Experimental and numerical investigation of conventional and stiffened re-entrant cell structures under compression

Abstract

A classical re-entrant cell is a type of metamaterial known as auxetic. While the most unusual and advantageous feature of auxetic materials is that they have negative Poisson’s ratios, having low stiffness—as seen in the classical re-entrant cell—may be a drawback. A study was conducted to increase the stiffness of the classical re-entrant cell while maintaining the negative Poisson's ratio. This paper reports the nonlinear experimental and numerical works of three re-entrant cells one of which is a well-known classical re-entrant cell, and the latter two were modified based on classical re-entrant cell. In the work, the cellular structure specimens were fabricated with a 3D printer using polylactic acid (PLA) material and crushing tests were conducted until the full crush phase. The specimens were also modelled using solid finite elements considering wall-to-wall frictional contacts and analysed. The linear mechanical properties of the cells were also determined by employing analytical expressions that were developed for modified cells. Thus, both the theoretical and the nonlinear numerical results were validated using experiments. In conclusion, the modified cells exhibited an increase in stiffness, energy absorption capacity, and plasticity, compared to the classical re-entrant cell. All benefits and drawbacks of the modifications to achieve stiff cells are reported in this paper.

Appendix

Note that while calculating the classical re-entrant cell's mechanical properties it should be employed with very low stiffness and low cross-sectional parameters for the strengthening walls of the modified cells in the following equations.

• (a) Type-I cell equations:

All equations of the Type-I cell may be derived from the deformation of the inclined walls. Employing inclined inner and outer walls on the quarter cell, the equations may be derived employed in this paper (for further information please refer to Baran and Ozturk [77]). Poisson’s ratio and elasticity modulus of the Type-I cell may be expressed as:

$${\nu }_{12}=-\frac{\left(2\frac{{A}_{1}{E}_{s1}{\mathrm{cos}\left({\theta }_{1}\right)}^{2}\mathrm{sin}\left({\theta }_{1}\right)}{{l}_{2}\mathrm{cos}\left({\theta }_{2}\right)}+\frac{{A}_{2}{E}_{s2}\mathrm{cos}\left({\theta }_{2}\right)\mathrm{sin}\left({\theta }_{2}\right)}{{l}_{2}}+\frac{{12E}_{s2}{I}_{2}\mathrm{sin}\left({\theta }_{2}\right)\mathrm{cos}\left({\theta }_{2}\right)}{{{l}_{2}}^{3}\left({\alpha }_{2}+1\right)}+\frac{{24E}_{s1}{I}_{1}\mathrm{sin}\left({\theta }_{1}\right){\mathrm{cos}\left({\theta }_{1}\right)}^{4}}{{{l}_{2}}^{3}{\mathrm{cos}\left({\theta }_{2}\right)}^{3}\left({\alpha }_{1}\frac{{\mathrm{cos}\left({\theta }_{1}\right)}^{2}}{{\mathrm{cos}\left({\theta }_{2}\right)}^{2}}+1\right)}\right)\left({l}_{2}\mathrm{cos}\left({\theta }_{2}\right)+\frac{{t}_{2}}{2}\right)}{\left(2\frac{{A}_{1}{E}_{s1}{\mathrm{sin}\left({\theta }_{1}\right)}^{2}\mathrm{cos}\left({\theta }_{1}\right)}{{l}_{2}\mathrm{cos}\left({\theta }_{2}\right)}+\frac{{A}_{2}{E}_{s2}{\mathrm{sin}\left({\theta }_{2}\right)}^{2}}{{l}_{2}}+\frac{{12E}_{s2}{I}_{2}{\mathrm{cos}\left({\theta }_{2}\right)}^{2}}{{{l}_{2}}^{3}\left({\alpha }_{2}+1\right)}+\frac{{24E}_{s1}{I}_{1}{\mathrm{cos}\left({\theta }_{1}\right)}^{5}}{{{l}_{2}}^{3}{\mathrm{cos}\left({\theta }_{2}\right)}^{3}\left({\alpha }_{1}\frac{{\mathrm{cos}\left({\theta }_{1}\right)}^{2}}{{\mathrm{cos}\left({\theta }_{2}\right)}^{2}}+1\right)}\right)\left(h+{l}_{2}\mathrm{sin}\left({\theta }_{2}\right)\right)}$$

