Normal Electric Displacement Research Articles

Rayleigh waves in viscoelastic piezoelectric half-space with cladding structures: An analytic Legendre-Laguerre polynomial method

Piezoelectric materials are widely used in surface acoustic wave devices. Many piezoelectric materials themselves have viscoelastic properties, and their surface wave characteristics, especially the attenuation characteristics still needs to be explored. This article proposes a Legendre-Laguerre orthogonal polynomial method to solve the Rayleigh wave problems in a viscoelastic piezoelectric half space with a covering layer. The proposed method compensates for the inherent shortcomings of traditional Laguerre polynomials in solving layered half spaces: the normal stress and electric displacement are discontinuous. The correctness of the method was verified through literature comparison and finite element simulation. At the same time, by utilizing the orthogonality of the Legendre-Laguerre polynomial, the integral analytical formula encountered in the solution process is derived, which improves the computational efficiency by more than ten times. Through the analysis and discussion of the dispersion and attenuation curves, it is found that the piezoelectric effect can suppress the attenuation of Rayleigh waves; the piezoelectric and viscous properties of the covering layer mainly affect attenuation at high frequencies, while those of the half space layer mainly affect attenuation at low frequencies of high-order modes.

A modified first-order plate theory of laminated piezoelectric plate actuators

Piezoelectric actuators with large output and high reliability are widely applied in electron device. In this paper, a modified first-order plate theory of laminated piezoelectric plates with upper and lower electrodes is presented. Then, based on the Mindlin first-order shear deformation theory, a quadratic function of electric potential across the thickness is used to describe the inverse piezoelectric effect. The laminated plate governing equations are derived in terms of the resultant force, moment and electric displacement. Further, supplementary equations are introduced through the continuity of electrical potential and the normal electric displacement across interface of each lamina, which are applied to solve the unknown electric functions in each lamina. The numerical results of displacement, deflection and electric potential are consistent with those obtained by the three-dimensional finite element method. Finally, with such a modified plate theory, regulation mechanisms of polarization direction and layer numbers are investigated on actuation properties of laminated piezoelectric actuators. It is shown that the polarization direction can change the driving displacement by an order of magnitude. It is expected that the modified theory can provide a more efficient and accurate approach for analysis of multilayer piezoelectric composite actuators.

Magnetic potential based formulation for linear and non-linear 3D RF sheath simulation

This paper reports a new numerical scheme to simulate the radio-frequency (RF) induced RF sheath, which is suitable for a large 3D simulation. In the RF sheath boundary model, the tangential component of the electric field () is given by the gradient of a scalar electric field potential. We introduce two additional scalar potentials for the tangential components of the magnetic field, which effectively impose the normal electric displacement (D n ) on the plasma sheath boundary condition via in-homogeneous Neumann boundary condition and constrain the tangential electric field on the surface as curl-free (). In our approach, the non-linear sheath impedance is formulated as a natural extension of the large thickness (or asymptotic) sheath limit (), allowing for handling both asymptotic and non-linear regimes seamlessly. The new scheme is implemented using the Petra-M finite element method analysis framework and is verified with simulations in the literature. The significance of non-linearity is discussed in various plasma conditions. An application of this scheme to asymptotic RF sheath simulation on the WEST ICRF antenna side limiters is also discussed.

Continuous and discontinuous contact problems of a functionally graded piezoelectric layer resting on a homogeneous piezoelectric half plane

In this study, the plane continuous and discontinuous contact problems between a graded piezoelectric layer and a homogeneous piezoelectric half plane are considered. The layer is pressed by a rigid conducting flat punch with an electrical load. The problem is converted using Fourier integral transform into a singular integral equation. This equation is then solved numerically by applying the Gauss-Chebychev integration formula. The primary objective of this article is to investigate the effect of varying several parameters including the contact length, the mechanical and electrical loads, and the graded layer non-homogeneity parameter on the contact stresses and on the normal electric displacement.

