Nonlinear static response of piezoelectric plates considering electro-mechanical coupling

Nonlinear static analysis of piezoelectric plates has been carried out using nonlinear finite element method considering electro-mechanical coupling,The geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the 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,Results are presented for piezoelectric plate under different mechanical boundary conditions,Numerical results for the plate are given in dimensionless graphical forms.Effects of boundary conditions on linear and nonlinear response of the plate are also studied.The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.

A generalized plane strain theory for transversely isotropic piezoelectric plates

Study of generalized plane strain has so far been limited to elasticity. The present is aimed at parallel development of transversely isotropic piezoelasticity. By assuming that the along depth distribution of electric potential is linear, and that com- monly used Kane-Mindlin kinematical assumption is valid, two dimensional solution systems were deduced, for which, explicit solutions of the out-of-plane constraint factor, as well as the stress resultant concentration factor around a circular hole in a transversely isotropic piezoelectric plate subjected to remote biaxial tension are obtained. Comparisons of these formulas with their counterparts for elastic case yielded suggestions that whether the piezoelectric effect exacerbates or mitigates the stress resultant concentration greatly depends on material properties, particularly, the piezoelectric coefficients; the effect of plate thickness was extensively investigated.

EXACT SOLUTIONS FOR FREE VIBRATION OF TRANSVERSELY ISOTROPIC PIEZOELECTRIC CIRCULAR PLATES

By employing the general solution for coupled three-dimensional equations of a transversely isotropic piezoelectric body, this paper investigates the free vibration of a circular plate made of piezoelectric material. Three-dimensional exact solutions are then obtained under two specified boundary conditions, which can be used for both axisymmetric and non-axisymmetric cases. Numerical results are finally presented.

Parity-time symmetric acoustic system constructed by piezoelectric composite plates with active external circuits

Researches on parity-time(PT)symmetry in acoustic field can provide an efficient platform for controlling the travelling acoustic waves with balanced loss and gain.Here,we report a feasible design of PT-symmetric system constructed by piezoelectric composite plates with two different active external circuits.By judiciously adjusting the resistances and inductances in the external circuits,we obtain the exceptional point due to the spontaneous breaking of PT symmetry at the desired frequencies and can observe the unidirectional invisibility.Moreover,the system can be at PT exact phase or broken phase at the same frequency in the same structure by merely adjusting the external circuits,which represents the active control that makes the acoustic manipulation more convenient.Our study may provide a feasible way for manipulating acoustic waves and inspire the application of piezoelectric composite materials in acoustic structures.

Nonlinear active control of damaged piezoelectric smart laminated plates and damage detection

Considering mass and stiffness of piezoelectric layers and damage effects of composite layers, nonlinear dynamic equations of damaged piezoelectric smart laminated plates are derived. The derivation is based on the Hamilton＇s principle, the higher- order shear deformation plate theory, von Karman type geometrically nonlinear straindisplacement relations, and the strain energy equivalence theory. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to realize the active control and damage detection with a closed control loop. Simply supported rectangular laminated plates with immovable edges are used in numerical computation. Influence of the piezoelectric layers＇ location on the vibration control is in- vestigated. In addition, effects of the degree and location of damage on the sensor output voltage are discussed. A method for damage detection is introduced.

Multi-field coupling and free vibration of a sandwiched functionally-graded piezoelectric semiconductor plate

Sandwiched functionally-graded piezoelectric semiconductor(FGPS)plates possess high strength and excellent piezoelectric and semiconductor properties,and have significant potential applications in micro-electro-mechanical systems.The multi-field coupling and free vibration of a sandwiched FGPS plate are studied,and the governing equation and natural frequency are derived with the consideration of electron movement.The material properties in the functionally-graded layers are assumed to vary smoothly,and the first-order shear deformation theory is introduced to derive the multi-field coupling in the plate.The total strain energy of the plate is obtained,and the governing equations are presented by using Hamilton’s principle.By introducing the boundary conditions,the coupling physical fields are solved.In numerical examples,the natural frequencies of sandwiched FGPS plates under different geometrical and physical parameters are discussed.It is found that the initial electron density can be used to modulate the natural frequencies and vibrational displacement of sandwiched FGPS plates in the case of nano-size.The effects of the material properties of FGPS layers on the natural frequencies are also examined in detail.

Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1：3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed. By numerical simulation, periodic vibration and quasi-periodic vibration responses of the Composite laminated piezoelectric plate are obtained.

LAMINATED PIEZOELECTRIC PLATE FOR ACTIVE UNDERWATER ACOUSTIC CONTROL

The pressure reflected from a bi-laminated piezoelectric plate hasbeen determined using the Thomson-Haskell matrix method. The plate iscomposed of a piezoelectric layer with grounded vacuum and An elasticlayer in contact with the fluid. An incident plane wave in the fluidmedium strikes the plate at dif- Ferent angles. The required electricpotential across the piezoelectric layer to cancel the reflectionfrom the Fluid/elastic boundary has been determined for thepiezoelectric material PZT-5 at various thickness parame- Ters andincident frequencies.

