Geometric Coupling Effects of Multiple Cracks on Fracture Behavior: Insights from Discrete Element Simulations

Abstract

Understanding the multi-crack coupling fracture behavior in brittle materials is particularly critical for aging dam infrastructure, where 78% of structural failures originate from crack network coalescence. In this study, we introduce the concepts of crack distance ratio (DR) and size ratio (SR) to describe the relationship between crack position and length and employ the discrete element method (DEM) for extensive numerical simulations. Specifically, a crack density function is introduced to assess microscale damage evolution, and the study systematically examines the macroscopic mechanical properties, failure modes, and microscale damage evolution of rock-like materials under varying DR and SR conditions. The results show that increasing the crack distance ratio and crack angle can inhibit the crack formation at the same tip of the prefabricated crack. The increase in the size ratio will promote the formation of prefabricated cracks on the same side. The increase in the distance ratio and size ratio significantly accelerate the rapid increase in crack density in the second stage. The crack angle provides the opposite effect. In the middle stage of loading, the growth rate of crack density decreases with the increase in crack angle. Overall, the size ratio has a greater influence on the evolution of microscopic damage. This research provides new insights into understanding and predicting the behavior of materials under complex stress conditions, thus contributing to the optimization of structural design and the improvement of engineering safety.

1. Introduction

In the realm of civil engineering, the capacity of structural materials to withstand fractures serves as a critical parameter for ensuring their long-term safety and durability—a principle particularly crucial for dam engineering, where structural failure could lead to catastrophic consequences. This holds especially true for brittle materials like concrete and reinforced concrete, which constitute over 80% of dam structures. These materials are susceptible to cracking during their service life due to factors such as hydraulic fracturing, thermal stresses, foundation settlement, and alkali–aggregate reactions, all of which are typical failure mechanisms observed in dam engineering. Should such cracks not be adequately managed, they can propagate, eventually leading to structural failure. Consequently, a comprehensive understanding of crack initiation, propagation, and their impact on structural integrity is vital to enhance the safety and longevity of civil engineering structures

Several scholars have investigated the issue of crack propagation in specimens featuring two cracks through experimental studies. Lin et al. [1] used acoustic emission (AE), and digital image correlation (DIC) techniques were used to study the crack merging process of two layers of different samples. Zhang et al. [2] examined the penetration mechanisms of two fissures positioned in various configurations, including echelon and coplanar alignments. Likewise, Yi Yongliang et al. [3] performed uniaxial compression tests on rock samples with dual prefabricated cracks to investigate the influence of crack and rock bridge inclinations on penetration modes. They noted that a single wing crack propagation typically occurs up to penetration when the prefabricated crack angle does not exceed 25° and the rock bridge angle does not surpass 45°. Lekan et al. [4] studied the crack behavior and coalescence process of brittle materials with two non-parallel overlapping defects using a high-speed camera. Their results show that the geometry of pre-existing defects affects the initiation and merging behavior of cracks. Jin et al. [5] conducted uniaxial compression tests on cubic samples with prefabricated discontinuous fractures using a rock mechanics servo-controlled testing machine, emphasizing the impact of fracture geometry on failure characteristics. Collectively, these studies provide valuable insights and reliable data regarding crack propagation in prefabricated double-crack specimens. However, physical experiments are constrained by available conditions and equipment, often entailing significant costs.

