Elastic Plate Research Articles

Research on the optimal excitation threshold interval of targeted energy transfer for piezoelectric nonlinear energy sink

In this paper, in order to design a piezoelectric nonlinear energy sink system (NES), a coupling system of elastic plate and piezoelectric nonlinear energy sink is established. The nonlinear stiffness and damping are realized by the shunt circuit of piezoelectric materials. By using the harmonic balance method to analyze the forced vibration characteristics of the coupling system, it is found that within a certain range of harmonic excitation amplitude, the targeted energy transfer (TET) phenomenon occurs in the system. There is a threshold interval for the TET phenomenon of the system, where the TET does not occur when the excitation is less than the lower limit of the threshold. When the excitation exceeds the upper limit of the threshold, the system response appears a new high amplitude branch. The definite solution to the lower limit of the excitation threshold is determined, but when the system parameters are different, the high amplitude branch of the system response will appear above or below the amplitude of the platform. The results of this paper give the solution of the optimal TET excitation threshold interval in these two different cases and are helpful to design a NES for different applications.

Influence of parameters of a piezoelectric nonlinear energy sink on targeted energy transfer characteristic

In the coupling system of elastic plate and piezoelectric nonlinear energy sink, the shunt circuit of piezoelectric material as a nonlinear energy sink (NES) is coupled with the main system, where the nonlinear stiffness and damping are realized by the shunt circuit of piezoelectric material. The targeted energy transfer (TET) characteristics of the system are analyzed by complexification-averaging method and harmonic balance method, the variation range of NES parameters is determined, and the slow invariant manifold, platform amplitude of the frequency responses, nonlinear normal modes and threshold interval of the system under different parameters are analyzed. The influences of piezoelectric NES parameters, such as mass ratio, damping and nonlinear stiffness, on the TET characteristics are obtained, and a general criterion for design of piezoelectric NES parameters is proposed.

Hydrodynamic stability analysis of shear-layered fluid flow over a porous bed in the presence of a floating elastic plate

The present research investigates the stability characteristics of external shear-driven horizontal fluid flows over a porous horizontal plane in the presence of a floating elastic plate. A two-domain approach is considered to describe the model. The study employs the Chebyshev spectral collocation method to generate and solve the Orr–Sommerfeld eigenvalue system numerically. The most unstable mode is detected through eigenspectrum in finite wavenumber region. The plate characteristic parameters contribute to stabilizing the surface wave instability by reducing the linear wave speed and perturbed growth rate, but compressive force follows the other way. However, the external shear has the potential to enhance the surface wave instability by increasing the bulk velocity of the fluid. The depth ratio between the fluid and porous layer and Darcy number shows a dual nature regarding stability. So, by selecting specific characteristic parameters for both the flexible plate and porous bed, it is possible to manage and control the instability of the shear-layered film flow.

FORCED OSCILLATIONS OF A PLATE CONSISTING OF ELLIPSES AND CIRCLES OF VARIOUS SIZES WITH COMPLEX CONTOURS ON MACHINE PARTS

The article considers forced oscillations of elastic plates in the multilinked area. The surface of the plate undergoes actions of the disturbing force directed along the normal. The task leads to solution of different kinds of differential equations satisfying to initial and boundary conditions of various type. These equations are solved by the Bessel functions and through the method of the theory of functions with complex variables. Thus, the article offers the general methodology to the solutions of the dynamics of the elastic plates that is under action of the disturbing force.

