Vibration and buckling analyses of FGM plates with multiple internal defects using XIGA-PHT and FCM under thermal and mechanical loads

In this paper, the vibration and buckling analyses of the FGM (functionally graded material) plates with multiple internal cracks and cutouts under thermal and mechanical loads are numerically investigated using the combined XIGA-PHT (extended isogeometric analysis based on PHT-splines) and FCM (finite cell method). Material properties are graded only in the thickness direction. The effective material properties are estimated by using either the rule of mixture or the Mori-Tanaka homogenization technique. The plate displacement field is based on the HSDT (higher-order shear deformation plate theory) without any requirement of the SCF (shear correction factor). The HSDT model can exactly represent the shear stress distribution and improve the accuracy of solutions. The PHT-splines can naturally fulfill the C¹-continuous requirement of the HSDT model. The representation of internal defects is mesh-independent. The discontinuous and singular phenomena induced by the cracks are captured using the enrichment pattern in the XIGA, and the influence of cutouts is implemented by the FCM. The geometries of cutouts are captured by means of adaptive quadrature procedure based on a simple unfitted structural mesh, which avoids the need for multiple patches to describe the complex geometry and eliminates the enforcement of C¹-continuity patch-coupling across the patch boundaries. The initial mesh density around the cracks and cutouts can be controlled flexibly utilizing the local refinement property of the PHT-splines. After validating the results of the developed approach with those available in the literature, the effects of material gradient index, side to thickness ratio, boundary conditions, cutout size and crack length on the normalized frequency and the critical buckling parameter are investigated. Numerical results illustrate the effectiveness and accuracy of the present approach.     

Finite element dynamic analysis of unsymmetric composite laminated beams with shear effect and rotary inertia under the action of moving loads

The finite element dynamic response of an unsymmetric composite laminated orthotropic beam, subjected to moving loads, has been studied. One-dimensional finite element based on classical lamination theory, first-order shear deformation theory, and higher-order shear deformation theory having 16, 20 and 24 degrees of freedom, respectively, are developed to study the effects of extension, bending, and transverse shear deformation. The theories also account for the Poisson effect, thus, the lateral strains and curvatures can be expressed in terms of the axial and transverse strains and curvatures and the characteristic couplings (bend–stretch, shear–stretch and bend–twist couplings) are not lost. The dynamic response of symmetric cross-ply and unsymmetric angle-ply laminated beams under the action of a moving load have been compared to the results of an isotropic simple beam. The formulation also has been applied to the static and free vibration analysis.     

Prediction of biaxial buckling behavior of single-layered graphene sheets based on nonlocal plate models and molecular dynamics simulations

The biaxial buckling behavior of single-layered graphene sheets (SLGSs) is studied in the present work. To consider the size-effects in the analysis, Eringen’s nonlocal elasticity equations are incorporated into the different types of plate theory namely as classical plate theory (CLPT), first-order shear deformation theory (FSDT), and higher-order shear deformation theory (HSDT). An exact solution is conducted to obtain the critical biaxial buckling loads of simply-supported square and rectangular SLGSs with various values of side-length and nonlocal parameter corresponding to each type of nonlocal plate model. Then, molecular dynamics (MD) simulations are performed for a series of armchair and zigzag SLGSs with different side-lengths, the results of which are matched with those obtained by the nonlocal plate models to extract the appropriate values of nonlocal parameter relevant to each type of nonlocal elastic plate model and chirality. It is found that the present nonlocal plate models with their proposed proper values of nonlocal parameter have an excellent capability to predict the biaxial buckling response of SLGSs.     

A vibration analysis of composite laminated plates

A finite element formulation of the equations governing laminated anisotropic plates using Reddy's higher-order theory is presented. This simple higher-order shear deformable theory takes into account the parabolic distribution of the transverse shear deformation through the thickness of the plate and contains the same unknowns as in the first-order shear deformation theory. Finite element solutions are presented for rectangular plates of different layups, such as cross-ply, antisymmetric angle-ply, and sandwich plates with various material properties, boundaries, and plate aspect ratios. The numerical results are compared with the available closed-form results, the 3-D linear elasticity theory results, and the other available numerical results. A comparison is also made with test data from a laminated cantilever plate.     

