Infinite Thin Plate Research Articles

Bound states in the continuum for antisymmetric lamb modes in composite plates made of isotropic materials

In this study, we report a numerical design and observation of bound states in the continuum (BICs) for Lamb waves. BICs for elastic-wave systems, especially in non-periodic configurations, are difficult to obtain due to their intricate polarization states. However, the study in this matter has become very important, especially in the field of opto-mechanics or other multi-field couplings at micro or nanoscales. To illustrate the design concept, we simulate the introduction of a piece of silica (SiO2) into a thin infinite Si plate and show that, for specific aspect ratios, BICs for elastic waves can be predicted. We present numerical results for both two-dimensional (2D) rectangular plates and three-dimensional (3D) disk structures. Moreover, we also investigate the modal contributions of both the background and inclusion media during the occurrence of BICs, further verifying the physical background of our design strategy. Although we have focused our work on asymmetric Lamb modes, the current method can also be applied to construct other types of elastic-wave BICs, providing a powerful tool for metamaterial device prototyping based on the control or guiding of elastic waves.

New Approach of Normal and Shear Stress Components for Multiple Curvilinear Holes Which Weakened a Flexible Plate

In this article, a thin infinite flexible plate weakened by multiple curvilinear holes is considered. The strength shapes are mapped outside a unit circle with the assistance of particular conformal mapping under certain conditions. The mathematical model that governs the rounded forces of the current physical problem is the boundary value problem of elastic media. This study is applicable to many phenomena throughout nature, like tunnels, caves, and excavations in soil or rock. The Cauchy method for complex variables is used to get the closed forms of Gaursat functions and change the problem to a second-type integrodifferential equation with a Cauchy kernel, which is used for a large area of the contact problems. Then, the normal and shear stress components that act on the model are derived. Afterward, some of the physical applications are studied, and different stress components at specific values in each application are calculated and plotted using Maple 2023.

Shear Waves in an Elastic Plate with a Hole Resting on a Rough Base

The article is devoted to the analytical and numerical study of the pattern of propagation and attenuation, due to Coulomb friction, of shear waves in an infinite elastic thin plate with a circular orifice of radius r0 lying on a rough base. Considering the friction forces and their influence on the sample of wave propagation in extended rods or thin plates is important for calculating the stress–strain state in them and the size of the area of motion. An exact analytical solution of a nonlinear boundary value problem for tangential stresses and velocities is obtained in quadratures by the Laplace transform, with respect to time. It turned out that the complete exhaustion of the wave front of a strong rupture occurs at a finite distance r* from the center of the orifice, and an elementary formula is given for this distance (the case of tangential shock stresses suddenly applied to the orifice boundary is considered). For various ratios of the magnitude of the limiting friction force to the amplitude of the applied load, the stopping (trailing) wave fronts are calculated. After passing them, a state of static equilibrium between the elastic and friction forces with a nonlinear distribution of residual stresses is established in the region r0≤r≤r*. For the first time, a precise analytical solution was obtained for the boundary value problem of the propagation of elastic shear waves in an infinite isotropic space with a cylindrical cavity, when a tangential shock load is set on its surface.

Numerical solution of the cavity scattering problem for flexural waves on thin plates: Linear finite element methods

Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution on the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem.

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.

