Three-dimensional thermoelastic analysis of finite length laminated cylindrical panels with functionally graded layers

Based on the 3D thermoelasticity theory, the thermoelastic analysis of laminated cylindrical panels with finite length and functionally graded (FG) layers subjected to three-dimensional (3D) thermal loading are presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The variations of the field variables across the panel thickness are accurately modeled by using a layerwise differential quadrature (DQ) approach. After validating the approach, as an important application, two common types of FG sandwich cylindrical panels, namely, the sandwich panels with FG face sheets and homogeneous core and the sandwich panels with homogeneous face sheets and FG core are analyzed. The effect of micromechanical modeling of the material properties on the thermoelastic behavior of the panels is studied by comparing the results obtained using the rule of mixture and Mori–Tanaka scheme. The comparison studies reveal that the difference between the results of the two micromechanical models is very small and can be neglected. Then, the effects of different geometrical parameters, material graded index and also the temperature dependence of the material properties on the thermoelastic behavior of the FG sandwich cylindrical panels are carried out.     

Accurate dynamic analysis of functionally graded beams under a moving point load

Based on the physical neutral surface, an N-node novel weak form quadrature beam element is proposed and the explicit formulas for computing the stiffness and mass matrices are given. The proposed element is then used to analyze the dynamic behavior of the functionally graded material (FGM) beams under a moving point load. Both elasticity modulus and mass density vary exponentially across the thickness. Investigations show that the maximum dynamic magnification factors are independent of the power-law exponent k at a fixed nondimensional parameter α. This finding may be useful in design and engineering applications.     

Exact solution for axial and transverse dynamic response of functionally graded nanobeam under moving constant load based on nonlocal elasticity theory

This paper aims to analyze the axial and transverse dynamic response of a functionally graded nanobeam under a moving constant load. The governing equations are obtained using the Hamilton principle and nonlocal Euler–Bernoulli beam theory. The mechanical properties vary in the thickness direction. The simply supported boundary condition is assumed and using the Laplace transform, the exact solution for the transverse and axial dynamic response is presented. Some examples were used to analyze nonlocal parameters such as power law index of FG materials, aspect ratio and the velocity of a moving constant load and also their influence on axial and transverse dynamic and maximum deflections. By obtaining a good agreement between the presented natural frequencies in this study and previous works, the results of this study are validated.     

Static analysis for functionally graded piezoelectric actuators or sensors under a combined electro-thermal load

Based on the theory of piezoelasticity, a functionally graded piezoelectric sandwich cantilever under an applied electric field and/or a heat load is studied. All materials may be arbitrary functional gradients in the thickness direction. The static solution for the mentioned problems is presented by the Airy stress function method. As a special case, assuming that the material composition varies continuously in the direction of the thickness according to a power law distribution, a comprehensive parametric study is conducted to show the influence of electromechanical coupling (EMC), functionally graded index, temperature change and thickness ratio on the bending behavior of actuators or sensors. The distribution of electric field or normal stress in present FGPM actuators is continuous along the thickness, which overcomes the problem in traditional layered actuators. The solution facilitates the design optimization for different piezoelectric actuators and has another potential application for material parameter identification.     

粘贴压电层功能梯度材料Timoshenko梁的热过屈曲分析

研究了上下表面粘贴压电层的功能梯度材料Timoshenko梁在升温及电场作用下的过屈曲行为。在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了压电功能梯度Timoshenko层合梁在热-电-机械载荷作用下的几何非线性控制方程。其中,假设功能梯度的材料性质沿厚度方向按照幂函数连续变化,压电层为各向同性均匀材料。采用打靶法数值求解所得强非线性边值问题,获得了在均匀电场和横向非均匀升温场内两端固定Timoshenko梁的静态非线性屈曲和过屈曲数值解。并给出了梁的变形随热、电载荷及材料梯度参数变化的特性曲线。结果表明,通过施加电压在压电层产生拉应力可以有效地提高梁的热屈曲临界载荷,延缓热过屈曲发生。由于材料在横向的非均匀性,即使在均匀升温和均匀电场作用下,也会产生拉-弯耦合效应。但是对于两端固定的压电-功能梯度材料梁,在横向非均匀升温下过屈曲变形仍然是分叉形的。     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

Free vibration of statically thermal postbuckled functionally graded material （FGM） beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electro- mechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.     

