Buckling of a pre-compressed or pre-stretched membrane in shear flow

Membranes enclosing capsules and biological cells undergo periodic compression and stretching due to an imparted hydrodynamic traction as they rotate in a shear flow. Compression may cause transient or permanent buckling manifested by the onset of wrinkled shapes. To study the effect of pre-compression and pre-stretching on the critical conditions for buckling, the response of an elastic circular plate flush mounted on a plane wall and deforming under the action of a uniform tangential load due to an over-passing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated resulting in a fourth-order partial differential equation with position-dependent coefficients parametrized by the Poisson ratio. Solutions are computed by applying Fourier series expansions to derive an infinite system of coupled ordinary differential equations, and then implementing orthogonal collocation. The solution space is illustrated, critical values for buckling are identified, the associated eigenfunctions representing possible modes of deformation are displayed, and the effect of the Poisson ratio is discussed.     

Nonlinear buckled states of rectangular plates

A constructive method is developed to establish the existence of buckled states of a thin, flat elastic plate that is rectangular in shape, simply supported along its edges, and subjected to a constant compressive thrust applied normal to its two short edges. Under the assumption that the stress function and the deformation of the plate are described by the nonlinear von Kármán equations, the approach used yields information regarding not only the number of buckled states near an eigenvalue of the linearized problem, but also the continuous dependence of such states on the load parameter and the possible selection of that buckled state “preferred” by the plate. In particular, the methods used provide a rigorous approach to studying the existence of buckled states near the first eigenvalue of the linearized problem (that is, near the “buckling load”) even when the first eigenvalue is not simple.     

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.     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

梯度弹性基础上正交异性薄板的屈曲分析

基于双参数弹性基础模型,研究了梯度弹性基础上正交异性薄板的屈曲问题. 首先,基于能量法与变分原理,给出了梯度弹性基础上正交异性薄板的屈曲控制方程,并得到了梯度弹性基础刚度系数K₁ 与K₂的计算式;进而,通过将位移函数采用三角函数展开的方法,给出了单向压缩载荷作用下、四边简支正交异性弹性基础板屈曲载荷的计算式;在算例中,通过将该文的解退化到单纯的正交异性板,并与经典弹性解比较,证明了理论的正确性;最后,求解了弹性模量在厚度方向上呈幂律分布的梯度基础上的薄板屈曲问题,分析了基础上下表层材料弹性模量比与体积分数指数对屈曲载荷的影响.     

Nonlinear stability and dynamics of composite skew plates under nonuniform loadings using differential quadrature method

The buckling, postbuckling and postbuckled vibration behaviour of composite skew plates subjected to nonuniform inplane loadings are presented here. The skew plate is modelled using first order shear deformation theory accounting for von-Kármán geometric nonlinearity and initial geometric imperfections. The different types of nonuniform loads that have been considered in this study are concentrated load, partial load and parabolic load. The explicit analytical expressions for prebuckling stress distributions within composite skew plate subjected to three different types of nonuniform inplane loadings are developed by solving plane elasticity problem using Airy's stress function approach. It is observed that the inplane normal stress distributions within the skew plate due to above nonuniform loadings do not become uniform even at mid-section. The generalized differential quadrature (GDQ) method is used to solve the nonlinear governing partial differential equations. It is observed that the postbuckled load carrying capacity of skew plate under concentrated loading is the lowest compared to other nonuniform and uniform loadings.     

ANALYTICAL RELATIONS BETWEEN EIGENVALUES OF CIRCULAR PLATE BASED ON VARIOUS PLATE THEORIES

Based on the mathematical similarity of the axisymmetric eigenvalue problems of a circular plate between the classical plate theory(CPT), the first-order shear deformation plate theory(FPT) and the Reddy's third-order shear deformation plate theory (RPT), analytical relations between the eigenvalues of circular plate based on various plate theories are investigated. In the present paper, the eigenvalue problem is transformed to solve an algebra equation. Analytical relationships that are expressed explicitly between various theories are presented. Therefore, from these relationships one can easily obtain the exact RPT and FPT solutions of critical buckling load and natural frequency for a circular plate with CPT solutions. The relationships are useful for engineering application, and can be used to check the validity, convergence and accuracy of numerical results for the eigenvalue problem of plates.     

