Stability analysis of functionally graded rectangular plates under nonlinearly varying in-plane loading resting on elastic foundation

In this research work, an exact analytical solution for buckling of functionally graded rectangular plates subjected to non-uniformly distributed in-plane loading acting on two opposite simply supported edges is developed. It is assumed that the plate rests on two-parameter elastic foundation and its material properties vary through the thickness of the plate as a power function. The neutral surface position for such plate is determined, and the classical plate theory based on exact neutral surface position is employed to derive the governing stability equations. Considering Levy-type solution, the buckling equation reduces to an ordinary differential equation with variable coefficients. An exact analytical solution is obtained for this equation in the form of power series using the method of Frobenius. By considering sufficient terms in power series, the critical buckling load of functionally graded plate with different boundary conditions is determined. The accuracy of presented results is verified by appropriate convergence study, and the results are checked with those available in related literature. Furthermore, the effects of power of functionally graded material, aspect ratio, foundation stiffness coefficients and in-plane loading configuration together with different combinations of boundary conditions on the critical buckling load of functionally graded rectangular thin plate are studied.     

Thermoelastic buckling behavior of thick functionally graded rectangular plates

Thermoelastic buckling behavior of thick rectangular plate made of functionally graded materials is investigated in this article. The material properties of the plate are assumed to vary continuously through the thickness of the plate according to a power-law distribution. Three types of thermal loading as uniform temperature raise, nonlinear and linear temperature distribution through the thickness of plate are considered. The coupled governing stability equations are derived based on the Reddy’s higher-order shear deformation plate theory using the energy method. The resulted stability equations are decoupled and solved analytically for the functionally graded rectangular plates with two opposite edges simply supported subjected to different types of thermal loading. A comparison of the present results with those available in the literature is carried out to establish the accuracy of the presented analytical method. The influences of power of functionally graded material, plate thickness, aspect ratio, thermal loading conditions and boundary conditions on the critical buckling temperature of aluminum/alumina functionally graded rectangular plates are investigated and discussed in detail. The critical buckling temperatures of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be served as benchmark results for researchers to validate their numerical and analytical methods in the future.     

面内阶跃载荷下矩形薄板的塑性动力屈曲

对于面内阶跃载荷作用下矩形薄板的塑性动力屈曲问题,将临界应力和屈曲惯性项指数参数作为双特征参数求解。由相邻平衡准则导出失稳控制方程,由动力屈曲发生瞬间的能量转换和守恒准则,导出波阵面上的屈曲变形补充约束条件。失稳控制方程、边界条件、塑性波阵面上的连续条件和补充约束条件构成了定量求解两个特征参数和动力屈曲模态的完备条件。研究了矩形薄板塑性动力屈曲过程中板的厚宽比、冲击载荷大小、屈曲模态和临界屈曲长度之间的关系。     

Magnetoelastic buckling of a rectangular block in plane strain

Of interest here is the stability of a rectangular block subjected to a uniform magnetic field perpendicular to its longitudinal axis. The two ends of the block are frictionless and kept parallel to each other. This boundary value problem is motivated by the classical problem of magnetoelastic buckling in which a cantilever beam subjected to a transverse magnetic field buckles when the applied field reaches a critical value.This work presents a finite strain continuum mechanics formulation of the stability problem of a homogeneous, compressible, magnetoelastic rectangular block in plane strain subjected to a uniform transverse magnetic field. The applied variational approach employs an unconstrained energy minimization recently proposed by the authors.The analytical solution for the critical buckling fields for both the antisymmetric and symmetric modes are obtained for three different constitutive laws. The corresponding result for thin beams is extracted asymptotically for a special material and the solution is compared to previously published results. The critical magnetic field is shown to increase monotonically with the block's aspect ratio for each material and mode type. Antisymmetric modes are always the critical buckling modes for stress saturated and neo-Hookean materials, except for a narrow range of moderate aspect ratios (about 0.25) where symmetric modes become critical. For strain-saturated solids no buckling is possible above a maximum aspect ratio.     

