Two-dimensional problem of periodic loading of an elastic plate floating on the surface of an infinitely deep fluid

The action of external periodic pressure on an elastic plate floating on the surface of a fluid assumed to be ideal and incompressible is examined by the method of normal modes in the linear formulation. The behavior of the matrix of coefficients of the hydrodynamic load on the plate is considered in detail for different frequencies. The behavior of the plate under localized periodic loading is compared for the cases of a heavy fluid with a finite or infinite depth and for a weightless infinite-depth fluid. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 61–72, May–June, 2005.     

Numerical and asymptotic study of the two-dimensional problem of the hydroelastic behavior of a floating plate in waves

The problem of the behavior of a floating elastic plate in waves is solved numerically. The normal mode method is used. For a fluid of finite depth, the hydrodynamic coefficients are obtained in explicit form. Numerical results are compared with experimental data for the stress distribution in the plate and also with numerical results of other authors. The results are in good agreement for not very short waves. For incident waves whose wavelength is comparable with the length of the plate, a long-wave approximation of the solution is proposed. Within the framework of this approximation, the solution is given in analytical form. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 90–96, March–April, 2000.     

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. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 153–161, July–August, 2006.     

Contact problem for a ring plate on an elastic half-space. Variational approach

A contact problem of an axisymmetrically loaded flexible ring plate lying frictionlessly on an elastic half-space is considered. The plate subsidences are represented as a power series with unknown coefficients, which are determined by the Rayleigh-Ritz method using the minimum condition for the total strain energy of the plate and the elastic foundation. The method of orthogonal polynomials is used implicitly. Belarussian State Polytechnical Academy, Minsk 220027. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 193–198, January–February, 1999.     

Plane linear problem of the immersion of an elastic plate in an ideal incompressible fluid

A plane unsteady-state linear problem of the immersion of an elastic plate of finite length in an ideal incompressible weightless fluid is considered. The deflection of the plate and the velocity of its points are known at the initial moment of time. The fluid occupies the lower halfplane, and its boundary outside the plate is free. The plate which is the bottom of a structure immersed in the fluid with a constant velocity is modeled by an Euler beam. At the initial stage of immersion, when the displacement of the structure is much smaller than the length of the plate, the plate deflection and the distribution of bending stresses in it are determined. The model used allows one to estimate the maximum stresses occurring in the elastic plate during its impact on water and to predict the moment and site of its occurrence. Calculations are performed under the conditions of the experiment carried out in MARINTEX (Norway). Qualitative agreement between the numerical and experimental results is shown. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 150–160, May–June, 1999.     

Coupled hydrodynamic and structural analysis of compressible jet impact onto elastic panels

This paper is concerned with the initial stage of a compressible liquid jet impact onto an elastic plate. The fluid flow is governed by the linear wave equation, while the response of the plate is governed by the classical linear dynamical plate equation. The coupling between the fluid flow and the plate deflection is taken into account through the dynamic and kinematic conditions imposed on the wetted part of the plate. The problem is solved numerically by the normal mode method. The principal coordinates of the hydrodynamic pressure and plate deflections satisfy a system of ordinary differential and integral equations. A time stepping method based on the Runge–Kutta scheme is used for the numerical integration of the system. Calculations are performed for two-dimensional, axisymmetric and three-dimensional jet impacts onto an elastic plate. The effects of the impact conditions and the elastic properties of the plate on the magnitudes of the elastic deflections and bending stresses are analysed.     

Wave impact on the center of an Euler beam

The problem of a symmetric wave impact on the Euler beam is solved by the normal modes method. The liquid is supposed to be ideal and incompressible. The initial stage of impact when hydrodynamic loads are very high and the beam is wetted only partially is considered. The flow of a liquid and the size of the wetted part of the body are determined by the Wagner approach with a simultaneous calculation of the beam deflection. The specific features of the developed numerical algorithm are demonstrated and the criterion of its stability is specified. In addition to a direct solution of the problem, two approximate approaches within the framework of which the dimension of the contact region is found ignoring the deformations of the plate are considered. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhnaika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 134–147, September–October, 1998.     

Edge waves in a shallow fluid beneath a fractured elastic plate

It is shown that a fracture in an elastic plate floating on the surface of a shallow liquid layer is a waveguide along which wave energy can be transported. The edge wave velocity is less than the velocity of flexural-gravity waves. The existence of an antisymmetric edge wave mode depends on the Poisson's ratio of the elastic plate. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 185–189, July–August, 1994.     

