Predicting Rock Failure in Wet Environments Using Nonlinear Energy Signal Fusion for Sustainable Infrastructure Design

Abstract

Moisture-induced instability in rock masses presents a significant threat to the safety and sustainability of underground infrastructure. This study proposes a nonlinear energy signal fusion framework to predict failure in moisture-affected limestone by integrating acoustic emission data with energy dissipation metrics. Uniaxial compression tests were carried out under controlled moisture conditions, with real-time monitoring of AE signals and strain energy evolution. The results reveal that increasing moisture content reduces the compressive strength and elastic modulus, prolongs the compaction phase, and induces a transition in failure mode from brittle shear to ductile tensile–shear behavior. An energy partitioning analysis shows a clear shift from storage-dominated to dissipation-dominated failure. A dissipation factor (η) is introduced to characterize the failure process, with critical thresholds η_min and η_f identified. A nonlinear AE-energy coupling model incorporating water-sensitive parameters is proposed. Furthermore, an energy-based instability criterion integrating multiple indicators is established to quantify failure transitions. The proposed method offers a robust tool for intelligent monitoring and predictive stability assessment. By integrating data-driven indicators with environmental sensitivity, the study provides engineering insights that support adaptive support design, long-term resilience, and sustainable decision making in groundwater-rich rock environments.

1. Introduction

The sustainability of underground infrastructure depends on understanding how geological materials respond to environmental conditions [1]. Moisture-induced weakening of rock masses threatens long-term structural safety, resilience, and lifecycle performance [2,3]. In water-rich environments, water–rock interaction accelerates material degradation and alters failure modes, leading to reduced serviceability and increased maintenance. These effects undermine sustainable design goals such as durability, adaptability, and risk mitigation [4]. Therefore, predictive approaches that account for environmental influence are essential for resilient and sustainable underground engineering.

Recent studies have shown that water content influences not only rock strength but also its energy conversion during loading [5,6]. Experimental research on energy evolution during rock failure has revealed clear links between strain energy partitioning and damage development [7,8]. Wang et al. [9] investigated the energy variation during rock failure, exploring the stress–energy mechanism of rock failure under compressive deformation. Meng et al. [10] quantified the rates and densities of stored, elastic, and dissipated energy at varying loading rates, showing that energy evolution is closely linked to axial stress levels. Notably, the regulatory effect of water–rock interactions on energy pathways has gradually become a research focus [11,12]. Ma et al. [13] demonstrated that prolonged immersion in water accelerates micro-pore expansion in gypsum rock, resulting in higher macroscopic energy dissipation. Liu et al. [14] introduced an energy-based brittleness index through uniaxial compression tests on water-bearing coal. However, current research still presents notable limitations. Most experimental studies focus on the static mechanical behavior of either dry or fully saturated rocks, lacking systematic and quantitative analyses of energy dissipation and damage accumulation under variable moisture conditions. Moreover, the critical thresholds that govern transitions in energy evolution and failure modes remain undefined, limiting the accuracy of instability prediction in water-bearing rocks. Few studies have connected such instability analysis to broader sustainability objectives. As a result, integrated models that couple environmental influences with energy-based indicators to support sustainable underground engineering remain largely absent.

In response to these challenges, the integration of acoustic emission (AE) technology into rock mechanics research has offered new insights into the internal damage evolution of water-saturated media. AE-based monitoring techniques have increasingly been employed to capture dynamic fracture processes in real time [15,16]. Xiao et al. [17] employed AE monitoring to elucidate the intrinsic correlation between internal and surface damage in water-bearing coal under uniaxial compression. Fan et al. [18,19] studied the evolution of AE signals during the loading process and analyzed the structural damage and mechanical properties of rock under high-pressure water conditions. However, despite some progress in this field [20,21], there remains a lack of systematic research integrating energy dissipation analysis with AE monitoring to quantitatively assess the instability of water-bearing rock. In particular, the correlation between AE signal evolution, energy release patterns, and failure mechanisms under different water contents remains inadequately understood.

In summary, this study systematically investigates the coupled effects of water content, energy dissipation, and acoustic emission characteristics on rock failure mechanisms. Uniaxial compression tests on hard limestone under different moisture conditions, combined with AE monitoring and energy analysis, reveal how water alters failure modes, energy evolution, and AE responses. A quantitative framework is established to assess failure transitions and signal behavior. Furthermore, an energy-based instability criterion is proposed to enhance predictive modeling of water-induced rock failure. These findings provide not only a theoretical foundation for risk assessment and hazard mitigation in moisture-affected rock masses, but also practical tools for early warning, adaptive support design, and long-term performance optimization, thereby contributing to more resilient and sustainable underground infrastructure.

