Elastic Plate Model Research Articles

Wave diffraction by multiple arbitrary shaped cracks in an infinitely extended ice sheet of finite water depth

Flexural-gravity wave interactions with multiple cracks in an ice sheet of infinite extent are considered, based on the linearized velocity potential theory for fluid flow and thin elastic plate model for an ice sheet. Both the shape and location of the cracks can be arbitrary, while an individual crack can be either open or closed. Free edge conditions are imposed at the crack. For open cracks, zero corner force conditions are further applied at the crack tips. The solution procedure starts from series expansion in the vertical direction based on separation of variables, which decomposes the three-dimensional problem into an infinite number of coupled two-dimensional problems in the horizontal plane. For each two-dimensional problem, an integral equation is derived along the cracks, with the jumps of displacement and slope of the ice sheet as unknowns in the integrand. By extending the crack in the vertical direction into the fluid domain, an artificial vertical surface is formed, on which an orthogonal inner product is adopted for the vertical modes. Through this, the edge conditions at the cracks are satisfied, together with continuous conditions of pressure and velocity on the vertical surface. The integral differential equations are solved numerically through the boundary element method together with the finite difference scheme for the derivatives along the crack. Extensive results are provided and analysed for cracks with various shapes and locations, including the jumps of displacement and slope, diffraction wave coefficient, and the scattered cross-section.

A hybrid method for linearized wave radiation and diffraction problem by a three dimensional floating structure in a polynya

A hybrid method is developed to solve the interaction problem of wave with a three dimensional floating structure in a polynya. The linearized velocity potential theory is used for fluid flow, and the thin elastic plate model is adopted for the infinitely extended ice sheet. Because of sudden change of the upper boundary of the computational domain, namely from the ice sheet to the free surface, the domain is divided into two sub-domains, one below free surface and the other below the ice sheet. The solution method is divided into three components. The first component is the integral equation over the structure surface and the interface of the two sub-domains. In the second component, the velocity potential is expanded into a series of eigenfunctions in the vertical directions, which avoids the numerical difficulty in calculation of the fifth derivatives. This is coupled with a series of integral equations along the edge of the ice sheet. In the third component of the method, two orthogonal inner products are used to impose the continuity conditions of the velocity and pressure on the interface, as well as the boundary conditions on the ice edge. The developed method is verified through comparison with the analytical solution for a circular cylinder. Case study is then made for a FPSO in a polynya with different shapes and floating positions. The hydrodynamic coefficients, wave exciting force and wave elevation in polynya are provided and analyzed.

Research on the First Breaking Mechanism of the Main Roof of Coal Seam with High Dip Angle

In order to study the mechanism and characteristics of the first breaking of the roof in thick coal seams with high dip angles, a mechanical model of elastic thin plate with four clamped edges of the roof was established, and the expressions of the deflection and stress of the roof were obtained by the energy method. The influence of the change in seam dip angle on the first breaking distance of the roof was presented. Based on the stress solution, the roof breaking criterion was proposed, and the breaking distance of the roof was calculated. Combined with numerical simulation, the stress distribution characteristics of the upper and lower surfaces of the stope roof were analyzed. The results show that the first breaking distance of the roof is inversely proportional to the seam dip angle. Broken morphology of roof in high dip seam is different from the “O‐X” morphology of horizontal roof. The roof breaking order of the seam with high dip angle is middle‐middle upper‐middle lower‐upper‐lower. These research results have certain theoretical guiding significance for the study of the first breaking mechanism of the main roof of highly inclined working face of coal seams.

