A unified formulation of PVD-based finite cylindrical layer methods for functionally graded material sandwich cylinders

A unified formulation of finite cylindrical layer methods (FCLMs) based on the principle of virtual displacements (PVDs) is developed for the quasi-three-dimensional (3D) bending and free vibration analyses of simply-supported, functionally graded material (FGM) sandwich circular hollow cylinders, in which the material properties of the FGM layer are assumed to obey the power-law distributions of the volume fractions of the constituents through the thickness coordinate. In this formulation, the cylinder is divided into a number of cylindrical finite layers, where the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the displacement components of each individual layer, respectively. Because an h-refinement is adopted in this article to yield the convergent solutions, the relative orders used for expansion of the displacement components remain variable, and can be freely chosen as linear, quadratic and cubic ones. The accuracy and convergence rate of a variety of PVD-based FCLMs developed in this article are assessed by comparing their solutions with the available 3D ones.     

The dynamic analysis of the functionally graded piezoelectric (FGP) shell panel based on three-dimensional elasticity theory

In this paper, three-dimensional elasticity solution is extended to investigate a FGPM finite length, simply supported shell panel under dynamic pressure excitation. The host panel is assumed to be of some functionally graded piezoelectric material (FGPM). The ordinary differential equations (o.d.e.) are derived from the highly coupled partial differential equations (p.d.e.) using series expansions of mechanical and electrical displacements. The resulting system of ordinary differential equations is solved by means of Galerkin finite element method. At last, numerical examples are presented for a FGPM shell panel. To verify the validity of code and formulation, the results of a FGM panel and a FGM plate are compared with the published results.     

Elasticity Solutions for Cylindrical Bending of Functionally Graded Piezoelectric Material Plates北大核心CSCD

功能梯度压电材料（FGPM）同时兼具功能梯度材料和压电材料特性,可为多功能或智能化轻质结构设计提供支撑,在诸多领域有着广泛的应用前景.将Mian和Spencer功能梯度板理论由功能梯度弹性材料推广到功能梯度压电材料,解析研究了FGPM板的柱面弯曲问题,其中,材料弹性常数、压电和介电参数沿板厚方向可以任意连续变化.最终,给出了FGPM板受横向均布荷载作用下柱面弯曲问题的弹性力学解.通过算例分析,重点讨论了压电效应对FGPM板静力响应的影响.     

Investigation on the 2D Contact of Multilayer Functionally Graded Piezoelectric Material Coating Under Conducting Indenters北大核心CSCD

考虑了材料参数可按照任意函数形式变化的功能梯度压电材料(FGPM)涂层在不同形状导电压头作用下的接触问题,研究了梯度系数对功能梯度压电涂层接触力学行为的影响.建立了多层功能梯度压电材料涂层模型,运用了Fourier积分变换和传递矩阵将多层功能梯度压电材料涂层的接触问题转化为奇异积分方程.利用GaussChebyshev数值计算方法,得到了多层功能梯度压电材料涂层-基底结构在刚性导电平压头和圆柱形压头作用下的表面应力分布和电荷分布.利用数值解,分析了材料参数按照不同变化形式的FGPM涂层对最大压痕和电势的影响,还分析了功能梯度压电涂层内部的应力和电位移分布.研究结果表明,功能梯度压电材料参数的不同变化形式对结构的接触性能具有重要的影响.     

Investigation on a rotating FGPM circular disk under a coupled hygrothermal field

In this paper, the distributions of the temperature, moisture, displacement and stress of a functionally graded piezoelectric material (FGPM) circular disk rotating around its axis at a constant angular velocity under a coupled hygrothermal field are presented by a numerical method. The material properties of the FGPM circular disk are assumed to vary along the radial coordinate exponentially. First, the coupled hygrothermal field along the radius of a rotating circular disk is achieved by solving the coupled hygrothermal equations, and then the dynamic equilibrium is solved by utilizing the finite difference method. Finally, numerical results show the effects of functionally graded index, inner radius, angular speed and hygrothermal index on the hygrothermal behaviors of the FGPM circular disk. The results can be useful for the optimal design of rotating FGPM circular disks under a coupled hygrothermal field.     

