There are many factors which affect the details of displacements and vibrations in a solid. But as a rough approximation one can perhaps assume that each element of a solid is either displaced or not, and that the displacements of neighboring elements interact by some definite rule—say a simple cellular automaton rule.
The pictures below show the behavior that one gets with a simple model of this kind. And even though there is no explicit randomness inserted into the model in any way, the paths of the cracks that emerge nevertheless appear to be quite random.
There are certainly many aspects of real materials that this model does not even come close to capturing. But I nevertheless suspect that even when much more realistic models for specific materials are used, the fundamental mechanisms responsible for randomness will still be very much the same as in the extremely simple model shown here.