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A generalised, multi-phase-field theory for dissolution-driven stress corrosion cracking and hydrogen embrittlement

C. Cui, R. Ma, E. Martínez-Pañeda

arXiv:2205.12096v1cond-mat.mtrl-scics.CEphysics.app-ph

TL;DR

Predicting failure in aggressive environments is challenging because stress corrosion cracking couples chemical and mechanical phenomena. This paper develops a multi-phase-field electro-chemo-mechanical theory combining dissolution and hydrogen embrittlement, and reports experimental agreement alongside potential-dependent changes in damage mechanism.

  • Problem

    Predicting failure in aggressive environments is challenging because SCC involves interacting chemical and mechanical phenomena.

    Source: §1 · p. 3

  • Method

    The paper formulates a multi-phase-field electro-chemo-mechanical theory for dissolution-driven SCC and hydrogen embrittlement.

    Source: §1 · p. 4, §2 · p. 5

  • Results

    The model agrees very well with experiments and captures potential-dependent damage-mechanism changes, including when hydrogen embrittlement ceases to dominate SCC in Al alloys.

    Source: §5 · p. 50

  • Takeaways & Limitations

    The authors conclude that the framework incorporates both hydrogen embrittlement and material dissolution-driven damage mechanisms in SCC.

    Source: §5 · p. 49

    Fully predictive estimations using first-principles hydrogen degradation laws are hindered by associated uncertainties.

    Source: §5 · p. 50

Abstract

from arXiv · show

We present a phase field-based electro-chemo-mechanical formulation for modelling mechanics-enhanced corrosion and hydrogen-assisted cracking in elastic-plastic solids. A multi-phase-field approach is used to present, for the first time, a general framework for stress corrosion cracking, incorporating both anodic dissolution and hydrogen embrittlement mechanisms. We numerically implement our theory using the finite element method and defining as primary kinematic variables the displacement components, the phase field corrosion order parameter, the metal ion concentration, the phase field fracture order parameter and the hydrogen concentration. Representative case studies are addressed to showcase the predictive capabilities of the model in various materials and environments, attaining a promising agreement with benchmark tests and experimental observations. We show that the generalised formulation presented can capture, as a function of the environment, the interplay between anodic dissolution- and hydrogen-driven failure mechanisms; including the transition from one to the other, their synergistic action and their individual occurrence. Such a generalised framework can bring new insight into environment-material interactions and the understanding of stress corrosion cracking, as demonstrated here by providing the first simulation results for Gruhl's seminal experiments.

1. Introduction

The introduction motivates a general framework for stress corrosion cracking (SCC) because dissolution- and hydrogen-driven mechanisms can interact. It presents a multi-phase-field formulation and case studies spanning different environments and failure mechanisms.

1. Introduction

  • Existing anodic- and hydrogen-driven SCC mechanisms have been investigated extensively but independently, despite often acting in concert.The authors identify the lack of a generalised model encompassing both mechanisms and their interactions.Source: §1 · pp. 3–4
  • SCC is difficult to predict because chemical and mechanical phenomena interact, while the defect shape influences local chemistry, mechanics, hydrogen uptake, and corrosion-front evolution.The introduction notes that dissolution pits can facilitate hydrogen uptake and act as stress concentrators.Source: §1 · pp. 3–4
  • The paper presents a generalised multi-phase-field formulation for dissolution-driven SCC and hydrogen embrittlement, including their synergistic interaction.It models evolving electrolyte-metal interfaces, metal-ion diffusion, passivation and film rupture, hydrogen transport, and mechanical deformation.Source: §1 · p. 4
  • The formulation accounts for mechanical-field effects on corrosion kinetics and bulk hydrogen diffusion, and links hydrogen content to material toughness.It uses an extended Fick's law for hydrogen transport and J2 flow theory for mechanical deformation.Source: §1 · pp. 4–5
  • The model shows very good agreement with experiments and captures changes between hydrogen-driven and dissolution-assisted SCC across environments.Case studies include stainless steel bending, seawater thresholds, and Gruhl's AlZn5Mg3 alloy experiments.Source: §1 · p. 5

2. Theory

The theory formulates a multi-phase-field description of electrolyte-solid interface evolution through corrosion, embrittlement, and fracture. It addresses electro-chemo-mechanical response under isothermal and isotropic assumptions.

