A homogeneous limit methodology and refinements of computationally efficient zigzag theory for homogeneous,laminated composite,and sandwich plates

The Refined Zigzag Theory (RZT) for homogeneous, laminated composite, and sandwich plates is revisited to offer a fresh insight into its fundamental assumptions and practical possibilities. The theory is introduced from a multiscale formalism starting with the inplane displacement field expressed as a superposition of coarse and fine contributions. The coarse displacement field is that of first‐order shear‐deformation theory, whereas the fine displacement field has a piecewise‐linear zigzag distribution through the thickness. The resulting kinematic field provides a more realistic representation of the deformation states of transverse‐shear‐flexible plates than other similar theories. The condition of limiting homogeneity of transverse‐shear properties is proposed and yields four distinct variants of zigzag functions. Analytic solutions for highly heterogeneous sandwich plates undergoing elastostatic deformations are used to identify the best‐performing zigzag functions. Unlike previously used methods, which often result in anomalous conditions and nonphysical solutions, the present theory does not rely on transverse‐shear‐stress equilibrium constraints. For all material systems, there are no requirements for use of transverse‐shear correction factors to yield accurate results. To model homogeneous plates with the full power of zigzag kinematics, infinitesimally small perturbations in the transverse shear properties are derived, thus enabling highly accurate predictions of homogeneous‐plate behavior without the use of shear correction factors. The RZT predictive capabilities to model highly heterogeneous sandwich plates are critically assessed, demonstrating its superior efficiency, accuracy, and a wide range of applicability. This theory, which is derived from the virtual work principle, is well‐suited for developing computationally efficient, C⁰ a continuous function of (x₁,x₂) coordinates whose first‐order derivatives are discontinuous along finite element interfaces and is thus appropriate for the analysis and design of high‐performance load‐bearing aerospace structures. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

不同理论下圆板特征值之间的解析关系

基于经典板理论(CPT)、一阶剪切变形板理论(FPT)以及Reddy三阶剪切变形板理论(RPT)之间,圆板轴对称特征值问题在数学上的相似性,研究了不同理论之间圆板特征值间的解析关系．将特征值问题的求解转化为代数方程的求解,并导出了不同理论之间圆板特征值的显式精确解析关系．从而,只要已知圆板特征值（临界屈曲载荷和固有频率）的经典结果,便很容易从这些解析关系中得到一阶和三阶理论下圆板特征值的相应结果,这便于工程应用,同时也可检验一阶和三阶理论下板特征值的数值结果的有效性、收敛性以及精确性等问题．     

应用4变量精确平板理论分析FG复合板的自由振动

应用4变量的精确平板理论,对矩形功能梯度材料(FGM)复合板进行自由振动分析.与其它的理论不同,该理论的未知函数数量只有4个,而别的剪变形理论的未知函数为5个.提出的4变量精确平板理论,协调条件有了改变,与经典的薄板理论相比,许多方面有着惊人的相似,无需引入剪切修正因数——当横向剪应力越过板厚后,为了满足剪应力自由表面条件,出现抛物线状的改变,导致横向剪应力的变化.考虑了两种常见类型的FGM复合板,即,FGM表面层和各向同性夹芯层的复合板,以及各向同性表面层和FGM夹芯层的复合板.通过Hamilton原理,得到了FGM复合板的运动方程.得到闭式的Navier解,然后求解特征值问题,得到自由振动的基本频率.将该理论得到的结果,与经典理论,一阶的及其它更高阶的理论所得到的结果进行比较,检验了该理论的有效性.研究发现,该理论在求解FGM复合板自由振动性能方面,既精确又简单.     

Analytical solution of piezolaminated rectangular plates with arbitrary clamped/simply-supported boundary conditions under thermo-electro-mechanical loadings

This paper extends an analytical method for static analysis of general cross-ply piezolaminated rectangular plates with any combination of clamped/simply-supported boundary conditions under uncoupled thermo-electro-mechanical loadings. This method is based on the novel superposition method and the first-order shear deformation theory (FSDT). The FSDT enables this expanded method to consider the effect of shear deformation of the plate. The process of applying electrical and thermal resultants causes some advantages due to its simplicity and less computational process. In this analysis displacement components are written in terms of unknown force and moment resultants. Using Fourier series for displacement components, mechanical, thermal, and/or electrical stress resultants, the complex governing differential equations of the plate are reduced to a set of linear algebraic equations with non-trivial solution. The obtained equations may be solved analytically to determine the unknown stress resultants. Several examples are proposed, and their obtained numerical results are compared with those available in the literature to verify the convergence, high accuracy, and the capability of the present method to analyze the static behavior of piezolaminated plates. It is found that there is high agreement between the present results with those obtained by other investigators.     

Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory

Buckling analysis of functionally graded micro beams based on modified couple stress theory is presented. Three different beam theories, i.e. classical, first and third order shear deformation beam theories, are considered to study the effect of shear deformations. To present a profound insight on the effect of boundary conditions, beams with hinged-hinged, clamped–clamped and clamped–hinged ends are studied. Governing equations and boundary conditions are derived using principle of minimum potential energy. Afterwards, generalized differential quadrature (GDQ) method is applied to solve the obtained differential equations. Some numerical results are presented to study the effects of material length scale parameter, beam thickness, Poisson ratio and power index of material distribution on size dependent buckling load. It is observed that buckling loads predicted by modified couple stress theory deviates significantly from classical ones, especially for thin beams. It is shown that size dependency of FG micro beams differs from isotropic homogeneous micro beams as it is a function of power index of material distribution. In addition, the general trend of buckling load with respect to Poisson ratio predicted by the present model differs from classical one.     

Analysis of Thick Laminated Composite Plates on an Elastic Foundation with the Use of Various Plate Theories

In this study, various theories of composite laminated plates are extended to rectangular composite laminates resting on an elastic foundation. First, an analysis based on the classical theory of laminated plates is employed. Then the first-order Reissner-Mindlin theory is used for analyzing the laminates. At last, the Reddy shear deformation theory, which allows for the transverse shear strains, is applied to the bending analysis of the laminates. In the analysis, the two-parameter Pasternak and Winkler foundations are considered. The accuracy of the present analysis is demonstrated by solving problems numerical results for which are available in the literature. Some numerical examples are presented to compare the three methods and to illustrate the effects of parameters of the elastic foundations on the bending of shear-deformable laminated plates. __________ Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 5, pp. 663–682, September–October, 2005.     

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.     

Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method

This paper presents a novel finite element formulation for static, free vibration and buckling analyses of laminated composite plates. The idea relies on a combination of node-based smoothing discrete shear gap method with the higher-order shear deformation plate theory (HSDT) to give a so-called NS-DSG3 element. The higher-order shear deformation plate theory (HSDT) is introduced in the present method to remove the shear correction factors and improve the accuracy of transverse shear stresses. The formulation uses only linear approximations and its implementation into finite element programs is quite simple and efficient. The numerical examples demonstrated that the present element is free of shear locking and shows high reliability and accuracy compared to other published solutions in the literature.     

韧性金属大变形拟流动角点理论及应用

本文提出韧性金属弹塑性大变形拟流动角点理论（ｑｕａｓｉ－ｆｌｏｗｃｏｒｎｅｒｔｈｅｏｒｙ）．该理论从塑性变形正交法则出发，将”模量衰减函数”及屈服面的尖点效应引入本构模型，从而实现了由正交法则本构模型向非正交法则本构模型以及从塑性加载向物理弹性却载的光滑过渡，使一般无角点各向异性硬化屈服函数与有角点硬化情形相结合成为可能．用于数值模拟各向异性金属薄板单向拉伸失稳与剪切带分析并与实验结果作比较，表明本文理论的有效性．     

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.     

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．     

A 4D Composite Reinforced along Cube Diagonals with a Small Content of Yarns under Large Shear Deformations

The deformation behavior of a 4D composite reinforced along cube diagonals under large shear deformations is examined. The investigation is based on an applied theory which allows one to perform a macromechanical analysis of composite materials with small volume contents of reinforcing yarns to an accuracy sufficient in practice. Qualitative differences between the properties of such composites under large and small shear deformations are revealed. The evolution of the structural geometry of the deformed composite material is traced.     

Analytical solutions of refined plate theory for bending,buckling and vibration analyses of thick plates

Analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates with various boundary conditions are presented using two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using shear correction factor. In addition, it contains only two unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion are derived from Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It is found that the deflection, stress, buckling load, and natural frequency obtained by the present theory match well with those obtained by the first-order and third-order shear deformation theories.     

Analysis of the dynamic response of 3D-braided rectangular plates on an elastic foundation

A new micromacromechanical model of unit cells is suggested. In this model, a 3D-braided composite plate can be a cell system, and the geometry of each cell strongly depends on its position in the cross section of the plate. Based on the Reddy higher-order shear deformation theory of plates and the general von Karman-type equations, analytical solutions for the dynamic response of simply supported 3D-braided rectangular plates resting on a two-parameter (Pasternak-type) elastic foundation and subjected to transverse dynamic loads are found by using the perturbation method. The numerical illustrations concern the effects of geometric parameters, fiber volume fraction, braiding angle, and foundation stiffness. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 861–880, November–December, 2008.     

