2.1 Experimental design

Ensuring strict confinement conditions for RockPSCT in laboratory experiments remains challenging. For instance, Huang et al. (Yin et al., 2009) developed RockPSCT equipment but identified several limitations (Huang et al., 2009): the installation of lateral deformation sensors is complex, the test data is unstable with poor accuracy, post-peak data of rock samples cannot be obtained, and the applied load on rock samples cannot be accurately measured. Similarly, Tang (Tang, 2007) designed an integral casting RockPSCT platform, but it also exhibits shortcomings, including insufficient monitoring space, a fixed sample placement size that cannot be adjusted, and the requirement for additional materials to fill the gap between the lateral confining device and the rock sample.

Given these challenges, the discrete element method (DEM), exemplified by Particle Flow Code (PFC), has been widely applied in rock mechanics and underground engineering research (Bao et al., 2023; Shi et al., 2022; Yang et al., 2022). PFC discretizes materials into numerous particles with varying diameters and stiffnesses, simulating the interaction and motion of granular media. A key advantage of using PFC in this study is its ability to provide ideal PSCT conditions, overcoming the limitations of physical experiments. In addition, PFC effectively monitors internal damage processes in rocks, making it a valuable tool for analyzing post-peak behavior under controlled conditions (Li, Li, Cao, et al., 2018, 2024).

The implementation of RockPSCT using DEM involves three main steps: Step 1—Model establishment and parameter calibration, Step 2—Condition setting, and Step 3—Monitoring operation. The used model is a standard sample with 100 mm × 50 mm × 50 mm dimensions. Since the microscopic parameters of the PFC model do not directly correspond to the physical material parameters, a commonly adopted approach involves adjusting these parameters using a trial-and-error method. This ensures that the macroscopic behavior calculated by the model aligns with the results of laboratory standard compression tests (Cui & Liang, 2019). The study's results from laboratory uniaxial compression tests serve as the target parameters (Cui & Liang, 2019; Xu, 2014). Following the calibration process and matching steps, the mesoscopic parameters for the PFC granite model were determined. Table 1 indicates that the calibrated macroscopic response of the rock model exhibits strong agreement with the laboratory uniaxial compression test results.

The constraint conditions (Figure 1) were then applied to the rock model (Figure 2), and the necessary monitoring functions were incorporated. During the PSCT process, the rock samples' axial stress and strain were monitored in real-time. Simultaneously, the stress variations in the confining wall were recorded. In addition, the number of microcracks within the rock samples was tracked. The progression and morphology of microcrack development and final fracture patterns were also observed and analyzed throughout the test.

2.2 Data feature points

The RockPSCT shows three key data feature points, as described below:

Feature Point A: At this point, axial stress transitions into the decelerated growth stage with lateral expansion, marking the onset of the platform process in the peak stage. It shows the beginning of nonlinear deformation in the rock sample. Simultaneously, confining wall stress accelerates, and the growth rate of microcracks begins to increase.

Feature Point B: The axial stress starts to decrease rapidly, indicating the end of the platform process. This point corresponds to the maximum rate of microcrack development, highlighting a critical phase of damage accumulation.

Feature Point C: At this stage, axial stress significantly declines. It represents the maximum confining wall stress and coincides with a marked deceleration in the microcrack development rate.

Figure 3c quantitatively represents these feature points, illustrating variations in axial stress, confining wall stress, and damage rate. The graphical depiction demonstrates that these feature points align across all three analytical dimensions. Point A is the transition point where the confining wall stress shifts from rapid to slower growth. Point B marks the onset of accelerated strength reduction and microcrack development. Point C represents the peak of confining wall stress and the subsequent deceleration of strength reduction and microcrack development.

These feature points collectively reflect the progression of rock samples from load-bearing to the gradual reduction and eventual loss of bearing capacity. This analysis highlights the significance of these data points in understanding the mechanical behavior of rocks under post-peak conditions. The mechanical properties of rocks under plane strain compression were preliminarily explored using a self-developed plane strain experimental platform (Lei, 2007; Tang, 2008). Physical experimental research indicated that the reduction process of the self-bearing capacity of rock mass occurs in two stages. A critical expansion rate (lateral strain) exists, and when this expansion rate exceeds the critical value, the bearing capacity of the rock mass enters a significant decline stage. This study verifies the findings of physical experimental research through an ideal PSCT.

