扁球壳在均布压力与均匀温度场联合作用下的屈曲

根据扁壳几何非线性理论,推导了均布压力与均匀温度场联合作用下的扁球壳的位移型几何非线性控制方程.考虑夹紧边界条件,采用打靶法得到了扁球壳轴对称弯曲与屈曲的数值结果.讨论了壳体几何参数对平衡路径、临界荷载的影响.给出了壳体临界几何参数.当几何参数大于临界几何参数时,上、下临界荷载都随几何参数增加而增加.给定几何参数时,考察了不同均匀温度场对壳体上、下临界荷载、临界几何参数以及平衡构型的影响.均匀升温会使上临界荷载显著增加,会使下临界荷载略有减小.均匀变温会使临界几何参数改变.     

Nonlinear in-plane buckling of fixed shallow functionally graded graphene reinforced composite arches subjected to mechanical and thermal loading

The nonlinear in-plane buckling analysis for fixed shallow functionally graded (FG) graphene reinforced composite arches which are subjected to uniform radial load and temperature field is presented in this paper. The arch is composed of multiple graphene platelet reinforced composite (GPLRC) layers with gradient changes of concentration of graphene platelets (GPLs) in each layer. The principle of virtual work, combined with the effective materials properties estimated by the Halpin-Tsai micromechanics model for GPLRC layer, is used to derive the nonlinear buckling equilibrium equations of the FG-GPLRC arch, and then the analytical solutions for the limit point and bifurcation buckling loads are obtained. Comprehensive parametric studies are conducted to explore the effects of various distribution patterns and geometries of GPL, temperature field and arch geometry on the nonlinear equilibrium path and buckling behavior of the composite arch. The influence of temperature on the geometric parameters which are defined as switches between limit point buckling, bifurcation buckling and no buckling are also discussed. It is found that a higher temperature field can increase the buckling loads of the FG-GPLRC arch but reduce the value of the minimum geometric parameters that switching the buckling modes. The results also show that even a small amount of GPLs filler content can increase the buckling loads of the FG-GPLRC arch considerably, and distributing more GPLs near the surface layers is the best pattern to enhance the buckling performances of FG-GPLRC arches.     

基于非局部应变梯度理论功能梯度纳米板的弯曲和屈曲研究

以纳米机器人等智能器件中的功能梯度纳米板结构为研究对象,基于非局部应变梯度理论,研究了其弯曲和屈曲问题.推导了一般情况下的功能梯度纳米板运动方程,弯曲和屈曲作为其特例可简化而成.分析了非局部尺度参数、材料特征尺度参数、梯度指数、纳米板尺寸等对弯曲挠度和临界屈曲载荷的影响.结果表明:不同高阶连续介质力学理论下的最大挠度都随梯度指数的增大而增大,正方形纳米板挠度较小,且板厚越大,弯曲挠度越小;最大挠度随非局部尺度参数的增大而增大,随材料特征尺度参数的增大而减小.临界屈曲载荷随梯度指数的增大而减小,随板厚、长宽比的增大而增大,随非局部尺度的增大而减小,随材料特征尺度的增大而增大.非局部应变梯度高阶弯曲和屈曲中存在结构软化与硬化机制,两个内特征参数之间具有耦合效应,当非局部尺度大于材料特征尺度时,非局部效应在功能梯度纳米板力学性能中占主导作用;当材料特征尺度大于非局部尺度时,应变梯度效应占主导作用.解析结果还证明了当非局部尺度等于材料特征尺度时,非局部应变梯度理论结果退化为经典结果.     

扁球壳在热-机械荷载作用下的稳定性分析

基于扁壳几何非线性理论,应用虚功原理和变分法推导了均匀变温场中圆底扁薄球壳在均布外侧压力作用下的位移型几何非线性控制方程.考虑周边不可移简支边界条件,运用打靶法计算获得了不同几何参数的扁球壳轴对称弯曲变形的数值结果.定义了壳体临界几何参数.考察了壳体几何参数对平衡路径和临界荷载的影响.当壳体几何参数大于壳体临界几何参数时,上临界荷载随几何参数的增加单调增加,下临界荷载在很小范围内随几何参数的增加而增加,之后随几何参数的增加而减小.给定几何参数时,考察了不同均匀温度变化对壳体临界几何参数、临界荷载和平衡构型的影响.均匀升温使上临界荷载显著增加,使下临界荷载和临界几何参数显著减小.     

Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory

In this article, an analytical approach for buckling analysis of thick functionally graded rectangular plates is presented. The equilibrium and stability equations are derived according to the higher-order shear deformation plate theory. Introducing an analytical method, the coupled governing stability equations of functionally graded plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function, called boundary layer function. Using Levy-type solution these equations are solved for the functionally graded rectangular plate with two opposite edges simply supported under different types of loading conditions. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. Furthermore, the effects of power of functionally graded material, plate thickness, aspect ratio, loading types and boundary conditions on the critical buckling load of the functionally graded rectangular plate are studied and discussed in details. The critical buckling loads of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be used as benchmark.     

Edge-buckling in stretched thin films under in-plane bending

The wrinkling instabilities of a stretched rectangular thin film subjected to in-plane bending are investigated within the framework of the linearised Donnell-von Kármán bifurcation equation for thin plates. One of our principal objectives is to assess the role played by the finite bending stiffness of the film on the linear wrinkling mechanism. To this end, we employ a non-homogeneous linear pre-bifurcation solution and cast the corresponding eigenvalue problem as a singularly-perturbed differential equation with variable coefficients. Numerical simulations of this problem reveal the existence of two different regimes for the behaviour of the lowest eigenvalue. Based on this observation, a WKB analysis is carried out in order to capture the dependence of the critical wrinkling load on the wavelength of the localised oscillatory buckling pattern and the stiffness of the elastic film. The validity of the analytical results is corroborated by independent numerical computations of the eigenvalues using the method of compound matrices.     

Edge-buckling in stretched thin films under in-plane bending

The wrinkling instabilities of a stretched rectangular thin film subjected to in-plane bending are investigated within the framework of the linearised Donnell-von Kármán bifurcation equation for thin plates. One of our principal objectives is to assess the role played by the finite bending stiffness of the film on the linear wrinkling mechanism. To this end, we employ a non-homogeneous linear pre-bifurcation solution and cast the corresponding eigenvalue problem as a singularly-perturbed differential equation with variable coefficients. Numerical simulations of this problem reveal the existence of two different regimes for the behaviour of the lowest eigenvalue. Based on this observation, a WKB analysis is carried out in order to capture the dependence of the critical wrinkling load on the wavelength of the localised oscillatory buckling pattern and the stiffness of the elastic film. The validity of the analytical results is corroborated by independent numerical computations of the eigenvalues using the method of compound matrices.     

弹性介质中受轴向冲击载荷作用的圆柱壳的屈曲临界载荷计算分析

建立并求解了弹性介质中圆柱壳的径向位移控制方程,考虑边界条件及相容条件,得到了应力波传播及反射过程中圆柱壳的动力屈曲分叉条件.通过计算得到了不同时间段屈曲临界载荷与应力波波阵面到达圆柱壳的位置、弹性介质的刚度、壳体未嵌入弹性介质部分的长度与总长之比的关系.数值计算结果表明,弹性介质中的圆柱壳发生轴对称屈曲和非轴对称屈曲趋势一致;嵌入弹性介质部分越深、弹性介质刚度越大圆柱壳越难屈曲;屈曲临界载荷随着弹性介质刚度的增大经历了增长缓慢、增长迅速以及增长较慢3个阶段;应力波反射前波阵面通过分界面后,屈曲仅发生在应力波传播区域,反射波波阵面通过分界面前,临界载荷较小时屈曲先发生在反射端部,随着轴向阶数增大,屈曲覆盖整个圆柱壳区域,反射波波阵面通过分界面后,壳体发生的屈曲始终覆盖整个圆柱壳区域.     

A general nonlocal nonlinear model for buckling of nanobeams

This study presents a unified model for the nonlocal response of nanobeams in buckling and postbuckling states. The formulation is suitable for the classical Euler–Bernoulli, first-order Timoshenko, and higher-order shear deformation beam theories. The small-scale effect is modeled according to the nonlocal elasticity theory of Eringen. The equations of equilibrium are obtained using the principle of virtual work. The stress resultants are developed taking into account the nonlocal effect. Analytical solutions for the critical buckling load and the amplitude of the static nonlinear response in the postbuckling state are obtained. It is found out that as the nonlocal parameter increases, the critical buckling load reduces and the amplitude of buckling increases. Numerical results showing variation of the critical buckling load and the amplitude of buckling with the nonlocal parameter and the length-to-height ratio for simply supported and clamped–clamped nanobeams are presented.     

