General question!!

This was the question asked by my grand daughter and i was thinking about it but could get no satisfying thoughts, it's a general question:
When we are folding a paper a stage is reached when the paper cannot get folded anymore.What is the reason for this(that the paper can not get folded after a certain point??)

 

[MadMango]

There are several things going on that can attribute to this. Taking a 8.5"x11" sheet of paper, I can only fold it in half about 6 times. Each time you fold the paper, you are increasing the material to be folded. The paper itself starts to crease on the inside bend, but the outer bend has a slight radius. As you continue to fold the paper, you are adding more and more layers, basically forming a laminated member, which is stiffer than the section before.

Just remember that paper is just a finer representation of wood, and you know how strong a laminated panel of plywood can be. "The attempt and not the deed confounds us."

 

[IRstuff]

A rule of thumb is that stiffness increases as the cube of the thickness, so 6 folds works out 24^3 increase in stiffness, which explains why the difficulty in folding does not go linearly.

TTFN

 

Thank you MadMango and IRStuff for the replies.MadMango i did not follow the significance of the sentence "The paper itself starts to crease on the inside bend, but the outer bend has a slight radius".Can you explain.
I follow that as with every fold i am adding oone more layer to be folded thus increasing the thickness each time.Hence finally with a certain maximum thickness i find it unable to fold further.What is the significance of the sentence"The paper itself starts to crease on the inside bend, but the outer bend has a slight radius"
IRStuff when yoy said "stiffness increases as the cube of the thickness do you mean moment of inertia?

 

[dvd]

Here's my two cents worth. If you fold a single piece of paper 6 times you get 2^6=64 thicknesses of paper. You can easily fold a stack of 64 sheets of paper in half however. Why? Because the paper can slide on itself and fan out at the edges. However when you fold the single sheet of paper six times the edges are locked in by the previous folds and movement of the individual layers relative to the paper folds becomes increasingly difficult. If the paper were capable of being compressed and deforming plastically about a neutral axis, as in the case of metals, then the folding might continue. But since paper is fairly incompressible there is a localized buckling normal to the previous fold which locks the paper surfaces in the stack against each other and makes the relative movement nearly impossible.

 

[IRstuff]

Stiffness is as it sounds, resistance to bending. The thicker a structure, the harder it is to bend.

As you can see, a fold is a large bend that distorts the structure, which is why stiffness is non-linearly proportional to thickness. The thicker a structure, the more deformation and stress occurs to make the same bend, which is why eacn folder is significantly hard than the previous one.

TTFN

 

[rhc]

There are two strengthening mechanisms present.
IRstuff is right ......"A rule of thumb is that stiffness increases as the cube of the thickness, so 6 folds works out 24^3 increase in stiffness, which explains why the difficulty in folding does not go linearly."......
But DVD is also right.
To form any regular solid material into a bend the outside radius is stretched and the surface which becomes the inside radius is compressed. That is why you will see creasing (buckling) of the paper on the inside radius.
If you think of each individual sheet as a slip plane it is easy to understand why a thick stack of paper such as a telephone book can easily be bent. Try to bend that same book if the pages were glued together!
By multiple folding you are mechanically locking the layers of paper (the slip planes) and preventing each layer from sliding in much the same way as glue. A simple example is a piece of soft (annealed) steel wire. It is easy to bend at first but as the wire is bent back and forth it becomes increasingly difficult to bend, Minute planes in the annealed steel permit the grains of steel to slip which facilitate bending etc. If the steel wire continues to be worked they progressively lock up and the wire becomes stiffer, yet, its diameter is nominally unchanged.

 

[GregLocock]

It is a furphy anyway:

1) if you start with a large enough sheet of paper you can scale the bending stiffness/wrinkling problem down to irrelevancy and fold it more than 7 times

2) if you use A4 or Letter size photocopying paper and you have a workshop then you will have access to many toys that will help you fold it 7 times or more. I've done it (OK, slow day at work)

3) you could probably pre-crease the paper into a mechanism that would fold more easily. An origami student might be the best to consult on this.








Cheers

Greg Locock

 

[Lcubed]

trouble!!!!

I think dvd gets close to the basics by pointing out that the paper in your example can't slide on itself. The rule that stiffness increases as the cube of thickness requires shear connectivity between the lamina, and there is some restraint, more or less equivalent to shear connectivity, in your example due to the inability of the paper to slide freely

Regards,
Lin Lawson