I was thinking about how the walls of a barrel is made then I realized it is someone like fitting a piece of wood of length $l$ in between some "gap" of length $m<l$. This would cause the piece of wood to bend such that it fits the gap.
My question would be to find a generalized equation of such a naturally occurring curve, formed by fitting a sufficiently flexible cardboard of length $l$ into a gap of width $m<l$.
What I have thought of is that I assume flexibility of the material wouldn't matter since I assumed the cardboard is sufficiently flexible yet it is hard and does not collapse.
To orientate myself, I create an $\mathbb R^2$ coordinate system $yOx$ and the 2 ends of the curve would be on the $x$-axis, touching the $x$-axis at $x=0$ and $x=m$. It will look something like an inverted cycloid.
I also tried to provide a few conditions: $$\int_{0}^m \sqrt{1+\left(\frac{dy}{dx}\right)^2} dx=l$$ for some constant $l$, and I assumed that a naturally occurring curve would only bend once hence, $$\frac{dy}{dx}=0$$ has one solution, that is, $x=\frac{m}{2}$.
However, I am unsure if these conditions will allow me to find the equation of such a curve. Perhaps some concepts of Physics can be borrowed over to create a mathematical model of my problem. Anyone care to help? Thank you! :)
It is the well known Elastica. (was investigated by Euler?)
Its differential equation has curvature proportional to $y$.
$ \dfrac{d^2y/dx^2}{(1+y{^\prime}^2)^{3/2}} = \dfrac {-P y}{EI} $
is for large deformations ( P applied axial force , EI bending rigidity), integrates to Elliptic integral arch form ...Includes 3 cases depending on P.
For small rotational deformations when $ \dfrac{dy}{dx}<< 1$ it tends to a sine curve.
As the name implies, it is in the elastic regime of bending of thin elastic strips made of plastics, pole-vaulting fiberglass ( a special S-glass is chosen to keep within elastic limit ) poles and the like. So thin plastic strips may be a better material to choose just like fiberglass that bends all the way into a U-shape without fracture... compared to cardboard in engineering usage with low deflection bending stress failure.