Formulation and numerical exploitation of mixed variational principles for coupled problems of Cahn–Hilliard‐type and standard diffusion in elastic solids

Abstract

SUMMARYThis work develops variational principles for the coupled problem of standard and extended Cahn–Hilliard‐type species diffusion in solids undergoing finite elastic deformations. It shows that the coupled problem of diffusion in deforming solids, accounting for phenomena like swelling, diffusion‐induced stress generation and possible phase segregation caused by the diffusing species, is related to an intrinsic mixed variational principle. It determines the rates of deformation and concentration along with the chemical potential, where the latter plays the role of a mixed variable. The principle characterizes a canonically compact model structure, where the three governing equations involved, that is, the mechanical equilibrium condition, the mass balance for the species content and a microforce balance that determines the chemical potential, appear as the Euler equation of a variational statement. The existence of the variational principle underlines an inherent symmetry of the coupled deformation–diffusion problem. This can be exploited in the numerical implementation by the construction of time‐discrete and space‐discrete incremental potentials, which fully determine the update problems of typical time stepping procedures. The mixed variational principles provide the most fundamental approach to the monolithic finite element solution of the coupled deformation–diffusion problem based on low‐order basis functions. They induce in a natural format the choice of symmetric solvers for Newton‐type iterative updates, providing a speedup and reduction of data storage when compared with non‐symmetric implementations. This is a strong argument for the use of the developed variational principles in the computational design of deformation–diffusion problems. Copyright © 2014 John Wiley & Sons, Ltd.

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