三维立方准晶椭球夹杂的Eshelby张量

准晶体颗粒复合智能材料具有优异的物化性能和应用前景。不同于传统的各向同性材料，三维立方准晶体材料包含声子场，相位子场，及声子-相位子耦合场。为更好地研究准晶体颗粒夹杂问题，揭示准晶体材料夹杂问题的物理现象，本文利用本征应变公式和柯西留数定理，考虑椭球体夹杂，获得了三维立方准晶材料夹杂问题的Eshelby张量，并给出了统一的表达式。进而，当三维立方准晶夹杂形状为球形、棒状、扁平状和带状时，获得了封闭形式的三维立方准晶Eshelby张量表达式。同时，给出了椭球体长径比变化时Eshelby张量的变化规律，这对研究准晶体颗粒夹杂问题具有重要的理论意义。

Eshelby Tensors for Three-dimensional Cubic Quasicrystal Materials with Ellipsoidal Inclusions

The Quasicrystal (QC) particle-mixed composite (QCPMC) is a new class of composite which combines the excellent comprehensive properties of QCs giving rise to many promising technological applications. However, due to its unique microstructure, QCs possess phonon field, phase field, and phonon-phase coupling field, which is different from traditional solid materials. In order to optimize the QCPMCs effectively, the Eshelby tensor of the 3D cubic QCs material with ellipsoid inclusion is obtained by using Green’s function and Cauchy’s residue theorem, which is further used to explore the physical phenomenon of the influence of the quasicrystal particles distribution in mesoscopic scale on the macroscopic properties of QCPMC. The obtained Eshelby tensors are validated by degrading QCs to isotropic materials. Furthermore, the closed-form expressions are given when the particle shape are spheroid, elliptic cylinder, rod-shaped, penny-shaped, and ribbon-like, respectively. These expressions are the function of the particle shape and material properties. Moreover, it is found that the number of the independent non-zero components of the Eshelby tensor is 48, 17, 12 and 6, when the particle shape is spheroid, elliptic cylinder, sphere and penny-shaped, respectively. Finally, numerical studies are given to investigate the effect of particle aspect ratio. With the increasing of the aspect ratio, the increase of S3333 and S6363 is much larger than others, that is, Eshelby tensors related to x3 are more sensitive with respect to the particle shape. It is worth noting that the variation of the Eshelby tensors trend to flat when the aspect ratio approaches 10, and the convergence value of each tensor is close to the fixed value when ρ→∞, respectively. Consequently, when the aspect ration is larger than 10, the effect of the particle shape to the macroscopic properties of QCPMC is limited. For further applications, the solutions obtained in this paper can serve as the theoretical basis to obtain the effective properties of QCPMC, and solve the more complicated problems, such as fracture problem and defect behavior of QCs.