# Fracture Closure Empirical Model and Theoretical Damage Model of Rock under Compression

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fracture Development Stages and Progressive Failure of Rock

_{cc}, fracture initiation stress σ

_{ci}, rock damage stress σ

_{cd}, and peak stress σ

_{p}. The post-peak curve starts at the peak stress σ

_{p}and ends at the residual stress σ

_{r}. The pre-peak curve can further be divided into five stages:

- Fracture closure stage. In the literal sense, fractures close during the initial loading process, and the stress–strain response is nonlinear, the regional extent of which is dependent on the initial fracture density and geometrical characteristics of the fracture population [49].
- Elasticity deformation stage. Rock can be regarded as a dense material in this stage, and the mechanical behavior is linear-elastic, reflected by a straight line in the stress–strain curve. The elastic mechanics constants of rock such as elastic modulus and Poisson’s ratio are usually determined by this line.
- Fracture initiation and stable fracture growth stage. New fractures initiate and propagate at a steady rate.
- Critical energy release and unstable fracture growth stage. Dilatancy occurs to the rock. The fractures grow precariously, and the old fractures and the new fractures interweave, causing macroscopical failure of rock.
- Failure and post-peak behavior stage. Rock stress drops rapidly but retains a certain bearing capacity provided by friction.

## 3. Establishment Process of Piecewise Constitutive Model of Rock

#### 3.1. Fitting of Empirical Model in Fracture Closure Stage

_{i}(ε

_{i}, σ

_{i}) is taken on the curve OA and point P

_{i}is connected to origin O, the slope of OP

_{i}can be calculated by the coordinates of point P

_{i}, k = σ

_{i}/ε

_{i}, which shows an increasing trend with the increase in axial strain. Based on the triaxial test data of various rock samples in previous studies [49,52,53], the variation law of k is presented in Figure 5. According to the variation characteristics, the change in the value of k from the fracture closure stage to the elasticity deformation stage is a gradually increasing process and the increment is also enlarging, which can be fitted by a power function as shown in Equation (1). All the values of R

^{2}in Figure 5 are greater than 0.95, demonstrating the effectiveness of this fitting. Therefore, the change in the value of k with strain ε can be expressed by Equations (1) and (2) and can be obtained by simple formula manipulation:

_{1}= 0 is namely the initial modulus E

_{ini}, and Equation (2) can be changed into the following:

_{1}= 0, the value of parameter c can be solved.

_{cc}(ε

_{1}= ε

_{cc}), the relationship of Equation (5) is obtained, and the derivative of the axial stress–strain curve equals elastic modulus E, which means:

#### 3.2. Derivation of Rock Damage Constitutive Model Underpinned by Fracture Growth

_{dam}is the damaged area in the total cross-sectional area of the rock; A

_{und}is the undamaged area in the total cross-sectional area of the rock; A

_{tol}is the total cross-sectional area of the rock.

_{nom}is the nominal stress applied on the damage rock material, σ

_{nom}= F/A

_{tol}; E

_{dam}is the elastic modulus of the damage rock material; σ is the actual stress the damaged rock material is subjected to, σ

_{act}= F/A

_{und}; E is the elastic modulus of the intact rock material; F is the applied axial force.

_{0},0). If elastic modulus E and closure stress σ

_{cc}are known, the value of ε

_{0}can be determined by Equation (12). Then, by shifting the vertical axis so that the origin O locates at point O’, a new coordinate system was set-up, as shown in Figure 7. Therefore, Equation (11) in the new coordinate system can be recast as

_{dam}and n

_{und}are the quantity of damaged units and undamaged units, respectively.

#### 3.3. Characterization of Residual Stage in Damage Constitutive Model

_{1}= 0 when D = 1. This obviously conflicts with the fact that the actual rock has residual strength. In this case, Shen [58] and Cao et al. [59] proposed that the damaged component of rock provided the residual stress σ

_{r}, as presented in Equation (22). In fact, there exist a great number of ways to consider the residual strength in the model such as using various correction factors with different definitions [34,35], while the core is to satisfy the relation that axial stress is equal to the residual strength when D is 1. By contrast, Equation (22) is more intuitive. That is, when the rock is completely damaged, the bearing capacity of the rock is provided by residual strength.