各向异性矩形板的剪切屈曲分析

根据各向异性矩形薄板剪切屈曲横向位移函数的微分方程建立了一般性的解析解。该一般解包括三角函数和双曲线函数组成的解,它能满足四个边为任意边界条件的问题;该一般解还包括代数多项式解,它能满足四个角的边界条件问题。因此,这一解析解可用于精确地求解任意边界的各向异性矩形板的剪切屈曲问题。其中待定常数可由四边和四角的边界条件来确定,由此得出的齐次线性代数方程系数矩阵行列式等于零可以求得各阶临界载荷及其屈型。结合配点法,利用变形的对称和反对称性,以及对称迭层正方形板均可使计算更简单。以四边平夹的对称角铺设复合材料迭层板为例进行了计算和讨论。     

基于新修正偶应力理论的Mindlin层合板热稳定性分析

以新修正偶应力理论为基础,首次提出了机械载荷与热载荷共同作用下的微尺度Mindlin层合板热稳定性模型,该模型只引入一个材料尺度参数,通过虚功原理推导出了控制方程和边界条件,以四边简支方板为例,进行了热稳定性分析,应用纳维叶解法得到解析解。结果表明,所建模型可以捕捉到尺度效应。材料尺度参数值越大,屈曲临界温度越高;当跨厚比增大时,屈曲临界温度下降;随着板几何参数的增大,模型将退化为宏观模型;温度变化量越大,考虑热载荷作用下的屈曲临界载荷越大,尺度效应体现越显著。     

正交各向异性压磁电弹性矩形板的稳定问题

从压磁电弹性力学的三维平衡方程和本构关系出发,推导得到了正交各向异性压磁电弹性矩形板稳定问题的状态方程。求解状态方程并结合边界条件得到了稳定问题的屈曲方程。给出了临界应力的算例。     

Elasto-plastic buckling analysis for perforated steel plates subject to uniform compression

This study investigates the elasto-plastic buckling behaviour of simply supported square and rectangular thin steel plates having elliptic cut-outs by means of finite element method. Plates with simply supported in the out-of-plane direction are applied uniform compression in long-edge direction. A50 steel was used in the analysis and the focus was on the effect of plate aspect ratio, elliptical hole size, elliptical hole angle, elliptical hole location and slenderness ratio on buckling behaviour. It was found in the study that as the plate slenderness ratio increases, the critical buckling stress decreases for all the perforated plates.     

弹性地基上多孔功能梯度材料矩形板的自由振动与临界屈曲载荷分析

多孔功能梯度材料(FGM)构件的特性与孔隙率和孔隙分布形式有密切关系。本文基于经典板理论,考虑不同孔隙分布形式时修正的混合率模型,研究Winkler弹性地基上四边受压多孔FGM矩形板的自由振动与临界屈曲载荷特性。首先利用Hamilton原理和物理中面的定义推导Winkler弹性地基上四边受压多孔FGM矩形板自由振动的控制微分方程并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程和边界条件进行变换,得到计算无量纲固有频率和临界屈曲载荷的代数特征方程。将问题退化为孔隙率为零时的FGM矩形板并与已有文献进行对比以验证其有效性。最后计算并分析了梯度指数、孔隙率、地基刚度系数、长宽比、四边受压载荷及边界条件对多孔FGM矩形板无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

Buckling analysis of pristine and defected graphene

The buckling behavior of monolayer graphene (pristine and vacancy-defected) and bilayer graphene (pristine) loaded in the armchair direction was simulated for different boundary conditions using a truss FE model, representing the exact atomic lattice of graphene, and a FE model of an equivalent 2D plate. The critical buckling stress of pristine monolayer graphene was derived as a function of aspect ratio. The results from the two FE models coincide and are in very good agreement with established analytical solutions. With increasing the aspect ratio, the critical buckling stress of monolayer graphene decreases until the value of 2 from which the effect starts to diminish. Using the truss FE model, the effect of randomly dispersed vacancies on the critical buckling stress and buckling mode of monolayer graphene was studied. It was found that the critical buckling stress decreases dramatically with increasing the defect density: for a defect density of 10%, the critical buckling stress decreases by almost 50%. Moreover, the presence of defects was found to affect the highest buckling modes (above 3) even at low densities. Bilayer graphene has a totally different critical buckling stress than monolayer graphene due to the effect of van der Waals forces which depends on the applied boundary conditions.     

