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 higher-order shear deformation theory.The scale effects of the nanoplates are captured by employing nonlocal strain gradient theory(NSGT).The motion equations are calculated in accordance with Hamilton’s principle.Finally,the dispersion characteristics of the nanoplates are numerically determined by using a harmonic solution.The results indicate that the nonlocal parameters(NLPs) and length scale parameters(LSPs) have exactly the opposite effects on the wave frequency.In addition,it is found that the effect of porosity volume fractions(PVFs) on the wave frequency depends on the gradient indices and damping coefficients.When these two values are small,the wave frequency increases with the volume fraction.By contrast,at larger gradient index and damping coefficient values,the wave frequency decreases as the volume fraction increases.

BENDING OF FUNCTIONALLY GRADED PIEZOELECTRIC RECTANGULAR PLATES

By introducing two displacement functions as well as two stressfunctions, two independent state equations with variable coefficientsare derived from the three-dimensional theory equations of piezo-elasticity for transverse isotropy. A laminated approximation is usedto transform the state equations to those with constant coefficientsin each sub-layer. The bending problem of a functionally gradedrectangular plate is then analyzed based on the state equations.Numerical results are presented and the effect of material gradi- entindex is discussed.

An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory

In this paper,we analytically study vibration of functionally graded piezoelectric(FGP)nanoplates based on the nonlocal strain gradient theory.The top and bottom surfaces of the nanoplate are made of PZT-5H and PZT-4,respectively.We employ Hamilton’s principle and derive the governing differential equations.Then,we use Navier’s solution to obtain the natural frequencies of the FGP nanoplate.In the first step,we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies.In the second step,we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded(FG)nanoplates.Finally,the effects of the FG power index,the nonlocal parameter,the aspect ratio,and the lengthto-thickness ratio,and the nanoplate shape on natural frequencies are investigated.

Electromagnetoelastic behaviors of functionally graded piezoelectric solid cylinder and sphere

Analytical studies on electromagnetoelastic behaviors are presented for the functionally graded piezoelectric material （FGPM） solid cylinder and sphere placed in a uniform magnetic field and subjected to the external pressure and electric loading. When the mechanical, electric and magnetic properties of the material obey an identical power law in the radial direction, the exact displacements, stresses, electric potentials and perturbations of magnetic field vector in the FGPM solid cylinder and sphere are obtained by using the infinitesimal theory of electromagnetoelasticity. Numerical examples also show the significant influence of material inhomogeneity. It is interesting to note that selecting a specific value of inhomogeneity parameter β can optimize the electromagnetoelastic responses, which will be of particular importance in modern engineering designs.

Free vibration analysis of functionally graded laminated sandwich cylindrical shells integrated with piezoelectric layer

An analytical method for the three-dimensional vibration analysis of a functionally graded cylindrical shell integrated by two thin functionally graded piezoelectric （FGP） layers is presented. The first-order shear deformation theory is used to model the electromechanical system. Nonlinear equations of motion are derived by considering the von Karman nonlinear strain-displacement relations using Hamilton＇s principle. The piezoelectric layers on the inner and outer surfaces of the core can be considered as a sensor and an actuator for controlling characteristic vibration of the system. The equations of motion are derived as partial differential equations and then discretized by the Navier method. Numerical simulation is performed to investigate the effect of different para- meters of material and geometry on characteristic vibration of the cylinder. The results of this study show that the natural frequency of the system decreases by increasing the non-homogeneous index of FGP layers and decreases by increasing the non-homogeneous index of the functionally graded core. Furthermore, it is concluded that by increasing the ratio of core thickness to cylinder length, the natural frequencies of the cylinder increase considerably.

Application and analysis of functionally graded piezoelectrical rotating cylinder as mechanical sensor subjected to pressure and thermal loads

The exact thermoelastic analysis of a functionally graded piezoelectrical （FGP） rotating cylinder is investigated analytically. The cylinder is subjected to a com- bination of electrical, thermal, and mechanical loads simultaneously. The structure is a simplified model of a rotational sensor or actuator. The basic governing differential equation of the system is obtained by using the energy method. A novel term, named as the additional energy, is introduced to exact the evaluation of the energy functional. The solution to the governing differential equation is presented for two types of boundary conditions including free rotating and rotating cylinders exposed to the inner pressure. The effect of the angular velocity is investigated on the radial distribution of various components. The mentioned structure can be considered as a sensor for measuring the angular velocity of the cylinder subjected to the pressure and temperature. The obtained results indicate that the electrical potential is proportional to the angular velocity.

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.

