Research on Strength Degradation and Crack Development in Defective Concrete

Abstract

Tunnel linings play a vital role in underground infrastructure, yet their performance can be severely affected by pre-existing cracks. This study investigates the mechanical behavior and failure mechanisms of C30 concrete with artificial cracks under uniaxial compression, simulating various crack conditions observed in tunnel linings. Specimens were designed with varying crack lengths and orientations. Acoustic emission (AE) monitoring was employed to capture the evolution of internal damage and micro-cracking activity during loading. Fractal dimension analysis was performed on post-test crack patterns to quantitatively evaluate the complexity and branching characteristics of crack propagation. The AE results showed clear correlations between amplitude characteristics and macroscopic crack growth, while fractal analysis provided an effective metric for assessing the extent of damage. To complement the experiments, discrete element modeling (DEM) using PFC3D was applied to simulate crack initiation and propagation, with results compared against experimental data for validation. The study demonstrates the effectiveness of DEM in modeling cracked concrete and highlights the critical role of crack orientation and size in strength degradation. These findings provide a theoretical and numerical foundation for assessing tunnel lining defects and support the development of preventive and reinforcement strategies in tunnel engineering.

1. Introduction

Concrete is the most widely used construction material in civil and geotechnical engineering. Its application in underground structures, such as tunnels and retaining systems, has been essential due to its high compressive strength and durability. However, in real-world conditions, concrete often contains pre-existing defects such as shrinkage cracks, thermal-induced fractures, or construction-induced voids. These imperfections significantly influence the mechanical performance and failure behavior of concrete structures, particularly under uniaxial compressive loads. Understanding the impact of these initial flaws is crucial for assessing structural safety, especially in seismic-prone or high-load environments. Tunnels play a critical role in transportation networks and infrastructure systems. Ensuring the structural integrity and safety of tunnel linings is essential for long-term serviceability, disaster mitigation, and public safety. However, tunnel linings are prone to various defects during their service life, such as cracks, voids, and delamination. These defects can originate from construction imperfections, long-term loading, environmental degradation, and seismic events. Among them, cracks are particularly concerning as they can compromise the load-bearing capacity of the lining, facilitate water ingress, and lead to progressive deterioration, ultimately threatening the stability of the entire tunnel structure.

In seismic regions, understanding the mechanisms of crack propagation in tunnel linings under dynamic loading is of paramount importance for seismic design, performance assessment, and post-earthquake repair strategies. The behavior of cracked linings under axial loads, simulating in situ stress conditions, provides valuable insights into their mechanical performance and failure modes. Therefore, studying the strength characteristics and crack development patterns in concrete with pre-existing defects is essential for optimizing design and maintenance strategies for tunnel structures.

Recent advances in the study of cracked concrete under uniaxial compression have integrated experimental monitoring techniques and numerical simulations to enhance our understanding of fracture evolution and mechanical degradation. Acoustic emission (AE) analysis has been extensively employed to capture real-time crack initiation and propagation. For instance, Bu et al. (2025) and Guoning et al. (2025) investigated AE energy and b-value trends in rock-like materials, demonstrating their efficacy in identifying damage stages [1,2]. Similarly, Fan et al. (2015) validated the utility of AE parameters for distinguishing between tensile and shear cracks in concrete [3]. Holsamudrkar and Banerjee (2025) further applied AE-based b-value localization in fiber-reinforced composites, highlighting AE’s adaptability in composite systems [4]. Fractal dimension (FD) analysis offers a quantitative measure of crack complexity and distribution. Studies by Huang et al. (2024) and Yin et al. (2024) applied FD to track micro-crack growth, revealing correlations between increasing FD values and damage accumulation [5,6]. Zhong et al. (2025) and Liu et al. (2021) confirmed that FD trends correspond to mechanical degradation and can be used for structural performance assessment [7,8]. Zhang et al. (2024) further demonstrated that fracture surface roughness and branching behavior can be effectively characterized using fractal geometry [9]. A multi-scale experimental framework combining X-ray tomography, digital volume correlation (DVC), and phase-field modeling has been proposed by Mishra et al. (2025) for comprehensive fracture characterization in concrete [10]. Zhou et al. (2019) applied a hybrid compressive–shear phase field model to simulate brittle fracture in rock-like materials, capturing complex crack morphologies under mixed loading [11]. Discrete element method (DEM) simulations complement experimental work by replicating crack initiation, propagation, and coalescence at the meso-scale. Feng et al. (2025) and Chen et al. (2023) used DEM to simulate crack networks in sandstone and concrete under different loading conditions [12,13]. Nitka and Tejchman (2020) and Bolander et al. (2021) advanced DEM models by incorporating interfacial transition zones and aggregate morphology, improving the realism of fracture modeling [14,15]. In the context of tunnel lining performance, Chen et al. (2025) utilized DEM-FDM coupling simulations to study the fracture evolution in steel fiber-reinforced concrete under both static and dynamic loading [16]. To simulate hydraulic fracture propagation in porous media, Ni et al. (2020) proposed a hybrid FEM-peridynamic approach, enabling fracture tracking in saturated environments [17]. Perera et al. (2022) introduced graph neural network models to predict crack coalescence and propagation in brittle materials, offering a novel data-driven simulation pathway [18]. Yu et al. (2021) developed predictive DEM-based models for concrete under uniaxial loading, providing a foundation for mechanical parameter calibration in simulation studies [19]. Zhu et al. (2025) and Li et al. (2021) demonstrated DEM’s ability to predict crack coalescence behavior, including the effects of crack orientation and length [20,21].

