On the elastic stability and positive definiteness of three-dimensional cubic quasicrystals

In this work, the elastic stability of three-dimensional cubic quasicrystals is investigated based on the positive definiteness of the elastic energy density. To this end, a normalized Voigt notation and the eigenvalue method are employed for the 12×12 constitutive matrix of the elastic constants. Six independent necessary and sufficient conditions for the positive definiteness of the elastic energy density imposed on the elastic constants are derived for three-dimensional cubic quasicrystals. The representation of the phonon and phason fields owing to the cubic symmetry offers an elegant application of the eigenvalue method, thereby delivering analytically simple explicit inequalities. Given the absence of experimentally determined elastic constants at present, the derived elastic stability conditions establish exact mechanical admissibility constraints imposed by cubic symmetry. They must be satisfied by any physically admissible set of elastic constants, whether obtained from future experiments, numerical simulations or approximant-based or engineering-oriented studies. In this sense, the derived elastic stability conditions provide a rigorous benchmark for the validation of experimental measurements, numerical simulations and engineering analyses.