Experimental and theoretical study on out-of-plane compression buckling properties of grid beetle elytron plate

Abstract

Based on the shell and plate theory, this paper uses the divide-and-combine method (DCM) to derive an analytical out-of-plane compression buckling limit expression of the basic unit of the core layer of a grid beetle elytron plate (GBEP); the accuracy of expression and the mechanism of GBEP unit buckling are investigated through experiments and finite element (FE) simulation. The results show that: (1) the theoretical expression of the out-of-plane compression buckling limit of a GBEP unit obtained by DCM is applicable and is closer to the test results than the classical solution of the out-of-plane compressive buckling of a cylindrical shell and the theoretical result of Flügge. Based on this, a modified expression of the theoretical result of Flügge’s compressive buckling limit load is proposed. (2) Given the significant differences between the theoretical values obtained in this paper and the experimental values, the out-of-plane compressive buckling resistance of the GBEP unit and its mechanism are investigated from the viewpoints of structural parameters $\eta$ (the ratio of the radius of the trabeculae to the width of the element) and deformation process; the synergistic mechanism of the honeycomb-trabeculae structure is also studied. This paper contributes to improving the bionic system derived from the beetle's forewing.

Appendix

Calculating the result of Eq. (11), we can have:

$$\begin{aligned}{p}_{{\rm cr}2}&=\frac{K{\pi }^{2}}{{\left(2l\right)}^{2}}\left[\frac{{m}^{2}{\left(2l\right)}^{2}}{{h}^{2}}+\frac{8{n}^{2}}{3}+\frac{16{n}^{4}{h}^{2}}{3{m}^{2}{\left(2l\right)}^{2}}\right]\\ &\quad+ \frac{{n}^{2}E{f}^{2}{\pi }^{2}t}{3({2l)}^{2}}\left[\frac{{n}^{2}{h}^{2}}{4{m}^{2}{(2l)}^{2}}+\frac{15{m}^{2}{(2l)}^{2}}{64{n}^{2}{h}^{2}}-\frac{{m}^{2}{n}^{2}{h}^{2}({2l)}^{2}}{{8(4{m}^{2}{l}^{2}+{n}^{2}{h}^{2})}^{2}}\right]. \end{aligned}$$

(27)

In this study, the post-buckling increase of the plate load is:

$$\begin{array}{c}\Delta {p}_{x}=\frac{{n}^{2}E{f}^{2}{\pi }^{2}t}{3({2l)}^{2}}[\frac{{n}^{2}{h}^{2}}{4{m}^{2}{(2l)}^{2}}+\frac{15{m}^{2}{(2l)}^{2}}{64{n}^{2}{h}^{2}}-\frac{{m}^{2}{n}^{2}{h}^{2}({2l)}^{2}}{{8(4{m}^{2}{l}^{2}+{n}^{2}{h}^{2})}^{2}}], \end{array}$$

(28)

According to Eq. (25), the deflection of the plate is:

$$\begin{array}{c}f=\sqrt{\frac{3{l}^{2}\left({p}_{{\rm cr}2}-{p}_{{\rm cr}x}\right)}{{n}^{2}{\pi }^{2}Et\left[{n}^{2}{h}^{2}/64{m}^{2}{l}^{2}+\frac{15{m}^{2}{l}^{2}}{64{n}^{2}{h}^{2}}-{m}^{2}{n}^{2}{h}^{2}{l}^{2}/8{\left(4{m}^{2}{l}^{2}+{n}^{2}{h}^{2}\right)}^{2}\right]}} \end{array}$$

(29)

Introduce Eq. (18) into Eq. (27):

$$\begin{array}{c}{f}^{2}=\frac{3{l}^{2}\left({f}_{y}t-{p}_{{\rm cr}x}\right)}{Et{n}^{2}{\pi }^{2}\left[\frac{{\lambda }_{2}^{2}}{16}+\frac{45}{256{\lambda }_{2}^{2}}+\frac{1}{{\left({\lambda }_{2}^{2}+1\right)}^{2}}\right]}, \end{array}$$

(30)

${p}_{{\rm cr}x}$ can be written as a function of ${\lambda }_{2}$, thus:

$$\begin{array}{c}{p}_{{\rm cr}2}=\frac{K{n}^{2}{\pi }^{2}}{{\left(2l\right)}^{2}}\left(\frac{1}{{\lambda }_{2}^{2}}+\frac{16{\lambda }_{2}^{2}}{3}+\frac{8}{3}\right)+\left[{f}_{y}t-\frac{K{n}^{2}{\pi }^{2}}{{\left(2l\right)}^{2}}\left(\frac{1}{{\lambda }_{2}^{2}}+\frac{16{\lambda }_{2}^{2}}{3}+\frac{8}{3}\right)\right]\frac{1+\frac{15}{16{\lambda }_{2}^{4}}-\frac{1}{{2\left({\lambda }_{2}^{2}+1\right)}^{2}}}{1+\frac{45}{16{\lambda }_{2}^{4}}+\frac{4}{{\left({\lambda }_{2}^{2}+1\right)}^{2}}}. \end{array}$$

(31)