(5)

$${E}_{1}=\frac{({{K}_{1}}^{2}-{K}_{2}{K}_{3})({l}_{2}\mathrm{cos}\left({\theta }_{2}\right)+\frac{{t}_{2}}{2})}{(2\frac{{A}_{1}{E}_{s1}{\mathrm{sin}\left({\theta }_{1}\right)}^{2}\mathrm{cos}\left({\theta }_{1}\right)}{{l}_{2}\mathrm{cos}\left({\theta }_{2}\right)}+\frac{{A}_{2}{E}_{s2}{\mathrm{sin}\left({\theta }_{2}\right)}^{2}}{{l}_{2}}+\frac{{12E}_{s2}{I}_{2}{\mathrm{cos}\left({\theta }_{2}\right)}^{2}}{{{l}_{2}}^{3}\left({\alpha }_{2}+1\right)}+\frac{{24E}_{s1}{I}_{1}{\mathrm{cos}\left({\theta }_{1}\right)}^{5}}{{{l}_{2}}^{3}{\mathrm{cos}\left({\theta }_{2}\right)}^{3}\left({\alpha }_{1}\frac{{\mathrm{cos}\left({\theta }_{1}\right)}^{2}}{{\mathrm{cos}\left({\theta }_{2}\right)}^{2}}+1\right)})(\left(h+{l}_{2}\mathrm{sin}\left({\theta }_{2}\right)\right){b}_{2})}$$

(6)

Appearing terms in equations may be given as:

$$ \begin{aligned} K_{1} = & \;2\frac{{A_{1} E_{s1} \cos \left( {\theta_{1} } \right)^{2} \sin \left( {\theta_{1} } \right)}}{{l_{2} \cos \left( {\theta_{2} } \right)}} + \frac{{A_{2} E_{s2} \cos \left( {\theta_{2} } \right)\sin \left( {\theta_{2} } \right)}}{{l_{2} }} \\ & \; + \frac{{12E_{s2} I_{2} \sin \left( {\theta_{2} } \right)\cos \left( {\theta_{2} } \right)}}{{l_{2}^{3} \left( {\alpha_{2} + 1} \right)}} + \frac{{24E_{s1} I_{1} \sin \left( {\theta_{1} } \right)\cos \left( {\theta_{1} } \right)^{4} }}{{l_{2}^{3} \cos \left( {\theta_{2} } \right)^{3} \left( {\alpha_{1} \frac{{\cos \left( {\theta_{1} } \right)^{2} }}{{\cos \left( {\theta_{2} } \right)^{2} }} + 1} \right)}} \\ \end{aligned} $$

(7a)

$$ K_{2} = 2\frac{{A_{1} E_{s1} \cos \left( {\theta_{1} } \right)^{3} }}{{l_{2} \cos \left( {\theta_{2} } \right)}} + \frac{{A_{2} E_{s2} \cos \left( {\theta_{2} } \right)^{2} }}{{l_{2} }} + \frac{{12E_{s2} I_{2} \sin \left( {\theta_{2} } \right)^{2} }}{{l_{2}^{3} \left( {\alpha_{2} + 1} \right)}} + \frac{{24E_{s1} I_{1} \sin \left( {\theta_{1} } \right)^{2} \cos \left( {\theta_{1} } \right)^{3} }}{{l_{2}^{3} \cos \left( {\theta_{2} } \right)^{3} \left( {\alpha_{1} \frac{{\cos \left( {\theta_{1} } \right)^{2} }}{{\cos \left( {\theta_{2} } \right)^{2} }} + 1} \right)}} $$

(7b)

$$ \begin{aligned} K_{3} = & \;2\frac{{A_{1} E_{s1} \sin \left( {\theta_{1} } \right)^{2} \cos \left( {\theta_{1} } \right)}}{{l_{2} \cos \left( {\theta_{2} } \right)}} + \frac{{A_{2} E_{s2} \sin \left( {\theta_{2} } \right)^{2} }}{{l_{2} }} \\ & \; + \frac{{12E_{s2} I_{2} \cos \left( {\theta_{2} } \right)^{2} }}{{l_{2}^{3} \left( {\alpha_{2} + 1} \right)}} + \frac{{24E_{s1} I_{1} \cos \left( {\theta_{1} } \right)^{5} }}{{l_{2}^{3} \cos \left( {\theta_{2} } \right)^{3} \left( {\alpha_{1} \frac{{\cos \left( {\theta_{1} } \right)^{2} }}{{\cos \left( {\theta_{2} } \right)^{2} }} + 1} \right)}} \\ \end{aligned} $$

