Analytical solution for functionally graded magneto-electro-elastic plane beams

This paper investigates the plane stress problem of generally anisotropic magneto-electro-elastic beams with the coefficients of elastic compliance, piezoelectricity, dielectric impermeability, piezomagnetism, magnetoelectricity, and magnetic permeability being arbitrary functions of the thickness coordinate. Firstly, partial differential equations governing stress function, electric displacement function and magnetic induction function are derived for plane problems of anisotropic functionally graded magneto-electro-elastic materials. Secondly, these functions are assumed in forms of polynomials in the longitudinal coordinate and can be acquired through a successive integral approach. The analytical expressions of axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced. Thirdly, problems of functionally graded magneto-electro-elastic plane beams are considered, with integral constants being completely determinable from boundary conditions. A series of analytical solutions are thus obtained, including the solutions for beams under tension and pure bending, for cantilever beams subjected to shear force applied at the free end, and for cantilever beams subjected to uniform load. These solutions can be easily degenerated into the solutions for homogenous anisotropic magneto-electro-elastic beams. Finally, a numerical example is presented to show the application of the proposed method.     

An exact analysis of sliding frictional contact of a rigid punch over the surface of magneto-electro-elastic materials

This article performs an exact analysis for a frictional triangular or cylindrical punch acting on the surface of magneto-electro-elastic materials. The punch moves relative to the surface of magneto-electro-elastic materials. Inside the contact area, the Coulomb friction law is applied. Eigenvalue distribution is analyzed, and then appropriate fundamental solutions are given. The stated problem is reduced to a system of singular integral equations of the second kind. The exact solution of the obtained singular integral equations makes it easy to get the explicit expressions of the surface physical quantities. Through plotting figures, the influences of the friction coefficient on contact behavior are shown and interesting results are observed. The in-plane stress, electric displacement and magnetic induction tend to be infinite near the leading edge of the frictional triangular punch, while having spikes at one edge of the frictional cylindrical punch, which may explain why surface damage occurs on the surface of magneto-electro-elastic materials.     

Thickness-shear vibrations of a quartz plate under time-dependent biasing deformations

Incremental thickness-shear vibrations of a Y-cut quartz crystal plate under time-harmonic biasing extensional deformations are studied using the two-dimensional equations for small fields superposed on finite biasing fields in an electroelastic plate. It is shown that the incremental thickness-shear vibrations are governed by the well-known Mathieu's equation with a time-dependent coefficient. Both free and electrically forced vibrations are studied. Approximate analytical solutions are obtained when the frequency of the biasing deformation is much lower than that of the incremental thickness-shear vibration. The incremental thickness-shear free vibration mode is shown to be both frequency and amplitude modulated, with the frequency modulation as a first-order effect and the amplitude modulation a second-order effect. The forced vibration solutions show that both the static and motional capacitances become time-dependent due to the time-harmonic biasing deformations.     

Extensional,thickness-stretch and symmetric thickness-shear vibrations of piezoceramic disks

A set of two-dimensional (2-D), second-order approximate equations for extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoelectric ceramic plates with electroded faces is extracted from the infinite system of 2-D equations deduced previously. The new truncation procedure developed recently is used for it improves the accuracy of calculated dispersion curves. Closed-form solutions are obtained for free vibrations of circular disks of barium titanate. Dispersion curves calculated from the present approximate 2-D equations are compared with those obtained from the 3-D equations, and the predicted resonance frequencies are compared with experimental data. Both comparisons show good agreement without any corrections. The frequencies of the edge modes calculated from the present 2-D equations are very close to the experimental data. Furthermore, mode shapes at various frequencies are calculated in order to identify the frequency segments of the spectrum at which one of the coupled modes-i.e., the radial extension (R), edge mode (Eg), thickness-stretch (TSt), and symmetric thickness-shear (s.TSh)-is predominant.     

