# PFC2D-Based Analysis of the Effect of the Axial Loading Rate on the Mechanical Properties of Backfill

Authors: Wang Kun; Fu Jian-Xin; Wang Yu
Journal: Advances in Materials Science and Engineering
 (2023)

---

## Abstract

The unloading failure process in mining engineering scenarios is similar to the loading failure process at different loading rates indoor. To clarify the relationship between the mechanical properties of backfill and the loading rate, a particle flow code 2D-based numerical simulation was performed to establish the backfill model, and tests involving five loading rates were conducted. The following results were obtained: (1) the compressive strength of the backfill body increases linearly with the increase in the loading rate. The peak strain increases in an S-shaped manner, and the modulus of elasticity first increases and then decreases. (2) The evolution of cracks is similar to that of damage energy consumption, and a smaller rate means that the curve’s inflection point arrives earlier. (3) Tensile failure is the dominant failure mode. As the rate increases, the model destruction mode transforms from single to multiple failures, and the crack distribution becomes denser. (4) The backfill body exhibits a uniform destruction form at all loading rates. However, the difference in the loading rate leads to different energy consumption growth rates and total energy consumption.


---

## Body

## 1. Introduction

With the steady transition from shallow to deep mining worldwide, filling mining has emerged as a popular technique owing to its high safety, low loss, and a low dilution rate. Given the complex environment in deep mines and to enhance the efficiency of mining, large-scale mechanized filling mining is being implemented to reduce the mining time and dilution rate and enhance the production capacity.Excavation and blasting disturbances in engineering influence the backfill support stability. Most of the existing studies on backfill have focused on its static properties, such as uniaxial compression, triaxial compression, Brazilian splitting, three-point bending moment testing, and rheological characteristics [1–5]. In addition, several scholars have examined the influence of dynamic loading and the loading rate on the mechanical properties, damage law, and energy dissipation law of the backfill. The mechanical properties of the backfill under dynamic loading reflect its ability to resist earthquakes, explosions, rock bursts, and engineering disturbances in the field. In actual scenarios, backfill deformation is attributable mainly to the unloading process. Specifically, the backfill transforms from a stable three-way stress state to a state in which a certain direction of restraint is lost. After unloading, the deformation of the backfill intensifies, and its characteristics are equivalent to those under loading in laboratory conditions. In other words, the loading rate in indoor tests can be reflected as the unloading rate under field conditions, such as the ore body recovery speed. For example, in two-step subsequent filling mining, the stress state of the first-step stope backfill body can be approximated as a three-way stress state. When the adjacent two-step ore body is mined back, the backfill is in the unloaded state. Therefore, the filling body damage caused by the two-step ore body recovery can be approximated as the loading process under different loading rates. The reasonable loading rate is determined through indoor testing, and then the corresponding ore body recovery rate is converted. These values can provide guidance for underground safety production.The existing studies on backfill can be summarized as follows: through uniaxial testing of a Hopkinson rod with four speeds, Cao et al. [6] concluded that the loading rate enhances the long-term strength of cemented paste backfill (CPB), the peak compressive strength has a power function relationship with the loading rate, and the failure form transforms from tensile-shear mixing to X-conjugate shear failure [7], and about energy, Hou et al. [8] constructed equations for energy and damage variables versus loading rates, respectively, and proposed that unlike high-strength rocks, the backfill had a critical loading rate. Zhou et al. [9] established the relationship between the damage variable and the fractal dimension versus the loading rate, respectively, and with the increase of the loading rate, the fractal dimension of CPB gradually decreases and the damage variable gradually increases. Xiu et al. [10] used the digital image correlation method and noted that at low loading rates, the difference between the peak strength of the backfill and true intensity is small. Li et al. [11] studied mechanic properties under different loading rates, and Gan et al. [12] analyzed the relation between the uniaxial compressive strength of backfill. Many theoretical predictions, laboratory tests, and numerical simulation studies have been performed on the macro meso-meso-characteristics of different tailing fills at different loading rates [13–17]. Experiments have shown that materials such as rock, concrete, and backfill exhibit different properties at different loading rates, compared with those at slow and static loading. These studies have different research directions, focusing on the mechanical properties of the loading rate, energy evolution, damage variables, etc.However, the technical level of nondestructive testing materials is low, and it is difficult to accurately characterize the failure process of the backfill in the loading stage in situ by mechanical tests at different loading rates [18]. Particle flow simulations can abstract the particles inside the material into particle units, construct the model geometry based on such abstractions, and ensure that the model mimics the real material through contact parameter assignments and iterative analyses [19, 20]. The numerical simulation results based on the particle flow code (PFC) are consistent with the field observations, and it provides several unique advantages in studying the development of internal joint fractures of loaded materials [21]. Song et al. [22] by using the PFC verified the results against experimental observations.Considering these aspects, this study was aimed at using the particle flow theory to quantitatively analyze the effects of the loading rate on the stress-strain, rupture state, energy, damage, and fracture evolution of the backfill during stress and strain cycles, rupturing, and failure by establishing a cementation model to simulate uniaxial compression experiments under different loading rates. The research results provide a reference for exploring the failure mechanism of CPB and ensuring safe production.


