Inhomogeneous Plate Research Articles

Hydroelastic analysis of Very Large Floating Structures in variable bathymetry regions by multi-modal expansions and FEM

A novel frequency domain numerical method for Very Large Floating Structure (VLFS) hydroelasticity is developed. The problem is formulated in the 2D ocean waveguide, featuring a realistic seabed bathymetry and the presence of inhomogeneous, elastic plates of varying thickness and negligible draft. An in vacuo modal expansion for the elastic body deflection, modelled as a structural plate, is employed to decouple the hydrodynamics from structural mechanics. The inhomogeneous plate is considered to undergo cylindrical bending, while depending on the structure slenderness and excitation wavelength the classical thin plate theory and Mindlin’s model, accounting for first order shear deformation effects are implemented. A weighted residual approach is employed to cast the formulated problems into a mixed weak form for which dimensionality reduction is sought. This is achieved by an enhanced vertical representation for the wave potential, able to accurately account for abrupt bathymetric changes, following Athanassoulis and Belibassakis (1999). The reduced two-field, weak problem is solved by means of the Finite Element Method (FEM). Finally, a series of comparisons are carried out against published results for a range of configurations.

Transformation Cloaking in Elastic Plates

In this paper we formulate the problem of elastodynamic transformation cloaking for Kirchhoff-Love plates and elastic plates with both the in-plane and out-of-plane displacements. A cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic plate (virtual problem) to that of an anisotropic and inhomogeneous elastic plate with a hole surrounded by a cloak that is to be designed (physical problem). For Kirchhoff-Love plates, the (out-of-plane) governing equations of the virtual plate is transformed to those of the physical plate up to an unknown scalar field. In doing so, one finds the initial stress and the initial tangential body force for the physical plate, along with a set of constraints that we call cloaking compatibility equations. These constraints involve the cloaking transformation, the unknown scalar field, and the elastic constants of the virtual plate. It is noted that the cloaking map needs to satisfy certain conditions on the outer boundary of the cloak and the surface of the hole. In particular, the cloaking map needs to fix the outer boundary of the cloak up to the third order. Assuming a generic radial cloaking map, we show that cloaking a circular hole in Kirchoff-Love plates is not possible; the cloaking compatibility equations and the boundary conditions that the cloaking map needs to satisfy are the obstruction to cloaking. Next, relaxing the pure bending assumption, the transformation cloaking problem of an elastic plate in the presence of in-plane and out-of-plane displacements is formulated. In this case, there are two sets of governing equations that need to be simultaneously transformed under the cloaking map. We show that cloaking a circular hole is not possible for a general radial cloaking map; the cloaking compatibility equations and the boundary conditions that the cloaking map needs to satisfy obstruct cloaking.

On non-axisymmetric buckling modes of inhomogeneous circular plates

Unsymmetrical buckling of nonuniform circular plates with elastically restrained edge and subjected to normal pressure is studied in this paper. The asymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value at which waves in the circumferential direction can appear. The effect of material heterogeneity and boundary on the buckling load is examined. For a plate with elastically restrained edge, the buckling pressure and mode number increase with a rise of spring stiffness. Increasing of the elasticity modulus to the plate edge leads to increasing of the buckling pressure, but the mode number does not change. If the translational flexibility coefficient is small, decreasing of the elasticity modulus to the shell (plate) edge leads to sufficient lowering of the buckling pressure.

Natural frequencies of an inhomogeneous square thin plate

Plates, which geometric and physical parameters slightly differ from constant and depend only on the radial coordinate, are analyzed. For free vibration frequencies of a plate, which thickness and/or Young’s modulus depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. As examples, free vibrations of a square plate with parameters linearly or parabolically depend on the radial coordinate, are examined. The double frequencies of square plates with similar edge support of all edges are of special interest, since any variation of the thickness or stiffness causes some loss of symmetry one may expect the split of double frequencies. The asymptotic formulas permit to determine, which of two equal unperturbed frequencies corresponding to wave numbers n and m increases faster with the small parameter. For a wide range of small parameter values, the asymptotic results for the lower vibration frequencies well agree with the results of finite element analysis with COMSOL Multiphysics 5.4.

Моделирование критического состояния слоистых сверхпроводящих структур с неоднородными слоями

An approach for calculating the critical state of layered structures consisting of inhomogeneous superconducting layers is proposed. The method is based on the numerical solution of the one-dimensional Ginzburg-Landau equations, generalized for an inhomogeneous plate. The method allows one to obtain the dependence of the critical current on the magnetic field, as well as the distribution of the current and magnetic field over the layers. A comparison is made of the averaged critical current of layered structures consisting of both inhomogeneous and homogeneous layers. It was found that with a relatively small number of layers and a small external magnetic field, the critical current of layered structures with homogeneous layers can exceed the critical current of structures with inhomogeneous layers. With an increase in the number of layers and / or the value of the external magnetic field, the critical current of layered structures with inhomogeneous layers begins, on the contrary, to exceed the critical current of structures with homogeneous layers. The pinning force in structures with inhomogeneous layers is higher than in the case of homogeneous ones.

