# Dynamic Response and Energy Absorption Characteristics of a Three-Dimensional Re-Entrant Honeycomb

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Finite Element Model and Parameters

#### 2.1. Digital Model

_{RVE}is the volume of three-dimensional honeycomb structure, and V

_{Total}is the total volume of representative structural cells in three-dimensional space in Formula (1).

#### 2.2. Model Parameters and Constraints

#### 2.3. Calculation Critical Velocity

- When the impact velocity is lower than the first critical impact velocity (i.e., the notch wave velocity), the whole specimen is slowly compressed, and the force is relatively uniform during the impact process. The material undergoes quasi-static deformation.
- When the impact velocity exceeds the first critical impact velocity (i.e., notch wave velocity), the impact process transits from the overall deformation to the local deformation and the local deformation band is formed. With the increase of impact velocity, the local deformation of the upper end of the specimen is more obvious.
- When the impact velocity is higher than the second critical impact velocity, the local deformation zone will propagate from the upper end to the lower end of the specimen in the mode of a shock wave.

_{1}is defined as the corresponding nominal strain (i.e., initial strain) when the stress reaches the stress peak for the first time in the process of impact fluctuation. σ(ε) represents the elastic modulus of honeycomb material in the online elastic stage, and ∆ρ is the relative density of honeycomb material. ρ

_{A}is the density of the honeycomb material.

_{p}is the plateau stress of honeycomb materials under quasi-static compression in Formula (3), and ε

_{3}is the locking strain, that is, the strain value at the beginning of the densification stage of honeycomb materials.

_{c1}≈11 m/s and the second critical impact velocity V

_{r2}≈ 62 m/s are calculated. This paper selects the impact velocity V

_{1}= 3 m/s (V

_{1}< V

_{cr1}), V

_{2}= 20 m/s (V

_{cr1}< V

_{2}< V

_{cr2}) and V

_{3}= 200 m/s (V

_{cr2}< V

_{3}) to study the impact deformation in order to observe the influence of different impact velocities on the dynamic response of three-dimensional re-entrant honeycombs.

## 3. The Result of Simulation and Discussion

#### 3.1. Deformation Mode

_{1}= 3 m/s), medium speed (V

_{2}= 20 m/s) and high speed (V

_{3}= 200 m/s) are shown in Figure 7, Figure 8 and Figure 9, respectively. The nominal strain (ε) in the Figures is the ratio of the displacement of the specimen in y-axis direction to the initial height.

_{1}= 3 m/s), the deformation process of three-dimensional honeycomb can be roughly divided into four stages. Phase I (ε = 0.096) is mainly the rotation of the inclined cell wall inside the three-dimensional honeycomb. The results show that the more the impact velocity is close to the velocity of quasi-static compression, the more uniform the stress is in the compression process. The specimen is uniformly deformed all the time in this stage and the upper and lower ends are close to the middle under the action of the x-axis force generated by the rotation of the cell wall, which shows that the specimen has a specific negative Poisson’s ratio. The middle part of the specimen in the y-axis direction has little force and almost no deformation, so the middle part of the specimen is convex during compression. In Phase II (ε = 0.277), the deformation is mainly caused by the continuous rotation of the inclined cell wall in the upper structure of the specimen in order to withstand the compression deformation in the y-axis direction. Therefore, under the action of the cohesion in the x-axis direction, the upper end of the specimen has obvious concave phenomenon. When the upper end is compressed to a certain extent, it enters Phase III (ε = 0.470). The internal inclined cell wall of the upper end of the specimen will maintain a certain angle in the process. The pressure is transferred from the upper end of the specimen to the lower end, which leads to the rotation of the internal inclined cell wall of the lower end of the specimen. The cohesive force begins to contract to the middle, and the concave shape appears to bear the impact force transmitted from the upper end. When the upper and lower ends are basically symmetrical and the upper and lower ends of the specimen are concave and the middle part is convex, forming a ‘barreling’ state, this stage is completed. Then the deformation enters Phase IV (ε = 0.782), the specimen continues compressing in the y-axis direction. Because the upper and lower ends of the specimen have been compressed to a certain extent, the inclined cell wall of the middle part of the specimen in the y-axis direction will rotate. Under the action of transverse force in the x-axis direction, the inclined cell wall begins to converge to the middle vertical plane. When the inner cell wall of the structure basically parallel, the adjacent cell walls reach full contact density, and the compression is stopped.

_{2}= 20 m/s), the deformation can be divided into two stages. The first stage includes the deformations where the strain is between 0 to 0.47. When ε = 0.096, the impact energy cannot be transmitted to the lower end of the specimen, resulting in the compression of the upper end of the specimen. The inclined cell wall begins to rotate inside the specimen. Because the impact velocity is higher than the one at low speed, the horizontal cell wall will produce buckling and the external impact energy is absorbed. Due to the rotation and buckling of the inner cell wall of the specimen structure, the upper part of the specimen will produce an obvious concave deformation, where negative Poisson’s ratio characteristics are shown. When ε = 0.277 and ε = 0.470, the deformation still presents in the first stage. The specimen is compressed layer by layer from top to bottom. In the second stage (ε = 0.782), the compression of the specimen is passed layer by layer to the lower of the specimen, and the internal cell walls begin to contact each other and produce dense compression. At this stage, the bottom layer of the specimen cannot bear the force in the x-axis direction due to the lessened friction force, so there is a slight rollover phenomenon.

_{3}= 200 m/s), the compression deformation can be seen as one stage. The inertia effect plays a leading role due to the fast impact speed. The inclined cell wall in the specimen structure cannot produce rotation and only buckling deformation occurs. The specimen is compressed layer by layer from upper to lower, until the compression reaches to the bottom and the inner cell wall of the structure is fully contacted and dense. Negative Poisson’s ratio can be hardly shown in this process.

