# Investigation of the Fatigue Life of Bottom-Up Cracking in Asphalt Concrete Pavements
## 3 Evaluating the Tensile Stress and Fracture Parameters \(A\) and \(n\)  
### Tensile Stress by Numerical Simulation  
In order to accurately calculate the fatigue cracking life of an asphalt mixture, we need to obtain the tensile stress at the bottom of each structural layer of the asphalt pavement. Therefore, the tensile stress-controlled fatigue cracking model recommended in Specifications for the Design of Highway Asphalt Pavement (JTG D50-2017) [22] was selected to calculate the fatigue cracking life. In this paper, commercial software ANSYS was used to establish a three-dimensional model of the pavement structure and obtain the tensile stress of each layer of the asphalt pavement. The viscoelastic element VISC089 was used for the model surface, which supported stress rigidization and was only suitable for small strain and small displacement analysis. The node SOLID95 with a 3D unit entity was used at the base level. According to the existing research results, incomplete continuity was assumed between the base layer and surface layer, and the friction coefficient was 0.5. The surface layer of the model was assumed to be not completely continuous, and the friction coefficient was 0.7 [23], which can accurately obtain the mechanical response of the pavement structure under a load. The load calculated by the model was the recommended standard axle load, BZ-100, in asphalt pavement design. The ANSYS calculation model is shown in Figure 4.  
Figure 4: Schematic diagram of 3D model.  
The pavement structure was established by a 3D coordinate system, and the model was divided into three layers of asphalt pavement, two layers of base, and a soil base. The soil base was used to limit the displacement in the X-Y-Z directions. Both sides of the model limited the displacement in the X direction. The displacement in the Y direction was restricted before and after the model, and the driving direction was consistent with the Y direction. Because the fatigue crack of asphalt pavement was the transverse crack perpendicular to the middle line of road, the maximum tensile stress at the bottom of asphalt layer in the Y direction (driving direction) was selected as the calculated stress of the fatigue life of the crack. Three pavement structures commonly used in practice were selected to investigate fatigue life, namely structures A, B, and C. The combination of pavement structure thickness was as follows:  
Structure A: 4 cm upper layer + 5 cm middle layer + 6 cm lower layer + 40 cm base layer + 600 cm soil foundation.  
Structure B: 4 cm upper layer + 6 cm middle layer + 8 cm lower layer + 40 cm base layer + 600 cm soil foundation.  
Structure C: 6 cm upper layer + 8 cm middle layer + 10 cm lower layer + 40 cm base layer + 600 cm soil foundation.  
When the numerical model was used to calculate the mechanical response of the pavement structure, the dynamic modulus obtained by the above test was selected as the modulus of the surface layer. If the base was semirigid, the modulus of the base was 4000 MPa and the modulus of the soil was 40 MPa. If the base was flexible, the modulus of the base was 1000 MPa, and the modulus of the soil was 40 MPa. The numerical results at the bottom of each structural layer of the flexible base of the asphalt pavement at 25 \({}^{\circ}\)C, 0 \({}^{\circ}\)C, and \(-\)10 \({}^{\circ}\)C are shown in Tables 5 and 6.