Collectively, these studies underscore the value of combining AE monitoring [22], CT imaging, fractal analysis, and DEM simulations [23,24] to comprehensively assess the fracture mechanics of cracked concrete under uniaxial loading. Such integrated approaches are crucial for understanding the failure mechanisms in tunnel linings and developing robust damage assessment frameworks [25,26].

This study aims to investigate the compressive behavior of pre-cracked concrete specimens through a combined experimental and numerical approach. A series of uniaxial compression tests were conducted on C30 concrete specimens containing various crack configurations, including straight and inclined cracks. The peak strengths and failure patterns were recorded and compared. Subsequently, the test configurations were replicated using PFC3D, and a bonded-particle model was employed to simulate the internal stress redistribution and crack propagation. The numerical results were validated against experimental data, and the influence of crack geometry on strength reduction and fracture mechanisms was systematically analyzed. This research contributes to a better understanding of the role of initial defects in the mechanical degradation of concrete. The findings also provide theoretical and numerical insights that can be used to improve the design and evaluation of cracked concrete structures in tunnel linings and other load-bearing systems.

2. Experimental Design

2.1. Specimen Design and Preparation

The concrete specimens were designed to simulate typical tunnel lining material properties, using C30 concrete mix proportions commonly adopted in tunnel engineering. Each specimen was cast into a standard cube of 100 mm × 100 mm × 100 mm dimensions, ensuring consistency with relevant test standards (e.g., JIS A 1108, ASTM C39). Prefabricated cracks were introduced into the specimens to mimic realistic defect conditions observed in tunnel linings. The cracks were created with varying lengths (25, 50, 75 mm). The crack was placed along the specimen centerline, oriented horizontally or inclined at 45° to simulate tensile and shear cracks, respectively. To ensure reproducibility, all prefabricated cracks were made with a uniform width of 5 mm, which was controlled using thin steel plates inserted into the fresh concrete during casting. Control specimens without pre-existing cracks were also prepared to establish a baseline for mechanical property comparisons.

The selection of crack lengths and orientations was based on their practical relevance to tunnel engineering. In practice, cracks of 25–75 mm represent a progression from small to large defects commonly detected during lining inspections. Inclination angles of 30°, 45°, and 60° reflect shear-related crack orientations typically observed under the effects of in situ stress redistribution. Among these, 45° cracks are particularly critical as they are associated with maximum shear stress conditions and are, therefore, expected to have the most significant influence on structural weakening. This specimen design thus provides a controlled framework for evaluating defect geometry effects on concrete behavior and for drawing implications applicable to tunnel lining inspection and reinforcement.

All specimens were cast and cured under controlled laboratory conditions (20 ± 2 °C, relative humidity 60 ± 5%). Environmental factors such as varying temperature and humidity were not included in the specimen preparation as the focus of this study was on geometric crack parameters.

2.2. Uniaxial Compression Test Setup

All specimens were tested under uniaxial compression using a servo-controlled universal testing machine with a capacity of 3000 kN. (In Figure 1) The loading rate was set to 0.5 mm/min to ensure quasi-static loading conditions. The load and displacement were continuously recorded, and acoustic emission (AE) sensors were attached to monitor the micro-crack evolution during loading. The AE data acquisition system was configured to capture parameters such as amplitude, energy, and hit count.

2.3. Experimental Matrix

This matrix enables a systematic investigation of crack geometry effects on mechanical response and failure patterns under uniaxial compression, forming the basis for subsequent AE and fractal dimension analyses.

Experimental Results: Uniaxial compression tests on concrete specimens with pre-existing cracks.

All tests were conducted under constant laboratory conditions without consideration of environmental factors (temperature and humidity).

2.4. Analysis of Experimental Results

As shown in

Table 1. Summary of peak strength results.

Group	Crack Type	Length (mm)	Angle (°)	Peak Stress (MPa)	Strength Reduction Compared to Intact Specimen (%)
1	Control (no crack)	-	-	37.33	-
2	Straight crack	25	0	29.94	−19.79%
3	Straight crack	50	0	25.46	−31.84%
4	Straight crack	75	0	21.03	−43.67%
5	Inclined crack	25	30	29.14	−21.93%
6	Inclined crack	25	45	22.62	−39.41%
7	Inclined crack	25	60	28.24	−24.34%

and Figure 2, the intact specimen without cracks exhibited the highest peak compressive strength of 37.33 MPa.

