Stress Based Insertion in L-Angle Specimen

Although, the bounding box cohesive element selection method is capable of large computational savings, the method has some drawbacks. This method is not self starting, instead it requires some cohesive elements to be present in regions where cohesive failure is expected. For fairly predictable problems, it can provide good results, but when the failure regions are not known a priori, the method is not optimal. An improved insertion method uses the average stresses of the neighbor volumetric elements to determine if the interface between these elements should be made cohesive. In effect, this allows us to begin a cohesive element free solution and only insert the elements as the stresses build to some predefined levels - defined by Equation 2.34.

As a test of the stress based insertion method, we use an L-angle problem presented in Figure 4.20. The top and right boundaries are fixed in place and a vertical velocity of $1.25$ $m/s$ is placed in shear along the left boundary. The bulk material of the domain is PMMA with a Young's Modulus $E = 3.24$ $GPa$, Poisson's Ratio, $\nu = 0.35$, and density $\rho = 1190$ $kg/m^3$. The cohesive elements have a maximum stress $\sigma_{max} = 32.4$ $MPa$, initial strength parameter $S_{init} = 0.995$ and a normal and tangential critical separations of $\Delta_{crit,N} = \Delta_{crit,T} = 2.2 \times 10^{-5}$ $m$. The domain is discretized into $3263$ nodes, $9531$ edges and $6269$ volumetric elements. Taking into account the instability of the cohesive elements to be inserted, the critical time step is reduced to $\Delta t = 3.0 \times 10^{-9} s$.

We run four different simulations for a duration of $60000$ time steps or $0.00018$ $s$, with the stress insertion selection occurring every $500$ time steps. The first represents the reference solution where cohesive elements are inserted everywhere in the domain at the start. The other three use the stress based insertion method for stress level of $15\%$, and $45\%$, respectively.

Figure 4.20: Schematic representation of L-angle test specimen with boundary conditions.

In order to verify the accuracy of the various solutions we observe the crack tip distance versus time, presented in Figure 4.22. From this figure, we can see that the crack tip distance, and indirectly the speed, are very close to the reference solution. Furthermore, from Figures 4.23 through 4.26, we can see that the crack profiles at the end of the simulation are very similar. In addition, to the crack profiles, we can see that the cohesive elements tend to concentrate in the high stress regions with the fewest elements present for the $45$ $\%$ stress insertion level. Even though we achieve the greatest savings for the larger stress insertion levels, these results contain greater instabilities in the solution. This is visually apparent by the greater number of failing cohesive elements on the fringes of the domain - represented by dashed lines. For lower stress levels, as well as for the reference solution, the failing cohesive elements tend to be limited to only the immediate vicinity of the crack.

The timing results for the reference and stress insertion cases are presented in Table 4.5. The largest savings, of $76\%$, occurs for the stress insertion of $45\%$, which uses the fewest number of cohesive elements to obtain the solution, as seen in Figure 4.21.

Table 4.5: Angle case timing results, in seconds, for various stress insertion levels.

Subroutine	Reference Case [s]	15%	30%	45%
$R^\textrm{co}$	2070.92	988.10	403.28	105.45
$R^\textrm{in}$	773.04	690.75	612.00	536.14
$Main$	1273.18	790.65	465.09	261.64
$Total$	4150.37	2615.30	1585.66	980.68
$\%$ $Total$ $Savings$		37%	62%	76%

Figure 4.21: Number of cohesive elements present in the domain over time for various stress insertion levels.
Figure 4.22: L-angle case crack tip distance versus time, for various stress insertion levels.

Figure 4.23: L-angle reference case with cohesive elements present from the beginning of the simulation (10x exaggeration).
Figure 4.24: L-angle case with stress based cohesive element insertion for a 15% stress level (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).

Figure 4.25: L-angle case with stress based cohesive element insertion for a 30% stress level (10x exaggeration).
Figure 4.26: L-angle case with stress based cohesive element insertion for a 45% stress level (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).