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.      

Interaction of surface and flexural-gravity waves in ice cover with a vertical wall

The problem of the interaction of surface and flexural-gravity waves with a vertical barrier is solved in a two-dimensional formulation. It is assumed that the fluid is ideal and incompressible, has infinite depth, and is partially covered with ice. The ice cover is modeled by an elastic plate of constant thickness. The eigenfrequencies and eigenmodes of oscillation of the floating elastic ice plate, the deflection and deformation of ice, and the forces acting on the wall are determined.     

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.     

Numerical investigation of the effect of boundary conditions on hydroelastic stability of two parallel plates interacting with a layer of ideal flowing fluid

The paper studies the hydroelastic stability of two parallel identical rectangular plates interacting with a flowing fluid confined between them. General equations describing the behavior of ideal compressible liquid in the case of small perturbations are written in terms of the perturbation velocity potential and transformed using the Bubnov–Galerkin method. The small deformations of elastic plates are defined using the first-order shear deformation plate theory. A mathematical formulation of the dynamic problem for elastic structures is developed using the variational principle of virtual displacements, which takes into account the work done by the inertial forces and hydrodynamic pressure. The numerical solution of the problem is carried out in three-dimensional formulation by means of the finite element method. A stability criterion is based on the analysis of complex eigenvalues of the coupled system of equations obtained for different values of flow velocity. The existence of different types of instability has been shown depending on the combinations of the kinematic boundary conditions defined at the edges of both plates. We considered both the symmetric and asymmetric types of clamping. It has been found that the dependence of the lowest eigenfrequency of two parallel plates on the height of quiescent fluid is nonmonotonic with a pronounced peak. At the same time, critical velocities of instability change insignificantly if the distance between plates is greater than half of the maximum linear dimensions of the structure. It should be noted that the critical velocities of divergence increase monotonically with growth of the height of the fluid layer, but critical velocities for the onset of flutter instability have sharp jumps. The cause of these jumps is a change in the mode shapes at which the system loses stability.     

Flutter of a rectangular panel as a part of a thin wedge surface

The stability of an elastic plate and an elastic panel placed in a supersonic gas flow is studied in a nonlinear formulation. The gas velocity vector is directed at a small angle to them. The critical velocity of the flow is determined for various parameter values. The numerical results obtained are compared.     

Stability analysis of a fractional viscoelastic plate strip in supersonic flow under axial loading

The stability of a viscoelastic plate strip, subjected to an axial load with the Kelvin–Voigt fractional order constitutive relationship is studied. Based on the classical plate theory, the structural formulation of the plate is obtained by using the Newton’s second law and the aerodynamic force due to the fluid flow is evaluated by piston theory. The Galerkin method is employed to discretize the equation of motion into a set of ordinary differential equations. To determine the stability margin of plate the obtained set of ordinary differential equations are solved using the Laplace transform method. The effects of variation of the governing parameters such as axial force, retardation time, fractional order and boundary conditions on the stability margin of fractional viscoelastic panel are investigated and finally some conclusions are outlined.     

Elastic nonlinear stability analysis of thin rectangular plates through a semi-analytical approach

A semi-analytical approach to the elastic nonlinear stability analysis of rectangular plates is developed. Arbitrary boundary conditions and general out-of-plane and in-plane loads are considered. The geometrically nonlinear formulation for the elastic rectangular plate is derived using the thin plate theory with the nonlinear von Kármán strains and the variational multi-term extended Kantorovich method. Emphasis is placed on the effect of destabilizing loads and on the derivation of the solution methodologies required for tracking a highly nonlinear equilibrium path, namely: parameter continuation and arc-length continuation procedures. These procedures, which are commonly used for the solution of discretized structural systems governed by nonlinear algebraic equations, are augmented and generalized for the direct application to the PDE. The boundary value problem that results from the arc-length continuation scheme and consists of coupled differential, integral, and algebraic equations is re-formulated in a form that allows the use of standard numerical BVP solvers. The performance of the continuation procedures and the convergence of the multi-term extended Kantorovich method are examined through the solution of the two-dimensional Bratu–Gelfand benchmark problem. The applicability of the proposed approach to the tracking of the nonlinear equilibrium path in the post-buckling range is demonstrated through numerical examples of rectangular plates with various boundary conditions.     