2. Theory

  • The multi-phase-field theory is aimed at predicting aqueous electrolyte-solid metal interface evolution due to corrosion, embrittlement, and fracture.The formulation concerns the electro-chemo-mechanical response of a domain.Source: §2 · p. 5
  • The theory assumes isothermal conditions and isotropic material behaviour.Its domain is specified in one to three dimensions.Source: §2 · p. 5
  • The formulation derives force balances, then establishes thermodynamically consistent constitutive choices and a generalised free-energy definition.The authors state that constitutive choices and governing-equation interactions are detailed in subsequent theory subsections.Source: §2 · p. 6

2.1. Kinematics and primal fields

The model uses displacement, species concentrations, and separate dissolution and fracture phase fields to represent the coupled system. Its kinematic description also specifies species transport and boundary conditions.

2.1. Kinematics and primal fields

  • The primal fields include displacement, metal-ion and hydrogen concentrations, and the dissolution and fracture phase fields.The species concentrations are defined in the electrolyte, while the phase fields characterize evolving interface morphology.Source: §2.1 · p. 6
  • The model uses separate phase fields for dissolution and fracture, distinguishing atom detachment from creation of new surfaces through bond breaking.Both fields are assigned 1 in solid, undamaged material and 0 in the electrolyte.Source: §2.1 · pp. 6–7
  • Species boundary conditions prescribe either fluxes or concentrations, and newly created corrosion and crack surfaces are treated as promptly exposed to the aqueous environment.Metal-ion concentration is normalized from 0 to 1 relative to the solid metal-ion concentration.Source: §2.1 · p. 7
  • Metal-ion and hydrogen transport are driven by chemical potential, with a scalar field η describing composition-change kinematics.The passage gives the evolution relation for η for both species.Source: §2.1 · p. 7

2.2. Principle of virtual work and force balances

The principle of virtual work equates internal and external virtual work for the model's kinematic fields. Applying it yields local equilibrium equations and corresponding boundary conditions.

2.2. Principle of virtual work and force balances

  • External virtual work is decomposed into non-inertial and inertial contributions, with the latter corresponding to solute number densities.The formulation separates the non-inertial and inertial parts of the metal-ion and hydrogen interactions.Source: §2.2 · p. 8
  • Equating internal and external virtual work for displacement, phase fields, and species fields provides the starting point for the force balances.The virtual-work statement is applied over an arbitrary part of the domain.Source: §2.2 · pp. 8–9
  • The resulting local equations cover mechanical equilibrium, dissolution and fracture phase-field balances, and metal-ion and hydrogen transport.The right-hand side of the virtual-work statement supplies the corresponding boundary conditions.Source: §2.2 · p. 9

2.3. Energy imbalance

The energy balance is formulated for a continuum body in an isothermal process. Replacing virtual fields with realizable rates and applying the divergence theorem yields a pointwise form.

2.3. Energy imbalance

  • The first and second laws are stated for a continuum body undergoing a dynamical process.The formulation identifies specific internal energy and specific entropy.Source: §2.3 · p. 10
  • The discussion is restricted to isothermal conditions, with T = T0.Source: §2.3 · p. 10
  • Replacing virtual fields with realizable velocity and chemical-potential fields produces an energy-balance expression in mechanical, phase-field, and transport terms.Source: §2.3 · p. 10
  • Applying the divergence theorem and local balance equations converts the energy imbalance to an equivalent pointwise form.Source: §2.3 · p. 10

2.4. Free energy density

The free-energy density combines mechanical, chemical, dissolution-interface, fracture-interface, and hydrogen-chemical contributions. Separate degradation and interface formulations couple these contributions to dissolution, fracture, and hydrogen content.

2.4. Free energy density

  • The total free energy is decomposed into mechanical, chemical, dissolution-interface, fracture-interface, and hydrogen-chemical terms.Source: §2.4 · p. 10

2.4.1. Mechanical free energy density

  • Mechanical free energy is degraded by separate dissolution and fracture functions, capturing their joint effect on the solid’s mechanical energy.The mechanical energy includes elastic and plastic parts, with hydrogen-related terms involving the partial molar volume and reference hydrogen concentration.Source: §2.4.1 · pp. 10–11

2.4.2. Electrochemical energy density

  • The electrochemical energy uses a KKS mixture of solid and liquid phases with distinct concentrations but similar chemical potentials.The phase-weighted energies include a double-well potential.Source: §2.4.2 · p. 11

2.4.3. Dissolution interface energy density

  • The dissolution-interface energy depends on the corrosion phase field’s gradient, with its coefficients related to interface energy and thickness.The interface region is defined by 0.05 < φ < 0.95.Source: §2.4.3 · p. 12

