Shock and Vibration

Volume 2016 (2016), Article ID 3016014, 9 pages

http://dx.doi.org/10.1155/2016/3016014

## A Collision Model for Protective Structure with Gradient Metallic Cellular Material under Low Velocity Impact

State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China

Received 19 May 2016; Revised 27 July 2016; Accepted 15 August 2016

Academic Editor: Mickaël Lallart

Copyright © 2016 Dengbao Xiao and Guiping Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

During quasi-static compressive process, the gradient metallic cellular material can be divided into deformed region and undeformed region before the material is compressed compactly. The deformation mode of the gradient metallic cellular material is not the same as that of homogeneous metallic cellular material. Meanwhile, no plateau stress is observed in the crushing stress versus strain curve of the gradient metallic cellular material. A formula of calculating crushing stress of the gradient metallic cellular material is presented. Subsequently, a collision model for protective structures with gradient metallic cellular material is presented to investigate the vibration isolation of the large mass protection under low velocity impact based on Lagrange’s equation of the second kind. The effects of the gradient metallic cellular material on the acceleration peak of inner protected structure are theoretically discussed. And the results show that adopting gradient metallic cellular material instead of homogeneous metallic cellular material with equivalent mass can decrease the peak acceleration value of the inner protected structure.

#### 1. Introduction

Because of the outstanding properties, such as the low weight and high efficient energy absorption [1, 2], the metallic cellular materials have been extensively applied in the civilian and aerospace fields (e.g., reentry capsule and airdrop platform) for impact mitigation applications. Among the metallic cellular materials, the gradient metallic cellular material has attracted many research interests [3, 4]. For the gradient metallic cellular material, its pore structures were graded through the thickness direction of the material resulting in varying mechanical properties. Meanwhile, the variation of the mechanical property may considerably influence its impact mitigation performances. Thus, the influence of the gradient metallic cellular material on the dynamic response of the protected structures is interesting [5–7].

The compressive properties of the gradient metallic cellular material are important to its impact resistance performances under impact loading. Ali et al. [8] investigated the crushing stress of the gradient hexagonal structure subjected to the low velocity impact. Their results showed that the deformation mode of the gradient hexagonal structure was “I” mode [9]. In “I” mode, compressing deformation began from the part of the gradient cellular material with the minimum plateau stress. And the deforming process carried on until the part of the gradient cellular material with the maximum plateau stress is crushed. Ajdari et al. [10] constructed the finite element (FE) model of the gradient Voronoi cellular structure and investigated its uniaxial deformation behavior under the quasi-static compression. The results also suggested the deformation mode was “I” mode. Hangai et al. [11] fabricated the gradient aluminum foam and stated that it had exhibited multiple plateau stresses. He et al. [12] investigated the effect of gradient structures on mechanical properties of the gradient aluminum foam under quasi-static compression. It was reported that the crushing stress increased with remarkable positive slope in the experimental nominal stress-strain curve of the gradient aluminum foam. Xiao et al. [13, 14] presented a simplified formula to calculate the quasi-static crushing stress of the gradient metallic cellular material. The RPPL (rigid-perfectly plastic-locking) model [15] was employed by Xiao et al. to model the metallic cellular material. In views of neglecting the strain hardening in RPPL model, the deformed cellular materials have been densified and their strains have reached densification strain during the compression.

In most of previous researches, the metallic cellular materials were used to absorb impact energy to prevent the inner protected structures/components from failure or destruction during collision process [16–19]. When the collision energy was exerted on the outer structure, the dynamic response (vibration isolation) of the inner protected structures/components (seat-occupant system) is an important issue. Joshi et al. [20] used a single degree of freedom mass-spring-damper model to discuss the vibration isolation of the seat-occupant. The dynamic response of the seated occupant was discussed. Toward and Griffin [21] developed a similar mass-spring-damper model to study the effects of sitting posture and vibration magnitude on the vertical apparent mass of the human body. Li et al. [22, 23] presented a double degree of freedom collision model to investigate the dynamic responses of the inner objects and the outer structure protected by metallic foam. The uniaxial compressive stress-strain curve of homogeneous foam was experimentally obtained by Li et al. [24]. In their studies, the effects of mass/damping ratio and initial impact velocity on peak acceleration of the protected structure were discussed. So far, the energy absorber in protective structure is mostly limited to homogeneous cellular material. Adopting the gradient metallic cellular material for energy absorber instead of homogeneous metallic cellular material in the protective structures, the vibration isolation of the inner protected structure and the outer structure under low velocity impact is an interesting research issue.

The outline of the present paper is as follows. Firstly, a formula considering hardening strain for cellular material is presented to predict the crushing stress of the gradient metallic cellular material. Secondly, based on Lagrange’s equation of the second kind, a collision model for protective structure with gradient metallic cellular material is developed for large mass protection structure under low velocity impact. Finally, the influences of gradient metallic cellular material on the dynamic response of the inner protected structure are investigated.

#### 2. The Collision Model for Protective Structure with Gradient Metallic Cellular Material under Low Velocity Impacting

When a reentry space capsule lands on the ground, the cushion material and the seat are the important protection system for the occupant. The process of the reentry capsule colliding with hard ground can be simplified as a low velocity impact problem with a large mass structure protected by the gradient metallic cellular material shown in Figure 1(a). The reentry capsule is simplified as a rigid body having mass ; the occupant and seat together are treated as a rigid structure having mass . Meanwhile, the seat-occupant structure is modeled as a mass-spring-damper system. Let the spring and damper be characterized by and , respectively. Therefore, the reentry capsule colliding problem can be modeled as a collision model for protective structure with gradient metallic cellular material as shown in Figure 1(b).