Why K? High order singularities and small scale yielding

Abstract

Singular terms in the crack tip elastic stress field of order σ ∼ r ^-3/2, r ^-5/2, ... are often neglected, thus rationalizing the use of the K field, σ ∼ r ^-1/2, as the dominant term for fracture mechanics. We find the common explanation for neglecting the more singular terms in the series solution for the crack tip stress field unsatisfying. Further, the more singular terms are non-zero and are needed to understand the energetics of fracture, i.e., J and {ie97-1}. Given that the singular terms are generally present, the rationale for the validity of the small scale yielding assumption (the basis of linear elastic fracture) is more subtle than any argument which depends on the elimination of terms with stress σ ∼ r ^-3/2, r ^-5/2, ... Our explanation for the validity of small scale yielding is as follows. First, with or without small scale yielding, the stress field outside of the nonlinear zone does contain more singular terms. In the limit as the nonlinear zone at the crack tip shrinks to zero size (SSY) we show that the r ^-1/2 term in the Williams expansion dominates both the more singular and the non-singular terms in an annular region somewhat removed from this zone. Further, in this limit the magnitude of the σ ∼ r ^-1/2 term is almost entirely determined by tractions on the outer boundary. Our theory and examples are for representative problems in mode III anti-plane shear fracture. We expect, however, that the general results also apply to mode I and mode II fracture.