(7c)

$${\alpha }_{1}=\frac{12{E}_{s1}{I}_{1}{k}_{1}}{{{l}_{1}}^{2}{G}_{s1}{A}_{1}}$$

(8a)

$${\alpha }_{2}=\frac{12{E}_{s2}{I}_{2}{k}_{2}}{{{l}_{2}}^{2}{G}_{s2}{A}_{2}}$$

(8b)

$${G}_{s1}=\frac{{E}_{s1}}{2(1+{\nu }_{s1})}$$

(8c)

$${G}_{s2}=\frac{{E}_{s2}}{2(1+{\nu }_{s2})}$$

(8d)

where ${\nu }_{12}$ is Poisson’s ratio, ${E}_{1}$ is the elasticity modulus of the re-entrant cell, ${A}_{1}$ and ${A}_{2}$ are the cross-sectional areas of the inclined outer walls and inner wall, respectively, ${E}_{s1}$ and ${E}_{s2}$ are the elasticity moduli of the inclined outer walls’ and inner wall’s materials, respectively, ${\theta }_{1}$ and ${\theta }_{2}$ are the angles of the inclined outer walls and inner wall, respectively, ${I}_{1}$ and ${I}_{2}$ are the second moment of inertias of the inclined outer walls and inner wall, respectively, ${l}_{2}$ is the length of the inclined inner wall, ${t}_{2}$ is the thickness of the inclined inner wall, $h$ is the height of the re-entrant cell (length of the horizontal wall), ${b}_{2}$ is the width of the inclined wall, ${K}_{1}$, ${K}_{2}$, and ${K}_{3}$ are the stiffness terms that governs cell deformation, ${\alpha }_{1}$ and ${\alpha }_{2}$ are the shear deflection coefficients of the inclined outer walls and inner wall, respectively, ${k}_{1}$ and ${k}_{2}$ are the cross-section shear coefficients of the inclined outer walls and inner wall, respectively, ${l}_{1}$ is the length of the inclined outer walls, ${G}_{s1}$ and ${G}_{s2}$ are the shear moduli of the inclined outer walls’ and inner wall’s materials, respectively, ${\nu }_{s1}$ and ${\nu }_{s2}$ are Poisson’s ratios of the inclined outer walls’ and inner wall’s materials, respectively.

• (b) Type-II cell equations:

All equations of the Type-II cell may be derived from the deformation of the inclined and circular walls. Employing inclined and circular walls of the quarter cell, the equations may be derived employed in this paper (for further information please refer to Tatlıer et al. [78]). Poisson’s ratio and elasticity modulus of the Type-II cell may be expressed as:

$${\nu }_{12}=-\frac{{\delta }_{21}(Lcos\left(\theta \right)+\frac{{t}_{2}}{2}}{{\delta }_{11}(h+Lsin\left(\theta \right))}$$

(9)

$$ E_{1} = \frac{{L\cos \left( \theta \right) + \frac{{t_{2} }}{2}}}{{\delta_{11} \left( {\left( {h + L\sin \left( \theta \right)} \right)} \right)b_{2} }} $$

(10)

Appearing terms in equations may be given as:

$${\delta }_{11}=\frac{{K}_{11}{K}_{23}\left({K}_{33}{{K}_{12}}^{2}+{K}_{11}{{K}_{23}}^{2}-{K}_{11}{K}_{22}{K}_{33}\right)-{K}_{11}{{K}_{12}}^{2}{K}_{23}{K}_{33}}{{{K}_{11}}^{2}{K}_{23}\left({K}_{33}{{K}_{12}}^{2}+{K}_{11}{{K}_{23}}^{2}-{K}_{11}{K}_{22}{K}_{33}\right)}$$

(11a)

$${\delta }_{21}=\frac{{K}_{12}{K}_{33}}{\left({K}_{33}{{K}_{12}}^{2}+{K}_{11}{{K}_{23}}^{2}-{K}_{11}{K}_{22}{K}_{33}\right)}$$

(11b)

$${K}_{11}=\frac{{A}_{2}{E}_{s2}{cos\left(\theta \right)}^{2}}{L}+\frac{12{E}_{s2}{I}_{2}{sin\left(\theta \right)}^{2}}{{L}^{3}(1+\alpha )} $$

(12a)

$${K}_{12}=\frac{{12{E}_{s2}I}_{2}{cos\left(\theta \right)sin\left(\theta \right)}}{{L}^{3}(1+\alpha )}-\frac{{A}_{2}{E}_{s2}cos\left(\theta \right)sin\left(\theta \right)}{L}$$

(12b)

$$ \begin{aligned} K_{22} = & \;\frac{{12E_{s2} I_{2} \cos \left( \theta \right)^{2} }}{{L^{3} \left( {1 + \alpha_{2} } \right)}} + \frac{{A_{2} E_{s2} \sin \left( \theta \right)^{2} }}{{L^{ } }} \\ & \; + \frac{{4\pi^{2} A_{1} E_{s1} G_{s1}^{2} I_{1}^{2} - 32A_{1}^{2} E_{s1} G_{s1}^{2} I_{1} R^{2} + 4\pi^{2} A_{1} E_{s1}^{2} G_{s1} I_{1}^{2} k^{\prime} + 4\pi^{2} A_{1}^{2} E_{s1} G_{s1}^{2} I_{1} R^{2} }}{D} \\ \end{aligned} $$

