Why is paper inelastic but flexible?

Question

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?

Answer 1

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.