Extended Kantorovich method for local stresses in composite laminates upon polynomial stress functions

The extended Kantorovich method is employed to study the local stress concentrations at the vicinity of free edges in symmetrically layered composite laminates subjected to uniaxial tensile load upon polynomial stress functions. The stress fields are initially assumed by means of the Lekhnitskii stress functions under the plane strain state. Applying the principle of complementary virtual work,the coupled ordinary differential equations are obtained in which the solutions can be obtained by solving a generalized eigenvalue problem. Then an iterative procedure is established to achieve convergent stress distributions. It should be noted that the stress function based extended Kantorovich method can satisfy both the traction-free and free edge stress boundary conditions during the iterative processes. The stress components near the free edges and in the interior regions are calculated and compared with those obtained results by finite element method（FEM）. The convergent stresses have good agreements with those results obtained by three dimensional（3D） FEM. For generality, various layup configurations are considered for the numerical analysis. The results show that the proposed polynomial stress function based extended Kantorovich method is accurate and efficient in predicting the local stresses in composite laminates and computationally much more efficient than the 3D FEM.

A nonlinear analysis of surface acoustic waves in isotropic elastic solids

With the fast evolution of wireless and networking communication technology,applications of surface acoustic wave(SAW),or Rayleigh wave,resonators are proliferating with fast shrinking sizes and increasing frequencies.It is inevitable that the smaller resonators will be under a strong electric field with induced large deformation,which has to be described in wave propagation equations with the consideration of nonlinearity.In this study,the formal nonlinear equations of motion are constructed by introducing the nonlinear constitutive relation and strain components in a standard procedure,and the equations are simplified by the extended Galerkin method through the elimination of harmonics.The wave velocity of the nonlinear SAW is obtained from approximated nonlinear equations and boundary conditions through a rigorous solution procedure.It is shown that if the amplitude is small enough,the nonlinear results are consistent with the linear results,demonstrating an alternative procedure for nonlinear analysis of SAW devices working in nonlinear state.

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.

Thickness-Shear Frequencies of an Infinite Quartz Plate with Graded Material Properties Across the Thickness

For quartz crystal resonators of thickness-shear type,the vibration frequency and mode shapes reflect the basic material and structural properties of a quartz crystal plate and the variation with time under attack by erosive gases and liquids that can cause surface and internal degradation in a graded form.Such gradual effects,in turn,will change surface conditions through elastic constants and stiffness and more importantly the gradient of such properties across the thickness with certain depth.Now we study the thickness-shear vibrations of functionally graded material plates with properties of the popular AT-cut quartz crystal varying across the thickness in a general pattern represented by both sine and cosine components of the thickness.The vibration solutions are obtained through the Fourier expansion of the plate deformation.

The Multi-field Coupled Vibration Analysis of AT-Cut Quartz Crystal Resonators with Parallelism Error

During the fabrication of quartz crystal resonators(QCRs),parallelism error is inevitably generated,which is rarely investigated.In order to reveal the influence of parallelism error on the working performance of QCRs,the coupled vibration of a non-parallel AT-cut quartz crystal plate with electrodes is systematically studied from the views of theoretical analysis and numerical simulations.The two-dimensional thermal incremental field equations are solved for the free vibration analysis via the coefficient-formed partial differential equation module of the COMSOL Multiphysics software,from which the frequency spectra,frequency–temperature curves,and mode shapes are discussed in detail.Additionally,the piezoelectric module is utilized to obtain the admittance response under different conditions.It is demonstrated that the parallelism error reduces the resonant frequency.Additionally,symmetry broken by the non-parallelism increases the probability of activity dip and is harmful to QCR’s thermal stability.However,if the top and bottom surfaces incline synchronously in the same direction,the influence of parallelism error is tiny.The conclusions achieved are helpful for the QCR design,and the methodology presented can also be applied to other wave devices.

Frequency–Temperature Analysis of Thickness-Shear Vibrations of SC-Cut Quartz Crystal Plates with the First-Order Mindlin Plate Equations

With the incremental thermal field theory by Lee and Yong,the Mindlin plate equations for the analysis of thickness-shear vibrations of SC-cut quartz crystal with the consideration of thermal effect have been obtained.By assuming straight-crested waves in a rectangular plate,the dispersion relations,frequency spectra,mode shapes,and frequency–temperature relations of the thickness-shear mode of SC-cut quartz crystal plates are calculated with plate configurations.The computational results from SC-cut quartz crystal plates are compared with those from AT-cut,which shows that SC-cut resonators have better frequency–temperature relation as a validation.For practical applications,such results will be useful in the precision design and improvement of SC-cut quartz crystal resonator structures.

Two-Dimensional Coupling Vibration Analysis of Laterally Acoustically Coupled Two-Port Thin-Film Bulk Acoustic Resonators

In this paper,we present an approach to studying the mode coupling vibrations in two-port thin-film bulk acoustic wave resonator(FBAR)devices with two pairs of electrodes deposited on the zinc oxide film.The two-dimensional plate theory established in our previous work is employed,which takes into account the coupling of the operating thickness-extensional mode with the extensional,flexural,fundamental and second-order thickness-shear modes.The propagation of straight-crested waves in the plate is studied,and the state-vector approach is successfully used to simplify the derivation process.For a structurally symmetric device,the modes are separated into quasi symmetric and antisymmetric ones.Frequency spectra and corresponding mode shapes are obtained under the stress-free boundary conditions,respectively,and then coupling effects and energy trapping phenomenon are discussed in detail.Some results for structures with asymmetric electrode distributions are also shown.It is found that the choice of aspect ratio has a great effect on mode couplings of FBAR devices.This study will be useful for the design of FBAR filters and sensors.