Gradient piezoelectricity for cracks under an impact load

The flexoelectric effect on elastic waves is investigated in nano-sized cracked structures. The strain gradients are considered in the constitutive equations of a piezoelectric solid for electric displacements and the higher-order stress tensor. The governing equations with the corresponding boundary conditions are derived from the variational principle. The finite element method (FEM) is developed from the principle of virtual work. It is equivalent to the weak-form of derived governing equations in gradient elasticity. The computational method can be applied to analyze general 2D boundary value problems in size-dependent piezoelectric elastic solids with cracks under a dynamic load. The FEM formulation is implemented for strain-gradient piezoelectricity under a dynamic load.     

SH wave propagation in a cylindrically layered piezoelectric structure with initial stress

Summary SH wave propagation in a cylindrically layered piezoelectric structure with initial stress is investigated analytically. By means of transformation, the governing equations of the coupled waves are reduced to Bessel and Laplace equations. The boundary conditions imply that the displacements, shear stresses, electric potential, and electric displacements are continuous across the interface between the layer and the substrate. The electrically open and short conditions at the cylindrical surface are applied to solve the problem. The phase velocity is numerically calculated for the electrically open and short cases, respectively, for different wavenumber and thickness of the layer. The effect of the initial stress on the phase velocity and the electromechanical coupling factor are discussed in detail for piezoelectric ceramics PZT-5H. We find that the initial stress has an important effect on the SH wave propagation in the cylindrically layered piezoelectric structures. The results also show that the ratio of the layer thickness to the wavelength has a remarkable effect on the SH wave phase velocity and electromechanical coupling factor.     

压电梯度薄壳的高阶理论解

压电功能梯度执行器能产生较大的位移、降低应力峰值并避免了粘结层带来的问题,压电梯度超声换能器能拓展频带宽度。本文作者提出了一个简单而有效的求解压电梯度薄壳力、电行为特性的高阶理论。设定位移分量为壳厚的线性函数,而电势沿厚度方向为二次分布。考虑了压电作动元的驱动信号不同时所具有的不同形式的电荷平衡方程。应用Fourier级数法得到压电系数沿厚度坐标变化的梯度壳的力电耦合的解析解。所得结果可退化至梁、板等多种特殊情况。利用所得方程分析了一非均匀简支压电层合板,并与三维精确结果作了对比,两者吻合得很好,表明该理论的正确性。最后具体求解了压电梯度圆柱壳的力、电特性,给出了位移、应力、电势沿厚度方向的变化规律。     

Free vibration of laminated piezoelectric composite plates based on an accurate theory

An accurate theory for laminated piezoelectric composite plates in cylindrical bending is developed for free vibration analysis. The displacement and electric potential fields are depicted approximately by the accurate displacement and electric potential distribution functions through thickness, respectively. The two functions are formulated according to particular solutions to the three-dimensional elasticity equilibrium equations and the electrostatics charge equation. The complicated electromechanical coupling relations and the interfacial continuity conditions are enforced. Accordingly the two functions are coupled and make the displacement and potential fields coupled. The governing equations use only four displacement and potential variables, the number of which is independent of the number of layers involved. A corresponding finite element model is also developed. Natural frequencies of piezoelectric laminates subjected to different sets of boundary conditions are given and parameter studies are conducted in numerical examples. The high accuracy of this theory is demonstrated by comparing the present results with the existing exact three-dimensional solutions.     

An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory

Based on third-order shear deformation plate theory of Reddy, the authors aim to provide an exact analytical solution for free vibration analysis of thick circular/annular plates, both upper and lower surfaces of which are in contact with a piezoelectric layer. Natural frequencies are determined by the solution of the coupled electromechanical governing equations for a combination of free, soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the plate. The electrodes on each piezoelectric layer are assumed to be short-circuited. The Maxwell electrostatics equation is satisfied by adopting a half-sine distribution of the electric potential in the transverse direction of the piezoelectric layers. A comparison of the present exact natural frequencies for piezoelectric coupled circular/annular plates with different boundary conditions is made with previously published results obtained by the Mindlin plate theory and 3-D modified finite element method. The effects of plate parameters such as host thickness to radius ratios, inner to outer radius ratios and piezoelectric to host thickness ratios on the natural frequencies of laminated circular/annular plates are investigated for different combinations of boundary conditions. Results obtained by the present exact closed-form solutions can be served as benchmark data for investigators to validate their numerical and analytical methods in the future.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.     

