Fan T.-Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications

Science Press Beijing and Springer-Verlag, Berlin, Heidelberg, 2011. — 377 p. — ISBN: 9783642146428This monograph is devoted to the development of a mathematical theory of elasticity of quasicrystals and its applications. Some results on elastodynamics and plasticity of quasicrystals are also included to document preliminary advances in recent years.
The first quasicrystal was observed in 1982 and reported in November 1984. At that time several physical and mathematical theories for quasicrystal study already existed. Soon after the discovery, the theory of elasticity of quasicrystals was put forward. Based on Landau-Anderson symmetry-breaking, a new elementary excitation - the phason - was introduced in addition to the well known phonon. The phason concept was suggested in the 1960’s in incommensurate phase theory. Group
theory and discrete geometry e.g. the Penrose tiling provide further explanations to quasicrystals and their elasticity from the standpoint of algebra and geometry.
The phonon and phason elementary excitations form the basis of the theory of elasticity of this new solid phase. Many theoretical (condensed matter) physicists have spent a great deal of effort on constructing the fundamental physical framework of the theory of elasticity of quasicrystals. Applications of group theory and group representation theory further enhance the physical basis of the description. On the basis of the physical framework and by extending the methodology of mathematical physics and classical elasticity, the mathematical theory of elasticity of quasicrystals
has been developed. Recent studies on the elasto-/hydro-dynamics and the plasticity of quasicrystals have made preliminary but significant progress. As regards the dynamics, there are various viewpoints in the quasicrystal community, which reveal the unusual characteristics of phason dynamics. Yet the effect of the phason degrees of freedom on plastic deformation is not well understood, and the basic plastic properties of the material are virtually unknown. Because of many unsolved critical issues, the study of quasicrystals has attracted many researchers. The contex of the
last few chapters in this book is a probe of the fascinating research area.