2. Materials and Methods

2.1. Sample Preparation and Moisture Conditioning

In this study, rock samples were collected from the sidewalls of the middle section of the Xia Ma Xi Tunnel in Guiyang, Guizhou Province. This location was mainly subjected to horizontal in situ stress and experiences minimal excavation-induced disturbance, making it representative of engineering conditions and suitable for mechanical and AE-energy-based testing. Based on the research objectives, intact rock blocks were retrieved on site and processed through coring, cutting, and grinding to produce standard cylindrical samples (Φ 50 mm × 100 mm). A total of twelve samples were prepared and divided into four moisture conditions: dry (D01~D03), natural (N01~N03), non-saturated (NS01~NS03), and saturated (S01~S03), with three samples in each group. To ensure material consistency, all samples were pre-screened using P-wave velocity measurements. Only samples with longitudinal wave velocities within the range of 5825–6625 m/s (±400 m/s) were selected for subsequent testing [22], thereby minimizing initial heterogeneity and improving the comparability of results across different moisture groups. The acoustic testing procedure and representative samples are illustrated in Figure 1.

To investigate groundwater effects on the mechanical properties of hard limestone, selected rock samples were prepared into four moisture-condition groups: (1) Dry sample preparation (D01~D03): The hard limestone samples were placed in an oven at a constant temperature of 105 °C and dried continuously for one day, cooled in a desiccator, and vacuum-sealed with near-zero moisture content (0% average). (2) Natural sample preparation (N01~N03): The hard limestone samples were sealed and stored for later use (0.13% average). (3) Saturated sample preparation (S01~S03): The hard limestone samples were achieved through 180 days of free water absorption, showing complete flaw saturation, with visible water-filled fissures after surface wiping (0.41% average). (4) Non-saturated sample preparation (NS01~NS03): The time point where the moisture content increases rapidly with time in the free water absorption method was selected as the preparation point of the non-saturated rock samples (0.28% average).

2.2. Microstructure Characterization

In geotechnical engineering, catastrophic failure of surrounding rock masses is closely associated with internal microstructures and inherent defects. These microstructural features directly influence the mechanical behavior, energy dissipation, and fracture modes of rock. Therefore, examining the microstructure of hard limestone provides a theoretical foundation for understanding its damage evolution mechanisms.

Figure 2 compares the SEM images of dry and saturated samples. The dry sample (Figure 2a) exhibits a dense microstructure with well-defined grain boundaries, sparse primary pores, and strong intergranular cementation. In contrast, the saturated sample (Figure 2b) shows hydration-induced degradation, including dissolution pits on mineral surfaces, roughened grain interfaces, and the presence of secondary calcite filling the original pores. In contrast, the saturated sample (Figure 2b) exhibits clear signs of hydration-induced degradation, such as dissolution pits on mineral surfaces, roughened grain boundaries, and secondary calcite filling previously open pores. These microstructural features are consistent with the observations reported by Wang et al. [23], further supporting the interpretation of water-driven deterioration.

These hydration effects weaken the cementation strength and alter the pore architecture. Water migration through pores and microcracks further exacerbates structural loosening, reducing the rock’s load-bearing capacity. As a result, microstructural deterioration promotes increased plasticity and accelerates progressive failure under mechanical loading.

2.3. Uniaxial Compression and Acoustic Monitoring

The experimental setup is shown in Figure 3. Uniaxial compression tests on hard limestone samples were performed using a WAW-600 servo-controlled testing system with a maximum axial load capacity of 600 kN and displacement control rates ranging from 0.005 to 25 mm/min. All tests were conducted under displacement control at a constant loading rate of 0.002 mm/s to ensure accurate AE acquisition and eliminate rate-dependent dynamic effects. The integrated control software enabled real-time acquisition of stress–strain data and automatic generation of deformation curves during testing.

To monitor fracture development, acoustic emission (AE) signals were recorded using an SAEU2S-1016-4 multi-channel system. AE sensors were attached to the Vaseline-coated sample surfaces and calibrated using standard pulse or tap testing to ensure signal fidelity. Four sample types, representing different moisture conditions (D01, N01, NS01, and S01), were tested. Each sample was loaded axially until failure while AE and mechanical parameters were continuously recorded. After testing, fragments were collected and fracture patterns were analyzed to assist in the identification of failure modes.

2.4. Energy and Damage Index Calculation

To investigate the mechanical behavior and failure mechanisms of hard limestone under different moisture conditions, strain energy components and AE-based damage indices were calculated during uniaxial compression. The analysis was grounded in the first law of thermodynamics and continuum damage mechanics. The total strain energy U is obtained from the area under the stress–strain curve, as represented in Equation (1):

$$U={\displaystyle \int \sigma d\epsilon },$$

(1)

where σ is the stress at any point on the stress–strain curve; and ɛ is the strain that corresponds to the stress site σ.

The elastic strain energy U^e, assuming linear elasticity in the pre-peak stage, is computed using the initial modulus, as shown in Equation (2):

$$U^e=\frac{1}{2}\sigma {\epsilon }^e=\frac{{\sigma }^2}{2E_u},$$

(2)

where E_u is the unloading modulus (the initial elastic modulus E₀ is used instead of E_u).

