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

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, $E_p^{\ast }(h)=E_{p,\mathrm{\infty }}^{\ast }+B{h}^{-3}$, 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.