High contrast materials in plasticity and magnetoelasticity

Metamaterials are artificially engineered composites whose microstructure is optimized in order to improve structural performances. The interest in metamaterials is due to their unusual properties, which arise as the by-product of their complex microstructures. Due to their importance for industrial applications, metamaterials are an increasingly active field of research at the interface between materials science, physics, engineering, and mathematics. The mathematical theory of homogenization is a fundamental tool to describe the structural properties of composites. In the periodic setting, the presence of a microstructure is modeled by considering the asymptotic behavior of a sequence of problems dependent on a small parameter. As this periodicity scale vanishes, the mathematical procedure of homogenization yields effective limit models, which are expected to capture the macroscopic behavior of the composite material. The project aims at developing the homogenization analysis of the effective behavior of a special class of metamaterials, that is those exhibiting a periodic strong-difference (high contrast) between the material properties of the components. The objective of the proposal is threefold. First, we aim at advancing the theory of periodic homogenization in the context of A-quasiconvexity and characterize high contrast materials in this framework. Our second and third goal are to develop the analysis of high contrast homogenization problems in the settings of elastoplasticity and magnetoelasticity, respectively. In particular, we will tackle the interplay of quasistatic time-evolution with microstructure formation due to the high contrast conformation. From a modeling point of view, the innovative nature of the project consists in aiming at a deeper understanding of high contrast problems, in connection with inelastic phenomena, and with the multiphysics settings. This may potentially have a strong impact in applications, especially concerning elastic cloaking and the modeling of advanced sensors and actuators. From a theoretical point of view, the mathematical modeling of inelastic phenomena, the understanding of the interplay between microstructure formation and time-evolution, and the analysis of constrained homogenization problems are currently very lively research areas in Mathematics. In these directions, the proposal will contribute to the current mathematical effort related to evolution problems for microstructures, and concur to the development of A-quasiconvexity techniques.

Metamaterials are artificially engineered composites whose microstructure is optimized in order to improve structural performances. The interest in metamaterials is due to their unusual properties, which arise as the by-product of their complex microstructures. Due to their importance for industrial applications, metamaterials are an increasingly active field of research at the interface between materials science, physics, engineering, and mathematics. The project aimed at developing a multiscale homogenization analysis of the effective behavior of a special class of metamaterials, that is those exhibiting a periodic strong-difference (high contrast) between the material properties of the components. One of the main results of the project concerns the establishment of a mathematically rigorous study for the emergence of chiral structures (magnetic skyrmions) in magnetic and magnetoelastic materials. Further, we have developed a very general framework, based on the mathematical theory of A-quasiconvexity, to treat high-contrast homogenization problems involving constraints expressed via a partial differential equation. Finally, we have identified an effective theory for composite metamaterials exhibiting possibly large elastoplastic deformations.