增强六手臂缺失支柱手性拉胀超材料力学性能理论研究

MECHANICAL MODELLING OF ENHANCED HEXA-MISSING RIB CHIRAL AUXETIC META-MATERIALS

前期研究工作中,基于有限元分析,作者发展了一种在大变形范围内具有可调恒定负泊松比的新型增强六手臂缺失支柱手性拉胀超材料.为了揭示微观结构-力学性能关系,并进一步指导超材料目标参数设计,本文在小变形框架下基于能量法建立了表征该拉胀材料等效泊松比和弹性模量的理论模型.增强六手臂缺失支柱手性拉胀材料由“Z”型手臂元件组成.“Z”型手臂可以被假设为两端简支的欧拉-伯努利梁.因此,本文首先推导了两端受集中力和力偶的任意形状欧拉-伯努利梁的应变能.然后,考虑平衡条件和变形协调条件进一步给出了材料等效泊松比和弹性模量的理论表达式.研究表明只有“Z”型梁的内外手臂比为2:1时,理论表达式才有简洁的形式.为了更好地利用所推导的理论表达,基于理论推导,本文开发了MATLAT图形用户界面(GUI).在GUI中输入可描述该超材料几何形状的独立几何参数,即可直接获取其等效泊松比和弹性模量.最后,基于理论结果,系统讨论了超材料微结构几何参数对其等效力学性能的影响,并将理论解与有限元计算结果进行了对比.结果表明,可以通过调控微结构几何参数获取大范围的目标力学性能.

In a previous work, the authors proposed novel enhanced hexa-missing rib chiral auxetic meta-materials(with straight ligaments and wavy ligaments), exhibiting tunable negative Poisson’s ratio and elastic modulus. The previous work, however, was limited in the finite element(FE) analysis. To facilitate the understanding of the underlying microstructure-property relationship and further provide guidelines of meta-material designs to yield target mechanical parameters, a mechanics model under infinitesimal deformation framework was developed by a simple energy-based approach. The considered chiral auxetic honeycombs consist of a set of zigzag ligaments, which can be assumed as simple supported Euler-Bernoulli beams. Therefore, the strain energy of an arbitrary shaped Euler-Bernoulli beam with concentrated forces and moments subjected to the end is derived firstly. Then, theoretical formulations of the effective Poisson’s ratio and elastic modulus for the enhanced hexa-missing rib chiral auxetics are further formulated with considering the equilibrium condition and displacement consistent condition. It is found that the theoretical formulations have a succinct form only if the length ratio between the outer and inner part of the zigzag ligaments is 2:1. To facilitate the application of the theoretical formulations, A graphical user interface(GUI) is developed based on MATLAB so that the effective Poisson’s ratio and elastic modulus of a specific design can be obtained directly by simply inputting the corresponding independent geometric parameters. The obtained analytic solutions, as compared with systematic FE calculations(conducted on one unit-cell with considering periodic boundary condition), elucidated different roles of the microstructure geometry on the effective mechanical parameters of the considered auxetic honeycombs. Results show that a wide range of targeted mechanical parameters can be obtained by adjusting the geometrical structure.