# Crashworthiness Optimization Design of Thin-Walled Tube Filled with Re-entrant Triangles Honeycombs

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## Abstract

A novel re-entrant triangles-filled tube (RTT) is proposed through decoupling structural stiffness and energy absorption. Inner re-entrant triangles are employed to satisfy energy absorption, and outer thin wall is used to acquire high stiffness. This paper starts from establishment of theoretical models between geometric parameters of re-entrant triangles and relative density, equivalent elastic modulus and energy absorption characteristics, which are validated by experiments. On this basis, the optimal geometric parameters of unit cell are sought to maximize unit volume energy absorption and minimize relative density by adopting NSGA-II method. Subsequently, the cross-section of tube with optimal stiffness is obtained with targets for maximizing axial stiffness and lateral stiffness by employing static topology optimization method. To verify the proposed optimization method, RTT is analyzed and compared with positive Poisson’s ratio foam-filled tube (PFT), non-filled traditionally optimized tube (NTT) and pre-optimized square tube (PST). The results show that the novel RTT can improve stiffness and energy absorption performance simultaneously. Compared with the positive Poisson’s ratio material, re-entrant triangles honeycomb shows superior advantages in energy absorption. In comparison with the PFT, energy absorption of the RTT increases by 17.23%, and the peak crush force reduces by 5.04%. Therefore, the proposed decoupling design method demonstrates superiority in satisfying various performance requirements simultaneously.

## Keywords

Thin-walled tube Re-entrant triangles honeycomb Multi-objective optimization Energy absorption Structural stiffness## 1 Introduction

With increased customer demands and tough vehicle safety regulations, increasing attentions have recently been drawn to achieving better crashworthiness of automobile [1, 2, 3]. Moreover, environmental sustainable developments and energy crisis have forced engineers to design increasingly lighter structures [4, 5, 6]. Structures that possess with characteristics and superior crashworthiness have drawn abundant attention for their better performance [7, 8]. Owing to outstanding capabilities of energy absorption, thin-walled structure is regarded as the most potential energy absorber.

To enhance energy absorption performance, substantial efforts on design shape optimization have been expended. Rossi [9] carried out the study on the crushing behavior among square, hexagonal and octagonal tubes. Gao et al. [10] carried out simulations on crushing impact of thin-walled tubes with various cross-sections, which provided reference for the selection of automotive cross-section. Thin-walled tubes should be equipped with energy absorption performance and possessed with stiffness requirements. For example, outer thin wall of front side rail should be equipped with high stiffness. It is, however, not always wise to enhance crashworthiness by sacrificing stiffness when using the aforementioned optimization methods. Therefore, this paper proposes the concept of decoupling to satisfy the contradictory requirements of structural stiffness and energy absorption, i.e., the outer thin wall needs high stiffness, while the thinner filler is responsible for strengthening energy absorption. Substantial research on thin-walled tubes with fillers has been conducted. Axial crushing analysis on foam-filled thin-walled structure was carried out by Ghamarian [11] and Zarei [12]. Ahmad et al. [13] emulate crushing behaviors of foam-filled conical tubes adopting numerical methods. Whereas, the above-mentioned studies only involve single performance and conventional positive Poisson’s ratio foam fillers. Positive Poisson’s ratio fillers usually possess unstable platform stress and higher peak stress which limit the improvement in energy absorption and further application of impact protection. Various re-entrant structures have also been studied. Ma et al. [14] proposed a type of re-entrant square cellular structure and illustrated the effects of cellular structural parameters on in-plane dynamic performance. Compressing this kind of structure results in structural contraction, and platform stress improves, which contributes greatly to enhancing the capability for energy absorption. Herein, re-entrant triangles honeycomb acts as a filler, and the octagonal and cruciform cross-section is regarded as the stiffener.

To satisfy the requirements of structural stiffness and energy absorption simultaneously, the novel tube utilizes re-entrant triangles as filler based on the concept of decoupling structural stiffness and energy absorption. Inner re-entrant triangles are utilized to meet the demands of energy absorption, while the octagonal and cruciform cross-section is designed to achieve higher stiffness.

