Thick Elastic Plate Research Articles

Mathematical study on the potential flow past a vertical submerged flexible plate of non-uniform thickness

The present work emphasizes on the mathematical study of two-dimensional potential fluid flow past a vertical flexible plate of non-uniform thickness submerged in a water domain of finite depth. The boundary value problem associated with the described model is solved by two mathematical techniques while considering Kirchhoff’s thin plate theory. The first method is based on a perturbation analysis involving a small parameter ϵ, while the second method directly employs the classical method of integration and Green’s integral theorem to the governing equations to obtain the solution. Both the techniques result into two different systems of integral equations which are solved numerically using appropriate approximations. It has been established numerically that for smaller thickness variations depicted by particular values of ϵ (ϵ≤0.001), the two methods yield the same numerical results for all the hydrodynamic quantities. The correctness of the present results is verified by comparing them graphically with the ones already existing in the literature. Effects of the non-uniform plate thickness on various hydrodynamic parameters are presented graphically by choosing various thickness profiles. Also, comparisons are made with the results of constant thickness elastic plate to distinguish its hydrodynamic behaviour from the one having non-uniform thickness.

Про вироджений випадок тривимірної задачі динаміки товстих пружних плит

Розвʼязано тривимірну задачу динаміки пружної мембрани довільного граничного контуру із заданим поверхнево розподіленим зовнішньо-силовим впливом на неї. Тривимірне поле пружних динамічних зміщень точок мембрани описується диференціальною моделлю, яка є виродженим випадком раніше побудованих рівнянь просторової теорії пружності стосовно товстих пружних плит. Припускається, що останні доповнені інформацією про початково-граничний стан мембрани, яка у загальному випадку має не силовий характер, а її кількості недостатньо для математично коректної постановки початково-крайової задачі. Розвʼязання цих неповно визначених початково-крайових задач динаміки мембрани стало можливим після математичного моделювання наявних просторово-динамічних спостережень за початковим станом та поточною динамікою її контуру системою дискретно і неперервно визначених моделюючих функцій. Побудоване в роботі поле пружно-динамічних зміщень точок мембрани, будучи точним розвʼязком її диференціальної математичної моделі, з наявною інформацією про початково-граничний стан останньої узгоджується за середньоквадратичним критерієм. У роботі оцінено точність такого моделювання та сформульовано умови його однозначності. Розглянуто випадки чисто поперечних та чисто площинних коливань мембрани. Чисельна реалізація отриманого розвʼязку розглядуваних задач, виконана навіть при мінімальній кількості спостережень за станом мембрани, підтверджує простоту і правильність підходу до вивчення цих математично складних задач еластодинаміки товстих пружних плит.

Комп’ютерне рішення для знаходження поперечних динамічних зміщень товстих пружних плит в умовах невизначеності

The article poses the task of finding the function of dynamic transverse displacements of thick elastic plates under the action of axisymmetric external dynamic disturbances acting on the surface of such plates. When setting the problem, it is indicated that some initial-boundary observations are known for the state of the plate, but their number is insufficient to solve this problem by classical methods of mathematical physics or numerical integration. Therefore, it is possible to apply to this problem the method of mathematical modeling of externally distributed dynamic processes under conditions of uncertainty, i.e. under insufficient number of initial and boundary conditions. For such a method, it is necessary to have an integral mathematical model and its main components - kernels, that is, integral functions for which there is a finding method. Or they can be calculated using numerical methods of computer algebra systems. With the built integral model of the dynamics of thick elastic plates, taking into account its kernels, the method of mathematical modeling of the dynamics of spatially distributed processes leads to the result - a set of solutions that accurately satisfy the integral and differential models and agree with the initial boundary conditions according to a certain criterion. The article selects one of the many solutions to the problem of finding the function of transverse dynamic displacements, which is found according to the methodology of mathematical modeling of the dynamics of spatially distributed processes and thanks to the calculation of the integral function of the mathematical integral model. For the problem, the elastic characteristics and density of the slab corresponding to some material are fixed, some specific external dynamic disturbances and initial-at-the-edge observations are determined, which are represented by certain conditions at specific points. Under such assumptions, graphs of the functions of transverse dynamic displacements are constructed at different values of the transverse coordinate z and at the value 0 of the radial coordinate r of the cylindrical coordinate system.

