A variational framework for Cahn–Hilliard-type diffusion coupled with Allen–Cahn-type multi-phase transformations in elastic and dissipative solids

This article presents a variational framework for coupled chemo-mechanical solids undergoing irreversible micro-structural changes at infinitesimal strains. The coupled problem is characterised by phenomena such as phase transitions, micro-structure coarsening and swelling. It is an extension of our previous work on variational inelasticity for a conserved chemo-mechanical setting to a unified conserved and non-conserved setting which include multi-phase transformations. The variational framework, again governed by continuous-time, discrete-time and discrete-space–time incremental variational principles, is outlined for coupled diffusion-phase transformation phenomena in elastic and dissipative solids. For the sake of simplicity, focus is restricted to isothermal conditions. It is shown that the governing macro- and micro-balance equations of the coupled problem appear as Euler equations of these minimisation and saddle point principles. In contrast to our previous work, extended variational principles (with the gradient of the chemical potential and phase fractions) are constructed that account for diffusion-phase transformation coupling. This is achieved by Legendre transformations. Note that the local–global solution strategy is still preserved and the resulting system of symmetric non-linear algebraic equations are solved by Newton–Raphson-type iterative methods. The applicability of the proposed framework is demonstrated by numerical simulations that qualitatively characterise lower bainitic micro-structure.