SCATTERING OF HARMONIC ANTI-PLANE SHEAR STRESS WAVES BY A CRACK IN FUNCTIONALLY GRADED PIEZOELECTRIC/PIEZOMAGNETIC MATERIALS

In this paper, the dynamic behavior of a permeable crack in functionally graded piezoelectric/piezomagnetic materials is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with the coordinate parallel to the crack. By using the Fourier transform, the problem can be solved with the help of a pair of dual integral equations in which the unknown is the jump of displacements across the crack surfaces. These equations are solved to obtain the relations between the electric filed, the magnetic flux field and the dynamic stress field near the crack tips using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter and the circular frequency of the incident waves upon the stress, the electric displacement and the magnetic flux intensity factors of the crack.

Active vibration control of functionally graded beams with piezoelectric layers based on higher order shear deformation theory

This paper reports on a study of active vibration control of functionally graded beams with upper and lower surface-bonded piezoelectric layers. The model is based on higher-order shear deformation theory and implemented using the finite element method （FEM）. The proprieties of the functionally graded beam （FGB） are graded along the thickness direction. The piezoelectric actuator provides a damping effect on the FGB by means of a velocity feedback control algorithm. A Matlab program has been developed for the FGB model and compared with ANSYS APDL. Using Newmark＇s method numerical solutions are obtained for the dynamic equations of FGB with piezoelectric layers. Numerical results show the effects of the constituent volume fraction and the influence the feedback control gain on the frequency and dynamic response of FGBs.

Three-dimensional free vibration analysis of multi-directional functionally graded piezoelectric annular plates on elastic foundations via state space based differential quadrature method

The three-dimensional free vibration analysis of a multi-directional func- tionally graded piezoelectric （FGP） annular plate resting on two parameter （Pasternak） elastic foundations is investigated under different boundary conditions. The material properties are assumed to vary continuously along the radial and thickness directions and have exponent-law distribution. A semi-analytical approach named the state space based differential quadrature method （SSDQM） is used to provide an analytical solution along the thickness using the state space method （SSM） and an approximate solution along the radial direction using the one-dimensional differential quadrature method （DQM）. The influence of the Winkler and shear stiffness of the foundation~ the material property graded variations, and the circumferential wave number on the nomdimensional natural frequency of multi-directional FGP annular plates is studied.

Transient thermal response of functionally graded piezoelectric laminates with an infinite row of parallel cracks normal to the bimaterial interface

In this paper,the problem of a functionally graded piezoelectric material strip(FGPM strip)containing an infinite row of parallel cracks perpendicular to the interface between the FGPM strip and a homogeneous layer is analyzed under transient thermal loading condition.The crack faces are supposed to be completely insulated.Material properties are assumed to be exponentially dependent on the distance from the interface.Using the Fourier transforms,the electro-thermoelastic problem is reduced to a singular integral equation,which is solved numerically.The stress intensity factors are computed and presented as a function of the normalized time,the nonhomogeneous and geometric parameters.

Transference of Love-type waves in a bedded structure containing a functionally graded material and a porous piezoelectric medium

The frequency of the Love-type surface waves in a bedded structure con- sisting of a porous piezoelectric (PP) medium and a functionally graded material (FGM) substrate is approximated. The FGM layer is assumed to have a constant initial stress. The Wentzel-Kramers-Brillouin (WKB) approximation technique is used for the wave solution in the FGM layer, and the method of separation of variables is applied for the solution in the porous piezoelectric medium. The dependence of the wave frequency on the wave number is obtained for both electrically open and short cases. The effects of the gradient coefficient of the FGM layer, the initial stresses (tensile stress and compressive stress), and the width of the FGM layer are marked distinctly and shown graphically. The findings may contribute towards the design and optimization of acoustic wave devices.

SCATTERING OF THE HARMONIC STRESS WAVE BY CRACKS IN FUNCTIONALLY GRADED PIEZOELECTRIC MATERIALS

The present paper considers the scattering of the time harmonic stress wave by a single crack and two collinear cracks in functionally graded piezoelectric material （FGPM）. It is assumed that the properties of the FGPM vary continuously as an exponential function. By using the Fourier transform and defining the jumps of displacements and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the influences of material properties on the dynamic stress and the electric displacement intensity factors.

Basic solutions of multiple parallel symmetric mode-III cracks in functionally graded piezoelectric/piezomagnetic material plane

The Schmidt method is adopted to investigate the fracture problem of multiple parallel symmetric and permeable finite length mode-III cracks in a functionally graded piezoelectric/piezomagnetic material plane. This problem is formulated into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. In order to obtain the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. The results show that the stress, the electric displacement, and the magnetic flux intensity factors of cracks depend on the crack length, the functionally graded parameter, and the distance among the multiple parallel cracks. The crack shielding effect is also obviously presented in a functionally graded piezoelectric/piezomagnetic material plane with mul- tiple parallel symmetric mode-III cracks.