## 2. Materials and Methods

### 2.1. Model Construction

Most mines adopt the subsequent filling method involving two-step back mining: first, the room is mined after pillars, and then, tail fillings are used. Therefore, the particle size must match the diameter of the tail sand particles, and appropriate mesoscopic mechanical parameters must be selected.A linear-bond model is used to model the contact between the backfill particles. This model is typically used to simulate engineering materials (such as rocks, backfill, cement, and coal) considering the transmission of both force and moment on the plane of a certain size. The contact model principle is shown in Figure1. The linear parallel bond model specifies the behavior of two interfaces, an infinitesimal, linear elastic (no-tension), and frictional interface that carries force and a finite, linear elastic, and bonded interface that carries force and moment (Figure 1). The first interface is equivalent to the linear model. It does not resist relative rotation, and slip is accommodated by imposing a Coulomb limit on the shear force. The second interface is termed a parallel bond because it acts in parallel with the first interface when bonded. Specifically, when the second interface is bonded, it resists relative rotation, and its behavior is linear elastic until the strength limit is exceeded and the bond breaks, rendering the interface unbonded. When the second interface is unbonded, it carries no load. The unbonded linear parallel bond model is equivalent to the linear model.Figure 1
Schematic of the contact model.The mesoscopic mechanical parameters of the backfill determine the degree of matching between the model and the real material. Table1 lists the mesoscopic mechanical parameters of the backfill selected for the simulation of the filling parameters. The numerical simulation results are similar to the macroscopic mechanical parameters obtained by the indoor experiment, which proves that the selected mesoscopic parameters are adequately realistic.Table 1
Meso-mechanical parameters of the backfill.
TermsParametersValuesLinear groupfric0.5emod1.1kratio1.0Dashpot groupdp_nratio0.5dp_sratio0.0Parallel-bond grouppb_emod1.1pb_kratio1.0pb_ten2.0pb_coh3.0kratio regulates Poisson’s ratio of materials. pb_ten and pb_coh regulate the peak stress of materials. pb_emod regulates the elastic modulus of the material. The mechanical parameters of the contact model between the wall and backfill particles are set as follows: emod = 1.1e9 and kratio = 1.0. The constructed backfill particle model has the following dimensions: diameter × length = 50 mm × 100 mm and particle radius: 0.5–1 mm. The radial particle number of the model satisfies the size relationship specified by ISRM. Figure 2 shows the numerical model of the backfill and particle contact model.Figure 2
The two-dimensional numerical model and contact model of the backfill body.


### 2.2. Experiment

The position of the wall is shown in Figure2. The uniaxial compression test of displacement loading control is simulated by assigning the wall velocity, with loading rates of 0.1 m/s, 1 m/s, 10 m/s, 50 m/s, and 100 m/s. The loading experiment is terminated after the failure of the specimen. The mechanical parameters of models with different rates are presented in Table 2, and the stress-strain curve is shown in Figure 3.Table 2
Numerical calculation results of the backfill model.
Specimen typeLoading rate (mm/s)Peak stress (MPa)Peak strain (%)Elastic modulus (GPa)Number of cracksCrack-initial stress (MPa)Backfill0.13.34470.22691.50805850.092813.34880.22711.50855880.0928103.47060.23661.51026880.0930503.85610.28341.525310390.09641004.27480.30321.515914460.0987Figure 3
Uniaxial compression stress-strain curve of the backfill model.


## 2.1. Model Construction

Most mines adopt the subsequent filling method involving two-step back mining: first, the room is mined after pillars, and then, tail fillings are used. Therefore, the particle size must match the diameter of the tail sand particles, and appropriate mesoscopic mechanical parameters must be selected.A linear-bond model is used to model the contact between the backfill particles. This model is typically used to simulate engineering materials (such as rocks, backfill, cement, and coal) considering the transmission of both force and moment on the plane of a certain size. The contact model principle is shown in Figure1. The linear parallel bond model specifies the behavior of two interfaces, an infinitesimal, linear elastic (no-tension), and frictional interface that carries force and a finite, linear elastic, and bonded interface that carries force and moment (Figure 1). The first interface is equivalent to the linear model. It does not resist relative rotation, and slip is accommodated by imposing a Coulomb limit on the shear force. The second interface is termed a parallel bond because it acts in parallel with the first interface when bonded. Specifically, when the second interface is bonded, it resists relative rotation, and its behavior is linear elastic until the strength limit is exceeded and the bond breaks, rendering the interface unbonded. When the second interface is unbonded, it carries no load. The unbonded linear parallel bond model is equivalent to the linear model.Figure 1
Schematic of the contact model.The mesoscopic mechanical parameters of the backfill determine the degree of matching between the model and the real material. Table1 lists the mesoscopic mechanical parameters of the backfill selected for the simulation of the filling parameters. The numerical simulation results are similar to the macroscopic mechanical parameters obtained by the indoor experiment, which proves that the selected mesoscopic parameters are adequately realistic.Table 1
Meso-mechanical parameters of the backfill.
TermsParametersValuesLinear groupfric0.5emod1.1kratio1.0Dashpot groupdp_nratio0.5dp_sratio0.0Parallel-bond grouppb_emod1.1pb_kratio1.0pb_ten2.0pb_coh3.0kratio regulates Poisson’s ratio of materials. pb_ten and pb_coh regulate the peak stress of materials. pb_emod regulates the elastic modulus of the material. The mechanical parameters of the contact model between the wall and backfill particles are set as follows: emod = 1.1e9 and kratio = 1.0. The constructed backfill particle model has the following dimensions: diameter × length = 50 mm × 100 mm and particle radius: 0.5–1 mm. The radial particle number of the model satisfies the size relationship specified by ISRM. Figure 2 shows the numerical model of the backfill and particle contact model.Figure 2
The two-dimensional numerical model and contact model of the backfill body.


## 2.2. Experiment

The position of the wall is shown in Figure2. The uniaxial compression test of displacement loading control is simulated by assigning the wall velocity, with loading rates of 0.1 m/s, 1 m/s, 10 m/s, 50 m/s, and 100 m/s. The loading experiment is terminated after the failure of the specimen. The mechanical parameters of models with different rates are presented in Table 2, and the stress-strain curve is shown in Figure 3.Table 2
Numerical calculation results of the backfill model.
Specimen typeLoading rate (mm/s)Peak stress (MPa)Peak strain (%)Elastic modulus (GPa)Number of cracksCrack-initial stress (MPa)Backfill0.13.34470.22691.50805850.092813.34880.22711.50855880.0928103.47060.23661.51026880.0930503.85610.28341.525310390.09641004.27480.30321.515914460.0987Figure 3
Uniaxial compression stress-strain curve of the backfill model.


