Static deformation of a multilayered one-dimensional hexagonal quasicrystal plate with piezoelectric effect

Quasicrystals (QCs) are sensitive to the piezoelectric (PE) effect. This paper studies static deformation of a multilayered one-dimensional (1D) hexagonal QC plate with the PE effect. The exact closed-form solutions of the extended displacement and traction for a homogeneous piezoelectric quasicrystal (PQC) plate are derived from an eigensystem. The general solutions for multilayered PQC plates are then obtained using the propagator matrix method when mechanical and electrical loads are applied on the top surface of the plate. Numerical examples for several sandwich plates made up of PQC, PE, and QC materials are provided to show the effect of stacking sequence on phonon, phason, and electric fields under mechanical and electrical loads, which is useful in designing new composites for engineering structures.     

Exact solutions for magneto-electro-elastic laminates in cylindrical bending

Analytical solutions are derived for the cylindrical bending of multilayered, linear, and anisotropic magneto-electro-elastic plates under simple-supported edge conditions. We construct the general solution in terms of a simple formalism for any homogeneous layer, from which any physical quantities can be solved for the given boundary conditions. For multilayered plates, we derive the solution in terms of the propagator matrices. A special feature of cylindrical bending, which distinguishes itself from the three-dimensional plate problem, is that the associated eigenvalues for any homogeneous layer are independent of the sinusoidal mode, and thus need to be solved only once. Typical numerical examples are also presented for a piezomagnetic plate, a two-layered piezoelectric/piezomagnetic plate, and a four layered piezoelectric/piezomagnetic plate, with different span-to-thickness ratios. In particular, the piezoelectric and piezomagnetic fields show certain interesting features, which give guidance on the development of piezoelectric/piezomagnetic thin-plate theories. Furthermore, it is shown that the variations of the elastic, electric, and magnetic quantities with thickness depend strongly upon the material property and layering, which could be useful in the analysis and design of smart composite structures with sensors/actuators.     

Vibration of a transversely isotropic,simply supported and layered rectangular plate

The propagator matrix method is used in this paper to study the vibration of a transversely isotropic, simply supported and layered rectangular plate. A new system of vector functions is constructed to deal with general surface loading, and general solutions and layer matrices of exact closed form are obtained in this system. The particular solution for forced vibration, and the characteristic equations for free vibration of various surface conditions are then obtained by simple multiplication of layer matrices. These results are presented in such a way that the dilatational and distortional modes of vibration are separated. As a special case of the layered plate, results for the corresponding homogeneous thick plate are also derived. They are presented in a very simple form, and contain the previous results for the static transversely isotropic and the dynamic isotropic plates.     

多层压电材料层合板的精确解

抛弃有关位移和应力的所有假设，直接从三维弹性力学理论的静电学理论，先导出正交各向异性压电材料板的状态方程，由此得到四边简支压电材料板的状态主程，再根据矩阵分析理论，建立了单层压电材料板的上下表面状态量之间的关系，进一步建立了多层压电板上，下表面状态量之间关系式，利用上下表面已知状态量，得到上表面未知状态的求解方程解。通过求解方程组，便得上表面未知状态量，最终可以得到任意位置处状态量，最后，同时给出了四边简支，两层不同压电材料组成，不同纵横比的层合板受正弦分布载荷作用下的精确解，其结果与现有解比较，吻合较好。     

Three-dimensional exact analysis of piezoelectric laminated plates via a sampling surfaces method

A paper focuses on the use of the efficient approach to three-dimensional (3D) exact solutions of electroelasticity for piezoelectric laminated plates. This approach is based on the new method of sampling surfaces (SaS) developed recently by the authors. We introduce inside the nth layer I_n not equally spaced SaS parallel to the middle surface of the plate and choose displacements of these surfaces as basic plate variables. Such an idea permits the representation of the proposed piezoelectric plate formulation in a very compact form. This fact gives the opportunity to derive the 3D exact solutions of electroelasticity for thick and thin piezoelectric laminated plates with a specified accuracy utilizing a sufficient number of SaS, which are located at interfaces and Chebyshev polynomial nodes.     

