## 3 Results and Discussion
\tag{4}\]  
At the stage of bending deformation, the \(\varepsilon_{m}\) of membrane tissue can be approximately regarded as zero, because the stress acting on the membrane is extremely small. Hence, the \(\varepsilon_{m}\) of membrane tissue can be further given by:  
\[\varepsilon_{m}=\left\{ \begin{array}{cc}ln\left(\frac{l_{m}}{w_{0}}\right),&\text{tension}\\ 0,&\text{bending}\\ ln\left(\frac{l_{m}}{2t_{m}}\right)&\text{compression}\end{array} \right. \tag{5}\]  
In addition, with the change of slit width, the strain, \(\varepsilon_{m}\), of membrane tissue can be further given by:  
\[\varepsilon_{m}=\left\{ \begin{array}{cc}ln\left(\frac{w}{l_{m0}}\right),&w \textless 2t_{m}\text{ or }w>\text{w}_{0}\\ 0,&2t_{m}\leq w\leq\text{w}_{0}\end{array} \right. \tag{6}\]  
here, \(w\) represents the width of the slit. Similarly, in the tensile or compressive stage, the strain, \(\varepsilon_{d}\), of the exoskeleton is given by:  
\[\varepsilon_{d=}\ln\left(\frac{l_{d}}{l_{d0}}\right) \tag{7}\]  
where \(l_{d}\) and \(d_{0}\) represent the final and initial slit widths, respectively. Based on the above analysis, Equation (2) can be given by:  
\[f_{m}=\frac{l_{m0}\exp(\varepsilon_{m})}{l_{m0}\exp(\varepsilon_{m})+l_{d0} \exp(\varepsilon_{d})}=\left[1+\gamma\exp\left(\frac{\sigma_{sig}}{E_{s}}- \frac{\sigma_{sig}}{E_{m}}\right)\right]^{-1} \tag{8}\]  
here, the applied stress \(\sigma_{sig}\) acting on the membrane and the epicuticle is given by: \(\sigma_{sig}=E_{m}\varepsilon_{m}=E_{s}\varepsilon_{d}\), and \(\gamma=d_{0}/l_{m0}\). In addition, \(f_{m}\) can be represented by:  
\[f_{m}=\frac{l_{m0}\exp(\varepsilon_{m})}{l_{m0}\exp(\varepsilon_{m})+l_{d0} \exp(\varepsilon_{d})}=\left\{1+\gamma exp\left[\varepsilon_{m}\left(\frac{E_{ m}}{E_{s}}-1\right)\right]\right\}^{-1} \tag{9}\]Substituting Equation (9) into Equation (1), the composite elastic modulus, \(E_{c}\), is then given by:  
\[E_{c}=\frac{E_{s}}{1+\big{(}\frac{E_{c}}{E_{m}}-1\big{)}\big{(}1+\gamma exp\big{[} \varepsilon_{m}\big{(}\frac{E_{m}}{E_{s}}-1\big{)}\big{]}\big{)}^{-1}} \tag{10}\]  
Based on Equation (10), the \(E_{c}\) as a function of \(\varepsilon_{m}\), as well as for different values of \(E_{m}\), is plotted in Figure 2c. The result indicates that there is a close relationship between \(E_{c}\) and \(E_{m}\). Specifically, under the same membrane strain, \(E_{c}\) decreases significantly for a decrease in \(E_{m}\) value 562 MPa to 5.62 MPa. In addition, the strain, \(\varepsilon_{c}\), of composite tissue as a function of applied stress, \(\sigma_{sig}\), can be given by \(\varepsilon_{c}=\sigma_{sig}/E_{c}\). Figure 2d shows that, under the same \(\sigma_{sig}\), the absolute value of \(\varepsilon_{c}\) is the smallest when \(E_{m}\) is equal to the measured value. The above comparative results clearly show that the composite tissue around the slit wake can efficiently absorb strain energy through small strain, \(\varepsilon_{c}\), for the reasonable configuration of \(E_{m}\). The membrane strain \(\varepsilon_{m}\) as a function of composite \(\varepsilon_{c}\) is [28]:  
\[\varepsilon_{m}=\varepsilon_{c}(l_{0}+l_{00})/l_{0} \tag{11}\]  
where, \(l_{0}\) is \(w_{0}\) and \(2t_{m}\), respectively, at the initial stage of tension and compression. The \(\varepsilon_{m}\) as a function of slit width, \(w\), can be given by: \(\varepsilon_{m}=(\omega-l_{0})/l_{0}\). Therefore, the width of the slit can be further expressed as:  
\[w=l_{0}+\varepsilon_{c}(l_{0}+l_{00}) \tag{12}\]  
Combining Equation (12) and relationship between \(\varepsilon_{c}\) and \(E_{m}\), it is clear that the change of slit width mainly depends on the mechanical property of the membrane. Specifically, in tensile or compressive stage, the surface membrane can reasonably control slit width through absorbing strain energy with small deformation. The crack-shaped slit will propagate from the slit tip with the excessive opening and closure of the slit wake because of the high stress intensity at the slit tip [29; 30].