Direct numerical simulation-based vibroacoustic response of plates excited by turbulent wall-pressure fluctuations

We use direct numerical simulation to study the vibroacoustic response of an elastic plate in a turbulent channel at $Re_\tau$ of 180 and 400 for three plate boundary conditions and two materials – synthetic rubber and stainless steel. The fluid–structure–acoustic coupling is assumed to be one-way coupled, i.e. the fluid affects the solid and not vice versa, and the solid affects the acoustic medium and not vice versa. The wall pressure consists of intermittent large-amplitude fluctuations associated with the near-wall, burst-sweep cycle of events. For stainless steel plates, displacement has similar large-amplitude peak events due to comparable time scales of plate vibration and near-wall eddies. For synthetic rubber plates, large-amplitude displacement fluctuations are observed only near clamped or simply supported boundaries. Away from boundaries, plate displacement resembles an amplitude-modulated wave, and no large-amplitude events are observed. We discuss displacement and acoustic pressure spectra over different frequency ranges. For frequencies much smaller than the first natural frequency, the product of plate-averaged displacement spectrum and bending stiffness squared collapses with Reynolds number and plate material in outer units. At high frequencies, displacement and acoustic pressure spectra scale better in inner units, and the scaling depends on the type of damping. For synthetic rubber plates, the spectra display an overlap region that collapses in both outer and inner units. Soft plate deformation displays a range of length scales. However, stiff plate deformation does not exhibit a similar range of scales and resembles plate mode shapes. The soft plate has two distinct deformation structures. Low-speed, large deformation structures with slow formation/break-up time scales are found away from boundaries. High-speed, small deformation structures with fast formation/break-up time scales formed due to boundary reflections exist near the boundaries.

Isolated disclination in an orthotropic von Kármán elastic plate

We investigate geometry and mechanics of the buckled orthotropic von Kármán elastic plate, with free boundary condition, in the presence of an isolated positive, or negative, disclination. The buckled shape of an isotropic elastic plate is cone-like for a positive disclination and saddle-like for a negative disclination. In either case, with increasing orthotropy, the shape of the buckled plate becomes more tent-like. The Gaussian curvature accordingly spreads along the ridge of the tent. The stress fields too remain focused in the neighbourhood of the defect point and the ridge, indicating that most of the stretching energy is accommodated in these singular regions.

Free vibration of thin, creased elastic plates: Optimization and scaling laws

Creasing, crumpling, and folding help to increase the flexural rigidity of thin elastic plates. In this study, the fundamental frequency is used as a measure of improvement in flexural stiffness. Firstly, an analysis of an orderly, creased, thin elastic plate with nine separate pyramidal crumples is presented. Such a plate shows an increase of 124% in the fundamental frequency with only a 0.5% increase in the total mass/ surface area when compared to that of a flat configuration. Further, scaling laws are introduced for such ordered creased structures for three separate cases- scaling based on (i) the total surface area of the structure, (ii) the amplitude/ height of the ordered pyramidal crumples, and (iii) the number of unit cells within a given edge length of the plate. Further, an optimization problem is posed to achieve maximum flexural stiffness under given constraints for an additively manufactured thin, elastic plate supported on all four edges. The objective is posed as a constrained optimization problem maximizing the fundamental frequency of the structure compared to that of a flat sheet. The results are promising, with a 176% increase in stiffness with only a 0.84% increase in the total mass/ surface area.

Buckling analysis of thin plates with circular cut-outs for sustainable ventilated food packaging design

Thin plates with circular cut-outs are widely used in engineering structures such as bridges, construction, and ventilated corrugated paperboard boxes in food packaging. This study investigates the critical buckling loads of thin elastic plates comprised of a single circular hole at different locations and with different sizes. To determine the critical loads, experimental tests and finite element method simulations are conducted. The vertical location and the diameter of the hole impact the buckling results significantly. The results indicate a significant correlation between the experimental and finite element data. The study develops second and third order polynomial formulas that predict buckling loads. In this study, corrugated paperboard plates were used as samples to gain insights into ventilated food packaging designs, aiming to enhance their buckling strength while reducing packaging material consumption for packaging sustainability purposes. The solution proposed in this study can be applied to predict the buckling of other elastic thin plate structures with a circular hole, relevant to industries such as construction and aerospace.

