# Effect of Stress Path on the Failure Envelope of Intact Crystalline Rock at Low Confining Stress

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Stress Paths Observed in the Field and in Typical Low Confinement Laboratory Tests

_{1}–σ

_{3}plot in Figure 1. In the direct tension and UCS tests, the principal stress monotonically increases from an initial state with zero stress until the sample fails. Similarly, in the triaxial test, a sample is initially taken to a hydrostatic condition and then σ

_{1}is monotonically increased until failure occurs.

_{1}–σ

_{3}plot in Figure 1 represents the in situ stress condition. Path ABCDE shows the change in σ

_{1}and σ

_{3}values as construction progresses towards point A and passes through it. The dashed line near point B, σ

_{1}− σ

_{2}= (1/3)UCS, represents the stress level around which crack initiates in the rock. When the stress path approaches point B, microcracks start forming in the rock and, thereafter, the damage process leading to failure depends upon the stress path it takes.

_{1}and σ

_{2}versus σ

_{3}. Comparing the initial and final stress states shows the presence of a stress rotation in addition to changes in the magnitudes of the principal stresses. Figure 1 and Figure 2 show that, during the stress path, rock can come across a condition where one of the principal stresses can be compressive and the other tensile. This is called a confined extension condition and is explored in the next section.

#### 2.2. Laboratory-Confined Extension Test on Rock

_{h}is the head area, and A

_{t}is the throat area. The confined extension test conducted is called the triaxial confined extension test when σ

_{2}= σ

_{1}. Rock tested in triaxial confined extension and triaxial tests in the same range of confinement shows that samples tested in triaxial confined extension fail at a considerably higher peak stress. The data available in the literature for confined extension are very limited. Two rocks subjected to extensive triaxial confined extension tests are Carrara marble [20] and Berea sandstone [21], shown in Figure 4a,b. Patel and Martin [22] compared the biaxial strength (σ

_{1}= σ

_{2}= compressive, σ

_{3}= 0) obtained from laboratory UCS testing of these two rocks and observed a clear impact of σ

_{2}on peak strength. If the confined extension test results (σ

_{1}= σ

_{2}> σ

_{3}) are compared with triaxial test results (σ

_{1}> σ

_{2}= σ

_{3}), there is a clear mismatch for both rock types (Figure 4). In the UCS tests, the difference is 41% and 22% for Carrara marble and Berea sandstone, respectively. The reason for this is yet unknown and could be due to the different stress path taken or the impact of σ

_{2}. This is explored in the next sections.

_{2}= 0, where increased confinement was tested based on samples with an increased depth of flattening. However, the maximum confinement achieved using the flattened Brazilian test was 37% of the UCS. The FJ BPM was therefore used to investigate confined extension.

#### 2.3. Flat-Jointed Bonded Particle Model

_{t}ratio of low porosity rocks [7] and requires cluster logic to increase the friction angles to more realistic values and increase the triaxial strength after calibrating the UCS [26]. Therefore, Potyondy [14] suggests the FJ BPM to address these limitations.

_{t}ratio. Patel and Martin [16] further investigated the role of initial microcracks present in a rock sample. They showed that an FJ sample with initial microcracks is essential to capture the initial nonlinearity, crack initiation stress, and tensile and compressive Young’s moduli in addition to the UCS/σ

_{t}ratio of intact rock.

## 3. Calibration of Microparameters for Lac du Bonnet Granite for the BPM

#### 3.1. Intergranular Stiffness Ratio and the Formation of Microcracks in the Sample

#### 3.2. Comparison of Laboratory Response of the Lac du Bonnet Granite Sample with the FJ Numerical Sample for the UCS Test

_{t}and E

_{c}) when stress-released microfractures were included in the model. The ratio between the simulated E

_{t}and E

_{c}was 0.69, which is within 5% of that obtained from laboratory testing. The crack initiation stress (σ

_{ci}) is the point in the axial stress vs. radial strain curve where the curve deviates from linearity after the initial elastic region; this value was 97.7 MPa for the numerical sample vs. 88.6 MPa in the laboratory. When the σ