Compared with traditional physical experiments, the numerical simulation method has lower research cost and can collect more data, meaning it becomes another effective means to study the problem of rock crack growth. The particle flow program PFC [6,7,8] based on the discrete element method (DEM) [9] treats materials as mutually bonded particles, which can simulate the initiation, expansion, and macro damage process in the rock interior [10,11,12]. It has unique advantages in studying the fracture of brittle materials. Ding Xiaobin et al. [13] used three-dimensional discrete element software (PFC3D) and the improved parallel crack structure model (IPBM) to study crack propagation in rock-like materials with double prefabricated cracks under biaxial compression. Jiang Mingjing et al. [14] used the discrete element method (DEM) to simulate the crack propagation rule and penetration mode of double-fracture rock samples with different prefabricated angles under uniaxial compression, and they found that the concentration of tensile stress between prefabricated cracks was the main cause of instability failure. Li Fan et al. [15] used DEM to study the crack propagation of two echelon prefabricated fissure samples with different bridge angles. Zhang et al. [8] studied the crack propagation process in parallel double-crack samples by using DEM and supplemented and modified the nine penetration modes observed by Wong et al. [16] in physical experiments. Huang Yanhua et al. [17] discussed the influence of confining pressure and rock bridge dip angle on the strength failure characteristics of red sandstone with intermittent double fractures. Zhang Han [18] simulated the uniaxial compression test of gypsum samples with parallel double cracks, classified and analyzed the crack propagation modes at different relative positions, and studied the force chains and particle displacement fields around the prefabricated cracks during their formation. In addition, realistic failure process analysis (RFPA) [19,20], phase-field simulation [21], the finite–discrete element method (FDEM) [22], smoothed particle hydrodynamics (SPH) [23], and near-field dynamics [24] have also been widely used in the study of crack propagation. However, most numerical simulation studies mainly focus on the crack morphology but ignore the effect of crack location distribution on the microscopic damage evolution of the specimen during loading. This oversight is significant because crack location directly governs the interaction between flaws, affecting stress redistribution and damage localization during loading. When flaw spacing and positioning are not properly considered, simulations may underestimate the role of stress amplification, shielding, and bridging, which collectively dictate the crack coalescence pathways and failure timing. Addressing this issue is essential for capturing the multiscale damage evolution processes observed in real materials, particularly in brittle geomaterials where spatial heterogeneity plays a dominant role in fracture propagation. By incorporating the spatial distribution of flaws through distance and size ratios, this study aims to reveal how geometric configurations influence both macroscopic failure patterns and microscopic damage accumulation.

In addition to DEM-based approaches, peridynamics (PD) has emerged as a powerful non-local numerical method for simulating crack initiation, propagation, and coalescence, especially in complex brittle materials. Unlike traditional continuum mechanics, PD naturally handles discontinuities and avoids singularities at crack tips. Recent works have successfully applied PD to studying multi-crack growth, branching behavior, and dynamic rupture processes under varying loading paths, providing insights that complement DEM simulations [25,26,27]. These studies demonstrate PD’s advantages in capturing crack interaction without the need for additional tracking schemes, offering new perspectives for analyzing fracture mechanisms in aging infrastructure.

In this study, the concepts of distance ratio and size ratio were introduced to describe the relationship between crack location and crack length, and crack propagation of specimens with different distance ratios and center ratios was simulated. To quantify damage characteristics more comprehensively, the concept of crack density was additionally introduced. In particular, the concept of crack density was introduced in this study, and the effects of these factors on macroscopic mechanical properties, crack propagation patterns, and microscopic damage evolution are discussed in depth. Compared with existing studies focusing on single flaw types or limited geometric variations, this work systematically investigates the coupled effects of multiple flaw parameters, offering a novel perspective on multi-crack interaction mechanisms. The research results herein provide important theoretical guidance for architectural and structural design and help to improve engineering safety.

2. Numerical Simulation

2.1. Model Description

The model used in this study is a rectangular specimen featuring two parallel opening cracks, as illustrated in Figure 1. The specimen is designed with a length of 150 mm and a width of 75 mm, providing a standardized geometry for investigating crack propagation and interaction under mechanical loading. To simulate the material’s granular structure, the specimen is filled with round particles, whose radii are uniformly distributed between 0.21 mm and 0.35 mm. This particle size distribution ensures a realistic representation of the material’s heterogeneity and mechanical behavior. Each sample contains approximately 38,000 particles, a quantity sufficient to capture the complex interactions and stress distributions within the material. Before introducing the cracks, the particles undergo an iterative process to achieve mechanical equilibrium, ensuring that the initial stress state of the specimen is stable and consistent across all simulations. Once equilibrium is reached, particles within a specified region are systematically removed to create prefabricated cracks.

Subsequently, a linear parallel bond model was used to simulate gypsum materials by adding a contact model between particles. When the force between particles exceeds the set bond strength, the bond will break, and the contact behavior between particles is described by the linear contact model. The influence of friction force is considered in both contact models. The numerical model is shown in Figure 1.

Next, a uniaxial compression simulation experiment was carried out. During the simulation process, the “wall” unit on the upper and lower sides of the model sample was used to simulate the loading plate to load the sample in a displacement-controlled manner at a loading rate of 0.05 m/s until the post-peak stress decays to 70% of the peak strength. This loading rate was selected based on previous DEM studies to ensure a balance between computational efficiency and simulation stability. To verify that the chosen rate did not introduce dynamic effects, a rate sensitivity analysis was conducted using lower rates of 0.01 m/s and 0.02 m/s. The resulting differences in peak strength, Young’s modulus, and crack evolution were found to be within 2%, confirming that the adopted loading rate yields quasi-static responses and does not compromise the accuracy of mechanical behavior or fracture patterns.

