# Hysteresis Behavior Modeling of Hard Rock Based on the Mechanism and Relevant Characteristics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Foundation for the Constitutive Model

#### 2.1. Rock Samples and Experimental Scheme

_{l}(0.6, 0.9, and 1.2 mm/min) were designed, and the unloading rate was uniformly set to 0.9 mm/min. After 8 cycles, loading monotonically increases until failure occurs.

#### 2.2. Results and Analysis of Mechanical Behavior

_{f}that prompts numerous AE signals is always below the peak stress σ

_{m}of the last cycle (i.e., the Felicity effect), and the schematic relationship is shown in Figure 4.

_{f}imply that the mechanical behaviors are different. When stress is on the unloading path P

_{u}and the reloading path P

_{r}before σ

_{f}, the AE signal is minimal because the source is the friction between inner fissures or cracks. However, when the reloading stress exceeds σ

_{f}, new plastic damage occurs, which will result in numerous AE signals.

_{m}

^{i−}

^{1}→ 0→ σ${}_{f}^{i}$, where i represents the number of cycles), the energy variation is primarily attributed to the release and accumulation of elastic energy. In this paper, the elastic stress is noted as f(ε), and friction is noted as h(ε). Then, the stress on P

_{u}can be expressed as f(ε) − h(ε) because P

_{u}is a process in which a rock sample performs work to the outside, and some energy is consumed by the friction of inner cracks. In contrast, the stress on P

_{r}can be expressed as f(ε) + h(ε) because the facility must overcome friction and perform work on the rock sample. As a result, P

_{r}is over P

_{u}with a difference of 2 h(ε), which could explain the massing effect to a certain extent.

_{m}

^{i}), the new plastic deformation begins to develop before the stress reaches σ

_{m}

^{i−}

^{1}due to the Felicity effect. Therefore, the loading path P

_{l}

^{i}will not extend along the trend of P

_{l}

^{i}

^{−1}completely but will enter the elastic–plastic stage in advance, which could explain the ratcheting effect to some extent.

## 3. Constitutive Model and Relative Numerical Implementation

#### 3.1. Basic Hypothesis

- (a)
- The elastic deformation and plastic deformation occur simultaneously during the initial loading and the reloading process from σ${}_{f}^{i}$ to σ
_{m}^{i}for the subsequent cycles. - (b)
- Only elastic deformation occurs during unloading and reloading from 0 to σ${}_{f}^{i}$ for each cycle.
- (c)
- The value of friction is dominated by roughness and contact pressure between surfaces in general; in this study, these two factors are considered to be representative of plastic deformation (damage degree of rock sample) and stress level, respectively. Therefore, the greater the damage and stress are, the greater the friction. In addition, friction performs negative work to the facility during the loading process but performs negative work on the rock sample during the unloading process.

#### 3.2. Constitutive Model

_{e}, plastic element E

_{p}, and friction element E

_{f}. The elastic and plastic elements are connected in series, and this assembly is connected with the friction element in parallel, as shown in Figure 5.

_{e}is only related to the elastic deformation ε

_{e}and is noted as f(ε

_{e}). The stress of E

_{p}is related to the accumulated plastic deformation ε

_{p}and is noted as g(ε

_{p}), and ε

_{p}is irreversible. The friction of E

_{f}is related to ε

_{p}and the stress level, in which the latter can be reflected by f(ε

_{e}) for simplicity; thus, the value of friction is noted as h(ε

_{p}, f).

_{m}

^{i}(when i = 1, σ${}_{f}^{i}$ = 0), the correlation laws of stress and strain between elements are given as Equation (1):

_{m}

^{i}to 0, the correlation laws of stress and strain between elements are given as Equation (2):

_{f}

^{i+}

^{1}, the correlation laws of stress and strain between elements are given as Equation (3):

#### 3.3. Procedure of Computer Implementation

#### 3.4. Constitutive Law of Elements in the Proposed Model

#### 3.4.1. Determination of f(ε_{e})

_{e}) is linear, but this linear relationship can hardly coincide well with the experimental curves of the rock sample. In reality, the tangential modulus always varies due to the tight or loose state of microscopic particles and damage development. Here, the constitutive law f

_{l}(ε

_{e}) of E

_{e}during loading (AE active period) is considered first, which is explored by extracting the elastic strain ε

_{e}(or recoverable strain) of the peak point in each cycle, as shown in Figure 7. The f(ε

_{e}) for each condition is well fitted by a polynomial (Equation (4)), and the corresponding parameters are shown in Table 3. It reveals that the elasticity increases linearly with ε

_{e}because the rock sample becomes denser during compression.

