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Homogenization of Reticulated Structures

verfasst von: Doina Cioranescu, Jeannine Saint Jean Paulin

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Professional Book Archive

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Über dieses Buch

This book presents recent works on lattice type structure. Some of the results discussed here have already been published in mathematical journals, but we give here a comprehensive and unified presentation. We have also added some new topics such as those contained in Chapter 4 treating elastic problems for gridworks. The aim of this book is to give continuous simple models for thin reticulated structures (which may have a very complex pattern). This means that we have to treat partial differential equations depending on several small parameters and give the asymptotic behavior with respect to these parameters (which can be the period, the thickness of the material, or the thickness of a plate or of a beam). This book is written from the point of view of the applied mathematician, atten tion being paid to the mathematical rigor, convergence results, and error estimates. It consists of six chapters and more than a hundred figures. The basic ideas are presented in the first two chapters, while the four last ones study some particular models, using the ideas of Chapters 1 and 2. Chapter 1 is an introduction to homogenization methods in perforated domains. Here the parameter to be taken into consideration is the period. After describing the multiple-scale method (which consists in asymptotic expansions), we focus our attention on the variational method introduced by Tartar, whose main idea is the construction of rapidly oscillating test functions.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Homogenization in Perforated Media

Abstract

We present here some model examples in homogenization of periodic media. To begin with, we consider a one-dimensional equation with oscillating coefficients. This example, due to Spagnolo (1967), shows that even in this simple case, the description of the global properties of the solution is rather complex. It also shows that in the periodic case, in order to give the limit equation, it not sufficient to take simply the average of the coefficients. The main difficulty comes from the fact that we have to pass to the limit in products of two weakly convergent sequences, and the limit of such a product may not be the product of the weak limits of these sequences.

Doina Cioranescu, Jeannine Saint Jean Paulin

Chapter 2. Lattice-Type Structures

Abstract

Our main interest in this work is lattice-type structures that are characterized by two properties: periodicity and small thickness of the material. We have several types of such structures that consist of identical cells periodically distributed in all directions, reticulated structures (see Figure 2.1); or in two directions, gridworks (see Figure 2.2); or in only one direction, tall structures (see Figure 2.3). If the material is concentrated along layers, then we speak about honeycomb structures (Figure 2.4), and if along bars, then reinforced structures (see Figure 2.5).

Doina Cioranescu, Jeannine Saint Jean Paulin

Chapter 3. Thermal Problem For Gridworks

Abstract

We now consider a particular case of three-dimensional lattice structures, the gridworks (or networks) that consist in regular arrays of thin wires.

The specific feature of such structures is that the periodicity occurs only in two directions. In the third direction their dimension e, in fact their thickness when considering these structures as thin plates, is small (see Figures 3.1, 3.2, 3.3). Consequently, in the asymptotic description of these structures we have this new small parameter to take into consideration, the other two being, of course, the period ε of the wires and their cross-section δ. In this chapter we consider thermal problems, and our aim is to study what happens when these three parameters tend to zero. Let us mention that in Chapter 4 we will study the linearized elasticity system for these structures.

Doina Cioranescu, Jeannine Saint Jean Paulin

Chapter 4. Elasticity Problem for Gridworks

Abstract

The structure we consider here is again the gridwork for which in Chapter 3 we studied the thermal problem (see Figure 4.1).

We now study the linearized elasticity system giving the displacement ofthis kind of structure. The displacement depends, of course, on the same three parameters: the thickness e of the gridwork, the period ε, and the thickness δ of the material in the reference cell.

Doina Cioranescu, Jeannine Saint Jean Paulin

Chapter 5. Thermal Problems For Thin Tall Structures

Abstract

This chapter is devoted to the study of very thin, tall structures. Such structures are called “tall” because one of their dimensions, the height, is much larger than the others and “thin” because the thickness of the material is much smaller than the characteristic dimensions of the cross section. Moreover, the material is periodically distributed along bars or layers. Examples of such structures are cranes and skyscrapers (see Figures 5.1–5.5).

Doina Cioranescu, Jeannine Saint Jean Paulin

Chapter 6. Elasticity Problems For Thin Tall Structures

Abstract

This chapter is again devoted to the study of very thin, tall structures, but a much more difficult problem is considered here, that of linearized elasticity. As can be expected from a physical point of view, the limit behavior is that of a beam. Thus we pass from the second-order three-dimensional elasticity system to two fourth-order and one second-order one-dimensional systems.

Doina Cioranescu, Jeannine Saint Jean Paulin

Backmatter

Titel: Homogenization of Reticulated Structures
verfasst von: Doina Cioranescu
Jeannine Saint Jean Paulin
Copyright-Jahr: 1999
Verlag: Springer New York
Electronic ISBN: 978-1-4612-2158-6
Print ISBN: 978-1-4612-7437-7
DOI: https://doi.org/10.1007/978-1-4612-2158-6