The dissipated energy U^d is then derived using Equation (3):

$$U^d=U-U^e,$$

(3)

To characterize the energy conversion efficiency and failure mode, the energy dissipation factor η is introduced, as expressed in Equation (4):

$$\eta =\frac{U^d}{U^e},$$

(4)

where η > 1 indicates dissipation-dominated energy behavior, and η < 1 indicates elastic energy dominance.

Additionally, based on continuum damage mechanics and AE monitoring, a damage variable D_AE is defined using cumulative AE energy [24], as shown in Equation (5):

$$D_{AE}=\frac{N_f}{N_g},$$

(5)

where D_AE represents the damage variable determined based on acoustic emission energy; N_f is the cumulative AE energy from zero to a given moment; and N_g is the total cumulative AE energy at sample failure.

These calculations provide the quantitative basis for analyzing the energy response and internal damage evolution of limestone under different moisture conditions, and serve as input for the nonlinear damage modeling developed.

3. Results

3.1. Mechanical Response Analysis

To further examine the mechanical behavior of hard limestone under different moisture conditions, Figure 4 shows the stress–strain curves and the failure crack morphology of the D01 (dry state), N01 (natural state), NS01 (non-saturated state), and S01 (saturated state) samples under axial stress. All samples exhibit similar curve shapes characterized by a gradual stress increase, rapid rise to a peak, and sharp post-peak drop. This mechanical behavior is consistent with the research law of Nejati and Ghazvinian [25]. The corresponding uniaxial compressive strengths are 91.39 MPa, 86.31 MPa, 75.86 MPa, and 65.95 MPa, respectively, showing a progressive strength reduction with increasing water content—down to 72.16% of the dry sample in the saturated state. Similar trends in moisture-induced strength weakening have also been reported in other studies [26], further confirming the softening effect of water–rock interaction.

The internal mechanism underlying this trend is closely related to pore water effects. Water-bearing samples (N01, NS01, S01) show a more gradual stress increase before the peak, indicating lower stiffness and extended deformation under load [27]. This can be attributed to water occupying pores and reducing intergranular friction, delaying internal resistance buildup. The elastic modulus also decreases consistently with water content, confirming that hydration weakens mineral bonding. Notably, localized stress drops are observed prior to the peak strength, indicating microcrack activity. To further support this mechanism, localized stress-drop values at pre-peak points are quantified for each sample, indicating microcrack initiation and coalescence: In the D01 sample, a stress drop from 76.11 MPa to 75.31 MPa at 0.83 σ_c, a reduction of 1.05%; in the N01 (natural) sample, from 64.39 MPa to 63.54 MPa at 0.74 σ_c (1.32%); in the NS01 (partially saturated) sample, from 48.78 MPa to 47.26 MPa at 0.63 σ_c (3.12%); and in the S01 (fully saturated) sample, from 40.37 MPa to 39.77 MPa at 0.61 σ_c (1.49%). These pre-peak stress fluctuations reflect different levels of microcrack evolution influenced by moisture content. In water-bearing samples, crack propagation is hindered by pore water pressure at the crack tip, requiring higher stress to achieve fracture penetration [5]. Overall, the results reveal a moisture-induced mechanical transition from stiff and brittle to more compliant and progressively weakening behavior.

To gain deeper insight into how the moisture content influences the fracture mode of hard limestone, the macroscopic failure morphology of samples under varying water conditions is analyzed. The failure morphology of hard limestone is significantly influenced by water content. Under uniaxial loading, dry (D01) and natural (N01) samples predominantly exhibit single shear failure with smooth fracture surfaces and visible rock spalling. In contrast, partially saturated (NS01) and fully saturated (S01) samples display tensile–shear composite failure, characterized by branching tensile cracks intersecting primary shear planes. This shift in failure mode is primarily governed by the water wedge effect. Infiltrated water accumulates at crack tips, generating localized tensile stress that promotes secondary crack formation. As moisture increases, this mechanism intensifies, leading to more diffuse and ductile-like failure patterns. An SEM analysis (Figure 2) further reveals hydration-induced dissolution of mineral cementation and weakening of grain boundaries. These microstructural changes correspond to the observed macroscopic transformation—from concentrated shear failure to distributed composite rupture [28]. Therefore, these findings demonstrate that water not only reduces rock strength but also alters fracture morphology by regulating crack propagation and promoting tensile–shear interactions.

3.2. Energy Evolution Laws

Figure 5 presents the evolution of total strain energy (U), elastic strain energy (U^e), and dissipated energy (U^d) for limestone samples under uniaxial compression in the different water states (D01, N01, NS01, and S01). All samples exhibit a continuous increase in U and U^e with strain, while U^d remains relatively low in early loading and rises more evidently near peak deformation. The energy evolution process can be divided into three stages: compaction (energy accumulation), elasticity (linear energy accumulation), and plasticity (energy dissipation). This is generally consistent with the findings of Yan [29] et al.

