The failure modes of different bedding slate under uniaxial compression

Slate, characterized by its layered structure, exhibits mechanical properties predominantly influenced by structural planes. To investigate the mechanical behavior and failure modes of slate under varying dip angles of structural planes, uniaxial compression tests were conducted on slate specimens. The study systematically examined the effects of dip angles on compressive strength, stress-strain curves, and failure patterns. Additionally, a particle flow model was established using PFC-2D to simulate the failure process of slate, with detailed analysis of stress responses under different dip angles. The results indicate that the compressive strength of slate exhibits significant anisotropy across different bedding inclinations. Overall, the strength distribution follows a “U-shaped” curve as the inclination angle varies. During uniaxial compression, the deformation process of slate specimens progresses through four distinct stages: compaction, elastic deformation, elastoplastic deformation, and failure. Three primary failure modes were identified: composite shear-tensile failure along bedding planes, shear failure along bedding planes, and compression-induced tensile failure involving both bedding planes and the rock matrix. Numerical simulations of uniaxial compression tests on slate with varying inclination angles were conducted using the particle flow code PFC-2D. The simulated stress response clearly reflects the mechanical behavior, and the failure modes observed in the models consistently match those from laboratory tests. Based on detailed analysis of stress evolution, the critical boundaries between different failure modes were accurately determined.

1 Introduction

Slate is widely distributed in stratigraphic formations, and its engineering geological properties are controlled by structural planes, exhibiting significant anisotropy characterized by low strength, weak interlayer cohesion, and susceptibility to water softening (Yang et al., 2024; Taotao et al., 2024). Such rock masses are prone to large deformations and failure, which can even lead to the destruction of supporting structures, directly impacting the safe construction and operation of engineering projects. Therefore, studying the failure modes of slate is of great significance for stability analysis and instability prevention of slope rock masses and surrounding rock in underground chambers.

Research on the mechanical properties of layered rock masses extends beyond basic mechanical parameters, often focusing on the influence of fissures on mechanical behavior and their evolution through experiments or numerical simulations. For instance, Haeri et al. (2014) investigated the effects of cracks at different angles on rock strength, stress, and strain under uniaxial compression, analyzing the failure process; Lee and Jeon (2011) studied the propagation process and evolution of fractures in rocks under uniaxial compression; Yin and Yang (2018) observed that the uniaxial compressive strength of bedded sandstone fluctuates with increasing bedding inclination; Lee and Pietruszczak (2008) deduced the influence of different structural plane inclinations on compressive strength in triaxial compression tests on layered rock masses; Li et al. (2018) revealed the failure evolution of typical layered rock specimens based on uniaxial compression tests; Feng et al. (2014) investigated the failure modes of phyllite under different moisture contents through triaxial compression tests; Tien and Tsao (2000) examined the varying effects of structural plane inclination on the compressive strength of layered rock masses; Hou et al. (2015) analyzed mesoscopic cracks in shale with different orientations during uniaxial compression tests using scanning electron microscopy. Zhengzhao et al. (2006)and Zhu et al. (2004) identified different fracture modes of heterogeneous rocks using RFPA software; Wang et al. (2014) obtained the failure evolution laws of schist through numerical simulation; Zhao et al. (2016) studied the crack propagation processes in different rocks using digital imaging technology. Regarding the anisotropy of slate, Chen et al. (2023) investigated the relationship between the triaxial peak strength, residual strength, and bedding angle of slate; Wang et al. (2016) found that the presence of weak bedding planes at different inclinations is the main cause of anisotropic failure mechanisms in slate; Liu (2013), (LIU, 2013) analyzed the relationship between uniaxial compressive strength, failure modes, and bedding angle in slate; Gao et al. (2011) derived the deformation and strength characteristics of sandy slate influenced by bedding inclination through uniaxial and triaxial compression tests; Liu et al. (2009) established a strength criterion for slate based on uniaxial and triaxial compression tests; Zhao et al. (2022) revealed the relationship between bedding angle and fracture modes of carbonaceous slate through Brazilian splitting tests; Ning et al. (2017) used PFC simulations to discover the influence of schistosity planes on the failure process of slate in Brazilian splitting tests; Hu et al. (2021) explored the response characteristics of uniaxial compressive strength in carbonaceous slate to multi-parameter systems using ABAQUS software; Ningning et al. (2022) simulated the failure behavior of layered slate under quasi-static compression using a bonded particle model; Kang et al. (2016) elucidated the crack evolution process during compressive failure of slate through laboratory tests and numerical simulations. Pham et al. (2024) utilizes the discrete element method (DEM) to explore the thermal–mechanical coupling behavior of slate. A thermal-degradation function for the parallel bond model is proposed, reasonably capturing the temperature’s effect on the strength and failure characteristics of the slate. Weng et al. (2022) samples three types of slate and two types of schist to investigate the mechanical properties of foliation. A series of pull-off tests and direct shear tests are conducted on the foliations to obtain the tensile and shear strengths, and a failure criterion for foliation is proposed. Park et al. (2018) presents the three-dimensional bonded-particle discrete element modeling of mechanical behavior of transversely isotropic rock as an extension to the previous work conducted in two dimensions (Park B, Min KB. Bonded-particle discrete element modeling of mechanical behavior of transversely isotropic rock.

