# Why is paper inelastic but flexible?

Recently I have been studying solid structural mechanics, and one of the points I find really confusing is how elasticity and flexibility are closely intertwined.

Consider an Euler-Bernoulli beam, for instance. The flexural rigidity of the beam is given by $$EI$$, where $$E$$ is the Young's Modulus of a beam, while $$I$$ is the second moment of area (basically moment of inertia, but with area instead of mass). This suggests that an object that is easy to bend (with a small flexural rigidity) must also be easy to stretch (and possess a small Young's Modulus).

I find it hard to wrap my head around this idea because there are so many objects in our everyday lives that are difficult to stretch but very easy to bend. Consider a thick string, for instance. Although it is effectively inextensible, it can be bent around objects. Paper, too, is very hard to stretch, but it can bend very easily.

Is there anything that I misunderstood from this concept? What other understanding should I need to make sense of this apparent contradiction?

• Why can it not be that I be made much smaller than E is huge, so that EI is small even though E is huge? Commented Dec 13, 2023 at 9:11
• I thought about this, but I was also wondering whether the anisotropic behavior of paper or a small moment of area plays a larger role... After all, thin metal plates are very hard to bend also, although they also have a small moment of area.
– FLP
Commented Dec 13, 2023 at 9:36
• Aluminium foil makes it clear that your argument cannot be correct. Thin metal plates are easy to bend. Commented Dec 13, 2023 at 9:47
• Real paper actually stretches a bit when conical singularities appear, see Tom Witten's page on paper crumpling. Commented Dec 13, 2023 at 10:15
• Aluminum foil bends easily, an aluminum I-beam not so much. Commented Dec 14, 2023 at 14:42

Paper-the-material is not very easily deformed, it does have a fairly large Young's modulus. A sheet of paper however is thin, i.e. it has a small 2nd moment of area (with respect to an axis parallel to the sheet), and moderate-times-small equals small.
The physics behind this boils down to the fact that the material on the inside of the curvature doesn't have to become much "smaller" than the material on the outside, because the radii of the curves formed by the inside and outside are almost the same.

Even a thick string consists of thin fibres that are very easy to bend by the same principle. These fibres are more or less parallel and therefore make the rope strong in tension, but because there are no cross-links they do not team up to a large moment of area. Instead, when bending the rope the fibres slide against each other, so (in particular for smooth synthetic rope) it only requires slightly more force than adding the small forces to bend each individual strand.

Incidentally, the same fibre-sliding effect also takes place in paper: when you bend it far enough, the cellulose fibres change their relative positions. But because these fibres have quite a lot of friction between them, they don't readily slide back again, which is why e.g. origami keeps the shape you bend it to. This is an inelastic deformation.

• Thanks for your answer! Just wondering - does the anisotropic behaviour of paper (that its young's modulus is different in different directions) a major factor in causing paper to be inelastic and flexible as well? Initially I thought this was the main reason but I guess the small moment of area plays a larger role.
– FLP
Commented Dec 13, 2023 at 9:32
• I don't think that's really relevant. Commented Dec 13, 2023 at 9:36
• @Davidmh I do think paper is anisotropic in the sense that the fibres are mostly tangential to the surface. It is thus theoretically "easier" to split paper into layers than to rip it into pieces - except there's nothing to grab on in that direction, it only happens when removing strong adhesive tape from a cardboard surface. Commented Dec 14, 2023 at 10:32
• A similar sliding effect can be observed bending a thick stack of paper (e.g. a whole ream). Held loosely, it bends quite easily as the sheets slide over each other. With each end of the stack clamped between 2 bars, the sheets can't slide and the stack doesn't bend - it roughly approximates a solid mass of paper. Even the wrapper is enough but that has sides so it's not a great test. Commented Dec 14, 2023 at 17:06
• @MaxW Paper should be fed parallel to the grain direction, so for normal copy paper, the grain is parallel to the long edge. This is because paper is stiffer perpendicular to the grain, you can see this yourself simply by grasping a sheet by the edge and seeing how it flops over. You can also see the anisotropy simply by tearing it, the tear parallel to the grain is cleaner and straighter. The "image this side first" is different, paper gains a curl as moisture is removed in the fuser. Good papers have a reverse curl built in, so when you image the designated side first, it will end up flat. Commented Dec 15, 2023 at 7:13