A MODEL OF A LINEAR VISCOELASTIC ANISOTROPIC PLATE, CONSIDERING AN ARBITRARY INHOMOGENEITY OF THE MATERIAL

The paper derives a model of an arbitrary inhomogeneous, anisotropic thin plate with linear viscoelastic components under a slowly varying in time loading; the components of a plate are also supposed anisotropic. The displacements of an inhomogeneous plate are developed following the integral formula approach of composite mechanics, by establishing an interrelation between Laplace transformants of displacement fields of the inhomogeneous plate and a homogeneous one with the corresponding type of anisotropy, so-called concomitant. The interrelation comes from an integral formula approximation by the structural functions technique; in this paper, the first-order approximation of the integral formula is considered. A system of partial differential equations for the first-order structural functions is adjusted for the case of linear viscoelastic components, both in the Laplace and time domain. For a concomitant body, Timoshenko plate equations are generalized onto the anisotropic case. An explicit form of displacements and strains of an inhomogeneous plate is obtained. An example−an instantly applied bending of a simply supported plate, composed of isotropic, asymmetrically laid layers−is considered.

Ginzburg–Landau Calculations for Inhomogeneous Superconducting Films

A method for modeling the superconducting state of an inhomogeneous film is proposed. The superconducting properties are changed by varying Ginzburg–Landau (GL) parameters over the film thickness. A system of 1-D equations was found using the standard minimization of the modified GL-free energy functional to describe the superconducting state of the inhomogeneous plate. Using the proposed method, the magnetic field dependencies of the critical current were calculated and compared with each other for the cases of inhomogeneous and homogeneous distributions of the superconducting properties over the film thickness. It was shown that the critical current in the zero magnetic field decreases and the critical magnetic field increases significantly when the degree of inhomogeneity of the film increases. Analysis of the calculated bulk pinning force as a function of the external magnetic field shows a shift of the maximum of this value toward stronger fields with increasing degree of inhomogeneity. It was detected the sharper change in properties at the boundaries of the plate and, consequently, the greater level of homogeneity in the middle, the critical current becomes noticeable larger. At the same time, the critical magnetic field changes slightly in this case.

Impact of Manufacturing Inaccuracies on the Acoustic Performance of Sound Insulation Packages with Plate-Like Acoustic Metamaterials

<div class="section abstract"><div class="htmlview paragraph">Thin plate-like metamaterials (e.g. membrane-type acoustic metamaterials or inhomogeneous plates) have a high potential for improving the sound transmission loss of sound insulation packages, especially in the challenging low-frequency regime. These types of metamaterials have been previously shown to achieve very high sound transmission loss values which can exceed the corresponding mass-law values considerably. However, like many other metamaterial realizations, their extraordinary acoustical performance relies on the periodicity of the sub-wavelength sized unit cells. In particular, for plate-like acoustic metamaterials (PAM) most theoretical and numerical investigations assume a perfect placement of equal added masses - an idealization which cannot be achieved in industrial manufacturing of these metamaterials. This contribution investigates the impact of randomized inaccuracies that can occur in manufacturing on the sound reduction behavior of PAM. First, a numerical model of a finite sized PAM is validated using sound transmission loss measurement data. Then, the simulation model is modified to incorporate randomly varying geometrical parameters of the added masses (such as mass placement, size, etc.). The parameters are randomized according to a Gaussian distribution and the impact of different standard deviations on the sound transmission loss is analyzed. The results indicate that while a random variation of the mass parameters does have an effect on the acoustic performance of the investigated PAM, it becomes significant only for relatively large standard deviations.</div></div>

Numerical‐analytical method for determining the dielectric constant of 1D inhomogeneous plate in a waveguide

The inverse diffraction problem is considered for a layered dielectric plate in a metal waveguide of arbitrary cross-section. The proposed solution is based on the Cauchy problem for the Riccati equation. Based on this solution, algorithm and code are developed for the dielectric permittivity profile reconstruction. The measured and mathematically simulated values of the complex reflection coefficient at finite number of frequencies are used as input data. Reconstructed dielectric permittivity profiles for layered plates are in good agreement with measured ones.

Effects of thermal stress, magnetic field and rotation on the dispersion of elastic waves in an inhomogeneous five-layered plate with alternating components.

