Springs are objects made from elastic materials which can be used to store mechanical energy.
The most famous helical springs are well described in terms of Hooke’s law, which states that the force with which the spring pushes back is linearly proportional to the distance from its equilibrium length: $F=k \Delta x$, where $k$ is the spring constant, $\Delta x$ is the displacement from equilibrium, and $F$ is the force [see Fig. 1]. However, elastic springs can have quite different shapes from the usual helical springs, and for larger deformations Hooke’s law does not generally apply. In this problem we measure the properties of a spring made from a sheet of elastic material, which is schematically illustrated in Fig. 2.
Suppose that we take an elastic ring and compress it as shown in Fig. 2. Its shape can be approximated by a stadium shape, consisting of two semicircles of radii $R_0$.
The force with which the spring pushes back depends on the curvature of the bent parts of the ring. The dependence of the force on the radius $R_0$ of the curved parts is approximately given by:
\[F = \dfrac{\varkappa \pi t}{2R_0^2},\]where $t$ denotes the ring width; $\varkappa$ denotes a parameter referred to as the bending rigidity, is determined by the elastic properties of the material and the thickness of the sheet.
The bending rigidity $\varkappa$ depends on the Young’s modulus $E=1.3~\text{GPa}$ and the thickness $d$ of the ring material according to:
\[\varkappa = \frac{Ed^3}{12(1-\nu^2)},\]where $\nu$ is the Poisson ratio for the material; for most materials $\nu\approx1/3$.