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The precondition of rock stress and deformation analysis is a reasonable rock constitutive model. Most of the previous studies have described the heterogeneous microdamage by Weibull distribution or normal distribution. However, both of them have limitations. Therefore, this paper intends to use the lognormal distribution as the probability distribution model of rock microunit strength. Based on the tensile failure of the single-fractured rock under the hydrodynamic force, the maximum tensile strain failure criterion is used as the distribution parameter of rock microunit strength. And, considering the multiphase properties of the filling fractured rock, the equivalent elastic modulus parameter is adopted in the model. We design a triaxial seepage test for the filled single-fractured rock and analyze the applicability and rationality of the modified lognormal statistical damage model for characterizing the fractured rock by using the test data. According to the comparison of the experimental stress-strain curve and the model stress-strain curve and the analysis of the damage value test curve and the model curve, the rationality of the established statistical damage constitutive model is verified, and the advantages and limitations of the model are proposed.

The precondition of rock stress and deformation analysis is a reasonable rock constitutive model, which is the theoretical basis for stability prediction and evaluation of the surrounding rock in geotechnical engineering and also the key to the development of rock mechanic science. Dougill [

Phenomenological statistical damage method.

Statistical method | Weibull distribution | Normal distribution | Lognormal distribution |
---|---|---|---|

Research achievements | Weibull [ | Yang et al. [ | Zhang et al. [ |

However, for brittle materials such as the rock, the Weibull theory cannot reflect the nonlocal characteristics of the damage, and the application of the Weibull theory to brittle materials will inevitably lead to size effect, resulting in a large difference between the theoretical results and the experimental results [

Based on the investigation of the factors affecting the strength of rock material RVE (representative volume element), Zhang et al. [

Therefore, on the basis of previous studies, this paper intends to use the lognormal distribution as the probability distribution model of rock microunit strength. According to the failure characteristics of the cracked rock in the seepage-stress field, the strength of the rock microunit is determined by the maximum tensile strain failure criterion. At the same time, considering the multiphase properties of the filling fractured rock, the equivalent elastic modulus parameter is adopted in the model. Finally, for the influence of confining pressure and water pressure, Hooke’s law and effective stress principle are used to modify. The statistical constitutive model of the lognormal distribution of seepage damage in the cracked rock is established. In this paper, the advantages and limitations of the model are obtained by comparing the experimental stress-strain curve with the model stress-strain curve. Based on the analysis of the damage variable test curve and the model curve, the rationality of the established statistical damage constitutive model is verified.

As shown in Figure

Mechanical model of the single-cracked rock under water pressure.

The rise of groundwater along the fault plane under pressure is called water wedge action. There are two manifestations. One is the splitting failure of rock cracks caused by water pressure. When the surrounding rock is in the state of compressive stress, the water pressure overcomes the surrounding rock stress and the rock tensile strength to induce the crack propagation. The other is the squeezing failure of rock cracks caused by water pressure. When the crack width in the rock mass is large or water-conducting fractures are formed, water pressure squeezes into the rock mass to promote the increase in the crack depth.

Therefore, water pressure enters the cracked rock mass, forming a large pore wall pressure, resulting in the formation of tensile cracks. The tensile strain rate in a certain direction first reaches an allowable value, and finally, the damaged rock in this environment contains more longitudinal tensile cracks.

Scholars carry out some studies on the mechanical properties of seepage in fractured rocks [

The natural rock mass structure is complex, and the rock samples taken at the site are random and nonrepeatable. When studying the influence of cracks on the rock mass seepage damage, it is difficult to control the variables to obtain effective conclusions. At present, in the indoor seepage test of the filled fractured rock mass, the joints are prefabricated by cutting and splitting and then filled. In this study, the single crack is made in the intact sandstone by cutting. According to the common types of fillings in natural fractured rock masses recorded in the Engineering Geology Site Manual, cement mortar is used to fill the prefabricated cracks to complete the sample.

First, select the complete sandstone rock sample (

Schematic of three cracked rocks.

In an indoor environment, the samples are dried for 18 days and then tested. The experiments are carried out on the MTS815 system. The axial loading is controlled by displacement. The confining pressure design values are 10 MPa and 20 MPa, and the water pressure is 7 MPa. The failure picture of the samples is shown in Figure

Failure of the cracked rock under seepage stress. (a) 2-1. (b) 2-2. (c) 2-3. (d) 2-4. (e) 2-5. (f) 2-6.

It can be seen that under the seepage stress, the macroscopic failure mode of the single-cracked sandstone with filling material shows that the tensile cracks, which develop along the dynamic hydraulic direction, and the mixed tensile shear cracks at the distal end originate from the tensile stress concentration area near the tip of the fracture. The failure is the overall instability caused by the tensile shear mixing cracks that originate from the crack tip and extend to the upper and lower end faces of the rock sample.

The study of rock structure models has been carried out for a long time, and the proposed models can be roughly divided into three categories [

Voigt’s equal strain model.

The model proposed by Voigt assumes that the various minerals that make up the rock are arranged in parallel along the direction of the force. The bulk modulus of each species mineral is

According to the relation between the elastic parameters, the calculation equation can be written as

The equivalent elastic parameters of the filled fractured rock.