## 4. Model Verifications

^{2}was used to assess the matching effect of the proposed model against the experimental data, which can be calculated by Equation (26).

_{test}and y

_{cal}are, respectively, the measured axial stress and the theoretical axial stress; y

_{ave}is the average of the y

_{test}.

#### 4.1. Uniaxial Compression Test Verification

_{cc}and fracture closure strain ε

_{cc}can be determined by the following methods: the fracture volume strain method [25], the axial strain curve method [61], the axial stiffness method [49], the axial strain response method [46], and the rock constitutive model method [62]. Any one of them, or other effective means, can be chosen according to the experimental conditions. Here, the fracture closure stress and the fracture closure strain have already been presented in the measured mechanical properties in the literature [52], and therefore, ε

_{cc}= 2.39 ×10

^{−3}and σ

_{cc}= 12.6 MPa were directly adopted.

_{ini}were determined from the axial stress–strain curve. In addition, E = 10.79 GPa and E

_{ini}= 1.99 GPa are given in the literature. By substituting ε

_{cc}, σ

_{cc}, E, and E

_{ini}into Equations (4), (7), and (8), the values of model parameters a, b, and c can be solved, wherein a = 0.759, b = 1.681, and c = 1.990. It should be noted that there exist certain differences between the calculated values and the fitted values in Figure 5 where a

_{fitted}= 0.957, b

_{fitted}= 1.479, and c

_{fitted}= 1.843, which are within the margin of error. Hence, the model of sandstone in the compaction stage is achieved, which is shown as:

_{p}= 3.970 × 10

^{−3}and σ

_{p}= 27.510 MPa) were gained for sandstone. The new origin ε

_{0}can be fixed by Equation (12) when the values of ε

_{cc}, σ

_{cc}, and E are known, and the calculated ε

_{0}= 1.220 × 10

^{−3}. According to Equations (20) and (21) and the coordinates of the peak point, the values of damage model parameters υ and λ can be gained, wherein υ = 5.046 and λ = 16.421. Therefore, the damage constitutive model of sandstone after the compaction stage is presented by Equation (28).

^{2}is calculated. By substituting the values of the tested axial strains into Equations (27) and (28) successively, the calculated axial stresses can be obtained. The result is R

^{2}= 0.999, verifying the validity of the proposed models. In addition, the fracture closure point where the damage curve and the compaction curve inosculate is smooth and continuous.

^{2}convincingly indicate significant agreements of the theoretical curves and test data.

#### 4.2. Triaxial Compression Test Verification

^{2}, are shown in Figure 10, Figure 11 and Figure 12. In addition, the necessary model parameters are presented in Table 2. It must be noted that the residual strength in reference [64] is ambiguous due to the high strength and low confining pressure.

^{2}are even up to 0.999 for these model curves and test data before the peak. By contrast, the agreements between the triaxial test data and the model curves after the peak are not so satisfactory, as shown in Figure 10b and Figure 11b,c. These theoretical curves significantly deviate from the test data in the post-peak stage, which causes the decrease in the value of R

^{2}. It is suspected that the practice of treating model parameter ν as a constant may be responsible for such a weak consistency between the model curve and test data after the peak. Although the proposed models cannot always agree well with the post-peak deformations of rock materials, they are still of great research significance and application value to the pre-peak characteristics of rock materials.

