Necessary and sufficient elastic stability conditions in 21 quasicrystal Laue classes

Mechanical stability is essential for solids and its stability criterion may date back to 75 years ago. Recently, the closed form necessary and sufficient conditions for elastic stability in all crystal classes have been investigated. Quasicrystals (QCs) are solids with long-range order and crystallographically forbidden rotational symmetries but without translational symmetry, attracting intense attentions in the last 30 years. In this work, we have explored the elastic constants and the elastic stability in detail for 1D, 2D and 3D QCs. All independent elastic constants and the closed form of necessary and sufficient conditions for elastic stability in all QCs classes are obtained, as a concise and pedagogical reference to stability criteria in aperiodic materials. Meanwhile, symmetry positions and stereographic projections of each QCs class are given as well.