On the solution of unstable fracture problems with non-linear cohesive laws

Fracture mechanics models have well-known numerical challenges when implemented within implicit quasi-static frameworks once the cumulative internal energy exceeds the capacity for dissipation through the fracture process. Although this finding has been mainly reported for linear softening traction-separation laws, this work comprehensively explores non-linear softening behaviours and proposes a more general instability criterion. The ratio of cohesive to internal power emerges as a crucial factor. As a result, even scenarios involving a single cohesive element undergoing monotonic loading may exhibit a limit point at any stage of crack propagation, not just during crack initiation. Two strategies for handling fracture problems with instabilities within an implicit solution are discussed: an arc-length technique and an extension of quasi-static formulation into a dynamic regime. A comparative assessment is performed, covering both simple single-element cases and more complex scenarios. Furthermore, the study delves into more intricate material responses, including transformation-induced plasticity effects. Notably, incorporating these dissipative phenomena in the bulk material mitigates the difficulties associated with snap-back-like behaviours.