_{ci}point was projected down in Figure 7, it matched the point where tensile microcracks started forming rapidly in the sample. Because only tensile cracks formed before σ

_{cd}, these cracks cannot be reflected in the axial stress versus axial strain curve. As shown in Figure 7, the axial stress versus axial strain curve deviates from linearity when the shear cracks start in the sample. These observations are in line with the laboratory findings of many researchers, such as Brace et al. [31] and Bieniawski [32]. The macroparameters for the numerical sample are compared with the Lac du Bonnet granite sample in Table 2.

## 4. Confined Extension Test Using the FJ BPM

_{1}is compressive and σ

_{3}is tensile. Schöpfer et al. [33] and Huang and Ma [34] use the parallel BPM to investigate rock under confined extension. The present study used the FJ code to investigate the intact rock behavior in confined extensions after obtaining acceptable results for the numerical sample compared to the typical laboratory test samples.

_{3}is then gradually reduced, keeping σ

_{1}and σ

_{2}constant until the sample fractures at point C. Samples that fail along such a stress path are influenced by σ

_{2}[35]. For the investigation of the confined extension on the numerical sample (Figure 3) with no σ

_{2}, the following steps were taken: (A) the cubical sample shown in Figure 5 was hydrostatically confined to a particular magnitude of σ

_{1}(point D, Figure 8); (B) keeping σ

_{1}constant, σ

_{2}and σ

_{3}were simultaneously reduced (point E, Figure 8); (C) the top and bottom 5 mm of the sample were then held and pulled in opposite directions keeping σ

_{1}constant and σ

_{2}at zero until the sample failed at point C (Figure 8). The x-coordinate gives the magnitude of σ

_{3}for the corresponding σ

_{1}; and (D) steps A–C were then repeated for different magnitudes of σ

_{1}to obtain different failure points in confined extension.

## 5. Results of FJ Modelling

#### 5.1. Influence of Stress Path on a Confined Extension Test

_{1}≥ σ

_{2}≥ σ

_{3}. Due to the excavation, i.e., change in the boundary condition, the rock at this point may fail in confined extension following path ABC. However, this stress path is case dependent and can be complex. The path followed by the conventional confined extension test using the dog-bone-shaped geometry is ADEC (Figure 8). The microfracture formation in rock and its peak strength are path dependent. To compare the impact of the stress path followed, paths ODEC and OEC were compared for the numerical FJ sample. As shown in Figure 9, more cracks form (533 vs. 470) in the sample when it follows stress path OEC (a low confinement stress path compared to ODEC), although the final stress conditions are similar (90 MPa at E). This influences the peak strength obtained from the samples due to the application of tensile load. As shown in Figure 10, stress paths ODEC and OEC have a 3.6% difference in peak strength.

#### 5.2. Results of the Confined Extension Test with σ_{2} = 0

_{1}. The stress–strain results for the tests between points E and C (Figure 8) at three stress levels—low (10 MPa), medium (90 MPa), and high (150 MPa)—are shown in Figure 12. At low confinement, the sample fails similar to a direct tension test sample. However, because of the confinement, the sample fails at a higher tensile stress (i.e., similar to the Brazilian tensile strength being greater than the direct tensile strength for Lac du Bonnet granite). With an increase in confinement, the peak strength of the sample increases (up to around 50% of the UCS value). As shown in Figure 13 and Figure 14, tensile cracking dominates at low confinement, while the percentage of shear cracks increases with increasing confinement. This finding is in agreement with observations by Brace [19] and Ramsey and Chester [20] using confined extension tests on dog-bone-shaped samples.

_{cd}of Lac du Bonnet granite. When a sample with confinement more than the crack damage stress is relaxed from point D to point E (Figure 8), equilibrium is not reached and the sample fails. This observation indicates the peak strength obtained after the crack damage stress depends upon the applied boundary condition.

