Post-buckling behaviour of carbon-nanotube-reinforced nanocomposite plate

The aim of the present paper is to investigate the buckling and post-buckling behaviour of nanocomposite plate having randomly oriented carbon nanotubes (CNTs) reinforced in magnesium (Mg) under uni-axial compression. The effect of non-bonded interaction at the interface between CNT and matrix is considered through a cohesive zone model, used to predict the elastic property of the interphase, while evaluating the elastic properties of the nanocomposite using a representative volume element. A special purpose program based on finite-element formulation is developed to study the buckling and post-buckling behaviour of nanocomposite plate. The formulation is based on first-order shear deformation theory in conjunction with geometrical non-linearity as per von Karman’s assumptions. A parametric study is conducted to investigate the effects of interphase between CNT and matrix, short-CNT and long-CNT reinforcements and boundary conditions on buckling and post-buckling response of nanocomposite plate. It is found that imperfect bonding between CNT and Mg results in the loss of buckling and post-buckling strength, as compared with perfect bonding, of CNT–Mg nanocomposite plate. It is also concluded that buckling and post-buckling strength is higher for long-CNT-reinforced nanocomposite plate than that of short-CNT reinforcement, irrespective of bonding between CNT and matrix material.     

Quasi-convex reproducing kernel meshfree method

A quasi-convex reproducing kernel approximation is presented for Galerkin meshfree analysis. In the proposed meshfree scheme, the monomial reproducing conditions are relaxed to maximizing the positivity of the meshfree shape functions and the resulting shape functions are referred as the quasi-convex reproducing kernel shape functions. These quasi-convex meshfree shape functions are still established within the framework of the classical reproducing or consistency conditions, namely the shape functions have similar form as that of the conventional reproducing kernel shape functions. Thus this approach can be conveniently implemented in the standard reproducing kernel meshfree formulation without an overmuch increase of computational effort. Meanwhile, the present formulation enables a straightforward construction of arbitrary higher order shape functions. It is shown that the proposed method yields nearly positive shape functions in the interior problem domain, while in the boundary region the negative effect of the shape functions are also reduced compared with the original meshfree shape functions. Subsequently a Galerkin meshfree analysis is carried out by employing the proposed quasi-convex reproducing kernel shape functions. Numerical results reveal that the proposed method has more favorable accuracy than the conventional reproducing kernel meshfree method, especially for structural vibration analysis.     

A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates

A Hermite reproducing kernel Galerkin meshfree approach is proposed for buckling analysis of thin plates. This approach employs the Hermite reproducing kernel meshfree approximation that incorporates both the deflectional and rotational nodal variables into the approximation of the plate deflection and the C ¹ continuous approximation requirement for the Galerkin analysis of thin plates can be easily achieved herein. The strain smoothing operation is consistently introduced to construct the smoothed rotation and curvature fields which appear in the weak form governing the thin plate buckling. The domain integration of the weak form is carried out by the method of sub-domain stabilized conforming integration with the smoothed measures of rotation and curvature, as leads to an efficient discrete meshfree formulation for the eigenvalue problem of thin plate buckling. A series of benchmark buckling problems are presented to assess the proposed algorithm and the results uniformly demonstrate the present approach is very effective and it performs superiorly compared to the conventional Galerkin meshfree formulations whose domain integration are performed by Gauss quadrature rules.     

Free vibration analysis of thin plates using Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration

A Hermite reproducing kernel (HRK) Galerkin meshfree formulation is presented for free vibration analysis of thin plates. In the HRK approximation the plate deflection is approximated by the deflection as well as slope nodal variables. The nth order reproducing conditions are imposed simultaneously on both the deflectional and rotational degrees of freedom. The resulting meshfree shape function turns out to have a much smaller necessary support size than its standard reproducing kernel counterpart. Obviously this reduction of minimum support size will accelerate the computation of meshfree shape function. To meet the bending exactness in the static sense and to remain the spatial stability the domain integration for stiffness as well as mass matrix is consistently carried out by using the sub-domain stabilized conforming integration (SSCI). Subsequently the proposed formulation is applied to study the free vibration of various benchmark thin plate problems. Numerical results uniformly reveal that the present method produces favorable solutions compared to those given by the high order Gauss integration (GI)-based Galerkin meshfree formulation. Moreover the effect of sub-domain refinement for the domain integration is also investigated.     