In addition, the damping ratio of the sample was set to 0.1. For each prefabricated crack geometry, four numerical samples were generated using different random seeds to initialize particle assemblies. These random seeds controlled the spatial packing and distribution of particles while keeping material parameters constant, allowing for the evaluation of response variability and robustness due to micro-structural heterogeneity. After the simulation, the typical failure modes of the four samples with different crack location distribution were summarized.

2.2. Parameter Correction

It is very important to correct the microscopic parameters of the contact model to ensure the reliability of the simulation results; thus, careful calibration of the mesoscopic parameters in the contact model was performed using a trial-and-error approach. The parallel bond model parameters were initially referenced from Zhang et al. [20] then iteratively refined to reproduce the mechanical behavior of gypsum as observed in the uniaxial compression experiments reported by Wong et al. [13].

The key parameters adjusted during calibration included the normal and shear stiff-ness of the parallel bond, bond tensile strength, cohesion, and friction coefficient. Each adjustment aimed to align the macroscopic mechanical responses of the numerical model with the experimental results, especially in terms of uniaxial compressive strength and Young’s modulus. The adjustment process was guided by two primary criteria: (1) reducing the deviation in peak stress and elastic modulus between the simulation and experiment to within 5% and (2) ensuring consistency in failure mode evolution, including crack initiation, propagation direction, and final rupture patterns.

To verify the robustness of the calibrated parameters, simulations were conducted with different random seed configurations to generate particle assemblies with slight het-erogeneity. The model exhibited stable mechanical responses and consistent crack evolution across multiple runs, demonstrating its robustness under minor variations.

Table 1. Microscopic parameters of materials.

Particle Parameters		Parallel Bond Parameters
$E_c$	0.668 GPa	$E_c$	0.668 GPa
$k_n/k_s$	2.2	${\overline{k}}_n/{\overline{k}}_s$	2.2
$\mu$	0.35	${\overline{\sigma }}_c$	5.3 MPa
$r_{hi}/r_{lo}$	1.66	$\overline{c}$	8.365 Mpa
$r_{lo}$	0.22	$\overline{\lambda }$	1.0
$\rho$	1830	$\overline{\varphi }$	0

summarizes the final selected material parameters.

2.2.1. Comparison of Macroscopic Mechanical Properties

By systematically comparing the compressive strength and elastic modulus obtained through physical experiments and numerical simulations (see

Table 2. Detailed parameter comparison of physical experiment and numerical simulation.

Parameters	Physical Experiment	Numerical Simulation
Density	1.54	1.54
Young’s modulus	5.96	6.01
Poisson’s ratio	0.15	0.149
Uniaxial compressive strength	33.85	33.61

), it can be observed that there is a remarkable agreement between the two sets of results, with the difference for all parameters being less than 5%. This close correspondence not only validates the accuracy of the experimental measurements but also confirms the robustness of the numerical model in capturing the material’s mechanical behavior. The minimal discrepancy between the experimental and simulated results suggests that the constitutive model, material parameters, and boundary conditions employed in the numerical simulations were appropriately selected and implemented.

2.2.2. Failure Mode Comparison

Subsequently, a comprehensive comparison was conducted between the physical experiment and the numerical simulation regarding the failure mode, with the comparative results presented in Figure 2. It can be observed from the figure that the initial crack in the failure mode consistently originates in the middle of the prefabricated crack and propagates in the direction perpendicular to the initial crack, a phenomenon that is consistently replicated in both physical experiments and numerical simulations. This alignment in crack initiation and propagation patterns underscores the accuracy of the numerical model in capturing the fundamental fracture mechanics of the material. Following the initial crack propagation, secondary cracks are generated at the crack tip, which similarly develop in the vertical direction, ultimately coalescing to form a penetrating crack that leads to the complete failure of the sample. The high degree of consistency in the failure mode between the physical experiments and numerical simulations highlights the effectiveness of the numerical model in simulating the complex crack evolution process. This agreement not only validates the numerical approach but also provides deeper insights into the failure mechanisms of the material under study. The consistent replication of crack initiation, propagation, and final failure patterns in both experimental and numerical domains reinforces the reliability of the findings and enhances the predictive capability of the numerical model for more complex fracture scenarios. Such a high level of agreement between experimental observations and numerical predictions is crucial for advancing the understanding of material failure and for developing more accurate and reliable computational tools for fracture analysis in engineering applications.