Complex band structure in viscoelastic thin plate phononic crystals

Phononic crystals (PnCs) are artificial composites designed by a periodic arrangement of two or more materials with different properties and/or geometry, in order to tailor the propagation of mechanical waves. The complex band structures of PnC thin plates, consisting of a hard matrix material reinforced by inclusions of soft viscoelastic material (HS-PnC), and vice-versa (SHPnC), are investigated. In general, elastic plate structures can be modelled using the Kirchhoff- Love theory, which is suitable for thin plates in low frequencies. However, there is a lack of knowledge concerning the effects of viscoelastic constituents on the complex band structure of flexural Bloch waves in PnC thin plates. In this study, an extended plane wave expansion (EPWE) method is used to compute the complex band structures of VPnC thin plates. The evanescent behaviour of periodic plates for the cases HS-PnC and SH-PnC is calculated, and the results show that the first creates flat bands for wave localization and the second opens up complete and partial Bragg scaterring band gaps.

Small-amplitude waves in a floating poroelastic plate forcing by vertical pitching plate

The linear two-dimensional problem of flexural-gravity waves generated by an oscillating rigid plate build-in a floating poroelastic plate is studied. The problem is coupled. The plate deflections and the hydrodynamics loads are determined at the same time. The liquid under the poroelastic plate is inviscid and incompressible. Dynamics of the floating plate is described by a thin elastic plate equation. Porosity of the floating plate is taken into account only through the liquid flux into the plate. The velocity of the inflow is assumed to be governed by Darcy's law being proportional to the hydrodynamic pressure at the plate/liquid interface. Two cases of the oscillating rigid plate with and without its part in the liquid are considered. The problems are solved by the Fourier transform method for non-zero porosity and by the vertical mode method for elastic plates with zero porosity. The deflection and strain distributions are analyzed depending on the excitation frequency and the porosity. Two models of floating plate porosity, where the hydrostatic pressure is included into Darcy's law (Zavyalova's model) and excluded (Meylan's model), are compared. Plate porosity induces damping to the system. It is shown that the damping rate is non-monotonic with respect to the plate porosity.

Three-Dimensional Bending-Gravitational Waves in a Floating Ice Sheet from a Moving Source of Disturbances

The ice cover is modeled by a thin elastic isotropic plate floating on the surface of a liquid of finite depth. The source of disturbances moves along the surface of the plate. The values of critical velocities at which the character of the wave disturbance changes are obtained. The angular zones in which the waves propagate are determined. The influence of the velocity of movement of the source of disturbances, the thickness of the ice plate, compression and stretching forces on the amplitudes of the waves formed is investigated.

Bounds of the technical constants for two-dimensional anisotropic elasticity

The complete set of bounds for the technical constants of an elastic layer, plate or laminate is given. The bounds are valid in general, also for completely anisotropic bodies. They are obtained transforming the polar bounds previously found. These bounds complete the knowledge of classical elasticity at least in the two-dimensional case and are useful in several situations, e.g. for determining the correct feasibility domain in design problems or as necessary conditions for accepting the results of laboratory tests on anisotropic layers.

Optimal Actuator Placement for the High-Precision Control of Quasi-Static Elastic Plates

Optimal Actuator Placement for the High-Precision Control of Quasi-Static Elastic Plates

An algorithm to couple smoothed particle hydrodynamics and finite element method for fluid–structure interaction problems

An algorithm for coupling the smoothed particle hydrodynamics (SPH) method and the finite element method (FEM) is proposed for fluid–structure interaction (FSI) problems. The SPH module solves the fluid motion, and the FEM module describes the deformation of structures. In the algorithm, a structure is considered as a solid boundary in the SPH module and dealt with using the dynamic boundary condition. Only three layers of SPH boundary particles are set to represent the structure surface, and the interaction forces between the SPH boundary and fluid particles, including the water pressure and the viscous shear stress, are quantified in a same way as the inter-fluid particle forces. Further, the computed forces on the SPH boundary particles exerted by the fluid are redistributed to the structure FEM nodes based on a well-defined projection scheme. In the FEM module, the deformation and motion of the structure under the exerted fluid forces are solved, and accordingly, the position and velocity of the SPH boundary particles are updated by interpolation from the new properties at the FEM nodes. The proposed SPH-FEM coupled model is validated in three FSI benchmarks. For the deformation of an elastic plate under hydrostatic water pressure, the relative difference between the computed and the analytical maximum vertical displacement at the plate midpoint is about 1.5%. For the deformation of an elastic gate subjected to a time-varying dam-break flow, the relative root mean squared errors in the computed vertical and horizontal displacement of the gate are, respectively, 5.5% and 2.2%. It is demonstrated that the proposed model performs well in simulating the deformation of structures under both hydrostatic water pressure and dynamic impacts of dam-break flows.