## 5. Conclusions

- (1)
- According to the increasing trend of the fracture closure stage of the axial stress–strain curve, the nonlinearity characteristic during fracture closure was fitted by the power function, which was then used to deduce the compaction empirical model. The model parameters were solved by the initial modulus of elasticity and fracture closure stress and strain.
- (2)
- The rock damage evolution was quantified by the fracture growth, and the damage constitutive model was derived based on the strain equivalence hypothesis to manifest the rock deformation after the fracture closure, which avoids the selection of the strength criterion for rock micro-units in damage statistical constitutive models. The model parameters were calculated by the derivative and the coordinate at the peak.
- (3)
- The compaction empirical model and the damage constitutive model consist of the piecewise constitutive model representing the whole axial stress–strain relation of rock materials. Through the comparisons between the test data of uniaxial tests and triaxial tests and the model curves, the model validity was demonstrated. The model curves perfectly agree with the test data before the peak. In addition, the models are continuous and smooth at the curve intersection.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Yu, W.; Li, K.; Liu, Z.; An, B.; Wang, P.; Wu, H. Mechanical characteristics and deformation control of surrounding rock in weakly cemented siltstone. Environ. Earth Sci.
**2021**, 80, 337. [Google Scholar] [CrossRef] - Zhang, Y.; Ren, B.; Hursthouse, A.; Deng, R.; Hou, B. Leaching and Releasing Characteristics and Regularities of Sb and As from Antimony Mining Waste Rocks. Pol. J. Environ. Stud.
**2019**, 28, 4017–4025. [Google Scholar] [CrossRef] [PubMed] - Feng, T.; Chen, H.; Wang, K.; Nie, Y.; Zhang, X.; Mo, H. Assessment of underground soil loss via the tapering grikes on limestone hillslopes. Agric. Ecosyst. Environ.
**2020**, 297, 106935. [Google Scholar] [CrossRef] - Zhang, X.; Lin, H.; Wang, Y.; Yong, R.; Zhao, Y.; Du, S. Damage evolution characteristics of saw-tooth joint under shear creep condition. Int. J. Damage Mech.
**2021**, 30, 453–480. [Google Scholar] [CrossRef] - Zhou, X.-P.; Wang, Y.-T.; Zhang, J.-Z.; Liu, F.-N. Fracturing Behavior Study of Three-Flawed Specimens by Uniaxial Compression and 3D Digital Image Correlation: Sensitivity to Brittleness. Rock Mech. Rock Eng.
**2019**, 52, 691–718. [Google Scholar] [CrossRef] - Ke, B.; Zhang, C.; Liu, C.; Ding, L.; Zheng, Y.; Li, N.; Wang, Y.; Lin, H. An experimental study on characteristics of impact compression of freeze-thawed granite samples under four different states considering moisture content and temperature difference. Environ. Earth Sci.
**2021**, 80, 661. [Google Scholar] [CrossRef] - Huang, Z.; Zhang, L.; Yang, Z.; Zhang, J.; Gao, Y.; Zhang, Y. Preparation and properties of a rock dust suppressant for a copper mine. Atmos. Pollut. Res.
**2019**, 10, 2010–2017. [Google Scholar] [CrossRef] - Wu, H.; Jia, Q.; Wang, W.; Zhang, N.; Zhao, Y. Experimental Test on Nonuniform Deformation in the Tilted Strata of a Deep Coal Mine. Sustainability
**2021**, 13, 13280. [Google Scholar] [CrossRef] - Chen, W.; Wan, W.; Zhao, Y.; Peng, W. Experimental Study of the Crack Predominance of Rock-Like Material Containing Parallel Double Fissures under Uniaxial Compression. Sustainability
**2020**, 12, 5188. [Google Scholar] [CrossRef] - Zhao, Y.; Tang, J.; Chen, Y.; Zhang, L.; Wang, W.; Wan, W.; Liao, J. Hydromechanical coupling tests for mechanical and permeability characteristics of fractured limestone in complete stress–strain process. Environ. Earth Sci.
