From doing some research I've learned that internal stresses inside an object can "concentrate" on sharp edges, as shown in the above picture. For that reason we often make fillets so that the change in diameter is smoother and we get a smaller stress concentration.
But why does the stress concentrate on sharp edges? From Googling I've found numerous inadequate answers, such as imagining "stress flow lines" getting packed at the sharp edge. Here I can see an analogy to electromagnetism, where we talk about flow lines of an electric field, but what is "flowing" here? What do the lines represent?
Intuitively, I can imagine this: If we press the object on the left in the above picture, there is a higher stress after the edge, since there is now a smaller area enduring the same force. Hence the stress is higher. But this does not explain why the stress concentrates at the edge. I mean, even with the fillet, the radius decreases anyway, so why is there not a similarly high stress on the right of the fillet?
Similar effect also occurs when we have a continuous piece of material with a hole in it. The stress is the highest around the hole.
Once again, force lines, what do they represent? Again I can see the similarity to electromagnetism. But what kind of a force field are we talking about here?
One source I found simply explained that we can use methods like finite element analysis to confirm this concentrating of stress does happen. But this is not an explanation either. Use of FEA implies there is some kind of differential equation to be solved, so what kind of theory is there behind it?