Engineering criterion for rupture of brittle particles in a ductile matrix including particle size and stress triaxiality effects

Catastrophic failure due to cleavage fracture is caused by the rapid propagation of a micro-crack in the vicinity of a macroscopicflaw. Micro-cracks are initiated at second-phase brittle particles, present in the steel in different sizes and distributed randomly inthe volume. The current understanding is that such particles rupture when overloaded by the plastically deforming matrix. Topredict the experimentally observed statistical nature of cleavage fracture under different constraint conditions, it is pertinent todevelop a particle size and constraint dependent criterion for the failure of a brittle particle in a ductile matrix.In this work the failure energy of an elastic-brittle spherical particle in a ductile matrix is analysed. Several loading conditionswere examined, from constrained-uniaxial through to plane strain with varying levels of constraint. To develop a size dependentcondition, results for multiple particle radii were investigated within a fixed matrix volume. The particle and matrix weredeformed initially; subsequently nodes along the particle mid-plane were released progressively imitating crack opening. Theenergy associated with particle rupture was determined from the change in reaction force before and after release andcorresponding opening displacements.The results for each loading case show clear linear relation between rupture energy and particle size. Further the results show thedependence of rupture energy on constraint, with a distinct increase of failure probability with increasing constraint. Finally, anexpression for particle rupture dependence on size, stress triaxiality, and plastic strain level is derived. It is intended that thismodel will then be used to advance continuum-based local approach models to cleavage and meso-scale models for distributedinteracting micro-cracks.