J. Cent. South Univ. (2020) 27: 2985-2998

DOI: https://doi.org/10.1007/s11771-020-4523-7

Experimental investigations on mechanical performance of rocks under fatigue loads and biaxial confinements

DU Kun(杜坤)^{1, 2}, LI Xue-feng(李雪锋)¹, YANG Cheng-zhi(杨成志)¹,ZHOU Jian(周健)¹, CHEN Shao-jie(陈绍杰)³, MANOJ Khandelwal⁴

1. School of Resources and Safety Engineering, Central South University, Changsha 410083, China;

2. Advanced Research Center, Central South University, Changsha 410083, China;

3. Mining Disaster Prevention and Control Ministry Key Laboratory, Shandong University of Science and Technology, Qingdao 266590, China;

4. School of Engineering, Information Technology and Physical Sciences, Federation University Australia, Ballarat 3350, Australia

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: In this research, a series of biaxial compression and biaxial fatigue tests were conducted to investigate the mechanical behaviors of marble and sandstone under biaxial confinements. Experimental results demonstrate that the biaxial compressive strength of rocks under biaxial compression increases firstly, and subsequently decreases with increase of the intermediate principal stress. The fatigue failure characteristics of the rocks in biaxial fatigue tests are functions of the peak value of fatigue loads, the intermediate principal stress and the rock lithology. With the increase of the peak values of fatigue loads, the fatigue lives of rocks decrease. The intermediate principal stress strengthens the resistance ability of rocks to fatigue loads except considering the strength increasing under biaxial confinements. The fatigue lives of rocks increase with the increase of the intermediate principal stress under the same ratio of the fatigue load and their biaxial compressive strength. The acoustic emission (AE) and fragments studies showed that the sandstone has higher ability to resist the fatigue loads compared to the marble, and the marble generated a greater number of smaller fragments after fatigue failure compared to the sandstone. So, it can be inferred that the rock breaking efficiency and rock burst is higher or severer induced by fatigue loading than that induced by monotonous quasi-static loading, especially for hard rocks.

Key words: biaxial confinements; fatigue loading; acoustic emission; fragments; intermediate principal stress

Cite this article as: DU Kun, LI Xue-feng, YANG Cheng-zhi, ZHOU Jian, CHEN Shao-jie, MANOJ Khandelwal. Experimental investigations on the mechanical performance of rocks under fatigue loads and biaxial confinements [J]. Journal of Central South University, 2020, 27(10): 2985-2998. DOI: https://doi.org/10.1007/s11771-020-4523-7.

1 Introduction

The underground engineering excavation changes the initial stress field of rocks in the vicinity to a secondary stress field [1-3]. The rocks located at the range of the secondary stress field is defined as engineering surrounding rock (ESR). The most affected part of the ESR by underground excavation is termed as excavation damaged zone (EDZ), which is the closest to the underground engineering boundaries, as shown in Figure 1(a).

The initial stress field is a three-dimensional compressive stress condition (σ₁≥σ₂≥σ₃>0), while the secondary stress field is a biaxial stress condition (σ₃=0) or a uniaxial stress condition (σ₂=σ₃=0) [4-6]. In general, most EDZ rocks are subjected to a biaxial stress condition. It is supposed that the stress path experienced by most EDZ rocks during the underground engineering excavations is referred to unloading of σ₃ to zero, with keeping σ₂ unchanged, and increasing σ₁ to a specific value [7], where σ₁, σ₂, and σ₃ are the maximum, intermediate, and the minimum principal stress, respectively [8]. It is found that σ₁ increases by 1.5-4.8 times to its initial value after the excavation through field testing [9].

