Anti-plane fracture problem of four nano-cracks emanating from a regular 4<i>n</i>-polygon nano-hole in magnetoelectroelastic materials

Abstract

According to the conformal mapping from the exterior region of the regular <i>n</i>-polygon hole to the exterior region of a unit circle and from the exterior region of four cracks emanating from a circle to the interior region of a unit circle, a new conformal mapping is constructed to map the exterior region of four cracks emanating from a regular 4<i>n</i>-polygon hole to the interior of a unit circle. Then, based on the Gurtin-Murdoch surface/interface model and complex method, the anti-plane fracture of four nano-cracks emanating from a regular 4<i>n</i>-polygon nano-hole in magnetoelectroelastic material is studied. The exact solutions of stress intensity factor, electric displacement intensity factor, magnetic induction intensity factor, and energy release rate are obtained under the boundary condition of magnetoelectrically impermeable with considering the surface effect. Without considering the effect of the surface effect, the exact solution of four cracks emanating from a regular 4<i>n</i>-polygon hole in a magnetoelectroelastic material can be obtained. The numerical results show the influences of surface effect and the size of defect on the stress intensity factor, electric displacement intensity factor, magnetic induction intensity factor and energy release rate under the magnetoelectrically impermeable boundary condition. It can be seen that the stress intensity factor, electric displacement intensity factor, and magnetic induction intensity factor are significantly size-dependent when considering the surface effects of the nanoscale defects. And when the size of defect increases to a certain extent, the influence of surface effect begins to decrease and finally tends to follow the classical elasticity theory. When the distance between the center and the vertex of the regular 4<i>n</i>-polygon nano-hole is constant, the dimensionless field intensity factor decreases gradually with the increase of the number of edges, and approaches to the conclusion of a circular hole with four cracks. With the increase of the relative size of the crack, the dimensionless field intensity factor increases gradually. The dimensionless energy release rate of the nanoscale cracked hole has a significant size effect. The increase of mechanical load will increase the normalized energy release rate. The normalized energy release rate first decreases and then increases with electrical load increasing. The normalized energy release rate decreases with magnetic load increasing.