In order to effectively describe the whole creep process of fractured rock mass, triaxial unloading creep tests were carried out on prefractured coal samples using constant axial pressure and graded unloading confining pressure, and the axial and lateral creep laws of fractured coal samples with different dip angles were studied. Combined with the characteristics of creep curve and based on Kachanov’s creep damage theory, the damage variable is introduced into the constitutive relation and creep equation, and the evolution equation of damage variable with time in the whole creep process is derived. At the same time, a new method to calculate the initial damage is proposed. The elastoplastic body with damage variable is connected with the Burgers model in series. Meanwhile, lade criterion and switch element are introduced into the creep model to establish a new fracture damage creep model. The one-dimensional and three-dimensional damage creep equations are derived. The damage creep equation is obtained according to the superposition principle. A simple and feasible method for parameter identification of the model is given based on the characteristics of creep curve. The applicability of the model is verified by comparing the creep test curve of fractured coal sample with the theoretical curve. The results show that the two models are in good agreement. The model can not only accurately reflect the nonlinear characteristics of creep curves in the attenuation and isokinetic stages but also describe the accelerated creep characteristics of fractured rocks.

Rock burst refers to the local high stress concentration in coal (rock) caused by the mining of working face and the tunneling of roadway, which makes the local elastic energy in coal (rock) accumulate stably in time and transfer to weak surface through easy propagation path, resulting in continuous generation of weak structure in coal and rock, causing local fracture, damage, instability, and sliding of coal (rock). The phenomenon that the elastic potential energy stored in coal (rock) is released instantaneously after some external disturbance [

The rheological property of rock is an important factor affecting the long-term stability of rock mass engineering. There are many factors affecting the creep process of rock, and one of the important factors is the influence of rock damage on creep. In the process of rock creep, the internal damage cracks of rock are produced continuously, which makes the bearing capacity of rock decrease continuously, making the rock to enter the accelerated creep stage. In addition, under the coupling effect of engineering disturbance and rock damage fracture penetration and fusion, roof falling, rock burst, and other disasters are likely to occur. Therefore, it is necessary and significant to introduce damage body to simulate the whole creep process curve of rock. At present, the existing rock creep model simulation has made great achievements, but there are still many problems in the simulation of accelerated creep process [

In order to simulate the accelerated creep process of rock, scholars at home and abroad have done a lot of research on nonlinear rheological theory and achieved good results: one is that three different material element models (such as Hooke body, Saint Venant body, and Newton body) can be combined in series and parallel to establish a new creep constitutive model, and a lot of rock creep can be established by this method of constitutive models, such as Maxwell model, Kelvin model, and Xiyuan model; Zhao Yanlin constructed a new rheological model, which was composed of components in series, and could well describe the creep law under the state of graded stress; Xia Caichu et al. proposed a unified rheological mechanical model including four basic rheological mechanical forms: viscoelastic, viscoplastic, viscous, and viscoelastic-plastic, and gave the identification method of the model.

Although there are many methods to construct traditional components, they still have some limitations. They can only reflect the characteristics of viscoelastic-plasticity and cannot simulate the process of accelerated creep well. Therefore, some nonlinear models are proposed. The others are based on the damage theory because the basic parameters of creep can be changed, so it can be realized by using the damage factor. The new component is combined with the traditional element in series, and the creep constitutive equation which can simulate the whole process of creep is established. Murakammi and Ohno proposed a damage model for jointed rock mass with multiple sets of planar fractures based on the description of the geometric size of the fracture system; Kawamoto et al. further hypothesized and improved the abovementioned model for jointed rock mass; Zhu Weishen and Qi Yinping introduced a second-order tensor to describe the damage of rock mass based on the geometric damage principle and the definition of the abovementioned damage variables. Based on Kachanov’s damage theory, Wu Zhulin et al. simulated the initial damage factor in series with Burgers element model and achieved good results. However, most of the existing creep models are based on the combination of original or improved components. Either there are many components, or the formula is complex, or cannot be solved. It can be seen that the simulation method of accelerated creep stage still has some shortcomings, which needs to be improved. Therefore, the damage factor is introduced into the creep model to simulate the whole creep process. There are two methods of damage factor: the first is to define the damage factor from the angle of energy damage; the second is to define the damage factor based on the effective bearing area of the structure in terms of geometric damage. However, the nonlinear characteristics of rock creep are not only in the accelerated creep stage but also in the whole creep process. However, most of the creep damage models focus on the accelerated creep process after the stress level exceeds the long-term strength of the rock, and there is little research on the nonlinear characteristics of the early stable creep. In this paper, the initial damage is defined on the basis of existing damage mechanics, and the evolution equation of damage variable is derived according to the Kachanov damage rate. At the same time, based on the lade failure criterion, the elastoplastic damage element is introduced to simulate the plastic damage body and then is combined with the Burgers model in series, a constitutive model with few parameters, easy to determine, uncomplicated components, and able to simulate accelerated creep is constructed. The research in this paper can be used for reference for the stability of underground engineering in the future, especially for the stability of support in deep mining and the prevention of rock burst [

