We are designing a button panel at work, made out of glass and are running into some issues. Glass, of course, can shatter. But we noticed that when dropping a steel ball in the center of the glass it is far less likely to shatter than when the same experiment is performed near the corners. It almost always shatters. Why is that? Is there something we can do with the geometry of the panel (currently rectangle) that will alleviate this issue? Maybe rounding the corners to be circular? Does anyone have a link to some research on the topic (i.e. how thick the glass should be to survive a ball of mass $m$ being dropped on it from height $h$)?
Glass tends to contain quite a lot of residual stresses from the manufacturing process and these will tend to be concentrated near corners (depending on how the sheet was manufactured). There is also the fact that for a panel supported at the edges the centre will be less stiff and thus can deform more before it fractures in the same way that a simple beam deforms most at the centre.
Similarly mounting the panel in a shock absorbing frame such as using a rubber or foam casket or an elastomer adhesive can help as this will allow the whole assembly to defect smoothing out the energy transfer from the impactor to the panel.
This sort of calculation in glass is complicated by the fact that it is a brittle material thus any local stress concentration which exceeds its UTS can cause it to fail completely even if the average stress is quite low where an equivalent ductile material would just be scratched or dented. This is why you can easily break a window with an impact from a hard sharp object with much less effort than just pushing against it with something blunt.
Equally any surface defects like pits or scratches can have a significant effect on the fracture toughness of glass, which is why scoring it with a diamond scribe is a very effective way to 'cut' it.
Some types of glass are designed to be much more impact resistant and to fail in a safe way, this include tempered glass which is heat treated to alleviate residual stresses and laminated glass which is composed of alternating layers of thin glass and a transparent polymer.
It sounds like you really need to talk to a glass supplier to find the appropriate material for your requirements.
There is no one generic formula for the impact toughness of glass (or really any material) any you would need to look at empirical data for a specific product and impact toughness metrics tend to only be applicable to quite specific conditions, especially for brittle materials.