**1. Introduction**

Cracking is a challenging topic which the researchers face. Insufficient and improper treatment of cracks often negatively impacts the pavement structure and its performance. Numerous studies on cracks' behavior have been conducted to reveal their influences on pavement performance. Initial cracks usually exist in the pavement structure in the form of micro-cracks, and these cracks grow due to temperature change, traffic loads, and other environmental factors. So, research on the mechanism of cracking growth will help monitor the cracks' behaviors from the beginning of their development and offer feasible ways of manufacturing and maintenance. Traditional fatigue methods explain the initiation and growth of cracks [1], but this only provides a rough approach that performs poorly while calculating longitudinal cracks. To solve this kind of problem, fracture mechanism theories were established. Paris established the approximate equation of crack growth rate under repeated load [2]. Majidzadeh expanded this theory, and normal forms of crack growth were obtained [3]. However, this equation needs four parameters related to material properties, which decrease its reliability and limit its usage. A life-related index was established based on engineering data, including traffic load, environment conditions, and embankment conditions [4]. Conditions included by this index are also limited, and it will take a long time to perform road surveys, which can be inconvenient.

**Citation:** Wang, H.; Wu, Y.; Yang, J.; Wang, H. Numerical Simulation on Reflective Cracking Behavior of Asphalt Pavement. *Appl. Sci.* **2021**, *11*, 7990. https://doi.org/10.3390/ app11177990

Academic Editors: Amir Tabakovic, Jan Valentin and Liang He

Received: 17 August 2021 Accepted: 27 August 2021 Published: 29 August 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

These methods are always based on numerous engineering data and cannot involve many working conditions. Correction factors are always acquired when using these methods, so a convenient way of researching crack behavior is needed.

Conventional research on crack behavior heavily relies on laboratory-scale experiments, which require considerable time and effort. These experiments always focused on the final status of cracks and pavement, and the processes of cracks' growth were ignored due to testing methods. Over these years, the finite element method has become a popular approach to simulate the cracks' behavior during their life cycle. During their growth, every kind of cracks' behavior can be calculated using finite element software, requiring much less time than local experiments. The two-dimensional plane strain model was established by Myers using finite element software ABAQUS [5]. This model revealed the growth of pre-set cracks under traffic loads and the direction of cracks' growth, which offered a mature way of pavement simulation. García used the finite element method combined with fracture mechanics theory to investigate the cracking behavior of the orthogonal layered pavement, and uses the method of preset interlayer cracks to predict the cracking of each layer of the pavement during use. The results show that the cracking performance of the orthogonal layered pavement is largely affected by the initial shape and location of the cracks. These initial cracks mostly occur at the joints between layers, thinner layers and the free boundaries of each layer [6]. Results show that behaviors of crack growth are related to the viscoelasticity and creep properties of asphalt deeply. However, compared with two-dimensional models, three-dimensional models always perform better, especially when the process of crack growth was taken into consideration [5]. Parameters of crack growth and their time history curve can be calculated using the finite element method and predictions of pavement life, degree of damage can be drawn. Road workers can also choose the appropriate asphalt mixture based on the finite element calculation results, thus promoting the development of the asphalt mixture.

J-integral and stress intensity factors are often used to describe the behaviors of crack growth, from which researchers can have a direct understanding of the process. Okada developed a 3D model to compute the J-integral of large deformation solids [7]. Results show that J-integral is unconditionally path independent, reflecting that it is not very important to set a specific path while simulating cracks using J-integral, thus decreasing the difficulties. Moreover, energy can be used to calculate the J-integral. Okada added the strain energy density into the J-integral formulation to calculate the deformation histories of the specimen [7]. Yu proposed a new model of calculating J-integral based on the energy density equivalence by introducing 3D constraint functions to describe the relationships between J-integral and load in mode-I cracks [8]. Sasan compared two criteria of stress intensity factor and fracture energy to investigate the behavior of asphalt mixtures under combined tensile shear loadings [9]. The SCB (Semi-Cycle Bending Test) fracture test results show that the fracture behavior of asphalt mixtures is significantly dependent on testing temperature and loading rate, which can reflect the working conditions of asphalt mixtures during engineering construction. Moreover, finite element calculation was conducted to verify the correctness of SCB tests. As for pavement structure, J-integral and stress intensity factors can be different from theoretical calculations and mechanical experiments due to the structural differences between theoretical calculation models or mechanical specimens and pavement structure. Researchers often use elastic layered continuum theory to describe the structure of asphalt pavement. Alae established a three-layer 3D model of asphalt pavement with top-down cracks [10]. Working temperature and vehicle speed were assumed to be the changing working conditions of the pavement structure. Results show that top-down cracks in pavement structure can be broken down into I + II fracture, and J-integral can be used to describe stress intensity on the crack tip. Ma developed a wave propagation-based analytical solution to calculate the mechanical responses of transversely isotropic viscoelastic multi-layered asphalt pavement subject to moving harmonic load [11]. This solution can be used for asphalt pavement design and analysis with consideration of realistic load and material parameters. However, a specimen of asphalt mixtures can

only reveal the crack resistance of the material, but the mixtures' behaviors in pavement structure are not widely studied. J-integral and stress intensity factors on the crack tip in pavement structures have not been researched systematically. Mechanical responses during the crack growth need to be calculated using the finite element method.

The research described in this paper is aimed to reveal the cracks' behaviors during the pavement's life cycle and under different working situations. The effects of traffic loads and Young's modulus of layers on the crack extension were investigated. This paper uses finite element software ABAQUS to simulate the pre-set crack's growth during the pavement's life cycle. J-integral, stress intensity factors, absolute stress, and strains were calculated to describe how the cracks extend while working. Moreover, the entropy method was used to analyze the relationship between these influencing factors aiming to help to deal with cracking problems efficiently.
