Pseudo‐reflection phenomena for singularities in thin elastic shells

We consider problems of statics of thin elastic shells with hyperbolic middle surface subjected to boundary conditions ensuring the geometric rigidity of the surface. The asymptotic behaviour of the solutions when the relative thickness tends to zero is then given by the membrane approximation. It is a hyperbolic problem propagating singularities along the characteristics. We address here the reflection phenomena when the propagated singularities arrive to a boundary. As the boundary conditions are not the classical ones for a hyperbolic system, there are various cases of reflection. Roughly speaking, singularities provoked elsewhere are not reflected at all at a free boundary, whereas at a fixed (or clamped) boundary the reflected singularity is less singular than the incident one. Reflection of singularities provoked along a non‐characteristic curve C are also considered. Copyright © 2003 John Wiley & Sons, Ltd.     

Sensitivity phenomena for certain thin elastic shells with edges

We consider two kinds of shells which are sensitive, i.e. they are geometrically rigid and as the thickness ϵ tends to zero the limit problem is unstable in the sense that there are very smooth loadings (belonging to the space 𝒟 of test functions of distributions) such that the corresponding solutions go out of the energy space. The first situation occurs when there is an edge and the middle surface is elliptic on both sides of it. The second one occurs when there is an edge Γ₀, the surface is respectively elliptic and hyperbolic on both sides of it and the ‘determination domain’ in the hyperbolic region issued from Γ₀ intersects another edge Γ₁. Copyright © 2000 John Wiley & Sons, Ltd.     

Boundary stabilization for a non‐linear beam on elastic bearings

In this paper, we study the equation under non‐linear boundary conditions which model the vibrations of a beam clamped at x=0 and supported by a non‐linear bearing at x=L. By adding only one damping mechanism at x=L, we prove the existence of a global solution and exponential decay of the energy. Copyright © 2001 John Wiley & Sons, Ltd.     

Second‐order finite‐volume schemes for a non‐linear hyperbolic equation: error estimate

We study second‐order finite‐volume schemes for the non‐linear hyperbolic equation u_t(x, t) + div F(x, t, u(x, t)) = 0 with initial condition u₀. The main result is the error estimate between the approximate solution given by the scheme and the entropy solution. It is based on some stability properties verified by the scheme and on a discrete entropy inequality. If u₀ ∈ L^∞ ∩ BV_loc(ℝ^N), we get an error estimate of order h^1/4, where h defines the size of the mesh. Copyright © 2000 John Wiley & Sons, Ltd.     

Some non‐local problems for the parabolic‐hyperbolic type equation with complex spectral parameter

In the present paper the unique solvability of two non‐local problems for the mixed parabolic‐hyperbolic type equation with complex spectral parameter is proved. Sectors for values of the spectral parameter where these problems have unique solutions are shown. Uniqueness of the solution is proved by the method of energy integral and existence is proved by the method of integral equations. In particular cases, eigenvalues and corresponding eigenfunctions of the studied problems are found. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A type of solution for thin shells in the form of an hyperbolic paraboloid

Résumé Pour le problème du paraboloïde hyperbolique une solution est présentée qui se base sur la théorie linéaire des voiles minces peu incurvés. La solution prend en considération l'effet de la flexion; c'est une extension du type de solution indiqué parM. Lévy pour la plaque rectangulaire (théorie classique). Un exemple d'application termine cette étude.     

Buckling localization in a non‐elastic rotating disk

Buckling localization of a rotating disk made of elastic‐perfectly plastic material is investigated using stress‐rate formulation of the stability boundary‐value problem. The phenomenon of plastic buckling localization and its analogy with elastic buckling localization is discussed. For a thin rotating disk, it is shown that buckling develops at a speed lower than one at which the disk passes to fully plastic state, or in other words, before the limit load has been attained. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

T‐complete functions for thin and thick plates on elastic foundation

This article deals with Trefftz functional systems for thin and thick plates on one‐ and two‐parameter Winkler foundation. The T‐complete set is derived by solving the homogeneous equations of the problem. This can be done with the method of separation of variables. For each separation parameter we deal with ordinary, linear differential equations (4th order for the Kirchhoff plate and set of fourth‐ and second‐order equations for the Reissner‐Mindlin plate) so the demanded number of fundamental solutions (linearly independent) is equal to the order of equation. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Impact of a cone on a thin elastic membrane

The problem of computation of parameters of motion of a thin elastic membrane under the Impact of a rigid body, was considered by various authors (see [1] together with bibliography) without, however, yielding a rational solution. This paper presents a full qualitative analysis of solution of this problem for the case of normal impact of a circular cone moving with constant velocity on an infinite elastic membrane of constant thickness. Although this is the simplest case, it is important, insofar as it brings to light the characteristic features of the problem. In the “membrane” approximation the thickness of the layer is found to be an unessential parameter, therefore the problem, as postulated by us, is self-similar and its solution is reducible to the problem for ordinary differential equations.     