Transient response of a thermo-diffusive elastic thick circular plate with variable conductivity and diffusivity

Transient response of a thermo-diffusive elastic thick circular plate with variable conductivity and diffusivity

О трехмерных начально-краевых задачах динамики толстых упругих плит

Решены сложные задачи трехмерной теории упругости для толстых упругих плит с произвольной геометрией их боковой поверхности. Построены аналитические зависимости компонент поля упругих динамических смещений внутренних точек плиты от предельно-поверхностных внешнединамических возмущающих факторов, определенных непрерывными функциями или векторами их значений. Предполагается, что эти возмущения имеют классически определенный силовой характер или задаются определенным количеством дифференциальных преобразований компонент поля динамических смещений точек плиты. Отсутствие количественных и качественных ограничений на названные преобразования первоначально-краевые задачи динамики рассматриваемых плит делает некорректными и неразрешимыми методами классической и вычислительной математики. предложена методика среднеквадратичного математического моделирования дискретно и непрерывно заданных наблюдений за начально-краевым состоянием плиты системой моделирующих функций и векторов их значений. Построенные при этом компоненты поля пространственно-динамических смещений точек плиты, точно удовлетворяя классическим уравнениям Ляме, с имеющейся информацией о начально-краевом состоянии ее согласовываются по среднеквадратическому критерию. Исследован вопрос однозначности полученных решений, проведена оценка их точности относительно информации о внешне-динамическом состоянии исследуемой плиты. Развернута динамика плиты в установившемся режиме, для случаев отсутствия информации о внешне-динамических воздействиях на нее и в условиях ее геометрической вырожденности по пространственным координатам. Компьютерная реализация полученных математических результатов инженерно проста и может быть легко реализована известными методами вычислительной математики.

Hydroelastic response of floating elastic plate in the presence of vertical porous barriers

ABSTRACT The attenuation of the incident wave interacting with a very large floating structure (VLFS) in the presence of vertical barriers is analysed considering small amplitude wave theory. The VLFS is considered to be articulated and is modelled based on Timoshenko-Mindlin plate theory. The eigenfunction expansion method along with the orthogonal mode-coupling relation is employed for the case of finite water depth. The numerical study is performed to analyse the wave reflection, transmission and dissipation characteristics due to the articulated floating plate for the case of bottom standing and surface piercing vertical porous barriers. The hydroelastic behaviour in terms of deflection and strain for an articulated floating thick elastic plate in the presence of porous barriers is analysed. The study reveals that the magnitude of wave attenuation is enhanced due to the presence of vertical porous barriers and also provides an understanding in mitigating the structural response.

Development of a track dynamics model using Mindlin plate theory and its application to coupled vehicle-floating slab track systems

Unlike track slabs in high speed railways, floating slabs in metro lines are usually relatively thicker and their thickness can often reach 0.5 m or more. When the thickness of the track slab is relatively large compared to the length and width, the influence of shear effect and rotatory inertia will become significant and it is more reliable to regard the track slab as an elastic thick plate than adopt the classical thin plate model in the simulation. This paper presents a three-dimensional dynamic model for coupled vehicle-floating slab track (CVFST) systems on the basis of Mindlin plate theory for the first time. The floating slab is described as an elastic rectangle Mindlin plate with free boundary conditions resting on steel springs, the mode shapes of the Mindlin plate are approximated by series of products of Timoshenko beam functions, and the corresponding vibration equations are solved by Rayleigh-Ritz method in time domain. The vehicle subsystem and track subsystem are coupled via wheel-rail nonlinear interactions. The effectiveness of the proposed model is validated by comparing with the measured data from an impact test and numerical results in the previous literature. Influence of size effect on the slab natural frequencies and dynamic responses of the CVFST system are investigated, and discrepancies of the calculation results between the developed model and the traditional thin plate model are simultaneously discussed. Some practical conclusions are drawn and the developed model may serve as a potent tool for more accurate assessment of train-induced vibrations in metro lines.

