Secondary Buckling Analysis of Thin Rectangular Plates Based on the Wavelet Galerkin Method北大核心CSCD

Application of the wavelet Galerkin method (WGM) to numerical solution of nonlinear buckling problems was studied with classical elastic thin rectangular plates. First, the discretized scheme of the von Kármán equation were introduced, then a simple calculation approach to the Jacobian and Hessian matrices based on the WGM was proposed, and the wavelet discretized scheme-based eigenvalue equation method, the extended equation method and the pseudo arc-length method for nonlinear buckling analysis were discussed. Second, the secondary post-buckling equilibrium paths of elastic thin rectangular plates and the effects of aspect ratios, boundary conditions and bi-directional compression on the mode jumping behaviors, were discussed in detail. Numerical results show that, the WGM possesses good convergence for solving buckling loads on rectangular plates, and the obtained equilibrium paths are in good agreement with those from the stability experiments, the 2-step perturbation method and the nonlinear finite element method. Given the feasibility of combination with different bifurcation computation methods, the WGM makes an efficient spatial discretization method for complex nonlinear stability problems of typical plates and shells. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Buckling Analysis of Re-Entrant Honeycomb Structures Under General Macroscopic Stress States北大核心CSCD

Based on the negative Poisson’s ratio effect of the re-entrant honeycomb, the finite element simulation of its buckling mechanical properties was carried out, and 2 buckling modes other than those of the traditional hexagonal honeycomb structures were obtained. The beam-column theory was applied to analyze the buckling strength and mechanism of the 2 buckling modes, where the equilibrium equations including the beam end bending moments and rotation angles were established. The stability equation was built through application of the buckling critical condition, and then the analytical expression of the buckling strength was obtained. The re-entrant honeycomb specimen was printed with the additive manufacturing technology, and its buckling performance was verified by experiments. The results show that, the buckling modes vary significantly under different biaxial loading conditions; the re-entrant honeycomb would buckle under biaxial tension due to the auxetic effect, being quite different from the traditional honeycomb structure; the typical buckling bifurcation phenomenon emerges in the analysis of the buckling failure surfaces under biaxial stress states. This research provides a significant guide for the study on the failure of re-entrant honeycomb structures due to instability, and the active application of this instability to achieve special mechanical properties. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Elastic Analysis of Anisotropic Functionally Graded Rotating Disks With Non-Uniform Thicknesses北大核心CSCD

研究了任意梯度变化的变厚度各向异性转动圆盘的弹性问题.假设圆盘绕刚性轴匀速转动,其材料性能和厚度沿径向任意梯度变化.考虑圆盘在中心转轴处受位移约束,外侧自由,根据各向异性转动圆盘的平衡微分方程,得到关于径向应力的Fredholm积分方程,继而通过对Fredholm积分方程进行数值求解,得到结构的位移场和应力场.对具体梯度变化情况仅需代入相应梯度变化进行求解即可.数值算例部分,通过假设厚度、弹性模量等参数为特殊的幂函数形式,将由Fredholm积分方程求出的数值解与对应的精确解进行对比,以及针对常见的Voigt模型,将由该方法算得的数值解和ANSYS有限元计算结果进行对比,验证了该方法的准确性和精度.其次,针对Voigt模型,重点分析了厚度变化、材料性能梯度参数、各向异性度等对应力场和位移场的影响.提出了针对材料性能和厚度沿径向呈任意梯度变化的圆盘结构弹性分析方法,将为优化功能梯度圆盘的结构和材料参数、有效调整构件应力分布、提高结构安全性,提供强有力的工具;算例分析结果对功能梯度圆盘在复杂条件下的结构安全设计有重要的理论指导意义.     

Thermal Shock Crack Propagation of Alumina Simulated With the Phase-Field Method Under Temperature-Dependent Damage Criteria北大核心CSCD

氧化铝陶瓷材料的力学性能受温度影响显著,因此使用相场法模拟热冲击裂纹的扩展时有必要考虑损伤判据的温度相关性.在现有热力学相场模型的基础上通过引入温度相关性损伤判据,修正了相场模型的控制方程.利用该模型对氧化铝陶瓷热冲击实验进行有限元模拟,并将模拟结果与氧化铝热冲击实验结果和不考虑温度相关性损伤判据的有限元模拟结果进行对比.结果表明,通过引入温度相关性损伤判据,可实现对热冲击裂纹的萌生和扩展过程更合理的模拟.     

夹层圆板轴对称非线性弯曲和屈曲的样条函数解法

以三次B样条函数为试函数,用配点法计算夹层圆板的非线性弯曲．支座可以是弹性的．夹层板采用Reissner模型．荷载可为多项式型的分布荷载、均布边缘力矩、均布径向压力或均布径向预应力及它们的组合．首次用非线性理论计算了夹层圆板的压曲临界荷载．在均布荷载作用下的结果同幂级数解的结果作了比较,说明样条配点法具有收敛范围大、精度高、编写程序通用的优点．     

复合载荷作用下夹层圆板的非线性振动和屈曲

基于von Karman薄板理论,建立了均布载荷和周边面内载荷联合作用下夹层圆板非线性振动问题的控制方程和滑动固定边界条件,给出了相应静力问题的精确解及其数值结果．基于时间模态假设和变分法,得到了空间模态的控制方程,并使用修正迭代法求解该方程,得到夹层圆板幅频-载荷特征关系．讨论了两种载荷对夹层圆板振动特性的影响规律．当周边面力使夹层圆板的最低固有频率为零时,就可获得临界载荷的值．     

周边面内压力作用下夹层圆板的非线性振动

基于von Krmn薄板理论,讨论了滑动固定基础上周边面内压力作用下夹层圆板的非线性振动问题,应用变分法导出了该问题的非线性特征方程和边界条件,给出了其精确静态解,并使用修正迭代法求解了该方程,导出了夹层圆板振幅和非线性振频的解析关系式．当周边面力使夹层圆板的最低固有频率为零时,就可获得临界载荷的值．     

夹层圆板的大幅度振动

本文利用哈密顿原理导出了夹层圆板轴对称大幅度自由振动的基本方程,并给出了表板很薄情况下的简化形式.作为算例,利用修正迭代法求出了具有滑动固定边界条件夹层圆板对轴称大幅度自由振动的一种解析解,并由此导出了夹层圆板振幅和振频的解析关系式.     