2.4.4. Fracture interface energy density

  • Hydrogen degrades fracture resistance by making the critical energy release rate concentration-dependent, with a linear coverage-based degradation option.For Fe-based materials, χ = 0.89 fits Jiang and Carter (2004), while χ = 0.41 fits the DFT simulations on Ni by Alvaro et al. (2015).Source: §2.4.4 · pp. 12–13

2.4.5. Hydrogen-chemical energy density

  • The hydrogen-chemical energy uses an occupancy-based mixing term and can represent lattice or diffusible hydrogen through the choice of diffusion coefficient.The formulation can account for microstructural trapping in fracture and diffusion, but does not capture evolving trap density.Source: §2.4.5 · p. 13

2.5. Constitutive theory

The constitutive theory derives mechanical, phase-field, and transport relations from the free-energy imbalance. Its choices specify how elastoplasticity, dissolution, ionic transport, fracture, and hydrogen diffusion enter the coupled model.

2.5. Constitutive theory

  • Dissolution and fracture microstresses are split into energetic and dissipative parts, with non-negative response functions enforcing the dissipation condition.The formulation assumes zero energetic response functions; setting the rate-dependent functions to zero corresponds to rate-independent processes.Source: §2.5 · p. 14

2.5.1. Chemo-elastoplasticity

  • The solid uses J2 plasticity with isotropic power-law hardening, while the stress response includes degradation by dissolution and fracture.Lattice dilation is omitted as having a negligible effect in hydrogen embrittlement; the framework can incorporate crystal anisotropy and strain-gradient hardening.Source: §2.5.1 · p. 15

2.5.2. A phase field description of metallic corrosion

  • Corrosion kinetics depend on mechanical fields through mobility, incorporating strain-enhanced corrosion and passive-film rupture and repassivation.Film rupture follows a competition between repassivation and crack-tip straining, occurring when accumulated equivalent plastic strain in a cycle reaches a critical failure strain.Source: §2.5.2 · pp. 16–17

2.5.3. Transport of ionic species

  • Metal-ion transport follows a Fick-type flux relation and supplies the governing equation for diffusion-controlled corrosion.Source: §2.5.3 · p. 17

2.5.4. A phase field description of fracture

  • Fracture is treated as rate-independent, and its phase-field balance incorporates a hydrogen-dependent toughness.The fracture formulation assumes a small concentration gradient at the interface.Source: §2.5.4 · p. 18

2.5.5. Diffusion of atomic hydrogen

  • Hydrogen transport is driven by chemical-potential gradients, with flux given by a linear Onsager relation under low occupancy.A transport-enhancing term from damaged to pristine regions is neglected when a penalty-based moving chemical boundary condition represents solution filling advancing cracks.Source: §2.5.5 · p. 18

2.6. Summary of the governing equations

The governing equations couple five physical processes through degradation, stress-sensitive transport, and hydrogen-dependent fracture resistance. Two length scales distinguish the dissolution and fracture interfaces and relate to their physical or regularizing roles.

2.6.1. Strong form and multi-physics interactions

  • The model resolves mechanical deformation, metal dissolution, ionic transport, fracture, and hydrogen diffusion through coupled governing equations.Dissolution degrades stiffness and affects corrosion through stress-sensitive kinetics and passive-film rupture; dissolution also prevents fracture nucleation in the electrolyte.Source: §2.6.1 · pp. 19–20
  • Mechanical fields drive hydrogen transport toward stressed regions, while hydrogen lowers fracture toughness and cracked regions are assigned environmental hydrogen content.The dissolution length scale ℓd reflects interface thickness and can be related to interface energy; fracture length scale ℓf regularizes the model and defines material strength.Source: §2.6.1 · p. 20, §2.6.2 · p. 20

2.6.2. A brief discussion on ℓd and ℓf

  • The dissolution scale ℓd is determined by the double-well height and gradient coefficient, whereas fracture scale ℓf is a regularizing parameter that defines material strength.The dissolution interface energy and thickness can, in principle, be estimated from experiments.Source: §2.6.2 · pp. 20–21

3. Numerical implementation

Section 3 describes the numerical implementation through weak-form derivation, finite-element discretisation, and residual and stiffness construction.