Failure characteristics of EDZ rocks can reflect the stability of underground engineering structures. The stress condition of EDZ rocks is a key factor affecting their failure characteristics and mechanism, which is coupled biaxial static stress and dynamic stress. The static stresses (σ₁, σ₂ and σ₃) are mainly formed from tectonic or gravitational forces, and the detailed initial magnitudes of σ₁, σ₂ and σ₃ in several underground hard rock engineering structures located in China are summarized by YANG [10] as shown in Figures 1(b)-(d). The initial values of σ₁, σ₂ and σ₃ increase with the buried depth of underground engineering structures. The stresses σ₁, σ₂, and σ₃ increase at a rate of 42.8, 25.3 and 16.3 MPa/km, respectively. So, the deep buried EDZ rocks are subjected to high in-situ static stresses. The dynamic stresses are induced mostly by earthquake, rock burst, drilling, blasting, or ore dropping in great quantity, and so on, which are repetitive cyclic disturbance with frequencies of 1-5 Hz [11-14].

The failure characteristics and mechanism of rocks under coupled static and dynamic stresses are studied mostly using split Hopkinson pressure bars (SHPB) and fatigue testing machines at a laboratorial scale [15, 16]. The difference between the two kinds of testing apparatuses is that SHPB can apply dynamic impact loads with medium high strain rates ranging from 10 to 10³ s^-1, and the fatigue testing machine can apply dynamic fatigue loads with frequencies of 0-20 Hz [15, 16]. In conclusion, the dynamic loads generated by fatigue testing machines are more consistent with the field dynamic loads [11-16]. Fatigue test is an important testing method to understand the failure characteristics of rocks under coupled dynamic (σ_F) and static stresses (σ_S). The fatigue failure is caused due to long-term repetitive loads with a peak stress level (σ_max) lower than their static strength (σ_ss) [17, 18], as shown in Figure 2. A clear understanding of the time-dependent fatigue mechanical performance of EDZ rocks is significant for support design and potential hazard prevention in underground engineering [16-18].

Figure 1 Stress types and values subjected by EDZ rocks: (a) Stress types in underground mine; Buried depth of hard rock underground engineering in China versus σ₁ (b); σ₂ (c); σ₃ (d)

Up to now, most previous fatigue tests were conducted on rock specimens under uniaxial or conventional triaxial compressive stress state [19-21], as shown in Figures 2(a) and (b). According to previous studies, the key factors affecting rock fatigue behaviours are fatigue loading parameters, e.g., waveform (w_f), frequency (f), amplitude (σ_a), mean stress (σ_mean), valley stress (σ_min), and peak stress (σ_max), besides rock lithology and confining stress (σ₃). The w_f used for repetitive loads in fatigue tests include sinusoidal waves, triangular waves, and square waves [22, 23], as shown in Figure 2(c). The most frequently used f of fatigue loads in lab tests was in a range of 0.1-10 Hz, in accordance with the frequency of dynamic loads generated during earthquakes, rock bursts, and blasting, etc. [13, 14]. In fatigue tests, the σ_a and σ_max are varied for different rock specimens, and the relationships σ_mean=(σ_max+σ_min)/2 and of σ_a=(σ_max-σ_min)/2 are applicable for fatigue loading. The studies conducted in uniaxial fatigue or conventional triaxial fatigue tests show that the strain increment caused by the sinusoidal loading waveform is larger than that induced by the triangle loading waveform. It is caused due to the fact that the input energy by sinusoidal loading was larger than that caused by triangle loading [14, 19]. With the increase of σ_a and σ_max, the rocks are easier to get final fatigue failure [24]. The fatigue resistance ability of rocks increased with the increase of f or the levels of confining stress [25, 26].

In summary, the stress condition of EDZ rocks is coupled biaxial static stress and dynamic stress. It is meaningful for stability analysis of underground engineering to study the failure mechanism and properties of rocks under the coupled static-dynamic stress state. Although number of studies have been conducted by different researchers on rock fatigue behaviours, but the biaxial compressive stress state of rocks in EDZ range was ignored in all the fatigue studies of rocks. So, in this research, a series of biaxial fatigue tests (BFT) have been conducted on marble and sandstone specimens to study the fatigue characteristics of rocks under biaxial confinements.

