Research Article  Open Access
LoadingRate Dependence of Rocks in Postfailure Region under Triaxial Compression
Abstract
To study the loadingrate dependence of four types of rocks in the peak and postfailure regions, the alternative strain rate test method is employed in conducting triaxial compressive tests under different confining pressure and moisture conditions. An index of the loadingrate dependence in the postfailure region is proposed, and the effects of confining pressure and moisture condition on the index are analyzed. The reliability of the test results is verified by comparing the experimental and calculated results of stress relaxation. Following are the results. (1) The peak strength and postfailure behavior significantly depend on the loading rate. The indices (n_{c} and n_{d}) of the loadingrate dependence in the postfailure region obtained based on the elastic and unloading moduli show a good linear relationship. It is feasible to use n_{c} to evaluate the loadingrate dependence of rocks in the postfailure region. (2) The loadingrate dependence of the rock strength increases with the increase in the confining pressure. The effect of confining pressure on the loadingrate dependence in the postfailure region is related to the moisture condition state. (3) Watersaturation treatment will increase the loadingrate dependence in the peak and postfailure regions, and the effect of water is greater than that of the confining pressure. (4) The experimental and calculated results of stress relaxation in the postfailure region are in good agreement, thus validating the experimental results.
1. Introduction
An understanding of the timedependent behavior of rocks is essential for the assessment of longterm deformation and stability of underground structures in mining engineering [1]. Loadingrate dependency of strength is one important aspect of the timedependent behavior of rocks. The relationship between uniaxial compression strength and loading rate has been thoroughly studied [2]. In contrast, studies on the time dependency of rocks under the confining pressure are lacking [3]. In practical projects, the formation of postfailure regions around underground structures is unavoidable [4]. Many rock mechanics problems in engineering, such as stability of underground roadways and pillars, are closely related to the mechanical properties and loadingrate dependence of rocks in the postfailure region. Therefore, studying the loadingrate dependence of rocks in this region has practical significance.
Only a few papers have been published concerning the loadingrate dependence of rocks in the postfailure region compared to the peakfailure region. Rummel and Fairhurst [5] investigated the postfailure behavior of brittle rocks under uniaxial and biaxial compression. The average slope in the postfailure region was found to decrease with the increase in the strain rate, indicating that the deterioration in the loadcarrying ability of brittle rocks at high strain rates is less than that at lower strain rates under uniaxial compression. Bieniawski [6] conducted compressive tests at a constant strain rate, changing strain rates, and constant loading conditions in the postfailure region. The results show that the slopes of the stressstrain curves in the postfailure region become increasingly horizontal with the decrease in the strain rate. This phenomenon helps improvise the stability of the rocks after failure. Peng and Podnieks [7] demonstrated that the stressstrain curves obtained at higher strain rates exhibited a lower negative slope, whereas those obtained at lower strain rates exhibited a higher negative slope. Later, Peng [8] conducted stress relaxation tests in the postfailure region to investigate the timedependence behavior of rocks. Hashiba et al. [9] investigated the loadingrate dependence of a stressstrain curve in the postfailure region using the alternative strain rate (ASR) test method. However, the results are only qualitative, and there is no quantitative method for studying the loadingrate dependence of rocks in the postfailure region under confined and wet conditions.
Previous studies have shown that the peak strength is proportional to the 1/(n + 1)^{th} power of the loading rate, and the creep life is inversely proportional to the n^{th} power of the creep stress level [10], where n is the index of time dependence, and is closely related to the crack growth rate [11]. The lower the value of n, the greater is the time dependence [3]. The index n can not only be used to evaluate the loadingrate dependence of the peak strength [12] and stresslevel dependence of the creep life [13] but can also be used as a parameter in the numerical simulation of constitutive equations wherein the time dependence is considered [14]. Moreover, it can be applied to rock mass rating [15]. Hashiba and Fukui [3] reviewed the results of previous studies on the index n of the peak strength under various loading conditions. However, relatively few studies have been conducted on the peak strength under confined and wet conditions. Furthermore, an index of the loadingrate dependence of rocks in the postfailure region has not been proposed. In fact, there are no clear conclusions concerning the relationship between the loadingrate dependence (in the postfailure region), confining pressure, and moisture condition, which is essential for assessing the stability of underground structures.
