微结构变形梯度理论及其在弹性板中的应用

本文将具有非局部效应的材料考虑为由一定体积的微元体构成的连续介质模型，构建了微结构系统并研究了其变形特性及相应的能量。从变形的角度定义微元体的微应变和微转动并与宏观变形联系起来，应用最小势能原理导出了具有两个长度参数的微结构变形梯度（Micro-Structural Deformation Gradient Theory，MSDG）理论，具有明确的变形物理意义。在 MSDG 理论中定义了广义应变能，从能量的角度给出了材料模量的新解释；引入了非局部效应参数定量地描述材料的非局部特性。在一定条件下，MSDG 理论可以退化为偶应力理论、应变梯度理论和经典连续介质理论。非局部变形可分为两种类型：非均匀材料的宏观变形和具有微结构材料的微观变形。对于颗粒增强复合材料的单轴拉伸，MSDG 理论成功预测和验证了微米尺度上弹性模量与颗粒尺寸的近似线性关系。此外，基于 MSDG 理论的微扭转解析解与实验结果吻合较好，并预测了更小直径圆柱体的抗扭刚度。

基于 MSDG 理论构建了具有两个长度参数的修正梯度弹性 Kirchhoff-Love 板模型。应用最小势能原理，结合 Cauchy 应力、偶应力和超应力，得到了适用于任意形状的六阶基本微分方程和对应的边界条件。给出了具有简支边界、固支边界和自由边界以及不同加载条件下梯度弹性 Kirchhoff-Love 板的弯曲和屈曲变形的解析解及数值分布。定义尺寸相关抗弯刚度（Size-Dependent Bending Stiffness，SDBS），直观地描述了弹性薄板弯曲变形的尺寸效应。另外，研究了梯度弹性 Kirchhoff-Love 圆形薄板的轴对称弯曲和屈曲问题，通过 Bessel 函数给出了轴对称弯曲的解析解，并构造高阶类 Bessel 函数给出了轴对称屈曲临界荷载的解析解。

结合 MSDG 理论中广义应变能和 Lyapunov 稳定性第二法研究了微结构弹性系统的动力稳定性。给出了基于微结构弹性系统的 Lyapunov 动力稳定性能量项（哈密顿量），除了经典弹性形变势能外，还包含了高阶梯度变形能以及与转动相关的变形能。非局部能量项中与材料特征长度相关的长度参数的引入使得本文给出的微结构弹性系统的动力稳定性研究可以用于描述稳定性问题的尺寸效应。文中分别对一维应变梯度弹性梁和二维变形梯度弹性薄板进行了动力稳定性的研究，得到了两种弹性体模型的尺寸相关临界荷载，并从动力学的角度给出了弹性系统在整个时域上的稳定性表述。

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