Sample | ||||||||
---|---|---|---|---|---|---|---|---|

2-1 | 8.53 | 4.44 | 96.1 | 0.057 | 0.027 | 3.9 | 4.27 | 6.15 |

2-2 | 8.88 | 4.62 | 96.1 | 0.087 | 0.041 | 3.9 | 4.44 | 7.20 |

2-3 | 8.53 | 4.44 | 92.1 | 0.057 | 0.027 | 7.9 | 4.09 | 6.14 |

2-4 | 8.88 | 4.62 | 92.1 | 0.087 | 0.041 | 7.9 | 4.27 | 7.67 |

2-5 | 8.53 | 4.44 | 88.1 | 0.057 | 0.027 | 11.9 | 3.92 | 6.11 |

2-6 | 8.88 | 4.62 | 88.1 | 0.087 | 0.041 | 11.9 | 4.08 | 6.85 |

One definition of the damage variable is a small face is taken inside the object and is evenly damaged by the

Let

Assuming that the rock failure criterion is

Let

The form of the rock’s failure mechanism and its failure criterion entirely depends on the microunit strength form. In this paper, considering the failure mode of the cracked rock, the theory of maximum tensile strain, which is simple and widely used in the rock media, is used to establish the rock microunit strength

The effective stress principle is generally used in the analysis of stress-seepage coupling problems. Terzaghi [

In cracked rock triaxial flow tests, assuming the rock matrix is impermeable and cracks are the seepage channel, when there is a seepage water pressure (

According to Hooke’s law and equation (

Substituting the above ones into equation (

This paper uses a lognormal distribution to represent the rock microunit strength:

The number of destroyed microunits under a certain load is

The damage variable can be expressed as

And, it is simplified as

From equations

The damage constitutive equation obtained by substituting (

According to the triaxial test process, confining pressure and pore water pressure are loaded before deviatoric stress, so that the axial strain is generated, and the strain in this section is not included in the test curve. The axial bias

And, there is an initial strain

So, the true axial strain

Substituting equations (

Similarly, by substituting equations (

The method of calculating the parameters of the rock damage constitutive equation is usually based on the rock triaxial stress-strain test curve using the graph method and the linear regression method. These methods have achieved good results, but the process is complicated, and it is difficult to achieve for some rocks, and the errors caused by insufficient data are large. In addition, the method of multivariate function extremum theory can be used to calculate parameters. According to the characteristic parameters of the test curve, such as peak intensity and peak strain, the parameters can be calculated. This method is not only simple, clear in physical meaning, and high in precision, but also suitable for various types of rocks.

The slope of the peak strength point

Therefore, the result of deriving on both sides of equation (

Deformation of equation (

The derivative result of equation (

Then, substituting equation (

Substituting the peak strength point

Calculating equation (

Substituting (

Deforming equation (

Substituting (

So, according to the peak strength point

The parameters

Based on the triaxial seepage tests of filled single-cracked sandstone (Table

Seepage tests results of single-cracked sandstone filled with cement mortar.

Conditions and results | 2-1 | 2-2 | 2-3 | 2-4 | 2-5 | 2-6 | |
---|---|---|---|---|---|---|---|

Sample size (mm) | D | 48.57 | 48.13 | 48.31 | 48.25 | 48.26 | 48.29 |

H | 98.64 | 98.41 | 99.21 | 98.80 | 98.17 | 99.02 | |

Crack length | 25% | 25% | 50% | 50% | 75% | 75% | |

Density | 2.34 | 2.32 | 2.31 | 2.32 | 2.30 | 2.31 | |

Confining pressure | 10.00 | 20.00 | 10.00 | 20.00 | 10.00 | 20.00 | |

Pore pressure (top) (MPa) | 7.00 | 7.00 | 7.00 | 7.00 | 7.00 | 7.00 | |

Pore pressure (bottom) (MPa) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | |

Osmotic pressure difference (MPa) | 5.38∼5.93 | 5.87∼5.93 | 5.76∼5.83 | 5.80∼5.92 | 5.76∼5.87 | 5.93∼6.05 | |

Peak stress | 43.99 | 71.19 | 40.01 | 70.62 | 32.80 | 64.88 | |

Peak strain | 0.0081 | 0.0098 | 0.0067 | 0.0090 | 0.0054 | 0.0097 | |

Permeability (Darcy) | 7.94∼8.67 | 3.51∼3.64 | 8.35∼9.84 | 4.72∼4.90 | 11.62∼11.85 | 5.85∼6.06 | |

Stress-strain model curve and the test curve of the cracked rock. (a) 2‐1. (b) 2‐2. (c) 2‐3. (d) 2‐4. (e) 2‐5. (f) 2‐6.

It can be seen that this model has the following advantages: (1) the model can reflect the whole process of rock deformation and rupture. The model curve shows a shape consistent with the test curve, and the elastic stage of the curve agrees well, which fully reflects the evolution process of rock seepage damage. (2) It better reflects the variation of rock strength with confining pressure and crack length. The strength of rock increases with the increase in confining pressure, and it decreases with the increase in crack length. (3) From the postpeak softening stage, the damage model can also better reflect the real softening characteristics of the fractured rock after failure. (4) The model has few parameters, and the parameters are easy to calculate.

