Analytical solutions of functionally graded piezoelectric circular plates subjected to axisymmetric loads

Based on the three-dimensional theory of piezoelectricity, this work analytically investigates the axisymmetric bending of circular plates whose material properties vary along the thickness. The transverse loads are expanded in terms of the Fourier–Bessel series, and the solutions corresponding to each item of the series are derived by a semi-inverse method. The overall results are then obtained through the superposition principle. The exact solutions are obtained for two unusual boundary conditions if the material properties obey an exponential law. Meanwhile, a layerwise model is employed for the case of arbitrary and independent distribution of each component of material properties and the approximate solutions are proposed for simply supported and clamped boundaries. Finally, the numerical examples are illustrated, and the results are compared with those of the finite element method to demonstrate the present method.     

单压电片悬臂梁式压电俘能器效能分析

以线性压电理论为基础,建立了单压电片悬臂梁式压电俘能器的输出功率与压电结构的构型、驱动频率以及外载荷阻抗等物理参数之间的解析关系。数值计算结果表明,优化负载阻抗、金属层厚度以及金属层材料可有效提高俘能器的俘能效率。文中还将单压电片悬臂梁式压电俘能器与双压电片悬臂梁式压电俘能器进行了对比,理论分析结果表明当外界机械振动的频率较稳定时,前者具有更高的俘能效率。     

Analytical solution for the axisymmetric plane strain electroelastic dynamics of a special non-homogeneous piezoelectric hollow cylinder

By virtue of the introduction of a dependent variable and the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of a special non-homogeneous piezoelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solutions of displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow cylinder with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are finally presented.     

Optimization of a circular thin-film piezoelectric actuator lying on a clamped multilayered elastic plate

A system consisting of a circular multilayered thin-film elastic plate and a piezoelectric actuator, which is generally used for ultrasound generation in air, is studied in this paper. Effects of the electrode dimension of a circular thinfilm piezoelectric actuator lying on a clamped multilayered elastic plate are discussed theoretically, while the first-order theory of asymmetrically laminated piezoelectric plates with consideration of coupled extension and flexure of the reference plane is used. Numerical results show that the deflection of the elastic plate can be optimized by adjusting the radius of the top electrode.     

Nonlinear characteristics of a circular plate piezoelectric harvester with relatively large deflection near resonance

Based on the von Karman thin circular plate theory, we report in this paper the analysis of the nonlinear behavior of a power harvester consisting of a circular piezoelectric plate and an electric resistance. Dependence of the output power of the harvester upon driving frequency for different electric loads and different applied forces is obtained. Numerical results show that the output power exhibits multi-valuedness and a jump phenomenon near resonance.     

考虑非局部应变梯度效应的轴对称压电纳米圆板热-力-电耦合振动EI北大核心CSCD

基于非局部应变梯度理论和Mindlin板理论,研究了热‐力‐电多场耦合下轴对称压电纳米圆板的振动特性。通过Hamilton原理推导了非局部应变梯度本构框架内的运动方程,采用微分求积法数值求解了理论模型微分方程组,分析了压电纳米圆板的振动固有频率受内尺度参数与外场参数的影响。压电纳米圆板的固有频率随着非局部参数的增大而减小,随着应变梯度特征参数的增大而增大。当非局部参数小于应变梯度特征参数时,纳米圆板表现出刚度硬化行为;当非局部参数大于应变梯度特征参数时,表现出刚度软化行为。当非局部参数等于应变梯度特征参数时,纳米圆板的刚度退化为相应的经典连续介质理论结果。此外,固有频率随着径向压力和正电压的增大而减小,随着径向拉力和负电压的增大而增大,随着温差的增加而小幅减小。特别地,研究发现当径向载荷和电压增大到一定程度时,纳米圆板出现了振动失稳现象,并分析了非局部参数与应变梯度特征参数对失稳临界径向载荷及临界电压的影响。     

Zernike polynomial based Rayleigh-Ritz model of a piezoelectric unimorph deformable mirror

Piezoelectric bimorph- or unimorph-type deformable mirrors are commonly used in adaptive optics to correct for time-dependent phase aberrations. In the optics community, the surface deformations that deformable mirrors are required to achieve, are routinely and conveniently described using Zernike polynomials. A Rayleigh-Ritz structural model, which uses Zernike polynomials directly to describe the displacements, is proposed in this paper. The proposed formulation produces a numerically inexpensive model that predicts deformations with remarkable accuracy. Since design variables, such as electrode layout, material properties, and mirror dimensions, are represented analytically, the model is well suited to optimization or sensitivity analysis applications. Furthermore, since the numerical implementation is very efficient, it could be employed in closed-loop control applications. Results achieved with the proposed model compare well with results from a traditional finite element analysis as well as experimental results of a representative design.     