A penny-shaped crack in a piezoelectric layer sandwiched between two elastic layers with the electrical saturation and mechanical yielding zones

This paper deals with an internal penny-shaped crack in a piezoelectric layer sandwiched by two outer elastic layers. It is assumed that the crack faces are electrically impermeable, and the thin annular electrical saturation and mechanical yielding zones appear around the crack front under the uniform distributed normal stress and electric displacement. Three different cases, i.e., the region of electrical saturation is longer, shorter than, or equal to the domain of mechanical yielding are respectively considered in this article. For such an axisymmetric penny-shaped crack problem, we simplify the mixed boundary value problem into coupling Fredholm integral equations of the second kind by employing Hankel transform technique and the Copson method. The problem is solved in the real domain by constructing real fundamental solutions and the numerical solutions for any layer-thickness are derived with appropriate formula derivation and numerical discretization. When the outer elastic layers vanish and the piezoelectric layer-thickness approaches infinity, the relation between the nonlinear zone-lengths and the loadings obtained in this paper reduces to the analytical expression in literature. The influence of the electromechanical loads, the thicknesses of the piezoelectric layer and the elastic layer, and the material parameter ratios on the nonlinear zones is discussed in details. The results of paper are believed to have some guiding references for the design of the piezoelectric sandwich plate.

A New BEM Modeling Algorithm for Size-Dependent Thermopiezoelectric Problems in Smart Nanostructures

The main objective of this paper is to introduce a new theory called size-dependent thermopiezoelectricity for smart nanostructures. The proposed theory includes the combination of thermoelastic and piezoelectric influences which enable us to describe the deformation and mechanical behaviors of smart nanostructures subjected to thermal, and piezoelectric loadings. Because of difficulty of experimental research problems associated with the proposed theory. Therefore, we propose a new boundary element method (BEM) formulation and algorithm for the solution of such problems, which involve temperatures, normal heat fluxes, displacements, couple-tractions, rotations, force-tractions, electric displacement, and normal electric displacement as primary variables within the BEM formulation. The computational performance of the proposed methodology has been demonstrated by using the generalized modified shift-splitting (GMSS) iteration method to solve the linear systems resulting from the BEM discretization. GMSS advantages are investigated and compared with other iterative methods. The numerical results are depicted graphically to show the size-dependent effects of thermopiezoelectricity, thermoelasticity, piezoelectricity, and elasticity theories of nanostructures. The numerical results also show the effects of the size-dependent and piezoelectric on the displacement components. The validity, efficiency and accuracy of the proposed BEM formulation and algorithm have been demonstrated. The findings of the current study contribute to the further development of technological and industrial applications of smart nanostructures.

Modeling and simulating the effects of a general imperfect interface in fibrous piezoelectric composites

Interfacial behavior within fibrous piezoelectric composites (PZCs) may dramatically degrade the overall electromechanical performance and simultaneously vary the effective coupling moduli of these intelligent materials. In the present paper, we first recapitulate a physics-based general interfacial relation, which is rigorously derived from an ideal three-phase configuration and thus owns explicit physical meaning, to characterize this behavior between two different piezoelectric media. Afterwards, the explicit strong and weak governing equations of fibrous PZC boundary value problems (BVPs) are constructed, and a computational approach is then developed with the help of the extended finite element method (XFEM) to account for the discontinuities within both the primary (i.e., the electric potential and displacement fields) and the secondary fields (i.e., the normal electric displacement and normal traction fields). To achieve a benchmark problem with analytical exact solution, a simplified general interfacial relation is introduced, and a fibrous PZC BVP involving a cylindrical simplified general interface is designated and analytically solved. Hereafter, the convergence performance and validity of the elaborated computational approach are tested in detail. Eventually, discussions are further made on the influence of material composition and inhomogeneity shapes on the electroelastic coupling behavior of PZCs, and a few concluding remarks are drawn.

Continuous and discontinuous contact problems of a homogeneous piezoelectric layer pressed by a conducting rigid flat punch

In this study, frictionless continuous and discontinuous contact problems between an electrically conducting rigid flat punch and a homogeneous layer are considered. The body force of the layer is considered, and the layer is lying on the rigid substrate without bond. Thus, if the external load is smaller than a certain critical value, the contact between the layer and substrate is continuous. However, when the external loads exceed the critical value, there is a separation between the layer and substrate on the finite region, that is, discontinuous contact. Using the Fourier integral transform technique, the general expressions of the stresses and displacements are derived in the presence of body force. Using the boundary conditions, the singular integral equations are obtained for both the continuous and discontinuous contact cases. The Gauss–Chebyshev integration formulas are utilized to convert the singular integral equations into a set of nonlinear equations which are solved using a suitable iterative algorithm to yield the lengths of the separation region and the associated contact pressure and normal electric displacement. The singular integral equations are solved numerically applying the appropriate Gauss–Chebyshev integration formulas. This is the first study that investigates the contact problem of the piezoelectric materials in the presence of the body force.