In this study, a series of triaxial compressive tests on four types of rocks are conducted under different confining pressure and moisture conditions (airdried and wet). The new test method (ASR method) proposed by Hashiba et al. [9] is employed to conduct the tests. Moreover, an index for evaluating the loadingrate dependence of the rocks in the postfailure region and its calculation method are developed, for which cyclic loadingunloading tests are conducted. The effects of the confining pressure and moisture condition on the index in the postfailure region are investigated, and the results are compared with those in the peakfailure region. Finally, the results of the ASR tests are validated by conducting stress relaxation tests in the postfailure region and from the calculation of a nonlinear constitutive equation.
2. Materials and Methods
2.1. Specimen Description
Following are the four types of rocks employed for testing: Tage tuff, Ogino tuff, Emochi andesite, and Jingkou sandstone. Tage tuff was obtained from Tochigi Prefecture, Japan. It mainly contains Albite, Feldspar, and small amounts of calcite. Ogino tuff was obtained from Fukushima Prefecture, Japan. The main components are zeolite, mullite, and clay, with small amounts of plagioclase and biotite. Emochi andesite was obtained from Fukushima Prefecture, Japan. It mainly comprises plagioclase, pyroxene, and a small amount of biotite. Jingkou sandstone was obtained from Chongqing, China. The main components are quartz, feldspar, vermiculite, and muscovite. Table 1 lists the basic physical and mechanical parameters of the rocks. The saturated water contents of Tage tuff and Ogino tuff are greater than those of Emochi andesite and Jingkou sandstone, whereas the peak strengths are lower.
 
UCS: uniaxial compression strength, E: Young’s modulus, : Poisson’s ratio. 
The specimens were obtained by drilling a block of rock and were then cut into cylindrical shapes with a diameter of 25 mm and a height of 50 mm. The specimen surfaces were polished to ensure the flatness, verticality, and parallelism standards as per ISRM. The specimens of the four rocks were airdried at room temperature for two weeks. For testing under wet conditions, the specimens were watersaturated using a vacuum pump. More than three specimens of each type of rock were employed for each testing condition.
Figure 1 shows the testing apparatus used for the tests. The maximum axial loading capacity was 500 kN, and the maximum axial displacement was 10 mm. The load was measured using a load cell, and the displacement was measured using a linear variable differential transformer (LVDT).
2.2. Test Method
The ASR test method, which was introduced by Hashiba et al. [9], was used to conduct the tests. In this section, a brief introduction about the method is given. Figure 2 shows the stressstrain curve for the ASR test. The strain rate is switched between high (C_{2}) and low values (C_{1}) at uniform strain intervals (△ε). Based on the experimental curve, indicated using thick lines in Figure 2, the stress increases abruptly with the increase in the strain rate at the location marked with •, whereas it decreases at the location marked with ○. Therefore, the stressstrain curves for C_{1} and C_{2} can be obtained by connecting the points • and ○ using a spline function, respectively, indicated using thin lines in Figure 2.
In this study, the confining pressures were set to 0, 3, 6, and 9 MPa for the ASR tests. Based on previous suggestions [9], the strain rate was switched between 10^{−4}/s (C_{2}) and 10^{−5}/s (C_{1}), while the strain interval (△ε) is set to 5 × 10^{−4} under uniaxial compression and 8 × 10^{−4} under triaxial compression. The peak strengths σ_{2f} and σ_{1f} corresponding to strain rates of C_{2} and C_{1}, respectively, can be obtained using this method. Hence, the loadingrate dependence index n of the peak strength can be calculated as follows:
The confining pressures in the cyclic loadingunloading tests were set to 3, 6, and 9 MPa. The strain rate was 10^{−4}/s.