This study explores the effects of the thermal stress, rotation, and magnetic fields on the propagation of surface waves in a symmetric five-layered inhomogeneous elastic plate. The harmonic wave solution was adopted to reduce the given partial differential equations to their corresponding ordinary differential equations, which were later solved using a classical method. The Mathematica software was also employed for the numerical simulations. The dispersion relations associated with both the symmetric and antisymmetric modes have been determined and analyzed with regard to the aforementioned external effects and rotation. The variations in thermal stress positively affect the wave propagation in the plate even for relatively smaller values of the temperature variation. Also, the same trend is observed in the case of the magnetic field, but for higher values. Finally, for the rotation, it was found that the angular velocity must reach at least 104 before a significant change is observed; which in fact differs from the situations of half-plane reflection and single-layered.

Free Vibration Frequencies of a Circular Thin Plate with Variable Parameters

We study transverse vibrations of an inhomogeneous circular thin plate in this work. Using the perturbation method, asymptotic formulas are obtained for free-vibration frequencies of a plate whose thickness and Young’s modulus linearly depend on radius. The effect of the boundary conditions on frequencies and the behavior of frequencies for a plate with the fixed mass are analyzed. For lower frequencies of the plate, the asymptotic results are compared with the results of analysis by finite elements.

Forced vibrations of functionally graded annular and circular plates by domain‐boundary element method

AbstractAxi‐symmetric dynamic response of functionally graded circular and annular Mindlin plates with through‐the‐thickness variations of physical properties is investigated by a new domain‐boundary element formulation. Three governing partial differential equations of motion of the inhomogeneous plate are converted to integral equations by utilizing the static fundamental solutions of the displacement components. These integral equations are then spatially discretized by dividing the entire domain into a number of cells and incorporating suitable shape functions to approximate the variation of unknown parameters across the cells. The resulting set of ordinary differential equations in time are then solved by the Houbolt method. To verify the accuracy of the developed procedure, dynamic responses generated for a homogeneous plate are compared to those calculated through an analytical solution. Different loading conditions such as step, harmonic, and impulsive loadings are considered for generation of numerical results of dynamic responses of functionally graded circular and annular plates. Convergence characteristics and effects of material inhomogeneity, and geometric parameters are extensively investigated. It has been shown that the presented method is a fast and accurate and technique in elastodynamic analysis of functionally graded plates.

Asymptotic analysis of an anti-plane shear dispersion of an elastic five-layered structure amidst contrasting properties

The present paper studies the anti-plane shear motion of an inhomogeneous elastic five-layered plate amidst the four contrasting material setups. The asymptotic analysis method in regard to the various material parameters is adopted for the study. The respective displacements, stresses and the Rayleigh-Lamb dispersion relation corresponding to the antisymmetric anti-plane motion with perfect interlayer and traction-free (on the outer faces) boundary conditions are determined. Furthermore, in order to analyze the said dispersion relation in the presence of these contrasts, a unification of parameters was proposed. The overall cutoff frequencies and the low-frequency estimates are determined for both the generalized and unified settings. A comparative analysis between the unified Rayleigh-Lamb dispersion relation and the optimal shortened polynomial dispersion relation is carried for each contrast. We also established some asymptotic formulae for the related unified displacements and stresses.

Investigating the viscous damping effects on the propagation of Rayleigh waves in a three-layered inhomogeneous plate

The present paper investigates the propagation of anti-plane Rayleigh waves in an elastic three-layered inhomogeneous plate in the presence of viscous damping. The core layer is sandwiched between skin layers with appropriate perfect interfacial boundary conditions, while traction-free conditions are prescribed on the outer faces. The eigenfunctions expansion method is employed for the study. Various cases of damped and undamped situations have been examined in connection to the obtained displacements and shear stresses. Finally, the effects of viscous damping on the propagation of Rayleigh waves are reported graphically using some physical available data.

Casimir energy due to inhomogeneous thin plates

We study the Casimir energy due to a quantum real scalar field coupled to two planar, infinite, zero-width, parallel mirrors with non-homogeneous properties. These properties are represented, in the model we use, by scalar functions defined on each mirror's plane. Using the Gelfand-Yaglom's theorem, we construct a Lifshitz-like formula for the Casimir energy of such a system. Then we use it to evaluate the energy perturbatively, for the case of almost constant scalar functions, and also implementing a Derivative Expansion, under the assumption that the spatial dependence of the properties is sufficiently smooth. We point out that, in some particular cases, the Casimir interaction energy for non-planar perfect mirrors can be reproduced by inhomogeneities on planar mirrors.

