General inverse-cube thickness scaling of projectile penetration energy in ultrathin films - Alessio Zaccone; Timothy W. Sirk (Proceedings of the National Academy of Sciences, 2026)
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, E p ∗ ( h ) = E p , ∞ ∗ + B h − 3 , 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.