The coal sample was taken from Wang JiaLing coal mine in Shanxi Province. The coal is hard and anthracite. In order to avoid the influence of anisotropy on the test results, rock samples were processed along the same direction when preparing coal samples. At the same time, in order to ensure the authenticity and comparability of the test results, the appearance of the processed coal samples was carefully observed before the test to determine that there were no obvious weak planes such as joints and cracks so as to ensure that there is no obvious difference between the test rock samples in macroscopic. According to the requirements of the International Society for Rock Mechanics (ISRM), the rock samples are processed into standard cylindrical rock samples with a diameter of 50 mm × 100 mm.

The precast fractured rock sample is shown in Figure

Geometry of the marble specimen with preexisting fissure.

The triaxial unloading creep test was performed using the ROCK 600-50 triaxial test system manufactured by Topouan, France (Figure

Testing machine of ROCK 600-50.

In the test, constant axial pressure is used to release confining pressure by stages, and load control mode is adopted. Firstly, according to the hydrostatic pressure condition _{1} = _{2} = _{3} = 10 MPa, confining pressure is applied to 10 MPa at the loading rate of 0.5 MPa/s, and then the axial load is continuously applied to 75% of the predicted ultimate bearing capacity under the confining pressure, which exceeds the uniaxial compressive strength of the coal sample and then maintains the principal stress difference _{1}–_{3}. Under the same condition, the confining pressure is gradually reduced by 0.5 MPa/s, and the unloading is carried out step by step from 10 MPa to 8 MPa, 6 MPa, 4 MPa, and 2 MPa; each stage is maintained for 24 h until the coal sample is completely destroyed. The specific control parameters are shown in Table

Testing schemes of grading unloading pressure in the triaxial creep test.

Number | Fracture characteristics | _{3} (MPa) | _{3} (MPa) | ||||
---|---|---|---|---|---|---|---|

Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | |||

A00 | Complete coal sample | 45 | 10 | 8 | 6 | 4 | 2 |

A30 | 30° angle | 40 | 10 | 8 | 6 | 4 | 2 |

A45 | 45° angle | 35 | 10 | 8 | 6 | 4 | 2 |

A60 | 60° angle | 33 | 10 | 8 | 6 | 4 | 2 |

In this section, the unloading creep test results of prefractured coal samples obtained by using the triaxial rheometer are analyzed, and the rheological deformation and failure law of coal samples under different fracture dip angles are studied. In order to ensure the reliability of the test data, the creep test is divided into three groups, four specimens are tested in each group, and the creep time of each stage is 24 h. The test results are shown in Table

Triaxial unloading creep test data of prefabricated cracked coal samples.