**2016**, 76, 24. [Google Scholar] [CrossRef] - Chen, Y.; Lin, H.; Xie, S.; Ding, X.; He, D.; Yong, W.; Gao, F. Effect of joint microcharacteristics on macroshear behavior of single-bolted rock joints by the numerical modelling with PFC. Environ. Earth Sci.
**2022**, 10, 276. [Google Scholar] [CrossRef] - Ali, S.; Xu, H.; Ahmed, W. Resolving strategic conflict for environmental conservation of glacial ecosystem: An attitudinal conflict resolution approach. Int. J. Glob. Warm.
**2019**, 18, 221–238. [Google Scholar] [CrossRef] - Liao, Y.; Yu, G.; Liao, Y.; Jiang, L.; Liu, X. Environmental Conflict Risk Assessment Based on AHP-FCE: A Case of Jiuhua Waste Incineration Power Plant Project. Sustainability
**2018**, 10, 4095. [Google Scholar] [CrossRef][Green Version] - Deng, Q.; Qin, Y.; Ahmad, N. Relationship between Environmental Pollution, Environmental Regulation and Resident Health in the Urban Agglomeration in the Middle Reaches of Yangtze River, China: Spatial Effect and Regulating Effect. Sustainability
**2022**, 14, 7801. [Google Scholar] [CrossRef] - Hu, K.; Zheng, J.; Wu, H.; Jia, Q. Temperature Distribution and Equipment Layout in a Deep Chamber: A Case Study of a Coal Mine Substation. Sustainability
**2022**, 14, 3852. [Google Scholar] [CrossRef] - Chen, T.; Feng, X.-T.; Cui, G.; Tan, Y.; Pan, Z. Experimental study of permeability change of organic-rich gas shales under high effective stress. J. Nat. Gas Sci. Eng.
**2019**, 64, 1–14. [Google Scholar] [CrossRef] - Fan, H.; Lu, Y.; Hu, Y.; Fang, J.; Lv, C.; Xu, C.; Feng, X.; Liu, Y. A Landslide Susceptibility Evaluation of Highway Disasters Based on the Frequency Ratio Coupling Model. Sustainability
**2022**, 14, 7740. [Google Scholar] [CrossRef] - Cook, N.G.W. The failure of rock. Int. J. Rock Mech. Min. Sci.
**1965**, 2, 389–403. [Google Scholar] [CrossRef] - Zhao, Y.; Luo, S.; Wang, Y.; Wang, W.; Zhang, L.; Wan, W. Numerical Analysis of Karst Water Inrush and a Criterion for Establishing the Width of Water-resistant Rock Pillars. Mine Water Environ.
**2017**, 36, 508–519. [Google Scholar] [CrossRef] - Li, Y.; Zeng, X.; Lin, Z.; Su, J.; Gao, T.; Deng, R.; Liu, X. Experimental study on phosphate rock modified soil-bentonite as a cut-off wall material. Water Supply
**2021**, 22, 1676–1690. [Google Scholar] [CrossRef] - Labuz, J.F.; Biolzi, L. Experiments with rock: Remarks on strength and stability issues. Int. J. Rock Mech. Min. Sci.
**2007**, 44, 525–537. [Google Scholar] [CrossRef] - Garaga, A.; Latha, G.M. Intelligent prediction of the stress–strain response of intact and jointed rocks. Comput. Geotech.
**2010**, 37, 629–637. [Google Scholar] [CrossRef] - Lu, Y.; Li, H.; Lu, J.; Shi, S.; Wang, G.G.X.; Ye, Q.; Li, R.; Zhu, X. Clean up water blocking damage in coalbed methane reservoirs by microwave heating: Laboratory studies. Process Saf. Environ. Prot.
**2020**, 138, 292–299. [Google Scholar] [CrossRef] - Zou, S.-H.; Li, K.-Q.; Han, Q.-Y.; Yu, C.W. Numerical simulation of the dynamic formation process of fog-haze and smog in transport tunnels of a hot mine. Indoor Built Environ.
**2017**, 26, 1062–1069. [Google Scholar] [CrossRef] - Martin, C.D.; Chandler, N.A. The progressive fracture of Lac du Bonnet granite. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1994**, 31, 643–659. [Google Scholar] [CrossRef] - Bieniawski, Z.T. Mechanism of brittle fracture of rock: Part II. Int. J. Rock Mech. Min. Ences Geomech. Abstr.
**1967**, 4, 407–423. [Google Scholar] [CrossRef] - Krajcinovic, D.; Silva, M.A.G. Statistical aspects of the continuous damage theory. Int. J. Solids Struct.
**1982**, 18, 551–562. [Google Scholar] [CrossRef] - Jian, D.; Gu, D. On a statistical damage constitutive model for rock materials. Comput. Geosci.