Numerical simulation of the interaction of gravity waves with elastic ice floes

The self-consistent motion of a fluid and elastically oscillating plates partially covering the fluid is simulated numerically in the linear approximation. The problem is reduced to the simultaneous solution of the Laplace equation for the fluid and the equation of elastic plate oscillations for the ice. The numerical and analytical solutions, the latter obtained from an integral equation containing the Green’s function, are compared. To solve the problem numerically, the boundary element method for the Laplace equation and the finite element method for the equation describing the elastic plate are proposed. The coefficients of transmission and reflection of surface gravity waves from the floating plates are calculated. It is shown that the solution may be quasi-periodic with characteristics determined by the initial values of the wave and ice-floe parameters. The ice floes may exert a filtering effect on the surface wave spectrum, essentially reducing its most reflectable components. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 123–131, May–June, 2000.     

Boundary film shear elastic modulus effect in hydrodynamic contact. Part I: theoretical analysis and typical solution

Boundary film shear elastic modulus effect is analyzed in a hydrodynamic contact. The contact is one-dimensional composed of two parallel plane surfaces, which are, respectively, rough rigid with rectangular micro projections in profile periodically distributed on the surface and ideally smooth rigid. The whole contact is consisted of cavitated area and hydrodynamic area. The hydrodynamic area consists of many micro Raleigh bearings which are discontinuously and periodically distributed in the contact. Analysis is thus carried out for a micro Raleigh bearing in this contact. The hydrodynamic contact in this micro Raleigh bearing consists of boundary film area and fluid film area which, respectively, occur in the outlet and inlet zones. In boundary film area, the film slips at the upper contact surface due to the limited shear stress capacity of the film–contact interface, while the film does not slip at the lower contact surface due to the shear stress capacity large enough at the film–contact interface. In boundary film area, the viscosity, density and shear elastic modulus of the film are varied across the film thickness due to the film–contact interactions, and their effective values are used in modeling, which depend on the film thickness. The analytical approach proposed by Zhang (J Mol Liq 128:60–64, 2006) and Zhang et al. (Int J Fluid Mech Res 30:542–557, 2003) is used for boundary film area. In fluid film area, the film does not slip at either of the contact surfaces, and the shear elastic modulus of the film is neglected. Conventional hydrodynamic analysis is used for fluid film area. The present paper presents the theoretical analysis and a typical solution. It is found that for the simulated case the boundary film shear elastic modulus effects on the mass flow through the contact, the overall film thickness of the contact and the carried load of the contact are negligible but the boundary film shear elastic modulus effect on the local film thickness of the contact may be significant when the boundary film thickness is on the 1 nm scale and the contact surfaces are elastic. In Part II will be presented detailed results showing boundary film shear elastic modulus effects in different operating conditions.

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On an inverse problem of bending of a physically nonlinear inhomogeneous plate

An elastic plate with a physically nonlinear inclusion of an arbitrary shape is considered. This plate is subjected to pure bending under the action of transverse forces and bending moments applied at the external boundary of the plate. There are no loads distributed over the surface. The problem of finding external actions that provide a necessary uniform moment state in the inclusion, i.e., prescribed constant moments and curvatures, is formulated and solved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 104–107, September–October, 2007.     

Elastic–plastic contact force history and response characteristics of circular plate subjected to impact by a projectile

A new elastic–plastic impact–contact model is proposed in this paper. By adopting the principle of minimum acceleration for elastic–plastic continue at finite deformation, and with the aid of finite difference method, the proposed model is applied in the problem of dynamic response of a clamped thin circular plate subjected to a projectile impact centrally. The impact force history and response characteristics of the target plate is studied in detail. The theoretical predictions of the impact force and plate deflection are in good agreements with those of LDA experimental data. Linear expressions of the maximum impact force/transverse deflection versus impact velocity are given on the basis of the theoretical results. The project supported by the National Natural Science Foundation of China (10532020).     

Stability of a viscoelastic plate in fluid flow

The stability of an infinite viscoelastic plate on an elastic foundation in a viscous incompressible flow is studied. The Navier-Stokes system is linearized for an exponential velocity profile. The problem is reduced by a Fourier-Laplace transform to a system of ordinary differential equations, whose solution is found in the form of convergent series. The roots of the dispersion relation that characterize the stability of the system are found numerically. The effect of the viscosities of the fluid and the plate on the stability of the waves propagating upstream and downstream is studied. The results are compared with available data on the stability of a viscoelastic plate in an ideal fluid flow. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 66–74, July–August, 2006.     

Time-dependent interaction between two inviscid fluid flows separated by a semi-infinite non-rigid plate in a flat-walled channel

The problem of the time-dependent interaction between two inviscid weightless fluids separated by a semi-infinite non-rigid plate in a channel with fixed rigid walls is solved in the linear approximation. The general case of deformation and harmonic oscillations of the plate (flapping mover) are considered. The time-dependent hydrodynamic reaction forces, the position of the interface, and the dynamic characteristics of the mover are determined. Kazan’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–76, January–February, 1997.     