2 Experimental methodology

2.1 Rock specimen preparation

The marble and sandstone cubical specimens with a side length of 50 mm were prepared and tested in this study, as shown in Figure 3(a). The mineral composition analysis through a light microscope using thin sections of rocks was conducted before the lab testing. From the mineral composition analysis, it was found that the marble was composed of magnesite (35%-45%), calcite　(40%-50%), and others (5%-25%) (Figure 3(b)), whereas the sandstone was composed of quartz (50%-70%), feldspar (10%-30%), and a little of muscovite and biotite (Figure 3(c)).

2.2 Testing system and loading procedure

BFT were conducted using the TRW-3000 true-triaxial testing apparatus at Central South University, China. The maximum load capacity of the apparatus is over 2000 kN. The apparatus can be used to apply load along three mutually perpendicular directions (defined as x, y and z, respectively) through different hydraulically driven pistons. The details of this apparatus can be referred to Refs.[4, 5, 27-29]. In this study, the cubic rock specimens of marble and sandstone specimens were placed in the y-z plane, as shown in Figures 4(a) and (b). Two LVDTs were used to monitor the axial and lateral deformation of specimens (Figure 5). Prior to the testing, the Vaseline and polythene sheets were applied evenly to the four surfaces of the specimens to minimize the frictional resistance of the solid loading platens.

Figure 2 Stress state and stress path of fatigue tests:

Figure 3 (a) Specimens and their size; Mineral components of (b) marble and (c) sandstone

The biaxial confinements and stress path of BFT are shown in Figures 4(c) and (d). The stress along the z direction of the true-triaxial testing apparatus is σ₁, whereas the stress along y direction is σ₂. The loading procedures are summarized as follows:

1) First, σ₁ and σ₂ were applied to the predefined values of σ₂ at a loading rate of 0.2 MPa/s;

2) Then, σ₁ was monotonically increased at a rate of 0.2 MPa/s to a predefined values of σ₁, and the valley value σ_min of fatigue loads equals to the predefined values of σ₁;

3) Finally, a sinusoidal fatigue load with peak stress σ_max of 90% or 95% σ_bcs and f of 5 Hz were applied on the specimens. σ_min and σ_max were the biaxial compression strength (σ_bcs) multiplying certain predefined ratios, as listed in Table 1.

In addition, several specimens in each of the rock types were tested in biaxial compression tests (BCT) to determine σ_bcs of rocks at different σ₂ values. In BCT, σ₁ was monotonically increased until the rock specimen failed after σ₁ and σ₂ applied to a predefined value of the σ₂. The peak value of the σ₁ in BCT was found to be σ_bcs.

3 Results and discussion

3.1 Biaxial compressive strength

The biaxial compressive strength (σ_bcs) of the cubical marble and sandstone specimens were determined and shown in Table 1. With the increase of σ₂, σ_bcs of both the rocks were increased initially, and subsequently decreased. An exponent strength criterion for the rocks in biaxial compression tests was put forward in this study, as shown in Eq. (1). The exponential equation fits appropriately with the strength results of the rock specimens under different σ₂, as shown in Figure 6.

Figure 4 TRW-3000 true-triaxial testing apparatus and testing procedure:

Figure 5 Measuring method of strain during biaxial compression loading

σ_bcs=σ_ucs+Aσ_ucsσ₂exp(1-Bσ₂)                 (1)

where σ_ucs is the uniaxial compressive strength of rocks; A and B are the material constants.

3.2 Fatigue failure properties

The detailed BFT parameters and results of both the rock types are presented in Table 2. In the study, the effective fatigue loading time was quite shorter and the fatigue lives of all specimens were lower than those in other uniaxial or conventional triaxial fatigue testing results presented in earlier published literatures, which was due to the fact that the σ_max of fatigue loads was set at a high level in this study. From the study results, several conclusions on different aspects are summarized as follows.

