In order to study the creep behavior of the surrounding rock of Hengda coal mine in Fuxin under different temperatures, the triaxial creep test of sandstone is carried out by the MTS815.02 test system. The relationship between damage variables and temperature is constructed based on the Weibull distribution of the meso-probability voxel intensity. Aiming at the nonlinear characteristics of rock creep, a nonlinear viscous pot element and a nonlinear spring element are proposed. The two linear viscous pot elements and one linear spring element in the Nishihara model can be replaced separately. Thus, an unsteady parameter creep model is established. The comparison between the Nishihara model curve and the model and the experimental curves in this article has been added to the article. Furthermore, the superiority of this model can be proved. The results show that the established variable-time aging creep model not only can describe the rock attenuation creep and stable creep deformation characteristics but also can make up for the shortcomings of the traditional creep model that cannot describe the accelerated creep characteristics. Moreover, it predicts the development law of creep deformation well. The model is in good agreement with the test curve, which shows the correctness and rationality of the model. It has guiding significance for actual engineering support and prediction of long-term deformation of surrounding rock.

After the deep coal mining, the surrounding rock of the roadway undergoes time-dependent deformation under the action of the supporting body, and this time-related deformation is generally called rheology [

In recent years, scholars had done a lot of research on the rheological properties of rocks, the mechanism of deformation and failure, and how to construct a suitable creep model [

Although the model in the above study can describe the acceleration of creep deformation of the rock into the cerebral blood vessels, the creep equation was finally formed to have a low degree of fitting for accelerated creep deformation. Therefore, it was very important to choose a simple transformation of creep parameters into time-dependent functions and then derived nonlinear creep equations. In this paper, the creep deformation characteristics of the surrounding rock of the roadway in Hengda coal mine of Liaoning Province under different temperatures were analyzed. The Nishihara model creep parameters were all converted into time-dependent functions. The rheological theory was used to recalculate the relationship between temperature and stress and time. A time-dependent creep constitutive model of deep surrounding rock under temperature-stress coupling was constructed. Finally, the correctness of the creep model was verified by comparing the experimental data with the model curve.

The Nishihara model consists of an elastomer, a viscoelastic body, and a viscoplastic body [

When _{s}

When _{s}_{0} is the elastic modulus of the elastomer, _{1} is the elastic modulus of the viscoelastic body, _{1} is the viscosity coefficient of the viscoelastic body, _{2} is the viscosity coefficient of the viscoplastic body, and _{s} is the yield stress.

The mechanical properties of roadway surrounding rock in a complex deep geological environment cannot be explained by conventional mechanics theory. At this time, the mechanical and the creep properties of surrounding rock have obvious nonlinear characteristics. In order to describe this nonlinear creep deformation characteristic, the creep parameter can no longer be used as a fixed value. It is a function of time variation. The creep parameter and time relationship function can be expressed as follows:

It is assumed that the variation law of damage variable and time product satisfies the following equation [

The distribution of microdefects in rock materials has significant self-similar characteristics, which can be characterized by the fractal dimension. The Weibull distribution function parameter

The relationship between damage variable and temperature and time is obtained by combining equations (

Liu and Zhang [

Equation (

The total strain _{e} is the elastic strain, _{ve} is the viscoelastic strain, and _{vp} is the viscoplastic strain.

The instantaneous elastic strain is only related to stress, not time. The change law of elastic modulus is only related to stress. The damage variable of elastic modulus satisfies equation (_{0} and the damage variable can be expressed as follows:_{0} is the damage influence factor related to stress; _{0}, _{0}, and _{0} are coefficients related to temperature.

The elastic strain of the rock is not affected by time, but it is affected by stress. In the one-dimensional state, the elastic strain _{e} of the rock satisfies the following equation:

In order to obtain the creep deformation of the viscoelastic body and make the calculation process conform to the laws of mathematics, this article will only improve the viscous pot element in the viscoelastic body and will not improve the spring element. In the one-dimensional state, the viscoelastic strain _{ve} of the rock satisfies the following equation:_{1} is the time coefficient of influence of the viscoelastic body; _{1}, _{1}, and _{1} are coefficients related to temperature.

Separate the variables to calculate the definite integral of Equation (

The exponential function in Equation (

By integrating Equations (

The viscoelastic deformation is as follows:

In the one-dimensional state, the viscoelastic strain _{vp} of the rock satisfies the following equation:_{2} is the time influence coefficient of the viscoplastic body, and _{2}, _{2}, and _{2} are coefficients related to temperature.

The exponential function is expanded by using the Taylor series. The integral solution of Equation (

In the actual project, the surrounding rock of the roadway is in a three-direction stress state, and the above model cannot describe the multidirectional force creep characteristics. This requires transforming the one-dimensional model into a three-dimensional model [

However, the total strain _{11} of the Nishihara model in the three-direction stress state satisfies the following equation:

The elastic strain in the three-dimensional state is not affected by time but by the stress state. The elastic strain can be expressed as a function of the stress state by the elastic model in the one-dimensional state as follows:_{0} is the elastomer shear modulus.

The expression of viscoelastic strain affected by time in the three-dimensional state is as follows:

The viscoplastic strain in the three-dimensional state cannot be directly converted by analogy. It is also affected by the plastic potential and the yield functions. Therefore, the viscoplastic strain in the three-dimensional state can be expressed as follows:

When _{s}

When _{s}_{0} is the initial reference value of the yield function of the rock.

