Implementation and application of some nonlinear models of diffusion/reaction in solids

Feulvarch et al. [5] defined an extended class of nonlinear models of diffusion/reaction in solids, applicable to both problems of diffusion of heat with phase change, and problems of diffusion of chemical elements with formation of simple, “stoechiometric” precipitate phases. They also presented an efficient finite element implementation of this class of models, based on a two-field formulation coupled with an implicit time-integration. This paper extends this earlier work in various ways. First it is shown that the class defined encompasses more elaborate models of diffusion of chemical elements with formation of complex, “non-stoechiometric” precipitate phases, consisting of solid solutions of “stoechiometric” constituents in variable proportions. Second, a more economical finite element implementation based on a one-field formulation - thus halving the number of nodal unknowns - is proposed. The keypoint in the new algorithm lies in an improved treatment of boundary conditions. Third, applications of this new algorithm pertaining to problems of internal oxidation of steel sheets are presented. Four distinct, practically significant situations are considered: (i) the case of a single, highly oxidizable element, with a reference to the seminal analytical solution of Wagner [27]; (ii) the case of a complex system involving 5 oxidizable elements and 9 a priori possible oxides; (iii) the case of a single oxidizable element but with formation of a non-stoechiometric oxide; (iv) a 2D case involving preferred diffusion along grain boundaries.