Transient Deformation of Anisotropic Timoshenko’s Plate

In this paper, we will present an approach to constructing of dynamical spatial Green’s function (elementary solutions, dominant function) for a thin infinite elastic plate of constant thickness. The plate material is anisotropic with a single plane of symmetry, geometrically coinciding with plate’s middle plane. The Timoshenko theory was used for describing the plate movement. Transient spatial Green’s functions for normal displacements and angles of orthogonal alteration to middle surface before deformation of material fiber are built in the Cartesian coordinate system. To construct Green’s function, direct and inverse Laplace and Fourier integral transformations are applied. The originals of Laplace Green’s functions were analytically found with the theorem of residues. To construct Fourier originals, a specific method was used based on Fourier series transformation inversion integral connection with Fourier series on a variable interval. Green’s function found for normal displacement made it possible to represent the normal transient function as three-fold convolution of Green function with distant load function. The functions of normal distant displacements were constructed in case of the impact of transient total loads concentrated and distributed across rectangular courts. The numerical method of rectangles was used to calculate the convolution integrals. The influence of the concentrated load speed on transient normal displacements of the anisotropic plate was analyzed. As a verification of constructed transient spatial Green’s functions, the results of numerical solutions were compared with the results found using known transient Green’s functions for isotropic thin elastic rectangular simply supported Timoshenko’s plate which solutions are constructed using Laplace integral transformation in time and its decomposition into Fourier series on coordinates. Besides, its confidence was proved analyzing the nature of waves in anisotropic, orthotropic and isotropic plate, found in the process of numerical calculations. The results are represented as diagrams. Examples of calculations are given.

Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity

In many practical engineering situations, a source of vibrations may excite a large and flexible structure such as a ship’s deck, an aeroplane fuselage, a satellite antenna, a wall panel. To avoid transmission of the vibration and structure-borne sound, radial or polar periodicity may be used. In these cases, numerical approaches to study free and forced wave propagation close to the excitation source in polar coordinates are desirable. This is the paper’s aim, where a numerical method based on Floquet-theory and the FE discretision of a finite slice of the radial periodic structure is presented and verified. Only a small slice of the structure is analysed, which is approximated using piecewise Cartesian segments. Wave characteristics in each segment are obtained by the theory of wave propagation in periodic Cartesian structures and Finite Element analysis, while wave amplitude change due to the changes in the geometry of the slice is accommodated in the model assuming that the energy flow through the segments is the same. Forced response of the structure is then evaluated in the wave domain. Results are verified for an infinite isotropic thin plate excited by a point harmonic force. A plate with a periodic radial change of thickness is then studied. Free waves propagation are shown, and the forced response in the nearfield is evaluated, showing the validity of the method and the computational advantage compared to FE harmonic analysis for infinite structures.

Coupling total Lagrangian SPH–EISPH for fluid–structure interaction with large deformed hyperelastic solid bodies

In this work, we propose a two-way coupling technique between a total Lagrangian smoothed particle hydrodynamics (SPH) method for Solid Mechanics and the explicit incompressible SPH (EISPH) to simulate fluid–structure interaction problems. In the solid part, the total Lagrangian framework guarantees that the particle distribution keep stable to correctly calculate the deformation gradient and thus the elastic forces. The constitutive model follows hyperelastic formulations, and the stability of the method is enforced by a Jameson–Schmidt–Turkel (JST) stabilization procedure. For the fluid part, we applied an EISPH formulation, which is a fully explicit incompressible scheme based on a projection method capable of providing accurate pressure distributions for free-surface flows, while avoiding costly linear equations. The coupling scheme follows the same manner as the fixed wall ghost particle (FWGP) approach, which was here adapted to include moving walls. In addition, the non-penetration condition is rigorously reinforced through a numerical algorithm to avoid penetration of every fluid particle, including free-surface particles. Our method for solid is then verified through a large deformed tension plate numerical test, and our coupling forces through a series floating tests and hydrostatic water column over a thin infinite plate. Then, the method is validated comparing it with experimental data of a dam break test in which the water column attacks a thin rubber plate.