Specimens with a single crack showed a reduction in peak strength; the reduction rate was influenced by the crack length. The experimental results clearly indicate that both crack length and crack angle have a significant influence on the peak compressive strength of concrete specimens. The specimen with a shorter crack (25 mm length) exhibited a moderate strength reduction, suggesting that crack length is a significant factor in strength degradation. For straight cracks aligned with the loading direction, an increase in crack length from 25 mm to 75 mm led to a progressive decrease in strength, with the peak stress dropping from 29.94 MPa to 21.03 MPa, representing a strength reduction of up to 43.67%. This trend reflects the intensification of stress concentration and an earlier onset of macro-crack propagation with longer flaws. In contrast, inclined cracks with a fixed length of 25 mm exhibited a non-monotonic response. The peak stress decreased most drastically at a 45° inclination (22.62 MPa), suggesting enhanced shear failure along the inclined crack plane. The specimen with a 60° crack angle showed a partial strength recovery (28.24 MPa), likely due to reduced alignment with the principal stress direction. These findings imply that both geometric attributes of cracks—length and orientation—play crucial roles in dictating failure mechanisms, and should be carefully considered in structural evaluation and reinforcement design for cracked concrete linings.

The presence of pre-existing cracks leads to a progressive reduction in peak compressive strength, with severity depending on crack geometry and orientation. Longer and thicker cracks cause more pronounced reductions, but inclined and multiple cracks have the most detrimental effect, suggesting the importance of crack configuration in structural integrity assessments.

These experimental findings serve as a valuable benchmark for validating DEM simulations, enabling a direct comparison of simulated crack growth and load-displacement behavior with physical test results.

The uniaxial compression results show that increasing crack length leads to a significant reduction in peak stress, with the 75 mm crack reducing strength by nearly 40% compared to the intact specimen. Additionally, crack inclination also affects structural integrity. The 45° inclined crack shows the lowest strength among all angled specimens due to alignment with maximum shear stress, while 30° and 60° cracks show moderate effects.

2.5. Acoustic Emission Monitoring and Data Analysis

Acoustic emission (AE) monitoring was employed as a non-destructive technique to capture the real-time damage evolution of the concrete specimens under uniaxial compression [27,28]. AE has been widely recognized as a sensitive method for detecting micro-crack initiation, crack growth, and eventual macro-fracture in cementitious materials, as demonstrated in studies on fiber-reinforced concrete and hybrid composites [4,9]. Shibano et al. (2025) experimentally evaluated the influence of crack orientation on AE activity in concrete compression tests, emphasizing directional sensitivity of AE signals [29]. In this study, the primary objective of AE monitoring was to investigate the fracture process of pre-cracked concrete specimens, correlate AE characteristics with stress–strain behavior, and quantify the influence of crack geometry on damage progression.

Two AE sensors (resonant frequency: 150 kHz) in Figure 1 were mounted on opposite sides of each specimen using coupling gel to ensure effective signal transmission. The sensor placement aimed to maximize signal detection for both tensile and shear fracture modes. The sensors were symmetrically installed on opposite faces of the cube specimens, approximately 50 mm away from the prefabricated crack centerline. This arrangement was selected to maximize the representativeness of signal acquisition and to ensure that both tensile- and shear-dominated cracking events could be effectively detected. The AE system (e.g., PCI-2, Physical Acoustics Corporation) was configured with a threshold of 40 dB to filter out background noise and a sampling rate of 5 MHz to capture detailed waveform information. AE parameters analyzed included:

① Amplitude (dB): representing the energy released during micro-cracking events; ② cumulative event count: reflecting the progression of damage; ③ event rate: indicating the intensity of active damage zones. ④ b-value analysis: a statistical method based on the Gutenberg–Richter relationship, used to distinguish fracture modes in concrete and rock-like materials [30,31].

For data interpretation, AE events were synchronized with the applied load and displacement data to identify critical points such as crack initiation stress, peak stress, and post-peak behavior. The time series of AE parameters was plotted alongside the stress–strain curves to reveal characteristic “AE bursts” corresponding to major crack propagation stages.

Additionally, frequency-domain analysis of AE waveforms was conducted to identify dominant crack frequencies, following approaches by Zhang et al. (2024) and Lv et al. (2023), who linked AE spectral features to crack mechanisms and thermal effects [9,22]. The statistical results were further complemented by fractal dimension analysis of the final crack patterns, enabling a comprehensive evaluation of crack complexity and damage distribution.

It should be noted that the AE monitoring in this study was performed under controlled laboratory conditions without consideration of environmental influences such as temperature and humidity, which may affect signal characteristics in field applications.

