Gao, Yang; Xu, Si-Peng; Zhao, Bao-Sheng A theory of general solutions of 3D problems in 1D hexagonal quasicrystals. (English) Zbl 1157.82395 Phys. Scr. 77, No. 1, Article ID 015601, 6 p. (2008). Summary: A theory of general solutions of three-dimensional (3D) problems is developed for the coupled equilibrium equations in 1D hexagonal quasicrystals (QCs), and two new general solutions, which are called generalized Lekhnitskii-Hu-Nowacki (LHN) and Elliott-Lodge (E-L) solutions, respectively, are presented based on three theorems. As a special case, the generalized LHN solution is obtained from our previous general solution by introducing three high-order displacement functions. For further simplification, considering three cases in which three characteristic roots are distinct or possibly equal to each other, the generalized E-L solution shall take different forms, and be expressed in terms of four quasi-harmonic functions which are very simple and useful. It is proved that the general solution presented by Peng and Fan is consistent with one case of the generalized E-L solution, while does not include the other two cases. It is important to note that generalized LHN and E-L solutions are complete in \(z\)-convex domains, while incomplete in the usual non-\(z\)-convex domains. Cited in 7 Documents MSC: 82D25 Statistical mechanics of crystals PDFBibTeX XMLCite \textit{Y. Gao} et al., Phys. Scr. 77, No. 1, Article ID 015601, 6 p. (2008; Zbl 1157.82395) Full Text: DOI