Efficient Higher-Order Shear Deformation Theories for Instability Analysis of Plates Carrying a Mass Moving on an Elliptical Path

The dynamic performance of structures under traveling loads should be exactly analyzed to have a safe and reasonable structural design. Different higher-order shear deformation theories are proposed in this paper to analyze the dynamic stability of thick elastic plates carrying a moving mass. The displacement fields of different theories are chosen based upon variations along the thickness as cubic, sinusoidal, hyperbolic and exponential. The well-known Hamilton’s principle is utilized to derive equations of motion and then they are solved using the Galerkin method. The energy-rate method is used as a numerical method to calculate the boundary curves separating the stable and unstable regions in the moving mass parameters plane. Effects of the relative plate thickness, trajectories radii and the Winkler foundation stiffness on the system stability are examined. The results obtained in this research are compared, in a special case, with those of the Kirchhoff’s plate model for the validation.

Wave transformation due to floating thick elastic plate over multiple stepped bottom topography

The wave transformation due to floating elastic plate is studied over multiple stepped bottom topography. The behaviour of the flexible floating plate is studied considering Timoshenko-Mindlin’s plate thick theory at finite water depth and the edge conditions for the floating plates are considered to be free-free support condition. The mathematical model is developed based on the eigenfunction expansion method to study the hydroelastic behaviour of a floating thick elastic plate at varying water depths, step thickness and plate sizes acted upon by ocean wave. The mode coupling relation is employed along with orthogonality relation and the continuity equations for velocity, pressure, deflection, bending moment, slope and shear force to determine the wave characteristics in reflection and transmission region due to variable bottom topography. The numerical computation is carried out to obtain the hydroelastic behaviour of floating elastic plate acted upon by ocean waves. Further, detail comparison of the numerical results is performed for different step bottom topography for the hydroelastic analysis of floating thick elastic plate. The present study will provide an insight into the effect of seabed profile due to the wave scattering from a large floating thick elastic plate at finite water depth.

Influence of Support Conditions on the Hydroelastic Behaviour of Floating Thick Elastic Plate

The hydroelastic response of very large floating structures (VLFS) under the action of ocean waves is analysed considering the small amplitude wave theory. The very large floating structure is modelled as a floating thick elastic plate based on Timoshenko-Mindlin plate theory, and the analysis for the hydroelastic response is performed considering different edge boundary conditions. The numerical study is performed to analyse the wave reflection and transmission characteristics of the floating plate under the influence of different support conditions using eigenfunction expansion method along with the orthogonal mode-coupling relation in the case of finite water depth. Further, the analysis is extended for shallow water depth, and the continuity of energy and mass flux is applied along the edges of the plate to obtain the solution for the problem. The hydroelastic behaviour in terms of reflection and transmission coefficients, plate deflection, strain, bending moment and shear force of the floating thick elastic plate with support conditions is analysed and compared for finite and shallow water depth. The study reveals an interesting aspect in the analysis of thick floating elastic plate with support condition due to the presence of the rotary inertia and transverse shear deformation. The present study will be helpful for the design and analysis of the VLFS in the case of finite and shallow water depth.

Experimental observation of theoretically predicted spectrum of edge waves in a thick elastic plate with facets

The existence of the edge waves spectrum predicted in a series of recent theoretical studies is investigated and confirmed experimentally. Wave motion in 20 mm thickness aluminium rectangular plate with two facets is generated by a circular piezoelectric transducer and measured by laser Doppler vibrometry. Eight edge modes (four symmetric and four antisymmetric) are clearly observed by means of wavenumber-frequency analysis applied to out-of-plane velocities measured at the edge. The semi-analytical method used by the authors in previous theoretical studies is modified to take into account the facets. By the numerical investigations of edge waves in the plate with facets, the sensitivity of attenuation of higher order edge waves to the form of edge profile is revealed. A good agreement between experimental data and theoretical predictions is demonstrated for dispersion curves and the waveforms of individual modes extracted by means of 2D bandpass filters.

Evidence for complete low-frequency vibration band gaps in a thick elastic steel metamaterial plate