Deformations and Wave Forces in the Motion of a Load on an Ice Cover in the Presence of a Current with Velocity Shear

The deformations of an ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a pressure domain that moves rectilinearly at a constant velocity in the presence of a current with velocity shift, as well as the wave forces exerted on a moving body, are studied. Fluid flow is not potential. The ice cover is modeled by a thin elastic plate with account for uniform compression. The motion of the load can occur at an arbitrary angle to the direction of current. It is assumed that the ice deflection is steady in the coordinate system moving with the load. The Fourier transform method is used within the framework of the linear wave theory. The maximum deformations of ice cover and the wave forces are studied depending on the current velocity gradient, the direction of motion, and the compression ratio.

Levy-type analytical series solution for the three-dimensional free vibrations of functionally graded material rectangular plates with piezoelectric layers

Analytical solutions founded on three-dimensional theories play a crucial role in evaluating the credibility and precision of different plate theories and numerical methodologies. While Levy-type analytical solutions are widely recognized, they have been primarily confined to purely elastic plates. This study introduces a Levy-type analytical series solution for three-dimensional vibrations in a sandwich rectangular plate featuring a functionally graded material (FGM) core, along with piezoelectric material (PM) layers on the top and bottom surfaces. The behaviors of the FGM and PM layers were described using three-dimensional elasticity and piezoelasticity theories, respectively. In this study, the displacement functions and electric potential of each layer were expanded by Fourier series and polynomial auxiliary functions. An analytical series solution was then established by satisfying the governing equations of each layer, the mechanical and electric boundary conditions on the six faces of the plate, and the continuity conditions on the interfaces between the PM and FGM layers. To validate the proposed solutions, in-depth convergence studies were conducted for the vibration frequencies of the first six modes of sandwich square plates with various boundary conditions on the other pair of side faces. The well-converged results were then compared with published data based on various plate theories to verify the accuracy of these published data. Finally, accurate nondimensional frequencies were tabulated for the first six modes of sandwich rectangular plates with various aspect ratios, thickness-to-width ratios, PM-to-FGM layer thickness ratios, power law indices for the FGM layer, and six combinations of boundary conditions. These new numerical results when piezoelectric coupling is considered should be very useful to future analytical and numerical studies.

SPH numerical model of wave interaction with elastic thin structures and its application to elastic horizontal plate breakwater

This paper develops a Smoothed Particle Hydrodynamics (SPH) model to simulate wave interaction with elastic thin structures. This is achieved by coupling the δ-SPH and the Total Lagrangian SPH (TL-SPH) models for solving the fluid and structure phases, respectively. To this aim, a multiresolution algorithm within the ghost particles framework is introduced in order to treat the thin elastic components of the structure. The main idea of the proposed technology is to evaluate the hydrodynamic forces acting on the structure and the elastic dynamic at the finest solid particle resolution. Then, the information is transferred to the coarse solid particles, which act as a non-penetrating boundary condition for the fluid phase. The present enforcement is introduced with the aim of overcoming the disadvantages in terms of CPU time for heavy SPH simulations, in which the choice of the initial spatial resolution is driven by the presence of elastic thin structures immersed in the fluid environment. The present elastic thin-wall boundary treatment is verified through several numerical cases, in which it is proved that the proposed technology can block non-physical interaction of fluid particles located at opposite sides of the elastic thin-wall boundary, ensures the stability of the fluid pressure field, and reasonably predicts the motion of elastic structures when these are subjected to hydrodynamic loads. Successively, the proposed numerical model is applied to simulate wave interaction with an elastic horizontal plate breakwater. The hydrodynamic performances of the breakwater, with focus on the reflection and transmission coefficients, are analyzed. The results of elastic and rigid horizontal plate breakwaters are compared in order to better understand the effects of the structure deformation on the hydrodynamics. Finally, an analysis of the structural deformation of the breakwater is also reported.

