An improved model for unsymmetric plates

In the absence of analytical expressions for the equivalent circuit parameters it is difficult to calculate the optimal design parameters and dimensions and to chose suitable materials. The influence of coupling between flexural and extensional deformation and coupling between structure and acoustic volume on the dynamic response of piezoelectrically actuated flextensional transducer is analyzed using three analytical methods: classical plate theory, Mindlin plate theory, and simple variational procedure. The transducer design is based on a flextensional transducer that excites the axisymmetric resonant modes of a clamped circular plate. It is constructed by depositing a thin piezoelectric annular plate onto a thin, edge clamped, circular plate. An ac voltage is applied accross the piezoelectric material to set the compound membrane into flexural vibration. The devices have a range of operating resonance frequencies starting from 450 kHz up to 4.5 MHz. The device is manufactured by silicon surface micromachining and implemented in the form of two-dimensional arrays. Individual elements are made of thin silicon nitride membranes covered by a coating of piezoelectric zinc oxide. Classical thin plate theory, Mindlin plate theory, and variational methods are applied to derive two-dimensional plate equations for the transducer, and to calculate the coupled electromechanical field variables such as mechanical displacement. These methods use classical kinematic relations for a plate and a variational equation for the coupled electromechanical field to reduce three-dimensional field equations to two-dimensional plate equations. As a result, three different exact solutions to corresponding systems are obtained. An equivalent circuit of the transducer is also obtained from these solutions.

9 - Relationships between vibration frequencies of higher-order plate theories and classical thin plate theory

Buckling behavior of functionally graded materials cylindrical shell under pure bending is studied in this paper. The stability equations of functionally graded materials cylindrical shell are derived using the classical plate and shell theory with Kirchhoff hypothesis, importing the bending boundary condition to obtain the critical buckling load. The result shows that the critical moment is linear with the radius, quadratic with the thickness and irrelevant with the length of the cylindrical shell. In addition, the critical moment is decreased by increasing the power law index of the material bulk, trending to a constant finally.

A non-classical Mindlin plate model is developed using a modified couple stress theory. The equations of motion and boundary conditions are obtained simultaneously through a variational formulation based on Hamilton’s principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Mindlin plate theory. In addition, the current model considers both stretching and bending of the plate, which differs from the classical Mindlin plate model. It is shown that the newly developed Mindlin plate model recovers the non-classical Timoshenko beam model based on the modified couple stress theory as a special case. Also, the current non-classical plate model reduces to the Mindlin plate model based on classical elasticity when the material length scale parameter is set to be zero. To illustrate the new Mindlin plate model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new model are smaller than those predicted by the classical Mindlin plate model, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significantly large when the plate thickness is small, but they are diminishing with increasing plate thickness.

Deviatoric couple stress theory and its application to simple shear and pure bending problems

The mode I crack problem in plates under membrane loading and bending is reconsidered. The purpose is to examine certain analytical features of the problem further and to provide some new results. The formulation and the results given by the classical and the Reissner plate theories for through and part-through cracks are compared. For surface cracks the three-dimensional finite element solution is used as the basis of comparison. The solution is obtained and results are given for the crack/contact problem in a plate with a through crack under pure bending and for the crack interaction problem. Also, a procedure is developed to treat the problem of subcritical crack growth and to trace the evolution of the propagating crack.

A nonclassical model for circular Mindlin plates subjected to axisymmetric loading is developed using a modified couple stress theory. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation based on Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike existing circular Mindlin plate models based on classical elasticity. In addition, both the stretching and bending of the plate are considered in the formulation. The current plate model reduces to the classical elasticity-based Mindlin plate model when the material length scale parameter is set to be zero. Additionally, the new circular Mindlin plate model recovers the circular Kirchhoff plate model as a special case. To illustrate the new model, the static bending problem of a clamped solid circular Mindlin plate subjected to an axisymmetrically distributed normal pressure is analytically solved by directly applying the new model and using the Fourier–Bessel series. The numerical results show that the deflection and rotation angle predicted by the new model are smaller than those predicted by the classical Mindlin plate model. It is further seen that the differences between the two sets of predicted values are significantly large when the plate thickness is small, but they are diminishing with the increase of the plate thickness.

A new non-classical model for circular Kirchhoff plates subjected to axisymmetric loading is developed using a modified couple stress theory, a surface elasticity theory and Hamilton’s principle. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation. The new plate model contains a material length scale parameter to capture the microstructure effect and three surface elasticity constants to describe the surface energy effect. The current non-classical plate model includes the plate models considering the microstructure influence or the surface energy effect alone as special cases and recovers the classical elasticity-based Kirchhoff plate model when both the microstructure and surface energy effects are suppressed. To demonstrate the new model, the static bending problem of a clamped solid circular Kirchhoff plate subjected to a uniform normal load is analytically solved by directly applying the general formulas derived. The numerical results reveal that the deflection of the clamped circular plate predicted by the current Kirchhoff plate model is smaller than that predicted by the classical elasticity-based model, but the former approaches the latter with the increase of the plate thickness. It is also found that the difference between the plate deflection predicted by the model incorporating the surface energy effect and that predicted by the classical model is significant when the plate thickness is sufficiently small. However, the difference is diminishing as the plate thickness increases. These predicted size effects at the micron scale agree with the general trends observed in experiments.

