Mindlin Plate Equations Research Articles

Free-Vibration Analysis for Truncated Uflyand–Mindlin Plate Models: An Alternative Theoretical Formulation

Plates are flat structural elements whose thickness is small in relation to the size of the surface. Their use may include engine foundations, reinforced concrete bridge elements or parts of various floating structures. Consequently, knowledge of their mechanical behavior under static and dynamic loads is of primary importance in engineering applications and of interest from a structural point of view. As a result, numerous works existing in the literature have investigated the mechanical properties of plates using various plate models, such as Reissner’s theory, Levinson’s theory, Kirchhoff’s theory and Mindlin’s theory, and their static and dynamic behavior has been examined. In the present paper the truncated Uflyand–Mindlin plate equation is proposed. According to Uflyand–Mindlin theory, an alternative theoretical formulation is presented for the free-vibration analysis of plates, and the equations of motion and the general corresponding boundary conditions are derived. This paper develops the truncated Uflyand–Mindlin plate equation, i.e., without the fourth-order derivative, by means of the direct method and variational formulation. The first-order shear deformable plate theory developed by Elishakoff, which takes into account rotational inertia and shear deformation and does not include a fourth-order time derivative, is variationally derived here. This derivation complements that performed by Mindlin some 70 years ago. The innovative aspect of the suggested strategy is that variational and direct methods for studying plate dynamics are analogous. Finding the third equation of the reduced Uflyand–Mindlin equations, the accompanying boundary conditions and their mathematical resemblance are the goals of the presented formulations. In order to solve the dynamic equilibrium problem of a truncated Uflyand–Mindlin equation via a variational formulation, it is demonstrated that the differential equations and the corresponding boundary conditions have the same form as those found using the direct technique. This paper successfully completes this task. Finally, in order to validate the effectiveness and correctness of the proposed procedure, a numerical example of the case of a plate simply supported at all four ends is proposed.

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.

Identification of Vibration Modes of Quartz Crystal Plates with Proportion of Strain and Kinetic Energies

For the design of quartz crystal resonators, finding and determining the vibration modes have always been very important and cumbersome. Vibration modes are usually identified through plotting displacement patterns of each coupled modes and making comparisons. Over the years, there is not much improvement in the identification procedure while tremendous efforts have been made in refining the equations of the Mindlin plate theory to obtain more accurate results, such as the adoption of the Finite Element Method (FEM) by implementing the high-order Mindlin plate equations for efficient analysis. However, due to the old fashioned mode identification method, the FEM application is still inadequate and cannot be fully automated for this purpose. To have this situation improved, a method using the proportions of strain and kinetic energies to characterize the energy level of each vibration mode is proposed. With solutions of displacements, the energy distribution of each vibration mode is calculated and the mode with the highest energy concentration at a specific frequency is designated as the dominant mode. The results have been validated with the traditional approach by plotting mode shapes at each frequency. Clearly, this energy approach will be advantageous with the FEM analysis for vibration mode identification automatically.

Examinations of vibration frequency and mode shape variations of quartz crystal plates in a thermal field with strain and kinetic energies

The frequency-temperature behavior has long been analyzed to ensure the smoothness of functioning frequency of a quartz crystal resonator during the temperature variation and maintain its operations in a stable state. It has been proven that vibrations of quartz crystal plates in the thickness-shear mode can be well described by the theory of incremental thermal field in conjunction with the Mindlin plate equations, demonstrating the possibility that vibration frequencies and mode shapes can also be well predicted with the temperature as a targeted variable. By studying the vibrations with temperature changes in a given range, the frequencies and vibration modes can be calculated as a function of temperature, then they can be used in the calculations of both strain and kinetic energies for vibration mode identification or characterization based on energy proportion and changes. The trends of changes of vibration frequency, amplitudes, and energy with temperature variation make it possible for characterization the variations of vibrations and particularly the weakening or enhancement of the dominance of specific vibration modes. With such relations between vibration properties and temperature known, possible analytical tools can be developed to aid the optimal design by avoiding significant mode changes or occurrences of mode conversion.

