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辐射对称金字塔型剪纸的力学行为

肖思 秦应霖 王慧 王鹏 马海铭 何军 王迎威

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辐射对称金字塔型剪纸的力学行为

肖思, 秦应霖, 王慧, 王鹏, 马海铭, 何军, 王迎威

Mechanical behaviors of radial symmetric pyramid kirigami

Xiao Si, Qin Ying-Lin, Wang Hui, Wang Peng, Ma Hai-Ming, He Jun, Wang Ying-Wei
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  • 通过在弹性薄板上引入切口, 构建了多边形辐射对称金字塔型剪纸结构. 利用伽辽金法求解的悬臂梁形变公式和悬臂梁组合的方法, 创建了用于解释形变过程的“梁模型”, 得到n个模块的正N边形金字塔结构的弹性系数与结构参数的关系公式, 并求出弹性系数线性阈值的表达式, 解释了该结构产生平面外扭曲的原因. 利用推导的“梁模型”公式, 并通过有限元仿真和实验的方法, 系统研究了辐射对称金字塔型剪纸结构的力学响应特征, 验证模型的准确性, 并用于已有报道的石墨烯剪纸结构的力学特征分析. 这项工作系统解释了竖直拉伸的金字塔型剪纸结构的力学响应.
    Kirigami, the art of cutting paper, recently emerged as a powerful tool to substantially modify, reconfigure and program the properties of material. The development of kirigami technology provides an effective solution for designing the inorganic flexible electronic devices. Pyramid kirigami, as a kind of kirigami structure, shows a large vertical extension characteristic. It has been widely used to demonstrate versatile applications, such as graphene kirigami spiral spring, three-dimensional stretchable supercapacitor, and wearable flexible sensors. In the present work, we construct a polygonal radial symmetric pyramid kirigami by introducing some cuts in the elastic sheet. The mechanical behavior of pyramid kirigami is investigated based on the cantilever formula solved by Galerkin method. In addition, a “beam model” is proposed to explain deformation process of pyramid kirigami, which consists of several “beam elements” containing two cantilever beams. The formula for the relationship between the elastic coefficient K and the structural parameters of the regular N-sided pyramid kirigami of n modules is obtained by combining several cantilever beams. The formula for the linear threshold of deformation DT is obtained based on the comparison between the approximate curve of small deflection and the theoretical curve of a cantilever beam. When the deformation of the structure exceeds the linear threshold, the structure cannot keep the elastic coefficient K value linear any more, and the mechanical behaviors become non-linear. The simple geometric relationship of a single module is used to explain the out-of-sheet distortion of the structure. The proposed theoretical model is confirmed by finite element method simulation and experimental methods, and it is used to analyze the mechanical characteristics of graphene krigami reported. The results indicate that the defined parameters can be adjusted to tailor or manipulate the ductility and mechanical behaviors. This work provides theoretical support for the application of pyramid kirigami in the field of flexible devices. In the macroscopic field, the pyramid kirigami structure is expected to be applied to the field of flexible devices as a flexible structure with controllable elastic coefficient. In the microscopic field, it is expected to use two-dimensional materials to make force measurement devices with a simple visual readout and femtonewton force resolution.
      通信作者: 何军, junhe@csu.edu.cn ; 王迎威, wyw1988@csu.edu.cn
    • 基金项目: 国家级-基于人工表面等离激元的功能集成型辐射器研究(61875232)
      Corresponding author: He Jun, junhe@csu.edu.cn ; Wang Ying-Wei, wyw1988@csu.edu.cn
    [1]

    Blees M K, Barnard A W, Rose P A, Roberts S P, McGill K L, Huang P Y, Ruyack A R, Kevek J W, Kobrin B, Muller D A, McEuen P L 2015 Nature 524 204Google Scholar

    [2]

    Chen B G, Liu B, Evans A A, Paulose J, Cohen I, Vitelli V, Santangelo C D 2016 Phys. Rev. Lett. 116 135501Google Scholar

    [3]

    Shyu T C, Damasceno P F, Dodd P M, Lamoureux A, Xu L, Shlian M, Shtein M, Glotzer S C, Kotov N A 2015 Nat. Mater. 14 785Google Scholar

    [4]

    陈珊珊, 刘幸, 刘之光, 李家方 2019 物理学报 68 248101Google Scholar

    Chen S S, Liu X, Liu Z G, Li J F 2019 Acta Phys. Sin. 68 248101Google Scholar

    [5]

    Han T, Scarpa F, Allan N L 2017 Thin Solid Films 632 35Google Scholar

    [6]

    Rafsanjani A, Pasini D 2016 Extreme Mech. Lett. 9 291Google Scholar

    [7]

    Rafsanjani A, Bertoldi K 2017 Phys. Rev. Lett. 118 084301Google Scholar

    [8]

    Rafsanjani A, Jin L, Deng B, Bertoldi K 2019 Proc. Natl. Acad. Sci. U.S.A. 116 8200Google Scholar

    [9]

    Hanakata P Z, Qi Z, Campbell D K, Park H S 2016 Nanoscale 8 458Google Scholar

    [10]