Elastic steel metamaterial plates can be used for noise- and vibration-reduction due to unique physical properties related to their vibration band gap. However, obtaining a complete low-frequency vibration band gap in a thick elastic steel metamaterial plate is difficult. In this paper, we simulate a complete low-frequency vibration band gap in a thick elastic steel metamaterial plate. The structure consists of periodic, double-sided, composite stepped resonators, which were deposited on a 2D locally resonant phononic crystal plate. The phononic crystal plate consists of an array of rubber fillers embedded in a thick steel plate. The dispersion relations, power-transmission spectra, and the displacement fields of the eigenmodes are calculated using the finite-element method. The results show that, for the proposed structure, the opening of the first complete vibration band gap is reduced by a factor of 9.5 compared to a conventional thick elastic steel metamaterial plate. This causes attenuation of low-frequency elastic waves. The formation mechanisms for the vibration band gap are also explored numerically. The results indicate that the formation mechanism for the new low-frequency vibration band gap can be attributed to coupling between a local resonance mode of the composite stepped resonators and the Lamb wave mode of the thick steel-plate. The location of the vibration band gap is determined by the resonator mode of the composite stepped resonators. The vibration band gap effects of the composite stepped resonators are also investigated in this paper. We find that the location of the complete vibration band gaps can be modulated with a relatively low frequency using different composite stepped resonators. Such an elastic steel metamaterial plate with a complete low-frequency vibration band gap can be used to reduce both vibration and noise in various commercial and research applications.

Experimental and theoretical investigation of transient edge waves excited by a piezoelectric transducer bonded to the edge of a thick elastic plate

The paper presents experimental and theoretical studies of wave phenomena accompanying transient edge waves (EW) excitation by a piezoelectric transducer attached at the edge of an elastic plate. Theoretical investigations are conducted on the basis of the three-dimensional elastodynamic theory. The boundary value problem describing non-stationary wave motion is solved with the use of integral transforms and the modal expansion technique. The asymptotic analysis with the wavenumber considered as a small or large parameter is applied to the three-dimensional problem. In the first case (long-wave vibrations) the approximate formulae for the fundamental EW's pole and corresponding residue are derived. These relations take into account the influence of the transverse shear load arising because of coupling between long-wave integral and Saint-Venant's boundary layer. In the second case (short-wave vibrations) an infinite series of poles corresponding to higher order EW is revealed. Results of numerical investigations of EW dispersion properties and waveforms are presented and used for an analysis of experimental data acquired with the use of Laser Doppler vibrometry. It is shown that the contribution of EW is predominant in the wave-field excited by the load under consideration. A good agreement between theoretical predictions and measurements is demonstrated. For the calculation of transient wave-field three different models of actuator-plate interaction are developed: in the low-frequency range the interaction is considered as a static one, while the lower and higher eigenfrequencies of the actuator are taken into account in the medium- and high-frequency ranges respectively. For the low-frequency range, an explicit analytical formula is derived for calculation of EW contribution into the transient wave-field. In the high-frequency range, the excitation of the first higher order EW is predicted by the numerical solution and observed experimentally.

Closed-form solutions for predicting the thick elastic plate behavior of CLT and timber panels with gaps

In a former paper by the authors (Franzoni et al., 2017), the elastic behavior of Cross Laminated Timber (CLT) and timber panels having periodic gaps between lateral lamellae has been analyzed. A thick plate homogenization scheme based on Finite Elements computations has been applied. The predicted behavior was in agreement with experimental results. In this paper, simplified closed-form solutions are derived in order to avoid FE modeling. Both cases of narrow gaps of CLT panels and wide gaps of innovative lightweight panels are investigated. CLT and timber panels with gaps are modeled as a space frame of beams connected with wooden blocks. The contribution of both beams and blocks to the panel’s mechanical response is taken into account, leading to closed-form expressions for predicting the panel’s stiffnesses and maximum longitudinal and rolling shear stresses. The derived closed-form solutions are in agreement with the reference FE results and they can be used for practical design purposes.

Three-Dimensional Integral Mathematical Models of the Dynamics of Thick Elastic Plates

The complex of problems related to constructing three-dimensional field of elastic dynamic displacements of flat elastic plate with arbitrary boundary-edge surface is solved. It is assumed that boundary condition of the plate is given in terms of powerful perturbation factors or displacement vector function. Problems solutions are based on classical Lame equations of spatial theory of elasticity under root-mean-square consistency of the solution with corresponding external-dynamic observations of the plate. The accuracy of such consistency is estimated. The uniqueness conditions for the solution of the considered problems are formulated.

