An examination of stability in cohesive zone modeling

Stable crack propagation hinges on the driving force and the resistance. In the context of a cohesive approach to fracture, properly resolving the cohesive zone ensures that the resistance is in equilibrium with the driving force. Additional restrictions on the mesh size can stem from crack stability. For high strength, low toughness (brittle) materials, the requirements for stability can exceed those for cohesive zone resolution. Examples in 1-D and 2-D reveal that decrements in the mesh size transition the system from indefinite to positive definite. Moreover, small decreases in the mesh size beyond the transition provide substantial reductions in the condition number. We contrast a physical instability resulting from material properties, geometry, or boundary conditions from a numerical instability resulting from the mesh size and propose that the selected discretization should ensure that the cohesive zone is both resolved and stabilized.