(12c)

$${K}_{23}=\frac{32{{A}_{1}}^{2}{{E}_{s1}{G}_{s1}}^{2}{I}_{1}{R}^{2}-8\pi {A}_{1}{E}_{s1}{{G}_{s1}}^{2}{{I}_{1}}^{2}-8\pi {{A}_{1}}^{2}{{E}_{s1}{G}_{s1}}^{2}{I}_{1}{R}^{2}+8\pi {A}_{1}{{E}_{s1}}^{2}{G}_{s1}{{I}_{1}}^{2}{k}_{1}}{D}$$

(12d)

$${K}_{33}=\frac{4{\pi }^{2}{A}_{1}{E}_{s1}{{G}_{s1}}^{2}{{I}_{1}}^{2}-32{{A}_{1}}^{2}{{E}_{s1}{G}_{s1}}^{2}{I}_{1}{R}^{2}+4{\pi }^{2}{A}_{1}{{E}_{s1}}^{2}{G}_{s1}{{I}_{1}}^{2}{k}_{1}+4{\pi }^{2}{{A}_{1}}^{2}{E}_{s1}{{G}_{s1}}^{2}{I}_{1}{R}^{2}}{D}$$

(12e)

${G}_{si}$, ${\alpha }_{i}$, R, and D that appear in equations above may be expressed as (where i is 1 for the circular wall and 2 for the inclined wall, respectively):

$${G}_{si}=\frac{{E}_{si}}{2(1+{\nu }_{si})}$$

(13a)

$${\alpha }_{i}=\frac{12{E}_{si}{I}_{i}{k}_{i}}{{{L}_{i}}^{2}{G}_{si}{A}_{i}}$$

(13b)

$$R=\frac{h}{2}+Lsin\left(\theta \right)$$

(13c)

$$ \begin{aligned} D = &\, \;32A_{1}^{2} G_{s1}^{2} R^{5} + 32A_{1} G_{s1}^{2} I_{1} R^{3} - 20\pi A_{1}^{2} G_{s1}^{2} R^{5} \\ & \; + \pi^{3} A_{1}^{2} G_{s1}^{2} R^{5} - 4\pi G_{s1}^{2} I_{1}^{2} R + \pi^{3} E_{s1}^{2} I_{1}^{2} Rk_{1}^{2} \\ & \; - 24\pi A_{1} G_{s1}^{2} I_{1} R^{3} + 2\pi^{3} A_{1} G_{s1}^{2} I_{1} R^{3} - 4\pi E_{s1}^{2} I_{1}^{2} Rk_{1}^{2} \\ & \; + 8\pi E_{s1} I_{1}^{2} G_{s1} Rk_{1} - 32A_{1} E_{s1} G_{s1} I_{1} R^{3} k_{1} + 2\pi^{3} E_{s1} G_{s1} I_{1}^{2} Rk_{1} \\ & \; + 2\pi^{3} A_{1} E_{s1} G_{s1} I_{1} R^{3} k_{1} - 8\pi A_{1} E_{s1} G_{s1} I_{1} R^{3} k_{1} \\ \end{aligned} $$

(13d)

where $\nu_{12}$ is Poisson’s ratio, $E_{1}$ is the elasticity modulus of the re-entrant cell, $\delta_{11}$ and $\delta_{21}$ are the axial and the transverse displacement of cell, $L$ is the length of inclined wall, $t_{2}$ is the thickness of the inclined wall, $\theta$ is the angle of the inclined wall, $h$ is the height of the re-entrant cell (length of the horizontal wall), $b_{2}$ is the width of the inclined wall, $K_{ij}$ terms are the stiffness terms that govern cell deformation (where i and j indicate the cell’s degrees of freedoms: 1 and 3 are along with 1 axis and 2 is along with 2 axis), $A_{1}$ and $A_{2}$ are the cross-sectional areas of the circular and inclined walls, respectively, $E_{s1}$ and $E_{s2}$ are the elasticity moduli of the circular and inclined walls’ materials, respectively, $I_{1}$ and $I_{2}$ are the second moment of inertias of the circular and inclined walls, respectively, $h$ is the height of the re-entrant cell (length of the horizontal wall), $b_{2}$ is the width of the inclined wall, $\alpha_{i}$ terms are the shear deflection coefficients of the circular and inclined walls, $k_{i}$ terms are the cross-sectional shear coefficients of the circular and inclined walls, $G_{si}$ terms are the shear moduli of the circular and inclined walls, $\nu_{{{\text{si}}}}$ terms are Poisson’s ratios of the circular and inclined walls’ materials, $R$ is the radius of the circular wall, and $D$ is the determinant of the structural matrix of the cell.