3.2.1 Fatigue life

Fatigue life (FL) is an important index to evaluate the resistance ability to fatigue loads of rocks, which is the applied cyclic number of fatigue loading until rock final failure. The larger the FL, the stronger the resistance ability of rocks. The main conclusions in biaxial fatigue tests are as follows:

1) The FL of specimen S-0-F1(198) is greater than that of specimen S-0-F595(98), which indicates that FL decreases with the increase of σ_max under the same σ_min and σ₂. FL of specimen S-20-F1(355) being smaller than that of specimen S-20-F790(612) indicates that FL values increase with the increase of σ_min under the same σ_max and σ₂. In conclusion, the change trend of FL in BFT is similar to that in uniaxial or conventional triaxial fatigue tests under the same confinements [20, 21];

Table 1 Predefined testing parameters and average biaxial compression strength

Figure 6 Biaxial compressive strength and fitting curves of marble and sandstone

2) FL values of the specimens M-0-F, M-10-F, and M-20-F are 55, 81, and 104, respectively; and FL of specimens S-0-F1, S-10-F1, and S-20-F1 are 198, 222, and 355, respectively, which shows a similar trend for each of rock types that FL values increase with the increase of σ₂. In the fatigue testing on the rock specimens, σ₂ can enhance the resistance capability to fatigue loads of rocks when the static strength increase phenomenon under biaxial confinements is considered, which is called the intermedium principal stress effect in BFT.

3) FL values of specimen M-0-F and S-0-595 are 55 and 98, respectively. The fatigue loads and stress confinements for the both specimens are similar, and FL of sandstone is larger than that of marble. So, it can be concluded that the soft and plastic rocks have higher ability to resist the fatigue loading.

3.2.2 Stress-strain curve

The stress-strain curves of the marble and sandstone specimens in biaxial compression and fatigue tests are shown in Figure 7. ε₁ is the strain along the σ₁ direction, whereas ε₂ is the strain along the σ₂ direction. From the study, the key conclusions can be drawn as follows:

1) Under the sinusoidal fatigue loading, the strain in hysteresis loop changes initially from greater to smaller and after few cyclic loadings, from smaller to greater. This trend is there until up to the specimen failure. While the strain hysteresis loop of the first cycle is always the largest under triangle fatigue loading [14].

2) The ultimate values of ε₁ and ε₂ in BFT tests are greater than the strain values at the peak strength in BCT under the same σ₂. The fatigue strain also equals to the post failure strain on the complete static stress-strain curve corresponding to σ_max according to Ref.[26].

Table 2 Fatigue testing parameters and results

3) With the increasing of σ₂, the ultimate ε₁ at the point of rock failure also increases. As shown in Figures 7 (a)-(c), the ultimate ε₁ values for M-0-F, M-10-F, and M-20-F are approximately 0.53, 1.02 and 1.12, respectively, whereas the final ε₁ values for S-0-F1, S-10-F1, and S-20-F1 are close to 0.76, 1.08 and 1.38, respectively (Figures 7(d), (f) and (h)).

3.2.3 Failure fragment

The main fracture plane angles of the marble specimen were vertical, whereas the sandstone specimen showed several inclined fracture planes. So, from the fracture pattern, it can be concluded that the failure pattern of the marble specimen under biaxial confinements was due to tension, while the failure pattern of the sandstone specimen was due to shearing, as shown in Figure 8.

