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首页> 外文期刊>Physica Scripta: An International Journal for Experimental and Theoretical Physics >A theory of general solutions of 3D problems in 1D hexagonal quasicrystals
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A theory of general solutions of 3D problems in 1D hexagonal quasicrystals

机译：一维六方准晶体中3D问题的一般解的理论

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摘要

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.
机译：针对一维六方准晶体（QC）中的耦合平衡方程，建立了三维（3D）问题的通用解的理论，并提出了两个新的通用解，分别称为广义Lekhnitskii-Hu-Nowacki（LHN）和Elliott-Lodge （EL）解决方案分别基于三个定理给出。作为一种特殊情况，广义LHN解是通过引入三个高阶位移函数从我们之前的一般解中获得的。为了进一步简化，考虑三个特征根彼此不同或可能相等的三种情况，广义的E-L解应采用不同的形式，并用四个非常简单和有用的准谐波函数表示。证明Peng和Fan提出的一般解与广义E-L解的一种情况是一致的，而没有包括另外两种情况。重要的是要注意，广义LHN和E-L解在z凸域中是完整的，而在通常的非z凸域中是不完整的。

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