Making the material as general as possible makes the problem very hard. E.g. rubber falling on an infinitely stiff surface is completely different than steel. To be completely general with respect to both material and velocity makes this problem into a PhD dissertation. At the very least, you will need to restrict yourself to a certain class of materials to have a tractable problem. The easiest is to stick with linear elastic materials. E.g. this would include most metals if the velocity is not too high, but would exclude polymers (i.e. plastics) and very high velocities.
Edit: adding how I would approach the problem if it were limited to linear elastic materials :
Short answer: use a transient dynamics simulation in ANSYS with the appropriate contact elements.
Long answer (disclaimer, although this isn’t a PhD. Dissertation level problem, it still may require a lot of time to pick up all of the necessary knowledge)
So the first thing you’ll need to think about is contact mechanics. For the case of the sphere, the problem is that the area of contact will change over the time of the impact. Initially, only a small area of the sphere is in contact. Then, the sphere deforms, and a larger area is in contact. As the sphere rebounds and moves back up again, the area of contact shrinks. The video that ja72 referenced gives some idea of that concept, an here is one of a tennis ball that is a little lower speed (https://www.youtube.com/watch?v=YTwDH-9rM7c) . If you stick with relatively stiff metals on a stiff surface, the applicable theory is called Hertz contact. For a sphere, it works out that the total force exerted on the sphere is proportional to the indentation to the 3/2 power. The pressure will vary over the region of contact, with the highest pressure in the center, tapering off to zero at the edge. Wikipedia has a pretty good page on contact mechanics https://en.wikipedia.org/wiki/Contact_mechanics. If you go with something like rubber, you can’t use Hertz contact, and probably need to use something more involved, like JKR contact.
If you consider a prismatic object, like a cube, you could get away without using contact mechanics if you are willing to assume that the object always lands perfectly flat. I.e. either the entire surface area is in contact, or none of it is in contact. The pressure is just the force divided by the area and is the same over the whole area. In practice, you will probably never get your object to land perfectly flat, but you said you wanted back of the envelope, so let’s just go with this. So ignore spheres for now and focus only on prismatic objects.
To solve for the internal stresses due to the contact forces, we will want to use the theory of linear elasticity (https://en.wikipedia.org/wiki/Linear_elasticity). However, it’s pretty complicated. If we are willing to make some assumptions on the geometry, we can simplify the problem greatly. Start by considering a rod or bar. i.e. the height of the object is much greater than the other two dimensions. Think of the aspect ratio of a pencil. With that assumption, we can simplify the three dimension theory of linear elasticity down to a single one dimension. In other words, the stresses will vary greatly along the length of the pencil, but won’t vary too much from the inside to the outside, so we can just assume that the stress is essentially identical throughout the cross section. We can write down a partial differential equation describing the motion of the bar. See equation 5 in this course handout from Texas A&M https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/93279/HD14%20vibes%20continuous%20systems%202008.pdf?sequence=1&isAllowed=y it’s a little too much for me to retype here.
Now, that’s the vibration of the bar. We will want to set that up with free-free boundary conditions, a gravity load across the entire bar, and finally a non-linear force at the end of the bar that represents the contact. i.e. the force is zero if the end of the bar is not in contact with the surface, and some non-zero value if it is contact. the easiest assumption is just a force that varies proportional with the indentation. i.e. don’t assume a perfectly rigid surface, just give it some fairly high stiffness.
Now you’ll need to solve that partial differential equation as a function of both time and space. Unfortunately, even this fairly simple 1-D equation is going to be pretty hard to solve analytically. What we need to do is to reduce the partial differential equation into a series of ordinary differential equations. If I were doing this, I’d use a Galerkin discretization. However, that is probably way more math that you are really looking to do. Most people would use the finite element method (i.e. ANSYS) to discretize the structure spatially.
The advantage of the Galerkin discretization is that it is analytical. So variables like the geometry of the bar or the Young’s modulus of the material will appear in the final answer. But the disadvantage is that it’s a lot of math. The advantage of a commercial finite element pacakge is that the computer does all the work for you. But, to study the effect of different variables you will have to run many different cases (i.e. a numerical experiement). Another advantage of ANSYS is that extending from a 1-D rod to a 2-D or even 3-D structure is pretty straightforward. Just add more elements. An analytical study will get much harder in 2-D. Also Ansys can automate the contact mechanics, so you can study spheres or other contacts
The only trick in ANSYS is that you will need to make sure you have a fairly small time step to capture the dynamics. Try some initial guess, and then run it again with the time step cut in half. If the answer changes, then your time step is too big. Keep making it smaller until you get the same answer even when you cut the step in half. Do the same thing for mesh density. Try a mesh, then refine it. If you get a different answer, you need more element.
The really serious impact guys, like those that simulate automotive crash tests, will probably use LS-DYNA instead of ANSYS. But LS-DYNA is not for the faint of heart and probably too much of a learning curve for what you want to do.