Based on the rock fragment size, the rock fragments in BCT and BFT are divided into four categories, i.e. >20 mm, 12-20 mm, 3-12 mm and <3 mm (Figure 9). The weight of fragments in each of the categories are weighed as shown in Figure 10. The weight of the marble fragments in each of the categories increased or decreased monotonously with the increase of σ₂. While the weight of the sandstone fragments showed a V-shaped change trend with the increase of σ₂, and the turning point was under σ₂=10 MPa. For the marble, the weight of fragments with size of >20 mm in BFT was lighter than that in BCT, and the weight of fragments with size of <20 mm in BFT was heavier than that in BCT. The fragment results of marble demonstrate that when the fatigue dynamic failure of hard rocks occurs, higher number of smaller rock pieces are formed, and the energy utilization rate is higher than that in static compressive failure. So, it can be inferred that the higher the smaller rock pieces formation is, the easier the rock piece ejection and rock burst occurrence are [4, 30, 31]. For the sandstone, the weight of fragments in each of the categories are similar in BFT and BCT. So, it can be concluded that the soft and plastic rocks have higher ability to resist the fatigue loading.

3.3 Acoustic emission characteristics

Two acoustic emission (AE) sensors were attached at the front and rear free surfaces of each of the specimens during testing, and the similar AE signals were received by both sensors. So, only one sensor’ data has been used in the following analysis. The AE hit rate, count rate and b-value for each of the specimens during loading process were calculated as shown in Figures 11 and 12.

The AE hit rate and the AE count rate mean the cumulative magnitude of the AE hit and the AE count in a minute during testing for each of the specimens, respectively, and the b-value reflects relationship between the frequency and magnitude of earthquakes proposed in Refs. [32-34], as shown in Eq. (2). In the case of the AE study, Eq. (2) can be modified into Eq. (3) using the AE parameters [33]. Moreover, the b-value represents the percentage of low amplitude AE hits in comparison to the high amplitude AE hits, which means that a large b-value indicates a larger proportion of low amplitude AE hits. In this study, b-value can be divided into two categories, which is the dynamic b-value and the overall b-value. The dynamic b-value is calculated using the AE data in 5-30 s before the specimen loaded to a certain axial stress or loading time, and the overall b-value is calculated using all the AE data of a specimen during testing. The main conclusions based on AE data analysis are as follows:

lgN=a-bM                              (2)

lgN(A/20)=a-b(A/20)                      (3)

where M is the magnitude of earthquakes; N is the cumulative number of earthquakes with a magnitude of M; A is the amplitude of the AE hits; and N is the cumulative number of AE hits with an amplitude of A; a and b are empirical constants.

Figure 7 Stress-strain curves in biaxial compression and fatigue tests:

Figure 8 Failure patterns of rock specimens in BCT:

1) The value of AE hit rate is approximately one-tenth of that of the AE count rate for all the specimens, and both of them show a similar change trend for the same specimens. In BCT, the larger the σ₂ is, the longer the period of AE hit rate and AE count rate maintain at a higher level. Besides, when σ₂ is larger, multiple peaks appear in the AE hit rate and AE count rate, which provides proofs for the multiple fracture planes of rocks under biaxial confinements. In BFT, the AE hit rate and AE count rate of most specimens appear the first peak on the point of fatigue loads applied original. In both of the test types, AE hit rate and AE count rate reach their maximum values when the specimens get final failure.

Figure 9 Failure fragments and their size distribution

2) In BCT, the overall b-value for marble and sandstone decrease with the increase of σ₂, such as the overall b-value for the specimen S-0-S, S-10-S, and S-20-S are 2.136, 1.698 and 1.402, respectively, which indicates that the proportion of low amplitude AE hits decrease with the increase of σ₂, and the size and the inducing energy of cracks producing AE signals is more larger for specimens under higher σ₂. The overall b-value in BFT is lower than that in BCT except marble specimens under σ₂ of 0 and 10 MPa, which indicates that fatigue loads can fracture rocks slowly, and the small size crack dominates the fatigue of rocks, which is in concert with the distribution of failure fragments, especially for marble.

3) The last dynamic b-value equals to the overall b-value for all specimens, and starts to decrease in 30-100 s before rock failure. If the rocks are under a stable condition, e.g., the axial stress being at a low level, the AE data is few, and the dynamic b-value is very small. When rocks are under a crack stable development state, the dynamic b-value maintains at a high level. When rocks are under a crack unstable development state, the dynamic b-value starts to decrease until to the overall b-value. So, the dynamic b-value can be used to predict the failure of rock specimens in BCT and BFT, as shown in Figures 11 and 12.