3. Results

For the sake of clarity, this paper refers to the left endpoints of the upper and lower pre-existing cracks in the model as the L-end and the right endpoints as the R-end. A schematic diagram illustrating the parameter definitions is provided in Figure 3.

The geometric configuration of the two pre-existing cracks in the specimen is controlled by three parameters. The crack length (L1) represents the length of the upper pre-existing crack, while the crack length (L2) denotes the length of the lower pre-existing crack. The vertical spacing (V) refers to the perpendicular distance between the two parallel pre-existing cracks. The analyzed ranges of distance ratio (DR = 0.80–2.80) and center ratio (CR = 0.25–1.00) are chosen to reflect realistic crack spacings and size variations commonly observed in concrete and other quasi-brittle materials. These values are consistent with prior experimental and numerical studies, ensuring practical relevance and comparability. To better evaluate the influence of crack configuration on the macro- and micro-mechanical properties of the specimen, two parameters are defined: the distance ratio (DR) and the size ratio (CR). The calculation methods are as follows:

$$VR=V/\mathrm{L}1$$

(1)

$$SR=\mathrm{L}2/\mathrm{L}1$$

(2)

By systematically adjusting the distance ratio and the center ratio of the model and conducting large-scale numerical simulations, the influence of these two parameters on the failure mode, macroscopic mechanical properties, and microscopic damage evolution of the specimen was investigated. The detailed analysis is as follows.

3.1. Effect of Crack Spacing

3.1.1. Effect of Crack Distance Ratio on Failure Mode

The failure modes of specimens with different distance ratios are illustrated in Figure 4a–g. Based on the observations from the figures, the initial failure of the specimens consistently initiates from the middle of the pre-existing cracks and eventually propagates approximately perpendicular to the direction of the pre-existing cracks. Secondary cracks emerge and begin to propagate from one tip of the pre-existing cracks, ultimately forming macroscopic cracks that connect the tips on the same side of the two pre-existing cracks. These are referred to as same-side cracks (LL cracks or RR cracks), and their angles are roughly parallel to the loading direction. When the distance ratio (DR) increases to 1.2, in addition to the main cracks propagating along the pre-existing cracks, macroscopic cracks also begin to appear in the intermediate region between the pre-existing cracks. As the distance ratio further increases to 2.8, no macroscopic cracks connecting the tips on the same side of the two pre-existing cracks are observed during specimen failure.

Based on the above observations, it can be inferred that, at a crack angle of 15°, there exists a threshold between DR = 1.4 and 2.8. When the DR exceeds this threshold, the coupling effect between the two pre-existing cracks significantly weakens, becoming insufficient to induce the formation of same-side cracks. Therefore, variations in the distance ratio within this range have a decisive influence on the presence or absence of same-side cracks. This phenomenon highlights the critical role of the distance ratio in determining crack propagation paths and the formation of same-side cracks, providing important insights into the understanding of material failure mechanisms. This observation is consistent with findings from true triaxial compression tests on flawed sandstone. In those studies, researchers reported that variations in flaw spacing and orientation could shift the dominant fracture mode, either promoting crack coalescence or leading to more independent failure behavior under complex stress conditions [28].

3.1.2. Effect of Crack Distance Ratio on Macroscopic Mechanical Properties

Figure 5 illustrates the stress–strain curves of specimens with different distance ratios (DR). The compressive strength exhibits a non-monotonic, inverted U-shaped trend with increasing DR. Specifically, the peak compressive strength decreases from 36.2 MPa at DR = 0.80 to a minimum of 31.7 MPa at DR = 1.60, then increases again to 35.6 MPa at DR = 2.20.

This variation reflects the competing effects of crack interaction and stress redistribution. At low DR values, adjacent cracks are in close proximity, leading to strong stress amplification that promotes early coalescence and reduces strength. As DR increases, these interactions weaken, and the bridging effect reaches its minimum around DR = 1.60, where the specimen becomes particularly susceptible to instability. Beyond this point, the cracks are sufficiently spaced so that interaction becomes negligible, and shielding effects dominate. This shift allows the material to behave more like an intact medium and recover part of its compressive strength.

The Young’s modulus decreases monotonically with DR, from 6.05 GPa at DR = 0.80 to 5.32 GPa at DR = 2.20. While the modulus continuously declines, the rate of decrease slows with increasing DR, indicating a saturation effect in stiffness degradation as the influence of flaw interaction diminishes.

