A structural reformulation of the local entropy inequality based on the thermodynamic admissibility of exchanged energy

The local entropy inequality in continuum thermodynamics is conventionally formulated by associating heat exchange with entropy transfer through the Clausius relation. This formulation, while appropriate for conventional thermal interactions, does not by itself establish that every energy-transfer contribution appearing in an internal-energy balance constitutes Clausius-effective heat. The present work addresses this unresolved thermodynamic attribution by introducing a structural decomposition of exchanged energy into Clausius-effective and non-Clausius contributions. The proposed framework establishes a necessary and sufficient criterion: an exchanged-energy contribution enters the local entropy-transfer term if and only if it admits a reversible thermodynamic representation consistent with the Clausius relation. Accordingly, the total energy balance remains unchanged and retains all energetic mechanisms, whereas the entropy-transfer term is restricted to thermodynamically admissible Clausius-effective exchanges. Non-Clausius contributions—including mechanically mediated dissipation, irreversible deformation, and internal structural energy conversion—remain fully represented in the energy balance and may contribute to internal entropy production through the irreversible evolution of the system, without being identified as Clausius heat. A general identification procedure is formulated based on the separation of thermal and non-thermal energy pathways, reversible and irreversible mechanical contributions, and independent and constrained energetic components. The framework is further connected to the structural viewpoint of Energy Structure Theory and General Energy Behavior (EST-GEB), providing a broader basis for classifying energy-transfer pathways across complex continua. For viscous heat-conducting fluids, the formulation distinguishes Fourier heat transfer from viscous dissipation without conflating a volumetric dissipative power with a heat-flux contribution. The resulting framework preserves the first law and the second law while sharpening the thermodynamic attribution of exchanged energy and providing a systematic basis for extending local entropy analysis to strongly coupled, multiphysics, and nonequilibrium systems.