THE REFINED THEORY OF ELASTIC PLATES AS ASYMPTOTIC APPROACH OF 3D PROBLEM

The refined theory of elastic thin and thick plates is constructed by the asymptotic method for reducing three-dimensional (3D) equations of linear elasticity to two-dimensional ones without the use of any assumptions. The resulting refined theory is much more complicated than the known classical Kirchhoff theory: the required values of the refined theory vary in thickness of the plate by more complex laws; the system of partial differential equations of the refined theory has a higher order than the system of equations of the classical theory. A comparison of the obtained theory with the popular refined theory of Timoshenko and E. Reissner, taking into account the transverse shear deformation is made. It is shown that the inclusion only of the transverse shear deformation is insufficient. In addition to the transverse shear deformation, many additional terms having the same order as the transverse shear deformation must be taken into account.

Free vibration analysis of thick viscoelastic composite plates on visco-Pasternak foundation using higher-order theory

Free vibration of laminated viscoelastic composite plates on Pasternak viscoelastic medium with simply supported edge conditions is investigated. The composite plate consists of linear viscoelastic polymeric matrix reinforced by transversely isotropic elastic fibers. Multi-cell micromechanical approach together with relaxation functions of bulk/shear moduli and Boltzmann superposition principle are adopted to establish time-dependent stiffness coefficients of laminates. The coupled integro-PDEs of motions are derived based on third-order shear deformation theory by Laplace transformation and Hamilton principle. Complex frequencies and closed form solutions for transient response are achieved by weighted residual method, iterative numerical algorithm and Fourier transform. To verify, the results are compared for thick elastic composite plates on Pasternak elastic foundation and thick viscoelastic composite sandwich plates without foundation that represent acceptable accuracy. By parametric study, effects of materials, lamination scheme, geometry and medium on dynamic characteristics are scrutinized.

Hydroelastic analysis of floating elastic thick plate in shallow water depth

Summary In the present study, the wave scattering due to very large floating structure (VLFS) is analysed considering the thick plate theory in shallow water depth. The effect of free-free boundary condition associated in the analysis of the floating thick plate in shallow water is analysed. The numerical computation is performed to analyse the wave reflection, wave transmission and plate deflection due to the action of incident wave. Further, the results are compared between thick plate theory and Kirchhoff's plate theory. The study reveals an interesting aspect of considering rotary inertia and transverse shear deformation in the analysis of floating elastic thick plate. The present study provides an understanding on design and analysis of VLFS based on thick plate theory in shallow water depth.

Methods of Pseudo-Inverse Algebra in State Identification Problems for Thick Elastic Plates

The authors solve identification problems for the three-dimensional function of transverse displacements of points of a thick elastic plate of finite dimensions, whose dynamics can be described by a linear differential model with discrete and continuous observations of the initial---boundary-value external dynamic perturbations of the plate. The working criterion is the RMS criterion of fitting the obtained solution and observations of the plate and methods of linear pseudo-inverse algebra. The conditions of accuracy and uniqueness of the identification results are formulated.

The time-harmonic antiplane elastic response of a constrained layer

Rib-reinforced structures are commonplace in many areas of engineering, especially in aircraft and marine applications, where the ribs or bulkheads are used to stiffen the exterior fuselage or hull. The wavelengths of vibrational modes of interest in such thin structures are usually long in comparison with the rib widths. However, it is found that the location and precise nature of the rib constraints (e.g. the size and imposed boundary conditions) have a significant effect on the overall vibrational properties of the body. Thus, it is essential that such stiffeners are modelled with sufficient accuracy when considering acoustical or vibrational aspects of the structure.The purpose of this paper is to examine an idealized model of an elastic structure that is simple enough to be amenable to analytical investigation yet offers a methodology that can be applied in more complex situations. A thick elastic plate, forced or constrained at a finite number of (parallel) discrete lines on one of its faces, is studied. Each constraint demands that the displacement vanishes over a small but finite-sized region of the surface, and asymptotic analysis is employed to determine the vibration field (consisting in this model of out-of-plane shear waves) both globally, away from the constraints, and in their vicinity. Analysis indicates that it is essential to model the constraint boundary condition accurately. However, perhaps surprisingly, this result is found to be asymptotically equivalent to taking a rib of vanishing thickness, but with the zero displacement condition applied at a single point, suitably shifted along the surface. It is noted that for point rather than distributed constraints, a discrete Fourier transform solution returns a finite, but erroneous, value at the constraint location. This value is not associated with any physical field, but instead depends only on the truncation employed. Finally, the effect of a number of constraints, and their location, on the propagation of shear waves, is investigated.