Two-dimensional Plate Equations Research Articles

Spectral and Evolution Analysis of Composite Elastic Plates with High Contrast

We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.

General decay estimate for a two-dimensional plate equation with time-varying feedback and time-varying coefficient

In this paper, we consider a plate equation as a model for a suspension bridge with a general nonlinear internal feedback and time-varying weight. Under some conditions on the feedback and the coefficient functions, we establish a general decay estimate for the associated energy functional, from which the exponential and the polynomial, are particular cases.

The Existence of Pullback Attractors for Non-autonomous Stochastic Plate Equations with Linear Memory

Consider the existence of a non-autonomous two-dimensional stochastic plate equation with linear memory term pullback the attractor on . Apply the Ornstein-Uhlenbeck process to deal with the random term, transform the original equation into a deterministic equation containing random variables, and then estimate its consistency by replacing the system solution with variables, and prove that the random dynamic system corresponding to the original system equation pullback the absorption set Existence, and finally proves the system's pullback asymptotically compaction , which leads to the existence of the pullback attractor of the original system.

The Ritz – Sublaminate Generalized Unified Formulation approach for piezoelectric composite plates

ABSTRACTThis paper extends to composite plates including piezoelectric plies the variable kinematics plate modeling approach called Sublaminate Generalized Unified Formulation (SGUF). Two-dimensional plate equations are obtained upon defining a priori the through-thickness distribution of the displacement field and electric potential. According to SGUF, independent approximations can be adopted for the four components of these generalized displacements: an Equivalent Single Layer (ESL) or Layer-Wise (LW) description over an arbitrary group of plies constituting the composite plate (the sublaminate) and the polynomial order employed in each sublaminate. The solution of the two-dimensional equations is sought in weak form by means of a Ritz method. In this work, boundary functions are used in conjunction with the domain approximation expressed by an orthogonal basis spanned by Legendre polynomials. The proposed computational tool is capable to represent electroded surfaces with equipotentiality conditions. Free-vibration problems as well as static problems involving actuator and sensor configurations are addressed. Two case studies are presented, which demonstrate the high accuracy of the proposed Ritz-SGUF approach. A model assessment is proposed for showcasing to which extent the SGUF approach allows a reduction of the number of unknowns with a controlled impact on the accuracy of the result.

Dynamic analysis of a transversely isotropic non-classical thin plate

This study investigates the dynamic analysis of a transversely isotropic thin plate. The plate is made of hyperelastic John\'s material and its constitutive law is obtained by taken the Frechect derivative of the highlighted energy function with respect to the geometry of deformation. The three-dimensional equation governing the motion of the plate is expressed in terms of first Piola-Kirchhoff\'s stress tensor. In the reduction to an equivalent two-dimensional plate equation, the obtained model generalizes the classical plate equation of motion. It is obtained that the plate under consideration exhibits harmonic force within its planes whereas this force varnishes in the classical plate model. The presence of harmonic forces within the planes of the considered plate increases the natural and resonance frequencies of the plate in free and forced vibrations respectively. Further, the parameter characterizing the transversely isotropic structure of the plate is observed to increase the plate flexural rigidity which in turn increases both the natural and resonance frequencies. Finally, this study reinforces the view that non-classical models of problems in elasticity provide ample opportunity to reveal important phenomena which classical models often fail to apprehend.

A non-local asymptotic theory for thin elastic plates

The three-dimensional dynamic non-local elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size. The integral constitutive relations, incorporating the variation of an exponential non-local kernel across the thickness, are adopted. Long-wave low-frequency approximations are derived for both bending and extensional motions. Boundary layers specific for non-local behaviour are revealed near the plate faces. It is established that the effect of the boundary layers leads to the first-order corrections to the bending and extensional stiffness in the classical two-dimensional plate equations.

A novel, single-layer model for composite plates using local-global approach

Abstract In structural analysis, many components are approximated as plates. More often that not, classical plate theories, such as Kirchhoff or Reissner-Mindlin plate theories, form the basis of the analytical developments. The advantage of these approaches is that they leads to simple kinematic descriptions of the problem: the plate's normal material line is assumed to remain straight and its displacement field is fully defined by three displacement and two rotation components. While such approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. For instance, it is well known from three-dimensional elasticity theory that the normal material line will warp under load for laminated composite plates, leading to three-dimensional deformations that generate complex stress states. To overcome this problem, several layer-wise plate theories have been proposed. While these approaches work well for some cases, they often lead to inefficient formulations because they introduce numerous additional variables. This paper presents a novel, single-layer theory using local-global Approach: based on a finite element semi-discretization of the normal material line, the two-dimensional plate equations are derived from three-dimensional elasticity using a rigorous dimensional reduction procedure. Three-dimensional stresses through the plate's thickness can be recovered accurately from the plate's stress resultants.

