On Elastic Stability and Positive Definiteness for One-Dimensional Quasicrystals

In this work, a proper formulation of the elastic stability and positive definiteness of the elastic energy density of one-dimensional quasicrystals is provided. Based on an appropriate Voigt notation, the Sylvester criterion is applied and the necessary and sufficient conditions for the positive definiteness of the elastic energy density imposed on the elastic constants are derived for each Laue class from 4 to 10 of one-dimensional quasicrystals. Comparisons of the obtained sets of the necessary and sufficient conditions of one-dimensional quasicrystals with the corresponding ones of crystals are made, revealing that the phason fields, which characterize the quasicrystalline structure in any case, considerably influence the number as well as the degree of the obtained inequalities.