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compression in a roll

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  compression in a roll Christopher Howard <christopher@librehacker.com> - 2026-10-14 07:13 -0800
    Re: compression in a roll ram@zedat.fu-berlin.de (Stefan Ram) - 2026-10-14 15:41 +0000
    Re: compression in a roll Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-10-14 22:17 +0200

#897299 — compression in a roll

Hi, I was wondering: when you roll up tightly something like sheets of
paper, toilet paper, or plastic, is there some kind of compression force
that makes sheets toward the center of the roll thinner than sheets
toward the outside of the roll? I was thinking about the mathematics of
the relationship between the radius of a roll and the length and
thickness of the roll, which I see is simple if you assume that there is
uniform thickness in a tight roll. But I was wondering if the roll tends
to compress more toward the center or if the tension of making the roll
is even throughout and not accumulating.

-- 
Christopher Howard

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#897300

Christopher Howard <christopher@librehacker.com> wrote or quoted:
>Hi, I was wondering: when you roll up tightly something like sheets of
>paper, toilet paper, or plastic, is there some kind of compression force
>that makes sheets toward the center of the roll thinner than sheets
>toward the outside of the roll?

  It seems there is more pressure near the center.

  The tension of the rolled-up layers is called "wound-in tension".

  There is a whole science of this called "web handling", "web
  winding", or "roll winding mechanics". Industrial winding machines
  do not use even tension throughout the process, to ensure the
  roll has a relatively uniform hardness and thickness throughout.

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#897302

Christopher Howard wrote:
> [...] when you roll up tightly something like sheets of
> paper, toilet paper, or plastic, is there some kind of compression
> force

There is if there is an external force in the first place.  Otherwise no.

> that makes sheets toward the center of the roll thinner than sheets
> toward the outside of the roll?

Barely, if at all.  Toilet paper being intentionally softer than other
paper, e.g. office paper, with its surface less even and each layer
containing more air, I would expect it to be better compressible than
the latter, but not by much.

The main factor appears to be how much air is _between_ each layer.  If
I consider (my current) toilet paper, due to the less even surface much
more air can be (is) between the layers than e.g. with office paper, and
so a roll of toilet paper is on the one hand thicker and on the other
more compressible than a stack of office paper with the same number of
layers because of that already.

> I was thinking about the mathematics of the relationship between the
> radius of a roll and the length and thickness of the roll, which I
> see is simple if you assume that there is uniform thickness in a
> tight roll. But I was wondering if the roll tends to compress more
> toward the center or if the tension of making the roll is even
> throughout and not accumulating.
If a force is exerted on the outer layers, and all layers are not or
barely compressible (which is the case for paper or plastic
/perpendicular/ to its surface), then the pressure on the inner layers
is greater because their (angular) surface area is smaller, but the
applied force is nearly the same:

                            _
                      .  '     '  .
                   '    .  ' '  .    `  A(R)
                 '   '     . .     `   `
               .   '   '         `   `   .
                  '  '   . ' ' . A(r) '
              :  .      '       '  '   .  :
              :  :  :  :    +----:--:--:--:
              :  '      .        r     '  R
                  .  '   ' . . '   '  .
               '   .   '.       .'   .   '
                 .   '     ' '     .   .
                   .    `  . .  '    .   __
                     `  .   _   .  '    |PE


  A(R) ∝ R > r ∝ A(r)  ⇒  p(r) = F/A(r) > F/A(R) = p(R).

  [N.B.: A(r) ∝ r² does not always hold.  Consider, for example,
   the formula for the lateral surface area of a cylinder.
   But A(r) ∝ r always holds 8-)]

We think that this explains why the inner terrestrial core is solid
despite the higher temperatures there than in the outer core (which is
liquid).  (The applied force there is the gravitational force.
Additionally, it increases towards the center because the mass above a
point in the interior increases towards the center.)

But whether the density of a layer increases because of that pressure
depends on the compressibility of the material which is given by

  κ = −1∕V (∂V∕∂p),

where the partial derivative depends on whether the process is
isothermic (probably the case here) or isentropic/adiabatic.  One can
see that (especially for solids) it can also depend on the direction in
which the force/pressure is applied.

<https://en.wikipedia.org/wiki/Compressibility>

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

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