China has confirmed coal reserves of 1.4Gt, accounting for approximately 97% of its total fossil energy (BP Amoco 2021). China’s primary energy consumption has consistently featured coal resources, maintaining a share of 60% in 2018 (China Mineral Resources 2019). In recent years, following the declaration of carbon peak and carbon neutrality targets (75th United Nations General Assembly 2020), the proportion of coal consumption in China has decreased. However, it still accounts for over 50% of primary energy consumption (BP Amoco 2022). However, with depletion of coal resources in the eastern regions, the coal mining in China is gradually shifting to the central and western regions which are extremely ecologically fragile (Yao et al. 2020; Zhang et al. 2021). Relevant data indicate that coal reserves in the western regions of China (incl. Shanxi, Shaanxi, Inner Mongolia, Xinjiang, and Ningxia) account for 76.2% of total reserves, while water resources make up only 1.6-6.6% of the total reserves (National Bureau of Statistics of China 2018; Han et al. 2022b), forming a unbalanced distribution pattern of coal-water resources.

The scarcity of water resources has become a significant limiting factor for mining operations in such arid regions. The strata movement can have a substantial impact on the groundwater storage (Yao et al. 2020). The high-intensity mining breaks the overlying strata, shifting the groundwater flow from interlayer to vertical, infiltrating fractures and cavity areas, leading to water resource losses (Yu et al. 2022a), see Fig. 1. Statistics show that for every ton of coal mined in China, 1.87 m³ of mine water is generated, with an average utilization rate of merely up to 35% (Gu et al. 2021). Moreover, out of 14 coal production bases in China, 9 are located in ecologically vulnerable western arid regions (Gu et al. 2021). Given the shallow burial depth of coal seams, severe strata weathering, high water content in the overlying strata (Yan et al. 2018), mining can cause severe ecological damage in these region (Verma 2014; Li 2016). How to store and utilize mine water in arid regions and preserve groundwater has become a crucial issue. The United States and Australia use the high-head pumps to discharge mine water directly to surface (Sivakumar et al. 1994). Belgium builds pumped hydroelectric storage (PHS) systems in the upper part of abandoned mines, which store the energy through the interconversion of potential energy and electricity, but this system is highly dependent on the existence of high terrain (Pujades et al. 2019). Eastern Africa, Brazil, India and Japan use subsurface dams to create aquifers for mine water storage, although this method is expensive (Ishida et al. 2011). Furthermore, numerous scholars have conducted research on “water conservation mining”, e.g., strip mining and backfill mining, which prevents the extension of fractures from deep coal seams to water aquifers. However, fractures in shallow coal seams will inevitably develop to surface (Gu 2015; Yao et al. 2020). Gu (2015) proposed the concept of coal-mine-based underground water reservoirs (CMUWR) with a principle of “transfer, store, use water”. This framework utilizes the goaf as storage spaces and embeds artificial dams in coal pillars located in roadways to create underground reservoir for mine water storage. The connections between CMUWR are through boreholes set in the dams, allowing for the mine water storage and transfer. Currently, the CMUWR has been applied in 32 arid mining areas in central and western China, with a maximum water storage capacity of 31 million cubic meters (Gu et al. 2021).

Schematic diagram of CMUWR. Groundwater flows across the overlying strata and penetrates vertically through the cracks into the caved areas, the CMUWR is mainly composed of coal pillars and artificial dams located in the roadway

As shown in Fig. 1. The coal pillars of CMUWR are subjected to long-term strata pressure, lateral water pressure, and water immersion (Hu et al. 2019). Therefore, many researchers have focused on exploring the interrelations between coal pillar stability and effects of water immersion (Poulsen et al. 2014; Hashiba and Fukui 2015; Chen et al. 2017; Wang et al. 2019; Yao et al. 2019; Ai et al. 2021; Han et al. 2022b) to investigate the damage process of coal due to water-rock interactions. Numerous studies have indicated that the action of water soaking can lead to change in composition of most rocks (sandstone, limestone, basalt, tuff, etc.). The generation of pores and the extension of microscopic fractures result in the weakening of mechanical properties of rocks (Yilmaz 2010; Daraei and Zare 2018). The intrusion of water has been proven to cause long-term damage to coal and rocks, including a decrease in tensile & compressive strengths, static friction coefficient, stiffness, and elastic modulus (Vergara and Triantafyllidis 2016; Zhou et al. 2016). The overlying strata pressure, lateral water pressure, and water immersion are the three primary factors influencing the stability of coal pillars (Hushmand et al. 2016; Hu et al. 2019). Prior research has emphasized predominantly the combined effects of one or two of these factors only. However, the relationships between these factors are more complex. For example, the studies often oversimplified the form of in-situ stress, typically representing them as monotonic loading, constant-amplitude or multi-level cyclic load (Elliott and Brown 1986; Jafari et al. 2003; Sorgi and Gennaro 2011; Meng et al. 2016; Bagde 2016; Li et al. 2019; Tang et al. 2019; Wang et al. 2022; Liang et al. 2022). Moreover, field monitoring has shown that strata pressure and lateral water pressure can be influenced by strata periodic weighting, water level fluctuations, and blasting procedures (Yerkes and Castle 1976; Ghamgosar et al. 2017; Yu et al. 2022b). As a result, the combination and variations of loading patterns, amplitudes, and frequencies for strata stress and lateral water pressure are complex (Bagde and Petroš 2005; Pan et al. 2010; Li et al. 2012; Zhong et al. 2019; Luo 2020; Liu et al. 2022; Liang et al. 2022). This study pays attention to the realistic in-situ stress of coal pillars, investigating the deformation characteristics, energy dissipation, and fatigue characteristics of dry and water-soaked coal under dynamic normal and shear cyclic loading with varying frequencies and amplitudes. This study aims to provide theoretical and practical insights for the design and long-term safety assessment of CMUWR from experimental insights.

2.1 Sample preparation and characterization

The samples in this study were obtained from Shendong coalfield in Inner Mongolia and are classified as bituminous coal. Shendong coalfield serves as a well-established application base for CMUWR. Hence, it is reasonable to assume that the coal pillars of CMUWR exhibit the similar physio-mechanical properties as the samples used in this study. 10 cubic bituminous coal samples, each with a dimension of 100 mm × 100 mm × 100 mm, were made for experimental tests (refer to Figs. 2a and c). The sample surface was hand-polished to meet the flatness proposed by International Society for Rock Mechanics (I.S.R.M.) (Bieniawski and Bernede 1979). To illustrate the distribution of cracks at the surface of coal samples before mechanical testing, two different surfaces of sample M6 (Views A and B, as shown in Figs. 2d and f) are presented. The corresponding sketches (Figs. 2e and g) reveal the cracks on both surfaces, with the main cracks having a maximum width of approximately 1.5 mm and sub-cracks about 0.5 mm wide. In Fig. 2e, the main-cracks intersect perpendicularly with the sub-cracks at the points M and N, indicating the potential for a preferential crack propagation path. Crack orientations mainly manifest in vertical or horizontal patterns, but predominantly are vertical, see sketches in Figs. 2e and g.