Beyond these general trends, the stress–strain curves display noticeable stress drops following peak strength, which can be interpreted as signatures of abrupt macrofracturing events. Similar phenomena have been observed in heated granite under true triaxial conditions, where internal crack coalescence and energy dissipation triggered global failure [29]. These patterns reinforce the interpretation that the simulated stress drops reflect localized damage accumulation followed by the rapid formation of a connected crack network.

The observed mechanical behavior aligns with the framework of stress amplification and shielding. At small flaw spacings, intense amplification encourages early fracture; at intermediate spacing, structural bridging is weakest; and at larger DR values, shielding enhances compressive resistance. This interpretation is consistent with previous studies emphasizing the critical role of flaw proximity on mechanical performance [30].

3.1.3. Effect of Crack Distance Ratio on Microscopic Damage Evolution

To further investigate the influence of the distance ratio on the microscopic damage evolution of the specimens, a crack density function was introduced to evaluate the microscopic damage in the samples. The crack density curves for specimens with different distance ratios are shown in Figure 6. Based on the observations from the figure, the crack density curves can be divided into four stages, detailed as follows.

Initial Stage: The crack density of all specimens remains relatively stable in the early phase, exhibiting consistent behavior.

Rapid Increase Stage (marked by a red ellipse in the figure): Subsequently, the crack density enters a phase of rapid increase, indicating the onset of significant microscopic damage accumulation.

Gradual Increase Stage: Following this, the crack density continues to increase at a slower rate over an extended period, suggesting a relatively slower rate of damage accumulation during this phase.

Failure Stage: Finally, the crack density rises sharply, accompanied by the overall failure of the specimen.

Of particularly note is that as the distance ratio (DR) increases, the rapid increase stage occurs earlier. This implies that a higher distance ratio accelerates the process of significant microscopic damage accumulation, leading to an earlier transition into the phase of rapid crack propagation and intensified damage.

3.2. Effect of Crack Size Ratio

3.2.1. Effect of Crack Size Ratio on Failure Mode

Figure 7 illustrates the failure modes of specimens with different crack size ratios. Based on the observations from Figure 4a–g and Figure 7a, the following results can be summarized.

When the size ratio is 0.2 (Figure 7a), cracks primarily initiate at the edges or middle regions of the two pre-existing cracks. On the longer side of the pre-existing cracks, the cracks propagate obliquely, eventually forming inclined cracks that connect the crack tips to the specimen corners. On the shorter side, the cracks propagate vertically. In the intermediate region between the two pre-existing cracks, a middle crack (M-type crack) forms, connecting the two pre-existing cracks.

When the size ratio increases to 0.6 (Figure 7b,c), as the size ratio increases, the failure mode in the intermediate region between the two pre-existing cracks changes. During failure, the specimen develops two fracture paths: one connecting the tips of the two pre-existing cracks (RR crack or LL crack), and the other starting from the tip of one pre-existing crack and eventually connecting to the middle region of the other crack (as shown in Figure 7b,c). These form the middle crack (M crack). This stage exhibits more complex crack propagation paths.

When the size ratio further increases to 0.8 (Figure 7d), cracks still primarily initiate at the edges or middle regions of the two pre-existing cracks. On the longer side, the cracks propagate obliquely, forming inclined cracks that connect the crack tips to the specimen corners. However, in the intermediate region between the two pre-existing cracks, no M-type cracks connecting the two pre-existing cracks are observed. Instead, two cracks connecting the tips of the pre-existing cracks are formed (as shown in Figure 7d). This indicates that an increase in the size ratio suppresses the formation of M-type cracks connecting the two pre-existing cracks.

When the size ratio is 1.0 (Figure 7e), the intermediate region between the two pre-existing cracks is connected by cracks extending from the tips of the pre-existing cracks. Cracks still initiate at the edges or middle regions of the two pre-existing cracks, but no inclined cracks connecting the crack tips to the specimen corners are formed. The increase in the size ratio inhibits their formation.

From these observations, we can conclude that an increase in the size ratio promotes the formation of cracks connecting the tips of the two pre-existing cracks in the intermediate region while suppressing the appearance of inclined cracks connecting the crack tips to the specimen corners.