Graded elastic metasurface for enhanced energy harvesting

In elastic wave systems, combining the powerful concepts of resonance and spatial grading within structured surface arrays enable resonant metasurfaces to exhibit broadband wave trapping, mode conversion from surface (Rayleigh) waves to bulk (shear) waves, and spatial frequency selection. Devices built around these concepts allow for precise control of surface waves, often with structures that are subwavelength, and utilise Rainbow trapping that separates the signal spatially by frequency. Rainbow trapping yields large amplifications of displacement at the resonator positions where each frequency component accumulates. We investigate whether this amplification, and the associated control, can be used to create energy harvesting devices; the potential advantages and disadvantages of using graded resonant devices as energy harvesters is considered. We concentrate upon elastic plate models for which the A0 mode dominates, and take advantage of the large displacement amplitudes in graded resonant arrays of rods, to design innovative metasurfaces that trap waves for enhanced piezoelectric energy harvesting. Numerical simulation allows us to identify the advantages of such graded metasurface devices and quantify its efficiency, we also develop accurate models of the phenomena and extend our analysis to that of an elastic half-space and Rayleigh surface waves.

Vibration of functionally graded Mindlin plate based on a modified strain gradient elasticity theory

Vibrational behavior of functionally graded (FG) microplates is investigated by a new modified strain gradient Mindlin plate (MSGMP) model. With the help of Hamilton’s principle, the dynamic equation is easily obtained. Furthermore, the general forms of boundary conditions are gotten by using coordinate transformation. The MSGMP model can be degenerated to a couple stress elastic Mindlin plate model or the classical Mindlin plate (CMP) model. Analytical solutions of vibrational problem of a rectangular microplate with four simply supported edges are gotten. Numerical results reveal significant effects of the dimensionless nonlocal parameters, the power law index and vibration mode on the free vibration behavior of FG plate.

Relationship between sound radiations resulting from airborne-sound and point-force excitations of a double-leaf infinite elastic plate

This paper presents an expanded theory for relating airborne-sound-excited and force-excited sound radiations from solid structures. Although the reduction of these two types of sound radiation is a fundamental issue in noise-control engineering, each of them has been historically treated as a separate issue. The reduction in the former is normally called airborne sound insulation and separated from the latter, especially in architectural acoustics. A previous study (M. Yairi et al., J. Acoust. Soc. Am., 140, 453–460 (2016)) established a fundamental relationship between the sound radiations from random-incidence sound-excited and point-force-excited vibrations of a single-leaf infinite elastic plate. A conversion function that relates the two excitation cases was presented in a simple closed form, not including the parameters of the plates, which included of only the specific impedance and the acoustic wavenumber of the medium surrounding the plate. In this paper, the applicability of the conversion function is expanded from a single-leaf infinite elastic plate model to a double-leaf infinite elastic plate model. The sound radiation from a double-leaf infinite elastic plate driven by random-incidence sound and that driven by point-force excitation are theoretically investigated. The conversion function derived from the present model successfully relates the two excitation problems at all analyzed frequencies and has been shown to agree with the previously established single-leaf theory.

Mechanical model of safety thickness for resource recovery of complex & difficult mining ore body

In view of the difficulty in calculating the safe thickness of complex and difficult-to-mine orebody in the process of resource recovery, based on the elastic-plastic theory and mechanics method, the mechanical models of the safe thickness of orebody recovery above the collapse area and that of orebody recovery at the side of the collapse area are constructed respectively. The results show that: 1. Using the fixed beam model to calculate the safety thickness, the additional load on the roof, the volume weight and span of the rock stratum, the width of the rock stratum and the water pressure are the important influencing factors. When the water pressure exists, the safety thickness can be reduced; 2. When the safety thickness is calculated by the elastic plate model with fixed support around, the safety thickness decreases with the increase of the tensile strength or shear strength of rock mass, and increases with the increase of the external load; 3. When the collapse area is located at the side of the ore body to be mined, the safety thickness decreases with the increase of the tensile or shear strength of rock mass, and with the increase of the external force. The load increases with the increase of load. The research can provide theoretical support for mining residual mineral resources.

Mechanical model of artificial roof overlay deformation elastic thin plate

After presupporting of roof, overlying roof bending deformation law has undergone greatly changed. In order to accurately calculate the top roof of artificial roof movement law after the roof pre-control top. Based on the analysis of the top moving characteristics of the roof and the theory of elastic foundation thin plate, the mechanics model of pre-control top-mining filling body-artificial roof-roof is established. The basic deflection, stress equations and the critical conditions of roof breaking are derived. The impact of roof bending deformation of the key factors affects the artificial roof thickness and elastic modulus. Combination of Wo Hu Shan specific engineering conditions, the calculation equation of tensile stress and sinking displacement of lying tiger hill is deduced and the critical roof thickness is 1.87m. To retain a certain safety factor, the artificial roof thickness of 2.5m is used. The monitoring results show that the maximum roof displacement is 15.80mm, which is consistent with the theoretical calculation.

