论文总字数：24090字

摘 要

在自然界中，板状和片状的结构往往有着许多很奇妙的失稳形貌，比如树叶和花瓣。并且，在实际工业应用中，有很多种方法来实现人工制造想要形貌的膜结构。利用膜内的压应力产生失稳自发地形成各种有用的形貌是一种直接高效的方法。在膜内出现压应力时，膜往往倾向于发生面内的变形来松弛这个压力，但是压应力超过某个阈值时，膜会倾向于发生离面变形来释放压力,整体的能量达到最低。利用温度的变化可以在薄膜状的物体内产生压应力，当这个压应力超过一定阈值时，薄膜就会发生失稳。理解日常生活和实际应用中薄膜结构的形貌形成机制，并且懂得如何制造和控制它们的形貌对相关的基础理论研究和许多实际应用都有着重要的意义。

本文回顾了关于薄膜屈曲的研究背景和概况，简述了国内外学者在相关领域的研究成果；回顾了大挠度板理论的发展历史，并详细介绍了板在发生大挠度时的研究理论（冯·卡门方程)和板在热应力作用下的基本微分方程；求出薄膜发生轴对称屈曲应满足的控制方程并对方程进行求解；分析薄膜非对称失稳应当满足的控制方程，并根据非线性微分方程理论分析薄膜非对称失稳控制方程的解，并确定薄膜屈曲的临界条件，明确临界失稳的依赖参数，研究非轴对称失稳形貌的产生机理。

关键词：温变应力；薄膜屈曲；大挠度板理论；非轴对称失稳

Abstract

In nature, plate-like and sheet-like structures often have many wonderfully distorted topographies, such as leaves and petals. And, in practical industrial applications, there are many ways to achieve the artificial manufacture of the membrane structure that it is desired to have. Using the compressive stress in the membrane to generate instability to spontaneously form various useful topographies is a straightforward and highly efficient method. In the presence of compressive stress in the film, the film tends to undergo in-plane deformation to relax this pressure, but when the compressive stress exceeds a certain threshold, the film tends to deform out of plane to release the pressure, because it will make the total system Lower energy. With the change of temperature, compressive stress can be generated in a film-like object. When the compressive stress exceeds a certain threshold value, the film will be unstable. Understanding the formation mechanism of thin-film structures in daily life and practical applications, and knowing how to manufacture and control their topography have important implications for related basic theoretical research and many practical applications.

This article reviews the research background and general situation of film buckling, briefly describes the research results of scholars at home and abroad in related fields, reviews the development history of the theory of large deflection plates, and details the theory of research on large deflection of the plate (Feng • Carmen equation) and the basic differential equation of the plate under thermal stress; the equations for the axisymmetrical buckling of the film should be solved and the equations solved; the analytical equations for the asymmetry of the film should be satisfied, and the nonlinear equations should be satisfied. The differential equation theory is used to analyze the solution of the asymmetric instability equation of the thin film, and the critical condition of the film buckling is determined. The dependence of critical instability parameters is defined and the mechanism of the non-axisymmetric destabilization morphology is studied.

Keywords：thermal stress, buckling, large-deflection plate theory，non-axisymmetrical instability

目 录

摘要 ............................................................Ⅰ

Abstract.............................................................Ⅱ

1. 绪论 ..................................................1
   1. 引言 ..................................................1
   2. 膜内夹杂引起的弹性效应研究............................2
   3. 大挠度板理论发展历程 .................................4
   4. 冯·卡门方程的求解及应用...............................5
   5. 本文主要研究工作.......................................7
2. 大挠度板存在温度应力的基本方程 ..........................8

2.1 引言...................................................8

2.2 板在大挠度弯曲下的应变-位移关系........................8

2.3 大挠度板的基本微分方程................................11

2.4 板内存在热应力时的微分方程............................13

1. 温变应力下的薄膜对称失稳研究....... .....................16

3.1 引言 ................................................ .16

3.2 温度应力下轴对称屈曲的基本方程........................16

3.3 基本方程的求解........................................19

3.4 结果分析与讨论........................................21

1. 温变应力下的薄膜非轴对称失稳研究.......... ..............22

4.1 温度应力下非轴对称屈曲的基本方程......................22

• 1. 基本方程的求解 ......................................23

4.2.1 时的解析解............................... ..24

4.2.2 时的解的讨论..................... ..........26

4.3 结果分析与讨论........................................29

1. 总结与展望............................................. .31

致谢...........................................................32

参考文献...................................................... 33

第一章 绪论

1.1 引言

在自然界中，板状和片状结构通常具有许多特殊和奇怪的不稳定特征，例如树叶和花瓣。在实际的工业应用中，人们已经提出了许多方法来制造希望形成的膜结构。这是一种非常直接和有效的方法，通过使用由膜中的压应力引起的不稳定性，自发地形成各种有用的结构。通常，当膜中的压应力大到某个临界值时，膜将不稳定并形成多种形态。膜中的压应力通常可能由于边界不对称约束导致的不对准而引起。不均匀特征的产生也可能由膜应变中不均匀的应变引起，这可能由各种机制引起。在膜中存在压缩应力的情况下，膜倾向于在平面内变形以缓解该压力，但是当压缩应力超过某个数值时，膜倾向于在平面外变形以释放压力，因为这会使系统总能量较低，这是面内变形能量与面外变形产生的弯曲能量之间竞争的结果，最终系统将达到总能量最低的状态^[1]。

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