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Open Access

Peer-reviewed

Research Article

Meso-structural evolution of sandstone under uniaxial loading: A study on microdefect compaction and transgranular crack formation mechanisms

  • Yaxin Zhang,

    Roles Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Visualization, Writing – original draft, Writing – review & editing

    Affiliations National Field Observation and Research Station of Landslides in Three Gorges Reservoir Area of Yangtze River, China Three Gorges University, Yichang, China, Key Laboratory of Geological Hazards on Three Gorges Reservoir Area, Ministry of Education, China Three Gorges University, Yichang, China, Hubei Key Laboratory of Disaster Prevention and Mitigation, China Three Gorges University, Yichang, China

    ⨯
  • Lixu Deng ,

    Roles Conceptualization, Funding acquisition, Investigation, Methodology, Resources, Validation, Writing – original draft

    Lixudeng@ctgu.edu.cn (LD); zgd@ctgu.edu.cn (GZ)

    Affiliations National Field Observation and Research Station of Landslides in Three Gorges Reservoir Area of Yangtze River, China Three Gorges University, Yichang, China, Key Laboratory of Geological Hazards on Three Gorges Reservoir Area, Ministry of Education, China Three Gorges University, Yichang, China, Hubei Key Laboratory of Disaster Prevention and Mitigation, China Three Gorges University, Yichang, China

    ⨯
  • Guodong Zhang ,

    Roles Funding acquisition, Methodology, Validation, Writing – review & editing

    Lixudeng@ctgu.edu.cn (LD); zgd@ctgu.edu.cn (GZ)

    Affiliations National Field Observation and Research Station of Landslides in Three Gorges Reservoir Area of Yangtze River, China Three Gorges University, Yichang, China, Key Laboratory of Geological Hazards on Three Gorges Reservoir Area, Ministry of Education, China Three Gorges University, Yichang, China, Hubei Key Laboratory of Disaster Prevention and Mitigation, China Three Gorges University, Yichang, China

    ⨯
  • Shuyuan Duan,

    Roles Formal analysis, Visualization

    Affiliations National Field Observation and Research Station of Landslides in Three Gorges Reservoir Area of Yangtze River, China Three Gorges University, Yichang, China, Key Laboratory of Geological Hazards on Three Gorges Reservoir Area, Ministry of Education, China Three Gorges University, Yichang, China, Hubei Key Laboratory of Disaster Prevention and Mitigation, China Three Gorges University, Yichang, China

    ⨯
  • Jianfei Yang

    Roles Data curation, Visualization

    Affiliation River Basin Hub Operation Management Center, China Three Gorges Corporation, Yichang, China

    ⨯

Abstract

Rock deterioration under uniaxial compression is significantly influenced by changes in meso-structure, which plays a key role in determining the mechanical behavior and stability of rock materials. Understanding how different loading stresses affect the evolution of meso-structure is crucial for assessing rock stability in engineering applications, such as tunneling and landslide prevention. This study investigates the damage mechanisms and meso-structural evolution of sandstone subjected to uniaxial compression at different loading stresses (0, 5, 15, 30, and 40 MPa). Utilizing Nuclear Magnetic Resonance (NMR), Scanning Electron Microscopy (SEM) and quantitative statistical analysis (e.g., Single-factor analysis of variance (ANOVA), Pearson correlation coefficients), the study analyzes how different stress levels influence the internal structural changes within the sandstone. The results revel that low loading stresses (5 and 15 MPa) primarily induce microdefect compaction and limited intergranular crack propagation, causing notable changes in failure strain without significant structural damage. In contrast, higher loading stresses (30 and 40 MPa) induce the formation of transgranular cracks, drastically reducing both failure strength and overall structural integrity. Meso-mechanical analysis identifies mineral rotation and crack propagation as critical factors driving these structural transformations. These findings demonstrate that rock deterioration is stress-dependent, with distinct characteristics at low versus high loading conditions. This research enhances the understanding of the underlying mechanisms of rock deterioration, providing valuable insights into rock stability evaluation. The findings are essential for predicting and mitigating geological hazards, offering critical implications for engineering practices aimed at enhancing rock stability and preventing disasters.

Citation: Zhang Y, Deng L, Zhang G, Duan S, Yang J (2025) Meso-structural evolution of sandstone under uniaxial loading: A study on microdefect compaction and transgranular crack formation mechanisms. PLoS One 20(6): e0325318. https://doi.org/10.1371/journal.pone.0325318

Editor: Fabio Trippetta, Sapienza University of Rome: Universita degli Studi di Roma La Sapienza, ITALY

Received: November 9, 2024; Accepted: May 9, 2025; Published: June 3, 2025

Copyright: © 2025 Zhang et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: “All relevant data are within the paper and its Supporting Information files.”

Funding: The work was supported by the National Natural Science Foundation of China (No. 41903023, Lixu Deng) and the Special Research Funds for National Field Observation and Research Station of Landslides in Three Gorges Reservoir Area of Yangtze River, Ministry of Science and Technology (No. Z2022106, Guodong Zhang). Lixu Deng was responsible for funding acquisition, investigation, methodology, resources, validation, and manuscript preparation, including writing the original draft and reviewing and editing the manuscript. Guodong Zhang contributed to funding acquisition, methodology, validation, reviewing and editing manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Rock masses are subjected to various stress conditions within the Earth, and failure is often induced by changes in these stress states, particularly in cases where rock masses are exposed to external forces, such as tectonic movement and engineering activities, which result in various geological disasters, including landslides, rock avalanches, and ground collapses [1–5]. More importantly, the failure of rock mass is significantly influenced by the initiation and propagation of pores and cracks due to changes in stress states, which are crucial for understanding the failure process of rocks and rock-like materials [6–8]. Therefore, the evolution of pore and crack characteristics triggered by stress state changes is important not only for ensuring the safety of rock engineering but also for mitigating the risks of various geological disasters [9,10].

