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

198

4

∫ Γ Γ =∫ Ω (21 2 + 2 |∇ | 2 ) Ω (2) where is the critical energy release rate. The approximated crack surface energy can be added to the bulk energy to form the total potential energy of the solid Ψ as Ψ = ∫ ((1 − ) 2 ( , ) Ω + (21 2 + 2 |∇ | 2 )) Ω (3) where the term (1− ) 2 describes the degradation of the stored energy with evolving damage. A regularization based on a viscous crack resistance that enhances the algorithm's robustness may easily be added. This work aims to solve the deviation and instability problems during crack propagation, especially in the chemo-mechanical case. To stabilize post-critical solution paths, we may add a viscous regularization. This stabilization process introduces a viscous term ( ̇ ) (the time derivative of the phase field variable) and viscosity parameter =1×10 −6 kNs/mm 2 that supplements the crack surface resistance. This feature of the proposed theory is a very important element of a robust numerical implementation, see Miehe et al. (2010b). The phase field equation with a viscous term reads (1 − Δ ) + ̇ − 2(1 − ) ( , ) = 0 (4) For the finite element implementation, the history variable, which represents the localized maximum energy corresponding to positive normal strains, is used in the formulation for the phase field variable update with satisfying the boundary condition ∇ ∙ = 0 on Ω . In the finite element method, the update of the phase field variable is based on a history variable that represents the localized maximum energy caused by positive normal strains. (1 − Δ ) + ̇ − 2(1 − )H + ( (t)) = 0 (5) H + is called the history variable, which for the current time t can be written as, H + =max tϵ[0,τ] + ( (t), c(t))) (6) By implementing a strain energy decomposition to prevent cracking in compression, as proposed by Amor et al. (2009) in their volumetric-deviatoric split approach, it is assumed that the strain energies associated with volume change and shear deformation can be susceptible to damage, while the volumetric strain energy under compression Here, ⟨ ⟩ + = 1 2 ( + | |) and is the bulk modulus. By ignoring the convection effect (i.e., the macroscopic motion of the ions is not considered (Moyne and Murad (2002)) and modified for multifield problems, the diffusion equation in Table 1 can be rewritten as, − ( ̇ − ̇ ℎ ) + 0 ̇ − 0 [(1− ) 2 + ]∇ 2 =0 (8) remains unaffected. Thus, the strain energy can be decomposed as + ( (t), ( )) = 12 ⟨tr( − ℎ )⟩ +2 + (( − ℎ ) : ( − ℎ ) ) (7)