A field theory of distortion incompatibility for coupled fracture and plasticity

The displacement discontinuity arising between the crack surfaces is assigned to smooth areal/tensorial densities of crystal defects referred to as disconnections, through the incompatibility of the continuous distortion tensor. In a dual way, the disconnections are defined as line defects terminating surfaces where the displacement encounters a discontinuity. A conservation argument for their strength (the crack opening displacement) provides a natural framework for their dynamics in the form of a transport law for the disconnection densities. Similar methodology is applied to the discontinuity of the plastic displacement arising from the presence of dislocations in the body, which results in the concurrent involvement of the dislocation density tensor in the analysis. The present model can therefore be viewed as an extension of the mechanics of dislocation fields to the case where continuity of the body is disrupted by cracks. From the continuity of the elastic distortion tensor, it is expected that the stress field remains bounded everywhere in the body, including at the crack tip. Thermodynamic arguments provide the driving forces for disconnection and dislocation motion, and guidance for the formulation of constitutive relationships insuring non-negative dissipation. The conventional Peach–Koehler force on dislocations is retrieved in the analysis, and a Peach–Koehler-type force on disconnections is defined. A threshold in the disconnection driving force vs. disconnection velocity constitutive relationship provides for a Griffith-type fracture criterion. Application of the theory to the slit-crack (Griffith–Inglis crack) in elastic and elasto-plastic solids through finite element modeling shows that it allows recovering earlier results on the stress field around cracks, and that crack propagation can be consistently described by the transport scheme. Shielding/anti-shielding of cracks by dislocations is considered to illustrate the static/dynamic interactions between dislocations and disconnections resulting from the theory. Sample size effects on crack growth are evidenced in solids encountering plastic yielding.