The AE findings from this study also have practical implications for field monitoring. Specifically, trends in b-value reduction and the occurrence of high-amplitude AE events may be integrated into real-time monitoring systems for tunnels. Such AE-based indicators could serve as early-warning signals of progressive lining deterioration, offering a practical pathway to improve tunnel safety management and maintenance strategies.

The acoustic emission (AE) monitoring results reveal clear precursors to specimen failure during uniaxial compression. As shown in the first figure, the AE b-value fluctuates in the early stage and gradually decreases during the stress ramp-up, with a sharp drop near peak stress (~65 s), indicating a shift toward high-energy, low-frequency events and increased damage localization. In the second figure, AE amplitude data show a burst of high-amplitude events (>100 dB) as the specimen approaches peak stress, consistent with rapid crack propagation. These patterns support the reliability of AE parameters as indicators of internal damage evolution and imminent failure in concrete specimens.

As illustrated in Figure 3, the AE behavior of specimens with varying crack lengths shows a clear correlation with structural damage development. As crack length increases from 25 mm to 75 mm, the AE b-value tends to decrease earlier and more significantly, indicating the localization of damage and the emergence of dominant crack propagation. Similarly, the amplitude distribution reveals that high-energy AE events become more frequent and concentrated, especially in the 50 mm and 75 mm groups. The 75 mm specimen, with a nearly through-going crack, exhibits early-stage energy release, suggesting that the fracture process is more sudden and unstable. In contrast, the intact specimen presents a more diffuse AE pattern, corresponding to a more distributed micro-cracking process.

As illustrated in Figure 3, the AE parameter evolution for specimens with different inclined cracks (30°, 45°, and 60°) demonstrates distinct damage development behaviors. The specimen with a 45° inclined crack exhibits the most significant decline in AE b-value and a concentration of high-amplitude events near peak stress, indicating rapid crack propagation and impending failure. In contrast, the 30° and 60° specimens show more stable b-values and fewer high-energy AE events, suggesting less critical crack growth. These observations confirm that cracks aligned with the principal shear direction (45°) pose the greatest risk of sudden failure under uniaxial compression.

2.6. Fractal Dimension Analysis of Crack Patterns

Fractal dimension analysis was conducted to quantitatively characterize the geometric complexity of the crack patterns observed on the failure surfaces of concrete specimens. Fractal theory has been widely applied in fracture mechanics to characterize scale-invariant crack networks in concrete, as shown in the work of Zhong et al. (2025) and Yin et al. (2024), who analyzed fractal dimensions in damage evaluation [6,7]. In situ X-ray mechanical testing by Stamati et al. (2021) revealed micro-concrete fracture evolution under triaxial loading, providing high-resolution insights into crack nucleation and branching [32]. The image processing workflow is as follows: The post-failure crack surfaces of each specimen were photographed under controlled lighting conditions. The images were first converted to grayscale to reduce color information and enhance contrast between the cracks and the background matrix. Subsequently, a binary thresholding operation was applied to produce a two-tone (black and white) image, isolating the crack networks (black) from the background (white). The threshold value was determined through iterative testing to ensure accurate crack boundary representation without losing fine details. This image preprocessing workflow was implemented using MATLAB R2018’s Image Processing Toolbox.

Box-counting method for fractal dimension calculation: The processed binary images were analyzed using the box-counting method to estimate the fractal dimension (D). A series of grids of varying sizes (ε) was superimposed onto the binary image, and the number of grid cells (N(ε)) containing part of the crack pattern was counted for each grid size. The relationship between N(ε) and ε follows a power-law distribution:

$$N\left(ϵ\right)\propto {ϵ}^{-D}$$

(1)

By plotting log(N(ε)) versus log(1/ε), the slope of the resulting linear fit corresponds to the estimated fractal dimension.

Figure 3. Comparison of AE b-value evolution and AE amplitude distribution with stress during uniaxial loading. (Left): b-value evolution; (Right): AE amplitude distribution.

Figure 3. Comparison of AE b-value evolution and AE amplitude distribution with stress during uniaxial loading. (Left): b-value evolution; (Right): AE amplitude distribution.

In this study, grid sizes were systematically varied from 2 × 2 to 64 × 64 pixels to ensure robust scaling analysis. For each specimen, fractal dimension values were calculated for both the entire surface and individual crack zones, allowing for a comparative assessment of damage distribution complexity.

CT scanning was performed to capture the internal crack geometry of selected specimens after failure. Due to limitations in image quality and resolution, only a representative cross-sectional slice is presented here to illustrate the general crack propagation pattern. This image provides qualitative support for the damage distribution trends observed in both AE monitoring and fractal dimension analyses.