On the theories used for the wave propagation in laminated composite thin elastic shells

In this paper, the problem of propagation of free harmonic waves in cross-ply laminated thin elastic shells is considered. For this problem, a theoretical unification of the most commonly used, in physical and engineering applications, thin shell theories which take into consideration transverse shear deformation effects is presented. In more detail, the problem is formulated in such a way that by using some tracers, which have the form of Kronecker's deltas, the obtained stress-strain relations, constitutive equations and equations of motion produce, as special cases, the corresponding relations and equations of the transverse shear deformable analogs of Donnell's, Love's and Sanders' theories. Using an eigenvalue form solution of the equations of motion, a comparison of corresponding numerical results obtained on the basis of all of the afore-mentioned theories is made. Comparisons with corresponding results obtained on the basis of the classical thin shell theories of Donnell, Love, Sanders and Flugge are also made.

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Resumé On étudie un problème de propagation d'ondes harmoniques libres dans une fine coque cylindrique, élastique, composée de lamelles croisées. On présente une unification des théories les plus courantes que les ingénieurs et physiciens appliquent aux problèmes de coques minces sous considération de la déformation due au cisaillement transverse. En détails, le problème est formulé de telle façon qu'en utilisant des opérateurs de trace (sous forme du symbole de Kronecker) les relations obtenues: contrainte-déformation, équations constitutives et équations de mouvement donnent comme cas spéciaux les relations correspondantes et les équations des problèmes analogues (déformation de coques minces par cisaillement transverse) des théories de Donnell, Love et Sanders. En utilisant une solution aux valeurs propres des équations de mouvement, on compare les résultats numériques obtenus grâce aux théories mentionées ci-dessus aux résultats correspondants sur la base des théories classiques de Donnell, Love, Sander et Flugge.

     

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.     

The strains and the energy in thin elastic shells of arbitrary shape for arbitrary deformation

Zusammenfassung Im ersten Teil (I) werden Formeln entwickelt, welche die kinetische und potentielle Energie dünner Schalen beliebiger Form explizit als Funktion eines beliebigen Verschiebungsfeldes ausdrücken. Dies geschieht auf rein analytische Weise ohne Zuhilfenahme geometrischer Darstellungen einzelner Schalenelemente. Von besonderem Wert für viele Anwendungen ist es, dass vollständige Freiheit in der Wahl der Flächenkoordinaten besteht.Im zweiten Teil (II) wird zunächst die Anwendung der Theorie auf zwei bekannte Beispiele gezeigt: 1. Platte in schiefwinkligen Koordinaten, 2. zylindrische Schale. Als neues Beispiel wird 3. die schraubenflächige Schale eingehend behandelt. Für einen besonderen Einspannungsfall wird die Frequenz der Grundschwingung als Funktion der Verwindung angenähert numerisch berechnet.     

Darcy's laws for non‐stationary viscous fluid flow in a thin porous medium

We consider a non‐stationary Stokes system in a thin porous medium Ω_? of thickness ? which is perforated by periodically solid cylinders of size a _? . We are interested here to give the limit behavior when ? goes to zero. To do so, we apply an adaptation of the unfolding method. Time‐dependent Darcy's laws are rigorously derived from this model depending on the comparison between a _? and ? . Copyright © 2016 John Wiley & Sons, Ltd.     

Equilibrium configurations of a thin elastic rod with self‐contacts

Spatial equilibria of a closed thin isotropic elastic rod are considered. The thin elastic rod is a classical model for the large‐scale structure of relatively long DNA molecules. Particular attention is paid to the shapes with self‐contacts which are assembled from the elementary loops.     

Loss of regularity for the solutions to hyperbolic equations with non‐regular coefficients—an application to Kirchhoff equation

We consider the Cauchy problem for second‐order strictly hyperbolic equations with time‐depending non‐regular coefficients. There is a possibility that singular coefficients make a regularity loss for the solution. The main purpose of this paper is to derive an optimal singularity for the coefficient that the Cauchy problem is C^∞ well‐posed. Moreover, we will apply such a result to the estimate of the existence time of the solution for Kirchhoff equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Well‐posedness theory for hyperbolic conservation laws

The paper presents a well‐posedness theory for the initial value problem for a general system of hyperbolic conservation laws. We will start with the refinement of Glimm's existence theory and discuss the principle of nonlinear through wave tracing. Our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the L₁(x) distance between the two solutions and is time‐decreasing. Moreover, the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition. It consists of a linear term measuring the L₁(x) distance, a quadratic term measuring the coupling of waves and distance, and a generalized entropy functional. © 1999 John Wiley & Sons, Inc.