High-frequency vibrations of circular and annular plates with the Mindlin plate theory

Circular and annular elastic plates have wide applications as essential elements in various engineering structures and products demanding accurate analysis of their vibrations. At higher frequencies, the analysis of vibrations needs appropriate equations, as shown by the Mindlin plate equations for rectangular plates with tailored applications for the analysis of quartz crystal resonators. Naturally, there are equally strong demands for the equations and applications in circular and annular plates with the consideration of higher-order vibration modes. By following the procedure established by Mindlin based on displacement expansion in the thickness coordinate, a set of higher-order equations of vibrations of circular and annular plates are derived and truncated for comparisons with classical and first-order plate equations of circular plates. By utilizing these equations, coupled thickness-shear and flexural vibrations of circular and annular plates are analyzed for vibration frequencies and mode shapes.

A unified approach to meshless analysis of thin to moderately thick plates based on a shear-locking-free Mindlin theory formulation

The conventional numerical approximation of Mindlin plate equations can lead to erroneous solutions for thin plates. The so-called shear-locking problem has been well studied in the context of the finite element method (FEM) whereas the development of numerical formulations for its successful elimination in meshfree methods is still a subject of intensive research. This paper studies the effectiveness of some of the most commonly adopted techniques for the reduction of shear-locking and presents the application of a shear-locking-free formulation based on first-order Mindlin plate theory. In this modified formulation, the shear strains are incorporated as degrees of freedom (DOFs) in lieu of the rotational DOFs in the conventional Mindlin theory formulation. A straightforward transformation technique is presented for the enforcement of boundary conditions and comparisons are made with available analytical and numerical solutions. The generalised reproducing kernel particle method (RKPM) is adopted as the numerical tool and a series of numerical examples are presented to demonstrate the accuracy and performance of the presented method.

An analysis of the TDNNS method using natural norms

The tangential-displacement normal-normal-stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress are the degrees of freedom of the finite elements. The TDNNS method was shown to converge of optimal order, and to be robust with respect to shear and volume locking. However, the method is slightly nonconforming, and an analysis with respect to the natural norms of the arising spaces was still missing. We present a sound mathematical theory of the infinite dimensional problem using the space {{mathbf {H}}}(mathbf {curl}) for the displacement. We define the space {underline{{mathbf {H}}}}({text {div}},mathbf {{div}}) for the stresses and provide trace operators for the normal-normal stress. Moreover, the finite element problem is shown to be stable with respect to the {{mathbf {H}}}(mathbf {curl}) and a discrete {underline{{mathbf {H}}}}({text {div}},mathbf {{div}}) norm. A-priori error estimates of optimal order with respect to these norms are obtained. Beside providing a new analysis for the elasticity equation, the numerical techniques developed in this paper are a foundation for more complex models from structural mechanics such as Reissner Mindlin plate equations, see Pechstein and Schöberl (Numerische Mathematik 137(3):713–740, 2017).

Equilibrium approach in the derivation of differential equations for homogeneous isotropic mindlin plates

In this paper, the differential equations of Mindlin plates are derived from basic principles by simultaneous satisfaction of the differential equations of equilibrium, the stress-strain laws and the strain-displacement relations for isotropic, homogenous linear elastic materials. Equilibrium method was adopted in the derivation. The Mindlin plate equation was obtained as a system of simultaneous partial differential equations in terms of three displacement variables (parameters) namely where w(x, y, z) is the transverse displacement and are rotations of the middle surface. It was shown that when where k is the shear correction factor, the Mindlin plate equations reduce to the classical Kirchhoff plate equation which is a biharmonic equation in terms of w(x, y, z = 0). http://dx.doi.org/10.4314/njt.v36i2.4

An analysis of nonlinear thickness-shear vibrations of quartz crystal plates using the two-dimensional finite element method

ABSTRACTA nonlinear analysis of high-frequency thickness-shear vibrations of AT-cut quartz crystal plates is presented with the two-dimensional finite element method. The Mindlin plate equations are truncated to the first-order ones as an approximation, and then they are used for the formulation of nonlinear finite element analysis with all zero- and first-order displacements. The matrix equation of motion is established with the first-order harmonic approximation, and the generalized nonlinear eigensystem is solved by a direct iterative procedure. A displacement amplitude versus frequency curve and corresponding mode shapes are obtained and examined. The nonlinear finite element program is developed based on the earlier linear edition and can be utilized to predict nonlinear characteristics of miniaturized quartz crystal resonators in the design process.

