Microscopic Nonaffine Deformation Theory of LAOS in Polymers

We develop a molecularly motivated framework connecting large-amplitude oscillatory shear (LAOS) nonlinearities in entangled polymers to frequency-dependent nonaffine relaxation in disordered solids. The central idea is that the first harmonic in LAOS measures the residual phase-locked elastic response, whereas the higher harmonics encode the Fourier signature of strain-dependent nonaffine relaxation. The finite-amplitude modulus is interpreted as a local tangent stiffness of the evolving microstructure, in the spirit of elastoplastic and incremental nonaffine models. For entangled polymers, the analogue of the decreasing coordination number in cage-breaking theories of glass mechanics is identified not with the tube-orientation tensor itself, but with the fraction of surviving tube constraints. This distinction leads naturally to a crossover description controlled by a characteristic strain amplitude $\gamma_c$, rather than by universal fixed power-law exponents. The fitted value $N_{\max}\simeq1.72$ indicates that the present experimental data approach a strong but not fully saturated nonlinear state, remaining below the ideal limiting value predicted for complete constraint collapse. Finally, a constraint-counting argument combining an eight-chain affine network representation with the central-force nonaffine isostatic threshold gives a limiting estimate $|\mathrm{NLI}|_{\max}=3$. The results support the interpretation of the NLI as a Fourier-resolved dynamic nonaffinity parameter and establish a bridge between tube-based polymer dynamics, LAOS harmonic analysis, elastoplastic rheology, and microscopic nonaffine lattice dynamics.