Wave propagation in piezoelectric layered structures of film bulk acoustic resonators

In this paper, we studied the wave propagation in a piezoelectric layered plate consisting of a piezoelectric thin film on an electroded elastic substrate with or without a driving electrode. Both plane-strain and anti-plane waves were taken into account for the sake of completeness. Numerical results on dispersion relations, cut-off frequencies and vibration distributions of selected modes were given. The effects of mass ratio of driving electrode layer to film layer on the dispersion curve patterns and cut-off frequencies of the plane-strain waves were discussed in detail. Results show that the mass ratio does not change the trend of dispersion curves but larger mass ratio lowers corresponding frequency at a fixed wave number and may extend the frequency range for energy trapping. Those results are of fundamental importance and can be used as a reference to develop effective two-dimensional plate equations for structural analysis and design of film bulk acoustic resonators.

Effects of nonlinearity on transient processes in AT-cut quartz thickness-shear resonators

We study the effects of mechanical nonlinearity arising from large thickness-shear deformation on the transient process of an AT-cut quartz plate resonator. Mindlin's two-dimensional plate equation is used, from which a system of first-order nonlinear differential equations governing the evolution of the vibration amplitude is obtained. Numerical solutions by the Runge-Kutta method show that in common operating conditions of quartz resonators the nonlinear effect varies from noticeable to significant. As resonators are to be made smaller and thinner in the future with about the same power requirement, nonlinear effects will become more important and need more understanding and consideration in resonator design.

Analysis of a monolithic, two-dimensional array of quartz crystal microbalances loaded by mass layers with nonuniform thickness

We study free thickness-shear vibrations of a monolithic, two-dimensional, and periodic array of quartz crystal microbalances loaded by mass layers with gradually varying thickness. A theoretical analysis is performed using Mindlin's two-dimensional plate equation. It is shown that the problem is mathematically governed by Mathieu's equation with a spatially varying coefficient. A periodic solution for resonant frequencies and modes is obtained and used to examine the effects of the mass layers. Results show that the vibration may be trapped or untrapped under the mass layers. The trapped modes decay differently in the two in-plane directions of the plate. The mode shapes and the decay rate of the trapped modes are sensitive to the mass layer thickness.

Existence of Global Attractors for the Nonlinear Plate Equation with Memory Term

A two‐dimensional nonlinear plate equation is revisited, which arises from the model of the viscoelastic thin rectangular plate with four edges supported. We establish that the system is exponentially decayed if the memory kernel satisfies the condition of the exponential decay. Furthermore, we show the existence of the global attractor by verifying the condition (C).

A few transient effects in AT-cut quartz thickness-shear resonators [Correspondence

We study a few transient effects of AT-cut quartz thickness-shear resonators, including resonator turning on and turning off as well as voltage amplitude and frequency fluctuations. Mindlin's two-dimensional plate equations are used and solved analytically. Both a sudden change and a gradual change of the driving voltage are studied. It is found that for a resonator with a frequency of 1.649430868 MHz and material quality factor of Q = 105, the characteristic time scale of the transient effects is of the order of 0.1 s.

Large Time Existence for Thin Vibrating Plates

We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large times under appropriate scaling of the initial values such that the limit system as h → 0 is either the nonlinear von Kármán plate equation or the linear fourth order Germain–Lagrange equation. In the case of the linear Germain–Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation.

Analytical Asymptotic Solutions for Rectangular Laminated Composite Plates

An analytical solution for rectangular laminated composite plates was obtained via a formal asymptotic method. From threedimensional static equilibrium equations, the microscopic one-dimensional and macroscopic two-dimensional equations were systematically derived by scaling of the thickness coordinate with respect to the characteristic length of the plate. The onedimensional through-the-thickness analysis was performed by applying a standard finite element method. The derived twodimensional plate equations, which take the form of recursive equations, were solved under sinusoidal loading with simplysupported boundary conditions. To demonstrate the validity and accuracy of the present method, various types of composite plates were studied, such as cross-ply, anti-symmetric angle-ply and sandwich plates. The results obtained were compared to those of the classical laminated plate theory, the first-order shear deformation theory and the three-dimensional elasticity. In the present analysis, the characteristic length of each composite was dependent upon the layup configurations, which affected the convergence rate of the method. The results shown herein are promising that it can serve as an efficient tool for the analysis and design of laminated composite plates.