_{e}for the unloading process may be different; relevant characteristics are validated by the experimental results in Figure 8, in which the abovementioned ε

_{e}-σ trend (Figure 7) is compared with the stress–strain curve of each unloading path. The curves seem to be parallel in general, except for the ending of unloading curves, in which the trend becomes gentler with the number of cycles. This phenomenon results from the elasticity loss due to the degradation of the rock sample; when rock changes from a dense state to a loose state with the release of stress, the effect of damage becomes more marked. In other words, the elasticity variation is dominated by ε

_{e}, while the damage controls the lower limit.

_{r}can reflect the damage degree of the rock sample [29,30], and there is a one-to-one correspondence relation between the residual strain and deformation modulus [31]. This paper investigates the tangential modulus k

_{13}at the endpoint of the unloading path in each cycle, at which the effect of accumulated plastic strain is sufficiently manifested. The value of k

_{13}and the corresponding residual strain ε

_{r}are plotted in Figure 9. It can be easily found that the tangential modulus decreases with the accumulation of plastic strain. This relationship can be approximately fitted by a linear equation (Equation (5)), and relevant parameters are derived by the regression method, as shown in Table 4.

_{u}(ε

_{e}) of E

_{e}for the unloading process, the gradient values of the entire unloading path are calculated and plotted in Figure 10; the gradient of the unloading path can be considered to be the derivative of f

_{u}(ε

_{e}) with respect to ε

_{e}approximately (i.e., f

_{u}′(ε

_{e})). In Figure 10, the relationship between f

_{u}′(ε

_{e}) and ε is also fitted by a linear function, and the fitting effect is acceptable according to the high correlation coefficients. As a result, f

_{u}(ε

_{e}) is considered a quadratic equation, which agrees with f

_{l}(ε

_{e}); the equation can be determined by the peak point and trough point of each cycle. The relevant boundary conditions are depicted as Equation (6).

_{u}(ε

_{e}) of E

_{e}for the unloading process can be obtained as Equation (7). In addition, considering that the reloading process from 0 to σ

_{f}

^{i+}

^{1}is also in the AE quiet period, the deformation characteristic and mechanism are the same as the unloading process; thus, Equation (7) is also applicable to this process as the constitutive law of E

_{e}. In summary, the constitutive law of E

_{e}for the loading and unloading process can be incorporated as Equation (8).

#### 3.4.2. Determination of g(ε_{p})

_{p}), a differential strain increment dε is considered first, and g(ε

_{p}) in dε can be considered to be a linear segment. According to the series relationship between the elastic element and plastic element, the corresponding stress equation can be written as Equation (9), in which g’ denotes the derivatives with respect to ε

_{p}; thus, the differential plastic strain increment dε

_{p}can be derived as Equation (10).

_{p}) is inversely proportional to ε

_{p}(i.e., the larger the gradient is, the smaller the plastic strain increment). This study extracted the plastic strain increment ∆ε

_{p}in each cycle from the experimental data, and Figure 11 plotted ∆ε

_{p}versus the ratio of peak stress to UCS σ/σ

_{UCS}; here, the σ/σ

_{UCS}was chosen due to the difference between rock samples. Figure 11 shows that ∆ε

_{p}exhibits a decreasing–stable–increasing trend throughout the deformation process, which means that the plastic strain increases rapidly at the initial stage and then develops slowly for a certain duration. Finally, the increment accelerates gradually as the stress approaches UCS. Similar results were also found in other studies [32,33]; therefore, g(ε

_{p}) was supposed to increase in a trend of acceleration–deceleration. In this study, the Logistic equation is used to describe g(ε

_{p}) by referring to Liu et al. [26], and the equation is formulated as Equation (11), in which k

_{2}, a, and b are parameters that need to be predicted.

_{p}(i.e., residual strain) of the peak point in each cycle is extracted, and the relationship with stress is fitted by Equation (11) in Figure 12. The corresponding parameters are given in Table 5, in which the high goodness of fit values reveal that the fitting effect is satisfactory.

