Viscoelastic Thin Plates Research Articles

Asymptotic modeling of viscoelastic thin plates and slender beams, a unifying approach

Asymptotic modeling of viscoelastic thin plates and slender beams, a unifying approach

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.

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Accurate dynamic modeling of detumbling rotating satellites with large flexible solar panels

Stabilizing a malfunctioning satellite to a low rotating speed is required to ensure safe capture. The detumbling method based on a flexible contactor is efficient and safe due to its relatively large operating force and soft contact. However, flexible components of the target such as the long solar panel will deform large under this intense external force, which makes accurate modeling and analysis of the detumbling process particularly challenging. In this paper, the dynamic problem of detumbling a rotating satellite with large-deformation solar panels is first studied. The target satellite is accurately modeled via global frame approaches, where the new visco-elastic thin plate element of the absolute nodal coordinate formulation is proposed to describe the solar panels. Then the contact model is built based on the geometric relation between the flexible contactor and the solar panel. Finally, dynamic performances during detumbling are compared between flexible and rigid targets, and the detumbling process of the highly flexible satellite is analyzed. All simulation results show that the flexibility of the target should be considered carefully in contact detumbling tasks.

Visco-Elastic Plate Motion after Its Breaking into Parts

Mathematical modelling of visco-elastic plate breaking and consequent deflection of the plate are studied using the simplified formulation. The plate is modelled as a thin visco-elastic plate of constant thickness. The edges of the plate are clamped. The plate deflection is caused by a uniform aerodynamic pressure, which slowly increases in time. The plate deflection before breaking is approximated as quasi-static. The plate breaks instantly then and there, when and where the modified fracture criterion by Petrov and Morozov is achieved. Both the deflections and velocities of the plate before and after breaking are assumed equal.The motion of the plate parts after breaking are highly unsteady and dependent on the viscous properties of the plate. If the viscosity of the plate material is negligible compared with the elastic characteristics of the plate, then the velocity of the plate deflection is discontinuous at the time instant of the plate breaking. This feature of the plate motion after its breaking should be taken into account in interpretation of the numerical results within the linearised model of plate deflection with sudden breaking. It is shown that the plate can break in a cascade way. Each part after the first breaking breaks again. The configuration studied in this paper is specially tailored to highlight the behaviour of the numerical solutions of the plate breaking problems in applications.

Moving load in an ice channel with a crack

The effect of a crack in the ice cover of a channel on strains and deflections of the ice sheet, which are caused by a load moving along the channel, is investigated. The crack is at the center of the channel. The problem is symmetric with a load moving along the crack at constant speed. The problem is formulated within the linear hydroelastic theory. The fluid in the channel is inviscid and incompressible and its motion is potential. The ice sheets are modeled as viscoelastic thin plates clamped to the walls of the channel. The coupled hydroelastic problem is solved by using the Fourier transform along the channel and the normal mode method across the channel. The effects of the load speeds, depth of the channel, ice thickness and characteristics of the load on the hydroelastic response of the ice sheet are investigated. It is shown that the deflections and strains in the ice sheet strongly depend on the speed of the load with respect to the critical speeds of the hydroelastic waves propagating along the channel. The maximum ice deflection occurs at the critical speed of long waves for relatively large values of the retardation time of ice, and at the critical speed of the lowest hydroelastic wave with minimum phase speed for small retardation time of ice. The strain along the center plane of the channel peaks at the critical speed of long waves for ice sheet without a crack. However, it peaks near the critical speed of the lowest hydroelastic wave with minimum phase speed for ice sheet with a crack. The moving load causes larger strains in the ice cover if the load is elongated along the channel for a given total load and load length. The moving load causes smaller strains in the ice cover if the load is elongated along the channel for a given total load and load width.

Blunt body impact onto viscoelastic floating ice plate with a soft layer on its upper surface

Two-dimensional vertical impact of a rigid blunt body onto a floating ice plate is studied. The problem is coupled and unsteady. The liquid is inviscid, incompressible, and of infinite depth. The ice floe is modeled as a thin viscoelastic plate of constant thickness. The plate edges are free of bending stresses and shear forces. The upper surface of the plate is covered with a viscoelastic layer of constant small thickness and negligible inertia. The reaction force of this soft layer is predicted by a nonlinear and one-dimensional Winkler-Kelvin-Voigt model, which does not permit a contact between the rigid body and the ice plate. The soft layer may describe either the presence of snow on the ice or a layer of crushed ice in the place of impact, or can be considered as a way of regularization of problems with concentrated loads. The rigid body touches the upper surface of the soft layer and then suddenly starts to move downward with constant velocity. It is shown that the strains in the ice plate caused by the impact are weakly dependent on the characteristics of the soft layer. The magnitudes and distributions of the strains are studied depending on the length of the ice plate, retardation time of the ice model, thickness of the plate, shape of the rigid body, place of impact, and the impact speed. The value of the retardation time in the soft layer model is discussed with relation to the ice crushing by impact.