Coal sample characterization a An overview of 10 cubic coal samples (M1-M10) b Geometry measurement of sample M2 c Geometry measurement of sample M4 d Surface of M6 from view A e Sketch of the surface of M6 from view A, where the main and sub cracks are indicated f Surface of M6 from view B e Sketch of the surface of M6 from view B, where the main and sub cracks are indicated

Before conducting mechanical tests, the detailed measurements of geometric and physical properties of the samples were performed and the results are presented in Table 1. The rightmost column in Table 1 indicates the water soaking treatment before mechanical tests. Samples M1-M6 were in a natural (dry) state, while M7-M10 underwent an approximately 45-day water immersion. The changes in sample mass due to water immersion were recorded for M7-M10. Coal with or without water immersion characterize the different states of coal pillars of CMUWR above and below the water table. Examining the mechanical behaviors of coal exposed to intricate combinations of normal and shear stresses is of paramount importance. Such investigations play a vital role in evaluating coal’s durability and its ability to withstand compressive and shear stress (Yao et al. 2020; Wang et al. 2020a; Zhang et al. 2021; Ma et al. 2023). This is especially critical when coal serves as the dam of CMUWR as it directly impacts the design of CMUWR and determines the maximum water storage capacity (Yao et al. 2021; Tang et al. 2022; Zhang et al. 2023).

2.2 Testing apparatus

The mechanical tests were conducted at Sun Yat-sen University using a DJZ-500 shear box, as elaborated in (Dang et al. 2022), where a detailed description of the device’s functionalities can be found. The diagram of shear box, as well as the schematic arrangement of the normal and shear LVDTs, are illustrated in Fig. 3. The horizontal shear displacement is measured at the red dot near the shear LVDT, as shown in Fig. 3b. For the measurement of normal displacement of the sample, the readings were recorded using 4 LVDTs positioned above the sample, as demonstrated in Fig. 3d. Figure 3c shows the structure of the shear box, in which the upper section of the shear box is fixed, while the lower part is horizontally movable and can be controlled using either mode of displacement (strain rate) or force (stress rate). The frame’s stiffness is approximately 5 GN/m. The normal load was applied to the specimen through the upper loading plate and the vertical piston, while the shear load was applied through the horizontal piston.

Configuration of shear box and layout of normal and shear LVDTs a An overview of DJZ-500 shear box b main measuring components including vertical and horizontal LVDTs c structure of shear box, the upper section is fixed, while the lower part is horizontally movable d layout of 4 normal LVDT

2.3 Test schemes: normal and shear stress paths

Given the intricate nature of applied stress paths, a tabular format may not effectively convey all details. To improve readability, we present stress paths through the schematic diagrams in Fig. 4. The stress paths are divided into two main categories: the “Pretest” (Sect. 2.3.1) and “Cyclic shear tests” (Sects. 2.3.2 and 2.3.3). Within cyclic shear tests, there are two distinct stages: the “stable stage” (Sect. 2.3.2) and “loading to failure stage” (Sect. 2.3.3).

2.3.1 Pretest: monotonic shear stress with constant & cyclic normal stress

Samples M1 and M2 underwent pretest to determine essential mechanical properties, such as shear strength and modulus. The stress paths for pretest are shown in Fig. 4a and b. The samples underwent a continuous increase in shear stress, following a monotonic manner. For sample M1, see Fig. 4a, the normal stress was fixed to 5 MPa, which is a value equivalent to the theoretical vertical in-situ stress for a coal pillar situated at a depth of approximately 250 m (a bulk weight of 20 kN/m³ for above strata is used). In case of sample M2, see Fig. 4b, the cyclic normal stress was applied, with the minimum normal stress set at 3 MPa, and the upper-stress limit was gradually increased by 1 MPa after every 10 cycles. When the initial monotonic normal load (at a rate of 30 kN/min) reached the lower limit of normal cyclic stress 3 MPa, a plateau period (lasting 100s) ensued. After this period, the shear stress for both M1 and M2 increased at a rate of 6 kN/min. For M2, the normal cyclic stress was applied in a form of triangular wave with a fixed frequency of 0.1 Hz. Ten cycles were applied at each cyclic loading stage (CLS). The monotonic shear strengths at the onset of failure are 5.58 MPa for M1 and 4.35 MPa for M2, respectively. It’s important to note that the calculation of normal and shear stresses was based on the cross-sectional area of the intact specimen: namely 100 mm × 100 mm.

2.3.2 Cyclic shear tests: stable stage

The remaining eight specimens were grouped into four distinct Groups, denoted by “G” in the subsequent sections to indicate the number of groups, as outlined below:

Group 1 (G1)

M3, M7.

Group 2 (G2)

M4, M8.

Group 3 (G3)

M5, M9.

Group 4 (G4)

M6, M10.

Each group consists of two samples, with one set in a dry (natural) state (M3, M4, M5, M6), and the other set in a water-soaked state (M7, M8, M9, M10). Two samples within each group underwent the identical stress path. In stable stage, the stress was applied at a low level. None of these specimens failed during the period, this allows researchers to observe the mechanical responses of coal samples to moderate variations in normal and shear stress. Four distinct groups (G1-G4) of stress patterns are detailed below.

2.3.2.1 Group 1 (G1): incremental multi-level normal and shear stresses

The normal and shear stresses for M3 and M7 followed a multi-level cyclic increase pattern. The incremental amplitude of maximum stress in the normal direction is 1 MPa, with a fixed lower limit stress of 3 MPa, while the incremental amplitude in shear stress was 0.5 MPa, with a constant lower limit of 0.5 MPa. The four sub-stress paths, denoted as G1-1 to G1-4, were applied sequentially, as plotted in Figs. 4c and f.

In G1-1, both normal and shear stresses were applied at a frequency of 0.1 Hz, with each CLS lasting 100 s (10 cycles). In G1-2, the frequency was increased to 0.2 Hz for both normal and shear stresses, indicating that 20 cycles (100 s) were applied for each CLS. In G1-3, the shear stress frequency was twice that of the normal direction, and the number of cycles per CLS was 10 for normal stress and 20 for shear stress. The frequencies in the normal and shear directions, as well as the number of cycles applied in G1-4, were opposite to the scheme employed in G1-3. The frequencies and applied cycles of the 4 sub-stress paths in G1 are as follows.

G1-1: normal stress 0.1 Hz (10 cycles each CLS), shear stress 0.1 Hz (10 cycles each CLS).

G1-2: normal stress 0.2 Hz (20 cycles each CLS), shear stress 0.2 Hz (20 cycles each CLS).

G1-3: normal stress 0.1 Hz (10 cycles each CLS), shear stress 0.2 Hz (20 cycles each CLS).

G1-4: normal stress 0.2 Hz (20 cycles each CLS), shear stress 0.1 Hz (10 cycles each CLS).