Compared to the dry sample (D01), water-bearing samples show a clear redistribution in energy components. With increasing water content, the gap between U and U^e widens, and the proportion of U^d increases. For instance, U^d accounts for 48.78% of total input energy in D01, but only 39.47% in S01 at comparable strain levels, indicating that pore water pressure inhibits early energy dissipation by reducing initial defect closure. Simultaneously, U^e increases in relative proportion, reaching 77.42–86.80% of U, likely due to the reduced energy threshold for elastic accumulation caused by hydration weakening of mineral bonding (Figure 2). At higher strain levels, U^d rises sharply, especially in dry and low-moisture samples, suggesting more concentrated energy release during crack propagation. In saturated samples, the growth of U^d is gentler, and crack development is more diffuse. This reflects a moisture-driven transition from energy-concentrated to energy-dissipative failure modes.

These results indicate that increasing water content alters the energy partitioning pathway by suppressing initial dissipation, enhancing elastic accumulation, and promoting spatially distributed failure, thereby providing quantitative insight into moisture-induced mechanical degradation in hard limestone.

Table 1. Energy changes in case of hard limestone failure.

Moisture Condition	Dry State	Natural State	Non-Saturated State	Saturated State
Total strain energy U (MJ·m−3)	0.1460	0.1412	0.1305	0.1224
Peak elastic strain energy Ue (MJ·m−3)	0.1347	0.1242	0.1107	0.0977
Peak dissipated strain energy Ud (MJ·m−3)	0.0109	0.0171	0.0199	0.0247
Energy storage ratio Ue/U	0.9226	0.8796	0.8483	0.7982
Dissipation ratio Ud/U	0.0747	0.1211	0.1525	0.2018

summarizes the peak energy parameters of hard limestone under different moisture conditions, including total strain energy (U), elastic strain energy (U^e), dissipated energy (U^d), and their ratios. As water content increases, U decreases from 0.1460 MJ/m³ (dry) to 0.1224 MJ/m³ (saturated), reflecting a 16.16% reduction in energy storage capacity. U^e similarly declines by 27.47% (0.1347 MJ/m³ to 0.0977 MJ/m³), indicating that hydration weakens the rock’s elastic deformation ability. In contrast, U^d increases markedly with water content, rising from 0.0109 MJ/m³ in the dry sample to 0.0247 MJ/m³ in the saturated state—an increase of 55.87%, consistent with the energy dissipation trend reported by Hu et al. [30]. This highlights the role of water–rock interaction in enhancing inelastic damage accumulation.

The brittleness index, defined as the elastic energy ratio (U^e/U), decreases from 0.9226 (dry) to 0.7982 (saturated), with intermediate values of 0.8796 and 0.8483 in the natural and partially saturated states, respectively. Simultaneously, the dissipation ratio (U^d/U^e) increases from 0.0747 to 0.2018, confirming that energy dissipation dominates the damage mechanism in water-rich conditions [31]. These trends provide a quantitative energy-based criterion for evaluating the stability of water-bearing strata in underground engineering. These trends confirm that water–rock interaction fundamentally alters the internal energy distribution of the rock during failure. The observed energy shifts provide a theoretical foundation for establishing multi-parameter instability criteria and offer a quantitative basis for evaluating the failure tendency of water-bearing strata in underground engineering.

To intuitively illustrate the influence of water on the energy evolution of hard limestone during compression, the energy dissipation factor η is adopted as a key indicator. Figure 6 illustrates the variation of η in samples under compressive deformation across different water states (D01, N01, NS01, and S01). The magnified view reveals that sample instability and failure involve a complex energy conversion process. As shown in Figure 6, the energy dissipation factor η exhibits a consistent decline–rise pattern across all samples, with clear distinctions as water content increases. Across all samples, high initial values of η reflect internal fracture closure and structural readjustment dominated by dissipative processes. With increasing load, η decreases and reaches a minimum value η_min, marking the transition to a relatively stable elastic state. In the dry sample, η_min = 0.046, indicating strong energy storage capacity. In the saturated sample, η_min = 0.328, representing an 85.98% increase compared to D01, suggesting that water reduces energy storage efficiency via pore water pressure effects. As stress continues to rise, crack propagation intensifies, and η begins to increase. In D01, η rises by 2.1 times (from 0.046 to 0.084), while in S01, the increase is only 1.34 times (from 0.328 to 0.338), due to water lubrication promoting more distributed, progressive damage rather than concentrated energy release.

These observations demonstrate that water inhibits elastic energy accumulation by reducing rock stiffness and facilitating microcrack development via pore pressure and lubrication effects. The decreasing gap between η_min and η_f with increasing water content reflects a transition from brittle to ductile failure. Establishing a relationship between η_min (stability threshold) and η_f (failure mode indicator) provides a quantitative basis for evaluating the instability risk of water-bearing limestone and offers theoretical support for energy-based early warning in rock engineering.

