Homogenization theory for structures of honeycomb and chessboard types

Abstract: This thesis consists of 5 papers. The first paper starts with an introduction to some of the most fundamental mathematical theory connected to homogenization. In particular, a portion of the classical theory in homogenization is presented, such as the convergence of monotone operators and homogenization in random domains. Moreover, also some later results in homogenization, such as degenerate cases, the four phase checkerboard and reiterated homogenization is reviewed and discussed. In the second paper stiffness properties of square symmetric unidirectional two-phase composites with given volume fractions are considered. The effective moduli of the stiffest possible of such materials, which satisfy transverse isotropy or square isotropy, is compared with the effective moduli of materials satisfying 3D-isotropy. Next, some numerical FEM computations of in plane stiffness properties of square honeycombs are presented. The results are compared with the effective moduli of the stiffest possible square symmetric composites. The calculations can be useful in connection with optimization of structural topology. In the third paper some methods of calculation of stiffness properties of periodic structures are considered. The theory concerning effective properties is made as simple as possible without involving complicated convergence processes and it is focused on the fact that the formulation of the effective properies are quite natural from a physical point of view, something that is often hidden in the modern mathematical literature of composite materials. In the fourth paper the effective conductivity of a checkerboard is considered. Due to the behavior of the solutions near the corner points in checkerboard structures it is difficult to solve the corresponding varational problems by using usual numerical methods, even for standard checkerboards. We focus on these difficulties, both theoretically and also by making some illustrative numerical experiments. Moreover, we consider a generalized version of the checkerboard and present a new numerical method for determining the corresponding field which converges in the energy norm independent of the local conductivities. A generalized version of the four-phase checkerboard is considered in the fifth paper. We meet the same problems here as in the fourth paper in solving the corresponding variational problems by using usual numerical methods. In this paper we are also able to show the convergence of the upper and lower bounds for the numerical estimates of the energy.