However, the model also has limitations: (1) the equivalent elastic modulus parameters used in the model cannot perfectly represent the initial compression stage of the cracked rock. Although the crack part of the rock is filled, the cement mortar still contains more pores, which makes the sample produce a larger strain during the compaction stage. Therefore, during the initial compaction stage of the filled fractured rock, the fit between the model curve and the test curve is poor. (2) The peak strength of the model curve is slightly lower than that of the test curve. In this test, the transient pulse technique is used to simultaneously test the permeability of the sample, causing the rock to creep. The viscoelastic parameters of the rock are not considered in the model, resulting in deviations in the calculated values. However, this error can be eliminated after changing the test method.

In order to verify the rationality of the rock damage statistical model established in this paper, the change law of the rock damage variable with strain is studied in combination with the test curve to analyze the correctness of the model damage variable (damage factor). Therefore, transforming equation (

And, substituting equations (

According to

According to the rock seepage softening damage model proposed in this paper and its parameter acquisition method, the damage factor in the model is calculated using equation (

The corresponding theoretical curve of

The damage variable

However, since the elastic parameters used in the calculation model do not well characterize the deformation characteristics of the compaction stage, the damage value of the rock in the compaction stage is not reflected in the theoretical curve, and the damage value in the initial stage is zero.

Based on the results of (

With the increase in the axial strain, when the stress exceeds a certain level, the rock cannot bear the load and begins to produce damage. It can be seen that the peak value of the test curve and the peak value of the theoretical curve correspond to the same strain value, and the final damage variable is the same.

As the confining pressure increases, the strain value increases when the rock is damaged; as the crack length increases, the strain value decreases at the same time. This rule is consistent in the test curve and the theoretical curve.

In the test curve, when the axial strain is small, the change of the damage variable is not regular, and

Therefore, the model established in this paper can reflect the characteristics of rock damage related to the stress state and can also reflect the damage state affected by the rock structural plane. The damage factor

Rock is a nonuniform quasibrittle material with complex mechanical properties. Considering the randomness of rock defect distribution and the nonuniformity of mechanical properties, the common research idea is to analyze the damage process by introducing statistical methods. The established statistical damage model introduces the failure criterion as a physical mechanic parameter into the nonuniformity of the material through a random distribution function. Lognormal distribution is widely used as the structural resistance probability distribution in structural reliability theory, and it can be considered that the random distribution of rock mechanical property parameters is similar to the resistance distribution of structural members. In describing the rock damage deformation curve, the lognormal model curve is more in line with the actual situation, and the curve shape and the final value are more accurate in this paper.

However, in geotechnical engineering, the types of filling in cracked rocks are very rich, such as hydraulic materials that can still play the role of bonding and strength in water like cement mortar, and there are also air-hardening materials that lose their cohesiveness and strength when encountering water. There are self-weight fillings as in this paper’s experiment, and there are also pressure cement fillings. At the same time, the fracture types of rock mass are more diverse, such as single fracture, multifractures, and en echelon joints. This study only reflects the applicability of the established lognormal statistical damage model in describing the seepage mechanical properties of single-fractured sandstone filled with cement mortar, which has little limitations in the experimental design and result data. In the future work, we will consider conducting more experimental studies on the seepage mechanical properties of fractured rocks with different fillings and fracture types under different stress environments to learn more about the applicability of the lognormal distribution statistical model.

In this paper, the permeability test of the fractured sandstone filled with cement mortar at a confining pressure of 20 MPa and 10 MPa and the water pressure at 7 MPa is completed. According to the tests data, the lognormal distribution is used as a probability distribution model of rock microunit strength to establish a rock stress-strain relationship constitutive model. According to the failure characteristics of the fractured rock in the seepage-stress field, the strength of the rock microunit is determined based on the maximum tensile strain failure criterion. And, considering the influence of confining pressure and water pressure, Hooke's law and the effective stress principle are used to improve the model parameter. The equivalent elastic modulus of the filled fractured rock is calculated by Voigt’s space average model. Finally, a statistical constitutive model of the lognormal distribution of seepage damage in the fractured rock is established. The following conclusions can be drawn:

The established rock damage statistical model agrees well with the experimental curve and can better reflect the stress-strain relationship of the filling fracture rock under different external load conditions and different structural plane types.

This model can fully reflect the characteristics of rock damage related to the stress state and also reflect the damage state affected by the characteristics of the rock structural plane.

The damage peak of the model curve is consistent with the actual situation, and its initial damage strain reflects the rock damage threshold effect, so the model is reasonable and meaningful.

This model has few parameters, simple acquisition, and clear physical meaning, which is extremely convenient in engineering application.

Demand data used to support the findings of this study are available from the first author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was supported by the National Natural Science Funds (41831278 and 51878249) of China, the Guangdong Province Water Resource Science and Technology Innovation Program (2017-30) of China, and the Postgraduate Research and Practice Innovation Program of Jiangsu Province (2018B661X14).