Electro-dynamic analysis of smart nanoclay-reinforced plates with integrated piezoelectric layers

Integrating engineering structures with piezoelectric layers as actuator and/or sensor offers smart sandwich structures with controllable static and dynamic deflections. In this paper, a smart sandwich plate consisting of a light nanoclay-reinforced composite core and two piezoelectric face sheets is considered. The static and dynamic behaviors of the proposed smart plate are obtained using their governing coupled electro-mechanical system of equations. In order to facilitate the governing equations, a mesh-free method based on moving least square (MLS) shape function and first order shear deformation (FSDT) is developed and implemented. Two morphologies of intercalated stack and exfoliated nanoclay dispersions are considered in the distribution of the nanoclay into the polymeric matrix. The effects of morphology and volume fraction of the nanoclay, time-dependent loading, and essential boundary condition on the static and dynamic behavior of the smart piezoelectric-integrated nanocomposite plates are examined. In the dynamic analysis, resonance and amplitude modulation phenomena are studied. It is observed that the use of nanoclay, especially with exfoliated morphology, improves the static and the free vibration responses of the smart sandwich plates. Moreover, the frequency of the applied mechanical load has a significant effect on the electro-dynamic response of the proposed smart sandwich plates.     

Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods

This article introduces a coupled methodology for the numerical solution of geometrically nonlinear static and dynamic problem of thin rectangular plates resting on elastic foundation. Winkler–Pasternak two-parameter foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the discrete singular convolution (DSC) and harmonic differential quadrature (HDQ) methods, respectively. Two different realizations of singular kernels such as the regularized Shannon’s kernel (RSK) and Lagrange delta (LD) kernel are selected as singular convolution to illustrate the present DSC algorithm. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse, an N-wave pulse, and a triangular load are investigated and the results are presented graphically. The effects of Winkler and Pasternak foundation parameters, influence of mass of foundation on the response have been investigated. In addition, the influence of damping on the dynamic analysis has been studied. The accuracy of the proposed DSC–HDQ coupled methodology is demonstrated by the numerical examples.     

Nonlinear flexural response of laminated composite plates under hygro-thermo-mechanical loading

The paper deals with Chebyshev series based analytical solution for the nonlinear flexural response of the elastically supported moderately thick laminated composite rectangular plates subjected to hygro-thermo-mechanical loading. 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 elastic and hygrothermal properties of the fiber reinforced composite material are considered to be dependent on temperature and moisture concentration and have been evaluated utilizing micromechanics model. 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 Winkler and Pasternak foundation parameters, temperature and moisture concentration on nonlinear flexural response of the laminated composite rectangular plate with different lamination scheme and boundary conditions are presented.     

损伤对粘弹性矩形板非线性动力特性的影响

研究了考虑损伤效应的粘弹性矩形板在横向周期载荷作用下的非线性动力学问题．基于Von Karman方程、Boltzmann叠加原理和连续损伤力学理论,建立了以中面位移表示的考虑损伤效应的粘弹性薄板的非线性动力学方程,然后,应用有限差分法和Newmark法进行求解,并与相应的文献作出了比较．具体讨论了外载荷参数和板的几何尺寸对含损伤效应的粘弹性板非线性动力响应的影响．数值结果表明,考虑损伤效应时,结构的非线性动力响应会发生显著的变化．     

线载和弹性支承作用面内运动薄板磁固耦合双重共振

针对磁场环境中具有线载荷和弹性支承作用的面内运动薄板,给出了系统的势能、动能及电磁力表达式,应用Hamilton变分原理,推得面内运动条形板的磁固耦合非线性振动方程.考虑边界为夹支-铰支的约束条件,利用变量分离法和Galerkin积分法,得到了含简谐线载力和电磁阻尼力项的两自由度非线性振动微分方程组.应用多尺度法对主-...     

Bending,buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory

In the present study, higher order shear and normal deformable plate theory is developed for analysis of incompressible functionally graded rectangular thick plates. Also, The effect of incompressibility is studied on the static, dynamic and stability responses of thick plate. It is assumed that plate is incompressible and the incompressibility condition is considered in addition to the governing equations for determining the unknowns. Since the plate is thick, higher order shear and normal deformable theory is applied so that the Legendre polynomials are used for expansion of displacement field components in the thickness direction. Also, it is supposed that material properties vary through the thickness based on the power law function. Utilizing the variational approach, governing equations for static, stability and dynamic analysis of plate are derived. Resulted equations are solved analytically for simply supported plates. Finally, the effects of material properties and dimensions on the response of incompressible plates are investigated in details.     

