Strength Estimation of Damaged Rock Considering Initial Damage Based on P-Wave Velocity Using Regression Analysis
Abstract
:1. Introduction
2. Materials and Methods
2.1. Test Instruments and Intact Rock Specimens
2.2. Test Procedure
3. Results
3.1. Analysis of the Typical Stress–Strain Curves
3.2. Multiscale Initial Damage Investigation
3.3. Multiscale Initial Damage Characterization Based on P-Wave Velocity
3.4. UCS and the 95% Confidence Interval
4. Strength Prediction Model
4.1. Model Fit and Parameter Analysis
- (1)
- Without considering the initial damage degree, the fitting curve slopes of the relationships between UCS and vp are essentially the same, but the intercepts are different. This means that the rock mass needs to be identified and classified in the field applications of engineering geology. The corresponding empirical formula can then be used for strength estimation.
- (2)
- The parameters of the empirical formulas, as shown in Figure 12, do not have any physical significance. Different empirical formulas can lead to prediction results with slightly different R2 values. It is particularly subjective and is similar to the direct empirical fitting of the strength and vp obtained in previous studies [11,12,13,14].
4.2. Adaptability Analysis of the Strength Prediction Model
4.2.1. Adaptability of the Prediction Model to Splitting Tensile Strength
4.2.2. Adaptability of the Prediction Model to Other Rocks
- (1)
- The fitting lines of the strength prediction model were highly consistent with the test data points of limestone, silty sandstone, fine sandstone, medium sandstone, mudstone, material similar to limestone, and intermittent jointed rock mass. More than 73% of the R2 values were greater than 0.80, indicating that the strength prediction model could offer good adaptability to other rocks.
- (2)
- Limited strength data for limestone and limestone rock masses were available in [31] (Figure 14), and the influence of data discreteness cannot be ruled out. The strength failure mode of mudstone in [34] was plastic failure, which differed from that of the brittle rock masses (Figure 15). The R2 value of the intermittent jointed rock mass reported by [22] was the lowest at 0.2655 (Figure 16). This is because Equation (6) was established on the assumption that the initial damage to the rock mass was macroscopically continuous. However, the initial damage to the intermittent jointed rock mass was macroscopically discontinuous. Therefore, the strength of mudstone and intermittently jointed rock mass cannot be predicted using the strength prediction model for brittle rock masses (Equation (6)).
- (3)
- Based on the comparability of the test results, only the three groups of rock specimens (silty sandstone, fine sandstone, and medium sandstone) in Figure 14 are compared. The larger the rock particle size, the smaller the correction parameter δ of the model (Equation (6)).
4.2.3. Comparison with Other Strength Prediction Models
5. Discussion
6. Conclusions
- (1)
- The strength prediction model for brittle rock masses considering multiscale initial damage based on P-wave velocity was proved to be reasonably practical, and its parameters had physical significance. The larger the rock particle size, the smaller the correction parameter of the strength prediction model.
- (2)
- The uniaxial compressive strength of the rock mass was linearly positively correlated with the square of the P-wave velocity and linearly inversely correlated with the initial damage. The line-fitted results were within the upper and lower limits of the 95% confidence interval of the UCS.
- (3)
- As the initial damage degree increased, the initial rock damage increased linearly, the P-wave velocity decay rate increased, the strength decreased monotonically, and the peak axial strain increased significantly. It was proved to be feasible to prepare damaged rock specimens with multiscale initial cracks under various uniaxial compressive stresses.
- (4)
- The strength prediction model, e.g., the uniaxial compressive strength prediction model and the splitting tensile strength prediction model, could offer good adaptability to other rocks, which can be used to obtain the strength of brittle rock masses directly in engineering geology projects.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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P-Wave Velocity, vp/m∙s−1 | Elastic Modulus, E/GPa | |
---|---|---|
Intact rock | 5737.45 | 30.3390 |
0.25σ0 damaged rock | 5730.97 | 30.0992 |
0.40σ0 damaged rock | 5638.75 | 28.7756 |
0.50σ0 damaged rock | 5458.41 | 26.6792 |
0.65σ0 damaged rock | 5270.98 | 23.4593 |
0.75σ0 damaged rock | 5166.67 | 22.5601 |
Label of Each Group of Specimens | Intact Rock | 0.25σ0 Damaged Rock | 0.40σ0 Damaged Rock | 0.50σ0 Damaged Rock | 0.65σ0 Damaged Rock | 0.75σ0 Damaged Rock |
---|---|---|---|---|---|---|
No. 1 | 86.381 | 85.923 | 83.246 | 77.379 | 72.583 | 69.581 |
No. 2 | 87.284 | 86.354 | 83.579 | 77.726 | 72.943 | 69.670 |
No. 3 | 88.065 | 86.927 | 84.623 | 78.534 | 73.058 | 70.693 |
No. 4 | 88.268 | 87.026 | 84.865 | 79.142 | 73.267 | 70.761 |
No. 5 | 90.697 | 87.461 | 85.137 | 79.390 | 73.852 | 71.310 |
No. 6 | 89.165 | 86.239 | 83.824 | 77.768 | 72.583 | 70.015 |
No. 7 | 85.971 | 86.532 | 84.116 | 77.674 | 72.807 | 70.417 |
No. 8 | 87.029 | 86.903 | 84.261 | 78.033 | 72.941 | 70.176 |
No. 9 | 88.775 | 87.064 | 84.586 | 78.131 | 73.367 | 70.482 |
No. 10 | 90.782 | 87.348 | 85.116 | 78.377 | 73.619 | 70.906 |
88.242 | 86.778 | 84.335 | 78.214 | 73.102 | 70.4011 | |
2.740 | 0.248 | 0.417 | 0.427 | 0.178 | 0.301 | |
[87.058, 89.426] | [86.421, 87.134] | [83.873, 84.797] | [77.748, 78.682] | [72.780, 73.404] | [70.009, 70.794] |
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Xu, X.; Xu, C.; Hu, J.; Ma, S.; Li, Y.; Wen, L.; Wen, G. Strength Estimation of Damaged Rock Considering Initial Damage Based on P-Wave Velocity Using Regression Analysis. Sustainability 2022, 14, 14768. https://doi.org/10.3390/su142214768
Xu X, Xu C, Hu J, Ma S, Li Y, Wen L, Wen G. Strength Estimation of Damaged Rock Considering Initial Damage Based on P-Wave Velocity Using Regression Analysis. Sustainability. 2022; 14(22):14768. https://doi.org/10.3390/su142214768
Chicago/Turabian StyleXu, Xiao, Chuanhua Xu, Jianhua Hu, Shaowei Ma, Yue Li, Lei Wen, and Guanping Wen. 2022. "Strength Estimation of Damaged Rock Considering Initial Damage Based on P-Wave Velocity Using Regression Analysis" Sustainability 14, no. 22: 14768. https://doi.org/10.3390/su142214768
APA StyleXu, X., Xu, C., Hu, J., Ma, S., Li, Y., Wen, L., & Wen, G. (2022). Strength Estimation of Damaged Rock Considering Initial Damage Based on P-Wave Velocity Using Regression Analysis. Sustainability, 14(22), 14768. https://doi.org/10.3390/su142214768