Some mathematical, numerical and engineering aspects of composite structures and perforated solids

Abstract: In this doctoral thesis in engineering mathematics we present some new modeling, numerical and engineering aspects and results concerning composite structures and perforated solids. The thesis consists of five papers, which are divided into three parts. Part A functions as an introduction where we look into some of the mathematical, numerical and engineering aspects of the effective in-plane elastic properties of periodic structures. We focus on one- and two-component composites and, in particular, we discuss locally anisotropic perforated structures. Moreover, we describe how the elastic properties of such and more general structures can be calculated. Finally, we present bounds and relations between different effective moduli. In Part B we study twelve of the simplest structures belonging to the class of one-component structures with holes and which, in addition, are three-fold rotational symmetric. We look into the effective elastic in-plane properties such as effective bulk and shear moduli and the so-called Vigdergauz constants. A comparative overview of all of these properties is given. Furthermore, we derive low-density asymptotics and present numerical data from finite element calculations. We also show that several of the studied structures have optimal elastic properties in the limit when the volume fraction of the connected material goes to zero. Finally, some remarks on the manufacturability of the structures are made. In Part C, which is divided into three papers, we continue to explore the effective properties of periodic structures. In paper [C1] we introduce a new model for estimating the elastic effective properties of regular hexagonal honeycombs. In contrast to previous cases, with this model we introduce a new formula, which match and complement the results obtained by Vigdergauz in 1999 in an essential way. Furthermore, by comparing some of the results with numerical finite element computations we find an improved formulae which enable us to estimate the effective properties with an accuracy better than one percent. In paper [C2] we study the effective elastic properties of regular triangular honeycombs. In particular, we obtain some simple approximate formulae for the corresponding Vigdergauz constants with accuracy better than one percent for all densities. In paper [C3] we study a scale of two-component composite structures of equal proportions with infinitely many microlevels. The structures are obtained recursively and we find that their effective conductivities are power means of the local conductivities.