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.

Nonlinear free vibration of piezoelectric semiconductor doubly-curved shells based on nonlinear drift-diffusion model

In this paper, the nonlinear free vibration behaviors of the piezoelectric semiconductor(PS) doubly-curved shell resting on the Pasternak foundation are studied within the framework of the nonlinear drift-diffusion(NLDD) model and the first-order shear deformation theory. The nonlinear constitutive relations are presented, and the strain energy, kinetic energy, and virtual work of the PS doubly-curved shell are derived.Based on Hamilton's principle as well as the condition of charge continuity, the nonlinear governing equations are achieved, and then these equations are solved by means of an efficient iteration method. Several numerical examples are given to show the effect of the nonlinear drift current, elastic foundation parameters as well as geometric parameters on the nonlinear vibration frequency, and the damping characteristic of the PS doublycurved shell. The main innovations of the manuscript are that the difference between the linearized drift-diffusion(LDD) model and the NLDD model is revealed, and an effective method is proposed to select a proper initial electron concentration for the LDD model.

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

Free vibration of statically thermal postbuckled functionally graded material （FGM） beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electro- mechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

Free vibration analysis of functionally graded laminated sandwich cylindrical shells integrated with piezoelectric layer

An analytical method for the three-dimensional vibration analysis of a functionally graded cylindrical shell integrated by two thin functionally graded piezoelectric （FGP） layers is presented. The first-order shear deformation theory is used to model the electromechanical system. Nonlinear equations of motion are derived by considering the von Karman nonlinear strain-displacement relations using Hamilton＇s principle. The piezoelectric layers on the inner and outer surfaces of the core can be considered as a sensor and an actuator for controlling characteristic vibration of the system. The equations of motion are derived as partial differential equations and then discretized by the Navier method. Numerical simulation is performed to investigate the effect of different para- meters of material and geometry on characteristic vibration of the cylinder. The results of this study show that the natural frequency of the system decreases by increasing the non-homogeneous index of FGP layers and decreases by increasing the non-homogeneous index of the functionally graded core. Furthermore, it is concluded that by increasing the ratio of core thickness to cylinder length, the natural frequencies of the cylinder increase considerably.

Free vibration analysis of a spinning piezoelectric beam with geometric nonlinearities

The linear and non-linear free vibrations of a spinning piezoelectric beam are studied by considering geometric nonlinearities and electromechanical coupling effect. The non-linear differential equations of the spinning piezoelectric beam governing two transverse vibrations are derived by using two Euler angles transformation and extended Hamilton principle, wherein an additional piezoelectric coupling term and different linear terms are present in contrast to the traditional shaft model. Linear frequencies are obtained by solving the standard eigenvalues of the linearized system directly, and the non-linear frequencies and non-linear complex modes are achieved by using the method of multiple scales. For free vibrations analysis of a spinning piezoelectric beam, the non-linear modal motions are investigated as forward and backward precession with different spinning speeds. The responses to the initial conditions for such a gyroscopic system are studied,and a beat phenomenon is found, which are then validated by numerical simulation. The influences of some parameters such as electrical resistance, rotary inertia and spinning speeds to the non-linear dynamics of a spinning piezoelectric beam are investigated.

FREE VIBRATION OF PIEZOELECTRIC CYLINDRICAL SHELLS

Three displacement functions are introduced to represent each mechanical displacement according to the 3-D theory in this paper. By expanding the displacement functions and the electric potential in orthogonal series, the free vibration equation of piezoelectric cylindrical shells satisfying SS3 boundary conditions can be obtained. The equation was solved by utilizing Bessel functions with complex arguments. Results are presented graphically as well as in table, and compared with those of other references. Some frequencies that were missing in Ref. [9] are discovered.

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.

Analytical modelling and free vibration analysis of piezoelectric bimorphs

An efficient and accurate analytical model for piezoelectric bimorph based on the improved first-order shear deformation theory （FSDT） is developed in this work. The model combines the equivalent single-layer approach for mechanical displacements and a layerwise-type modelling of the electric potential. Particular attention is devoted to the boundary conditions on the outside faces and to the interface continuity conditions of the bimorphs for the electromechanical variables. Shear correction factor （k） is introduced to modilfy both the shear stress and the electric displacement of each layer. And the detailed mathematical derivations are presented. Free vibration problem of simply supported piezoelectric bimorphs with series or parallel arrangement is investigated for the closed circuit condition, and the results for different length-to-thickness ratios are compared with those obtained from the exact 2D solution. Excellent agreements between the present model prediction with k=-8/9 and the exact solutions are observed for the resonant frequencies.