BASIC CURVILINEAR COORDINATE EQUATIONS OF ELECTROELASTIC PLATES UNDER BIASING FIELDS WITH APPLICATIONS IN BUCKLING ANALYSIS

The authors have developed a two-dimensional model for the extension and flexure response of electroelastic plates under biasing fields in a curvilinear coordinate system. Applications of the model in analyzing buckling of two circular piezoelectric plates, one single-layered and the other double-layered, are included. The analysis indicates that the piezoelectric coupling has a strengthening effect against buckling.

Analysis on vibration characteristics of large-size rectangular piezoelectric composite plate based on quasi-periodic phononic crystal structure

Based on the theory of composite materials and phononic crystals(PCs),a large-size rectangular piezoelectric composite plate with the quasi-periodic PC structure composed of PZT-4 and epoxy is proposed in this paper.This PC structure can suppress the transverse vibration of the piezoelectric composite plate so that the thickness mode is purer and the thickness vibration amplitude is more uniform.Firstly,the vibration of the model is analyzed theoretically,the electromechanical equivalent circuit diagram of three-dimensional coupled vibration is established,and the resonance frequency equation is derived.The effects of the length,width,and thickness of the piezoelectric composite plate at the resonant frequency are obtained by the analytical method and the finite element method,the effective electromechanical coupling coefficient is also analyzed.The results show that the resonant frequency can be changed regularly and the electromechanical conversion can be improved by adjusting the size of the rectangular piezoelectric plate.The effect of the volume fraction of the scatterer on the resonant frequency in the thickness direction is studied by the finite element method.The band gap in X and Y directions of large-size rectangular piezoelectric plate with quasi-periodic PC structures are calculated.The results show that the theoretical results are in good agreement with the simulation results.When the resonance frequency is in the band gap,the decoupling phenomenon occurs,and then the vibration mode in the thickness direction is purer.

Time-varying nonlinear dynamics of a deploying piezoelectric laminated composite plate under aerodynamic force

Using Reddy’s high-order shear theory for laminated plates and Hamilton’s principle, a nonlinear partial differential equation for the dynamics of a deploying cantilevered piezoelectric laminated composite plate, under the combined action of aerodynamic load and piezoelectric excitation, is introduced. Two-degree of freedom(DOF)nonlinear dynamic models for the time-varying coefficients describing the transverse vibration of the deploying laminate under the combined actions of a first-order aerodynamic force and piezoelectric excitation were obtained by selecting a suitable time-dependent modal function satisfying the displacement boundary conditions and applying second-order discretization using the Galerkin method. Using a numerical method, the time history curves of the deploying laminate were obtained, and its nonlinear dynamic characteristics,including extension speed and different piezoelectric excitations, were studied. The results suggest that the piezoelectric excitation has a clear effect on the change of the nonlinear dynamic characteristics of such piezoelectric laminated composite plates. The nonlinear vibration of the deploying cantilevered laminate can be effectively suppressed by choosing a suitable voltage and polarity.

A continuum damage model for piezoelectric materials

In this paper, a constitutive model is proposed for piezoelectric material solids containing distributed cracks. The model is formulated in a framework of continuum damage mechanics using second rank tensors as internal variables. The Helrnhotlz free energy of piezoelectric mate- rials with damage is then expressed as a polynomial including the transformed strains, the electric field vector and the tensorial damage variables by using the integrity bases restricted by the initial orthotropic symmetry of the material. By using the Talreja＇s tensor valued internal state damage variables as well as the Helrnhotlz free energy of the piezoelectric material, the constitutive relations of piezoelectric materials with damage are derived. The model is applied to a special case of piezoelectric plate with transverse matrix cracks. With the Kirchhoff hypothesis of plate, the free vibration equations of the piezoelectric rectangular plate considering damage is established. By using Galerkin method, the equations are solved. Numerical results show the effect of the damage on the free vibration of the piezoelectric plate under the close-circuit condition, and the present results are compared with those of the three-dimensional theory.

Nonlinear thickness-shear vibration of an infinite piezoelectric plate with flexoelectricity based on the method of multiple scales

This paper presents a nonlinear thickness-shear vibration model for onedimensional infinite piezoelectric plate with flexoelectricity and geometric nonlinearity.The constitutive equations with flexoelectricity and governing equations are derived from the Gibbs energy density function and variational principle.The displacement adopted here is assumed to be antisymmetric through the thickness due to the thickness-shear vibration mode.Only the shear strain gradient through the thickness is considered in the present model.With geometric nonlinearity,the governing equations are converted into differential equations as the function of time by the Galerkin method.The method of multiple scales is employed to obtain the solution to the nonlinear governing equation with first order approximation.Numerical results show that the nonlinear thickness-shear vibration of piezoelectric plate is size dependent,and the flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro-size thin plates.The geometric nonlinearity also affects the thickness-shear vibration frequencies greatly.The results show that flexoelectricity and geometric nonlinearity cannot be ignored in design of accurate high-frequency piezoelectric devices.

