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.

Figures (1)

• Figure 1: Universal inverse--cube thickness scaling of the specific penetration energy. (a) Multilayer graphene: specific penetration energy $E_p^*$ as a function of the number of graphene layers $N$, digitized from Fig. 4b of Bizao et al.Bizao2018. (b) Graphene oxide films under $1000~\mathrm{m/s}$ impact ImpactData2025. (c) Polymer thin films under $800~\mathrm{m/s}$ impact from Hyon2018. Solid lines are fits to Eq. \ref{['eq:Ep_hm3']}. Dashed lines in the log-log insets indicate the pure power-law trend $h^{-3}.$