3.2.2. Effect of Crack Size Ratio on Macroscopic Mechanical Properties

The stress–strain curves of specimens with different size ratios are systematically presented in Figure 8, revealing significant insights into the size-dependent mechanical behavior of the material. As can be observed from the figure, the compressive strength of the specimens exhibits a pronounced monotonically decreasing trend with increasing size ratio, demonstrating a clear size effect phenomenon. This indicates that the larger the size ratio, the lower the maximum load-bearing capacity of the material under uniaxial compression, which can be attributed to the increased probability of defects and stress concentration in larger specimens. The observed trend aligns well with the classical Weibull statistical theory of strength, where larger volumes of material are more likely to contain critical flaws that initiate failure. Similarly, the Young’s modulus of the specimens also decreases monotonically as the size ratio increases, suggesting that the material’s stiffness is sensitive to dimensional scaling effects. This reduction in elastic modulus with increasing size ratio may be explained by the potential changes in microstructural characteristics and the influence of boundary conditions that become more pronounced at larger scales. The consistent decrease in both compressive strength and Young’s modulus with increasing size ratio underscores the importance of considering size effects in the design and analysis of structural components, particularly when extrapolating laboratory-scale test results to real-world engineering applications. These findings contribute to the fundamental understanding of size-dependent material behavior and have significant implications for the development of reliable scaling laws in structural engineering.

Moreover, the stress drops observed after peak strength in Figure 8 correspond well with the crack density surges seen in Figure 9. This correlation highlights that macroscopic weakening is intrinsically linked to microcrack accumulation, especially in specimens with larger size ratios, where internal damage evolves more rapidly. Integrating the insights from Figure 8 and Figure 9 provides a coherent multiscale perspective on how crack size ratio governs both the global mechanical response and the underlying damage processes. Furthermore, the size ratio also influences the crack initiation process. At lower SR values, initial cracks tend to appear simultaneously along both pre-existing flaws. In contrast, higher SR values lead to earlier initiation at the longer flaw, where stress concentration becomes more dominant. This shift suggests that the asymmetry in crack size not only alters the damage accumulation rate but also governs the spatial onset of fracture.

3.2.3. Effect of Crack Size Ratio on Microscopic Damage Evolution

The evolution of microscopic damage is further characterized by the crack density curves of specimens with different crack size ratios, as shown in Figure 9. Crack density is defined here as the total length of generated microcracks normalized by the specimen area, providing a quantitative measure of internal damage during loading.

As shown in the figure, the progression of crack density during the loading process can be interpreted in three distinct stages, each corresponding to a particular damage evolution mechanism. In the initial phase, the crack density remains relatively constant, reflecting an elastic response with minimal microcrack activity. As loading progresses, the curve enters a rapid growth phase, where the crack density increases sharply over a short strain interval. This sudden escalation indicates the onset of substantial microcrack nucleation and linkage and often coincides with observable stress fluctuations in the stress–strain curve. Following this phase, the crack density continues to rise, but at a slower and more stable rate, suggesting that damage accumulates in a more distributed manner throughout the specimen after the main fracture pathways have developed.

Notably, specimens with larger crack size ratios display an earlier and more intense acceleration in crack density, especially in the intermediate strain range. This behavior implies that an increased size ratio promotes faster development of internal damage, likely due to enhanced stress concentration and flaw interaction, and consequently leads to reduced macroscopic strength and stiffness. This interpretation is further supported by the trend in post-loading observations, where specimens with larger size ratios exhibit higher peak crack densities. This pattern shows a clear inverse correlation with the observed variation in compressive strength, underscoring the intrinsic link between microscopic damage accumulation and overall mechanical degradation.

Additionally, when loading is stopped, the peak crack density of the specimens gradually increases with increasing size ratio. This trend is inversely correlated with the variation in compressive strength, indicating a certain relationship between the two.

Moreover, the acceleration phase preceding failure, as seen in the crack density curves (Figure 9), exhibits a pattern similar to the power-law growth commonly associated with critical failure phenomena. This progression aligns with the theoretical framework of time-reversed Omori-type acceleration, where the rate of internal damage accumulation increases significantly before final rupture. Such behavior has been demonstrated in recent studies as a potential indicator for forecasting failure time, particularly through the monitoring of dissipated energy and plastic strain increments in true triaxial tests [29] or through acoustic emission analysis in quasi-brittle materials [31]. These findings suggest that the microscopic damage evolution observed in our simulations holds potential relevance for failure prediction models under brittle failure conditions [32,33].

3.3. The Influence of Crack Angle

In this section, we investigate the influence of the crack angle (θ), which is defined as the inclination of the pre-existing cracks relative to the horizontal direction. A crack angle of 0° corresponds to horizontal cracks, while larger angles indicate counterclockwise rotation from the horizontal.

3.3.1. Effect of Crack Angle on Failure Mode

The failure modes of specimens with different crack angles are shown in Figure 10. Based on the observations from Figure 10a–e, the following results can be summarized.