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.     

多孔饱和半空间上弹性圆板的动力分析

用解析方法研究多孔饱和半空间上弹性圆板的低垂直振动，首先用Ｈａｎｋｅｌ变换求解多孔饱和介质动力问题控制方程，然后按混合边值条件建立多孔饱和半空间上弹性板的垂直振动的对偶积分方程，用Ａｂｅｌ变换化对偶积分方程为第二类Ｆｒｅｄｈｏｌｍ积分方程，并给出了数值算例。     

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.     

Oscillations of a cylindrical body submerged in a fluid with ice cover

The linear plane problem of oscillations of an elliptic cylinder in an ideal incompressible fluid of finite depth in the presence of an ice cover of finite length is solved. The ice cover is modeled by an elastic plate of constant thickness. The hydrodynamic loads acting on the body are determined as functions of the oscillation frequency and the positions of the cylinder and plate.     

On the stability of a plate under longitudinal compression on a two-layer half-space with lower layer prestressed by gravity forces

In the plane (plane strain) and axially symmetric statements, we study the problem of stability, under the action of longitudinal compressing forces, of an infinite elastic plate in two-sided contact with an elastic half-space. The upper layer of finite depth is described by the usual equations of linear theory of elasticity; the lower layer, which is geometrically nonlinear, incompressible, and infinite in depth, is prestressed by gravity forces. The total adhesion between the layer of finite depth and the lower half-space is realized. It is also assumed that the same adhesion takes place between the upper layer of the half-space and the plate with the contact tangential stresses taken into account.The results can be used to calculate the working capacity of coated bodies and layered composites and in problems of geophysics.The problem of stability of an infinite elastic plate under longitudinal compression under conditions of two-sided contact with an elastic base was studied earlier in the monograph [1] (Fuss-Winkler base) and in [2–4].     

Linear interactions affecting the propogation of waves in a linear elastic fluid

Small linear interactions affecting the propogation of waves in a linear elastic fluid are investigated. These linear interactions may occur as a result of impurities on the surface of a linear elastic fluid. These interactions are imposed on the linear wave equations which were investigated in Momoniat (Propogation of waves in a linear elastic fluid, submitted for publication) using the non-classical contact symmetry method. The occurrence of a small parameter in the wave equations under consideration in this paper makes the problem ideal for analysis using an approximate non-classical contact symmetry method. Approximate contact symmetries and approximate solutions are determined and discussed for the problems under consideration. Comparisons are made with the case of no interaction.     

Nonlinear oscillations of a floating elastic plate

The multi-scales method is used to derive third-order equations of gravitational bending oscillations of a thin elastic plate floating on the surface of a homogeneous perfect incompressible fluid of finite depth. The equations incorporate the compressive force and nonlinear acceleration of vertical displacements of the plate. Based on these equations, the deflection of the plate and the velocity potential of the fluid induced by a traveling periodic wave of finite amplitude are expanded into asymptotic series to terms of the third order of smallness. The dependence of the oscillation characteristics on the elastic modulus and thickness of the plate, compressive force, the initial length and steepness of the wave is analyzed     