New analytical solutions for modified polarization saturation models in piezoelectric materials

In this paper, we present new analytical solutions for modified polarization saturation (PS) models for arbitrary polarized and semipermeable 2-D piezoelectric media. The PS model is modified to various other non-linear fracture models by varying the normal electric displacement saturated conditions in place of a constant saturated value. These variations are hereby defined as interpolating linear, quadratic and cubic type’s polynomials into saturated value interpolated on the basis of possible electric displacement values at the centre of the crack, actual crack tip and saturated zone tip. To present the analytical study, a centre crack problem in 2-D semipermeable piezoelectric media under arbitrary poling direction and combined in-plane mechanical and electric displacement loadings is considered. Using extended Stroh formalism and complex variable approach, these models are mathematically reduced into generalized non-homogeneous Riemann–Hilbert problems. Applying Riemann–Hilbert technique, the solution of these problems is obtained in terms of generalized complex potential functions representing traction forces and electric displacement components. These lead to obtain the explicit expressions for saturated zone length, crack opening displacement, crack opening potential and local intensity factor in each case. Also, the results of saturation zones and local intensity factor are obtained in each case and compared with the established numerical results. The studies show that saturated zone length increases with increasing electric loading for each case and also depends on the polynomial varying saturation condition. For a particular electric loading applied within the range of critical value, it increases with increasing degree of polynomial type saturated condition i.e. from constant to cubic type’s normal electric displacement saturated condition.

Surface effect on the contact problem of a piezoelectric half-plane

Based on the theory of the surface piezoelectricity, the two-dimensional frictionless contact problem of the piezoelectric half-plane under rigid flat and cylindrical punches is investigated in this paper. The Fourier integral transform method is adopted to derive the fundamental solution for the contact problem of piezoelectric materials with the surface effect, including the residual surface stress, surface elastic constant, surface piezoelectric constant and surface dielectric constant. Then, the contact problem is solved numerically to determine the distribution of the normal contact stress, in-plane stress and electric displacement at the contact surface. The surface effect on the normal contact stress, in-plane stresses and in-plane electric displacement is discussed in detail. We find that the surface effect plays a significant role on contact behaviors of piezoelectric materials at micro/nano-scales.

Three-dimensional analytical solution for the instability of a parallel array of mutually attracting identical simply supported piezoelectric microplates

Three-dimensional analytical solutions are derived for the structural instability of a parallel array of mutually attracting identical simply supported orthotropic piezoelectric rectangular microplates by means of a linear perturbation analysis. The two surfaces of each plate can be either insulating or conducting. By considering the fact that the shear stresses and the normal electric displacement (or electric potential) are zero on the two surfaces of each plate, a \(2 \times 2\) transfer matrix for a plate can be obtained directly from the \(8 \times 8\) fundamental piezoelectricity matrix without resolving the original Stroh eigenrelation. The critical interaction coefficient can be determined by solving the resulting generalized eigenvalue problem for the piezoelectric plate array. Also considered in our analysis is the in-plane uniform edge compression acting on the four sides of each piezoelectric plate. Our results indicate that the stabilizing influence of the piezoelectric effect on the structural instability is unignorable; the edge compression always plays a destabilizing role in the structural instability of the plate array with interactions.

Complex variable approach in studying modified polarization saturation model in two-dimensional semipermeable piezoelectric media

A modified polarization saturation model is proposed and addressed mathematically using a complex variable approach in two-dimensional (2D) semipermeable piezoelectric media. In this model, an existing polarization saturation (PS) model in 2D piezoelectric media is modified by considering a linearly varying saturated normal electric displacement load in place of a constant normal electric displacement load, applied on a saturated electric zone. A centre cracked infinite 2D piezoelectric domain subject to an arbitrary poling direction and in-plane electromechanical loadings is considered for the analytical and numerical studies. Here, the problem is mathematically modeled as a non-homogeneous Riemann-Hilbert problem in terms of unknown complex potential functions representing electric displacement and stress components. Having solved the Hilbert problem, the solutions to the saturated zone length, the crack opening displacement (COD), the crack opening potential (COP), and the local stress intensity factors (SIFs) are obtained in explicit forms. A numerical study is also presented for the proposed modified model, showing the effects of the saturation condition on the applied electrical loading, the saturation zone length, and the COP. The results of fracture parameters obtained from the proposed model are compared with the existing PS model subject to electrical loading, crack face conditions, and polarization angles.