Effect of Li content on the microstructure and properties of lead-free piezoelectric(K_(0.5)Na_(0.5))_(1-x)Li_xNbO_3 ceramics prepared by SPS

Lead-free piezoelectric （K0.5sNa0.5）1-xLixNbO3 （x = 0at%-20at%） ceramics were synthesized by spark plasma sintering （SPS） at low temperature and the effects of LiNbO3 addition on its crystal structure and properties were also studied. When the Li content was less than 6at%, a single proveskite phase with the similar structure of （K0.5Na0.5）NbO3 was formed; and a secondary phase with K3Li2Nb5O15 structure was observed in the 6at% 〈 x 〈 20at% compositional range. Furthermore, LiNbO3 existed as the third phase when the Li content was higher than 8at%. The grain sizes increased from 200-500 nm to 5-8 μm when the K3Li2Nb5O15 and LiNbO3 like phases were formed. With increasing Li content, the relative density of the ceramics first decreased from 97% to 93% and then kept constant. The piezoelectric coefficient d33, dielectric constant, and planner electromechanical coupling factor exhibited a decreasing tendency with increasing Li content because of the decrease in density and the formation of the secondary phase such as K3Li2Nb5O15 and LiNbO3. The formation of dense microstructure with a single phase is necessary in improving the properties of the （K0.5Na0.5）1-xLixNbO3 ceramics.

Preparation and Piezoelectric Properties of CuO-added(Ag_(0.75)Li_(0.1)Na_(0.1)K_(0.05_)NbO_3 Lead-free Ceramics

CuO-doped（Ag0.75Li0.1Na0.1K0.05）NbO3（ALNKN-xCuO, x = 0.2mol%） lead-free piezoelectric ceramics were prepared by the solid-state reaction method in air atmosphere. The effects of CuO addition on the phase structure, microstructure, and piezoelectric properties of the ceramics were investigated. The experimental results show that the ALNKN ceramics without doping CuO possess rhombohedral phase along with K2NbrO16- type phase and metallic silver phase. For all of the CuO-doped ALNKN ceramics, a pure perovskite structure with the orthorhombic phase was obtained by enclosing the samples in a corundum tube. A homogeneous microstructure with＇the grain size of about 1 ~tm was formed for the ceramics with 0.5mo1% CuO. The grain size increases with increasing amount of CuO. The temperature dependence of dielectric properties indicates that the ferroelectric phase of the ALNKN-xCuO ceramics becomes less stable with the addition of CuO. The ceramics with x = lm01% exhibit relatively good electrical properties along with a high Curie temperature. These results will provide a helpful guidance to preparing other AN-based ceramics by solid-state reaction method in air atmosphere.

Manufacture and Cytotoxicity of a Lead-free Piezoelectric Ceramic as a Bone Substitute—Consolidation of Porous Lithium Sodium Potassium Niobate by Cold Isostatic Pressing

Aim The piezoelectric properties and cytotoxicity of a porous lead-free piezoelectric ceramic for use as a direct bone substitute were investigated. Methodology Cold isostatic pressing （CIP） was applied to fabricate porous lithium sodium potassium niobate （Li0.06Na0.5K0.44） NbO3 specimens using a pore-forming method. The morphologies of the CIP-processed specimens were characterized and compared to those of specimens made by from conventional pressing procedures. The effects of the ceramic on the attachment and proliferation of osteoblasts isolated from the cranium of 1-day-old Sprague- Dawley rats were examined by a scanning electron microscopy （SEM） and metbylthiazol tetrazolium （MTT） assay. Results The results showed that CIP enhanced piezoelectricity and biological performance of the niobate specimen, and also promoted an extracellular matrix-like topography of it. In vitro studies showed that the CIP-enhanced material had positive effects on the attachment and proliferation of osteoblasts. Conclusion Niobate ceramic generated by CIP shows a promise for being a piezoelectric composite bone substitute.