Variational principles for the equations of porous piezoelectric ceramics

The governing equations of a porous piezoelectric continuum are presented in variational form, though they were well established in differential form. Hamilton's principle is applied to the motions of a regular region of the continuum, and a three-field variational principle is obtained with some constraint conditions. By removing the constraint conditions that are usually undesirable in computation through an involutory transformation, a unified variational principle is presented for the region with a fixed internal surface of discontinuity. The unified principle leads, as its Euler-Lagrange equations, to all the governing equations of the region, including the jump conditions but excluding the initial conditions. Certain special cases and reciprocal variational principles are recorded, and they are shown to recover some of the earlier ones.     

Scattering of the SH wave from a crack in a piezoelectric substrate bonded to a half-space of functionally graded materials

The scattering of the SH wave from a crack in a piezoelectric substrate which is bonded to a half-space of functionally graded materials (FGM) is studied. The governing equations along with permeable crack boundary, regularity and continuity conditions across the interface are reduced to a coupled set of Cauchy singular integral equations which are solved approximately by applying Chebyshev polynomials. Numerical results for the normalized dynamic stress intensity factors (NDSIF) and the normalized electric displacement intensity factors (NEDIF) are presented. The effects of the geometric and physical parameters, and the effects of the frequency and the angle of incidence on NDSIF and NEDIF are discussed.     

Variational principles for generalized thermodiffusion theory in pyroelectricity

The universal thermodynamic variational principle proposed in the previous papers for nonlinear dielectrics and thermopiezoelectricity is extended to the thermodiffusion theory in pyroelectricity, and it is used as a fundamental physical principle to derive the simple complete governing equations of the generalized thermo-electro-diffuso-elastic theory in this paper. In the generalized thermo-electro-diffuso-elastic theory it is assumed that the variation of temperature needs the extra heat which introduces the inertial entropy, and the variation of chemical potential also needs the extra heat which introduces the inertial concentration, etc. The electro-chemical Gibbs function variational principle, the electric Gibbs function variational principle and the internal energy variational principle are derived in this paper.     

Variational‐based modeling of micro‐electro‐elasticity with electric field‐driven and stress‐driven domain evolutions

Recently, increasing interest in so‐called functional or smart materials with electromechanical coupling has been shown such as ferroelectric piezoceramics. These materials are characterized by microstructural properties, which can be changed by external stress and electric field stimuli, and hence find use as the active components in sensors and actuators. The electromechanical coupling effects result from the existence and rearrangement of microstructural domains with uniformly oriented electric polarization. The understanding and efficient simulation of these highly nonlinear and dissipative mechanisms, which occur on the microscale of ferroelectric piezoceramics, are a key challenge of the current research. This paper does not offer a substantially new physical model of these phenomena but a new mathematical modeling approach based on a rigorous exploitation of rate‐type variational principles. This provides a new insight in the structure of the coupled problem, where the governing field equations appear as the Euler equations of a variational statement. We outline a variational‐based micro‐electro‐elastic model for the microstructural evolution of both electrically and mechanically driven electric domains in ferroelectric ceramics, which also incorporates the surrounding free space. To this end, we extend recently developed multifield incremental variational principles of electromechanics from local to gradient‐extended dissipative response and specialize it by a Ginzburg–Landau‐type phase field model, where the thickness of the domain walls enters the formulation as a length scale. This serves as a natural starting point for a canonical compact, symmetric finite element implementation, considering the mechanical displacement, the microscopic polarization, and the electric potential induced by the polarization as the primary fields. The latter is defined on both the solid domain and a surrounding free space. Numerical simulations treat domain wall motions for electric field‐driven and stress‐driven loading processes, including the expansion of the electric potential into the free space. Copyright © 2012 John Wiley & Sons, Ltd.     

