Isotropic Elastic Plate Research Articles

WIND EFFECTS ON SLOSHING OF A FLOATING ROOF IN AN OPEN-TOPPED CYLINDRICAL LIQUID STORAGE TANK

Sloshing of a floating roof in an open-topped cylindrical liquid storage tank under wind loads is investigated analytically. Wind tunnel test in a turbulent boundary layer is carried out to measure the wind pressure distributing over the roof surface. The measured data for the wind pressure is utilized to predict the wind-induced dynamic response of the floating roof, which is idealized herein as an isotropic elastic plate of uniform stiffness and mass. The dynamic interaction between the liquid and the floating roof is taken into account exactly within the framework of linear potential theory. Numerical results are presented which illustrate the significant effect of wind loads on the sloshing response of the floating roof.

Finite-amplitude elastic waves interacting with temperature and nonequilibrium atomic defect fields

Nonlinear dynamics of one-dimensional longitudinal waves in isotropic elastic plates was studied taking into account the interaction of displacement fields, temperature, and concentration of nonequilibrium (relaxing) atomic point defects. A nonlinear evolution equation for describing the self-consistent field of longitudinal thermoelastic strain was derived. The effect of generation-recombination processes on the evolution of nonlinear localized and periodic waves was analyzed. In the single-wave approximation, an equation was derived which describes the amplitude variation of nonlinear waves; based on this equation, characteristic features of damping of these waves were considered taking into account low-and high-frequency losses. The interaction of counterpropagating waves is briefly discussed taking into account dissipative effects.

Bending of piezoelectric plates with a circular hole

Departing from the refined theory of transversely isotropic piezoelectric plates, an analytical solution for the bending of an infinite piezoelectric plate with a circular hole is obtained. Expressions of moment, stress and electric displacement concentration factors are presented in closed form. When the piezoelectric coupling is absent, the results reduce to the corresponding solutions for the transversely isotropic elastic plates. Some numerical results for PZT-6B piezoelectric ceramics are obtained and illustrated by figures. These results show that the effect of piezoelectric coupling on the concentration factors is not negligible.

Simulation and measurement of the optical excitation of the S1 zero group velocity Lamb wave resonance in plates

Recent reports on the thermoelastic generation of Lamb waves in isotropic elastic plates show that a laser source efficiently excites a resonance that occurs at the minimum frequency of the first order symmetric (S1) Lamb mode. The group velocity of the Lamb wave goes to zero at this frequency while the phase velocity remains finite, and the resonance is referred to as the S1 zero group velocity (S1 ZGV) resonance. The S1 ZGV resonance can be employed for the nondestructive evaluation of the elastic properties of plates or plate thickness. A model for the generation of elastic waves in plates using an intensity-modulated continuous wave laser source is developed and used to study the behavior of the S1 ZGV resonance. The effects of the laser source parameters on the generation of the S1 ZGV resonance are explored, and the spatial distribution of the displacement produced at the resonance frequency is determined. The predicted displacement spectrum of Lamb waves generated in micron scale plates is found to compare well with experimental measurements. In addition, experimental measurements demonstrate that the S1 ZGV resonance can be used to map subsurface features in thin (4μm) membranes at high ultrasonic frequencies (700MHz).

Vibration of an incompressible isotropic linear elastic rectangular plate with a higher-order shear and normal deformable theory

We use a mixed higher-order shear and normal deformable plate theory (HOSNDPT) of Batra and Vidoli with Poisson's ratio equal to 0.49 and the finite element method to analyze vibrations of a homogeneous isotropic rectangular plate made of an incompressible linear elastic material. Through-the-thickness integrals are evaluated exactly, and those over an element in the midplane of the plate are evaluated by using the 2 × 2 Gauss quadrature rule. The plate theory equations are used to ascertain frequencies of a clamped–clamped and a clamped–free square/rectangular plate of different aspect ratios. Through-the-thickness modes of vibration valid for compressible and incompressible materials and missed by previous investigators are also identified. Computed frequencies are found to match well with those deduced from the analytical solution.

Elastoplastic model of cylindrical projectile-plate interaction

Elastoplastic models of the transverse impact of a cylindrical solid projectile on an isotropic elastic Uflyand-Mindlin plate have been studied using numerical simulation methods. Displacements of the plate points outside the contact region are caused by a transverse shear wave propagating at a finite velocity. This wave is formed at the moment of impact and represents a surface of strong discontinuity. Behind the transverse wave front, the unknown dynamic quantities are represented in the form of single-term ray expansions. The local crumpling of the target material proceeds in a quasi-static manner and its dependence on the contact force is described using an elastoplastic model. The dependence of the contact force on the elastoplastic properties of a target plate at the site of impact and on the contact disk inertia has been considered.

