The Theorem of Topological Entropic Rigidity (TER): A Computational Theory for Thermo-Mechanical Response of Disordered Solids

We present a comprehensive computational theory for predicting the thermo-mechanical response of disordered solids based on the Theorem of Topological Entropic Rigidity (TER) . The theory establishes deterministic relationships between the average number of mechanically active topological constraints per atom 𝒩𝑐, the configurational entropy 𝑆𝑐, and the effective shear modulus 𝐺eff(𝑇). We derive closed-form expressions for (i) the temperature-dependent shear modulus 𝐺eff(𝑇,𝒩𝑐) = 𝐺0[1 − 𝑇/𝑇𝑔(𝒩𝑐)]𝛼(𝒩𝑐), (ii) the glass transition temperature 𝑇𝑔(𝒩𝑐) = Δ𝐻conf/[Δ𝑆𝑐 + 𝑘𝐵ln⁡((𝒩max − 𝒩𝑐)/(𝒩𝑐 − 𝒩min))], (iii) the topological yield criterion 𝜎𝑦(𝒩𝑐) = 𝜎0[(𝒩𝑐 − 𝒩min)/(𝒩max − 𝒩𝑐)]𝛽 with 𝛽 = 0.5, and (iv) the nanoscale size effect 𝑇𝑔(𝐿) = 𝑇𝑔bulk ⋅ (1 − 𝛿/[𝐿(𝒩𝑐bulk − 𝒩min)]). We provide complete numerical algorithms, complexity analyses, convergence proofs, and a validated Python implementation. The computational framework achieves 𝑂(𝑁 ⋅ deg) scaling for constraint counting, runs on standard laptop hardware, and reproduces all theoretical predictions with relative errors <10−6. All source code is provided as supplementary material. PACS numbers: 61.43.Fs, 62.20.Fe, 64.70.pv, 81.05.Kf