3. Results
3.1. Alternative Strain Rate Test Results
Figure 3 shows the typical ASR test results of the airdried rocks under different confining pressures. The existence of the confining pressure can effectively improve the bearing deformation capacity of the rocks in the postfailure region. When no confining pressure is applied, the stress decreases rapidly to a very low level after the peak strength point, and it is very difficult to quantitatively describe the loadingrate dependence in this stage. Nevertheless, the method described by Hashiba et al. [9] is found to be more effective. Hence, the index of the loadingrate dependence in the postfailure region was not calculated when the confining pressure was 0 MPa.
(a)
(b)
(c)
(d)
The stressstrain curves for the airdried Tage tuff and Ogino tuff samples (Figures 3(a) and 3(b)) are similar. With the increase in the confining pressure, the peak strength gradually increases. After the peak strength, the stressstrain curves gradually transition from strain softening to approximately plastic behavior. Although no fluctuation in the stress is observed with respect to the change in the strain rate from the start of the test, the stress varies with the change in the strain rate as the stress increases up to the peak strength. The magnitude of the fluctuation is maximum near the peak strength point and gradually decreases thereafter. The stress in the postfailure region fluctuates with the change in the strain rate.
The stressstrain curves for the airdried Emochi andesite and Jingkou sandstone samples (Figures 3(c) and 3(d)) are also similar. Compared to the curves for the Tage tuff and Ogino tuff samples, the difference is that the stress decreases abruptly after the peak strength and thereafter stabilizes in the postfailure region. In addition, the stress fluctuates with the change in the strain rate in the peak and postfailure regions.
The stressstrain curves for the four rock samples, shown in Figure 4, under the wet condition are similar to those under the airdried condition. This stress fluctuation is closely related to the loadingrate dependence in the peak and postfailure regions. From the fluctuation, the index of the loadingrate dependence can be obtained using data analysis.
(a)
(b)
(c)
(d)
3.2. Cycle LoadingUnloading Test Results
The cyclic loadingunloading test was conducted to obtain the unloading modulus (E_{d}) in the postfailure region. This modulus will be used to calculate the index in the postfailure region. Therefore, the analyses of the mechanical properties of the rocks under the cyclic loadingunloading condition were excluded from the study.
Considering the results of Tage tuff under a confining pressure of 3 MPa as an example, the method of calculating E_{d} is introduced as follows: Figure 5(a) shows the typical stressstrain curve obtained for the Tage tuff. In the cyclic loadingunloading test, hysteresis loops are generated, and E_{d} is obtained from the hysteresis loop, as shown in Figure 5(b).
(a)
(b)
The points A and B are the inflection points for unloading and reloading, respectively, and the slope of the line AB is defined as the unloading modulus E_{d}. For the airdried and wet Tage tuff and Ogino tuff samples, the change in the stressstrain curves in the postfailure region is gradual. Therefore, the average slope of all the lines AB after the peak strength is calculated as E_{d}. For the airdried and wet Emochi andesite and Jingkou sandstone samples, the average slope of the line AB is calculated as E_{d} when the stressstrain curves become horizontal after the decrease in the stress in the postfailure region. Table 2 lists the statistical results of E_{d} for the four airdried and wet rock samples.

3.3. Index Calculation Method and Results
In this study, the index of the loadingrate dependence is denoted as n for the peakfailure region and n_{i} for the postfailure region. The value of n can be easily calculated based on Equation (1) and ASR test results. To calculate n_{i}, the authors adopted four methods (a, b, c, and d) which were proposed by Hashiba et al. [16], as shown in Figure 6.
Figure 6 shows two stressstrain curves corresponding to strain rates of C_{1} and C_{2} obtained from the ASR test results. In the postfailure region, point A (the stress is ) is located on the curve of C_{2}. Line a passes through A and is perpendicular to the strain axis; line b joins the origin and point A; line c passes through A with a slope equal to Young’s modulus (E); and line d passes through A with a slope equal to E_{d} obtained from the cyclic tests (given in the previous section). All the lines intersect with the curve of C_{1}, and the stresses at the intersection points are denoted by , , , and , respectively. The index of the loadingrate dependence is calculated using
Although method “a” is the easiest to understand, a theoretical background is lacking. The theoretical background of method “b” is the variable compliancetype constitutive equation [17]. However, the nonlinear strain during the compression is not considered in this equation. Therefore, we calculated the index based on methods “c” and “d” and discussed their relationship.
