A Generalized Mechanical Model for the Cycle Rank Dependence of Stretch at Break in Phantom Chain Star Polymer Networks

A simple mechanical model was recently proposed to explain the universality of stretch at break ({\lambda}_b) as a function of cycle-rank density ({\xi}) in phantom-chain network simulations [J Non-Newtonian Fluid Mech., 349, 105620 (2026)]. Here, that model is reformulated as a series of the bottleneck strand and the surrounding network, yielding {\lambda}_b-1=({\lambda}_bs-1)[1+{\nu}_h/(1+c{\xi})]. In this formula, {\lambda}_bsis the stretch at break of the bottleneck strand, {\nu}_h is the number of stiff units in series along the rupture path, and c is a geometric constant for the parallel redundancy of the medium. Since c and {\nu}_hare difficult to separate over the examined range of {\xi}, c is fixed, and {\lambda}_bs and {\nu}_hare treated as fitting parameters. The formula is applied to phantom-chain simulations of networks with various conditions. In all cases, it reasonably captures the data, and the two parameters represent network characteristics.