Is there a relatively inexpensive (let's say under $500US) crack-detection device and related mobile app that would be easy enough for a non-engineer/non-metallurgist to use to determine whether a bicycle frame was forming cracks that would be undetectable to the human eye?
Solar Mike is exactly right, the method he cites is called dye penetrant and it works for ferrous metals, aluminum alloys, and composites as well. all you need is a bottle of the fluorescent dye, the spray-on developer powder, and a UV light. This will cost less than $100.
There are three dye-pene types. The cheapest one is likely good enough ;1- A dye is applied ( usually red), 2- allow to soak ,3 - clean off surface dye( May use a cleaner for this ), 4- spray on a white developer: Depending on experience ,very small cracks can be found in any material. The second type of dye-pene uses a fluorescent dye and UV light ; faster than the red dye but requires a dark room and UV light. The third type uses magnetic fluorescent dye , a magnet and UV light but only works with steel . Cracks revealed by the red dye viewed days later. Also , I think red dye is the best method to reveal porosity or other non-linear imperfections . Also best if results need to be viewed by several people to make a judgement. Not likely to be your problem , but I remember well several people tying to squeeze under the curtain to view the UV light findings. The red dye only requires three aerosol cans ( cleaner, dye, and developer) but I have no idea what the price is today....PS - The method I call "magnetic fluorescent dye" is commonly called "wet mag".
Dye penetrant is a good solution but it's finite, once gone you'll need to buy more. A long term solution is an Ultrasonic Flaw Detection device. Unfortunately your looking at around 2k for a cheap one.