^{i}is introduced to Equation (11), representing that the rock enters the yield state in advance when reloading for each cycle. The relationship between the stress and yield surface throughout the cyclic loading process is then schematically plotted in Figure 14. In this way, the ratcheting effect and Felicity effect can be easily reflected in the model; thus, Equation (11) is modified to Equation (12).

^{i}is considered to be dominated by the historical maximum stress and rock strength, so it is expressed as Equation (13) for simplicity, where d is also a parameter to be predicted. In addition, the occasion of σ

_{f}

^{i}is regarded as the Kaiser point, so the value is expressed as Equation (14), where R

_{F}is the Felicity ratio. The experimental values of R

_{F}for each cycle in different conditions are shown in Table 6, in which subtle changes are observed; the small difference in R

_{F}only causes a negligible effect on the entire posture of the stress–strain curve; thus, this paper simplifies R

_{F}as a constant equal to 0.95.

#### 3.4.3. Determination of h(ε_{p}, f)

_{p}. More ε

_{p}means more friction surface; thus, the roughness is simplified to be k

_{31}∙ε

_{p}, where k

_{31}is the coefficient. On the other hand, the pressure between the contact surfaces is dominated by f(ε

_{e}), and higher f(ε

_{e}) means that the contact is closer between surfaces. Thus, the pressure is simplified to be k

_{32}∙f, where k

_{32}is also the coefficient. In summary, h(ε

_{p}, f) is formulated as Equation (15), where k

_{3}= k

_{31}∙k

_{32}, and also needs to be predicted in the following section.

## 4. Parameter Calibration

#### 4.1. Calibration Procedure

_{11}, k

_{12}, k

_{13}′, b

_{13}′, k

_{2}, a, b, d, k

_{3}) that need to be determined. It is noted that although Table 3, Table 4 and Table 5 give the preliminary values of some parameters, these values are derived by decoupling the parameters; when the coupling effect is considered, the simulated results may deviate from the experimental curves. Therefore, this study uses the BP neural network to calibrate further the parameters. The neural network is a data-driven method that can simulate any nonlinear relationship in theory; however, too many parameters may make the neural network fall into the disaster of dimensionality. Fortunately, mechanism analysis can relieve the adverse effect of multiple parameters. In other words, the reference values in Table 3, Table 4 and Table 5 can help us narrow the number of parameters, as well as the range of parameter values, and create a superior sample set.

_{11}, k

_{12}, k

_{2}, a, and b; the posture of the unloading curve is dominated by k

_{13}’ and b

_{13}’; and the massing effect and ratcheting effect are primarily controlled by d and k

_{3}. Among these parameters, k

_{13}’ and b

_{13}’ are relatively independent, particularly for the residual strain. Therefore, the values of k

_{13}’ and b

_{13}’ are set as shown in Table 4, and the other seven parameters are the objects to be determined by the neural network. In this study, the maximum strain and residual strain in every cycle are considered as inputs with a total of 17 features, and the 7 variable parameters are considered as outputs. The regression procedure is specified as follows, and the corresponding flow is shown in Figure 15.

- (a)
- Based on the reference values, three different values of each parameter are designated as the labels of the sample set (Table 7). The combination of different parameters results in a total of 2187 samples, of which 90 percent are used for training, and 10 percent are used for validation. By substituting these parameters into the algorithm of the constitutive model, the corresponding strain values can be derived as the features of the sample set.
- (b)
- Considering an order-of-magnitude difference between each parameter, the Min-Max normalization is used for data preprocessing.
- (c)
- Constructing a regression model using the BP neural network, which contains one input layer with 17 neurons, one hidden layer with 12 neurons, and one output layer with 7 neurons. The ReLU activation function is used in the hidden layer so as to simulate the nonlinear relation between the input and output, and the Adam optimizer is used to improve the quality of backpropagation. After each iteration, the training loss and validation loss are recorded and saved.
- (d)
- Setting termination criteria in terms of loss value (magnitude of loss< 0.1 and the change of loss <0.005). If the criteria are satisfied, we can conduct the calibration by introducing the experimental strain data into the qualified regression model. If not, a new iteration begins.

#### 4.2. Results

## 5. Conclusions

- (1)
- By combining the uniaxial cyclic compression on the granite sample with AE monitoring, it is found that the loading process exhibits both elastic deformation and plastic deformation regardless of the stage that the rock sample is currently in. In addition, the mechanical behaviors during the unloading process and reloading process before the Kaiser point are primarily elastic deformation, accompanied by the obstruction of friction between inner cracks. When the loading exceeds the Kaiser point, the plastic strain will continue to develop.
- (2)
- The cyclic behavior of granite is simulated by a proposed comprehensive body that consists of an elastic element E
_{e}, a plastic element E_{p}, and a friction element E_{f}, in which E_{f}is connected in parallel with the serial combination of E_{e}and E_{p}. The opposite effect of E_{f}during the unloading process and reloading process produces the massing effect, and the plastic deformation during reloading is prompted prior to the historical maximum stress, which brings about the ratcheting effect and Felicity effect. - (3)
- In terms of hard rock such as granite, the elasticity reflected by the tangential modulus is affected by elastic strain and plastic strain. The elastic strain dominates the variation process of elasticity, while plastic strain determines the lower limit. Specifically, the tangential modulus of the elastic element exhibits a linear positive correlation with elastic strain, while the lower limit decreases linearly with the plastic strain. In addition, the plastic deformation grows from fast to slow throughout the deformation process, which can be simulated by the Logistic equation.
- (4)
- The proposed model and corresponding parameters are validated by comparison with the experimental stress–strain curves under three different conditions, and the strain (peak strain and residual strain in each cycle) differences between simulated results and experimental results for different conditions are basically less than 18%, 6%, and 13%, respectively, which strengthens the reliability. In addition, it is worth noting that even though the proposed model holds for the uniaxial compression case, the modeling flow could be extended to other applications, such as triaxial compression, and this paper provides a reference for future research on the cyclic behavior of other materials.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bagde, M.; Petroš, V. Fatigue properties of intact sandstone samples subjected to dynamic uniaxial cyclical loading. Int. J. Rock Mech. Min. Sci.
**2005**, 42, 237–250. [Google Scholar] [CrossRef] - Chen, H.; Cong, T.N.; Yang, W.; Tan, C.; Li, Y.; Ding, Y. Progress in electrical energy storage system: A critical review. Prog. Nat. Sci.
**2009**, 19, 291–312. [Google Scholar] [CrossRef] - Li, C.; Liu, N.; Liu, W.; Feng, R. Study on Characteristics of Energy Storage and Acoustic Emission of Rock under Different Moisture Content. Sustainability
**2021**, 13, 1041. [Google Scholar] [CrossRef] - Gong, F.-Q.; Wu, C.; Luo, S.; Yan, J.-Y. Load–unload response ratio characteristics of rock materials and their application in prediction of rockburst proneness. Bull. Eng. Geol. Environ.
**2019**, 78, 5445–5466. [Google Scholar] [CrossRef] - Zhu, X.; Li, Y.; Wang, C.; Sun, X.; Liu, Z. Deformation Failure Characteristics and Loading Rate Effect of Sandstone Under Uniaxial Cyclic Loading and Unloading. Geotech. Geol. Eng.
**2019**, 37, 1147–1154. [Google Scholar] [CrossRef] - Imre, B.; Räbsamen, S.; Springman, S.M. A coefficient of restitution of rock materials. Comput. Geosci.
**2008**, 34, 339–350. [Google Scholar] [CrossRef] - Chen, Y.P.; Wang, S.J.; Wang, E.Z. Quantitative Study on Stress-strain Hysteretic Behaviors in Rocks. Chin. J. Rock Mech. Eng.
**2007**, 26, 4066–4073. [Google Scholar] - Wen, T.; Tang, H.; Wang, Y.; Ma, J.; Fan, Z. Mechanical Characteristics and Energy Evolution Laws for Red Bed Rock of Badong Formation under Different Stress Paths. Adv. Civ. Eng.
**2019**, 2019, 8529329. [Google Scholar] [CrossRef] - Xi, D.Y.; Chen, Y.P.; Tao, Y.Z.; Liu, Y.C. Nonlinear Elastic Hysteric Characteristics of Rocks. Chin. J. Rock Mech. Eng.
**2006**, 25, 1086–1093. [Google Scholar] - Meng, Q.; Zhang, M.; Han, L.; Pu, H.; Nie, T. Effects of Acoustic Emission and Energy Evolution of Rock Specimens Under the Uniaxial Cyclic Loading and Unloading Compression. Rock Mech. Rock Eng.
**2016**, 49, 3873–3886. [Google Scholar] [CrossRef] - Meng, Q.; Zhang, M.; Han, L.; Pu, H.; Chen, Y. Acoustic Emission Characteristics of Red Sandstone Specimens Under Uniaxial Cyclic Loading and Unloading Compression. Rock Mech. Rock Eng.
**2018**, 51, 969–988. [Google Scholar] [CrossRef] - Wang, T.; Wang, C.; Xue, F.; Wang, L.; Teshome, B.H.; Xue, M. Acoustic Emission Characteristics and Energy Evolution of Red Sandstone Samples under Cyclic Loading and Unloading. Shock Vib.
**2021**, 2021, 8849137. [Google Scholar] [CrossRef] - Li, Y.; Zhao, T.; Li, Y.; Chen, Y. A five-parameter constitutive model for hysteresis shearing and energy dissipation of rock joints. Int. J. Min. Sci. Technol. 2022, in press. [CrossRef]
- McCall, K.R.; Guyer, R.A. Equation of state and wave propagation in hysteretic nonlinear elastic materials. Geophys. Res.