2.3 Microcracks development

Figure 4 depicts a detailed monitoring visualization of the rock sample's behavior from Feature Point A to Feature Point C under PSCT conditions. It includes perspectives from the constraint surface (X) and the free surface (Y). From top to bottom, it displays the microcrack development map, the contact force distribution map, and the particle displacement map.

Microcrack development. In the pre-peak stage (from crack initiation to Point A), microcracks are sparsely distributed, primarily developing at the upper and lower ends of the sample. These dispersed microcracks correspond to the linear growth phase of axial stress, indicating minimal interaction between cracks. During the peak stage (from Point A to Point B), microcracks become more localized, concentrating within specific zones that can eventually form fracture channels. This stage coincides with the rock sample's maximum bearing capacity as axial stress peaks. In the post-peak stage (from Point B to rupture), microcrack clusters coalesce and deepen, eventually forming macrocracks that connect along the fracture path. This process results in a rapid loss of the core bearing capacity of the rock sample. The deepening and connectivity of microcracks reflect the progressive reduction of axial stress post-peak, signaling the onset of instability. The data and feature points in Figure 4 effectively illustrate the stages of bearing capacity evolution and the damage process in the rock sample. The dispersed development of microcracks corresponds to the linear growth of axial stress, while localized development indicates the rock's capacity to bear load. The deepening and interconnection of microcracks correspond to the decline in axial stress after the peak and the eventual loss of bearing capacity.

Particle contact force. The particle contact force characterizes the transmission and evolution of forces among particles during the compression process. It transitions from a uniform to an uneven distribution and from a dense to a loose configuration, with the thickness of the contact force chain representing the contact force's magnitude. The sparsest regions of contact force correspond to the formation of the shear fracture surface, ultimately resulting in the complete loss of the specimen's bearing capacity.

Particle displacement. Changes in particle displacement provide a clear depiction of the microcrack evolution process, including its initiation, accumulation, and gradual development into macrocracks. During the “AB” stage, particle displacement exhibits a uniform pattern under axial loading, with larger displacements observed at the ends and smaller ones in the middle of the rock sample. In the “BC” stage, displacement within the fracture channel becomes minimal, while displacement on both sides of the fracture surface increases significantly.

The data feature points from the PSCT effectively determine the staged damage process of rock materials. Microcracks evolve from sporadic and scattered formations to localized development, eventually deepening and interconnecting. The particle contact force transitions from a uniform to a nonuniform distribution, reflecting the meso-fracturing of the sample. Particle displacement initially decreases from the ends to the center of the sample, but at a critical stage, it forms a low-displacement zone that defines the fracture surface.

3.1 Experimental design

Two types of lithology and differently shaped samples were incorporated into the new tests to investigate and define the damage characteristics of rock masses in a plane strain state. Cuboid samples, consistent with those used in the first section, served as the parameter calibration standard, while flat and square samples were newly designed. All three sample shapes were designed to have the same volume. The shape factor was defined as the cross-sectional area divided by the height of the sample and normalized by its minimum value. The shape factors for the flat, square, and cuboid samples were 6.25, 2.50, and 1.00, respectively, as illustrated in Figure 4. In addition to granite, a set of sandstone samples was also calibrated (Huang et al., 2009). Therefore, six groups of rock samples were subjected to PSCT. This section focuses on analyzing variations in the bearing capacity of rocks during lateral expansion and their damage processes under PSCT.

3.2 Bearing capacity-damage characteristics

Curves of axial stress, confinement wall stress, and the number of microcrack developments were plotted for all rock samples as functions of lateral strain and axial strain, respectively (Figure 5a,b). The results indicate that samples with different lithology and shapes exhibit relatively consistent data feature points under PSCT. In the axial stress process, three feature points are identified: the stress growth deceleration point, marking the start of the bearing platform; the stress decrease acceleration point, denoting the end of the bearing platform; and the stress decrease deceleration point. Two key feature points are observed for confinement wall stress: the acceleration point of stress growth and the maximum stress point. In the development trend of microcracks, three significant feature points are noted: the initiation point, indicating the onset of crack formation; the maximum point of crack development rate; and the deceleration point of crack development. The definitions of the critical feature points are as follows:

Damage Starting Point (DSP). This point marks the onset of significant microcrack development. The DSP occurs at approximately 75% of the peak bearing capacity (normalized axial stress, used hereafter), where the axial and confining wall stresses remain in the linear growth phase.