Buckling strength analysis of adhesively bonded aluminum hat sections

This paper provides an analytical solution for the critical buckling stress of adhesively bonded aluminum hat sections under static axial compression. The governing rectangular plate member of the structure is treated based on the differential equation for out-of-plane deflections of thin plates. Finite element eigenvalue buckling analysis is performed to verify the assumed simply supported boundary conditions for common edges between adjacent plate elements. Elastic restraint is applied to the two loaded edges of the rectangular plate, and the relative critical buckling stress is computed according to the transcendental equations. It is found from experiments that there is no adhesive bonding failure in the elastic buckling stage. The analytical solution yields buckling stress predictions which are in reasonable agreement with measured values.     

Buckling of a circular plate made of a shape memory alloy due to a reverse thermoelastic martensite transformation

We solve the axisymmetric buckling problem for a circular plate made of a shape memory alloy undergoing reverse martensite transformation under the action of a compressing load, which occurs after the direct martensite transformation under the action of a generally different (extending or compressing) load. The problem was solved without any simplifying assumptions concerning the transverse dimension of the supplementary phase transition region related to buckling. The mathematical problem was reduced to a nonlinear eigenvalue problem. An algorithm for solving this problem was proposed. It was shown that the critical buckling load under the reverse transition, which is obtained by taking into account the evolution of the phase strains, can be many times lower than the same quantity obtained under the assumption that the material behavior is elastic even for the least (martensite) values of the elastic moduli. The critical buckling force decreases with increasing modulus of the load applied at the preliminary stage of direct transition and weakly depends on whether this load was extending or compressing. In shape memory alloys (SMA), mutually related processes of strain and direct (from the austenitic into the martensite phase) or reverse thermoelastic phase transitions may occur. The direct transition occurs under cooling and (or) an increase in stresses and is accompanied by a significant decrease (nearly by a factor of three in titan nickelide) of the Young modulus. If the direct transition occurs under the action of stresses with nonzero deviator, then it is accompanied by accumulation of macroscopic phase strains, whose intensity may reach 8%. Under the reverse transition, which occurs under heating and (or) unloading, the moduli increase and the accumulated strain is removed. For plates compressed in their plane, in the case of uniform temperature distribution over the thickness, one can separate trivial processes under which the strained plate remains plane and the phase ratio has a uniform distribution over the thickness. For sufficiently high compressing loads, the trivial process of uniform compression may become unstable in the sense that, for small perturbations of the plate deflection, temperature, the phase ratio, or the load, the difference between the corresponding perturbed process and the unperturbed process may be significant. The results of several experiments concerning the buckling of SMA elements are given in [1, 2], and the statement and solution of the corresponding boundary value problems can be found in [3–11]. The experimental studies [2] and several analytic solutions obtained for the Shanley column [3, 4], rods [5–7], rectangular plates under direct [8] and reverse [9] transitions showed that the processes of thermoelastic phase transitions can significantly (by several times) decrease the critical buckling loads compared with their elastic values calculated for the less rigid martensite state of the material. Moreover, buckling does not occur in the one-phase martensite state in which the elastic moduli are minimal but in the two-phase state in which the values of the volume fractions of the austenitic and martensite phase are approximately equal to each other. This fact is most astonishing for buckling, studied in the present paper, under the reverse transition in which the Young modulus increases approximately half as much from the beginning of the phase transition to the moment of buckling. In [3–9] and in the present paper, the static buckling criterion is used. Following this criterion, the critical load is defined to be the load such that a nontrivial solution of the corresponding quasistatic problem is possible under the action of this load. If, in the problems of stability of rods and SMA plates, small perturbations of the external load are added to small perturbations of the deflection (the critical force is independent of the amplitude of the latter), then the critical forces vary depending on the value of perturbations of the external load [5, 8, 9]. Thus, in the case of small perturbations of the load, the problem of stability of SMA elements becomes indeterminate. The solution of the stability problem for SMA elements also depends on whether the small perturbations of the phase ratio and the phase strain tensor are taken