Active control of dynamic behaviors of graded graphene reinforced cylindrical shells with piezoelectric actuator/sensor layers

This work investigates the active vibration control and vibration characteristics of a sandwich thin cylindrical shell whose intermediate layer is made of the graphene reinforced composite that is bonded with integrated piezoelectric actuator and sensor layers at its outer and inner surfaces. The volume fraction of graphene platelets in the intermediate layer varies continuously in the shell's thickness direction, which generates position-dependent effective material properties. The constitutive relations of the graphene reinforced composite and piezoelectric materials are given by taking one-dimensional steady thermal field into account. Considering Donnell's shell theory, a final equation of motion in terms of the generalized radial displacement is derived by using Hamilton's principle and Galerkin method. Shell's natural frequencies are derived considering influences of the thermo-electro-elastic field. Introducing a constant velocity feedback control algorithm, active vibration control of the sandwich cylindrical shell is presented by employing the Runge-Kutta method. The feedback control gain has a pronounced effect on the damping, as well as the inertia of the system. Comparisons between the present results and those in other papers are done to validate the present solutions. Influences of weight fractions, distribution patterns and geometrical sizes of graphene platelets, temperature variations, thicknesses of layers and the feedback control gain on the vibration characteristics and active vibration control behaviors of the novel sandwich cylindrical shell are discussed.     

Analytical solution for electromagnetothermoelastic behaviors of a functionally graded piezoelectric hollow cylinder

Analytical study for electromagnetothermoelastic behaviors of a hollow cylinder composed of functionally graded piezoelectric material (FGPM), placed in a uniform magnetic field, subjected to electric, thermal and mechanical loads are presented. For the case that the electric, magnetic, thermal and mechanical properties of the material obey an identical power law in the radial direction, exact solutions for electric displacement, stresses, electric potential and perturbation of magnetic field vector in the FGPM hollow cylinder are determined by using the infinitesimal theory of electromagnetothermoelasticity. Some useful discussions and numerical examples are presented to show the significant influence of material inhomogeneity, and adopting a certain value of the inhomogeneity parameter β and applying suitable electric, thermal and mechanical loads can optimize the FGPM hollow cylindrical structures. This will be of particular importance in modern engineering design.     

RMVT- and PVD-based finite cylindrical layer methods for the three-dimensional buckling analysis of multilayered FGM cylinders under axial compression

The unified formulations of finite cylindrical layer methods (FCLMs) based on the Reissner mixed variational theorem (RMVT) and the principle of virtual displacements (PVD) are developed for the three-dimensional (3D) linear buckling analysis of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders and laminated composite ones under axial compression. The material properties of the FGM layer are assumed to obey the power-law distributions of the volume fraction of the constituents through the thickness coordinate. In these formulations, the cylinder is divided into a number of finite cylindrical layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the primary variables of each individual layer, respectively, as well as the related order of each primary variable can be freely chosen, such as the layerwise linear, quadratic or cubic function distribution through the thickness coordinate. The accuracy and convergence of the RMVT- and PVD-based FCLMs developed in this article are assessed by comparing their solutions with the exact 3D solutions available in the literature.     

Effect of material in-homogeneity on electro-thermo-mechanical behaviors of functionally graded piezoelectric rotating shaft

In this article, a hollow circular shaft made from functionally graded piezoelectric material (FGPM) such as PZT_5 has been studied which is rotating about its axis at a constant angular velocity ω. This shaft subjected to internal and external pressure, a distributed temperature field due to steady state heat conduction with convective boundary condition, and a constant potential difference between its inner and outer surfaces or combination of these loadings. All mechanical, thermal and piezoelectric properties except for the Poisson’s ratio are assumed to be power functions of the radial position. The governing equation in polarized form is shown to reduce to a system of second-order ordinary differential equation for the radial displacement. Considering six different sets of boundary conditions, this differential equation is analytically solved. The electro-thermo-mechanical stress and the electric potential distributions in the FGPM hollow shaft are discussed in detail for the piezoceramic PZT_5. The presented results indicate that the material in-homogeneity has a significant influence on the electro-thermo-mechanical behaviors of the FGPM rotating shaft and should therefore be considered in its optimum design.     

Electro-thermo-mechanical vibration and stability analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow

New sandwich panels and tubes have widely applications in nanotechnology such as transportation, naval, aerospace industries, micro and nanoelectromechanical systems and fluid storage. For example, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In the current study, stability and free vibration analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow embedded in an orthotropic foundation under electro-thermo-mechanical loadings are presented. In fact, this work can be provided a valuable background for more research and further experimental investigation. It is assumed that the micro tubes are made of flexible material and smart piezoelectric composites reinforced by carbon nanotubes as core and face sheets, respectively. Energy method and Hamilton's principle are applied to derive the governing equations of motions based on Euler–Bernoulli beam model and using modified strain gradient theory. Moreover, generalized differential quadrature method is used to discretize and solve the governing equations of motions. Numerical results are investigated to predict the influences of length-to-radius, thickness of face sheets-to-thickness of core ratio, temperature changes, orthotropic elastic medium, Knudsen number, and carbon nanotubes volume fraction on the dimensionless natural frequencies and critical flow velocity of sandwich double-bonded piezoelectric micro composite tubes. The results of this article show that increasing the thickness ratio, volume fraction carbon nanotubes and orthotropic elastic constants lead to enhance the dimensionless natural frequency and stability of system, while decrease these parameters with increasing the temperature and length-to-radius ratio.     