A Hermite reproducing kernel approximation for thin‐plate analysis with sub‐domain stabilized conforming integration

A Hermite reproducing kernel (RK) approximation and a sub‐domain stabilized conforming integration (SSCI) are proposed for solving thin‐plate problems in which second‐order differentiation is involved in the weak form. Although the standard RK approximation can be constructed with an arbitrary order of continuity, the proposed approximation based on both deflection and rotation variables is shown to be more effective in solving plate problems. By imposing the Kirchhoff mode reproducing conditions on deflectional and rotational degrees of freedom simultaneously, it is demonstrated that the minimum normalized support size (coverage) of kernel functions can be significantly reduced. With this proposed approximation, the Galerkin meshfree framework for thin plates is then formulated and the integration constraint for bending exactness is also derived. Subsequently, an SSCI method is developed to achieve the exact pure bending solution as well as to maintain spatial stability. Numerical examples demonstrate that the proposed formulation offers superior convergence rates, accuracy and efficiency, compared with those based on higher‐order Gauss quadrature rule. Copyright © 2007 John Wiley & Sons, Ltd.     

A Lagrangian reproducing kernel particle method for metal forming analysis

A Meshless approach based on a Reproducing Kernel Particle Method is developed for metal forming analysis. In this approach, the displacement shape functions are constructed using the reproducing kernel approximation that satisfies consistency conditions. The variational equation of materials with loading-path dependent behavior and contact conditions is formulated with reference to the current configuration. A Lagrangian kernel function, and its corresponding reproducing kernel shape function, are constructed using material coordinates for the Lagrangian discretization of the variational equation. The spatial derivatives of the Lagrangian reproducing kernel shape functions involved in the stress computation of path-dependent materials are performed by an inverse mapping that requires the inversion of the deformation gradient. A collocation formulation is used in the discretization of the boundary integral of the contact constraint equations formulated by a penalty method. By the use of a transformation method, the contact constraints are imposed directly on the contact nodes, and consequently the contact forces and their associated stiffness matrices are formulated at the nodal coordinate. Numerical examples are given to verify the accuracy of the proposed meshless method for metal forming analysis.     

Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation

In this article, an exact analytical solution for buckling analysis of moderately thick functionally graded (FG) sector plates resting on Winkler elastic foundation is presented. The equilibrium equations are derived according to the first order shear deformation plate theory. Because of the coupling between the bending and stretching equilibrium equations of FG plates, these plates have deflection under in-plane loads lower than the critical buckling load acting on the mid-plane. The conditions under which FG plates remain flat in the pre-buckling configuration are investigated and the stability equations are obtained based on the flat plate assumption in the pre-buckling state. The stability equations are simplified into decoupled equations and solved analytically for plates having simply supported boundary condition on the straight edges. The critical buckling load is obtained and the effects of geometrical parameters and power law index on the stability of functionally graded sector plates are studded. The results for the critical buckling load of moderately thick functionally graded sector plates resting on elastic foundation are reported for the first time.     

Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures

A dispersion analysis is carried out to study the dynamic behavior of the Hermite reproducing kernel (HRK) Galerkin meshfree formulation for thin beam and plate problems. The HRK approximation utilizes both the nodal deflectional and rotational variables to construct the meshfree approximation of the deflection field within the reproducing kernel framework. The discrete Galerkin formulation is fulfilled with the method of sub-domain stabilized conforming integration. In the dispersion analysis following the HRK Galerkin meshfree semi-discretization, both the deflectional and rotational nodal variables are expressed by harmonic functions and then substituted into the semi-discretized equation to yield the characteristic equation. Subsequently the numerical frequency and phase speed can be obtained. The transient analysis with full-discretization is performed by using the central difference time integration scheme. The results of dispersion analysis of thin beams and plates show that compared to the conventional Gauss integration-based meshfree formulation, the proposed method has more favorable dispersion performance. Thereafter the superior performance of the present method is also further demonstrated by several transient analysis examples.     

A generalized approximation for the meshfree analysis of solids

This paper presents a new approach in the construction of meshfree approximations as well as the weak Kronecker‐delta property at the boundary, referred to as a generalized meshfree (GMF) approximation. The GMF approximation introduces an enriched basis function in the original Shepard's method. This enriched basis function is introduced to meet the linear or higher order reproducing conditions and at the same time to offer great flexibility on the control of the smoothness and convexity of the approximation. The construction of the GMF approximation can be viewed as a special root‐finding scheme of constraint equations that enforces that the basis functions are corrected and the reproducing conditions with certain orders are satisfied within a set of nodes. By choosing different basis functions, various convex and non‐convex approximations including moving least‐squares (MLS), reproducing kernel (RK), and maximum entropy (ME) approximations can be obtained. Furthermore, the basis function can also be translated or blended with other functions to generate a particular approximation for a special purpose. One application in this paper is to incorporate a blending function at the boundary based on the concept of local convexity for the non‐convex approximation, such as MLS, to acquire the weak Kronecker‐delta property. To achieve the higher order GMF approximation, two possible methods are also introduced. Several examples are presented to examine the effectiveness of various GMF approximations. Copyright © 2010 John Wiley & Sons, Ltd.     

