# Dynamic Crushing Analysis of a Three-Dimensional Re-Entrant Auxetic Cellular Structure

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}to 120 s

^{−1}. The dynamic behaviors of the auxetic structure, including the crushing modes, the crushing strength and the energy absorption capacity, have been further investigated with respect to the crushing velocity and the structure relative density.

## 2. Geometry and Modeling

#### 2.1. Geometry

#### 2.2. Description of Finite Element Modeling

^{®}, as shown in Figure 2a. The 3D re-entrant auxetic structure adopted for simulation have been given with 12 × 9 × 6 representative cells (twelve unit cells in x-direction, nine unit cells in y-direction and six unit cells in z-direction). The geometrical parameters of the 3D specimens in crushing analysis are set as $\mathrm{L}=20\text{}\mathrm{mm}$, $\mathrm{H}=30\text{}\mathrm{mm}$, $\mathsf{\theta}=60\xb0$ and $\mathrm{t}$ varies from 1 mm to 4 mm, featuring a variable relative density closely between 1% and 10%. The dimensions of whole model are then given as 416 mm by 360 mm by 208 mm in Figure 2a.

^{®}has been adopted to model the contact between surfaces with a friction coefficient of 0.35.

## 3. Dynamic Crushing Modes

^{−1}to 120 s

^{−1}in this study. Figure 4, Figure 5, Figure 6 and Figure 7 depict the deformation processes of the auxetic structures subjected to dynamic crushing along the y-direction. As seen from the simulation results, similar to the studies for regular hexagonal honeycombs [26], three specific crushing modes have also been observed from the numerical simulations for auxetic structures, namely quasi-static, transition and dynamic mode. But due to the auxetic effect, these specific crushing modes all exhibit different deformation shapes compared to the crushing patterns for regular hexagonal honeycombs reported as ‘X,’ ‘V’ and ‘I’ types [26]. Figure 4, Figure 5 and Figure 6 show the deformation processes under the crushing velocities of 5, 15 and 50 m/s, representing the three specific crushing modes, respectively. Each process recorded is given with six typical deformation stages with the longitudinal ${\epsilon}_{y}$= 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5.

## 4. Dynamic Crushing Strength

## 5. Energy Absorption Capacity

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) 3D re-entrant auxetic cellular structure; (

**b**) Representative volume element (RVE); (

**c**) Geometrical features of the RVE.

**Figure 2.**The numerical simulation model (12 × 11 × 6 type): (

**a**) Overview of the mesh model; (

**b**) The boundary conditions when crushing along the y direction.

**Figure 3.**Convergence of the simulations under a constant velocity ${v}_{c}$ of 30 m/s with the basic element size varying from 4 mm to 10 mm: (

**a**) Crushing reaction force; (

**b**) Hourglass error.

**Figure 4.**Deformation process of the 3D re-entrant auxetic structure under the crushing velocity ${v}_{c}$ = 5 m/s.

**Figure 5.**Deformation process of the 3D re-entrant auxetic structure under the crushing velocity ${v}_{c}$ = 15 m/s.

**Figure 6.**Deformation process of the 3D re-entrant auxetic structure under the crushing velocity ${v}_{c}$ = 50 m/s.

**Figure 7.**Deformation processes of the3D re-entrant auxetic structure (${\rho}_{R}=1\%$) under different crushing velocities.

**Figure 8.**Mode classification map for the 3D re-entrant auxetic structures under dynamic crushing. The markers are the simulation results showing different crushing modes. The dashed lines represent the linear fits of boundaries associated with the transition among three specific crushing modes.

**Figure 9.**Dynamic crushing strength of the 3D re-entrant auxetic structure: (

**a**) Crushing stresses at the proximal end; (

**b**) Crushing stresses at the distal end; (

**c**) Dynamic densification strains ${\epsilon}_{D}$ under different crushing velocities; (

**d**) Crushing stresses (proximal end) of structures with different relative densities.

**Figure 10.**The quasi-static plateau stress ${\sigma}_{0}$ and the densification strain ${\epsilon}_{D}$ with respect to relative density.

**Figure 11.**The Dynamic plateau stress ${\sigma}_{D}$ from numerical simulation and theoretical prediction using Equation (6).

**Figure 12.**(

**a**) Plastic energy dissipation ${U}_{p}$ versus the crushing strain $\epsilon $; (

**b**) Frictional energy dissipation ${U}_{f}$ versus the crushing strain $\epsilon $.

**Figure 13.**(

**a**) Specific plastic energy dissipation ${U}_{p}^{s}$ versus the normalized crushing velocity $\overline{\mathrm{V}}$; (

**b**) Normalized plastic energy dissipation ${\overline{U}}_{p}$ versus the normalized crushing velocity $\overline{\mathrm{V}}$.

**Table 1.**Dynamic plateau stress ${\sigma}_{D}$ from numerical simulation and theoretical prediction using Equations (2)–(4).

Relative Density ${\mathit{\rho}}_{\mathit{R}}$ | Crushing Velocity ${\mathit{v}}_{\mathit{c}}$ (m/s) | Dynamic Plateau Stress ${\mathit{\sigma}}_{\mathit{D}}\text{}$ (MPa) | ||
---|---|---|---|---|

Theoretical | Simulation | Deviation (%) | ||

0.01 | 5 | 0.081 | 0.080 | 1.11 |

15 | 0.145 | 0.142 | 2.03 | |

30 | 0.337 | 0.311 | 8.36 | |

50 | 0.804 | 0.733 | 9.66 | |

70 | 1.708 | 1.521 | 12.31 | |

0.03 | 5 | 0.731 | 0.715 | 2.26 |

15 | 0.718 | 0.692 | 3.81 | |

30 | 1.286 | 1.216 | 5.77 | |

50 | 2.294 | 2.127 | 7.86 | |

70 | 4.015 | 3.611 | 11.20 | |

0.08 | 5 | 4.431 | 4.291 | 3.26 |

15 | 4.616 | 4.398 | 4.96 | |

30 | 5.058 | 4.713 | 7.32 | |

50 | 6.716 | 6.112 | 9.88 | |

70 | 8.608 | 7.612 | 13.08 |

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**MDPI and ACS Style**

Wang, T.; Li, Z.; Wang, L.; Ma, Z.; Hulbert, G.M.
Dynamic Crushing Analysis of a Three-Dimensional Re-Entrant Auxetic Cellular Structure. *Materials* **2019**, *12*, 460.
https://doi.org/10.3390/ma12030460

**AMA Style**

Wang T, Li Z, Wang L, Ma Z, Hulbert GM.
Dynamic Crushing Analysis of a Three-Dimensional Re-Entrant Auxetic Cellular Structure. *Materials*. 2019; 12(3):460.
https://doi.org/10.3390/ma12030460

**Chicago/Turabian Style**

Wang, Tao, Zhen Li, Liangmo Wang, Zhengdong Ma, and Gregory M. Hulbert.
2019. "Dynamic Crushing Analysis of a Three-Dimensional Re-Entrant Auxetic Cellular Structure" *Materials* 12, no. 3: 460.
https://doi.org/10.3390/ma12030460