CT and optical images were processed to extract fracture patterns for fractal analysis, consistent with the methodologies of Liu et al. (2021) and Wang et al. (2025), who correlated FD changes with crack complexity and strength loss [8,33]. Initial CT scans (first image) confirmed the intact condition of the specimen, while post-compression CT (second image) revealed internal crack development. The third image illustrates a grid-based segmentation of the CT data with marked crack paths, which facilitates box-counting for fractal dimension computation. Subsequent binary images of specimen surfaces display the development of visible cracks across various conditions, including different crack lengths and angles, before and after compression. These binarized fracture maps serve as input for calculating the fractal dimension by quantifying the number of crack intersections across grid scales.

Figure 4 presents image processing sequence for crack pattern extraction and fractal dimension analysis. From left to right:

Figure 4a shows the surface image of the specimen before compression, where no visible cracks are present. Figure 4b illustrates the surface image after compression, revealing multiple crack branches generated during loading. Figure 4c presents the CT scan image after compression, providing an internal view of crack propagation not observable on the surface. The CT image corresponds to a vertical cross-sectional scan taken through the central axis of the specimen. Figure 4d demonstrates the CT crack extraction with grid overlay, which was used in the box-counting method to calculate the fractal dimension of the crack pattern.

This sequence clearly demonstrates the transition from an intact specimen to a fractured state and shows how surface and internal crack features were jointly utilized in the fractal analysis. By aligning the text with Figure 4 subfigures, the correspondence between experimental images, CT data, and fractal dimension calculations is clarified.

The obtained fractal dimensions served as a supplementary indicator of crack complexity, providing a quantitative link between macroscopic crack patterns and the damage evolution observed through acoustic emission monitoring.

According to

Table 2. Summary of FD value results.

Group	Crack Type	Length (mm)	Angle (°)	Peak Stress (MPa)	FD Before	FD After	ΔFD (After-Before)
1	Control (no crack)	-	-	37.33	1.003	1.465	+0.462
2	Straight crack	25	0	29.94	1.245	1.540	+0.295
3	Straight crack	50	0	25.46	1.384	1.621	+0.237
4	Straight crack	75	0	21.03	1.450	1.946	+0.496
5	Inclined crack	25	30	29.14	1.293	1.425	+0.132
6	Inclined crack	25	45	22.62	1.333	1.708	+0.375
7	Inclined crack	25	60	28.24	1.215	1.419	+0.204

, the fractal dimension (FD) analysis reveals the evolution of geometric complexity in cracked concrete specimens before and after uniaxial compression. Across all cases, the FD increases after loading, indicating damage accumulation and crack propagation. The initial FD rises with crack length, reflecting higher pre-existing crack complexity. The 0 mm crack specimen exhibits a significant FD increase (+0.462), suggesting that damage is primarily initiated and distributed under loading. In contrast, the 75 mm crack case shows the highest post-compression FD (1.946) and the greatest increase (+0.496), signifying severe crack coalescence and a highly connected fracture network. For intermediate cases (25 mm and 50 mm), the FD increments are moderate, implying that while cracks grow, they are somewhat confined along the initial flaw. These results confirm that both crack presence and length significantly influence the spatial evolution and complexity of the failure pattern in concrete.

As illustrated in Figure 5, the fractal dimension (FD) analysis for specimens with inclined cracks at 30°, 45°, and 60° reveals notable differences in damage complexity before and after compression. Prior to loading, the FD values range from approximately 1.215 to 1.333, indicating moderate initial geometric complexity of the inclined cracks. After compression, all specimens show increased FD values, reflecting crack propagation and structural damage. The 45° crack exhibits the highest increase in FD (+0.375), confirming its critical role in guiding shear failure along the principal stress direction. In contrast, the 30° crack results in a minimal FD increase (+0.132), suggesting limited crack extension and more localized damage. The 60° crack shows a moderate rise in FD (+0.204), indicating a combination of shear and tensile mechanisms. These findings demonstrate that crack inclination significantly affects the evolution of fracture complexity, with 45° being the most critical angle for inducing extensive and connected fracture networks.

As illustrated in Figure 6, a clear negative linear correlation was observed between the peak stress and the post-test fractal dimension (FD), suggesting that increased crack complexity directly corresponds to mechanical degradation. Furthermore, ΔFD increased with crack length among straight-crack specimens, highlighting the effect of initial defect size on the extent of crack propagation. In inclined-crack specimens, ΔFD peaked at 45°, indicating that shear-induced cracking leads to more complex fracture networks. These findings establish FD variation as a potential damage quantification tool, correlating well with both strength reduction and crack geometry.

The relationships between FD variation and crack parameters were visualized and quantitatively fitted. A negative linear trend was found between post-failure FD and peak strength, suggesting that higher FD reflects more severe damage and strength degradation. The variation in FD (ΔFD) showed a quadratic relationship with crack length and a cubic trend with crack angle, indicating that both crack geometry factors nonlinearly influence the degree of fracture complexity. These results confirm the feasibility of using FD as an indicator for damage evolution and mechanical response in cracked concrete specimens.