_{2}and σ

_{3}gradually relaxed until the sample failed at point G (Figure 8). The results are plotted and a discussion is presented in Section 6. These results are also compared with the conventional triaxial stress (stress path OFG vs. OHG) in Section 6.

#### 5.3. Impact of σ_{2} on the Confined Extension Test

_{2}on the confined extension test of rock. A confined extension test on a numerical sample was conducted by loading the sample hydrostatically to point D (Figure 8). After point D, σ

_{3}was gradually reduced while keeping σ

_{1}and σ

_{2}constant. The sample was then extended in the axial direction to fail the sample in confined extension. Figure 15 compares the formation of cracks in the sample after σ

_{3}is unloaded (point E; Figure 8) and the sample fails in extension (point C; Figure 8) at a σ

_{1}of 90 MPa and three intermediate stress levels (90, 45, and 0 MPa). The cracks formed at point E (Figure 8) in the samples are random. The number of cracks formed is less when additional confinement is provided by σ

_{2}. σ

_{2}also prevents the formation of tensile cracks in the model. The shear crack percentage in the case where σ

_{2}is equal to σ

_{1}is greater than when σ

_{2}= 0. As shown in Figure 16, the peak strength increases from 11.0 MPa at σ

_{2}= 0 to 19.2 MPa when σ

_{2}is kept the same as σ

_{1}. Figure 17 compares the crack formation from points E to F (with an increase in extensile stress in the numerical sample). The number of shear cracks formed in the sample is greater when σ

_{2}is applied to the sample. This is in agreement with confined extension results on dog-bone-shaped Carrara marble [20].

## 6. Failure Envelope for Lac du Bonnet Granite

_{3}decreases sharply with the increase in σ

_{1}. However, as shown in Figure 18, the confinement in the laboratory sample could not be applied beyond 80% of the UCS, which is around σ

_{cd}for Lac du Bonnet granite. The numerical confined extension test was continued on the compression side (σ

_{1}compressive, σ

_{3}= σ

_{2}compressive, stress path OFG; Figure 8). Comparing these results to the conventional triaxial test on the numerical sample (stress path OAG; Figure 8) results in a good match and indicates that the influence of the stress path is negligible. When the Hoek–Brown failure envelope was extended into the confined extension region, it was found out that it underestimates actual strength compared to the laboratory direct tension, Brazilian, flattened Brazilian, and confined extension test results on numerical samples. The laboratory direct tension strength was 10.6 MPa, whereas the Hoek–Brown gave a value of 6.8 MPa.

## 7. Conclusions

- By changing the code, it was possible to study the behavior of the rock in confined extension at σ
_{2}= 0. When compared with the Hoek–Brown failure criterion, the actual strength values obtained from the numerical samples in confined extension were higher than data points from the Hoek–Brown criterion; however, the Hoek–Brown failure envelope overestimated the peak strength obtained from the laboratory tests in confined extension, e.g., as in the case of Carrara marble (Figure 4a). - The data points obtained from the Brazilian and flattened Brazilian tests were close to the values from the numerical analysis.
- Limited data available from laboratory testing (Brazilian and flattened Brazilian tests) and data from the numerical analysis indicate a tensile cutoff for Lac du Bonnet granite, as suggested by Hoek and Martin [2].
- For the numerical investigation on confined extension, the stress paths investigated have a minor impact on peak strength. However, microfracture formation was found to be path dependent.
- The numerical sample shows a clear impact of σ
_{2}in the confined extension test. Rock with 90 MPa confinement for both σ_{1}and σ_{2}produced ~76% higher strength compared to the sample with σ_{2}= 0. This requires a review of the present methodology to test dog-bone-shaped specimens in confined extension. - Further investigations should consider simultaneous stress rotation along with the stress path to better understand its impact on strength reduction in field conditions compared to the laboratory.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

σ_{ci} | Crack initiation stress |

σ_{cd} | Crack damage stress |

σ_{1}, σ_{2}, σ_{3} | Major, intermediate, and minor principal stress |

DT | Direct tension test |

BT | Brazilian tensile test |

FB | Flattened Brazilian test |

UCS | Uniaxial compressive strength |

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**Figure 1.**Stress path obtained for a point 50 mm above the crown of a mine-by tunnel using 3D numerical modelling [9]. The Hoek–Brown (H–B) failure envelope and the stress path for direct tension, UCS, and triaxial tests are also shown. The thick dashed line represents the stress level above which crack initiates in rock.

