  A basic paradigm underlying the Hookean mechanics of amorphous, isotropic
solids is that small deformations are proportional to the magnitude of external
forces. However, slender bodies may undergo large deformations even under
minute forces, leading to nonlinear responses rooted in purely geometric
effects. Here we study the indentation of a polymer film on a liquid bath. Our
experiments and simulations support a recently-predicted stiffening response
[Vella & Davidovitch, Phys. Rev. E 98, 013003 (2018)], and we show that the
system softens at large slopes, in agreement with our theory that addresses
small and large deflections. We show how stiffening and softening emanate from
nontrivial yet generic features of the stress and displacement fields.