**2011**, 37, 122–128. [Google Scholar] - Liu, J.; Zhao, Y.; Tan, T.; Zhang, L.; Zhu, S.; Xu, F. Evolution and modeling of mine water inflow and hazard characteristics in southern coalfields of China: A case of Meitanba mine. Int. J. Min. Sci. Technol.
**2022**, 32, 513–524. [Google Scholar] [CrossRef] - Li, H.; Liao, H.; Xiong, G.; Han, B.; Zhao, G. A three-dimensional statistical damage constitutive model for geomaterials. J. Mech. Sci. Technol.
**2015**, 29, 71–77. [Google Scholar] [CrossRef] - Liu, H.; Zhang, L. A Damage Constitutive Model for Rock Mass with Nonpersistently Closed Joints Under Uniaxial Compression. Arab. J. Sci. Eng.
**2015**, 40, 3107–3117. [Google Scholar] [CrossRef] - Xu, X.L.; Gao, F.; Zhang, Z.Z. Thermomechanical coupling damage constitutive model of rock based on the Hoek–Brown strength criterion. Int. J. Damage Mech.
**2018**, 27, 1213–1230. [Google Scholar] [CrossRef] - Xu, X.L.; Karakus, M. A coupled thermo-mechanical damage model for granite. Int. J. Rock Mech. Min. Sci.
**2018**, 103, 195–204. [Google Scholar] [CrossRef] - Xu, X.L.; Karakus, M.; Gao, F.; Zhang, Z.Z. Thermal damage constitutive model for rock considering damage threshold and residual strength. J. Cent. South Univ.
**2018**, 25, 2523–2536. [Google Scholar] [CrossRef] - Feng, W.L.; Qiao, C.S.; Wang, T.; Yu, M.Y.; Jia, Z.Q. Strain-softening composite damage model of rock under thermal environment. Bull. Eng. Geol. Environ.
**2020**, 79, 4321–4333. [Google Scholar] [CrossRef] - Chen, Y.; Lin, H.; Wang, Y.; Xie, S.; Yong, W. Statistical damage constitutive model based on the Hoek–Brown criterion. Arch. Civ. Mech. Eng.
**2021**, 21, 117. [Google Scholar] [CrossRef] - Xie, S.; Lin, H.; Chen, Y. New constitutive model based on disturbed state concept for shear deformation of rock joints. Arch. Civ. Mech. Eng.
**2022**, 23, 26. [Google Scholar] [CrossRef] - Zhu, C.; Long, S.; Zhang, J.; Wu, W.; Zhang, L. Time Series Multi-Sensors of Interferometry Synthetic Aperture Radar for Monitoring Ground Deformation. Front. Environ. Sci.
**2022**, 10. [Google Scholar] [CrossRef] - Xie, S.J.; Han, Z.Y.; Hu, H.H.; Lin, H. Application of a novel constitutive model to evaluate the shear deformation of discontinuity. Eng. Geol.
**2022**, 304, 106693. [Google Scholar] [CrossRef] - Cao, W.G.; Zhao, H.; Xiang, L.; Zhang, Y.J. Statistical damage model with strain softening and hardening for rocks under the influence of voids and volume changes. Can. Geotech. J.
**2010**, 47, 857–871. [Google Scholar] [CrossRef] - Liu, H.H.; Rutqvist, J.; Berryman, J.G. On the relationship between stress and elastic strain for porous and fractured rock. Int. J. Rock Mech. Min. Sci.
**2009**, 46, 289–296. [Google Scholar] [CrossRef] - Zhao, Y.; Liu, Q.; Zhang, C.; Liao, J.; Lin, H.; Wang, Y. Coupled seepage-damage effect in fractured rock masses: Model development and a case study. Int. J. Rock Mech. Min. Sci.
**2021**, 144, 104822. [Google Scholar] [CrossRef] - Zhao, Y.; Zhang, C.; Wang, Y.; Lin, H. Shear-related roughness classification and strength model of natural rock joint based on fuzzy comprehensive evaluation. Int. J. Rock Mech. Min. Sci.
**2021**, 137, 104550. [Google Scholar] [CrossRef] - Yuan, Z.; Zhao, J.; Li, S.; Jiang, Z.; Huang, F. A Unified Solution for Surrounding Rock of Roadway Considering Seepage, Dilatancy, Strain-Softening and Intermediate Principal Stress. Sustainability
**2022**, 14, 8099. [Google Scholar] [CrossRef] - Wang, X.; Ren, B.; Zhou, Y.; Shi, X. Study on the mechanism and kinetics of manganese release from waste manganese ore waste rock under rainfall leaching. Environ. Sci. Pollut. Res.
