Pressure model and scaling laws in jammed bidisperse granular packings

Abstract

This investigation delves into the scaling laws governing pressure and key mean variables throughout the first and second jamming transitions previously observed in asymmetric bidisperse granular packings. Motivated by a theoretical model integrating crucial parameters—size ratio, \(\delta\), concentration of small particles, \(X_{\mathrm{S}}\), packing fraction, \(\phi\), mean contact number, \(\langle Z \rangle\), mean overlap, \(\langle \alpha ^{c}_{n} \rangle\), and mean branch vector length \(\langle \ell ^{c}_{n} \rangle\)—we employ molecular dynamics simulations to validate the model. Our findings reveal a non-linear relationship between pressure and \(\phi\) stemming from the dynamic interaction of mean variables with \(\phi\) during compression. Regardless of \(X_{\mathrm{S}}\) for δ = 0.73, the scaling exponent \(c_{Z}\) characterizing \(\langle Z \rangle\) with \(\phi\) consistently approximates 0.5, holding true for δ = 0.73 and high \(X_{\mathrm{S}}\) values. Intriguingly, for δ = 0.15 and low \(X_{\mathrm{S}}\), where the two jamming transitions are observed, \(c_{Z}\) exhibits distinct values. At the first transition, where large particles jam, \(c_{Z}\) slightly exceeds 0.5, while it diminishes to approximately 0.3 at the second transition following the jamming of small particles. Additionally, the exponents associated with the scaling of \(\langle \alpha ^{c}_{n} \rangle\) and \(\langle \ell ^{c}_{n} \rangle\) with \(\phi\) consistently converge around \(c_{\alpha } = c_{\ell } \sim 0.92\) varying with changes in \(X_{\mathrm{S}}\) and \(\delta\). Moreover, the pressure model aligns seamlessly with simulation trends, exhibiting a consistent exponent around \(c_{p} \sim 1.1\)–1.3 throughout the first and second jamming transitions. These results offer valuable insights into the compression behavior of highly asymmetric bidisperse packings, emphasizing the substantial influence of \(\delta\) and \(X_{\mathrm{S}}\) on the system’s macroscopic properties.