# Crasworthiness Analysis and Multi-Objective Optimization for Concave I-Shaped Honeycomb Structure
## 1 Introduction  
Materials with positive Poisson's ratio expand laterally when subjected to axial pressure and contract laterally when subjected to axial tension. In contrast, materials with negative Poisson's ratio compress laterally under axial pressure. These materials were first studied in 1944 by the scientist [PERSON] [1], who observed a Poisson's ratio of \(-0.14\) while studying the mechanical deformation behavior of a natural pyrite cubic single crystal structure and found that it had a significant internal shrinkage effect. Later on, dozens of natural negative Poisson's ratio crystal structures were discovered one after another, and negative Poisson's ratio materials were widely studied and applied. [PERSON] [2] referred to these substances and structures as auxetics. Because of this result, materials with negative Poisson's ratio are also called pull-up materials. The equivalent modulus of elasticity was researched by [PERSON] [3], beginning with material deformation. [PERSON] [4] modified the size and angle of the typical re-entrant hexagon structure to make Poisson's ratio \(-1\). [PERSON] [5] investigated the impact of transverse cracking on the elastic modulus and Poisson's ratio using both analytical approximation results and finite element technique solutions. [PERSON] [6] optimized the structure of materials with a negative Poisson's ratio to improve their energy absorption abilities. When [PERSON] et al. [7] added irregularity to the negative Poisson's ratio honeycomb structure, they discovered that the attributes of the negative Poisson's ratio under high-speed impact were accelerated. To confirm that the theoretical derivation of the negative Poisson's ratio equations matched the numerical results, [PERSON] et al. [8] investigated the dynamic response of a novel type of negative Poisson's ratio honeycomb structure with multiple internal concave angles under in-plane impacts. By merging the circular element with the folded plates, [PERSON] et al. [9]created a cross-circular honeycomb model. The deformation mode of the cross honeycomb has a zero Poisson's ratio and a strong energy absorption capability. In recent years, the research aimed at developing a material model which is equivalent to the mechanical response of a lattice structure has provided a good reference for how to calculate the equivalent properties. In [10], the mechanical behavior of a body-centered cubic (BBC) configuration under compression and within the elastic limit is considered. The finite element analysis approach and theoretical calculations are used on a single unit cell BBC for several cases to predict equivalent solid properties. The results are then used to develop a neural network model so that the equivalent solid properties of a BCC lattice of any configuration can be predicted. In [11; 12], for the aim of capturing the behavior of the entire heterogeneous cellular structure based on the analyses of the InsideBCC unit cell, only the periodic boundaries would suffice. Regarding both the elastic modulus and Poisson's ratio, the lattice unit cell faces are free to move, including all translational and rotational degrees of freedom for rigid plates. These articles provide an idea on how to calculate the equivalent properties and how to investigate the size of the specimens. Cellular materials and constructions with negative Poisson's ratio fall under the category of porous materials and have qualities such as low weight, high energy absorption, high sound insulation, and high damping. Scholars have thoroughly analyzed the deformation process of cellular structures, and the present research on cellular structures has increasingly moved from two-dimensional to three-dimensional [13]. [PERSON] et al. [14] tested the mechanical properties of several cellular architectures and ran dynamic simulations using additive manufacturing technology.