What determines the elastic limit of a substance?
First let me clarify that there is no way to calculate the elastic limit of a substance - it has to be measured. Viscoelastic theory is pretty well-developed for only being about 150 years old, but it's not that well developed.

To answer your question, though, lots of factors affect elasticity in general and the elastic limit in particular.

It would probably be the most helpful to think about what is happening on the molecular scale. In any viscoelastic material, there is a range of deformations (strain, or the response to stress) and timescales over which molecules will go back to pretty much the same relative position that they were in before the deformation occurred. That's elasticity. What makes this happen? Each atom or molecule sits in a little potential energy well that exists because of its position relative to other atoms in the substance. Atoms like to be close to each other, but not too close. As a result, when given the opportunity they will arrange themselves so that as many atoms as possible are the optimal distance apart(resulting in the lowest energy).

When a material is deformed, we are moving atoms relative to each other. When one atom moves, its energy increases. This also changes the potential energy surface seen by its neighbors. If it moves a little bit, then the change is not usually great enough to cause the neighbors to move by much. On the other hand, for big movements, the effect can be great enough that the other nearby atoms rearrange themselves in order to find the next lowest energy level.

When the relative motion of atoms due to mechanical deformation is small enough that most of them go back to their original positions once the applied stress is removed, then we say the stress/strain was within the elastic limit.

When the atoms are permanently rearranged, the shape of the material is changed and we say that we have exceeded the elastic limit.

The ease with which atoms can move out of position (you can think of this like the depth of the wells), and their tendency to move back to the lowest energy (which can be thought of as the "steepness" of the wells) are the two main factors that affect how much stress/strain is needed to permanently deform a material. There are lots of things that can affect each of these, which is why there is no comprehensive theoretical model that can predict the elastic limit of materials based on molecular structure alone.

I can summarize a few important factors here, though:


degree of crystallinity in polymeric materials
type and number of dislocations in crystalline materials
length of time that the stress/strain is applied, in conjunction with the temperature of the material (this related to a phenomenon called time-temperature superposition)
structure of the molecules making up the material (e.g. linear vs. crosslinked or branched polymers, fcc vs. bcc crystal structure, or type of atoms in an alloy)
mesoscopic structure (e.g. crystalline grain size and shape, aggregation of nanoparticles, or long-range order in polymers)
macroscopic structure (e.g. fiber orientation in composite materials)


There are many more! And each of the ones I listed is practically an entire field of its own in materials science.

To answer the second part of your question - the cloudiness you are referring to is called "crazing" in polymer science. It is basically caused by molecules pulling apart from each other, without covalent bonds breaking. It can be reversed when it happens before the elastic limit is reached because the molecules can still go back to their original positions. If it doesn't happen until after the elastic limit, then the material is permanently deformed, and the molecules cannot return to their original position.