The axisymmetric torsional contact problem of a functionally graded piezoelectric coated half-space

In this article, we study the axisymmetric torsional contact problem of a half-space coated with functionally graded piezoelectric material (FGPM) and subjected to a rigid circular punch. It is found that, along the thickness direction, the electromechanical properties of FGPMs change exponentially. We apply the Hankel integral transform technique and reduce the problem to a singular integral equation, and then numerically determine the unknown contact stress and electric displacement at the contact surface. The results show that the surface contact stress, surface azimuthal displacement, surface electric displacement, and inner electromechanical field are obviously dependent on the gradient index of the FGPM coating. It is found that we can adjust the gradient index of the FGPM coating to modify the distributions of the electric displacement and contact stress.

Effects of an attached functionally graded layer on the electromechanical behaviors of piezoelectric semiconductor fibers

In this paper,we propose a specific two-layer model consisting of a functionally graded(FG)layer and a piezoelectric semiconductor(PS)layer.Based on the macroscopic theory of PS materials,the effects brought about by the attached FG layer on the piezotronic behaviors of homogeneous n-type PS fibers and PN junctions are investigated.The semi-analytical solutions of the electromechanical fields are obtained by expanding the displacement and carrier concentration variation into power series.Results show that the antisymmetry of the potential and electron concentration distributions in homogeneous n-type PS fibers is destroyed due to the material inhomogeneity of the attached FG layer.In addition,by creating jump discontinuities in the material properties of the FG layer,potential barriers/wells can be produced in the middle of the fiber.Similarly,the potential barrier configuration near the interface of a homogeneous PS PN junction can also be manipulated in this way,which offers a new choice for the design of PN junction based devices.

Analytical solution of multiple moving cracks in functionally graded piezoelectric strip

The dynamic behaviors of several moving cracks in a functionally graded piezoelectric （FGP） strip subjected to anti-plane mechanical loading and in-plane elec- trical loading are investigated. For the first time, the distributed dislocation technique is used to construct the integral equations for FGP materials, in which the unknown variables are the dislocation densities. With the dislocation densities, the field intensity factors are determined. Moreover, the effects of the speed of the crack propagation on the field intensity factors are studied. Several examples are solved, and the numerical results for the stress intensity factor and the electric displacement intensity factor are presented graphically finally.

Thermo-electromechanical behavior of functionally graded piezoelectric hollow cylinder under non-axisymmetric loads

This paper presents an analytical solution of a thick walled cylinder composed of a functionally graded piezoelectric material （FGPM） and subjected to a uniform electric field and non-axisymmetric thermo-mechanical loads. All material properties, except Poisson＇s ratio that is assumed to be constant, obey the same power law. An exact solution for the resulting Navier equations is developed by the separation of variables and complex Fourier series. Stress and strain distributions and a displacement field through the cylinder are obtained by this technique. To examine the analytical approach, different examples are solved by this method, and the results are discussed.

Dynamic behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips

The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is jump of displacement across the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter, the circular frequency of the incident waves and the thickness of the strip upon stress, electric displacement and magnetic flux intensity factors of cracks.

Effects of thermo-magneto-electro nonlinearity characteristics on the stability of functionally graded piezoelectric beam

Due to the increasing interests in using functionally graded piezoelectric materials(FGPMs) in the design of advanced micro-electro-mechanical systems, it is important to understand the stability behaviors of the FGPM beams. In this study, considering the effects of geometrical nonlinearity, temperature, and electricity in the constitutive relations and the effect of the magnetic field on the FGPM beam, the Euler-Bernoulli beam model is adopted, and the nonlinear governing equation of motion is derived via Hamilton's principle. A perturbation method, which can decompose the deflection into static and dynamic components, is utilized to linearize the nonlinear governing equation. Then,a dynamic stability analysis is carried out, and the approximate analytical solutions for the nonlinear frequency and boundary frequencies of the unstable region are obtained.Numerical examples are performed to verify the present analysis. The effects of the static deflection, the static load factor, the temperature change, and the magnetic field flux on the stability behaviors of the FGPM beam are discussed. From the proposed analytical solutions and numerical results, one can easily and clearly find the effects of various controlled parameters, such as geometric and physical properties of the system, on the mechanical behaviors of structures, and the conclusions are very important and useful for the design of micro-devices.