3.3. Acoustic Emission (AE) Energy and Damage Evolution

3.3.1. AE Energy Characteristics Analysis

To investigate the influence of water content on damage evolution, acoustic emission (AE) monitoring is used to track energy release during deformation. By analyzing AE energy peaks and cumulative energy under different moisture conditions, the failure mechanisms associated with energy evolution are further clarified. As shown in Figure 7, the AE response varies significantly with water content. In the dry sample (D01), AE energy surges sharply at 0.31% strain (45.74 × 10⁴ J) and peaks at 0.37% strain (158.80 × 10⁴ J), with a cumulative energy of 133.42 × 10⁵ J at failure—indicating intense, brittle fracture, driven by rapid microcrack coalescence. As the water content increases, the AE energy peak shifts toward higher strain levels (up to 0.44% in S01) while its magnitude declines markedly, from 108.32 × 10⁴ J (N01) to 68.60 × 10⁴ J (NS01) and 9.03 × 10⁴ J (S01). Cumulative energy also drops exponentially, decreasing by 39%, 57%, and 97%, respectively, relative to the dry sample.

These differences reflect the regulatory role of water–rock interaction on energy release and damage evolution [32]. Specifically, pore water pressure at crack tips acts to buffer stress concentration, delaying crack initiation and suppressing abrupt propagation. Water films reduce inter-particle friction, making fracture advancement more stable and distributed. Additionally, hydration-induced mineral dissolution weakens cementation strength, promoting heterogeneous damage over time rather than sudden structural collapse. These mechanisms explain the observed reduction in the AE energy peak, the exponentially decreasing cumulative AE energy, and the shift in AE activity toward the post-peak phase with increasing moisture content. The AE response thus captures the transition from brittle, energy-concentrated failure to ductile, energy-dissipative failure induced by water, providing a micro-mechanical interpretation for the cross-scale effects of water–rock–energy coupling.

3.3.2. AE-Based Damage Variable and Evolution Law

The cumulative acoustic emission (AE) energy reflects both energy evolution and damage accumulation during rock loading. It provides essential parameters for damage constitutive modeling. Integrating AE energy analysis offers deeper insight into failure mechanisms and supports quantitative stability assessment in rock engineering [33].

Figure 8 presents the variation in the AE-based damage variable D with normalized strain under different moisture conditions. The results show that water–rock interaction enhances damage rates, with higher water content leading to earlier onset and faster accumulation of damage. The damage evolution can be divided into three stages based on the curve patterns of D01, N01, NS01, and S01. In the incubation stage, D remains low and increases slowly. Drier samples exhibit delayed damage due to stronger cementation, while saturated samples show earlier crack initiation driven by water lubrication. During the stable propagation stage, D grows in a staircase pattern, indicating intermittent cracking. This growth becomes smoother and more continuous as water content increases, reflecting enhanced damage continuity. In the final stage, all curves rise sharply and converge, suggesting that macroscopic failure dominates near peak strain, while the micro-scale influence of water diminishes [13]. These trends highlight water’s dual role in both promoting early microcrack activation and weakening crack-tip resistance during propagation. The D-strain curves provide a visual and quantitative representation of how water–rock interaction alters the damage path from delayed brittle failure to progressive ductile breakdown.

Based on the above results, the damage evolution curve of the rock under different water-content states is determined as Equation (6):

$$D=W_0(1+kw)+A{e}^{(R_0+kw)\epsilon },$$

(6)

where W₀ is the initial damage evolution parameter; k is the correction coefficient of the moisture content; w is the moisture content; and A and R₀ are the damage parameters.

Based on Equation (6), the fitted results of damage evolution under different water states are obtained (see Figure 9), along with the specific values of parameters in the equation (see

Table 2. Test piece damage evolution equation parameters under different moisture states.

Moisture Condition	Dry State	Natural State	Non-Saturated State	Saturated State
W0	0.026	0.030	0.033	0.036
k	295.38	306.62	314.83	321.25
A	5.35 × 10−8	9.94 × 10−9	3.69 × 10−11	7.17 × 10−13
R0	4403	4679	5613	6029
Variance	0.9901	0.9434	0.9729	0.9284

). Figure 9 demonstrates that the damage evolution equation effectively describes the damage evolution behavior of samples under different water states, with the theoretical curves closely matching the experimental results. The 95% confidence interval indicates that in the low-strain stage (ε < 0.3%), the confidence band width of ±0.04~0.06 represents the consistency of the damage initiation mechanism. In the high-strain stage (ε > 0.7%), the band width expands to ±0.15~0.20, reflecting the high randomness of the crack penetration process. The 95% prediction interval (band width ±0.22~0.28) covers most of the observed data, validating the model’s extrapolation capability in damage evolution prediction. Overall, the fitted curves, 95% confidence band, and 95% prediction band comprehensively illustrate the variation and uncertainty of rock damage variables with strain, providing a theoretical foundation for further investigating rock damage mechanisms and predicting failure behavior.

