Let’s use paper from your example to explain this.
Paper is not one solid, contiguous thing on a microscopic level. Paper is really made from layers and layers of interlocking plant fibers. Those fibers are made of layers and layers of interlocking cellulose molecules, which [look like this, more or less](https://upload.wikimedia.org/wikipedia/commons/f/f9/Cellulose-Ibeta-from-xtal-2002-3D-balls.png).
Those cellulose molecules form fibers because sometimes hydrogen (the white balls) on the outer side of one strand of cellulose will bond to an oxygen (the red balls) on a neighboring strand. The fibers form the paper due to the process in which the paper’s made leaving them physically interlocked, and some of the hydrogen bonding between fibers. It’s a very weak bond, which is why paper’s so easy to tear and bend.
~~As to why it stays bent, though. As you bend paper, it requires you put energy into the act, and that energy breaks some of the hydrogen bonds, changing the orientation of some of the fibers within the paper, which then form new hydrogen bonds.~~EDIT: It was pointed out that what I said wasn’t quite correct. Creasing *does* break some of the fibers, and that does add up over time.
So when you’re bending the paper, you’re changing its structure at a microscopic level.
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