Buckling and lateral buckling interaction in thin-walled beam-column elements with mono-symmetric cross sections

Effects of axial forces on beam lateral buckling strength are investigated here in the case of elements with mono-symmetric cross sections. A unique compact closed-form is established for the interaction of lateral buckling moment with axial forces. This new equation is derived from a non-linear stability model. It includes first order bending distribution, load height level and effect of mono-symmetry terms (Wagner’s coefficient and shear point position). Compared to the so-called three-factors (C₁–C₃) formula commonly employed in beam lateral buckling stability, another factor C₄ is added in presence of axial loads. Pre-buckling deflection effects are considered in the study and the case of doubly-symmetric cross sections is easily recovered. The proposed solutions are validated and compared to finite element simulations where 3D beam elements including warping are used. The agreement of the proposed solutions with bifurcations observed on the non-linear equilibrium paths is good. Dimensionless interaction curves are dressed for the beam lateral buckling strength and the applied axial load, where the flexural-torsional buckling axial force is a taken as reference.     

Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory

In the present work, attention is focused on the prediction of thermal buckling and post-buckling behaviors of functionally graded materials (FGM) beams based on Euler–Bernoulli, Timoshenko and various higher-order shear deformation beam theories. Two ends of the beam are assumed to be clamped and in-plane boundary conditions are immovable. The beam is subjected to uniform temperature rise and temperature dependency of the constituents is also taken into account. The governing equations are developed relative to neutral plane and mid-plane of the beam. A two-step perturbation method is employed to determine the critical buckling loads and post-buckling equilibrium paths. New results of thermal buckling and post-buckling analysis of the beams are presented and discussed in details, the numerical analysis shows that, for the case of uniform temperature rise loading, the post-buckling equilibrium path for FGM beam with two clamped ends is also of the bifurcation type for any arbitrary value of the power law index and any various displacement fields.     

Thermal Buckling Analysis of FGM Sandwich Circular Plates Under Transverse Nonuniform Temperature Field Actions北大核心CSCD

Based on the von Kármán geometric nonlinear plate theory, the displacement⁃type geometric nonlinear governing equations for FGM sandwich circular plates under transverse nonlinear temperature field actions were derived. With the immovable clamped boundary condition, the analytical formula for dimensional critical buckling temperature differences of the system was obtained from the solution of the linear eigenvalue problem. Moreover, the 2⁃point boundary value problem of ordinary differential equations was solved with the shooting method. The effects of geometric parameters, constituent material properties, gradient indexes, temperature field parameters and layer⁃thickness ratios on the critical buckling temperature differences, the thermal postbuckling equilibrium paths, and the buckling equilibrium configurations of FGM sandwich circular plates, were investigated. The results show that, with the increases of the thickness⁃radius ratio, the relative thickness of the FGM layer and the gradient index, the FGM sandwich circular plate's critical buckling temperature difference will increase monotonically. Given a fixed radius and a fixed total thickness, the postbuckling deformation of the FGM sandwich circular plate will decrease significantly with the relative thickness of the FGM layer. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

基于非局部弹性应力场理论的纳米尺度效应研究:纳米梁的平衡条件、控制方程以及静态挠度

该文成功地解答了3个关于非局部应力理论用于纳米梁的问题:(ⅰ) 在绝大多数研究中,非局部效应增加导致纳米结构体刚度下降,其现象表现为弯曲挠度增加,固有频率减少,屈曲载荷下降,但为什么Eringen 的非局部弹性理论给出了完全相反的结论;(ⅱ) 为什么在某些研究结果中,非局部效应消失或是对研究结果无影响,比如纳米悬臂梁在集中载荷作用下的弯曲挠度; (ⅲ) 在高阶控制方程中,为什么高阶边界条件不存在．通过应用非局部弹性理论和精确变分原理分析纳米梁的弯曲问题,推导出全新的平衡条件、控制方程、边界条件和静态响应．这些方程和条件包含了与之前的相关研究结果符号相反的高阶微分项,这一差别导致了纳米效应对结构体的影响结果完全相反． 还证明之前为大家所公认的纳米梁静态或动态平衡条件实际上没有达到平衡,只有用等效弯矩代替非局部弯矩时,才可达到平衡．这些结论通常是可以被其它方法,比如应变梯度理论、耦合应力模型以及相关实验所证明．     

