Cohesive Damping

One possible approach to reduce the potentially detrimental oscillations associated with the "blind" insertion of cohesive elements involve the introduction of some form of damping in the cohesive response. Schematically, this approach corresponds to adding a dash-pot in the cohesive element description (Figure 2.16).

Figure 2.16: Schematic representation of a damped 1-D cohesive element.

In its simplest form, the "damped" cohesive element response can be characterized by a multiplicative term $(1 + \eta \dot{\delta})$ to the cohesive failure law described in Section 2.1, with $\eta$ denoting the damping coefficient, $\dot{\delta}$, 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 1-D 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 $\eta$) is strongly problem dependent. For the simple 1-D problem at hand, the optimal damping coefficients for the $1000th$ and $2000th$ time step insertion cases are $\eta = 20$ and $\eta = 30$, respectively.

Figure 2.17: Effect of cohesive damping: evolution of the displacement jump across the cohesive element for the simple 1-D test problem shown in Figure 2.14 and resulting from ``blind'' cohesive element insertion with damping at time 0, $1000\Delta t$ ($33.3$ $s$) and $2000\Delta t$ ($66.6$ $s$).

Furthermore, in 2-D, 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 pre-stretching of the dynamically inserted cohesive elements. This approach is described next.