Cohesive zone modeling in load – unload situations

In some circumstances, loss of uniqueness of the stress/displacement solution may arise in cohesive modeling, preventing the use of cohesive elements in simulations involving loading and unloading of solids containing cracks. In these cases, it may be erroneously predicted that the damage of cohesive cracks increases when the external forces acting on the solid containing the cracks are reduced due to a prescribed “load reversal”. This issue is illustrated with various practical examples, including a finite element model of a double cantilever beam fracture specimen. Further numerical experiments are presented, that help to explain the causes of the observed phenomenon and shed some light on possible solutions. It is shown that lack of thermodynamic consistency of the cohesive model is not the cause of the observed behavior. The cause of the unphysical response is finally identified in its relationship with features of the nonlinear solution procedure and material stress update routines, by means of the formulation of a custom one-dimensional cohesive element. Two solutions are proposed based on a redesign of the loading/unloading criterion and a custom material model using one of them is presented. It is used as the material definition for Abaqus cohesive elements via UMAT subroutine, and it is shown that it solves the problem found during unloadings for all the considered numerical examples.