Introduction

Cyan sandstone, as a common geological and engineering material, its mechanical properties, especially the damage evolution and failure mechanism under uniaxial compression loading, that constitute a critical basis for stability evaluation and design in geotechnical engineering1. In deep resource exploitation (such as oil and gas, minerals), tunnel construction, and major geotechnical engineering projects, accurately grasping the strength and deformation characteristics of rocks is of vital importance. It is noteworthy that rock mechanical behaviors often exhibit significant size effects, meaning the geometric dimensions of specimens have a non-negligible impact on mechanical parameters such as strength, deformation modulus, and failure modes2,3,4,5. In engineering practice, test results from small-sized laboratory specimens need to be extrapolated to large-scale rock masses. Neglecting the size effect will lead to deviations in strength prediction, which in turn may trigger risks such as instability in support design and failure in surrounding rock control. Therefore, revealing the mechanism of the size effect is an important guarantee for the safety of geotechnical engineering.

Extensive studies have addressed the size effect in rocks6,7,8. Zhai and Hao et al. examined weak rocks with uniaxial compressive strength (UCS) less than 10 MPa and reported that UCS and elastic modulus initially increased and then decreased with increasing size, while tensile strength and point load index consistently declined9. Chen et al. considered the combined influence of size and end effects, showing that UCS followed a power-law trend, increasing before plateauing10. Li et al. investigated the size effect and anisotropy in slate, a transversely isotropic rock, and found a significant size dependency in Young’s modulus measured parallel to bedding planes11.

Acoustic emission (AE) monitoring has become a powerful tool for studying rock damage processes. AE captures transient elastic waves generated by microcrack initiation, growth, and coalescence during deformation12,13,14,15,16,17,18. Yang et al. used staged AE source localization and velocity tomography to reveal phases of crack development and stress redistribution, highlighting the role of localized damage and internal heterogeneity in failure acceleration19. Zhang et al. analyzed spatial and temporal AE patterns in granite under uniaxial and triaxial loading, using fractal analysis to quantify the complexity and chaotic nature of failure processes20.

In terms of theoretical models for size effects, existing research has established a multi-dimensional framework21. Originating from the perspective of random distribution of internal material defects, Weibull statistical theory has developed a correlation model between strength and volume, which posits that rock strength decreases with increasing specimen volume, with its probability distribution characteristics quantifiable via shape and scale parameters22. Additionally, damage mechanics models incorporate size factors into damage evolution equations, elucidating the intrinsic mechanism of size effects by coupling internal defect propagation with macroscopic mechanical responses. For instance, the Lemaitre model quantifies the influence of size on strength attenuation through equivalent damage variables23. However, most existing models are tailored to single lithologies or specific loading conditions, exhibiting limited applicability to rocks like bluestone sandstone that possess both brittle and porous characteristics—particularly with room for improvement in prediction accuracy across multiple size ranges.

Despite these advancements, few studies have comprehensively examined the size effect in cyan sandstone using a multi-scale approach. In this study, uniaxial compression tests were performed on five groups of cylindrical cyan sandstone specimens with varying High-diameter ratios. AE monitoring was used to track damage evolution, while SEM was employed to observe post-failure microstructural characteristics. The influence of specimen size on UCS, failure mode, and AE activity was investigated. Based on these observations, a size effect law for UCS and a damage-based constitutive model were developed and validated. These findings provide a theoretical basis and experimental support for tunnel design and stability evaluation in engineering applications involving cyan sandstone.

Experimental design

Sample preparation

Cylindrical cyan sandstone specimens were collected from a tunnel construction site in Xinjiang, China. This naturally cemented sedimentary rock is predominantly composed of quartz, feldspar, and other silicate minerals. It exhibits high compressive strength, good durability, and a relatively homogeneous structure. Initial visual inspections were performed to exclude samples with apparent surface microcracks. Longitudinal wave velocities were measured along three orthogonal directions using ultrasonic equipment to assess internal anisotropy. Only blocks exhibiting minimal wave velocity anisotropy were retained for testing to reduce data scatter.