An XFEM/level set strategy for simulating the piezoelectric spring-type interfaces with apparent physical background

The piezoelectric spring-type interface is widely applied to describe the physical phenomenon that the displacement vector and the electric potential suffer jumps across an interface in a certain intelligent material, while both the normal traction vector and the normal electric displacement stay continuous across the same interface. This work is dedicated to accurately depicting the effects of the physics-based piezoelectric spring-type interfaces in arbitrary shapes and to predicting the effective electroelastic moduli of composites containing such interfaces. To achieve this two-fold goal, a computational approach combining the extended finite element method (XFEM) and the level set method (LSM) is developed and interpreted in detail. The accuracy and convergence performance of the elaborated approach are assessed with a benchmark problem for which the exact analytic solution is derived. Eventually, the approach validated is further utilized to explore the size- and shape-dependent effects induced by these spring-type interfaces on the overall couple-field moduli of fibrous piezoelectric composites.

Axisymmetric solutions for the multi-layered transversely isotropic piezoelectric medium

By introducing the precise integration algorithm (PIA) and the technique of the mixed variable formalism accounting for any number of layers, variations of components of the axisymmetric piezoelectric field including elastic displacement, vertical normal stress, electric potential and electric displacement along the axis of symmetry in the cylindrical coordinate system are explored. The piezoelectric field is induced by the vertical loadings of mechanical or electric types uniformly distributed over a circular region. With the aid of the mixed variable formulations, along with the Hankel integral transform and the corresponding matrix algebraic operations, the governing partial differential equations of equilibrium expressed in terms of displacements and electric potential are reduced to the first order ordinary differential matrix equations. As a highly accurate algorithm, the PIA is provided to evaluate the obtained ordinary differential matrix equations in the transformed domain. Both mechanical and electrical quantities in the physical domain can be acquired by taking the inversion of the Hankel integral transform. An example as benchmark is illustrated to examine the applicability and performance of the proposed technique compared with results in the literature. Numerical examples are used to demonstrate the influence of the degree of the material anisotropy and stratified parameters.

Boundary element formulation for plane problems in size-dependent piezoelectricity

A new boundary element formulation is developed to analyze two-dimensional size-dependent piezoelectric response in isotropic dielectric materials. The model is based on the recently developed consistent couple stress theory, in which the couple-stress tensor is skew-symmetric. For isotropic materials, there is no classical piezoelectricity, and the size-dependent piezoelectricity or flexoelectricity effect is solely the result of coupling of polarization to the skew-symmetric mean curvature tensor. As a result, the size-dependent effect is specified by one characteristic length scale parameter l, and the electromechanical effect is specified by one flexoelectric coefficient f. Interestingly, in this size-dependent multi-physics model, the governing equations are decoupled. However, the problem is coupled, because of the existence of a flexoelectric effect in the boundary couple-traction and normal electric displacement. We discuss the boundary integral formulation and numerical implementation of this size-dependent piezoelectric boundary element method, which provides a boundary-only formulation involving displacements, rotations, force-tractions, couple-tractions, electric potential, and normal electric displacement as primary variables. Afterwards, we apply the resulting BEM formulation to several computational problems to confirm the validity of the numerical implementation and to explore the physics of the flexoelectric coupling. Copyright © 2016 John Wiley & Sons, Ltd.

On the total electro-mechanical potential energy and energy release rate at the interface crack tips in an initially stressed sandwich plate-strip with piezoelectric face and elastic core layers

An investigation into the values of the total electro-mechanical potential energy and energy release rate (ERR) at the interface crack tips in an initially stressed sandwich plate-strip is carried out for the opening crack mode. It is assumed that the face layers of the considered sandwich plate are made from piezoelectric materials but that the core is made of elastic material, and that this sandwich plate has initial stresses caused by the uniformly distributed normal forces acting on the planes of the two parallel grounded ends of the plate. The plate has two parallel interface cracks symmetrically located with respect to the middle planes and additional uniformly distributed normal opening mechanical forces acting on the cracks’ edges. Moreover, the face planes of the face layers and the cracks’ edges are unelectroded with vanishing normal electric displacement and are free of electrical charge. The investigations are made by utilizing the three-dimensional linearized theory of electro-elasticity in a plane-strain state within the framework of the piecewise homogeneous body model. The solution to the corresponding boundary value problems are found numerically by employing the finite element method. Numerical results, which are presented and analyzed, illustrate the influences of the geometrical parameters and the initial stresses, as well as the coupling effects between the electro-mechanical fields, on the values of the total electro-mechanical potential energy and of the ERR.

