# A Novel Design Method for Energy Absorption Property of Chiral Mechanical Metamaterials

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Topology Optimization of the Rotating Properties

**α**of the level set function during the interpolation by the compactly supported radial basis function. The optimization goal is to maximize ${D}_{ijkl}$ in the coupling matrix $D$. The constraint is the volume fraction of the physical material. The specific optimization equation is as follows:

**x**, and $\delta (\varphi )$ is the Dirac function. This section uses the optimization criterion method to perform the topology optimization analysis of the above model.

## 3. Parametric Optimization of Chiral Metamaterials

#### 3.1. Parametric Modeling

#### 3.2. Parametric Optimization Based on a Surrogate Model

^{3}mm/s, and the impact direction is shown in Figure 5. (3) The structure was constrained at the bottom of the thin plate. Table 2 shows the parameters of the chiral structural properties during the simulation analysis. Figure 6 shows a nephogram of the dynamic simulation of chiral metamaterials in the parametric optimization.

#### 3.3. Analysis of the Parametric Optimization Results

- 1.
- The surge state of the initial compression reaction force is the change in stress caused by the impact load on the chiral structure. At this stage, the compression displacement produced by the structural deformation lags behind the compression reaction force, which leads to a sharp increase in the compression reaction force in the initial stage. The final surge state of the compression reaction force is the stage when the walls of solid holes in the chiral structure are in full contact. At this stage, the structural deformation is very small, the external force is close to lossless transmission, and the slight increase in compression displacement causes the compression load to rise sharply. The slow increase in the compression reaction force is the effective stage of the chiral structure. Due to the rotational deformation of the chiral structure, the stress generated by compression of the partial structures will be consumed and transferred. The macroscopic expression is the slow increase in the compression reaction force.
- 2.
- As the impact velocity increases, the compression reaction force corresponding to the same compression displacement increases, the compression reaction force leads to increased growth in the slow increase phase, and the compression displacement corresponding to the final surge state of the compression reaction force increases. This is because the greater the impact speed, the lower the efficiency of internal stress consumption and transfer through structural rotation deformation. The macroscopic manifestation is an increase in the compression reaction force and its related state.

- 1.
- As the compression displacement continues to increase, the energy absorbed by the chiral structure and its acceleration continues to increase. If we compare the energy absorption curves of chiral metamaterials at different impact speeds, it can be found that the energy absorbed by the chiral structure under the same compression displacement increases with an increase in the impact speed. The energy absorption efficiency of the chiral structure is highest in the middle of the compression displacement (80–140 mm).
- 2.
- Chiral metamaterials under high-speed impact will enter the nonlinear deformation stage more quickly, and the proportion of this stage is larger. These phenomena show that it is necessary to pre-select the appropriate impact velocity to make full use of the energy absorption properties of chiral structures. The safety interval of the chiral structure can be set, and its utilization efficiency can be improved according to this phenomenon.

_{in}corresponding to different structures (No. 1, No. 41, No. 81, and No. 88) and different speeds (2–16 m/s). Figure 9 shows the mechanical properties of chiral metamaterials during parametric optimization. For the curve of v-F (Figure 9a), the average compression reaction force in the slow increase stage during impact deformation of the corresponding chiral metamaterial is collected as F. For the curve of v-E

_{in}(Figure 9b), the absorbed energy corresponding to the end of the force stabilization phase of the compression reaction during the impact deformation process of the corresponding chiral metamaterial is collected as E

_{in}. The following conclusions can be drawn from Figure 9:

- 1.
- For the optimized chiral metamaterials, the compression reaction force F and the absorbed energy E
_{in}under the same impact velocity were greatly improved. This indicates the effectiveness of parametric optimization for improving the mechanical properties of chiral metamaterials. - 2.
- For any chiral metamaterial structure in the parametric optimization, the compression reaction force F and the absorbed energy E
_{in}increase continuously when the impact velocity increases. However, the growth rate of the compression reaction force F gradually decreases to zero, and the growth rate of the absorbed energy E_{in}continues to increase. These indicate that chiral metamaterials have a specific impact velocity range. The energy absorption properties of the chiral metamaterials can be maximized within a suitable range. If the application range is exceeded, the structure will fail, and thus, the energy absorption properties of the chiral metamaterials will be greatly reduced.

## 4. Experimental Analysis of Impact Compression

#### 4.1. Specimen Manufacturing and Experimental Design

#### 4.2. Impact Compression Experiment

#### 4.3. Feedback Adjustment

_{in}under the same impact velocity and different impact velocities was conducted with the same setup as the previous impact experiment. Comparisons of the experimental and simulation curves of the optimized structure are shown in Figure 17 to illustrate the effectiveness of the feedback adjustment. Figure 17a,b and show the x-F curve and the x-E

_{in}curve under the same impact velocity. Figure 17c,d show the v-F curve and the v-E

_{in}curve under different impact velocities.

_{in}during the impact process were slightly improved. These changes in the mechanical curves shown in Figure 17 illustrate that the feedback adjustment is of great significance for improving the structural performance and ensuring the effectiveness of parametric optimization. It is also an indispensable link in the FIP design.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Couple stress analysis of the plane structure. ((

**a**) shows the couple stress in the rectangular coordinate system, and (

**b**) shows the microstructure unit cell considering the couple stress).

**Figure 3.**Analysis of the energy absorption properties. ((

**a**) shows a uniform porous structure with the same volume fraction as the optimized structure shown in (

**b**)).

**Figure 4.**Schematic diagram of the dimensions of the chiral structure. ((

**a**) represents the sectional view, and (

**b**) represents the stereograph).

**Figure 5.**Schematic diagram of the practical chiral structure. ((

**a**,

**b**) show the schematic cross-sectional view of the practical structure and the three-dimensional view of the practical structure, respectively).

