Nonlinear Dynamic Constitutive Model of Frozen Sandstone Based on Weibull Distribution

To obtain the dynamic mechanical properties of frozen sandstone at different temperatures (i.e., 20°C, −10°C, −20°C, and −30°C), dynamic uniaxial compression tests of saturated sandstone are conducted using a split-Hopkinson pressure bar.-e experimental results demonstrated that the brittleness of the saturated sandstone increased and its plasticity weakened with a decrease in temperature. -e peak strength and dynamic elastic modulus of the sandstone were positively correlated with its strain rate. -e peak stress was most sensitive to the strain rate at −10°C, and the elastic modulus was most sensitive to the strain rate at −30°C. According to the evident segmentation characteristics of the obtained stress-strain curve, a viscoelastic dynamic constitutive model considering the strain rate effect and temperature effect is developed; this model combines a nonlinear (or linear) body and aMaxwell body in parallel with a damage body.-e applicability of the constitutive model is also verified using experimental data. -e fitting results were demonstrated to be in good agreement with the experimental results. Furthermore, the fitting results serve as reference for the study of the constitutive model of weakly cemented soft rock and the construction of roadway freezing methods.


Introduction
e Jurassic coal seam in western China is covered in thick cretaceous water-rich sandstone. Owing to a special diagenetic process, the sandstone has the mechanical properties of weak cementation, low strength, and large porosity [1][2][3]. A freezing method is used to stop water and improve the strength of the weak surrounding rocks. However, the freezing wall is often disturbed by drilling and blasting during its construction process, which can result in water seepage and water gushing accidents [4][5][6]. e main reason for this phenomenon is the damage to the frozen wall caused by blasting impact loads. For example, the failure of the frozen wall of the auxiliary shaft of the no. 1 coal mine in the Shanghai temple, Inner Mongolia, resulted in a shutdown of nearly six months. erefore, it is necessary to study the dynamic mechanical properties of this sandstone under freezing conditions and to develop a dynamic constitutive model to provide the basic mechanical parameters for engineering design.
Several studies have investigated the mechanical properties, damage evolution, and constitutive models in the statics of this rock. Yang et al. [7][8][9] systematically studied the static mechanical properties of water-rich soft rock. A common conclusion is that the strength and softening coefficient of such soft rock were low in unfrozen conditions, while the compressive strength and elastic modulus increased with a decrease in temperature in the frozen state. Wang [10][11][12] simulated the change process of the stress state of cretaceous frozen soft rock in underground construction via a testing method. In addition, a nonlinear creep constitutive model of frozen soft rock was proposed. Zhao et al. [13,14] used the strain equivalent principle to study the damage evolution of such soft rocks under the triaxial compression test. Furthermore, a statistical damage constitutive model considering the residual deformation of sandstone was developed. However, in terms of dynamics, Liu and Xu [15,16] conducted uniaxial impact tests on sandstone samples under different confining pressures. ey found that the stress-strain curves under confining pressure presented typical elastic-plastic characteristics. Zhu et al. [17,18] investigated the failure process of sandstone from the perspective of microscopic crack propagation and energy absorption by using a split-Hopkinson pressure bar (SHPB) test device. In addition, the damage evolution was analyzed based on the statistical damage theory of the Weibull distribution. According to the characteristics of the stress-strain curves of soft rocks under impact loading, Zhao et al. [19,20] modified the Zhu-Wang-Tang (Z-W-T) model by replacing the elastic element with a damaged body, after which a damage viscoelastic dynamic constitutive equation was derived. Yang et al. [21,22] analyzed the variation in strength with temperature by combining SHPB systems with SEM. e microfracture mechanisms of saturated frozen red sandstone are explored. However, these studies mainly applied impact loading to sandstone samples at room temperature. Few studies have conducted experiments on rock behavior under low temperature, which is important for determining the significance and reference value for further research on the dynamic mechanical properties and constitutive model of cretaceous water-rich sandstone in the frozen state.
is paper aims to investigate the dynamic mechanical properties of saturated sandstone at different temperatures (20°C, −10°C, −20°C, and −30°C) using an SHPB pressure device. e dynamic constitutive model is established based on the stress-strain curves characteristics of the specimens. In addition, the applicability and rationality of the constitutive model are also verified using the experimental data. e results obtained can provide a reference and basis for the dynamic response, safety prediction, and support optimization of infrastructure under varying impact loads in cold regions.