Number | _{1} (MPa) | _{2} (MPa) | _{i1} (%) | _{i3} (%) | _{c1} (%) | _{c3} (%) | ||
---|---|---|---|---|---|---|---|---|

A00 | 45 | 10 | 1.682 | 0.615 | 0.452 | 0.080 | 0.011 | 0.058 |

8 | 3.444 | 1.248 | 0.948 | 0.085 | 0.016 | 0.054 | ||

6 | 5.211 | 1.900 | 1.411 | 0.089 | 0.027 | 0.035 | ||

4 | 6.982 | 2.605 | 1.773 | 0.095 | 0.032 | 0.031 | ||

2 | 8.759 | 3.363 | 2.032 | — | — | — | ||

A30 | 40 | 10 | 1.793 | 0.644 | 0.506 | 0.102 | 0.020 | 0.062 |

8 | 3.690 | 1.361 | 0.968 | 0.111 | 0.022 | 0.067 | ||

6 | 5.595 | 2.151 | 1.293 | 0.121 | 0.029 | 0.064 | ||

4 | 7.510 | 3.043 | 1.424 | 0.137 | 0.034 | 0.069 | ||

2 | 9.469 | 4.034 | 1.401 | — | — | — | ||

A45 | 35 | 10 | 1.970 | 0.699 | 0.572 | 0.119 | 0.019 | 0.081 |

8 | 4.129 | 1.466 | 1.198 | 0.131 | 0.038 | 0.055 | ||

6 | 6.299 | 2.292 | 1.716 | 0.142 | 0.083 | −0.023 | ||

4 | 8.487 | 3.204 | 2.079 | — | — | — | ||

A60 | 33 | 10 | 2.084 | 0.757 | 0.570 | 0.188 | 0.012 | 0.164 |

8 | 4.286 | 1.534 | 1.217 | 0.196 | 0.061 | 0.073 | ||

6 | 6.501 | 2.380 | 1.741 | 0.209 | 0.096 | 0.016 | ||

4 | 8.751 | 3.309 | 2.134 | — | — | — |

_{1} and _{3} are the axial pressure and confining pressure; _{i1}, _{i3}, and _{c1}, _{c3}, and

The creep law of the prefractured coal sample is obtained through the creep test of unloading confining pressure step by step under constant axial pressure, as shown in Figure

Axial and lateral creep curves of coal samples with different fracture dip angles. (a) A00 (_{1} = 45 MPa). (b) A30 (_{1} = 40 MPa). (c) A45 (_{1} = 35 MPa). (d) A60 (_{1} = 33 MPa).

The creep curve obtained under the condition of step unloading creep is ladder shape, which cannot be directly used and needs to be transformed. According to Chen’s loading treatment method, the whole curve is transformed into a creep curve under various loads, as shown in Figures

Unloading rheological curve cluster of the complete coal sample. (a) Cluster of axial creep curves. (b) Cluster of lateral creep curves.

Unloading rheological curve cluster of coal samples with 30° fracture angle. (a) Cluster of axial creep curves. (b) Cluster of lateral creep curves.

Unloading rheological curve cluster of coal samples with 45° fracture angle. (a) Cluster of axial creep curves. (b) Cluster of lateral creep curves.

Unloading rheological curve cluster of coal samples with 60° fracture angle. (a) Cluster of axial creep curves. (b) Cluster of lateral creep curves.

It can be seen from Figures

Taking A00 sample as an example, the instantaneous deformation and creep deformation of A00 sample increased with the increase of deviator stress (Figures ^{−2}, and the lateral increment is about 3.5 × 10^{−4}. Since the second stage unloading, the axial and lateral creep deformation of each stress level is significantly increased, and the axial strain increase is about 0.5 × 10^{−2}. When the confining pressure is 2 MPa, the axial and lateral creep deformation increases significantly, and the accelerated creep phenomenon appears. Within 1 hour, the axial strain increased by 4.05 × 10^{−4} in the direction almost parallel to the axial strain axis, and the lateral deformation increased by 0.2 × 10^{−4}. The results show that the internal bearing capacity of the sample is weakening with time, and the through fracture surface under the level of fracture confining pressure leads to the failure of the sample.