Carbonaceous slate is developed in the water source area of the Bailong River Water Diversion Project, and its mechanical behavior directly affects slope stability. Therefore, Based on uniaxial compression tests, this study employs PFC-2D software to innovatively conduct a comprehensive multi-angle simulation analysis of slate specimens. The research focuses on determining the precise angular boundaries of failure modes and investigating the propagation patterns of cracks under uniaxial compression conditions. aiming to provide references for stability prediction and disaster prevention of reservoir bank slopes.

2 Test scheme

2.1 Specimen preparation

The Bailongjiang Water Diversion Project is a large-scale inter-basin water transfer project designed to divert water from the Bailong River in Gansu Province to the Jinghe and Weihe River basins. The primary objectives of the project are to address industrial and domestic water demands in regions such as Qingyang, Pingliang, and Tianshui, meet water supply needs for energy development in the Longdong area, and resolve water resource constraints in the Tianshui-Guanzhong Economic Zone. By transferring water resources from the relatively water-abundant Bailong River basin to the arid and semi-arid regions in the east, the project aims to alleviate water scarcity, support socio-economic development, and enhance regional water security in northwestern China. The slate samples were collected from the dam site area of the Daigu Temple Reservoir in the Bailong River Water Diversion Project, located in Diebu County, Gansu Province. The samples are Lower Silurian carbonaceous slate, with an average spacing of approximately 3 mm between adjacent schistosity planes. According to the “Standard for Test Methods of Engineering Rock Masses,” the samples were processed into standard cylindrical specimens with a diameter of 50 mm and a height of 100 mm. Due to the distinct layered structure of the obtained slate samples, the angle between the schistosity plane and the base of the cylinder in the uniaxial compression test is defined as θ. Specimens with θ = 0°, 15°, 30°, 45°, 60°, 75°, and 90° were prepared accordingly (as shown in Figure 1). The experiment involved the preparation of fourteen slate specimens in total. For each designated angle, two samples were manufactured: one maintained in a dry state and the other in a saturated state.

Figure 1

Figure 1. Diagram of slates and their structural plane dip angles.

The specimens were placed in an oven at 100 °C for 2 days for drying. After cooling to room temperature, water absorption tests were conducted until the specimens reached saturation. The measured parameters of the slate specimens are as follows: Natural Density: 2.762–2.77 g/cm³; Saturated Density: 2.779–2.786 g/cm³; Dry Density: 2.75–2.761 g/cm³; Water Absorption: 0.100–0.114.

2.2 Testing procedure

Uniaxial compression tests were conducted using a YAW6206 servo-controlled compression testing machine (Figure 2), which has a loading capacity of approximately 2000 kN. The loading rate was set to 0.06 mm/min. Axial load was applied until the slate specimens were completely failed. Throughout the loading process, data points were recorded at 5-s intervals until the test concluded.

Figure 2

Figure 2. YAW6206 servo pressure testing machine.

3 Analysis of test results

3.1 Stress-strain process

Based on the uniaxial compression tests, the stress-strain process of the slate specimens can be broadly divided into four stages: the compaction stage, the elastic deformation stage, the elastoplastic deformation stage, and the failure stage (Figures 3, 4).

Figure 3

Figure 3. Stress-strain curves of slate specimens in dry state.

Figure 4

Figure 4. Stress-strain curves of slate specimens in saturated state.

The stress-strain process of the slate specimens exhibits significant differences depending on the dip angle of the structural plane.

The difference in the first stage, compaction, is particularly remarkable. As shown in Figure 3, under dry conditions, specimens with θ = 15° and 0° show a distinct compaction stage, characterized by a noticeably concave-upward stress-strain curve. For specimens with other dip angles, the compaction stage is inconspicuous, and the elastic stage is reached rapidly after loading begins. This indicates that the compression of internal structural planes is more pronounced, resulting in a longer compaction stage, when θ is 15° and 0°. Water saturation also significantly influences the stress-strain process. As shown in Figure 4, under saturated conditions, the interlayer cementitious material in specimens at all dip angles is more readily compressed, leading to a clearly observable compaction stage. Furthermore, the compaction stage generally becomes more pronounced with smaller θ values.