## 3. Numerical Calculation Results and Analysis

Figure3 shows that the variational law of the stress-strain curve of the backfill is not affected by the loading rate. The curve can be divided into four stages:(1)
Compaction stage: the internal cracks of the backfill are filled, and the pores are compressed(2)
Linear elastic stage: the stress-strain curve is approximately straight, and the fracture develops steadily(3)
Unstable rupture development stage: the rupture becomes unstable, and the backfill exhibits plasticity(4)
Post-failure stage: after reaching the peak stress, the backfill rapidly disintegrates, and the fissure penetrates the macroscopic fracture surfaceAs the rate increases, the area of the complete stress-strain curve gradually increases, the peak strain shifts backward, and the peak stress increases.


### 3.1. Effect of Peak Stress and Strain

Figure4 shows that the peak stress and loading rate exhibit a polynomial relationship, with R2 = 0.9997. When the rate increases from 0.1 mm/s to 1 mm/s, 10 mm/s, 50 mm/s, and 100 mm/s, the peak stress increases by 0.123%, 3.764%, 15.29%, and 27.868%, respectively. When the rate increases from 0.1 mm/s to 1 mm/s and 50 mm/s, the uniaxial compressive strength (UCS) increases by 0.122% and 15.29%, respectively, indicating that the UCS is more sensitive to high strain rates. The results show that the loading rate increases the UCS of the CPB. When the loading rate increases, the time required for the material to respond to the strain decreases. Consequently, the strain becomes more localized, and additional stress is required for the sample to be destroyed. Moreover, as the loading rate increases, the development of microcracks inside the sample is limited (Table 3). Therefore, the force area of the sample increases, which increases the sample strength. Specifically, as the loading rate increases, the strength of the backfill body first increases and then decreases. The strength of the backfill body is maximized at a critical rate [11, 12]. The curve is slightly convex, and it can be estimated that the critical loading rate is between 50 mm/s and 100 mm/s. The peak stress starts to decrease when the rate exceeds the critical load rate.Figure 4
Peak stress change curve of the backfill model.Table 3
Backfill destruction mode at different loading rates.
Axial strain (mm/s)2e − 32.5e − 33e − 33.5e − 34e − 3Analysis of the law of destruction0.1The model fails mainly through tensile failure. Over time, the tensile strain reaches a peak, and the specimen is violently destroyed1The compressive strength of the model increases, and the crack develops slowly compared with that at 0.1 mm/s. After reaching the peak strain, the crack rapidly develops10Tensile failure occurs, and the cracking stress increases. After the peak strain is reached, the crack rapidly develops50The crack distribution gradually densifies, no longer limited to the axial direction. The destruction mode is tensile failure accompanied by shear fracture100Tension failure is accompanied by secondary fracture. The destruction is more severe at high speeds, and the cracks are densely distributedFigure5 shows that the peak strain increases first rapidly and then slowly, and the model exhibits plastic characteristics. When the loading rate increases from 0.1 mm/s to 1 mm/s, 10 mm/s, 50 mm/s, and 100 mm/s, the strain increases by 0.881%, 4.275%, 24.901%, and 33.627%, respectively. The curve is S-shaped. The growth rate first increases and then decreases. The sigmoidal curve fits with the Boltzmann function curve, with R2=0.999. Furthermore, as the strain rate increases, the peak strain growth retards. This finding indicates that a critical value exists for the strain and that it does not increase infinitely. After the rate increases to a certain value, the load inhibits subsequent fracture growth, and thus, the strain growth decelerates.Figure 5
Peak strain change curve of the backfill model.


### 3.2. Elastic Modulus Analysis

Figure6 shows that the elastic modulus change curve is consistent with the Gaussian fit curve, with the complex correlation coefficient being R2=0.999. The fitted curve agrees with the test data. As the loading rate increases, the value first increases to a peak and then gradually decreases.Figure 6
Elastic modulus change curve.A critical loading intensity exists for the peak stress, and the peak stress decreases after reaching a peak as the loading rate increases. Nevertheless, the peak strain continues to increase, and the elastic modulus of the backfill decreases. The critical loading rate is between 50 mm/s and 100 mm/s, consistent with the abovementioned estimate. The modulus of elasticity decreases after reaching a peak value as the loading rate increases. The peak point location is experimentally obtained to analyze the optimal backfill physical strength.


### 3.3. Analysis of Crack Evolution Laws

#### 3.3.1. Analysis of Number of Cracks and Crack Stress

Figure7 shows that the curve of the number of cracks exhibits a satisfactory fitting relationship with the linear fitting curve, and the composite correlation coefficient is R2 = 0.999. As the loading rate increases, the number of cracks linearly increases. At low rates, the number of cracks increases by a limited value (by 17.606% from 0.1 mm/s to 10 mm/s), and it increases by a large extent (147.18%) as the rate increases to 100 mm/s. In other words, the damage suffered by the backfill at high strain rates is more severe. A higher strain rate corresponds to greater energy accumulated in the backfill, and as the model is destroyed, a considerable amount of energy is released. The energy release leads to the rapid destruction of the backfill and a significant increase in the number of cracks, especially at peak stress.Figure 7
Curve of the number of crack variations.Figure8 shows that the cracking stress increases with the increase in the loading rate, and the speed slightly decreases. For linear and polynomial curve fitting, R2=0.983 and R2=0.991, respectively. In other words, polynomial fitting is more suitable for the data. The increased strain rate reduces the time available to respond to the strain. Consequently, the strain becomes more localized, causing the model to crack with a greater loading force. The high loading rate inhibits the generation of cracks to a certain extent, and the ductility of the model increases. However, the slowdown of the curve indicates that the cracking stress is subjected to certain limits as the rate increases.Figure 8
Curve of the cracking stress variation.