According to the parameter variation in Table 2, the initial damage parameter W₀ increases from 0.026 to 0.036 as the water content rises from 0% to 0.41%, indicating that the water environment intensifies initial damage accumulation. The water content correction coefficient k increases from 295.38 to 321.25, reflecting the enhanced nonlinear modification effect of water–rock interaction on damage propagation. The damage parameter A sharply decreases from 5.35 × 10⁻⁸ to 7.17 × 10⁻¹³, indicating a transition in the damage mode from exponential growth to progressive dissipation under high water content. The rate parameter R₀ increases from 4403 to 6029, confirming that crack propagation accelerates significantly with higher water content.

3.4. Nonlinear Damage Model

3.4.1. Model Established

By analyzing the damage evolution patterns during the compressive failure of rocks under different water states, the strain at which damage initiation begins is defined as ε_c, leading to the expression for damage evolution during the compressive deformation process shown in Equation (7):

$$D=\left\{\begin{array}{ll}0\hfill & \hfill 0\le \epsilon \le {\epsilon }_c\hfill \\ W_0(1+kw)+A{e}^{(R_0+kw)\epsilon }\hfill & \hfill \epsilon >{\epsilon }_c\hfill \end{array}\right.,$$

(7)

Based on damage mechanics theory and the strain equivalence principle, the stress–strain relationship of rocks under different water states is divided into two stages, as expressed in Equation (8):

$$\sigma =\left\{\begin{array}{ll}{E}_0\epsilon \hfill & \hfill 0\le \epsilon \le {\epsilon }_c\hfill \\ E_0\left[1-W_0(1+kw)+A{e}^{(R_0+kw)\epsilon })\right]\epsilon \hfill & \hfill \epsilon \le {\epsilon }_c\hfill \end{array}\right.,$$

(8)

By integrating Equations (1)–(3), the relationship between energy and damage evolution is established, yielding the elastic strain energy U^e, as shown in Equation (9):

$$U^e=\left\{\begin{array}{ll}\frac{1}{2}{E}_0{\epsilon }^2\hfill & \hfill 0\le \epsilon \le {\epsilon }_c\hfill \\ \frac{1}{2}{E}_0(1-W_0(1+kw)-A{e}^{(R_0+kw)\epsilon }){\epsilon }^2\hfill & \hfill \epsilon \le {\epsilon }_c\hfill \end{array}\right.,$$

(9)

The total input energy equals the integral area under the stress–strain curve, while the dissipated energy is the difference between the total input energy and the elastic strain energy. By analyzing the damage evolution equation and energy balance relationship, the dissipated strain energy U^d is obtained as Equation (10):

$$U^d=\left\{\begin{array}{ll}0\hfill & \hfill 0\le \epsilon \le {\epsilon }_c\\ \frac{1}{2}{E}_0{\epsilon }^2-\frac{1}{2}{E}_0(1-W_0(1+kw)-A{e}^{(R_0+kw)\epsilon }){\epsilon }^2\hfill & \hfill \epsilon \le {\epsilon }_c\end{array}\right.,$$

(10)

Based on the above research, an energy evolution model for rocks under different water states during the loading process is established. This model captures the accumulation and release of internal rock energy while depicting the dynamic process of damage evolution. The model regulates the damage evolution pathway and energy dissipation characteristics through four key parameters (W₀, k, A, and R₀), effectively simulating the critical state characteristics prior to rock failure.

3.4.2. Parameter Analysis

As shown in Table 2, the model parameters W₀, k, A, and R₀ exhibit strong sensitivity to water content. With increasing moisture, W₀, k, and R₀ show a monotonic increase, indicating enhanced energy thresholds and damage resistance, while A decreases, reflecting reduced energy accumulation rates. To quantify these trends, nonlinear regression was performed, and the resulting Equations (11)–(14) achieved high correlation coefficients (0.9890, 0.9734, 0.9985, and 0.9632), confirming the reliability of the parameter–water content relationships. As illustrated in Figure 10, the fitted curves align well with the inversion results, validating the model’s ability to capture parameter evolution across different moisture states. This sensitivity analysis enhances the physical interpretability of the parameters and supports the robustness and transferability of the calibration framework to varying rock types and geological settings.

$$W_0(w)=0.03e^{0.77w},$$

(11)

$$k(w)=300.22e^{0.16w},$$

(12)

$$A(w)=5.54\times {10}^{-8}{e}^{-12.47w},$$

(13)

$$R_0(w)=4331e^{0.83w},$$

(14)