Bending analysis of functionally graded piezoelectric plates via quasi-3D trigonometric theory

Abstract

This work is devoted to the bending analysis of functionally graded piezoelectric (FGP) material plate by using a simple quasi-3D sinusoidal shear deformation theory under simply supported edge conditions. The governing equations and boundary conditions are derived by using the principle of virtual work. The impact of piezoelectric, electric loading, and gradient index on the displacement, electric displacement, electric potential, and stresses are explored numerically presented and discussed in detail. To check the accuracy and validity of bending results obtained from this analysis of FGP plates, results are compared with the analytical solution obtained by 3D, quasi-3D, and higher order shear deformation theories. Parametric studies are then performed to examine the effects of the thickness of the plate and the electric field on the overall electro-mechanical response of the FGP plates. The presented results are useful in design processes of smart structures and analysis from piezoelectric materials.     

金属芯压电纤维/聚合物复合材料压电-黏弹-塑性行为的细观力学模型

为了表征金属芯压电纤维增强聚合物基（MPF/PM）复合材料非线性、时变的压电-黏弹-塑性行为,基于变分渐近理论建立MPF/PM增量形式的细观力学模型。首先分别导出聚合物和MPF增量型本构方程,基于汉密尔顿扩展原理推导出MPF/PM压电-黏弹-塑性变分原理的能量泛函。考虑材料的时变和非线性特征,建立与求解瞬时有效机-电耦合矩阵有关的增量过程,并通过有限元技术实现模型的数值模拟。利用构建模型研究了不同铝芯体积分数、电场变化率和加载条件对MPF/PM有效全局应力-应变和单轴纵向拉伸性能的影响。结果表明,构建的模型能准确模拟MPF/PM多场耦合作用下的非线性、时变行为,为该新型智能材料的实际工程应用奠定理论基础。     

Electromechanical modeling and analytical investigation of nonlinearities in energy harvesting piezoelectric beams

Piezoelectric materials are extensively applied for vibrational energy harvesting especially in micro-scale devices where other energy conversion mechanisms such as electromagnetic and electrostatic methods encounter fabrication limitations. A cantilevered piezoelectric bimorph beam with an attached proof (tip) mass for the sake of resonance frequency reduction is the most common structure in vibrational harvesters. According to the amplitude and frequency of applied excitations and physical parameters of the harvester, the system may be pushed into a nonlinear regime which arises from material or geometric nonlinearities. In this study nonlinear dynamics of a piezoelectric bimorph harvester implementing constitutive relations of nonlinear piezoelectricity together with nonlinear curvature and shortening effect relations, is investigated. To achieve this goal first of all a comprehensive fully-coupled electromechanical nonlinear model is presented through a variational approach. The governing nonlinear partial differential equations of the proposed model are order reduced and solved by means of the perturbation method of multiple scales. Results are presented for a PZT/Silicon/PZT laminated beam as a case study. Findings indicate that material nonlinearities of the PZT layer has the dominant effect leading to softening behavior of the frequency response. At the primary resonance, different frequency responses of the extracted power can be distinguished according to the excitation amplitude, which is due to harmonic generation as a result of piezoelectric nonlinearity. The extracted power is analytically computed and validated with a good agreement by a numerical solution.     

Two collinear unequal cracks in a poled piezoelectric plane: Mode I case solved by a new approach of real fundamental solutions

The purpose of the present work is to study the problem of two collinear unequal cracks in a piezoelectric plane under mode I electromechanical loadings via a new approach. For the first time, real fundamental solutions are derived for in-plane piezoelectric governing equations. The cracks are simulated by continuously distributed generalized dislocations and Cauchy singular integral equations are established from the solution of a generalized point dislocation. Both the theorectical derivation and numerical computations are validated by the exact solution in a special case. Parametric studies are conducted to reveal the effects of crack space, crack length, electric loading and remanent electric displacement on energy release rate. It is found that negative electric displacement loading can decrease both the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR), implying that it has a shielding effect on cracks definitely. Positive electric displacement loading can enhance MSERR, but meanwhile it can enhance or reduce TERR depending on the magnitude of the electric loading factor. The effect of a remanent electric displacement along the poling direction is equivalent to that of a positive electric field loading and should be considered in engineering design.     

Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory

ABSTRACT

This article investigates the nonlinear vibration of piezoelectric nanoplate with combined thermo-electric loads under various boundary conditions. The piezoelectric nanoplate model is developed by using the Mindlin plate theory and nonlocal theory. The von Karman type nonlinearity and nonlocal constitutive relationships are employed to derive governing equations through Hamilton's principle. The differential quadrature method is used to discretize the governing equations, which are then solved through a direct iterative method. A detailed parametric study is conducted to examine the effects of the nonlocal parameter, external electric voltage, and temperature rise on the nonlinear vibration characteristics of piezoelectric nanoplates.     

Fatigue crack growth driven by electric fields in piezoelectric ceramics and its governing fracture parameters

Fatigue crack growth test for piezoelectric ceramics was performed under cyclic electric loading. Double cantilever beam specimen, which was made of two different piezoelectric ceramics, with a through notch was used. The specimens were, varying the amplitude and the mean value, subjected to various cyclic electric fields. It was found that crack growth behavior is greatly dependent on the amplitude and mean value of cyclic electric field and materials. Crack growth rate decreased as electric field increased and finally stopped. Crack growths under the positive, the negative and the shifted electric field were very slow compared to that under fully reversed electric field. However, threshold for the crack propagation did not depend greatly on materials. Then, as possible governing fracture parameters, CED and electric displacement intensity factor were chosen based on the results of electromechanical finite element analysis within linear framework and their closed form equations were also obtained considering the influences of electric boundary conditions inside the notch. Finally, the parameters were correlated with crack growth rate measured experimentally by employing Paris law type equation.     

Finite element analysis for geometrically nonlinear deformations of smart functionally graded plates using vertically reinforced 1-3 piezoelectric composite

In this paper, a finite element model has been developed for the geometrically nonlinear static analysis of simply supported functionally graded (FG) plates integrated with a patch of vertically reinforced 1-3 piezoelectric composite material acting as a distributed actuator. The material properties of the functionally graded substrate plate are assumed to be graded only in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. The analysis of the electro-elastic coupled problem includes the transverse deformations of the overall plate to utilize the transverse normal actuation by the distributed actuator for counteracting the nonlinear deformations of smart functionally graded plates. The nonlinear governing equations of equilibrium are solved by using direct iteration method with under-relaxation. The numerical illustrations suggest the potential use of the distributed actuator made of vertically reinforced 1-3 piezoelectric composite material for active control of nonlinear deformations of smart functionally graded plates. The effect of variation of piezoelectric fiber orientation in the distributed actuator on its control authority for counteracting the nonlinear deformations of smart functionally graded plates has also been investigated.     

An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics

An equivalent single-layer model for the dynamic analysis of magnetoelectroelastic laminated plates is presented. The electric and magnetic fields are assumed to be quasi-static and the first-order shear deformation theory is used. The formulation of the model provides for a preliminary fulfillment of the electro-magnetic governing equations, which allows to determine the electric and magnetic potential as functions of the mechanical variables. Then, by using this result, the equations of motion are written leading to the problem governing equations. They involve the same terms of the elastic dynamic problem weighted by effective stiffness coefficients, which take the magneto-electro-mechanical couplings into account. Additional terms, exclusively arising in force of the piezoelectric and piezomagnetic behavior, appear. The electromagnetic inputs are treated as equivalent external distributed axial forces and bending moments. Free and forced vibrations solutions for simply-supported plates are presented to validate the model by comparing the present results with benchmark solutions found in the literature.     

Electromechanical responses of piezoelectric nanoplates with flexoelectricity

Flexoelectricity, representing the coupling between electrical polarizations and strain gradients, should be taken into account in the analysis of electromechanical responses of nanostructures where large strain gradients are expected. In this paper, we will explore the influence of flexoelectricity on the electromechanical coupling behavior of a simply supported piezoelectric nanoplate by using the Kirchhoff plate theory. The governing equations and corresponding boundary conditions are deduced from Hamilton’s principle, and the analytical solutions are obtained for the deflection and natural frequency. The results indicate that the deflections predicted by the present model are smaller than those calculated by the classical one which only considers piezoelectricity, while the frequencies exhibit the opposite trend. In addition, the flexoelectric effect is more prominent for thinner plates; the differences of the deflections or frequencies between the two models are gradually diminishing with an increase in the plate thickness. The current work may contribute to the understanding of the higher-order electromechanical coupling mechanism. Moreover, the modified plate model can be utilized to accurately design novel piezoelectric nanoplate-based sensors in nanoelectromechanical systems.