Study On The Anti-plane Problem Of Nano-elliptic Holes In One-dimensional Hexagonal Piezoelectric Quasicrystals With Surface Effect

Due to the particularity of quasi-periodic structure,quasicrystalline materials have many excellent properties and broad application prospects in engineering practice,such as as coatings and composites reinforcement elements.But the brittleness of quasicrystals at room temperature greatly limits their application as structural materials.The actual strength of the quasicrystalline material is lower than the theoretical strength due to the existence of dislocations,holes and cracks.In this paper,the anti-plane shear problem of a nano-elliptical hole in one-dimensional hexagonal quasicrystals with surface effect at micro and nano scales is studied,and the exact solution of fracture mechanical behavior is given,which provides theoretical reference for the preparation,design and application of nano-quasicrystalline materials.Firstly,based on the Gurtin-Murdoch surface/interface model,the anti-plane shear problem of a nano-elliptic hole in one-dimensional hexagonal quasicrystals is studied.The exact solutions of phonon stress and phason stress near the nano-elliptic hole are obtained by using the complex function method and mapping technique.When the geometrical parameters of the elliptical hole are adjusted,the size-dependent analytical expressions of the phonon and phason stress intensity factors at the nano-crack tip and the phonon and phason stress around the nano-circular hole are obtained.Numerical examples are used to analyze the stress concentration,stress intensity factor at the crack tip and the stress distribution at the edge of the circular hole.The results show that the surface effect has a great influence on the phonon stress,phason stress at the edge of the elliptical hole and the field intensity factor at the crack tip at the micro and nano scale.The stress distribution of phonon and phason fields at the edge of the circular hole changes periodically.Secondly,based on the Gurtin-Murdoch surface/interface model,the antiplane shear problem of a nano-elliptic hole in one-dimensional hexagonal piezoelectric quasicrystals subjected to in-plane electrical and anti-plane mechanical loads at infinity is studied.By considering the dielectric properties of nano-elliptical holes,the exact solutions of phonon stress,phason stress and electric displacement of matrix are obtained under the electrically permeable and electrically impermeable boundary conditions by using conformal mapping and complex function method,and the special cases such as nano-cracks and nano-circular holes are also discussed.Numerical examples are given to analyze the effects of surface material properties,aspect ratio,crack length and dielectric coefficient on phonon stress,phason stress,electrical displacement of elliptic hole and field intensity factor of nano-crack tip field under two kinds of electrical boundary conditions.The results show that the surface effect and the electrically permeable boundary have great influence on the field concentration of the nano-hole and the field intensity factor of the nano-crack.Finally,in order to effectively reduce the field concentration near nano-elliptic holes and nano-cracks,the antiplane shear problem of reinforced nano-elliptic holes in one-dimensional hexagonal piezoelectric quasicrystals is studied.By using the technique of conformal mapping and complex function method,the exact solutions of phonon stress,phason stress and electric displacement in the matrix and reinforcement layer under the electrically permeable and electrically impermeable boundary are obtained with considering the surface effect and dielectric characteristics in the nano-hole.The field intensity factors near the crack tip are obtained when the nano-elliptical hole is degenerated into a nano-crack.The effects of surface material properties,reinforcement layer material properties,aspect ratio of elliptical holes and coating thickness on the field concentration of the nano-elliptical hole and the field intensity factor at the crack tip are analyzed by numerical examples.The results show that the surface material properties and the reinforcement material properties have obvious effects on the field concentration of nano-hole and nano-crack.