Curing Stress Influences the Mechanical Characteristics of Cemented Paste Backfill and Its Damage Constitutive Model

Abstract

As mechanical characteristics are one of the most important indexes that represent the backfill effect of CPB, curing stress is less considered, thus, establishing a damage constitutive model under the effect of curing stress has great significance for the stability of CPB. Firstly, a multifield coupling curing experiment was developed, and a uniaxial pressure testing experiment was used to test the mechanical parameters. Then, the evolution rule of mechanical characteristics of CPB, considering the effect of curing stress, was analyzed. Secondly, combined with elastic mechanics and damage mechanics theory, a damage constitutive model of CPB was explored. Thirdly, based on the laboratory results, an established damage constitutive model was verified. The results indicate that uniaxial compressive strength (UCS) of the CPB was significantly improved because of increasing curing stress and was also influenced by curing age. It was also shown that there existed four stages for the stress-strain curve of the CPB specimens. Moreover, the stress-strain curves of the model and the experiment’s results were the same. There were also good validity and rationality for the established two-stage damage constitutive model, which can provide a good reference for engineering applications of CPB.

1. Introduction

As lots of mined-out areas have been generated in the process of exploitation and utilization of mineral resources [1] and the tailings present on the surface can easily cause environmental pollution and ecological safety problems [2], there is a major hidden threat in mine safety. With the development of backfill mining technology, the tailings which have been disposed of on the surface can be filled in the mined-out stopes. On the one hand, the tailings which are disposed of in the tailing dams are decreased, and on the other hand, the safety of the underground stope is improved, therefore, backfill mining technology has important significance for the safety of mined-out areas, and tailings can be disposed of well [3,4]. Meanwhile, cemented paste backfill (CPB) technology brings the positive characteristics of safety, high efficiency and economic and environmental protection, and its application is increasingly wide [5,6].

The mass concentration of CPB is in the range of 70–85%, including the deep cone thickening, mixing, transportation and curing processes [7,8,9,10]. The direct index that reflects the effect of CPB is the UCS of the CPB material, which is important to the stability of mines. Mechanical characteristics and the damage constitutive model of CPB have been studied deeply by scholars at home and abroad. In terms of UCS characteristics of CPB, the effect of mix proportion and tailing fineness on the strength of CPB has been studied by Yilmaz [11], Li [12] and Qiu [13]. The results show that UCS is significantly influenced by them. Andrew [14] and Jiang [15] have shown that the hydration reaction rate of cement is influenced by a high curing temperature, thus affecting the strength of the CPB. Wei [16], Benzaazoua [17], Wu [18] and Huang [19] have studied the influences of cement type, mixing water quality, curing time and strain rate on the mechanical characteristics of CPB, respectively, and UCS is influenced by a combination of multiple factors. In terms of the damage constitutive model for CPB, based on theory analysis, Jin [20] established a damage constitutive model considering the impact of the cement-tailing ratio. Moreover, the data mining approach was adopted when establishing a damage constitutive model by Qi [21], and a numerical simulation was adopted to verify its rationality. Wang [22] carried out a laboratory test of CPB under the effect of sulfate. Moreover, the impact of structure characteristics has been studied by Wang [23] and Fu [24], and a damage constitutive model of CPB considering structural characteristics has been established. Finally, dynamic loading has aroused wide public concern and was therefore considered by Chen [25], Qiu [26] and Lu [27]. The damage constitutive model of CPB, which can reflect the damage evolution law of CPB to a certain extent, was also investigated. It can be seen that current studies are mostly based on laboratory standard curing tests [28,29,30,31], while the mechanical characteristics and the damage constitutive model under the influence of curing stress are seldom considered.

To sum up, based on standard laboratory tests, a multifield coupling experiment was developed, and it was used for curing CPB. The mechanical parameters were also established for all CPB specimens. Moreover, the evolution law of the mechanical properties of the CPB was studied. Finally, a damage constitutive model of the CPB was investigated based on the results above and then verified.

