Existence results in the linear dynamics of quasicrystals with phason diffusion and non-linear gyroscopic effects

TL;DR

This paper investigates the linear and non-linear dynamics of quasicrystals, establishing existence and uniqueness of weak solutions for models including phason diffusion and gyroscopic effects, advancing the understanding of their mechanical behavior.

Contribution

It introduces a refined mathematical framework for quasicrystal mechanics, handling phason diffusion and non-linear gyroscopic effects, with rigorous existence and uniqueness results.

Findings

Proved existence of weak solutions for linear quasicrystal dynamics with phason diffusion.

Established uniqueness of solutions under general boundary and initial conditions.

Demonstrated existence of solutions for non-linear models with phason rotational inertia.

Abstract

Quasicrystals are characterized by quasi-periodic arrangements of atoms. The description of their mechanics involves deformation and a (so called phason) vector field accounting at macroscopic scale of local phase changes, due to atomic flips necessary to match quasi-periodicity under the action of the external environment. Here we discuss the mechanics of quasicrystals, commenting the shift from its initial formulation, as standard elasticity in a space with dimension twice the ambient one, to a more elaborated setting avoiding physical inconveniences of the original proposal. In the new setting we tackle two problems. First we discuss the linear dynamics of quasicrystals including a phason diffusion. We prove existence of weak solutions and their uniqueness under rather general boundary and initial conditions. We then consider phason rotational inertia, non-linearly coupled with the curl of the macroscopic velocity, and prove once again existence of weak solutions to the pertinent balance equations.