Effective theories for quasi-crystalline microstructures

A class of materials widely used by the practitioners of Materials Science is the one of heterogeneous media. At the microscopic level, they are made of two or more constituents but at the macroscopic scale, they behave as a homogeneous material whose properties can be obtained by averaging out the ones of the constituents. The rigorous study of the effective or macroscopic behaviour of heterogeneous media is the core of the homogenization theory. Due to the complexity and the huge variety of microstructures, a crucial modelling assumption is periodicity. In other words, we assume that the fine structure consists of a unit cell which is periodically repeated. Such an assumption has allowed us to provide a satisfactory analysis of the effective behaviour of many heterogeneous materials and by now, the periodic homogenization is a well-established theory. However, periodicity is not the only possible arrangement. Indeed, randomness plays a key role in characterizing the macroscopic properties of a heterogeneous material whose microstructure lacks an ordered pattern. In the recent years, a new arrangement of fine structures has emerged: the quasi- crystalline or quasi-periodic microstructure. It differs from the random since the microstructure is ordered, but it is not periodic since it exhibits forbidden symmetries for periodic structures, such as 5-, 7-, 8- and 10-fold symmetries. Thanks to such a peculiar arrangement, quasi-periodic composites enjoy extraordinary features and find applications in many different fields ranging from engineering to Materials Science. In view of the wide range of applications, it is of primary interest to understand the effective response of quasi-periodic materials. The aim of the project is to lay the foundations for the investigation of aperiodic microstructures in different scenarios. We will investigate double porosity problems, bilayered composites and phase transitions. These settings have been chosen for their relevance for applications and the high level of challenges they pose. Indeed, the tools developed in the periodic and stochastic homogenization are inadequate to characterize the effective response of quasi-periodic materials. This calls for peculiar and novel methods to be tackled, and to describe the possible new phenomena arising in the three scenarios described above.