Assessing the mechanical responses for anisotropic multi-layered medium under harmonic moving load by Spectral Element Method (SEM)

The article is concerned with the mechanical responses of anisotropic multi-layered medium under harmonic moving load. An analytical solution for two-dimensional anisotropic multi-layered medium subjected to harmonic moving load is devoted via Spectral Element Method (SEM), while the anisotropic property is approximated as transverse isotropy. Starting with the constitutive equations of transversely isotropic body and the governing equations of motion based on the loading properties. The analytical spectral elements in the wavenumber domain are obtained according to the principle of wave superposition and Fourier transformation. Then, the spectral global stiffness matrix of the multi-layered medium is derived by assembling the nodded stiffness matrices of all layers depended on the different interlayer conditions between the adjacent layers, i.e. sliding and bonded. The corresponding analytical solutions are achieved by taking the Fourier series and Inverse Fast Fourier Transform (IFFT) algorithm. Finally, some examples are given to validate the accuracy of the proposed analytical solution, and to demonstrate the impact of both anisotropy, top layer thickness, interlayer conditions, and loading properties (velocity and natural frequency) on the mechanical response of the multi-layered medium.     

Rectangular anisotropic (orthotropic) plates on a tensionless elastic foundation

The behavior of anisotropic (orthotropic) elastic plates of rectangular shape on a tensionless Winkler foundation is analyzed. The tensionless character of the foundation is taken into account by using an auxiliary function. The displacement function of the plate is approximated by using the eigenfunctions of the completely free beam. The difference between the free-end boundary conditions of the plate and the beam is compensated for by considering a differential operator in addition to the governing equation of the plate. Using Galerkin's method, the problem is reduced to the solution of a system of algebraic equations. The governing equations of the plate are derived under action of external uniformly distributed load, concentrated load, and moments. However, the influence of the mechanical properties on the configurations of the contact region and on the distribution of the displacements is investigated for concentrated load and moments for various values of the mechanical properties characterizing the anisotropy of the plate material. Considered problems are solved within the framework of Kirchhoff-Love hypothesis.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 3, pp. 378–386, May–June, 1995.     

粘性流体两相多孔介质非线性动力问题的罚有限元法

采用基于混合物理论的多孔介质模型,给出粘性流体饱和两相多孔介质非线性动力问题的控制场方程以及相应边值和初值问题的提法,用Galerkin加权残值法导出罚有限元公式,并给出该非线性方程组的迭代求解方法。考虑了体积分数和渗透率与变形相关的情况。用编制的有限元程序计算分析了一维多孔柱体在脉冲载荷作用下的瞬态响应,数值结果表明文中方法正确有效。     

四边简支对称正交层合矩形板的非线性弯曲问题

本文在von Kármán型板理论的基础上,采用双重Fourier级数方法,研究了对称正交层合矩形板在简支边条件下,承受任意分布横向载荷和面内载荷联合作用的非线性弯曲问题,得到了满足控制方程和边界条件的解.     

Influence of nonlinear terms on dynamical behavior of graphene reinforced laminated composite plates

This research is focused on the effects of nonlinear terms on the dynamical behavior of graphene reinforced laminated composite plates. Firstly, the governing equations of the graphene reinforced composite thin plate subjected to transverse excitations are derived by using the Hamilton's principle and the von Karman deformation theory. Then numerical method is applied to investigate the nonlinear behaviors of graphene reinforced composite plates. Bifurcation diagram, waveform and phase portrait are demonstrated to analyze the nonlinear dynamics of the graphene reinforced laminated composite plates. Furthermore, the effects of nonlinear terms on the dynamical behavior are discussed in detail, where both the stronger and weaker nonlinear characteristics of lower modes of the plate are presented. Moreover, some interesting phenomena are obtained in numerical simulation.     

Optimization of dynamic mechanical response of a composite plate using multi-field coupling with thermal constraints

We consider the problem of optimizing the dynamic response of a mechanically loaded, rectangular, electrically conductive anisotropic composite plate by applying an electromagnetic field, which exploits the electro-magneto-mechanical field coupling phenomenon. An important aspect of the formulated nonlinear partial differential equation (PDE)-constrained optimization model is the presence of a thermal constraint that prevents polymer matrix degradation in the composite material due to Joule heating produced by the electromagnetic field. A black-box optimization approach based on the active set algorithm is employed. A system of governing PDEs is solved using a series of sequential numerical procedures that includes the method of lines, Newmark time-stepping scheme, quasilinearization, integration of two-point boundary-value problems, and a superposition method followed by orthonormalization. Implementation in hyper-dual arithmetics facilitated automatic differentiation and computation of the gradient. Optimization results show that application of an electromagnetic field with optimal characteristics enables one to significantly reduce the amplitude of the plate vibrations while controlling for Joule heating.     