An analytical solution for buckling of moderately thick functionally graded sector and annular sector plates

In this article, an analytical solution for buckling of moderately thick functionally graded (FG) sectorial plates is presented. It is assumed that the material properties of the FG plate vary through the thickness of the plate as a power function. The stability equations are derived according to the Mindlin plate theory. By introducing four new functions, the stability equations are decoupled. The decoupled stability equations are solved analytically for both sector and annular sector plates with two simply supported radial edges. Satisfying the edges conditions along the circular edges of the plate, an eigenvalue problem for finding the critical buckling load is obtained. Solving the eigenvalue problem, the numerical results for the critical buckling load and mode shapes are obtained for both sector and annular sector plates. Finally, the effects of boundary conditions, volume fraction, inner to outer radius ratio (annularity) and plate thickness are studied. The results for critical buckling load of functionally graded sectorial plates are reported for the first time and can be used as benchmark.     

求解矩形夹层板在集中载荷下的稳定问题

在考虑了横向切应力和横向正应力对夹层板稳定影响的情况下,给出了矩形夹层板结构屈曲失稳的控制方程、基本解以及边界条件。应用功的互等定理求解了在均布载荷作用下的矩形夹层板的屈曲失稳问题。     

多层正交异性矩形薄板弯曲振动稳定解析解

构造了带有补充项的双重正弦傅里叶级数通解来求解各种边界条件的多层正交各向异性矩形薄板的弯曲、振动和稳定问题.将坐标轴取在中性面上,求出用挠度表示的应力表达式,然后由横截面上每单位宽度的应力合成板的内力;再将层合板的内力代入板的平衡方程中得到板的控制方程,将多层板的物理参数折算为等价的单层板物理参数;最后联立控制方程与边界条件,求得未知量的系数并代入本文的通解中.本文的通解不需要叠加即可求解各种边界条件的板的弯曲、振动和稳定问题;现有的对于单层板的研究都可以用本文的方法拓展到多层板领域;对于复杂边界条件的板,也可以使用该通解分析.     

任意边界条件下双模量矩形薄板的弯曲

将双模量板等效为两个各向同性小矩形板组成的层合板,假定该层合板的中性面即为两个小矩形板的交界面。根据中性面上应力为零且薄板全厚度上应力的代数和为零,推导了双模量矩形薄板的中性面位置。本文采用严宗达提出的带补充项的双重正弦傅里叶级数通解,该通解可以适用于任意边界条件的矩形薄板且不需要叠加或者重新构造。联立边界条件和控制方程,求得通解中的待定系数并代入到通解中,即可得到任意边界条件下双模量矩形薄板的弯曲解析解。与有限元结果比较,本文结果符合工程精度要求。     

THERMAL BUCKLING ANALYSIS OF CIRCULAR PLATE EMBEDDED IN AN INFINITE ELASTIC PLATE 1)

分析了嵌入无限大弹性板中的圆板在变温时的热屈曲问题。由于圆板的热膨胀系数与无限大弹性板的热膨胀系数不同，温度变化时圆板中会产生压应力。当压应力达到其临界值时，圆板会发生热屈曲。首先，基于弹性力学平面应力问题的基本理论，得到圆板和无限大弹性板的应力和位移；然后建立圆板热屈曲的控制微分方程，求得临界屈曲温度的解析解和数值解，着重讨论圆板和无限大弹性板的材料物性参数的关系对圆板临界屈曲温度的影响。     

Buckling delamination of a rectangular plate containing a rectangular crack and made from elastic and viscoelastic composite materials

