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.

Behavior of two parallel symmetry permeable cracks in functionally graded piezoelectric materials subjected to an anti-plane shear loading

The behavior of two parallel symmetry permeable cracks in functionally graded piezoelectric materials subjected to an anti-plane shear loading was investigated. To make the analysis tractable, it was assumed that the material properties varied exponentially with coordinate vertical to the crack. By using the Fourier transform, the problem could be solved with the help of two pairs of dual integral equations, in which the unknown variables were the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces was expanded in a series of Jacobi polynomials. The normalized stress and electrical displacement intensity factors were determined for different geometric and property parameters for permeable electric boundary conditions. Numerical examples were provided to show the effect of the geometry of the interacting cracks and the functionally graded material parameter upon the stress intensity factors of cracks.

Flutter and Thermal Buckling Properties and Active Control of Functionally Graded Piezoelectric Material Plate in Supersonic Airflow

This paper is devoted to investigate the flutter and thermal buckling properties of the functionally graded piezoelectric material(FGPM)plate in supersonic airflow,and the active flutter control is carried out under different temperature fields.The piezoelectric material component of the FGPM plate has gradient changes along the thickness,such as piezoelectricity and dielectric coefficients.The supersonic piston theory is used to evaluate the aerodynamic pressure.Based on the first-order shear deformation theory and Hamilton’s principle with the assumed mode method,the equation of motion of the structural system is deduced.The effect of aerodynamic pressure on the frequency and damping ratio of the FGPM plate is analyzed.Moreover,the flutter and thermal buckling properties of the FGPM and pure metal plates are compared to show the superior aerothermoelastic properties of the FGPM plates.The influences of volume fraction exponent and temperature on the flutter and thermal buckling properties of the FGPM plate are also examined in detail.The LQR controller is adopted to achieve active flutter control.The simulation results show that the present control method can largely improve dynamic stability of the FGPM plate in supersonic airflow and high-temperature environment.

Peculiarities of surface acoustic waves,propagation in structures with functionally graded piezoelectric materials,coating from different ceramics on the basis of PZT

A model of a piezoelectric structure with an inhomogeneous coating is considered.The structure is a homogeneous half-space made of PZT-5H ferroelectric ceramics with a functionally graded coating.The properties of coating vary continuously in thickness from parameters of one material to parameters of another material in a continuously nonmonotonic or piecewise-continuous manner.As coating materials,various combinations of ceramics of different stiffness based on PZT are considered.Using the example of the problem of the propagation of sh-waves in a piezoelectric structure,we studied the influence of the ratio of the physical parameters of the coating materials,the localization region,and the size of the transition zone of one material to another on the propagation features of surface acoustic waves(SAWs)and the structure of the wave field.

Two coplanar cracks in a functionally graded piezoelectric material strip under mechanical and transient thermal loadings

In this paper,the fracture problem of a functionally graded piezoelectric material strip(FGPM strip) containing two coplanar cracks perpendicular to its boundaries is considered.The problem is solved for an FGPM strip that is suddenly heated from the bottom surface under static mechanical loading.The top surface is maintained at the initial temperature.The crack faces are supposed to be completely insulated.Material properties are assumed to be exponentially dependent on the distance from the bottom surface.By using the Laplace and Fourier transforms,the thermoelectromechanical fracture problem is reduced to a set of singular integral equations,which are solved numerically.The stress intensity factors for the cases of the two embedded cracks,two edge cracks,and an embedded crack and an edge crack are computed and presented as a function of the normalized time,the nonhomogeneous and geometric parameters.

Three-dimensional Analysis of Functionally Graded Piezoelectric Plate with Arbitrarily Distributed Material Properties

An orthotropic functionally graded piezoelectric rectangular plate with arbitrarily distributed material properties was studied, which is simply supported and grounded（electrically） on its four lateral edges. The state equations of the functionally graded piezoelectric material were obtained using the state-space approach, and a Peano-Baker series solution was obtained for the coupled electroelastic fi elds of the functionally graded piezoelectric plate subjected to mechanical and electric loading on its upper and lower surfaces. The influence of different distributions of material properties on the structural response of the plate was studied using the obtained solutions.

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.

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.

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.

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.

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.

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.

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.

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.

THE PERIODIC CRACK PROBLEM IN BONDED PIEZOELECTRIC MATERIALS

The problem of a periodic array of parallel cracks in a homogeneous piezoelectric strip bonded to a functionally graded piezoelectric material is investigated for inhomogeneous continuum. It is assumed that the material inhomogeneity is represented as the spatial variation of the shear modulus in the form of an exponential function along the direction of cracks. The mixed boundary value problem is reduced to a singular integral equation by applying the Fourier transform, and the singular integral equation is solved numerically by using the Gauss-Chebyshev integration technique. Numerical results are obtained to illustrate the variations of the stress intensity factors as a function of the crack periodicity for different values of the material inhomogeneity.

Specific features of SH-waves propagation in structures with prestressed inhomogeneous coating made of piezoceramics based on LiNbO_(3)

Within the framework of the linearized theory of electroelastic wave propagation,a model of a piezoelectric structure with a prestressed functionally graded coating made of piezoceramics of a trigonal system with a symmetry class of 3m is considered.The ferroelectric LiNbO_(3) is used as the main material of the structure.The initial deformed state of the coating material is homogeneous,induced by the action of initial mechanical stresses and an external electrostatic field,the properties of the coating continuously change in thickness.By the example of the problem of the propagation of SH-waves from a remote source for structures with an inhomogeneous prestressed coating in the case of an electrically free and short-circuited surface,the influence of the nature and localization of the inhomogeneity of the coating on the features of SAW propagation is studied.The separate and combined effects of initial actions on changes in the physical properties of the structure,the transformation of the surface wave field,and the change in the SAW velocities in a wide frequency range is studied.The results obtained in this work are useful for understanding the dynamic processes in prestressed piezoelectric structures,in the optimization and design of new structures and devices on SAW with high performance characteristics.