4. Discussion

4.1. Multi-Parameter Energy Instability Criterion Construction

Based on η_min and η_f, an energy evolution criterion is established to quantitatively characterize rock energy accumulation, release, and failure modes. The energy accumulation index E_d is defined to represent the capacity of rock to accumulate and dissipate energy during loading. The energy dissipation evolution index D_η is introduced to describe the dynamic evolution of energy dissipation during the loading process. Additionally, the energy evolution mode coefficient C_η is defined to characterize the energy evolution pattern of rock. The expressions are as follows in Equations (15)–(17):

$$E_d=\frac{1-{\eta }_{min}}{1-{\eta }_f},$$

(15)

$$D_{\eta }=\frac{{\eta }_f-{\eta }_{min}}{{\eta }_f},$$

(16)

$$C_{\eta }=E_d\times D_{\eta }=\frac{(1-{\eta }_{min})({\eta }_f-{\eta }_{min})}{(1-{\eta }_f){\eta }_f},$$

(17)

To investigate the regulatory effect of water on the failure mode of hard limestone, a multi-parameter analysis involving the E_d, D_η, and C_η is conducted (Equations (15)–(17)), as summarized in

Table 3. Modes of energy evolution at different water contents.

Moisture Condition	Dry State	Natural State	Non-Saturated State	Saturated State
Ed	1.041	1.040	1.038	1.015
Dη	0.452	0.200	0.170	0.030
Cη	0.471	0.208	0.177	0.030
Failure mode	Semi-brittle failure	Transition from brittle failure to ductility	Semi-ductile failure	Typical ductile failure

. As water content increases, E_d slightly decreases from 1.041 to 1.015, indicating a moderate decline in the rock’s elastic energy accumulation. In contrast, both D_η and C_η drop sharply, by approximately 93%, revealing a significant shift toward energy dissipation. This transformation reflects moisture-induced changes in rock fracture mechanics. Elevated pore water pressure reduces the effective stress at crack tips, delaying fracture propagation and suppressing sudden energy release [26,27]. Simultaneously, hydration leads to mineral dissolution that weakens intergranular bonding, facilitating earlier microcrack development [13]. In saturated states, water films further lower internal friction, promoting gradual, spatially distributed damage [14]. These effects collectively shift the failure mode from brittle and energy-concentrated in dry samples to ductile and dissipation-dominated in moist conditions. The evolution of AE signals supports this trend and confirms the effectiveness of η-based indicators in capturing moisture-sensitive failure behavior.

Figure 11 further visualizes this evolution through the spatial distribution of E_d, D_η, and C_η across different moisture states. The failure trajectory, forming a declining curve on the energy surface, moves from a high-C_η-D_η zone to a low-C_η-D_η zone. This progression corresponds to a transition from high-energy brittle fracture to progressive tensile–shear composite failure. The system demonstrates a nonlinear evolution path, with increasing moisture driving the rock mass into a more unstable regime characterized by delayed but more ductile failure. These trends align with observed AE activity and fracture patterns in saturated laboratory tests. The E_d-D_η-C_η framework thus provides not only a theoretical model to describe energy transformation and failure mode transitions, but also a quantitative basis for classifying rock mass stability.

The color-gradient surface in Figure 11 is shaded according to the energy mode coefficient C_η, with a continuous transition from purple to red, intuitively reflecting the stability level of the rock mass. Regions with high C_η values (purple) correspond to brittle failure modes dominated by elastic energy accumulation and rapid release, indicating higher rigidity and energy storage capacity. In contrast, low-C_η zones (red) represent failure processes dominated by dissipation and gradual damage evolution, typically associated with lower stability. As moisture content increases, the system’s trajectory on this energy surface shifts toward the low-C_η-D_η region, revealing an evolution from energy-concentrated brittle failure to dissipation-controlled ductile failure. This color-mapped surface not only visualizes the dominant energy mechanisms but also serves as an auxiliary diagnostic tool for assessing rock mass instability risk. The smooth variation in color provides a continuous scale of stability indicators, offering a quantitative and intuitive reference for real-time monitoring and early warning in moisture-sensitive underground engineering scenarios.

4.2. Model Validation

Substituting Equations (11)–(14) into Equations (9) and (10) enables the prediction of rock energy dissipation behavior under different water-content conditions (see

Table 4. Measurement results of sample parameters under different water-content states.

Moisture Condition	Parameter				R2
	W0	k	A	R0
Dry state	0.030	300.22	5.54 × 10−8	4331	0.9922
Natural state	0.033	306.53	1.10 × 10−8	4824	0.9418
Non-saturated state	0.037	313.98	1.69 × 10−9	5464	0.9720
Saturated state	0.041	320.57	3.33 × 10−10	6087	0.9236

). Figure 12 presents a comparison between experimental data and model predictions for U, U^e, and U^d in rock under different water states (D01, N01, NS01, and S01). The predicted curves align well with the experimental results, effectively capturing the energy accumulation, dissipation, and failure modes of rock under different water states. From Figure 12, the evolution trends of U and U^e are consistent. As water content increases, the model’s inflection points appear progressively earlier than the experimental data (0.034%, 0.059%, 0.08%, and 0.162% strain), highlighting the model’s sensitivity to instability thresholds. The proportion of U^d also increases significantly. Although the predicted curve deviates in the high-strain stage, it still captures the trend in energy dissipation transitioning from localized to non-uniform diffusion. This deviation is attributed to the increased randomness of plastic dissipation caused by pore water pressure and the nonlinear effect of damage accumulation due to grain boundary dissolution.