2.3.2.2 Group 2 (G2): constant amplitude cyclic normal stress and incremental multi-level cyclic shear stress

For M4 and M8, Figs. 4g and j illustrate four sub-stress paths which are denoted as G2-1 to G2-4 in sequence. In contrast to the stress paths presented in Figs. 4c and f, the amplitude of the cyclic normal stress was fixed within the range 3-7 MPa, while the amplitude of cyclic shear stress increased by an amplitude of 0.5 MPa. The lower limit for cyclic shear stress remained fixed at 0.5 MPa. The frequency and the number of cycles in each CLS were consistent with G1, namely the scheme described in Sect. 2.3.2.1.

2.3.2.3 Group 3 (G3): incremental multi-level cyclic normal stress and constant amplitude cyclic shear stress

For M5 and M9, Figs. 4k and n show four consecutive sub-stress paths of G3, denoted as G3-1 to G3-4. In these paths, the amplitude of the cyclic shear stress was fixed within the range of 0.5-2 MPa, while the cyclic normal stress followed a step-up pattern with an incremental amplitude of 1 MPa and a fixed lower limit of 3 MPa. The frequency and the number of cycles in each CLS remained consistent with the scheme described in Sect. 2.3.2.1.

2.3.2.4 Group 4 (G4): incremental multi-level cyclic normal stresses and incremental multi-level cyclic shear stress

For M6 and M10, Figs. 4o and r show four consecutive sub-stress paths of G4, G4-1 to G4-4. Notably, the normal and shear stresses followed distinct patterns with the former schemes: the cyclic normal stress exhibited a step-down pattern with a decremental amplitude of 1 MPa and a fixed lower limit of 3 MPa, while the cyclic shear stress featured a step-up amplitude of 0.5 MPa with a fixed lower limit of 0.5 MPa. The frequency and the duration of each CLS remained consistent with the scheme described in Sect. 2.3.2.1.

2.3.3 Cyclic shear tests: loading to failure

After stable cyclic loading, as discussed in Sect. 2.3.2, the samples were loaded under specific stress paths until reaching failure. Based on the type of stress paths, the 8 samples (M3-M10) can be categorized into two groups: M3, M5, M7, and M9 followed the first stress path, while M4, M6, M8, and M10 followed the second one. In both stress paths, the normal stress was fixed at 5 MPa, and each CLS contains 10 cycles of shearing. The major distinction between the two stress paths lies in the frequency of cyclic shear stress: the first stress path held a frequency of 0.1 Hz, while the second one escalated the shear stress frequency to 0.2 Hz. The stress path is illustrated in Figs. 4s and t, with different colorful markers indicating the failure shear stress for each of the eight samples.

Normal and shear stress paths: Pretest: a M1 b M2; Cyclic shear tests, stable stage: G1: c G1-1 d G1-2 e G1-3 f G1-4; G2: g G2-1 h G2-2 i G2-3 j G2-4; G3: k G3-1 l G3-2 m G3-3 n G3-4; G4: o G4-1 p G4-2 q G4-3 r G4-4; Cyclic shear tests, loading to failure: s Stress path for M3, M5, M7, M9 t Stress path for M4, M6, M8, M10

This chapter conducts an analysis of four key aspects: 1. deformation characteristics, 2. energy dissipation, 3. secant loading modulus, 4. roughness of shear surface.

3.1 Deformation behavior

The definitions of terminologies used in this study are as follows:

Normal strain:

Normal strain is defined as the ratio of the average of the four vertical displacements recorded by 4 LVDT to the original height of the cubic coal sample.

Nominal shear strain (subsequently referred to as shear strain): Shear strain is defined as the ratio of the shear displacement the original width of the cubic coal sample.

Normal stress:

Normal stress is determined by dividing the normal force by the cross-sectional area of the cubic coal sample.

Shear stress:

Shear stress is determined by dividing the shear force by the cross-sectional area of the cubic coal sample.

SP:

An abbreviation for “Stress Path” experienced during stable cyclic shear tests, including SP1-SP4, SP1 comprises G1-1, G2-1, G3-1, G4-1. Similarly, SP2, SP3, and SP4 follow this pattern.

3.1.1 Deformation behaviors under cyclic shear tests: stable stage

3.1.1.1 Group 1 (M3 and M7)

Figures 5a-5h illustrate the evolution of normal and shear strains observed at the highest (peak strain) and lowest (residual strain) normal and shear stresses across the G1-1 to G1-4. For M3 (dry) and M7 (wet), the corresponding strain rates for these samples are shown in Figs. 5i-5p. The strain rate is calculated using the formula (ε_n-ε₁)/n, where ‘n’ represents the number of cycles of a CLS. We take n as 10 when the frequency is 0.1 Hz and n as 20 when the frequency is 0.2 Hz.

The normal and shear deformations in wet sample, M7, consistently exhibits a higher value than that observed in the dry sample, M3, across G1-1 to G1-4 with the varying loading frequencies. Notably, the peak and residual strains of M3 (dry) are approximately 30-50% of those in M7 (wet). Moreover, the strain rates in M7 (wet) are generally higher than in M3 (dry), particularly during the initial two CLS. This finding highlights the substantial influence of water soaking on coal deformation under G1, with the water-soaking treatment inducing more pronounced and rapid deformations, particularly in the first two CLS.

The presence of micropores and cracks in the initial coal samples result in more significant normal and shear strains in the initial SP (G1-1) compared to the other three SP. Notably, the normal strain rate exhibited an inverse trend with respect to shear strain rate, indicating a clear competitive relationship between normal and shear deformations. This phenomenon is most evident during the first two SP in G1 (G1-1 and G1-2). For instance, in Figs. 5i and m, the normal strain rate (red and green bars) for coal sample M3 (dry) exhibits a decreasing and then increasing trend, while the corresponding shear strain rate showed an opposite trend, see the red arrows.

In G1-2, the loading frequency is twice that of G1-1. The strain rate of M3 (dry) in G1-2 notably decreases in both normal and shear directions, while the normal strain rate of M7 (wet) experiences a significant decrease, and even remains negative, with a simultaneous sharp increase in the corresponding shear strain rate. This suggests that when normal and shear frequencies are equal, an increase in frequency can accelerate the shear deformation in soaked coal samples under G1. The frequencies in G1-3 and G1-4 are not the same in normal and shear stresses. Specifically, the shear frequency in G1-3 is twice as the normal frequency, and the normal frequency in G1-4 is twice as the shear frequency. Under these conditions, the strain rate of the dry coal sample (M3) demonstrates a clear monotonic increase (indicated by black arrows) in the high-frequency loading direction, as seen in the red and green bars in Figs. 5l and o. This higher strain rate is observed compared to the low-frequency direction, as depicted in Figs. 5k and p. This indicates that in G1, when the frequencies of normal and shear stresses do not coincide, the higher frequency promotes an increase in strain rate for dry samples, and this trend applies to both normal and shear directions, with more pronounced strain increase observed at the peak stress.