3.1. Weak form

  • The conditioning parameter κ is 1 × 10−5, and fracture-driving weighting factors βe and βp are both set to 1.κ is chosen to keep the equation system well-conditioned.Source: §3.1 · p. 22
  • The weak form uses a spherical/deviatoric split to prevent damage under compression and a history-based fracture drive enforcing irreversibility only on the elastic contribution.The text notes that effective plastic work increases monotonically, so irreversibility is applied only to the elastic part.Source: §3.1 · p. 22

3.2. Finite element discretisation

  • The finite-element discretisation solves coupled equations (61)-(65) by interpolating displacement, corrosion and fracture order parameters, metal-ion concentration, and hydrogen concentration with nodal shape functions.The nodal fields use shape functions Ni.Source: §3.2 · p. 22

3.3. Residuals and stiffness matrices

  • Residuals and tangent stiffness matrices feed an incremental time-parametrised Newton–Raphson solution implemented in ABAQUS through a UEL; Abaqus2Matlab preprocesses input files.The hydrostatic stress gradient is computed from shape-function derivatives and integration-point hydrostatic stress values.Source: §3.3 · pp. 23–24
  • Moving chemical boundary conditions mimic prompt exposure of newly created defect surfaces; a penalty enforces cH=cenv as φdφf approaches zero, with a transition from 0.5 and kp=10^5 reported stable.The penalty factor is chosen to enforce cH=cenv near the corrosion-fracture interface while maintaining numerical stability.Source: §3.3 · p. 23

4. Results

The results assess the model across dissolution-driven corrosion, hydrogen embrittlement, and their combined action, using benchmark cases and studies of environmental effects. They report cracking and threshold behavior across materials and environments, including the transition between failure mechanisms in stainless-steel bending and Gruhl’s tube tests.

4.1. Growth of a semi-circular pit

  • Pit depth follows a parabolic time trend typical of diffusion-controlled corrosion, and predictions show excellent agreement with Mai et al. (2016).The pit grows uniformly with time, consistent with experiments and previous theoretical studies.Source: §4.1 · p. 26, Figure 3 · p. 27
  • With a passive film, pit growth initially proceeds at approximately the same rate across film-stability cases, then shifts from diffusion- to activation-controlled corrosion until passivation prevents further corrosion.The film progressively reduces corrosion kinetics; no mechanical loading is applied in this study.Source: §4.1 · p. 27, Figure 4 · p. 28

4.2. Hydrogen assisted cracking of a notched square plate

  • At cenv = 0, plate force–displacement predictions agree very well with Miehe et al. (2010); increasing hydrogen lowers load capacity and critical load and initiates cracking earlier.The reported cause is hydrogen-induced degradation of material toughness.Source: §4.2 · pp. 29–30, Figure 6 · p. 30
  • Hydrogen concentration is elevated immediately ahead of the crack tip, where hydrostatic stresses are large, and the results capture interplay among hydrogen diffusion, fracture, and deformation.At the crack tip, cH = cenv under the moving chemical boundary condition.Source: §4.2 · p. 30, Figure 7 · p. 31

4.3. Bending of stainless steel samples: transition from anodic dissolution to hydrogen embrittlement

  • The stainless-steel bending study investigates the transition from anodic dissolution to hydrogen-assisted cracking.The supplied passage introduces this investigation but does not provide its findings.Source: §4.3 · p. 31
  • For t ⩽ 5 h, SCC predictions are relatively similar, consistent with anodic-dissolution-dominated growth; after hydrogen ingress and diffusion, differences become significant, with crack length twice as large when hydrogen interplay is unmodeled.Source: §4.3 · p. 33
  • As time passes, pit sharpening, passive-film rupture, and increasing load raise local stress, hydrogen content, and energy release rate, shifting SCC growth toward hydrogen embrittlement.The hydrogen plots show a drop in hydrogen content where passivation impedes ingress, although ingress continues during the interval before the film stabilises.Source: §4.3 · p. 34
  • Lower film failure strain accelerates SCC growth, while larger failure strain delays hydrogen-embrittlement dominance; hydrogen-driven growth can ultimately become unstable.Larger failure strain also reduces hydrogen exposure time and the size of the hydrogen-degraded region, giving it a dual influence on crack length at transition.Source: §4.3 · pp. 35–36, Figure 11 · p. 36