Thermoelastic analysis of functionally graded graphene reinforced rectangular plates based on 3D elasticity

Thermoelastic bending behaviour of novel functionally graded polymer nanocomposite rectangular plate reinforced with graphene nanoplatelets (GPLs) whose weight fraction varies continuously and smoothly along the thickness direction is investigated. The generalized Mian and Spencer method is utilized to obtain the analytical solutions of nanocomposite rectangular plate with two opposite edges simply supported and under a uniformly distributed transverse load and a temperature change. Three GPL distribution patterns are considered. Comparison between the present analytical solutions and those available in literature is carried out to verify the accuracy of our analytical solutions. A parametric study is conducted to examine the effects of GPL’s weight fraction, distribution pattern, geometry and size as well as the temperature change and plate boundary conditions on the stress and deformation fields of the nanocomposite plates. Numerical results show that the addition of GPLs at a very low content can have a significant reinforcing effect on the thermo-mechanical response of the plate.     

Buckling analysis of functionally graded nanobeams under non-uniform temperature using stress-driven nonlocal elasticity

In this work, the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction. By utilizing the variational principle of virtual work, the governing equations and the associated standard boundary conditions are systematically extracted, and the thermal effect, equivalent to the induced thermal load, is explicitly assessed by using the nonlocal heat conduction law. The ...     

3D elasticity solution for static analysis of functionally graded piezoelectric annular plates on elastic foundations using SSDQM

Three-dimensional elasticity solution for static analysis of functionally graded piezoelectric (FGP) annular plates with and without elastic foundations through using state-space based differential quadrature method (SSDQM) at different boundary conditions is presented in this paper. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is utilized to obtain the mechanical behavior of FGP annular plates. The state variables include a combination of electric potential, electric displacement, three mechanical displacement parameters and three stress parameters. Numerical results are given to demonstrate the convergency and accuracy of the present method. Both closed circuit and open circuit effects are studied and the influences of the Winkler and shearing layer elastic coefficients of the foundations, the material property graded index, radius, thickness, mechanical load and boundary conditions on the deflection response of the FGP annular plates are investigated. The new results can be used as a benchmark solutions for future researches.     

The wave propagation and dynamic response of rectangular functionally graded material plates with completed clamped supports under impulse load

In this paper, the wave propagation and dynamic response of the rectangular FGM plates with completed clamped supports under impulse load are analyzed. The effective material properties of functionally graded materials (FGMs) for the plate are assumed to vary continuously through the plate thickness and be distributed according to a volume fraction power law along the plate thickness. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. A complete discussion of dispersion of the FGM plates is given. Using the dispersion relation and integral transforms, exact integral solutions for the FGM plates under impulse load are obtained. The influence of volume fraction distributions on wave propagation and dynamic response of the FGM plates is analyzed.     

Free vibration analysis of functionally graded panels and shells of revolution

The aim of this paper is to study the dynamic behaviour of functionally graded parabolic and circular panels and shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to study these moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when the materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. For the first power-law distribution, the bottom surface of the structure is ceramic rich, whereas the top surface is metal rich and on the contrary for the second one. The governing equations of motion are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system equations by means of the Generalized Differential Quadrature (GDQ) method leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. Numerical results concerning eight types of shell structures illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behaviour of parabolic and circular shell structures. Preliminary results were presented by the authors at the XVIII° National Conference of Italian Association of Theoretical and Applied Mechanics (AIMETA 2007) (Tornabene and Viola 27).     