A hysteresis model for a stacked-type piezoelectric actuator

The hysteresis nonlinearity in piezoelectric materials brings difficulties in controlling the systems. In order to mitigate the effect of hysteresis, such nonlinearity needs to be characterized and modeled under different load circumstances. For this purpose, the actuator is modeled in terms of a mass-spring-damper system utilizing the stop operator as one of the operators of the Prandtl-Ishlinskii (PI) model. Merging the structural model with the nonlinear hysteresis model, we observe that the results demonstrate better correspondence to the measured output compared to that of the classical PI model for a wide range of working conditions, i.e., different input frequency and amplitude.     

Analytical solutions describing the consolidation of a multi-layered soil under circular loading

Analytical solutions describing the consolidation of a multi-layered soil under circular loading are presented. From the governing equations of saturated poroelastic soil in a cylindrical coordinate system, the eighth-order state-space equation of consolidation is obtained by eliminating the variation of time t using the Laplace transform together with the technique of Fourier expansions with respect to the coordinate θ and the Hankel transform with respect to coordinate r. The solution of the eighth-order state-space equation is derived directly by using the Laplace transform and its inversion of the z-domain. Based on the continuity between layers and the boundary conditions, the transfer-matrix method is utilized to derive the solutions for the consolidation of a multi-layered soil under circular loading in the transformed domain. By the inversion of the Laplace transform and the Hankel transform, the analytical solutions in the physical domain are obtained. A numerical analysis based on the solutions is carried out by a corresponding program.     

Closed form solutions for partially debonded circular inclusion in piezoelectric materials

Summary This paper deals with the problem of a partially debonded piezoelectric circular inclusion in a piezoelectric matrix. This boundary value problem is reduced to two Riemann-Hilbert problems through the use of the analytical continuation theory.Closed form solutions are obtained by considering the behavior of the complex field potentials at origin and infinity. The formulae for the electro-elastic field intensity factors of the interfacial crack are derivedexplicitly. Several particular cases are provided to show the effect of the crack angle, the mechanical and electrical properties and the loads on the electroelastic field singularities.     

压电复合材料中III型唇形裂纹问题的解析解

采用复变函数方法和保角映射技术,研究了压电复合材料中含唇形裂纹的无限大体远场受反平面机械载荷和面内电载荷作用下的反平面问题,利用复变函数中的留数定理和Cauchy积分公式,分别获得了电不可通和电可通两种边界条件下裂纹尖端场强度因子和机械应变能释放率的解析表达式。当唇形裂纹的高度趋于零时,可得到无限大压电复合材料中Griffith裂纹的解析解。若不考虑电场作用,所得解退化为经典材料的已知结果。数值算例显示了裂纹的几何尺寸和机电载荷对机械应变能释放率的影响规律。结果表明: 唇形裂纹高度的增加会阻碍裂纹的扩展;机械载荷总是促进裂纹的扩展;电载荷对裂纹扩展的影响与裂纹面电边界条件有关。     

In-plane P-SV waves from a piezoelectric strip actuator: exact versus effective boundary condition solutions

A piezoelectric strip of finite width and thickness is placed on top of an isotropic elastic half space. It operates in actuator mode, and a time harmonic voltage is thus applied across it. The piezoelectric material is of type 6 mm oriented so that a two-dimensional (2-D) in-plane (P-SV) problem results. By Fourier series expansions, the problem is solved exactly, and this result is compared to the case when the piezoelectric strip is replaced by an effective boundary condition, which is derived by series expansions in the thickness coordinate in the piezoelectric strip. At low frequencies, the results agree very well, and this corresponds to the situation often met in practice. In general, the effective boundary condition should be much easier to apply, for example when a finite element method (FEM) program is used.     

Analytical solutions of heterogeneous rectangular plates with transverse small periodicity

We consider the cylindrical bending of a simply-supported orthotropic rectangular plate, which is small-periodically heterogeneous in the thickness direction. A homogenized plate model is first established by using the two-scale asymptotic expansion method. The state-space method is then adopted to analyze the homogenized plate exactly. Analytical expressions for two sets of approximate stresses, i.e. the homogenized model stresses and the zeroth-order two-scale model stresses, are presented. To check the accuracy of the approximate stresses, the state-space method is also applied to the original laminate plate or the approximate laminate model of the original heterogeneous plate with continuously varying material properties to obtain an analytical solution. In the latter case, the analytical solution is approximate but approaches the exact solution gradually when the number of layers increases. In order to avoid numerical instability, the joint coupling matrix is utilized. Numerical results illustrate that the two-scale model can predict accurately the realistic stress field in the original plate if it contains enough repeated units along the thickness.     

