Finally, a question involving my PhD! In my work I studied the nano-rigidity of DNA duplexes to see if they were good materials for making nano-walkers. (Spoiler alert: duplexes aren't great; origami is probably rigid enough.)
You are absolutely right to be skeptical about applying continuum equations carelessly to nanostructures. If a nanostructure is soft / small enough that significant deformations only require energies on the order of a few $k_BT$, then the canonical ensemble of the nanostructure will include significant contributions from deformed states, and you need polymer theoretical physics to inform you about the free energy of deformation.
As a practical example, the lower graph from my paper shows the free energy released upon bending a DNA duplex "back in on itself", thanks to "sticky ends" on each side, with the sticky ends glued together in two different ways. Note that both graphs are very nonlinear, with the bending energy hardly changing with DNA duplex length after about 30-40 base pairs -- this is more consistent with a thermalized worm-like chain than any rigid theory.
EDIT: from there, the best way to proceed is (1) quantify just how deformation is measured in the nanosystem (2) define that in terms of a reaction co-ordinate measurable in an MD simulation (3) use importance sampling methods, like umbrella sampling or forward flux sampling, to obtain a free energy surface in that reaction co-ordinate, and thus the energy of deformation.