Vibrations of asymptotically and variationally based Uflyand–Mindlin plate models

In this paper, we provide alternative Uflyand–Mindlin's plate equations taking into account rotary inertia and shear deformation, based on both asymptotic expansion and variational arguments. The aim is to derive truncated versions of Uflyand–Mindlin's equations, specifically without the fourth order derivative term with respect to time. The truncated version of Uflyand–Mindlin's plate model may be derived starting from three-dimensional elasticity equations, by using asymptotic arguments based on expansion of displacements with respect to a small geometrical parameter. This expansion method also leads to a proper identification of the shear correction factor. It is shown that suitably modified variational derivation leads to an additional term which is shown to be negligible for determination of the fundamental natural frequency of the all-round simply supported plates, but may contribute significantly in estimation of higher natural frequencies. It is argued that the proposed version of Uflyand–Mindlin's plate equations is simpler and more consistent than the original Uflyand–Mindlin equations. Likewise, it is advantageous over the equation that stems from neglecting the fourth order time derivative in original Uflyand–Mindlin equations. The two alternative truncated models serve as intermediate theories between the classical plate theory and the original Uflyand–Mindlin theory their usefulness depending on the problem at hand.

Free and forced vibrations of SC-cut quartz crystal rectangular plates with the first-order Mindlin plate equations

Mindlin plate theory was used to provide accurate solutions to thickness-shear vibrations of plates, which have a much higher frequency than usual flexural vibrations and are the functioning modes of quartz crystal resonators. The vibration frequency solutions obtained with the Mindlin plate theory are proven being accurate along with mode shapes. In this paper, straight-crested wave solutions of free and forced vibrations of doubly rotated SC-cut of quartz crystal plates of rectangular shapes with four free edges are obtained with validated Mindlin plate equations. A procedure has been established for the calculation of dispersion relations, frequency spectra, mode shapes, and capacitance ratios of forced vibrations needed in resonator design.

Forced vibrations of SC-cut quartz crystal rectangular plates with partial electrodes by the Lee plate equations

Lee plate equations for high frequency vibrations of piezoelectric plates have been established and perfected over decades with the sole objective of obtaining accurate predictions of frequency and mode shapes to aid the analysis and design of quartz crystal resonators. The latest improvement includes extra terms related to derivatives of the flexural displacement to provide much accurate solutions for vibrations of the thickness-shear mode, which is the functioning mode of resonators and has much higher frequency than the flexural mode. The improved Lee plate equations have been used in the analysis of high frequency vibrations of quartz crystal plates as an essential step for analysis of AT- and SC-cut quartz crystal resonators after validations with fully electrode quartz crystal piezoelectric plates. In this study, closed-form solutions of free and forced vibrations of SC-cut quartz plates with partial electrodes are obtained. A procedure has been established for the calculation of dispersion relations, frequency spectra, selected vibration modes, and capacitance ratios of forced vibrations. The vibration solutions obtained with the first-order Lee plate equations are proven to be close to solutions from the Mindlin plate equations. It is now clear that both the Mindlin and Lee plate equations can be used in the analysis and design of quartz crystal resonators.

A new locking-free finite element method based on more consistent version of Mindlin plate equation

A finite element (FE) approach is presented for the dynamic analysis of the Mindlin plates considering both shear deformation and rotary inertia effects. The model is based on the consistent version of the Mindlin equations, which neglects the higher-order time derivative contribution. The approach provides a new class of interdependent Hermite shape polynomials by the definition of a fictitious deflection that takes into account the effective interdependence between the generalized displacements in both the continuous and FE discretized schemes. This implies that the proposed approach is free-shear-locking and is characterized by a good accuracy even for low-order FEs. Several examples are considered whose results are compared with analogous ones proposed in the literature with other approaches.

An analysis of thickness-shear vibrations of doubly-rotated quartz crystal plates with the corrected first-order mindlin plate equations

The Mindlin plate equations have been widely used in the analysis of high-frequency vibrations of quartz crystal resonators with accurate solutions, as demonstrated by the design procedure based on analytical results in terms of frequency, mode shapes, and optimal parameters for the AT-cut quartz crystal plate, which is the core element in a resonator structure. Earlier studies have been focused on the AT-cut (which is one type of rotated Y-cut) quartz crystal plates because it is widely produced and has relatively simple couplings of vibration modes at thickness-shear frequencies of the fundamental and overtone modes. The simplified equations through the truncation, correction, and modification of the Mindlin plate equations have been widely accepted for practical applications, and further efforts to expand their applications to similar problems of other material types, such as doubly-rotated quartz crystals, with the SC-cut being a typical and popular one, are also naturally expected. We have found out that the Mindlin plate theory can be truncated and corrected for the SC-cut quartz crystal plates in a manner similar to the AT-cut plates. The analytical results show that the corrected Mindlin plate equations are equally accurate and convenient for obtaining essential design parameters of resonators for the thickness-shear vibrations of SC-cut quartz crystal plates.

