### Failure and deterioration mechanism of white sandstone

Rock is a complex mineral collection with its own particular internal material composition and spatial structure^{27}. The fracture extension direction, fine fracture characteristics, and the relative displacement direction of rock masses on both sides of the section were judged by the morphological characteristics of different fractures^{28}. By comparing the fracture morphology and fracture extension of creep specimens of peak-unloading damaged white sandstone, the creep damage mechanism of peak-unloading damaged white sandstone is analyzed. Microscopic analysis adopts the Hitachi High-tech SU8000 Scanning Electronic Microscope, and the SEM images of white sandstone fracture with varying damage is shown in Fig. 15.

Figure 15a–f show the scanning electron micrographs of creep fracture at 2000 magnification for the nondestructive specimen and the pre-peak unloading damaged specimen of white sandstone respectively. The fine fracture observation surface of the nondestructive specimen shows obvious deconstruction steps, and the fracture joints of each deconstruction platform are relatively flat. In addition, a small amount of crystal debris is present on the fracture observation surface. Figure 15b shows that the fracture observation surface is rough, obvious scale-like brittle fractures and along-crystal cracks can be observed, and a few tough nests are produced on the fracture surface. Figure 15c shows that the fine view fracture observation surface of the pre-peak unloading damaged white sandstone also shows an obvious deconstruction platform. The fracture observation surface is relatively flat with only a few crystal debris particles.

In contrast, the deconstruction platform’s extension direction is consistent with the deconstruction damage on the fracture observation surface. Figure 15d shows that the brittle-tough transition phenomenon begins to appear in the fine fracture morphology. The fine fracture observation surface has obvious tough nest groups and is rougher. Figure 15e shows that the fracture surface has obvious transgranular cracks in the low concave area. The along-crystal cracks and transgranular cracks coexist and develop together. Figure 15f shows that many tough nests are distributed on the fine fracture observation surface. The transgranular cracks and along-crystal cracks intersect, forming an obvious macroscopic fracture zone on the fine fracture observation surface of the specimen.

In the creep process of white sandstone, the internal crystal structure of the rock is in the joint action of constant axial force and peak front unloading damage effect; when the damage degree is low, the relative displacement degree between crystals is low. During the relative motion of the crystal particles, the incompatibility of the deformation of the crystal particles will cause local stress concentration and then form obvious deconstruction platforms. When the degree of damage is higher, the dislocations in the internal structure of the crystal increase rapidly, the crystal particles’ strength decreases, and cracks form transgranular cracks from inside the crystal. Under the continuous effect of long-term stress, the crystal particles are not in equilibrium due to the pre-peak unloading damage effect, and the cracks develop rapidly. The fracture observation surface shows that the transgranular cracks and along-crystal cracks are staggered through. Overall, with the increase of the pre-peak unloading damage degree, the internal cracks of the white sandstone specimens develop chaotically, the meso-structure damage intensifies, and the overall performance is more “soft” due to the weakening of local properties. It can be seen that the white sandstone experiences a higher pre-peak unloading damage effect, the damage caused by creep is more complex, and the cracks are more abundant.

The deformation and strain of the white sandstone with pre-peak unloading damage were measured by PMLAB DIC-2D, and the moment of failure of the specimen was selected after shooting, the transient failure form of the specimen is shown in the upper part of Table 4. Using the crack detection system for rock specimens based on Matlab-GUI, several real-time damage crack detection tests for white sandstone specimens were carried out. The image processing results are shown in the lower part of Table 4.

Table 4 shows that the white sandstone in the nondestructive natural state exhibits splitting tensile damage, and the damage form of white sandstone gradually changes from “splitting”-“single bevel shear damage”-“splitting-shear damage” with the increase of unloading damage. DR0-0 images show that multiple cracks are generated at the rupture initiation point of the specimen, and the expansion of the primary cracks triggers the generation of secondary cracks. DR1-0 images show that the damage of the specimen is in the form of splitting damage, which leads to the destruction of the rock end by dilation phenomenon. DR2-0 and DR3-0 images show that the damage of the specimen is in the form of single bevel shear damage, a main crack penetrates the whole specimen from top to bottom, and the fracture angles of the specimen are about 68° and 71° respectively, and the overall shape of the specimen remains basically intact. DR4-0 image shows that the specimen still exhibits significant shear damage with a primary shear fracture crack angle of 72° and secondary cracks appears in the middle of the rock specimen. DR5-0 image shows that the specimen damage is in the form of mixed shear-splitting damage with a primary crack angle of 70°.In addition, splitting cracks are generated from the top and eventually intersect with shear cracks, and there are fewer secondary cracks at the top of the specimen.