Table 3 lists the calculation results of n_{c} and n_{d} obtained under airdried and wet conditions. For the four rocks, n_{c} is approximately equal to n_{d} under the same test condition. Figure 7 shows the relationship between n_{c} and n_{d} based on the values listed in Table 3; n_{c} is proportional to n_{d}, and the ratio is approximately 1. Furthermore, the relationship between n_{c} and n_{d} was not affected by the rock type, moisture condition, or confining pressure. Therefore, the index n_{c}, which is calculated using E, was used to assess the loadingrate dependence of the rocks in the postfailure region. Hence, the cyclic loadingunloading test was not conducted to reduce the effort and cost.

3.4. Effects of Confining Pressure and Moisture Condition on Index
Based on Equation (1) and ASR test results, the index n of the peak strength is calculated. Table 4 lists the results. Water has a significant weakening effect on the peak strength and n of all the rocks. For Tage tuff and Ogino tuff, the values of n under the wet condition are lower than those under the airdried condition, indicating that the loadingrate dependence of the peak strength under the wet condition is greater than that under the airdried condition. Similarly, for Emochi andesite and Jingkou sandstone, the values of n under the wet condition are lower than those under the airdried condition. However, the difference in n between the airdried and wet conditions is lower than that for Tage tuff and Ogino tuff. This shows that the effect of water on the loadingrate dependence of the peak strengths of Tage tuff and Ogino tuff is greater than that on the loadingrate dependence of the peak strengths of Emochi andesite and Jingkou sandstone. The effects of water on n_{c} and n_{d} in the postfailure region, as listed in Table 3, are similar to that on n.

To investigate the effect of the confining pressure on the loadingrate dependence in the peak and postfailure regions, Figures 8 and 9 show the indices n and n_{d} with respect to the confining pressure, plotted based on the values listed in Tables 3 and 4, respectively. Figure 8 shows that the values of n under the airdried and wet conditions increase with the increase in the confining pressure, indicating that the increasing confining pressure reduces the loadingrate dependence of the peak strength. The effect of the confining pressure on the n values of Tage tuff and Ogino tuff is relatively lower than that on the n values of Emochi andesite and Jingkou sandstone.
(a)
(b)
(a)
(b)
Figure 9 shows that the value of n_{d} under the airdried condition decreases with the increase in the confining pressure, except for the Jingkou sandstone sample. Under the wet condition, the value of n_{d} does not show a clear variation with respect to the confining pressure. From Table 3, the value of n_{d} under the wet condition changes very slightly under the different confining pressures. Therefore, it can be considered that the confining pressure does not affect the loadingrate dependence in the postfailure region. Moreover, from Tables 3 and 4, it can be concluded that the effect of water on the loadingrate dependence of the rocks in the peak and postfailure regions is greater than that of the confining pressure.
3.5. Comparison between Peak and Postfailure Regions
Comparing Tables 3 and 4, the value of n is lower than n_{d} in all cases, regardless of the condition (airdried or wet), indicating that the loadingrate dependence in the peakfailure region is greater than that in the postfailure region. The ratio of n under the wet condition to that under the airdried condition (n(wet)/n(airdried)) was calculated for the range of confining pressures employed in this study. The ratios (n) for the Tage tuff (under different confining pressures), Ogino tuff, Emochi andesite, and Jingkou sandstone are in the ranges of 71–80%, 63–70%, 80–89%, and 97–100%, respectively. The results of n_{d} (n_{d}(wet)/n_{d}(airdried)) for the Tage tuff, Ogino tuff, Emochi andesite, and Jingkou sandstone are in the ranges of 66–76% (under different confining pressures), 52–71%, 77–95%, and 84–100%, respectively. Therefore, water has a greater effect on the loadingrate dependence in the postfailure region.