Some Features of Critical Parameters Calculations of Nonhomogeneous Superconducting Films Using the Ginzburg–Landau Theory

A method for calculating the critical state of inhomogeneous superconducting films using the Ginzburg–Landau (GL) theory is proposed. From the condition of minimizing the functional of the GL free energy, a system of equations describing the superconducting state of an inhomogeneous plate is derived. The dependences of the critical current of the plate on the external magnetic field are numerically calculated. It is shown that the critical current in a zero external magnetic field decreases with increasing degree of film inhomogeneity, while the critical magnetic field (at a zero transport current) increases. An analysis of the calculated dependences of the product of the critical current and the external magnetic field (an analogue of the volume pinning force) of superconducting films on the external magnetic field shows that the maximum of this value shifts to stronger fields with increasing degree of inhomogeneity, i.e., an increase in inhomogeneity leads to an increase in the pinning force.

Free vibrations of anisotropic rectangular plate laying on a heterogeneous viscouselastic basis

The aim of the work. Free, transverse vibrations are considered heterogeneous along the three spatial coordinates of rectangular plates lying on an inhomogeneous viscoelastic base. It is assumed that the boundary conditions are homogeneous. A closed solution for the problem of free vibration of an inhomogeneous rectangular orthotropic plate based on an inhomogeneous viscoelastic foundation is developed in the article. Young's moduli and the density of the orthotropic plate continuously change with respect to three spatial coordinates, while the characteristics of a viscoelastic base change depending on the coordinates in the plane. Methods. The corresponding equation of motion is obtained using the classical theory of plates. The solution to the problem was constructed using the method of separation of variables and the Bubnov - Galerkin method. Results. Explicit formulas of the fundamental tone of the frequency of the transverse vibration of an anisotropic plate lying on an inhomogeneous viscoelastic base are determined. The influence of heterogeneity of orthotropic materials, viscosity inhomogeneities, inelastic and elastic substrates at dimensionless plate frequencies have been studied in detail.

Прохождение звука через упругую пластину с неоднородным покрытием, граничащую с вязкими жидкостями

В статье рассматривается задача об отражении и прохождении плоской звуковой волнычерез однородную упругую пластину с непрерывно-неоднородным упругим покрытием,граничащую с вязкими жидкостями. Полагается, что законы неоднородности покрытияпластины описываются дифференцируемыми функциями.Распространение малых возмущений в вязкой жидкости в случае установившихся коле-баний описывается скалярным и векторным уравнениями Гельмгольца. Распространениеупругих волн в однородной изотропной упругой пластине описывается двумя волновымиуравнениями для продольных и поперечных волн. Колебания неоднородного изотропногоупругого покрытия описываются общими уравнениями движения сплошной среды.Для нахождения поля смещений в неоднородном покрытии построена краевая задачадля системы обыкновенных дифференциальных уравнений второго порядка.Получено аналитическое описание отраженного и прошедшего через пластину акусти-ческих полей.Представлены результаты численных расчетов зависимостей коэффициентов отраже-ния и прохождения продольных волн от угла падения плоской волны.

Infinite Kirchhoff plate on a compact elastic foundation may have arbitrary small eigenvalue

An inhomogeneous Kirhhoff plate composed from semi-infinite strip-waveguide and a compaсt resonator which is in contact with the Winkler foundation of small compliance, is considered. It is shown that for any 0, it is possible to find the compliance coefficient O(2) such that the described plate possesses the eigenvalue 4embedded into continuous spectrum. This result is quite surprising because in an acoustic waveguide (the spectral Neumann problem for the Laplace operator) a small eigenvalue does not exist for any unsubstantial perturbation. A reason of this dissension is explained as well.

Estimating the material parameters of an inhomogeneous poroelastic plate from ultrasonic measurements in water.

The estimation of poroelastic material parameters based on ultrasound measurements is considered. The acoustical characterisation of poroelastic materials based on various measurements is typically carried out by minimising a cost functional of model residuals, such as the least squares functional. With a limited number of unknown parameters, least squares type approaches can provide both reliable parameter and error estimates. With an increasing number of parameters, both the least squares parameter estimates and, in particular, the error estimates often become unreliable. In this paper, the estimation of the material parameters of an inhomogeneous poroelastic (Biot) plate in the Bayesian framework for inverse problems is considered. Reflection and transmission measurements are performed and 11 poroelastic parameters, as well as 4 measurement setup-related nuisance parameters, are estimated. A Markov chain Monte Carlo algorithm is employed for the computational inference to assess the actual uncertainty of the estimated parameters. The results suggest that the proposed approach for poroelastic material characterisation can reveal the heterogeneities in the object, and yield reliable parameter and uncertainty estimates.