Frictional contact of an anisotropic piezoelectric plate

The purpose of this paper is to derive and study a new asymptotic model for the equi- librium state of a thin anisotropic piezoelectric plate in frictional contact with a rigid obstacle. In the asymptotic process, the thickness of the piezoelectric plate is driven to zero and the convergence of the unknowns is studied. This leads to two-dimensional Kirchhoff-Love plate equations, in which mechanical displacement and electric potential are partly decoupled. Based on this model numerical examples are presented that illustrate the mutual interaction between the mechanical displacement and the electric potential. We observe that, compared to purely elastic materials, piezoelectric bodies yield a significantly different contact behavior.

Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls

In this work, we derive a result of exact controllability for a structural acoustic partial differential equation (PDE) model, comprised of a three-dimensional interior acoustic wave equation coupled to a two-dimensional Kirchoff plate equation, with the coupling being accomplished across a boundary interface. For this PDE system, we show that by means of boundary controls, the interior wave and Kirchoff plate initial data can be steered to an arbitrary finite energy state. In this work, key use is made of recent, microlocally-derived, $L^{2}\times H^{-1}$ "recovery" estimates for wave equations with Dirichlet boundary data. Moreover, the coupling of the disparate acoustic wave/Kirchoff plate dynamics is reconciled by means of sharp regularity estimates which are valid for hyperbolic equations of second order.

Plate equations for piezoelectrically actuated flexural mode ultrasound transducers

This paper considers variational methods to derive two-dimensional plate equations for piezoelectrically actuated flexural mode ultrasound transducers. In the absence of analytical expressions for the equivalent circuit parameters of a flexural mode transducer, it is difficult to calculate its optimal parameters and dimensions, and to choose suitable materials. The influence of coupling between flexural and extensional deformation, and coupling between the structure and the acoustic volume on the dynamic response of piezoelectrically actuated flexural mode transducer is analyzed using variational methods. Variational methods are applied to derive two-dimensional plate equations for the transducer, and to calculate the coupled electromechanical field variables. In these methods, the variations across the thickness direction vanish by using the stress resultants. Thus, two-dimensional plate equations for a stepwise laminated circular plate are obtained.

Piezoelectrically actuated flextensional micromachined ultrasound transducers--I: theory.

This series of two papers considers piezoelectrically actuated flextensional micromachined ultrasound transducers (PAFMUTs) and consists of theory, fabrication, and experimental parts. The theory presented in this paper is developed for an ultrasound transducer application presented in the second part. In the absence of analytical expressions for the equivalent circuit parameters of a flextensional transducer, it is difficult to calculate its optimal parameters and dimensions and difficult to choose suitable materials. The influence of coupling between flexural and extensional deformation and that of coupling between the structure and the acoustic volume on the dynamic response of piezoelectrically actuated flextensional transducer are analyzed using two analytical methods: classical thin (Kirchhoff) plate theory and Mindlin plate theory. 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.

Micromachined Piezoelectrically Actuated Flextensional Transducers For High Resolution Printing

In this paper, we present a technique for the deposition of inks, toners, organic polymers, fuels, small solid particles, biological and chemical fluids, using a fluid ejector. The ejector 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. Liquids or solid-particles are placed behind one face of the plate which has a small orifice at its center. By applying an ac signal across the piezoelectric element, continuous or drop-on-demand ejection of fluids has been achieved. The ejected drop size ranges in diameter from 4 μm at 3.5 MHz to 150 μm at 7 kHz, the corresponding ejected drop volume ranges from 34 fl to 1.5 nl, and the corresponding flow rate ranges from 117 nl/s to 10 μl/s. The unique features of the device are that the fluid is not pressurized, the fluid container is chemically or biologically compatible with most fluids, and the vibrating plate contains the orifice as the ejection source. 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 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.

Propagation of fluid-loaded structural waves along a duct with smoothly varying bending characteristics

This article is concerned with the sound field generated by a fluid-loaded structural wave travelling along the boundary of a two-dimensional duct with walls that have smoothly varying elastic properties. It is the field within the duct that is of interest and the exterior region is assumed to be in vacuo. By recourse to a modified form of the two-dimensional plate equation the boundary-value problem is posed in terms of the fluid velocity potential. A functional difference equation method is used to obtain an explicit integral solution from which the reflection and transmission coefficients are determined. Various limiting forms of the modified plate equation are discussed together with the implications each has on both the analytic method used herein and the physical situation it describes.