This paper presents novel micromachined two-dimensional array piezoelectrically actuated flextensional transducers that can be used to generate sound in air or water. Micromachining techniques to fabricate these devices are also presented. Individual unimorph array elements consist of a thin piezoelectric annular disk and a thin, fully clamped, circular plate. We manufacture the transducer in two-dimensional arrays using planar silicon micromachining and demonstrate ultrasound transmission in air at 2.85 MHz with 0.15 /spl mu/m/V peak displacement. The devices have a range of operating resonance frequencies starting from 450 kHz up to 4.5 MHz. Such an array could be combined with on-board driving and addressing circuitry for different applications. Classical thin plate theory and Mindlin plate theory are applied to derive two-dimensional plate equations for the transducer, and to calculate the coupled electromechanical field variables such as mechanical displacement and electrical input impedance. In these methods, the variations across the thickness direction vanish by using the bending moments per unit length or stress resultants. Thus, two-dimensional plate equations for a step-wise laminated circular plate are obtained as well as two different solutions to the corresponding systems. An equivalent circuit of the transducer is also obtained from these solutions.

This paper is concerned with free vibration analysis of plate/shell coupled structures with two opposite edges simply supported by the method of reverberation-ray matrix. The equations of motion of the flat plate and the open circular cylindrical shell, respectively based on the classical thin plate theory and the Flügge thin shell theory, are introduced. Analytical solutions of the combination of a traveling wave form along the circumferential direction and a standing wave form along the axial direction are obtained. The method of reverberation-ray matrix is applied to derive the equation of the natural frequencies for the plate/shell coupled structures. The semi-analytical natural frequencies are obtained with the employment of the golden section search algorithm. The semi-analytical calculation results of three typical plate/shell coupled structures are presented and the results are compared with those obtained by the finite element method. The comparison shows that the calculation results obtained in this paper are of high accuracy and that the formulation presented in this manuscript are validated for free vibration analysis of plate/shell coupled structures.

Acoustic radiation from a fluid-loaded laminated composite plate with two sets of periodic stiffeners is investigated, in which the expressions for the stress tensors due to the transverse shear deformation are included on the basis of Reddy’s layerwise plate theories. By employing the Fourier transform method and the stationary phase approximation, far field pressure expressions are presented to address the acoustic characteristics of the stiffened laminated plate. Sound pressure obtained from the layerwise theory are compared with existing results from the classic thin plate theory, and increasingly remarkable differences among them are found from medium to high frequency range. Numerical calculations are subsequently carried out to explore the influences of the spacing of the stiffeners, the thickness of the plate, and the lamination schemes on the sound pressure. Practical concerns are addressed which would be beneficial to the optimization design of stiffened plates.

To develop a new analytical model for vibration analysis of cracked-submerged orthotropic micro-plate affected by fibre orientation and thermal environment. The proposed analytical model is based on Kirchhoff’s classical thin plate theory and the size effect is introduced using the modified couple stress theory. Effect of crack is deduced using appropriate crack compliance coefficients based on line spring model while the effect of thermal environment is introduced in terms of thermal in-plane moments and forces. The coupling of shear and normal stresses for fibre orientation is represented using the coefficient of mutual influence. The fluid forces associated with its inertial effects are added in the governing differential equation to incorporate the fluid–structure interaction effect. The results are presented for frequency response as affected by different fibre orientation, crack length, crack location, level of submergence, temperature variation and material length-scale parameter for simply supported boundary condition. Furthermore, to study the phenomenon of shifting of primary resonance in a cracked micro-plate, the classical relations for central deflection of plate is also proposed. The results show that the fundamental frequency of micro-plate decreases by the presence of crack and thermal environment and this decrease in frequency is further intensified by the presence of surrounding fluid medium in present study. Another important conclusion is that with increase in temperature variation the reduction in frequency at 45° of fibre orientation is less when compared to 0 and 90° for both intact and cracked orthotropic plates.

Frequency analysis of corner point supported mindlin plates by a finite difference energy method

Based on Mindlin’s theory of elastic thick plates and Hamiltonian formulism to structural vibration of cantilever plates, the solution of the problem is given. Using Lagrangian density function and Hamiltonian function, the flexural wave equation of plates is derived, and the relation of the transverse eigenvalue and longitudinal eigenvalue is given. Different from traditional engineering analysis, dispersion equations of the propagation mode of the Mindlin plate are deduced from the eigenfunction expansion method. By satisfying the boundary conditions of plates, the wavenumber of vibration modes in the plate structure can be obtained. The independent modal space control is applied to study the active vibration control of cantilever plates. Feedback gain is adjusted to change the eigenstructure of the original vibration system in order to improve the modal damping and stiffness of the system. The applicability of plates based on two theories (Mindlin’s theory and classical thin plate theory) is studied. Finally, the results are analyzed and discussed by numerical simulations.

On vibrations of thin plates with one-dimensional periodic structure