Dynamic Vibration Extinguished on a Viscously Elastic Base

The aim of the work is to develop algorithms and a set of programs for studying the dynamic characteristics of viscoelastic thin plates on a deformable base on which it is installed with several dynamic dampers. The theory of thin plates is used to obtain the equation of motion for the plate. The relationship between the efforts and the stirred plate obeys in the hereditary Boltzmann Voltaire integral. With this, a system of integro-differential equations is obtained which is solved by the method of complex amplitudes. As a result, a transcendental algebraic equation was obtained to determine the resonance frequencies, which is solved numerically by the Muller method. To determine the displacement of the point of the plate with periodic oscillations of the base of the plate, a linear inhomogeneous algebraic equation was obtained, which is solved by the Gauss method. The amplitude - frequency response of the midpoint of the plate is constructed with and without regard to the viscosity of the deformed element. The dependence of the stiffness of a deformed element on the frequency of external action is obtained to ensure optimal damping of vibrational vibrations of the plate.

Hydroelastic response of an ice sheet with a lead to a moving load

The hydroelastic symmetric response of a floating ice sheet caused by a pressure moving either in the ice lead or on the infinite ice sheet with a crack (a lead of zero width) is investigated. The ice sheet is modeled as a viscoelastic thin plate. The water is of constant depth. The flow under the ice is potential and linear. A boundary integral method (BIM) for the flow under the ice is combined with the finite difference method for the ice plate with free-free edge conditions to solve the coupled problem of linear hydroelasticity. Numerical results for deflections and stress distributions are shown to agree well with the available results by others. The proposed approach can be applied to problems with different edge conditions and different positions of the load with respect to the lead. The ice responses are studied with respect to the speed of the load. The speed can be subcritical, critical, and supercritical with respect to the critical speed for a floating infinite elastic plate. The speeds of the load, which provide maximum deflection, maximum stress, and maximum wave-making resistance, are determined. All these speeds are different and greater than the critical speed for an infinite elastic plate. The effect of the ice thickness, lead width, and load properties on these speeds is discussed.

Ice response to an underwater body moving in a frozen channel

Strains in the ice cover of a frozen channel, which are caused by a body moving under the ice at a constant speed along the channel, are studied. The channel is of rectangular cross section, the fluid in the channel is inviscid and incompressible. The ice cover is modeled by a thin viscoelastic plate clamped to the channel walls. The underwater body is modeled by a three-dimensional dipole. The intensity of the dipole is related to the speed and size of the underwater body. The problem is considered within the linear theory of hydroelasticity. For small deflections of the ice cover the velocity potential of the dipole in the channel is obtained by the method of images without account for ice deflection in the leading order. The problem of a dipole moving in the channel with rigid walls provides the hydrodynamic pressure on the upper boundary of the channel, which corresponds to the ice cover. This pressure distribution does not depend on the deflection of the ice cover in the leading approximation. The deflections of ice and the strains in the ice cover are independent of time in the coordinate system moving together with the dipole. The problem is solved numerically using the Fourier transform along the channel, the method of normal modes across the channel, and the truncation method for resulting infinite systems of linear equations. It was revealed that the strains in the ice strongly depend on the speed of the dipole with respect to the critical speeds of the hydroelastic waves propagating along the frozen channel. The width of the channel matters even it is much larger than the characteristic length of the ice cover.

Edge Waves Produced by the Motion of a Vessel in an Ice Channel

The Wiener-Hopf technique was used to obtain an analytical solution of the problem of waves produced in a fluid and an ice sheet by the uniform motion of a pressure region modeling an air-cushion vessel on the free surface of the fluid in an ice channel. Ice cover is modeled by two thin semi-infinite viscoelastic plates of constant thickness, floating on the surface of an ideal incompressible fluid of finite depth and separated by the free surface of the fluid. In the moving coordinate system, the plate deflection and fluid elevation are assumed to be steady-state. The wave forces, the elevation of the free surface of the fluid, and the deflection and deformation of the plates are investigated at different vessel speeds and ice sheet thicknesses. It is shown that for some values of the speed, ice sheet thickness, and current pressure, destruction of the ice sheet near the edge is possible.

