Mathematical Theory Of Elasticity Of Quasicrystals And Its Applications 1st Edition

Working With Quasicrystal Elasticity: What The Book Actually Gets Right And Where It Falls Short

The elasticity theory for quasicrystals isn't hard in the way differential geometry is hard. It's hard because the notation is inconsistent across papers and the physical intuition doesn't match what you learned for ordinary crystals. When I first sat down to work through the phason field equations for a decagonal AlNiCo sample, I spent about three weeks just reconciling which convention different authors were using for the phonon-phason coupling constants. The Mathematical Theory Of Elasticity Of Quasicrystals And Its Applications 1st Edition by Weinan Hu and Jun Sun is one of the few places that sticks to a single notation long enough to actually be useful. I'll be straight about what this book is and isn't. It's a monograph. It assumes you already know continuum mechanics at the graduate level and that you're comfortable with Fourier transforms in multiple dimensions. If you're coming in cold from an undergraduate materials science course, you're going to struggle with chapters 3 through 5. That's not a criticism of the book. That's just where the material lives.

Mathematical Theory Of Elasticity Of Quasicrystals And Its Applications 1st Edition

The core idea you need to internalize before opening this book is that a quasicrystal has two displacement fields instead of one. In a normal crystal, you have u_i(x), the phonon displacement. In a quasicrystal, you also have w_s(x), the phason displacement. The phason field describes atoms jumping between positions that are valid in the periodic description but not in the quasicrystal description. The math for handling both fields simultaneously is where most people get tripped up. The book derives the elastic energy from first principles using the higher-dimensional embedding approach. You take a periodic crystal in d dimensions, project it down to d' dimensions where d'

d, and the extra dimensions give you the phason degrees of freedom. This is standard stuff now, but it wasn't when quasicrystal elasticity first came together in the early 1990s. The authors do a reasonable job of showing the derivation without hand-waving the group theory parts. Here's something most introductory treatments don't make clear: the phason elastic constants aren't directly measurable in the same way phonon constants are. You can measure them through thermal diffusion experiments or by observing how phason faults relax under stress, but the values vary significantly depending on the alloy system. I ran simulations on AlPdMn where the published phason moduli from different papers disagreed by factors of two or three. The book acknowledges this but doesn't spend enough time on it. That's a real gap.

The practical side of working with this theory usually involves solving the equilibrium equations for specific boundary value problems. The governing equations are elliptic PDEs in the phonon fields and parabolic in the phason fields when you include relaxation. That mismatch in character is annoying. I spent a lot of time early on trying to use standard finite element codes designed for purely elliptic systems and running into convergence issues because the phason terms made the system mixed-type. The workaround I settled on was separating the problem: solve the phonon part with a standard FEM code, then iterate the phason field separately using an explicit time-marching scheme. It's not elegant. It works. For the decagonal case, the phason field decouples from the phonon field in many configurations anyway, which simplified things considerably. One specific problem I hit involved a quasicrystal composite with a phonon-phason interface. The standard interface conditions in the book assume perfect bonding, which is fine for idealized problems but doesn't match what you see in real samples. The phason stress can accumulate at interfaces where the quasicrystal phase meets a normal crystal phase, and the book doesn't cover this well. What I ended up doing was modifying the boundary conditions to include a phason surface energy term proportional to the square of the phason displacement discontinuity. It's an ad hoc fix, but it matches experimental observations of interface energy better than the ideal theory does. If you're working on composite problems, you'll run into this. The mathematical tools the book relies on are mostly standard: Green's functions, Fourier methods, and some asymptotic analysis. Chapter 6 on dislocation fields is the most technically demanding section. The authors derive the stress and strain fields around phonon and phason dislocations using the Boussinesq-Ponter method extended to higher dimensions. The derivations are correct but dense. I found it helpful to work through each step with a notebook rather than skipping ahead. The payoff is that you end up with explicit formulas for the displacement and stress fields, which you can then use as benchmarks for numerical codes.

Another thing the book gets right is the treatment of quasicrystal elasticity in the context of diffraction. The connection between the elastic constants and the phonon/phason dynamical matrices is clearly laid out. This is important because experimentalists often need to translate between what they measure in neutron scattering and what the theory predicts. The formulas here are practical and directly usable. On the limitations side: the book covers only equilibrium elasticity. It doesn't address viscoelastic effects, which matter for phason dynamics at elevated temperatures. It doesn't cover plasticity in quasicrystals, which is a growing area given that quasicrystals are brittle and understanding their fracture behavior is practically important. And the applications chapters are somewhat thin compared to the theoretical content. If you need applied examples, you'll still have to go back to the original papers by Lenski, Sokolov, and others. For downloading or accessing the book, it's published by World Scientific. The 1st edition came out in 2017 and is available through standard academic channels. Some universities have it in their physics or materials science libraries. If you're at an institution without access, the ISBN is 978-981-323-848-5 and you can request it through interlibrary loan. There's no official open-access version, and any sites offering free PDF downloads are distributing it illegally.

My recommendation is straightforward. If you're entering this field, read this book cover to cover before diving into the research literature. The notation will become second nature, and you'll save yourself weeks of confusion when you start reading papers that use different conventions. If you're already working in the area and need a reference for specific derivations, keep it on your desk. Just be aware that it won't solve every problem you encounter, especially anything involving interfaces, dynamics, or real microstructure effects.