Analytical geometrical responses in large deflection of simply supported piezoelectric layered plate under initial tension

Abstract

The analytical geometrical responses in large deflection of a simply supported and layered piezoelectric circular plate under initial tension due to lateral pressure are presented. The approach follows von Karman plate theory for large deflection with a consideration of a symmetrically laminated case including a piezoelectric layer. The related nonlinear governing equations are derived in a non‐dimensional form and are simplified by neglecting the arising nonlinear terms, yielding a modified Bessel equation or a standard Bessel equation for the lateral slope. The associated analytical solutions are developed by imposing the simply supported edge conditions of the problem. For a 3‐layered nearly monolithic plate under a low pretension and a low applied voltage upon the piezoelectric layer, the results agree well with those obtained by using the classical plate theory for a single‐layered plate under pure mechanical loading, and thus the developed approach is validated. Typical 3‐layered piezoelectric plates are then implemented and the results show that, no apparent edge effect was found for the present problem. In additions, a piezoelectric effect appears to be present only up to a moderate initial tension. For a relatively high pretension, the tension effect tends to be dominant, resulting in nearly the same results for the geometrical responses, regardless of the magnitude of the applied voltage.     

Control of mechanical deformation of a laminate by electrical load to piezoelectric actuator considering the effects of damping and transverse shear

In this paper, we treat the control of dynamic deformation of a laminate by applying electrical load to piezoelectric actuators. Dynamic behavior of the laminate is analyzed considering the effect of damping due to interlaminar shear and the effect of transverse shear. The analytical model is a rectangular laminate composed of fiber-reinforced laminae and piezoelectric layers. The model is assumed to be a symmetric cross-ply laminate with all edges simply supported and to be subjected to unavoidable mechanical load and to electrical loads to piezoelectric actuators. Behavior of the laminate is analyzed based on the first-order shear deformation theory. The effect of damping due to interlaminar shear is incorporated into our analysis by introducing the interlaminar shear stresses which satisfy the Newton’s law of viscosity. The following quantities are obtained: (1) natural frequencies of the laminate, (2) weight functions for the deflection and rotations and (3) transient deflection due to loads varying arbitrarily with time. Moreover, the methods to control the deflection due to mechanical load by applying electrical voltage to the piezoelectric actuator are shown.     

Analytical solutions of reduced rigidity plastic bifurcation of circular plates

Summary Two analytical solutions of reduced rigidity plastic bifurcations, respectively for simply supported and clamped circular plates, are found in the way of a post-confirmed presumption for the distribution of unloading area at the bifurcation. TheJ ₂ flow theory is used. While the solutions, by their bifurcation modes and distributions of the unloading areas, recall many aspects of Engesser bifurcation, the corresponding bifurcation loads lie above the true Engesser loads.     

Analytical solutions for functionally graded incompressible eccentric and non-axisymmetrically loaded circular cylinders

We study analytically plane strain static deformations of functionally graded eccentric and non-axisymmetrically loaded circular cylinders comprised of isotropic and incompressible linear elastic materials. Normal and tangential surface tractions on the inner and the outer surfaces of a cylinder may vary in the circumferential direction. The shear modulus is taken to vary either as an exponential function or as a power law function of the radius only. The radial and the circumferential displacements, and the hydrostatic pressure are expanded in Fourier series in the angular coordinate, and expressions for their coefficients are derived from equations expressing the balance of mass (or the continuity equation) and the balance of linear momentum. Boundary conditions are satisfied in the sense of Fourier series. For the exponential variation of the shear modulus, the method of Frobenius series is used to solve 4th-order ordinary differential equations for coefficients of the Fourier series. It is shown that the series solutions for displacements and the hydrostatic pressure converge rapidly. Results for eccentric cylinders and non-axisymmetrically loaded circular cylinders are computed and exhibited graphically. Effects on stress distributions of the eccentricity in the cylinders and of the gradation in the shear modulus are illuminated. It is found that in a thin cylinder subjected to cosinusoidally varying pressure on the inner surface, segments of the cylinder between two adjacent cusps in the pressure deform due to bending rather than stretching.