In its simplest form, the "damped" cohesive element response can be characterized by a multiplicative term to the cohesive failure law described in Section 2.1, with denoting the damping coefficient, , the norm of the velocity jump vector.
To illustrate the effect of this additional damping term on the response of the inserted cohesive element, we reconsider the simple 1D problem described in Figure 2.14. As shown in Figure 2.17, the introduction of a damping term in the cohesive element response eliminates all oscillations after just a few time steps. However, it was found that the amount of damping (i.e., the value of the coefficient ) is strongly problem dependent. For the simple 1D problem at hand, the optimal damping coefficients for the and time step insertion cases are and , respectively.

Furthermore, in 2D, the presence of the damping term was found to be much less effective in reducing the oscillations. These shortcomings forced us to adopt another approach based on the prestretching of the dynamically inserted cohesive elements. This approach is described next.