By Tian-You Fan
This inter-disciplinary paintings overlaying the continuum mechanics of novel fabrics, condensed subject physics and partial differential equations discusses the mathematical conception of elasticity of quasicrystals (a new condensed topic) and its functions by way of developing new partial differential equations of upper order and their ideas lower than complex boundary price and preliminary price stipulations. the recent theories built the following dramatically simplify the fixing of advanced elasticity equation structures. huge numbers of complex equations regarding elasticity are decreased to a unmarried or a couple of partial differential equations of upper order. Systematical and direct tools of mathematical physics and intricate variable features are constructed to resolve the equations lower than applicable boundary price and preliminary price stipulations, and lots of particular analytical recommendations are built.
The dynamic and non-linear research of deformation and fracture of quasicrystals during this quantity provides an leading edge technique. It supplies a simple, strict and systematic mathematical assessment of the sphere. entire and particular mathematical derivations consultant readers in the course of the paintings. through combining mathematical calculations and experimental information, theoretical research and sensible functions, and analytical and numerical reports, readers will achieve systematic, accomplished and in-depth wisdom on continuum mechanics, condensed subject physics and utilized mathematics.
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Additional info for Mathematical Theory of Elasticity of Quasicrystals and Its Applications
Phys Rev Lett, 1986, 56(25): 2740–2743  Sutherland B. Simple system with quasiperiodic dynamics: a spin in a magnetic ﬁeld. Phys Rev B, 1986, 34(8): 5208–5211  Fujiwara T, Yokokawa T. Universal pseudogap at Fermi energy in quasicrystals. Phys Rev Lett, 1991, 66(3): 333–336 Chapter 4 The physical basis of elasticity of quasicrystals The physical background on elasticity of quasicrystals is quite diﬀerent from that of the classical elasticity, the discussion about this provides a basis of the subsequent contents of the book.
The unit-cell description based on the Penrose tiling is adopted too, but the density-wave description based on the Laudau phenomenological theory on symmetry-breaking of condensed matter has played the central role and been widely acknowledged. This means there are two elementary excitations of low-energy, phonon u and phason w for quasicrystals, in which vector u is in the parallel space E 3 and vector w is in the perpendicular 3 , respectively. 1-1), where ⊕ represents the direct sum. 1-2). 8.
How to describe mathematically the behaviour of the quasicrystal deformation? To answer these questions, it is necessary to consider the physical background of elasticity of quasicrystals. The study in this regard was conducted soon after discovery of the new solid phase. Because the quasicrystal is a new structure of solid, theoretical physicists have proposed various descriptions of its elasticity. The majority agrees that the Landau density wave theory (Refs[1-25]) is the physical basis of elasticity of the quasicrystals.