A Variational Formulation for Irreversible Thermodynamics with Path Dependence

This work introduces a path-dependent energy Lagrangian for irreversible thermomechanics that embeds heat and entropy accounting directly into the action. The formulation requires neither Lagrange multipliers nor Rayleigh potentials. An explicit θs term enforces Helmholtz conjugacy and positive heat capacity; writing heat as a divergence produces the natural flux; nonnegative dissipative productions are collected in a single modular term; and a history integral supplies an upper-limit variation that converts instantaneous power into entropy production. Stationarity yields the standard field equations together with a global entropy balance and a channel-wise power identity by placing each production once in entropy and once, with opposite sign, in its own channel. Classical closures-including Fourier and non-Fourier heat conduction, diffusion, and viscous mechanics-arise as special cases of the same functional. Compact examples show how the framework provides a unified action, a single entropy audit, and consistent positive production across coupled dissipative mechanisms.