Resonant frequency function of thickness-shear vibrations of rectangular crystal plates

The resonant frequencies of thickness-shear vibrations of quartz crystal plates in rectangular and circular shapes are always required in the design and manufacturing of quartz crystal resonators. As the size of quartz crystal resonators shrinks, for rectangular plates we must consider effects of both length and width for the precise calculation of resonant frequency. Starting from the three-dimensional equations of wave propagation in finite crystal plates and the general expression of vibration modes, we obtained the relations between frequency and wavenumbers. By satisfying the major boundary conditions of the dominant thickness-shear mode, three wavenumber solutions are obtained and the frequency equation is constructed. It is shown the resonant frequency of thickness-shear mode is a second-order polynomial of aspect ratios. This conforms to known results in the simplest form and is applicable to further analytical and experimental studies of the frequency equation of quartz crystal resonators.     

Extended displacement discontinuity Green's functions for three-dimensional transversely isotropic magneto-electro-elastic media and applications

A versatile method is presented to derive the extended displacement discontinuity Green's functions or fundamental solutions by using the integral equation method and the Green's functions of the extended point forces. In particular, the three-dimensional (3D) transversely isotropic magneto-electro-elastic problem is used to demonstrate the method. On this condition, the extended displacement discontinuities include the elastic displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, while the extended forces include the point forces, the point electric charge and the point electric current. Based on the obtained Green's functions, the extended Crouch fundamental solutions are derived and an extended displacement discontinuity method is developed for analysis of cracks in 3D magneto-electro-elastic media. The extended intensity factors of two coplanar and parallel rectangular cracks are calculated under impermeable boundary condition to illustrate the application, accuracy and efficiency of the proposed method.     

An analysis of nonlinear vibrations of coupled thickness-shear and flexural modes of quartz crystal plates with the homotopy analysis method

We investigated the nonlinear vibrations of the coupled thickness-shear and flexural modes of quartz crystal plates with the nonlinear Mindlin plate equations, taking into consideration the kinematic and material nonlinearities. The nonlinear Mindlin plate equations for strongly coupled thickness- shear and flexural modes have been established by following Mindlin with the nonlinear constitutive relations and approximation procedures. Based on the long thickness-shear wave approximation and aided by corresponding linear solutions, the nonlinear equation of thickness-shear vibrations of quartz crystal plate has been solved by the combination of the Galerkin and homotopy analysis methods. The amplitude frequency relation we obtained showed that the nonlinear frequency of thickness-shear vibrations depends on the vibration amplitude, thickness, and length of plate, which is significantly different from the linear case. Numerical results from this study also indicated that neither kinematic nor material nonlinearities are the main factors in frequency shifts and performance fluctuation of the quartz crystal resonators we have observed. These efforts will result in applicable solution techniques for further studies of nonlinear effects of quartz plates under bias fields for the precise analysis and design of quartz crystal resonators.     

Three-dimensional analysis of piezoelectric/piezomagnetic elastic media

This paper presents an exact three-dimensional method of solution for a transversely isotropic piezoelectric/piezomagnetic elastic media. The control partial differential equation set that is expressed by the elastic displacement, electric potential and magnetism potential function are established. By means of drawing two-displacement potential functions the general solution for three-dimensional transversely isotropic magneto-electro-elastic is derived. As an illustrative example, the analysis solutions of a half space body of magneto-electro-elastic acted by a point force at the origin directed along the z-axis and a point charge and a point electric current at the origin are obtained.     

Three-dimensional free vibration of functionally graded fiber orientation and volume fraction cylindrical panels

Elasticity solution for free vibrations analysis of functionally graded fiber orientation and volume fraction cylindrical panel is presented, using differential quadrature method. The orthotropic panel is simply supported at the edges and assumed to have arbitrary variations of fiber orientation and volume fraction in the radial direction. Suitable displacement functions that identically satisfy the simply supported boundary conditions are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by differential quadrature method to obtain the natural frequencies. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. Numerical results are presented for an orthotropic cylindrical panel with arbitrary variations of fiber orientation and volume fraction in the shell’s thickness and compared with discrete laminates composite panels. The interesting and new results show that normalized natural frequency of the functionally graded fiber orientation cylindrical panel is smaller than that of a discrete laminate composite panel and close to that of a 4-layer. In contrast, the normalized natural frequency of a functionally graded fiber volume fractions is larger than that of a discrete laminated and close to that of a 2-layer.     