2. Experimental Schemes

2.1. Materials

2.1.1. Copper Tailings

Tailings are one of the most important components that impact the properties of CPB, which was selected from one copper mine. According to laboratory tests, the density of the tailings was obtained, which was 2.74 g/cm³, and based on tests with a laser grain sizer, the particle size distribution (PSD) curve was obtained, which is shown in Figure 1. It can be seen that 50% of the particles of the tailings were smaller than 40 μm, and the d₁₀, d₃₀ and d₆₀ were 6.32, 20.53 and 50.85 μm, respectively. Studies show that the non-uniform coefficient and coefficient of curvature can reflect the quality of the total tailings [32,33]. The theory’s calculation indicates that the non-uniform coefficient (d₆₀/d₁₀) was 8.05, while the coefficient of curvature ((d₃₀·d₃₀)/(d₆₀·d₁₀)) was 1.31, indicating that the total tailing was graded well.

Additionally, an X-ray fluorescence analyzer was adopted to explore the chemical elements of the copper tailings. The results showed that CaO, Fe₂O₃, MgO, SiO₂ and Al₂O₃ were included, and the tailing were low-sulfur tailings, therefore, the influence of sulfur was ignored, and the selected tailings met the requirements of this study.

2.1.2. Binders

Cement is one of the most commonly used binder agents. When cement is mixed with water and tailings, it can react with them and produce the cement hydration product. Then the strength of a CPB is formed gradually [34,35]. In this study, the commonly used ordinary Portland cement (R42.5) was selected as the cement material. Figure 1 shows the PSD of the cement. It reveals that the PSD of the cement was finer when compared with that of the total tailings.

2.1.3. Water

As water is essential to the components of CPB [36], tap water from the laboratory was applied as the mixing water. It is an important part of mixing CPB because the cement can react with the water and the tailings.

2.2. Laboratory Equipment

2.2.1. Curing Experiment Equipment of CPB

In this study, a multifield coupling curing experiment was developed. The curing experiment device is shown in Figure 2. The experiment simulated the effect of curing stress, curing humidity, cement-tailing ratio, curing temperature, mass concentration and other influencing factors. The curing stress system was composed of a motor controller, a planetary gear reducer, a torque motor and a ball screw. Moreover, the main principles of the curing stress system are shown below: (1) Torque was produced by the torque motor under the effect of an electric current. (2) The output torque from the torque motor was amplified by the planetary gear reducer. (3) Vertical curing stress was obtained by the torque transformed from the ball screw. The range of the curing stress that could be applied by the experimental device was between 0 and 1000 kPa, and the accuracy was ±5 kPa.

2.2.2. Uniaxial Compressive Equipment

UCS is a primary index that represents the mechanical characteristics of CPB. To obtain uniaxial compressive strength (UCS) and stress-strain curves for the CPB [37], the equipment of the WDW-50 servo-controlled uniaxial compression testing experiment was adopted. The uniaxial compression testing experiment could be controlled by displacement and velocity, its maximum range was 50 kN, and the control accuracy was ±0.2%. According to studies by Egorova [38], when specimens are cured for 3, 7, 14 and 38 days, they make standard specimens with a diameter of 50 mm and length of 100 mm. Then, the specimens were placed into the WDW-50 servo-controlled uniaxial compression testing experiment. The loading rate was 0.1 mm/m, and the UCS was determined by the average value of three specimens with the same conditions.

2.3. Schemes of the Experiment

The main influencing factor that needs to be considered is curing stress, therefore, the mass concentration (76%) and the cement-tailing ratio (1:6) of the CPB were fixed, and the stope height of 30 m was defined as the basis of curing stress. The CPB’s density was 1.80 kg/m³, and a maximum curing stress of 540 kPa was considered. Figure 3 shows the distribution of the curing stress at different positions in a single stope. Moreover, it illustrates the application process of the curing stress. According to earlier studies, the hydration reaction mainly occurs in the first 48 h of the curing process, therefore, the curing stress was applied within 48 h for all the specimens, which then were cured for 3 days. In addition, the specimens were placed on a curing box with the standard curing conditions (with a curing temperature of 20 ± 2 °C and humidity of 95 ± 1%) until they were cured for 7, 14 and 28 days.

