A phase field formulation for dissolution-driven stress corrosion cracking

• Despite decades of research, SCC mechanisms remain poorly understood, and their characteristics vary across materials and environments while mechanisms may act together.The introduction notes that such variability hinders critical evaluation of mechanistic interpretations.Source: §1 · p. 3
• The passage distinguishes anodic crack propagation driven by the corrosion reaction at the crack tip, including active path dissolution and FRDR models, from cathodic-driven growth associated with hydrogen ingress and transport.Source: §1 · p. 3
• Tracking evolving interfaces and complex pit-to-crack morphologies challenges coupled diffusion, electrochemistry, and mechanics models; phase fields replace interface conditions with an evolving field equation.This avoids explicit interface treatment; the introduction also notes limitations of existing moving-boundary methods in handling arbitrary 3D geometries, cumbersome implementations, or computational cost.Source: §1 · pp. 3–4
• The new phase field model simulates pitting corrosion and stress corrosion cracking using the FRDR mechanism, with mechanics driving interface kinetics and local film breakage.Source: §1 · p. 5
• The 2D and 3D problems, including cases with analytical or experimental data, show remarkable agreement with those data.Source: §1 · p. 5

2. Theory

• The theory derives governing equations from coupled field equations and free-energy imbalance, with thermodynamically consistent constitutive equations.Source: §2 · p. 6

2.1. Phase field FRDR model

• Localized damage is represented as cyclic film rupture, dissolution, and repassivation, with further mechanical work required to rupture each newly deposited film.The cycle is governed by competition between filming and mechanical-straining kinetics.Source: §2.1 · p. 6
• Mechanical straining and hydrostatic stress enter a local mechanochemical corrosion-current term, extending mechanics beyond film rupture to accelerate corrosion kinetics.The mechanochemical term is a local function of effective plastic strain and hydrostatic stress.Source: §2.1 · pp. 8–9
• Corrosion becomes diffusion-controlled when interfacial dissolved-ion concentration reaches saturation, after which diffusion away from the pit boundary controls interface velocity.Below saturation, the model distinguishes activation-controlled corrosion from the diffusion-controlled regime.Source: §2.1 · p. 9
• The FRDR model uses a phase-field variable to approximate interface evolution implicitly, avoiding boundary-condition complications for activation- and diffusion-controlled corrosion.The formulation replaces the need to apply Robin and Dirichlet boundary conditions for the two corrosion regimes.Source: §2.1 · pp. 9–10

2.2. Principle of virtual work and balance equations

• The mechanical and electrochemical systems are coupled because mechanics accelerates pit evolution while metal dissolution redistributes stress; virtual fields include displacement, phase field, and composition.The coupled system is formulated using the principle of virtual work.Source: §2.2 · pp. 10–11
• Virtual work links mechanical traction, phase-field microtraction and microstress, and mass flux to their respective variations; arbitrary variations yield local balance equations.The fields include displacement, phase field, and concentration.Source: §2.2 · p. 12

2.3. Energy imbalance

• The energy imbalance is formulated for separate mechanical and electrochemical free energies under an isothermal-process assumption, with the resulting Clausius-Duhem inequality constraining the constitutive laws.The pointwise inequality includes phase-field evolution, chemical-potential-driven transport, and dissolution.Source: §2.3 · p. 13

2.4. Constitutive theory

• The constitutive theory defines free-energy functionals for each system and derives thermodynamically consistent relations from them.Source: §2.4 · p. 13

2.4.1. Constitutive prescriptions for the mechanical problem

• The mechanical law uses von Mises J2 plasticity and degrades elastic and plastic free-energy components with the phase field; strain-gradient plasticity is left for future work.The phase field represents the transition from undissolved solid to electrolyte.Source: §2.4.1 · p. 14

