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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?

FLP
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1 Answers1

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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.

leftaroundabout
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    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 Dec 13 '23 at 09:32
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    I don't think that's really relevant. – leftaroundabout Dec 13 '23 at 09:36
  • @FLP normal paper can be folded equally well in any direction, and with similar properties, so the anisotropy isn't that great. In fact, in origami you don't have to find a preferred orientation, at least not that I am aware of. – Davidmh Dec 14 '23 at 08:16
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    @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. – leftaroundabout Dec 14 '23 at 10:32
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    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. – Chris H Dec 14 '23 at 17:06
  • @ChrisH - I don't remember which copier, but one that I used made a point to load the ream of paper a certain way. So put your hand under the ream, then flip the stack over. For one of the two way the paper bends easier. – MaxW Dec 14 '23 at 19:02
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    @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. – user71659 Dec 15 '23 at 07:13
  • @user71659 but the same paper can be loaded in different printers, some of which feed landscape (e.g. A4 paper in an A3-capable printer) and some of which feed portrait (most desktop printers) so this in-plane anisotropy can't have a very big effect – Chris H Dec 15 '23 at 10:24
  • @user71659 - I should have mentioned that I was referring to the ream of paper bending along the long axis. Pretty sure the reason is because of the curl of the paper on the roll before it is trimmed to the short axis. – MaxW Dec 15 '23 at 11:21
  • @ChrisH For professional printing applications, you specify the grain orientation, either long grain or short grain. Printers almost always feed short edge first, the major exception is when you run letter/A4 in a tabloid/A3 printer. Grain also affects how it folds, that's why office-made booklets often don't fold tightly. In the US system, the second dimension is the grain orientation. – user71659 Dec 15 '23 at 17:02
  • @user71659 I'm asking about common office printers and MFDs (many of which are A3 capable and feed A4 long edge first) and supplies. – Chris H Dec 15 '23 at 18:06
  • @ChrisH maybe you should ask that as a separate question... – leftaroundabout Dec 15 '23 at 18:10
  • @leftaroundabout "asking" was dodgy autocorrect and should have read "talking" in reference to my previous comment – Chris H Dec 15 '23 at 20:14