Quasi-three-dimensional theory for solution of dynamic thermoelastic problem of laminated composite plates

An uncoupled dynamic thermoelastic problem for laminated composite plates has been considered. The hypotheses used take into account the nonlinear distribution of temperature and displacements over the thickness of a laminated plate. On the basis of these hypotheses a quasi-three-dimensional (layerwise) theory is constructed that makes it possible to investigate the internal thermal and stress-strain states, as well as the edge effects of the boundary layer type for laminated plates. Systems of the heat conduction and motion equations are derived using the variational method. The order of the equations depends on the number of layers and terms in expansions of temperature and displacements of each layer. An analytical solution of the dynamic thermoelastic problem is presented for a cross-ply laminated rectangular plate with simply supported edges. The reliability of the results is confirmed by a comparison with the known exact solutions. The results based on the proposed theory can be used for verifying various two-dimensional plate theories when solving the dynamic thermoelastic problems for laminated composite plates.     

Free vibration analysis of piezoelectric coupled annular plates with variable thickness

This paper presents the free vibration analysis of piezoelectric coupled annular plates with variable thickness on the basis of the Mindlin plate theory. No work has yet been done on piezoelectric laminated plates while the thickness is variable. Two piezoelectric layers are embedded on the upper and lower surfaces of the host plate. The host plate thickness is linearly increased in the radial direction while the piezoelectric layers thicknesses remain constant along the radial direction. Different combinations of three types of boundary conditions i.e. clamped, simply supported, and free end conditions are considered at the inner and outer edges of plate. The Maxwell static electricity equation in piezoelectric layers is satisfied using a quadratic distribution of electric potential along the thickness. The natural frequencies are obtained utilizing a Rayleigh–Ritz energy approach and are validated by comparing with those obtained by finite element analysis. A good compliance is observed between numerical solution and finite element analysis. Convergence study is performed in order to verify the numerical stability of the present method. The effects of different geometrical parameters such as the thickness of piezoelectric layers and the angle of host plate on the natural frequencies of the assembly are investigated.     

Heat transfer and Couette flow of a chemically reacting non‐linear fluid

In this paper we study the flow and heat transfer in a chemically reacting non‐linear fluid between two long horizontal parallel flat plates that are at different temperatures. The top plate is sheared, whereas the bottom plate is fixed. The fluid is modeled as a generalized power‐law fluid whose viscosity is also assumed to be a function of the concentration. The effects of radiation are neglected. The equations are made dimensionless and the boundary value problem is solved numerically; the velocity and temperature profiles are obtained for various dimensionless numbers. Published in 2009 by John Wiley & Sons, Ltd.     

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.     

New benchmark solutions for free vibration of clamped rectangular thick plates and their variants

It is of significance to explore benchmark analytic free vibration solutions of rectangular thick plates without two parallel simply supported edges, because the classic analytic methods are usually invalid for the problems of this category. The main challenge is to find the solutions meeting both the governing higher order partial differential equations (PDEs) and boundary conditions of the plates, i.e., to analytically solve associated complex boundary value problems of PDEs. In this letter, we extend a novel symplectic superposition method to the free vibration problems of clamped rectangular thick plates, with the analytic frequency solutions obtained by a brief set of equations. It is found that the analytic solutions of clamped plates can simply reduce to their variants with any combinations of clamped and simply supported edges via an easy relaxation of boundary conditions. The new results yielded in this letter are not only useful for rapid design of thick plate structures but also provide reliable benchmarks for checking the validity of other new solution methods.     

On the Cauchy problem for thermoelastic plates

The Cauchy problem for an infinite thermoelastic plate with a non‐homogeneous governing system and homogeneous initial conditions is solved by means of an area potential. This is the first step in the construction of a potential theory for time‐dependent problems for thermoelastic plates, enabling the reduction of various initial‐boundary value problems to their versions for the homogeneous system of equations with homogeneous initial conditions, which, in turn, may then be solved by means of dynamic potentials. Copyright © 2006 John Wiley & Sons, Ltd.     

On the heat transfer from a finite plate in channel flow for vanishing Prandtl number

Summary The temperature field produced by a finite, hot plate at zero incidence in uniform channel flow is solved exactly for the limiting case of zero prandtl number by means of the Wiener-Hopf technique. The heat transfer on the plate is found to agree with the corresponding boundary layer result over most of the plate for Péclét numbers as low as ten. Extensions to similar Ossen-flow problems are indicated.     

