# Experimental Study of Mechanical Properties and Fracture Characteristics of Conglomerates Based on Mohr–Coulomb Criteria

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Methods and Numerical Models

#### 2.1. Triaxial Compression Experiments Based on Moorcullen’s Criterion

_{1}and σ

_{3}acting at a point in the rock are known, the positive stress σ and the shear stress τ in any plane can be derived from Equation (1). In the plane right-angle coordinate system, the curve of Equation (1) as shown in Figure 1d is a circle, i.e., the Mohr stress circle. A series of Mohr stress circles can be plotted by performing the strength curves of the rock under different stress states as shown in Figure 1e, and the line tangent to these Mohr stress circles is the damage envelope, whose expression is expressed by Equation (2). According to the damaged envelope, parameters such as cohesion C and internal friction angle φ of different conglomerates can be obtained, through which the mechanical properties of conglomerates can be analyzed.

_{1}and σ

_{3}are principal stress, MPa. C is cohesion, MPa. φ is the internal friction angle, °.

_{1}tilt β angle, the normal stress and shear stress as shown in Equations (3) and (4), combined with Equation (2) and the introduction of the Moorcullen rupture criterion to obtain the damage criterion Equation (5), Equation (5) in the plane of the main stress of σ

_{1}and σ

_{3}is a straight line, then the uniaxial compressive strength σ

_{c}as shown in Equation (6), that is, in the case of the same cohesion uniaxial compressive strength is proportional to the rupture interface dynamic friction coefficient μ.

#### 2.2. Linear Parallel Bond Model

_{n}, the tangential stiffness k

_{s}, and the friction coefficient μ; the damping force is determined by the normal damping ratio β

_{n}, and the tangential damping ratio β

_{s}(Figure 2a), it does not resist relative rotation, and slip is accommodated by imposing a Coulomb limit on the shear force. The parallel bonding model acts in parallel with the linear model to transfer forces and moments between the contacting particles (Figure 2b). Unlike the linear model, the parallel bonding model resists the relative rotation between particles and its behavior is linear elastic until the strength limit is exceeded and the bond breaks, making it unbonded. If the parallel bonding model is activated when the surface gap of the object g

_{s}≤ 0, the bond breaks when it exceeds the strength limit, as shown in Figure 2c,d the particles will produce translation or rotation between them, at which point the parallel bonding model is equivalent to the linear model.

_{c}is the contact force, N. F

^{l}is the linear force, N. F

^{d}is the dashpot force, N. M

_{c}is the contact moment, N·m. $\overline{F}$ is the parallel-bond force, N. $\overline{M}$ is the parallel-bond moment, N·m. E

_{k}is the strain energy, N·m/pa. ${F}_{n}^{l}$ is the linear normal force, N. ${F}_{s}^{l}$ is the linear shear force, N. ${k}_{n}$ is the normal stiffness, N/m. ${k}_{s}$ is the shear stiffness, N/m. E

_{μ}is the slip energy, N·m/pa. ${\left({F}_{s}^{l}\right)}_{0}$ is the linear shear force at the beginning of the timestep, N. $\Delta {\delta}_{s}^{\mu}$ is the slip component of relative shear-displacement increment, m. E

_{β}is the dashpot energy, N·m/pa. $\delta $ is the relative translational velocity, m/s. $\Delta t$ is the time step, s.

#### 2.3. 3D Numerical Model of Conglomerate

## 3. Results

#### 3.1. Experimental Results of Conglomerate

#### 3.2. Numerical Simulation Results of Conglomerate

#### 3.2.1. Effect of Gravel Morphology and Spatial Location on Mechanical Behavior

#### 3.2.2. Effect of Gravel Content on Mechanical Behavior

#### 3.2.3. Effect of Surrounding Pressure on the Mechanical Behavior of Gravels

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

σ | normal stress, MPa |

τ | shear stress, MPa |

σ_{1,} σ_{3} | principal stress, MPa |

C | cohesion, MPa |

φ | internal friction angle, ° |

F_{c} | contact force, N |

F^{l} | linear force, N |

F^{d} | dashpot force, N; |

M_{c} | contact moment, N·m |

$\overline{\mathit{F}}$ | parallel-bond force, N |

$\overline{\mathit{M}}$ | parallel-bond moment, N·m |

E_{k} | strain energy, N·m/pa |

${\mathit{F}}_{\mathit{n}}^{\mathit{l}}$ | linear normal force, N |

${\mathit{F}}_{\mathit{s}}^{\mathit{l}}$ | linear shear force, N |

${\mathit{k}}_{\mathit{n}}$ | normal stiffness, N/m |

${\mathit{k}}_{\mathit{s}}$ | shear stiffness, N/m |

E_{μ} | slip energy, N·m/pa |

${\mathbf{\left(}{\mathit{F}}_{\mathit{s}}^{\mathit{l}}\mathbf{\right)}}_{\mathbf{0}}$ | linear shear force at the beginning of the timestep, N |

$\mathbf{\Delta}{\mathit{\delta}}_{\mathit{s}}^{\mathit{\mu}}$ | slip component of relative shear-displacement increment, m |

E_{β} | dashpot energy, N·m/pa |

$\mathit{\delta}$ | relative translational velocity, m/s |

$\mathbf{\Delta}\mathit{t}$ | time step, s |

β | failure angle |

μ | dynamic friction coefficient |

${\mathit{\sigma}}_{\mathit{c}}$ | uniaxial compressive strength, MPa |

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**Figure 1.**Conglomerate specimens and typical stress-strain curves.(