#### 3.3.2. Law of Crack Evolution

Figure9 shows that the increase in the number of cracks is gradual in the early stage. It increases sharply and then gradually in an S-shaped curve with two inflection points. A lower loading rate corresponds to an earlier arrival of the first inflection point of the crack evolution curve. This finding indicates that a more rapid growth of the crack corresponds to a higher probability of destruction of the backfill. Moreover, a higher loading rate corresponds to earlier cracking and more crack generation. The later the second inflection point arrives, the more complete the fracture.Figure 9
Graph of the evolution of cracks in the backfill.Figure10 shows the inflection point at which the number of cracks sharply increases. This point appears after the peak strain is reached. The second inflection point corresponds to the 70% peak stress after destruction. At a low rate, the compressive strength of the model is low. Early destruction occurs, strain is released, rapid expansion and cracking occur, and the number of cracks sharply increases. Therefore, the inflection point appears earlier. At larger rates, the compressive strength increases and the peak strain increases. Because the energy accumulated in the model is higher, the number of cracks increases more than that at low loading rates after failure occurs.Figure 10
Crack evolution and stress variation curves.


### 3.4. Analysis of the Law of Destruction

The backfill destruction images at strains of 2e − 3, 2.5e − 3, 3e − 3, 3.5e − 3, and 4e − 3 are examined to analyze the development of fractures inside the backfill. In the images, red and yellow mark tensile and shear cracks, respectively. The backfill specimen fails mainly through tensile failure.The failure form is mainly tensile failure, and this is likely because precompression is performed in advance, which is equivalent to lateral pressure unloading. Owing to the effect of Poisson’s ratio, when the lateral tensile stress exceeds the tensile limit of the model, the fracture failure form gradually changes. The number of cracks increases and interconnects through the weak surface, and finally, macroscopic tensile failure is generated.At low loading rates, the crack distribution pertains mainly to tension failure close to the axis. The ductility of the backfill increases at high loading rates. The model begins to crack at higher strains, and the number of initial cracks decreases. In contrast, at high loading rates, cracks develop rapidly after the peak strain is reached, and the number of cracks is considerably higher than that at low loading rates. Furthermore, the distribution of cracks is no longer limited to the vicinity of the axis, and the cracks in the model are equally dense. Consequently, fragmentation occurs along with more extensive damage. This phenomenon is attributable to the increased loading rate. Therefore, as the energy accumulated under the unit strain intensifies, the energy in the total volume of the backfill rises, and when the peak strain is reached, the backfill violently disintegrates.


## 3.1. Effect of Peak Stress and Strain

Figure4 shows that the peak stress and loading rate exhibit a polynomial relationship, with R2 = 0.9997. When the rate increases from 0.1 mm/s to 1 mm/s, 10 mm/s, 50 mm/s, and 100 mm/s, the peak stress increases by 0.123%, 3.764%, 15.29%, and 27.868%, respectively. When the rate increases from 0.1 mm/s to 1 mm/s and 50 mm/s, the uniaxial compressive strength (UCS) increases by 0.122% and 15.29%, respectively, indicating that the UCS is more sensitive to high strain rates. The results show that the loading rate increases the UCS of the CPB. When the loading rate increases, the time required for the material to respond to the strain decreases. Consequently, the strain becomes more localized, and additional stress is required for the sample to be destroyed. Moreover, as the loading rate increases, the development of microcracks inside the sample is limited (Table 3). Therefore, the force area of the sample increases, which increases the sample strength. Specifically, as the loading rate increases, the strength of the backfill body first increases and then decreases. The strength of the backfill body is maximized at a critical rate [11, 12]. The curve is slightly convex, and it can be estimated that the critical loading rate is between 50 mm/s and 100 mm/s. The peak stress starts to decrease when the rate exceeds the critical load rate.Figure 4
Peak stress change curve of the backfill model.Table 3
Backfill destruction mode at different loading rates.
Axial strain (mm/s)2e − 32.5e − 33e − 33.5e − 34e − 3Analysis of the law of destruction0.1The model fails mainly through tensile failure. Over time, the tensile strain reaches a peak, and the specimen is violently destroyed1The compressive strength of the model increases, and the crack develops slowly compared with that at 0.1 mm/s. After reaching the peak strain, the crack rapidly develops10Tensile failure occurs, and the cracking stress increases. After the peak strain is reached, the crack rapidly develops50The crack distribution gradually densifies, no longer limited to the axial direction. The destruction mode is tensile failure accompanied by shear fracture100Tension failure is accompanied by secondary fracture. The destruction is more severe at high speeds, and the cracks are densely distributedFigure5 shows that the peak strain increases first rapidly and then slowly, and the model exhibits plastic characteristics. When the loading rate increases from 0.1 mm/s to 1 mm/s, 10 mm/s, 50 mm/s, and 100 mm/s, the strain increases by 0.881%, 4.275%, 24.901%, and 33.627%, respectively. The curve is S-shaped. The growth rate first increases and then decreases. The sigmoidal curve fits with the Boltzmann function curve, with R2=0.999. Furthermore, as the strain rate increases, the peak strain growth retards. This finding indicates that a critical value exists for the strain and that it does not increase infinitely. After the rate increases to a certain value, the load inhibits subsequent fracture growth, and thus, the strain growth decelerates.Figure 5
Peak strain change curve of the backfill model.