To understand the failure process of rock, the uniaxial compression stress–strain curve can be effectively used to reflect the development of microdefects (e.g., micropores, microcracks, grain boundaries, and bedding planes). The development of these microdefects can be classified into five stages: the crack closure stage, the elastic deformation stage, the crack stable growth stage, the crack unstable growth stage, and the post-peak deformation stage [11,12]. In addition, the crack stable and unstable growth stages are the primary stages in the development of microdefects in rocks subjected to externally applied stresses. These microdefects provide energy dissipation, resulting in acoustic emission phenomena and subsequent damage to the rock [13–15]. Therefore, energy dissipation is the root cause of microdefect development in rocks under loading [16,17]. During the crack stable and unstable growth stages, most of the stored energy is dissipated through the formation, extension, and coalescence of internal pores and cracks [18]. The initiation, propagation, and coalescence of these pores and cracks determine the meso-structure and further influence rock deterioration [19,20]. Thus, the damage characteristics of the meso-structures in rocks have been investigated by examining pore and crack evolution during the crack stable and unstable growth stages [21–23]. However, compared with the crack stable and unstable growth stages, the crack closure and elastic deformation stages are ubiquitous in rocks due to engineering activities [24], temperature changes [25–27], and repeated traffic loading [28,29]. During the crack closure stage, the extent of damage to the rock is often underestimated because the elastoplastic damage increment is not obvious, even though the mechanisms during the crack stable and unstable growth stages are well recognized. Nevertheless, the evolution of pores and cracks in rocks during the crack closure stage should also be investigated. At the beginning of loading, pore collapse and intergranular shear cracks are commonly observed in rocks [30–32]. Moreover, the major meso-structure characteristics of rocks during the crack closure stage involve a decrease in intergranular spacing and an increase in grain boundary contacts [33]. However, the propagation process of intergranular cracks remains unclear. Additionally, the evolution of intergranular cracks during the crack closure stage affects not only the main meso-structure of rocks but also the damage mechanisms during the crack stable and unstable growth stages. Therefore, the damage mechanisms of rocks under different loading stresses should be thoroughly investigated to fully understand the failure process.

Compared with igneous and metamorphic rocks, sedimentary rocks such as sandstone are more widely distributed on the Earth's surface are the main rock types associated with many geological hazards [2,34]. Meanwhile, the loose internal structure of sandstone, which is composed of skeleton minerals and filler materials, is highly sensitive to changes in stresses [35]. The uniaxial compressive stress–strain curve of sandstone shows complex plastic–elastic–plastic deformation characteristics [36]. A stress–strain curve with a high degree of concavity indicates considerable plastic deformation of sandstone during the crack closure stage [23]. Therefore, sandstone provides a unique opportunity to investigate the mechanical damage characteristics of rocks at the crack closure stage. Furthermore, many landslides in the Three Gorges Reservoir Area (TGRA) are related to sandstone [2,37]. Consequently, this study selected Jurassic sandstone, which is widely exposed in the TGRA. This primary objective of this study is to comprehensively investigate the evolution of pores and cracks in sandstone during the crack closure stage, which has not been as widely studies compared to the later stages of crack growth. This research is novel because it focuses on the early damage stages in rocks, particularly intergranular crack propagation and the evolution of meso-structures under different loading stresses. By utilizing Nuclear Magnetic Resonance (NMR), Scanning Electron Microscopy (SEM), quantitative statistical analysis (e.g., Single-factor analysis of variance (ANOVA), Pearson correlation coefficients), and meso-mechanics modeling, this study aims to provide deeper understanding of the mechanical damage mechanisms at these early stages, which are crucial for predicting and mitigating the effects of geological hazards, such as landslides and rock avalanches.

Experimental materials, equipment, and process

Preparation of experimental materials

Sandstone specimens were collected from TGRA in Yangtze River, China. The petrophysical and mechanical parameters of the sandstone specimens used in this research were determined in accordance with the International Society for Rock Mechanics standards, and they are shown in Table 1. The sandstone specimens are cylindrical with 100 mm in height and 50 mm in diameter. All specimens were polished with fine sandpaper to achieve an end surface flatness of 0.002 mm. The sandstone specimens are shown in Fig 1a. All specimens were cored from the same rock mass to reduce the dispersion between specimens. According to the petrographic characteristics of the sandstone (Fig 1b), skeleton minerals (quartz and feldspar) make up the skeleton structure, and filler materials (carbonate and clay minerals) are filled in between the skeleton minerals. X-Ray Diffraction reveals that mineralogically, the sandstone specimens consist of quartz (44.29%), feldspar (microcline and albite) (16.29%), carbonate minerals (calcite) (18.86%), and clay minerals (smectite, illite, and chlorite) (20.56%), as shown in Fig 1c.

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Table 1. Basic petrophysical and mechanical parameters of the sandstone specimens.

https://doi.org/10.1371/journal.pone.0325318.t001

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Fig 1. Sandstone specimens: (a) sandstone specimens for the experiment; (b) meso-structure characteristics obtained by optical microscopy; and (c) X-ray diffraction patterns and mineral contents.

https://doi.org/10.1371/journal.pone.0325318.g001

On the basis of the effect of porosity on P-wave velocities in sandstone [38,39], specimens with a P-wave velocity of 2535–2555 m/s were tested (Fig 2) to further ensure uniformity of specimens. Only intact and non-destructive sandstone specimens were selected to ensure specimen consistency.

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Fig 2. Measured P-wave velocity of the sandstone specimens.

https://doi.org/10.1371/journal.pone.0325318.g002

Experimental equipment

Uniaxial compression tests were conducted with the RMT-150C rock mechanics digitally controlled electro-hydraulic servo testing machine designed by the Wuhan Institute of Rock and Soil Mechanics (WIRSM), Chinese Academy of Sciences (CAS). The tests were performed in a displacement-control mode, and the displacement rate was 0.002 mm/s.

The P-wave velocity of rock specimens, especially in minerals that do not undergo obviously changed, can effectively reflect internal defects, such as pores/cracks [38,39]. The P-wave velocity test was conducted with the RSM-RCT(B) acoustic logging device developed by the WIRSM, CAS. The test data were recorded using the RSM-RCT(B) transient digital signal recording and processing equipment.

NMR, as a rapid and nondestructive technology extensively employed in geotechnical engineering, can effectively reflect the changes in the size and number of intrinsic flaws in materials/rocks [21,22,40]. The T2 spectrum in NMR employs an external magnetic field to align the protons in pore water within rock, causing the hydrogen nuclei to resonate and absorb energy. Upon removal of the magnetic field, the coil detects the energy released during relaxation [23]. In saturated rocks, pores are filled with water, and the relaxation process varies significantly with pore size. The T2 spectrum, derived through mathematical fitting, effectively reflects this relationship: smaller pores correspond to shorter relaxation times [41]. Simultaneously, the peak area of the T2 spectrum provides quantitative information about the number and distribution of pores across different sizes. Additionally, Nuclear Magnetic Resonance Imaging (MRI) can show the deterioration area in rock [21,42]. Here, the MacroMR12-150H-I NMR analyzer and MRI system made by Suzhou NiuMag Analytical Instrument Corporation with a magnetic field strength of 0.3 ± 0.05 T was used to analyze the pore structures of the specimens. A 60 × 60 mm coil was also used to conduct imaging tests on the specimens' porosity, and pore size distribution. The specific parameters for the Carr-Purcell-Meiboom-Gill (CPMG) sequence used in this experiment are shown in Table 2.