Strain and corresponding strain rates of samples M3 (dry) and M7 (wet) in G1. Normal strain: a G1-1 b G1-2 c G1-3 d G1-4; Shear strain: e G1-1 f G1-2 g G1-3 h G1-4; Normal strain rate: i G1-1 j G1-2 k G1-3 l G1-4; Shear strain rate: m G1-1 n G1-2 o G1-3 p G1-4

3.1.1.2 Group 2 (M4 and M8)

Figures 6a and h show the evolution of normal and shear strains observed in samples M4 (dry) and M8 (wet) across G2-1 to G2-4. The normal cyclic stress amplitude for this group remains constant within the range of 3-7 MPa, while cyclic shear stress increases gradually. The lower limit is set to 0.5 MPa, with the upper limit incrementally rising by an amplitude of 0.5 MPa between two adjacent CLS.

From the incremental cyclic normal stress in G1 to the constant amplitude cyclic normal stress in G2, a noticeable change in the pattern of normal strain becomes apparent, which is characterized by a gradual and steady alteration. In the case of dry sample (M4), a linear increase in strain rate is observed in the high-frequency loading direction for G2-3 and G2-4, as indicated by the red and green bars (marked by black arrows) in Figs. 6l and o. As for shear strains, the majority of CLS show notably positive rates for both peak and residual strains, attributed to the gradual increase in the upper limit of cyclic shear stress. However, for normal strains, some CLS exhibit negative strain rates, which is a phenomenon observed in the last three SP. The positive or negative peak strain rate is a result of competition between deformations from two orthogonal (normal and shear) directions. Compared to the fixed amplitude normal stress, the incremental cyclic shear stress dominates. The increasing shear stress leads to a progressive increase in shear deformation. Under peak stress conditions, the increase in shear strain results in a slight dilatancy in the normal direction (indicated by negative strain rate). This suggests that normal stress is relatively weaker in competition with shear stress during this phase. Nonetheless, this phenomenon is not evident at the minimum stress, namely the residual strain as seen in the positive green bar graphs in Figs. 6i and l. This is because both minimum normal and shear stresses are fixed, and in this state, the competition between normal and shear strains reaches a “quasi-equilibrium”. In the G2, the difference in deformation in normal direction between dry and wet samples is not manifest. However, the water immersion continues to enhance the deformation of coal samples in the shear direction.

Strain and corresponding strain rates of samples M4 (dry) and M8 (wet) in G2. Normal strain: a G2-1 b G2-2 c G2-3 d G2-4; Shear strain: e G2-1 f G2-2 g G2-3 h G2-4; Normal strain rate: i G2-1 j G2-2 k G2-3 l G2-4; Shear strain rate: m G2-1 n G2-2 o G2-3 p G2-4

3.1.1.3 Group 3 (M5 and M9)

Figures 7a and h present the evolution of normal and shear strains in G3, M5 (dry) and M9 (wet). Within this group, the shear cyclic stress amplitude remains fixed within the range of 0.5-2 MPa, while the normal cyclic stress amplitude increases. It is evident that both the normal and shear deformations of the wet sample (M9) exceed those of the dry sample (M5), indicating that the water immersion significantly enhances coal sample deformations under G3. As the normal cyclic stress amplitude gradually increases, there is an overall upward trend in normal deformation, with all strain rates remaining positive.

Across all four SP, the shear strain rate exhibits significantly higher values during the first CLS, followed by a substantial decrease in subsequent SP, resulting in the fluctuation of strain rate around zero, as observed in Figs. 7m and p. This behavior is attributed to the gradual increment in normal stress. Due to the presence of a competition between normal and shear stress, the difference between normal stress and shear stress is minimal during the first CLS. At this stage, the strain rate is higher, but as the upper limit of normal stress gradually increases, shear stress begins to lose the advantage, and in some cases, even results in dilatancy as indicted by the negative strain rate. Through the results in G1-G3, it is evident that the deformations of water-soaked samples in both normal and shear directions are significantly greater than those of dry samples when subjected to an incremental cyclic normal stress, regardless of the shear stress path.

Strain and corresponding strain rates of samples M5 (dry) and M9 (wet) in G3. Normal strain: a G3-1 b G3-2 c G3-3 d G3-4; Shear strain: e G3-1 f G3-2 g G3-3 h G3-4; Normal strain rate: i G3-1 j G3-2 k G3-3 l G3-4; Shear strain rate: m G3-1 n G3-2 o G3-3 p G3-4

3.1.1.4 Group 4 (M6 and M10)

Figures 8a and h illustrate the evolution of normal and shear strains for M6 (dry) and M10 (wet) in G4. In this group, the cyclic normal stress amplitude undergoes a stepwise decrease, while the cyclic shear stress amplitude experiences a stepwise increase.

Strain and corresponding strain rates of samples M6 (dry) and M10 (wet) in G4. Normal strain: a G4-1 b G4-2 c G4-3 d G4-4; Shear strain: e G4-1 f G4-2 g G4-3 h G4-4; Normal strain rate: i G4-1 j G4-2 k G4-3 l G4-4; Shear strain rate: m G4-1 n G4-2 o G4-3 p G4-4

As CLS increases, the normal strain displays a noticeable decreasing trend, and the normal strain rate predominantly remains negative, especially for the strain at peak stress, as shown in Figs. 8a, d, i and l. This normal dilation is caused by the dual effect of reduced normal stress and increased shear stress. In all four SP (G4-1 to G4-4), the shear strain rates most exhibit a large positive value, with an overall trend of initial increase between SP1 and SP2 then followed by a decrease. The decreasing shear strain rate after SP2 suggests that the coal samples were gradually compacted along the shear direction, resulting in an enhancement of shear stiffness. The strain of the water-soaked sample M10 is significantly lower than that of coal sample M6 both in the shear and normal direction, see Figs. 8a and h. This differs from the deformation results obtained from G1-G3. The decrease in normal stress weakens the influence of water-soaking on deformation.

The effects of water-soaking, stress paths, and loading frequency on normal and shear strains can be drawn as follows based on the results of G1-G4.

1. (1)

   When normal stress is maintained without reduction, water soaking leads to a significant increase in both the normal and shear deformations of coal samples.

2. (2)

   There is a noticeable competition mechanism between normal and shear stresses. The strain rate on the dominant stress direction exhibits significantly higher values, while the strain rate on the weaker stress direction is lower or even negative, resulting in the dilation.

3. (3)

   The strain rate in the direction of high-frequency stress continues to grow with the CLS in the case that shear and normal stresses are not decreased.

3.1.2 Deformation behaviors under cyclic shear tests: loading to failure

To reach the failure of each sample, a specific loading phase was employed after the stable stage, involving the application of a constant normal load along with stepwise increased cyclic shear stress. The stress paths are shown in Fig. 4q-r. Figure 9 illustrates the evolution of normal and shear strains under cyclic stresses for both dry and wet samples, namely, M3-M10. To clearly demonstrate the strain evolution and because it is convenient for comparison, the strain for the first cycle for each sample is zeroed. The plots exhibit four types of strains: peak normal strain, residual normal strain, peak shear strain, and residual shear strain, where ‘peak’ and ‘residual’ denote the strains recorded at the maximum and minimum stresses, respectively. Due to the constant normal stress, the peak normal strain corresponds to peak shear stress. Samples with and without water soaking are marked by different symbols, with dry samples represented by circles and water-soaked samples represented by triangles.

