Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory

Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail.     

Geometrically nonlinear bending analysis of laminated composite plate

In this work, a transverse bending of shear deformable laminated composite plates in Green–Lagrange sense accounting for the transverse shear and large rotations are presented. Governing equations are developed in the framework of higher order shear deformation theory. All higher order terms arising from nonlinear strain–displacement relations are included in the formulation. The present plate theory satisfies zero transverse shear strains conditions at the top and bottom surfaces of the plate in von-Karman sense. A C⁰ isoparametric finite element is developed for the present nonlinear model. Numerical results for the laminated composite plates of orthotropic materials with different system parameters and boundary conditions are found out. The results are also compared with those available in the literature. Some new results with different parameters are also presented.     

Application of the Haar wavelet-based discretization technique to problems of orthotropic plates and shells

The Haar wavelet discretization technique for solving the elastic bending problems of orthotropic plates and shells is proposed. Free transverse vibrations of orthotropic rectangular plates with a variable thickness in one direction are considered as a model problem. In the case of constant plate thickness, the numerical results are validated by comparing them with an exact solution. The results obtained are found to be in good agreement with those available in the literature.     

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．     

Refined Orthotropic Plate Motion Equations for Acoustoelasticity Problem Statement

The formulation of the acoustoelasticity problem is given on the basis of refined motion equations of orthotropic plates. These equations are constructed in the first approximation by reducing the three-dimensional equations of the theory of elasticity to the two-dimensional equations of the theory of plates, where the approximation of the transverse tangential stresses and the transverse reduction stress is made with the help of trigonometric basis functions in the thickness direction. Wherein at the points of the boundary (front) surfaces, the static boundary conditions of the problem for tangential stresses are satisfied exactly and for transverse normal stress — approximately. Accounting for internal energy dissipation in the plate material is based on the Thompson—Kelvin—Voigt hysteresis model. In case of formulating problems on dynamic processes of plate deformation in vacuum, the equations are divided into two separate systems of equations. The first of these systems describes non-classical shear-free, longitudinal-transverse forms of movement, accompanied by a distortion of the flat form of cross sections, and the second system describes transverse bending-shear forms of movement. The latter are practically equivalent in quality and content to the analogous equations of the well-known variants of refined theories, but, unlike them, with a decrease in the relative thickness parameter, they lead to solutions according to the classical theory of plates. The motion of the surrounding the plate acoustic media is described by the generalized Helmholtz wave equations, constructed with account of energy dissipation by introducing into consideration the complex sound velocity according to Skudrzyk.     

Mechanics of Axially Moving and Vibrating in Transverse Direction Orthotropic Thermoelastic Web

We consider deformations and stability of axially moving orthotropic thermoelastic web. The web is modelled by a thin continuous plate moving with a constant velocity with small transverse vibrations and supported by a system of rollers. It is supposed that the plate is subjected to a combined thermomechanical loading including pure mechanical in-plane tension and also centripetal forces. Thermal strains corresponding to thermal tension and bending of the moving plate are taken into account. We formulate and analytically investigate the problem of out-of-plane thermomechanical divergence of orthotropic plate.     

Transverse vibration of nonhomogeneous orthotropic viscoelastic circular plate of varying parabolic thickness

An analysis of the transverse vibration of nonhomogeneous orthotropic viscoelastic circular plates of parabolically varying thickness in the radial direction is presented. The thickness of a circular plate varies parabolically in a radial direction. For nonhomogeneity of the circular plate material, density is assumed to vary linearly in a radial direction. This paper used the method of separation of variables in solving the governing differential equation. In this paper, an approximate but quite convenient frequency equation is derived by using the Rayleigh–Ritz technique with a two‐term deflection function. Deflection, time period and logarithmic decrement for the first two modes of vibration are computed for the nonhomogeneous orthotropic viscoelastic circular plates of varying parabolic thickness with clamped edge conditions for various values of nonhomogeneity constants and taper constants and these are shown in tabular form for the Voigt–Kelvin model. Copyright © 2011 John Wiley & Sons, Ltd.     

均布载荷下正交异性变厚度圆薄板大挠度问题

本文导出了正交各向异性变厚度圆薄板大挠度问题的基本方程,用修正迭代法求解了正交各向异性变厚度圆薄板在均布载荷下的大挠度问题.作为特例,令ε=0,则由本文结果得到的表达式与J.Nowinski用摄动法得到的正交各向异性等厚度圆薄板大挠度问题的解完全一致.     

The dynamic stressed state of a rectangular composite plate rigidly fixed along its edges

Using the generalized dynamic theory of bending of plates, which takes account of the compliance of the material to transverse shear strains, we obtain an approximate solution of the problem of the dynamic stressed state of a composite plate with rigidly fixed edges subject to an impact load. We exhibit the contribution of the physico-mechanical and geometric parameters of the plate to the magnitude of the computed stresses and their time variation for different coefficients of internal dissipation of mechanical energy.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 129–133.     

剪切变形对直线型正交异性层合圆板大幅度受迫振动的影响*

本文研究了计及横向剪切变形的直线型正交异性层合圆板在简谐载荷q₀cosωt作用下的非线性受迫振动问题.采用伽辽金方法得到强振频率与振幅关系的解析解.最后,分析了横向剪切对板振动的影响,并给出了板的非线性自由振动的非线性周期对线性周期的比值.     

First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich-Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two-dimensional orthotropic thick plate theories with stress or mixed edge data.     