2.4.2. Constitutive prescriptions for the electrochemical problem

• The electrochemical free energy combines composition and interface-gradient contributions, with the KKS model treating each material point as coexisting solid and liquid phases; a Fick-type law defines flux.The mixture assumption assigns different concentrations but similar chemical potentials to the two phases.Source: §2.4.2 · pp. 15, 16, 19
• The interface kinetics coefficient incorporates mechanical straining through a mechanochemical factor linked to corrosion current, while coupling remains one-way: phase-field damage reduces stiffness, but mechanics does not drive phase-field evolution.Crack advance is therefore attributed to localized metal loss; the framework could be extended to include mechanical free-energy contributions.Source: §2.4.2 · p. 18
• Metal-ion diffusion is governed by an extension of Fick’s second law and occurs only along the interface and in the electrolyte.Source: §2.4.2 · p. 19

3. Numerical implementation

• The finite element formulation discretizes the coupled problem and uses displacement, phase field, and normalized concentration as nodal variables.The weak forms are formulated for the displacement, phase-field, and concentration equations.Source: §3 · pp. 19–20
• The weak-form balances for displacement, phase field, and concentration are discretized in both time and space.Source: §3 · p. 20
• The tangent stiffness matrices use the elastic-plastic consistent material Jacobian.Source: §3 · pp. 20–21
• The system is solved simultaneously with an incremental-iterative Newton–Raphson scheme, achieving good convergence without staggered schemes.Residual and stiffness components use current values, and all equations are solved at once.Source: §3 · p. 21
• The numerical model is implemented in ABAQUS using a user element subroutine (UEL).Abaqus2Matlab is used to preprocess input files.Source: §3 · p. 21

4.1. Analytical verification under uniform corrosion: pencil electrode test

• At the metal-electrolyte interface, normalised concentration reaches csat/csolid = 0.036, and the model captures saturation, which prevents any further increase in pit-surface concentration.Source: §4.1 · p. 24
• For the pencil electrode test, the model’s pit-depth evolution as a function of time shows excellent agreement with the analytical solution (51)-(52).Figure 5 compares model predictions for pit depth versus time.Source: Figure 5 · p. 26, §4.1 · p. 26
• Small L0 produces activation-controlled corrosion with pit depth increasing linearly over time, whereas sufficiently large L0 yields diffusion-controlled, parabolic growth and a transition the model captures.Pit-depth sensitivity to increasing L0 is nonlinear and saturates; at sufficiently large L0, further increases do not affect corrosion kinetics.Source: §4.1 · p. 27

4.2. Chemo-mechanical validation with experiments: C-ring test on Q345R steel

• The C-ring test on Q345R steel is used to validate chemo-mechanical coupling by quantifying pit and crack growth under different applied stresses in hydrofluoric acid.The applied strain is held constant, and film passivation has negligible influence in this environment.Source: §4.2 · p. 29
• C-ring simulations show good agreement with experimental defect morphology and SCC-region length over time at σc = 0.55σy and σc = 0.8σy.The morphology comparison is shown at σc = 0.8σy after 210 min; plasticity can develop at SCC surface notches even while the circumferential stress remains below yield.Source: §4.2 · p. 32, Figure 8 · p. 32, Figure 9 · p. 33

4.3. Parametric study: cracking from a semi-elliptical pit

• At low remote displacement, hydrostatic stress has negligible influence and pits grow nearly equally in all dimensions; increasing displacement produces a pit-to-crack transition.The study varies remote displacement with passivation disabled.Source: §4.3 · p. 36, Figure 11 · p. 36
• At u∞ = 0.125 µm, stresses trigger plastic deformation, and defect shape is governed by hydrostatic stress and plastic-strain distribution.Mechanical straining also sharpens the SCC defect.Source: §4.3 · pp. 36–37
• Mechanical contributions significantly affect localized-corrosion rates and increase SCC-region length as applied displacement rises.The length evolution is shown for three remote displacement values.Source: §4.3 · p. 37, Figure 12 · p. 37
• The repassivation study applies a fixed remote displacement of u∞ = 0.1 µm and examines sensitivity to passive-film stability parameter k under the FRDR formulation.The supplied passages state the setup but not the study's results.Source: §4.3 · p. 38
• Higher film stability parameter k accelerates SCC growth at the pit base relative to its mouth, sharpening the pit and triggering pit-to-crack transition.As the defect sharpens and plasticity localises, tip corrosion rates rise while the protective film weakens in areas of high mechanical straining.Source: Figure 13 · p. 38, §4.3 · p. 38
• Repassivation slightly reduces SCC-region length early, but later growth accelerates with increasing k as film rupture and equivalent plastic strain increase.The length is shown for multiple k values, including k = 0 with no film.Source: §4.3 · p. 39, Figure 14 · p. 40
• At the SCC tip, L/L0 rises over time before plateauing, while damage becomes more localised as k increases and interface kinetics are higher where mechanical fields are larger.The initial L is smaller at higher k due to film passivation, and the three-dimensional study extends the framework to a hemispherical pit in a tension-loaded steel plate to address observed cross-section crack initiation (Turnbull et al., 2010).Source: §4.3 · pp. 40–41