Zeroed strain in the phase of loading to failure. Zeroed normal strain: a Peak strains for M3, M5, M7 and M9; b Residual strains for M3, M5, M7 and M9; c Peak strains for M4, M6, M8 and M10; d Residual strains for M4, M6, M8 and M10; Zeroed shear strain: e Peak strains for M3, M5, M7 and M9; f Residual strains for M3, M5, M7 and M9; g Peak strains for M4, M6, M8 and M10; h Residual strains for M4, M6, M8 and M10

The results indicate that the samples exhibit a longer fatigue life under a higher frequency (0.2 Hz). From Figs. 9a and d, it is noticed that the normal deformation finally remains negative associated with a decreasing trend (see the black box), which is an indication of dilation. Notably, the dilation is more pronounced particularly at peak normal stress, when the load is applied at a higher frequency (0.2 Hz). The shear strain gradually increases with a quasi-linear trend under progressively increased shear stress. Notably, the water soaking treatment of the samples significantly impacts the shear peak strain, with the strains of water-soaked samples generally larger than those of dry samples under the same SP, as observed in Figs. 9e and g. For instance, in Fig. 9e, the strains of water-soaked sample M7 (blue triangles) are consistently higher than those of dry sample M3 (red circles) under the same SP. These findings underscore the significant influence of factors such as frequency and water content on the fatigue properties of coal samples.

3.2 Energy dissipation

Cyclic loading can lead to the accumulation of energy dissipation, which is of great importance in fatigue analysis. The accumulation of energy may cause the development of microscopic cracks in rocks and eventually lead to catastrophic failure. Therefore, energy as a key parameter in rock cycling experiments has been intensively studied by many scholars (Takarli et al. 2008; Moradian et al. 2010; Momeni et al. 2015; Liu et al. 2016). In rock fatigue tests, three primary energy parameters are typically considered: input energy (E_i), elastic recovery energy (E_r), and dissipation energy (E_d). Figure 10 illustrates a stress-strain curve, showing the calculation of these energy parameters. The input energy (shaded in red) represents the cumulative energy input during the loading phase of a cycle, while the elastic recovery energy (shaded in blue) reflects the elastic energy recovered during the unloading phase. The dissipated energy (shaded in green) is the portion of energy that is not fully released as strain energy during the unloading phase of the cycle. The formulas for calculating these energy parameters are presented in Eqs. (1)–(3). This section is primarily focused on the analysis of dissipated energy in the samples, to unveil the impacts of loading path, frequency, and water soaking on energy evolution.

Illustration of different types of energy during cyclic loading: Input energy (E_i), recoverable strain energy (E_r), and dissipated energy (E_d), where σ_min and σ_min’ are respectively the stress levels at beginning and end of a cycle, σ_max is the peak stress of a cycle

3.2.1 Energy dissipation under cyclic shear tests: stable stage

Figure 11 illustrates the normal stress induced cumulative dissipated energy (refer to normal cumulative dissipated energy hereafter) for both wet and dry samples under the G1-G4. In G1 and G3, the growth rate of normal cumulative dissipated energy exhibits an upward trend with CLS, see in Figs. 11a, d, i and l. This trend is directly incurred by the stepwise increase in normal stress. Due to constant normal stress amplitude, the normal cumulative dissipated energy in G2 follows an approximately linear growth pattern, as indicated in Figs. 11e and h. Meanwhile, in G4 with decremental normal stress, the growth rate of normal cumulative dissipated energy demonstrates a distinct decreasing trend, as depicted in Figs. 11m and p. Figure 11 also reveals that water soaking exerts a significant influence on the normal cumulative dissipated energy. In most of plots, the normal cumulative dissipated energy of water-soaked sample exhibits a lower value, as indicated by blue points in Figs. 11d and p.

For G1-G4, when a higher frequency (0.2 Hz) is applied in the normal stress, the normal cumulative dissipated energy exhibits a higher value, roughly doubling those at the lower frequency (0.1 Hz). Specifically, SP2 and SP4 were subjected to a normal stress with a frequency of 0.2 Hz, while SP1 and SP3 were loaded with a normal stress with a frequency of 0.1 Hz. The total normal cumulative dissipated energy for SP2 and SP4 is approximately twice that of SP1 and SP3. For the higher frequency (0.2 Hz), the number of normal loading cycles was 20 cycles for a fixed loading time of 100 s, which is twice the number of loading cycles at the lower frequency (0.1 Hz). With twice the number of cycles, the total cumulative dissipated energy under high-frequency normal stress approximately doubled the energy at the lower frequency. This suggests that the impact of frequency on cumulative dissipated energy in the normal direction is relatively unpronounced. This is because the total normal cumulative dissipated energy is proportional to the number of cycles applied, the moderate variation in frequency applied in this work (0.1–0.2 Hz) doesn’t significantly affect the normal dissipated energy of a single cycle.

Normal cumulative dissipated energy. G1: Samples M3 (dry) and M7 (wet): a G1-1 b G1-2 c G1-3 d G1-4; G2: Samples M4 (dry) and M8 (wet): e G2-1 f G2-2 g G2-3 h G2-4; G3: Samples M5 (dry) and M9 (wet): i G3-1 j G3-2 k G3-3 l G3-4; G4: Samples M6 (dry) and M10 (wet): m G4-1 n G4-2 o G4-3 p G4-4

In Fig. 12, the evolution of cumulative dissipated energy induced by shear stress is shown (refer to shear cumulative dissipated energy hereafter). The growth rate of shear cumulative dissipated energy exhibits a consistent upward trend with CLS in G1, G2, and G4. This trend is directly related the stepwise increase in shear stress. The shear cumulative dissipated energy in the first CLS shows a relatively low value, and the growth trend is non-pronounced. This is due to the initial shear stress (1 MPa) being at a disadvantage in competition with normal stress (5 MPa, 7 MPa, and 8 MPa), making it hard to accumulate plastic deformation in shear direction. As the upper limit of shear stress increases, the growth rate of shear cumulative dissipated energy exhibits a stepwise increase. In G3, where constant amplitude stress is applied, the shear cumulative dissipated energy displays an approximately linear growth trend. It is noteworthy that the shear cumulative dissipated energy also exhibits a higher value under high-frequency (0.2 Hz), roughly doubling the dissipated energy at low-frequency. Similarly, it is shown that moderate variation in shear load frequency (0.1–0.2 Hz) has a quiet limited effect on the shear dissipated energy of a single cycle.

