I am investigating the pull-off behavior of epoxy nodules bonded to a thin layer of silicone elastomer. There is a sharp ~90 degree corner around the epoxy-silicone junction. The epoxy is pulled perpendicular to the plane of the silicone. With an elastomer, an elasto-plastic analysis is not very suitable to handle the singularity / theoretically infinite stress in the corner.
I've seen a similar study using h-elements that had a nice method to deal with the singularity. That method used finer and finer elements in the corner to progressively simulate stress vs. distance from corner. Based on results from very fine meshes, they choose a coarser mesh that would reliably find the stress 5 microns away from the corner. I've tried this with Mechanica (WF 5.0) with some success, but the p-elements give a oscillatory stress that makes interpretation of the real stress curve difficult (see pic attached). With h-elements, as the corner is approached, finer meshes simply cause the stress to level-off at higher values.
Attached is an example graph of 5 mesh refinement levels at the corner. I don't care what the oscillations are very near the corner (< ~.003mm) but I would like to reduce the oscillations >0.003mm away. I've tried a number of things with limited success:
- Isolate and exclude the corner elements -- no effect, with various settings (stress, stress and disp, max poly on/off and variations).
- Finer meshes approaching the corner (max element size on corner edge) -- improvement, but oscillations remain even "far" away from the corner. 3 micron elements are better than 10 micron, but 1 micron is not necessarily better than 2-3 micron.
- Reduce max polynomial order (2, 4): lack of convergence, oscillations larger, lower period.
- Increase max polonomial order (9): no effect as only 5-6 was needed.
- Tighter AutoGEM > settings > Limits > Angle controls. I usually spec 150 deg and 25 deg, which give an aspect ratio of < 3. I tried 27 deg which gave AR < 2.2 and a very even mesh, but oscillations were similar. 20 deg was worse.
- Average the results from left/right sides, and multiple cutting planes -- not much improvement, as the average tends to pick up the extremes.
- Tigher and looser % convergence on the analysis. 10% was better than both 3% and 20%.
Are there any known guidelines for reducing the stress oscillations approaching (but not extremely near) a singularity? The only thing I can think of next is to use max polynomial order = 1 (h-elements); I'd do my own convergence study, and spec a much finer mesh everywhere, but I'm guessing Mechanica isn't optimized to run order = 1, and maybe other issues?
I realized the way I made the plots grabbed data from both the stiff epoxy and flexible silicone. When the stress gradients are smooth, the stress in the 2 materials normal to their interface is the same, of course, but clearly not in high gradient regions. That caused most of the oscillation "far" from the corner. Thanks to Tad Doxsee for pointing out that there were 2 stress datapoints for each x-coordinate (1 epoxy and 1 silicone).
So I made a cutting surface 1 micron below the interface to just plot the normal stress in the silicone. This gives much smoother data. Now any oscillation is confined to the corner element, and often just a small portion of the element. Also 3% convergence gives a bit smoother results than 10%.