Optimal piezoelectric shunt dampers for non-deterministic substructure vibration control: estimation and parametric investigation

Piezoelectric (PZT) shunt damping is an effective method to dissipate energy from a vibrating structure; however, most of the applications focus on targeting specific modes for structures vibrating at low-frequency range, i.e. deterministic substructure (DS). To optimally attenuate structures vibrating at high-frequency range, i.e. non-deterministic substructure (Non-DS) using a PZT shunt damper, it is found that the impedance of the PZT patch’s terminal needs to be the complex conjugate of its inherent capacitance paralleled with the impedance ‘faced’ by its non-deterministic host structure underline moment actuation. The latter was derived in terms of estimation of the effective line moment mobility of a PZT patch on a Non-DS plate by integrating the expression of driving point moment mobility of an infinite thin plate. This paper conducts a parametric investigation to study the effect of changing the size, quantity and configuration of the PZT patch to the performance of the optimal PZT shunt dampers in dissipating the energy of its non-deterministic host structure. Results are shown in terms of energy reduction ratio of the thin plate when attached with optimal PZT shunt damper(s).

Buckling-induced interaction between circular inclusions in an infinite thin plate.

Design of slender artificial materials and morphogenesis of thin biological tissues typically involve stimulation of isolated regions (inclusions) in the growing body. These inclusions apply internal stresses on their surrounding areas that are ultimately relaxed by out-of-plane deformation (buckling). We utilize the Föppl-von Kármán model to analyze the interaction between two circular inclusions in an infinite plate that their centers are separated a distance of 2ℓ. In particular, we investigate a region in phase space where buckling occurs at a narrow transition layer of length ℓ_{D} around the radius of the inclusion, R (ℓ_{D}≪R). We show that the latter length scale defines two regions within the system, the close separation region, ℓ-R∼ℓ_{D}, where the transition layers of the two inclusions approximately coalesce, and the far separation region, ℓ-R≫ℓ_{D}. While the interaction energy decays exponentially in the latter region, E_{int}∝e^{-(ℓ-R)/ℓ_{D}}, it presents nonmonotonic behavior in the former region. While this exponential decay is predicted by our analytical analysis and agrees with the numerical observations, the close separation region is treated only numerically. In particular, we utilize the numerical investigation to explore two different scenarios within the final configuration: The first where the two inclusions buckle in the same direction (up-up solution) and the second where the two inclusions buckle in opposite directions (up-down solution). We show that the up-down solution is always energetically favorable over the up-up solution. In addition, we point to a curious symmetry breaking within the up-down scenario; we show that this solution becomes asymmetric in the close separation region.

Determination of the sheet resistance of an infinite thin plate with five point contacts located at arbitrary positions

In this paper, a five-probe method of sheet resistance measurement that is independent of probe positions is reported. The method is strict for an infinite homogeneous plane. It has potential applications as a sheet resistance standard based on planar molecular layers. The method can be used to measure the sheet resistance of layers covering objects with a spherical topology, particularly on micro- and nanometric scales, where it is difficult to control probe positioning.

Dynamic response of an infinite thin plate loaded with concentrated masses

Thin plates with attached concentrated masses are considered. Time-harmonic flexural waves are generated by a force applied over a finite region of the plate. The problem of calculating the resulting plate response reduces to calculating the displacement of the masses, and this is done by solving a finite system of linear algebraic equations in the manner of previous work by Evans and Porter. Numerical results for two masses are presented and compared with finite element computations. Analytical results for N masses arranged around a circle concentric with a circular forcing region are obtained; it is shown that results for a corresponding thin solid ring are recovered as N goes to infinity. Similar results are obtained for a scattering problem, with an incident plane wave.

Performance of Turbulence Models in Simulating Wind Loads on Photovoltaics Modules