Estimation of mechanical material properties using an isotropic three-dimensional elastic plate model

This paper develops an inverse method to simultaneously estimate complex Young’s and shear moduli of a rectangular-shaped solid material. This new method exploits a relatively simple experiment and is designed for a frequency range of approximately 1 kHz–10 kHz. A fully elastic spatial domain model of a plate is derived and corresponding experimental measurements are collected. An error function is defined using the residual between the model displacements and the measured displacements, which allows a metric for a search over a space where the complex model values of Young’s and shear moduli are allowed to vary. When this error function is minimized, the parameters used to create the model are considered the best estimate possible. An experiment is included to illustrate the estimation process, and it is shown that the real parts of both moduli compare favorably to previously available measurements. Finally, the model is compared to the experimental measurements to illustrate its accuracy when it is formulated using the material property estimates obtained using this method.

Theoretical Analysis and Experimental Evaluation of Road Dynamic Behavior on Different Subgrades

The dynamic characteristics of a road on a subgrade filled with different materials under traffic loads are studied. An elastic double‐layer plate model on a viscoelastic subgrade is established to describe the dynamic behavior of the road. The load diffusion path in the form of a cone in the stabilised layer under single vibration source and three vibration sources is considered, and the equations for the displacement and effective soil pressure of the elastic double‐layer plate on the viscoelastic subgrade are derived. The relevant empirical formulae are then properly revised to reflect the dynamic behavior of the road. The dynamic response of the road is analysed by adjusting factors such as the excitation load, excitation frequency, and subgrade filling materials. The results show that gravel as a subgrade filling material has a better vibration reduction effect than sand bag. In addition, the test data not only have good consistency with the theoretical formulae but also validate the reliability of the empirical formulae.

The piezoelectric road status sensor using the frequency scanning method and machine-learning algorithms

This paper introduces a piezoelectric sensor for measuring the thickness of ice and water film on road surfaces, utilizing the frequency scanning method and machine-learning algorithms. With a constant elasticity alloy plate and a three-electrode piezoelectric transducer disc, this sensor detects ice and water by vibrating. During the research, a model of elastic thin plate with small deflections was built to describe the sensor’s vibration characteristics and the sensing unit’s amplitude frequency response curves of different ice and water film thickness were recorded. The curves show a high correlation between the thickness of loads and the curve shapes. In order to expand the measurement range, 425 sets featuring vectors for the machine-learning model were extracted, and some specific features such as median, variance, area, centroid and energy under the curves were utilized. The performance of conventional methods and two regression models based on support vector regression (SVR) and an artificial neural network (ANN) were evaluated with cross-validation results. The measurement range of the sensor turned out to be 0–10 mm. Based on the ANN model, on an average 85.5% and 100% of the neural network outputs had a regression error less than 0.5 mm and 1.0 mm respectively. Finally, the result of an 18-day field test on the roads of the Norwegian University of Science and Technology is presented.

Head-on collision between two hydroelastic solitary waves with Plotnikov-Toland's plate model

ABSTRACT Head-on collision between two hydroelastic solitary waves propagating at the surface of an incompressible and ideal fluid covered by a thin ice sheet is analytically studied by means of a singular perturbation method. The ice sheet is represented by the Plotnikov-Toland model with the help of the special Cosserat theory of hyperelastic shells and the Kirchhoff-Love plate theory, which yields the nonlinear and conservative expression for the bending forces. The shallow water assumption is taken for the fluid motion with the Boussinesq approximation. The resulting governing equations are solved asymptotically with the aid of the Poincare-Lighthill-Kuo method, and the solutions up to the third order are explicitly presented. It is observed that solitary waves after collision do not change their shapes and amplitudes. The wave profile is symmetric before collision, and it becomes, after collision, unsymmetric and titled backward in the direction of wave propagation. The wave profile significantly reduces due to greater impacts of elastic plate and surface tension. A graphical comparison is presented with published results, and the graphical comparison between linear and nonlinear elastic plate models is also shown as a special case of our study.