3. Mechanical Properties of CPB

3.1. Correlation between UCS and Curing Stress

Based on the lab experiments, Figure 4 shows the evolution curves between UCS and curing stress. It indicates that there was an increasing trend between UCS and curing stress as the stress during curing increased. In addition, as curing age increased, there was also an increasing trend for UCS when the stress during curing was fixed. That is to say, curing stress was beneficial to the improvement of UCS for all CPB specimens. Moreover, the linear, logarithmic, exponential and quadratic functions were applied to investigate the correlation between UCS and curing stress. The correlation coefficient was larger than 0.95 and indicated that the quadratic function could characterize the relationship between them best.

We estimated the correlation between the uniaxial compressive strength’s growth rate and curing stress. The curves are shown in Figure 5. Uniaxial compression strength’s growth rate showed a different change rule with different curing ages when the stress during curing was increased. The uniaxial compression strength’s growth rate showed a decreasing trend, and the slope of the curve decreased gradually, that is to say, a larger curing stress had less impact on the UCS of all the CPB specimens. Secondly, when curing age was 3 days, the uniaxial compression strength’s growth rate was larger than that with 28 days, meaning that the effect of curing stress was larger on early UCS than on later UCS.

3.2. Evolution Law of Stress-Strain for CPB Specimens

As the stress-strain curve can reflect the deformation characteristics of CPB [39,40], thus, it is very important to estimate its evolution law with different curing stresses. Based on the lab experiment results, stress-strain curves of the CPB specimens were obtained and are shown in Figure 6. When curing age remained unchanged, there was a considerable difference in the curves for all CPB specimens, while the change trend was extremely similar.

According to the stress-strain curves of the CPB specimens, when a specimen was tested, as the first stage, the curve tended to be elastic, then it became steeper with larger curing stress, meaning that the elastic modulus was larger with a larger curing stress. Secondly, at the first stage, more pores existed in the structure of the CPB, thus, there was an initial compaction stage, and maximum strain in this stage showed a decreasing trend when curing stress was increased. This was because there were many micropores and microcracks in the CPB structure that were squeezed when the curing stress was applied during the testing; then, the space of the micropores and microcracks showed a decreasing trend, thus, the spaces between the pores were decreased and the compactness degree was significantly improved, resulting in the diminution of the maximum strain under the effect of uniaxial compression.

In addition, the CPB would be damaged when the stress of the CPB reaches its peak. It was also found that peak stress showed an increasing trend when curing stress was increased, while strain showed a decreasing trend when it corresponded to the peak stress when stress during curing was increased. In other words, CPB had greater stiffness when stress during curing was increased. Moreover, when curing time was constant, the deformation of the CPB specimens decreased gradually with the increase of stress during curing. When it occurred from peak stress to complete loss of its bearing capacity, the failure process of the CPB was more sudden, and it showed the characteristics of brittle failure. Similarly, the stress-strain curves for all CPB specimens showed a similar rule as curing stress changed when curing age was 7, 14 and 28 days, respectively.

3.3. Damage Process of the CPB

Based on the UCS’s evolution law and the stress-strain curves of the CPB specimens under the impact of curing stress, Figure 7 shows its typical stress-strain curve. By analyzing its shape and evolution law, we could see that there were four stages for the typical stress-strain curve:

(1) The initial compaction stage of the CPB: As shown in Figure 7, the first stage belonged to the OA segment, as the CPB was composed of tailing, cement and water. Inside the CPB there were microscopic pores and fractures in its porous material. The internal microscopic pores and fractures were closed gradually with UCS, its pore structure was compacted and it could be seen that the slope of the curve was 0 at point O. Then, the slope of the curve gradually increased for the CPB specimen, and strain was increased at the same time.

(2) The linear elastic deformation stage: This was the second stage for the stress-strain curve, shown as the AB segment in Figure 7. After the initial compaction stage, the internal microscopic pores and fractures expanded gradually under the effect of uniaxial compression; as the stress increased, the strain also increased. Moreover, the increasing trend between the stress and strain presented an approximately linear relationship, i.e., the slope of the curve in this stage was constant. Additionally, the elastic modulus could be calculated through the stress-strain curve, and it was larger as curing stress increased.

