See the attached file for an illustration of this problem.
Consider an elastic sheet S (orange) that’s laying atop a solid base B (blue). The sheet is under tension (arrows) and is held to the base by surface tension due to a fluid between S and B (arrowhead). I’m looking for an expression of how closely the top of the sheet S reflects the shape of the base B.
If the base has a gentle wave to it (middle panel), then the top surface of the sheet will reflect what’s below it, though the curvature will change due to the added thickness, t. It’s straightforward to estimate how closely the top surface of the sheet will reflect what’s below it. But what if the change in shape is abrupt, or very narrow, like shown in the bottom panel? How do I best estimate the limits of how closely the sheet follows that shape?
at the point where you've broken contact as in your abrupt change example, is there anything else causing the sheet to deflect downward (other than maybe it's own gravitational pull if not negligible)? if not, why would it deflect downward at all? seems the tension in the sheet would just pull it across the gap. even if there is gravity, at that point that segment would be modeled as a 1D suspended beam or possibly a diaphragm (if there's an unshown depth).
There is a thin film of fluid between the sheet and the base (red arrowhead) that will tend to keep them in contact.
and that film fills the white space in the abrupt gap? Is that downward force going to be the same on the membrane over that gap, or will it have changed? Since that is the only force causing a deflection, it will need to be estimated. I still think the way to go is to assume a suspended membrane or beam and calculate the elongation/deflection caused by the fluid applied force. Once a force profile is somehow acquired those deflection formula should be available. (But for significant elongation may not be closed form).
If it's elastic it would want to relax, and technically, if it never does, it's no longer elastic (at those highly curved areas). To fill that space downwards is possible when a higher deformation mode. Then, it could relax too, if a blanket, since fabrics has much smaller limit in compressive direction (almost none, only little bcz there is a thickness, so that is why they withstand shear, also). Then you have to get it there; it actually embraces snap-through, if speaking in terms of deformation modes, but if a glue, it's not the same, in fact. Interesting question, in many ways, glue is more hard than a fluid.
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