This might seem like a super vague and ambiguous question, but I'm not sure how to translate it in a mathematical language! So apologies. I'm not a mathematician but a metallurgist who wants to approach a problem mathematically.
Background I'm studying an observed undulatory behaviour of arched thin metallic foils under a compression load. When you apply a compression force to this thin arch, it buckles and forms two new archs. The behaviour can be somewhat describe by standing wave theory, but i falls short in the transition state between one harmonic to another. Please see the attachment "1" to get a picture o1
This is the so called "undulatory" response of this thin material. However, if he aspect ratio of the initial arch is over a certain ratio (0.5), such behaviour does not occur. Instead of wavy forms (akin to standing waves), you get localised permanent deformation. If you're interested, the reason why the wavy behaviour is preferred- we are investigating potential uses in micro electronic switches. The material these foils are made forms allows the progression up to 9 waves before it permanently deforms. A more traditional material only manages 5-6 at best.
Problem
finding a relationship between the aspect ratio (see image 2) 2 which is h1/L and the number of waves it can form (like nth harmonic). I have tried approaching the problem from a mechanics point of view using curved beam theory. But this might be too tedious. Instead, I was advised to find the relationship empirically. That is, find relationships between geometrical feature and loading conditions. This is where my question arises. Which is the best way I could go about doing that? What kind of topic does this come under in mathematics?
Apologies if my explanation is vague! At the very least, I hope i can learn how to identify the right search terms when approaching a problem mathematically.
Thanks
Picture $1$:
Picture $2$:
