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.

Decay estimate in a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity

In this article, we consider a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. We prove explicit and general decay rate results of the solution to the viscoelastic plate equation with past history. Convex properties, logarithmic inequalities, and generalized Young’s inequality are mainly used to prove the decay estimate.

A linear two-dimensional mathematical model for thin two-layer plates with partial shear interaction, with a view towards application to laminated glass

This work presents a consistent derivation, from three-dimensional linear elasticity, of a two-dimensional mathematical model describing the bending and in-plane stretching behaviours, under a general system of quasi-static distributed loads and prescribed support displacements, of thin two-layer plates with partial shear interaction. This layerwise model is specifically tailored for the requirements posed by the analysis of laminated glass plates commonly used in building structures (consisting of two thin glass layers bonded together by an adhesive interlayer). Our approach, based on Podio-Guidugli’s method of internal constraints, avoids mutually contradictory assumptions (not uncommon in the literature on structural mechanics) and yields a complete two-dimensional characterisation of displacement, strain and stress fields that exactly satisfy the field equations of three-dimensional linear elasticity and the boundary conditions at the end faces. The choice of generalised variables is designed to bring to light the following fundamental conclusion and physical insight: the resulting two-dimensional boundary value problem is a combination of the equations of Kirchhoff and Mindlin plates (with specified rigidities).Two examples illustrate the application of the proposed model: (i) the cylindrical bending of plate strips and (ii) a family of problems with Navier-type analytical solution. The solutions exhibit continuity across the whole range of zero, partial and full interaction between the layers. Moreover, the Navier-type solutions are consistently in close agreement with the results of three-dimensional finite element analyses. On the contrary, analogous results previously reported in the literature exhibit considerable deviations. An explanation for these discrepancies is discussed in detail.

Axisymmetric large deformation problems of thin shallow shells with different moduli in tension and compression

When materials are in tension or in compression, their mechanical properties are essentially different, however, in existing studies it is generally ignored because of the complexity of the analysis. In this paper, a theoretical study of the problem of axisymmetric large deformation of thin shallow shells with different moduli in tension and compression (bimodular thin shallow shells) under uniformly vertical loading is presented. Based on the Föppl–von Kármán equations of flexible thin plates, the governing equation of the axisymmetric large deformation of bimodular thin shallow shells are established by introducing an appropriate curvature modification term from thin shallow shells and a bending stiffness term from bimodular plates. By selecting the central deflection as a perturbation parameter, the perturbation solution of the axisymmetric large deformation problem is derived using the perturbation method. The applications of variation method as well as the implementation of numerical simulation verify, from the theoretical and numerical perspectives, the correctness of the perturbation solution obtained. This study shows that the introduction of different moduli in tension and compression not only affects the relationship of loads vs. central deflection, but also affects the critical values of the rise-thickness ratio of bimodular thin shallow shells when the snapping phenomenon occurs. In particular, the bimodular effect may accelerate or slow down the occurrence of the snapping phenomenon of thin shallow spherical shells, according to the relative magnitude of tensile and compressive moduli. The work presented here will be helpful to analyze the mechanical response of heavily loaded thin shallow shells with obvious bimodular effect.

Stability of inverse scattering problem for the damped biharmonic plate equation

This paper is concerned with the stability of the inverse scattering problem for the damped biharmonic plate equation in a bounded domain with the Cauchy data. A sharp estimate for the mass density is established using a priori information concerning Sobolev norms and a priori information about the support of the inhomogeneity. Our results improve previous estimates and explicitly depend on the damping coefficient. The proof mainly relies on the complex geometric optics solution method and the Fourier analysis.

Bilinear boundary optimal control of a Kirchhoff plate equation

This work shows that a problem of boundary optimal control of a Kirchhoff plate equation has a solution that we characterized using the differentiability of a functional cost. The Kirchhoff plate equation is governed by bilinear control acting on the boundary, in which non-linear terms are constructed by multiplication of the control vector and the state one. The question is to obtain a distributed control which minimizes a function cost constituted of the deviation between a desired state and the reached one, and the energy term. The purpose of this study is to prove that an optimal control exists, and it is characterized as a solution to an optimality system. Thus, We show that the problem of bilinear boundary control has a unique solution if the final time is sufficiently small.

A uniformly-valid asymptotic plate theory of growth with numerical implementation

In this paper, we aim to develop a simplified and uniformly-valid finite-strain plate theory of growth. First, starting from a consistent finite-strain plate theory of growth proposed in our previous work, we specify the magnitudes of growth functions in five different cases and conduct systematic asymptotic analyses, from which the original plate theory can be reduced to some classical plate and membrane theories. Based on the asymptotic analyses, it is found that some terms in the original plate equations (which contain the high-order stress tensor S(2)) always have higher asymptotic orders. By dropping these terms, the plate equations can be simplified significantly. Then, a reduced finite-strain plate theory of growth is established, which is valid in a wide range of growth and mechanical loading conditions. The associated weak form of this uniformly-valid plate theory has also been derived for numerical implementation. To demonstrate the efficiency of this plate theory, it is implemented into a finite element software and applied to study four typical examples. To verify the accuracy of the 2D plate models, the corresponding 3D volume models have also been constructed for these examples. Through some comparisons, it is found that the numerical results obtained from the 2D and 3D models can fit each other quite well. Furthermore, the numerical calculations based on the plate models show obvious advantages in the aspects of computational efficiency, convergence rate and stability. In our opinion, the uniformly-valid asymptotic plate theory of growth proposed in the current work is adequate for studying the complex growth behaviors of thin hyperelastic plates.

A monolithic finite element formulation for the hydroelastic analysis of very large floating structures

AbstractIn this work we present a novel monolithic Finite Element method for the hydroelastic analysis of very large floating structures (VLFS) with arbitrary shapes that is stable, energy conserving, and overcomes the need of an iterative algorithm. The new formulation enables a fully monolithic solution of the linear free‐surface flow, described by linear potential flow, coupled with floating thin structures, described by the Euler–Bernoulli beam or Poisson–Kirchhoff plate equations. The formulation presented in this work is general in the sense that solutions can be found in the frequency and time domains, it overcomes the need of using elements with continuity by employing a continuous/discontinuous Galerkin approach, and it is suitable for finite elements of arbitrary order. We show that the proposed approach can accurately describe the hydroelastic phenomena of VLFS with a variety of tests, including structures with elastic joints, variable bathymetry, and arbitrary structural shapes.

A general multi-layered hyperelastic plate theory for growth-induced deformations in soft material samples

In this paper, we propose a general multi-layered hyperelastic plate theory to study the growth-induced deformations of soft material samples. First, by considering a multi-layered plate sample with an arbitrary geometrical shape and adopting a general hyperelastic constitutive model, the 3D governing system is established, which incorporates the growth effects of the different layers in the plate. Then, a series expansion-truncation approach is adopted to eliminate the thickness variables in the 3D governing system. An elaborate calculation scheme is applied to derive the iteration relations of the coefficient functions in the series expansions. Through some further manipulations, a 2D vector plate equation system with the associated boundary conditions is established. To show the efficiency of the plate theory, three typical examples regarding the growth-induced deformations and instabilities of multi-layered plate samples are studied, where the incompressible neo-Hookean constitutive model is adopted. Some analytical and numerical solutions to the plate equation system are obtained, which can provide accurate predictions on the growth behaviors of the plates. Furthermore, the problem of ‘shape-programming’ of multi-layered plates through differential growth is studied and the explicit formulas for some typical examples are derived. By using these formulas, the shape evolutions of the plates during the growing processes can be controlled accurately. The results obtained in the current work are applicable for the design of intelligent soft devices with multi-layered plate structures.

Axisymmetric buckling of annular plate with internal source of heat

In this article, the axisymmetric stability of an elastic annular plate with distributed internal heat source under uniform constant compressive radial loading at the outer edge is investigated. The governing and compatibility equations of the annular plate are taken from modified Karman’s equations, which are solved by the shooting method. The effect of the density of distributed internal heat source on critical load is investigated and discussed in detail. Numerical results of the plate without an internal heat source are in good agreement with the results from the literature, validating the effectiveness and advantages of the current method. The critical buckling load increases with a rise of the density of distributed internal heat sources. Moreover, an increase of wave number increases the critical buckling loads of the plate. The results of this research can be considered as benchmark solutions for buckling of the annular plates with an internal heat source.

Influence of circular cutouts and elastic foundation on vibration characteristics of porous sandwich FGM plates under hygro-thermal environments

The present article investigates the effect of cutouts, porosity inclusion, and hygro-thermal environment on the vibrational characteristics of sandwich functionally graded material (FGM) plates resting on Pasternak elastic foundation. The governing equations for the sandwich FGM plates have been developed using the finite element method (FEM) with the conjunction of non-polynomial-based higher-order shear deformation theory (HSDT). The trigonometric shear-strain shape function has been used for the first time for sandwich FGM plates. The effective material properties with porosity inclusions have been computed using modified power–law distribution. The present results have been acquired by employing a C0 continuity isoparametric finite element (FE), using 4-noded elements with 7-DOF's per node. The accuracy of the current solution has been confirmed by convergence and validation studies. The influence of circular cutout, elastic foundations, hygro-thermal conditions, and geometric and boundary conditions on the vibrational frequency of porous sandwich FGM plates have been discussed in detail.