The macroscopic damage morphology of white sandstone damaged by pre-peak unloading is significantly different, and the reason is attributed to (1) The mineral composition of the white sandstone contains a large number of clay minerals, such as montmorillonite, kaolinite and chlorite, etc. Different minerals have uneven deformation under the action of pre-peak unloading damage, causing certain damage to the sample; (2) The microscopic closed cracks in the white sandstone gradually accumulate inside the sample, and under long-term loading conditions, aging damage helps the new cracks and the original cracks to penetrate each other to form a macroscopic fracture zone. The propagation of the cracks in the lower specimen will gradually stabilize. Under the condition of yield stress, due to the reduction of the effective area, a stress concentration effect is formed inside the sample, the stress exceeds the strength of the rock micro-element, and the micro-cracks continue to develop and extend, and finally the sample ruptures along the main failure surface (see Fig. 16).

The above test results show that the pre-peak unloading damage white sandstone creep test shows obvious weakening phenomenon, and the pre-peak unloading damage effect involves all stages of rock creep. In order to further study the damage effect of white sandstone in each creep stage, from the perspective of rock creep constitutive relationship, a creep constitutive model considering the pre-peak unloading damage effect is constructed to further quantitatively explore the pre-peak unloading damage. Creep properties of sandstone under action.

### Creation and analysis of damage components

The effect of pre-peak unloading damage on rock creep characteristics can be qualitatively expressed as under the constant stress of external load, the rock attenuation creep, isokinetic creep and accelerated creep all change with the weakening effect of pre-peak unloading damage^{29}.

The quantitative expression can be based on the non-quantitative characteristics of rock creep mechanical parameters changing with the weakening degree of pre-peak unloading damage, and the long-term creep action of rock will cause fatigue damage, and the parameters based on aging characteristics themselves have attenuation. Therefore, in terms of macroscopic mechanical behavior, this parameter can describe the coupling effect of pre-peak unloading damage and creep damage.

First, considering that the viscoelastic parameters have unsteady eigenvalues that vary with pre-peak unloading damage, the details are shown below.

$$ E(U) = K_{e} (U)E_{0} $$

(4)

$$ \eta (U) = K_{\eta } (U)\eta_{0} $$

(5)

where: \(E(U)\), \(\eta (U,t)\) is the elastic model and viscosity coefficient after the weakening of the pre-peak unloading damage; \(E_{0}\), \(\eta_{0}\) is the elastic modulus and viscosity coefficient of the rock in the nondamaged state; \(K_{e}\), \(K_{\eta }\) is the elastic modulus damage coefficient and the viscosity damage coefficient of the pre-peak unloading damage effect.

During long-term creep, the difference in long-term mechanical properties between microscopic crystal particles reflects that the accumulation of microscopic damage has a random distribution characteristic based on the aging effect, which is expressed by a statistical continuous distribution function as:

where, \(f(t)\) is the damage density function; \(D\) is the long-term load damage variable based on aging effect; Considering that the aging damage accumulation is a nonlinear incremental behavior, a two-parameter Weibull distribution function is introduced to define the damage density function i.e.:

$$ f(t) = \frac{\theta }{\lambda }\left( {\frac{t}{\lambda }} \right)^{\theta – 1} e^{{ – (t/\lambda )^{\theta } }} $$

(7)

$$ D_{R} = \int_{0}^{t} {\frac{\theta }{\lambda }\left( {\frac{t}{\lambda }} \right)^{\theta – 1} e^{{ – (\frac{t}{\lambda })\theta }} } dt = 1 – e^{{ – (t/\lambda )^{\theta } }} $$

(8)

where, \(\theta\), \(\lambda\) are long-term load damage parameters.

The viscosity coefficient is an aging parameter with pre-peak unloading damage effect, so the variation law of viscosity coefficient has obvious aging attenuation and pre-peak unloading damage effect characteristics. load-damage coupling, and then the coupling equation can be obtained:

$$ \eta (U,t) = (1 – D)K_{\eta } (U)\eta_{0} = K_{\eta } (U)\eta_{0} e^{{ – (\frac{t}{\lambda })\theta }} $$

(9)

The above creep mechanical parameters are introduced into the Nishihara model, and the Nishihara damage constitutive model considering the damage effect of pre-peak unloading is established, which can better describe the uniaxial creep change law of white sandstone with pre-peak unloading damage.