A method of asymptotic expansions of the solutions of the steady heat conduction problem for laminated non-uniform anisotropic plates

Outer asymptotic expansions of the solutions of the steady heat conduction problem for laminated anisotropic non-uniform plates for different boundary conditions on the faces are constructed. The two-dimensional resolvents obtained are analysed and the asymptotic properties of the solutions of the heat-conduction problem are investigated. Estimates are obtained of the accuracy with which the temperature in the plate outside the limits of the boundary layer can be assumed to be piecewise-linearly or piecewise-quadratically distributed over the thickness of the laminated structure. A physical justification for certain features of the asymptotic expansions of the temperature is given.     

On the boundary value problem for the spectrally loaded heat conduction operator

We consider the boundary value problems in a quarter-plane for a loaded heat conduction operator (one-dimensional in the space variable). A peculiarity of the operator in question is as follows: first, the spectral parameter is the coefficient of the loaded summand; second, the order of the derivative in the loaded summand is equal to that of the differential part of the operator, and third, the load point moves with a variable velocity. We demonstrate that the boundary value problem under study is Noetherian.     

Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported

England (2006) [13] proposed a novel method to study the bending of isotropic functionally graded plates subject to transverse biharmonic loads. His method is extended here to functionally graded plates with materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the three-dimensional theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable, in which the unknown constants can be determined from the boundary conditions similar to that in the classical plate theory. The elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations. A comparison of the present results for a uniform load with existing solutions is made and good agreement is observed. The influence of boundary conditions, material inhomogeneity, and thickness to length ratio on the plate deflection and stresses for the load x²yq are studied numerically.     

On the three-dimensional thermostressed state of rectangular plates under nonuniform heating

We consider three-dimensional boundary-value problems of the stationary theory of heat conduction and thermoelasticity for rectangular homogenous isotropic plates of arbitrary thickness. It is assumed that the temperature or heat flux density prescribed on the top and bottom surfaces admit a representation in the form of double trigonometric series. A closed-form analytic solution is obtained for the boundary-value problems of thermoelasticity in the case of plates with contacting edges along the lateral faces. Numerical computations are given for three types of boundary-value problems using the software package mathcad PLUS 6.0 for thin and thick plates. We construct the graphs of variation of the temperature, deflection, and normal stresses over the thickness of the plate. Three figures, 1 table. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp 18–26.     

The null field equations for flexural oscillations of elastic plates

A null field method is constructed to solve the exterior Dirichlet, Neumann, and Robin boundary value problems associated with the high‐frequency harmonic oscillations of Mindlin‐type plates. The case of an infinite plate with a bounded elastic inclusion is also considered. Additionally, the completeness of certain sets of wavefunctions is investigated. Copyright © 2012 John Wiley & Sons, Ltd.     

Inelastic buckling of skew and rhombic thin thickness-tapered plates with and without intermediate supports using the element-free Galerkin method

In this paper, skew and rhombic isotropic plates subjected to in-plane loadings are analyzed using the element-free Galerkin method. Inelasticity effect is included in the buckling analysis while plates are thin thickness-tapered type. The governing differential equation for a plate in plastic range of response is numerically solved using the Galerkin method. The shape functions are constructed using the moving least squares (MLS) approximation and the essential boundary conditions are introduced into the formulation through the use of the Lagrange multiplier method and the orthogonal transformation techniques. The Stowell theory for the plastic buckling of flat skew plates with variable thicknesses is used. The inelastic analysis is based on the Ramberg–Osgood representation of the stress–strain curve which is used in the deformation theory of plasticity. Using this method the initial inelastic local buckling of skew plates with or without intermediate line supports is studied. Stiffness and geometric matrices are formulated by weak form of the Galerkin method. Finally, the inelastic local buckling loads of these plates are obtained and the results are compared with known solutions in the literature.     

正交各向异性功能梯度层合板稳态热传导问题的精确解

对轴对称正交各向异性功能梯度层合圆板稳态热传导问题进行精确分析.假设材料热传导率沿板厚方向按指数函数形式梯度分布,从正交各向异性功能梯度圆板稳态热传导的基本方程出发,利用分离变量法,获得了在上、下表面作用任意热分布情况下的精确解.通过数值算例的分析,指出材料性质的梯度变化、板厚边界条件等分析了对温度场分布的影响.所获得的精确结果,可以作为评价其它近似方法的标准解答.     

