Asymptotic Formulas for the Stress Field of a Crack by Nonlocal Elasticity

Abstract

A new mathematical approach is adopted to deal with the crack tip stress field. By considering the parameter α, which is proportional to the reciprocal of the distance between atoms, being large, we construct two asymptotic expansions for the stress field, which are uniformly valid for ra (r is the distance to the crack tip) being bounded and unbounded respectively. The results show that the classical singularity is eliminated and a finite value at the crack tip is found. We define this value as Nonlocal Boundary Residual (NBR) which is microscopic mechanics quantity and disappears in macroscopic mechanics theory. It is also found that to small angle stress there is a maximum stress near to the crack tip.