**2022**, 29, 5541–5551. [Google Scholar] [CrossRef] - Peng, J.; Cai, M.; Rong, G.; Zhou, C.B.; Zhao, X.G. Stresses for crack closure and its application to assessing stress-induced microcrack damage. Chin. J. Rock Mech. Eng.
**2015**, 34, 1091–1100. [Google Scholar] - Li, M.; Lv, H.; Lu, Y.; Wang, D.; Shi, S.; Li, R. Instantaneous discharge characteristics and its methane ignition mechanism of coal mine rock damage. Environ. Sci. Pollut. Res.
**2022**, 29, 62495–62506. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.; Lin, H.; Wang, Y.; Zhao, Y. Damage Statistical Empirical Model for Fractured Rock under Freezing-Thawing Cycle and Loading. Geofluids
**2020**, 2020, 8842471. [Google Scholar] [CrossRef] - Eberhardt, E.; Stead, D.; Stimpson, B.; Read, R.S. Identifying crack initiation and propagation thresholds in brittle rock. Can. Geotech. J.
**1998**, 35, 222–233. [Google Scholar] [CrossRef] - You, M. Strength and damage of marble in ductile failure. J. Rock Mech. Geotech. Eng.
**2011**, 3, 161–166. [Google Scholar] [CrossRef][Green Version] - Lin, Y.; Zhou, K.; Gao, F.; Li, J. Damage evolution behavior and constitutive model of sandstone subjected to chemical corrosion. Bull. Eng. Geol. Environ.
**2019**, 78, 5991–6002. [Google Scholar] [CrossRef] - Gao, F.; Xiong, X.; Xu, C.; Zhou, K. Mechanical property deterioration characteristics and a new constitutive model for rocks subjected to freeze-thaw weathering process. Int. J. Rock Mech. Min. Sci.
**2021**, 140, 104642. [Google Scholar] [CrossRef] - Chang, S.H.; Lee, C.I. Estimation of cracking and damage mechanisms in rock under triaxial compression by moment tensor analysis of acoustic emission. Int. J. Rock Mech. Min. Sci.
**2004**, 41, 1069–1086. [Google Scholar] [CrossRef] - Lemaitre, J. A continuous damage mechanics model for ductile fracture. Trans. Asme J. Eng. Mater. Technol.
**1985**, 107, 83–89. [Google Scholar] [CrossRef] - Kachanov, L.M. On creep rupture time. Izv. Acad. Nauk SSSR Otd. Techn. Nauk.
**1958**, 8, 26–31. [Google Scholar] - Rabotnov, Y.N. Creep rupture. In Applied Mechanics; Springer: Berlin/Heidelberg, Germany, 1969. [Google Scholar]
- Liu, D.Q.; Wang, Z.; Zhang, X.Y. Characteristics of strain softening of rocks and its damage constitutive model. Rock Soil Mech.
**2017**, 38, 2901–2908. [Google Scholar] - Shen, Z.J. An elasto-plastic damage model for cemented clays. Chin. J. Geotech. Eng.
**1993**, 15, 21–28. [Google Scholar] - Cao, W.; Zhao, H.; Li, X.; Zhang, L. A statistical damage simulation method for rock full deformation process with consideration of the deformation characteristics of residual strength phase. Tumu Gongcheng Xuebao/China Civ. Eng. J.
**2012**, 45, 139–145. [Google Scholar] - Zhao, X.G.; Cai, M.; Wang, J.; Ma, L.K. Damage stress and acoustic emission characteristics of the Beishan granite. Int. J. Rock Mech. Min. Sci.
**2013**, 64, 258–269. [Google Scholar] [CrossRef] - Cai, M.; Kaiser, P.K.; Tasaka, Y.; Maejima, T.; Morioka, H.; Minami, M. Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. Int. J. Rock Mech. Min. Sci.
**2004**, 41, 833–847. [Google Scholar] [CrossRef] - Zhang, C.; Cao, W.G.; Xu, Z.; He, M. Initial macro-deformation simulation and determination method of micro-crack closure stress for rock. Yantu Lixue/Rock Soil Mech.
**2018**, 39, 1281–1288. [Google Scholar] - Deng, J.; Li, S.; Jiang, Q.; Chen, B. Probabilistic analysis of shear strength of intact rock in triaxial compression: A case study of Jinping II project. Tunn. Undergr. Space Technol.
**2021**, 111, 103833. [Google Scholar] [CrossRef] - Yin, Z.; Li, R.; Lin, H.; Chen, Y.; Wang, Y.; Zhao, Y. Analysis of Influencing Factors of Cementitious Material Properties of Lead-Zinc Tailings Based on Orthogonal Tests. Materials
**2023**, 16, 361. [Google Scholar] [CrossRef]

