Modelling of dynamic networks of thin elastic plates

The purpose of this paper is to derive junction conditions for networks of thin elastic plates and to analyse the dynamic equations of such networks. Junction conditions for networks of Kirchhoff plates and networks of Reissner–Mindlin plates are derived based on geometric considerations of the deformation at a junction. It is proved that the dynamic system which describes the Reissner–Mindlin network is well-posed is an appropriate energy space. It is further established that the Kirchhoff network is obtained in the limit of the Reissner–Mindlin network as the shear moduli go to infinity.     

Random vibrations of thin elastic plates

Zusammenfassung Das Verfahren des mittleren Fehlerquadrates für Zufallsfunktionen zweiter Ordnung wird auf das Problem der Querschwingungen dünner elastischer Platten unter Zufallsbelastung angewandt. Es werden Ausdrücke gefunden für die Kovarianz und die Spektraldichte der seitlichen Verschiebung und der Spannung in Abhängigkeit der Kovarianz und Spektraldichte der Zufallsfunktion für die Belastung.     

Asymptotic analysis of contact problems between an elastic material and thin-rigid plates

We consider an elastic material in contact with a three-dimensional rigid plate of varying thickness. We suppose that a perfect adhesion occurs along thin zones disposed in a self-similar way on the interface between the two materials. We suppose that the elasticity coefficients in the plate depend on its thickness and tend to infinity as this thickness tends to zero. We derive the effective material properties using Γ-convergence methods.     

Static and dynamic stability for geometrically nonlinear governing equations of elastic thin shallow shells

In this study, the governing equations for large deflection of elastic thin shallow shells are deduced into an algebraic cubic equation to determine the unknown coefficient of the assumed deflection by applying Galerkin's method in combination with the algebraic polynomial and Fourier series. For the dynamic problem, the coefficient is replaced by an unknown function of time; after the same process is applied, the governing equations are deduced to be a nonlinear ODE of order two called the Duffing equation, and its analytical solution is known. The combination of the algebraic polynomial and Fourier series gives very rapid convergence in the asymptotic solutions.     

Asymptotic analysis of 3D thin anisotropic plates with a piezoelectric patch

We consider a thin elastic plate with piezo patches mounted on top of it. Electrodes are located on the upper and, depending on the devices, at the lower surface of the patches. This piezo actuator is coupled to an elastic body. We develop an asymptotic procedure to derive a two‐dimensional approximation of the entire structure. As a result, we obtain an inhomogeneous fourth‐order plate equation with piecewise smooth coefficients for the vertical displacement coupled to a second‐order in‐plane problem. The analysis and the resulting asymptotic limits help clarifying the modeling issue concerning active piezo devices in multidimensional smart structures. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.     

The null field equations for flexural oscillations of elastic plates

A null field method is constructed to solve the exterior Dirichlet, Neumann, and Robin boundary value problems associated with the high‐frequency harmonic oscillations of Mindlin‐type plates. The case of an infinite plate with a bounded elastic inclusion is also considered. Additionally, the completeness of certain sets of wavefunctions is investigated. Copyright © 2012 John Wiley & Sons, Ltd.     

Variational treatment of exterior boundary-value problems for thin elastic plates

Existence, uniqueness and continuous dependence on the dataare investigated for weak solutions of exterior problems inthe theory of bending of plates with transverse shear deformation.     

Asymptotic analysis for linear difference equations

We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.

     

Asymptotic behavior of dynamic equations ontime scales

As a way to unify a discussion of many kinds of problems for equations in the contionous and discrete case(but also in order to reveal discrepancies between both cases), a theory of "time scales" was proposed and developed by Sulbach and Hilger. In our paper we investigate the asymptoic behaviour of so-called dynamic equations on time scales, and sych dynamic equations are differentialequations in the continous case and difference equations in the discrete case. We offer a perturbation result that leads to a time scales version of Levinson's Fundamental Lemma. Crucial are a dichotomy condition and a growth condition on the perturbation. Also, in the case that Levinson's result cannot be applied immediately, we suggest several preliminary transformations that might lead to a situation where Levinson's lemma is applicable. Such tranformations have been suggested by Harris and Lutz in the continuous case and by Benzaid and Lutz in the discrete case. Both those cases are covered by our theory, plus cases "in between". Examples for such cases will also be discussed in this paper.     

Asymptotic analysis of three-dimensional dynamic equations of a thin plate : PMM vol. 38, n 6, 1974, pp. 1072–1078

Two-dimensional dynamic equations of thin plate vibrations are obtained from the three-dimensional dynamic equations of elasticity theory on the basis of an asymptotic method [1 – 3], Such an approach permits establishing the limits of applicability of the two-dimensional dynamic equations and the corresponding boundary and initial conditions, and indicating the means of obtaining refined results.The question of the construction of an inner state of stress of a thin plate under dynamic conditions is examined herein. The possibility of considering states of stress with distinct variability in time and in the coordinates and with a distinct relationship between the displacement intensities, is taken into account.     

Asymptotic solutions of coupled dynamic problems of thermoelasticity for isotropic plates

The asymptotic method of solving boundary-value problems of the theory of elasticity for anisotropic strips and plates is used to solve coupled dynamic problems of thermoelasticity for plates, on the faces of which the values of the temperature function and the values of the components of the displacement vector or the conditions of the mixed problem of the theory of elasticity are specified. Recurrence formulae are derived for determining the components of the displacement vector, the stress tensor and for the temperature field variation function of the plate.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.     

Asymptotic analysis of elastic curved rods

We consider a sequence of curved rods which consist of isotropic material and which are clamped on the lower base or on both bases. We study the asymptotic behaviour of the stress tensor and displacement under the assumptions of linearized elasticity when the cross‐sectional diameter of the rods tends to zero and the body force is given in the particular form. The analysis covers the case of a non‐smooth limit line of centroids. We show how the body force and the choice of the approximating curved rods can affect the strong convergence and the limit form of the stress tensor for the curved rods clamped on both bases. Copyright © 2006 John Wiley & Sons, Ltd.     

Asymptotic analysis of delay differential equations

We present an asymptotic analysis for the solution of period 4 of , where f is an odd function and a positive parameter.This work was supported by a C.N.R. fellowship during the period in which the author was visiting Rutgers University.     

Asymptotic properties of solutions of certain third-order dynamic equations

In this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation

(r₂(t)((r₁(t)x^Δ(t))^Δ)^γ)^Δ+q(t)f(x(t))=0     

Asymptotic analysis of heat conducting elastic materials

In this paper, an asymptotic one dimensional model for a clamped curved rod is rigorously derived as a limit of respective three dimensional models. The rod is made of isotropic elastic, heat conductive linearly responding material. Asymptotic analysis is used with respect to the thickness of domains.     

A group analysis of the equations of the dynamic transversely isotropic elastic model

Group foliation of the system of equations of motion of a transversely isotropic elastic model of geomaterials, satisfying the Gassmann conditions, is carried out. A linear system of first-order differential equations is obtained, equivalent to the equations of this model. A number of theorems describing its properties are proved. A fundamental Lie transformation and an optimum system of its subgroups are obtained, which enable all the invariant, partially invariant and differentially invariant submodels of the dynamic model of a transversely isotropic elastic medium to be obtained. Some exact solutions are derived and their physical meaning is indicated.