**1994**, 99, 23887–23897. [Google Scholar] [CrossRef] - Scalerandi, M.; Gliozzi, A.; Idjimarene, S. Power laws behavior in multi-state elastic models with different constraints in the statistical distribution of elements. Commun. Nonlinear Sci. Numer. Simul.
**2014**, 19, 3628–3641. [Google Scholar] [CrossRef] - Chen, X.; Huang, Y.; Chen, C.; Lu, J.; Fan, X. Experimental study and analytical modeling on hysteresis behavior of plain concrete in uniaxial cyclic tension. Int. J. Fatigue
**2017**, 96, 261–269. [Google Scholar] [CrossRef] - Wang, X.; Song, L.; Gao, X.; Chang, X. Effect of loading rate on the nonlinear elastic response of concrete. Eur. J. Environ. Civ. Eng.
**2021**, 25, 909–923. [Google Scholar] [CrossRef] - Hua, D.; Jiang, Q. Three-Dimensional Modeling of Hysteresis in Rocks. J. Geophys. Res. Solid Earth
**2022**, 127, e2021JB023230. [Google Scholar] [CrossRef] - Hashiguchi, K. Generalized plastic flow rule. Int. J. Plast.
**2005**, 21, 321–351. [Google Scholar] [CrossRef] - Kong, L.; Zheng, Y.R.; Yao, Y.P. Subloading surface cyclic plastic model for soil based on Generalized plasticity (I): Theory and model. Rock Soil Mech.
**2003**, 24, 141–145. [Google Scholar] [CrossRef] - Kong, L.; Zheng, Y.R.; Yao, Y.P. Subloading surface cyclic plastic model for soil based on Generalized plasticity (II): Constitutive equation and identification. Rock Soil Mech.
**2003**, 24, 349–354. [Google Scholar] [CrossRef] - Zhou, Y.Q.; Sheng, Q.; Zhu, Z.Q.; Fu, X.D. Subloading surface model for rock based on modified Drucker-Prager criterion. Rock Soil Mech.
**2017**, 38, 400–418. [Google Scholar] [CrossRef] - Zhou, Y.; Sheng, Q.; Li, N.; Fu, X.; Zhang, Z.; Gao, L. A constitutive model for rock materials subjected to triaxial cyclic compression. Mech. Mater.
**2020**, 144, 103341. [Google Scholar] [CrossRef] - Valanis, K.C.; Fan, J. A numerical algorithm for endochronic plasticity and comparison with experiment. Comput. Struct.
**1984**, 19, 717–724. [Google Scholar] [CrossRef] [Green Version] - Cerfontaine, B.; Charlier, R.; Collin, F.; Taiebat, M. Validation of a New Elastoplastic Constitutive Model Dedicated to the Cyclic Behaviour of Brittle Rock Materials. Rock Mech. Rock Eng.
**2017**, 50, 2677–2694. [Google Scholar] [CrossRef] [Green Version] - Liu, D.; He, M.; Cai, M. A damage model for modeling the complete stress–strain relations of brittle rocks under uniaxial compression. Int. J. Damage Mech.
**2018**, 27, 1000–1019. [Google Scholar] [CrossRef] - Wu, Y.L.; Chen, J.; Zeng, S.M. The Acoustic Emission Technique Research on Dynamic Damage Characteristics of the Coal Rock. Procedia Eng.
**2011**, 26, 1076–1082. [Google Scholar] [CrossRef] [Green Version] - Kong, X.; Wang, E.; He, X.; Zhao, E.; Zhao, C. Mechanical characteristics and dynamic damage evolution mechanism of coal samples in compressive loading experiments. Eng. Fract. Mech.
**2019**, 210, 160–169. [Google Scholar] [CrossRef] - Xiao, J.-Q.; Ding, D.-X.; Jiang, F.-L.; Xu, G. Fatigue damage variable and evolution of rock subjected to cyclic loading. Int. J. Rock Mech. Min. Sci.
**2010**, 47, 461–468. [Google Scholar] [CrossRef] - Liu, E.; He, S. Effects of cyclic dynamic loading on the mechanical properties of intact rock samples under confining pressure conditions. Eng. Geol.
**2012**, 125, 81–91. [Google Scholar] [CrossRef] - Wang, Z.; Li, S.; Qiao, L.; Zhao, J. Fatigue Behavior of Granite Subjected to Cyclic Loading Under Triaxial Compression Condition. Rock Mech. Rock Eng.
**2013**, 46, 1603–1615. [Google Scholar] [CrossRef] - Chen, Y.; Zuo, J.; Guo, B.; Guo, W. Effect of cyclic loading on mechanical and ultrasonic properties of granite from Maluanshan Tunnel. Bull. Eng. Geol. Environ.
**2020**, 79, 299–311. [Google Scholar] [CrossRef] - Li, X.W.; Yao, Z.S.; Huang, X.W.; Liu, Z.X.; Zhao, X.; Mu, K.H. Investigation of deformation and failure characteristics and energy evolution of sandstone under cyclic loading and unloading. Rock Soil Mech.
**2021**, 42, 1693–1704. [Google Scholar]