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Table 2. Parameters for CPMG experiments in NMR.

https://doi.org/10.1371/journal.pone.0325318.t002

SEM can be used to study the mineral shape and microdefect distribution in rocks [43–45]. The JSM-IT100 high-vacuum scanning electron microscope at Wuhan Sample Solution Analytical Technology Co. Ltd, was used to examine the morphology and distribution of the mineral structure of the specimens. Before the test, the small particles on the surface of each specimen were sprayed off. Then, the specimens were coated with gold using an ion-sputtering device.

Experimental process

On the basis of the characteristics of the stress–strain curve determined by the RMT-150C rock mechanics testing machine, the failure process of the sandstone specimens is divided into five stages, namely, (I) crack closure stage, (II) elastic deformation stage, (III) crack stable growth stage, (IV) crack unstable growth stage, and (V) post-peak deformation stage. Several important stress thresholds, including crack closure stress (), crack initiation stress (), crack damage stress (), and peak stress (), are associated with these stages [11,12].

According to the stress–volumetric strain curve [46], the reversal point (crack damage stress ) is about 44 MPa (Fig 3a). Here, it is the key to determining the crack initiation stress (), and defining the elastic deformation stage and the crack stable growth stage. In this study, with the relative compression strain method [12], the crack initiation stress () is calculated to be about 27 MPa (Fig 3b). At the elastic deformation stage, the crack closure stress () is calculated to be 18 MPa due to axial stress () having a linear relation with axial strain () (Fig 3c). In addition, axial strain stiffness is characterized by uniform fluctuation and a basic level of the elastic deformation stage [47], and crack closure stress () is calculated to be about 18 MPa (Fig 3c). Furthermore, the peak point (peak stress ) is determined to be about 60 MPa on the basis of the stress–strain curve of the specimen (Fig 3d). Thus, the five stages of the uniaxial stress–strain curves of the sandstone specimens can also be characterized by the axial stress (Fig 3d) as follows: (Ⅰ) crack closure stage (0–18 MPa), (Ⅱ) elastic deformation stage (18–27 MPa), (Ⅲ) crack stable growth stage (27–44 MPa), (Ⅳ) crack unstable growth stage (44–60 MPa), and (Ⅴ) post-peak deformation stage.

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Fig 3. Determination of the stress threshold: (a) curve between axial stress () and volumetric strain (), i.e., , where is the axial strain and is the lateral strain; (b) curve between axial stress () and relative compression strain (), i.e., , where is the lateral strain corresponding to the crack damage stress ; (c) curve between axial stress () and axial strain (), and (d) uniaxial stress–strain curve of the specimens.

https://doi.org/10.1371/journal.pone.0325318.g003

The uniaxial compressive stress–strain curve of the specimens shows typical plastic–elastic–plastic deformation characteristic (Fig 3d). Under uniaxial loading, meanwhile, the specimens experienced two plastic deformation stages: low-stress stage (0–18 MPa) and high-stress stage (27–60 MPa). Consequently, the sandstone specimens are loaded to plastic deformation stage at the low loading stresses (5 MPa [σ||] and 15 MPa [σ|||]) and high loading stresses (30 MPa [σ|V] and 40 MPa [σV]) and then unloaded to 0 MPa (Fig 4). Meanwhile, no obvious changes in the appearance of the specimens are found. Therefore, the uniaxial loading method can be used to investigate changes in the specimens' meso-structure.

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Fig 4. Loading damage process of the specimens: (a) axial stress-loading time curve of sandstone under uniaxial loading; and (b) axial stress-axial strain curve for sandstone under different loading stress.

https://doi.org/10.1371/journal.pone.0325318.g004

The detail investigation procedures involves the following steps: (1) The specimens were dried in a 25 °C oven until their weight became constant. (2) Compression tests were conducted on 15 specimens, which were divided into five groups. The five groups of specimens were uniaxially loaded to different stresses (0, 5, 15, 30, and 40 MPa) and then slowly unloaded to 0 MPa (Fig 4). (3) P-wave velocity tests were performed. After the specimens were unloading to 0 MPa, P-wave velocity tests were conducted immediately to investigate the degree of damage of the specimens due to stresses. (4) NMR tests were applied. After the P-wave velocity tests, one specimens from each group were saturated in a vacuum saturator for more than 24 h and then tested using NMR to determine the total porosity, pore size distribution, and MRI characteristics. (5) SEM tests were conducted after the P-wave velocity. One specimen from each group (excluding those used in the NMR test) were selected for SEM analysis. To eliminate the influence of moisture absorbed from the air on the SEM instrument, the specimens were dried in a 105 °C oven for 24 h using a slow heating method (2°C/min) to avoid any potential impact on the specimens caused by rapid temperature change. After drying, the specimens were cooled to ambient temperature (25 °C) before the SEM tests were performed to obtain the petrographic characteristics of the five groups of specimens under different loading stresses. (6) Mechanical tests were performed. After these non-destructive tests were carried out, P-wave velocity, NMR, and SEM, uniaxial compression tests were conducted to obtain the mechanical parameters of the five groups of specimens.