Closed-form elasticity solution for smart curved sandwich panels with soft core

Due to the rapid development of intelligent engineering structure, the demand for high performance smart structures has been increased in recent years. In this paper, an exact elasticity solution for bending analysis of sandwich panel with generally orthotropic facings and core is developed. Two electro-orthotropic piezoelectric layers in the top and bottom surfaces of sandwich panel as sensor and actuator are considered. From practical point of view, an initially curved sectorial geometry is considered for smart sandwich panel. In this regard, the constitutive relations and governing equations are considered in polar coordinate and coupled Euler–Cauchy Equations are derived. The characteristic equations are determined and closed-form basis functions of displacements and stresses are achieved for various material and geometrical conditions.Furthermore, based on classical and first-order shell theories, the governing equations of smart curved sandwich panels are derived. The governing equations are solved analytically and compared with exact elasticity solution.Several parametric studies are performed on both material and geometrical properties such as angular span, facing and core thickness, and external electrical voltage.     

Numerical analysis of the dispersion of acoustoelectric waves in a layer with a thickness axis of symmetry for its physical propertes

A numerical method is developed for studying normal electroelastic waves in a layer of piezoelectric materials with mm2 rhombic symmetry, and a second order thickness atris of symmetry. The main system of equations is reduced to eight hamiltonian-type equations in the thickness coordinate. For harmonic waves, the generalized spectral problem is solved numerically taking into account the even (odd) character of the solution with respect to the central plane of the layer. Some solutions of specific problems are analyzed. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 162–169, 1999.     

Multilayer beams: A geometrically exact formulation

Summary We review and extend our recent work on a new theory of multilayer structures, with particular emphasis on sandwich beams/1-D plates. Both the formulation of the equations of motion in the general dynamic case and the computational formulation of the resulting nonlinear equations of equilibrium in the static case based on a Galerkin projection are presented. Finite rotations of the layer cross sections are allowed, with shear deformation accounted for in each layer. There is no restriction on the layer thickness; the number of layers can vary between one and three. The deformed profile of a beam cross section is continuous, piecewise linear, with a motion in 2-D space identical to that of a planar multibody system that consists of three rigid links connected by hinges. With the dynamics of this multi (rigid/flexible) body being referred directly to an inertial frame, the equations of motion are derived via the balance of (1) the rate of kinetic energy and the power of resultant contact (internal) forces/couples, and (2) the power of assigned (external) forces/couples. The present formulation offers a general method for analyzing the dynamic response of flexible multilayer structures undergoing large deformation and large overall motion. With the layersnot required to have equal length, the formulation permits the analysis of an important class of multilayer structures with ply drop-off. For sandwich structures, an approximated theory with infinitesimal relative outer-layer rotations superimposed onto finite core-layer rotation is deduced from the general nonlinear equations in a consistent manner. The classical linear theory of sandwich beams/1-D plates is recovered upon a consistent linearization. Using finite element basis functions in the Galerkin projection, we provide extensive numerical examples to verify the theoretical formulation and to illustrate its versatility. Dedicated to the memory of Professor Juan Carlos Simo, whose early demise is a great loss for the applied and computational mechanics community This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

A new reformulation of vibration suppression equations of functionally graded magnetorheological fluid sandwich beam

This article presented a new reformulation of governing equations of functionally graded magnetorheological fluid (FGMRF) sandwich beams using new auxiliary functions. This technique led to the decoupled vibration equations of the FGMRF sandwich beams and also to the analytical solutions for the in-plane and out of plane displacement fields by considering the Euler-Bernoulli beam theory (EBBT). The material properties of top and bottom layers were changed through the layer thickness according to a power-law distribution of the volume fraction of the constituents. Complex shear modulus of the magnetorheological fluid was varied continuously as a quadratic function of magnetic field intensity. Natural frequencies and corresponding loss factors were calculated with high accuracy in comparison with those available in literature. The effects of boundary conditions, geometric and material properties and the magnetic field intensity on vibrational modes were investigated. Results revealed that unlike the natural frequencies, the loss factors were more affected by the magnetic field.     

