Piecewise Linear Strength Models for Analyzing Multiple Failure Mechanisms in Rocks Materials
Abstract
:1. Introduction
2. Related Work
2.1. PMC Model
2.2. Extensional-Strain Criterion and Brittle Failure
3. Methodology
3.1. Proposal of the PMC Model Considering Extensional Strain
3.2. Unified Equation of the Multiple Failure Mechanism Model
3.3. Parameter Fitting Method
4. Experiments
4.1. Experimental Conditions
4.2. Experimental Results
4.3. Applicability of the PMC Model for Strength with Multiple Failure Mechanisms
5. Conclusions
- (1)
- In light of the limitations of traditional rock strength theories, which consider only the impact of a single failure mechanism, and recognizing that tensile and splitting failures of rocks under low-stress conditions conform to the maximum elongation strain theory while shear-type failures occur at high-pressure levels, a multiple failure mechanism theory model for rocks has been proposed based on the PMC piecewise linear model. This model establishes a unified expression for multiple failure modes in the form of inequalities. Experimental verification has demonstrated that this criterion can effectively achieve nonlinear fitting for granite, sandstone, and other rock types.
- (2)
- The rock multiple failure mechanism theory model’s failure criterion can encompass the MC strength theory, the PMC strength model, and the 12-sided PMC model. It can also be extended to derive the asymmetric PMC-BPMC model, which considers isotropic yield. By integrating the principles of plastic flow rules and hardening laws, the model can be expanded to analyze and explain the mechanisms of pore collapse phenomena in rocks under high pressure, corresponding to the specialized “cap” yield model mentioned in the text.
- (3)
- Using experimental data from Jinchuan granite and Sichuan yellow sandstone, the rock multiple failure mechanism theory model was fitted for n = 1 (considering only shear failure), n = 2 (considering the transition from shear failure to tensile failure), and n > 2 (coexistence of tensile and shear failure). The results indicate that the rock strength exhibits distinct segmented strength characteristics, consistent with the strength patterns under the influence of multiple failure mechanisms. This validates the applicability of the multiple failure mechanism theory model to rock materials. However, since this study only considers generalized shear stress, mean stress, and stress Lode angle parameters in the rock failure process, other minor stress influences and environmental factors were not discussed, presenting certain limitations.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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σ/MPa | P/kN | σ/MPa | P/kN | σ/MPa | P/kN |
---|---|---|---|---|---|
5 | 3.5 | 65 | 45.5 | 125 | 87.5 |
10 | 7 | 70 | 49 | 130 | 91 |
15 | 10.5 | 75 | 52.5 | 135 | 94.5 |
20 | 14 | 80 | 56 | 140 | 98 |
25 | 17.5 | 85 | 59.5 | 145 | 101.5 |
