1. Introduction
Asphalt concrete is a non-homogeneous composite material with a specific spatial structure. It is a mixture of asphalt mortar, aggregates, and voids. Traditional laboratory tests were carried out to reveal the relationship between fatigue cracking of the asphalt concrete and various factors, such as different additives [
1], fiber content [
2], and reclaimed asphalt pavement (RAP) content [
3]. Moreover, combined with ENDB (Edge Notched Disc Bend) and SEM (Scanning Electron Microscopy), it revealed the effect of asphalt-to-cement ratio (A/C) on the stress intensity factor of materials [
4]. Moreover, aiming at warm mix asphalt (WMA), the viscoelastoplastic continuum damage (VEPCD) model was established to illustrate the cracking sensitivity of asphalt concrete [
5]. And the four-point bending test [
6] was carried out to analyze the effect of mineral additives on the asphalt concrete viscoelastic behavior from the density, gradation, and strength properties, respectively.
However, because of the microcrack randomness in the spatial distribution of asphalt concrete, the stress field is different from the stress field of asphalt concrete with a single crack, ultimately leading to changes in the fracture behavior of asphalt concrete [
7,
8]. The Muskhelishvili complex function method [
9,
10], the small parameter method [
11,
12,
13], the distribution dislocation method [
14], the averaging method [
15], and the crack line method [
16] have been used to obtain the analytical solution of the stress intensity factor of co-linear double cracks and cyclic co-linear multi-cracks in an infinitely large flat plate under uniform load and concentrated load. These methods have also been used to determine the connection between the interaction effect and crack propagation from the perspective of the special distribution of microcracks. However, in practical engineering, the randomness of the spatial distribution of microcracks hinders the calculation of the analytical solution of the stress intensity factor.
Therefore, in studying the impact of penetrating crack clusters and surface crack clusters on macrocrack extension evolution, cracks have been mainly assumed to be in-plane cracks or surface cracks under plane stress or plane strain conditions. Jiang [
17,
18] and Renshaw [
19] used the projection method to establish a wing-like double-crack model by consolidating the penetrating crack group into a single crack. The model was used to investigate the effect of double-crack interactions and the interaction mechanism between secondary crack populations and macrocracks, considering parallel equal-length edge cracks and parallel unequal-length centering cracks, respectively. Moreover, the wing-like double-crack model has been applied to parallel-biased double cracks [
20], co-linear or biased double cracks [
21], short cracks [
22], and triple co-linear cracks [
23], respectively, where the interaction mechanism between penetrating crack groups and main cracks was determined by analyzing the influence of penetrating crack groups on the main crack propagation. Additionally, Kamaya [
24] merged surface crack clusters into a single crack using the envelope method; the author considered the relative position of the double cracks to reveal the shielding effect of parallel surface-biased double cracks on the macrocrack propagation. Moussa [
25] considered the size of double cracks to analyze the influence of the load type on the shielding effect of the parallel surface-biased double cracks. Meanwhile, from the perspective of secondary surface cracks, studies have revealed the influence of the depth of the secondary surface crack on the interaction effect between the secondary surface crack and the deflected primary crack [
26,
27,
28].
The Finite Element Method (FEM) and lab tests have also been combined to study the shielding effect of coplanar double cracks on macrocrack extension, considering various factors, such as microcrack spacing [
29,
30], the relative spacing between cracks [
21,
31], crack size [
32], and crack depth [
33]. The local fracture energy index was utilized to characterize the transient impact of a specific number of randomly distributed microcracks on macrocrack propagation through the integration of Digital Image Technology (DIT) and Acoustic Emission (AE) [
34]. The FEM was used to introduce microcracks with a consistent orientation in the constant and crack extension regions to establish the correlation between macroscopic crack extension and toughening mechanisms from the perspective of the number of microcracks [
35]. Furthermore, from the perspective of the toughening modulus of fiber-reinforced composites, the toughening effect of different microcracks with a unidirectional distribution was determined by analyzing the effect of unidirectionally distributed microcracks on the macrocrack propagation considering different fiber spatial orientations [
36].
In summary, most studies have focused on revealing the effect of the interactions between the penetrating double crack, the surface (buried) double crack, and the main crack. These studies have primarily used the projection method or envelope method, which combines a group of microcracks into a single crack. Meanwhile, a combination of lab experiments and the FEM has been used to describe the toughening effect of microcracks in terms of microcrack amount and orientation. However, changes in the stress field at the macrocrack tip resulting from the microcracks’ randomness in the spatial distribution of asphalt concrete influence the evolutionary behavior of the macrocrack’s extension.
Therefore, according to previous research results, the AC-13 suspended dense structural asphalt concrete at −20 °C was taken as the research object in this study. The Talyor medium method and the discrete element method (DEM) were combined to establish a meso-structural model of asphalt concrete with different microcrack densities; crack density was introduced to characterize the spatial distribution of the model’s internal microcracks. Additionally, the virtual semi-circular bend (SCB) was used to analyze the influence of different crack densities on the propagation behavior of the macrocrack considering macro and micro parameters, such as stress–strain curve, effective modulus change, and crack zone stress field. Moreover, we established the relationship between the distribution characteristics of microcracks and the interaction effect between cracks.
