PSI - Issue 80

M. Elkhodbia et al. / Procedia Structural Integrity 80 (2026) 187–194

189

Author name / Structural Integrity Procedia 00 (2023) 000–000

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be included, they are neglected here. This is justified by the fact that, in the DCB test, plastic deformation primarily occurs during wedge insertion—prior to exposure to the solution and thermal boundary conditions—rendering its influence on thermal evolution negligible for the modeled stage.

2.2. Constitutive Relations

2.2.1. Specific Free Energy Formulation The total Helmholtz free energy density ψ is defined as the sum of stored mechanical energy (elastic and plastic), fracture surface energy, and chemical energy: ψ ( ϵ e , ϵ p , T , C ,ϕ, ∇ ϕ ) = ψ 0 ( ϵ e , ϵ p ,ϕ ) + ψ frac ( ϕ, ∇ ϕ ) + ψ C ( T , C ) (4) here, ψ 0 is the elastic-plastic strain energy density, which combines the recoverable elastic part ψ e and the irrecoverable plastic work ψ p , both degraded by damage: ψ 0 = g e ( ϕ ) ψ e ( ϵ e ) + g p ( ϕ ) ψ p ( ε p ) (5) The elastic strain energy ψ e is given by: ψ e = ϵ e : C : ϵ e (6) where C is the fourth-order elasticity tensor, and ϵ e = ϵ − ϵ p − ϵ th is the elastic strain ( ϵ th = α ( T − T function g e ( ϕ ) ensures energy vanishes in fully damaged regions and is defined as: g e ( ϕ ) = (1 − ϕ ) 2 + k (7) where ϕ ∈ [0 , 1] is the phase-field variable and k ≪ 1 is a small regularization parameter. Next, the plastic work density ψ p is described by isotropic von Mises plasticity with power-law hardening: 0 ) I ). The degradation where σ y 0 is the initial yield stress, E is Young’s modulus, ε y 0 / E is the reference yield strain, n p is the hard ening exponent, and ε p is the equivalent plastic strain. The corresponding flow stress, representing the instantaneous resistance to plastic flow, is given by: σ f ( ε p ) = σ y 0  1 + ε p ε 0  n p (9) To control how plastic energy contributes to fracture, a plastic degradation function g p ( ϕ ) is introduced Cupertino Malheiros et al. (2024); Mandal et al. (2024): g p ( ϕ ) = β g e ( ϕ ) + (1 − β ) , 0 ≤ β ≤ 1 (10) This form allows a tunable fraction of the plastic work to drive damage. A value of β = 0 . 1 was taken following Taylor and Quinney (1934). Moreover, the fracture energy is modeled as: ψ frac = (11) where G c ( C ) is the hydrogen-dependent critical energy release rate, and ℓ is the regularization length scale. Finally, the chemical energy is given by: ψ C = µ 0 C + RTN [ θ L ln θ L + (1 − θ L ) ln(1 − θ L )] (12) where C is the hydrogen concentration, µ 0 is the reference chemical potential, R is the universal gas constant, T is the absolute temperature, N is the number of lattice sites, and θ L = C / N is the lattice occupancy. 2.2.2. Stress, Phase-Field, and Hydrogen-Transport Relations With the free-energy density specified in the previous subsection, the constitutive relation for the stress is: σ = g e ( ϕ ) C : ϵ e − 3 K α ( T − T 0 ) I (13) Correspondingly, the evolution of the phase-field variable ϕ , which captures the damage state, is governed by:  ϕ ℓ − ℓ ∇ 2 ϕ  − 2(1 − ϕ ) H = 0 (14) 0 = σ 1 2 G c ( C )  ϕ 2 ℓ + ℓ |∇ ϕ | 2  1 2 ψ p ( ε p ) = σ y 0 ε 0 n p + 1     1 + ε p ε 0  n p + 1 − 1    (8)