Experimental results analysis

Macro-mechanical characteristics of specimens

Uniaxial compression tests were performed on the sandstone specimens under different loading stresses. The stress–strain curve (Fig 5) is concave during the crack closure stage because the microdefects gradually close under axial loading. Compared with the degree of concavity in the specimens without loading stress (0 MPa), the degree of concavity in the specimens with 5 MPa loading stress decreases due to the closure of initial microdefects. However, the degree of concavity in the specimens gradually increases as the loading stresses increase to 15, 30, and 40 MPa. Li. et al. [36] reported that the concavity of the stress–strain curve represents the degree of microdefect development, and the development of microdefects is remarkably enhanced under uniaxial loading. After the crack closure stage, the stress–strain curve becomes linear, representing the elastic deformation stage during which stiffness remains constant. As shown in Table 3, compared with the specimens without loading stress (0 MPa), the elastic modulus of the treated specimens initially increases from 12.80 GPa to 24.52 GPa, then decreases to 10.44 GPa, and further decreases to 8.84 GPa with increasing loading stresses (5, 15, 30, and 40 MPa). The initial increase and subsequent decrease in elastic modulus are attributed to the changes in rock stiffness, which significantly affect the rocks' ability to resist deformation [48,49]. When the axial stress reaches its peak value, the specimens are destroyed. The failure strength of the treated specimens initially increases from 60.13 MPa to 61.45 MPa, then decreases to 57.28 MPa, and further decreases to 44.28 MPa, compared with the specimen without loading stress (0 MPa) (Fig 5). This result directly demonstrates that the sandstone specimens first undergo compaction and then experience damage during the uniaxial compression.

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Table 3. Mechanical properties of specimens subjected to different loading stresses.

https://doi.org/10.1371/journal.pone.0325318.t003

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Fig 5. Uniaxial stress–strain curves of specimens subjected to different loading stresses.

https://doi.org/10.1371/journal.pone.0325318.g005

Furthermore, P-wave velocity (assuming the minerals in the rock remain unchanged) and total porosity are effective methods for assessing the degree of damage in rocks [25,38,50]. In this study, the P-wave velocity and total porosity of the specimens subjected to different loading stresses were determined using the RSM-RCT(B) acoustic logging device and the MacroMR12-150H-I NMR analyzer, respectively. With increasing loading stresses, the P-wave velocity initially increases from 2545 m/s to 2576 m/s, then gradually decreases from 2492 m/s to 2425 m/s, and further to 2315 m/s (Table 4). The total porosity initially decreases from 9.834% to 9.696%, then gradually increases from 10.371% to 11.252%, and further to 11.720% for specimens subjected to 5, 15, 30, and 40 MPa of loading stress (Table 4). These results reconfirm that the sandstone exhibits different characteristics at various stages during uniaxial loading.

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Table 4. P-wave velocity of specimens subjected to different loading stresses.

https://doi.org/10.1371/journal.pone.0325318.t004

Based on the characteristics of the stress–lateral strain curves (Fig 5) and failure strength (Table 3), the specimens subjected to different loading stresses can be divided into two categories: Low Stress Specimens (LSS) with stresses up to 5 and 15 MPa and High Stress Specimens (HSS) with stresses up to 30 and 40 MPa.  The classification is based on the theoretical basis that 0.5 serves as a dividing point for irreversible damage, where the appearance of microdefects becomes significant [22]. According to this criterion, loading stresses below 0.5 (5 MPa and 15 MPa) are classifies as low stress, while loading stresses above 0.5 (30 MPa and 40 MPa) are classified as high stress. This division reflects the onset of irreversible damage and the formation of microdefects under increasing loading stresses. Compared with the specimens without loading stress (0 MPa), LSS with low loading stresses () can be divided into two stages: the initial microdefect compaction exhibits a lower failure strain (3.64‰ vs. 6.51‰), while the propagation of microdefects exhibits a higher failure strain (9.21‰ vs. 6.51‰). However, they have similar failure strength (57.28–61.45 MPa vs. 60.13 MPa) (Table 3 and Fig 5), indicating that the changes in sandstone caused by low loading stresses are manifested in failure strain rather than failure strength. In addition, this finding reveals the differences in plastic deformation of the specimens during low-stress loading. However, compared with the specimens subjected to 15 MPa loading stress, HSS with high loading stresses () has lower failure strength (44.28–48.13 MPa vs. 57.28 MPa) but exhibits a relatively similar failure strain (9.37–9.38‰ vs. 9.21‰) (Table 3 and Fig 5), indicating that the damage in HSS is mainly reflected in its failure strength. In addition, HSS has a lower elastic modulus (8.84–8.85 GPa vs. 10.44 GPa) (Table 3 and Fig 5), lower P-wave velocity (2315–2425 m/s vs. 2492 m/s), and higher total porosity (11.252–11.720% vs. 10.371%) (Table 4) compared to the specimens subjected to 15 MPa loading stress, indicating that high loading stresses () led to considerable sandstone deterioration.

Mesoscopic phenomena of specimens

Meso-testing methods (NMR and SEM) are used to visualize the pore and crack distribution characteristics under different loading stresses for further investigation into the damage mechanisms. As a new nondestructive testing method, NMR has become an important physical test to determine the size and distribution of pores in materials [51]. Based on the relationship between relaxation time, pore size, and peak area with the porosity of the T2 spectrum from NMR [21–23,40], the pore size distribution can effectively indicate changes in the size and number of pores. Thus, the distribution of different-sized pores can be determined by examining the pore size distribution from NMR for specimens subjected to different loading stresses (Table 5 and Fig 6). At the same time, based on the use of capillary pressure to measure pore radius classification in the laboratory [52] and combine with the pore diameter distribution features of the sandstone specimens [53], all pores (Fig 6) can be divided into four categories: micro-pores (0–0.08μm), small-pores (0.08–1μm), medium-pores (1–8μm), and large-pores (above 8μm) (Table 5). In addition, RMI (Fig 7) can effectively reveal the spatial distribution characteristics of microdefects in rocks [21,42]. SEM, a meso-morphological observation technique, can directly determine the mineral morphology characteristics on the surface of specimens [43–45]. Therefore, in this study, the mesoscopic morphology of the microdefects and minerals in the specimens is examined by SEM to identify the mesoscopic evolution processes during external loading under different stresses (Figs 8 and 9).

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Table 5. Pore size distribution of specimens under different loading stresses.

https://doi.org/10.1371/journal.pone.0325318.t005

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Fig 6. NMR characteristics of specimens under different loading stresses: (a) pore size distribution of specimens without loading stress (0 MPa) and with 5 and 15 MPa loading stresses; and (b) pore size distribution of specimens subjected to 15, 30, and 40 MPa loading stresses.

https://doi.org/10.1371/journal.pone.0325318.g006

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Fig 7. MRI images of specimens under different loading stresses.

Higher water saturation indicates a higher concentration of microdefects in the rock, and the size of the arrows indicates the loading stresses applied to the sandstone specimens.

https://doi.org/10.1371/journal.pone.0325318.g007

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Fig 8. SEM images showing the petrographic characteristics of specimens under loading stresses of 0, 5, and 15 MPa.

https://doi.org/10.1371/journal.pone.0325318.g008

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Fig 9. SEM images showing the petrographic characteristics of specimens under loading stresses of 30 and 40 MPa.

https://doi.org/10.1371/journal.pone.0325318.g009

Microdefect propagation under low loading stresses.