into account. According to this, the problem of stability of SMA elements can be solved in the framework of several statements (concepts, hypotheses) which differ in the set of quantities whose perturbations are admissible (taken into account) in the process of solving the problem. The variety of these statements applied to the problem of buckling of SMA elements under direct martensite transformation is briefly described in [4, 5]. But, in the problem of buckling under the reverse transformation, some of these statements must be changed. The main question which we should answer when solving the problem of stability of SMA elements is whether small perturbations of the phase ratio (the volume fraction of the martensite phase q) are taken into account, because an appropriate choice significantly varies the results of solving the stability problem. If, under the transition to the adjacent form of equilibrium, the phase ratio of all points of the body is assumed to remain the same, then we deal with the “fixed phase atio” concept. The opposite approach can be classified as the “supplementary phase transition” concept (which occurs under the transition to the adjacent form of equilibrium). It should be noted that, since SMA have temperature hysteresis, the phase ratio in SMA can endure only one-sided small variations. But if we deal with buckling under the inverse transformation, then the variation in the volume fraction of the martensite phase cannot be positive. The phase ratio is not an independent variable, like loads or temperature, but, due to the constitutive relations, its variations occur together with the temperature variations and, in the framework of connected models for a majority of SMA, together with variations in the actual stresses. Therefore, the presence or absence of variations in q is determined by the presence or absence of variations in the temperature, deflection, and load, as well as by the system of constitutive relations used in this particular problem. In the framework of unconnected models which do not take the influence of actual stresses on the phase ratio into account, the “fixed phase ratio” concept corresponds to the case of absence of temperature variations. The variations in the phase ratio may also be absent in connected models in the case of specially chosen values of variations in the temperature and (or) in the external load, as well as in the case of SMA of CuMn type, for which the influence of the actual stresses on the phase compound is absent or negligible. In the framework of the “fixed phase ratio” hypothesis, the stability problem for SMA elements has a solution coinciding in form with the solution of the corresponding elastic problem, with the elastic moduli replaced by the corresponding functions of the phase ratio. In the framework of the supplementary phase transition” concept, the result of solving the stability problem essentially depends on whether the small perturbations of the external loads are taken into account in the process of solving the problem. The point is that, when solving the problem in the connected setting, the supplementary phase transition region occupies, in general, not the entire cross-section of the plate but only part of it, and the location of the boundary of this region depends on the existence and the value of these small perturbations. More precisely, the existence of arbitrarily small perturbations of the actual load can result in finite changes of the configuration of the supplementary phase transition region and hence in finite change of the critical values of the load. Here we must distinguish the “fixed load” hypothesis where no perturbations of the external loads are admitted and the “variable load” hypothesis in the opposite case. The conditions that there no variations in the external loads imply additional equations for determining the boundary of the supplementary phase transition region. If the “supplementary phase transition” concept and the “fixed load” concept are used together, then the solution of the stability problem of SMA is uniquely determined in the same sense as the solution of the elastic stability problem under the static approach. In the framework of the “variable load” concept, the result of solving the stability problem for SMA ceases to be unique. But one can find the upper and lower bounds for the critical forces which correspond to the cases of total absence of the supplementary phase transition: the upper bound corresponds to the critical load coinciding with that determined in the framework of the “fixed phase ratio” concept, and the lower bound corresponds to the case where the entire cross-section of the plate experiences the supplementary phase transition. The first version does not need any additional name, and the second version can be called as the "all-round supplementary phase transition" hypothesis. In the present paper, the above concepts are illustrated by examples of solving problems about axisymmetric buckling of a circular freely supported or rigidly fixed plate experiencing reverse martensite transformation under the action of an external force uniformly distributed over the contour. We find analytic solutions in the framework of all the above-listed statements except for the case of free support in the “fixed load” concept, for which we obtain a numerical solution.     