Vibration energy harvesting from roads under traffic loads

In this study, piezoelectric energy harvesting using depressions in roads generated by traffic loads was investigated theoretically and experimentally. The elastic double-layer plate model was used to analyse piezoelectric energy harvesting under traffic loads. The asphalt road layer and cement-stabilized layer were regarded as elastic bearing plates. Based on the vibration differential equations of an elastic double-layer plate, the expressions for the vertical displacements of the plates were obtained using the Fourier transform and contour integral theorem. Then the expressions for output voltage and power were derived. An indoor test of a layered road for piezoelectric transformation was carried out using a traffic load model groove. The results show that the effect of the piezoelectric transducer position on the voltage and power should be considered, and that the excitation load, frequency, and thickness of the asphalt layer can affect the output voltage and power.

Classroom demonstration of acoustic landmine detection using a clamped soil plate oscillator

In landmine detection some mines (constructed out of plastic) are difficult to detect using ground penetrating radar. In acoustic landmine detection, one can detect a vibration profile across the surface of the soil (sand) due to the interaction between the soil column and the elastic top plate. Table top classroom demonstrations can bring to life idealized field experiments. The soil plate oscillator (SPO) apparatus models an idealization of an acoustic landmine detection experiment. The clamped circular elastic plate models the top plate of a plastic buried mine while the soil column supported by the elastic plate models the burial depth. An amplified sinusoidal chirp from a sweep spectrum analyzer drives an AC coil placed below a small magnet attached to the underside of the clamped plate—causing vibration. A laser Doppler vibrometer measured the rms particle velocity at 13 scan locations across the 4.5 inch diameter soil surface in sweeps between 50 Hz and 850 Hz. Results show a fundamental mode shape...

Flexural Isostasy: Constraints From Gravity and Topography Power Spectra

AbstractWe have used spherical harmonic coefficients that describe Earth's gravity anomaly and topography fields to quantify the role of isostasy in contributing to crustal and upper mantle structure. Power spectra reveal that the gravity effect of topography and its flexural compensation contribute significantly to the observed free‐air gravity anomaly spectra for spherical harmonic degree 33 < n < 400, which corresponds to wavelength 100 < λ < 1200 km. The best fit is for an elastic plate (flexure) model with an elastic thickness, Te, of 34.0 ± 4.0 km. Smaller values underpredict the observed gravity spectra while higher values overpredict. The best fit Te is a global average and so there will be regions where Te is lower and higher. This is confirmed in studies of selected regions such as the Hawaiian‐Emperor seamount chain and the Himalaya fold and thrust belt where we show that flexural isostatic anomalies are near zero in regions where Te~34.0 km and of large amplitude in regions of lower and higher Te. Plate flexure may also contribute at higher (n > 400) and lower (n < 33) degrees, but topography appears either uncompensated or fully compensated at these degrees, irrespective of the actual Te. All isostatic models underpredict the spectra at 2 < n < 12 and so we interpret the low‐order Earth's gravity field as caused, at least in part, by nonisostatic processes due to dynamic motions such as those associated with convective upwellings and downwellings in Earth's mantle.

Development of efficient size-dependent plate models for axial buckling of single-layered graphene nanosheets using molecular dynamics simulation

In the current paper, axial buckling characteristics of nanoscaled single-layered graphene sheets (SLGSs) are investigated on the basis of Eringen’s nonlocal elasticity continuum and different plate theories namely as classical plate theory and first-order shear deformation theory. Through implementing of the nonlocal equations into the different types of plate theory, nonlocal plate models are developed to consider the small-scale effects in the axial buckling analysis of SLGSs. Generalized differential quadrature method is utilized to discretize the governing differential equations of the nonlocal elastic plate models along simply-supported and clamped boundary conditions. Afterward, molecular dynamics (MD) simulations are performed for a series of SLGS with various values of side-length and chiralities, the results of which are matched with those of nonlocal plate models to extract the appropriate values of nonlocal parameter. It is found that among the type of boundary conditions, chirality, and nonlocal plate theory, boundary conditions have the most significant influence on the recommended values of nonlocal parameter to predict the axial buckling behavior of SLGSs.