Similar deviations and energy dispersion effects under high-moisture conditions have also been observed in previous studies [34,35]. These comparisons further confirm the validity and generalizability of the proposed model in characterizing the failure behavior of water-sensitive rocks. Despite showing some deviations under high-water-content conditions, the overall trend remains reasonable, demonstrating the validity and applicability of the established energy evolution model. These findings provide a theoretical basis for rock instability warning and contribute to improved engineering safety.

4.3. Engineering Implications

The findings of this study offer a robust foundation for advancing the design, monitoring, and long-term stability of underground structures in moisture-affected rock masses. The observed transition in failure mode, from brittle and energy-concentrated under dry conditions to ductile and dissipation-dominated in saturated states, emphasizes the necessity of integrating environmental moisture effects into geotechnical risk assessment and infrastructure design frameworks.

The energy dissipation factor η, especially the relationship between η_min and η_f, serves as a robust and interpretable metric for identifying critical instability thresholds. A larger gap between these values indicates greater elastic energy accumulation and higher sudden-failure risk, while a smaller gap reflects stable energy release and delayed failure onset. This quantitative indicator offers strong potential for real-time failure prediction and supports the development of early warning systems in water-rich geological settings. The AE-energy coupled damage model further enables continuous tracking of internal rock degradation by linking acoustic emission characteristics with energy dissipation behavior. Its parameter calibration under different moisture conditions (W₀, k, A, R₀) supports adaptive interpretation of damage states and enhances failure-time prediction. These features are especially valuable for smart sensing systems that aim to automate hazard identification and optimize structural responses. These components jointly provide a basis for practical implementation in tunnel and underground engineering projects. Based on the AE–energy coupled model and energy indicators (η_min, η_f, E_d, D_η, C_η), an integrated framework for moisture-sensitive rock instability monitoring and early warning is proposed, as illustrated in Figure 13. From an engineering sustainability perspective, the proposed framework contributes to proactive risk management, energy-efficient support design, and lifecycle performance optimization. When embedded into tunnel, cavern, or slope monitoring systems, it can reduce reliance on reactive maintenance, extend service life, and support decision making under climate-sensitive or groundwater-vulnerable conditions. Overall, this study advances the integration of environmental awareness, intelligent diagnostics, and sustainability objectives in underground engineering practice.

While the proposed energy-based framework shows strong theoretical and practical value, its broader application still faces limitations. The model is based on lab tests of a single lithology and lacks validation across diverse rock types and field conditions. Real-time AE monitoring in complex environments may be affected by noise, equipment constraints, and deployment challenges. Moreover, the current analysis only considers moisture effects, omitting other coupled factors like temperature, chemistry, and creep. Future work should focus on multi-lithology validation, field testing, and integration of multi-physical process monitoring to enhance model robustness and scalability.

5. Conclusions

This study explored how moisture content affects the mechanical behavior and failure characteristics of hard limestone through uniaxial compression, acoustic emission monitoring, and energy-based analysis. Findings show that increasing water content reshapes the energy distribution, intensifies AE activity, and promotes a brittle-to-ductile transition. The main conclusions are summarized as follows:

(1)

The presence of moisture has a pronounced effect on the mechanical properties and failure modes of hard limestone. As water content increases, peak strength, elastic modulus, and energy storage capacity decrease significantly. The failure mode transitions from brittle shear to ductile tensile–shear composite behavior, indicating a clear shift toward ductility.

(2)

Water content alters the energy evolution pathway during deformation. Pore water pressure suppresses the closure of initial defects, while lubrication effects reduce crack propagation resistance. These factors increase energy dissipation during the plastic stage and lower the proportion of elastic energy, marking a transformation from concentrated release to dissipation-dominated failure.

(3)

Acoustic emission (AE) parameters effectively characterize the damage evolution and critical failure state. Dry samples exhibit intense and abrupt AE energy release, while saturated samples show more gradual and progressive signals, reflecting moisture-induced ductile behavior. The AE-based damage variable model captures a three-stage damage process: initiation, stable propagation, and accelerated failure.

(4)

An energy-based instability criterion incorporating the dissipation factor η, together with E_d, D_η, and C_η, is proposed to quantify the transition from stable to unstable deformation. A nonlinear energy–damage coupling model is also developed, with water-sensitive parameters (W₀, k, A, R₀), enabling unified prediction of both moisture state and energy response.

From an engineering sustainability perspective, the proposed framework provides a tool for intelligent monitoring, early warning, and energy-efficient support design in moisture-influenced rock masses. These outcomes contribute to proactive risk control, extended infrastructure service life, and the realization of sustainable underground engineering systems.