# Investigation of the Fatigue Life of Bottom-Up Cracking in Asphalt Concrete Pavements
## 3 Evaluating the Tensile Stress and Fracture Parameters \(A\) and \(n\)
### Calculation of Fracture Parameters \(A\) and \(n\)  
As mentioned in the previous introduction, the Paris formula (proposed for fracture mechanics) is used to estimate the fatigue life at the crack propagation stage, as shown in Equation (1):  
\[\frac{dc}{dN}=A\left(\Delta K\right)^{n} \tag{1}\]  
The integral Equation (2) can be obtained by the transformation of the above Equation (1):  
\[\int_{C_{0}}^{C_{i}}\frac{dc}{A\left(\Delta K\right)^{n}}dc=\int_{0}^{N_{p}}N=N _{p} \tag{2}\]  
where, \(c\) is the crack length; \(N\) is the number of load cycles; \(A\) and \(n\) are the parameters related to materials; \(\Delta K\) is the variation range of the stress intensity factor. The OT model is shown in Figure 5. The base layer is assumed to be nondeformable, and this OT model aims to study the crack propagation of the AC layer. According to Equation (2), the extension life of the cracks can be obtained when the fracture parameters \(A\) and \(n\) are known. However, in order to obtain the fracture parameters of an asphalt mixture, the relationship between the stress intensity factor (SIF) and fracture length should be obtained first. Therefore, the finite element analysis software ANSYS was adopted to establish the calculation model of the OT method, as shown in Figure 6. The model used SOLID95 with intermediate nodes to establish cracks in the middle, and SOLID45 was used to establish other parts (except the cracks). The left side of the model limited the displacement in the X, Y, and Z directions, while the right side limited the displacement in the Y and Z directions. Fixed constraints were used between the upper and lower layers. Figure 5: Schematic diagram of the OT model. Figure 6: Numerical calculation model based on the OTs. The model was used to analyze the variation in SIF with crack propagation under test conditions. According to the study of [PERSON] et al. [24], when the dynamic modulus was 1 MPa and the maximum opening displacement was 1 mm, the relationship between SIF and crack length can be obtained, as shown in Figure 7.  
Certainly, for any combination of the maximum opening displacement and dynamic modulus, the SIF is proportional to the magnitude of dynamic modulus; therefore, SIF can be obtained by regression calculation, as shown in Equation (3). \[SIF=0.2911\times E\times MOD\times c^{-0.4590} \tag{3}\]  
where \(SIF\) is the stress intensity factor; \(E\) is the dynamic modulus; \(MOD\) is the maximum crack opening displacement; \(c\) is the crack length. Before using OT to analyze the relationship between the maximum load of a single circle and crack length, the relationship curve obtained by the OTs should be analyzed first, as shown in Figure 8. It can be found that the initial crack was generated from the 2 mm gap reserved at the bottom, and the reduction in the load for each cycle was due to crack generation and expansion. Figure 8: The relationship between the maximum load of a single circle and the number of load cycles. \[\int_{C_{0}}^{C_{i}}\frac{dc}{A\left(\Delta K\right)^{n}}dc=\int_{0}^{N_{p}}N=N _{p} \tag{2}\]0 The variation in SIF with crack propagation. On the basis of the above discussion, in the OTs, the load required for each crack opening to the maximum tensile displacement in the OTs was proportional to the modulus of the asphalt mixture and inversely proportional to the length of the crack propagation. Therefore, if the maximum load of a single circle was assumed to be 1 kN, and the corresponding crack length was 0 mm, then this meant that the crack had not yet occurred. When the load became 0 kN, the crack penetrated the specimen. As shown in Figure 9, the relationship curve between the maximum load of a single circle and crack length can be obtained. According to Equation (3), the relationship between the maximum load of a single circle and number of load cycles and the relationship between the maximum load of a single circle and crack length have been obtained.