**Figure 9.**The comparison between uniaxial test data and model curves of various rocks. (

**a**) Beishan Granite; (

**b**) 130 m LdB Granite; (

**c**) Hwangdeung Granite; (

**d**) Yeosan Marble.

**Figure 10.**The comparison between partial triaxial test data and model curves of Jinping Marble. (

**a**) Confining stress = 5 MPa; (

**b**) confining stress = 10 MPa; (

**c**) confining stress = 30 MPa.

**Figure 11.**The comparison between partial triaxial test data and model curves of fine sandstone. (

**a**) Confining stress = 1 MPa; (

**b**) confining stress = 2 MPa; (

**c**) confining stress = 3 MPa.

**Figure 12.**The comparison between partial triaxial test data and model curves of coarse sandstone. (

**a**) Confining stress = 1 MPa; (

**b**) confining stress = 2 MPa; (

**c**) confining stress = 3 MPa.

Test Rocks | a | b | c | ε_{cc}/10^{−3} | ε_{0}/10^{−3} | λ | υ |
---|---|---|---|---|---|---|---|

Beishan Granite | 75.579 | 3.729 | 49.045 | 0.404 | 0.064 | 16.416 | 6.059 |

130 m LdB Granite | 14.739 | 1.463 | 32.652 | 1.036 | 0.332 | 23.527 | 6.019 |

Hwangdeung Granite | 3.865 | 2.051 | 16.399 | 1.535 | 0.654 | 116.688 | 31.069 |

Yeosan Marble | 21.860 | 3.996 | 15.582 | 0.765 | 0.442 | 4.216 | 1.962 |

Test Rocks | a | b | c | ε_{cc}/10^{−3} | ε_{0}/10^{−3} | λ | υ | σ_{r}/MPa |
---|---|---|---|---|---|---|---|---|

Jinping Marble 5 MPa | 6.490 | 5.642 | 31.714 | 0.857 | 0.265 | 8.153 | 2.106 | 43.495 |

Jinping Marble 10 MPa | 15.585 | 0.849 | 14.021 | 1.222 | 0.409 | 5.209 | 1.115 | 69.607 |

Jinping Marble 30 MPa | 10.184 | 1.270 | 7.784 | 1.411 | 0.655 | 5.573 | 1.018 | 112.321 |

Fine sandstone 1 MPa | 1.352 | 1.011 | 1.817 | 4.794 | 2.119 | 48.527 | 5.566 | - |

Fine sandstone 2 MPa | 1.129 | 1.046 | 5.198 | 4.198 | 1.430 | 15.920 | 1.604 | - |

Fine sandstone 3 MPa | 3.169 | 0.777 | 1.668 | 3.892 | 1.542 | 28.800 | 3.509 | - |

Coarse sandstone 1 MPa | 0.539 | 1.039 | 1.342 | 6.608 | 2.875 | 72.564 | 5.716 | - |

Coarse sandstone 2 MPa | 0.755 | 0.892 | 1.384 | 7.423 | 3.011 | 56.586 | 4.469 | - |

Coarse sandstone 3 MPa | 0.890 | 0.904 | 1.886 | 7.200 | 2.882 | 45.500 | 4.137 | - |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Lin, H.; Xie, S.; Cao, R.; Sun, S.; Zha, W.; Wang, Y.; Zhao, Y.; Hu, H. Fracture Closure Empirical Model and Theoretical Damage Model of Rock under Compression. *Materials* **2023**, *16*, 589.
https://doi.org/10.3390/ma16020589

**AMA Style**

Chen Y, Lin H, Xie S, Cao R, Sun S, Zha W, Wang Y, Zhao Y, Hu H. Fracture Closure Empirical Model and Theoretical Damage Model of Rock under Compression. *Materials*. 2023; 16(2):589.
https://doi.org/10.3390/ma16020589

**Chicago/Turabian Style**

Chen, Yifan, Hang Lin, Shijie Xie, Rihong Cao, Shuwei Sun, Wenhua Zha, Yixian Wang, Yanlin Zhao, and Huihua Hu. 2023. "Fracture Closure Empirical Model and Theoretical Damage Model of Rock under Compression" *Materials* 16, no. 2: 589.
https://doi.org/10.3390/ma16020589