7R24. Homogenization of Reticulated Structures. - D Cioranescu (Lab d’Analyze Numerique, Univ Paris VI et CNRS, 4, place Jussieu, Paris Cedex 05, 75252, France) and J Paulin (Dept de Mathematique, Univ de Metz, Ile du Saulcy, Metz, Cedex 01, 57045, France). Springer-Verlag, NY. 1999. 343 pp. ISBN 0-387-98634-0. $74.95.
Reviewed by JJ Telega (Inst of Fund Tech Res, Polish Acad of Sci, Swietokrzyska 21, 00-049 Warsaw, Poland).
Many books have already been published on various mathematical and applied problems of homogenization which provide rigorous tools for micro-macro passage. The books which have appeared in recent years include Homogenization of Multiple Integrals, by A Braides and A Defranceschi, (Oxford Univ Press, Oxford, 1998); Variational Methods for Structural Optimization, by A Cherkaev, (Springer-Verlag, New York, 2000); An Introduction to Homogenization, by D Cioranescu and P Donato, (Oxford Univ Press, Oxford, 1999); Homogenization and Porous Media, by U Hornung, (Springer-Verlag, New York, 1997); Plates Laminates and Shells: Asymptotic Analysis and Homogenization, by T Lewinski and JJ Telega, (World Scientific, Singapore, 2000); Theory of Composites, by GW Milton, (Laminated Publishers, in press); and Macroscale Models of Flow Through Highly Heterogeneous Porous Media, by M Panfilov, (Kluwer Academic Publishers, Dordrecht, 2000).
The current book under review is written by applied mathematicians and presents a rigorous study of a significant class of linear problems related to homogenization of lattice-type structures characterized by two or three small parameters. The homogenization means passing with these parameters to zero, thus finding macroscopic models.
The book consists of six chapters. Chapter 1 introduces the reader to homogenization problems of stationary second-order linear differential equations posed on microperiodically perforated domains. The boundary conditions imposed on microholes are the homogeneous Neumann condition, the Dirichlet condition, the Fourier condition. Each case is carefully examined since the homogenized behavior in these three cases differs. First- and second-order correctors are also introduced. Having in mind applications, not only regular microperforated domains were investigated, but also domains where the holes meet the boundary. The last topic presented in the first chapter concerns the homogenization problem for the Neumann eigenvalue problem in perforated domains. Such a study pertains to convergence of eigenvalues and eigenvectors.
In Chapter 2, two- and three-dimensional lattice-type structures are investigated. Such structures are characterized by two small parameters: the periodicity parameter ε and the thickness of the elements of the structure δ. The passage with ε utilizes the results of the previous section. More difficult is the passage to zero with the second small parameter. The authors studied carefully both two-dimensional and three-dimensional cases. To obtain the final results, one has to derive a lot of estimates. In essence, here and in subsequent chapters, the part of the unit cell occupied by the material is suitably decomposed and next transformed to domains independent of small parameter. Both transport equation, ie, the heat equation, and the linear elasticity equations posed on such structures (domains) have been investigated. In the case of rod-like structure, referred to in the book as a reinforced structure, the ellipticity is lost. Then the passage with δ is more difficult since one has to resort to singular perturbations method. The eigenvalue problem for the transport equation is also briefly discussed.
The next two chapters are concerned with rigid and elastic gridworks with periodic microstructure. Now, similarly to the case of thin plates, its thickness 2e is also small. In Chapter 3, the authors studied only stationary thermal problems. It was shown that the order of passage to the limit with the three small parameters; ε, δ, and e plays generally an important role. Linear elastic gridworks characterized by three small parameters have been studied in Chapter 4. The passage to the limit (zero) with these parameters is carefully studied. In the limit, one obtains plate models. Here also the phenomenon of loss of stability for certain structures has also been observed. It has also been shown how to avoid this unpleasant property; for instance, if the gridwork contains oblique bars then this phenomenon does not occur. Time-dependent plate models are also briefly discussed. Now, however, only a simplified problem has been investigated since one starts at once from a two-dimensional plate model. The reader lacks a discussion of problems similar to the static case where the starting point was a 3D problem.
The last two chapters are concerned with thin tall structures. More precisely, in Chapter 5 thermal problems for such structures, characterized by three small parameters, have been carefully examined. Both two- and three-dimensional problems are discussed. In the last chapter thin three-dimensional, linear elastic tall structures have been investigated. Then in the limit a beam model is obtained.
The book is well written. However, throughout it plate-like structures are misleadingly called honeycomb structures. Sometimes the authors refer to references not listed in the list of references. Also, linear elasticity is always called linearized elasticity.
Homogenization of Reticulated Structures is addressed to specialists in micromechanics familiar with mathematical tools of homogenization and asymptotic analysis methods. It will also be useful to researchers elaborating models of cellular solids. Specialists in structural mechanics, exploiting asymptotic methods, will appreciate its rigor. Finally, applied mathematicians interested in mathematical problems of solid mechanics will profit from the book since it offers a lot of new problems.
The book is nicely edited and its price is acceptable even for those who are not from rich countries.