The performance of five conventional turbulence models, commonly used in the wind industry, are examined in predicting the complex wake of an infinite span thin normal flat plate with large pressure gradients at Reynolds number of 1200. This body represents a large array of Photovoltaics modules, where two edges of the plate dominate the flow. This study provided a benchmark for capabilities of conventional turbulence models that are commonly used for wind forecasting in the wind energy industry. The results obtained from Reynolds Averaged Navier-Stokes (RANS) k - ε , Reynolds Normalization Group (RNG) k - ε , RANS k - ω Shear Stress Transport (SST) and Reynolds Stress Model (RSM) were compared with existing Direct Numerical Simulations (DNS). The mean flow features and unsteady wake characteristics were used as testing criteria amongst these models. All turbulence models over-predicted the mean recirculation length and under-predicted the mean drag coefficient. The major differences between numerical results in predicting the mean recirculation length, mean drag and velocity gradients, leading to deficits in turbulence kinetic energy production and diffusion, hint at major difficulties in modeling velocity gradients and thus turbulence energy transport terms, by traditional turbulence models. Unsteadiness of flow physics and nature of eddy viscosity approximations are potential reasons. This hints at the deficiencies of these models to predict complex flows with large pressure gradients, which are commonly observed in wind and solar farms. The under-prediction of wind loads on PV modules and over-estimation of the recirculation length behind them significantly affects the efficiency and operational feasibility of solar energy systems.

Superlensing effect for flexural waves on phononic thin plates composed by spring-mass resonators

This paper demonstrates the superlensing effect of flexural waves by phononic plates with the negative index of refraction. The phononic plate consists of a square lattice of spring-mass resonators attached to an infinite thin plate. The periodic resonator array induces a resonant band gap between the first and second dispersion curves of band structures calculating by a plane wave expansion method. All-angle negative refraction phenomenon has been found for a propagation mode under specific elastic parameters of spring-mass resonators. Furthermore, a flat lens composed by a finite number of spring-mass resonators is designed to focus elastic fields of a point-like excitation operating at this propagating mode. Multiple scattering simulations show that the image resolution of the designed flat lens is about 0.15λ, overcoming the Rayleigh diffraction limit of traditional imaging systems.

Flexural wave scattering by varying-thickness annular inclusions on infinite thin plates

This paper proposes a semi-analytical method for flexural wave scattering by cylindrical annular inclusions of varying thickness on an infinite thin plate. The thickness of inclusions is assumed to vary in arbitrary power form along the radial direction. In the numerical method, analytically linear relationships between expansion coefficients of exciting and scattered fields around each inclusion are firstly derived by means of closed form solutions of a hypergeometric governing equation of varying-thickness thin plates, and boundary conditions at the interface between different media. Then, linear algebraic equations containing only expansion coefficients of exciting fields are set up with the aid of foregoing linear relations. Expansion coefficients of exciting fields are finally solved by a numerical collocation method. Numerical examples of single and multiple scattering are designed to demonstrate the applicability of the method. The results of single scattering show the change of thickness variations allows tuning local resonances of inclusions to specified frequencies. Furthermore, displacement amplitude spectra of flexural waves for multiple scattering have captured the details of stop band formation of square and hexagonal arrangements of varying-thickness inclusions. It is also demonstrated that stop bands of flexural wave propagation can be controlled by changing stiffness gradients of varying-thickness inclusions.

Analytical Study of the Head-On Collision Process between Hydroelastic Solitary Waves in the Presence of a Uniform Current

The present study discusses an analytical simulation of the head-on collision between a pair of hydroelastic solitary waves propagating in the opposite directions in the presence of a uniform current. An infinite thin elastic plate is floating on the surface of water. The mathematical modeling of the thin elastic plate is based on the Euler–Bernoulli beam model. The resulting kinematic and dynamic boundary conditions are highly nonlinear, which are solved analytically with the help of a singular perturbation method. The Poincaré–Lighthill–Kuo method is applied to obtain the solution of the nonlinear partial differential equations. The resulting solutions are presented separately for the left- and right-going waves. The behavior of all the emerging parameters are presented mathematically and discussed graphically for the phase shift, maximum run-up amplitude, distortion profile, wave speed, and solitary wave profile. It is found that the presence of a current strongly affects the wavelength and wave speed of both solitary waves. A graphical comparison with pure-gravity waves is also presented as a particular case of our study.