Firstly, relationships among geometric parameters and relative density, equivalent elastic modulus and energy absorption characteristics are established, which are validated by experiments. The optimal geometric parameters of unit cell are then sought to maximize unit volume energy absorption and minimize relative density utilizing the NSGA-II method. Subsequently, the cross-section of tube with optimal stiffness is obtained by employing static topology optimization method with targets for maximizing axial stiffness and lateral stiffness. To certify the optimization method, RTT, PFT, NTT and PST are analyzed and compared in light of stiffness and energy absorption. The proposed RTT demonstrates superior energy absorption and structural stiffness, and this work proposes a design method of decoupling structural stiffness and energy absorption and applying re-entrant triangles on tube inside. The novel RTT presents prominent effects on energy absorption and structural stiffness. The results are encouraging in the sense that it offers another potential material for engineers to address the issue and the novel-type RTT with re-entrant triangles can be widely promoted and achieve further applications on new energy absorbent structures.

## 2 Theoretical Models of Re-entrant Triangles Honeycomb

In order to set up the relationship among structural parameters and relative density, equivalent elastic modulus and energy absorption, theoretical models of the honeycomb are established. On the hypothesis that cellular material is continuous, parameters can be utilized to represent performance of cellular structure. The equivalent description is called as asymptotic homogenization method, put forward by Benssousan [15] and Sanchez-Palentia [16]. Asymptotic homogenization methods [17, 18, 19] have been regarded as effective means for multiscale modeling, while these methods are always neglected in the aspect of mechanics during the homogenization process and widely used in the prediction of equivalent performance of composites. Asymptotic homogenization method focuses on the theory where main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with the unit cell. It acts as a bridge-linking microscale analysis and macroscopic properties of cellular structures; structural response is obtained from the perspective of macroscopic scale. If size of macroscopic structure is larger compared with microscopic unit cell and the number of unit cell is sufficient, more accurate results can be expected by utilizing homogenization method.

### 2.1 Relationship Between Relative Density and Geometric Parameters

It can be seen that \(\rho_{{ 2 {\text{D}}}}\) is dominated by \(S_{1}\) and \(S_{2}\), which represent the areas of internal cell wall and 2D unit cell, respectively.

### 2.2 Mathematical Descriptions of Equivalent Elastic Modulus and Plateau Stress

Average stress of plateau region is regarded as plateau stress in stress–strain curve [22]. Plateau stress acts as a key factor to improve capabilities of energy absorption effectively, that is, energy absorption attributes significantly to plateau stress. Plateau stress also has a close relationship with the structural failure mechanism. Hence, it is of significance to further study the predominant deformation mode of these re-entrant triangles. Because the cellular structure is subjected to axial compression, the structural deformation is uniform in the elastic stage. As loading stress exceeds the stress limit, a series of failure modes including elastic buckling, plastic collapse and brittle fracture will occur.

### 2.3 Characterization of Energy Absorption Under Quasi-Static Compression

The stress–strain curve can be utilized to analyze energy absorption characteristics and evaluate cell shape of the re-entrant cellular structure. Effective unit volume energy absorption \(W\) comprises three parts: the elastic region \(W_{1}\), plateau region \(W_{2}\) and plateau stress enhancement region \(W_{3}\).

## 3 Certification of Theoretical Models

Three groups of geometric parameters

Thickness coefficient \(\alpha\) | Length coefficient \(\beta\) | Cellular angle \(\varphi\) | Scaling factor of length \(K\) | |
---|---|---|---|---|

First group | 0.1 | 0.2 | 35° | 0.6 |

Second group | 0.08 | 0.2 | 35° | 0.7 |

Third group | 0.06 | 0.2 | 30° | 0.7 |

Comparisons of theoretical and experimental results for unit volume energy absorption

Theoretical results (J/m | Experimental results (J/m | Relative error (%) | |
---|---|---|---|

First group | 9347.872 | 10,162 | 8.01 |

Second group | 13,720.62 | 14,318 | 4.17 |

Third group | 10,470.34 | 10,792 | 2.98 |

## 4 Optimization of Unit Cell

### 4.1 Single Objective Optimization on Maximizing Unit Volume Energy Absorption

With the purpose of achieving the optimal geometric parameter, multi-objective optimization on unit cell is executed. The NSGA-II method is adopted with the objectives for maximum \(W_{u}\) and minimum \(\rho_{RD}\).