3. Discrete Element Modeling Using PFC3D

3.1. Significance of DEM Modeling

To further investigate the fracture behavior of pre-cracked concrete specimens and validate experimental observations, discrete element method (DEM) simulations were conducted using PFC3D. The influence of aggregate distribution on crack initiation and propagation has been effectively analyzed using DEM, showing that heterogeneity plays a critical role in fracture localization and strength variation [34]. Tensile fracture processes in concrete were further explored by Zhu et al. (2025) using mesoscale DEM models, revealing detailed crack patterns under uniaxial tension [20]. Yang et al. (2022) combined experimental data with PFC2D simulations to analyze uniaxial compression of cracked rock masses, validating numerical results against observed failure patterns [35].

Each concrete specimen was represented as an assembly of spherical particles, generated within a cubic domain matching the experimental dimensions (100 mm × 100 mm × 100 mm). The particles were bonded using a linear parallel bond model to simulate the cohesive behavior of the cement matrix. The contact stiffness, bond strength, and friction coefficients were calibrated against the stress–strain curves obtained from the control group (uncracked specimens), ensuring the numerical model could reasonably reproduce the compressive strength and deformation characteristics of C30 concrete. As shown in Figure 7, pre-existing cracks were introduced in the DEM model by removing bonds along specified planes, replicating the same geometries (length, width, depth) as the experimental specimens. The loading conditions, including axial compression and boundary constraints, were applied to mimic the uniaxial compression test setup.

The DEM approach allows for meso-scale exploration of fracture processes via particle interaction modeling and bond failure evolution, as detailed by Nitka and Tejchman (2025) and Zhu et al. (2025) [23,24].

Yu et al. (2021) developed predictive DEM-based models for concrete under uniaxial loading, providing a foundation for mechanical parameter calibration in simulation studies [19]. Micro-crack evolution, as visualized through bond breakage events, aligned well with AE monitoring results, showing a transition from distributed micro-damage to localized macro-crack formation.

In particular, the simulations revealed distinct fracture mechanisms influenced by crack geometry: Long crack promoted early localization of damage along the pre-existing plane, leading to brittle splitting failure. Inclined cracks induced mixed-mode fracture behavior, combining tensile and shear failure characteristics. These observations provided a more granular understanding of how crack geometry influences stress redistribution and failure propagation within the material, supplementing the experimental findings with insights into the underlying meso-scale mechanics.

The integration of experimental testing, AE monitoring, fractal dimension analysis, and DEM simulations offers a comprehensive approach to studying tunnel lining defects. By bridging the gap between observed macroscopic behavior and meso-scale fracture mechanics, this study demonstrates the potential of DEM as a powerful tool for damage assessment and future reinforcement design in tunnel engineering applications.

3.2. Model Setup and Parameter Calibration

3.2.1. Elastic Model Selection for DEM Simulation

In this study, the bonded-particle model (BPM) is employed to simulate the behavior of concrete with pre-existing cracks under uniaxial compressive loading. BPM is a widely used contact model in the discrete element method (DEM) framework, particularly suitable for brittle materials such as concrete and rock. The key features of BPM include the following:

Particles are represented as rigid spheres (balls) that interact via a linear elastic contact law. Each contact between particles is assigned bond properties, representing cohesive forces within the material. Bonds can transmit both normal and shear forces, as well as moments, enabling the model to capture tensile, compressive, and shear failure modes.

When bond strengths are exceeded, the bond breaks irreversibly, simulating micro-crack initiation and propagation.

The mechanical behavior of the bonded particles follows a linear elastic constitutive law until bond breakage occurs.

3.2.2. Parameter Calibration and Selection

Simulation parameters were calibrated using C30 concrete test data and literature references such as Nitka et al. (2020),Wang et al. (2022) and Wang et al. (2025), who provided validated DEM input for crack analysis [14,33,36]. The main parameters are included in

Table 3. Parameters of DEM simulation.

Parameter	Symbol	Typical Value	Notes
Particle density	$\rho$	2400 kg/m3	Consistent with concrete density
Particle size (radius)	$r$	0.5–1.5 mm	Adjusted for model resolution
Normal stiffness	$k_n$	1.0 × 109 N/m	Calibrated to match the elastic modulus
Shear stiffness	$k_s$	0.5 × 109 N/m	Typically $k_s/k_n$ = 0.5
Normal bond strength	${\sigma }_n$	4.75 MPa	Calibrated to match peak compressive strength
Shear bond strength	${\sigma }_s$	2.85 MPa	Typically, ${\sigma }_s$ = 0.6 × ${\sigma }_n$
Friction coefficient	$\mu$	0.5	Typical value for concrete contact
Bond radius multiplier	$\lambda$	1.0–1.2	Slightly larger than the particle radius for stability

.

