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A phase field model for elastic-gradient-plastic solids undergoing hydrogen embrittlement
Philip K. Kristensen, Christian F. Niordson, Emilio Martínez-Pañeda
TL;DR
Hydrogen embrittlement motivates a continuum model that can represent near-tip dislocation hardening while explicitly predicting fracture. The paper combines strain-gradient plasticity, hydrogen diffusion, and hydrogen-sensitive phase-field fracture, and concludes that sufficiently strong crack-tip strengthening can produce brittle fracture in otherwise ductile metals.
Problem
Existing continuum frameworks had not explicitly predicted cracking while accounting for the dislocation-hardening mechanisms governing crack-tip deformation.
Method
The framework couples strain-gradient plasticity, deformation–diffusion–fracture theory, and hydrogen-dependent fracture modeling.
Results
Brittle fracture occurs when strength is on the order of 10σy or larger, and is predicted in otherwise ductile metals under the modeled conditions.
Takeaways & Limitations
The model provides physical insight into hydrogen-induced fracture mechanisms, including the transition from ductile to brittle fracture.
Takeaways & Limitations
The framework is presented as addressing crack-tip dislocation hardening, but the supplied scope does not establish its applicability beyond the modeled elastic-plastic setting.
Abstract
from arXiv · showhide
We present a gradient-based theoretical framework for predicting hydrogen assisted fracture in elastic-plastic solids. The novelty of the model lies in the combination of: (i) stress-assisted diffusion of solute species, (ii) strain gradient plasticity, and (iii) a hydrogen-sensitive phase field fracture formulation, inspired by first principles calculations. The theoretical model is numerically implemented using a mixed finite element formulation and several boundary value problems are addressed to gain physical insight and showcase model predictions. The results reveal the critical role of plastic strain gradients in rationalising decohesion-based arguments and capturing the transition to brittle fracture observed in hydrogen-rich environments. Large crack tip stresses are predicted, which in turn raise the hydrogen concentration and reduce the fracture energy. The computation of the steady state fracture toughness as a function of the cohesive strength shows that cleavage fracture can be predicted in otherwise ductile metals using sensible values for the material parameters and the hydrogen concentration. In addition, we compute crack growth resistance curves in a wide variety of scenarios and demonstrate that the model can appropriately capture the sensitivity to: the plastic length scales, the fracture length scale, the loading rate and the hydrogen concentration. Model predictions are also compared with fracture experiments on a modern ultra-high strength steel, AerMet100. A promising agreement is observed with experimental measurements of threshold stress intensity factor $K_{th}$ over a wide range of applied potentials.
1. Introduction
Hydrogen embrittlement severely reduces metal ductility and fracture toughness, while existing models do not simultaneously resolve near-tip length scales, complex cracking, and dislocation hardening. The paper introduces a coupled framework combining strain-gradient plasticity, hydrogen transport, and phase-field fracture to address these limitations.
- Hydrogen can reduce the fracture toughness of modern steels by up to 90%.
- Hydrogen diffuses toward high-hydrostatic-stress regions, where damage mechanisms remain under debate.
- Conventional continuum models fail to resolve the critical length scale of hydrogen-assisted fracture, which occurs within 1 µm or less of the crack tip.
- Large plastic strain gradients near crack tips increase geometrically necessary dislocation storage and produce stresses much higher than conventional plasticity predicts.
- The framework couples strain-gradient plasticity, stress-assisted hydrogen diffusion, phase-field fracture, and a first-principles-informed hydrogen-dependent fracture-energy law.
- The authors aim to predict the hydrogen-induced ductile-to-brittle transition and reproduce measurements across a wide range of applied potentials.
2. Theory
The theory formulates a coupled deformation–damage–hydrogen-transport response for elastic-plastic solids occupying an arbitrary domain.
- The theory couples deformation, damage, and hydrogen transport in elastic-plastic bodies.
- The solid is modeled over an arbitrary domain Ω in one, two, or three dimensions.
2.1. Kinematics
The kinematic formulation uses displacement, plastic strain, a damage phase field, and hydrogen concentration, with diffuse fracture and chemically driven transport represented continuously.
- The model uses displacement, plastic strain, damage phase field, and hydrogen concentration as kinematic fields.
- The phase field ranges from 0 for intact material to 1 for fully broken material, representing discrete cracks diffusely.
- The phase-field length scale controls the smearing of cracks and approximates fracture energy over a discontinuous surface.
- The diffuse fracture representation avoids tracking discrete crack surfaces in numerical models.
- Hydrogen concentration evolves through mass conservation, with flux driven by gradients of chemical potential through an Onsager relation.
2.2. Principle of virtual work. Balance of forces
The virtual-work formulation introduces work-conjugate stresses and higher-order quantities for displacement, plastic strain, its gradient, damage, and hydrogen flux.
- The coupled boundary-value problem derives balance equations from external and internal virtual work.
- The plastic response uses a micro-stress tensor conjugate to plastic strain and a higher-order stress tensor conjugate to its gradient.
- Damage is represented by a scalar stress-like quantity conjugate to the phase field and a micro-stress vector conjugate to its gradient.
- The phase field is driven by the displacement problem alone, so no external traction is associated with it.
- Hydrogen boundary flux is work-conjugate to chemical potential in the diffusion problem.
2.3. Energy imbalance
The model formulates a thermodynamically consistent free energy incorporating chemo-elasticity, fracture, plastic-strain gradients, and hydrogen-dependent fracture resistance. Fracture resistance is explicitly defined as a function of hydrogen concentration.
- The Helmholtz free energy depends on strain, plastic-strain gradients, the phase field and its gradient, and hydrogen concentration.It combines chemo-elastic bulk energy, chemical free energy, and gradient contributions.