Two-dimensional equations for electroelastic plates with relatively large shear deformations

A set of two-dimensional, nonlinear equations for electroelastic plates in moderately large thickness-shear deformations is obtained from the variational formulation of the three-dimensional equations of nonlinear electroelasticity by expanding the mechanical displacement vector and the electric potential into power series in the plate thickness coordinate. As an example, the equations are used to study nonlinear thickness-shear vibrations of a quartz plate driven by an electrical voltage. Nonlinear electrical current amplitude-frequency behavior near resonance is obtained. The equations and results are useful in the study and design of piezoelectric crystal resonators and the measurement of nonlinear material constants of electroelastic materials.     

Axisymmetric analytical solutions for a heterogeneous multi-ferroic circular plate subjected to electric loading

In the present article, responses of an electrically loaded heterogeneous circular plate of transversely isotropic multi-ferroic materials are investigated. The axisymmetric problem is solved by adopting the direct displacement method developed by the second author. The corresponding generalized displacement functions are obtained through a step-by-step integration procedure. The coupling magneto-electro-elastic fields in the plate, which exactly satisfy the upper and lower boundary conditions and approximately meet the cylindrical boundary condition, are explicitly expressed. Numerical calculations are performed for two particular plates to investigate the influence of material heterogeneity.     

Stochastic minimax optimal time-delay state feedback control of uncertain quasi-integrable Hamiltonian systems

In this paper, free vibrations of symmetric angle-ply laminated cylindrical shells are presented under different thickness variations. The thickness variations of the shells are linear, exponential, and sinusoidal. Spline function techniques are adopted to approximate the displacement functions, and the point collocation method is used to obtain the generalized eigenvalue problem with appropriate boundary conditions. The effects of the geometric parameters, thickness parameters, and material parameters on the frequencies of symmetric angle-ply laminated cylindrical shells with two kinds of material properties under different boundary conditions are analyzed.     

On the accuracy of Mindlin plate predictions for the frequency-temperature behavior of resonant modes in AT- and SC-cut quartz plates

The frequency spectra of resonant modes in AT- and SC-cut quartz plates and their frequency-temperature behavior were studied using Mindlin first- and third-order plate equations. Both straight-crested wave solutions and two-dimensional plate solutions were studied. The first-order Mindlin plate theory with shear correction factors was previously found to yield inaccurate frequency spectra of the modes in the vicinity of the fundamental thickness-shear frequency. The third-order Mindlin plate equations without correction factors, on the other hand, predict well the frequency spectrum in the same vicinity. In general, the frequency-temperature curves of the fundamental thickness-shear obtained from the first-order Mindlin plate theory are sufficiently different from those of the third-order Mindlin plate theory that they raise concerns. The least accurately predicted mode of vibration is the flexure mode, which results in discrepancies in its frequency-temperature behavior. The accuracy of other modes of vibrations depends on the degree of couplings with the flexure mode. Mindlin first-order plate theory with only the shear correction factors is not sufficiently accurate for high frequency crystal vibrations at the fundamental thickness-shear frequency. Comparison with measured resonant frequencies and frequency-temperature results on an AT-cut quartz plate shows that the third-order plate theory is more accurate than the first-order plate theory; this is especially true for the technically important fundamental thickness shear mode in the AT-cut quartz plate.     

Nonlinear modes of vibrations for simply supported cylindrical shell with geometrical nonlinearity

The system of three partial differential equations with respect to displacements (Donnell equations) is used to analyze nonlinear vibrations of a cylindrical shell. The Galerkin method is applied to every partial differential equation to obtain a finite-degree-of-freedom model of the shell. The system of ordinary differential equations with respect to the general coordinates of the radial shell displacements is derived. The nonlinear modes of free vibrations are calculated using the harmonic balance method. The stability analysis of periodic motions is performed.     

Vibrations of an asymmetrically electroded piezoelectric plate

Two-dimensional equations for coupled extensional, flexural, and thickness-shear motions of a piezoelectric plate are obtained systematically from the three-dimensional equations of linear piezoelectricity by retaining lower order terms in power series expansions in the plate thickness coordinate. The plate can have asymmetric electrodes on its major surfaces. The equations are specialized to crystals of 6-mm symmetry and are used to analyze thickness-shear vibrations of an asymmetrically electroded plate. Energy trapping, a behavior of thickness-shear modes crucial to applications, is examined.     