According to the discussion above, the changes in the specimens under low loading stresses () are mainly reflected in the failure strain, which is related to changes in the rock's pores [23]. The inherent spatial distribution of different-sized pores in specimens without loading stress (0 MPa) and with 5 and 15 MPa loading stresses (LSS) was examined using NMR to investigate the change patterns at low loading stresses (). As shown in Fig 6a, compared with the specimens without loading stress (0 MPa), the specimens subjected to 5 MPa loading stress show a significant increase in the first peak (0.02μm) and second peak (0.25μm), while the third peak (2μm) and forth peak (10μm) remain largely unchanged. Additionally, within the micro-pores range (0–0.08μm), there is a notable reduction in pore sizes between 0.03–0.08μm, while the pore sizes between 0–0.03μm increase. This result directly demonstrates that under uniaxial loading, larger pores in the specimens are compressed and transformed into smaller pores during the early stage of crack closure. On the other hand, the specimens subjected to 15 MPa loading stress shows a slight increase and a rightward shift in the first peak (0.02μm), while the second peak (0.25μm), third peak (2μm), and four peak (10μm) all show a noticeable increase (Fig 6a). More significantly, the porosity of micro-pores (0–0.08μm) shows a slight increase, while the porosity of small-pores (0.08–1μm) shows a slight decrease (Table 5). Additionally, the porosity of medium-pores (1–8μm) and large-pores (>8μm) increases significantly (Table 5). These changes reveal that larger pores (medium-pores and large-pores) are generated by the propagation of small-pores, while smaller pores (micro-pores) are formed by the compression of small-pores during the later stage of crack closure in uniaxial loading. This result is consistent with the findings of Bi. et al. [21]. It indicates that microdefect compaction and propagation are responsible for the changes in the specimens subjected to low loading stresses ().

To explore the locations of the microdefects, the meso-textures of the specimens without loading stress (0 MPa) and with 5 and 15 MPa loading stresses (LSS) were scanned using SEM (Fig 8). For the specimens without loading stress (0 MPa), skeleton minerals form the structural framework, while filler materials are located between the skeleton minerals (Fig 8a). In addition, the initial pores are primarily located between the filler materials (Fig 8b), indicating that the porous structure is concentrated within the filler materials. Furthermore, the initial cracks are sporadically distributed between the skeleton minerals and filler materials (Fig 8c), demonstrating the presence of initial micro-cracks along the boundaries of the skeleton minerals. Compared with the specimens without loading stress (0 MPa) (Fig 8a, 8b, and 8c), the specimens subjected to 5 MPa loading stress shows that the pores between the filler materials have been compacted (Fig 8e), and the cracks between the skeleton minerals and filler materials have also undergone compaction (Fig 8f), resulting in a noticeable reduction in microdefects (Fig 7b). The compaction of the pores and cracks leads to a substantial decrease in failure strain (3.64‰ vs. 6.51‰). However, the specimens subjected to 15 MPa loading stress shows larger pores located between the filler materials (Fig 8h) and many intergranular cracks along the edges of the skeleton minerals (Fig 8i), implying that microdefect propagation has significantly extended between the skeleton minerals and filler materials. This propagation leads to an increase in failure strain (9.21‰ vs. 6.51‰). This observation reconfirms the results of previous studies that observed intergranular cracks at the onset of loading [6,30,31]. Meanwhile, the spatial distribution of the microdefects in the LSS is similar to that in the specimens without loading stress (0 MPa) (Fig 7a, 7b, and 7c), indicating limited damage from microdefect compaction and propagation. This result may be related to the modification of the specimens' meso-structures. In this study, the skeleton structure of the sandstone specimens is composed of skeleton minerals (mainly quartz and feldspar), while filler materials (mainly clay minerals and carbonate minerals) are located between the skeleton minerals (Fig8a, 8d, and 8g), indicating that the strength of sandstone mainly depends on the stability of the skeleton structure. Therefore, the similar failure strength of the LSS and the specimens without loading stress (57.28–61.45 MPa vs. 60.13 MPa) indicates that microdefect compaction and propagation are not sufficient to damage the skeleton structure of the sandstone.

Meso-structure transformation under high loading stresses.

Specimens must experience a low degree of deterioration before a high degree of deterioration as the loading stress is gradually increased (). This suggests that HSS may inherit the deterioration characteristics of LSS. In this study, compared with the specimens subjected to 15 MPa loading stress, the HSS remains similar in terms of the size and distribution of medium-pores and large-pores (Fig 6b and Table 5). Similar to LSS, many intergranular cracks exist between the skeleton minerals and filler materials in HSS (Fig 9b and 9e). Thus, microdefects are also found, indicating that microdefect propagation is an important factor in the deterioration of HSS. However, compared with the specimens subjected to 15 MPa loading stress, HSS exhibits lower failure strength (44.28–48.13 MPa vs. 57.28 MPa) and higher total porosity (11.252–11.720% vs. 10.371%), showing an obvious difference in the meso-structures between HSS and LSS. This finding suggests that, aside from microdefect propagation between the skeleton minerals and filler materials, another key factor affects the skeleton structure of the specimens. Fortunately, the pore size distribution (Fig 6b) reveals the formation of numerous smaller micro-pores (0.004–0.02μm) and an increase in small-pores in HSS, reflecting the initiation of micro-pores and medium-pores, which is similar to the experimental observations reported by Cheng. et al. [22]. In other words, these newly formed micro-pores and small-pores may be related to the high-degree deterioration of HSS, although microdefect propagation also influences the meso-structure of the rock.

The meso-structure of HSS in Fig 9c and 9f shows the presence of many transgranular cracks in the skeleton minerals, indicating that smaller micro-pores (0.004–0.02μm) and additional small-pores are primarily formed within the skeleton minerals. The newly formed micro-pores and small-pores connect to form transgranular cracks in the skeleton minerals (Fig 9c and 9f). Meanwhile, these transgranular cracks connect to the intergranular cracks, eventually forming continuous penetrating fractures that considerably transform the meso-structure of HSS. In addition, the spatial distribution of microdefects throughout HSS (Fig 7d and 7e) confirms this scenario. This characteristic reconfirms previous results, which indicate that grains primarily exhibit splitting failure, with transgranular cracks being evident [30,54]. Thus, the formation of transgranular cracks is a key factor in transforming the skeleton structure of HSS.