Stability of hydrostatic-pressured FGM thick rings with material nonlinearity

Buckling behaviors of elastoplastic ceramic/metallic functionally graded material (FGM) rings are investigated by using the first order shear deformation theory. The hydrostatic-pressured rings are assumed to be in both the plane-stress case and the plane-strain case, which lead respectively to a uniaxial and a biaxial elastoplastic stress states in prebuckling stage. A uniform strain hypothesis helps to deal with the elastoplastic stress states. By introducing in the graded material properties, the constitutive model of FGMs is formulated under the framework of J₂ deformation theory. By considering the kinetic relations of von-Kárman type and employing the principle of virtual displacement, the equilibrium equations and the buckling governing equations of FGM circular rings are formulated, and the analytical solution of the anisotropic rings is obtained. Finally, the elastoplastic buckling problem is numerically solved through a semi-analytical method, which is proposed to seek the real circumferential strain of FGM rings at the buckling point and determinate the elastoplastic buckling critical hydrostatic pressure. The effects of the inhomogeneous and geometrical parameters on the buckling critical load and the position of the elastoplastic interface are discussed. Results show that, in both the plane-stress and the plane-strain cases, the elastoplastic critical loads are generally lower than their elastic counterparts due to material flow, and the plane-strain critical load is generally larger than the plane-stress one. The elastoplastic critical load does not always decrease monotonously with the increase of the inhomogeneous parameters, which is quite different from their elastic counterparts.     

Stability analysis of graphene based laminated composite sheets under non-uniform inplane loading by nonlocal elasticity

Size dependent buckling of composite laminates made of isotropic graphene layers interlaid with bonding agents is considered. Nonlocal theory of elasticity is used in the buckling analysis to reflect the size scale effects on the critical buckling loads which is discussed in detail. The method is capable of predicting the relative buckling modes for non-uniform inplane loading applied through the thickness of the laminate. All modes of buckling in which the layers may displace together or opposite one another are investigated to study their scale dependent effects. Displacement or load controls are implemented through independent parameters as constraints to form special combination of buckling modes. Each graphene sheet is considered as a Kirchhoff plate model. The interlaid bonding agent is laterally treated as Winkler elastic foundation between graphene layers while neglecting their other load carrying capacities. Various numerical results are obtained reflecting the nonlocality effects. It is observed that in cases of higher load ratios and simpler buckling modes, the effect of nonlocality tends to drastically increase. The results of simpler examples studied are verified by another reference.     

Instability of optimal equilibria in the minimum mass design of uniform shallow arches

Necessary and sufficient conditions for the minimum mass design of arbitrarily loaded uniform shallow arches are derived. The problem is posed as an optimal control problem with mass as the criterion, initial curvature and axial load as design variables, and with the differential equations of axial and transverse equilibrium of the arch as side conditions. Thus, an optimal equilibrium is associated with each optimal design, and the stability of these equilibria becomes an integral part of the problem solution. As an example, the design process is carried out for the sinusoidally loaded hinged-hinged arch with a fixed span. It turns out that, depending on the given load amplitude, the optimal equilibrium can be unstable, stable after snap-through, and nonunique with one equilibrium unstable and the other stable after snap-through, at the design load of the arch. In addition, a necessary condition for a local minimum is the same as the usual critical point condition in stability analysis, thus assuring the instability of the arch at the optimum. A brief survey of earlier work on the optimal design of arches and curved beams is also included.     

Thermal effects on buckling of shear deformable nanocolumns with von Kármán nonlinearity based on nonlocal stress theory

Thermal buckling of nanocolumns considering nonlocal effect and shear deformation is investigated based on the nonlocal elasticity theory and the Timoshenko beam theory. By expressing the nonlocal stress as nonlinear strain gradients and based on the variational principle and von Kármán nonlinearity, new higher-order differential governing equations with corresponding higher-order nonlocal boundary conditions both in transverse and axial directions for instability of nanocolumns are derived. New analytical solutions for some practical examples on instability of nanocolumns are presented and analyzed in detail. The paper concluded that the critical buckling load is significantly increased in the presence of nonlocal stress and the results confirm that nanocolumn stiffness is enhanced by nanoscale size effect and reduced by shear deformation. The critical temperature change is increased with larger diameter to length ratio and higher nonlocal nanoscale. It is also concluded that at low and room temperatures the buckling load of nanocolumns increases with increasing temperature change, while at high temperature the buckling load decreases with increasing temperature change.     

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.     

Mechanical stability of functionally graded stiffened cylindrical shells

This paper is concerned with the elastic buckling of stiffened cylindrical shells by rings and stringers made of functionally graded materials subjected to axial compression loading. The shell properties are assumed to vary continuously through the thickness direction. Fundamental relations, the equilibrium and stability equations are derived using the Sander’s assumption. Resulting equations are employed to obtain the closed-form solution for the critical buckling loads. The results show that the inhomogeneity parameter and geometry of shell significantly affect the critical buckling loads. The analytical results are compared and validated using the finite element method.