Universal inverse-cube thickness scaling of projectile penetration energy in ultrathin films

Abstract: Ultrathin films of widely different materials exhibit a dramatic enhancement of projectile penetration resistance under high--velocity impact. Despite extensive simulations and experiments, a unifying physical explanation has remained elusive. Here we show that the thickness dependence of the specific penetration energy obeys a universal law, $E_p^(h)=E_{p,\infty}^+B h^{-3}$, independent of chemical composition and degree of disorder. The inverse--cube scaling is traced back to a finite--size correction to the effective shear modulus arising from the suppression of long--wavelength nonaffine deformation modes in confined solids. The scaling quantitatively describes impact data for multilayer graphene, graphene oxide, and polymer thin films, revealing a common elastic origin for nanoscale impact resistance.