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**Published:** Jan 1993

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**Source: **STP1189-EB

In this paper, asymptotic crack-tip fields of steady-state crack extension in materials with linear plastic hardening under combined in-plane (Mode I) and anti-plane shear (Mode III) loading conditions are investigated. Only the power singularity form of the in-plane and the anti-plane shear stresses is assumed, and the coupled differential equations are solved by means of the perturbation method. The governing equations of the asymptotic crack-tip field are formulated from two groups of angular functions: one for the in-plane mode and the other for the anti-plane shear mode. All stresses and deformations are of variable-separable forms of *r* and ϑ that represent the polar coordinates centered at the actual crack tip. Perturbation solutions of the governing equations have been carried out. The perturbation solutions predict that, regardless of the mixity of the crack-tip fields and strain-hardening, the in-plane stresses under the combined Mode I and Mode III loading conditions are generally more singular than the anti-plane shear stresses. The anti-plane shear stresses perturbed from the plane-strain Mode I field lose their singularity for small strain hardening, whereas the angular variation of stresses perturbed from the plane-stress Mode I behaves quite similarly to the pure Mode III solution. An obvious deviation between the perturbed and the unperturbed solutions can be observed under combined plane-strain and anti-plane Mode III loading conditions, but not under the plane-stress and Mode **III** conditions. The results imply that there exist no uniformly singular crack-tip fields under combined in-plane Mode I and anti-plane Mode **III** loading conditions.

**Keywords:**

asymptotic analysis, mixed-mode loading conditions, crack-tip field, plane-strain Mode I, plane-stress Mode I, anti-plane Mode III, steady-state crack extension, perturbation analysis, fracture mechanics, fatigue (materials)

**Author Information:**

Yuan, H *Research engineer and section head of Materials Mechanics, Institute of Materials Research, GKSS Research Center, Geesthacht,*

Cornec, A *Research engineer and section head of Materials Mechanics, Institute of Materials Research, GKSS Research Center, Geesthacht,*

**Committee/Subcommittee:** E08.08

**DOI:** 10.1520/STP24270S