PSI - Issue 61

İmren Uyar et al. / Procedia Structural Integrity 61 (2024) 195 – 205 İ. Uyar, E. Gürses / Structural Integrity Procedia 00 ( 2019) 000 – 000 5 In the equation (8), the chemical diffusivity 0 is allowed to degrade by [(1− ) 2 + ] with the evolution of the phase field variable and k is a numerical parameter chosen as 1×10 −6 to stabilize the solution. 199

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Fig. 1. (a) A phase field approximation of the discrete discontinuities (b) Proposed solution scheme for coupled phase field algorithm The linear momentum balance equation in Table 1 is modified for phase field coupling where ( , c) is the stress tensor as a function of strain ( ( )) and concentration ( ). [(1− ) 2 + ]∇ ∙ ( , c) = 0 (9) In the solution algorithm, diffusion equation (8) and linear momentum balance equation (9) are simultaneously solved for the variables and while the phase field state remains fixed. Once the convergence criterion is met, a history variable is computed. This variable retains essential information concerning the phase transition's evolution and defects or cracks within the fiber. The phase field equation shown in (5) is subsequently solved for the variable , utilizing the frozen values of and , along with the assistance of the history variable. This integration of the history variable into the model enables it to consider the system's previous states and behavior, particularly under electrochemical loading conditions. After the convergence criterion is met, the simulation continues for the next time step, see Fig. 1 (b). In the absence of remote tractions and body forces, the resulting weak form can be obtained by considering the trial ሺ ǡ ǡ ሻ and test ( ǡ ̂ , ) functions ∫ [(1 − ) 2 + ]( ( − ℎ ) + 2 ( − ℎ )): =∫ ∙ (10) ∫(− ( − ) + 0 − ) ̂ − 0 ∫ [(1 − ) 2 + ] ∙ ̂ = ∫ 0 ( ∙ ) ̂ (11) ∫ { ∙ + ( − ) + ( +2 + ) − 2 + } = ∫ ∙ ∙ (12) By taking advantage of the symbolic differentiation capabilities of FENICS, the mechanical, electrochemical, and phase field weak forms, i.e., (10), (11), and (12), are directly transferred to the FENICS code.