This paper studies a three-dimensional buckling delamination problem for a rectangular plate made from elastic and viscoelastic composite material. It is assumed that the plate contains a rectangular band-crack (Case 1) and a rectangular edge-crack (Case 2) and that the edge-surfaces of these cracks have an initial infinitesimal imperfection. The evolution of this initial imperfection with an external compressive loading, acting along the crack (for an elastic composite) or with duration of time (for a viscoelastic composite under fixed external loading) is investigated within the framework of three-dimensional geometrically non-linear field equations of the theory of the viscoelasticity for anisotropic bodies. To determine the values of the critical force or critical time as well as the buckling delamination mode, the initial imperfection criterion is used. The corresponding boundary-value problems are solved by employing boundary form perturbation techniques, Laplace transform and FEM (Finite Element Method). The influence of the materials and/or the geometrical parameters of the plate on the critical values are discussed. In particular, it is established that for the considered change range of the problem parameters, the buckling form depends only on the initial infinitesimal imperfection mode of the crack edges.     

Structural stability and failure analysis using peridynamic theory

The peridynamic theory has been successfully utilized for damage prediction in many problems. However, the elastic stability of structures has not been studied using the peridynamic theory. Therefore, this paper investigates the elastic stability of simple structures to determine buckling characteristics of the peridynamic theory by considering two sets of problems. The first set of problems involves rectangular columns under compression to find the effects of the cross-sectional area and boundary conditions on buckling load. The second set involves rectangular plates under a uniform temperature load to establish the effects of plate dimensions and material properties on the critical buckling temperature. The predictions of the peridynamic theory agree with those published in the literature. The solution method is based on reducing the peridynamic equations of motion to discrete forms by using collocation points. These discrete equations are then solved using adaptive dynamic relaxation. Furthermore, perturbation method using geometrical imperfections is utilized to trigger lateral displacements in the numerical solutions.     

Numerical solution to the buckling problem for a sandwich plate under uniaxial compression

A thin rectangular sandwich plate with isotropic linear elastic layers is considered. The plate is in a plane-strain state under uniaxial compression. An exact statement of the buckling problem is given. Its approximate solution is found by the finite-difference method. The concept of base scheme is used to formulate discrete problems in explicit and compact form. As an example, the critical parameters of the plate are calculated using a computation optimization procedure. Its efficiency is demonstrated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 98–105, September 2006.     

Thermoelastic buckling response of thick functionally graded plates

The thermoelastic buckling behavior of a thick plate made of a functionally graded material is investigated in this paper by using an exponential shear deformation plate theory. A simple power law based on the rule of mixtures is used to estimate the effective material properties as functions of the plate thickness. The neutral surface position for such functionally graded plates is determined on the basis of the nonlinear strain-displacement relations. Uniform, linear, and nonlinear temperature distributions across the plate are considered. An analytical approach is presented to find the critical buckling temperature, which can be used in engineering calculations. A numerical solution of the problem with the use of an exponential dependence for shear strains is presented. The results obtained are compared with available data.     

冲击载荷作用下矩形薄板的弹性动力屈曲

为了研究冲击载荷作用下考虑应力波效应弹性矩形薄板的动力屈曲,根据动力屈曲发生瞬间的能量转换和守恒准则,导出板的屈曲控制方程和波阵面上的补充约束条件,真实的屈曲位移应同时满足控制方程和波阵面上的附加约束条件。满足上述条件,建立了该问题的完整数值解法,对屈曲过程中冲击载荷、屈曲模态和临界屈曲长度之间的关系进行研究,定量计算了横向惯性效应对提高薄板动力屈曲临界应力的贡献。研究表明:板的厚宽比一定时,临界屈曲长度随冲击载荷的增大而减小;由于屈曲时的横向惯性效应,应力波作用下薄板一阶临界力参数是相应边界板的静力失稳临界力参数的1.5倍;随着边界约束逐渐减弱,板临界力参数逐渐减小,动力特征参数逐渐增大。