Discussion

Quantitative analysis of the mechanical properties of the sandstone

Statistical analysis were performed to quantitatively assess the influence of uniaxial loading stress and its relationship with the physical and mechanical properties of sandstone specimens. Single-factor analysis of variance (ANOVA) was used to evaluate the significance of uniaxial loading stress on the deterioration of these properties, providing insight into how different loading conditions affect the strength and deformation behavior of the sandstone. Additionally, Pearson correlation coefficients were used to quantitatively analyze the correlations among various physical and mechanical parameters of the sandstone specimens, offering a clear understanding of the interrelationships between these factors.

Single-factor analysis of variance (ANOVA) was used to examine the significance of uniaxial loading stress on the physical and mechanical properties of the sandstone specimens. Table 6 shows that all the -values are less than 0.05, with -values of 91.14 for failure strength, 1367.08 for failure strain, and 3279.15 for elastic modulus, indicating that uniaxial loading stresses have a significant effect on the deterioration of the mechanical properties of the sandstone specimens. These results demonstrate that the loading stress significantly affects the failure strength, failure strain, and elastic modulus, highlighting the critical role of loading stress in altering the mechanical properties of the sandstone specimens.

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Table 6. Single-factor ANOVA results of specimens subjected to different loading stresses.

https://doi.org/10.1371/journal.pone.0325318.t006

Pearson correlation coefficients were used to quantitatively analyze the correlations of P-wave velocity and total porosity with mechanical parameters of sandstone specimens. As shown in Table 7, the Pearson correlation test results indicate significant correlations between the P-wave velocity and total porosity with the mechanical properties of sandstone specimens. Among these, the correlation coefficient between P-wave velocity and failure strength () is 0.973, with a -value of <0.01, indicating a strong positive correlation between P-wave velocity and the failure strength of the sandstone specimens. In contrast, the correlation coefficient between P-wave velocity and failure strain () is −0.777, with a -value of 0.122, showing a moderate negative correlation. However, the -value is greater than 0.05, meaning this relationship is not statistically significant. Similarly, the correlation coefficient between P-wave velocity and elastic modulus () is 0.721, with a -value of 0.169, indicating a moderate positive correlation, but the high -value suggests that the relationship is not statistically significant.

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Table 7. Correlations of P-wave velocity and total porosity vs. mechanical properties of sandstone.

https://doi.org/10.1371/journal.pone.0325318.t007

Table 7 also shows that the correlation coefficients between total porosity and mechanical properties () are all between −0.996 and −0.730, indicating a strong negative linear relationship between total porosity and mechanical properties () of sandstone. Specifically, the correlation coefficient between total porosity and failure strength () is −0.996, with a -value of <0.01, suggesting a very strong negative correlation. This implies that as the total porosity of the sandstone increases, its failure strength () decreases significantly. The correlation between total porosity and elastic modulus () is −0.730, with a -value of 0.162, indicating a moderate negative correlation, but due to the higher -value, this relationship is not statistically significant. Similarly, the correlation between total porosity and failure strain () is 0.801, with a -value of 0.103, showing a moderate positive correlation, but again, this is not statistically significant.

Based on the Pearson correlation test results, it can be concluded that P-wave velocity and total porosity can accurately reflect the trend of changes in failure strength () of the sandstone specimens. Specifically, the strong positive correlation between P-wave velocity and failure strength, along with the very strong negative correlation between total porosity and failure strength, suggests that both parameters are reliable indicators of these relationships, supported by the -value of <0.01, further confirms the robustness of these correlations.

P-wave velocity vs. total porosity for failure strength

All microdefects in rocks are distributed discretely under loading. However, macroscopic parameters (failure strength, P-wave velocity, total porosity, etc.) can effectively reflect the overall state of microdefects in rocks and are used to indicate the degree of rock deterioration [36]. Based on the conclusions drawn from the previous analysis, both P-wave velocity and total porosity are reliable indicators of failure strength of the sandstone. As the most direct and observable indicator of damage for rocks, failure strength is valuable for understanding rock deterioration. Therefore, the degree of deterioration of the specimens, , can be expressed in terms of failure strength as:

(1)

where is the failure strength of the specimens without loading stress (0 MPa), and is the failure strength of the specimens subjected to different loading stresses. Compared with the of the specimens without loading stress (0 MPa), the decrease in for the specimens subjected to different loading stresses is approximately −2.20%, 4.74%, 19.96%, and 26.36% (Fig 10a).

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Fig 10. Damage parameters based on mechanical data: (a) degree of deterioration in failure strength; (b) degree of deterioration in P-wave velocity; (c) degree of deterioration in total porosity; and (d) comparison of the degrees of deterioration in failure strength, P-wave velocity, and total porosity.

https://doi.org/10.1371/journal.pone.0325318.g010

Aside from failure strength, P-wave velocity and total porosity are effective parameters for evaluating rock damage [25,38,50]. Here, the degree of deterioration of the specimens can be represented by and , which relate to the P-wave velocity and total porosity as follows:

(2)(3)

where and are the P-wave velocity and total porosity of the specimens without loading stress (0 MPa), respectively; and are the P-wave velocity and total porosity of the specimens subjected to different loading stresses. Compared with the of the specimens without loading stress (0 MPa), the decrease in of the specimens subjected to different loading stresses is approximately −1.22%, 2.08%, 4.72%, and 9.04% (Fig 10b). Meanwhile, with the increase in loading stresses (5, 15, 30, and 40 MPa), the increase in of the specimens is approximately −1.40%, 5.46%, 14.42%, and 19.18% (Fig 10c).

Consequently, according to the discussion above, the deterioration of the specimens subjected to different loading stresses can be expressed by , , and . However, the degree of deterioration exhibits obvious differences when expressed for the damaged specimens (Fig 10d). Given that the degree of deterioration of failure strength () has been assigned as a key parameter [55], it is used as a benchmark to evaluate other parameters in measuring the degrees of deterioration (i.e., and ). The comparison of the degrees of deterioration in failure strength, P-wave velocity, and total porosity (, , and , respectively) is shown in Fig 10d. Compared with the points of deterioration in P-wave velocity (), the points of deterioration in total porosity () closely align with the points of deterioration in failure strength (), while the points of deterioration in P-wave velocity () deviate more from this trend (Fig 10d). This result confirms that total porosity provides a stronger correlation with failure strength than P-wave velocity, making it a more sensitive indicator of rock deterioration. This observation is consistent with Grindrod et al. [56], who reported that the bulk mechanical properties of specimens are primarily controlled by the sample porosity. Generally, the strength increases because the porosity decreases. In other words, total porosity is more appropriate to be used as a measure of damage in rock than P-wave velocity. This phenomenon may be related to the changes in the microdefects of rocks under different loading stresses.