A Size-Dependent Functionally Graded Higher Order Plate Analysis Based on Modified Couple Stress Theory and Moving Kriging Meshfree Method

A size-dependent computational approach for bending, free vibration and buckling analyses of isotropic and sandwich functionally graded (FG) microplates is in this study presented. We consider both shear deformation and small scale effects through the generalized higher order shear deformation theory and modified couple stress theory (MCST). The present model only retains a single material length scale parameter for capturing properly size effects. A rule of mixture is used to model material properties varying through the thickness of plates. The principle of virtual work is used to derive the discrete system equations which are approximated by moving Kriging interpolation (MKI) meshfree method. Numerical examples consider the inclusions of geometrical parameters, volume fraction, boundary conditions and material length scale parameter. Reliability and effectiveness of the present method are confirmed through numerical results.     

刚性基底上弹性约束矩形板的屈曲行为分析

对刚性基底上非加载边弹性约束矩形板在固支边均匀面内压力作用下的屈曲行为进行了分析研究。用里兹能量变分法建立确定板屈曲强度的本征值问题,利用满足边界条件的屈曲位移函数导出板屈曲系数计算公式。将该方法用于矩形钢管混凝土钢板局部屈曲的分析和计算,提出了目标板边界约束系数计算公式。试验结果表明,方形钢管混凝土柱钢板局部屈曲强度计算值与试验值吻合良好,钢板边界弹性约束的处理和约束系数的计算方法是有效的。     

Post-buckling analysis of skew plates subjected to combined in-plane loadings

Post-buckling behavior of laminated composite, sandwich and functionally graded skew plates is analyzed in the present work. The problem formulation is based on higher-order shear deformation theory and von Kármán’s nonlinear kinematics. Linear mapping is used to transform the physical domain into the computational domain. Chebyshev polynomials are used for spatial discretization of governing differential equations and boundary conditions. The nonlinear terms are linearized using quadratic extrapolation technique. The effect of the skew angle on the buckling and post-buckling response of the composite, sandwich and FGM-clamped skew plates is investigated for different combinations of in-plane compressive loadings.     

A point collocation method based on reproducing kernel approximations

A reproducing kernel particle method with built‐in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations. The design and implementation of a Galerkin‐based reproducing kernel particle method, however, faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions. In this paper we present a point collocation method based on reproducing kernel approximations. We show that, in a point collocation approach, the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach. The point collocation method based on reproducing kernel approximations, however, requires the calculation of higher‐order derivatives that would typically not be required in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point collocation method are derived. The point collocation method is shown to be accurate for several one and two‐dimensional problems and the convergence rate of the point collocation method is addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

A size-dependent third-order shear deformable plate model incorporating strain gradient effects for mechanical analysis of functionally graded circular/annular microplates

In this paper, we develop a novel size-dependent plate model for the axisymmetric bending, buckling and free vibration analysis of functionally graded circular/annular microplates based on the strain gradient elasticity theory. The displacement field is chosen by using a refined third-order shear deformation theory which assumes that the in-plane and transverse displacements are partitioned into bending and shear components and satisfies the zero traction boundary conditions on the top and bottom surfaces of the microplate. Besides, the present model contains three material length scale parameters to capture the size effect. The material properties of the microplate are assumed to vary in the thickness direction and estimated through the classical rule of mixture. By using Hamilton's principle, the equations of motion and boundary conditions are obtained. Afterward, the differential quadrature method is adopted to discretise the governing differential equations along with various types of edge supports and therefore the deflection, critical buckling load and natural frequency can be determined. Convergence and comparison studies are carried out to establish the reliability and accuracy of the numerical results. Finally, a parametric study is conducted to investigate the influences of material length scale parameters, gradient index, thickness-to-outer radius ratio, outer-to-inner radius ratio and boundary conditions on the mechanical characteristics of the microplate.     

The influence of mechanical couplings on the compressive stability of anti-symmetric angle-ply laminates

The compressive stability of anti-symmetric angle-ply laminated plates with particular reference to the degrading influence of membrane–flexural coupling is reported in this paper. The degree of membrane–flexural coupling in the laminated composite plates is varied, essentially, by altering the ply-angle and the number of plies in the laminated stack for a given composite material system. The coupled compressive buckling solutions are determined in the paper using the finite strip method of analysis and the buckling displacement fields of the strip formulation are those which are able to provide zero in-plane normal movement at the edge boundaries of the laminated plates.