## 3.2. Elastic Modulus Analysis

Figure6 shows that the elastic modulus change curve is consistent with the Gaussian fit curve, with the complex correlation coefficient being R2=0.999. The fitted curve agrees with the test data. As the loading rate increases, the value first increases to a peak and then gradually decreases.Figure 6
Elastic modulus change curve.A critical loading intensity exists for the peak stress, and the peak stress decreases after reaching a peak as the loading rate increases. Nevertheless, the peak strain continues to increase, and the elastic modulus of the backfill decreases. The critical loading rate is between 50 mm/s and 100 mm/s, consistent with the abovementioned estimate. The modulus of elasticity decreases after reaching a peak value as the loading rate increases. The peak point location is experimentally obtained to analyze the optimal backfill physical strength.


## 3.3. Analysis of Crack Evolution Laws

### 3.3.1. Analysis of Number of Cracks and Crack Stress

Figure7 shows that the curve of the number of cracks exhibits a satisfactory fitting relationship with the linear fitting curve, and the composite correlation coefficient is R2 = 0.999. As the loading rate increases, the number of cracks linearly increases. At low rates, the number of cracks increases by a limited value (by 17.606% from 0.1 mm/s to 10 mm/s), and it increases by a large extent (147.18%) as the rate increases to 100 mm/s. In other words, the damage suffered by the backfill at high strain rates is more severe. A higher strain rate corresponds to greater energy accumulated in the backfill, and as the model is destroyed, a considerable amount of energy is released. The energy release leads to the rapid destruction of the backfill and a significant increase in the number of cracks, especially at peak stress.Figure 7
Curve of the number of crack variations.Figure8 shows that the cracking stress increases with the increase in the loading rate, and the speed slightly decreases. For linear and polynomial curve fitting, R2=0.983 and R2=0.991, respectively. In other words, polynomial fitting is more suitable for the data. The increased strain rate reduces the time available to respond to the strain. Consequently, the strain becomes more localized, causing the model to crack with a greater loading force. The high loading rate inhibits the generation of cracks to a certain extent, and the ductility of the model increases. However, the slowdown of the curve indicates that the cracking stress is subjected to certain limits as the rate increases.Figure 8
Curve of the cracking stress variation.


### 3.3.2. Law of Crack Evolution

Figure9 shows that the increase in the number of cracks is gradual in the early stage. It increases sharply and then gradually in an S-shaped curve with two inflection points. A lower loading rate corresponds to an earlier arrival of the first inflection point of the crack evolution curve. This finding indicates that a more rapid growth of the crack corresponds to a higher probability of destruction of the backfill. Moreover, a higher loading rate corresponds to earlier cracking and more crack generation. The later the second inflection point arrives, the more complete the fracture.Figure 9
Graph of the evolution of cracks in the backfill.Figure10 shows the inflection point at which the number of cracks sharply increases. This point appears after the peak strain is reached. The second inflection point corresponds to the 70% peak stress after destruction. At a low rate, the compressive strength of the model is low. Early destruction occurs, strain is released, rapid expansion and cracking occur, and the number of cracks sharply increases. Therefore, the inflection point appears earlier. At larger rates, the compressive strength increases and the peak strain increases. Because the energy accumulated in the model is higher, the number of cracks increases more than that at low loading rates after failure occurs.Figure 10
Crack evolution and stress variation curves.


## 3.3.1. Analysis of Number of Cracks and Crack Stress

Figure7 shows that the curve of the number of cracks exhibits a satisfactory fitting relationship with the linear fitting curve, and the composite correlation coefficient is R2 = 0.999. As the loading rate increases, the number of cracks linearly increases. At low rates, the number of cracks increases by a limited value (by 17.606% from 0.1 mm/s to 10 mm/s), and it increases by a large extent (147.18%) as the rate increases to 100 mm/s. In other words, the damage suffered by the backfill at high strain rates is more severe. A higher strain rate corresponds to greater energy accumulated in the backfill, and as the model is destroyed, a considerable amount of energy is released. The energy release leads to the rapid destruction of the backfill and a significant increase in the number of cracks, especially at peak stress.Figure 7
Curve of the number of crack variations.Figure8 shows that the cracking stress increases with the increase in the loading rate, and the speed slightly decreases. For linear and polynomial curve fitting, R2=0.983 and R2=0.991, respectively. In other words, polynomial fitting is more suitable for the data. The increased strain rate reduces the time available to respond to the strain. Consequently, the strain becomes more localized, causing the model to crack with a greater loading force. The high loading rate inhibits the generation of cracks to a certain extent, and the ductility of the model increases. However, the slowdown of the curve indicates that the cracking stress is subjected to certain limits as the rate increases.Figure 8
Curve of the cracking stress variation.


## 3.3.2. Law of Crack Evolution

Figure9 shows that the increase in the number of cracks is gradual in the early stage. It increases sharply and then gradually in an S-shaped curve with two inflection points. A lower loading rate corresponds to an earlier arrival of the first inflection point of the crack evolution curve. This finding indicates that a more rapid growth of the crack corresponds to a higher probability of destruction of the backfill. Moreover, a higher loading rate corresponds to earlier cracking and more crack generation. The later the second inflection point arrives, the more complete the fracture.Figure 9
Graph of the evolution of cracks in the backfill.Figure10 shows the inflection point at which the number of cracks sharply increases. This point appears after the peak strain is reached. The second inflection point corresponds to the 70% peak stress after destruction. At a low rate, the compressive strength of the model is low. Early destruction occurs, strain is released, rapid expansion and cracking occur, and the number of cracks sharply increases. Therefore, the inflection point appears earlier. At larger rates, the compressive strength increases and the peak strain increases. Because the energy accumulated in the model is higher, the number of cracks increases more than that at low loading rates after failure occurs.Figure 10
Crack evolution and stress variation curves.


