# Characterization and Theoretical Analysis of the Venus Flytrap Trigger Hair
## 3 Results and Discussion
### Mechanical Characteristics of the Venus Flytrap Trigger Hair  
Inspired by the Venus flytrap trigger hair, the mechanical characteristics of the cantilever beam is a key factor in determining the sensitivity of the trigger hair. To further discuss the relationship between the external tactile load and the maximum tensile strain, a simulated trigger hair cantilever beam model (STHCBM) was created for theoretical simulation analysis. The first step was to create two models of the STHCBM, each consisting of two sections: a podium and a lever. It is worth noting that the podium of the STHCBM has a notch structure. The schematic diagram in Figure 3 illustrates the STHCBM and how it responds to external force \(F\). The STHCBM consists of two parts: a fixed podium and a free lever. The lever consists of a length \(L_{a}\), diameter \(D_{2}\), and elastic modulus \(E_{2}\). The podium consists of a length \(L_{b}\), diameter \(D_{1}\), and elastic modulus \(E_{1}\). The STHCBM is a two-stage cantilever beam structure with variable stiffness, and the elastic modulus of the trigger hair lever and the basal podium are different. The elastic coefficient \(K\) of the STHCBM is:  
\[K=\frac{F}{\Phi}=\frac{3E_{1}I_{1}E_{2}I_{2}}{L_{b}E_{2}I_{2}\big{(}L_{b}^{2}+ 3L_{a}^{2}+3L_{b}L_{a}\big{)}+L_{a}^{3}E_{1}I_{1}} \tag{8}\]  
In addition, when the external force causes the bending and deforming of the STHCBM, the internal STHCBM will also produce the corresponding stress. According to solids mechanics, when any part of the lever is acted on by \(F\), the stress at any section of \(x\) on it can be expressed as:  
\[\sigma=\left\{ \begin{array}{ll}\frac{64y\big{(}x_{f}-x\big{)}}{\pi D_{1}^{2} }F,&0\leq x\leq L_{b}\\ \frac{64y\big{(}x_{f}-x\big{)}}{\pi D_{2}^{2}}F,&L_{b}\leq x\leq x_{f}\\ 0,&x_{f}\leq x\leq L_{b}+L_{a}\end{array} \right. \tag{9}\]  
The \(y\) is the distance between the point where the stress is calculated and the neutral axis. It can be seen from Equation (9) that the maximum stress in the section of STHCBMoccurs at the periphery of the section, and the stress reaches the maximum at the fixed end of the base, which decreases linearly with the increase in \(x\). For the STHCBM cantilever beam, the podium is the main area of deformation, which is the area of stress concentration, and the lever is the area of obvious displacement change. When the external force is applied to the free end of the STHCBM, the surface stress at the extrados and intrados of the podium section is:  
\[\sigma=\frac{32(L_{b}+L_{a}-x)}{\pi D_{1}^{3}}F \tag{10}\]  
By combining Equation (10) and substituting it into Equation (10), the relation between the maximum stress of the section at any \(x\) at the base of the STHCBM and the displacement of the free end can be obtained:  
\[\sigma=\frac{3\pi E_{1}E_{2}D_{1}D_{2}^{4}(L_{b}+L_{a}-x)}{2\pi[L_{b}E_{2}D_{2 }^{4}(L_{b}^{2}+3L_{a}^{2}+3L_{b}L_{a})+L_{a}^{3}E_{1}D_{1}^{4}]}\Phi \tag{11}\]  
Further, a notch structure with a gradually decreasing sectional size is designed at the base of the STHCBM. The diameter of the smallest size on the notch structure is \(d\), and the distance from the fixed end is \(X_{d}\), and other geometric parameters remain unchanged. According to engineering mechanics, STHCBM has a further stress concentration effect on the notch structure, and the maximum stress at the notch structure is:  
\[\sigma_{d}=\frac{32(L_{b}+L_{a}-x_{d})}{\pi d^{3}}F \tag{12}\]  
According to Equation (13), the maximum stress at the fixed end of the STHCBM is:  
\[\sigma_{b}=\frac{32(L_{b}+L_{a})}{\pi D_{1}^{3}}F \tag{13}\]  
The stress amplification factor \(K_{\sigma}\) of the notch structure is defined as the ratio of \(\sigma_{d}\) and \(\sigma_{b}\), so the stress amplification effect of the notch structure is:  
\[K_{\sigma}=\frac{\sigma_{d}}{\sigma_{b}}=\frac{(L_{b}+L_{a}-x_{d})D_{1}^{3}}{( L_{b}+L_{a})d^{3}} \tag{14}\]  
As per Equation (14), the stress amplification effect of the notch structure is stronger when its minimum size \(d\) is smaller and closer to the base's bottom.