In contrast to normal direction, the shear cumulative dissipated energy is also influenced by water soaking. This is primarily reflected by the higher value of shear cumulative energy for soaked coal samples (indicated by yellow dots in Fig. 12), especially noticeable in G3, as shown in Figs. 12i and l. In G4, the normal cumulative dissipated energy demonstrates a high growth rate in the first CLS, whereas the counterpart in shear cumulative dissipated energy exhibits a growth rate nearly approaching zero, as indicated by the red boxed line in Fig. 12. This is a consequence of the substantial difference between the initial normal and shear stress. The higher normal stress (maximum value of 8 MPa) holds a significant advantage in competition with the shear stress (maximum value of 1 MPa), resulting in a notable amount of damage deformation in the normal direction. As the upper limit value of shear stress incrementally rises, the impact of shear stress on energy dissipation is gradually dominant in competition with normal stress. The shear dissipated energy increases, while the normal dissipated energy decreases to a significant low level, a flat trend is notably observed in CLS 4 of G4, as indicated by the black box in Fig. 11. The contrasting growth patterns of cumulative dissipated energy in CLS 1 and CLS 4 of G4 effectively illustrate the competitive relationship between normal and shear stresses.

Overall, the influence of frequency on dissipated energy of a single cycle under the same stress level is relatively unpronounced both for normal and shear stress. The water soaking exerts a significant impact on cumulative dissipated energy and demonstrates an opposite result in energy dissipation in normal and shear directions.

Shear cumulative dissipated energy. Samples M3 (dry) and M7 (wet) in G1 mode: a G1-1 b G1-2 c G1-3 d G1-4; Samples M4 (dry) and M8 (wet) in G2 mode: e G2-1 f G2-2 g G2-3 h G2-4; Samples M5 (dry) and M9 (wet) in G3 mode: i G3-1 j G3-2 k G3-3 l G3-4; Samples M6 (dry) and M10 (wet) in G4 mode: m G4-1 n G4-2 o G4-3 p G4-4

3.2.2 Shear energy dissipation under cyclic shear tests: loading to failure

Figure 13 presents the evolution of shear cumulative dissipated energy for samples during the phase of loading to failure. Each figure illustrates both dry and wet samples exposed to the identical stress path and frequency. Dry samples are represented by green symbols, while water-soaked samples are denoted by yellow symbols. All four samples (M3, M5, M7, and M9) with a lower frequency (0.1 Hz) have the lower shear failure strengths, see Figs. 13a and b, indicating that the samples are more susceptible to shear stress with a lower frequency.

During the first two CLS (0–20 cycles), the shear cumulative dissipated energy is mainly distributed around zero. However, as the upper limit of shear stress gradually increases, the shear stress gradually prevails over the influence of normal stress (fixed to 5 MPa), leading to an increase in shear deformation. Consequently, the shear cumulative dissipated energy exhibits an upward trend. Before the failure, the cumulative dissipated energy shows a surge, as evidenced by black box in Fig. 13.

Shear cumulative dissipation energy in the failure phase. 0.1 Hz: a Dry sample M3 and wet sample M7 b Dry sample M5 and wet sample M9; 0.2 Hz: c Dry sample M4 and wet sample M8 d Dry sample M6 and wet sample M10

3.3 Secant modulus

3.3.1 Secant modulus under cyclic shear tests: stable stage

The evolution of both normal and shear secant modulus for samples in G1-G4 is presented in Fig. 14. Here the secant modulus is calculated as the slope of the line from the beginning of each cycle pointing to the peak stress. The results are displayed in the form of box plots, where the top and bottom of the I-beam legend signify the maximum and minimum modulus for each CLS, respectively. The bars within the box plots represent the mean values of secant moduli, with line segments connecting the mean values of each CLS to illustrate a trend indicating the changes in mean values over time. Dry samples are represented in black, while water-soaked samples are in red.

Four groups of moduli in G1: a Normal of samples M3(dry) and M7(wet) b Shear of samples M3(dry) and M7(wet); Four groups of moduli in G2 mode: c Normal of samples M4(dry) and M8(wet) d Shear of samples M4(dry) and M8(wet); Four groups of moduli in G3 mode: e Normal of samples M5(dry) and M9(wet) f Shear of samples M5(dry) and M9(wet); Four groups of moduli in G4 mode: g Normal of samples M6(dry) and M10(wet) h Shear of samples M6(dry) and M10(wet)

In general, the normal secant modulus remains within the range of 2-15 GPa. In contrast, the shear secant modulus exhibits significant fluctuations in certain CLS, and even negative shear moduli are observed. This occurs when shear stress is in a disadvantaged position in orthogonal competition with normal stress, resulting in reduced shear deformation or even dilation.

In G1, the average normal secant modulus experiences a stepwise decrease with ongoing of CLS. This trend is also observed in G3. This is attributed to step increase of normal stress. Conversely, in G2 and G4, where the normal stress remains constant amplitude (G2) or decreases (G4), the average normal secant modulus exhibits a stepwise increase with CLS. It can be known from above conclusions that the evolution of the normal secant modulus is mainly affected by the normal stress and has little relationship with the shear stress. The increase in normal stress can impose obvious damage to coal’s normal stiffness. When the normal stress decreases or remains a constant-amplitude, the normal deformation will be affected by a combination of the normal and shear stresses, leading to the strain hardening due to unloading or dilation, which increases the normal secant modulus. The normal secant modulus displays a consistent trend across all four different stress paths in the G1-G4 modes, here refers to Figs. 14a, c, e and g, indicating that frequency has unpronounced impact on secant modulus. Notably, the water-soaking significantly affects the secant modulus. Specifically, in G1, G2, and G4, where shear stresses incrementally increase, the mean modulus of water-soaked samples (red bars) is higher than that of dry samples (black bars). In contrast, the reverse occurs in G3, where the shear stress remains at a constant amplitude. In this case, the mean secant modulus of dry samples surpasses that of water-soaked samples. This suggests that the water soaking leads to a higher normal secant modulus in terms of mean value when the shear stress increases incrementally, while it results in a lower normal secant modulus mean value when shear stress remains at a constant amplitude.

3.3.2 Secant modulus under cyclic shear tests: loading to failure

Figure 15 illustrates the evolution of shear secant modulus under the phase of loading to failure. In this stage, as the normal stress is set to constant, only the shear secant modulus can be determined. The figure is divided into two sections: A and B. Section A provides an overview of the modulus across the complete range of cycles, with a gray box that is enlarged to create Section B. Triangles and circles in the figure represent dry and water-soaked samples, respectively, and the colored bars in the legend denote the magnitudes of the modulus.

It is evident that the shear secant modulus exhibits great fluctuations in the first two CLS. It underscores the presence of a competitive mechanism between normal and shear directions. When the normal stress remains fixed at 5 MPa, the lower shear stress (1–2 MPa) has huge difficulty causing the effective deformation, resulting in substantial fluctuations in shear secant modulus that spanning positive and negative ranges. As shear stress increases, the competition between normal and shear stresses reaches a quasi-equilibrium or even favors the shear. During this period, shear deformation gradually rises, and the variability of shear secant modulus tends to stabilize. In the stabilization phase, the modulus undergoes gradual and slow reductions, primarily falling within the range of 0.5-3 GPa. When close to the shear failure, all shear secant loading moduli are below 1 GPa.