_{1}= 3 m/s), the nominal stress–strain curve of the specimen is shown in Figure 11. When the three-dimensional honeycomb parameters are α = 45°, L = 5 mm and t = 0.3 mm, the ε represents the nominal strain in the horizontal coordinate, that is, the ratio of the compression reaction of the upper rigid plate to the initial contact area of the specimen, and the σ represents the nominal stress in the vertical coordinate.

_{1}), the stress begins to fluctuate and finally tends to be stable. In the platform region, the compressive stress of the specimen after reaching the initial strain ε

_{1}fluctuates around a certain value and remains relatively stable. The specimen undergoes great compressive deformation in this stage, so it is the main stage of energy absorption. After the end of the platform stage, it enters the platform enhancement region. With the continuous increase of the compressive strain of the specimen, the stress no longer remains relatively stable, but gradually increases with a specific slope and exceeds the platform stress value to a certain extent. After the strain at the end of the enhancement stage reaches ε

_{3}, the cells in the specimen begin to contact with each other in dense region. At this region, the stress value of the specimen rises sharply in a small strain stage until the inner wall of the cells in the specimen is completely bonded together and the dense stage ends.

#### 3.2. Platform Stress

_{1}to ε

_{2}, the stress in this region is called the platform stress (σ

_{p}). It is an important indicator for describing the dynamic response characteristics of the honeycomb and can be calculated by the following Formulas (4) and (5):

_{1}is the initial strain, that is, the corresponding strain value when the initial stress is just stable and reaches the platform stress, so the value of ε

_{1}is very small. In order to achieve a high calculation accuracy, the value of ε

_{1}in this paper is set as 0.013. ε

_{3}is dense strain, that is, the strain corresponding to the contact between adjacent cell walls within the specimen. In Formula (5), the value of F(ε) is derived from the average value of the force of the upper rigid plate in the platform area obtained by the simulation. L

_{x}is the length of the specimen in the x-axis direction, and L

_{z}is the length of the specimen in the z-axis direction.

_{s}represents the yield stress of the matrix material, Δρ represents the relative density of the designed honeycomb material, ρ

_{s}represents the density of the matrix material and m and n are the coefficients to be calculated or fitted.

#### 3.3. Energy Absorption

_{u}is the internal energy of the material, E

_{k}is the kinetic energy of the material, E

_{f}is the energy of the contact friction loss, E

_{w}is the work done by the external load and E

_{qb}is the energy dissipated by the surrounding medium damping.

_{1}= 3 m/s), the ability to absorb energy increases with the increase of cell angle shown Figure 14b. Therefore, the energy absorption ability of three-dimensional honeycomb can be improved by changing the impact velocity and cell angle.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Evolution of three-dimensional re-entrant honeycomb structure. (

**a**) Traditional re-entrant honeycomb; (

**b**) Representative structural cell (RSC); (

**c**) Array of the RSC.

**Figure 2.**Structural parameters of the traditional re-entrant honeycomb (2L is the length of upper and lower cell walls, L is the length of the ligament, α is the angle between the cell walls and t is the thickness of cell wall.).

**Figure 4.**Comparison of deformation at speed of 100 m/s. Reprinted with permission from ref. [24]. Copyright 2022 Elsevier.

**Figure 13.**The relationship between nominal stress and strain of specimen. (

**a**) The stress and strain of different velocities. (

**b**) The stress and strain of different cell angles.

**Figure 14.**Relationship between energy absorption and strain of specimens. (

**a**) E (total energy) with different velocities. (

**b**) E (total energy) with different angles.

**Figure 15.**Relationship between energy distribution coefficient and strain. (

**a**) Φ (internal energy distribution coefficient) with different velocities. (

**b**) Φ (internal energy distribution coefficient) with different cell angles).

Material | ρ/(Kg·m^{−3}) | E/GPa | σ_{s}/MPa | v |
---|---|---|---|---|

Aluminum | 2700 | 69 | 76 | 0.3 |

v/(m/s) | σp/(MPa) | |||
---|---|---|---|---|

Δρ = 0.037 | Δρ = 0.02 | Δρ = 0.012 | Δρ = 0.008 | |

3 | 0.037 | 0.026 | 0.023 | 0.019 |

7 | 0.038 | 0.031 | 0.028 | 0.022 |

20 | 0.081 | 0.053 | 0.039 | 0.035 |

35 | 0.167 | 0.109 | 0.058 | 0.055 |

70 | 0.623 | 0.303 | 0.192 | 0.171 |

100 | 1.450 | 0.615 | 0.378 | 0.314 |

200 | 3.778 | 2.158 | 1.403 | 1.079 |

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**MDPI and ACS Style**

Zhang, J.; Shi, B.; Han, T.
Dynamic Response and Energy Absorption Characteristics of a Three-Dimensional Re-Entrant Honeycomb. *Electronics* **2022**, *11*, 2725.
https://doi.org/10.3390/electronics11172725

**AMA Style**

Zhang J, Shi B, Han T.
Dynamic Response and Energy Absorption Characteristics of a Three-Dimensional Re-Entrant Honeycomb. *Electronics*. 2022; 11(17):2725.
https://doi.org/10.3390/electronics11172725

**Chicago/Turabian Style**

Zhang, Jun, Boqiang Shi, and Tian Han.
2022. "Dynamic Response and Energy Absorption Characteristics of a Three-Dimensional Re-Entrant Honeycomb" *Electronics* 11, no. 17: 2725.
https://doi.org/10.3390/electronics11172725