Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

Elastic equilibrium of an anisotropic plate of arbitrary thickness with an elliptic cavity

We study the stressed state of an anisotropic plate of arbitrary thickness weakened by a cylindrical elliptic cavity and subject to forces that are independent of the thickness coordinate. The results of numerical computations are described. Four figures. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 76–80.     

The three-dimensional expansion-contraction problem for an orthotropic plate with elliptic cavities

On the basis of functions of generalized complex variables that are exact solutions of the three-dimensional equations of the theory of elasticity of an orthotropic body, we construct the solution for studying the stress state of a plate with elliptic cavities. We use the projection-grid method on the transverse coordinate. As basis functions we have chosen functions of compact support. We have carried out numerical studies for a plate with one elliptic cavity. Three tables. Bibliography: 8 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 20–27.     

The inverse scattering problem for cavities with impedance boundary condition

We consider the inverse scattering problem of determining the shape of a cavity with impedance boundary condition from sources and measurements placed on a curve inside the cavity. It is shown that both the shape ?D\partial D of the cavity and the surface impedance λ are uniquely determined by the measured data and numerical methods are given for determining both ?D\partial D and λ where neither one is known a priori. Numerical examples are given showing the viability of our method.     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

Magnetohydrodynamic shear flow along a flat plate with uniform suction or blowing

An analysis is made of the steady shear flow of an incompressible viscous electrically conducting fluid past an electrically insulating porous flat plate in the presence of an applied uniform transverse magnetic field. It is shown that steady shear flow exists for suction at the plate only when the square of the suction parameter S is less than the magnetic parameter Q. In this case the velocity at a given point increases with increase in either the magnetic field or suction velocity. The shear stress at the plate increases with increase in either S or the free-stream shear-rate parameter σ₁ or Q. The analysis further reveals that solution exists for steady shear flow past a porous flat plate subject to blowing only when the square of the blowing parameter S₁ is less than Q. It is found that the induced magnetic field at a given location decreases with increase in Q. Further the wall shear stress decreases with increase in S₁. No steady shear flow is possible for blowing at the plate when S₁² > Q. Received: June 16, 2004; revised: October 24, 2004     

Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

We consider coupled PDE systems comprising of a hyperbolic and a parabolic-like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A specific prototype consists of a wave equation defined on a three-dimensional bounded domain Ω coupled with a thermoelastic plate equation defined on Γ ₀—a flat surface of the boundary \partial Ω . Thus, the coupling between the wave and the plate takes place on the interface Γ ₀. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the ``minimal amount' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to Γ <sub>0</sub>, suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity is accounted for in the plate model; (ii) the plate model does not account for any type of mechanical damping, including the structural damping most often considered in the literature; (iii) there is no mechanical damping placed on the interface Γ <sub>0</sub>; (iv) the boundary damping is nonlinear without a prescribed growth rate at the origin; (v) the undamped portions of the boundary \partial Ω are subject to Neumann (rather than Dirichlet) boundary conditions, which is a recognized difficulty in the context of stabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold. The main mathematical challenge is to show how the thermal energy is propagated onto the hyperbolic component of the structure. This is achieved by using a recently developed sharp theory of boundary traces corresponding to wave and plate equations, along with the analytic estimates recently established for the co-continuous semigroup associated with thermal plates subject to free boundary conditions. These trace inequalities along with the analyticity of the thermoelastic plate component allow one to establish appropriate inverse/ recovery type estimates which are critical for uniform stabilization. Our main result provides ``optimal' uniform decay rates for the energy function corresponding to the full structure. These rates are described by a suitable nonlinear ordinary differential equation, whose coefficients depend on the growth of the nonlinear dissipation at the origin. \par Accepted 12 May 2000. Online publication 6 October 2000.     

Calculation of microstrains and microstresses in a thick non-symmetric heterogeneous plate by homogenization

We consider the thermoelastic behaviour of a thick heterogeneous plate containing in its thickness a large number of periodically distributed transverse holes or inclusions. We use the Reissner-Mindlin thermoelastic linear model of thick plates with a known temperature and we distinguish displacements in the upper and lower part of the plate with respect to the middle plane. Due to the structure of the plate, thermal and elastic coefficients are non-uniformly and rapidly oscillating functions of the space variable. Two-scale convergence, which is the state of the art in mathematical homogenization technics, is used and gives convergence results and formulae allowing to calculate the distribution of microstrains and microstresses inside the plate when a macroscopic behaviour is given.     

Computation of anisotropic plates with curvilinear holes reinforced by prestressed rods

We consider the stressed state of an anisotropic plate with an elliptic hole whose boundary is reinforced by a prestressed curvilinear rod of arbitrary cross section symmetric with respect to the middle plane of the plate. The elastic equilibrium of the rod is described by the equations of the theory of curvilinear rods. The solution of the problem is reduced to a system of linear algebraic equations. We show the influence of prestressing on the stressed states of the bodies.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 76–80.     

