A nanoscale assembled plate with surface elasticity containing an interface crack subject to bending moments and shear forces

Abstract

This article studies a nanoscale plate assembled by two nanoplates with surface elasticity containing an interface crack when the crack surfaces are loaded by bending moment and shear force. The classical (Kirchhoff) plate theory incorporating Gurtin–Murdoch (GM) surface elasticity is utilized. A bimaterial nanoplate with an interface through-thickness crack is converted to a mixed boundary-value problem, and is solved by using the Fourier integral transform. A singular integral equation with two Cauchy kernels for either uniform bending moment or constant effective shear force is derived. The closed-form solution is given and the exact expressions for the bending moment and shear force are determined. Specially, the oscillatory singular behavior of the crack-tip field is found. When two dissimilar nanoplates are identical, the oscillatory singularity disappears but singularity dominated by $r^{-3/2}$ and $r^{-1/2}$ ($r$ being the distance from the crack tip) remains. Additionally, when the surface elasticity is neglected, the results of a large-scale assembled plate with a through-thickness interface crack are directly derived from the present. The obtained results show that surface elasticity has a significant effect on the energy release rate of the oscillatory crack-tip field.