Static response and free vibration analysis of FGM plates using higher order shear deformation theory

Free vibration and static analysis of functionally graded material (FGM) plates are studied using higher order shear deformation theory with a special modification in the transverse displacement in conjunction with finite element models. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. The fundamental equations for FGM plates are derived using variational approach by considering traction free boundary conditions on the top and bottom faces of the plate. Results have been obtained by employing a continuous isoparametric Lagrangian finite element with 13 degrees of freedom per node. Convergence tests and comparison studies have been carried out to demonstrate the efficiency of the present model. Numerical results for different thickness ratios, aspect ratios and volume fraction index with different boundary conditions have been presented. It is observed that the natural frequency parameter increases for plate aspect ratio, lower volume fraction index n and smaller thickness ratios. It is also observed that the effect of thickness ratio on the frequency of a plate is independent of the volume fraction index. For a given thickness ratio non-dimensional deflection increases as the volume fraction index increases. It is concluded that the gradient in the material properties plays a vital role in determining the response of the FGM plates.     

A new sinusoidal shear deformation theory for bending,buckling, and vibration of functionally graded plates

A new sinusoidal shear deformation theory is developed for bending, buckling, and vibration of functionally graded plates. The theory accounts for sinusoidal distribution of transverse shear stress, and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. Unlike the conventional sinusoidal shear deformation theory, the proposed sinusoidal shear deformation theory contains only four unknowns and has strong similarities with classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of plate are assumed to vary according to power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of simply supported plates are obtained and the results are compared with those of first-order shear deformation theory and higher-order shear deformation theory. It can be concluded that the proposed theory is accurate and efficient in predicting the bending, buckling, and vibration responses of functionally graded plates.     

A new refined simple TSDT-based effective meshfree method for analysis of through-thickness FG plates

In this paper a novel numerical method based on the Moving Kriging (MK) interpolation meshfree method, integrated with a simple higher-order shear deformation plate theory for analysis of static bending, free vibration and buckling of functionally graded (FG) plates is presented. In the proposed technique, the shape functions are built by the Kriging technique which possesses the property of Kronecker delta function which makes it easy to enforce essential boundary conditions. The present formulation is based on a refined simple third-order shear deformation theory (R-STSDT), which is based on four variables and it still accounts for parabolic distribution of the transverse shearing strains and stresses through the thickness of the plate present in the original simple third-order shear deformation theory (STSDT). In this theory, instead of assuming a specific distribution for the displacement field, the theory of elasticity is used for obtaining the kinematics of the plate deformation. We first propose the formulation, and then several numerical examples are provided to show the merits of the proposed approach.     

A simple four-unknown refined theory for bending analysis of functionally graded plates

In the present paper, a refined trigonometric higher-order plate theory is simply derived, which satisfies the free surface conditions. Moreover, the number of unknowns of this theory is the least one comparing with other shear theories. The effects of transverse shear strains as well as the transverse normal strain are taken into account. The number of unknown functions involved in the present theory is only four as against six or more in case of other shear and normal deformation theories. The bending response of FG rectangular plates is presented. A comparison with the corresponding results is made to check the accuracy and efficiency of the present theory. Additional results for all displacements and stresses are investigated through-the-thickness of the FG rectangular plate.     

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C⁰ continuity. A direct iterative based C⁰ nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).     

