Homogenization for periodic media: from microscale to macroscale
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ELEMENTARY PARTICLES AND FIELDS Theory
Homogenization for Periodic Media: from Microscale to Macroscale* G. P. Panasenko** Saint Etienne University, France Received August 13, 2007
Abstract—The mathematical tools for the up-scaling from micro to macro in periodic media are considered: the homogenization techniques, the multiscale modelling, etc. The main applications of these tools are the mechanics of composite materials, flows in porous media, and discrete models for nanostructures. PACS numbers: 68.65.-k DOI: 10.1134/S106377880804008X
1. WHAT IS A COMPOSITE MATERIAL? We define here a composite material as a heterogeneous material constituted of alternating volumes of some homogeneous materials at the “supermolecular scale.” The shape classification of these materials can be represented by stratified (laminated) media constituted of (periodically) alternating homogeneous thin layers; fiber reinforced materials: a periodic system of unidirectional parallel fibers of one compound (or a union of some systems of different directions) separated by a background material called a matrix which fills the space between the fibers; granular composite: the periodic structure of a three-dimensional periodic system of grains and a material which fills the space between the grains (the matrix). This list can be continued. It is assumed throughout that the dimensions of the periodic cell are much smaller than the characteristic macroscopic spacial size of the problem. If we take this macroscopic size as a space dimension unity, then the period of the structure becomes a dimensionless small parameter denoted ε. Composite materials were widely applied in the 20th century in spacecraft, aircraft, and sports industries. These materials allowed one to combine such properties as high stiffness, small density, highfailure resistance, low heat conductivity, radio transparency, and so on. The necessity of computation of the macroscopic constants and properties starting from the microscopic shape information generated a ∗ **
The text was submitted by the authors in English. E-mail: [email protected]
lot of theories of the passing from the microscopic to the macroscopic scale (this procedure was called “averaging” and later “the homogenization” or “upscaling”). Apart from composite materials, there are a lot of other examples of heterogeneous media (which can also be considered as a special case of composite materials with a vacuum as one compound): porous media, frames, grids, lattice structures, and atomic lattices. In particular, the modeling of flows in porous media is an important mathematical tool in oil and gas recovery engineering. The standard numerical methods (such as the finite-element method, the finite-differences method, the finite-volume method) are inapplicable to these materials because all these methods use meshes which should be much finer than ε and it leads to memory and time overflow for the majority of practical problems. The analytical methods are applicable mainly in the case of constant coefficients (which is n
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