A field theory of strain/curvature incompatibility for coupled fracture and plasticity
Résumé
The displacement and rotation discontinuities arising between crack surfaces are assigned to smooth areal/tensorial densities of crystal defects referred to as disconnections and rotational disconnections (r-disconnections), through the incompatibility of the smooth strain and curvature tensors. In a dual way, the disconnections and r-disconnections are defined as line defects terminating surfaces where the displacement and rotation encounter a discontinuity. Relationships for the conservation across arbitrary patches of their strength (the crack opening displacement and opening angle) provide a natural framework for crack dynamics in terms of transport laws for the line defect densities. Similar methodology is applied to the discontinuities of the plastic displacement and rotation arising from the presence of dislocations and standard disclinations in the body, which results in the concurrent involvement of the dislocation/disclination density tensors in the analysis. The present model can therefore be viewed as an extension of the mechanics of dislocation and disclination fields to the case where continuity of the body is disrupted by cracks. From the continuity of the elastic strain and curvature tensors, it is expected that the stress/couple stress fields remain bounded everywhere in the body, including at the crack tip and in dislocation/disclination cores. Thermodynamic arguments provide the driving forces for the crystal defects motion, and guidance for the formulation of mobility laws insuring non-negative dissipation. The Peach–Koehler-type forces on dislocations and disclinations are retrieved in the analysis, and similar Peach–Koehler-type forces are defined for the disconnections and r-disconnections. A threshold in the (r-)disconnection driving force vs. (r-)disconnection velocity constitutive relationship provides for a Griffith-type fracture criterion. Application of the theory to a pencil-shaped notch configuration in elastic and elasto-plastic solids through finite element modeling shows that it provides elastic fields with lower energy level around crack tips than conventional singular solutions, and that crack propagation can be consistently described by the transport scheme. Interactions between crack growth and tilt boundary mobility are explored.