Experimental and theoretical evaluation of influence of hole size on deformation and fracture of elastic perforated plate

As an ideal benchmark problem for the researches of solid mechanics and fracture mechanics, the influence of the hole size on the deformation and the fracture behaviors of perforated plates has been studied for more than sixty years. However, the seemingly simple problem has not been thoroughly answered yet. To gain more understanding of the hole-size dependence of the strain/stress concentration, the present study measured the stress and the strain concentration factors by using tensile tests together with the digital image correlation analysis. A noticeable hole-size dependence was observed for both concentration factors. The experimental observations can be reasonably explained in terms of a new formulation of strain energy density which considers the contributions of both the conventional strain and the high-order strain. It is shown that the strain energy density gets more uniformly distributed when the hole gets smaller. Accordingly, the maximum strain decreases as well. Due to the positive contribution of the higher-order strain energy, the material around the hole hardens and results in a concentration factor lower than three. Because of the influence of the length scale parameter, the stress cannot be scaled by a reference strain and this may lead to a discrepancy between the stress and the strain concentration factors.

Full-field dynamic response of low-velocity impact on an elastic plate using 3D digital image correlation

Measurement of the impact process is challenging due to its short duration and high frequency response. Due to the very small strain in low-velocity impacts, the requirements for the temporal and spatial resolution of test methods are extremely high. Traditionally, strain gauges or lasers are commonly used to measure the dynamic response of a single point, or quasi-static loading is used to achieve full-field response. To finely describe the transient full-field deformation process of low-velocity impacts, a high-speed imaging system combined with 3D digital image correlation (DIC) technique is set up to investigate the instantaneous responses of impact on a polyvinyl chloride (PVC) plate at the ms and με level. The full-field deformation of the contact surface and the whole process of elastic wave propagation are measured. The measurement by DIC closely matches the measurement by strain gauges and agrees well with the numerical simulation results. The measurement reveals that the “multi-micro-impacts” phenomenon may happen during a complete impact process due to the complex strain wave propagation. This work provides a benchmark case for low-velocity impact full-field testing, and the results show that 3D DIC can accurately evaluate dynamic behaviors of low-velocity impact between flexible bodies and can accurately describe the spatiotemporal propagation characteristics of strain waves.

A mechanical model for thin sheet straight cutting in the presence of an elastic support

We study the mechanics of sheet straight cutting in terms of a linear elastic fracture mechanics (LEFM) problem for a infinite thin elastic Kirchhoff plate partly supported by a Winkler foundation. The plate features a semi-infinite crack that is located at the edge of the supported zone and that is subjected to shear and bending loads, representing the action of the cutting tool (e.g. scissors blades). The fact that the plate is only partly supported by the foundation significantly complicates the analysis for it creates a non-symmetric framework, both locally and globally. Yet, a semi-analytical solution is obtained through casting the matrix Wiener–Hopf problem in terms of a pair of convolution integral equations defined on a semi-infinite domain. Stress intensity factors (SIFs) are obtained which converge to the known limits for a symmetric and skew-symmetric free plate. This analysis reveals the fundamental role played by the support in affecting the SIFs in an opposing manner, by enhancing/decreasing the symmetric/skew-symmetric components. Consequently, changing the support stiffness is capable of shifting the failure mechanism, from bending to shear. This observation may be taken advantage of when cutting materials which are more sensitive to either of these failure mechanisms. Also, it proves that the role of the support cannot be neglected when developing mechanical models of any cutting process.