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If we fold a paper and then apply pressure on the newly formed crease, it seems that the paper's surface gets a permanent deformation but what exactly has happened to the paper at a molecular scale?

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    $\begingroup$ This isn't specific to origami (the Japanese art form) - paper behaves the same way under folding whether the end result is art or just a tent card. $\endgroup$ – J... Jan 13 at 23:03
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    $\begingroup$ Closely related: physics.stackexchange.com/questions/579314/… $\endgroup$ – Allure Jan 13 at 23:11
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    $\begingroup$ @J... Alternatively, you could just think of a tent card as really trivial origami. From the physics perspective, the esthetic quality of the result is irrelevant. $\endgroup$ – Barmar Jan 14 at 15:58
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    $\begingroup$ @Barmar At the same time, I think we have sufficient historical evidence to show that tent cards evolved independently from origami. All origami is paper folding. Not all paper folding is origami. $\endgroup$ – J... Jan 14 at 16:56
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    $\begingroup$ @J... Convergent evolution :) $\endgroup$ – Barmar Jan 14 at 17:07
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Basically, a fold or crease in paper will remain because the structure of the fibers in the paper have become irreversibly damaged. This happens because the paper is bent/compressed beyond its elastic limit.

Chemically, paper is mainly composed of cellulose from plant fibres. It is an organic polymer, which has D-glucose units connected through hydrogen bonds. These bonds form between the oxygen atom of the one-hydroxyl group belonging to the glucose and the hydrogen atom of the next glucose unit. These are microscopic properties of paper, but to understand what happens when we fold paper or do Origami, it is sufficient to learn what is happening macroscopically.

All materials have what is called an elastic limit and a plastic region. The elastic limit is the point at which a material will bend but still return to its original position without any change or damage to its structure. Further deforming the material beyond this limit takes it to its plastic region. At this point any structural or physical changes become permanent and the paper will not return to its original form.

Every material has a different elastic limit/yield and plastic region. Imagine holding a piece of paper slightly bent but not folding or creasing it. The plant fibers that make up the paper will not have exceeded their elastic limit. So as soon as you let go of the paper sheet it will quickly return to its noncreased original flat state. However, if you were to roll that piece of paper into a cylinder and hold it for a few minutes, some of these fibers will be pushed beyond the elastic limit which is evident since it will not lie flat anymore and slight deformations have occurred to this sheet.

Now, when you properly fold a piece of paper as you would during Origami, the plant fibers along the crease will be pushed into the plastic region of the paper, and will reach a fracture point at the actual line of the fold. A pratical example of this is if you were to fold a piece of paper, you will note that if you stretch the paper evenly on both sides of the fold, the paper will tear right on the fold (a quick way to "cut" paper if you have no scissors). The fold then becomes a structural failure irreversibly and the fibers in the paper will never regain their original state.

Because of the damage to its structure, the paper will from then on have this fold. And no matter how hard you try to flatten out the fold it will never return to its original state. This is why Origami models continually retain their shape.

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    $\begingroup$ A great illustration of the damage being done is how easy paper becomes to tear after you bend it back and forth over the same crease a few times. $\endgroup$ – Luaan Jan 13 at 19:40
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    $\begingroup$ it seems like it's possible to reverse the rolled paper example by rolling it the opposite way or pressing it flat with a book for a while (while it seems impossible to reverse a solid fold). Or is that just an illusion and the paper is still filled with permanent deformities? $\endgroup$ – eps Jan 13 at 20:33
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    $\begingroup$ @eps You can get rid of the deformities, but doing so does not eliminate the damage done at the molecular level. Essentially, it’s the same as metal fatigue in this respect, the weakness remains even if the object has been returned to it’s original shape. $\endgroup$ – Austin Hemmelgarn Jan 13 at 22:51
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    $\begingroup$ @DarrelHoffman that suggests an idea for a trick. I wonder if it would work. Crease until the paper weakens, iron out the crease, but it's still much easier to tear at that point - "I can tear a piece of paper exactly in half, bet you can't" $\endgroup$ – Chris H Jan 14 at 17:40
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    $\begingroup$ I used to reverse folds by applying the exactly opposite fold on top. The result was something that readily bends in either direction. $\endgroup$ – Joshua Jan 14 at 22:09
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Curved creases are sometimes used in origami – a practical example being the French-fry box used in fast food restaurants. However, little is understood about the mechanics of such structures. Now, Marcelo Dias, Christian Santangelo and colleagues at the University of Massachusetts, Amherst and Harvard University are the first to develop a set of equations to describe the physics of curved-crease structures. As well as providing a better understanding of origami, the team hopes that the work will lead to practical 3D materials that are both strong and flexible.

Santangelo and colleagues focused on a ring because it is a relatively simple example of how a 2D structure can be transformed into 3D object by creating a curved crease. To gain a basic understanding of the physics, the team built a few origami saddles out of paper – from which they deduced which physical properties are key to understanding the mechanics of the curved crease.

At the heart of the transition from a 2D sheet to a 3D object are the planar stresses created in the ring when it is folded. These stresses are relieved by the sheet wrapping around itself to create a saddle-like structure. If the ring is cut, then the stresses are relieved and the saddle will collapse to a ring that will lie flat – albeit with a smaller radius.

(Source)

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    $\begingroup$ This is very interesting, but I do not see how it answers the question that was asked about changes to the structure of the paper at the location of the crease. SE is not for general discussion, but for answers to specific questions being asked. $\endgroup$ – arp Jan 13 at 20:34

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