**Figure 2.**Stress path obtained from field measurements using CSIRO HI stress cells for σ

_{1}and σ

_{2}versus σ

_{3}in the hanging wall of 565#6 stope, Winston Lake Mine [4]. Note: the rock in situ stress changes from initially confined compression to confined extension.

**Figure 4.**Results of confined extension test for (

**a**) Carrara marble [23] and (

**b**) Berea sandstone [21]. Direct tension (DT), UCS, triaxial compression, and confined extension test results as well as the Hoek–Brown failure envelopes are shown. The biaxial strength of the Berea sandstone was calculated using interpolation.

**Figure 5.**(

**a**) The cubical FJ BPM made up of 3618 grains and 17,110 FJ contacts used in the investigation and (

**b**) representation of a partially fractured FJ [24]. In the fractured FJ, 14 of the 16 elements are cracked due to bending.

**Figure 6.**Comparison of microcrack formation in the numerical sample with krat = 1.6 or 1.2 and the laboratory Lac du Bonnet granite sample. The sample-1 result is from Martin [29] and the sample-2 result is from Eberhardt et al. [30]. Note the numerical samples with krat = 1.6 or 1.2 are calibrated by the same macroparameters as those for Lac du Bonnet granite.

**Figure 7.**Results of the UCS test for the numerical sample. (

**a**) shows the axial stress versus the axial and radial strain and (

**b**) shows the generation of tensile and shear cracks with axial strain.

**Figure 8.**A typical failure envelope for rock showing the stress path during probable field-confined extension (ABC, unloading), confined extension (ODEC and OFG), and triaxial (OHG) tests. Note: the location of point A depends upon the boundary condition to which the rock is subjected in the field.

**Figure 9.**Cracks formed in the sample at point E when the sample follows stress path (

**a**) OEC, σ

_{1}= 90 MPa, σ

_{2}= 0 and (

**b**) ODEC, σ

_{1}= σ

_{2}= 90 MPa. (

**a**) 533 tensile and 3 shear, (

**b**) 470 tensile and 5 shear.

**Figure 10.**Tensile axial stress–strain curve obtained due to the application of tensile stress for stress paths ODEC and OEC (from point E to point C, Figure 8) at σ

_{1}= 90 MPa.

**Figure 11.**Comparison of crack development in the sample with an increase in the tensile strain from point E to point C for stress paths ODEC and OEC: (

**a**) tensile cracks and (

**b**) shear cracks.

**Figure 12.**Stress–strain curves obtained for numerical confined extension tests at low (10 MPa), medium (90 MPa), and high (150 MPa) confinements (from point E to point C; Figure 8). In all cases, σ

_{2}= 0.

**Figure 13.**Number of tensile cracks formed with an increase in σ

_{3}for numerical confined extension tests at low (10 MPa), medium (90 MPa), and high (150 MPa) confinements (from point E to point C; Figure 8).

**Figure 14.**Number of shear cracks formed with an increase in σ

_{3}for numerical confined extension tests at low (10 MPa), medium (90 MPa), and high (150 MPa) confinements (from point E to point C; Figure 8).