Two simple finite element methods for Reissner–Mindlin plates with clamped boundary condition

We present two simple finite element methods for the discretization of Reissner–Mindlin plate equations with clamped boundary condition. These finite element methods are based on discrete Lagrange multiplier spaces from mortar finite element techniques. We prove optimal a priori error estimates for both methods. The first approach is based on a so-called standard Lagrange multiplier space for the mortar finite element method, where the Lagrange multiplier basis functions are continuous. The second approach is based on a so-called dual Lagrange multiplier space, where the Lagrange multiplier basis functions are discontinuous. The advantage of using the second approach is that easy static condensation of degrees of freedom corresponding to the Lagrange multiplier is possibly leading to a symmetric positive definite formulation.

The fifth-order overtone vibrations of quartz crystal plates with corrected higher-order mindlin plate equations

Higher-order overtone resonators have been widely used in various electronic products for their higher vibration frequencies, which are in the much-needed frequency range beyond the reach of the fundamental mode. However, the existing designs of higher-order overtone resonators and further improvement for meeting more precise requirements are largely based on empirical approaches. As an analytical effort, we have derived the corrected fifth-order Mindlin plate equations with the consideration of electric potential and overtone displacements. The elimination and truncation of the infinite two-dimensional equations has been done to ensure the exact cut-off frequencies of the fundamental, the third-order overtone, and fifth-order overtone thickness-shear modes in comparison with the three-dimensional equations. The frequency spectra are plotted in the vicinity of overtone thickness-shear modes for analysis of couplings and interactions with spurious modes, and the optimal design of quartz crystal blanks for overtone vibrations has been suggested. The equations, solutions, and method will be important in design of the higher-order overtone thickness-shear vibration resonators.

Correction factors of the mindlin plate equations with the consideration of electrodes [Correspondence

The Mindlin plate theory plays a vital role in the analysis of high-frequency vibrations of quartz crystal resonators in the thickness-shear mode. The coupled equations with an infinite number of vibration modes have been truncated to retain essential modes with strong couplings for simplified analysis, and correction factors have been introduced to ensure exact solutions at cut-off frequencies as part of the validation procedure. Starting from plates without complications, correction factors have also been modified to include electrodes for their mass effect. Such results have been important for higher-order correction factors to ensure accurate consideration of electrodes in AT-cut quartz crystal plates. The findings in this systematic study of correction factors are readily available for the analytical equations with selected modes with strong couplings, and equally convenient for the implementation of the finite element analysis with the Mindlin plate equations.

A locking-free meshfree method for the simulation of shear-deformable plates based on a mixed variational formulation

The problem of shear-locking in the thin-plate limit is a well known issue that must be overcome when discretising the Reissner–Mindlin plate equations. In this paper we present a shear-locking-free method utilising meshfree maximum-entropy basis functions and rotated Raviart–Thomas-Nédélec elements within a mixed variational formulation. The formulation draws upon well known techniques in the finite element literature. Due to the inherent properties of the maximum-entropy basis functions our method allows for the direct imposition of Dirichlet (essential) boundary conditions, in contrast to methods based on moving least squares basis functions. We present benchmark problems that demonstrate the accuracy and performance 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.

Strong stabilization of models incorporating the thermoelastic Reissner–Mindlin plate equations with second sound

In this article we are concerned with the strong stabilization of models for the Reissner–Mindlin plate equations with second sound, that is, models that include thermal effects described according to Cattaneo's law of heat conduction instead of Fourier's law in classical thermoelasticity. Two models will be considered which are distinct with respect to the property of compactness or non-compactness of the resolvent of the generator of the underlying semigroup. In accordance with the compactness or non-compactness of the resolvent operator, a different criterion for strong stability is implemented to achieve the strong stabilization of each model. In the compact resolvent case we avail ourselves of a result given by Benchimol [C.D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim. 16 (1978), pp. 373–379] and in the non-compact case we resort to a stability criterion Tomilov [Y. Tomilov, A resolvent approach to stability of operator semigroups, J. Operator Theory 46 (2001), pp. 63–98].