In the second stage, elastic deformation, specimens under both conditions experience a prolonged period of elastic deformation, which constitutes a major portion of the entire uniaxial compression test. The stress-strain curves in Figure 3 show slopes that are largely linear, indicating clear linear elastic characteristics.

The third stage, elastoplastic deformation, shows considerable variation between dry and saturated states. The stress-strain curves for dry specimens exhibit significant fluctuations, particularly at θ = 75°, where repeated oscillations occur. This suggests a substantial influence of the internal structural planes on the mechanical behavior at this angle. Under saturated conditions, the rock strength decreases, and the influence of the structural planes diminishes.

In the fourth stage, the failure stage, the post-peak curves for dry and saturated states show little difference. After the stress reaches its peak, it begins to drop rapidly. Most specimens failed directly, with the stress quickly decreasing to zero, demonstrating relatively clear brittle characteristics.

Overall, under uniaxial compression, whether in a dry or saturated state, the slate specimens exhibit significant elasticity with negligible plasticity. After the peak stress is reached, it drops rapidly to zero. This is especially true for specimens with θ = 15° and 0°, which fragmented completely upon failure, characterizing the typical brittle failure behavior of rock.

3.2 Strength variation analysis

As shown in Figure 5, the compressive strength of the layered rock mass exhibits significant anisotropy with the variation in the dip angle θ of the structural plane. When θ increases from 0° to 45°, the compressive strength of the slate decreases rapidly under both dry and saturated states. As θ continues to increase from 45° to 90°, the compressive strength under both states shows a continuous increase. It is noteworthy that the compressive strength reaches its minimum value at θ = 45° for both states, representing an approximately 83% decrease compared to the strength at θ = 0°. This indicates that the influence of different structural plane dip angles on the mechanical properties of the layered slate has a distinct interval effect.

Figure 5

Figure 5. Plot of compressive strength vs. laminar inclination angle.

Test data reveal that water exerts a certain influence on the mechanical performance of the layered rock mass. Comparison between dry and saturated specimens shows that the compressive strength of saturated specimens is generally lower than that of their dry counterparts. As θ increases from 0° to 90°, the softening coefficients are 0.85, 0.96, 0.91, 0.76, 0.75, 0.96, and 0.54, respectively.

3.3 Elastic modulus variation analysis

The elastic modulus, a mechanical parameter representing the ability of a solid material to resist deformation, intrinsically reflects the material’s inherent properties. A smaller elastic modulus indicates that the solid material is more susceptible to deformation. As illustrated in Figure 6, the elastic modulus shows a significant trend of variation with the increasing dip angle of the layered structural plane. The elastic modulus continuously decreases in the range of 0°–45°, and then continuously increases from 45° to 75°, exhibiting a U-shaped variation pattern of initial decline followed by a rise. The elastic modulus is at its minimum at θ = 45°. Compared to the baseline state at θ = 0°, the elastic modulus at θ = 45° shows a decrease of 65%.

Figure 6

Figure 6. Plot of modulus of elasticity versus laminar inclination.

At relatively small angles (e.g., 0° or 15°): The schistosity planes of the rock specimen are nearly orthogonal to the uniaxial loading direction. The specimen exhibits high densification characteristics during the compaction stage. Its axial compressive strength is enhanced, and the loading rate maintains good synchronization with the axial deformation, demonstrating typical linear elastic characteristics.

When θ is between 30° and 60°: The elastic modulus in this range is relatively smaller compared to the previous case. This is because the shear-slip effect along the structural planes dominates within this interval, granting the rock a greater capacity for deformation.

Notably, at θ = 90°: Although the schistosity planes are theoretically parallel to the loading direction, the measured elastic modulus value is smaller than that at θ = 75°. This is likely because the weakly cemented material along the bedding planes becomes ineffective. The specimen essentially behaves as a stack of independent vertical plates, which undergo significant bending under applied axial stress. This substantial bending deformation leads to a smaller measured elastic modulus.

3.4 Failure modes

By extracting the cracks generated on the surface of the slate specimens and projecting them along the specimen height, the relationship between crack distribution and the schistosity planes was compared. Under uniaxial compression, the failure modes of the slate specimens can be broadly categorized into three types:

1. Composite Failure (θ = 0° and θ = 15°): This is a composite failure involving tensile-shear cracks penetrating through the schistosity planes and shear slippage along them.

