Fracture in quasicrystals: vistas

Quasicrystals are aluminium-based alloys – inter-metallic solids, indeed – characterized by long-range order and absence of periodicity in the distribution of atomic maximum probability locations. And periodicity is typical of standard crystals. In contrast, quasi-periodicity is the characteristic feature of quasicrystal lattices: we have, in fact, prevailing atomic clusters – typically with icosahedral symmetry in 3D space, which is incompatible with lattice translations – and additional atomic arrangements determining the quasi-periodic structure. When we evaluate Fourier representation of mass density over such a quasi-periodic lattice of atoms in 3D ambient space, the related wave vectors result six dimensional to render compatible the expansion with the quasi-periodic geometry. Three of these six degrees of freedom can be attributed to finer spatial scale flips of atoms which annihilate and reconstruct quasi-periodicity in going from metastable states to stable ones and vice versa, depending on the interaction with the surrounding environment. These microstructural events influence even drastically the macroscopic mechanical behaviour of quasicrystals and non-standard actions (in the sense that they do not admit a representation in terms of standard Cauchy’s stress alone) are power conjugated with them. The micro-to-macro coupling is essentially stochastic [1]. In presence of fractures growing dynamically, such microstructural actions influence the force driving the crack tip. Here, we present an overview of questions connected with the dynamic propagation of fractures in quasicrystals. 1. In finite strain setting, we show (by following the general theoretical structure presented in [2] for the mechanics of simple and complex materials undergoing macroscopic mutations such as fractures) how the laws of dynamics crack propagation in quasicrystals derive from a requirement of invariance of what is called relative power. The procedure puts in evidence the existence of a bulk conservative self-action, in contrast with common formulations of the mechanics of quasicrystals where the conservative part of the self-action is neglected. 2. In the limit of vanishing conservative self-action, and within the limitations of small strain regime, by using Stroh’s formalism we provide solutions to boundary value problems involving dynamic steady-state crack propagation in quasicrystals [3]. 3. Finally, in small strain regime we present solutions to boundary value problems involving subsonic crack propagation between dissimilar quasicrystals [4] and allow comparison between the cases of presence and absence of conservative microstructural self-action.