4 Conclusions

The fatigue failure behaviors of rocks, i.e.,marble and sandstone, under biaxial stress conditions were firstly examined using a true-triaxial loading testing apparatus. It is more meaningful for engineering practice that the study methods used in this paper are promoted and applied to the jointed rocks or rocks under different water or temperature conditions [35-37]. The main conclusions in this study are summarized as follows:

Figure 10 Weight of fragments with different sizes of:

Figure 11 AE hit rate, AE count rate and b-value of rock specimens in biaxial compression tests:

Figure 12 AE hit rate, AE count rate and b-value of rock specimens in biaxial fatigue tests:

1) The biaxial compressive strength (σ_bcs) in biaxial compression test firstly increased, and then subsequently decreased with the increase of the intermediate principal stress σ₂, which is consistent with that in true triaxial compression tests [5].

2) There is a clear σ₂ effect on the fatigue failure of rocks in biaxial fatigue tests, even after considering σ₂ effect on σ_bcs, in which a novel　 finding in rock mechanics. So, σ₂ enhances the resistance capability to fatigue loads of rocks.

3) For hard rocks as marble in this study, the severe failure, such as rock burst, can be induced by fatigue loads easier than that induced by quasi-static loads. For soft rocks as sandstone in this study, the fatigue failure is similar to the compressive failure, and the soft rocks have higher ability to resist to the fatigue loading than hard rocks.

4) Before the point of rock failure, the value of AE hit rate, AE count rate and the dynamic b-value decrease. So, these parameters from AE data can be used to predict the failure of rocks in biaxial compression tests and biaxial fatigue tests.

Contributors

DU Kun and CHEN Shao-jie provided the concept and edited the draft of manuscript. LI Xue-feng, YANG Cheng-zhi and ZHOU Jian conducted the tests, the literature review and wrote the first draft of the manuscript. DU Kun, ZHOU Jian, CHEN Shao-jie and MANOJ Khandelwal edited the draft of manuscript.

Conflict of interest

DU Kun, LI Xue-feng, YANG Cheng-zhi, ZHOU Jian, CHEN Shao-jie and MANOJ Khandelwal declare that they have no conflict of interest.

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(Edited by ZHENG Yu-tong)

中文导读

双轴应力边界和疲劳载荷作用下岩石的力学特性

摘要：本文通过开展双轴压缩试验和双轴疲劳试验，研究了大理岩和砂岩的破裂特性。双轴压缩试验结果表明，随着中间主应力的增大，岩石的双轴抗压强度呈先增大后减小的变化趋势。双轴应力边界内岩石的疲劳破坏与疲劳载荷峰值水平、中间主应力大小和岩性有关。随着疲劳载荷峰值水平的增加，岩石的疲劳寿命降低。中间主应力对岩石抗疲劳性能有明显的强化作用，在相同的应力比下，岩石的疲劳寿命随着中间主应力的增大而增大。通过试验中声发射以及破坏后岩石碎片分析，发现相同试验条件下砂岩具有更高的抗疲劳性能，而大理岩产生更多的小尺寸碎片。疲劳载荷作用下的破岩效率更高，也更容易诱发剧烈的硬岩岩爆灾害。

关键词：双轴应力边界；疲劳载荷；声发射；岩石碎片；中间主应力

Foundation item: Projects(51774326, 41807259) supported by the National Natural Science Foundation of China; Project(MDPC201917) supported by Mining Disaster Prevention and Control Ministry Key Laboratory at Shandong University of Science and Technology, China

Received date: 2020-06-10; Accepted date: 2020-08-10

Corresponding author: CHEN Shao-jie, PhD, Professor; Tel: +86-532-86057948; E-mail: chensj@sdust.edu.cn; ORCID: https://orcid. org/0000-0003-1377-0808