4.4. SCC driven by the interaction between multiple pits

• After approximately 4 hours, the upper and bottom pits merge, predicting complete failure of the plate.The three pits deepen, the upper two branch, and the bottom pit coalesces with the central defect before the branches merge.Source: §4.4 · p. 43, Figure 17 · p. 44
• At multiple pits, higher initial interface kinetics L0 produces more uniform growth and reduces the mechanical contribution, which is greater at lower L0.Mechanical effects enhance interface kinetics and facilitate the transition from activation-controlled to diffusion-controlled corrosion; once diffusion-limited, rates become independent of L and k_m.Source: §4.4 · p. 45, Figure 18 · p. 46

4.5. Three-dimensional predictions of SCC defect growth

• The framework is extended to three-dimensional SCC predictions for a hemispherical pit in a tension-loaded steel plate.The case study aims to bring insight into defect-growth mechanics and address cracks initiating at the cross-section of uniaxial tension samples.Source: §4.5 · pp. 46–47
• The pit evolves from hemispherical to ellipsoidal, growing more along y and z than x as mechanical effects increase interface kinetics, consistent with deeper pits and lateral-crack initiation.Plastic straining is larger at the pit base and near parts of the pit mouth, where the passive film weakens and interface kinetics increase.Source: §4.5 · p. 48, Figure 21 · p. 49
• Areas with greater mechanical straining continue corroding faster until L reaches its saturation value everywhere, after which corrosion becomes uniform.Above its saturation value, L makes the mechanical contribution negligible.Source: §4.5 · pp. 49–50
• Pit growth is linear along x but transitions from linear to parabolic along y and z, as mechanical enhancement triggers diffusion-controlled corrosion.Figure 22 shows pit depth along the three axes over time; as L exceeds its saturation value, the mechanical contribution becomes negligible.Source: §4.5 · p. 49, Figure 22 · p. 50

• The formulation combines a KKS phase-field model for activation- and diffusion-controlled corrosion, mechanics-enhanced interface kinetics grounded in mechanochemical theory, and strain-rate-governed film rupture and repassivation via FRDR.Source: §5 · pp. 50–51
• Five boundary-value problems benchmark the model, spanning a pencil-electrode test, C-ring experiments, ellipsoidal-pit SCC, multiple-pit interaction, and 3D pit growth.Source: §5 · p. 51
• Increasing interface kinetics coefficient L raises corrosion rates from activation-controlled toward diffusion-controlled corrosion, until a diffusion-limited regime makes rates insensitive to further increases.Source: §5 · p. 51
• Mechanical stresses and strains enhance corrosion kinetics, promote localized damage and pit-to-crack transition, but their effect saturates at the diffusion-controlled limit and is strongest under activation control.Source: §5 · pp. 51–52
• Passive films exacerbate localization and mechanical effects, especially when film stability is high; cracks are more likely to nucleate under activation-controlled corrosion.Source: §5 · p. 52
• Future extensions could include cathodic-reaction-driven cracking and account for material microstructure and microchemistry, which may matter for accurate predictions of initial development stages.Source: §5 · p. 52