# Optimization Design of Lattice Structures in Internal Cooling Channel of Turbine Blade
## 2 Functions of Heat Transfer and Mechanical Performances
### The Function of Bearing Load Performance  
Bearing load performance was improved in turbine blades using LS cores. Elastic modulus (\(E\)) of the LS channel was used to evaluate its bearing load capacity. The relevance  
Figure 4: Printed XLS channel and vibration validation: (**a**) Experimental samples of XLS; (**b**) Model experiment value and calculated value; and (**c**) Vibration mode of modal experiment and simulation result.  
between \(E\) and geometric parameters of LS core was derived using PLS core, and one member of PLS core was analyzed, as shown in Figure 5. An outer load \(F\) was added at the upper end of the member, and the member had a displacement \(\delta\), which caused axial force \(F_{A}\) and shear force \(F_{S}\). As all forces in the member were balanced, \(F\) can be obtained along the vertical direction:  
\[F=F_{A}\sin\omega+F_{S}\cos\omega \tag{17}\]  
According to the compression and bending function of the member, \(F_{A}\) and \(F_{S}\) can be obtained:  
\[F_{A}=E_{c}\frac{\pi D^{2}}{4}\frac{\delta\sin\omega}{l} \tag{18}\]  
\[F_{S}=\frac{3}{4}E_{c}\pi\frac{D^{4}}{16l^{3}}\delta\cos\omega \tag{19}\]  
where \(E_{c}\) is the elastic modulus of the material of PLS channel.  
The equivalent plane compressive stress and strain can be defined as:  
\[\sigma_{e}=\frac{4F}{A} \tag{20}\]  
\[\epsilon_{e}=\frac{\delta}{l\sin\omega} \tag{21}\]  
where \(l\) is the length of the member, as shown in Figure 5, \(l=H/\sin\omega\). \(A\) is the area of bottom surface of the PLS core, \(A=\left(\sqrt{2}l\cos\omega+2t\right)^{2}\); \(D\) is the diameter of the member; \(t\) is a small spacing between two members at the top wall of the PLS core and also is a small spacing between one of the side walls of the PLS core and the member at the bottom wall of the core.  
Combining Equations (20) and (21), the plane elastic modulus of PLS can be obtained:  
\[E=E_{c}\frac{\sigma_{e}}{\epsilon_{e}}=E_{c}\overline{\rho}_{p}\sin^{4}\omega +\frac{3}{4}E_{c}\overline{\rho}_{p}\left(\frac{D}{2l}\right)^{2}\cos^{2} \omega\sin^{2}\omega \tag{22}\]  
where \(\overline{\rho}_{p}\) is the relative density of the PLS core, and \(\overline{\rho}_{p}=\pi D^{2}/\sin\left(\sqrt{2}l\cos\omega+2t\right)^{2}\). As shown in Figure 5b, \(D/(2l)\) is generally a small value; therefore, for Equation (22), the second part can be ignored.  
Thus, the equivalent elastic modulus can be expressed as:  
\[E=E_{c}\overline{\rho}_{p}\sin^{4}\omega \tag{23}\]  
The function of \(E\) can be obtained as:  
\[f_{3}(x)\ =E=E_{c}\overline{\rho}_{p}\sin^{4}\omega \tag{24}\]  
where \(f_{3}(x)\) is the third objective of the present study. Equation (24) is also used for the \(E\) value of XLS channel. The Euler buckling critical load of the member is:  
\[\sigma=\frac{k^{2}\pi^{2}E_{c}D^{2}}{l^{2}} \tag{25}\]  
where \(k\) is the end restraint factor for buckling.  
For LS, due to the large slenderness ratio of the member and the thin panel, the constraint at the end of the member was weak. Therefore, \(k\) was taken as 2, and \(f_{3}(x)\) was not used as an objective of the present study, while it was used as a constraint in order to simplify the optimization calculation.