# Effect of Impact and Bearing Parameters on Bird Strike with Aero-Engine Fan Blades
## 2 Bird and Fan Modeling  
### Bird Geometry and Material Modeling  
To better simulate the real situation of bird strikes, a real bird was scanned using computed tomography (CT) to obtain the body geometry. In a recent report from FAA [1], during 1990-2020, 1191 mallard strikes were reported in total, accounting for 20% of waterfowl strikes. Huge economic losses have been caused by mallards, so the mallard bird model was chosen for this research. An anesthetized mallard was passed through a CT scanning device to obtain the mallard three-dimensional (3D) geometry. The mallard was 40 cm in length and 45 cm in wingspan weighing 820 g. The SPH method was adopted for bird body modeling based on the mallard 3D model. The model used in the numerical simulation consisted of 41,685 SPH elements each having a mass of 0.0191 g. The SPH bird model is shown in Figure 1.  
According to previous studies [27], birds behave similarly to fluids during high-speed impacts where the inhomogeneity and nonuniformity due to other components such as feathers and bones can be neglected. Based on this assumption that all parts of the bird body exhibit the same mechanical behavior, a single material model can effectively predict the impact behavior of the bird body under different conditions [18]. To simulate this material behavior of the bird body, a material model accompanied with an equation of states (EOS) is usually used. The null material model is often used to simulate fluids in high-velocity conditions. The strain of the fluid could be related to its stress by:  
\[\sigma_{ij}=-P\delta_{ij}+2\rho\gamma\dot{e}_{ij} \tag{1}\]  
where \(\sigma_{ij}\) is the viscous stress, \(P\) is the pressure on the fluid, \(\delta_{ij}\) is the identity tensor, \(\rho\) is the fluid density, \(\gamma\) is the dynamic viscosity coefficient, and \(\dot{e}_{ij}\) is the rate-of-deformation tensor. The material properties of the mallard bird model are listed in Table 1.  
The null material model must be used with an EOS. The EOS is linked to the material's physical state variables and is often applied in interpreting the properties of fluids and solids. According to Reference [28], the Gruneisen EOS and the Tillosten EOS are mainly used in numerical simulation studies of high-speed collision problems. The Gruneisen EOS was implemented in this study. There are two forms of the Gruneisen equation, corresponding to two different states of compression and expansion. Pressure for materials in compression is expressed as:  
\[p=\frac{\rho_{0}C^{2}\mu\left[1+(1-\frac{\gamma_{0}}{2})\mu-\frac{\alpha}{2} \mu^{2}\right]}{\left[1-(S_{1}-1)\mu-S_{2}\frac{\mu^{2}}{\mu+1}-S_{3}\frac{ \mu^{3}}{(\mu+1)^{2}}\right]^{2}}+(\gamma_{0}+a\mu)E \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Density & 938 kg/m\({}^{3}\) \\ Relative volumetric strain for erosion in tension & 1 \\ Relative volumetric strain for erosion in compression & 0.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Null material parameters for the bird [18].  
Figure 1: The SPH model of the mallard.  
and in tension is given by:  
\[p=\rho_{0}C^{2}\mu+(\gamma_{0}+a\mu)E \tag{3}\]  
where \(C\) is the intercept of the \(v_{s}-v_{p}\) curve; \(S_{1}\), \(S_{2}\), and \(S_{3}\) are the unitless slope coefficients; \(\gamma_{0}\) is the Gruneisen constant; \(a\) is the unitless first-order correction of \(\gamma_{0}\). The parameter \(\mu\) is expressed by:  
\[\mu=\frac{\rho}{\rho_{0}}-1 \tag{4}\]  
where \(\rho\) and \(\rho_{0}\) are density of the material and the reference density, respectively. The parameters used in this study are listed in Table 2.