## 3.4. Analysis of the Law of Destruction

The backfill destruction images at strains of 2e − 3, 2.5e − 3, 3e − 3, 3.5e − 3, and 4e − 3 are examined to analyze the development of fractures inside the backfill. In the images, red and yellow mark tensile and shear cracks, respectively. The backfill specimen fails mainly through tensile failure.The failure form is mainly tensile failure, and this is likely because precompression is performed in advance, which is equivalent to lateral pressure unloading. Owing to the effect of Poisson’s ratio, when the lateral tensile stress exceeds the tensile limit of the model, the fracture failure form gradually changes. The number of cracks increases and interconnects through the weak surface, and finally, macroscopic tensile failure is generated.At low loading rates, the crack distribution pertains mainly to tension failure close to the axis. The ductility of the backfill increases at high loading rates. The model begins to crack at higher strains, and the number of initial cracks decreases. In contrast, at high loading rates, cracks develop rapidly after the peak strain is reached, and the number of cracks is considerably higher than that at low loading rates. Furthermore, the distribution of cracks is no longer limited to the vicinity of the axis, and the cracks in the model are equally dense. Consequently, fragmentation occurs along with more extensive damage. This phenomenon is attributable to the increased loading rate. Therefore, as the energy accumulated under the unit strain intensifies, the energy in the total volume of the backfill rises, and when the peak strain is reached, the backfill violently disintegrates.


## 4. Damage Variable and Energy Analysis

### 4.1. Analysis of Damage Variables

#### 4.1.1. Method for Defining Damage Variables

According to the damage model and method of determining damage under one-way load on the filling material [23], the degree of fracture damage of the filling material is expressed by variable D (i.e., the damage variable) during loading [24]:(1)D=1−exp−εαm,where D is the damage variable, ε is the strain (m), α is the shape parameter (m), and α is a nonnegative number. A novel method is used to determine the parameters of the damage constitutive model. The Weibull distribution parameters in (1) are calculated as follows in the uniaxial case:(2)m=1lnEεc/σc,(3)α=εc1/m1/m,where E is the initial deformation modulus, εc is the peak strain, and σc is the peak stress. Table 4 lists the parameters obtained using (2) and (3).Table 4
Weibull distribution parameters.
Loading rate (mm/s)Initial modulus (GPa)Peak stress (MPa)Peak strain (%)mα/10−30.11.507993.3450.226944.1502.47211.508493.3490.227144.11582.47455101.510233.4700.236634.1202.624501.525333.8560.28348.7513.6311001.515874.2750.303213.8073.667Substituting the data in Table4 into equation (1), the damage variable change curve is obtained as shown in Figure 11.Figure 11
Damage variables and strain rates at different loading rates.The damage to the backfill at different loading rates is of the same type. The damage variable is positively correlated with crack development and exhibits an S-curve. First, the variable gradually increases, then it sharply increases after reaching the inflection point, and the growth finally slows down. The number of cracks inside the backfill increases with the increase in the damage variable. At large damage variables, the internal crack distribution of the large-rate model is uniform. As the strain energy is released, the model is destroyed more violently, and the large block rate is small.According to the curve analogy, the damage curves at the loading rates of 0.1 mm/s and 1 mm/s are consistent. The difference in the damage curves at low rates is not significant, and the last three curves gradually move backward with the increase in the loading rate, consistent with the trend of the peak strain. At low strain rates, the strain required for the backfill to occur from the inflection point to complete destruction is reduced, and the ductility of the backfill increases at high strain rates. The inflection point of the damage variable is the same as that of the crack growth curve. After continuous application of the peak stress, the damage variable first increases sharply and then gradually, and the number of cracks on the unit axial cross section increases, consistent with the degree of damage to the backfill.The backfill loaded at 100 mm/s is completely destroyed after the inflection point more rapidly than that at a loading rate of 50 mm/s. Moreover, the elastic modulus of the model is lower than that at 50 mm/s, which highlights that the critical loading speed lies between 50 mm/s and 100 mm/s. The peak stress is achieved at the critical loading rate, and then the stress decreases, corresponding to a decrease in the modulus of elasticity.


### 4.2. Energy Analysis

Assuming no heat exchange occurs between the filling material and external environment, the energy of the material element in a complex stress state satisfies the following relationship, according to the first law of thermodynamics [25, 26], where W is the total energy generated by the external force work, MJ/m3; Wd is the dissipation energy of the unit, MJ/m3; We is the released elastic strain energy, MJ/m3:(4)W=Wd+We.The energy absorbed by the specimen under uniaxial conditions is the total area of the stress-strain curve, which can be solved through integration. Owing to the uncertainty of the curve equation, the surface is infinitely divided into rectangular bars by the definite integral method:(5)W=∫σ1dε1=∑i=1nσi+1εi+1−εi,where σi (MPa) is any point on the stress-strain curve and σ1andε1 are the axial stress and strain corresponding to that point, respectively. When i = 0, then σi = εi=0.The elastic strain energyWe is(6)We=12σ1ε1e=σ122E,where Ei is the modulus of relief elasticity. For the convenience of calculation, the initial elastic modulus E is typically used instead of Ei. The rationality of this approach has been discussed [27] and is thus not repeated here. The dissipation energy Wd is(7)Wd=W−We,∫σ1dε1−12σ1ε1e=Wd.This term can take into account the energy inside the model during loading. Figure12 shows the energy evolution curves under different loads.Figure 12
Schematic of energy evolution at different loading rates.The energy initially absorbed by the backfill is converted to internal strain energy. A higher loading rate corresponds to greater total energy absorbed and greater corresponding strain energy. After passing through the elastic stage, unstable rupturing gradually occurs inside the sample. The dissipation energy gradually increases, and finally, the strain energy is sharply released after the peak stress is attained. The released energy is converted to dissipation energy, and the dissipation energy increases. The evolution curves of the energy at different rates are qualitatively similar, although certain quantitative differences exist.