(3) The plastic deformation stage: Figure 7 shows the plastic deformation stage of the CPB as the BC segment. In this stage, the internal microscopic pores and fractures expanded continuously until the CPB was damaged, and the curve showed a concave shape. The stress of the CPB showed an increasing trend as strain rose; however, the growth rate of the curve decreased gradually.

(4) The post-failure stage: This stage is on the CD segment in Figure 7. In this stage, strain continuously increased, while the stress of the CPB was decreased because the internal microscopic pores and fractures were further expanded and the CPB was damaged. While there was still some bearing capacity for the CPB, the slope of the curve decreased as the strain also showed a decreasing trend. Moreover, combined with Figure 6, as stress during curing became larger, the decrease of the stress-strain curve for the CPB specimens in the post-failure stage became faster.

4. Establish of the Damage Constitutive Model of CPB Specimens

According to lab experiment results and the theoretical analysis, there was a phenomenon with the stress-strain curves where they presented a concave shape in the initial compactness stage. That is to say, there was a nonlinear relationship between stress and strain in this stage. Therefore, the stress-strain curves under the influence of curing stress were analyzed according to the initial compactness stage and its later stages. Then, two-stage damage constitutive models of the CPB specimens considering the impact of curing stress can be established. It was assumed that there was no damage to the CPB in the initial compactness stage, while continuous damage occurred in the elastic and plastic stages. According to studies which have been done by scholars at home and abroad [41], the relationship of stress and strain for the CPB in the initial compactness stage can be expressed by Formula (1):

$$\sigma ={\sigma }_A{(\frac{\epsilon }{{\epsilon }_A})}^2$$

(1)

where $\sigma$ is UCS, MPa; $\epsilon$ is the strain; ${\sigma }_A$ (MPa) and ${\epsilon }_A$ are the maximum stress and strain in the initial compactness stage.

According to Lematre’s equivalent strain hypothesis [42], effective stress is equal to the damage deformation of the CPB under external loading. Assuming that CPB is an isotropic material, the damage to the CPB under external loading is also the isotropic, therefore, the constitutive relationship of the CPB in one-dimensional conditions is shown by Formula (2):

$$\sigma =E\cdot \epsilon (1-D)$$

(2)

where E is the CPB’s elastic modulus, MPa; D is the damage variable, its range is between 0 and 1; when D = 1, it is equivalent to the damage of all the infinitesimals for the CPB; when D = 0, it shows that CPB belongs to no-damage material.

According to the internal structure of the CPB specimens, it can be assumed that there exists a power function for the distribution of the microelement failure probability, and combined with continuum mechanics, the microelement size of the CPB can be regarded as that of particles. It also contains many microscopic pores and fractures, then, the function of the probability density for the CPB can be expressed by Formula (3) [43]:

$$p(F)=\frac{m}{F_0}{(\frac{F}{F})}^{m-1}$$

(3)

where p(F) is the distribution function of the microelement strength; F is the random distribution variance of the microelement strength; m and F₀ are the distribution functions of the CPB and can reflect the response characteristics of the CPB under external loading.

The CPB contains many internal microscopic pores and fractures that are prone to being damaged under the influence of external loading. As the external loading reaches one stress level, Formula (4) can represent the number of the damaged microelements [44]:

$$n={\displaystyle {\int }_0^{F}NP(x)}dx=N{(\frac{F}{F_0})}^m$$

(4)

where n is the number of the damaged microelements corresponding to the condition where external loading reaches a certain stress level; N is the total number of microelements in the CPB.