According to the triaxial unloading creep test results of coal samples with different fracture dip angles, the influence of fractures on the creep deformation of coal samples with different inclination angles is analyzed, and the creep mechanical properties of coal samples with different fracture dip angles are discussed. In Figures

According to Figures ^{−2}, 2.17 × 10^{−2}, and 2.19 × 10^{−2}, and the increment of lateral instantaneous strain is 2.4 × 10^{−4}, 2.6 × 10^{−4}, and 2.9 × 10^{−4}. Compared with the intact coal sample, the deformation increases significantly. When the confining pressure is 10 MPa, the axial strain increased by 1188.8 × 10^{−6}, and the lateral strain increased by 5.88 × 10^{−6} after 24 hours at a constant stress level; when the confining pressure drops to 8 MPa, the axial strain increased by 1306 × 10^{−6}, and the lateral strain increased by 11.84 × 10^{−6} after 24 hours at the constant stress level. When the confining pressure is reduced to 6 MPa, the axial strain and lateral strain increased by 1422 × 10^{−6}, and 25.94 × 10^{−6}, respectively, after 24 hours at constant stress level. When the confining pressure is 4 MPa, the creep deformation characteristics of the samples are very obvious. There are three typical creep stages in the whole stage, and the creep deformation characteristics of nonlinear acceleration are presented. In a short time, the axial and lateral creep deformation increased rapidly by 739.6 × 10^{−6} and 184.2 × 10^{−6}.

In the process of rock formation, there are a lot of microdefects (joints, holes, etc.) in the process of rock formation, so the initial damage is not zero before the creep test. Damage variable is used to measure the damage state of materials, and it is an important link to establish the damage mechanics model of materials. It can be established from microscopic (number of voids, length, area, volume, etc.) and macroscopic (elastic constant, yield stress, tensile strength, etc.). Considering that the angle and length of some joints and fissures in rock mass can be observed in practical engineering, this paper describes the initial damage of rock mass from the geometric damage principle.

The fracture geometry distribution of the rock sample is shown in Figure _{1}, and the projection area _{30} is calculated; secondly, the distance _{30} between the center normal vectors of the two projection planes is calculated; thirdly, considering the influence of the fracture spacing on the mechanical properties of the rock sample, the spacing _{0} with the minimum influence of the fracture on the mechanical properties of the coal sample is calculated; and fourthly, the damage coefficient _{30} is the damage value of 30° fractured coal sample; _{30} is the projected volume of crack in the _{1} direction; _{30}/_{0}; and

Coordinate axis of jointed rock mass.

According to the nonlinear characteristics of rock creep stage, many researches on the rock creep model have been carried out in recent years. There are four methods to establish the nonlinear creep model:

① According to the indoor or field creep test and measured data, the empirical model can be obtained by using the creep formula or empirical formula fitting regression, but its universality is poor

② The parameters of the linear element are treated unsteady to make it a nonlinear element, and then the modified nonlinear creep model is obtained

③ Based on the damage theory of creep and damage quantity, the damage model is established

④ The creep model is obtained considering the external environment, such as the nonlinear creep model coupled with temperature and chemical conditions

Under the action of stress, the accumulation and evolution of internal crack damage is the essential reason for the final deformation and failure of rock. However, the fractured rock mass itself has some damage before the test, and then the creep test is carried out to further expand the damage. Therefore, the creep damage of fractured rock mass does not only exist in the accelerated creep stage but also exist throughout the whole creep process. The research idea of this section is based on the time-dependent evolution mechanism of unloading creep damage of fractured rock mass, introducing the damage factor and combining with the existing creep element model, to establish a creep model which can describe the nonlinear deformation characteristics of each stage of unloading creep process of fractured rock mass, including attenuation creep, steady-state creep, and accelerated creep.

According to the Kachanov damage rate [

The fractured rock mass itself contains different degrees of initial cracks, and the rock mass contains initial damage during loading. Therefore, the initial damage is not 0 at the moment of loading.

By introducing equation (

By introducing equation (

The evolution law of damage variable of fractured rock mass in the whole creep process is as follows:

The Burgers model is composed of the Maxwell model and the Kelvin model in series. The Kelvin body describes the viscoelastic properties of materials and reflects the attenuation creep of rocks; the Maxwell volume describes the elastic and viscous characteristics of the material, reflecting the instantaneous deformation of rock in the loading stage and the constant rate creep deformation in the steady-state creep stage. The Burgers model has the characteristics of elastic viscoelastic viscosity, which can better describe the attenuation and steady-state creep process of general rock mass under low stress level. In this paper, the Lade criterion considering plastic deformation is introduced into the Burgers model, and the damage variable

Damage creep model of fractured rock mass.