Stabilization of the damped plate equation under general boundary conditions

We consider a damped plate equation on an open bounded subset of ℝ d , or a smooth manifold, with boundary, along with general boundary operators fulfilling the Lopatinskiĭ-Šapiro condition. The damping term acts on an internal region without imposing any geometrical condition. We derive a resolvent estimate for the generator of the damped plate semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first through microlocal estimates, then local estimates, and finally a global estimate.

Nonlinear flutter analysis of arbitrary functionally graded plates using Isogeometric approach

The main objective of the present study is to analyze the nonlinear aeroelastic behavior of Functionally Graded (FG) plates with arbitrary shapes using an isogeometric approach. Hence, the aeroelastic equations of such plates under simultaneous supersonic airflow and temperature gradient have been established using extended Hamilton’s Principle. The structural dynamics equations were derived using First-order Shear Deformation Theory (FSDT) and considering von Karman’s nonlinear strains. Also, the first-order Piston theory has been used to incorporate aerodynamic loads into aeroelastic equations. The material properties of the FG plate have been assumed to be temperature independent through the thickness and follow the linear rule of mixtures, and change according to the simple power-law distribution. Finally, the obtained equations have been discretized using Isogeometric Analysis (IGA). The natural frequencies and the flutter boundaries of the linear aeroelastic model have been obtained by eigenvalue analysis. Moreover, the post-flutter behavior plate of the FG plate has been conducted using Newton–Raphson and Newmark methods. The flutter and post-flutter behavior of the arbitrary shapes of the plate have been evaluated and compared with validated references. The effects of the shape and the boundary condition on such behaviors have been studied.

Bending of FGM sandwich plates with tunable auxetic core using DQM

Static bending of FGM sandwich plates with auxetic core is considered in the present work. The plate model is composed of FGM skins at top and bottom and auxetic core at the middle. Poisson's ratio of the core is tunable, which has different values for different angle parameters of the unit cell. It means the auxetic core can be metamaterials or traditional materials. The governing equations of the FGM sandwich plate with tunable auxetic core are derived using variational principle. Solution for the FGM sandwich plate is obtained using differential quadrature method (DQM). The effects of the geometry parameters, gradient index, parameters of the auxetic core and boundary conditions on the deflection and stresses are investigated.

Quantitative analysis of MEMS piezoresistive squared diaphragm pressure sensor for biomedical applications

Now a days MEMS based for biomedical applications with high performance parameters like sensitivity and linearity are required for low pressure range.The low-pressure range (1–5 kPa) is a significant range that comprises intra-body pressures like intracranial pressure (ICP) and intraocular pressure (IOP).A lot of work on piezoresistive pressure sensor with different shapes diaphragms like squared, circular and rectangular has been carried out.In MEMS Piezoresistive Pressure sensors the most important pressure-sensitive component is the squared diaphragm. The performance of the Micro-sensor is directly determined, under uniform applied pressure with its deformation. In pressure sensor design, the accurate relationship between pressure and deformation plays a very consequence role. In this paper, squared diaphragm of silicon material is modeled and simulate using the analytical equations for a thin plate, along with the theory of small-deflection to providing quick insight regarding important parameter like stress and Deformation for the operating pressure range of 0 kPa–5 kPa. The mechanical behavior of micro-machined silicon diaphragm in terms of stress and deflection generation has been simulated.The piezoresistivity principle is used to assess the sensor's sensitivity. The mechanical change in the thin diaphragm in the form of deflection and stress is calculated and presented. In this Modeling both the analytical and simulated(FEA) results for the performance parameters such as stress, Deflection and the output voltage is computed and compared, it is found that the error between analytical and simulation results is no more than 5.6–6.5%.

Numerical study on shear interaction between the track plate of deep-sea mining vehicle and the seafloor sediment based on CEL method

Deep-sea mining vehicles can easily slip when walking on the surface of thin and soft sediments, so it is necessary to study the shear interaction between vehicles and sediments. The track plate is an important part of the travelling mechanism, thus, the study of the interaction characteristics of track plates and sediments can improve the traction force and the travelling performance of mining vehicles. In this study, the effects of various parameters on the shear interaction between the track plate and the seafloor sediment were studied. Firstly, a quasi-static equation of the single track plate–sediment interaction was established based on elastoplastic mechanics. Then, a numerical model of the track plate–sediment interaction was established by using the CEL method. Based on the model, the simulation results under different track plate structures were studied, and a comparison to the quasi-static equation was obtained. It is found that the calculated difference between the two methods is small, but the numerical simulation has wider applicability. Thus, the influence of different parameters which consisted of different velocities, numbers of grousers, and sediment layers was evaluated through CEL method, and the sediment thrust, stress change, disturbance were obtained.