4.4. SCC threshold predictions in artificial and biologically-active seawater

  • The Steel 690 threshold study simulates Robinson and Kilgallon’s experiments in sterile artificial and biologically active seawater across cathodic protection potentials.Bacteria in biologically active seawater increase sulfide concentration and thereby promote hydrogen uptake; the study measures stress-intensity thresholds, Kth, in each condition.Source: §4.4 · pp. 36–37
  • The analysis reproduces hydrogen-embrittlement test data, then extends applied potentials beyond the tests to predict a transition to dissolution-driven SCC.Source: §4.4 · pp. 38–40
  • Both hydrogen degradation laws predict a significant reduction in Kth as hydrogen content increases, but the atomistically informed law slightly underpredicts embrittlement.Source: Figure 13 · p. 39, §4.4 · p. 39
  • The atomistic toughness relation saturates at higher Gc as χ approaches 1 and is sensitive to interface binding energy, whose uncertain characterization limits parameter-free modeling.Source: §4.4 · pp. 39–40, Figure 14 · p. 40
  • The model agrees quantitatively with cathodic experiments and captures Kth rising to a peak at Ecorr before falling as anodic SCC dominates.The predictions cover cathodic, corrosion, and higher-potential regimes in sterile artificial and biologically active seawater.Source: §4.4 · pp. 42–43, Figure 15 · p. 42

4.5. SCC tube testing of Al alloys (Gruhl, 1984): hydrogen embrittlement vs anodic dissolution

  • Gruhl (1984)’s AlZn5Mg3 tests showed brittle cracks initiating near the outer notch, far from the corrosive environment, implicating hydrogen embrittlement.Hydrogen can diffuse through the lattice and accumulate near stress concentrators such as notches.Source: §4.5 · pp. 44–45
  • After 11 h of loading, simulations showed hydrogen accumulating at the notch tip near peak stress, supporting hydrogen-driven SCC initiation there.The predicted hydrogen concentration became comparable to that at the inner surface.Source: §4.5 · p. 46, Figure 17 · p. 47
  • At cenv = 72 wppm, the results shown at the left edge of Fig. 18 attain good agreement with the experimentally reported results.Source: §4.5 · p. 47
  • Increasing L0 shifts failure from hydrogen embrittlement through hybrid hydrogen-and-anodic damage to anodic dissolution, while failure time drops.The results extend Gruhl’s experiments to higher corrosion currents, where anodic dissolution can dominate.Source: §4.5 · pp. 48–49
  • Under hydrogen-driven failure, time to failure is initially insensitive to L0: 10.6 h are needed for hydrogen to accumulate at the notch tip, where cracks nucleate and propagate inward.Failure is governed by diffusion in this regime.Source: §4.5 · pp. 47–48
  • Above L0 = 10^-7 mm^2/(N·s), pitting lowers sample stiffness and triggers earlier hybrid failure; above 10^-6 mm^2/(N·s), significant pre-charging corrosion causes an abrupt loss of load capacity.In the hybrid regime, pitting initiates first and the main crack extends from the notch pit; at higher L0, dissolution dominates.Source: §4.5 · pp. 48–49

5. Discussion and summary

The paper presents a coupled framework for hydrogen embrittlement and dissolution-driven damage, then uses simulations to examine their competition across materials and environments. Results include agreement with experiments, potential-dependent mechanism changes, and the first simulation results for Gruhl’s Al-alloy experiments.

5. Discussion and summary

  • The formulation couples mass-conserved metal-ion diffusion, KKS phase-field corrosion, extended Fickian hydrogen diffusion, elastic-plastic response, and Griffith-based fracture.These elements capture diffusion- and activation-controlled corrosion, film rupture and repassivation, and mechanically enhanced corrosion kinetics.Source: §5 · p. 49
  • The finite-element model uses displacement, corrosion and fracture order parameters, metallic-ion concentration, and hydrogen concentration as degrees of freedom.It was used to reproduce benchmarks for semi-circular pit growth and hydrogen-assisted failure of a notched square plate.Source: §5 · p. 49
  • Very good agreement with experiments accompanies potential-dependent damage-mechanism changes; the simulations provide the first Gruhl Al-alloy results and quantify where hydrogen embrittlement stops dominating.The results extend the interpretation to more aggressive environments.Source: §5 · p. 50
  • Mechanics and dissolution localize corrosion; defect sharpening can rupture passive films and raise dissolution rates, while accumulated hydrogen can later drive failure.The authors also associate toughness drops above the corrosion potential with increased dissolution, hydrogen generation at crack faces, or both.Source: §5 · p. 50
  • Uncertainty in trap-binding-energy estimates hinders fully predictive hydrogen-degradation estimates, whereas quantitative dissolution estimates can use corrosion current density.The paper notes that first-principles hydrogen laws avoid calibration but remain affected by this uncertainty.Source: §5 · p. 50
  • Film passivation reduces hydrogen uptake but requires higher failure loads, and can favor a transition from anodic-dissolution-driven cracking to hydrogen-assisted fracture.The study examines this transition in stainless steel under pure bending.Source: §5 · pp. 49–50
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