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A phase field formulation for dissolution-driven stress corrosion cracking
Chuanjie Cui, Rujin Ma, Emilio Martínez-Pañeda
TL;DR
The paper addresses the need for predictive models of mechanically assisted, dissolution-driven corrosion and SCC that can represent evolving interfaces and complex defect morphologies. It develops a phase field formulation combining film rupture–dissolution–repassivation with mechanical effects, and reports remarkable agreement with analytical and experimental data. The concluding results identify enhanced corrosion kinetics from mechanical stresses and strains, while the paper notes limited accuracy during initial development stages.
Problem
Predictive SCC modelling is hindered by incomplete mechanistic understanding and the difficulty of representing coupled diffusion, electrochemistry, mechanics, and evolving interfaces.
Method
The paper develops a phase field formulation for dissolution-driven pitting and SCC that incorporates the film rupture-dissolution-repassivation mechanism and mechanical effects on interface kinetics.
Results
The model shows remarkable agreement with analytical solutions and experimental measurements across relevant 2D and 3D problems.
Takeaways & Limitations
Mechanical stresses and strains enhance corrosion kinetics, and their influence contributes to changes in SCC defect shape, including the pit-to-crack transition.
Takeaways & Limitations
The formulation provides accurate prediction of the initial development stages only within the stated scope.
Abstract
from arXiv · showhide
We present a new theoretical and numerical framework for modelling mechanically-assisted corrosion in elastic-plastic solids. Both pitting and stress corrosion cracking (SCC) can be captured, as well as the pit-to-crack transition. Localised corrosion is assumed to be dissolution-driven and a formulation grounded upon the film rupture-dissolution-repassivation mechanism is presented to incorporate the influence of film passivation. The model incorporates, for the first time, the role of mechanical straining as the electrochemical driving force, accelerating corrosion kinetics. The computational complexities associated with tracking the evolving metal-electrolyte interface are resolved by making use of a phase field paradigm, enabling an accurate approximation of complex SCC morphologies. The coupled electro-chemo-mechanical formulation is numerically implemented using the finite element method and an implicit time integration scheme; displacements, phase field order parameter and concentration are the primary variables. Five case studies of particular interest are addressed to showcase the predictive capabilities of the model, revealing an excellent agreement with analytical solutions and experimental measurements. By modelling these paradigmatic 2D and 3D boundary value problems we show that our formulation can capture: (i) the transition from activation-controlled corrosion to diffusion-controlled corrosion, (ii) the sensitivity of interface kinetics to mechanical stresses and strains, (iii) the role of film passivation in reducing corrosion rates, and (iv) the dependence of the stability of the passive film to local strain rates. The influence of these factors in driving the shape change of SCC defects, including the pit-to-crack transition, is a natural outcome of the model, laying the foundations for a mechanistic assessment of engineering materials and structures.
1. Introduction
The paper motivates a phase field framework for dissolution-driven pitting and stress corrosion cracking, addressing unresolved mechanisms and the computational difficulty of evolving metal–electrolyte interfaces. It combines film rupture–dissolution–repassivation with mechanical effects and demonstrates agreement with analytical and experimental results.
- Stress corrosion cracking mechanisms remain not well understood, while material and environmental variability hinders critical evaluation of competing interpretations.
- Predictive SCC modelling must couple diffusion, electrochemistry, and mechanics while resolving evolving interfaces through pit growth, pit-to-crack transition, and crack propagation.
- Existing moving-boundary methods can be limited by arbitrary 3D geometries, cumbersome implementations, and computational cost.
- The phase field method replaces explicit interface boundary conditions with differential equations for an auxiliary field, allowing the whole system to be solved without explicitly treating interface conditions.
- The proposed formulation models pitting corrosion and SCC using the film rupture-dissolution-repassivation mechanism, with mechanics driving interface kinetics and local film breakage.
- The model is evaluated on relevant 2D and 3D problems, including cases with analytical or experimental data, and shows remarkable agreement.
- The framework is intended to address complex engineering problems involving localized corrosion and stress corrosion cracking.
2. Theory
The theory couples mechanical deformation, electrochemical dissolution, and phase-field interface evolution to model film rupture, corrosion kinetics, and localized damage. It incorporates strain-accelerated corrosion, passivation, and transitions between activation- and diffusion-controlled regimes.
- Coupled formulation: The formulation couples mechanical and electrochemical systems through deformation, diffusion, material dissolution, and phase-field variables.The coupled behavior includes pit evolution accelerated by mechanics and stress redistribution caused by metal dissolution.
- Film rupture-dissolution-repassivation: Film rupture, dissolution, and repassivation proceed cyclically, with rupture governed by accumulated effective plastic strain reaching a critical value.The critical strain for film rupture is reported as approximately 0.1%.
- Film rupture-dissolution-repassivation: Passive-film formation reduces corrosion current density, while rupture restores the bare-metal current before dissolution and repassivation begin.One cycle contains a current-decay period and a post-rupture interval, with ti = t0 + tf.
