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增强六手臂缺失支柱手性拉胀超材料力学性能理论研究

朱一林 江松辉 于超

朱一林, 江松辉, 于超. 增强六手臂缺失支柱手性拉胀超材料力学性能理论研究. 力学学报, 待出版 doi: 10.6052/0459-1879-22-188
引用本文: 朱一林, 江松辉, 于超. 增强六手臂缺失支柱手性拉胀超材料力学性能理论研究. 力学学报, 待出版 doi: 10.6052/0459-1879-22-188
Zhu Yilin, Jiang Songhui, Yu Chao. Mechanical modelling of enhanced hexa-missing rib chiral auxetic meta-materials. Chinese Journal of Theoretical and Applied Mechanics, in press doi: 10.6052/0459-1879-22-188
Citation: Zhu Yilin, Jiang Songhui, Yu Chao. Mechanical modelling of enhanced hexa-missing rib chiral auxetic meta-materials. Chinese Journal of Theoretical and Applied Mechanics, in press doi: 10.6052/0459-1879-22-188

增强六手臂缺失支柱手性拉胀超材料力学性能理论研究

doi: 10.6052/0459-1879-22-188
基金项目: 国家自然科学基金( 11702036), 四川省科技厅国际合作项目(2020YFH0127), 国家留学基金委公派访问学者项目(202008515100)和成都市科技局国际合作项目基金(2019-GH02-00060-HZ)
详细信息
    作者简介:

    朱一林, 副研究员, 主要研究方向: 力学超材料设计, 性能表征及应用; 材料多场耦合循环本构理论. E-mail: lin5210feng@163.com

  • 中图分类号: O34

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

  • 摘要: 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. 为了更好的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, 本文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.

     

  • 图  1  传统手性拉胀材料: (a) 三手臂; (b) 反三手臂; (c) 四手臂; (d) 反四手臂; (e) 六手臂

    Figure  1.  (a) Tri-; (b) anti-tri-; (c) tetra-; (d) anti-tetra-; (e) hexa-chiral honeycombs:

    图  2  缺失支柱手性拉胀材料: (a) 三手臂; (b) 反三手臂; (c) 四手臂; (d) 反四手臂; (e) 六手臂

    Figure  2.  (a) Tri-; (b) anti-tri-; (c) tetra-; (d) anti-tetra-; (e) hexa-missing rib honeycombs

    图  3  增强六手臂缺失支柱手性拉胀材料元胞图: (a) 直臂; (b)曲臂

    Figure  3.  Unit-cells of enhanced hexa-missing rib honeycomb: (a) with straight ligaments (b) with wavy ligaments

    图  4  (a) 阵列结构受远程应力示意图; (b) 最小代表单元受力图;(c) 等效最小代表单元

    Figure  4.  (a) Schematic diagram of an enhanced hexa-missing rib lattice under far field uniaxial loading; (b) the related free boundary diagram of the smallest repeated unit; (c) the effective smallest repeated unit

    图  5  任意欧拉−伯努利梁结构受力图

    Figure  5.  Force diagram of an Euler–Bernoulli beam with arbitrary Shape

    图  6  最小代表单元各“Z”型梁受力图

    Figure  6.  Force diagram of the zigzag ligaments of the smallest repeated unit

    图  7  元胞中心节点区受力图

    Figure  7.  Force diagram of an area surrounding the central point of the unit-cell

    图  8  最小代表单元变形示意图

    Figure  8.  Schematic diagram of the deformation of smallest repeated unit

    图  9  (a) 直臂与 (b) 曲臂结构计算模型

    Figure  9.  Calculation diagram of the enhanced hexa-missing rib auxetic honeycombs with (a) straight ligament and (b) wavy ligament

    图  10  图形用户界面: (a) 直臂结构计算; (b) 曲臂结构计算

    Figure  10.  Graphical User Interface: (a) calculation of auxetics with straight ligaments; (b) calculation of auxetics with wavy ligaments

    图  11  结构等效泊松比云图: (a) 不同内角$ \varphi $和增强率p (q =1);(b) 不同手臂比q和增强率p ($ \varphi = {90^ \circ } $)

    Figure  11.  Contour plots the effective Poisson’s ratio over a wide range of (a) interior angle $ \varphi $ and reinforcement ratio p (q = 1) and (b) ligament ratio q and reinforcement ratio p ($ \varphi = {90^ \circ } $).

    图  12  结构等效弹性模量云图: (a) 不同内角$ \varphi $和增强率p (q = 1);(b) 不同手臂比q和增强率p ($ \varphi = {90^ \circ } $)

    Figure  12.  Contour plots the effective elastic modulus over a wide range of (a) interior angle $ \varphi $ and reinforcement ratio p (q = 1) and (b) ligament ratio q and reinforcement ratio p ($ \varphi = {90^ \circ } $)

    图  13  等效应变为5%时最小代表单元最大面内主应变云图:(a) p = 0.4, q = 1, $ \varphi = {90^ \circ } $; (b) p = 0.8, q = 1, $ \varphi = {90^ \circ } $

    Figure  13.  Contour plots the maximum in-plane principal strain of the smallest repeated units at an effective strain of 5%: (a) p = 0.4, q = 1, $ \varphi = {90^ \circ } $; (b) p = 0.8, q = 1, $ \varphi = {90^ \circ } $

    图  14  不同几何参数下结构等效泊松比的理论和有限元模拟结果对比图: (a) 不同内角$ \varphi ;$ (b) 不同增强率p; (b) 不同手臂比q

    Figure  14.  Theoretical and FE results of the effective Poisson’s ratio of the enhanced hexa-missing rib auxetic honeycomb over a range of (a) interior angle $ \varphi $, (b) reinforced ratio p and (c) ligament ratio q

    图  15  不同几何参数下结构等效弹性模量的理论和有限元模拟结果对比图: (a) 不同内角$ \varphi ;$ (b) 不同增强率p; (b) 不同手臂比q

    Figure  15.  Theoretical and FE results of the effective elastic modulus of the enhanced hexa-missing rib auxetic honeycomb over a range of (a) interior angle $ \varphi $, (b) reinforced ratio p and (c) ligament ratio q

    B1  对一个单胞施加周期性边界条件以及对阵列结构(包含n × n 单胞)直接建模所得泊松比对比图 (p = 0.6, q = 1, $ \varphi = {90^ \circ } $)

    B1.  The effective Poisson’s ratio obtained by the PBCs-based homogenization method and calculated from lattice structures consisting of n × n unit-cell array (p = 0.6, q = 1, $ \varphi = {90^ \circ } $).

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