Analytical and numerical approaches to piezoelectric bimorph

We propose an efficient and accurate approach to piezoelectric bimorph based on a refined expansion of the elastic displacement and electric potential. The field approximation of the through-the-thickness variation accounts for a shear correction and a layerwise modelling for the electric potential. A particular attention is devoted to the boundary conditions on the bottom and top faces of the plate as well as to the interface continuity conditions for the electromechanical variables. The continuity condition on the electric potential imposes some restrictions on the approximation of the electric potential. Moreover, the continuity condition on the normal component of the electric induction at the bimorph interface is ensured by a Lagrange multiplier. The equations of the piezoelectric bimorph are obtained by using variational formulation involving the appropriate boundary and continuity conditions.A selection of numerical illustrations is presented for the series and parallel piezoelectric bimorphs simply supported under cylindrical bending conditions. Two types of electromechanical load are considered (i) a surface density of force applied on the top face and (ii) an electric potential applied on the bottom and top faces of the bimorph. The results thus obtained are compared to those provided by finite element computations performed for the full 3D model and by a simplified model without shear effect. At last, the problem of piezoelectric bimorph vibration is also examined for both closed and open circuit conditions. Excellent predictions with low error estimates of the local (profile) and global responses as well as resonant frequencies are observed. The comparisons assess of the effectiveness of the present approach to piezoelectric bimorph.     

Postbuckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings

A postbuckling analysis is presented for a simply supported, shear deformable functionally graded plate with piezoelectric actuators subjected to the combined action of mechanical, electrical and thermal loads. The temperature field considered is assumed to be of uniform distribution over the plate surface and through the plate thickness and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The initial geometric imperfection of the plate is taken into account. Two cases of the in-plane boundary conditions are considered. A two step perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, geometrically mid-plane symmetric FGM plates with fully covered or embedded piezoelectric actuators under different sets of thermal and electric loading conditions. The effects played by temperature rise, volume fraction distribution, applied voltage, the character of in-plane boundary conditions, as well as initial geometric imperfections are studied.     

Elastic field due to an edge dislocation in a multilayered composite

A numerical procedure is presented for the analysis of the elastic field due to an edge dislocation in a multilayered composite. The multilayered composite consists of n perfectly bonded layers having different material properties and thickness, and two half-planes adhere to the top and bottom layers. The stiffness matrices for each layer and the half-planes are first derived in the Fourier transform domain, then a set of global stiffness equations is assembled to solve for the transformed components of the elastic field. Since the singular part of the elastic field corresponding to the dislocation in the full-plane has been extracted from the transformed components, regular numerical integration is needed only to evaluate the inverse Fourier transform. Numerical results for the elastic field due to an edge dislocation in a bimaterial medium are shown in fairly good agreement with analytical solutions. The elastic field and the Peach–Kohler image force are also presented for an edge dislocation in a single layered half-plane, a two-layered half-plane and a multilayered composite made of alternating layers of two different materials.     

层状层电介质空间轴对称问题的状态空间解

从横观各向同性压电介质空间轴对称问题的基本方程出发,建立了压电介质空间轴对称问题的状态变量方程,对状态变量方程进行Hankel变换,得到以状态变量表示的单层压电介质在Hankel变换空间中的解,讨论了3种不同特征根的情况,利用提出的解得到了半无限压电体在垂直集中载荷和点电荷作出下的Boussinesq解。利用传递矩阵方法导出了多层压电介质空间轴对称问题解一般解析式。     

Properties of Love waves in layered piezoelectric structures

Love waves propagating in a layered structure with an elastic layer deposited on a piezoelectric substrate are analytically investigated. We present a general dispersion equation that describes the properties of Love waves in the structure. A detailed discussion regarding the dispersion equation is presented, and the parameters for Love-mode sensors are also introduced. The properties of Love waves are illustrated by means of sample results for a layered structure with an SiO₂ layer sputtered on an ST-cut 90°X-propagating quartz substrate. Interestingly, we found that a threshold-normalized layer thickness existed for the fundamental Love mode in such a structure.     

