Bending and Buckling of Functionally Graded Nanoplates Based on the Nonlocal Strain Gradient Theory
-
摘要: 以纳米机器人等智能器件中的功能梯度纳米板结构为研究对象,基于非局部应变梯度理论,研究了其弯曲和屈曲问题.推导了一般情况下的功能梯度纳米板运动方程,弯曲和屈曲作为其特例可简化而成.分析了非局部尺度参数、材料特征尺度参数、梯度指数、纳米板尺寸等对弯曲挠度和临界屈曲载荷的影响.结果表明:不同高阶连续介质力学理论下的最大挠度都随梯度指数的增大而增大,正方形纳米板挠度较小,且板厚越大,弯曲挠度越小;最大挠度随非局部尺度参数的增大而增大,随材料特征尺度参数的增大而减小.临界屈曲载荷随梯度指数的增大而减小,随板厚、长宽比的增大而增大,随非局部尺度的增大而减小,随材料特征尺度的增大而增大.非局部应变梯度高阶弯曲和屈曲中存在结构软化与硬化机制,两个内特征参数之间具有耦合效应,当非局部尺度大于材料特征尺度时,非局部效应在功能梯度纳米板力学性能中占主导作用;当材料特征尺度大于非局部尺度时,应变梯度效应占主导作用.解析结果还证明了当非局部尺度等于材料特征尺度时,非局部应变梯度理论结果退化为经典结果.Abstract: The bending and buckling of functionally graded nanoplates in intelligent devices (e.g., nanorobots) were studied based on the nonlocal strain gradient theory. The motion equations in general cases were derived, and then reduced to bending and buckling in special cases. The effects of the nonlocal scale parameter, the material characteristic scale parameter, the gradient index and the geometric size on the bending deflection and the critical buckling load were acquired and analyzed in detail. The results show that, the maximum bending deflections under different higherorder continuum mechanics theories increase with the gradient index. The deflection goes lower for the square nanoplate. The thicker the nanoplate is, the smaller the bending deflection will be. The maximum deflection increases with the nonlocal scale parameter but decreases with the material characteristic scale parameter. The critical buckling load decreases with the gradient index, and increases with the thickness and the aspect ratio. When the nonlocal scale parameter increases, the critical buckling load will decrease, but will increase with the material characteristic scale parameter. The softening and hardening mechanisms exist in higherorder bending and buckling of the functionally graded nanoplates, and the coupling effect between 2 internal characteristic parameters also occurs. When the nonlocal scale is greater than the material characteristic scale, the nonlocal effect will dominate in the mechanical properties of functionally graded nanoplates, otherwise the strain gradient effect will play a leading role. The analytical solutions also show that, when the nonlocal scale is equal to the material characteristic scale, the results based on the nonlocal strain gradient theory will degenerate into the corresponding classical ones.
-
Key words:
- nonlocal strain gradient theory /
- functionally graded nanoplate /
- bending /
- buckling /
- critical load
-
[1] KAFADAR C B, ERINGEN A C. Micropolar media Ⅰ: the classical theory[J]. International Journal of Engineering Science,1971,9(3): 271-305. [2] TOUPIN R. Elastic materials with couple-stresses[J]. Archive for Rational Mechanics and Analysis,1962,11(1): 385-414. [3] MINDLIN R D, ESHEL N N. On first strain-gradient theories in linear elasticity[J]. International Journal of Solids and Structures,1968,4(1): 109-124. [4] ERINGEN A C, EDELEN D G B. On nonlocal elasticity[J]. International Journal of Engineering Science,1972,10(3): 233-248. [5] 徐晓建, 邓子辰. 非局部因子和表面效应对微纳米材料振动特性的影响[J]. 应用数学和力学, 2013,34(1): 10-17. (XU Xiaojian, DENG Zichen. Surface effects of adsorption-induced resonance analysis of micro/nanobeams via nonlocal elasticity[J]. Applied Mathematics and Mechanics,2013,34(1): 10-17. (in Chinese)) [6] LIM C W, ZHANG G, REDDY J N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation[J]. Journal of the Mechanics and Physics of Solids,2015,78: 298-313. [7] LI L, LI X, HU Y. Free vibration analysis of nonlocal strain gradient beams made of functionally graded material[J]. International Journal of Engineering Science,2016,102: 77-92. [8] EBRAHIMI F, BARATI M R. A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams[J]. Composite Structures,2017,159: 174-182. [9] XU X J, ZHENG M L, WANG X C. On vibrations of nonlocal rods: boundary conditions, exact solutions and their asymptotics[J]. International Journal of Engineering Science,2017,119: 217-231. [10] SAHMANI S, AGHDAM M M. Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells[J]. Composites Part B: Engineering,2018,132: 258-274. [11] WANG J, SHEN H M, ZHANG B, et al. Complex modal analysis of transverse free vibrations for axially moving nanobeams based on the nonlocal strain gradient theory[J]. Physica E: Low-Dimensional Systems and Nanostructures,2018,101: 85-93. [12] SAHMANI S, AGHDAM M M, RABCZUK T. Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory[J]. Composite Structures,2018,186: 68-78. [13] LU L, ZHU L, GUO X M, et al. A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells[J]. Applied Mathematics and Mechanics(English Edition),2019,40(12): 1695-1722. [14] MIR M, TAHANI M. Graphene-based mass sensors: chaotic dynamics analysis using the nonlocal strain gradient model[J]. Applied Mathematical Modelling,2020,81: 799-817. [15] BARRETTA R, FAGHIDIAN S A, DE SCIARRA F M, et al. On torsion of nonlocal Lam strain gradient FG elastic beams[J]. Composite Structures,2020,233: 111550. [16] LI J X, AVILA B E F, GAO W, et al. Micro/nanorobots for biomedicine: delivery, surgery, sensing, and detoxification[J]. Science Robotics,2017,2(4): eaam6431. DOI: 10.1126/scirobotics.aam6431. [17] 石振海, 李克智, 李贺军, 等. 航天器热防护材料研究现状与发展趋势[J]. 材料导报, 2017,21(8): 15-18. (SHI Zhenhai, LI Kezhi, LI Hejun, et al. Research status and application advance of heat resistant materials for space vehicles[J]. Materials Reports,2007,21(8): 15-18. (in Chinese)) [18] LI X B, LI L, HU Y J, et al. Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory[J]. Composite Structures,2017,165: 250-265. [19] MAHINZARE M, ALIPOUR M J, SADATSAKKAK S A, et al. A nonlocal strain gradient theory for dynamic modeling of a rotary thermo piezo electrically actuated nano FG circular plate[J]. Mechanical Systems and Signal Processing,2019,115: 323-337. [20] KARAMI B, JANGHORBAN M. A new size-dependent shear deformation theory for free vibration analysis of functionally graded/anisotropic nanobeams[J]. Thin-Walled Structures,2019,143: 106227. [21] CHEN P J, PENG J, ZHAO Y C, et al. Prediction of the adhesive behavior of bio-inspired functionally graded materials against rough surfaces[J]. AIP Advances,2014,4(6): 067143. [22] EBRAHIMI F, SHAFIEI N. Application of Eringen’s nonlocal elasticity theory for vibration analysis of rotating functionally graded nanobeams[J]. Smart Structures and Systems,2016,17(5): 837-857. [23] DANESHMEHR A, RAJABPOOR A, POURDAVOOD M. Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions[J]. International Journal of Engineering Science,2014,82: 84-100. [24] BARATI M R, SHAHVERDI H. An analytical solution for thermal vibration of compositionally graded nanoplates with arbitrary boundary conditions based on physical neutral surface position[J]. Mechanics of Advanced Materials and Structures,2017,24(10): 840-853. [25] THAI C H, TRAN T D, PHUNG-VAN P. A size-dependent moving Kriging meshfree model for deformation and free vibration analysis of functionally graded carbon nanotube-reinforced composite nanoplates[J]. Engineering Analysis With Boundary Elements,2020,115: 52-63. [26] ZHANG D G, ZHOU Y H. A theoretical analysis of FGM thin plates based on physical neutral surface[J]. Computational Materials Science,2008,44(2): 716-720.
点击查看大图
计量
- 文章访问数: 713
- HTML全文浏览量: 131
- PDF下载量: 241
- 被引次数: 0