Stress and mixed boundary conditions for two-dimensional dodecagonal quasi-crystal plates

For plate bending and stretching problems in two-dimensional (2D) dodecagonal quasi-crystal (QC) media, the reciprocal theorem and the general solution for QCs are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order. The method developed by Gregory and Wan is used to generate necessary conditions which the prescribed data on the edge of the plate must satisfy in order that it should generate a decaying state within the plate; these decaying state conditions are obtained explicitly for axisymmetric bending and stretching of a circular plate when stress or mixed conditions are imposed on the plate edge. They are then used for the correct formulation of boundary conditions for the interior solution. For the stress data, our boundary conditions coincide with those obtained in conventional forms of plate theories. More importantly, appropriate boundary conditions with a set of mixed edge-data are obtained for the first time. Furthermore, the corresponding necessary conditions for transversely isotropic elastic plate are obtained directly, and their isotropic elastic counterparts are also obtained.

Fractional-derivative viscoelastic model of the shock interaction of a rigid body with a plate

The impact of a rigid body upon an elastic isotropic plate is investigated for the case when the equations of motion take rotary inertia and shear deformation into account. The impactor is considered as a mass point, and the contact between it and the plate is established through a buffer involving a linear-spring–fractional-derivative dashpot combination, i.e., the viscoelastic features of the buffer are described by the fractional-derivative Maxwell model. It is assumed that a transient wave of transverse shear is generated in the plate, and that the reflected wave has insufficient time to return to the location of the spring’s contact with the plate before the impact process is completed. To determine the desired values behind the transverse-shear wave front, one-term ray expansions are used, as well as the equations of motion of the impactor and the contact region. As a result, we are led to a set of two linear differential equations for the displacements of the spring’s upper and lower points. The solution of these equations is found analytically by the Laplace-transform method, and the time-dependence of the contact force is obtained. Numerical analysis shows that the maximum of the contact force increases, tending to the maximal contact force when the fractional parameter is equal to unity.

Optimal design of a compound plate weakened by a periodic crack system

We use the minimax criterion to perform atheoretical analysis of determining the optimal interference for fitting elastic inclusions into holes in an isotropic elastic plate weakened by a periodic system of rectilinear cracks. We construct a closed system of algebraic equations that allows us to minimize the fracture parameters depending on the geometric and mechanical characteristics of the inclusions. The obtained fitting interference for the inclusions ensures an increase in the bearing strength of the compound plate under bending.

The decomposed form of magnetoelastic beams with free faces

The decomposed form of an isotropic elastic plate is extended to a bending magnetoelastic beam with free faces, and the corresponding decomposition theorem is presented. It is shown that the stress state of the magnetoelastic beam without transverse surface loadings can be decomposed into three parts: the interior state, the Papkovich-Fadle state (in shortened form the P-F state) and the magnetic state. Based on the three new lemmata, a rigorous mathematical proof of the decomposition theorem is given concisely and directly, which is independent of the Papkovich-Fadle eigenfunction expansion of biharmonic functions. In the proof course, some basic mathematical methods are used only, so the proof is more convenient for being understood. More importantly, these three states correspond to the three equations derived in the refined theory of magnetoelastic beams: the fourth-order equation, the transcendental equation and the magnetic equation, respectively. Therefore, this work virtually proves the consistency between the decomposition theorem and the refined theory of bending magnetoelastic beams.

Analytical solution for vibration of an incompressible isotropic linear elastic rectangular plate, and frequencies missed in previous solutions

An analytical solution is given for free vibration of a simply supported rectangular plate made of an incompressible linear elastic isotropic material. Through-the-thickness modes of vibration valid for compressible and incompressible materials and missed by previous investigators are also identified.

A problem of the bending of a plate for a doubly connected domain with a partially unknown boundary

The problem of the bending of an isotropic elastic plate, bounded by two rectangles with vertices lying on the same half-line, drawn from the common centre, is considered. The vertices of the inner rectangle are cut by convex smooth arcs (we will call the set of these arcs the unknown part of the boundary). It is assumed that normal bending moments act on each rectilinear section of the boundary contours in such a way that the angle of rotation of the midsurface of the plate is a piecewise-constant function. The unknown part of the boundary is free from external forces. The problem consists of determining the bending of the midsurface of the plate and the analytic form of the unknown part of the boundary when the tangential normal moment acting on it takes a constant value, while the shearing force and the normal bending moments and torques are equal to zero. The problem is solved by the methods of the theory of boundary-value problems of analytical functions.

センターポンツーン付シングルデッキ型浮屋根を有する円筒液体貯槽の地震時スロッシング応答

An analytical solution is presented to predict the sloshing response of a cylindrical liquid storage tank with a single-deck type floating roof under seismic excitation. The floating roof is considered to be composed of an inner deck stiffened with outer and center pontoons. The deck is idealized as an isotropic elastic plate with uniform stiffness and mass, while the outer and center pontoons are modeled as an elastic curved beam and a rigid circular plate, respectively. The contained liquid is assumed to be inviscid, incompressible and irrotational. The dynamic interaction between the floating roof and the liquid is taken into account exactly within the framework of linear potential theory. By expanding the response of the coupled deck-pontoon system into free vibration modes in air and employing the Fourier-Bessel expansion technique in cylindrical coordinates, the solution is obtained in an explicit form which will be useful for parametric understanding of the sloshing behavior and preliminary study in the early design stage. Numerical results are provided to investigate the effect of the presence of the center pontoon on the sloshing response of the floating roof.

