What kind of residual displacement are you expecting? Plastic? Cohesive elements have only an elastic followed by a damage-induced softening behavior. Their displacement is dictated by the relative displacement of the two surfaces they're bounding. If these go back to the initial position, the cohesive elements' displacement will go to 0. Furthermore, upon crack front propagation, cohesive elements fail and are effectively removed from the computation, leaving the surfaces in a contact pair interaction to avoid interpenetration (with eventually friction).

You should define better your expected outcomes. Maybe you just need to use friction in the contact pair interaction. Or maybe you need to change the element type.

Thanks Luca. Yes, I did mean plastic displacement. I guess I should use another element type then.

I was also thinking that I could write a script defining this material behavior and incorporate that in ABAQUS, although I have no idea how to do that.

If you want to implement a custom material, you need a UMAT subroutine. For a custom element, the UEL subroutine. Subroutines in Abaqus are written in Fortran, but they all have predefined templates.

Check out this link for an introduction:

http://imechanica.org/files/Writing%20User%20Subroutines%20with%20ABAQUS_0.pdf

Thanks a lot! I will take a look.