Frictional contact problem between a functionally graded magnetoelectroelastic layer and a rigid conducting flat punch with frictional heat generation

ABSTRACTIn this study, we consider the frictional sliding contact problem between a functionally graded magnetoelectroelastic (MEE) layer resting on a perfectly insulated rigid half plane and a perfectly conducting rigid flat punch with frictional heat generation. The punch is subjected to magnetoelectromechanical loads. The graded layer is modeled as a nonhomogeneous medium with a transversely isotropic stress–strain law and an exponential variation of the magnetoelectrothermoelastic properties along the thickness direction. Neglecting inertia effects and assuming a constant friction coefficient, the solution is obtained within the framework of steady-state plane magnetoelectrothermoelasticity under plane strain conditions. The heat equation is first solved using Fourier transform to yield the temperature field in the layer which is then substituted in the MEE governing equations. These equations are solved analytically using the same transform leading to three coupled Cauchy-type singular integral equations in which the main unknowns are the normal contact stress, the electric displacement, and the magnetic induction. These equations are then solved numerically to obtain the distributions of the normal contact stress, electric displacement, and magnetic induction at the surface of the graded medium. The main objective of this paper is to study the effect of the nonhomogeneity parameter; the friction coefficient; and the elastic, electric, and magnetic coefficients on the surface contact pressure, electric displacement, and magnetic induction distributions for the case of flat punch profile.

The contact problem of a rigid stamp with friction on a functionally graded magneto-electro-elastic half-plane

We consider in this study the frictional sliding contact problem between a functionally graded magneto-electro-elastic material and a perfectly conducting rigid punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the unknowns are the normal contact stress, the electric displacement and the magnetic induction. These integral equations are then solved numerically to obtain the distributions of the normal contact stress, electric displacement and magnetic induction at the surface of the graded medium. The main objective of this paper is to study the effect of the non-homogeneity parameter, the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat and circular punch profiles.

Coupled electro-elastic analysis of functionally graded piezoelectric material plates

A unified formulation of finite layer methods (FLMs), based on the Reissner mixed variational theorem (RMVT), is developed for the three-dimensional (3D) coupled electro-elastic analysis of simply-supported, functionally graded piezoelectric material (FGPM) plates with open- and closed-circuit surface conditions and under electro-mechanical loads. In this formulation, the material properties of the plate are assumed to obey an exponent-law varying exponentially through the thickness coordinate, and the plate is divided into a number of finite rectangular layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-plane variations of the primary field variables of each individual layer, respectively, such as the elastic displacement, transverse shear and normal stress, electric potential, and normal electric displacement components. The relevant orders used for expanding these variables in the thickness coordinate can be freely chosen as the linear, quadratic and cubic orders. Four different mechanical/electrical loading conditions applied on the top and bottom surfaces of the plate are considered, and the corresponding coupled electro-elastic analysis of the loaded FGPM plates is undertaken. The accuracy and convergence rate of the RMVT-based FLMs are assessed by comparing their solutions with the exact 3D piezoelectricity ones available in the literature.

Homogenization of fibrous piezoelectric composites with general imperfect interfaces under anti-plane mechanical and in-plane electrical loadings

The present paper aims mainly to estimate the size-dependent effective properties of fibrous piezoelectric composites with general imperfect interfaces. The interface model used states that the displacement, traction, electric potential, and normal electric displacement all suffer jumps across an interface. In addition, it can degenerate into the well-known special ones by employing appropriate high-contrast interfacial parameters. To achieve our objective, an auxiliary inhomogeneity problem of a circular fiber embedded in an infinite cylindrical reference phase via general imperfect interface under anti-plane mechanical and in-plane electrical boundary conditions is analytically solved. This solution allows us to apply the well-known micromechanical schemes such as the dilute, Mori–Tanaka to obtain the closed-form expressions for the size-dependent overall properties of composites under consideration. Some numerical examples are provided for illustrating the features of the obtained general results.