As shown in Figure 7. Vertical shear failures of varying scales occur within the specimen. Because the schistosity planes are nearly perpendicular to the loading direction, the failure surfaces predominantly cut across them. Simultaneously, the thickness of individual rock layers is much smaller than the specimen diameter. Incoherent deformation begins to develop horizontally between different layers, inducing strain concentration and subsequent crack initiation. As loading progresses, these microcracks coalesce and penetrate the structural planes, forming a composite shear-tensile failure mode. This eventually leads to violent bursting, fragmenting the specimen into pieces.

1. 2. Shear Slip (θ = 30°–60°): This failure mode encompasses shear failure both penetrating through and occurring along the schistosity planes.

Figure 7

Figure 7. θ = 0°and θ = 15° The distribution characteristics of cracks on the surface of slate samples. (a) θ = 0°. (b) θ = 15°.

As shown in Figure 8. At θ = 30°–45°: Failure primarily involves penetration through the schistosity planes. The fracture surfaces at these angles are more complex compared to those at smaller angles, exhibiting irregular cracks, some of which traverse the top and bottom ends. However, a general trend of development along the structural planes is observed. Cracks in saturated specimens are more complex than in dry ones. This is attributed to the presence of pore water within the saturated slate’s microstructure. Under axial stress, water molecules act as a lubricant, facilitating the sliding of mineral particles. This seepage effect increases the deformability of the slate and promotes crack generation.

Figure 8

Figure 8. θ = 30°–60° The distribution characteristics of cracks on the surface of slate samples. (a) θ = 30°. (b) θ = 45°. (c) θ = 60°.

At θ = 60°: Failure occurs primarily as shear slip along the schistosity planes. The failure mode is quite typical. Failure initiates within the structural plane regions of the specimen and then propagates towards the upper and lower ends. As fissures generated by the axial stress continually intersect the pre-existing structural planes, a main crack eventually forms. When propagating cracks encounter these pre-existing planes, a portion of the energy is released. This mechanism explains why the compressive strength is lower than that of rock at θ = 0°, a phenomenon observed at other angles as well. In this state, the shear strength of the schistosity planes governs the uniaxial compressive strength, resulting in its lower value.

1. 3. Compression-Induced Tensile Splitting (θ = 75°–90°): Failure occurs as tensile splitting along the schistosity planes.

As shown in Figure 9. Under these schistosity plane angles, the rock cracks exhibit noticeable tensile branching. Since the schistosity planes are parallel to the loading direction, axial load application readily generates transverse tensile stress perpendicular to these planes. Consequently, tensile cracks initiate and propagate along the schistosity planes, progressively developing until the specimen ultimately delaminates and splits.

Figure 9

Figure 9. θ = 75°-90° The distribution characteristics of cracks on the surface of slate samples. (a) θ = 75°. (b) θ = 90°.

4 PFC-2D numerical simulation analysis of crack propagation in slate

4.1 Parameter configuration

Uniaxial compression tests on slate specimens with varying structural plane dip angles were simulated using PFC-2D. The specimen dimensions were set to 50 mm × 100 mm. The simulation was terminated when the slate specimen was completely crushed. The primary objective of the numerical simulation experiments was to analyze the failure modes and crack propagation characteristics of the specimens. The parameters adopted for the simulation are listed in Table 1.

Table 1

Table 1. Main control parameters of slate model.

By comparing the crack distribution patterns (Table 2) from laboratory tests and numerical simulations for θ values of 0°, 15°, 30°, 45°, 60°, 75°, and 90°, it is observed that the results are fundamentally consistent. The compressive strength and stress-strain curves also show substantial agreement, thereby validating the accuracy of the parameter selection for the PFC-2D numerical simulation.

Table 2

Table 2. Comparison of crack distribution between laboratory tests and numerical simulations.

4.2 Peak stress variation

Based on the established model framework, PFC-2D numerical simulations were conducted across the full range of θ ∈ [0°, 90°] to investigate the continuous evolution of the mechanical response in the rock mass under different structural plane dip angles. The simulation results indicate that the peak stress initially decreases and then increases with the dip angle of the weak interlayer, exhibiting an approximately “U-shaped” distribution (Figure 12).