Modified phase transition and electrical properties of [Li_(0.05)(Na_(0.535)K_(0.48))_(0.95)]- (Nb_(0.94)Sb_(0.06))O_3 lead-free piezoelectric ceramic by sintering temperature

Li/Sb-doped (Na,K)NbO3 with a nominal composition of [Li0.05(Na0.535K0.48)0.95](Nb0.94Sb0.06)O3 ceramic was synthesized by normal sintering. The phase structure, microstructure, and electrical properties were investigated with a special emphasis on the influence of the sintering temperature. A polymorphic phase transition (PPT) from orthorhombic to tetragonal symmetry was observed when the sintering temperature was raised from 1040 to 1050 ℃, whereby the piezoelectric coefficient d33 and the electromechanical coupling coefficient kp reached the peak values of 245 pC·N-1 and 41.2%, respectively. The PPT induced by varying the sintering temperature is due to the different volatilization extents of alkali metals and appears to a lower sintering temperature with increasing Li content. The trace modifying of alkali metal content is more effective than doping B site element to enhance the d33 value.

Piezoelectric and dielectric properties of Li_x(K_(0.46)Na_(0.54))_(1-x)Nb_(0.86)Ta_(0.1)Sb_(0.04)O_3 lead-free ceramics

Lead-free piezoelectric ceramics Lix(K0.46Na0.54)1-xNb0.86Ta0.1Sb0.04O3 (with x ranging from 0 to 0.1) were synthesized by conventional solid state sintering method. The effect of cationic substitution of Li for K and Na in the A sites of perovskite lattice on the structure, phase transition behavior and electrical properties were investigated. Morphotropic phase boundaries(MPB) between orthorhombic and tetragonal phase are found in the composition range of 0.06≤x≤0.08. Analogous to Pb(Zr,Ti)O3, the dielectric and piezoelectric properties are enhanced for the composition near the morphotropic phase boundary. The Li0.06(K0.46Na0.54)0.94- Nb0.86Ta0.1Sb0.04O3 ceramics show excellent electrical properties, that is, piezoelectric constant d33=215 pC/N, planar electromechanical coupling factor kp=41%, dielectric constant ε 3T3 /ε0=1 303, and dielectric loss tan δ=2.45%. The results indicate that Lix(K0.46Na0.54)1-x Nb0.86 Ta0.1Sb0.04O3 ceramic is a promising lead-free piezoelectric material.

Microstructure and electrical properties of Ti-modified (Na_(0.5)K_(0.5))(Ti_xNb_(1-x))O_3 lead-free piezoelectric ceramics

Ti-Modified (Na0.5K0.5)(TixNb1-x)O3 (NKNT) piezoelectric ceramics were fabricated by double-layer buried powder process at 1020°C for 2 h. The microstructures,and piezoelectric and dielectric properties of the lead-free NKNT ceramics were investigated. X-ray diffraction re-sults indicated that Ti4+ had diffused into the (Na0.5K0.5)NbO3 lattices to form a solid solution with a perovskite structure. The introducing of Ti into the (Na0.5K0.5)NbO3 solid solution effectively reduced the sintering temperature and...

Synthesis and Piezoelectric Properties of Lead-free (Bi_(0.5)Na_ (0.5))_ 1-x(Ba_aSr_b)_xTiO_3 Ceramics

A new group of lead-free piezoelectric ceramics,（Bi0.5 Na0.5）1-x（BaaSrb）xTiO3（abbreviated as BNBST[100x-100a/100b],0〈x〈1,a＋b=1）,was synthesized.The ceramics were prepared by conventional ceramic sintering technique,and the ceramics with density of 95% of the theoretical one can be sintered without the atmosphere control during the sintering process.The results of the X-ray diffraction（XRD） data show that the ceramics possess a single perovskite phase.The measurements of dielectric and piezoelectric properties reveal that the ceramics provide relatively high piezoelectric charge constant d33 and high planar electromechanical coupling factor kp.For the BNBST6-95/5 ceramics,d33 is equal to 170pC/N,and kp is equal to 32.0%.The fabrication technique for these ceramics is conventional and stable.

Effects of Na/K ratio on the phase structure and electrical properties of Na_xK_(1-x)NbO_3 lead-free piezoelectric ceramics

Lead-free piezoelectric NaxK1-xNbO3（x = 0.3-0.8）（NKN） ceramics were fabricated by normal sintering at 1060°C for 2 h.Microstructures and electrical properties of the ceramics were investigated with a special emphasis on the influence of Na content.The grain size of the produced dense ceramic was decreased by increasing Na content.A discontinuous change in the space distance was found at the composition close to Na0.7K0.3NbO3 ceramic, which indicates the presence of a transitional composition between two different orthorhombic phases, which is similar to the behavior of morphotropic phase boundary（MPB） in NaxK1-xNbO3 ceramics.Such MPB-like behavior contributes to the enhanced piezoelectric coefficient d33 of 122 pC/N, planar-mode electromechanical coupling coefficient kP of 28.6%, and dielectric constant εr of 703, respectively for the Na0.7K0.3NbO3 ceramic.Cubic temperature TC and the transitional temperature TO-T from orthorhombic to tetragonal phase are observed at around 420°C and 200°C, respectively.

