柔性电子和压缩屈曲引导的三维微结构组装技术是近年来的研究热点，平面曲梁作为一类基本结构，在其中发挥着重要作用。本文旨在建立平面曲梁大变形理论，深入研究相关工程应用中常见平面曲梁构型的大变形力学行为，得到加载与变形之间的定量关系，为实际应用提供指导。主要研究内容和结论包括： 根据Koiter的渐进后屈曲理论，若想得到正确的后屈曲行为，几何关系中的变形分量至少要展开到位移的3次方，而现有的平面曲梁的几何关系多是经验性和线性的，无法满足该要求。本文建立了一般形状平面曲梁的弹性三维大变形理论框架。首先通过严格的几何推导，得到了适用于求解后屈曲行为的非线性几何关系。然后分别通过平衡分析和虚功率原理得到了适用于求解弹性三维大变形的平衡方程和本构关系。 对于面内大变形问题，本文建立了蛇形结构的理论模型及基于分量传递的隐式求解算法，分析了几何非线性和几何形状参数对蛇形结构拉伸极限的影响。理论和有限元结果表明，对于很多常用的几何形状，采用线性理论得的结果会对拉伸极限造成明显的高估（~50%）。此外，本文提出了蛇形结构加载应变和最大应变之间的非线性经验公式，可用来指导蛇形结构的工程设计。 利用平面曲梁的三维大变形基本关系式，通过引入变形摄动小参数，建立了解析求解平面曲梁后屈曲行为的理论框架，给出了三个典型算例的显式解析解：圆环受静水压力面内失稳，圆弧形曲梁在两端弯矩作用下弯扭失稳，圆弧形曲梁受分布载荷弯扭失稳。其中圆环面内失稳算例的结果与前人一致，后两个算例说明了曲率-位移关系中的非线性项对后屈曲行为影响较大，如：圆弧受分布载荷弯扭失稳算例中，若不取这些非线性项，所得到的后屈曲行为是不稳定的，若取这些非线性项，则是稳定的。 通过同时引入变形和几何摄动小参数，构造出双摄动展开的方法，进而建立了解析求解条带状平面曲梁的后屈曲问题的理论框架。解析求解了三种典型形状条带受对压的后屈曲行为：正弦形、多项式形、圆弧形，其中，正弦形平面曲梁的结果表明几何摄动展开项数越多，结果越精确。将理论与实验、有限元结果对比，发现理论结果能较好地预测三种形状条带的后屈曲变形。
Stretchable electronics and three-dimensional (3D) assembly of microstructures guided by compressive buckling represent two hot topics of research in recent years. As a basic form of structures, planar curved beams played an important role in these areas. This paper aims to formulate a fundamental finite-deformation theory of planar curved beams, and to analyze the mechanical behavior of several representative configurations widely used in relevant engineering applications. It also focuses on the quantitative relationships between the applied load and the deformation components, in order to provide guidelines for practical applications. The main research contents and conclusions are as follows: To obtain the accurate postbuckling behavior, the deformation components need to be expanded at least to the 3rd order in terms of displacements, according to the stability theory of Koiter. However, the existing geometrical relations of planar curved beams are mostly empirical and linear, which do not meet well the above requirement. In this thesis, a theoretical framework has been established for the general 3D finite deformation of planar curved beams. In particular, the nonlinear geometrical relationships suitable for postbuckling analyses are obtained through rigorous derivation. The equilibrium equations and constitutive relationships are derived through the equilibrium analyses and principle of virtual rate of work. In the condition of in-plane finite deformation, this thesis developed a theoretical model of serpentine structures and an implicit nonlinear analytical algorithm based on the transforming of components. The effects of geometrical nonlinearity and various shape parameters on the stretchability have been studied. The results of theoretical model and finite element analyses (FEA) indicate that the linear model can induce a considerable overestimation (~ 50%) of the stretchability for many representative shapes. Furthermore, an approximate nonlinear model was proposed to obtain the stretchability explicitly, which can serve as guidelines in the engineering design of serpentine structures. For the postbuckling of planar curved beams, a perturbation method was built by introducing a small parameter, which is related to the deformation, in the theoretical framework. Explicit analytical solutions of three typical examples, namely thin ring under uniform compression, arch under uniform bending and arch under uniform compression, were obtained. The analytic results on the postbuckling of the ring under uniform compression agree well with the previous studies. The nonlinear terms in curvature-displacement relationships have been proved to have prominent effects in the postbuckling analysis, according to the solutions of the latter two flexural-torsional postbuckling problems. For example, the postbuckling behavior would be unstable without the consideration of these nonlinear terms, while the correct solution is stable, for the arch under uniform compression. A double-perturbation method was constructed by introducing simultaneously two small parameters, related to the deformation and initial geometry, respectively. A theoretical framework was then developed for the flexural-torsional postbuckling of ribbons. Closed-form solutions were obtained for three representative types of ribbons under compression at two ends, including sine, polynomial and circle shapes. The results of sine-shape ribbons show that the accuracy of the theoretical model increases with increasing the number of terms of expansion for the shape-related small parameter. By comparing to the results of experiments and FEA, the theoretical model was found to offer accurate predictions on the postbuckling deformations.
