【作者】 马源；

【导师】 徐新生；

【作者基本信息】 大连理工大学 ， 固体力学， 2007， 博士

【摘要】 结构的稳定性问题是近代固体力学中的一重要研究内容和热门话题，也是目前工业装备特别是航空航天工业非常关注的课题之一。由于圆柱壳是最基本的结构之一，其屈曲问题在结构稳定理论中占有重要的地位。目前虽然已有大量的研究成果并解决了不少的实际问题，然而在耦合应力波效应下研究圆柱壳的动态屈曲问题是困难的，特别是在对称结构和载荷下的非轴对称屈曲问题。因此研究和提出一种新的方法是必要的。冲击载荷是一种常见的动载形式，由于时间效应和结构的惯性及可变形性，冲击载荷会以应力波的形式在结构内传播及反射等。在耦合冲击载荷作用下，注意到各种应力波(如轴向压缩应力波和扭转应力波)的波速不同，结构被分成若干区域，各区域的内力不同，且这些区域的边界随时间变化。在这种复杂的应力情况下研究圆柱壳的动态屈曲是本文的主要特色之一。壳体结构在耦合冲击载荷作用下动态屈曲研究工作的难点在于高阶偏微分方程的求解困难，对于高阶偏微分方程，传统的半逆求解法有很大的局限性，它依赖于具体问题而缺乏一般性，且往往只能找到某些解而遗漏部分解。将哈密顿体系引入结构的动态屈曲研究中，其意义在于将结构的动态屈曲研究从传统的欧几里得几何空间进入到由原变量和对偶变量组成的辛几何空间之中，从而使辛本征函数展开的直接解析法得以实施，这样就可以求解许多以往半逆凑合法无法求解的问题，并且所得到的解空间是完备的。本文针对弹性圆柱壳在轴向冲击、扭转冲击、轴扭耦合冲击及径向均布压力和轴向冲击载荷耦合作用下的动态屈曲问题，借助于系统的能量构造出问题的原变量和对偶变量，建立了系统的哈密顿体系。在辛几何空间中，将圆柱壳的临界屈曲载荷和屈曲模态归结为辛本征值和本征解问题。其中零本征值、本征解和非零本征值、本征解分别揭示圆柱壳轴对称屈曲问题和非轴对称屈曲问题。根据哈密顿正则方程及其辛空间的特点，利用对偶变量表示的边界条件和相容条件，导出前屈曲分叉条件，将偏微分方程的初边值问题化为代数方程问题。通过求解，得到了辛本征值和本征解的解析表达式，即屈曲临界载荷和对应的屈曲模态。数值结果给出了弹性圆柱壳在轴向冲击、扭转冲击、轴扭耦合冲击及径向均布压力和轴向冲击载荷等耦合作用下动态屈曲的临界屈曲载荷和屈曲模态。结果表明不同边界条件会对圆柱壳的临界屈曲载荷和屈曲模态产生很大影响；冲击载荷耦合的形式直接影响临界载荷耦合曲线(或曲面)及其屈曲模态形状；应力波传播和反射的特点使屈曲模态具有特殊的形式；此外，材料常数和几何参数也直接影响临界载荷的规律。由于辛体系存在辛共轭正交关系、叠加原理及展开定理，因此任意的屈曲形式都可以由所得到的辛本征解和共轭辛本征解叠加得到。这种辛方法和研究成果对具有非线性大变形的后屈曲问题研究是有益的。同时该方法也可推广到其它领域和方向中。更多还原

【Abstract】 The stability of structure is an important research subject in solid mechanics, and much attention is focused on the problem in industry equipments, especially in aerospace now. Since cylindrical shell is a basal structure, its buckling problem is playing an important role in the theories of structural stability. The research of instability has achieved a lot of results and solved a lot of practical problems. But the solution and mechanism for dynamic buckling of cylindrical shells in the effect of coupling stress wave is very hard to research, especially for non-axisymmetric buckling in symmetric structure and load. So it is necessary to present a new research method.Impact load is a familiar dynamic load. It propagates and reflects as a stress wave in structure under the affect of inertia and transform of structures and time effect. The speeds of different kinds of stress waves(such as compressed stress wave and torsional stress wave) are unequal under coupled impact load. So the internal forces in structure are different between different areas, and the boundaries of these areas are changing with time. It is a characteristic of this paper that dynamic buckling of cylindrical shells at complicated stress is researched.The research of dynamic buckling is so difficult because the high-order partial differential equations are very hard to be solved. The traditional method for high-order partial differential equations is limited. It does not fit universal problems, and the solution space is not complete. When the Hamiltonian system is introduced into the research of dynamic buckling in cylindrical shells, the problem uses dual variables in Symplectic space instead of variables in traditional Euclidean space. The symplectic eigenfunction expansion method gives out an exact solution space. So many problems are solved, which can not find the solution in traditional methods.In this paper, the fundamental problems of the dynamic buckling in cylindrical shells are axial impact, torsion impact and coupling impact. The dual variables in symplectic space are gotten from energy function, and the Hamiltonian system is established. The critical buckling loads and buckling modes are described by eigenvalues and eigensolutions in symplectic space. The zero-eigensolutions describe axisymmetric buckling and non-zero eigensolutions describe non-axisymmetric buckling. The bifurcation condition of front buckling is obtained from the boundary and continuous conditions of dual variables, and partial differential equation is converted to linear equations. The eigenvalues and eigensolutions are given out to describe the critical buckling loads and buckling modes.Numerical results show the critical buckling loads and buckling modes of cylindrical shells under axial impact, torsion impact and coupling impact. The pictures show that the critical buckling loads and buckling modes of cylindrical shells are affected by boundary conditions. The form of coupled impact load decides the critical buckling load curves and critical buckling modes. The critical buckling modes have special forms that according to characteristics of propagation and reflection of stress wave. Material constants and geometry parameters also have effects on the rule of critical buckling load.The symplectic eigensolutions are conjugated and orthogonal in Hamiltonian system and the solution space is complete. So all buckling forms can be assembled by symplectic eigensolutions and their conjugated eigensolutions. The symplectic method and its results are useful for post-buckling questions that are non-linear and large-deformation. This method can also be extended to other research fields.更多还原