Stability of a flexible insert in one wall of an inviscid channel flow

A hybrid of computational and theoretical methods is extended and used to investigate the instabilities of a flexible surface inserted into one wall of an otherwise rigid channel conveying an inviscid flow. The computational aspects of the modelling combine finite-difference and boundary-element methods for structural and fluid elements respectively. The resulting equations are coupled in state-space form to yield an eigenvalue problem for the fluid–structure system. In tandem, the governing equations are solved to yield an analytical solution applicable to inserts of infinite length as an approximation for modes of deformation that are very much shorter than the overall length of the insert. A comprehensive investigation of different types of inserts – elastic plate, damped flexible plate, tensioned membrane and spring-backed flexible plate – is conducted and the effect of the proximity of the upper channel wall on stability characteristics is quantified. Results show that the presence of the upper-channel wall does not significantly modify the solution morphology that characterises the corresponding open-flow configuration, i.e. in the absence of the rigid upper channel wall. However, decreasing the channel height is shown to have a very significant effect on instability-onset flow speeds and flutter frequencies, both of which are reduced. The channel height above which channel-confinement effects are negligible is shown to be of the order of the wavelength of the critical mode at instability onset. For spring-backed flexible plates the wavelength of the critical mode is much shorter than the insert length and we show very good agreement between the predictions of the analytical and the state-space solutions developed in this paper. The small discrepancies that do exist are shown to be caused by an amplitude modulation of the critical mode on an insert of finite length that is unaccounted for in the travelling-wave assumption of the analytical model. Overall, the key contribution of this paper is the quantification of the stability bounds of a fundamental fluid–structure interaction (FSI) system which has hitherto remained largely unexplored.     

Domain Decomposition Methods for Sensitivity Analysis of a Nonlinear Aeroelasticity Problem

Abstract

We consider the nonlinear aeroelasticity problem of the interaction between a viscous, incompressible fluid and Lin elastic solid undergoing large displacement. The non-linearities of the problem formulation include the solid and fluid governing equations. as well as thc dependence of the How geometry on the solid deformation. The resulting coupling is thus two-way. We develop domain-decomposition methods for solution and sensitivity analysis of the coupled problem. The domain decomposition is in the form of a block-Gauss-Seidel-like prcconditioncr that decomposes ihc coupled-domain problem into distinct nonovcrlapping fluid and solid subdotnain problems. The preconditioner thus enables exploitation or single-domain algorithms for solid and fluid mechanics discretization and solution. On the other hand, two-way fluid-solid coupling is retained within the residuals, which is essential for correct sensitivities. Sensitivities of field quantities can be found with little additional work beyond that required for solving the coupled fluid-solid system. The methodology developed here is illustrated by the solution of a problem of viscous incompressible flow about an infinite clastic cylinder. Sensitivities of the resulting velocity and displacement fields with respect to elastic modulus and fluid viscosity are computed.     

Bending vibrations of a rectangular poroelastic plate

This paper presents the equations of motion of an air saturated rectangular porous plate. The model is based on a mixed displacement–pressure formulation of the Biot–Allard theory [1]. We obtain a system of equations which describe the coupling beetween the solid and fluid phases of the plate. This system is solved by applying the Galerkin method.     

Dynamics of interaction between a squeezed layer of a viscous incompressible fluid and an elastic three-layer plate

We study the hydrodynamic response of a thin layer of a viscous incompressible fluid squeezed between impermeable walls. We consider the distribution of pressure and force dynamic characteristics of the fluid layer in the case of forced flows along the gap between a vibration generator (which is a rigid plane) exhibiting harmonic vibrations and a stator (which is an elastic freely supported three-layer plate). The inertial forces of the viscous fluid motion and the stator elastic properties are taken into account. The amplitude and phase frequency characteristics of the elastic three-layer plate are found from the solution of the plane problem.     

Equilibrium of an Elastic Spherical Shell Filled with a Heavy Fluid under Pressure

This paper considers the problem of equilibrium of a nonlinearly elastic spherical shell filled with a heavy fluid and resting on a smooth, absolutely rigid, flat surface. The weight of the shell is assumed to be negligible in comparison with the weight of the fluid filling it. The contact region with the supporting plane is one of the unknowns in the problem. Equilibrium equations for a membrane shell are obtained in an accurate nonlinear formulation. Stresses and strains of a shell made of a Mooney–Rivlin material are numerically investigated. The results are compared with calculation results for the case of inflation of a spherical shell ignoring the weight of the fluid filling. The effect of the fluid weight on shell strains and stresses is estimated.     

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.