Test
Equipment. An SHPB device was used to study the dynamic mechanical properties of the specimens. e schematic and the main structures of the experiment device are shown in Figure 1. e pressure bar of the system is alloy steel. e rod diameter is 50 mm, the elastic modulus is 210 GPa, and the longitudinal wave velocity is 5172 m/s. Pulse signals are collected using strain gauges on the incident and transmitted bars. e data acquisition system comprises a PC and an LK2107B ultrasonic dynamic strain gauge. e system can investigate the dynamic properties of rock, coal, concrete, and other materials under strain rates in the range 10 1 ∼10 3 s −1 .
e rocks on-site were transported to the laboratory to process a standard rock sample with a diameter of 50 mm and a height-diameter ratio of 0.5, as shown in Figure 2. e basic physical and mechanical parameters are listed in Table 1. e specimens were selected first based on uniformity and integrity. en, the UTA-2000A ultrasonic analyzer was used to select rock samples with similar longitudinal wave velocities as the test samples. All samples were placed inside a vacuum saturation machine to saturate the samples for 2 h, and the samples were then soaked in distilled water for more than 48 h to fully saturate the pores within the samples. After the saturation process, the specimens were placed into a lowtemperature test freezer in subgroups for freezing at different temperatures.

Test Principle.
e basic principle of the SHPB system is the propagation theory of stress waves in an elastic rod, which is based on the one-dimensional (1D) elastic wave hypothesis and uniformity hypothesis. Each cross section in the elastic rod is assumed to always remain planar when the stress wave propagates in the elastic rod in one dimension. According to the assumption of stress uniformity, the stress and strain are generally analyzed using the three-wave method: where ε I (t), ε R (t), and ε T (t) are the incident strain, the reflected strain, and the transmitted strain during impact, respectively; E c , A c , and C c are the modulus of elasticity, the cross-sectional area, and the longitudinal wave velocity of the bar, respectively; and A s and l s are the cross-sectional area and the length of the sample, respectively.

Test
Program. e range of blasting and impact loading strain rates for rock materials was 10∼10 3 s −1 [26]. e temperature of low-temperature brine in the freezing system can reach −30°C in the western shaft freezing construction. Four temperatures were set for the test considering the temperature field distribution of the frozen wall, i.e., 20°C, −10°C, −20°C, and −30°C. e test results obtained at 20°C were considered to be the results of the comparative analysis group. e impact velocities of the striker were determined to be 3.0 m/s, 3.5 m/s, 3.8 m/s, 4.6 m/s, 5.2 m/s, and 6.0 m/s based on the monitoring data at different positions of the frozen wall during tunneling at the construction site. e impact tests on samples were repeated six times to ensure the accuracy of the test results considering the influence of the discreteness of the test data. For clear identification, the specimens were marked from A to D. For example, samples from −10°C were numbered B1-1∼B1-6, . . ., B6-1∼B6-6. When the test was completed, the test data near the mean value were analyzed.

Dynamic Stress-Strain Curve.
A series of impact tests was conducted for different strain rates at different temperatures, i.e., 20°C, −10°C, −20°C, and −30°C. e stressstrain curves of the specimen can be obtained from these tests, as shown in Figure 3. Figure 3 shows that the stress-strain curves of sandstone under impact loading can be divided into three distinct stages: (i) the first stage is the linear elastic deformation stage, where the relationship between the stress and strain is approximately linear. e slope represents the elastic deformation modulus of the rock. Figure 3 shows that the elastic deformation of sandstone is extremely small. (ii) e second stage is the plastic development stage. During this stage, the internal cracks of the sandstone begin to sprout and expand under impact loading. e slope of the curves in this stage is notably smaller than that of the first stage. en, it decreases to 0 gradually. e results show that the stress reaches the maximum value at this stage. (iii) e third stage is the strain softening stage, in which the crack in the rock expands nonlinearly, and the stress decreases rapidly. en, the specimen loses the capability of resistance to deformation.
A comparison of the stress-strain curves of sandstone at different tested temperatures demonstrates that the decreasing temperatures significantly influence the mechanical properties of the sandstone. (i) e strain at the translate point between the first and the second stages continues to decrease, while the strength continues to increase, indicating that the elasticity and brittleness of the rock are enhanced. In addition, the ability to resist the deformation of the rock is improved. (ii) In the second stage, the slope of the curves continues to decrease with temperature, indicating that the plasticity of the rock weakens and the brittleness increases. (iii) In the third stage, gradually, the rate of decline accelerated and the curves steepen. is indicates that the deformation ability of the rock is weakened from the peak strength point to the ultimate failure point at low temperatures. Evidently, the rock changes from extended brittleness to brittleness. Figure 4 shows the relationship between the peak stress and strain rate of sandstone under different temperatures. Figure 4 shows that there is an increase in the peak stress with an increase in the strain rate. In addition, it indicates that the relationship between the peak stress and strain rate is a linear relationship, revealing that the relationship is an evident rate correlation. e peak stress of the specimen is the most sensitive to the strain rate at −10°C and the least sensitive to the strain rate at 20°C. It can be determined that the translate point occurs at 20°C, and this is believed to be because the water-rich cretaceous sandstone is soft and loose and has high porosity. e incident wave carries too much energy under high speed impact, resulting in the failure of the rock before the wave is fully passed into the transmission bar.

