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Contact phase-field modeling for chemo-mechanical degradation processes. Part I: Theoretical foundations

As phase-field modelling is booming across various disciplines and has been proven fitted for numerically modeling interfacial problems, we aim at taking a step back to revisit its fundamental validity, especially for far-from-equilibrium problems. This questioning is relevant for rapid phase change and pattern formation, which is often the case in the processes modelled by PFM, although it is designed for slow dynamics and therefore not to deviate too much from local thermodynamic equilibrium. Starting from the main assumptions of PFM, non-equilibrium thermodynamics and maximum dissipation principle, the gradient flow equation usually obtained from variational formulation can be extended to generalized relaxation equations, based on a generalized free energy, which long term manifolds correspond to dynamic equilibria. For that, a general contact thermodynamic framework is derived from contact geometry, thus extending Gibbs seminal geometrical representation of thermostatics. The obtained viscous Allen-Cahn equation allows both the PFM kinematic degrees of freedom, the order parameter and its gradient, to be fully dissipative. The model is also extended to include chemo-mechanical coupling, corresponding respectively to endothermic and exothermic processes, leading to a phase change bidirectionality. This contact phase-field model will be applied in the second part of this work to irregular microstructures like geomaterials, valid for porous media in general.