Initial Critical Expansion Point (CEP0). This point signifies the transition where axial stress growth enters a deceleration phase, indicating the onset of nonlinear deformation in the rock sample. At CEP0, the confining wall stress growth and the rate of microcrack development begin to increase significantly. Statistical analysis reveals that the bearing capacities at CEP0 are relatively consistent across samples: granite exhibits an average bearing capacity of 87.4%, while sandstone averages 89.3%.

First Critical Expansion Point (CEP1). This point is characterized by a decrease in axial stress and the peak rate of microcrack development. Statistical data indicate uniformity in the bearing capacity of samples at CEP1: the average post-peak bearing capacity is 95.1% for granite and 95.3% for sandstone.

Second Critical Expansion Point (CEP2). This point depicts a significant deceleration in the decrease of axial stress, the maximum confining wall stress, and a considerable slowdown in microcrack development. Statistical analysis shows that the bearing capacities at CEP2 are also relatively consistent: granite averages 69.0% of its post-peak bearing capacity, while sandstone averages 64.6%.

The CEPs represent key mechanical stages of rock specimens under PSCT, capturing the transition from exertion to the gradual loss of bearing capacity. During this process, the specimen undergoes expansion deformation on the free surface while constrained by the confining wall. The stage before the acceleration of confining wall stress growth, culminating at its peak, is critical for the specimen to exert its bearing capacity. Once the confining wall stress reaches its maximum, the core bearing capacity of the specimen is entirely lost.

The relationship between bearing capacity and lateral expansion of rock materials under PSCT is encapsulated in the Rock Bearing Expansion Model (RockBEM), as illustrated in Figure 6. This model identifies four key feature points, DSP, CEP0, CEP1, and CEP2, and integrates four analytical dimensions: (1) the trend of bearing capacity variation with free surface expansion, (2) the trend of stress variation in the confining wall, (3) the trend of microcrack quantity, and (4) the development and evolution of microcracks. RockBEM provides a comprehensive framework for understanding changes in the bearing capacity of rock materials under PSCT. The first and second critical expansion points (CEP₁ and CEP₂) are mechanical indicators of the internal development of microcracks and the phased transitions in post-peak bearing capacity. These indicators capture the process of “bearing platform → bearing capacity decline → bearing capacity loss” in rock materials, providing valuable insights into their mechanical behavior.

3.3 Evaluation method of rock bearing capacity

How can the CEPs be employed to evaluate rock samples' bearing capacity under PSCT?

Figure 7 depicts the BCI results, with $$ of 1000. The analysis reveals the following key findings: (1) Among rock samples of different lithologies, the BCI of granite is significantly higher than that of sandstone. (2) Within samples of the same lithology, the BCI2 values follow the ranking: flat samples > cube samples > cuboid samples. This indicates that at CEP2, a larger shape factor corresponds to greater bearing capacity. (3) A comparison of the BCI1 results exhibits that the cube sample of granite exhibits the lowest value, while the flat sample of sandstone also has the lowest value. This indicates that the BCI values calculated using the two CEPs differ, as they represent the bearing capacity of the samples at distinct stages. A comprehensive analysis of the BCI for the two lithologies indicates that cuboid samples demonstrate superior bearing performance during the peak stage, whereas flat samples exhibit better bearing performance in the post-peak stage prior to fracture. The lithology and shape of rock samples significantly impact the values of the indicators. Regarding lithological factors, the “hardness and softness” of granite and sandstone differ, allowing the optimal bearing shape of a specific lithological rock to be identified through experimental results. For the shape factor of the specimen, the superior bearing capacity of cuboid samples at the peak position is attributed to the sporadic development of internal cracks at the upper and lower ends, which slows down the degradation rate of the rock mass. Flat specimens have an advantage at the post-peak rupture position because the crack clusters deepening from the end to the center of the specimen require a larger connecting section. These analytical results provide valuable insights for optimizing underground engineering design.

The BPI is a dimensionless index that reflects the relationship between the DB and DF stages of rocks under PSCT. When the BPI is greater than 1, it indicates that the sample exhibits a robust bearing process during the peak stage; however, the subsequent failure is likely to be more abrupt and severe. In contrast, when the BPI is less than 1, the sample's bearing capacity during the damage process is weaker, and it transitions more quickly into the stage of bearing capacity loss. Figure 7c compares the BPI results, showing that four samples have BPI values greater than 1, with the granite cuboid sample exhibiting the highest value. In contrast, the two sample types with BPI values less than 1 are flat samples. This indicates that a larger shape factor accelerates the transition to the stage of bearing capacity loss.