The selected blocks were cored and machined into cylindrical specimens with a constant diameter of 50 mm and varying heights of 50 mm, 75 mm, 100 mm, 125 mm, and 150 mm, corresponding to height-to-diameter (H/D) ratios of 1.0, 1.5, 2.0, 2.5, and 3.0 (Marked as S1 to S5, respectively). As shown in Fig. 1b–c, five specimens were prepared for each size group, resulting in a total of 25 samples. Due to the loading of 27 samples, the test data of 2 samples deviated significantly and were not taken. Therefore, the test data of 25 samples were retained. The flatness and parallelism of specimen ends conformed to relevant ISRM and ASTM standards.

Fig. 1
Fig. 1
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Experimental testing system.

Testing system and setup

Uniaxial compression tests on cyan sandstone were performed using a microcomputer-controlled electro-hydraulic servo testing machine, with a load range of 0–2000 kN, load accuracy of ± 0.01 kN, and displacement resolution of 0.001 mm (Fig. 1a). The system enabled simultaneous recording of load, stress, and strain values. Loading was conducted in a uniaxial displacement-controlled mode: initial preloading under stress control was applied to 0.2 kN to eliminate contact gaps, followed by loading at a constant rate of 0.05 mm/min until specimen failure.

A DS5-8B acoustic emission (AE) monitor (dynamic range: 40–100 dB) was employed, with four AE probes (frequency response range: 50–400 kHz) (Fig. 1d). The AE signal acquisition threshold was set to 35 dB to filter ambient noise. Specimens were ground such that the flatness error at both ends was ≤ 0.02 mm before being centered on the testing machine’s pressure plate. At the start of each test, the loading system and AE acquisition equipment were activated synchronously, with parameters including load, deformation, AE ringing counts, and energy monitored and recorded throughout the process.

Results

Stress–strain curve

Figure 2 presents the typical stress–strain curves of cyan sandstone under uniaxial compression with different sizes, where four stages can be identified: initial compaction (concave-up curve), linear elastic (approaching a straight line), plastic yield (concave-down curve), and post-peak failure (post-peak curve). Smaller specimens exhibit a shorter compaction stage and transition more rapidly to the elastic stage. This is attributed to relatively fewer internal pores and microcracks in smaller specimens (consistent with Weibull statistical theory) and the end effect, which facilitates densification during compaction. In the plastic failure stage, larger specimens show lower compressive strength because more internal defects promote crack propagation and coalescence into macroscopic fractures, leading to failure. In contrast, smaller specimens, significantly affected by the end effect, have restricted microcrack development and thus higher strength. Additionally, the peak strain of cyan sandstone increases with increasing specimen size.

Fig. 2
Fig. 2
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Stress–strain curve.

AE characteristics

Acoustic emission (AE) refers to the release of elastic waves with varying energy and frequency, generated by microstructural changes within rock under mechanical loading24. Monitoring AE parameters during uniaxial compression of cyan sandstone provides valuable insight into internal defect development and damage evolution, enabling early prediction of failure.

AE ringing counts

The relationship between AE ringing counts and stress during uniaxial compression of cyan sandstone specimens with different sizes is shown in Fig. 3. As observed from Fig. 3a–e, the ringing counts of all specimens reached their maximum values near the peak stress. This was because, after the initiation and propagation of microcracks in the early stage, the internal microcracks in cyan sandstone specimens connect to form macroscopic through-cracks when approaching the peak stress, leading to a sudden release of accumulated energy and a sharp increase in AE ringing count signals to peak levels. Thus, the abrupt change in AE ringing counts could be used as a precursor to determine the impending failure of cyan sandstone.

Fig. 3
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The AE ringing counts of cyan sandstone of different sizes.