Assuming that the CPB is a continuous medium, according to the damage mechanics theory [45], Formula (5) can characterize the CPB’s damage variable D as follows:

$$D^{\prime }=\frac{n}{N}={(\frac{F}{F_0})}^m$$

(5)

It is generally assumed that the CPB loses its bearing capacity when the internal microelement unit is damaged [46]. In fact, according to the stress strain of the CPB, when peak stress was reached, although the CPB was damaged, it still had a certain bearing capacity. Therefore, the correction coefficient of a (its range is between 0 and 1) is proposed to reflect the bearing capacity of the microelement for the CPB corresponding to the peak stress. The number of the microelement can be obtained during the stage of peak stress to complete damage, which is a·n. For the damage variable of the CPB, it can be expressed by Formula (6):

$$D=a{(\frac{F}{F_0})}^m$$

(6)

Combined with Formulas (2) and (6), the damage statistical constitutive model of the CPB specimen that excludes the initial compaction stage can be obtained, which is shown in Formula (7):

$$\sigma =E\epsilon (1-a{(\frac{F}{F_0})}^m)$$

(7)

According to the expressions above, under the impact of curing stress, the two-stage damage constitutive models of the CPB specimens are characterized by Formula (8):

$$\sigma =\left\{\begin{array}{ll}{\sigma }_A(\frac{\epsilon }{{\epsilon }_A}{)}^2& \epsilon <{\epsilon }_A\\ {\sigma }_A+E(\epsilon -{\epsilon }_A)\left[(1-a{(\frac{F}{F_0})}^m\right]& \epsilon \ge {\epsilon }_A\end{array}\right.$$

(8)

According to the studies above, by characterizing the microelemental strength of the CPB specimen under the impact of curing stress [47], the Drucker–Prager failure criterion is adopted. Then, its expression can be shown by Formula (9):

$$f=\alpha I_1+J_2{}^{1/2}$$

(9)

where $\alpha =\frac{2\sin \varphi }{\sqrt{3}(3-\sin \varphi )}$; $\varphi$ is the internal friction angle; I₁ and J₂ are the first invariant of the stress and the second invariant deviation, respectively. The I₁ and J₂ can be expressed by Formulas (14) and (15) [48]:

$$I_1={\sigma }_1{}^{\ast }+{\sigma }_2{}^{\ast }+{\sigma }_3{}^{\ast }$$

(10)

$$J_2=\frac{1}{6}[{({\sigma }_1{}^{\ast }-{\sigma }_2{}^{\ast })}^2+{({\sigma }_2{}^{\ast }-{\sigma }_3{}^{\ast })}^2+{({\sigma }_3{}^{\ast }-{\sigma }_1{}^{\ast })}^2]$$

(11)

where ${\sigma }_1{}^{\ast }$, ${\sigma }_2{}^{\ast }$ and ${\sigma }_3{}^{\ast }$ are the effective stresses.

According to Formula (10), the effective stress of ${\sigma }_n{}^{\ast }$ can be expressed by the nominal stress of ${\sigma }_n{}^{}$, which is shown in Formula (12):

$${\sigma }_n{}^{\ast }=\frac{{\sigma }_n}{1-D}(n=1,2,3)$$

(12)

Combined with Formulas (9)–(12), an expression of the statistical distribution of microelemental strength is performed and shown in Formula (13):

$$F=\frac{(a_0{I}_1^{}+{(J_2)}^{1/2})E{\epsilon }_1}{{\sigma }_1-\mu ({\sigma }_2+{\sigma }_3)}$$

(13)

where $I_1={\sigma }_1+{\sigma }_2+{\sigma }_3$, $J_2=\frac{1}{6}[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2]$.

Under the effect of uniaxial compression, the confining stress of the CPB is 0, therefore, it can be obtained that: $\sigma ={\sigma }_1$, ${\sigma }_2={\sigma }_3=0$, ${\epsilon }_1=\epsilon$. According to the Drucker–Prager failure criterion, the statistical distribution of the microelement for a CPB specimen is shown in Formula (14):

$$F=(a_0+\frac{1}{\sqrt{3}})E\epsilon$$

(14)

Therefore, the statistical damage formulation of the CPB is shown in Equation (15):

$$\sigma =E\epsilon \left\{1-a{\left[\frac{(a_0+\frac{1}{\sqrt{3}})E\epsilon }{F_0}\right]}^m\right\}$$

(15)

For the stress strain of the CPB, when it reaches peak stress, ${\sigma }_c$, it must meet the requirement of Formulas (16) and (17):

$$\epsilon ={\epsilon }_c,\sigma ={\sigma }_c$$

(16)

$$\epsilon ={\epsilon }_c,d\sigma /d\epsilon =0$$

(17)