Analysis of functionally graded plates using higher order shear deformation theory

This work addresses a static analysis of functionally graded material (FGM) plates using higher order shear deformation theory. In the theory the transverse shear stresses are represented as quadratic through the thickness and hence it requires no shear correction factor. The material property gradient is assumed to vary in the thickness direction. Mori and Tanaka theory (1973) [1] is used to represent the material property of FGM plate at any point. The thermal gradient across the plate thickness is represented accurately by utilizing the thermal properties of the constituent materials. Results have been obtained by employing a C° continuous isoparametric Lagrangian finite element with seven degrees of freedom for each node. The convergence and comparison studies are presented and effects of the different material composition and the plate geometry (side-thickness, side–side) on deflection and temperature are investigated. Effect of skew angle on deflection and axial stress of the plate is also studied. Effects of material constant n on deflection and the temperature distribution are also discussed in detail.     

波形钢腹板箱梁的约束扭转剪应力分析

基于周边不变形理论,结合闭口薄壁杆件约束扭转的计算分析,研究了波形钢腹板箱梁在约束扭转时混凝土悬臂板上扭转剪应力的分布,并进行了计算.通过对悬臂板在约束扭转中剪力流计算公式的推导,进一步阐述了其自由扭转剪应力及翘曲扭转剪应力的分布,指出了相关文献在这部分计算中存在的问题.通过一个简支波形钢腹板组合箱梁算例,将该文方法计算结果与ANSYS有限元计算结果进行比较.结果表明:在波形钢腹板箱梁截面中,主要由波形钢腹板承受扭转剪应力,其次是混凝土底板,底板剪应力最大值发生在底板中心处,其数值近似等于腹板剪应力的一半,而混凝土顶板和悬臂部分的扭转剪应力很小;该文计算的扭转剪应力结果在总体上符合有限元得到的扭转剪应力分布规律,在悬臂自由端为0,随着离开悬臂自由端距离的增大,扭转剪应力逐渐增大并达到峰值.     

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.     

考虑高阶横向剪切正交各向异性板非线性弯曲的微分求积分析

采用微分求积方法（DQ方法）讨论了计及高阶横向剪切的正交各向异性弹性板的非线性弯曲问题．导出了非线性控制方程的DQ形式,利用推广的DQWB技巧处理了高阶矩的边界条件．进一步推广并运用新的分析技术简化了非线性方程的计算．为说明该方法的可靠性和有效性,将考虑剪切变形及不计剪切变形的薄板的数值结果与三维弹性解析解及其它数值解进行了比较,同时研究了数值结果的收敛性,并考察了不同的节点分布对收敛速度的影响A·D2还考察了几何、材料参数及横向剪切效应对正交各向异性板非线性弯曲的影响．分析结果表明横向剪切效应对正交各向异性中厚板的影响是显著的．     

Thermoelastic stability analysis of laminated composite plates: An analytical approach

The paper presents Chebyshev series based analytical solutions for the postbuckling response of the moderately thick laminated composite rectangular plates with and without elastic foundations. The plate is assumed to be subjected to in-plane mechanical, thermal and thermomechanical loadings. In-plane mechanical loading consists of uniaxial, biaxial, shear loadings and their combinations. The temperature induced loading is due to either uniform temperature or a linearly varying temperature across the thickness. The mathematical formulation is based on higher order shear deformation theory (HSDT) and von-Karman nonlinear kinematics. The elastic foundation is modeled as shear deformable with cubic nonlinearity. The thermal and mechanical properties of the composites are assumed to be temperature dependent. The quadratic extrapolation technique is used for linearization and fast converging finite double Chebyshev series is used for spatial discretization of the governing nonlinear equations of equilibrium. The effects of plate parameters and foundation parameters on buckling and postbuckling response of the plate are presented.     

Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.     