4. DEM Results Analysis
The SCB virtual test was conducted for asphalt concrete containing macrocracks with different microcrack densities
f3 by the discrete element analysis platform (PFC2D). The results are shown in
Figure 8, where the solid orange line indicates the crack extension in the material.
It is obvious that there is a large difference in the spatial distribution of the internal crack propagation (see
Figure 8), which indicates that the interaction effect between microcracks and the macrocrack changes as the density of microcracks changes.
Moreover, the numerical simulation results show the variation curve of the effective modulus of asphalt concrete with a microcrack density of
f3, as shown in
Figure 9. It can be seen in
Figure 9 that the effective modulus of asphalt concrete with a macrocrack has a nonlinear decreasing trend as the microcrack density
f3 increases. A good correlation exists between the numerical calculation results and the theoretical calculation results of the Talyor medium method, which indicates that the numerical calculation is accurate.
To further analyze the influence of the interaction effect between the microcracks and macrocrack cracks on the macrocrack propagation, the stress–strain relationship, crack extension process, and crack tip stress field are analyzed in the following subsections.
4.1. Stress–Strain Curve
The stress–strain variation was monitored by stress measurement circles, and the result is illustrated in
Figure 10.
Figure 10 shows that the interaction between microcracks and a macrocrack influences crack propagation until fracture failure occurs in the asphalt concrete, and the interaction effect is substantial. When the crack density
f3 ≤ 0.4, no obvious changes are observed in the fracture stress of the asphalt concrete, meaning that the interaction effect is weak, i.e., the interaction has no obvious effect on the macrocrack propagation (see
Figure 8). Moreover, the result also indicates that the interaction at this crack density mainly reduces the effective modulus of the asphalt concrete (see
Figure 9).
On the other hand, the interaction effect is enhanced when the crack density
f3 ≥ 0.6, resulting in changes in the crack propagation (see
Figure 8); meanwhile, the stress during the damage of asphalt concrete increases and then decreases, that is, the interaction has a shielding effect and an acceleration effect on the internal crack propagation of the material. The shielding and acceleration effects are clearly observed from the parameters related to the crack propagation.
4.2. Crack Propagation Process
On the discrete element analysis platform (PFC2D), the crack propagation monitoring procedure was used to determine the crack behavior during the asphalt concrete fracturing in real time. The number of cracks was characterized by the number of particle contact failures in the material; this means that the more particle contact failures, the more the material cracks, indicating that greater energy is required to destroy the asphalt concrete, i.e., the cracking strength of the asphalt concrete is greater.
Therefore, the logarithm and increment of the time step represent the time elapsed at the crack incubation stage, expansion stage, and fracture stage, respectively [
47], as shown in
Table 6.
Table 6 shows that the increase in the crack density results in a gradual decrease in the crack incubation time, meaning the interaction effect shortens the crack incubation time and speeds up the crack nucleation. Moreover, the table also shows that at the crack extension stage, no significant difference exists in the crack propagation and failure process times when the crack density
f3 ≤ 0.4, further indicating that the interaction effect is weak.
Meanwhile, when the crack density
f3 ≥ 0.6, the time elapsed at the crack extension stage decreases with the increase in crack density; the time elapsed at the fracture stage increases and then decreases (see
Table 6). Moreover, the longest elapsed time at the crack extension stage was at the crack density
f3 = 0.6, indicating that the interaction delays the crack extension process; when the crack density
f3 = 0.8, the time elapsed at the fracture stage reached a maximum, meaning the interaction prevents the formation of macroscopic cracks. This result indicates that under these crack densities, the interaction has a shielding effect. Otherwise, at the crack density
f3 = 1.0, a significant decrease occurs in the time elapsed at the crack extension and fracture stages, indicating that the interaction accelerates the crack propagation to macroscopic crack formation, i.e., it has an acceleration effect.
To reveal the generation mechanism of the interaction between microcracks and macrocracks, we analyze the changes in the stress field on the crack tip domain in the following subsection.
4.3. Stress Field Analysis for Crack Tip Domain of Main Crack
The subroutine was designed using FISH to identify the stress field of asphalt concrete. The stress field force chain diagrams were obtained, as shown in
Figure 11: the red line represents the average tensile stress field and the blue line represents the average shear stress field. The area ratio of the stress field was extracted, as shown in
Figure 12.
Figure 11 and
Figure 12 show significant changes in the tensile field, unlike the shear field, which only changes slightly. This result indicates that as the crack density increases, the interaction causes the asphalt concrete failure to change from tensile stress failure to shear stress failure. However, when the crack density
f3 = 0.6 and 0.8, the increments of the tensile field and shear field (see
Figure 12b) indicate that the crack shielding effect predominantly causes tensile stress damage to the asphalt concrete, delaying the macroscopic crack formation (see
Table 6) and increasing the low-temperature crack resistance of the asphalt concrete (see
Figure 10).
Meanwhile,
Figure 11 and
Figure 12 also show that the area ratio of the tensile field and the area ratio between the tensile and shear fields decreases considerably when the crack density
f3 = 1.0. This illustrates that the interaction predominantly induces shear stress damage to the asphalt concrete, which accelerates the crack propagation to a macroscopic crack formation (see
Table 6).