#### 4.2.1. Effect of the Loading Rate on the Energy

Figure13 shows the energy evolution curve under different loading rates of the backfill model. In Figure 13(a), the strain energy change curve is similar to the full stress-strain curve. The strain energy gradually rises through the elastic stage as the strain increases, peaks, and then rapidly decreases. The evolution form of the elastic energy does not change with the loading rate but is related to the magnitude of the peak stress. As the loading rate increases, the compressive strength and ductility of the backfill increase, resulting in an increase in the overall strain energy. In Figure 13(b), the law of dissipation energy evolution is different from that of elastic energy, which increases with an increase in the rate and strain. When the loading rate is low, the microfissure inside the backfill model expands to produce dissipation energy. When the rate is high, the microcracks inside the backfill cannot easily expand, and the energy is stored in the form of strain energy inside the material, resulting in a reduced proportion of dissipative energy in the initial stage.Figure 13
Schematic of energy evolution at different loading rates. (a) Elastic energy evolution. (b) Dissipative energy evolution map.
(a)(b)


## 4.1. Analysis of Damage Variables

### 4.1.1. Method for Defining Damage Variables

According to the damage model and method of determining damage under one-way load on the filling material [23], the degree of fracture damage of the filling material is expressed by variable D (i.e., the damage variable) during loading [24]:(1)D=1−exp−εαm,where D is the damage variable, ε is the strain (m), α is the shape parameter (m), and α is a nonnegative number. A novel method is used to determine the parameters of the damage constitutive model. The Weibull distribution parameters in (1) are calculated as follows in the uniaxial case:(2)m=1lnEεc/σc,(3)α=εc1/m1/m,where E is the initial deformation modulus, εc is the peak strain, and σc is the peak stress. Table 4 lists the parameters obtained using (2) and (3).Table 4
Weibull distribution parameters.
Loading rate (mm/s)Initial modulus (GPa)Peak stress (MPa)Peak strain (%)mα/10−30.11.507993.3450.226944.1502.47211.508493.3490.227144.11582.47455101.510233.4700.236634.1202.624501.525333.8560.28348.7513.6311001.515874.2750.303213.8073.667Substituting the data in Table4 into equation (1), the damage variable change curve is obtained as shown in Figure 11.Figure 11
Damage variables and strain rates at different loading rates.The damage to the backfill at different loading rates is of the same type. The damage variable is positively correlated with crack development and exhibits an S-curve. First, the variable gradually increases, then it sharply increases after reaching the inflection point, and the growth finally slows down. The number of cracks inside the backfill increases with the increase in the damage variable. At large damage variables, the internal crack distribution of the large-rate model is uniform. As the strain energy is released, the model is destroyed more violently, and the large block rate is small.According to the curve analogy, the damage curves at the loading rates of 0.1 mm/s and 1 mm/s are consistent. The difference in the damage curves at low rates is not significant, and the last three curves gradually move backward with the increase in the loading rate, consistent with the trend of the peak strain. At low strain rates, the strain required for the backfill to occur from the inflection point to complete destruction is reduced, and the ductility of the backfill increases at high strain rates. The inflection point of the damage variable is the same as that of the crack growth curve. After continuous application of the peak stress, the damage variable first increases sharply and then gradually, and the number of cracks on the unit axial cross section increases, consistent with the degree of damage to the backfill.The backfill loaded at 100 mm/s is completely destroyed after the inflection point more rapidly than that at a loading rate of 50 mm/s. Moreover, the elastic modulus of the model is lower than that at 50 mm/s, which highlights that the critical loading speed lies between 50 mm/s and 100 mm/s. The peak stress is achieved at the critical loading rate, and then the stress decreases, corresponding to a decrease in the modulus of elasticity.


## 4.1.1. Method for Defining Damage Variables

According to the damage model and method of determining damage under one-way load on the filling material [23], the degree of fracture damage of the filling material is expressed by variable D (i.e., the damage variable) during loading [24]:(1)D=1−exp−εαm,where D is the damage variable, ε is the strain (m), α is the shape parameter (m), and α is a nonnegative number. A novel method is used to determine the parameters of the damage constitutive model. The Weibull distribution parameters in (1) are calculated as follows in the uniaxial case:(2)m=1lnEεc/σc,(3)α=εc1/m1/m,where E is the initial deformation modulus, εc is the peak strain, and σc is the peak stress. Table 4 lists the parameters obtained using (2) and (3).Table 4
Weibull distribution parameters.
Loading rate (mm/s)Initial modulus (GPa)Peak stress (MPa)Peak strain (%)mα/10−30.11.507993.3450.226944.1502.47211.508493.3490.227144.11582.47455101.510233.4700.236634.1202.624501.525333.8560.28348.7513.6311001.515874.2750.303213.8073.667Substituting the data in Table4 into equation (1), the damage variable change curve is obtained as shown in Figure 11.Figure 11
Damage variables and strain rates at different loading rates.The damage to the backfill at different loading rates is of the same type. The damage variable is positively correlated with crack development and exhibits an S-curve. First, the variable gradually increases, then it sharply increases after reaching the inflection point, and the growth finally slows down. The number of cracks inside the backfill increases with the increase in the damage variable. At large damage variables, the internal crack distribution of the large-rate model is uniform. As the strain energy is released, the model is destroyed more violently, and the large block rate is small.According to the curve analogy, the damage curves at the loading rates of 0.1 mm/s and 1 mm/s are consistent. The difference in the damage curves at low rates is not significant, and the last three curves gradually move backward with the increase in the loading rate, consistent with the trend of the peak strain. At low strain rates, the strain required for the backfill to occur from the inflection point to complete destruction is reduced, and the ductility of the backfill increases at high strain rates. The inflection point of the damage variable is the same as that of the crack growth curve. After continuous application of the peak stress, the damage variable first increases sharply and then gradually, and the number of cracks on the unit axial cross section increases, consistent with the degree of damage to the backfill.The backfill loaded at 100 mm/s is completely destroyed after the inflection point more rapidly than that at a loading rate of 50 mm/s. Moreover, the elastic modulus of the model is lower than that at 50 mm/s, which highlights that the critical loading speed lies between 50 mm/s and 100 mm/s. The peak stress is achieved at the critical loading rate, and then the stress decreases, corresponding to a decrease in the modulus of elasticity.