It can be assumed that the initial yield function value of the rock is 1. Moreover, according to the flow law in plasticity theory, Equation (

In general, the yield function selects the generalized Drucker–Prager yield function. The function expression is as follows:_{1} is the first invariant of stress, _{2} is the second invariant of stress bias, and

The test parameters

The internal friction angle and cohesion will also deteriorate under the action of time and temperature. The function expressions are as follows:

Therefore, the viscoplastic strain of the rock can be obtained as follows:

The time-dependent constitutive equation for obtaining rock in a three-dimensional state is as follows:

When _{s}

When _{s}

In this paper, the surrounding rock (sandstone) of the roadway in Hengda coal mine of Fuxin is shown in Figure

Part of the samples.

According to the requirements of the International Rock Mechanics Society for standard test pieces, the surrounding rock was made into a cylindrical sample with a height of 100 mm and a diameter of 50 mm. It must be ensured that the nonparallelism and unevenness of both ends of the specimen are less than 0.05 mm. Prior to the test, Vaseline was evenly applied to both end faces of the rock to eliminate the end effect during the test. The test equipment used was the MTS815.02 rock test system (shown in Figure

MTS815.02 rock three triaxial test machine.

In this paper, the indoor triaxial creep test was carried out by the single specimen gradual loading method. First, the confining pressure was applied to a predetermined value, and the confining pressure was selected to be 10 MPa. The temperature of the creep test is set to 100 and 200°C. The stress levels were 50, 60, 70, and 80 MPa. After the confining pressure was stabilized, the axial pressure was applied. The applied load rate was set to 500 N/s. The temperature is loaded to the predetermined temperature at a rate of 0.5°C/s. When applying axial stress, it must be ensured that the confining pressure had been changed within a controllable range of the predetermined value. After the stress level creep deformation entered the stable creep, the next level of load application began. This cycle was repeated until the rock sample was destroyed. Finally, the test data was saved at intervals of 3 s. After unloading the confining pressure and the axial pressure, the sample is taken out and stored.

The single test piece is gradually loaded to complete the creep test, and the creep data under each load is affected by the historical load. Therefore, the data need to be processed by Chen’s superposition method. The axial creep deformation-time curve of the surrounding rock of the roadway under different temperatures is shown in Figure

Axial creep duration curve: (a)

It can be seen from Figure

Before verifying the rock creep aging model, the long-term strength of the rock under different confining pressures needs to be determined [

Isochronic stress-strain curve.

It can be seen from Figure

The improved creep model was fitted by the least-squares method, and the creep parameters are obtained as shown in Table

Fitting values of creep parameters.

_{1} (MPa) | 50 | 60 | 70 | 80 |
---|---|---|---|---|

_{0} (GPa) | 8.588 | 8.411 | 7.661 | 7.346 |

7.188 | 7.030 | 6.409 | 5.157 | |

_{1} (GPa) | 12.966 | 12.700 | 11.566 | 11.083 |

_{1} (GPa·h) | 914.594 | 1181.396 | 1421.250 | 1966.222 |

_{2} (GPa·h) | — | 9833.979 | 12079.614 | 16337.142 |

_{0} | 1.440 | 1.203 | 0.917 | 0.740 |

_{1} | 2.041 | 2.002 | 1.795 | 1.696 |

_{2} | — | 0.454 | 0.493 | 0.740 |

_{0} | 0.030 | 0.069 | 0.118 | 0.227 |

_{1} | 0.108 | 0.237 | 0.316 | 0.454 |

_{2} | 0.513 | 0.562 | 0.641 | 0.769 |

_{0} | 0.897 | 1.006 | 1.163 | 1.449 |

_{1} | 0.375 | 0.404 | 0.572 | 0.828 |

_{2} | — | 0.878 | 0.937 | 1.075 |

_{0} | 0.404 | 0.552 | 0.878 | 1.380 |

_{1} | 1.400 | 1.637 | 1.933 | 2.258 |

_{2} | — | 0.878 | 1.321 | 1.893 |

^{2} | 0.975 | 0.987 | 0.951 | 0.923 |

By substituting parameters of different stress levels into the model, the model curves of the rock under different stress levels are obtained. The comparison between the model curve and the experimental data is shown in Figure

Comparison curves: (a)

It can be seen from Figure

In order to verify the correctness of the model established in this paper better, the creep test curve under different temperature conditions and the curve of the traditional Nishihara model and the model curve of this paper are compared and analyzed as shown in Figure

Comparison of the Nishihara model and this model: (a)

It can be seen from Figure

In this paper, the creep behavior of the surrounding rock of Hengda coal mine in Fuxin under different temperatures is analyzed. The creep parameters of the Nishihara model are transformed into time-dependent functions. Furthermore, a time-dependent creep constitutive model of deep surrounding rock under temperature-stress coupling is constructed.

The instantaneous strain and creep strain increases with the increase of stress level, and the ratio of instantaneous strain to total strain first decreases and then increases. This is due to the compaction of the internal voids of the rock under the initial stress level.

The rock model curve and the experimental data had good fitness, and the correlation coefficients are all above 0.90. It is shown that the variable parameter time-dependent creep model established in this paper not only can describe the rock attenuation creep and stable creep deformation characteristics but also can make up for the shortcomings of the traditional creep model that cannot describe the accelerated creep characteristics.

It is a good predictor of the development of creep deformation. The model is in good agreement with the experimental curve, which indicates the correctness and rationality of the model. It has guiding significance for actual engineering support and prediction of long-term deformation of surrounding rock.

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

The authors declare that this paper has been presented as a preprint in Research Square.

The authors declare that they have no financial and personal relationships.

The authors gratefully acknowledge the National Natural Science Foundation of China (51774173) for funding this work. Special thanks are due to Professor Xiangzhi Yin for providing technical support.