Secant loading modulus in the failure phase. 0.1 Hz: a M3(dry) and M7(wet) b M4(dry) and M8(wet); 0.2 Hz: c M5(dry) and M9(wet) d M6(dry) and M10(wet)

3.4 Roughness of failure surface

3.4.1 Characterization of failure surface roughness

Surface roughness plays a crucial role in influencing the mechanical properties of jointed rocks, e.g., peak shear strength (Homand et al. 2001; Grasselli and Egger 2003; Yang et al. 2016), the characteristics of shear fractured zone (Hayward et al. 2019; Wang et al. 2023), and nonlinear fluid flow in rock fractures (Chen et al. 2015; Wang et al. 2016b). Initially, Barton and his colleagues (Barton 1973; Barton and Choubey 1977) introduced the concept of Joint Roughness Coefficient (JRC) and developed the JRC-Joint Compressibility Softening (JCS) model based on rock shear tests. As engineering applications developed, various methods have been proposed to characterize surface roughness, e.g., straight-edge graphical method (Barton and Bandis 1982), fractal dimension (D) (Turk et al. 1987), mean square (MS), root mean square roughness index (RMS), autocorrelation function (ACF), structural function (SF), root mean square of the first-order deviation of the profile (Z₂) (Myers 1962; Tse and Cruden 1979), standard deviation of the angle (SD_i) (Yu and Vayssade 1991), and other statistical parameters. Many researchers have explored the effects of roughness on shear strength (Zandarin et al. 2013; Renaud et al. 2019; Wang et al. 2023), hydraulic properties (Jiang et al. 2006; Luo et al. 2022), excavation damage zones (EDZ) (Tsang et al. 2005; Hao et al. 2016; Zhao et al. 2020), stress wave propagation (Li et al. 2018; Yuan et al. 2023), damage patterns (Hong et al. 2016; Asadizadeh et al. 2018; Han et al. 2022a), and shear rates (Wang et al. 2016a; Liu et al. 2023) in jointed rocks. However, there has been limited research on the roughness of failure surfaces resulting from the cyclic shearing of intact coal samples under complex normal and shear stress paths.

Through the analysis of 112 digitized rock joint profiles, Li and Zhang (2015) introduced 15 indexes for the Joint Roughness Coefficient (JRC), among which the root mean square of first-order profile deviation, Z₂ offers a good characterization of the roughness correlation coefficient. In our study, we employ Z₂ as an index to quantify the roughness of failure surface. The equation for Z₂ (Tse and Cruden 1979) is presented below:

Where, N represents the total number of discrete points acquired along the joint contour; Z_i is the height of the joint surface, mm; and ∆x is the sampling interval. The DaVinci 3D® scanner (refer to Fig. 16a) (Wang et al. 2020b) was employed to perform the 3D scanning of failure surface in a scale of 100 × 100 mm. Subsequently, the 3DMine® software was used to create a 3D morphological model of the damaged surface of coal sample (depicted in Figs. 16b and c). The failure surface was divided into multiple profiles along the shear direction, with each profile spaced 9 mm apart, thereby 10 parallel surface contours (illustrated as green wireframes). The coordinate values of these contour lines were then digitized for analysis. Ultimately, the roughness index Z₂ was computed for each contour line using Eq. (4). Figure 16b provides a visual representation of the 3D roughness morphology on the failure surfaces of samples M4, M5, and M7. To ensure uniformity, the elevation of the highest point of the surface was set to zero. A color gradient legend was utilized to show the elevation, with red indicating the highest point and blue representing the lowest point, see Fig. 16c. Regrettably, due to the brittle nature of coal, many of the samples were not able to measure the roughness of the failure surface. Consequently, we selected M4, M5, and M7 as representatives for analysis.

Measurement of surface roughness based on 3D scanning a 3D scanning device used in this work: DaVinci 3.0® and scanning process b Rough models of M4, M5, and M7 after scanning c The roughness of the failure surface with the elevation indicated, the elevation is normalized where the highest point was set to zero

3.4.2 Size effects of sampling intervals

To reveal the influence of sampling interval on roughness, various sampling intervals ranging from 2 to 20 mm were applied. Through the analysis of the scanned model, it is observed that the individual resolution scale constituting the model (side length of a small equilateral triangle) is approximately 1.15 mm, see Fig. 17a. To ensure the accuracy of the data, the sampling interval should be greater than the scale of resolution. Therefore, we recommend a sampling interval of 2 mm in this study, balancing data processing and accuracy. Figure 17b illustrates the change in Z₂ under different sampling intervals. The upper and lower boundaries of I-beam legend represent the maximum and minimum values of Z₂ of selected ten parallel profiles, respectively. The square symbols correspond to their average values, connected by line segments to create a trend line depicting the variation of mean values.

a Size of scan resolution, approximately 1.15 mm b Variation of roughness coefficient Z₂ with sampling interval c Contour lines for ∆x = 2 d Contour lines for ∆x = 3 e Contour lines for ∆x = 5

Z₂ exhibits a decrease as the sampling interval increases. This trend can be divided into two parts. In the first stage (Part I) spanning from 2 to 5 mm, Z₂ experiences a rapid decrease, while the second stage (Part II) between 5 and 20 mm displays a slower, more fluctuating trend. This observation, as shown in Fig. 17b, highlights a direct correlation between the roughness of the cyclic shear damage surface and the chosen sampling interval. To illustrate this relationship, we chose the L6 contour line of M4 as a representative, highlighted as the blue contour line in Fig. 18f. In Figs. 17c and e, we compare the regenerated M4-L6 contour lines at different sampling intervals of 2, 3, and 5 mm (see red dotted line) with the scanned M4-L6 contour line (the black line is generated by scanner with a default resolution around 1.15 mm). The smaller sampling intervals allow for a more accurate reproduction of the original surface contour, but also incur larger processing time. Conversely, increasing the sampling interval to a greater level can introduce obvious errors between the regenerated contour line and the scanned contour line, such as illustrated by the blue boxed line in Fig. 17e. Therefore, the critical threshold for sample interval is identified as 5 mm. Beyond this value, the roughness coefficient no longer exhibits a consistent decrease with the increase in sampling interval but instead shows a fluctuation.