30 | 21 | 90 | 63 | 150 | 105 |
35 | 24.5 | 95 | 66.5 | 155 | 108.5 |
40 | 28 | 100 | 70 | 160 | 112 |
45 | 31.5 | 105 | 73.5 | 165 | 115.5 |
50 | 35 | 110 | 77 | 170 | 119 |
55 | 38.5 | 115 | 80.5 | 175 | 122.5 |
60 | 42 | 120 | 84 | 180 | 126 |
Number | Wave Velocity/km/s | σ2/MPa | σ3/MPa | σ1/MPa | p/MPa | q/MPa | /° |
---|---|---|---|---|---|---|---|
SY-0-1 | 1.43 | 0 | 0 | 49 | 16 | 49 | −30 |
SY-10-1 | 1.58 | 10 | 10 | 107 | 42 | 97 | −30 |
SY-20-1 | 1.69 | 20 | 20 | 173 | 71 | 153 | −30 |
SY-30-1 | 1.42 | 30 | 30 | 222 | 94 | 192 | −30 |
SY-40-1 | 1.54 | 40 | 40 | 219 | 100 | 179 | −30 |
SY-50-1 | 1.43 | 50 | 50 | 267 | 122 | 217 | −30 |
SY-60-2 | 1.5 | 60 | 60 | 274 | 131 | 214 | −30 |
SY-65-1 | 1.52 | 65 | 65 | 322 | 151 | 257 | −30 |
SY-70-1 | 1.46 | 70 | 70 | 361 | 167 | 291 | −30 |
SY-90-1 | 1.46 | 90 | 90 | 398 | 193 | 308 | −30 |
SY-110-2 | 1.72 | 110 | 110 | 377 | 199 | 267 | −30 |
SY-140-2 | 1.43 | 140 | 140 | 430 | 237 | 290 | −30 |
SY-145-1 | 1.61 | 145 | 145 | 477 | 256 | 332 | −30 |
SY-150-1 | 1.46 | 150 | 150 | 460 | 253 | 310 | −30 |
SY-160-2 | 1.62 | 160 | 160 | 502 | 274 | 342 | −30 |
SY-170-1 | 1.45 | 170 | 170 | 490 | 277 | 320 | −30 |
HG-0-1 | 8.78 | 0 | 0 | 148 | 49 | 148 | −30 |
HG-10-1 | 8.33 | 10 | 10 | 127 | 49 | 117 | −30 |
HG-20-1 | 8.56 | 20 | 20 | 221 | 87 | 201 | −30 |
HG-30-1 | 8.33 | 30 | 30 | 296 | 119 | 266 | −30 |
HG-40-2 | 8.45 | 40 | 40 | 341 | 140 | 301 | −30 |
HG-50-1 | 8.78 | 50 | 50 | 381 | 160 | 331 | −30 |
HG-60-1 | 8.36 | 60 | 60 | 413 | 178 | 353 | −30 |
HG-100-1 | 7.89 | 100 | 100 | 559 | 253 | 459 | −30 |
HG-120-1 | 8.33 | 120 | 120 | 547 | 262 | 427 | −30 |
HG-130-2 | 8.38 | 130 | 130 | 561 | 274 | 431 | −30 |
HG-140-1 | 8.82 | 140 | 140 | 597 | 292 | 457 | −30 |
HG-150-1 | 7.14 | 150 | 150 | 687 | 329 | 537 | −30 |
HG-160-1 | 8.33 | 160 | 160 | 658 | 326 | 498 | −30 |
Material | φc/° | φe/° | V0/kPa |
---|---|---|---|
Berea Sandstone | 37.7 | 44.2 | 21.7 |
Failure Mechanism Number n | φc/° | φe/° | V0/kPa |
---|---|---|---|
1 | 42.853 | 42.853 | 27.712 |
2 | 28.020 | 28.020 | 130.953 |
Failure Mechanism Number n | V0/MPa | φc/° | φe/° |
---|---|---|---|
1 (low-pressure failure segment) | 14.66 | 40 | 40 |
2 (transition segment) | 76.87 | 27 | 27 |
3 (high-pressure failure segment) | 403.31 | 13 | 13 |
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Li, S.; Li, Y.; Fan, D.; Zhao, L.; Zhang, L. Piecewise Linear Strength Models for Analyzing Multiple Failure Mechanisms in Rocks Materials. Materials 2024, 17, 4102. https://doi.org/10.3390/ma17164102
Li S, Li Y, Fan D, Zhao L, Zhang L. Piecewise Linear Strength Models for Analyzing Multiple Failure Mechanisms in Rocks Materials. Materials. 2024; 17(16):4102. https://doi.org/10.3390/ma17164102
Chicago/Turabian StyleLi, Shiqi, Yuan Li, Dongjue Fan, Liang Zhao, and Litian Zhang. 2024. "Piecewise Linear Strength Models for Analyzing Multiple Failure Mechanisms in Rocks Materials" Materials 17, no. 16: 4102. https://doi.org/10.3390/ma17164102