Electromagnetoelastic behaviors of functionally graded piezoelectric

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 can optimize the electromagnetoelastic responses, which will be of particular importance in modern engineering designs.     

Thermomagnetic viscoelastic responses in a functionally graded hollow structure

This paper presents an analytical solution for the interaction of electric potentials,electric displacements,elastic deformations,and thermoelasticity,and describes electromagnetoelastic responses and perturbation of the magnetic field vector in hollow structures(cylinder or sphere),subjected to mechanical load and electric potential.The material properties,thermal expansion coefficient and magnetic permeability of the structure are assumed to be graded in the radial direction by a power law distribution.In the present model we consider the solution for the case of a hollow structure made of viscoelastic isotropic material,reinforced by elastic isotropic fibers,this material is considered as structurally anisotropic material.The exact solutions for stresses and perturbations of the magnetic field vector in FGM hollow structures are determined using the infinitesimal theory of magnetothermoelasticity,and then the hollow structure model with viscoelastic material is solved using the correspondence principle and Illyushin’s approximation method.Finally,numerical results are carried out and discussed.     

Stresses distributions in a rotating functionally graded piezoelectric hollow cylinder

An analytic solution to the axisymmetric problem of a long, radially polarized, hollow cylinder composed of functionally graded piezoelectric material (FGPM) rotating about its axis at a constant angular velocity is presented. For the case that electric, thermal and mechanical properties of the material obey different power laws in the thickness direction, distributions for radial displacement, stresses and electric potential in the FGPM hollow cylinder are determined by using the theory of electrothermoelasticity. Some useful discussions and numerical examples are presented to show the significant influence of material nonhomogeneity, and adopting suitable graded indexes and applying suitable geometric size and rotating velocity ω may optimize the rotating FGPM hollow cylindrical structures. This will be of particular importance in modern engineering application.     

Electromagnetic twisting of a penny-shaped crack in an elastic conducting cylinder

This paper deals with the electromagnetoelastic problem of an elastic, conducting circular cylinder with a penny-shaped crack under a uniform axial current flow and a constant axial magnetic field. The current flow is disturbed by the presence of the crack and the torsional stresses are caused by the interactions between the magnetic field and the disturbed current. Two problems concerning the electric current density field and the electromagnetoelastic field are formulated by means of integral transform techniques and reduced to two Fredholm integral equations of the second kind. Numerical calculations are carried out and stress intensity factors are obtained for several values of the geometric parameters.     

Free vibration of a FGPM circular plate placed in a uniform magnetic field

In this paper, free vibration of a circular plate composed of a transversely isotropic functionally graded piezoelectric material (FGPM) placed in a uniform magnetic field is investigated. Material properties are assumed to depend on the thickness of the circular plate and they are expressed as the same exponential function of h. The problem of free vibration for the transversely isotropic FGPM circular plate with clamped and simply supported boundary conditions is solved by means of the state space method. Numerical examples and some useful discussions are given to demonstrate the significant influence of material inhomogeneity, and adopting a certain value of the graded index can optimize structures of the circular plate. This will be of particular importance in modern engineering design.     

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

This paper presents an analytical solution of a thick walled cylinder com- posed 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.     

The scattering of P-waves by a piezoelectric particle with FGPM interfacial layers in a polymer matrix

Propagation of P-wave in an unbounded elastic polymer medium which contains a set of nested concentric spherical piezoelectric inhomogeneities is formulated. The polymer matrix is made of Epoxy and is isotropic; each phase of the inhomogeneity is made of a different piezoelectric material and is radially polarized and has spherical isotropy. Note that the individual phases are homogeneous, and all interfaces are perfectly bonded. The scattered displacement and electric potentials in the matrix are expressed in terms of spherical wave vector functions and Legendre functions, respectively. The transmitted displacement and electric potentials within each phase of the piezoelectric particle are expressed in terms of Legendre functions. The equations of motion and electrostatics in each phase of the piezoelectric inhomogeneity lead to a system of coupled second order differential equations, which is solved using the generalized Frobenius series. The present theory is extended to the case where the core of the inhomogeneity is made of PZT-4 and its coating is made of functionally graded piezoelectric material (FGPM) whose microstructural composition varies smoothly from PZT-4 at the core–coating interface to Epoxy at the coating–matrix interface. The effects of different types of variation in the electro-mechanical properties of FGPM on scattering cross-section and other electro-mechanical fields are addressed. The present theory is valid for arbitrary coating thickness, and arbitrary frequencies.     