Meso-mechanical mechanisms of sandstone under uniaxial loading

The sandstone specimens experienced two plastic deformation stages during uniaxial loading: the crack closure stage and the crack stable (and/or unstable) growth stage. According to the discussion above, the skeleton structure of the specimens exhibited considerable differences across different plastic deformation stages. For instance, the modification of the skeleton structure was primarily limited to microdefect propagation during the crack closure stage, while newly created pores and cracks were the key factor in the transformation of the skeleton structure during the crack stable (and/or unstable) growth stage. In addition, the evolution of pores and cracks was entirely determined by the mineral evolution behavior associated with these pores and cracks. Thus, whether due to microdefect propagation or newly created pores and cracks, rock deterioration depended on changes in the physical state of the associated minerals under loading. Therefore, this section will analyze the evolution characteristics of skeleton minerals at different stages through granular mechanics theory, and how these characteristics influence the macro-mechanical properties.

Theoretical analysis based on granular mechanics.

Rocks are solid aggregates of skeleton minerals, filler materials, pores, and cracks (Fig 11a). The skeletal structure of sandstone consists of skeleton minerals, such as quartz and feldspar, which form the main load-bearing structure of the rock and can be regarded as a continuous medium (Fig 11a). The filler materials in sandstone, including calcareous cement and clay minerals, adhere primarily to the surfaces of skeletal minerals and occupy the spaces between them (Fig 11a). Due to the strong bonding between the minerals in sandstone, the particles are not as loosely arranged as in granular materials like sandy or soil-rock mixtures. However, under external loading, there is still some degree of relative movement between the skeleton minerals, primarily in the form of rotation or sliding along grain boundaries. This movement is constrained by the surrounding matrix, such as the clay minerals and cementing materials, but it can still contribute to deformation at the meso-structure. Therefore, the meso-structure of the sandstone can be expressed as the particle system model shown in Fig 11b. Although the model is typically applied to granular materials, it can provide valuable insights into the behavior of tightly bonded materials like sandstone. In this case, the rotation of the skeleton minerals, though limited by the surrounding matrix, still plays a role in the rock's overall deformation under stress. This constrained movement of the grains, along with the interaction between the skeleton minerals and filler materials (Fig 11c), ultimately lead to changes in the macro-mechanical characteristics of the rock. Therefore, this section applies granular mechanics theory to explain the underlying mechanisms behind these mineral behaviors and their contribution to the macro-mechanical response of sandstone under uniaxial loading.

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Fig 11. Generalized model of the meso-structure of sandstone: (a) arrangement of skeleton minerals; (b) generalized model of the skeleton mineral system; and (c) schematic diagram of stress distribution on skeleton minerals.

https://doi.org/10.1371/journal.pone.0325318.g011

We assume that the external loading on the skeleton minerals is uniformly distribution, i.e., . Meanwhile, according to the mineral morphology characteristics of sandstone, the particle system model with a volume of and an exterior boundary of a closed surface () shown in Fig 11b. Sun. et al. [57,58] derived the expression for granular materials under external uniform loading based on the Cauchy stress tensor:

(4)

where is the average meso-stress tensor of the skeleton minerals; is the unit volume composed of the skeleton minerals; and is the contact force at contact point of skeleton mineral , including normal force and tangential friction force; is the unit normal vector at contact point ; is the moment at contact point , generated by eccentric force; is the unit vector associated with contact point , used to describe the direction of the moment; is the volume element associated with contact point .

At the same time, the presence of the moment causes the rotation of the skeleton minerals, and the moment can be expressed as:

(5)

where is the moment arm vector from the center of mass of the skeleton mineral to the contact point , and is the normal contact force. Therefore, the average meso-stress tensor of skeleton minerals can be expressed as

(6)

Meso-structure changes and damage evolution under uniaxial loading.

The evolution of meso-stress is an essential feature in the movement of minerals and the transformation of meso-structures in rocks (Fig 12a). In this study, the meso-stress evolution of the specimens at different stress states is as follows. During the crack closure stage () (Fig 12b and 12e), the skeleton mineral particles rotate under the influence of the normal contact force . This rotation of the mineral particles leads to the compaction of the pores between the filler materials, significantly enhancing the propagation of intergranular cracks between the skeleton minerals and filler materials. At this stage, the rotation of mineral particles induces localized compression within the pore spaces, primarily affecting the grain boundaries, which results in the closure of larger pores (Fig 6a) and the initiation of microdefects primarily in the form of intergranular cracks (Fig8g, 8h, and 8i). This process leads to an early-stage reduction in the porosity of the rock, which is crucial for understanding the early stages of damage formation. This phenomenon explains why the initial pores are compacted and why the microdefects in rocks are primarily intergranular cracks at the onset of loading [30]. Furthermore, although the meso-structure of the specimens is modified, the skeletal structure's modification during the crack closure stage is limited to the rearrangement of mineral grains and compaction at the interfaces, rather than large-scale structural changes.

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Fig 12. Diagram of the intergranular interaction of the specimens under loading: (a) generalized model of the particle system; (b) particle system model of the crack closure stage; (c) particle system model of the elastic deformation stage; (d) particle system model of the crack stable (and/or unstable) growth stage; and (e) stress-strain curve.

https://doi.org/10.1371/journal.pone.0325318.g012

During the elastic deformation stage () (Fig 12c and 12e), the friction meso-stress between the skeleton mineral particles increases as the external loading increases, leading to a reduction in the moment arm vector () (Fig 12b1 and 12b2) until it approaches 0 (Fig 12c1). As a result, becomes the dominant meso-stress in the minerals. At this stage, the minerals particles undergo elastic deformation, with the intergranular cracks from the previous stage becoming more pronounced, but they are not yet transitioning into larger-scale cracks. The energy dissipation during this stage is relatively low, and the microdefects remain primarily localized, contributing to the elastic deformation behavior of the rock. When the meso-stress is less than the bearing capacity of the skeleton minerals (), the particles undergo elastic deformation (Fig 12c), and most of the mechanical energy is stored as elastic energy [18,36].