**a**) Side view of conglomerate specimen; (

**b**) End view of conglomerate samples; (

**c**) Typical stress-strain curve; (

**d**) Mohr stress circle; (

**e**) Mohr stress circle under different confining pressure.

**Figure 2.**Rheological components (

**a**,

**b**) and behavior (

**c**,

**d**) of the linear model and linear parallel bond model.

**Figure 3.**Electron microscope scan of conglomerate (

**a**,

**c**) and its corresponding elemental analysis (

**b**,

**d**).

**Figure 4.**The conglomerate specimen (

**a**),real gravel (

**b**) and numerical model of conglomerates (

**f**) and different shapes of gravel models (

**c**–

**e**).

**Figure 5.**(

**a**) Stress-strain curves of uniaxial compression physical experiments on conglomerates; (

**b**) Relationship between gravel content and UCS; (

**c**) Stress-strain curves of conglomerate triaxial compression physical experiments; (

**d**) Mohr stress circle.

**Figure 7.**Gravel edge stress diagram with chain gravel (

**a**), block gravel (

**b**), and spherical gravel (

**c**).

**Figure 8.**Stress-strain curve of numerical simulation. (

**a**) Stress-strain curves at different gravel contents; (

**b**) Relationship between gravel content and UCS.

**Figure 10.**Stress-strain curves for different gravel contents under different confining pressures. (

**a**) Stress-strain curve under 10 MPa circumferential pressure; (

**b**) Stress-strain curve under 20 MPa circumferential pressure; (

**c**) Stress-strain curve under 30 MPa circumferential pressure; (

**d**) Stress-strain curve under 40 MPa circumferential pressure.

**Figure 11.**Mohr stress circle and damaged envelope. (

**a**) Mohr stress circle with 10% gravel content; (

**b**) Mohr stress circle with 20% gravel content; (

**c**) Mohr stress circle with 30% gravel content; (

**d**) Mohr stress circle with 40% gravel content; (

**e**) Mohr stress circle with 50% gravel content; (

**f**) Mohr stress circle with 60% gravel content; (

**g**) Mohr stress circle with 70% gravel content; (

**h**) Mohr stress circle with 80% gravel content; (

**i**) Mohr stress circle with 90% gravel content.

**Figure 12.**Conglomerate fracture map at different scales. (

**a**,

**b**) Electron microscope scan of conglomerate; (

**c**) Conglomerate diagram under optical microscope.

**Figure 13.**Fracture rosette diagram for different gravel contents. (

**a**) Fracture rosette diagram with 10% gravel content; (

**b**) Fracture rosette diagram with 20% gravel content; (

**c**) Fracture rosette diagram with 30% gravel content; (

**d**) Fracture rosette diagram with 40% gravel content.

**Figure 14.**Conglomerate fracture rupture process diagram. (

**a**) The first type of fracture rupture process; (

**b**) The second type of fracture rupture process.

Contact Property | Effective Young’s Modulus (Pa) | Tensile Strength(Pa) | Cohesion (Pa) | Friction Coefficient |
---|---|---|---|---|

Ball to ball | 1 × 10^{9} | 1 × 10^{7} | 1 × 10^{7} | 0.57 |

Ball to clump | 1 × 10^{6} | 1 × 10^{6} | 5 × 10^{6} | 0.3 |

Gravel Content (%) | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | ||
---|---|---|---|---|---|---|---|---|---|---|---|

UCS (MPa) | Confining Stress | 10 MPa | 64.8 | 61 | 57.5 | 56.6 | 53.4 | 52.9 | 51.4 | 49.8 | 48.5 |

20 MPa | 102.4 | 95 | 89.2 | 86.7 | 82.7 | 81.3 | 79.3 | 76.9 | 74.8 | ||

30 MPa | 130.5 | 121.7 | 112.3 | 109 | 105.4 | 103.4 | 100.2 | 97.2 | 94.6 | ||

40 MPa | 155.2 | 144.6 | 132.1 | 127.8 | 124.5 | 121.2 | 116.6 | 113.4 | 110.7 | ||

C (MPa) | 10.9 | 10.7 | 11.25 | 11.52 | 10.41 | 10.75 | 10.89 | 10.64 | 10.43 | ||

Φ (°) | 30.121 | 28.19 | 25.27 | 24.08 | 24.05 | 23.05 | 21.85 | 21.16 | 20.54 |

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## Share and Cite

**MDPI and ACS Style**

Liu, P.; Shen, Y.; Meng, M.; Luo, S.; Zhong, Y.; Cen, Q.
Experimental Study of Mechanical Properties and Fracture Characteristics of Conglomerates Based on Mohr–Coulomb Criteria. *J. Mar. Sci. Eng.* **2023**, *11*, 1219.
https://doi.org/10.3390/jmse11061219

**AMA Style**

Liu P, Shen Y, Meng M, Luo S, Zhong Y, Cen Q.
Experimental Study of Mechanical Properties and Fracture Characteristics of Conglomerates Based on Mohr–Coulomb Criteria. *Journal of Marine Science and Engineering*. 2023; 11(6):1219.
https://doi.org/10.3390/jmse11061219

**Chicago/Turabian Style**

Liu, Pengyu, Yinghao Shen, Mianmo Meng, Senlin Luo, Yi Zhong, and Qiming Cen.
2023. "Experimental Study of Mechanical Properties and Fracture Characteristics of Conglomerates Based on Mohr–Coulomb Criteria" *Journal of Marine Science and Engineering* 11, no. 6: 1219.
https://doi.org/10.3390/jmse11061219