3.4.3 Relationship between roughness and dissipated energy

Figures 18d and f present a 3D model about the ten parallel profiles spaced 9 mm apart along failure surface, the corresponding average roughness coefficients Z₂ under the interval of 2 mm for these profiles are displayed in Figs. 18a and c. Notably, it is observed that Z₂ exhibit higher values at profile L1 and L10, as evident in Figs. 18a and c. This phenomenon is attributed to the splitting of coal pieces from the shear surface. In Fig. 18h, the upper and lower boundaries in the legend represent the maximum and minimum values of Z₂, while the solid box denotes the average roughness coefficient Z₂, and the circles denote the Z₂ values for the ten profiles. As indicated by the green arrows, the average Z₂ of three samples (M7, M5, and M4) exhibit an increasing trend with failure strength (50, 60 and 90 MPa). A lower failure strength (e.g., M7, 50 MPa) is associated with a smaller average Z₂, whereas a higher failure strength (e.g., M4, 90 MPa) corresponds to a larger average Z₂. A higher shear strength will result in the extension of the shear surface in a more divergent morphology. Figure 18g presents the distribution of energy dissipated of the cycle prior to failure and the average Z₂ for the selected three samples exposed to cyclic shear tests. It is evident that the Z₂ follows an opposite evolutionary trend as the dissipated energy, with a lager value of Z₂ associated the lower dissipater energy. A higher Z₂ indicates a rougher shear surface. In comparison to a relatively smooth shear surface, the elevated roughness is advantageous in constraining shear displacements of specimens, thereby reducing the accumulation of shear deformation and effectively lowering the energy dissipation during the shear.

Average roughness coefficient Z₂ for three samples under the case when the sampling interval ∆x = 2: a M7 b M5 c M4; 3D outline of the selected 10 profiles with a spacing of 9 mm: d M7 e M5 f M4; g Relation of energy dissipated and average roughness coefficient Z₂ versus the cycle number at failure h Average roughness coefficient Z₂ for M4, M5 and M7, when sampling interval is 2 mm

In the dynamic cyclic shear test, with the change of normal and shear stresses, the rock surface was found deviating from the horizontal, or rotated (Dang et al. 2016). Dang et al. (2016) conducted cyclic shear tests on jointed rocks and revealed that the rotation is caused by inhomogeneous force distribution. As the shear continues, the reaction force is gradually uneven, and the contact area of shear surfaces gradually decreases. This causes a difference in diagonal lengths of the upper and lower jointed rock masses, leading to the rotation. The rotation angle (φ) grows with an increase in normal load while decreasing with an increase in shear velocities. In the localized shear range, the rotation can lead to high stress concentration, which can result in extra damage. Existing research on rotation angle in shear tests primarily has focused on jointed rocks, with almost no investigations on intact coal can be found in cyclic shear tests. In our tests, the vertical displacements of sample were recorded by four high-precision LVDTs positioned at four corners above the shear box, see Fig. 19a. These four sets of vertical displacements for coal samples in the normal direction are denoted as y₁, y₂, y₃, and y₄, as shown in Fig. 19b. The four LVDTs measure the change in vertical displacement of the upper shear box with respect to the lower one at the four corners. The value is close to sample normal displacement at the respective corners.

In Fig. 19b, a schematic illustration of rotation angle is given. When the averaged displacements on one side (y₁ + y₄)/2 are larger or smaller than those on the opposite side, (y₂ + y₃)/2, the sample surface deviates from the horizontal, resulting in rotation. It’s important to note that here the rotation angle only considers its magnitude, without specifying the direction of rotation. It is expressed through the following equation:

Where, l represents the length of sample, while (y₁ + y₄)/2 signifies the average displacement of one side of the sample in the horizontal direction. Similarly, (y₂ + y₃)/2 denotes the average displacement of the other side of the sample in the horizontal direction. These values are from the four LVDTs positioned above the sample.

Schematic diagram of the calculation of the rotation angle. a Four LVDTs of the shear box b Schematic of the calculation of the rotation angle

The evolution of rotation angle is shown through the data of M7, M8, M9, M10, respectively. To account for the impact of stress path on the rotation angle, the analysis focuses on the SP1 of G1-G4 (namely G1-1, G2-1, G3-1, G4-1) because of no prior loading phase, see Fig. 20. The normal and shear stresses for each sample are shown on the x-coordinate axes of the left and right figures, while the y-coordinate representing the magnitude of the rotation angle. It is evident in both G1 and G2 (Figs. 20a and d), where the change in the rotation angle is directly proportional to the variations in both normal and shear cyclic stress. The rotation angle holds a larger magnitude at the upper bound of stress than the lower bound. As cyclic stress increases, the rotation angle shows a stepwise rise. In G3, where the constant amplitude cyclic shear stress and incremental cyclic normal stress was applied, the rotation angle doesn’t exhibit a pronounced step-increase in the normal direction. Notably, the rotation angle shows a smaller value at the upper bound of shear peak stress. In G4, with gradual decreasing normal stress, the rotation angle only shows an evident increase in the first CLS, indicating that the decrease in normal stress restricts the growth of rotation. To sum up, the rotation angle is sensitive to the change of normal load, the rotation is more obvious with the increase in normal load, which is consistent with Dang et al. (2016).

The above results demonstrate that when the normal and shear stresses are applied at the same time, the magnitude of rotation angle is influenced by both stresses, exhibiting a roughly positive correlation. The relationship between the rotation angle and normal stress is more prominent. An increase in normal stress can result in an increased rotation angle, while a decrease in normal stress can mitigate the growth of rotation, leading to a gradually reduced rotation angle.

Evolution of the sample’s rotation angle with cyclic stress under the SP1. a M7 Normal b M7 Shear c M8 Normal d M8 Shear e M9 Normal f M9 Shear g M10 Normal h M10 Shear

We conducted experimental tests to investigate the mechanical responses of coal subjected to complex normal and shear stress paths. The experiments revealed the influence of stress paths, water soaking and frequency on deformation, energy dissipation, secant modulus, and failure surface roughness, the following conclusions can be drawn:

1. (1)

   When normal and shear stresses are applied simultaneously, a clear competition exists between these two types of stresses. On the direction (refer to normal or shear stress) where stress dominates, the strain rate is notably higher, while on the other side, a much lower strain rate is observed or even demonstrates a tendency to expand namely the dilatancy occurs.

2. (2)

   Samples subjected to a higher frequency (0.2 Hz in this work) exhibit a longer fatigue life than the case of 0.1 Hz. When normal and shear stresses hold distinct frequencies, the strain rate in the direction with a high-frequency continues to increase in cases where shear and normal stresses are not reduced. The dissipated energy of a single cycle under both normal and shear stresses are almost frequency independent within the range of [0.1–0.2 Hz]. During the phase of loading to failure under cyclic shear stress, the coal is more likely to exhibit dilation in the normal direction under the influence of high-frequency shear stress (0.2 Hz).

3. (3)

   Water soaking significantly increases both the normal and shear deformation of coal when normal stress is not reduced. During the stable phase of cyclic shear tests, the shear strains for water-soaked samples are generally higher than the dry samples. Water soaking is more likely to result in a higher normal secant modulus when shear stress follows a stepwise increasing pattern.

4. (4)

   The average roughness coefficient Z₂ of failure surface exhibits an increasing pattern with increase in shear strength under cyclic shear test, the elevated roughness of a shear surface is advantageous in constraining shear displacements of specimens, thereby reducing the accumulation of shear deformation and effectively lowering the energy dissipation during the shear.