## 4 Axial Fan Blade Crack Propagation Simulation  
During operation, the blade experiences the effects of aerodynamic and centrifugal forces. These forces significantly affect the performance and stability of the blade. Due to stress concentration, surface fatigue fractures readily develop at the root area of the blade. The crack's propagation zone will exhibit an exponential type of propagation, in line with the fundamental theory of fatigue failure, resulting in blade breaks due to the rapid propagation of the microfracture.  
Through the co-simulation method (ABAQUS and FRANC3D), this section investigates the surface crack propagation path under aerodynamic, centrifugal, and a combination of both loads, with an emphasis on the crack propagation growth rate for the axial fan blade for different growth steps. The fan blade used in this study is made of AL2024t861 aluminum, where the main mechanical properties are as follows: [PERSON]'s modulus \(E=71\) GPa, Poisson's ratio \(\upsilon=0.33\), density \(\rho=2768\) Kg/m\({}^{3}\). The C3D10 10-node quadratic tetrahedral element was selected in this work, primarily for its compatibility with mesh refinement techniques and its high accuracy in representing the stress gradient at the crack tip.  
The crack propagation flow chart using FRANC3D code is illustrated in Figure 7. Four steps are required for the analysis. During the first step, the initial FEA model built in ABAQUS was imported in FRANC3D in order to create a sub-model for the investigation of crack growth. An initial semi-elliptical crack front shape was assumed, with a surface length \(c=5\) mm and a ratio \(a/c\) of 0.5. A template radius \(r_{0}=0.1\) mm was inserted into the sub-model at the conjunction between the blade and the blade roots. The sub-model was then meshed and reinserted into the original FEA model, preserving the boundary nodes and the remaining original FEA mesh. The crack growth parameters are specified in step 3. For crack propagation, the cyclic load is applied, and the parameters of the Paris growth rate model are as follows: \(C=8\times 10^{-12}\), \(m=3\), \(K_{C}=1750\) MPa\(\sqrt{mm}\), and \(\Delta K=10\) MPa\(\sqrt{mm}\).  
The crack fronts are defined by multiple quantities of points N\({}_{n}\) to evaluate the correlation among the nodes of different 3D crack fronts. Using p as the independent variable, several polynomial functions can be generated based on this to fit the data on the crack front. These polynomials can be represented as follows:  
\[\mathbf{P}_{n}(\mathbf{p})=\left\{\mathbf{K}_{n}(\mathbf{p}),\mathbf{S}_{n}(\mathbf{p}),\mathbf{e}_{n}(\bm {p})\right\} \tag{11}\]  
where n is the number of crack propagation steps, \(K_{n}\) is the cartesian coordinates of the mesh nodes corresponding to the nth crack front, and \(S_{n},e_{n}\) are the SIF and local coordinate systems corresponding to \(K_{n}\).  
With trial tests and to avoid complicated propagation of the crack tip, a fifth-order polynomial (Equation (11)) is fitted to the crack tip nodes, allowing for fitting adjustment during crack growth. The M-integral approach is used to compute SIFs. When calculating the SIFs, the maximum circumferential stress (MTS) is used. At each crack growth increment, SIFs and crack inclination angles were generated to update the new crack front. Therefore, the cracked sub-model is re-meshed and re-inserted into the remaining FEA model. Consequently, the crack propagation path can be computed.