### Creep modelling

The Nishihara model consists of a Hooker body, a Kelvin body, and an ideal viscous body in series, which can fully respond to the elastic, viscous, and plastic properties of the rock^{30}. Its mechanical model is shown in Fig. 17.

When \(\sigma < \sigma_{s}\), the Y body is rigid, at this time the model rheological properties have creep properties. When \(\sigma \ge \sigma_{s}\), at this time, the deformation of the creep model dissipates continuously with the increase of time. According to the series–parallel relationship between components in the previous section, it is known that \(E(U)\) and \(\eta (U,t)\) are introduced into the H-body and N-body in the West original model, while the damage due to the short-term effect is not obvious, so \(\eta (U,t)\) is only introduced into the ideal viscoplastic body describing the accelerated creep stage. Based on the above content, a creep constitutive model of white sandstone considering pre-peak unloading damage was established:

$$\left\{\begin{array}{l}{\sigma }_{1}={\sigma }_{2}={\sigma }_{3}\\ \varepsilon ={\varepsilon }_{1}+{\varepsilon }_{2}+{\varepsilon }_{3}\\ {\sigma }_{1}={E}_{H}(U){\varepsilon }_{1}\\ {\sigma }_{2}={E}_{Y}\left(U\right){\varepsilon }_{2}+{\eta }_{Y}(U)\frac{d{\varepsilon }_{2}}{dt}\\ {\sigma }_{3}={P}_{t}\left(t\right)[{\sigma }_{s}+{\eta }_{N}(U,t)\frac{d{\varepsilon }_{2}}{dt}\end{array}\right.$$

(10)

where \({{\sigma_{1} } \mathord{\left/ {\vphantom {{\sigma_{1} } {\varepsilon_{1} }}} \right. \kern-\nulldelimiterspace} {\varepsilon_{1} }}\), \({{\sigma_{2} } \mathord{\left/ {\vphantom {{\sigma_{2} } {\varepsilon_{2} }}} \right. \kern-\nulldelimiterspace} {\varepsilon_{2} }}\) and \({{\sigma_{3} } \mathord{\left/ {\vphantom {{\sigma_{3} } {\varepsilon_{3} }}} \right. \kern-\nulldelimiterspace} {\varepsilon_{3} }}\) are the stress/strain corresponding to the Hooke body, Kelvin body, and ideal viscoplastic body in the West Plains model, respectively, \(E_{H} (U)\) is the modulus of elasticity of the spring component in the Hooke body, \(E_{Y} (U)\) is the modulus of elasticity of the spring component in the Kelvin body, \(\eta_{Y} (U)\) is the coefficient of viscosity of the slider component in the Kelvin body, \(\sigma_{s}\) is the creep yield strength, \(\eta_{N} (U)\) is the coefficient of viscosity of the slider component in the ideal viscoplastic body, and \(P(t)\) is the determination function, as follows:

$$ P(t) = \left\{ \begin{gathered} 0,t \le t_{s} \hfill \\ 1,t > t_{s} \hfill \\ \end{gathered} \right. $$

(11)

where, \(t_{s}\) is the accelerated creep onset time point in the rock creep process. The first-order linear partial differential equation of formula (9) is obtained as

$$ \varepsilon (t) = \left\{ \begin{aligned} & \frac{\sigma }{{E_{H} (U)}} + \left( {1 – e^{{ – \frac{{\varepsilon_{Y} (U)}}{{\eta_{Y} (U)}}t}} } \right)\frac{\sigma }{{E_{Y} (U)}},t \le t_{s} \hfill \\ & \frac{\sigma }{{E_{H} (U)}} + \left( {1 – e^{{ – \frac{{\varepsilon_{Y} (U)}}{{\eta_{Y} (U)}}t}} } \right)\frac{\sigma }{{E_{Y} (U)}} + \frac{{\sigma – \sigma_{s} }}{{\eta_{N} (U,t)}}t,t > t_{s} \hfill \\ \end{aligned} \right. $$

(12)

\(\varepsilon (t)\) is the principal equation of the Nishihara model considering the weakening effect of pre-peak unloading damage.