Study of a perforated thin plate according to the relative sizes of its different parameters

We are interested in the study of a thin plate, periodicially perforated by cylindrical holes, the axes of which are perpendicular to the plane of the plate. A horizontal section of the plate specifies its geometry, and shows a periodicity in the order of ?. The thickness of the plate is equal to e. The ratio of material is small, and is characterized by the parameter δ, the thickness of the bars being equal to ?δ. In this paper, we study the dependence of displacements on e, ? and δ, and to give equivalent limits when e, then ?, and finally δ, tend towards zero. An interesting result obtained in this work is the negative Poisson coefficient of the final equivalent material. Although this coefficient is theoretically between ?½ and 1, most materials encountered in practice have a positive one.     

Numerical analysis of 3D vortical cavity flow

The vortical flows of an incompressible fluid in a rectangular three-dimensional container with a large spanwise aspect ratio driven by a moving solid lid are studied using a combined compact finite difference (CCD) scheme with high accuracy and high resolution. The study focuses on the change of the steady flow structures in the cavity with Reynolds numbers ranging from 100 to 850. The results of the flow in the cavity with a spanwise aspect ratio 6.55 show that several stable closed streamlines localized near the symmetric plane are found at Re ≥500, while a closed stable streamline is found near the side wall at Re ≤300. The change of the flow pattern present in this system affects the diffusion properties in the flow but seems to have no qualitative effect on the global flow properties which include energy dissipation in the cavity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich-Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two-dimensional orthotropic thick plate theories with stress or mixed edge data.     

Vertical oscillations of multimass system on elastic half space with a cylindrical cavity

A problem on oscillations of a multimass system (MS) is considered on an elastic half space with a cylindrical cavity. Equations of motions of an MS are given, which are modeled by masses that are connected by springs and dampers. A motion of the half space with a cavity is characterized by a transmitting function,which is known from a solution of a contact problem with vertical oscillations of a die on the half space given. The conditions of interrelation of the MS with the base close the system of algebraic linear equations for determining amplitudes of oscillations of each element of the MS. Translated from Dinamicheskie Sistemy, No. 7, pp. 13–18, 1988.     

Thin elastic and periodic plates

This paper is devoted to the study of a three dimensional model of elastic periodic plate when the thickness e of the plate and the size ω of the periods are small. In the three studied limits (e → 0 then ω → 0), (ω → 0 then e → 0) and lately (e and ω → 0 together) the three dimensional equation of elasticity are approached by the two dimensional general equations of a linear anisotropic plate, the stretching and bending being coupled. This study points out the importance of the ratio of the two small parameters, indeed the moduli occuring in the two dimensional equations are different in the three limits. In each case a convergence proof is carried out.     

hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.     

Propagation of high-frequency harmonic elastic waves excited by surface perturbation of a noncircular cylindrical cavity

A ray method based on geometrical optics is applied to solve the problem of propagation of harmonic elastic waves excited by the perturbation of the surface of a noncircular cylindrical cavity. Stresses are computed under plane strain conditions for a cavity in the form of a parabolic cylinder and for a cylindrical cavity with a Munger oval section subjected to a uniform, surface load or a surface load which is a cosinusoidal function of the angle.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 75–82, 1987.     

The dynamics of structures on an elastic base

The problem of the dynamics of structures interacting with an elastic homogeneous half-plane through a rigid unyielding plate is reduced to the construction of the matrix of transmission momentum functions (the Green's matrix), which establishes the dependence between the generalized coordinates of the oscillations of the plate and the force characteristics. The elements obtained for this matrix are represented by graphs as the result of numerical solution of the nonsteady-state dynamic contact problem. We conclude that approximating a Green's matrix of exponential type on an infinite time interval is unjustified.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 60–63.     

Γ-convergence for incompressible elastic plates

We derive a two-dimensional model for elastic plates as a Γ-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The resulting model describes plate bending, and is determined from the isochoric elastic moduli of the three-dimensional problem. Without the constraint of incompressibility, a plate theory was first derived by Friesecke et al. (Comm Pure Appl Math 55:1461–1506, 2002). We extend their result to the case of p growth at infinity with p ϵ [1, 2), and to the case of incompressible materials. The main difficulty is the construction of a recovery sequence which satisfies the nonlinear constraint pointwise. One main ingredient is the density of smooth isometries in W ^2,2 isometries, which was obtained by Pakzad (J Differ Geom 66:47–69, 2004) for convex domains and by Hornung (Comptes Rendus Mathematique 346:189–192, 2008) for piecewise C ¹ domains.     

Unsteady MHD Couette flow in a rotating system

The unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system has been considered. An exact solution of the governing equations has been obtained by using a Laplace transform. Solutions for the velocity distributions as well as shear stresses have been obtained for small times as well as for large times. It is found that for large times the primary velocity decreases with increase in the rotation parameter K² while it increases with increase in the magnetic parameter M². It is also found that with increase in K², the secondary velocity v₁ decreases near the stationary plate while it increases near the moving plate. On the other hand, the secondary velocity decreases with increase in the magnetic parameter.     

Vibration of an elastic plate under action of an incompressible fluid in case of N = 0 approximation of I. Vekua's hierarchical models

This article deals with the study of the interaction between an elastic plate with constant thickness and an incompressible fluid when the plate is approximated by the N?=?0 order term of Vekua's hierarchical model.