In the post-peak failure stage, compared with smaller-sized specimens, larger-sized cyan sandstone exhibited a greater increase in cumulative ringing counts, which was attributed to larger internal crack scales and more intense internal activities within larger specimens. Additionally, as shown in Fig. 3f (cumulative AE ringing counts), the five specimen sizes followed a similar trend in cumulative ringing counts over time. At the initial stage of the experiment, the cumulative counts of all specimens increased slowly, remaining low with little difference between 0 and 100 s. As the experiment proceeded, the cumulative counts gradually increased between 100 and 400 s, while differences among specimens of different sizes began to emerge, with larger specimens showing faster growth. A sharp rise in cumulative counts occurs between 400 and 500 s, during which the H/D = 3 specimens exhibited significantly higher cumulative counts than those of other sizes, whereas the H/D = 1 specimens show the lowest. This indicated that larger specimens exhibit more intense AE activities in the later stage, generating more ringing signals, which suggested more active internal damage or crack propagation processes.

The differences in AE ringing counts among specimens of different sizes gradually expand over time. This may be because, with the continuous accumulation of internal damage in the material, larger specimens have a higher probability of internal crack interconnection, forming more complex damage networks, thereby resulting in more detected AE ringing signals. In contrast, smaller specimens, with relatively fewer internal defects, undergo less severe internal damage accumulation and propagation under the same experimental conditions, leading to lower cumulative AE ringing counts. Overall, larger specimens show a more pronounced increasing trend in cumulative AE ringing counts over time, reflecting the influence of specimen size on internal damage accumulation and AE signal generation in the material.

AE energy

Figure 4 illustrates the relationship between AE energy and stress for cyan sandstone specimens of different sizes under uniaxial compression. As shown in Fig. 4a–e, the energy reached its maximum near the peak stress. This arises from the initiation and propagation of microcracks during loading, which eventually connect to form macroscopic fractures, leading to the concentrated release of accumulated energy and a consequent sharp increase in AE signal intensity. Thus, abrupt changes in AE energy could be regarded as a precursor to failure25. In the post-peak stage, large-sized specimens, containing numerous initial defects that form complex crack networks, exhibited rapid growth in cumulative AE energy due to significant energy release from crack propagation and penetration, resulting in a violent failure process.

Fig. 4
Fig. 4
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The AE energy of cyan sandstone of different sizes.

Figure 4f reveals similar trends in cumulative AE energy over time for the five sizes: For specimens with H/D = 1, frequent but low-magnitude energy releases occur in the early loading stage, with slow cumulative growth, reflecting weak microcrack activities (dominated by closure/initiation). Specimens with H/D = 2 show increased frequency and magnitude of energy releases, with multiple high-energy events near the peak stress and accelerated growth in cumulative energy, corresponding to accelerated crack propagation and connection. Specimens with H/D = 3 exhibited the most intense energy releases in the late loading stage (especially the post-peak stage), with continuous high-energy events resulting in the largest total cumulative energy, indicating substantial energy release from the expansion and connection of internal complex crack networks. In summary, the size of cyan sandstone significantly affects its mechanical behavior and failure process: Increasing size led to reduced peak stress and more complex crack evolution and energy release. Analysis of AE energy and cumulative energy can effectively capture the dynamic evolution of internal damage.

Failure characteristics

Significant variations in failure characteristics were observed among cyan sandstone specimens of different sizes. Representative failure surfaces for each size category post-testing are presented in Fig. 5, with quantitative crack angle measurements (via image analysis software) added to enhance objectivity. For specimens with H/D = 1, failure was predominantly by splitting, characterized by dominant through-going vertical or near-vertical cracks with measured angles ranging from 85° to 90° relative to the horizontal.

Fig. 5
Fig. 5
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Failure characteristics of cyan sandstone specimens.

