The review is devoted to introduce the recent development of the study in mathematical theory and methods of mechanics of quasicrystals, respectively. The mechanics of quasicrystalline materials includes elasticity, plasticity, defects, dynamics, fracture etc. In the article some relevant measured data are collected for some important quasicrystal systems, which are necessary for understanding physics and applications of the materials. It is very interesting that the mathe-matical theory and solving methods of the mechanics of quasicrystals have developed rapidly in recent years, which is strongly supported by the experiments and applications. The theoretical development strongly enhances the understanding in-depth the physics including mechanics of the materials. The mathematical theory and computational methods provide a basis to the applications of quasicrystals as functional and structural materials in practice as well. More recently the quasicrystals in soft matter are observed, which challenge the study of based on the quasicrystals of binary and ternary alloys and greatly enlarge the scope of the materials and have aroused a great deal attention of researchers, an introduction about this new phase and its mathematical theory is also given in the review.
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