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

and facilitating brittle fracture Djukic et al. (2019); Negi et al. (2024a). The NACE TM0177 TM0177 (2016) Method D Double Cantilever Beam (DCB) test is used to measure the SSC-specific fracture toughness, denoted as K ISSC , under sour service conditions containing H 2 S. The test involves preloading a pre-cracked specimen and exposing it to a sour environment for a specified period. The crack growth is then measured, and K ISSC is calculated. This value provides a quantitative measure of a material’s resistance to SSC and is critical for material selection in sour service applications. Moreover, temperature plays a multifaceted role in SSC susceptibility, a ff ecting both hydrogen transport and embrittlement mechanisms. For ferritic and martensitic steels, increasing temperature (e.g., 25–96 ◦ C) generally reduces SSC severity by enhancing hydrogen di ff usivity and potentially forming protective corrosion layers that slow hydrogen ingress Omura et al. (2009); Asmara (2018). However, current design standards often overlook these temperature dependencies, relying on conservative assumptions or empirical thresholds MR0175 (2021). Computational techniques, particularly the phase-field method (PFM), provide a powerful framework for modeling HAC. PFM handles complex crack evolution without explicit tracking and naturally incorporates multiphysics phe nomena. Building on foundational work Mart´ınez-Pan˜eda et al. (2018), HAC-focused PFM has evolved to include trapping Isfandbod and Mart´ınez-Pan˜eda (2021), fatigue and corrosion e ff ects Golahmar et al. (2022); Cui et al. (2022), and microstructural heterogeneity Valverde-Gonza´lez et al. (2022); Elkhodbia et al. (2025). Recent e ff orts also leveraged PFM to calibrate SSC test conditions and interpret DCB results Cupertino-Malheiros et al. (2024); Negi et al. (2024b); Negi and Barsoum (2024); Elkhodbia and Barsoum. In particular, the thermo-chemo-mechanical model introduced in Elkhodbia and Barsoum successfully captured temperature-dependent SSC behavior by incorporating temperature e ff ects into hydrogen di ff usion and fracture en ergy degradation. However, that formulation assumed purely elastic material response. Since SSC mechanisms often involve localized plasticity—even in cases of macroscopically brittle fracture—excluding plastic deformation lim its predictive fidelity. This study extends the model in Elkhodbia and Barsoum by incorporating plasticity into the coupled framework. The updated model targets the DCB test configuration, aiming to assess how temperature and plasticity jointly influence crack initiation and growth in sour environments. This section summarizes the governing equations and constitutive relations of a coupled phase-field model developed to capture SSC under di ff erent applied temperatures and chemically aggressive environments. The model accounts for the interplay between mechanical deformation, hydrogen di ff usion, temperature fields, and progressive material degradation using a thermodynamically consistent phase-field approach. Building upon the previous chemo-thermo mechanical framework Elkhodbia and Barsoum, the current formulation introduces the e ff ects of plastic deformation to enhance the model’s fidelity for fracture processes where elastic assumptions alone may be insu ffi cient. 2.1. Governing Equations The coupled system is governed by field equations derived using the principle of virtual power, ensuring thermody namic consistency. Neglecting body forces, the local forms of the governing equations for mechanical equilibrium, phase-field damage evolution, and mass conservation of hydrogen are: ∇· σ = 0 , ∇· ξ − ω = 0 , ˙ C + ∇· J = 0 in Ω (1) with boundary conditions: σ · n = t , ξ · n = 0 , − J · n = ρ on ∂ Ω (2) Here, σ denotes the Cauchy stress tensor, ξ is the microstress vector associated with the gradient of the phase field ϕ , and ω is a scalar microforce driving the damage evolution. C is the hydrogen concentration field, and J is the hydrogen flux vector. n is the outward unit normal vector on the domain boundary, t is the applied traction, and ρ is the prescribed hydrogen flux on the boundary. Extensive derivation and detailed thermodynamic formulation of these governing equations can be found in Elkhodbia and Barsoum. Moving forward, the heat equation used in this study simplifies to: c ˙ T = ∇· ( k ∇ T ) (3) Here, T is the temperature field, c is the specific heat capacity, and k is the (possibly damage-degraded) thermal con ductivity. While additional terms representing dissipation due to inelastic strain (e.g., plastic work) should theoretically 2. Modeling framework