The torsional buckling of a cruciform column under compressive load with a vertex plasticity model

The torsional buckling of a plastically deforming cruciform column under compressive load is investigated. The problem is solved analytically based on the von Kármán shallow shell theory and the virtual work principle. Solutions found in the literature are extended for path-dependent incremental behaviour as typically found in the presence of the vertex effect that is present in metallic polycrystals.At the critical load for buckling the direction of straining changes by an additional shear component. It is shown that the incremental elastic–plastic moduli are spatially nonuniform for such situations, contrary to the classical J₂ flow and deformation theories. The critical shear modulus that governs the buckling equation is obtained as a weighted average of the incremental elastic–plastic moduli over the cross-section of the cruciform.Using a plasticity model proposed by the authors, that includes the vertex effect, the buckling-critical load is computed for a aluminium column both with the analytical model and a FEM-based eigenvalue buckling analysis. The stable post-buckling path is determined by the energy criterion of path-stability. A comparison with the experimentally obtained classical results by Gerard and Becker (1957) shows good agreement without relying on artificial imperfections as necessary in the classical J₂ flow theory.     

Buckling and postbuckling of magnetoelastic flat plates carrying an electric current

The basic equations of a fully nonlinear theory of electromagnetically conducting flat plates carrying an electric current and exposed to a magnetic field of arbitrary orientation are derived. The relevant equations have been obtained by considering that both the elastic and electromagnetic media are homogeneous and isotropic. The geometrical nonlinearities are considered in the von-Kármán sense, and the soft ferromagnetic material of the plate is assumed to feature negligible hysteretic losses. Based on the electromagnetic and elastokinetic field equations, by using the standard averaging methods, the 3-D coupled problem is reduced to an equivalent 2-D one, appropriate to the theory of plates. Having in view that the elastic structures carrying an electric current are prone to buckling, by using the presently developed theory, the associated problems of buckling and postbuckling are investigated. In this context, the problem of the electrical current inducing the buckling instability of the plate, and its influence on the postbuckling behavior are analyzed. In the same context, the problem of the natural frequency–electrical current interaction of flat plates, as influenced by a magnetic field is also addressed.     

Stability of an annular plate of a shape memory alloy

Analytical solutions using various hypotheses are obtained for the problem of buckling of an annular circular plate of a shape memory alloy in the forward or reverse thermoelastic phase transformations under uniform radial compression. It is established that, despite the axial symmetry of the body geometry, loads, and boundary conditions, the minimum critical loads for annular plates correspond to nonaxisymmetric buckling modes, in contrast to continuous plates.     

Imperfection sensitivity of ultimate buckling strength of elastic-plastic square plates under compression

The mechanism of imperfection sensitivity of elastic-plastic plates under compression is complex as they undergo elastic and/or plastic buckling, dependent on their width-thickness ratio. For elastic buckling, the Koiter power law is an established means to describe the imperfection sensitivity. Yet, for plastic buckling, there is no such an established way to describe it. In this paper, the quadratic power law is advanced to describe imperfection-insensitive plastic buckling behavior. The Koiter power law is extended by implementing the quadratic law so as to describe the elastic and plastic buckling in a synthetic manner. The finite-displacement, elastic-plastic analysis was conducted on simply-supported square plates under compression by varying the plate thickness and the initial deflection of a sinusoidal form. In association with an increase of the plate slenderness parameter (decrease of plate thickness), the predominant buckling is shown to change from (1) plastic buckling to (2) unstable elastic-plastic buckling and to (3) elastic stable bifurcation followed by a maximum point of load. In accordance with the change of the mechanism of buckling, the power law is changed pertinently to describe the complex imperfection sensitivity of the compression plates in a synthetic manner. The extended imperfection sensitivity law is thus advanced as a simple and strong tool to describe the ultimate buckling strength of elastic-plastic plates.     

On out-of-plane buckling of anisotropic elastic plate subjected to a simple shear

Out-of-plane buckling of anisotropic elastic plate subjected to a simple shear is investigated. From exact 3-D equilibrium conditions of anisotropic elastic body with a plane of elastic symmetry at critical configuration, the eqution for buckling direction (buckling wave direction) parameter is derived and the shape functions of possible buckling modes are obtained. The traction free boundary conditions which must hold on the upper and lower surfaces of plate lead to a linear eigenvalue problem whose nontrivial solutions are just the possible buckling modes for the plate. The buckling conditions for both flexural and barreling modes are presented. As a particular example of buckling of anisotropic elastic plate, the buckling of an orthotropic elastic plate, which is subjected to simple shear along a direction making an arbitrary angle of θ with respect to an elastic principal axis of materials, is analyzed. The buckling direction varies with θ and the critical amount of shear. The numerical results show that only the flexural mode can indeed exist. Project supported by the National Natural Science Foundation of China (No. 19772032).     