高阶剪切变形板理论K■rm■n型方程及在热后屈曲分析中的应用

本文基于Reddy高阶剪切变形板理论导出Karman型非线性大挠度方程并用于层合板热后屈曲分析．分析中计及板初始几何缺陷和热效应．给出了四边简支．对称正交铺设层合板在均匀或非均匀抛物型热分布作用下的后屈曲分析．采用摄动-Galerkin混合法确定板的热屈曲载荷与热后屈曲平衡路径．同时讨论了横向剪切变形，板长宽比，铺层数以及初始几何缺陷等各种参数变化的影响．     

Analysis of adiabatic shear bands in thermo-elasto-viscoplastic materials by using piece-wise discontinuous basis functions

An adiabatic shear band (ASB) is a narrow region of intense plastic deformation that forms when some metallic alloys and some polymers are deformed at high strain rates and there is not enough time for the heat generated by plastic deformations to diffuse away. The study of ASBs is important because an ASB is a precursor to shear/ductile fractures. Initial-boundary-value problems simulating the initiation and propagation of an ASB have been analyzed usually using the finite element method (FEM). Because of the large plastic strains involved, the FE mesh needs to be refined several times to delineate the ASB width. Each refinement requires, in turn, interpolation of data from the previous mesh to the new one which causes a smoothening of the sharp gradients of the deformation fields, and affects characteristics of the ASB. In this paper, we propose the application of the finite element method with piecewise discontinuous basis functions for studying the occurrence of ASBs in simple shearing deformations of a body composed of an isotropic and homogeneous thermo-elastoviscoplastic material. The mathematical model of the problem is defined by a system of coupled nonlinear partial differential equations and an inequality constraint associated with the plastic strain rates admissibility.     

Size-Dependent Effects of Micro-Nano Mindlin Plates Based on the Couple Stress Theory北大核心CSCD

基于偶应力理论,建立了适用于微纳米结构的Mindlin板理论。考虑横向剪切变形和材料的尺度效应并引入长度尺寸参数,推导了各向同性微纳米Mindlin板的本构方程。根据板的平衡条件,进一步推导出用位移函数和转角函数表示的板的屈曲和振动控制方程。通过对位移和转角变量进行空间和时间域上的分离,得出了四边简支（SSSS）和对边简支、对边固支（SCSC）两种边界情况下微纳米板的屈曲和振动问题的解析解。然后利用MATLAB软件进行算例分析,获得了不同尺寸参数、长宽比、厚长比等情况下板的临界屈曲荷载和固有频率。研究结果与已有文献中的结果以及ABAQUS有限元仿真解进行对比,结果表明,不同参数下的三种方法得到的结果均十分接近。算例分析发现,尺度效应对屈曲载荷和固有频率都有显著影响。     

A two-variable first-order shear deformation theory considering in-plane rotation for bending,buckling and free vibration analyses of isotropic plates

This paper presents a two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analyses of isotropic plates. In recent studies, a simple first-order shear deformation theory (S-FSDT) was developed and extended. It has only two variables by separating the deflection into bending and shear parts while the conventional first-order shear deformation theory (FSDT) has three variables. However, the S-FSDT provides incorrect predictions for the transverse shear forces on the insides and the twisting moments at the boundaries except simply supported plates since it does not consider in-plane rotation. The present theory also has two variables but considers in-plane rotation such that it is able to correctly predict the responses of plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with two opposite edges that are simply supported, with the other edges having arbitrary boundary conditions. Numerical results of deflections, stress resultants, buckling loads and natural frequencies are presented with the FSDT, the S-FSDT and the present theory. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the bending, buckling and free vibration responses of isotropic plates.     

A two-dimensional layerwise-differential quadrature static analysis of thick laminated composite circular arches

In this paper, a high accuracy and rapid convergence hybrid approach is developed for two-dimensional static analyses of circular arches with different boundary conditions. The method essentially consists of a layerwise technique in the thickness direction in conjunction with differential quadrature method (DQM) in the axial direction. Hence, the high accuracy and fast convergence of DQM with generality of layerwise formulations for modeling the transverse deformations of arbitrary laminated composite thick arches are combined. This results in superior accuracy with fewer degrees of freedom than conventional finite element method (FEM) or finite difference method (FDM). The convergence behavior of the method is shown and to verify its accuracy, the results are compared with those obtained based on the first order shear deformation Reissner–Naghdi type shell theory and also higher order shear deformation theory. The effects of opening angles, ply angle, boundary conditions, and thickness-to-length ratio on the stress and displacement components are studied.