## 4.2. Energy Analysis

Assuming no heat exchange occurs between the filling material and external environment, the energy of the material element in a complex stress state satisfies the following relationship, according to the first law of thermodynamics [25, 26], where W is the total energy generated by the external force work, MJ/m3; Wd is the dissipation energy of the unit, MJ/m3; We is the released elastic strain energy, MJ/m3:(4)W=Wd+We.The energy absorbed by the specimen under uniaxial conditions is the total area of the stress-strain curve, which can be solved through integration. Owing to the uncertainty of the curve equation, the surface is infinitely divided into rectangular bars by the definite integral method:(5)W=∫σ1dε1=∑i=1nσi+1εi+1−εi,where σi (MPa) is any point on the stress-strain curve and σ1andε1 are the axial stress and strain corresponding to that point, respectively. When i = 0, then σi = εi=0.The elastic strain energyWe is(6)We=12σ1ε1e=σ122E,where Ei is the modulus of relief elasticity. For the convenience of calculation, the initial elastic modulus E is typically used instead of Ei. The rationality of this approach has been discussed [27] and is thus not repeated here. The dissipation energy Wd is(7)Wd=W−We,∫σ1dε1−12σ1ε1e=Wd.This term can take into account the energy inside the model during loading. Figure12 shows the energy evolution curves under different loads.Figure 12
Schematic of energy evolution at different loading rates.The energy initially absorbed by the backfill is converted to internal strain energy. A higher loading rate corresponds to greater total energy absorbed and greater corresponding strain energy. After passing through the elastic stage, unstable rupturing gradually occurs inside the sample. The dissipation energy gradually increases, and finally, the strain energy is sharply released after the peak stress is attained. The released energy is converted to dissipation energy, and the dissipation energy increases. The evolution curves of the energy at different rates are qualitatively similar, although certain quantitative differences exist.


### 4.2.1. Effect of the Loading Rate on the Energy

Figure13 shows the energy evolution curve under different loading rates of the backfill model. In Figure 13(a), the strain energy change curve is similar to the full stress-strain curve. The strain energy gradually rises through the elastic stage as the strain increases, peaks, and then rapidly decreases. The evolution form of the elastic energy does not change with the loading rate but is related to the magnitude of the peak stress. As the loading rate increases, the compressive strength and ductility of the backfill increase, resulting in an increase in the overall strain energy. In Figure 13(b), the law of dissipation energy evolution is different from that of elastic energy, which increases with an increase in the rate and strain. When the loading rate is low, the microfissure inside the backfill model expands to produce dissipation energy. When the rate is high, the microcracks inside the backfill cannot easily expand, and the energy is stored in the form of strain energy inside the material, resulting in a reduced proportion of dissipative energy in the initial stage.Figure 13
Schematic of energy evolution at different loading rates. (a) Elastic energy evolution. (b) Dissipative energy evolution map.
(a)(b)


## 4.2.1. Effect of the Loading Rate on the Energy

Figure13 shows the energy evolution curve under different loading rates of the backfill model. In Figure 13(a), the strain energy change curve is similar to the full stress-strain curve. The strain energy gradually rises through the elastic stage as the strain increases, peaks, and then rapidly decreases. The evolution form of the elastic energy does not change with the loading rate but is related to the magnitude of the peak stress. As the loading rate increases, the compressive strength and ductility of the backfill increase, resulting in an increase in the overall strain energy. In Figure 13(b), the law of dissipation energy evolution is different from that of elastic energy, which increases with an increase in the rate and strain. When the loading rate is low, the microfissure inside the backfill model expands to produce dissipation energy. When the rate is high, the microcracks inside the backfill cannot easily expand, and the energy is stored in the form of strain energy inside the material, resulting in a reduced proportion of dissipative energy in the initial stage.Figure 13
Schematic of energy evolution at different loading rates. (a) Elastic energy evolution. (b) Dissipative energy evolution map.
(a)(b)


## 5. Conclusion

(1)
According to the UCS simulation test of the backfill model at different loading rates, the increase in the loading rate increases the UCS. The UCS at different loading rates satisfies the polynomial function relationship, the peak strain increases in an S-shaped curve, the elastic modulus is distributed along the Gaussian curve, and a critical loading rate exists. These findings can facilitate the control of the unloading process in practical engineering.(2)
The crack evolution law at different loading rates is similar, and the number of cracks increases linearly with the loading rate. A higher rate corresponds to an earlier cracking time and more intense destruction. As the loading rate increases, the main form of model failure gradually changes from single to multiple failures, and the cracks are no longer limited to the vicinity of the axis. These findings can provide a reference for engineering support.(3)
The variation in the damage variable of the backfill under different loading rates is consistent with that of the number of cracks. The damage formula based on the material is suitable for numerical simulation, and increasing the loading rate within a certain range can help improve the deformation ability of the backfill.(4)
The elastic energy is mainly related to the peak stress-strain relationship. The elastic performance and total energy of the backfill decrease more rapidly after failure at low loading rates. The internal energy is higher as the loading rate increases. The degree of deformation is greater during failure, and the increased range of dissipated energy is consistent with that of the damage variable.



---

*Source: 4398285-2023-02-22.xml*