**Figure 3.**Variations in stress and AE hits with time (the blue horizontal dashed lines denote the peak stress in each cycle, while the vertical dashed lines denote the moment when AE hits increase abruptly). (

**a**) v

_{l}= 0.6 mm/min. (

**b**) v

_{l}= 0.9 mm/min. (

**c**) v

_{l}= 1.2 mm/min.

**Figure 7.**Elastic strain of the peak point in each cycle and the corresponding fitting relationship between ε

_{e}and σ.

**Figure 8.**ε

_{e}-σ relationship of the loading process throughout the test and the stress–strain curve of the unloading process in each cycle. (

**a**) v

_{l}= 0.6 mm/min. (

**b**) v

_{l}= 0.9 mm/min. (

**c**) v

_{l}= 1.2 mm/min.

**Figure 9.**Tangential modulus k

_{13}and residual strain ε

_{r}at the endpoint of the unloading path in each cycle and the corresponding fitting relationship.

**Figure 10.**Gradient variation of the unloading path for each cycle and the corresponding fitting relationship between the gradient and ε. (

**a**) v

_{l}= 0.6 mm/min. (

**b**) v

_{l}= 0.9 mm/min. (

**c**) v

_{l}= 1.2 mm/min.

**Figure 12.**Plastic strain of the peak point in each cycle and the corresponding fitting relationship between ε

_{p}and σ.

**Figure 13.**Schematic stress–strain curve without the ratcheting effect and curve with the ratcheting effect. (

**a**) Defective stress–strain curve. (

**b**) Real stress–strain curve.

**Figure 17.**Comparison of the stress–strain curve under variable amplitude loading between the simulated results and experimental results. (

**a**) v

_{l}= 0.6 mm/min. (

**b**) v

_{l}= 0.9 mm/min. (

**c**) v

_{l}= 1.2 mm/min.

**Figure 18.**Comparison of peak strain and residual strain in each cycle between the simulated and experimental results. (

**a**) v

_{l}= 0.6 mm/min. (

**b**) v

_{l}= 0.9 mm/min. (

**c**) v

_{l}= 1.2 mm/min.

