An Analysis of the Roles of J and Q in the Assessment of Fracture for Quasi-Statically Extending Cracks in Residual Stress Fields

It is well known that the crack tip stress and strain fields for a crack in an elastic-plastic body depend on the crack tip contour integral J, the Q-stress, and the elastic-plastic properties of the material. This dependence is the fundamental basis of conventional two-parameter J-Q fracture mechanics assessments. It is normally assumed that the crack is created in an unstressed body, or else is inserted concurrently into an existing non-zero stress and strain field such that the crack tip fields build up monotonically and dominate at the crack tip. In such cases, the crack may be regarded as stationary and the J-Q procedure is valid provided that care is taken to calculate J and Q properly when initial stress and/or strains exist. When a crack is introduced progressively and quasi-statically into a component, the location of the crack tip will move along a distinct path. If the component contains residual stress and this is of a significant size along the crack tip path, a re-distribution of the residual stress will occur as the crack tip moves. Specifically, the stress field ahead of the crack tip will unload as the crack tip advances so that non-proportional loading will occur behind the advancing crack tip. In elastic-plastic materials, a wake of plasticity will usually be deposited in the material behind the moving or growing crack tip. Similar effects will also occur when a stationary crack extends due to critical or sub-critical processes. The presence of a plastic wake alters the stress and strain fields at the crack tip so that they do not generally match the fields of a stationary crack. Moreover, J and Q may not describe the stress and strain fields, invalidating the use of the fracture mechanics procedure for such cases. In this paper, a Finite Element analysis of J and Q is carried out for a quasi-statically extending crack inserted in a strip of elastic-plastic material containing an initial residual stress field. Care is taken to model the crack tip conditions appropriately as the crack extends and J is determined using the JEDI post-processing program which can allow for the effects of initial plastic strains and non-proportional loading. An assessment is made of the crack tip field and the likelihood of further extension or fracture is made using local approach models. The analysis considers both cleavage and ductile fracture. The extent of the relationship between J and Q and the crack tip fields is established and the validity of the J-Q procedure to such cases is discussed. The paper considers whether the procedure is conservative when J and Q are determined from an analysis of a stationary crack of the same size inserted into the same initial field.