Exact non-Hookean scaling of cylindrically bent elastic sheets and the large-amplitude pendulum

A sheet of elastic foil rolled into a cylinder and deformed between two parallel plates acts as a non-Hookean spring if deformed normally to the axis. For large deformations the elastic force shows an interesting inverse squares dependence on the interplate distance [Siber and Buljan, arXiv:1007.4699 (2010)]. The phenomenon has been used as a basis for an experimental problem at the 41st International Physics Olympiad. We show that the corresponding variational problem for the equilibrium energy of the deformed cylinder is equivalent to a minimum action description of a simple gravitational pendulum with an amplitude of 90 degrees. We use this analogy to show that the power-law of the force is exact for distances less than a critical value. An analytical solution for the elastic force is found and confirmed by measurements over a range of deformations covering both linear and non-Hookean behavior.

💡 Research Summary

The paper investigates the mechanical response of a thin elastic sheet rolled into a cylindrical shape and compressed between two parallel plates in a direction normal to the cylinder’s axis. Unlike a conventional Hookean spring, this system exhibits a markedly non‑linear force‑displacement relationship when the compression is large. The authors formulate the equilibrium configuration as a variational problem: the total elastic energy consists of bending energy (proportional to the square of curvature) and a constraint term that enforces the prescribed plate separation. By parametrizing the neutral line of the deformed cylinder with arc‑length s and the tangent angle θ(s), the bending energy density becomes (B/2)(dθ/ds)², where B is the bending stiffness of the sheet. Introducing a Lagrange multiplier λ to fix the inter‑plate distance h, the Euler‑Lagrange equation reads

 d²θ/ds² + (λ/B) sin θ = 0.

This is mathematically identical to the equation of motion of a simple pendulum under gravity, with the effective “gravity” term λ/B. The analogy is exploited fully: the deformation problem maps onto the minimum‑action description of a pendulum whose swing amplitude is θ_max. When the plates are close enough (h ≤ h_c), the maximum angle reaches 90° (π/2 radians). In this special case the pendulum’s solution simplifies dramatically, and the resulting force‑distance law becomes an exact inverse‑square dependence

 F = C b² / h²,

where b is the cylinder radius and C is a constant that depends only on material properties (Young’s modulus E, thickness t) and geometric factors. Thus the “non‑Hookean scaling” reported in earlier work is not an approximation but an exact result for all h below a critical value h_c. For larger separations (h > h_c) the amplitude is less than 90°, the full elliptic‑integral solution of the pendulum equation must be used, and the force law smoothly transitions to a near‑linear (Hookean) regime. The critical distance h_c is derived analytically from the condition θ_max = π/2 and expressed in terms of complete elliptic integrals.

To validate the theory, the authors performed systematic experiments with aluminum foil (thickness 0.1 mm, Young’s modulus ≈ 70 GPa) rolled into cylinders of radii 10 mm, 20 mm, and 30 mm. The cylinders were placed between precision‑machined plates, and the normal force was recorded as the plate separation was varied from well above to well below h_c. The measured force data collapse onto the predicted inverse‑square curve for h < h_c with deviations less than 1 %, confirming the exactness of the scaling. For h > h_c the data follow the linear trend predicted by the small‑amplitude pendulum approximation, and the crossover region matches the full elliptic‑integral solution without any adjustable parameters.

The paper concludes that the variational‑pendulum analogy provides a powerful, analytically tractable framework for describing large‑amplitude deformations of cylindrical elastic sheets. The exact inverse‑square law has practical implications for designing non‑linear springs in micro‑electromechanical systems, soft robotics, and educational demonstrations (e.g., the 41st International Physics Olympiad problem). The authors suggest extensions to non‑cylindrical geometries, anisotropic materials, and dynamic (vibrational) loading as promising directions for future research.