Stress and strain-inertia gradient elasticity in free vibration analysis of single walled carbon nanotubes with first order shear deformation shell theory

A gradient-enriched shell formulation is introduced in the present study based on the first order shear deformation shell model and the stress gradient and strain-inertia gradient elasticity theories are used for dynamic analysis of single walled carbon nanotubes. It provides extensions of the first order shear deformation shell formulation with additional higher-order spatial derivatives of strains and stresses. The higher-order terms are introduced in the formulation by using the Laplacian of the corresponding lower-order terms. The proposed shell formulation includes two length scale size parameters related to the strain gradients and inertia gradients. The effects of the transverse shear, aspect ratio, circumferential and half-axial wave numbers and length scale parameters on different vibration modes of the single-walled carbon nanotubes are elucidated. The results are also compared with those obtained from a classical shell theory with Sanders–Koiter strain-displacement relationships.     

夹芯梁的精确解法

夹芯梁与普通梁的本质区别在于剪切引起芯层横截面严重的而又不均匀的翘曲变形,其应力分布已远非初等理论所能描述,而正在广泛应用的经典夹层理论却都建立在平面假设基础上,尤其不能正确反映弱芯的轻质夹层结构的行为,本文放弃了不合理的假设,将夹芯梁视为一般层状弹性体,严格按弹性理论导出了既满足控制方程又同时满足全部边界条件、层间的应力及位移的连续条件的封闭解.它可确切地反映夹芯梁的位移形态和应力分布,并从不同角度,包括多种实验和FEM数值解,验证了它的正确性.     

Application of nonclassical models of shell theory to study mechanical parameters of multilayer nanotubes

Stress-strain state of multilayer anisotropic cylindrical shells under a local pressure is studied. Such a problem may model the bending of an asbestos nanotube under the action of a research probe. In earlier works, these authors showed that the application of classical shell theories yields results far from experimental data. More accurate results are obtained by taking into account additional factors, such as the change of the transverse displacement magnitude (according to the Timoshenko-Reissner theory) or the layered structure of asbestos and cylindrical anisotropy (according to the Rodinova-Titaev-Chernykh theory). In the present paper, yet another shell theory, the Palii-Spiro theory, is applied to solve the problem; this theory was developed for shall of average thickness and is based on the following assumptions: (a) the rectilinear fibers of the shell perpendicular to its middle surface before deformation remain rectilinear after deformation; (b) the cosine of the angle between the shell of such fibers and the middle surface of the deformed shell equals the averaged angle of the transverse displacement. Deformation field are studied with the use of nonclassical (the Rodinova-Titaev-Chernykh and Palii-Spiro) shell theories; a comparison with results obtained for three-dimensional models with the use of the Ansys 11 package is performed.     

Transverse shear stress relaxed zigzag theory for cylindrical bending of laminated composite and piezoelectric panels

Cylindrical bending is studied by developing a new zigzag theory which relaxes the zero transverse shear stress condition on the outer surfaces of the panels subjected to transversely applied electromechanical load. The mechanical portion of the transverse displacement approximation in this new shear deformation theory is considered constant as well as non-constant through the development of three models. Unlike the existing zigzag theories which enforce the condition of vanishing transverse shear stresses on outer surfaces of laminates, these new theories relax it. Though the number of primary mechanical variables get increased by four or five or six, the computational cost does not increase appreciably. Approximating the electric potential in each piezoelectric layer as sublayerswise linear, variational principle is applied in deriving equilibrium equations and boundary conditions. Accuracy of the new base model as well as two augmented models is assessed by comparing with elasticity and piezoelasticity solutions. While it is observed that the new base model is highly accurate than the existing zigzag model, the two augmented models do not aid in its further improvement. This is attributed to the fact that layerwise consideration of the transverse displacement, not global consideration, is needed to correctly establish the effect of transverse normal deformation in the laminated composite and smart panel.     

复合材料层合板的分析

在复合材料层合板静力和动力分析方面,本文提出了一个层合板理论.此理论从板的总挠度中分开了由弯曲所产生的挠度W_b与由剪切产生的挠度W_s,因而使得求控制方程的解变得容易了.而且便于讨论和分析横向剪切变形对层合板弯曲,振动和稳定性的影响.     

Analysis of elastomeric composites based on fiber-reinforced systems 3. Two-directional composites

Results of investigation of deformation of elastomeric composite materials with a two-directional reinforcement scheme are presented. The study is performed on the basis of a structural macroscopic theory. The matrix of the composites analyzed is of a poorly compressible material. The fibers of both reinforcing systems are simulated as compressible bodies. Dependences of the parameters of tensile and shear strains on the strain values for different geometries of fiber arrangement are obtained.State Metallurgical Academy of Ukraine, Dnepropetrovsk, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 479–492, May–June, 1999.