4. Discussion
4.1. Stress Increment
To study the loadingrate dependence of the rocks in the peak and postfailure regions, it is necessary to obtain the stress increment values with tenfold increase in the strain rate. The equations to calculate the same are as follows:
The following equations [3] are obtained by substituting Equations (3) and (4) in Equations (1) and (2), respectively:where △σ and △σ_{d} are the increments with tenfold increase in the strain rate in the peak and postfailure regions, respectively. From Equations (5) and (6), △σ and △σ_{d} are found to be proportional to σ_{1f} (deviator stress) and σ_{1d} (deviator stress), respectively.
Figure 10 shows the changes in △σ for the four rocks under the airdried and wet conditions. The results show that △σ increases with the increase in σ_{1f} (deviator stress) under the airdried and wet conditions. The values of △σ for the Tage tuff and Ogino tuff samples are lower; however, the increment in △σ is greater than those for Emochi andesite and Jingkou sandstone, based on the comparison of the slopes of the fitting lines. The data pertaining to the sandstone samples under the wet condition are relatively scattered.
(a)
(b)
Figure 11 shows the changes in △σ_{d} for the four rocks under the airdried and wet conditions. Similar to △σ, △σ_{d} increases with the increase in σ_{1d} (deviator stress) under the airdried and wet conditions.
(a)
(b)
Overall, both △σ and △σ_{d} have a good linear relationship with σ_{1f} and σ_{1d}, respectively, under both airdried and wet conditions. As mentioned in the previous section, it is difficult to quantitatively study the loadingrate dependence of rocks in the postfailure region when the confining pressure is zero. Based on the results obtained in this section, △σ_{d} can be estimated using Equations (4) and (6). The index n_{d} corresponding to a confining pressure of 0 MPa can be calculated using Equation (2).
4.2. Verification of Test Results
In the previous section, we found that n_{c} calculated using E was approximately equal to n_{d} calculated using E_{d}. Therefore, n_{d} can be replaced by n_{c} to assess the loadingrate dependence of the rocks in the postfailure region. However, only few quantitative results on the loadingrate dependence in the postfailure region are available in literature. Therefore, we need to verify the results of the ASR tests. Hashiba and Fukui [14] demonstrated that the various timedependent behaviors, such as the loadingrate dependence, creep, and stress relaxation, are closely related. Hence, stress relaxation tests were conducted in the postfailure region, and a nonlinear constitutive equation, the parameters of which are obtained from the ASR tests, was employed to determine the stress relaxation. Based on the comparison of the experimental and calculated results, the reliability of the ASR test results was verified.
The stress relaxation tests were conducted, and the airdried Tage tuff was taken as the example under confining pressures of 3, 6, and 9 MPa. In addition, stress relaxation tests on the airdried Ogino tuff, Emochi andesite, and Jingkou sandstone samples were conducted under a confining pressure of 6 MPa. The relaxation time was 10^{5}/s. The strain was controlled with a strain rate of 10^{−4}/s for loading up to the start point of the stress relaxation. The start points are located in the stable region of the stressstrain curves in the postfailure region. Table 5 lists the strain and stress values at the start points.

The Nonlinear Maxwell model proposed by Okubo and Fukui [18] was used to simulate the stress relaxation of rocks in postfailure region. For conventional Maxwell model, σ is proportional to the rate of dashpot strain (). For the Nonlinear Maxwell model in Figure 12, σ follows the powerlaw like a nonNewtonian fluid. Accordingly, the constitutive equations can be written as follows:where ε_{1} and ε_{2} denote the strain values of the spring and dashpot, respectively, η is the viscosity coefficient of the dashpot, and σ is the stress at the start point of stress relaxation. n was obtained from the ASR test results. E and E_{d} were used for the calculation in Equation (8). Table 5 lists the values of the parameters.