Results are given for anti-symmetric angle-ply laminated plates subjected to uniaxial compression and these have been obtained from fully converged finite strip structural models. Validation of the finite strip formulation is indicated in the paper through comparisons with exact solutions where appropriate. Increasing the number of plies in the laminated system is seen to reduce the degree of coupling and the critical stress levels are noted to tend towards the plate orthotropic solutions. The ply-angle corresponding to the optimised buckling stress for any particular laminate is noted in the paper to be influenced by the support boundary conditions at the plates unloaded edges. For any particular laminate the minimum critical buckling stress and corresponding natural half-wavelength of the buckling mode are shown to be highly sensitive to ply-angle variation.

Some post-buckling results are presented in the paper and these have been determined using the finite element method of analysis. The influence of membrane–flexural coupling is shown to be significant throughout the compressive post-buckling history of the laminated plates. The optimised ply-angle with regard to the critical compressive buckling stress of square simply supported anti-symmetric angle-ply laminates is shown to be less effective in the post-buckling range with regard to post-buckled compressional stiffness.     

剪切可变形梁热过屈曲解析解

该文导出了面内热载荷作用下,梁过屈曲问题的精确解。首先基于非线性一阶剪切变形梁理论,推导了控制轴向和横向变形的基本方程。然后,将3 个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。该方程与相应的边界条件构成了微分特征值问题。直接求解该问题,得到了热过屈曲构形的闭合解,这个解是外加热载荷的函数。利用精确解,得到了临界屈曲载荷的一阶结果与经典结果的解析关系。为考察热载荷、横向剪切变形以及边界条件的影响,根据得到的精确解给出了两端固定、两端简支以及一端固定一端简支边界条件下的具体数值算例,讨论了梁在面内热载荷作用下的过屈曲行为,并与经典结果进行了比较。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。     

Nonlinear analysis of bimodular composite plates under compression

The present work deals with the buckling and post-buckling behaviour of bimodular laminated composite plates. The fiber-governed bimodular constitutive model is adopted for the analysis and the geometrical nonlinearities are accounted for by using the consistent small strain and moderate rotation theory for a shear deformable Mindlin plate. Governing equations of the problem are developed and the finite element formulation is given using lagrangian C°-elements. Some numerical examples are developed to analyze the buckling and post-buckling behaviour of laminated bimodular plates.     

Filters, reproducing kernel, and adaptive meshfree method

 Reproducing kernel, with its intrinsic feature of moving averaging, can be utilized as a low-pass filter with scale decomposition capability. The discrete convolution of two nth order reproducing kernels with arbitrary support size in each kernel results in a filtered reproducing kernel function that has the same reproducing order. This property is utilized to separate the numerical solution into an unfiltered lower order portion and a filtered higher order portion. As such, the corresponding high-pass filter of this reproducing kernel filter can be used to identify the locations of high gradient, and consequently serves as an operator for error indication in meshfree analysis. In conjunction with the naturally conforming property of the reproducing kernel approximation, a meshfree adaptivity method is also proposed. Received: 31 July 2002 / Accepted: 3 March 2003 The support of this work by the NSF/DARPA OPAAL Program under the grant DMS 98-74015 to UCLA is greatly acknowledged.     

Mechanical and thermal postbuckling of higher order shear deformable functionally graded plates on elastic foundations

This paper presents an analytical investigation on the buckling and postbuckling behaviors of thick functionally graded plates resting on elastic foundations and subjected to in-plane compressive, thermal and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation plate theory taking into account Von Karman nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, closed-form relations of buckling loads and postbuckling equilibrium paths for simply supported plates are determined. Analysis is carried out to show the effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the plates.     

Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties

As a first endeavor, the thermal buckling and postbuckling analysis of functionally graded (FG) annular plates with material properties graded in the radial direction is presented. The formulation is derived based on the first-order shear deformation theory (FSDT) and the geometrical nonlinearity is modeled using Green’s strain in conjunction with von Karman’s assumptions. The material properties are temperature-dependent and graded according to the power law distribution. It is assumed that the temperature varies along the radial direction. Using the virtual work principle, the pre-buckling and postbuckling equilibrium equations and the related boundary conditions are derived. Differential quadrature method (DQM) as an efficient numerical technique is adopted to solve the governing equations. The presented formulation and the method of solution are validated by performing convergence and comparison studies with available results in the literature. Finally, the effects of volume fraction index, geometrical parameters, mechanical/thermal properties of the constituent materials and boundary conditions on the thermal buckling and postbuckling behavior of the radially graded annular plate are evaluated and discussed.