- Mechanochemical corrosion: Mechanical straining and residual stresses are incorporated as electrochemical drivers through a local mechanochemical corrosion term dependent on plastic strain and hydrostatic stress.The mechanochemical coefficient km is treated as a local function of εp and σh.
- Corrosion regimes: Corrosion is activation-controlled at lower interfacial concentrations and becomes diffusion-controlled when the surface concentration reaches saturation.The two regimes use different moving-boundary descriptions, with diffusion away from the pit controlling interface velocity in the saturated regime.
3. Numerical implementation
The coupled electro-chemo-mechanical equations are discretized with finite elements and implicit time integration. A monolithic Newton–Raphson solution advances displacement, phase field, and normalized concentration together.
- Finite element discretization: The finite element implementation discretizes the coupled displacement, phase-field, and concentration equations in space and time.Nodal interpolation uses shape functions for displacement, phase field, and normalized concentration variables.
- Time integration: The discrete system advances from time step n to n + 1 using an implicit increment dt.The resulting equations are written as residuals for the coupled balance problems.
- Linearization: Consistent tangent stiffness matrices are constructed using the elastic-plastic consistent material Jacobian Cep.The tangent operators support linearization of the coupled finite element system.
- Nonlinear solution: An incremental-iterative Newton–Raphson scheme solves all coupled equations simultaneously without staggered schemes.The reported convergence is good when residual and stiffness components are assembled from current values.
- Software implementation: The numerical model is implemented in ABAQUS through a user element subroutine, with Abaqus2Matlab used for preprocessing.The code, examples, and documentation are reported as downloadable from the authors’ website.
4. Results
Five case studies validate the phase-field framework against analytical and experimental benchmarks and examine how corrosion kinetics, mechanics, passivation, and pit interactions shape SCC evolution.
- Five case studies compare model predictions with analytical solutions and experimental measurements, including pencil-electrode, C-ring, SCC, multiple-pit, and three-dimensional problems.The studies assess pure corrosion, anodic dissolution-driven growth, mechanical effects, pit interaction, and large-scale 3D pit corrosion.
- Pencil electrode test: The pencil-electrode predictions show excellent agreement with the analytical pit-depth evolution under diffusion-controlled corrosion.The benchmark assumes high reaction rates and saturation surface concentration, producing diffusion-controlled behavior.
- Pencil electrode test: Varying interface kinetics captures the transition from linear activation-controlled growth to parabolic diffusion-controlled growth.At sufficiently large L, corrosion becomes diffusion-limited and independent of the interface reaction parameter.
- Chemo-mechanical SCC: Mechanical stresses and strains accelerate localized corrosion, sharpen SCC defects, and drive the pit-to-crack transition through stress concentration and plastic-strain localization.Higher interface kinetics at pit tips coincide with larger plastic strain and hydrostatic stress, while weaker protective films promote faster dissolution.
- Film passivation: Film passivation initially restrains SCC growth, but later mechanical straining and film rupture increase interface kinetics and localize damage.Increasing film-stability parameter k reduces early SCC growth and produces more localized damage; after rupture, sharper defects and higher plastic strain accelerate growth.
- Multiple-pit interaction: Multiple pits deepen, branch, coalesce, and are predicted to cause complete plate failure after approximately 4 hours.The upper pits branch, the bottom pit coalesces with the central defect, and the upper and bottom pits eventually merge.
5. Concluding remarks
The paper presents a phase field framework for dissolution-driven corrosion that couples mechanics, interface kinetics, and film rupture–repassivation to model pitting, SCC, and pit-to-crack transitions. Benchmark problems show how kinetics, mechanical loading, and passive-film stability shape corrosion localisation and defect evolution.
- Formulation: The formulation combines a KKS-based phase field, mechanics-enhanced interface kinetics, and strain-rate-governed film rupture and repassivation.Displacements, phase field, and dissolved-ion concentration are used as computational degrees of freedom.
- Validation and applications: Five boundary value problems benchmark the model against analytical solutions, experimental data, and paradigmatic two- and three-dimensional corrosion scenarios.The cases include pencil-electrode corrosion, C-ring experiments, ellipsoidal-pit SCC, interacting pits, and three-dimensional pit growth.
- Main findings: Increasing the interface kinetics coefficient L raises corrosion kinetics from activation-controlled toward diffusion-controlled corrosion, after which rates become diffusion-limited and insensitive to further increases.Mechanical enhancement of L likewise reaches a saturation stage once the diffusion-controlled limit is reached.
- Main findings: Mechanical stresses and strains enhance corrosion kinetics, favour localised damage, and trigger the pit-to-crack transition, with the strongest influence under activation-controlled conditions.The mechanical contribution becomes less influential after diffusion-controlled corrosion is reached.
- Main findings: Passive films exacerbate localisation and amplify the role of mechanics, especially when high film stability restricts rupture to strongly strained regions.Cracks are more likely to nucleate under activation-controlled corrosion conditions.
- Scope and future work: Future extensions should include cathodically driven cracking and material microstructure and microchemistry to improve predictions of early development stages.The authors identify these factors as important in many material systems.