A Bayesian analysis of the experimental evidence underpinning the fourth-power law in shock loading

Abstract

The Swegle–Grady fourth-power law, a proposed universal scaling for shock compression in materials, has long guided our understanding of how strain rates relate to peak stresses under extreme conditions. Yet, its empirical basis remains narrow, and experimental data often suffer from variability that obscures the true material response. This study re-examines the law using expanded datasets for fcc Al and bcc Fe, spanning decades of strain rates, to test its universality. We employ a Bayesian framework, combining Gaussian process regression to regularize shock wave profiles with a hierarchical model and Student’s t-likelihood to account for inter-run biases and outliers. Our results reveal distinct power-law exponents—2.911 for Al and 2.06 for Fe—challenging the notion of a universal fourth-power scaling and highlighting the influence of material-specific properties like crystal structure. This work underscores the need for robust statistical methods in shock compression studies, offering a reproducible approach that can be applied to other materials and experimental contexts. By clarifying the limits of universality, our findings pave the way for more accurate models of material behavior under extreme conditions, with implications for fields from planetary science to high-speed impact engineering.