    王沅倩, 林才纺, 张景迪, 何军, 肖思 2015 物理学报 64 034214Google Scholar

    Wang Y Q, Lin C F, Zhang J D, He J, Xiao S 2015 Acta Phys. Sin. 64 034214Google Scholar

    [11]

    Lyu J, Hammig M D, Liu L, Xu L, Chi H, Uher C, Li T, Kotov N A 2017 App. Phys. Lett. 111 161901Google Scholar

    [12]

    Zhang X J, Yuan Z H, Yang R X, He Y L, Qin Y L, Xiao S, He J 2019 J. Cent. South Univ. 26 2295Google Scholar

    [13]

    Xiao S, Wang H, Liu S, Li M, Wang Y W, Chen J Z, Guo L H, Li J B, He J 2018 Chin. Phys. Lett. 35 067801Google Scholar

    [14]

    Zhao Y, Wang C, Wu J, Sui C, Zhao S, Zhang Z, He X 2017 Phys. Chem. Chem. Phys. 19 11032Google Scholar

    [15]

    韩同伟, 李攀攀 2017 物理学报 66 066201Google Scholar

    Han T W, Li P P 2017 Acta Phys. Sin. 66 066201Google Scholar

    [16]

    He Z, Xiong J, Dai Q L, Yang B C, Zhang J, Xiao S 2020 Nanoscale 12 6767Google Scholar

    [17]

    He S, Qiu L, Wang L, Cao J, Xie S, Gao Q, Zhang Z, Zhang J, Wang B, Peng H 2016 J. Mater. Chem. A 4 14968Google Scholar

    [18]

    Yang C, Zhang H, Liu Y, Yu Z, Wei X, Hu Y 2018 Adv. Sci. 5 1801070Google Scholar

    [19]

    Tsien H S 1953 J. Am. Rocket Soc. 23 14Google Scholar

    [20]

    赵则昂, 邓宗白, 宋安平 2014 力学与实践 3 341Google Scholar

    Zhao Z A, Deng Z B, Song A P 2014 Mech. Eng. 3 341Google Scholar

    [21]

    王冬梅 2008 纸和造纸 2 82Google Scholar

    Wang D M 2008 Paper and Paper Making 2 82Google Scholar

    [22]

    韩同伟, 贺鹏飞, 骆英, 张小燕 2011 力学进展 41 279Google Scholar

    Han T W, He P F, Luo Y, Zhang X Y 2011 Adv. Mech. 41 279Google Scholar

    [23]

    Zhu Y, Wang P, Xiao S, He S, Chen J, Jiang Y, Wang Y, He J, Gao Y 2018 Nanoscale 10 21782Google Scholar

    [24]

    Lamoureux A, Lee K, Shlian M, Forrest S R, Shtein M 2015 Nat. Commun. 6 8092Google Scholar

  • 图 1  典型金字塔型剪纸结构 (a) 边数N = 4, 模块数n = 3的金字塔结构; (b) 模型在竖直拉力F作用下产生竖直形变

    Fig. 1.  Typical pyramid kirigami structure: (a) Pyramid structure with number of edges N = 4 and number of modules n = 3; (b) pyramid model produces vertical deformation under the action of vertical tension F.

    图 2  由“梁单元”构成的“梁模型” (a) 金字塔结构一个形变区域简化成的“梁模型”; (b) 悬臂梁组成的“梁单元”

    Fig. 2.  “Beam model” consisting of “beam elements”: (a) Simplified “beam model” of a deformed area of the pyramid structure; (b) “beam element” consisting of cantilever beams.

    图 3  基于(1)式和(2)式的悬臂梁理论曲线与小挠度近似曲线对比

    Fig. 3.  Theoretical curve of cantilever beam compared with the approximate theoretical curve of small deflection based on Eq. (1) and (2).

    图 4  FEM模拟和理论计算验证弹性系数与结构参数关系 (a)−(c) 弹性系数K分别与梁宽w、厚度t的三次方及边数N值呈线性变化关系; (d) 取不同模块切口长度L的增加值b, 验证K值与模块数n的关系, 点为模拟值, 虚线为计算值

    Fig. 4.  Verify the relationship between elastic coefficient and structural parameters through FEM simulation and theoretical calculation: (a)−(c) The elastic coefficient K varies linearly with the beam width w, the cube of thickness t, and the number of sides N; (d) take different values b to verify the relationship between the elastic coefficient K and the number of modules n. The points are simulation values, and the dotted lines are calculated values.

    图 5  利用实验对K, DT的计算公式(11)和(14)式进行验证 (a) 实验图; (b) 四边形实验数据, 点为测量结果, 虚线红色为线性区域拟合结果, 黑色虚线为计算出的线性阈值; (c) Nature上发表的石墨烯剪纸弹簧在激光驱动下的形变-受力结果[1]

    Fig. 5.  The K and DT formulas (11) and (14) are verified experimentally: (a) Experimental picture; (b) the experimental data of the quadrangular pyramid structure, the points are the measurement results, the red dotted line is the linear region fitting result, and the black dotted line is the calculated linear threshold; (c) laser-driven deformation of graphene kirigami springs published in Nature[1].

