非局部应变梯度—速度梯度理论及其在纳米结构振动与屈曲问题中的应用

A large amount of experimental and numerical simulation results show that the deformation behaviors of materials at micro-/nano-scale have remarkable size effects. In addition, it has been found that there is an association between the dynamic behaviors of such materials and higher- order inertia effects induced by its motion. Due to neglecting the influences of material constants related to structural scale parameters in the constitutive relations, the classical continuum theory fails to yield consistent results with actual situations when it is applied to analyze the above mentioned problems. Against this background, this project will established a nonlocal strain gradient combined velocity gradient theory in which the strain gradient and velocity gradient are introduced into the deformation energy and kinetic energy of materials respectively and the nonlocality of the stress, momentum, higher-order stress and higher-order momentum fields is taken into account. The mathematical forms of thermodynamics equations, linear and angular momentum equations, virtual work equation followed by conventional linear elastic materials will be reformulated in forms suitable to cope with size effects combined higher-order inertia effects of micro-/nano-scale materials, from which the constitutive relations, governing equations and boundary conditions can be obtained. Afterwards, the uniqueness of the solutions of the boundary value problems will be proved. Based on the newly proposed theory, size-dependent continuum models of nano-beams, nano-plates and cylindrical nano-shells will be set up, and then they will be applied for analyzing the vibration and bucking characteristics of carbon nanotubes and graphenes. The research results of this project are hoped to provide theoretical reference for the design and application of nanodevices based on carbon nanotubes and graphenes.

众多实验以及数值模拟结果表明，微纳米尺度下材料的变形行为具有尺度效应。另外，研究者发现某些材料的动力学行为与运动所引起的高阶惯性效应存在关联性。采用经典连续介质力学分析以上问题时，由于忽略材料微观结构的影响，所得结果将与物理事实不符。本课题以尺度效应和高阶惯性效应为背景，在材料变形和运动中分别引入应变梯度和速度梯度的作用，假定应力、动量、高阶应力、高阶动量均具有非局部性，拟构建反映两类效应共同作用的非局部应变梯度-速度梯度理论模型。 对传统线弹性材料所遵循的热力学方程、动量方程、角动量方程以及虚功方程的数学形式进行重构，建立相应的本构方程、平衡方程以及边界条件，证明边值问题解的唯一性。基于所发展的理论，建立纳米梁、板和圆柱壳等的连续统模型，运用它们分析碳纳米管和石墨烯的振动与屈曲特性。研究成果有望为以碳纳米管和石墨烯为基础的纳米器件的设计和应用提供理论参考依据。

当结构或材料的几何特征尺寸减小至微纳米尺度时，由于材料在空间分布上的不连续性致使约束面内变形的形式不同, 因而微构件的力学行为将呈现出显著的尺度依赖性。事实上, 当材料或结构的特征长度与其内禀长度相当时，材料变形的尺度效应将不可忽略。另外，研究者发现某些微构件的动力学行为与运动所引起的高阶惯性效应存在关联性，因而构件振动频率会发现违反常规的增大现象。通过引入反映材料或结构特征尺寸的参数来表征尺度效应以更好地描述其力学特性，建立材料或结构的尺度效应的连续介质力学理论具有重要的理论和现实意义。在该项目资助下，课题围绕微尺度梁板结构的振动及屈曲问题进行理论建模和数值方法相关的研究，取得的研究成果如下: (1) 本文基于非局部应变梯度理论，利用多尺度建模思想，采用变分方法建立了轴向运动纳米梁的非线性动力学控制方程和相应的边界条件。通过引入两个尺度参数，非局部参数和材料特征长度参数，考察了系统多种振动行为的尺度效应问题，其中包括自由振动、自激振动、参激振动、强迫振动、内共振和联合振动; (2) 建立了高阶梯度理论下分析微尺度Euler-Bernoulli梁、Timoshenko梁、Reddy梁、kirchhoff 板以及Mindlin 板结构的振动与屈曲的微分求积有限元方法; (3) 非局部-应变梯度-速度梯度本构模型构造失败。以上成果为微构件工程实际中的应用提供若干有价值的参考结果。