基于高阶位移模式的压电层合板动力有限元模型

建立了含压电片层合板的有限元动力学模型。以位于压电层上下表面处的电场强度和层间电压为未知量,给出了三次函数的电势分布模式,采用Reddy的高阶剪切理论描述板的位移场,假设板厚度方向的正应力为零给出了减缩的本构方程,采用有限元方法,基于Hamilton原理导出结构的动力学方程,然后用静态缩聚的方法压缩掉电场自由度和次要的位移自由度。最后用四边形矩形单元求解了一对称铺层和非对称铺层悬臂板的固有频率,并与ANSYS结果对比验证了本文模型的精确性。     

The state vector methods for space axisymmetric problems in multilayered piezoelectric media

The state vector equations for space axisymmetric problems of transversely isotropic piezoelectric media are established from the basic equations. Using the Hankel transform, the state vector equations are reduced to a system of ordinary differential equations. An analytical solution of the problems in the Hankel transform space is presented in the form of the product of initial state vector and transfer matrix. The transfer matrices are given for the three distinct eigenvalues. Applications of the solutions are discussed. An analytical solution for the transversely isotropic semi-infinite piezoelectric media subjected to concerted point loads on the surface z=0 is presented in the Hankel transform space. Using transfer matrix and the continuity conditions at the layer interfaces, the general solution formulation of N-layered transversely isotropic piezoelectric media is given. A selected set of numerical solutions is presented for a layered semi-infinite piezoelectric solid.     

带剪切型压电激励器的板的高阶剪切变形理论

利用压电材料固有的正，逆压电效应可以对结构变形和振动进行控制。与外加电场与极化方向平行于板厚度的压电材料的拉伸作动机制相比，外加电场与极化方向垂直的压电材料的剪切作动机制可以在作动器内产生较小的应力，从而降低作动器边界产生分层破坏的危险。本文对于压电材料的剪切作动机制进行研究，应用三阶剪切变形理论建立带剪切型压电激励器的智能层合板模型。采用哈密顿原理导出带剪切型压电激励器的层合板的控制方程。采用空间法得到了各种边界条件组合条件下板的解析解。数值算例对一三层板采用高阶和一阶剪切变形理论进行计算，结果表明两种理论所得的变形曲线很相似。但对于厚度剪切型激励器而言，由于激励器是引起板的剪切变形，而高阶剪切变形理论比一阶剪切变形理论能更好地反映结构的剪切应变能，因此高阶剪切变形理论可以提供板变形的更为精确的解。因此，对于厚度剪切型激励器，剪切变形理论的选取对于板变形结果的好坏有重要的作用。     

Nonlinear dynamic response and fatigue damage evolution for piezoelectric laminated plates with matrix cracks

Based on the Talreja’s damage model with tensor valued internal state variables, considering the effects of piezoelectricity, geometrical nonlinearity and the constitutive relations of the composite uni-ply plate with matrix cracks, the nonlinear dynamic equations of piezoelectric laminated plates with damage are established and the problems are solved by using the finite difference method. In numerical calculating examples, the nonlinear dynamics responses and the fatigue damage evolution curves of the plates are obtained, and the effects of piezoelectric materials, damage and external load on the nonlinear dynamic response and fatigue damage evolution of the plates are discussed.     

Modelling of finite piezoelectric patches: Comparing an approximate power series expansion theory with exact theory

Plate equations for a plate consisting of one elastic layer and one piezoelectric layer with an applied electric voltage have previously been derived by use of power series expansions of the field variables in the thickness coordinate. These plate equations are here evaluated by the consideration of a time harmonic 2D vibration problem with finite layers. The boundary conditions at the sides of the layers then have to be considered. Numerical comparisons of the displacement field are made with solutions from two other theories; a solution with equivalent boundary conditions for a thin piezoelectric layer applied on an elastic plate and an exact solution based on Fourier series expansions. The two approximate theories are shown to be equally good and they both yield accurate results for low frequencies and thin plates.     

Green’s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces

By virtue of two systems of vector functions and the propagator matrix method, Green’s functions for transversely isotropic, piezoelectric functionally graded (exponentially graded in the vertical direction), and multilayered half spaces are derived. It is observed that the homogeneous solution and propagator matrices for each functionally graded layer in the transformed domain are independent of the choice of the two systems of vector functions. For a point force and point charge density applied at any location of the functionally graded half space, Green’s functions are expressed in terms of one-dimensional infinite integrals. To carry out the numerical integral involving Bessel functions, an adaptive Gauss quadrature approach is introduced and modified. The piezoelectric functionally graded Green’s functions include those in the corresponding elastic functionally graded media as special results with the latter being also unavailable in the literature. Two piezoelectric functionally graded half-space models are analyzed numerically: one is a functionally graded PZT-4 half space, and the other a coated functionally graded PZT-4 layer over a homogeneous BaTiO₃ half space. The effects of different exponential factors on Green’s function components are clearly demonstrated, which could be useful in the design and manufacturing of piezoelectric functionally graded structures.     