Free vibration of piezoelectric annular plate

General solutions for coupled three dimensional equations of piezoelectric media were used in this work to obtain some analytical solutions for free vibration of piezoelectric annular plates. These solutions not only satisfy the governing equations at every point in the concerned region but also satisfy the prescribed boundary conditions at every point on the boundaries. Therefore, they are three-dimensional exact. Numerical results are finally tabulated.

Microstructures and Electrical Properties of Bi_(0.5)-(Na_(1-x-y)K_xLi_y)_(0.5)TiO_3 Lead-free Piezoelectric Ceramics

The microstructures and electrical properties of Bi0.5（Na1-x-yKxLiy）0.5TiO3 lead-free piezoelectric ceramics were studied.These ceramics were prepared by conventional ceramic technique.XRD analysis reveals that the ceramics possess almost pure perovskite phase when y≤0.2.The SEM results show that,with more amounts of Li＋,the crystalline grain growing speed is accelerated,and the sintering temperature can effectively be decreased.The measurements of piezoelectric properties indicate that the ceramics with relatively low amount of Li＋ and high amount of K＋ have comparatively large piezoelectricity.The dielectric measurements show that the ceramics have properties like relaxor ferroelectrics and diffuse phase transition（DPT） at Td and Tc,respectively.The results of ferroelectric measurements reveal the system has relatively higher remanent polarization Pr（27.6 μC/cm2） and lower coercive field Ec（37.5 kV/cm）.

Effects of A-site non-stoichiometry on the structural and electrical properties of 0.96K_(0.5)Na_(0.5)NbO_3-0.04LiSbO_3 lead-free piezoelectric ceramics

Effects of A-site non-stoichiometry on the structural and electrical properties of 0.96K0.5＋xNa0.5＋xNbO3- 0.04LiSbO3 lead-free piezoelectric ceramics were examined for 0 ≤ x ≤0.02. The piezoelectric coefficients exhibited a maximum, d33 = 187 pC/N at x = 0.0075, coinciding with the maximum of the grain size and the apparent density at x = 0.0075. The apparent density and the piezoelectric coefficients decreased with increasing x at higher x which was likely due to the crystal geometrical distortion of 0.96K0.5＋xNa0.5＋xNbO3-0.04LiSbO3. In addition, super-large grains were found and this may be due to liquid phase sintering. Excess （K＋＋Na＋） attracted a sum of space charges to keep the charge neutral, resulting in charge leakage during the course of ceramic polarization, influencing the piezoelectric and ferroelectric properties. These findings are of importance for guiding the design of Ko.sNao.sNbO3-based lead-free ceramics with enhanced electrical properties.

Enhanced Ferroelectric and Converse Piezoelectric Properties of Dense Lead-Free Na_0.4K_0.1Bi_0.5TiO₃Ceramics for Actuator Applications

In this work, lead-free Na0.4Bi0.5 TiO3-K0.1Bi0.5TiO3 (NKBT) piezoelectric ceramics were prepared by the solid-state reaction method and their structural, ferroelectric and dielectric properties were systematically studied. NKBT sintered at 1150℃ for 4 h exhibited highly dense (96% - 98% of the theoretical density) and uniform grains of size 1.1 μm. The coexistence of tetragonal (P4bm) and rhombohedral (R3c) phases due to the presence of morphotrophic phase boundary for the chosen composition has been confirmed by the Rietveld refinement studies. Enhanced ferroelectric properties such as remnant polarization (Pr) and coercive field (EC) are found to be 37 μC/cm2 and 30 kV/cm, respectively. The optimized synthesis procedure for NKBT ceramics resulted in enhanced strain (%) and converse piezoelectric coefficient (d33*) of 0.3 % and 554 pm/V which is attributed to smaller grain size and switching of non 180o domains. NKBT ceramics with such excellent piezoelectric properties can be considered as a promising candidate for actuator applications.