Buckling Analysis of a Composite Honeycomb Reinforced Sandwich Embedded with Viscoelastic Damping Material

In this study, the buckling loads of a composite sandwich structure, which is reinforced by a honeycomb layer and filled with viscoelastic damping material, are analyzed. By applying von Karman anisotropic plate equations for large deflection, the governing equation of the composite sandwich structure is determined, and the deflection of the structure is further defined by a double triangular series. According to the dynamic equivalent effective stiffness obtained by the homogenous asymptotic method and Hill’s generalized self-consistent model based on the Halpin–Tsai model, limiting the dynamic load buckling of the composite honeycomb reinforced sandwich structure embedded with viscoelastic damping material under axial compression can be achieved. The factors that influence the composite sandwich’s buckling loads are discussed and compared, such as the load and geometry parameters, the thickness of the honeycomb reinforcement layer and the honeycomb’s width. Finally, the results obtained by the present method are validated by the existing literature.

Hydrodynamic analysis of the magnetic field dependent viscous fluid flow and thermosolutal convection between rotating channels

According to research, exposing a person to a magnetic field enhances blood flow and minimizes their risk of suffering a heart attack. Ferrohydrodynamics is the study of fluid motion mechanics that is affected by strong magnetic polarisation forces (FHD). Ferrofluids may transmit heat in a variety of ways by using magnetic fluids. This behaviour is demonstrated by liquid-cooled speakers, which utilise less ferrofluid to prevent heat from reaching the speaker coil. This modification boosts the coil’s ability to expand, which enables the loudspeaker to create high-fidelity sound. It is investigated how the fluid dynamics of spinning, squeezing plates are affected by thermosolutal convection and a magnetic field dependent (MFD) viscosity. Standard differential equations are used to represent the equations of the modified form of Navier Stokes, Maxwell’s, and thermosolutal convection. The magnetic field, modified velocity field equations, and thermosolutal convection equations all yield suitable answers. Additionally computed and thoroughly detailed are the MHD torque and fluid pressure that are imparted to the top plate. To create a technique with quick and certain convergence, the resulting equations for uniform plates are solved using the Homotopy Analysis Method (HAM) with appropriate starting estimates and auxiliary parameters. The validity and reliability of the HAM outcomes are shown by comparing the HAM solutions with the BVP4c numerical solver programme. It has been found that a magnetic Reynolds number lowers the temperature of the fluid as well as the tangential and axial components of the velocity field. Additionally, when the fluid’s MFD viscosity rises, the axial and azimuthal components of the magnetic field behave in opposition to one another. This study has applications in the development of new aircraft take-off gear, magnetorheological airbags for automobiles, heating and cooling systems, bio-prosthetics, and biosensor systems.

Some closed form series solutions to peridynamic plate equations

Peridynamics is a generalized continuum theory which is developed to account for long range internal force/moment interactions. Peridynamic equations of motion are of integro-differential type and only several closed-form analytical solutions are available for elementary structures. The aim of this paper is to derive analytical solutions to peridynamic Kirchhoff type (shear rigid) plates equations for both static and dynamic cases. Applying trigonometric series with respect to in-plane coordinates, solutions for the deflection function of a rectangular simply supported plate are derived. The coefficients in the series are presented in a closed analytical form such that the equations of motion are exactly satisfied. For the dynamic case the solution is derived applying the variable separation with respect to the time and the in-plane coordinates. Several numerical cases are presented to illustrate the derived solutions. Furthermore, results of peridynamic plate equations are compared against results obtained from the classical plate theory. A very good agreement between these two theories is obtained for the case of the small horizon sizes which shows the capability of the presented approach.

Asymptotic behavior for stochastic plate equations with memory in unbounded domains

Abstract In this paper, we investigate the dynamics of stochastic plate equations with memory in unbounded domains. More specifically, we obtain the uniform in time estimates for solutions of the problem. Based on the estimates above, we prove the existence and uniqueness of random attractors in unbounded domains. Finally, we show the upper semicontinuity of the attractors when stochastic perturbations approaches zero.