During the crack stable and unstable growth stages () (Figs 12d and 12e), when the meso-stress reaches or exceeds the bearing capacity of the skeleton minerals, new microdefects initiation, propagate, and coalesce, forming transgranular cracks within most skeleton mineral particles (Fig 12d1) [30,54]. At this stage, the transition from intergranular to transgranular cracks becomes significant. The development of transgranular cracks is facilitated by the cumulative effect of localized microdefects that extend into the bulk of the mineral grains, effectively breaking the bounds between grains and leading to the formation of larger, through-going fractures. From an energy perspective, the stored elastic energy is converted into dissipated energy (e.g., kinetic, surface, frictional heat, and radiant energy), facilitating the formation, extension, and coalescence of internal cracks [18]. This conversion of elastic energy into dissipated energy at the meso-scale directly contributes to the formation of transgranular cracks, which then link up with existing intergranular cracks, eventually leading to the development of penetration fractures. This process transforms the meso-structure of the rocks (Fig 12d and 12e). The transgranular crack formation is the pivotal mechanism that leads to significant rock failure, as the transgraular cracks weaken the structural integrity of the grains, allowing for more extensive crack growth through the rock.

Although energy conversion is an essential factor in rock failure under loading [16–18], the deterioration process, especially during the crack closure and elastic deformation stages with minimal energy dissipation, is difficult to reveal. Furthermore, the deterioration is often underestimated due to the inconspicuous damage to the rock at the crack closure stage [22,23]. From a mesoscopic perspective, in this study, the rotation of mineral particles during the crack closure stage is the key factor in the mechanical damage of the rock. During this stage, microdefects primarily involve the compression and rotation of particles at grain boundaries, leading to the initial development of intergranular cracks. The development of pores and cracks due to mineral particle rotation sets the stage for the transition into larger-scale transgranular cracks during the crack stable and unstable growth stages, affecting the extension and coalescence of internal microdefects. This progression provides insights into the rock's deterioration under high stress loading.

The stress-strain curve of rocks illustrates the changes in their internal structure during uniaxial loading. The characteristics of these curves indicate how pores and cracks within the rock change and develop, allowing the degree of macro-deformation to be used to assess the extent of rock damage. Based on the deformation characteristics of the curve, the sandstone selected for this study exhibits a "plastic-elastic-plastic" deformation type, suggesting that irreversible deformation occurs at various plastic deformation stages. The evolution of minerals within the rock under different loading stresses determines the degree and location of pores and cracks development. This process highlights significant differences in the development of pores and cracks at different plastic deformation stages. Therefore, the mineral evolution behavior, especially the transition from intergranular to transgranular cracks, plays a crucial role in influencing the stress-strain behavior of rocks. This provides a new understanding of the macro-failure mode and variations in the mechanical characteristics of the rock. The analysis of meso-mechanical evolution shows that the mineral behavior (rotating and breaking) influences the development of microdefects, providing a new perspective on the damage mechanism of rocks.

Conclusion

In this study, the evolution of meso-structure of sandstone under uniaxial compression was investigated, focusing on the plastic damage characteristics and the role of microdefect compaction and transgranular crack formation mechanisms under different loading stresses. The main findings and theoretical conclusions of this research are summarized as follows:

  1. (1). Through detailed quantitative analysis of the mechanical properties of sandstone, including single-factor analysis of variance (ANOVA) and Pearson correlation coefficients, this study revealed significant changes in macro-mechanical properties, such as failure strain, and failure strength, under different loading stresses. Specifically, low loading stresses (5 and 15 MPa) resulted in significant changes in failure strain (3.64–9.21‰ vs. 6.51‰), especially during the crack closure stage (). On the other hand, at higher loading stresses (30 and 40 MPa), failure strength (44.28–48.13 MPa vs. 60.13 MPa) was the primary change observed during the crack stable growth stage (). The correlation analysis also showed a strong relationship between total porosity and failure strength, emphasizing the reliability of total porosity as an indicator of rock damage. This quantitative analysis provides valuable insights into the statistical relationship between loading stresses and sandstone's mechanical behavior, advancing the understanding of how meso-structural changes affect macro-mechanical properties.
  2. (2). The study introduced an innovative perspective to understanding the evolution of the meso-structure, focusing on the role of microdefect compaction, crack propagation, and transgranular crack formation under different loading stresses. Under low loading stresses (5 and 15 MPa), microdefects primarily manifested as pore compaction and intergranular crack propagation, with limited structural changes. However, at higher loading stresses (30 and 40 MPa), the formation of transgranular cracks within the skeleton minerals led to significant alterations in the rock's structure, extending beyond the effects of microdefect propagation. The meso-mechanical analysis reveals novel insights into how mineral behavior (such as particle rotation and cracking) influences the meso-structure evolution of rocks under stress.
  3. (3). From a meso-mechanical perspective, the fundamental mechanisms behind meso-structural changes varied according to the applied stress. Theoretical analysis based on granular mechanics reveals that at low loading stresses (), particle rotation and limited microdefect propagation result in minimal damage, while at higher loading stresses (), the exceeding of the bearing capacity of the skeleton minerals leads to the formation of new micro-pores and the initiation of transgranular cracks. These findings contribute to the understanding of the complex relationship between meso-structural changes and macro-mechanical properties in rocks, providing deeper insights into how different loading conditions impact rock behavior. The insights gained from this research can contribute to more accurate predictions of rock failure and stability under stress, particularly for engineering applications such as tunnel construction, mining, and landslide prevention.
  4. (4). This study presents a novel approach by integrating mesoscopic techniques like NMR and SEM to visualize pore evolution and crack development under different loading stresses. The analysis of meso-structure evolution using these techniques offers a deeper understanding of the underlying rock deterioration mechanisms, which have previously been underexplored, especially in the early stages of crack closure. The findings provide valuable theoretical and practical implications for assessing the mechanical properties and stability of rocks under different loading conditions, particularly in the context of geological hazard mitigation.

Supporting information

S1 Data. Zhang-Raw data of test results of sandstone specimens under different loading stresses.

https://doi.org/10.1371/journal.pone.0325318.s001

(XLSX)

Acknowledgments

The authors would like to thank the staff of the Wuhan Sample Solution Analytical Technology Co., Ltd for providing SEM testing services for this experiment.

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