Local radial basis function collocation method for bending analyses of quasicrystal plates

The local radial basis function collocation method (LRBFCM) is proposed for plate bending analysis in orthorhombic quasicrystals (QCs) under static and transient dynamic loads. Three common types of the plate bending problems are considered: (1) QC plates resting on Winkler foundation (2) QC plates with variable thickness and (3) QC plates under a transient dynamic load. According to the Reissner–Mindlin plate bending theory, there is allowed to simulate the behavior of the two excitations in QC plates, phonon and phason, by 2D strong formulations for the system of governing equations. The governing equations, which describe the phason displacements, are based on Agiasofitou and Lazar elastodynamic model. Numerical results demonstrate the effect of the elastic foundation, as well as plate thickness on the phonon and phason characteristics in this paper. For the transient dynamic analysis, the influence of the phason friction coefficients on the responses of QC plate to transient dynamic loads is also studied.     

主被动阻尼层合板结构的自由振动和阻尼特性分析

给出了含主被动阻尼非对称复合材料层合板结构的振动微分方程;得到了在压电材料和高粘弹材料作主被动阻尼层情况下,简支层合板结构自由振动的自然频率和损失因子的解析解;分析了正逆向压电效应对自然频率和损失因子的影响     

Delamination growth of laminated circular plates under in-plane loads and movable boundary conditions

An investigation is made on interlaminar delamination growth of composite laminated circular plates under in-plane loads and movable delamination boundary conditions. A four-dissociated-region model is developed on the basis of von-Karman plate theory. The model is geometrically nonlinear and the laminated circular plate considered is subjected to axisymmetrical delamination. The effects of transverse shear deformation and contact effect of the delamination on the laminated plates are taking into account in the development of the governing equations of the laminated circular pates with random axisymmetrical delamination. The formulas for describing the total energy release rate and its individual mode components along the delamination front are also derived with considerations of Griffith criterion for fracture. Based on the model established, the delamination growth is numerically studied; and the influences of the parameters such as delamination radii and depths, together with material properties of the plates on the energy release rate are analyzed in detail.     

Geometrically nonlinear static and dynamic analysis of functionally graded skew plates

The present paper deals with nonlinear static and dynamic behavior of functionally graded skew plates. The equations of motion are derived using higher order shear deformation theory in conjunction with von-Karman’s nonlinear kinematics. The physical domain is mapped into computational domain using linear mapping and chain rule of differentiation. The spatial and temporal discretization is based on fast converging finite double Chebyshev series and Houbolt’s method. Quadratic extrapolation technique is employed to linearize the governing nonlinear equations. The spatial and temporal convergence and validation studies have been carried out to establish the efficacy of the present solution methodology. In case of dynamic analysis, the results are obtained for uniform step, sine, half sine, triangular and exponential type of loadings. The effect of volume fraction index, skew angle and boundary conditions on nonlinear displacement and moment response are presented.     

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.     

Thermo-mechanical post-critical analysis of nonlocal orthotropic plates

A closed-form analytical solution for critical temperature and nonlinear post-critical temperature-deflection behaviour for nonlocal orthotropic plates subjected to thermal loading is presented. The long-range molecular interactions are represented by a nonlocal continuum framework, including orthotropy. The Von-Karman nonlinear strains are employed in deriving the governing equations. An approximate solution to the system of nonlinear partial differential equations is obtained using a perturbation type method. Series expansions up to second order of the associated field variables and the load parameter, dictating nonlinearity are employed. The behaviour in the post-critical regime is illustrated numerically by adopting an example of orthotropic Single Layer Graphene Sheet (SLGS), a widely acclaimed nano-structure, often modelled as plate. Post-critical temperature-deflection paths are presented with special emphasis on their post-critical reserve in strength and stiffness. Influence of aspect ratio and behaviour in higher modes are demonstrated. Implications of nonlocal interactions on the redistribution of in-plane forces are presented to show striking disparity with the classical plates. The obtained solution may serve as benchmark for verification of numerical solutions and may be useful in formulating simple design guidelines for plate type nanostructures liable to the thermal environment.     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

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.