This chapter provides a very brief summary of the types of heterogeneous materials considered in this monograph: fiber-reinforced composites, particulate composites, nanocomposites, porous composites, and so on. A succinct summary is given of analytical homogenization methods to determine the overall properties of particulate composites based on the upper and lower bounds of Hashin and Shtrikman; the Eshelby ellipsoidal inclusion theory and the Self-consistent Method of Eshelby; and the Mori-Tanaka Method and some other semi-analytical methods. Numerical methods such as the finite element method, the boundary element method, XFEM, and so on to model a representative volume element (RVE) of a heterogeneous material are reviewed, and thus the motivation for the Computational Grains method discussed in the rest of this book is presented.

This chapter discusses the role of a representative volume element (RVE) in the computational homogenization of heterogeneous materials. The use of the finite element method in modeling an RVE is discussed. The role of using the Hill-Mandel boundary conditions, and the use of periodic displacement and aperiodic traction boundary conditions on an RVE are discussed. The advantages of using the present Computational Grains method in modeling an RVE, not only to determine the macro properties of a heterogeneous material but also to determine the detailed interfacial stress states which are damage precursors at the micro level are discussed.