A class of shear deformable isotropic elastic plates with parametrically variable warping shapes

AbstractA homogeneous shear deformable isotropic elastic plate model is addressed in which the normal transverse fibers are allowed to rotate and to warp in a physically consistent manner specified by a fixed value of a real non‐negative warping parameter ω. On letting ω vary continuously (at fixed load and boundary conditions), a continuous family of shear deformable plates is generated, which spans from the Kirchhoff plate at the lower limit , to the Mindlin plate at the upper limit ; for , identifies with the third‐order Reddy plate. The boundary‐value problem for the generic plate is addressed in the case of quasi‐static loads, for which a principle of minimum total potential energy is given. Taking the Kirchhoff deflection wK and the shear angle potentials as basic unknown fields, the problem turns out to be governed by three uncoupled fourth‐order PDEs, of which one coincides with the classical PDE for the Kirchhoff plate, the other two are PDEs of Helmholtz type. A closed‐form representation of the general solution for the plate family is given in terms of to within an arbitrary biharmonic function and an arbitrary pair of conjugate harmonic functions. These arbitrary functions are available to enforce the inherent boundary conditions. The connections between the present theory and other existing theories are pointed out. An application to a family of simply supported circular plates under uniform load is analytically worked out.

Thermo-mechanical vibration of a single-layer graphene sheet and a single-walled carbon nanotube on a substrate

The thermo-mechanical vibration of a single-layer graphene sheet (SLGP) and a single-walled carbon nanotube (SWCNT) on a substrate is studied by using a nonlocal elastic plate model and two nonlocal elastic beam models (including Timoshenko-beam model and Euler-beam model) with quantum effects, respectively. The effect of the van der Waals (vdW) interactions between the SLGP (or the SWCNT) and the substrate on the vibration is obtained. Checking against our molecular dynamics simulations shows that the present models are reasonable. In particular, the radial vibration of the SWCNT on the substrate with quantum effects is further derived through the continuum shell model due to the different vdW forces on each point of the SWCNT circumference. The present models show that the normalized transverse frequency decreases and the normalized radial frequency increases with increasing SWCNT radius, respectively. The radial amplitude of the SWCNT (or the amplitude of the SLGP) nonlinearly increases and the transverse amplitude of the SWCNT linearly increases with increasing temperature for a given distance, respectively. The obtained analytical solution should be of great importance for understanding the thermo-mechanical vibration of nanoelectronic devices on a substrate.

Modelling and analysis of acoustic field in a rectangular enclosure bounded by elastic plates under the excitation of different point force

The coupled acoustic field of fully elastic plate model is described by the modal analysis method. The acoustic potential energy resonance peaks of the fully elastic plate model are significantly more than that of the one elastic plate model due to the influence of the vibration of multi elastic plates. The acoustic field characteristics of the fully elastic plate model are analyzed when the primary excitation source is applied on the different elastic plates. The results show that the coupled acoustic field of the fully elastic plate model is dominated by the structural mode of the elastic plate with primary excitation, and the acoustic mode of the enclosure, and the structural-acoustic coupling between the plate and the enclosure; the structure modes of the other elastic plates have less effects on the acoustic field in the enclosure except the first ones of them.

Model of attenuation of long waves under continuous ice layer

In this work new mathematical model of long wave propagation on water surface with ice cover is proposed. The model of thin elastic plate is used to describe ice layer movement. Equation for ice cover contain additional term to takes into account dissipation effects in the ice cover to explain wave attenuation. Proposed model was reduced to one nonlinear evolution equation for water level perturbation. The expression for wave energy was obtained under assumption of long waves. Proposed model is numerically studied, energy of system is computed. Obtained results are compared with results of suggested before model that takes into account the flow law of Glen.