Static analysis of functionally graded doubly-curved shells and panels of revolution

The Generalized Differential Quadrature (GDQ) Method is applied to study four parameter functionally graded and laminated composite shells and panels of revolution. The mechanical model is based on the so-called First-order Shear Deformation Theory (FSDT), in particular on the Toorani-Lakis Theory. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The generalized strains and stress resultants are evaluated by applying the Differential Quadrature rule to the generalized displacements. The transverse shear and normal stress profiles through the thickness are reconstructed a posteriori by using local three-dimensional elasticity equilibrium equations. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with semi-analytical formulations and with 3D finite element method. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of functionally graded shell structures made of a mixture of ceramics and metal.     

Thermomechanical vibration analysis of a functionally graded shell with flowing fluid

This paper reports the results of an investigation into the vibration of functionally graded cylindrical shells with flowing fluid, embedded in an elastic medium, under mechanical and thermal loads. By considering rotary inertia, the first-order shear deformation theory (FSDT) and the fluid velocity potential, the dynamic equation of functionally graded cylindrical shells with flowing fluid is derived. Here, heat conduction equation along the thickness of the shell is applied to determine the temperature distribution and material properties are assumed to be graded distribution along the thickness direction according to a power-law in terms of the volume fractions of the constituents. The equations of eigenvalue problem are obtained by using a modal expansion method. In numerical examples, effects of material composition, thermal loading, static axial loading, flow velocity, medium stiffness and shell geometry parameters on the free vibration characteristics are described. The new features in this paper are helpful for the application and the design of functionally graded cylindrical shells containing fluid flow.     

Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads

We present a generalized shear deformation theory in combination with isogeometric (IGA) approach for nonlinear transient analysis of smart piezoelectric functionally graded material (FGM) plates. The nonlinear transient formulation for plates is formed in the total Lagrange approach based on the von Kármán strains, which includes thermo-piezoelectric effects, and solved by Newmark time integration scheme. The electric potential through the thickness of each piezoelectric layer is assumed to be linear. The material properties vary through the thickness of FGM according to the rule of mixture and the Mori–Tanaka schemes. Various numerical examples are presented to demonstrate the effectiveness of the proposed method.     

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials subjected to harmonic antiplane shear waves is investigated using the Schmidt method. The present problem can be solved using the Fourier transform and the technique of dual integral equations, in which the unknown variables are jumps of displacements across the crack surfaces, not dislocation density functions. To solve the dual integral equations, the jumps of displacements across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the electric, magnetic flux, and dynamic stress fields near crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter, the distance between the two parallel cracks, and the circular frequency of the incident waves upon the stress, electric displacement, and magnetic flux intensity factors at crack tips.     

Exact solution of orthotropic cylindrical shell with piezoelectric layers under cylindrical bending

An exact elasticity solution for an orthotropic cylindrical shell with piezoelectric layers is obtained in this paper. The stress and displacement distributions are presented. The influence of the piezoelectric layers on the mechanical behavior of structures is studied. Both the direct piezoelectric effect and the converse piezoelectrical effect of the piezoelectric material are investigated. Results presented in this paper can be used to study various approximate shell theories used in the numerical simulations of piezoelectric structures.     

Wave propagation analysis of porous functionally graded piezoelectric nanoplates with a visco-Pasternak foundation

This study investigates the size-dependent wave propagation behaviors under the thermoelectric loads of porous functionally graded piezoelectric(FGP) nanoplates deposited in a viscoelastic foundation. It is assumed that(i) the material parameters of the nanoplates obey a power-law variation in thickness and(ii) the uniform porosity exists in the nanoplates. The combined effects of viscoelasticity and shear deformation are considered by using the Kelvin-Voigt viscoelastic model and the refined hi...     

Surface effect and non-local elasticity in wave propagation of functionally graded piezoelectric nano-rod excited to applied voltage

A non-local solution for a functionally graded piezoelectric nano-rod is presented by accounting the surface effect. This solution is used to evaluate the characteristics of the wave propagation in the rod structure. The model is loaded under a two-dimensional (2D) electric potential and an initially applied voltage at the top of the rod. The mechanical and electrical properties are assumed to be variable along the thickness direction of the rod according to the power law. The Hamilton principle is used to derive the governing differential equations of the electromechanical system. The effects of some important parameters such as the applied voltage and gradation of the material properties on the wave characteristics of the rod are studied.