Insertion Analysis

As a first step in our insertion analysis we select the two simple 2-D test problems, presented in Figure 4.6. Each domain is $2.5$ $m$ tall and $1.5$ $m$ wide, with a fixed base and an applied velocity of $0.056$ $m/s$ pulling on the top. The material properties for the system are selected such that the dilatational wave speed is $10$ $m/s$ and a conservative time step value of $\Delta t = 0.0004$ $s$ is used to ensure that any instabilities are a direct result of the insertion and not the cohesive elements.

Figure 4.6: Simple 2-D meshes with three cohesive elements inserted along (a) "horizontal" and (b) "mixed" interfaces.

The two cases differ in the orientation of the cohesive elements with respect to the applied load. In the first case, all three cohesive elements are perpendicular to the applied force, so that only normal tractions and separations will be present on these cohesive elements, with the second having an inclined cohesive element that will have both normal and shear tractions and separations.

In the upcoming sections we will present the three main insertion methods, which include blind insertion, insertion with damping, and pre-stretched insertion. From our 1-D analysis, we have determined that the solution for a dynamic insertion problem is dependent on the stress level of the cohesive element at the time of its insertion. Using the average stresses for volumetric elements $\char93 23$ and $\char93 14$, the cohesive elements are inserted at time steps 0 ($0.0$ $s$), $2500 \Delta t$ ($1.0$ $s$) or $5000 \Delta t$ ($2.0$ $s$) of a $10000$ ($4.0$ $s$) time step simulation. corresponding to stress levels of $0\%$, $23\%$ and $43\%$, respectively, as seen in Figure 4.7. In each case we will follow the nodal separation of the tracking node presented in Figure 4.6.

Figure 4.7: Normalized average stress levels for the volumetric elements of the middle cohesive element. Vertical lines at the $0th$ ($0.0$ $s$), $2500th$ ($1.0$ $s$) and $5000th$ ($2.0$ $s$) time step represent dynamic insertion times.