A moving interface crack between two dissimilar functionally graded piezoelectric layers under electromechanical loading

The dynamic propagation of an interface crack between two dissimilar functionally graded piezoelectric material (FGPM) layers under anti-plane shear is analyzed using the integral transform method. The properties of the FGPM layers vary continuously along the thickness. The properties of the FGPM layers vary differently and the two layers are connected weak-discontinuously. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate (DERR) are presented for the FGPM to show the effects on the electric loading, gradient of material properties, crack moving velocity, and thickness of layers. Followings are helpful to increase of the resistance of the interface crack propagation of FGPM: (a) certain direction and magnitude of the electric loading; (b) increase of the gradient of material properties; (c) increase of the material properties from the interface to the upper and lower free surface; (d) increase of the thickness of FGPM layer. The DERR increases or decreases with increase of the crack moving velocity.     

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

In this article, we study the axisymmetric tor-sional contact problem of a half-space coated with func-tionally graded piezoelectric material (FGPM) and subjected to a rigid circular punch. It is found that, along the thick-ness direction, the electromechanical properties of FGPMs change exponentially. We apply the Hankel integral trans-form technique and reduce the problem to a singular integral equation, and then numerically determine the unknown con-tact 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.     

Wave characteristics in functionally graded piezoelectric hollow cylinders

Based on linear three-dimensional piezoelasticity, the Legendre orthogonal polynomial series expansion approach is used for determining the wave characteristics in hollow cylinders composed of the functionally graded piezoelectric materials (FGPM) with open circuit. The displacement and electric potential components, expanded in a series of Legendre polynomials, are introduced into the governing equations along with position-dependent material constants so that the solution of the wave equation is reduced to an eigenvalue problem. Dispersion curves for FGPM and the corresponding non-piezoelectric hollow cylinders are calculated to show the piezoelectric effect. The influence of the ratio of radius to thickness is discussed. Electric potential and displacement distributions are used to show the piezoelectric effect on the flexural torsional mode. The influence of the polarizing direction on the piezoelectric effect is illustrated. For the radial and axial polarization, the piezoelectric effect reacts mostly on the longitudinal mode. For circumferential polarization, the piezoelectric effect reacts mostly on the torsional mode. In the FGPM hollow cylinder, piezoelectricity can weaken the guided wave dispersion.     

Magnetothermoelastic stress and perturbation of magnetic field vector in a functionally graded hollow sphere

In this article, a closed-form solution for one-dimensional magnetothermoelastic problem in a functionally graded material (FGM) hollow sphere placed in uniform magnetic and temperature fields subjected to an internal pressure is obtained using the infinitesimal theory of magnetothermoelasticity. Hyper-geometric functions are employed to solve the governing equation. The material properties through the graded direction are assumed to be nonlinear with an exponential distribution. The nonhomogeneity of the material in the radial direction is assumed to be exponential. The temperature, displacement and stress fields and the perturbation of magnetic field vector are determined and compared with those of the homogeneous case. Hence, the effect of inhomogeneity on the stresses and the perturbation of magnetic field vector distribution are demonstrated. The results of this study are applicable for designing optimum FGM hollow spheres.     

Transient response of an elastic conductor with a penny-shaped crack under electromagnetic force

The axisymmetric dynamic response of a penny-shaped crack in an elastic conductor under an impulsive electric current flow and a constant axial magnetic field is analyzed. The axial current flow is disturbed by the presence of the crack and the torsional shear stresses are caused by the interactions between the magnetic field and the disturbed current. Laplace and Hankel transforms are used to reduce the electromagnetoelastic problem to a Fredholm integral equation of the second kind in the Laplace transform plane. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical results on the dynamic stress intensity factor are obtained and are presented in a graphical form.     