圆板在局部Gauss温度场作用下的响应分析

基于薄板热弯曲理论,推导了圆板在Gauss(高斯)温度场作用下的挠度和热应力解析表达式,分析了边界条件和局部温度参数对圆板挠度和热应力的影响,为局部温度变化薄板结构的热力学分析提供理论依据.研究结果表明:圆板中心处的挠度和压应力有最大值;在热影响区内,圆板内一点的挠度随着该点到板中心距离的增大呈Gauss型减小趋势;在热影响区外,圆板挠度的变化趋势与圆板边界约束形式和辐照因子有关,辐照因子越大,边界简支圆板挠度越先呈线性减小趋势;圆板挠度的解析解与有限元解一致.在热影响区内,圆板内一点的热应力随着该点到板中心距离的增大呈Gauss型减小趋势,两种边界约束圆板的热应力变化趋势相似;在热影响区外,圆板热应力的变化趋势与圆板边界约束形式和辐照因子有关.     

环形夹层板的屈曲状态

本文根据Reissner假设讨论了内外边界固支并在外边界上受面内径向均匀压力作用的环形夹层板的轴对称屈曲状态.首先给出了屈曲问题的基本方程;其次,用打靶法对一些参数值给出了最小的临界载荷;最后讨论了在临界载荷附近屈曲状态的存在性及其渐近形式.     

Stability of a Circular Sandwich Ring under an Axially Symmetric Temperature Field Inhomogeneous across the Thickness

The linearized problems on the stability of a circular sandwich ring of symmetric structure under an axially symmetric temperature field inhomogeneous across the core thickness are stated and analytical solutions to them are given. The first problem deals with the mixed flexural buckling form (BF) of the ring as a whole, realized as a result of buckling in one of the load-carrying layers due to formation of precritical pressure stresses in the layer. The second problem considers purely shear BFs when one load-carrying layer is rotated relative to the other. The deformation processes for the load-carrying layers are described by the Kirchhoff-Love model, and for the core of arbitrary thickness - by two models, namely by the equations of the plane problem of elasticity theory and by the model of a transversely soft layer of arbitrary thickness (the same equations simplified by the assumption of zero circumferential normal stresses). Within the frames of the first model adopted for the core, the shear BF is theoretically possible but practically unrealizable, since the mixed flexural BF arises earlier than the shear BF.     

Shear and Flexural Buckling Modes of a Spherical Sandwich Shell in a Centrosymmetric Temperature Field Inhomogeneous across the Thickness

Problems on buckling modes (BMs) are considered for a spherical sandwich shell with thin isotropic external layers and a transversely soft core of arbitrary thickness in a centrosymmetric temperature field inhomogeneous across the shell thickness. For their statement, the two-dimensional equations of the theory of moderate bending of thin Kirchhoff–Love shells are used for the external layers, with regard for their interaction with the core; for the core, maximum simplified geometrically nonlinear equations of thermoelasticity theory, in which a minimum number of nonlinear summands is retained to correctly describe its pure shear BM, are utilized. An exact analytical solution to the problem on initial centrosymmetric deformation of the shell is found, assuming that the temperature increments in the external layers are constant across their thickness. It is shown that the three-dimensional equations for the core, linearized in the neighborhood of the solution, can be integrated along the radial coordinate and reduced to two two-dimensional differential equations, which supplement the six equations that describe the neutral equilibrium of the external layers. It is established that the system of eight differential equations of stability, upon introduction of new unknowns in the form of scalar and vortical potentials, splits into two uncoupled sets of equations. The first of them has two kinds of solutions, by which the pure shear BM is described at an identical value of the parameter of critical temperature. The second system describes a mixed flexural BM, whose realization, at definite combinations of determining parameters of the shell and over wide ranges of their variation, is possible for critical parameters of temperature by orders of magnitude exceeding the similar parameter of shear BM.     

Shear Buckling Form of a Circular Sandwich Ring under Uniform External Pressure

The problem on the stability of a circular sandwich ring under uniform external pressure is considered. It is shown that, along with a mixed flexural-shear buckling form (BF), a pure shear BF can be realized in the core. This form is accompanied by rotation of the load-carrying layers at the cost of the transverse shear strain (constant in the circumferential direction) in the core. It is found that the simplified equations of the theory of shallow shells cannot describe this nonclassical BF. The critical loads corresponding to the shear BF may prove to be smaller than the critical load of the classical mixed flexural BF for a circular sandwich ring of medium thickness at a low shear modulus of the core. The results obtained contribute greatly to the understanding of buckling mechanisms of sandwich structures and supplement the existing classification of BFs.     

均布载荷作用下夹层圆板的非线性振动

给出了均布载荷作用下夹层圆板的大幅度振动方程.按假设的时间模态导出了该问题的非线性耦合的代数和微分特征方程组,并利用修正迭代法求出了该方程组的近似解析解,得到了周边固定夹层圆板振动的幅频-载荷特征关系.讨论了载荷对非线性振动性态的影响.