Stress analysis of multi-layered hollow anisotropic composite cylindrical structures using the homogenization method

This paper presents a general and efficient stress analysis strategy for hollow composite cylindrical structures consisting of multiple layers of different anisotropic materials subjected to different loads. Cylindrical material anisotropy and various loading conditions are considered in the stress analysis. The general stress solutions for homogenized hollow anisotropic cylinders subjected to pressure, axial force, torsion, shear and bending are presented with explicit formulations under typical force and displacement boundary conditions. The stresses and strains in a layer of the composite cylindrical structures are obtained from the solutions of homogenized hollow cylinders with effective material properties and discontinuous layer material properties. Effective axial, torsional, bending and coupling stiffness coefficients taking into account material anisotropy are also determined from the strain solutions for the hollow composite cylindrical structures. Examples show that the material anisotropy may have significant effects on the effective stiffness coefficients in some cases. The stress analysis method is demonstrated with an example of stress analysis of a 22-layer composite riser, and the results are compared with numerical solutions. This method is efficient for stress analysis of thin-walled or moderately thick-walled hollow composite cylindrical structures with various multiple layers of different materials or arbitrary fiber angles because no explicit interfacial continuity parameters are required. It provides an efficient and easy-to-use analysis tool for assessing hollow composite cylindrical structures in engineering applications.     

A semi-analytical solution for stress analysis of moderately thick laminated cylindrical panels with various boundary conditions

Stress analysis of moderately thick laminated cylindrical panels with different loading and boundary conditions is presented. Boundary conditions include clamped, simply supported and free while uniform and sinusoidal distributed loadings are considered. Assuming effects of shear deformation and initial curvature, governing equations of the problem are derived. The governing partial differential equations (PDEs) in terms of three displacement components, two rotations and ten stress resultants include a system of 15 first order PDEs. Application of the extended Kantorovich method (EKM) to the governing equations yields to a double set of algebric-differential equations in terms of x and θ. The resulted systems are then solved iteratively with very fast convergence. It is demonstrated that the method converges rapidly independent of initial guess functions. Comparisons of the EKM predictions with other analytical and FEM analyses are in close agreement. More results for panels with particular boundary conditions are presented for future studies.     

阶梯式板条的振动

用奇异函数建立非单一材质的阶梯式板条自由振动和强迫振动的微分方程并求得其通解,用Ｗ 算子给出主振型函数的表达式及常见支承条件下板条的频率方程,用广义函数讨论此类板在不同形式载荷作用下的强迫振动响应。     

Buckling and postbuckling response of sandwich panels under non-uniform mechanical edge loadings

The buckling and postbuckling responses of cylindrical sandwich panels, subjected to non-uniform in-plane loadings are investigates in this paper by analytical method. A fourth and fifth order expansions are used respectively for the transverse and tangential displacement of the core to model the core compressibility effect. The stress distribution within the panels due to the applied non-uniform in-plane edge loadings are determined by prebuckling analysis. The governing partial differential equations describing the buckling and postbuckling behavior of cylindrical sandwich panels are derived using the principle of minimum total potential energy. Galerkin’s method is used to reduce the governing partial differential equations to a set of non-linear algebraic equations. Newton–Raphson method in conjunction with Riks approach is employed to solve the algebraic equations. Numerical results are presented for both flat and cylindrical sandwich panels subjected to various non-uniform in-plane edge loadings. The sandwich panels used in the present investigation are made up of isotropic and composite materials.     

Three-dimensional vibration analysis of a transversely isotropic piezoelectric cylindrical panel

Summary. In this work, based on three-dimensional piezoelectric elasticity, an exact analysis of the free vibrations of a simply supported, homogeneous, transversely isotropic cylindrical panel is presented. Three displacement potential functions are introduced so that the equations of motion and Gauss equation are uncoupled and simplified. It is noticed that a purely transverse (SH) mode is independent of piezoelectric effects and the rest of the motion. The equations for free vibration problems are further reduced to four second-order ordinary differential equations, after expanding the displacement and electric potential functions with an orthogonal series. The dispersion relations for an electrically shorted and charge free simply supported cylindrical panel with stress free edges have been obtained and discussed. A modified Bessel function solution with complex arguments is directly used for complex eigenvalues. In order to clarify the developed method and to compare the results to the existing areas, numerical examples are presented and the computed functions are illustrated graphically.