Figure 13 shows the comparison of the experimental and calculated results of stress relaxation for the four rocks under the different confining pressures on a semilogarithmic chart. The calculated and experimental results are in good agreement. There is no obvious difference between the calculations with E and E_{d}. This demonstrates the reliability of the parameters obtained from the ASR test results.
(a)
(b)
4.3. Research Limitations and Suggestions
Although we investigated the loadingrate dependence of various rocks in the peak and postfailure regions under different confining pressure and moisture conditions, there are some limitations in this study. Therefore, following are the suggestions given to improve this study and the direction for future research:(i)The mechanism of the effect of confining pressure or moisture condition on n and n_{d} values for various rocks should be studied. For example, the physicochemical effects of water on the loadingrate dependence of the rocks in the peak and postfailure regions can be investigated.(ii)The evolution of the stressstrain curve is related to the evolutionary morphology and propagation path of cracks [19]. Therefore, it is important to investigate the relationship between crack propagation and fracture mechanism and the loadingrate dependence of rocks.(iii)The ASR test results show that brittle rocks (e.g., Jingkou sandstone) fail abruptly after the peak strength, and the stress decreases almost instantaneously. This failure process cannot be controlled using the conventional strain control mode. Therefore, special control methods need to be considered to accurately obtain the stressstrain curve after the peak strength, such as the linear combination of stress and strain [20, 21] or constant lateralstrain rate method [22]. Based on these methods, the loadingrate dependence of brittle rocks in the postfailure region can be more accurately investigated.
5. Conclusion
In this study, with Tage tuff, Ogino tuff, Emochi andesite, and Jingkou sandstone as the research subjects, ASR and cyclic loadingunloading tests were conducted to investigate the loadingrate dependence of these rocks in the peak and postfailure regions. An index of the loadingrate dependence in the postfailure region was proposed. The effects of confining pressure and moisture condition on the indices n and n_{d} were discussed. Finally, stress relaxation tests were conducted in the postfailure region and later compared with the results obtained using the nonlinear Maxwell model to verify the reliability of the ASR test results. Following are the main conclusions drawn from this study:(i)The stress fluctuated with the change in the strain rate, which is closely related to the loadingrate dependence. Both the airdried and wet rocks exhibit loadingrate dependence in the pre, peak, and postfailure regions.(ii)A method to calculate the loadingrate dependence index in the postfailure region was proposed. The value of n_{c} calculated using E is largely equal to the value of n_{d} calculated using E_{d}. In the future, n_{c} can be used to assess the loadingrate dependence in the postfailure region. This would put an end to the timeconsuming and costly cyclic tests.(iii)For Tage tuff and Ogino tuff, the loadingrate dependence in the peak and postfailure regions under the wet condition was greater than that under the airdried condition. The effects of water on Emochi andesite and Jingkou sandstone were minor. The effect of water on the loadingrate dependence of the rocks was greater than that of the confining pressure. The loadingrate dependence in the peakfailure region was greater than that in the postfailure region, regardless of the condition employed (airdried or wet).(iv)The loadingrate dependence of the peak strength decreased with the increase in the confining pressure. The effects of the confining pressure on the loadingrate dependence of the peak strengths of Tage tuff and Ogino tuff were relatively lower than those on the loadingrate dependence of the peak strengths of Emochi andesite and Jingkou sandstone. The increase in the confining pressure increased the loadingrate dependence of Tage tuff, Ogino tuff, and Emochi andesite in the postfailure region under the airdried condition. The loadingrate dependence of the wet rocks in the postfailure region did not exhibit any significant variation with the increase in the confining pressure.
Data Availability
The data of alternative strain rate test, cycle loadingunloading test under different confining pressure, and water content used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
The authors would like to thank the National Natural Science Foundation of China (51434003), Basic and Frontier Research Projects of Chongqing (cstc2016jcyjA0117), and Project of Science and Technology for Chongqing Education Commission (no. KJ1711271) for the financial support.
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Copyright © 2018 Yang Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.