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。     

Stiffness and buckling optimization of thin plates with BEM

The thickness optimization is used to maximize the stiffness or the buckling load of a Kirchhoff plate having constant volume. The shape of the plate is arbitrary and it is subjected to any type of admissible boundary conditions. The optimization consists in establishing the thickness variation law, for which either the stiffness of the plate or the buckling load is maximized. Beside the equality constraint of constant volume, the thickness variation is subjected also to inequality constraints resulting from serviceability requirements (upper and lower thickness bounds) as well as from the condition that the Kirchhoff plate theory remains valid. The latter constraint is new and it is derived herein by approximating the plate with a three-dimensional prismatic elastic body having curved upper and lower surface. The optimization problem is solved using the sequential quadratic programming algorithm. The bending and the plane stress problem of a plate with variable thickness, required for the evaluation of the objective function, are solved using the analog equation method. The thickness is approximated using integrated radial basis functions that approximate accurately not only the thickness function but also its first and second derivatives involved in the plate equation and in the constraints. Several plate optimization problems have been studied giving realistic and meaningful optimum designs without violating the validity of the thin plate theory.     

Mechanical buckling and free vibration of thick functionally graded plates resting on elastic foundation using the higher order B-spline finite strip method

In this study, the mechanical buckling and free vibration of thick rectangular plates made of functionally graded materials (FGMs) resting on elastic foundation subjected to in-plane loading is considered. The third order shear deformation theory (TSDT) is employed to derive the governing equations. It is assumed that the material properties of FGM plates vary smoothly by distribution of power law across the plate thickness. The elastic foundation is modeled by the Winkler and two-parameter Pasternak type of elastic foundation. Based on the spline finite strip method, the fundamental equations for functionally graded plates are obtained by discretizing the plate into some finite strips. The results are achieved by the minimization of the total potential energy and solving the corresponding eigenvalue problem. The governing equations are solved for FGM plates buckling analysis and free vibration, separately. In addition, numerical results for FGM plates with different boundary conditions have been verified by comparing to the analytical solutions in the literature. Furthermore, the effects of different values of the foundation stiffness parameters on the response of the FGM plates are determined and discussed.     

基于物理中面FGM板屈曲的有限元分析

基于物理中面的概念,根据最小势能原理推导了功能梯度材料FGM薄板屈曲的有限元控制方程,求得临界荷载的有限元解。利用FGM板的屈曲方程与参考均匀板的屈曲方程之间的相似性,建立了FGM板的临界荷载与参考均匀板的临界荷载之间的相似转换关系式,从而将FGM板临界荷载的计算转变为参考均匀板的临界荷载和材料不均匀系数的计算,极大地提高了计算效率,为FGM的推广起积极的推动作用。通过数值算例,将由有限元法和转换关系式得到的临界荷载进行了比较,并讨论了边界条件、荷载工况、材料组成和几何尺寸等对FGM板临界荷载的影响。     

ANALYSIS OF SYMMETRIC LAMINATED RECTANGULAR PLATES IN PLANE STRESS

Symmetric laminated plates used usually are anisotropic plates. Based on the fundamental equation for anisotropic rectangular plates in plane stress problem, a general analytical solution is established accurately by method of stress function. Therefore the general formula of stress and displacement in plane is given. The integral constants in general formula can be determined by boundary conditions. This general solution is composed of solutions made by trigonometric function and hyperbolic function, which can satisfy the problem of arbitrary boundary conditions along four edges, and the algebraic polynomial solutions which can satisfy the problem of boundary conditions at four corners. Consequently this general solution can be used to solve the plane stress problem with arbitrary boundary conditions. For example, a symmetric laminated square plate acted with uniform normal load, tangential load and nonuniform normal load on four edges is calculated and analyzed.     

Exact solutions for the stability of viscoelastic rectangular plate subjected to tangential follower force

An exact solution procedure is formulated for the stability analysis of viscoelastic rectangular plate with two opposite edges simply supported and other two edges clamped as well as the viscoelastic rectangular plate with one edge clamped and other three edges simply supported under the action of tangential follower force. Firstly, by assuming the transverse displacement (W) as independent functions which automatically satisfies the simply supported boundary conditions, the governing partial differential equation is reduced to an ordinary differential equation with variable coefficients. Then, by the normalized power series method and applying the boundary conditions yield the eigenvalue problem of finding the roots of a fourth-order characteristic determinant. The results show that the aspect ratio λ and the dimensionless delay time H have great effects on the types of instability and the critical loads of the viscoelastic plates.