Application of the thermodynamic state index: Mapping the Unified Mechanics Theory onto the critical phenomena of the 2D Ising model

Abstract

Current predictive models for material failure rely heavily on empirical curve-fitting with derivations that do not start from first-principles' physics. This study proposes a physics-informed framework for reliability by validating the Unified Mechanics Theory (UMT) against the exact analytical solution of the 2D Ising Model. The Thermodynamic State Index (TSI), a normalised measure of entropic disorder, was mapped onto the Onsager solution for the Ising lattice. The result demonstrates that the phase transition, representing the system's "structural" collapse, occurs at a specific critical state index Φ_c ≈ 0.264 for the square lattice. At this threshold, the magnetic susceptibility diverges, confirming that the TSI accurately tracks the onset of system-wide instability. This work bridges the gap between statistical mechanics and continuum damage mechanics, offering a rigorous, non-empirical metric for quantifying the proximity of a system to catastrophic failure or critical phenomena.