When the crack angle is 0° (Figure 10a), cracks initially nucleate at the tips of the pre-existing cracks and eventually form tip-to-tip cracks (RR-type or LL-type cracks) connecting the upper and lower pre-existing cracks. Subsequently, cracks nucleate in the middle or edges of the upper and lower pre-existing cracks and propagate vertically.

When the crack angle increases to 15° and 30° (Figure 10b,c), the failure mode in the intermediate region between the two pre-existing cracks changes. A middle crack (M-type crack) connecting the two pre-existing cracks forms in the intermediate region. This stage exhibits more complex crack propagation paths.

When the crack angle further increases to 45° and 60° (Figure 10d,e), at this point, no macroscopic cracks connecting the tips on the same side of the two pre-existing cracks are observed during specimen failure (as shown in Figure 10e). This indicates that an increase in the crack angle suppresses the formation of cracks connecting the tips on the same side of the two pre-existing cracks.

3.3.2. Effect of Crack Angle on Macroscopic Mechanical Properties

Figure 11 shows the stress–strain curves of specimens with different crack angles. It can be observed from the figure that as the crack angle increases, the compressive strength of the specimen exhibits a monotonic increasing trend. This indicates that the larger the crack angle, the greater the maximum load-bearing capacity of the material under uniaxial compression. Similarly, the Young’s modulus of the specimens also increases monotonically with an increase in crack angle. The observation that both compressive strength and Young’s modulus increase monotonically with crack angle suggests significant changes in how stress is distributed within the material. Larger crack angles may lead to more favorable stress redistribution around the crack tip, effectively reducing stress concentration and delaying the onset of failure. This could explain why materials with higher crack angles exhibit greater load-bearing capacities. However, it is essential to consider the underlying microstructural mechanisms contributing to this behavior, such as crack deflection, branching, or bridging, which can enhance the material’s toughness and stiffness. Understanding these mechanisms can guide the design of materials with tailored properties for specific applications, optimizing their performance under uniaxial compression. Further research into the precise interaction between crack geometry and mechanical properties will provide deeper insights and potential improvements in material durability and reliability.

The observed increase in compressive strength with larger crack angles can also be attributed to stress shielding effects. As the crack angle increases, the orientation of flaws promotes more favorable redistribution of applied loads, reducing the stress concentration at flaw tips. This weakens the driving force for crack propagation and delays the onset of failure. In contrast, smaller crack angles tend to align more closely with the loading direction, intensifying stress amplification and promoting early failure. These behaviors align well with analytical and experimental studies showing how flaw geometry governs the balance between stress amplification and shielding [30].

3.3.3. Effect of Crack Angle on Microscopic Damage Evolution

The crack density curves of specimens with different distance ratios are shown in Figure 12. As can be observed from the figure, the crack density curves of all specimens can generally be divided into four stages, detailed as follows.

Initial Stage: The crack density of all specimens remains relatively stable during the early loading phase.

Rapid Increase Stage: Subsequently, the crack density enters a phase of rapid increase, indicating significant accumulation of microscopic damage.

Gradual Increase Stage: Following this, the crack density continues to increase at a slower rate over an extended period.

Failure Stage: Finally, the crack density rises sharply, accompanied by the overall failure of the specimen.

It is particular noteworthy that as the crack angle increases, the rate of crack density growth gradually decreases, and the transition point at which the crack density begins to increase rapidly is delayed. This suggests that an increase in the crack angle slows down the accumulation of microscopic damage and postpones the onset of rapid crack propagation.

Quantitative analysis of Figure 12 reveals that the specimen with a crack angle of 0° exhibits the highest peak crack density, reaching 0.042 mm/mm². In contrast, the specimen with a 60° angle shows the lowest value, around 0.030 mm/mm². The onset of rapid crack growth also occurs earlier in specimens with smaller angles. For example, the transition appears at a strain of 0.18 for 0°, whereas for 60°, it begins at 0.23. These findings support the interpretation that increasing the crack angle delays crack nucleation and reduces overall damage intensity, reinforcing its stabilizing role in fracture evolution.

Furthermore, the evolution of crack density shown in Figure 12 is consistent with the post-peak stress drops observed in Figure 11. This agreement highlights the close relationship between internal damage accumulation and macroscopic mechanical degradation. Larger crack angles not only enhance the material’s stiffness and strength but also suppress early-stage microcrack coalescence. These complementary observations reinforce the understanding that geometric configuration plays a critical role in regulating failure timing and damage localization. A combined reading of Figure 11 and Figure 12 thus reveals a multiscale interaction between crack angle, strength development, and internal fracture evolution.

