Dynamic of a bent sheet of paper

Question

Take a standard printer sheet of paper (dimension $L \times H$ say), wrap the sheet of paper around a cylinder of radius $r$ so that the sheet of paper takes its form. Now put the sheet of paper down on a table and hold it by the two sides, then leave one side: you can remark that the sheet of paper automatically wraps on itself toward the other side.

Is it possible to describe the dynamic of the motion of the sheet?

I am not a physicist, so excuse me if this is basic physics.

I think it might be elastic deformation, perhaps a (stress-strain?)model/Hooke's law would be a step in that direction? — Emil, Apr 04 '18 at 20:36
@sammygerbil this is not a duplicate even if the topic you are referring to is interesting! — Jacques Mardot, Apr 06 '18 at 12:38
Sorry you are right, I misunderstood the other question, which is asking for a static solution. — sammy gerbil, Apr 07 '18 at 09:20

score 0 · Answer 1 · answered Apr 04 '18 at 20:47

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the problem you pose here is not a basic one- it's a hard one to solve!

the easy part involves the sheet of paper's tendency to uncurl itself to flatness. this can be modeled as a weak, nonlinear, elastic spring with low mass moving through large deflections, for which formulas could be derived.

The difficult part involves

1) the movement of air around the sheet as it unfurls, which tends to oppose the unfurling. because the sheet weighs little and has a large surface area in contact with the air, the motion of the sheet will be strongly affected by this, and the equations describing it will be complicated, and approximate and

2) the friction present inside the sheet of paper, which will be hard to encompass in equations.

answered Apr 04 '18 at 20:47

niels nielsen

92,630

I'm glad this problem is of interest! Could you give me some good reference to read for what you said about nonlinear elastic spring? This exercise was proposed in an oral examination in France (second year undergraduate competitive exams), so the examiner was not expecting a complete solution I think. – Jacques Mardot Apr 06 '18 at 12:40
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@jacquesmardot, you will find extensive (no pun intended) treatment of nonlinear, large deflection springs in the literature pertaining to the mechanical engineering field of dynamical systems modeling. the hard parts of modeling the time response of an unfurling sheet of paper will be that the deflections are very large (no small-angle simplifications of the geometry can be used), the paper as a spring is itself highly nonlinear (it is "constituitively" nonlinear) and the geometry of the unfurling process makes that geometry displacement-dependent. Very hard to solve! – niels nielsen Apr 06 '18 at 16:57

Dynamic of a bent sheet of paper

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