Rubber contains long molecules which are tangled up in its resting state. This is due to different charges in different parts of the molecule causing it to be attracted to each other and other parts of itself. When you stretch the rubber you are stretching out all the tangled molecules making them straight. When you release the rubber the molecules gets attracted to each other again and snap back to its former tangled configuration.
Everything is elastic. If you put a flea on a slab of granite and measured it very precisely, you’d see the granite flex and come back when you remove the flea. It’s just that the amount of force something can take before it breaks or permanently deforms is different for every material. In the case of something like a rubber band, it can take a lot of stretch before it permanently deforms or breaks
To add onto other comments- atoms in rubber bands are long chains (you can think of it like spaghetti noodles or a long paper clip chain), but not all materials are like this. Metals, for example, have atoms stacked together in organized patterns and structures. Rubber bands are like noodles, metals are like Legos stacked together. You can’t pull on these organized structures and have them go back to their original shapes like you can do with long chains (the reason is because of entropy, and you can do a lot of math to prove that’s the case).
Depending on the bond and the molecular geometry, molecular bonds can bend a bit (even “rigid” steel has a noticeable sag over a long span, and thus can be used to make springs). Past a certain point, the molecules in something like steel flow past each other and don’t go back to their previous position. Brittle materials like glass have very little ability to flex and shatter instead.
Some molecules (like elastic polymers) are long chains that are folded and held in a specific position by weaker electromagnetic bonds between adjacent parts of the molecule that can be overcome, and this allows the folded molecule to take on a different folded shape without breaking any of the stronger covalent bonds. But when the pressure is released, the polymer chain tends to fold up into the initial position.
Atoms can be thought of like magnets. Put two near each other and they attract til they bond. You can try to pull them apart but requires energy. With enough force, they can separate. Think of this as pulling sides of an elastic band. If you remove this force while they are still close enough, the attraction takes over and they bond again. What we call plastic deformation is when the atoms are too far apart and the attractiveness isn’t enough. The degree of this attractiveness is dependent on the type atoms present.
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