Quartz (%) | K-Feldspar (%) | Plagioclase (%) | Biotite (%) | Others (%) |
---|---|---|---|---|

35.11 | 28.21 | 30.64 | 5.45 | 0.59 |

Water Content (%) | Density (g/cm^{3}) | Dry Density (g/cm^{3}) | Velocity of Longitudinal Waves (m/s) | Schmidt Hardness | UCS (MPa) | Uniaxial Tensile Strength (MPa) |
---|---|---|---|---|---|---|

0.09 | 2.7 | 2.6 | 3077 | 69 | 150.0 | 4.8 |

k_{11} | k_{12} | R^{2} | |
---|---|---|---|

v_{l} = 0.6 mm/min | 8391.26 | 1,075,623.84 | 0.9977 |

v_{l} = 0.9 mm/min | 9152.61 | 1,270,575.23 | 0.9974 |

v_{l} = 1.2 mm/min | 13,657.93 | 452,678.39 | 0.9986 |

k_{13}’ | b_{13}’ | R | |
---|---|---|---|

v_{l} = 0.6 mm/min | −4,487,920 | 12,859.92 | −0.8751 |

v_{l} = 0.9 mm/min | −3,298,220 | 10,145.63 | −0.9333 |

v_{l} = 1.2 mm/min | −5,555,450 | 17,057.99 | −0.9681 |

k_{2} | a | b | R^{2} | |
---|---|---|---|---|

v_{l} = 0.6 mm/min | 183.34 | 5.02 | 2337.40 | 0.9967 |

v_{l} = 0.9 mm/min | 192.88 | 4.87 | 2401.32 | 0.9944 |

v_{l} = 1.2 mm/min | 702.44 | 7.16 | 2896.71 | 0.9244 |

Cycle 2 | Cycle 3 | Cycle 4 | Cycle 5 | Cycle 6 | Cycle 7 | Cycle 8 | Cycle 9 | |
---|---|---|---|---|---|---|---|---|

v_{l} = 0.6 mm/min | 1.16 | 0.98 | 0.97 | 0.95 | 0.93 | 0.91 | 0.89 | 0.87 |

v_{l} = 0.9 mm/min | 1.13 | 0.98 | 0.97 | 0.97 | 0.96 | 0.95 | 0.94 | 0.91 |

v_{l} = 1.2 mm/min | 1.08 | 0.98 | 0.98 | 0.97 | 0.97 | 0.95 | 0.95 | 0.93 |

k_{11} | k_{12} | k_{2} | a | b | d | k_{3} | |
---|---|---|---|---|---|---|---|

v_{l} = 0.6 mm/min | 7000, 8000, 9000 | 9 × 10^{5}, 1.0 × 10^{6}, 1.1 × 10^{6} | 150, 200, 250 | 3, 5, 7 | 2000, 3000, 4000 | 0.0001, 0.0005, 0.0009 | 10, 20, 30 |

v_{l} = 0.9 mm/min | 7000, 8000, 9000 | 1.1 × 10^{6}, 1.2 × 10^{6}, 1.3 × 10^{6} | 150, 200, 250 | 3, 5, 7 | 2000, 3000, 4000 | 0.0001, 0.0005, 0.0009 | 10, 20, 30 |

v_{l} = 1.2 mm/min | 12,000, 13,000, 14,000 | 4 × 10^{5}, 5 × 10^{5}, 6 × 10^{5} | 650, 700, 750 | 5, 7, 9 | 2000, 3000, 4000 | 0.0001, 0.0005, 0.0009 | 10, 20, 30 |

k_{11} | k_{12} | k_{2} | a | b | d | k_{3} | |
---|---|---|---|---|---|---|---|

v_{l} = 0.6 mm/min | 7279 | 1,039,693 | 164.6 | 5.7 | 3356.2 | 0.00051 | 17.8 |

v_{l} = 0.9 mm/min | 8509 | 1,142,430 | 241.3 | 6.2 | 3573.3 | 0.00024 | 15.2 |

v_{l} = 1.2 mm/min | 12,237 | 569,263 | 702 | 7.2 | 3300 | 0.00013 | 13.7 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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**

Fu, H.; Li, J.; Li, G.; Li, D.
Hysteresis Behavior Modeling of Hard Rock Based on the Mechanism and Relevant Characteristics. *Sustainability* **2022**, *14*, 10412.
https://doi.org/10.3390/su141610412

**AMA Style**

Fu H, Li J, Li G, Li D.
Hysteresis Behavior Modeling of Hard Rock Based on the Mechanism and Relevant Characteristics. *Sustainability*. 2022; 14(16):10412.
https://doi.org/10.3390/su141610412

**Chicago/Turabian Style**

Fu, Helin, Jie Li, Guoliang Li, and Dongping Li.
2022. "Hysteresis Behavior Modeling of Hard Rock Based on the Mechanism and Relevant Characteristics" *Sustainability* 14, no. 16: 10412.
https://doi.org/10.3390/su141610412