In H/D = 2 specimens, while splitting features remained evident (cracks at 80°–85°), a tendency toward inclined cracks (68°–75°) was noted in some cases, indicating a transitional mode. As specimen size increased to H/D = 3, failure morphology clearly transitioned to a primary single inclined shear fracture plane with angles concentrated between 50° and 58°, signifying shear failure as the dominant mode. The evolution of failure patterns across H/D = 1 to H/D = 3, visually documented in Fig. 5, demonstrates a distinct transition from splitting-dominated to shear-dominated modes with increasing height-to-diameter ratio. This shift provides quantitative and visual evidence of the size effect inherent in the failure mechanism of cyan sandstone.

Size effect degree of UCS

The measured uniaxial compressive strength (UCS) of cyan sandstone specimens IS presented in Fig. 6a. It can be observed that the UCS of each group exhibited low dispersion, and the UCS decreased with increasing cylindrical specimen size. The UCSs of specimens with dimensions 50 × 75 mm, 50 × 100 mm, 50 × 125 mm, and 50 × 150 mm are 94.77%, 88.13%, 83.11%, and 77.46% of that of the 50 × 50 mm cylindrical specimen, respectively.

Fig. 6
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(a) UCS (b) Size effect degree.

For quantitative analysis of the size effect on the UCS of specimens, the 50 × 50 mm cylindrical specimen was designated as the reference-sized specimen, while the 50 × 75 mm, 50 × 100 mm, 50 × 125 mm, and 50 × 150 mm cylindrical specimens were defined as non-reference specimens. The size effect degree was defined as the percentage of the difference between the UCS of the reference-sized specimen and that of the non-reference specimen relative to the compressive strength of the reference-sized specimen. Thus, the size effect degree of the UCS for a cube with diameter d was expressed as:

$$\gamma_{{\text{d}}} = \frac{{f_{1,0} - f_{{{\text{cu}}}} }}{{f_{1,0} }} \times 100\%$$
(1)

where \(\gamma_{{\text{d}}}\) was the size effect degree of UCS, \(f_{1,0}\) was the UCS of a cylindrical specimen with a diameter of 50 mm and a height of 50 mm, and \(f_{{{\text{cu}}}}\) was the UCS of specimens with other h/d ratios.

By substituting the data from Fig. 6a into Eq. (1), the size effect degrees of UCS for cyan sandstone specimens were obtained, as shown in Fig. 6b. It can be observed that the size effect degree of UCS increased with specimen size, while the increasing trend gradually slowed down as the size continued to increase.

Size effect law of UCS

Bažant21,26 proposed that crack propagation under uniaxial compression leads to stress redistribution within a specimen and the release of fracture energy. Importantly, the amount of energy released during fracture increases with the specimen size. By applying the principles of energy conservation and deformation compatibility, a relationship between the compressive strength and the specimen size of cyan sandstone can be established as follows:

$$\sigma_{{\text{N}}} = \frac{{Bf_{{\text{t}}} }}{{\sqrt {1 + \beta } }}$$
(2)

Among them, \(\beta = h/d_{0}\), B and \(d_{0}\) were geometric parameters; \(f_{{\text{t}}}\) represented the splitting tensile strength; and \(\beta\) is the material’s brittleness parameter. To determine the combined parameter \(Bf_{{\text{t}}}\) and \(d_{0}\), Eq. (2) can be rearranged as follows:

$$y = Ax + C$$
(3)

In the formula, \(y = \frac{1}{{\sigma_{N}^{2} }}\), \(C = \frac{1}{{(Bf_{t} )^{2} }}\), \(A = C/d_{0}\).

Regression analysis of the experimental data was performed to estimate the size effect parameters for cyan sandstone. The resulting values were summarized in Table 1.

Table 1 Fitting parameters for the size effect of cyan sandstone.
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By substituting the parameters listed in Table 1 into Eq. (2), the relationship between specimen size and UCS for cyan sandstone was obtained as follows: \(\sigma_{{\text{N}}} = \frac{{{83}{\text{.49}}}}{{\sqrt {1 + H/104.13} }}\), R2 = 0.998.