### 4.2 Single Objective Optimization on Minimizing Relative Density

### 4.3 Multi-objective Optimization of Unit Cell

The optimal geometric parameters

Optimal geometric parameters | Values |
---|---|

Thickness coefficient \(\alpha\) | 0.1 |

Length coefficient \(\beta\) | 0.2 |

Cellular angle \(\varphi\) | 33.5° |

Scaling factor of length \(K\) | 0.56 |

## 5 Topology Optimization of Thin-Walled Tube

Based on above analysis, the re-entrant triangles honeycombs with the optimal parameters will be embedded into thin-walled tube to improve energy absoprtion performance, whereas the cross-section of tube should be determined firstly. An automotive front side rail is a critical component of load bearing and safety protection. This paper takes front side rail as an example. Based on the decoupling thought, filler re-entrant triangles honeycomb is utilized to satisfy energy absorption performance, and outer thin wall should meet high structural stiffness. In order to obtain the cross-section shape profile, static topology optimization is conducted with targets for maximizing axial stiffness and lateral stiffness. On the basis of static topology optimization, dynamic topology optimization of impact energy absorption of thin-walled tube is also conducted to compare the RTT with the NTT design.

### 5.1 Theories of Static Topology Optimization

### 5.2 Static Topology Optimization

### 5.3 Dynamic Topology Optimization of Thin-Walled Tube Without Re-entrant Triangles

## 6 Analysis Results and Discussion

To validate the superiority of proposed optimization design method, the RTT designed based on the decoupling concept, NTT designed using traditional optimization method and PST are analyzed and compared in light of structural stiffness and energy absorption performance. To verify outstanding energy absorption capabilities of the proposed re-entrant triangles structure, RTT is compared with PFT herein. The most widely used filler with positive Poisson’s ratio and most excellent performance is aluminum foam [29]. Here, structural parameters of PFT are the same with that of RTT, and the density of filled foam is similar to that of re-entrant triangles. During simulation analysis, the foam material model is established based on the isotropic uniform material model proposed by Deshpande and Fleck [30, 31].

### 6.1 Stiffness Analysis

Stiffness analysis results of four thin-walled tubes

PST (mm) | NTT (mm) | PFT (mm) | RTT (mm) | |
---|---|---|---|---|

Axial stiffness (maximum displacement) | 14.60 | 16.21 | 13.37 | 12.95 |

Lateral stiffness (maximum displacement) | 13.99 | 15.72 | 13.30 | 13.55 |

### 6.2 Energy Absorption Capabilities

Comparisons of peak crush force for four tubes

PST | NTT | PFT | RTT | |
---|---|---|---|---|

Peak crush force (KN) | 212.10 | 176.32 | 156.06 | 148.19 |

It can be seen from above results that although traditional optimization design method can enhance the capabilities of energy absorption and reduce peak crush force effectively, structural stiffness is weakened, while the proposed decoupling method can improve both energy absorption and structural stiffness simultaneously. In comparison with positive Poisson’s ratio material, re-entrant triangles designed herein exhibit outstanding advantages in energy absorption. In the aspect of stiffness, RTT and PFT show little difference. The reason why lateral stiffness of RTT is slightly lower is that re-entrant triangles are possessed with anisotropic characteristics; stiffness along \(Y\) direction is higher than that along \(X\) direction. When the strain exceeds 0.57 and plateau stress increases, energy absorption of the RTT has been significantly enhanced. In comparison with the PFT, energy absorption of RTT increases by 17.23% and peak crush force reduces by 5.04%, which is mainly because the re-entrant triangles structure is more regular and impact deformation process is more stable relative to that of aluminum foam.

## 7 Conclusions

This paper takes front side rail to introduce decoupling thought. Filler re-entrant triangles are utilized to satisfy the requirements of energy absorption, and outer thin wall should meet high structural stiffness. This work initially exhibits theoretical models of re-entrant triangles in terms of relative density, equivalent elastic modulus and energy absorption. Then, experimental test is executed to validate the accuracy and reasonability of mechanical models. On this basis, the optimal geometric parameters of unit cell are achieved by utilizing NSGA-II method in terms of maximum unit volume energy absorption and minimum relative density.

The results show that the novel RTT designed based on decoupling thought can improve stiffness and energy absorption performance simultaneously. Compared with the positive Poisson’s ratio material, re-entrant triangles show superior advantages in energy absorption; this is mainly due to the plateau stress enhancement region. In comparison with PFT, energy absorption of RTT increases by 17.23% and peak crush force reduces by 5.04%. The results are encouraging in the sense that it offers another potential material for engineers to address the issue and the novel-type RTT with re-entrant triangles can be widely promoted and achieve further applications on new energy absorbent structures.

## Notes

### Acknowledgements

The authors are highly appreciated for the financial support from the National Nature Science Foundation of China (No. 2016YFB0101601), Jilin Province Scientific Research Program (No. SXGJQY2017-7). In addition, we also express our great gratitude to ZD Ma’s team support in University of Michigan, Ann Arbor, USA.

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