Thermal Buckling Analysis of FGM Sandwich Circular Plates Under Transverse Nonuniform Temperature Field Actions北大核心CSCD

Based on the von Kármán geometric nonlinear plate theory, the displacement⁃type geometric nonlinear governing equations for FGM sandwich circular plates under transverse nonlinear temperature field actions were derived. With the immovable clamped boundary condition, the analytical formula for dimensional critical buckling temperature differences of the system was obtained from the solution of the linear eigenvalue problem. Moreover, the 2⁃point boundary value problem of ordinary differential equations was solved with the shooting method. The effects of geometric parameters, constituent material properties, gradient indexes, temperature field parameters and layer⁃thickness ratios on the critical buckling temperature differences, the thermal postbuckling equilibrium paths, and the buckling equilibrium configurations of FGM sandwich circular plates, were investigated. The results show that, with the increases of the thickness⁃radius ratio, the relative thickness of the FGM layer and the gradient index, the FGM sandwich circular plate's critical buckling temperature difference will increase monotonically. Given a fixed radius and a fixed total thickness, the postbuckling deformation of the FGM sandwich circular plate will decrease significantly with the relative thickness of the FGM layer. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

State vector solutions for nonaxisymmetric problem of multilayered half space piezoelectric medium

A state space formulation is established for the nonaxisymmetric space problem of transversely isotropic piezoelectric media in a system of cylindrical coordinate by introducing the state vector. Using the Hankel transform and the Fourier series, the state vector equations are transformed into ordinary differential equations. By the use of the matrix methods, the analytical solutions of a single piezoelectric layer are presented in the form of the product of initial state variables and transfer matrix. The applications of state vector solutions are discussed. An analytical solution for a semiinfinite piezoelectric medium subjected to the vertical point forceP _z, horizontal point forceP _x along x-direction and point electric charge Q at the origin of the surface is presented. According to the continuity conditions at the interfaces, the general solution formulation forN-layered transversely isotropic piezoelectric media is given. Project supported by the National Natural Science Foundation of China (Grant No. 59648001).     

Two-dimensional solution for electro-mechanical behavior of functionally graded piezoelectric hollow cylinder

In this paper, the general theoretical analysis for a hollow cylinder made of functionally graded piezoelectric material subjected to two-dimensional electromechanical load, is developed. The material properties, except the Poisson’s ratio, are assumed to vary with the power law function through the thickness of the cylinder. The mechanical and electrical displacements are assumed to be a function of radial and circumferential directions. By using the separation of variables method and complex Fourier series, the Navier equations in terms of displacements are derived and solved.     

Static plane-strain deformation of transversely isotropic magneto-electro-elastic and layered cylinders to general surface loads

Anisotropic and layered cylinders are important composite structures; however, their system of governing equations is usually solved numerically due to the complicated geometry and material anisotropy involved. In this paper, we analytically solve the plane-strain equations for general static deformation of a cylindrically anisotropic, layered magneto-electro-elastic (MEE) cylinder. We assume that the layers are perfectly bonded at the interfaces. We solve the equations through separation of variables and eigenfunction expansion. Results for each mode shape (2π periodic) are solved independently. Because the eigenspace of the mode shapes is the set of all continuous functions on the interval, any continuous loading can be applied and the corresponding solution can be found analytically through superposition of the mode-shape results. To check our formulation, we consider a cylinder with two isotropic-elastic layers under simple radial loading and reproduce the known, exact results. Then, we compare our formulation to an FEA solution for a layered piezo-electric (PE) cylinder. Finally, we apply a radial stress to three comparable MEE cylinders (one uniform MEE cylinder and two layered cylinders made of alternating piezo-electric (PE) and piezo-magnetic (PM) materials). Deformation and stress amplitudes are plotted for the first six mode shapes of each cylinder as benchmarks for further reference.     

Electro-thermo-mechanical behaviors of FGPM spheres using analytical method and ANSYS software

A hollow sphere made from functionally graded piezoelectric material (FGPM) such as PZT_4 has been considered. One-dimensional analytical method for electro-thermo-mechanical response of symmetrical spheres is used. For asymmetric three-dimensional analysis, ANSYS finite element software is employed in this study. Loading is combination of internal and external pressures, a distributed temperature field due to steady state heat conduction and a constant electric potential difference between its inner and outer surfaces for analytical solution. In three-dimensional solutions closed and open spheres with different boundary conditions subjected to an internal pressure and a uniform temperature field are studied. All mechanical, thermal and piezoelectric properties except the Poisson’s ratio are assumed to be power functions of radius. It has been found from analytical solution that the induced radial and circumferential stresses of an imposed electric potential is similar to the residual stresses locked in the homogeneous sphere during the autofrettage process of these vessels. It has been concluded from the three-dimensional analysis that the magnitudes of effective stresses at all node points are higher for the clamped-clamped boundary condition and are lower for the simply-simply supported condition.     

采用新方法研究加层压电材料中平行界面共线双裂纹的断裂问题

利用新的方法(Schmidt方法)研究加层压电材料中含共线并与材料界面平行的双裂纹在稳态弹性波作用下的动态问题,经富立叶变换使问题的求解转换为求解两对三重对偶积分方程.这些方程可以采用Schmidt方法来求解,这个方法不同与以前求解所利用的方法.结果表明应力强度因子不仅与裂纹的几何尺寸、入射波频率、加层厚度有关,而且与材料性质有关.