3.2.3. Model Assumptions and Limitations

High-performance computing methods have been explored for damage analysis, including GPU-accelerated DEM simulations for SCC-rock composites by Wang et al. (2025) [33]. Early foundational work by Marooden (2018) contributed to the development of discrete element models tailored for concrete behavior simulation [34]. The model assumes isotropic and homogeneous material properties at the particle scale, which simplifies the heterogeneous nature of concrete. Aggregate effects and pore structures are not explicitly modeled but are implicitly represented through particle interactions. The loading is applied under quasi-static conditions to replicate uniaxial compression tests. Scale effects due to particle size selection are acknowledged, and particle size is chosen as a compromise between computational efficiency and accuracy in capturing crack behavior.

3.2.4. Relevance to Tunnel Lining Simulation

By establishing a calibrated DEM model for cracked concrete specimens, the framework can be extended to simulate tunnel lining segments with pre-existing cracks. The calibrated contact and bond parameters enable modeling of crack initiation, coalescence, and propagation under compressive and bending stresses, providing insights into the mechanical behavior of tunnel linings under service and seismic conditions.

As illustrated in Figure 8, the displacement vector plots illustrate the influence of increasing crack length (0 mm, 25 mm, 50 mm, and 75 mm) on the deformation behavior of concrete specimens under uniaxial compression. In the intact specimen (0 mm), the displacement field shows a typical butterfly-shaped pattern with symmetrical diagonal shear zones, indicating uniform deformation and relatively high structural integrity. With a 25 mm crack, displacement becomes concentrated around the crack tips, breaking the overall symmetry and revealing the onset of localized damage driven by stress concentration. At 50 mm, the crack begins to dominate the displacement field, guiding deformation along its length and leading to asymmetric shear zones and crack propagation. In the 75 mm case, the crack nearly divides the specimen, with intense, localized displacement observed along the crack plane, indicating a transition to tensile–shear dominated failure. The particle movement becomes strongly decoupled above and below the crack, reflecting the reduced stiffness and early failure of the specimen. Overall, the results confirm that longer cracks significantly alter the displacement field and reduce structural resistance.

The figure presents displacement vector fields (left) and DFN-based failure particle distributions (right) for various pre-cracked concrete configurations under uniaxial compression. Notably, the green particles on the right subfigures represent fractured or failed particles identified based on local bond breakage or contact rupture, corresponding to damage localization and crack propagation zones. A key observation from this figure is the inverse relationship between the number of visible DFN particles and the extent of structural integrity. Specifically, a lower number of green particles often indicates larger zones of coalesced or concentrated failure, where extensive bond breakage has caused particles to be ejected or displaced beyond the central failure region.

As illustrated in Figure 9, the displacement vector plots illustrate the influence of crack inclination angles (0°, 30°, 45°, 60°, 75°, and 90°) on particle movement under uniaxial compression in PFC3D simulations. At 0°, the horizontal crack inhibits vertical displacements directly above and below the crack plane, indicating crack closure under compression. At 30°, noticeable shear displacement begins to emerge, with particles moving diagonally in response to stress concentrations near the crack tips. The 45° case shows the most distinct shear–slip behavior, with symmetric displacement vectors aligned along the crack, representing maximum displacement and damage localization due to alignment with the principal shear direction. In the 60° case, the displacement pattern becomes more asymmetric and suggests a mixed-mode failure, combining shear and tensile effects. At 75°, the displacement vectors indicate a transitional failure mode. Compared with 60°, the shear component becomes less dominant, and the crack inclination is nearly aligned with the vertical loading axis. This leads to a pronounced asymmetric displacement distribution, where localized shear slip initiates along the crack plane but gradually transforms into tensile opening near the crack tips. Finally, the 90° vertical crack shows a relatively symmetric displacement field with minimal interference from the crack itself, indicating that vertically aligned cracks tend to close and exert minimal influence on displacement propagation. These findings confirm that 45° is the most critical angle for facilitating displacement and failure propagation under compressive loading.

In contrast, a higher number of dispersed green particles suggests that the damage is more distributed or less catastrophic, potentially reflecting micro-crack accumulation rather than dominant macro-failure. Higher numbers of dispersed DFN particles suggest distributed micro-damage, consistent with observations by Wang et al. (2022) and Zhu et al. (2025), where severe failure showed localized DFN zones, while ductile damage showed scattered, numerous points [23,36]. DFN visualizations under progressive loading reveal that more severe, unstable fractures result in fewer but more localized and connected failed zones, while more ductile or less critical failure modes result in fragmented, numerous DFN points.

Furthermore, the corresponding displacement fields support this conclusion. In images such as Figure 9a,d, the concentrated large displacement zones and clear vortex structures suggest shear localization or tensile breakout, often correlated with bulk collapse. These behaviors are consistent with lower particle count in the DFN image because particles in these regions are fully mobilized or expelled and no longer registered as “intact or marginally failed”.

