Surface Effect and Size Dependence on the Energy Release Due to a Nanosized Hole Expansion in Plane Elastic Materials

This paper deals with the surface effect and size dependence on the $M$-integral representing the energy release due to a nanodefect expansion in plane elasticity. Due to the high surface-to-volume ratio for reinforcing particles in the nanometer scale, the surface effect along the nanosized hole may be induced from the residual surface stress and the surface Lamé constants. The invariant integrals such as the $Jk$-integral vector and the $M$-integral customarily used in macrofracture mechanics are extended to treat plane elastic materials containing a nanosized hole. It is concluded that both components of the $Jk$-integral vanish when the contour selected to calculate the integral encloses the whole nanosized hole. This leads to the independence of the $M$-integral from the global coordinate shift. It is concluded that the surface effect and the size dependence on the energy release due to the nanohole expansion are significant especially when the hole size is less than 40 nm. This present study reveals that the discrepancies of the $M$-integral value with the surface effect from the referenced value $M0$ without the surface effect are mainly induced from the residual surface stress $τ0$ rather than from the surface Lamé constants $μs$ and $λs$⁠.