As shown in Figure 10, the numerical simulation demonstrates that the compressive strength of the slate specimen reaches its minimum at θ = 45°, with the overall pattern confirming the “U-shaped” distribution. However, a notable strength transition is observed: the compressive strength is 108 MPa at θ = 30° but drops sharply to 50 MPa at θ = 31°. Similarly, at θ = 69° the strength is 62 MPa, and it jumps to 84 MPa at θ = 70°. The sharp strength increase and decrease across these specific angular thresholds are likely attributable to a fundamental change in the dominant failure mechanism. This suggests that 31° and 70° can be inferred as the critical boundaries differentiating the three primary failure modes.

Figure 10

Figure 10. Simulation curves of slate compressive strength at different structural plane angles.

4.3 Failure modes

Similar to the laboratory uniaxial compression test results, the failure modes observed in the numerical simulations of slate specimens can also be categorized into three types: composite tensile-shear failure along the schistosity planes, shear failure along the schistosity planes, and compression-induced tensile splitting along the schistosity planes.

1. Composite Tensile-Shear Failure along Schistosity Planes (θ < 31°):

The failure manifests as composite tensile-shear failure along the schistosity planes. During loading, stress concentration initially appears at both ends of the structural planes and propagates along them. As the load increases, extensive stress concentration occurs along the structural planes in the central and lower-right regions of the specimen, leading to crack formation. With continued loading, the cracks ultimately propagate through the entire specimen, resulting in failure characterized as typical brittle failure. Figure 11 shows the stress distribution evolution process at θ = 19°.

1. 2. Shear Failure along Schistosity Planes (31° < θ < 70°):

Figure 11

Figure 11. Simulation results of stress distribution at θ = 19°.

Stress concentration first occurs along the structural planes. The internal condition during the compaction stage is largely consistent with that observed for θ < 30°, with no significant crack formation in the central region and only minor fracturing observed at the two end faces. As the load increases, numerous stress concentration zones develop along the structural planes in the central part of the specimen, accompanied by the subsequent emergence of a large number of cracks. This eventually leads to shear failure along the schistosity planes. Figure 12 illustrates the stress evolution process under uniaxial compression at θ = 68°.

1. 3. Compression-Induced Tensile Splitting along Schistosity Planes and Matrix (θ > 70°):

Figure 12

Figure 12. Simulation results of stress distribution at θ = 68°.

Stress initially concentrates in the lower-right corner of the specimen, causing tensile cracks to initiate from that area. Subsequently, stress concentration phenomena appear at various locations in the central region. As the load gradually increases, the stress concentration zones progressively spread towards the periphery of the model, inducing secondary cracks and scattered microcracks on both sides of the schistosity planes. The tensile cracks continue to propagate steadily. As the cracks develop, failure begins to occur on both sides of the slate specimen, ultimately forming splitting cracks parallel to the axial direction. Figure 13 depicts the stress change process under uniaxial compression at θ = 89°.

Figure 13

Figure 13. Simulation results of stress distribution at θ = 89°.

5 Conclusion

1. Laboratory uniaxial compression tests on slate specimens reveal that the compressive strength initially decreases and then increases with the dip angle of the schistosity planes, exhibiting an approximately “U-shaped” distribution. The compressive strength of saturated specimens is lower than that of dry specimens. For both states, the compressive strength decreases as θ approaches 45°, and the specimens become more prone to fragmentation.

2. The uniaxial compression process of slate was simulated using the PFC-2D numerical simulation software. The results demonstrate that the crack propagation paths, failure modes, and stress-strain curves from the numerical simulations are fundamentally consistent with the laboratory test results, validating the accuracy of the numerical simulation parameters. The numerical simulations further elucidate the stress concentration areas and crack propagation laws in slate under different structural plane dip angles.

3. Integrating findings from both laboratory tests and numerical simulations, the failure modes of the slate specimens can be categorized into three primary types: composite tensile-shear failure along the schistosity planes when θ < 31°; shear failure along the schistosity planes when 31° < θ < 70°; and compression-induced tensile splitting along the schistosity planes and matrix when θ > 70°.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

XS-L: Writing – original draft, Data curation, Investigation, Writing – review and editing. HZ-Q: Writing – original draft, Software, Investigation, Writing – review and editing. CP: Supervision, Project administration, Writing – review and editing, Formal Analysis, Software. YG-X: Funding acquisition, Validation, Formal Analysis, Supervision, Resources, Writing – review and editing, Visualization, Project administration.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This research is supported by the Key Research and Development Program of Henan Province (Grant No. 221111321500). Sponsored by National Natural Science Foundation of China; the funding number:1.

Conflict of interest

Author XS was employed by Henan Yudi Science and Technology Group Co., Ltd.

The remaining author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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