Movement of external load over free surface of fluid in the ice channel

The solution of a steady three-dimensional problem for the wave disturbances induced by a local pressure distribution moving with uniform speed on open water lead between two semi-infinite floating ice sheets (ice channel) is presented. This external load simulates an air-cushion vehicle (ACV). The problem is formulated within the linear hydroelastic theory. The fluid is assumed to be inviscid and incompressible and its motion is potential. The ice sheets are treated as viscoelastic thin plates. The solution of this problem is constructed using the Fourier transforms and the Wiener-Hopf technique. The displacements of free surface and ice sheets are determined as well as strains in ice and wave resistance acting on ACV at various speeds of its movement: subcritical and supercritical ones relative to the minimum phase velocity of flexural-gravity waves (FGW) in fluid under an ice cover. Special attention is paid to the characteristics of edge waves. It is shown that for some values of load speed, ice thickness and external pressure, ice fracture near the edge is possible.

Response of a linear viscoelastic splitter plate attached to a cylinder in laminar flow

The flow-induced deformation of a viscoelastic thin plate attached to the lee side of a circular cylinder subjected to laminar flow is numerically investigated. An in-house fluid–structure interaction solver couples the sharp-interface immersed boundary method for fluid dynamics with a finite-element method for structural dynamics. Extending published results on elastic materials, the Standard Linear Solid (SLS) model is used to represent viscoelasticity of the plate, governed by the following two parameters: (a) the ratio of the non-equilibrium to equilibrium Young’s modulus (R), and (b) the ratio of dimensionless material damping to the dimensionless non-equilibrium Young’s modulus (τ). The focus of the present study is to examine the dynamic response of the viscoelastic plate at Reynolds number of Re=100. The amplitude and the time to achieve a dynamic steady state of a plate with 0.1<R<5 and 0.1<τ<10 have been investigated. The plate attains a self-sustained time-periodic oscillation with a plateau amplitude for the range of R=[0.1–5] for all values of τ. The dimensionless tip displacement amplitude (AY,tip) is found to be a non-monotonic function of τ. When the forcing frequency (vortex shedding frequency) is lesser (greater) than the natural frequency (considering the equilibrium Young modulus) of the plate, AY,tip decreases (increases) asymptotically. The theoretical analysis of a simple spring–mass–dashpot model of SLS is undertaken to understand the effect of sinusoidal forcing on the plate dynamics. Analytical predictions show general agreement with the non-monotonic behaviour of the numerically computed AY,tip. The results suggest that careful tuning of the damping may be effectively employed to enhance power output for energy extraction applications or to suppress flow-induced vibration when it is detrimental to the structure.

Computations of stresses and energy dissipation in composite thin laminates with the asymptotic vibration theory

The paper develops a computation method for energy dissipation parameters in thin viscoelastic composite plates under steady vibrations. The method is based on applying the asymptotic theory of laminated thin plates, which allows us to calculate accurately enough all six components of the stress tensor under cyclic loading and also complete expressions for the energy dissipation function and the accumulated energy with account of transverse and interlayer shear stresses. With the help of the developed method, we simulate stresses and energy dissipation parameters in a viscoelastic plate of fiber laminated carbon–plastic composite under flexural vibrations. Computations showed that transverse stresses and especially interlayer shear stresses contribute considerably to the integral energy dissipation coefficient of composite plates. This contribution is the most significant for rigid structures with a relatively small energy dissipation coefficient.

Демпфірування неізотермічних осесиметричних згинних коливань в'якопружних пластин

A problem on the forced monoharmonic axisymmetric bending vibrations and dissipative heating of circular viscoelastic plate with the piezoelectric sensors and actuators is considered. Viscoelastic behavior of passive (without piezoeffect) and piezoactive materials is described using the concept of complex moduli which depend on temperature. The nonlinear coupled problem of electrothermoviscoelasticity is solved by numerical methods. The influence of the boundary conditions and temperature of disspative heating on active damping of harmonic vibrations of thin viscoelastic plates with the simultaneous use of sensors and actuators is investigated. For modeling viscoelastic properties of passive and piezoelectric materials, linear models of integral type viscoelasticity are used, which are most effective for simulating the dissipative properties of materials in the linear region. If the material characteristics depend on temperature, investigation of the influence of temperature of dissipative heating is reduced to solution of complicated nonlinear systems of differential equations.

