Abstract
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.
Highlights
An icosahedral structure in Al-Mn alloy with five-fold rotation symmetry was first discovered in April 1982, and reported in November 1984, refer to [1] and Figure 1
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 mathematical theory and solving methods of the mechanics of quasicrystals have developed rapidly in recent years, which is strongly supported by the experiments and applications
Summary
An icosahedral structure in Al-Mn alloy with five-fold rotation symmetry was first discovered in April 1982, and reported in November 1984, refer to [1] and Figure 1. After the work, reference [2] claimed that the structure is a quasicrystal, which presents quasiperiodic translational symmetry and orientational symmetry that is not allowed by the well-known crystal symmetry law. This has aroused a great attention of scientists in chemistry, crystallography, physics, mathematics, materials science etc. Many of the quasicrystals observed so far present thermodynamically stability, they become a new class of functional and structural materials, and present current and potential applications. For three-dimensional quasicrystals, along all three directions the atom arrangement is quasiperiodic. The unusual structures of quasicrystals lead to some novel phenomena different from those of crystals and conventional structural materials, we have to provide some additional knowledge before the subsequent presentation
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