**Figure 15.**Comparison of crack formation in the numerical sample for stress path ODEC (Figure 8): (

**a**–

**c**) after the relaxation of σ

_{3}(point E, Figure 8) and (

**d**–

**f**) at peak strength (point C, Figure 8) for σ

_{2}= 0, σ

_{2}= σ

_{1}/2 = 45 MPa, and σ

_{2}= σ

_{1}= 90MPa. Red = tensile crack, green = shear crack. (

**a**) σ

_{1}= 90 MPa; σ

_{2}= 0 (964 cracks); (

**b**) σ

_{1}= 90 MPa; σ

_{2}= 45 MPa (323 cracks); (

**c**) σ

_{1}= σ

_{2}= 90 MPa (204 cracks); (

**d**) σ

_{1}= 90 MPa; σ

_{2}= 0; (

**e**) σ

_{1}= 90 MPa; σ

_{2}= 45 MPa; (

**f**) σ

_{1}= σ

_{2}= 90 MPa.

**Figure 16.**Confined extension test at σ

_{2}= σ

_{1}= 90 MPa, σ

_{2}= σ

_{1}/2 = 45 MPa, and σ

_{2}= 0 (stress–strain curve showing the impact of σ

_{2}in the numerical sample).

**Figure 17.**Comparison of crack formation in the numerical sample with an increase in σ

_{2}: (

**a**) tensile cracks and (

**b**) shear cracks. In all cases, σ

_{1}was kept constant at 90 MPa.

**Figure 18.**Results of laboratory and numerical samples on Lac du Bonnet granite. DT = direct tension, BR = Brazilian, FB = flattened Brazilian.

**Table 1.**List of microparameters obtained after calibration for the numerical rock sample. Parameters required for grain and material genesis are also listed [27].

Parameter | Value |
---|---|

Associated with grain size distribution: | |

Minimum grain diameter | 2.2 mm |

Grain-size ratio | 2.3 |

Associated with material genesis: | |

Width of sample | 54 mm |

Height–width ratio | 1 |

Associated with FJ material group: | |

Installation gap | 1.31 mm |

Bonded fraction | 0.65 |

Gapped fraction | 0.35 |

Slit fraction, derived | 0 |

Initial surface-gap distribution, mean | 0.002 mm |

Initial surface-gap distribution, standard deviation | 0 |

Elements in radial direction | 1 |

Elements in circumferential direction | 3 |

Radius-multiplier code | 0 |

Radius-multiplier value | 0.577 |

Effective modulus | 135.8 GPa |

Stiffness ratio | 1.2 |

Friction coefficient | 1.4 |

Tensile-strength distribution, mean | 41.6 MPa |

Tensile-strength distribution, standard deviation | 0 |

Cohesion distribution, mean | 203 MPa |

Cohesion distribution, standard deviation | 0 |

Friction angle | 43.2° |

Associated with the linear material group: | |

Effective modulus | 135.8 GPa |

Stiffness ratio | 1.2 |

Friction coefficient | 2.2 mm |

**Table 2.**Comparison of macroresponses observed in the laboratory Lac du Bonnet granite and numerical samples.

Case | E_{c} (GPa) | ν_{c} | σ_{ci} (MPa) | σ_{cd} (MPa) | UCS (MPa) | E_{t} (GPa) | σ_{t} (MPa) | E_{t}/E_{c} |
---|---|---|---|---|---|---|---|---|

Lab | 70.5 | 0.26 | 88.6 | 163.3 | 221.7 | 45.8 | 10.6 | 0.65 |

Numerical | 69.3 | 0.1 | 97.7 | 208.1 | 219.6 | 47.7 | 10.8 | 0.69 |

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Patel, S.; Martin, C.D. Effect of Stress Path on the Failure Envelope of Intact Crystalline Rock at Low Confining Stress. *Minerals* **2020**, *10*, 1119.
https://doi.org/10.3390/min10121119

**AMA Style**

Patel S, Martin CD. Effect of Stress Path on the Failure Envelope of Intact Crystalline Rock at Low Confining Stress. *Minerals*. 2020; 10(12):1119.
https://doi.org/10.3390/min10121119

**Chicago/Turabian Style**

Patel, Shantanu, and C. Derek Martin. 2020. "Effect of Stress Path on the Failure Envelope of Intact Crystalline Rock at Low Confining Stress" *Minerals* 10, no. 12: 1119.
https://doi.org/10.3390/min10121119