Bending solution of third-order orthotropic Reddy plates with asymmetric interfacial crack

In this paper Reddy’s third-order shear deformable plate theory is applied to asymmetrically delaminated orthotropic composite plates under antiplane–inplane shear fracture mode. A double-plate system is utilized to capture the mechanical behavior of the uncracked plate portion. An assumed displacement field is used and modified in order to satisfy the traction-free conditions at the top and bottom plate boundaries. Moreover, the system of exact kinematic conditions was also implemented into the novel plate model. An important improvement of this work compared to previous papers is the continuity condition of the shear strains at the interface of the double-plate system. Applying these conditions it is shown that the nineteen parameters of the third-order displacement field can be reduced to nine. Using the simplified displacement field the governing equations are derived, as well. The solution of a simply-supported delaminated plate is presented using the state-space model and the displacement, strain and stress fields are determined, respectively. The energy release rate and mode mixity distributions are calculated using the 3D J-integral. The analytical results are compared to those by finite element computations and it is concluded that the present model is the most accurate one among the previous plate theory-based approaches.     

A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches

A modified mixed variational principle for piezoelectric materials is established and the state-vector equation of piezoelectric plates is deduced directly from the principle. Then the exact solution of the state-vector equation is simply given, and based on the semi-analytical solution of the state-vector equation, a realistic mathematical model is proposed for static analysis of a hybrid laminate and dynamic analysis of a clamped aluminum plate with piezoelectric patches. Both the plate and patches are considered as two three-dimensional piezoelectric bodies, but the same linear quadrilateral element is used to discretize the plate and patches. This method accounts for the compatibility of generalized displacements and generalized stresses on the interface between the plate and patches, and the transverse shear deformation and the rotary inertia of the plate and patches are also considered in the global algebraic equation system. Meanwhile, there is no restriction on the thickness of plate and patches. The model can be also modified to achieve a semi-analytical solution for the transient responses to dynamic loadings and the vibration control of laminated plate with piezoelectric patches or piezoelectric stiffeners.     

High Frequency Measurements of Viscoelastic Properties of Hydrogels for Vocal Fold Regeneration

This report describes a torsional wave experiment used to measure the viscoelastic properties of vocal fold tissues and soft materials over the range of phonation frequencies. A thin cylindrical sample is mounted between two hexagonal plates. The assembly is enclosed in an environmental chamber to maintain the temperature and relative humidity at in vivo conditions. The bottom plate is subjected to small oscillations by means of a galvanometer driven by a frequency generator that steps through a sequence of frequencies. At each frequency, measured rotations of the top and bottom plates are used to determine the ratio of the amplitudes of the rotations of the two plates. Comparisons of the frequency dependence of this ratio with that predicted for torsional waves in a linear viscoelastic material allows the storage modulus and the loss angle, in shear, to be calculated by a best-fit procedure. Experimental results are presented for hydrogels that are being examined as potential materials for vocal fold regeneration.     

Non-linear modulation of SH waves in a two-layered plate and formation of surface SH waves

Non-linear modulation of shear horizontal (SH) waves in a two-layered elastic plate of uniform thickness is considered. Both layers are assumed to be homogeneous, isotropic and incompressible elastic and having different mechanical properties. The problem is investigated by a perturbation method and in the analysis it is assumed that between the linear shear velocities of the top layer, c₁, and the bottom layer, c₂, the inequality c₁<c₂ is valid. In the layered structure then an SH wave exists if the wave velocity c of the wave satisfies either the condition c₁<c?c₂ or the one c₁<c₂?c. Here the problem is examined under the former condition and it is shown that the non-linear modulation of SH waves is governed by a non-linear Schrödinger equation. In this case the formation of surface SH (Love) waves is also revealed if the top layer is thinner when compared with the bottom layer. Then the stability condition is discussed and the existence of bright (envelope) and dark solitons are manifested.