Analytical bending solutions of clamped rectangular thin plates resting on elastic foundations by the symplectic superposition method

In this paper, the analytical bending solutions of clamped rectangular thin plates resting on elastic foundations are obtained by a rational symplectic superposition method which is based on the Hamiltonian system. The proposed method is capable of solving the plate problems with different boundary conditions via a step-by-step derivation without any trial solutions. The presented solution procedure can be extended to more boundary value problems in engineering.     

A rigid inclusion in the contact problem for elastic plates

A family of problems under consideration describes the contact of elastic plates situated at a given angle to each other and, in the natural condition, touching along a line. The plates are subjected only to bending. The limiting process from the elastic inclusion to the rigid one is studied. It is demonstrated that the limit problems precisely describe the contact of an elastic plate with a rigid beam and the problem of the equilibrium of an elastic plate with a rigid inclusion. The solvability of the problems is established; the boundary conditions holding on the possible contact set are found as well as their precise interpretation.     

An analytical solution for bending of transversely isotropic thick rectangular plates with variable thickness

In this study, the bending solution of simply supported transversely isotropic thick rectangular plates with thickness variations is provided using displacement potential functions. To achieve this purpose, governing partial differential equations in terms of displacements are obtained as the quadratic and fourth order. Then, the governing equations are solved using the separation of variables method satisfying exact boundary conditions. The advantage of the purposed method is that there is no limitation on the thickness of the plate or the way the plate thickness is being varied. No simplifying assumption in the analysis process leads to the applicability and reliability of the present method to plates with any arbitrarily chosen thickness. In order to confirm the accuracy of the proposed solution, the obtained results are compared with existing published analytical works for thin variable thickness and thick constant thickness plate. Also, due to the lack of analytical research on thick plates with variable thickness, the obtained results are verified using the finite element method which shows excellent agreement. The results show that the maximum displacement of the plates with variable thickness is moved from the center toward the thinner plate edge. In addition, results exhibit the profound effects of both thickness and aspect ratio on stress distribution along the thickness of the plate. Results also show that varying thickness has not a profound impact on bending and twisting moments in transversely isotropic plates. Five different materials consist of four transversely isotropic and one isotropic, as a special case, are considered in this paper, which it is shown that the material properties have a more considerable impact on higher thickness plate.     

Application of Neumann-Ulam scheme to solving the first boundary value problem for a parabolic equation

Statistical estimates of the solutions of boundary value problems for parabolic equations with constant coefficients are constructed on paths of random walks. The phase space of these walks is a region in which the problem is solved or the boundary of the region. The simulation of the walks employs the explicit form of the fundamental solution; therefore, these algorithms cannot be directly applied to equations with variable coefficients. In the present work, unbiased and low-bias estimates of the solution of the boundary value problem for the heat equation with a variable coefficient multiplying the unknown function are constructed on the paths of a Markov chain of random walk on balloids. For studying the properties of the Markov chains and properties of the statistical estimates, the author extends von Neumann-Ulam scheme, known in the theory of Monte Carlo methods, to equations with a substochastic kernel. The algorithm is based on a new integral representation of the solution to the boundary value problem.     

基于简化的应变梯度理论下Kirchhoff板模型边值问题的提法及其应用北大核心CSCD

考虑应变梯度和速度梯度的影响,建立薄板控制微分方程及给出其边值问题的提法,修正了前人给出的薄板角点条件.采用Levy法,给出受分布力作用下简支板的挠度及自由振动频率的解析解.通过与文献中分子动力学数据对比,验证了该文模型的有效性并提出校核材料参数的一种方法.研究结果表明,增大弹性地基和应变梯度参数可以有效提高板的等效刚度,而速度梯度参数则相反.该文提出的板的边值问题为研究薄板在复杂支撑边界及外荷载等条件提供了理论依据.同时,有望为其有限元法、有限差分法和基于能量原理的Galerkin法等数值方法提供理论依据.     

Solution of the Basic Boundary Value Problems of Stationary Thermoelastic Oscillations for Domains Bounded by Spherical Surfaces

The boundary value problems of stationary thermoelastic oscillations are investigated for the entire space with a spherical cavity, when the limit values of a displacement vector and temperature or of a stress vector and heat flow are given on the boundary. Also, consideration is given to the boundary-contact problems when a nonhomogeneous medium fills up the entire space and consists of several homogeneous parts with spherical interface surfaces. Given on an interface surface are differences of the limit values of displacement and stress vectors, also of temperature and heat flow, while given on a free boundary are the limit values of a displacement vector and temperature or of a stress vector and heat flow. Solutions of the considered problems are represented as absolutely and uniformly convergent series.