Boundary Conditions for Plate Bending in One-dimensional Hexagonal Quasicrystals

For plate bending in one-dimensional (1D) hexagonal quasicrystals (QCs), the reciprocal theorem and the general solution for QCs media are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order for plates of general edge geometry and loadings. Through generalizing the method developed by Gregory and Wan, a set of necessary conditions on the edge-data for the existence of a rapidly decaying solution is established. The prescribed data must satisfy these conditions in order that they should generate a decaying state. When a set of stress edge-data or mixed edge-data is imposed on the plate edge, these decaying state conditions for the case of axisymmetric deformation of 1D hexagonal QC plates are derived explicitly. They are then used for the correct formulation of boundary conditions for the plate theory solution (or the interior solution). Furthermore, in the absence of phonon–phason fields coupling effect, corresponding necessary conditions for the case of transversely isotropic elastic plates are derived subsequently, and their isotropic elastic counterparts are also obtained.

Bending of elastic plates with a physically nonlinear inclusion

An infinite elastic isotropic plate with an elliptical, physically nonlinear inclusion loaded at infinity by uniformly distributed moments is considered. Surface loads are absent. The problem of the stress-strain state of the plate is solved in a closed form. It is shown that, for reasonably general stress-strain relations for the inclusion, the bending-moment field (and the corresponding curvatures) in the inclusion is homogeneous.

Hydroelastic analysis of a floating plate of finite draft

In this paper the diffraction of incident surface water waves by a very large floating structure of finite thickness and draft is under consideration. The floating structure (platform) of finite, but small, draft is modelled by a thin elastic isotropic plate. An analytical and numerical study of the hydroelastic behavior of the plate is presented. The problem is solved for the main case of finite water depth. To simplify the equations involved, the half-plane problem is considered. Recommendations are given for other horizontal planforms of the plate and possible extensions of the method developed. The deflection of the plate is represented as a series of solutions with respect to the plate draft; we present and use first and second terms. Each of these terms can be written as a series of exponential functions multiplied by coefficients. The integro-differential formulation and equation obtained, the dispersion relations in the plate region and open water, the Green’s theorem, the geometrical-optics approach and the Lindstedt method are used to determine reduced wavenumbers of zero and first draft order. Further, sets of equations are derived to find the corresponding wave-mode amplitudes. Results obtained for the plate of finite draft are compared to those for a plate with zero draft.

Analysis of thick plates by using a higher-order shear and normal deformable plate theory and MLPG method with radial basis functions

Infinitesimal deformations of a homogeneous and isotropic thick elastic plate have been analyzed by using a meshless local Petrov–Galerkin (MLPG) method and a higher-order shear and normal deformable plate theory (HONSDPT). Radial basis functions (RBF) are employed for constructing trial solutions, while a spline function is used as the weight function over a local subdomain. The present method uses a number of randomly distributed nodes in the domain and is truly meshless. Two types of RBFs, i.e. multiquadrics (MQ) and thin plate splines (TPS), are employed and effects of their shape parameters on the quality of the computed solution are examined for deformations of thick plates under different boundary conditions. It is found that the present MLPG formulations give results very close to those obtained by other researchers. A benefit of using RBFs is that no special treatment is needed to impose the essential boundary conditions, which substantially reduces the computational cost.

Rayleigh lamb waves in micropolar isotropic elastic plate

The propagation of waves in a homogeneous isotropic micropolar elastic cylindrical plate subjected to stress free conditions is investigated. The secular equations for symmetric and skew symmetric wave mode propagation are derived. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduces to Rayleigh surface wave frequency equation. Thin plate results are also obtained. The amplitudes of displacements and microrotation components are obtained and depicted graphically. Some special cases are also deduced from the present investigations. The secular equations for symmetric and skew symmetric modes are also presented graphically.

Sloshing in a Cylindrical Liquid Storage Tank With a Floating Roof Under Seismic Excitation

An analytical solution is presented to predict the sloshing response of a cylindrical liquid storage tank with a floating roof under seismic excitation. The contained liquid is assumed to be inviscid, incompressible, and irrotational, while the floating roof is idealized as an isotropic elastic plate with uniform stiffness and mass. The dynamic interaction between the floating roof and the liquid is taken into account exactly within the framework of linear potential theory. By expanding the response of the floating roof into free-vibration modes in air and employing the Fourier–Bessel expansion method in cylindrical coordinates, the solution is obtained in an explicit form, which will be useful for parametric understanding of the sloshing behavior and preliminary study in the early design stage. Numerical results are also provided to investigate the effect of the stiffness and mass of the floating roof on the sloshing response.

Inverse doubly periodic problem of the theory of bending of a plate with elastic inclusions

Based on the balanced strength principle, a problem of determining the optimal interference for fitting elastic inclusions into holes of an isotropic elastic plate weakened by a doubly periodic system of circular holes is solved. A closed system of algebraic equations is derived, which allows solving this problem. The resultant interference increases the load-carrying capacity of the composite plate being bent.