According to the series parallel relationship of the model, we can know that

In order to compare with the test results, the following reasonable assumptions are made firstly:

① The rock is isotropic, and the damage is assumed to be consistent in all directions

② The damage time and creep time are consistent in the accelerated creep stage

The evolution law of the one-dimensional Burgers model based on damage variable in the whole creep process is as follows:

In the three-dimensional stress state, the stress tensor inside the material can be decomposed into the spherical stress tensor _{m} and the deviatoric stress tensor _{ij}:_{ij} is the Kronecker function. It is generally believed that the spherical stress tensor can only change the volume of the material, but not its shape, while the partial stress tensor can only cause the shape change. Similarly, the strain tensor can be decomposed into spherical strain tensor and deviatoric strain tensor, as shown in the following formula:

For the Hooke body under three-dimensional stress state,_{1} and _{2} are the shear modulus. _{1}, _{2}, _{1}, and _{2} are all the model parameters, which are determined by fitting the test results:_{1}, and _{2} that need to be fitted can be transformed into _{1} and _{2}:

The three-dimensional creep equation of the damaged plastic body is as follows:_{0} is the initial value of the rock yield function, which is generally taken as 1 for the convenience of calculation;

The Mohr–Coulomb yield criterion and the von Mises yield criterion are widely used. However, the Mohr–Coulomb criterion cannot consider the effect of intermediate principal stress on rock yield, and von Mises yield criterion ignores the influence of spherical stress on rock creep characteristics. In this paper, the yield criterion is adopted, which has the following advantages:

The results show that the three curves are smooth on the meridian plane, which is conducive to the numerical calculation and the determination of the direction of plastic strain increment

The S-D effect of different tensile and compressive strengths of geotechnical materials is considered

The material parameters are few and easy to be tested

It can be widely used in sandy soil, rock-soil, concrete, and over consolidated clay

It reflects the nonlinear relationship between the shear expansion yield curve and the hydrostatic pressure

The effect of hydrostatic pressure

The yield criterion is expressed as_{1} and _{3} are the first and third invariants of the stress tensor; _{a} is the atmospheric pressure; and

For conventional creep tests,

The creep equation of the nonlinear creep model of fractured rock mass under three-dimensional stress state is obtained by using simultaneous equations (

In the process of model parameter identification, the selection of the mathematical model identification method is very important. At present, the main methods to determine rock creep model parameters are as follows: least squares method, regression inversion method, and displacement back analysis method. The least squares fitting curve is widely used to determine the relevant parameters, but the disadvantage is that the unreasonable selection of the initial value will lead to divergence of results or slow convergence speed. In this paper, the parameters of the rock creep model are identified by the mathematical optimization analysis software 1stOpt. The software does not need the user to provide appropriate initial parameters, which overcomes the shortcomings of ordinary least squares method. Levenberg–Marquardt + general global optimization algorithm is adopted to find the optimal model parameters. Before fitting, equation (

According to the fitting results, the coefficients

In order to verify the effect of the creep damage model established in this paper on the whole creep stage, the traditional Burgers model and the F-D model considering the initial damage are used to fit the test curve at the same time. The fitting results are shown in Table

Identification results of creep model parameters.

Model | _{1} (MPa) | _{1} (MPa·h) | _{2} (MPa) | _{2} (MPa·h) | _{3} (MPa) | ^{2} | |||
---|---|---|---|---|---|---|---|---|---|