3.3.4. Coupling Effect of Crack Angle and Distance Ratio on Macroscopic Mechanical Properties

The compressive strength of specimens with different crack angles and coupled crack distance ratios is systematically presented in Figure 13, providing valuable insights into the complex interaction between crack geometry and material failure behavior. As can be observed from the figure, the compressive strength of specimens with different crack angles exhibits a distinct U-shaped trend with increasing crack distance ratio, where the strength is higher at both ends and lower in the middle. This non-monotonic relationship suggests the existence of a critical interaction zone between the cracks, where stress concentration and crack-tip interaction are most pronounced. Upon closer comparison, it is found that for specimens with a crack angle of 15°, the compressive strength reaches its minimum value at a crack distance ratio of 1.2, indicating the point of maximum crack interaction for this specific geometry. In contrast, for specimens with a crack angle of 45°, the compressive strength reaches its minimum value at a crack distance ratio of 1.6, demonstrating the significant influence of crack orientation on the stress field distribution and failure mechanisms.

Although direct stress field visualization is not provided in this study, the U-shaped strength trend indirectly reflects the dynamic evolution of stress field interaction. A smaller crack spacing promotes overlapping stress zones, which intensify crack-tip interaction and reduce compressive strength. As spacing increases, these zones gradually decouple, and the material begins to behave as though the cracks are isolated. This interpretation is consistent with prior studies in fracture mechanics where stress shielding and amplification effects dominate at intermediate distances.

From these observations, we can conclude that as the crack distance ratio increases, the weakening effect of the double cracks on the compressive strength of the specimens gradually diminishes due to the reduced stress field overlap and decreased crack–tip interaction. Beyond a critical crack distance ratio, the double cracks no longer exert a weakening effect on the specimens, effectively behaving as independent flaws. This critical distance represents the threshold beyond which crack interaction becomes negligible, and the material’s strength is governed by the individual crack behavior rather than their combined effect. Additionally, an increase in the crack angle leads to an increase in this critical value, which can be attributed to the more extensive stress field redistribution associated with inclined cracks compared to straight cracks. This trend further supports the hypothesis that crack inclination modifies the local stress trajectories, leading to less focused stress accumulation and more dispersed energy release. These findings have important implications for fracture mechanics analysis and structural integrity assessment, particularly in the context of predicting the strength reduction caused by multiple flaw interactions in engineering materials. The results also highlight the need to consider both crack orientation and spacing when evaluating the residual strength of cracked structures, as these parameters significantly influence the material’s load-bearing capacity and failure characteristics.

4. Conclusions

This study reconstructed a DEM-based numerical model to investigate the effects of crack distance ratio, size ratio, and angle on the mechanical behavior and damage evolution of rock-like materials. The main conclusions are as follows.

(1)

Crack morphology and failure modes: Increasing the crack distance ratio and angle suppresses the formation of same-side tensile cracks, while a higher size ratio enhances internal crack coalescence. When the crack distance ratio exceeds a threshold (e.g., 2.4 at 15°), flaw interaction becomes negligible, altering failure patterns.

(2)

Macroscopic mechanical properties: Compressive strength shows a U-shaped trend with increasing distance ratio and decreases monotonically with size ratio. Young’s modulus declines with both parameters. Crack angle positively correlates with strength and stiffness due to improved stress redistribution.

(3)

Crack density and damage evolution: All specimens follow a four-stage crack density evolution. Larger crack sizes and smaller angles lead to faster damage accumulation and earlier failure. These trends align with post-peak stress drops, indicating multiscale coupling between internal damage and global weakening.

(4)

Mechanistic insights and design implications: The strength variations stem from stress amplification and shielding effects. Closely spaced or large flaws amplify stress and promote early failure, while wider spacing or higher angles reduce stress concentration and delay damage. These findings offer guidance for flaw configuration in brittle materials.

In summary, the distance ratio and size ratio are critical factors influencing the macroscopic mechanical properties, failure modes, and microscopic damage evolution of rock-like materials. The findings of this study provide a theoretical basis for optimizing material design and predicting material behavior under different conditions, thereby contributing to the improvement of material performance and reliability.

This study is subject to several limitations. The numerical model adopts a two-dimensional representation and considers idealized flaw shapes under uniaxial loading, which may not fully reflect the complexity of three-dimensional fracture processes in natural rocks. Future studies are encouraged to explore 3D modeling, incorporate spatial heterogeneity of flaws, and consider field-relevant loading paths to enhance applicability in rock engineering contexts.