According to the boundary conditions, the formula above can be calculated; then, the following expression can be obtained:

$$F_0=(a_0+\frac{1}{\sqrt{3}})E{\epsilon }_c{[(m+1)a]}^{\frac{1}{m}}$$

(18)

$$m=\frac{{\sigma }_c}{E{\epsilon }_c-{\sigma }_c}$$

(19)

By combining the expressions above, the following expressions (20) and (21) can be obtained:

$$D=a{(\frac{F}{F_0})}^m=\frac{1}{m+1}{(\frac{\epsilon }{{\epsilon }_c})}^m$$

(20)

$$\sigma =E\epsilon [1-\frac{1}{m+1}{(\frac{\epsilon }{{\epsilon }_c})}^m]$$

(21)

Combined with Formulas (1) and (21), a two-stage damage constitutive model of CPB is obtained, as shown in Formula (22):

$$\sigma =\left\{\begin{array}{ll}{\sigma }_A(\frac{\epsilon }{{\epsilon }_A}{)}^2& \epsilon <{\epsilon }_A\\ {\sigma }_A+E(\epsilon -{\epsilon }_A)\left[1-\frac{1}{m+1}{(\frac{\epsilon -{\epsilon }_A}{{\epsilon }_c-{\epsilon }_A})}^m\right]& \epsilon \ge {\epsilon }_A\end{array}\right.$$

(22)

According to Formula (22), a two-stage damage constitutive model of the CPB has been established. However, the distribution forms of internal microdefects of the CPB are different, and the parameters in Formula (22) are also different. They are affected by tailing fineness, cement type, cement tailing ratio, curing stress, etc.

5. Model Validation and Analysis

As the two-stage damage constitutive model of the CPB has been established, it is necessary to verify its rationality based on experimental results. Firstly, the theoretical stress-strain curve was calculated by the established model, while the experimental curve of the CPB was obtained based on laboratory experiments. The theoretical curves and experimental curves of the CPB are shown in Figure 8, which indicates that the theoretical stress-strain curve and the experimental curves were extremely consistent with each other, especially in the first three stages. In the post-failure stage, there were some differences between the theoretical and experimental stress-strain curves. The reason was that the structure of the CPB was assumed to be an isotropic homogeneous medium. There are many influencing factors that can affect the physical and mechanical characteristics of the CPB, such as mixing time, curing time, tailing fineness, tailing–cement ratio, etc. There are many internal microscopic pores and fractures in the CPB structure, leading to certain randomness in the CPB during the damage process under external loading.

6. Summary and Conclusions

In this study, a multifield coupling curing experiment has been developed. According to lab test results, the UCS, stress-strain curves, and the damage processes of the CPB specimens were investigated. A damage constitutive model of the CPB specimen was also created, and as a result, the following conclusions could be drawn:

(1) There was an increasing trend for UCS when curing stress was increased. It showed a decreasing trend for UCS’s growth, and the relationship between UCS and curing stress was characterized well by the quadratic function. In addition, the effect of curing stress was larger on early UCS than on later UCS.

(2) Under the impact of curing stress, the changing trend of the stress-strain curves was extremely similar, and there were four phases for the stress-strain curves. Peak stress showed an increasing trend, while strain showed a decreasing trend when it corresponded to peak stress when stress during curing was increased.

(3) The results indicate that the theoretical curves of the CPB that were calculated by the established two-stage damage constitutive model were consistent with the experimental curves, which means that the established damage constitutive model considering the influence of curing stress could describe UCS and the deformation evolution law of CPB well. It also had good rationality and effectiveness, which can provide references for similar mechanical characteristics’ analysis and engineering applications of CPB.

According to the studies above, it could be found that curing stress is beneficial to the improvement of UCS for CPB specimens. A two-stage damage constitutive model was also established. However, more in-depth work needs to be done, such as the consideration of more influencing factors, such as the cement-to-tailing ratio, mass concentration and cement type. In addition, our work also needs to be extended to a field application. More research needs to be done about CPB characteristics in the future.