Burgers | 1405.173 | 1282.984 | 970486.977 | 24829.455 | 69348.095 | 0.997 | |||

690.772 | 630.705 | 943793.638 | 25684.545 | 67545.498 | 0.994 | ||||

460.658 | 420.601 | 928555.343 | 26136.347 | 63703.257 | 0.991 | ||||

347.036 | 316.859 | 842878.145 | 25978.658 | 55542.856 | 0.990 | ||||

265.569 | 249.237 | 798621.687 | 24036.263 | 49851.892 | 0.969 | ||||

F-D | 982.525 | 897.283 | 635384.554 | 16248.256 | 45454.447 | 191574.563 | 7.713 × 10^{−9} | 0.055 | 0.998 |

485.155 | 443.064 | 630126.484 | 17148.671 | 45070.881 | 104343.923 | 7.929 × 10^{−9} | 0.015 | 0.996 | |

325.072 | 296.869 | 631538.602 | 17776.032 | 43292.905 | 68274.135 | 8.471 × 10^{−9} | 0.003 | 0.994 | |

245.983 | 224.642 | 582988.970 | 17966.628 | 38380.009 | 51212.810 | 2.469 × 10^{−9} | 0.292 | 0.993 | |

198.108 | 180.921 | 704302.959 | 15649.894 | 27820.801 | 55882.069 | 1.040 × 10^{−3} | 1.093 | 0.994 |

Comparison between test data and model fitting curve.

Combined with Table ^{2} of model identification is more than 0.99, and the fitting accuracy of the model meets the requirements. The experimental curve and the theoretical curve are in good agreement; the theoretical curve of the model is basically consistent with the test curve. The F-D model can not only accurately describe the creep characteristics of attenuation and constant velocity stage but also describe the accelerated creep characteristics of fractured rock mass. The deformation characteristics of three typical stages of rock rheology can be well described, which verifies the applicability of the model.

Rock burst is one of the major disasters in coal mines because of its great destructiveness. Because of the mining depth, creep-type rock burst occurs frequently, and the number of coal mines and coal seams is increasing. This kind of rock burst disaster has seriously restricted the safety production of coal mines in China. The prediction and prevention of this kind of rock burst is different from the conventional type. If the mining boundary is complex, the self-weight stress field is large, and the residual tectonic stress field exists, and the superposition of these multiple factors will make the coal (rock) body produce “strong” creep failure. If disturbed by the dynamic stress field of the working face, it is easy to produce large-scale and serious rock burst. The existing prevention methods of rock burst are to release the energy accumulated in coal and rock mass by releasing confining pressure, so as to reduce the stress concentration in coal and rock mass and prevent rock burst phenomenon. Based on this, triaxial unloading confining pressure creep tests are carried out on fractured coal samples with different angles, and the fracture damage creep model is established. The conclusions are as follows:

Under the same stress state, the different fracture dip angle significantly affects the creep deformation of coal samples. With the decrease of confining pressure, the axial instantaneous strain and creep strain increase with the increase of fracture dip angle, each curve is basically straight line, and the deformation increase amount of the same type coal sample under each stress level is basically the same. The order of axial instantaneous strain and creep strain from large to small are 60° fracture dip angle, 45° fracture dip angle, 30° fracture dip angle, and complete coal sample. Under the same stress state, with the decrease of confining pressure, the order of lateral instantaneous strain from large to small is 60° fracture angle, 45° fracture angle, 30° fracture angle, and complete coal sample; compared with complete coal sample, due to the existence of prefabricated cracks, the bearing capacity of fractured coal sample in creep process decreases, and the lateral expansion deformation increases. The larger the fracture angle, the greater the lateral strain.

Based on the geometric damage principle, a second-order tensor is introduced to describe the initial damage of rock mass. According to Kachanov’s creep damage theory, the damage variable with initial damage is introduced into the constitutive relation and creep equation, and the evolution equation of damage variable with time in the whole creep process is derived.

The elastoplastic body with damage variable is connected with the Burgers model in series. Meanwhile, the

The applicability of the fracture damage creep model is verified by using the creep test data of fractured coal samples. The results show that the creep model can not only accurately reflect the creep characteristics of the decay and constant velocity creep curves of fractured coal samples but also describe the accelerated creep characteristics and effectively simulate the whole creep process of fractured coal samples.

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

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (grant number 51574156) and the Key Development Program for Research of Shandong Province (grant number 2018GNC110023). This research was also partially funded by the Shandong Province Higher Educational Science and Technology Program (grant number J18KA195).