Π

_{cal(}

_{j)}

*is the calculated value of parameter in the*

_{}*j*th sample,

*j*= 1, 2,...,

*n*, is the number of tested samples, here is 50.

### 3.2.2 Relationship between *M*_{R} and maximum volumetric strain *e*_{cd} for all studied samples

The observed values between *M*_{R} and ε_{cd} are plotted in Fig. 12. As it is clear, these parameters are partially related (*R*^{2} = 0.2) for studied rock samples. Palchik [7] however, found a good relationship (*R*^{2} = 0.85) between these two parameters for carbonated rock samples.

### 3.2.3 Relationship between *M*_{R} and crack damage stress *s*_{cd} for all studied samples

Fig. 13 shows the relationship between *M*_{R} and crack damage stress (σ_{cd}) for all studied rock samples. As it can be seen, there is a relationship (*R*^{2} = 0.41) between these two parameters.

### 3.2.4 Relationship between *M*_{R} and$\frac{{\sigma}_{\text{cd}}}{{\sigma}_{\text{c}}}$ for all studied samples

The relationship between *M*_{R} and *R*^{2} = 0.0009) between them.

### 3.2.5 Relationship between *M*_{R} and $\frac{{\epsilon}_{\text{a,max}}}{{\epsilon}_{\text{cd}}}$ for all studied samples

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

### 3.2.6 Relationship between $\frac{{\sigma}_{\text{cd}}}{{\sigma}_{\text{c}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\epsilon}_{\text{a,max}}}{{\epsilon}_{\text{cd}}}$ for all studied samples

The relationship between *R*^{2} = 0.4). Palchik [22], however, found the relationship (*R*^{2} = 0.7) for carbonated rock samples.

### 3.2.7 Relationship between *M*_{R} and crack initiation stress (σ_{ci})

The relationship between *M*_{R} and crack initiation stress (σ_{ci}) is presented in Fig. 17. As it can be seen, there is practically no relationship between them (*R*^{2} = 0.08).

### 3.2.8 Relationship between *M*_{R} and crack initiation strain (ε_{ci})

Fig. 18 shows the relationship between *M*_{R} and crack initiation strain (ε_{ci}). As it is shown, there is a relationship (*R*^{2} = 0.13).

### 3.2.9 Relationship between ${M}_{R}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\frac{{\sigma}_{\text{ci}}}{{\sigma}_{\text{cd}}}$

Fig. 19 presents the relationship between *R*^{2} = 0.03).

### 3.2.10 Relationship between ${M}_{R}and\text{\hspace{0.17em}}\frac{{\epsilon}_{\text{ci}}}{{\epsilon}_{\text{cd}}}$

Fig. 20 shows the relationship between *R*^{2} = 0.06).

### 3.2.11 Relationship between $\frac{{\sigma}_{\text{ci}}}{{\sigma}_{\text{cd}}}and\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\epsilon}_{\text{cd}}}{{\epsilon}_{\text{ci}}}$

Fig. 21 shows the relationship between *R*^{2} = 0.54).

## 4 Results and discussions

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_{a}_{max} < 0.19, 0.04 < ε_{cd} < 0.14). From the results of this study, the following main conclusions are made:

- – The mean value of
*M*_{R mean}= 439 for all granitic rock samples observed in this study and the mean value of*M*_{R mean}= 420 obtained by Deere [15] for limestone and dolomite and the mean value of*M*_{R mean}= 380.5 obtained by Palchik [7] for carbonated rock samples are similar. However, the range of*M*_{R}= 326.42–597.42 obtained in this study is narrower than the range of*M*_{R}= 250–700 obtained by Deere [15] and the range of*M*_{R}= 60–1,600 obtained by Palchik [7]. - – The observation confirms that there is no general empirical correlation (with reliable
*R*^{2}) between elastic modulus (*E*) and uniaxial compressive strength (σ_{c}),*M*_{R}and maximum volumetric strain (ε_{cd}),*M*_{R}and crack damage stress σ_{cd}. - – The analytical
*l*relationship (Eq. 1) between ε_{a max}and*M*_{R}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. - – The observed correlation between
*M*_{R}and ε_{cd}for studied granitic rock sample is*R*^{2}= 0.2 . Palchik [7], however, found a good relationship (*R*^{2}= 0.85 ) between these two parameters for carbonated rock samples. - – It is established that there is a correlation between
and$\frac{{\sigma}_{\text{ci}}}{{\sigma}_{\text{cd}}}$ $\frac{{\epsilon}_{\text{cd}}}{{\epsilon}_{\text{ci}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}^{2}=\mathrm{0.54.}$ - ‒ Based on the obtained results, there is practically no relationship between
*M*_{R}and however, there is a relationship between$\left(\frac{{\epsilon}_{\text{a,max}}}{{\epsilon}_{\text{cd}}}\right),\left(\frac{{\sigma}_{\text{cd}}}{{\sigma}_{\text{c}}}\right),\left({\sigma}_{\text{ci}}\right),\left(\frac{{\sigma}_{\text{ci}}}{{\sigma}_{\text{cd}}}\right),\text{\hspace{0.17em}}\frac{{\epsilon}_{\text{cd}}}{{\epsilon}_{\text{ci}}};$ *M*_{R}and (ε_{cd}),(σ_{cd}) and (ε_{ci}).

- ‒ 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].

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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