Figure 10 illustrates a comparative analysis of peak compressive strength between experimental and numerical results across various crack configurations, including changes in crack length and inclination angle. The overall trend reveals that both experimental and numerical data demonstrate a consistent decrease in strength with increasing crack length and more critical inclination angles (e.g., 45°). The control specimen (no crack) shows the highest strength, with only a minor difference between simulation and test results (~0.2 MPa), indicating the reliability of the model under intact conditions.

For specimens with straight cracks (0° inclination), the numerical results tend to slightly overestimate strength, though the error remains within a reasonable range. Notably, the 75 mm crack case exhibits the most significant strength reduction, highlighting the detrimental effect of long, surface-aligned flaws. For inclined cracks, especially at 45°, the simulation underestimates the strength compared to experimental values, suggesting potential limitations in modeling shear-induced crack bridging or particle rearrangement under inclined failure planes.

Overall, the trends between simulation and experiment align well, supporting the model’s predictive capability. However, deviations in specific cases (e.g., inclined 45°) underscore the importance of incorporating more advanced contact laws or microstructure representations in future numerical approaches.

The DEM simulations provide important insights into the crack propagation process and its implications for tunnel lining performance. By calibrating the bond strength parameters, the numerical model was able to reproduce the general stress–strain response observed in the laboratory tests. More importantly, the simulations captured the qualitative failure patterns associated with different crack geometries.

Specifically, the DEM simulations showed that specimens with longer prefabricated cracks exhibited splitting-type failures characterized by dominant tensile fracture planes, consistent with the experimental observations of brittle splitting and surface spalling. In contrast, specimens containing 45° inclined cracks developed shear-dominated failure mechanisms, where sliding along the crack plane triggered premature loss of load-bearing capacity. This was in strong agreement with the experimental results, where 45° cracks led to the greatest reduction in peak strength and distinct shear fracture surfaces.

Furthermore, the displacement vector fields from the simulations matched well with the measured deformation patterns in the tests. For example, localized displacements near the crack tips and the development of diagonal shear bands in inclined-crack specimens were consistently reproduced in both approaches. These correspondences highlight the reliability of the bonded-particle model in representing crack propagation and failure mechanisms in pre-cracked concrete.

Taken together, the alignment between DEM simulations and laboratory experiments provides confidence that the numerical model can be extended to tunnel-scale analyses. The ability to capture both tensile- and shear-dominated crack mechanisms suggests that DEM is a suitable tool for evaluating the impact of defect geometry on tunnel lining safety.

4. Conclusions

This study comprehensively investigated the mechanical response of concrete specimens with varying crack lengths and orientations under uniaxial compression, through both laboratory experiments and discrete element method (DEM) simulations. The key findings are summarized as follows:

• Crack length and inclination significantly affected the strength and failure behavior of concrete. Longer cracks and higher inclination angles (e.g., 45°) led to greater reductions in peak stress and earlier onset of macroscopic failure.
• Acoustic emission (AE) activity effectively captured the internal damage evolution, with the b-value and high-amplitude AE events showing clear correlations with crack growth and stress drops.
• Fractal dimension (FD) analysis of post-failure crack patterns revealed a consistent trend, where specimens with more complex and branched cracks exhibited higher FD values. Notably, a negative correlation between FD and peak strength was observed.
• DEM simulations reproduced the general crack propagation paths and displacement fields observed in experiments, validating the effectiveness of the bonded-particle model in simulating pre-cracked concrete behavior.

These results provide important insights into the damage mechanisms of tunnel lining concrete with structural defects and demonstrate the value of combining AE monitoring, fractal analysis, and numerical modeling to evaluate and predict structural performance.

4.1. Practical Implications for Tunnel Engineering

The findings have direct relevance to tunnel lining assessment and maintenance. In particular, cracks oriented at approximately 45° to the loading axis should be regarded as critical defects as they lead to the most severe strength reduction. Reinforcement measures such as FRP wrapping, steel plate bonding, or pressure grouting should, therefore, be prioritized for inclined cracks near 45° since these are the most likely to trigger premature failure.

Moreover, the demonstrated influence of crack length and orientation suggests that future tunnel design codes could incorporate crack-orientation-based safety factors. This would provide engineers with more reliable criteria for assessing the residual strength of cracked linings. From a repair perspective, localized strengthening strategies should prioritize long and inclined cracks, which accelerate failure more rapidly than shorter or horizontal ones.

4.2. Limitations and Future Work

This study has several limitations. First, only single-crack specimens were investigated to isolate the effects of crack length and orientation. The synergistic influence of multiple interacting cracks, which is highly relevant for tunnel linings, was not considered and will be the subject of future work. Second, environmental factors such as temperature and humidity were not incorporated in the experimental program, although these conditions can significantly influence the long-term durability of tunnel concrete. Finally, discrepancies remain between numerical and experimental results, especially for inclined cracks at 45°, which highlights the need for more advanced modeling approaches. Future research will extend the current framework to address these limitations and provide more comprehensive guidance for tunnel design and maintenance.