| 基于非局部应变梯度理论功能梯度纳米板的弯曲和屈曲研究 |
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| 引用本文： | 王平远,李成,姚林泉.基于非局部应变梯度理论功能梯度纳米板的弯曲和屈曲研究[J].应用数学和力学,2021,42(1):15-26. |
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| 作者姓名： | 王平远 李成 姚林泉 |
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| 作者单位： | 1苏州大学 轨道交通学院 车辆工程系, 江苏 苏州 215131;2暨南大学 重大工程灾害与控制教育部重点实验室, 广州 510632 |
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| 基金项目： | 国家自然科学基金（11972240；11572210） |
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| 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.
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| 关 键 词： | 非局部应变梯度理论 功能梯度纳米板 弯曲 屈曲 临界载荷 |
| 收稿时间： | 2020-06-22 |
Bending and Buckling of Functionally Graded Nanoplates Based on the Nonlocal Strain Gradient Theory |
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| Affiliation: | 1Department of Vehicle Engineering, School of Rail Transportation, Soochow University, Suzhou, Jiangsu 215131, P.R.China;2MOE Key Lab of Disaster Forecast and Control in Engineering, Jinan University, Guangzhou 510632, P.R.China |
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| 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 higher order 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 higher order 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. |
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