Figure 7 compares the experimental UCS data of cyan sandstone specimens with the theoretical prediction curve, providing a reference for estimating UCS across different specimen sizes.

Fig. 7
Fig. 7
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The theoretical UCS curve and test values of the cyan sandstone specimens.

Discussion

Microstructural mechanisms

In contrast to concrete or backfill materials, cyan sandstone exhibited intrinsically high compressive strength and mechanical stability, rendering it a suitable candidate for tunnel support systems and composite lining applications. To investigate its microstructural characteristics, selected specimens were mounted with conductive adhesive, sputter-coated with gold to enhance surface conductivity, and examined using a JSM-IT200 scanning electron microscope (SEM). The sampling location is derived from the center of the failure surface.

The size-dependent mechanical response of cyan sandstone under uniaxial compression can be directly attributed to its intrinsic microstructural characteristics, as revealed by SEM analysis. SEM micrographs at multiple magnifications (Fig. 8) reveal a complex internal structure comprising densely packed, irregularly shaped grains interspersed with microvoids, intergranular pores, and pre-existing microfractures. These heterogeneities act as preferential sites for stress concentration and crack initiation. Under applied load, localized deformation begins in these weak zones, evolving into microcracks that propagate and coalesce into macroscopic fractures. This process aligns with observations from AE monitoring, where ringing counts and energy surges were consistently recorded near the peak stress, indicating the abrupt release of stored elastic energy during fracture localization.

Fig. 8
Fig. 8
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Microscopic morphology of cyan sandstone.

A clear size effect was observed in both mechanical and AE responses: as specimen height increased, uniaxial compressive strength decreased, while cumulative AE activity intensified. This trend can be explained by the Weibull statistical theory, which predicts that larger volumes have a greater probability of containing critical-sized flaws. SEM observations further support this: larger specimens, with more spatial volume, are more likely to contain interconnected microdefect networks, which facilitate crack nucleation and accelerate failure. This is evidenced by the higher cumulative AE ringing counts and energy levels recorded in larger specimens, particularly in the post-peak phase, indicating sustained internal cracking and energy dissipation.

Furthermore, the transition from tensile splitting failure in smaller specimens to dominant shear failure in larger ones (Fig. 5) can also be traced back to microstructural influences. In small specimens, limited internal flaw networks confine damage to tensile fracture planes, while larger specimens, with more extensive defect connectivity, favor oblique shear bands where energy can be more effectively dissipated through distributed cracking. The pronounced AE energy release and longer post-peak response in large specimens are consistent with this shift in failure mode.

These findings collectively suggest that the microstructure of cyan sandstone governs its size-dependent mechanical behavior through two primary pathways: (1) the spatial distribution and density of microdefects influence the strength and mode of failure; (2) larger specimens exhibited more complex fracture processes, reflected in more intense and prolonged AE activity. Thus, integrating SEM analysis with AE monitoring offered a robust multiscale framework to understand rock failure.

Damage model and constitutive equations for multi-scale cyan sandstone

Treating cyan sandstone failure as a continuum damage evolution process, damage-coupled constitutive relationships were established based on Lemaitre’s strain equivalence principle23:

$$\sigma = E\varepsilon (1 - D)$$
(4)

where, \(\sigma\), \(E\), and \(\varepsilon\) denoted the effective stress, elastic modulus, and strain of cyan sandstone, respectively. The damage variable D characterized material degradation, where D = 0 corresponded to an intact state and D = 1 signified complete failure.