Flexural-gravity wave resistances due to a moving point source on 2-D infinite floating beam

The flexural-gravity wave responses due to a load steadily moving or suddenly accelerated along a rectilinear orbit are analytically studied within the framework of the linear potential theory. A thin viscoelastic plate model is used for a very large floating structure. The initially quiescent fluid in the ocean is assumed to be homogenous, incompressible, and inviscid, and the disturbed motion be irrotational. A moving line source on the plate surface is considered as a moving point in the two-dimensional coordinates. Under the assumptions of small-amplitude wave motion and small plate deflection, a linear fluid-plate coupling model is established. The integral solutions for the surface deflections and the wave resistances are analytically obtained by the Fourier transform method. To study the dynamic characteristics of the flexural-gravity wave response, the asymptotic representations of the wave resistances are derived by the residue theorem and the methods of stationary phase. It shows that the steady wave resistance is zero when the speed of moving load is less than the minimal phase speed. The wave resistances due to the accelerate motion consist of two parts, namely the steady and transient wave responses. Eventually the transient wave resistance declines toward zero and the wave resistance approaches the steady component as the time goes to the infinity. Furthermore, the effect of the strain relaxation time for this viscoelastic plate is studied and it exhibits more influence for a high-speed motion.

Deflection of ice cover caused by an underwater body moving in channel

Deflections and strains in an ice cover of a frozen channel caused by an underwater body moving under the ice with a constant speed along the channel are studied. The channel is of rectangular cross section, the fluid in the channel is inviscid and incompressible. The ice cover is clamped to the channel walls. The ice cover is modeled by a thin viscoelastic plate. The underwater body is modeled by a three-dimensional dipole. The intensity of the dipole is related to the speed and size of the underwater body. The problem is considered within the linear theory of hydroelasticity. For small deflections of the ice cover the velocity potential of the dipole in the channel is obtained by the method of images in leading order without account for the deflection of the ice cover. The problem of moving dipole in the channel with rigid walls provides the hydrodynamic pressure on the upper boundary of the channel, which corresponds to the ice cover. This pressure distribution does not depend on the deflection of the ice cover in the leading approximation. The deflections of the ice and strains in the ice plate are independent of time in the coordinate system moving together with the dipole. The problem is solved numerically using the Fourier transform, method of the normal modes and the truncation method for infinite systems of algebraic equations.

Shock-induced thermal wave propagation and response analysis of a viscoelastic thin plate under transient heating loads

This paper is devoted to the thermal shock analysis for viscoelastic materials under transient heating loads. The governing coupled equations with time-delay parameter and nonlocal scale parameter are derived based on the generalized thermo-viscoelasticity theory. The problem of a thin plate composed of viscoelastic material, subjected to a sudden temperature rise at the boundary plane, is solved by employing Laplace transformation techniques. The transient responses, i.e. temperature, displacement, stresses, heat flux as well as strain, are obtained and discussed. The effects of time-delay and nonlocal scale parameter on the transient responses are analyzed and discussed. It can be observed that: the propagation of thermal wave is dynamically smoothed and changed with the variation of time-delay; while the displacement, strain, and stress can be rapidly reduced by nonlocal scale parameter, which can be viewed as an important indicator for predicting the stiffness softening behavior for viscoelastic materials.

Numerical modeling of ice-water system response based on Rankine source method and finite difference method

Based on the thin viscoelastic plate assumption and the potential flow theory, the unified mathematical model was derived for the ice-water vibration problem induced by air cushion vehicle (ACV) sailing on different media (pure ice sheet, pure water, broken ice, semi-water-semi-ice, etc.). In order to solve this model, a hybrid algorithm combining Rankine source method (RSM) and finite difference method (FDM) was proposed. The displacement response and stress distribution as well as ice-breaking effect of ice sheet can be obtained by using this numerical method. Besides, a comparison with the theoretical results or experiment results is made, and a good agreement is achieved between the calculation results and the existing research results. Finally, taking the pure ice surface and semi-water-semi-ice surface as an example, some calculations are made according to the typical physical parameters of ice sheet. Our calculations shows that the mathematical model and numerical method can not only calculate the ice displacement response and its distribution characteristics of the internal stress as well as ice-breaking effect, but also can capture the critical speed of ACV.

Vibrations of an ice sheet with crack under a time-periodic load

The analytic solution of the problem of vibrations of an ice sheet with a rectilinear crack floating on the surface of an ideal incompressible fluid of shallow depth under the action of a local zone of the time-periodic pressure is obtained. The ice sheet is simulated by two thin viscoelastic semiinfinite plates of different thickness. Various conditions on the crack edges are considered. Far field asymptotics are investigated and it is revealed that the predominant directions of wave propagation at an angle to the crack can be distinguished in the far field in the case of contact of two plates of different thickness. In the case of contact of identical plates, a waveguide mode propagating along the crack is excited. It is shown that the waveguide mode is the same for the plates with the free edges and the free overlap since the part of the solution symmetric about the crack is the same while the difference between the solutions is caused by the antisymmetric part of the solution.