Effective Properties of Composite Materials, Reinforced Structures and Smart Composites: Asymptotic Homogenization Approach

Asymptotic homogenization is a powerful mathematically rigorous technique for analyzing composite materials and structures. The proof of the possibility of homogenizing a composite material of a regular structure, i.e. of examining an equivalent homogeneous material instead of the original inhomogeneous composite material, is one of the principal results of this theory. Method of asymptotic homogenization has also indicated a procedure of transition from the original problem for the inhomogeneous composite solid to a problem for a homogeneous solid. The effective properties of this equivalent homogeneous solid are determined through the solution of the unit cell problems, derived in the process of asymptotic homogenization. In the present chapter, the basics of the asymptotic homogenization method and its applications to the analysis of a wide range of composite and smart materials and structures are presented. The asymptotic homogenization technique is applied to the analysis of three-dimensional composite materials and smart composite structures, thin-walled composite reinforced structures, and smart composite orthotropic shells and plates. The analytical solutions of the corresponding unit cell problems are obtained and the explicit analytical formulae are derived for the effective properties of three-dimensional generally orthotropic grid-reinforced smart composites of various structures, smart grid-reinforced composite shells and plates, rib- and wafer-reinforced shells and plates, sandwich composite shells with cellular cores of different geometry. Finally, the analytical expressions for the effective mechanical properties of carbon nanotubes are presented. For many problems that are analyzed in the present chapter, any other analytical or numerical approaches are not as effective as the asymptotic homogenization method.