Shape of a compressed wristband What is the curve of the top part of this wristband after I squish the two ends closer? It's a curve of fixed length with given start $(x_1, 0)$ and end $(x_2, 0)$, and zero slope at these points, where my nails are. I believe it might be the curve that minimizes the maximum curvature subject to these constraints, but I'm not sure.


 A: Nice question! Here are some ideas so you can solve it yourself. Check out section 2.3.1 in this pdf. The Euler-Bernoulli bending beam equation, plus the balance of forces on the band, assuming small deformations yields the equation
$$y'''' + (P/EI)y'' = 0,$$
where $y(x)$ is the vertical deflection, $E$ is the elastic modulus, $P$ is the stress, and $I$ is a shape parameter (see the reference for more about them, they're all constants though). The general solution (eq. 2.3.14) is
$$y(x) = A \sin k x + B \cos k x + C x + D,$$
where $k = \sqrt{P/EI}$ and $A,B,C,D$ are constants. This means that when you hold the pieces at the same height, with flat angles at your fingers, then these boundary conditions imply $A = C = 0$, so the curve is a cosine.
The bending beam equation only makes sense when the deformations are small enough that the curve can be described as the graph of a function $y(x)$ and the slope is small enough that the stress-strain curve remains linear. If the latter assumption fails then we need to introduce nonlinear terms into the differential equation. In the first case, we can still get by cutting up our curve and using rotated coordinate systems for each piece to solve for the shape, so it will look like a bunch of $y(x)$ as above but rotated and glued together to be smooth up to the fourth derivative. This might be hard to find out in practice though.
You mentioned it might be trying to minimize the curvature, and yes it is doing something like that. In Landau-Lifschitz vol 10 around eqn 17.7 they derive the free energy of a bent rod to be
$$F(\gamma) = \int_\gamma ds E I_y(s)/R(s)^2,$$
where $s$ is an arc-length parameter, $R(s)$ is the radius of curvature at $s$ , $E$ is the elastic modulus, and $I_y(s)$ depends on the angle at $s$ unless the cross section has equal principal moments of inertia (like a square or circular cross section). In the end of the day, the shape is whatever curve $\gamma_0$ for which (Hamilton's principle of least action)
$$\frac{\delta F}{\delta \gamma}|_{\gamma_0} = 0.$$
A: Here is another possible way to determine the shape of the wristband.
First of all, the shape obviously has the left-right symmetry. 

Secondly, the slope of the wristband at point C is zero, i.e., the same as the slope at point A, which implies that the top and the bottom sections of the wristband are also symmetric.
At point A, the wristband is constrained in two ways: it has a horizontal force $F_A$ acting on it and keeping it from spreading outward and a moment $M_A$ keeping it from unbending. I would note that force $F_A$ has to be horizontal, because otherwise the two external forces, $F_A$ and $F_E$, acting on the wristband from the left and right sides would not be able to balance each other.
The bending moment at point D, $M_D$, could be found as a difference between the bending moment $M_A$ at point A and the bending moment created by force $F_A$ at point D. As the bending moment due to force $F_A$ will be increasing with height $h$, the resulting bending moment at D, $M_D=M_A-F_Ah$, will be decreasing and, correspondingly, the radius $R$ of the wristband curviture will be increasing. The radius will become infinity at the inflection point, B, where the bending moment due to force $F_A$ will cancel the bending moment $M_A$.
As we move further up from point B, the bending moment $M=M_A-F_Ah$ will change its sign and its magnitude will start increasing, so the radius of the curviture will start decreasing until, at point C, it will reach the same value as the radius at point A (due to symmetry). The magnitude of the bending moment at point C will also be the same as at point A, but it will have the opposite, counterclockwise, direction.
In summary, the shape of the wristband will be characterized by the radius, which will be infinity at B and will be decreasing as inverse of the vertical distance from B, going up and down, reaching its minimum value at A and C
