PSI - Issue 80

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

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

where a history variable H is introduced to prevent non-physical crack healing: H = max τ ∈ [0 , t ] Ψ + e + βψ p G c ( C )

(15)

Here, Ψ + e is the tensile part of the elastic energy, derived using volumetric-deviatoric decomposition per Amor et al. Amor et al. (2009). It includes the tensile volumetric and deviatoric contributions: Ψ + 0 = + + µ ϵ dev : ϵ dev (16) ± = ( •±| • | ) / 2 extracts the positive and negative parts, respectively. Finally, the stress-assisted Fickian hydrogen flux relation for low lattice occupancy ( θ L ≪ 1) and a uniform trap density reads: J = − D ∇ C + D ¯ V H C RT ∇ σ H + DC ln C T ∇ T (17) where D = D 0 1 + k d ⟨ ϕ − ϕ th ⟩ is the damage-enhanced di ff usivity, following Cupertino-Malheiros et al. (2024), ¯ V H the partial molar volume of hydrogen, and σ H = 1 3 tr σ . Hydrogen reduces the fracture energy of steels by accumulating at microstructural trap sites, promoting embrittlement. In this study, we operate within a temperature range of 25°C to 96°C, which corresponds to the upper-shelf regime for low-alloy steels (within the typical ductile to brittle transition behavior of metals). Within this range, the direct influence of temperature on the intrinsic fracture toughness is considered negligible, allowing us to isolate the degra dation e ff ect due to hydrogen concentration. Hence, the fracture energy degradation is assumed to depend primarily on hydrogen concentration C , following a phenomenological exponential form: G c ( C ) = d C ( C ) · G c (0) (18) with the degradation function defined as: d C ( C ) = G min c G c (0) + 1 − G min c G c (0) exp( − mC ) (19) Here, the parameters m = 0 . 50, G min c = 1N / mm, and G c (0) = 78 . 7N / mm correspond to P110 steel properties from Zhang et al. Zhang et al. (2020), consistent with K ISSC measurements and hydrogen permeation data. G c values were derived from fracture toughness ( K Ic ) through the plane strain relationship K Ic = EG c / (1 − ν 2 ). This section validates the extended chemo-thermo-mechanical phase-field model by simulating DCB tests for P110 steel under sour service conditions. The model’s predictions for the threshold stress intensity factor for SSC, K ISSC , are compared against experimental data reported by Vera et al. Vera and Case (1997). We evaluate both the original elastic formulation Elkhodbia and Barsoum and the newly implemented elasto-plastic framework, considering the coupled e ff ects of mechanical loading, hydrogen di ff usion, temperature, and material deformation. Fig. 1 illustrates the specimen and wedge geometry, with dimensions specified in Table 1. The test involves wedge insertion to achieve a constant arm displacement ( δ ), followed by immersion in an H 2 S-saturated solution for 14 days (336 hours) to allow crack growth. Finally, the wedge lift-o ff load ( P ) is determined, from which K ISSC is calculated using the standard formula for grooved specimens: 1 2 K ⟨ tr( ϵ e ) ⟩ 2 where µ is the shear modulus and ϵ dev = ϵ e − 1 3 tr( ϵ e ) I is the deviatoric strain. The operator ⟨•⟩ 2.3. Fracture Toughness Degradation in Hydrogen-Assisted Cracking 3. Double Cantilever Beam (DCB) test

Pa f 2 √ 3 + 2 . 38 h

a f

B B n

1 √

3

K ISSC = (20) where a f is the final crack length, h is the half-specimen height, B is the thickness, and B n is the web thickness. Bh 3 2