Study of wave processes in shells and plates with regard to transverse normal and shear deformation

A variant of the theory of orthotropic plates and cylindrical shells taking account of transverse normal and shear deformation was examined. Independent approximations were adopted for distribution of displacements and stresses over the thickness of the shell. One of the requirements for constructing the theory is physical correctness, which is achieved by utilizing variational methods for formulating the mathematical model. The Reissner principle for dynamic processes was used for derivation of the equations. The elliptical part of the starting differential operator was shown to be symmetrical and positive in the space of the integrate of square functions. We examined the problem of the propagation of axially symmetric harmonic waves in the cylinder using the starting differential equations. These results were compared with those obtained equations derived in elasticity theory. Analysis of induced vibration was carried out for the case of a square plate upon the action of a suddenly applied load.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, Latvia, October, 1995.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6. pp. 816–823, November–December, 1995.     

On the theory of thermoelastic bending of reinforced shells and plates

The author examines an approach to the construction of a theory of thermoelastic bending of arbitrarily reinforced shells and plates that takes into account the actual structure, the deformation characteristic, and the actual thermomechanical properties of the elements which make up the composite. The final equations are obtained with and without allowance for transverse shear strains. The problem of the thermoelastic bending of a thin, arbitrarily reinforced rectangular plate hinged at the edges, when transverse shear strains can be neglected, is considered as an example.Institute of Hydrodynamics, Siberian Division, Academy of Sciences of the USSR, Novosibirsk. Translated from Mekhanika Polimerov, No. 5, pp. 861–873, September–October, 1972.     

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.     

On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates

Exact bending solutions of fully clamped orthotropic rectangular thin plates subjected to arbitrary loads are derived using the finite integral transform method. In the proposed mathematical method one does not need to predetermine the deformation function because only the basic governing equations of the classical plate theory for orthotropic plates are used in the procedure. Therefore, unlike conventional semi-inverse methods, it serves as a completely rational and accurate model in plate bending analysis. The applicability of the method is extensive, and it can handle plates with different loadings in a uniform procedure, which is simpler than previous methods. Numerical results are presented to demonstrate the validity and accuracy of the approach as compared with those previously reported in the bibliography.     

Damping sources interactions in impact of viscoelastic composite plates with damping treated SMA wires,using a hyperbolic plate theory

In the present paper, a comprehensive study is made on effects of the viscoelastic and the phase-transformation-based dissipations and their interactions on impact responses of viscoelastic composite plates with damping treated (structural hierarchy) shape memory alloy (SMA) wires, for the first time. In contrast to almost all of the available researches, a high-order hyperbolic plate theory that includes not only odd but also even functions of the transverse coordinate, is proposed and employed here. While a hierarchical viscoelastic constitutive law is employed for both the orthotropic and SMA materials, Brinson's constitutive law is refined to include the loading fluctuations and structural hierarchy of the SMA wire, simultaneously. The traditional Hertz and Yang-Sun contact laws are modified accordingly. The resulting highly nonlinear piecewise-defined integro-differential finite element governing equations are solved by an iterative algorithm within each time step. The presented discussions show that in contrast to the common belief, the zero-shear traction condition on the top and bottom surfaces of the viscoelastic orthotropic plate cannot be satisfied by the available plate theories, even for the symmetric lamination schemes. Results show that the viscoelasticity and phase-transformation effects on the resulting dynamic responses are more pronounced for the low and high energy impacts, respectively.     

功能梯度板的柱面弯曲弹性力学解

在推广后的England-Spencer功能梯度板理论基础上,研究了功能梯度板在不同荷载作用下的柱面弯曲问题.采用该理论中的位移展开公式,并且材料参数沿板厚方向可以任意连续变化,并将材料由各向同性推广到正交各向异性.假设板在y方向无限长,最终建立了一个从弹性力学理论出发的正交各向异性功能梯度板在横向分布荷载作用下柱面弯曲问题的板理论.通过算例分析,讨论了边界条件、材料梯度及板厚跨比等因素对功能梯度板静力响应的影响.     

Bending of plates made of oriented glass-reinforced plastics

A plate made of oriented glass-reinforced plastic (GRP) is treated as a regular medium with numerous anisotropic layers. The problem of bending is solved in the elastic formulation of the problem without the use of the hypothesis of undeformed normals and a preliminary specification of the law of distribution of shearing stresses in transverse planes. The important effect of shear strains on the deflections and stresses in plates of oriented GRP is illustrated in the example of a simply supported square plate with transverse isotropy. An estimate is given of the errors introduced by arbitrary specification of the law of distribution of shearing stresses.Mekhanika Polimerov, Vol. 1, No. 3, pp. 129–136, 1965     

Stress and fracture analysis in delaminated orthotropic composite plates using third-order shear deformation theory

The third-order shear deformable plate theory is applied in this work to calculate the stresses and energy release rates in delaminated orthotropic composite plates with straight crack front. The delaminated parts are modeled by the general third-order plate theory, while a double-plate model with interface constraint is developed for the uncracked portion of the plate. The governing equations of the uncracked part are formulated by considering the equilibrium and the displacement continuity along the interface. As an example, a simply-supported delaminated orthotropic plate subjected to a point force is solved adopting Lévy plate formulation and the state-space approach. The mode-II and mode-III energy release rate distributions along the crack front were calculated by the J-integral. To verify the analytical results the 3D finite element model of the plate was constructed and the energy release rates were calculated by the virtual crack-closure technique. A previous second-order plate theory solution was also utilized in the course of the comparison. The results indicate a good agreement between analysis and numerical computation and that third-order theory is better in some cases than the second-order approximation.