According to reference27, the stress–strain relationship curve of cyan sandstone can be represented by the Weibull statistical distribution density function. Therefore, the damage value D of cyan sandstone during uniaxial compression can be expressed as:

$$D = 1 - \exp [ - (\varepsilon /a)^{b} ]$$
(5)

In the formula: \(\varepsilon\)、\(a\) and \(b\) were respectively the axial strain, Weibull statistical distribution scale parameter, and Weibull statistical distribution shape parameter of the cyan sandstone. Among them, both a and b were greater than 0.

$$\sigma = E\varepsilon \exp [ - (\varepsilon /{\text{a}})^{{\text{b}}} ]$$
(6)

By differentiating the strain ε in Eq. (6), we can obtain:

$$\frac{{d_{\sigma } }}{{d_{\varepsilon } }} = E \times \exp [ - (\varepsilon /{\text{a}})^{{\text{b}}} ][1 - {\text{b}}(\varepsilon /{\text{a}})^{{\text{b}}} ]$$
(7)

After the calculation, it is obtained that:

$$a = \varepsilon_{p} /[(1/b)^{1/b} ]$$
(8)
$$b = 1/\ln (E\varepsilon_{p} /\sigma_{p} )$$
(9)

By substituting Eqs. (8) and (9) into Eq. (5), the damage evolution equation and damage constitutive model of sandstone can be obtained, as shown in Eqs. (10) and (11). The relationship curve between damage variables and axial strain of different sizes of sandstone is shown in Fig. 9.

$$D = 1 - \exp [ - 1/b(\varepsilon /\varepsilon_{p} )^{b} ]$$
(10)
$$\sigma = E\varepsilon \exp [ - 1/b(\varepsilon /\varepsilon_{p} )^{b} ]$$
(11)
Fig. 9
Fig. 9
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Damage variable curve of cyan sandstone.

Figure 9 illustrates that both the magnitude of damage and its accumulation rate during loading diminish as specimen size increases. Size-dependent damage constitutive models, formulated based on Eq. (6) in combination with Eqs. (8) and (9), are compiled in Table 2. As shown in Fig. 10, the theoretical stress–strain curves derived from these models exhibit strong agreement with experimental results across all deformation stages, except for the initial compaction phase. The predicted deviation of peak intensity points is controlled within ± 1.5% (for example, when H/D = 2.5, the measured peak value was 56.52 MPa and the calculated value was 56.57 MPa, with a deviation of 0.1%). However, since this early-stage behavior is typically negligible in real-world applications, such as tunnel engineering, the proposed models remain highly relevant for practical design considerations.

Table 2 Constitutive model parameters and corresponding equations for damage characterization of cyan sandstone under uniaxial compression.
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Fig. 10
Fig. 10
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Stress–strain test and theoretical curves of cyan sandstone.

Conclusions

Through uniaxial compression tests combined with AE monitoring and SEM observations, this study investigates the damage evolution laws of cyan sandstone with different sizes. The main conclusions are as follows:

  1. (1)

    The UCS of cyan sandstone decreased with increasing size, while peak strain increased. Notably, this study is the first to establish specific formulas for the size effect degree and law of UCS for cyan sandstone, which effectively quantify and predict its size-dependent strength behavior—filling the gap in size effect research for this unique lithology.

  2. (2)

    Small-sized specimens failed mainly by splitting, transitioning to shear failure with increasing size. Through SEM analysis, this study innovatively links macroscopic failure mode evolution to microscopic internal defects (e.g., microcracks), clarifying their key role in governing size-dependent strength and failure mechanisms of cyan sandstone.

  3. (3)

    Size significantly influences AE characteristics: small specimens showed more intense AE activity in the compaction stage, while large ones had more overall AE events. Uniquely, this study confirms that abrupt changes in AE ringing counts and energy near peak stress universally serve as failure precursors for cyan sandstone across all sizes, providing a novel early-warning indicator.

  4. (4)

    Based on peak stress, strain, elastic modulus, and shape parameters, size-specific damage models and constitutive equations for cyan sandstone were established. These models, which capture the reduced damage accumulation rate with increasing size, offer a new theoretical basis for tunnel support design in cyan sandstone formations, addressing the lack of tailored models for this rock type.