### Creep model validation

Based on the uniaxial creep test data of white sandstone with pre-peak unloading damage, using Matlab software and using the improved Leweinberg-Marquardt algorithm to analyze and simulate the model parameters in the Nishihara model considering the pre-peak unloading damage, and the simulation results are shown in Tables 5, 6, and 7. In order to more intuitively compare the changes between the theoretical curve and the actual creep test curve, the test data of the pre-peak unloading damage \(\sigma_{\mu l}\) = 70%, 80% and 90% are selected for analysis, as shown in Fig. 18.

Tables 5, 6, and 7 show the data tables of the identification parameters of the Nishihara model considering the effect of pre-peak unloading damage, from which it can be seen that the elastic modulus and viscosity coefficients in the Nishihara model decrease with the increase of the degree of pre-peak unloading damage. This is because the rock undergoes pre-peak unloading damage and weakening, and the internal structure is damaged, resulting in the continuous weakening of mechanical properties. Table 5 shows that at \(\sigma_{\mu l} = {5}0\%\), with the increase of creep stress, the elastic modulus and viscosity coefficient of the pre-peak unloading damage creep model both increase, and the coefficient increases by a similar amount. Table 6 shows that the creep model parameters of white sandstone with the same degree of damage increased with the creep stress level at \(\sigma_{\mu l} = 70\% ,\;80\% ,\;90\%\); however, the elastic modulus and viscosity parameters gradually decreased with the increase of the peak front unloading damage at the same creep stress level.

Figure 18a–c shows the comparison of experimental data and theoretical curves for DR3-0, DR4-0 and DR5-0 under non-yield stress conditions, while Fig. 18d shows the comparison between the experimental data and the theoretical curve of the unloading damage degree before the peak under the condition of yield stress (including the accelerated creep stage). Figure 18 shows the theoretical curve of the Nishihara creep model of white sandstone considering pre-peak unloading damage, whether the white sandstone is in the yield stress state or not, it can better describe the uniaxial creep process of the white sandstone with pre-peak unloading damage. Taking Fig. 18d as an example, when the peak front unloading damaged white sandstone is in the yield stress state, although the accelerated creep curve is difficult to describe, the Nishihara creep model of white sandstone considering pre-peak unloading damage can still accurately describe its change process. And the experimental curve is highly consistent with the theoretical curve, the linear correlation regression coefficient R^{2} ≥ 0.99, which shows that the creep model can better describe the whole process of creep of peak front unloading damage.

Figure 18 shows: (1) The theoretical simulation of the Nishihara creep model considering the pre-peak unloading damage of white sandstone has a very high degree of fit, which shows that the creep model can describe the stability of the specimen when it is in a state of non-yield stress. In the creep stage, the strain changes from nonlinear growth to linear growth viscoelasticity, which can also accurately describe the viscoplastic features of the unstable creep stage when the specimen is in a state of yield stress, and the specimen undergoes an accelerated creep stage to produce plastic deformation, and compared with previous studies, it can be found that the Nishihara model considering the pre-peak unloading damage effect can more accurately describe the characteristics of the accelerated creep stage when the rock is in the state of yield stress. (2) Fig. 18d and Table 7 show that the model is based on the aging damage effect, thus introducing a two-parameter Weibull distribution function, the parameters \(\theta\), \(\lambda\) in the function have an important effect on the curve description of the accelerated creep stage of the rock under the yield stress state, and the changes of the two are inversely proportional. When the damage parameter \(\theta\) is larger and \(\lambda\) is smaller, the accelerated creep stage is more obvious, the degree of curve change is larger, and the axial accelerated creep strain rate of the specimen is larger, so that the internal structure of the rock specimen changes from crack quantity to qualitative change. That is, from the increase in the number of cracks to the formation of macroscopic crack bands, the shorter the specimen is required for aging. When \(\lambda\) is larger, \(\theta\) is smaller, indicating that the curve change of rock accelerated creep stage under the yield stress state has obvious time effect, that is, with the increase of creep time, the axial creep deformation rate of rock increases nonlinearly and slowly with time showing a non-linear trend.

To sum up, the mechanism of action is that under the condition of yield load, the crystals inside the rock rub and slip with each other, and a new stress structure is continuously formed to resist the external load. With the gradual accumulation of microscopic defects, the effective area corresponding to the stress gradually decreases, resulting in local stress concentration, the surrounding micro-cracks rapidly expand into macro-sections, and the rock becomes unstable and its strain surges. Considering the pre-peak unloading damage effect, the Nishihara model can better describe the creep deformation of the surrounding rock in the deep well roadway during the re-mining process, and provides a certain reference value.

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