**Figure 7.**The change trends of the structural parameters and output variables. ((

**a**) shows the change trends of the structural parameters during the optimization of chiral metamaterials with the surrogate model. (

**b**) shows the change tendency of the output variable P during the optimization process of the surrogate model.).

**Figure 8.**The mechanical characteristic curves of the optimized structures of chiral metamaterials at different impact speeds. ((

**a**) shows the compression reaction force-compression displacement curves of the chiral metamaterials at different impact speeds. (

**b**) shows the energy absorption curve of chiral metamaterials at different impact speeds).

**Figure 9.**Mechanical properties of chiral metamaterials during parametric optimization. ((

**a**,

**b**) show the curves of v-F and v-E

_{in}, respectively).

**Figure 10.**Flow chart of the process for additive manufacturing of the chiral metamaterials. ((

**a**) is the basic introduction model, (

**b**) is the structure during the forming process, and (

**c**) shows some of the final structures).

**Figure 11.**Materials and devices for packaging Fiber Bragg Grating sensors. ((

**a**–

**e**) show the jumper wires, bare Fiber Bragg Grating, demodulator, fiber welding machine, and fiber cleaver).

**Figure 12.**Calibration of the Fiber Bragg Grating pressure sensor. ((

**a**) represents the operation interface, and (

**c**) is a partly enlarged view of the compression site in (

**b**)).

**Figure 13.**Setup of the specimen compression experiment. ((

**a**) is the overall layout of the experimental setup, and (

**b**,

**c**) are partly enlarged views of the test area and the impact area).

**Figure 14.**The deformation of the chiral metamaterial during the impact process. ((

**1**–

**9**) represented the impact compression views which were taken every 300 consecutive photos).

**Figure 15.**Comparative analysis of the experiment and simulation results for optimized structural. ((

**a**,

**b**) compare the experimental and simulation results of the optimized structure at the same impact speed (4 m/s). (

**c**,

**d**) compare the experimental and simulated optimized structure at different speeds from 2–16 m/s).

**Figure 16.**The change trends of the structural parameters (

**a**) and output variables (

**b**) from the feedback adjustment.

**Figure 17.**A comparison of the curves of experimental and simulation results for the optimized structure in the feedback adjustment. ((

**a**,

**b**) show the x-F curve and the x-E

_{in}curve under the same impact velocity. (

**c**,

**d**) show the v-F curve and the v-E

_{in}curve under different impact velocities).

Type | Structural Parameters | |||||||
---|---|---|---|---|---|---|---|---|

L1 | L2 | L3 | L4 | L5 | L6 | L7 | α1 | |

Range (mm/°) | 80–100 | 20–35 | 5–15 | 45–60 | 10–20 | 10–20 | 20–40 | 15–20 |

Type | Property Parameters | ||||
---|---|---|---|---|---|

Density $\mathit{\rho}$ | $\mathbf{Elastic}\mathbf{Modulus}\mathit{E}$ | $\mathbf{Poisson}\u2019\mathbf{s}\mathbf{Ratio}\mathit{\mu}$ | $\mathbf{Yield}\mathbf{Stress}\mathit{\sigma}$ | $\mathbf{Shear}\mathbf{Modulus}\mathit{G}$ | |

Value | $7.8\times {10}^{3}\mathrm{kg}/{\mathrm{m}}^{3}$ | $2.1\times {10}^{11}\mathrm{Pa}$ | 0.3 | $2.3\times {10}^{9}\mathrm{Pa}$ | $6\times {10}^{10}\mathrm{Pa}$ |

Type | Structural Parameters | |||||||
---|---|---|---|---|---|---|---|---|

L1 | L2 | L3 | L4 | L5 | L6 | L7 | $\mathit{\alpha}1$ | |

Value (mm/°) | 100 | 23 | 11 | 52 | 10 | 12 | 40 | 16 |

Type | Structures | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Initial Structure | Optimized Structures | Optimal Structure | |||||||||

1 | 41 | 46 | 51 | 56 | 61 | 66 | 71 | 76 | 81 | 88 | |

Number | A | B | C | D | E | F | G | H | I | J | K |

Type | Structural Parameters | |||
---|---|---|---|---|

L2 | L3 | L6 | L7 | |

Ranges (mm/°) | 18.4–27.6 | 8.8–13.2 | 9.6–14.4 | 32–40 |

Results (mm/°) | 21.9 | 11.6 | 13.1 | 40 |

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**MDPI and ACS Style**

Ye, M.; Gao, L.; Wang, F.; Li, H.
A Novel Design Method for Energy Absorption Property of Chiral Mechanical Metamaterials. *Materials* **2021**, *14*, 5386.
https://doi.org/10.3390/ma14185386

**AMA Style**

Ye M, Gao L, Wang F, Li H.
A Novel Design Method for Energy Absorption Property of Chiral Mechanical Metamaterials. *Materials*. 2021; 14(18):5386.
https://doi.org/10.3390/ma14185386

**Chicago/Turabian Style**

Ye, Mengli, Liang Gao, Fuyu Wang, and Hao Li.
2021. "A Novel Design Method for Energy Absorption Property of Chiral Mechanical Metamaterials" *Materials* 14, no. 18: 5386.
https://doi.org/10.3390/ma14185386