Interpretation of gravity anomalies over the eighty-five East Ridge in the Bay of Bengal

Several profiles of the negative gravity anomaly over the submerged 85o E ridge in the Bay of Bengal of the Indian Ocean have been interpreted in two dimension and results are combined together to give a detailed picture of its morphology and thickness of surrounding sediments. The observed negative gravity anomaly has been explained as the combined effect due to the positive mass anomaly caused by the replacement of sediments by high dense igneous rocks of the ridge, and the negative mass anomaly caused by replacing the high dense upper mantle material by the oceanic crust, which has been bended and sunk into the upper mantle due to the pressure exerted by the ridge. Downward migration of the oceanic crust has been also calculated assuming that the oceanic crust is behaving as a thin infinite elastic plate resting on inviscid fluid half space and is found to closely agree with the results of the gravity study. Both studies indicate that the thickness of the ridge vary from 11 km to 16 km while the oceanic crust has undergone a depression of 9 km to 14 km.

Wave Motion due to a Ring Source in Two Superposed Fluids Covered by a Thin Elastic Plate

The problem of wave generation by a horizontal ring of wave sources of the same time-dependent strength present in any one layer of a two-layer fluid is investigated here. The upper fluid is of finite height above the interface and is covered by a floating thin infinite elastic plate (modeling a thin sheet of ice) while the lower fluid extends infinitely downwards. Assuming linear theory, the problem is formulated as an initial value problem and the Laplace transform in time is employed to solve it. For time-harmonic source strength, the asymptotic representations of the potential functions describing the motion in the two layers for large time and distance are derived. In these representations, the two different coefficients for each of the surface and interface wave modes have the same numerical values although it has not been possible to prove their equivalence analytically. This shows that the steady-state analysis of the potential functions produces outgoing progressive waves at the surface and at the interface. The forms of the surface and interface waves are depicted graphically for different values of the flexural rigidity of the elastic plate and the ring source being submerged in the lower or upper layer.

Acoustic Wave Propagation Through a Plate Fixed on a Rigid Frame Via Elastic Spacers and Located Between Two Barriers

The propagation of a stationary acoustic wave through an infinite thin plate stiffened on two sides by a system of absolutely rigid, crossed ribs and located between two absolutely rigid barriers. It is assumed that the plate and the ribs evenly distributed along rectangular Cartesian axes are connected through elastic spacers (supports) without slip. The dynamic deformation of the plate is described by the linearized Kirchhoff–Love equations of the classical theory of plates, the dynamic deformation of the spacers is described by two-dimensional and one-dimensional relations based on linear approximations of displacements of points of the coating and spacers along the thickness and taking into account only transverse compression and transverse shear, and the motion of acoustic media by the well-known wave equations. The solution of the problem is obtained using the Ritz method. The constructed solution was used to investigate how the physico-mechanical and geometric parameters of the mechanical system and the frequency of acoustic waves incident on the plate influence the sound-insulating parameters and the stress–strain state of the plate.

Analytic solutions of thermoelectric materials containing a circular hole with a straight crack

The two-dimensional problem of thermoelectric materials under the action of uniform electric current and uniform total energy flux is studied by means of the complex variable function method and the conformal mapping technique. The model of infinite thin plate containing a circular hole with a straight crack is considered, and the boundary conditions of the inner surface are assumed to be electrically and thermally impermeable. The analytic solutions of the electric current density and total energy flux are derived. In addition, the electric current density intensity factor (EIF), the total energy flux density intensity factor (UIF) and the stress intensity factor (SIF) near the crack tip are obtained, which have a great significance for the engineering application of thermoelectric materials. It is found that, on the one hand, the EIF and UIF are in proportion to the applied far-field electric current loads and the total energy flux loads, respectively. On the other hand, all these field's intensity factors are closely related to the geometrical characteristics of the inner boundary (i.e. the circular hole radius R and crack length L). When the radius tends to zero, the degenerated results are identical to the pre-existing solutions in case of the Griffith crack. Numerical results are also made to mainly discuss the effects of radius and crack length on these nondimensionalized field's intensity factors.