    图 6  模块的横向收缩 (a) 模块形变的实物图; (b) 某一模块形变的简单几何关系

    Fig. 6.  Transverse strain of a module: (a) Experimental diagram of module deformation; (b) simple geometric relationship of deformation of a single module.

    图 7  不同模块切口长度L对横向应变${\varepsilon _{\rm{T}}}$的影响

    Fig. 7.  Influence of different module cut length L on transverse strain ${\varepsilon _{\rm{T}}}$.

  • [1]

    Blees M K, Barnard A W, Rose P A, Roberts S P, McGill K L, Huang P Y, Ruyack A R, Kevek J W, Kobrin B, Muller D A, McEuen P L 2015 Nature 524 204Google Scholar

    [2]

    Chen B G, Liu B, Evans A A, Paulose J, Cohen I, Vitelli V, Santangelo C D 2016 Phys. Rev. Lett. 116 135501Google Scholar

    [3]

    Shyu T C, Damasceno P F, Dodd P M, Lamoureux A, Xu L, Shlian M, Shtein M, Glotzer S C, Kotov N A 2015 Nat. Mater. 14 785Google Scholar

    [4]

    陈珊珊, 刘幸, 刘之光, 李家方 2019 物理学报 68 248101Google Scholar

    Chen S S, Liu X, Liu Z G, Li J F 2019 Acta Phys. Sin. 68 248101Google Scholar

    [5]

    Han T, Scarpa F, Allan N L 2017 Thin Solid Films 632 35Google Scholar

    [6]

    Rafsanjani A, Pasini D 2016 Extreme Mech. Lett. 9 291Google Scholar

    [7]

    Rafsanjani A, Bertoldi K 2017 Phys. Rev. Lett. 118 084301Google Scholar

    [8]

    Rafsanjani A, Jin L, Deng B, Bertoldi K 2019 Proc. Natl. Acad. Sci. U.S.A. 116 8200Google Scholar

    [9]

    Hanakata P Z, Qi Z, Campbell D K, Park H S 2016 Nanoscale 8 458Google Scholar

    [10]

    王沅倩, 林才纺, 张景迪, 何军, 肖思 2015 物理学报 64 034214Google Scholar

    Wang Y Q, Lin C F, Zhang J D, He J, Xiao S 2015 Acta Phys. Sin. 64 034214Google Scholar

    [11]

    Lyu J, Hammig M D, Liu L, Xu L, Chi H, Uher C, Li T, Kotov N A 2017 App. Phys. Lett. 111 161901Google Scholar

    [12]

    Zhang X J, Yuan Z H, Yang R X, He Y L, Qin Y L, Xiao S, He J 2019 J. Cent. South Univ. 26 2295Google Scholar

    [13]

    Xiao S, Wang H, Liu S, Li M, Wang Y W, Chen J Z, Guo L H, Li J B, He J 2018 Chin. Phys. Lett. 35 067801Google Scholar

    [14]

    Zhao Y, Wang C, Wu J, Sui C, Zhao S, Zhang Z, He X 2017 Phys. Chem. Chem. Phys. 19 11032Google Scholar

    [15]

    韩同伟, 李攀攀 2017 物理学报 66 066201Google Scholar

    Han T W, Li P P 2017 Acta Phys. Sin. 66 066201Google Scholar

    [16]

    He Z, Xiong J, Dai Q L, Yang B C, Zhang J, Xiao S 2020 Nanoscale 12 6767Google Scholar

    [17]

    He S, Qiu L, Wang L, Cao J, Xie S, Gao Q, Zhang Z, Zhang J, Wang B, Peng H 2016 J. Mater. Chem. A 4 14968Google Scholar

    [18]

    Yang C, Zhang H, Liu Y, Yu Z, Wei X, Hu Y 2018 Adv. Sci. 5 1801070Google Scholar

    [19]

    Tsien H S 1953 J. Am. Rocket Soc. 23 14Google Scholar

    [20]

    赵则昂, 邓宗白, 宋安平 2014 力学与实践 3 341Google Scholar

    Zhao Z A, Deng Z B, Song A P 2014 Mech. Eng. 3 341Google Scholar

    [21]

    王冬梅 2008 纸和造纸 2 82Google Scholar

    Wang D M 2008 Paper and Paper Making 2 82Google Scholar

    [22]

    韩同伟, 贺鹏飞, 骆英, 张小燕 2011 力学进展 41 279Google Scholar

    Han T W, He P F, Luo Y, Zhang X Y 2011 Adv. Mech. 41 279Google Scholar

    [23]

    Zhu Y, Wang P, Xiao S, He S, Chen J, Jiang Y, Wang Y, He J, Gao Y 2018 Nanoscale 10 21782Google Scholar

    [24]

    Lamoureux A, Lee K, Shlian M, Forrest S R, Shtein M 2015 Nat. Commun. 6 8092Google Scholar

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出版历程
  • 收稿日期:  2020-01-15
  • 修回日期:  2020-02-24
  • 刊出日期:  2020-05-05

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