Relationship between Peak Stress and Strain Rate.
erefore, the extreme value appears because the transmitted wave signal collected by the system is less than the normal value. e peak strength of the rock increases first and then decreases with the decrease in temperature, reaching a maximum at −20°C. e strength at −30°C is significantly lower than that of the samples at −20°C when the impact pressure is large, that is, under the action of high strain rates. e rule is the same as Yang's [27], which is due to the "frostbite effect" of sandstone under high strain rates.

Dynamic Elastic Modulus Analysis.
e elastic modulus of the rock has an evident correction to the strain rate in the impact test. erefore, the elastic modulus can be defined as the secant slope of 40%∼60% or a tangent slope at 50% of the peak stress. Figure 3 shows that the stress-strain curve of frozen sandstone has an evident piecewise property and changes at different temperatures. erefore, the methods above cannot be used for calculation. e tangent modulus of the elastic phase is used for analysis in this part, as shown in Figure 5. Figure 5 shows that the dynamic elastic modulus increases with the decrease in temperature when T ≤ −10°C, where the growth rate from −10°C to −20°C is slow, while it is fast from −20°C to −30°C. With the increase in impact loading, the difference of the elastic modulus between −10°C and −20°C decreases to 0 and then increases gradually, while there is a gradual increase in the difference of elastic modulus between −20°C and −30°C. Comparing the range of variation of the elastic modulus of sandstone at different temperatures, it can be seen that the dynamic elastic modulus has the greatest sensitivity to the strain rate at −30°C, while the sensitivity is basically the same at −10°C and −20°C.
Based on the above analysis of the stress-strain curves, the brittleness of saturated sandstone increases, while the plasticity decreases with a decrease in temperature. According to the previous research results on the dynamic constitutive model of sandstone under room temperature, the dynamic constitutive model is established considering the temperature effect, strain rate effect, and damage effect

Establishment of Constitutive Model considering Temperature Effects
Tang et al. proposed the Z-W-T model [28], which is a dynamic constitutive model capable of describing the nonlinear viscoelastic responses of materials. is model was established based on homogeneous and isotropic simple material under 1D conditions. Based on the impact of polymer viscoelastic material, the Z-W-T model is composed of two different Maxwell bodies and a nonlinear spring. e constitutive model equation is as follows: where E 0 is the elastic modulus of the nonlinear spring; χ and κ are the nonlinear coefficients; E 1 and η 1 are the elastic modulus and viscosity coefficient of low-frequency linear spring, respectively; and E 2 and η 2 are the elastic modulus and viscosity coefficient of high-frequency linear spring, respectively. e two different Maxwell bodies in the model describe the viscoelastic response under different strain rates, which act only on their own "effective responsible area" [29]. e response range of η 1 is 10 1 ∼10 2 s −1 , and the response range of η 2 is 10 −4 ∼10 −6 s −1 . e relaxation time of the low-frequency Maxwell element is 10 1 ∼10 2 s −1 .
e loading time in the impact test is measured with a resolution of the order of several µs, there is therefore insufficient time to relax, and it is simplified as a simple spring.
According to the quasistatic compression test of saturated frozen sandstone, it can be obtained as follows: (i) the curves increase nonlinearly when −10°C ≤ T ≤ 20°C, so the polynomial part of the equilibrium stress independent of the strain rate in the Z-W-T equation can remain and represents the nonlinear elasticity; (ii) the curve increases linearly when −30°C ≤ T < − 10°C, so only one term is considered in the polynomial part of the equilibrium stress independent of the strain rate in the Z-W-T equation, which is simplified in linear elasticity. erefore, a viscoelastic constitutive model considering temperature effects is established. According to the response range of each Maxwell body in the Z-W-T model, the low-frequency part is replaced by a spring. e damaged body with elastic modulus E D is increased. e simplified damage model is shown in Figure 6.
e spring E 0 is nonlinear when −10°C ≤ T ≤ 20°C. e constitutive model of the nonlinear elastomer I is where E 0 � E ′ + E ″ in which E ′ is the elastic modulus of low-frequency linear spring, E ″ is the elastic modulus of the nonlinear elastic spring, and χ and κ are the nonlinear coefficients. e spring E 0 is linear when −30°C ≤ T < − 10°C. e nonlinear relation for elastomer I is To facilitate the analysis, the following assumptions are made regarding the model: (i) the rock interior is continuous, and the damage to the rock is homogeneous; (ii) the rock impact process is considered to be completed under constant strain rate loading conditions; and (iii) the strength of each microelement is subject to the Weibull distribution, and its probability density is as follows:

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where m and α are distributed parameters, and the size is related to the nature and shape of the material. e degree of damage to the test piece is the sum of the defects contained in each microelement. e relationship between the damage variable D and the probability density of microelement failure can be given as follows: After integration, From equation (7), the constitutive model of the damaged body II is e Maxwell body comprises an elastic element and a damping element connected in series. e stress-strain relationship can be obtained as follows: σ 12 � η_ ε 12 (t). (12) en, the derivative of equation (10) is found. Combining equations (9) and (11) with (12) yields e Laplace transform of equation (13) gives where σ(0) � 0. By substituting equation (15), the following can be obtained: Laplace inverse transformations of equation (16) yield Substituting t � (ε 1 /_ ε 1 ) into equation (18) gives Combined with equations (3), (4), and (19), the dynamic constitutive equation of saturated sandstone considering the damage effect, strain rate effect, and temperature effect is as follows:

Test Verification of Constitutive Equation
ere are eight unknown parameters in equation (20). e elastic modulus of each temperature sandstone can be first obtained from the static uniaxial test [30]. en, the leastsquares approximation theory is applied to fit the quasidynamic stress-strain curves at different temperatures in order to obtain E 0 , E 1 , χ, and κ from the constitutive equation. Finally, the stress-strain curves of impact tests under a high strain rate are fitted to determine the values of E D , η, α, and m. Here, two sets of data at each temperature are selected for analysis. e fitting curves for the prepeak stress and whole stress are given when −10°C ≤ T ≤ 20°C, and the prepeak stress fitting curves are given when −30°C ≤ T < − 10°C. e fitting results are shown in Figure 7. Figure 7 shows that the test curves are in good agreement with the fitting curves, where the correlation coefficient R 2 is 0.91487∼0.99336. is indicates that the model is suitable for describing the dynamic mechanical constitutive model of e fitting results of the whole stress-strain curves have an evident segmental phenomenon when T ≥ −10°C. erefore, it is possible to deduce the dynamic mechanical properties of sandstone during the whole impact process. e findings can give a better understanding of the shortcomings of the modified model, which can only partially address the prepeak stress. Generally, the fitting accuracy of the stress part before the peak is slightly greater than the entire stress-strain curves. e stress yielding platform that appears during the impact process leads to a decrease in the fitting accuracy. e parameters obtained by fitting the above constitutive equation are listed in Table 2.
From analyzing the values of each fitting parameter in Table 2, the value of E 0 is found to be 10 3 , indicating that the elastic modulus obtained under static uniaxial loading is reasonable. E 1 is 10 4 , which represents the elastic coefficient of the high-frequency Maxwell body, and is one order of magnitude larger than the numerical value of E 0 , indicating that the response of sandstone under dynamic impact tests is more sensitive. e parameter m is always less than 1, and α is always less than 0.1. e relaxation time η 1 of sandstone is not constant, and it is correlated with temperature and strain rate. However, the range of the variation is small.

Conclusions
is study investigated the dynamic mechanical properties and constitutive model of frozen sandstone using a series of impact tests. e main conclusions drawn are summarized as follows: (1) e dynamic stress-strain curves of low-temperature sandstone can be divided into three distinct stages: linear elastic deformation stage, plastic development stage, and strain softening stage. e peak stress is positively correlated with the strain rate, and the peak stress of the sample is most sensitive to the strain rate at −10°C. However, it is least sensitive to the strain rate at 20°C. e "frostbite effect" of the sandstone under high strain rates occurred at −30°C. is mitigates the shortcomings of the modified model, which only addresses the prepeak stress. By analyzing the rationality of each fitting parameter value in the constitutive equation, the applicability of the dynamic viscoelastic constitutive model to saturated sandstone is proven considering the effects of temperature, strain rate, and damage. e results reflect well the dynamic mechanical response of water-rich sandstone at different freezing temperatures, and they serve as reference for the related mechanical calculations.
Data Availability e data sets generated and analyzed during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments is work was supported by the National Natural Science Foundation of China (nos. 51404193 and 41702339), the