ABSTRACT
The purpose of this chapter is twofold: first we present, in a didactic form, the main ideas underlying the method of homogenization (also called homogenization theory) and, second, we use the method as a tool to develop a multiscale modeling approach, able to analyze a wide spectrum of transport phenomena in random heterogeneous media (media whose microstructures may be described appropriately by non-trivial joint probability density functions [JPDFs]). The approach is also based on variational calculus and the finite element method and leads to the prediction of macroscopic effective properties of heterogeneous media. Here, the multiscale approach is exposed in the context of the heat conduction problem in composite materials, whose components are all thermally conducting. An expression for the tensorial effective thermal conductivity of such materials is derived, and some properties of the effective conductivity are shown. In this chapter, we present in detail the continuous formulations of the heat conduc-
tion problems, which are part of the multiscale approach. On the other hand, we only summarize the main steps for numerical solution of these problems via the finite element method. Sample numerical results for the effective thermal conductivity of the 2D square array of circular cylindrical fibers and of the 3D simple cubic array of spheres are presented up to maximum packing. The reader is referred to the works by Cruz and Patera (1995), Cruz et al. (1995), Cruz (1997, 1998), Machado and Cruz (1999), Matt (1999, 2003), Rocha (1999), Machado (2000), Rocha and Cruz (2001), Matt and Cruz (2001, 2002a, 2002b, 2004, 2006, 2008), and Pereira et al. (2006) for more details of the numerical solutions and for the presentations and analyses of numerical results for the effective thermal conductivities of 2D and 3D, ordered and random composites. Various computational techniques developed to address the heat conduction problem in composite materials are reviewed by Pereira et al. (2006), Matt and Cruz (2006, 2008), and Cruz (2001). It should be remarked that there are several other approaches to analyze transport
phenomena in heterogeneous and multiphase systems. Phenomenological effective medium approaches (see Torquato 2002) do not tackle the underlying physics at the microstructural level, such that they attempt to establish the macroscopic properties by proposing ad hoc assumptions. Another much employed technique is volume averaging, as discussed in Chapter 1 and in the monograph by Whitaker (1999). The main objective of volume averaging is to formulate the spatially smoothed governing equations that are valid everywhere in the heterogeneous medium of interest. The development of closure problems is then necessary to permit the prediction of the medium’s effective transport properties, which relate macroscopic fluxes to intensity gradients. Regarding both volume averaging and homogenization approaches, it appears that much more research effort has been devoted to formulating several different classes of transport problems in heterogeneous media than to computing the associated macroscopic properties. Therefore, a comparative analysis of effective property results arising from these alternative methods is beyond the scope of the present work. The outline of this chapter is as follows. In Section 2.2, the method of homogenization is
introduced didactically. We first offer a formal definition and then illustrate with physical examples the mathematical problems involved in the definition. Next, we give a brief overview of the analytical techniques that may be employed in the homogenization procedure. Finally, we apply the method to a general elliptic model problem in strong form.
interest, adopting a variational approach and exploiting the analysis of Section 2.2. Although some of our results are also shown, in a different form, in Auriault (1983), we not only present a more detailed derivation here, but also the variational treatment makes the final expressions directly suitable for subsequent numerical treatment using the finite element method. In Sections 2.4 through 2.7, we describe the multiscale modeling approach, which decomposes the original multiscale problem into the macroscale, mesoscale, and microscale (sub)problems. In Section 2.8, we briefly discuss the numerical treatment of the pertinent problems, and in Section 2.9 we present some representative results stemming from solutions to the mesoscale and microscale problems. Finally, in Section 2.10, we state the conclusions.