Boundary film shear elastic modulus effect in hydrodynamic contact. Part II: influence of operational parameters

The present paper is the subsequent research of the first part (Theor Comput Fluid Dyn, 2009). It investigates the boundary film shear elastic modulus effect in a hydrodynamic contact in different operating conditions. The hydrodynamic contact is one-dimensional, composed of two parallel plane surfaces, which are respectively rough rigid with rectangular micro projections in profile periodically distributed on the surface and ideally smooth rigid. The whole contact consists of cavitated area and hydrodynamic area. The hydrodynamic area consists of many micro Raleigh bearings which are discontinuously and periodically distributed in the contact. The hydrodynamic contact in a micro Raleigh bearing consists of boundary film area and fluid film area which, respectively, occur in the outlet and inlet zones. In boundary film area, the film slips at the upper contact surface due to the limited shear stress capacity of the film–contact interface, while the film does not slip at the lower contact surface due to the shear stress capacity of the film–contact interface large enough. In boundary film area, the viscosity, density, and shear elastic modulus of the film are varied across the film thickness due to the film–contact interactions, and their effective values are used in modeling which depends on the film thickness. In fluid film area, the film does not slip at either of the contact surfaces, and the shear elastic modulus of the film is neglected. It is found from the simulation results that the boundary film shear elastic modulus influences are normally negligible on the mass flow through the contact, the carried load of the contact and the overall film thickness of the contact, and the boundary film shear elastic modulus would normally influence the local film thickness in an elastic contact when the local film thickness is on the film molecule diameter scale. It is also found that the boundary film shear elastic modulus effect has the tendency of being increased with the reduction of the width of a micro contact. It is increased with the reduction of the boundary film–contact interfacial shear strength or with the increase of the critical boundary film thickness, while it is strongest at certain values of the contact surface roughness, the width ratio of fluid film area to boundary film area, and the lubricant film shear elastic modulus.

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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. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 152–157, November–December, 2006.     

Dynamics of a prestressed incompressible layered half-space under moving load

The paper addresses a plane problem: a concentrated force acts on a plate resting on an elastic half-space with homogeneous prestrain. The equations of motion of the plate incorporate shear and rotary inertia. The half-space is assumed to be incompressible and isotropic in the natural state. The elastic potential is given in general form and is only specified for numerical purposes. The dependence of the critical velocity of the load and the stress-strain state on the prestresses is analyzed for different ratios between the stiffnesses of the layer and half-space and different contact conditions. The calculations are carried out for a half-space with Bartenev-Khazanovich potential __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 36–54, March 2008.     

Diffraction of surface waves on an inhomogeneous elastic plate

The oblique incidence of small-amplitude waves on an elastic semi-infinite composite plate floating on the free surface of finite-depth water is studied. The front part of a constant-width plate is highed to the basic part and has characteristics different from those of the basic part. The reflection and transmission coefficients of the waves and the vertical displacements of the plate are determined. It is shown that the heterogeneity of the plate material exerts a strong effect on surface-wave diffraction. Methods for decreasing the elastic deformations of the basic part of the plate are proposed. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 42–48, July–August, 2000.     

Stability of a rectangular plate under biaxial tension

The stability problem of a rectangular plate undergoing uniform biaxial in-plane tensile strain is solved using the three-dimensional equations of nonlinear elasticity. The surfaces of the plate are stress-free, and special boundary conditions that allow one to separate variables in the linearized equilibrium equations are specified on the lateral surfaces. For three particular models of incompressible materials, the critical curves are constructed and the instability region is determined in the plane of the loading parameters (the multiplicities of elongations of the plate material in the unperturbed equilibrium state). The numerical results show that for thin plates loaded by tensile stresses, the size and shape of the instability region depend only slightly on the relation among the length, width, and thickness of the plate. Based on the results obtained, a simple approximate stability criterion is proposed for an elastic plate under tensile loads. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 94–103, July–August, 2007.     

On a Fluid-Structure Model in Which the Dynamics of the Structure Involves the Shear Stress Due to the Fluid

In this paper we consider a model for fluid-structure interaction. The hybrid system describes the interaction between an incompressible fluid in a three-dimensional container with interior a fixed domain and a thin elastic plate, the interface, which coincides with a flexible flat part of the surface of the vessel containing the fluid. The motion of the fluid is described by the linearized Navier–Stokes equations and the deformation of the plate by the classical plate equations for in-plane motions, modified to include the viscous shear stress which the fluid exerts on the plate as well as damping of Kelvin–Voigt type. We establish the existence of a unique weak solution of the interactive system of partial differential equations by considering an appropriate variational formulation. Uniform stability of the energy associated with the model is shown under the assumption that the potential plate energy is dominated by the dissipation induced by the viscosity of the fluid. The retention of the physical parameters in the problem is an a priori requirement in this physical condition.