Mathematical Analysis of a Thermo–Mechanical System With Phase Change and Melt–Layer Dynamics: Stability, Regime Transition, and Bifurcation Perspective

This study develops a nonlinear thermo–mechanical dynamical model describing the coupled interaction among mechanical motion, heat transfer, viscous dissipation, and melt-layer evolution during thermally induced phase change. The formulation addresses the limited availability of unified mathematical frameworks capable of simultaneously capturing thermal activation, interfacial evolution, and mechanical response within a single system of coupled nonlinear ordinary differential equations. The proposed model incorporates lubrication-based viscous resistance together with a temperature-activated melting mechanism, resulting in a strongly coupled piecewise-smooth dynamical system. A systematic non-dimensionalization is performed to identify the governing dimensionless parameters controlling thermal interaction, viscous dissipation, and phase-change dynamics. The mathematical analysis combines equilibrium analysis, Jacobian linearization, eigenvalue decomposition, Lyapunov energy arguments, and parameter-dependent stability investigation to characterize the qualitative behaviour of the coupled system. Numerical simulations are performed to illustrate the predicted thermo–mechanical evolution, energy dissipation, and regime-transition behaviour under representative parameter conditions. The analysis demonstrates that the coupled thermo–mechanical system possesses a block-structured Jacobian whose eigenvalue spectrum consists of one dissipative mode together with multiple neutral modes, establishing marginal stability of the equilibrium configuration. The results further show that variations in viscous resistance and melt-layer evolution govern continuous transitions between weakly damped, strongly dissipative, and melt-dominated operating regimes without classical eigenvalue crossing. The principal contribution of the present work is the development of a mathematically consistent nonlinear dynamical framework that unifies thermo–mechanical coupling, phase-change dynamics, stability analysis, and regime-transition behaviour within a single analytical formulation. The present model is intentionally formulated as a reduced-order dynamical system and therefore does not incorporate spatial heat conduction, temperature-dependent material properties, or fully distributed phase-interface evolution. Nevertheless, it provides a rigorous analytical foundation for future extensions involving higher-dimensional thermo–mechanical models and more general multiphysics coupling.