Vibration of a Two-Layer“Metal+PZT”Plate Contacting with Viscous Fluid

The present work investigates the mechanically forced vibration of the hydro-elasto-piezoelectric system consisting of a two-layer plate“elastic+PZT”,a compressible viscous fluid,and a rigid wall.It is assumed that the PZT(piezoelectric)layer of the plate is in contact with the fluid and time-harmonic linear forces act on the free surface of the elastic-metallic layer.This study is valuable because it considers for the first time the mechanical vibration of the metal+piezoelectric bilayer plate in contact with a fluid.It is also the first time that the influence of the volumetric concentration of the constituents on the vibration of the hydro-elasto-piezoelectric system is studied.Another value of the present work is the use of the exact equations and relations of elasto-electrodynamics for elastic and piezoelectric materials to describe the motion of the plate layers within the framework of the piecewise homogeneous body model and the use of the linearized Navier-Stokes equations to describe the flow of the compressible viscous fluid.The plane-strain state in the plate and the plane flow in the fluid take place.For the solution of the corresponding boundary-value problem,the Fourier transform is used with respect to the spatial coordinate on the axis along the laying direction of the plate.The analytical expressions of the Fourier transform of all the sought values of each component of the system are determined.The origins of the searched values are determined numerically,after which numerical results on the stress on the fluid and plate interface planes are presented and discussed.These results are obtained for the case where PZT-2 is chosen as the piezoelectric material,steel and aluminum as the elastic metal materials,and Glycerin as the fluid.Analysis of these results allows conclusions to be drawn about the character of the problem parameters on the frequency response of the interfacial stress.In particular,it was found that after a certain value of the vibration frequency,the presence of the metal layer in the two-layer plate led to an increase in the absolute values of the above interfacial stress.

Dynamic responses of a piezoelectric cantilever plate under high-low excitations

In this paper,the nonlinear dynamic responses of a piezoelectric cantilever plate near the first-order and second-order natural frequencies under the action of electromechanical coupling are studied by experiments and finite element(FE)methods.The influence of different excitation frequencies on the dynamical characteristics of piezoelectric cantilever plates is analyzed with the fixed excitation amplitude.First,an experimental setup is built,including a carbon fiber cantilever plate attached to a macro fiber composite(MFC)sheet.Then,the electromechanical coupling excitations are subjected to the plate with different frequencies,which are chosen near the first and second-order natural frequencies of the plate.The piezoelectric cantilever plate has periodical motions under a lower frequency excitation,and the motions of the plate become more complex after another high frequency excitation added in the physical field.The experimental results show that the motion of the piezoelectric cantilever plate changes from stable to unstable with high-low coupled resonant frequencies.At last,the FE study is carried out to compare and verify the experimental results and the effects of isotropic and orthotropic materials on the accuracy of natural frequencies results are also compared.

Multi-pulse orbits dynamics of composite laminated piezoelectric rectangular plate

The multi-pulse orbits and chaotic dynamics of a simply supported laminated composite piezoelectric rectangular plate under combined parametric excitation and transverse excitation are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motion for the laminated composite piezoelectric rectangular plate are derived from von Karman-type equation and third-order shear deformation plate theory of Reddy. The two-degree-of-freedom dimensionless equations of motion are obtained by using the Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plate. The four-dimensional averaged equation in the case of primary parametric resonance and 1:3 internal resonances is obtained by using the method of multiple scales. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on the normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plate. The analysis of the global dynamics indicates that there exist multi-pulse jumping orbits in the perturbed phase space of the averaged equation. Based on the averaged equation obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the laminated composite piezoelectric rectangular plate are also found by numerical simulation. The results obtained above mean the existence of the chaos in the Smale horseshoe sense for the simply supported laminated composite piezoelectric rectangular plate.

ElasticWaves in Generalized Thermo-Piezoelectric Transversely Isotropic Circular Bar Immersed in Fluid

In this paper,a mathematical model is developed to study the wave propagation in an infinite,homogeneous,transversely isotropic thermo-piezoelectric solid bar of circular cross-sections immersed in inviscid fluid.The present study is based on the use of the three-dimensional theory of elasticity.Three displacement potential functions are introduced to uncouple the equations of motion and the heat and electric conductions.The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied based on Lord-Shulman,Green-Lindsay and Classical theory theories of thermo elasticity.The frequency equations of the coupled system consisting of cylinder and fluid are developed under the assumption of perfectslip boundary conditions at the fluid-solid interfaces,which are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid.The computed non-dimensional frequencies are compared with Lord-Shulman,Green-Lindsay and Classical theory theories of thermo elasticity for longitudinal and flexural modes of vibrations.The dispersion curves are drawn for longitudinal and flexural modes of vibrations.Moreover,the dispersion of specific loss and damping factors are also analyzed for longitudinal and flexural modes of vibrations.

Further studies on thickness vibrations of piezoelectric plate excited by parallel field

In this paper, the thickness vibrations of the piezoclectric plate excited by a parallel field are further analysed. The new formula of resonant frequency equation and electromechanical coupling factor of the coupling modes are obtained. The characteristics of piezoelectrically-tunable frequency for the pure and coupling modes are investigated and ap plied to two examples.