Based on the Gurtin-Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping, the electrically permeable boundary condition with surface effect is established, and the antiplane fracture problem of three nanocracks emanating from a hexagonal nanohole in one-dimensional hexagonal piezoelectric quasicrystals with surface effect is studied. The exact solutions of the stress intensity factor of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate are obtained under the two electrically permeable and the electrically impermeable boundary conditions. The numerical examples show the influence of surface effect on the stress intensity factors of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate under the two boundary conditions. It can be seen that the surface effect leads to the coupling of the phonon field, phason field, and electric field, and with the decrease of cavity size, the influence of surface effect is more obvious.

中文翻译：

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一维六方压电准晶体中电可渗透的六方纳米孔引起的三个纳米裂纹的反平面破裂问题

基于Gurtin-Murdoch表面/界面模型和复数势理论，通过构造新的共形映射，建立了具有表面效应的电可渗透边界条件，并从六边形纳米孔中产生了一维三个纳米裂纹的反平面破裂问题研究了具有表面效应的六角形压电准晶体。在两个电导和电导边界条件下，获得了声子场和声子场的应力强度因子，电位移强度因子和能量释放速率的精确解。数值示例表明了表面效应对声子场和声子场的应力强度因子，电位移强度因子的影响，以及两个边界条件下的能量释放率。可以看出，表面效应导致声子场，声子场和电场的耦合，并且随着腔尺寸的减小，表面效应的影响更加明显。