Variational formulation of Cahn–Hilliard-type diffusion coupled with crystal plasticity

Research output: Contribution to journalArticleResearchpeer-review


This article presents a relatively general framework for the description of the Cahn–Hilliard-type diffusion in solids undergoing infinitesimal elastic and plastic deformations. The coupled chemo-mechanical problem, characterised by phenomena such as phase segregation, microstructure coarsening and swelling, is treated using the variational framework which is governed by continuous-time, discrete-time and discrete-space–time incremental variational principles. It is shown that the governing equations of the coupled problem can be derived as Euler equations of minimisation and saddle point principles. A point of departure from the existing works is the coupling of crystal plasticity to the problem of diffusion and optimising the potential with respect to the plastic variables such that they are solved locally at the integration points. This is done using a return map algorithm which results in a reduced global problem. The variational framework results in a system of symmetric non-linear algebraic equations that are solved by Newton–Raphson-type iterative methods. This is a novel and attractive feature with respect to numerical implementation, as models resulting from the proposed variational framework are computationally less expensive in comparison with non-symmetric formulations. The numerical simulations presented at the end predict the applicability of models resulting from the proposed variational framework for multiple scenarios.
Original languageEnglish
Article number103652
Number of pages22
JournalInternational journal of plasticity
Issue numberAugust
Publication statusPublished - Aug 2023

Bibliographical note

Publisher Copyright: © 2023 The Authors


  • Cahn–Hilliard diffusion
  • Chemo-mechanical coupling
  • Monolithic solution scheme
  • Phase-field model
  • Variational formulation

Cite this