A cohesive element fomulation connects via nonlinear spring elements the relative displacements between the upper and lower surface to a force per unit area. The element is really two dimensional. Instead of strains, the deformation is in terms of the relative displacements between the upper and lower surfaces interpolated to the Gauss points. Unlike strains, the incoming deformations have units of length. The output of the material model is the force per unit area (LS-DYNA manual: traction) at the Gauss points, acting to oppose the displacement.
In LS-DYNA V971, solid formulations ELFORM 19/20 and material models *MAT_138, *MAT_184, *MAT_185, *MAT_186 (*MAT_COHESIVE_MIXED_MODE, *MAT_COHESIVE_ELASTIC, *MAT_COHESIVE_TH, *MAT_COHESIVE_GENERAL) correspond to cohesive elements, whereby *MAT_138 is not available until LS-DYNA V971 R3.
There are two element formulations in LS-DYNA, which can be used with cohesive material models: ELMORM 19/20. ELFORM 20 will transfer moments between the bonded parts, whereas ELFORM 19 will not. The order of the nodes in defining the element is important. If the cohesive element bonds Element A to Element B, nodes 1-2-3-4 of the cohesive element should be shared by Element A or by Element B. In the first case, the normal of face 1-2-3-4 should point towards Element B and nodes 5-6-7-8 should be shared by Element B. In the second case, the normal of face 1-2-3-4 should point towards Element A and nodes 5-6-7-8 should be shared by Element A.
See also: Cohesive material models