The Method of Homogenization
It is common in engineering and scientific problems to have to deal with materials that are formed from multiple constituents. Some examples are shown in Fig. 5.1 and include laminated wood, a fluid-filled porous solid, an emulsion, and a fiber-reinforced
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The Method of Homogenization
5.1 Introduction It is common in engineering and scientific problems to have to deal with materials that are formed from multiple constituents. Some examples are shown in Fig. 5.1 and include laminated wood, a fluid-filled porous solid, an emulsion, and a fiber-reinforced composite. Solving a mathematical problem that includes such variations in the structure can be very difficult. It is therefore natural to try to find simpler equations that effectively smooth out whatever substructure variations there may be. An example of this situation occurs when describing the motion of a fluid or solid. One usually does not consider them as composites of discrete interacting molecules. Instead, one uses a continuum approximation that assumes the material to be continuously distributed. Using this approximation, material parameters, such as the mass density, are assumed to represent an average. In this chapter, we investigate one approach that can be used to smooth out substructure variations that arise with spatially heterogeneous materials. In this sense, our objective differs from what we have done in the other chapters. Usually we have been interested in deriving an approximate solution, but in this chapter our primary goal is to derive an approximate problem. The substructural geometric and material variations will not be present in the reduced problem, but they will be used to determine the coefficients of the problem. The procedure we will be developing is called homogenization. It goes by other names, including effective media theory, bulk property theory, and the two-space scale method.
5.2 Introductory Example To introduce the method of homogenization, we will examine the boundaryvalue problem M.H. Holmes, Introduction to Perturbation Methods, Texts in Applied Mathematics 20, DOI 10.1007/978-1-4614-5477-9 5, © Springer Science+Business Media New York 2013
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5 The Method of Homogenization
Figure 5.1 Examples of composite systems. This includes plywood, a 400-μm view of quartz grains with water-filled pores (Haszeldine, 2010), an 80-μm view of an oil and water emulsion (Akay, 2010), and a 20-μm view of a fiber-reinforced ceramic composite (MEIAF, 2010)
d du D = f (x) for 0 < x < 1, dx dx
(5.1)
u(0) = a and u(1) = b.
(5.2)
where
Of interest here is when the function D includes a relatively slow variation in x as well as a faster variation over a length scale that is O(ε). Two examples of this are illustrated in Fig. 5.2. The function in Fig. 5.2a is an example of the type of variation that might be found in a laminated structure such as plywood since D has jump discontinuities but is constant over intervals that are O(ε). The function in Fig. 5.2b has a similar variation to that in Fig. 5.2a, but it is continuous. We will concentrate on examples of the latter type, but the discontinuous problem is not that much harder (Exercise 5.9). It is worth pointing out that a problem with rapidly varying coefficients, as in (5.1), is not easy to solve, even numerically. This makes the smoothing
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