The Stress Response of Functionally Graded Isotropic Linearly Elastic Rotating Disks

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic solid circular disks or cylinders, rotating at constant angular velocity about a central axis. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The analog of the classic problem for a homogeneous isotropic rotating solid disk or cylinder is considered. The special case of a body with Young"s modulus depending on the radial coordinate only, and with constant Poisson"s ratio, is examined. For the case when the Young"s modulus has a power-law dependence on the radial coordinate, explicit exact solutions are obtained. It is shown that the stress response of the inhomogeneous disk (or cylinder) is significantly different from that of the homogeneous body. For example, the maximum radial and hoop stresses do not, in general, occur at the center as in the case for the homogeneous material. Furthermore, for the case where the Young"s modulus increases with radial distance from the center, it is shown that radially symmetric solutions exist provided the rate of growth of the Young"s modulus is, at most, cubic in the radial variable. It is also shown for the general inhomogeneous isotropic case how the material inhomogeneity may be tailored so that the radial and hoop stress are identical throughout the disk. This revised version was published online in August 2006 with corrections to the Cover Date.     

Crack in functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading

Solved is the problem of a crack in a functionally graded piezoelectric material (FGPM) bonded to two elastic surface layers. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric permittivity of the FGPM vary continuously along the thickness of the strip. The outside layers are under antiplane mechanical loading and in-plane electric loading. The solution involves solving singular integral equations by application of the Gauss–Jacobi integration formula. Numerical calculations are carried out to obtain the energy density factors. Their variations with the geometric, loading and material parameters are shown graphically.     

Some features of the dynamics of ferroelectric (ferromagnetic) heterostructures

This paper considers the dynamic coupled problem of harmonic vibrations of a layeredinhomogeneous electromagnetoelastic medium subjected to an oscillating mechanical or electrical load under various electric and magnetic field conditions specified on the surface and internal boundaries of this medium. Green’s function of the medium is constructed. Dispersion curves and phase velocities for different boundary conditions and materials are obtained.     

Magneto-thermo-elastic problem of a rotating nonhomogeneous anisotropic solid cylinder

In the present paper, we investigate the generation of thermal stresses in a nonhomogeneous anisotropic solid cylinder rotating about the z-axis at a constant angular velocity in the presence of a magnetic field. The governing equations are solved numerically using the boundary-element method (BEM) and numerical results are obtained for the variation of the temperature, displacements, and stresses along x-axis. The effect of nonhomogeneity is investigated.     

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.     

Magneto-thermoelastic problem in non-homogeneous isotropic cylinder

The present paper concerns the investigation of the stress, temperature and magnetic fields in an isotropic elastic cylinder in a primary magenetic field when the curved surface of the cylinder subject to certain boundary conditions.The system of fundamental equations is solved by means of a finite difference method and the numerical calculations are carried out for the temperature, the components of displacement and the components of stresses with time and through the thickness of the cylinder. The results indicate that the effects of inhomogeneity and magnetic field are very pronounced.     

Second order effects in fluidmagnetic buoyancy

A non-magnetic solid object placed in a magnetically responsive fluid in the presence of a magnetic field gradient experiences a net buoyancy force of magnetic origin. A procedure is developed to account for the effects of magnetic field distortion due to the difference of magnetic permeability between the fluid and the solid and non-zero dependence of fluid magnetization on magnetic field strength. This procedure gives an expression for the magnetic buoyancy force correct to first order in the dimensionless magnetization of the fluid and in the dimensionless variation of fluid magnetization across the object. Calculations are performed for a sphere, cylinder and plate in an applied magnetic field where the field and field gradient are either aligned or at right angles in order to give an indication of the range of force variation due to a change of shape and due to a change of applied field geometry. Variations on the order of 10% can be expected in typical applications.     

<Emphasis Type="Italic">T</Emphasis>-stress analysis for a Griffith crack in a magnetoelectroelastic solid

T-stress as an important parameter characterizing the stress field around a cracked tip has attracted much attention. This paper concerns the T-stress near a cracked tip in a magnetoelectroelastic solid. By applying the Fourier transform, we solve the associated mixed boundary-value problem. Adopting crack-faces electromagnetic boundary conditions nonlinearly dependent on the crack opening displacement, coupled dual integral equations are derived. Then, the closed-form solution for the T-stress is obtained. A comparison of the T stresses for a cracked magnetoelectroelastic solid and for a cracked purely elastic material is made. Obtained results reveal that in addition to applied mechanical loading, the T-stress is dependent on electric and magnetic loadings for a vacuum crack.