General 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 specific penetration energy follows a general inverse-cube scaling law, [Formula: see text], across chemically and structurally distinct systems. The inverse-cube scaling is traced to a finite-size correction to the effective shear modulus caused by suppression of long-wavelength nonaffine modes. The scaling describes impact data for multilayer graphene, graphene oxide, and polymer thin films, revealing a common elastic contribution to nanoscale impact resistance.