- The chemical contribution uses lattice-site occupancy, hydrogen concentration, partial molar volume, and a reference chemical potential.The chemical free energy includes a lattice-mixture term based on θL = C/N.
- The elastic strain energy density provides the driving force for fracture, while LE quantifies energetic gradient hardening.Energetic gradient hardening is associated with long-range back-stresses from geometrically necessary dislocations.
- The critical energy release rate is defined as a function of hydrogen concentration, establishing direct hydrogen degradation of fracture resistance.This makes hydrogen content part of the fracture-energy formulation rather than an independent post-processing quantity.
2.4. Constitutive relations
The constitutive framework couples phase-field fracture, chemo-elasticity, strain-gradient plasticity, and hydrogen transport through a thermodynamically based set of relations. Fracture is driven by elastic energy, while hydrogen affects fracture resistance through the concentration dependence of Gc.
- Chemo-elasticity: The phase-field variable degrades elastic stiffness through a quadratic degradation function, with elastic energy defined from elastic strain and the isotropic stiffness tensor.The formulation omits the lattice-dilation term in hydrogen embrittlement analyses because its influence is considered negligible.
- Strain gradient plasticity: Higher-order strain-gradient plasticity includes energetic and dissipative gradient contributions characterized by plastic length scales.LD quantifies dissipative strengthening, while the reference plastic length scale controls the gradient contribution.
- Strain gradient plasticity: The effective stress is work-conjugate to the effective plastic rate, and micro-stresses and higher-order stresses are decomposed into energetic and dissipative parts.Plastic deformation is assumed purely dissipative, whereas plastic-strain gradients contain both energetic and dissipative terms.
- Hydrogen-sensitive fracture: Fracture in the elastic-plastic solid is driven solely by the elastic component of the material strain energy density.The local phase-field balance is reformulated after neglecting concentration gradients across the narrow region where the phase field varies.
- Hydrogen transport: Hydrogen transport is driven by chemical-potential gradients and is enhanced by hydrostatic tensile stress through lattice dilatation.Hydrogen diffuses from high to low chemical potential, with the stress-dependent chemical-potential contribution included in the model.
3. Numerical implementation
The model is implemented with a mixed finite element framework that solves displacement, plastic strain, phase field, and hydrogen concentration fields. The implementation uses staggered implicit integration, viscoplastic regularization, and penalty-based moving chemical boundary conditions.
- Finite element formulation: The finite element framework uses a history field and strain-energy split to prevent damage reversibility and damage under compressive loading.A mixed finite element problem is discretized and its residuals are formulated within an Abaqus user-element implementation.
- Moving chemical boundary: Penalty-based chemical boundary conditions model prompt environmental occupation of newly created crack space.The penalty term modifies the hydrogen concentration residual so concentration at the crack surface approaches the environmental value.
- Finite element formulation: The displacement, plastic strain, phase field, and hydrogen concentration are discretized in a weakly coupled finite element system.The global system is solved using an implicit time-integration framework and a staggered approach.
- Finite element formulation: A small residual stiffness parameter k = 1 × 10^-7 maintains algebraic conditioning in fully broken states.The parameter prevents complete degradation of the energy while preserving the broken-state formulation.
- Viscoplasticity: The viscoplastic formulation bounds ∂Σ/∂∆Ep as the plastic rate approaches zero and is calibrated to reproduce rate-independent behavior.The effective stress combines the current flow stress with a viscoplastic function, and time sensitivity is studied in all computations.
4. Results
The results use boundary-layer simulations to examine stationary crack-tip fields and crack-growth resistance in hydrogen-affected elastic-plastic solids. Plastic strain gradients elevate crack-tip stresses, while hydrogen reduces fracture resistance and can drive a ductile-to-brittle transition consistent with experiments.
- Numerical framework: The boundary-layer formulation applies a remote mode-I K_I field to investigate stationary crack-tip fields and crack-growth resistance.The simulations cover plastic length scales, fracture length scale, hydrogen concentration, and loading rate.
- Stationary crack-tip fields: Nonzero plastic length scales produce an inner elastic K_I field with the linear elastic r^-1/2 stress singularity.This inner field appears for any nonzero energetic or dissipative plastic length scale.
- Stationary crack-tip fields: Hydrostatic stress near the crack tip is roughly four times larger than the conventional plasticity prediction when plastic length-scale effects are included.The elevated stresses also produce large hydrogen concentrations within a few microns of the crack-tip surface.
- Crack growth resistance: Larger L_p/R_0 values magnify gradient effects, elevate crack-tip stresses, and reduce fracture resistance.Energetic and dissipative contributions affect the resistance curves differently depending on the hardening exponent.
- Crack growth resistance: Hydrogen significantly reduces the work of fracture G_c and enables a ductile-to-brittle transition at parameter values that remain ductile without hydrogen.For hydrogen-free cases, ductile fracture is predicted below L_p/R_0 = 0.03, whereas with hydrogen L_p/R_0 ≈ 0.01 can intersect the brittle-fracture threshold at σ̂/σ_y = 10.
- Crack growth resistance: Slower loading emphasizes embrittlement, while increasing loading rate reduces hydrogen-induced degradation because less hydrogen diffuses into the fracture process zone.The model also captures decreasing fracture resistance with increasing environmental hydrogen concentration.
- Comparison with experiments: The model shows very good agreement with AerMet100 crack-initiation measurements over applied potentials where fracture is quasi-brittle.The best fit uses a hydrogen damage coefficient χ = 0.97.
5. Conclusions
The paper presents a coupled deformation–diffusion–fracture framework incorporating stress-assisted hydrogen transport, strain-gradient plasticity, and a hydrogen-sensitive phase field. Its predictions link plastic strain gradients and hydrogen-dependent fracture energy to brittle fracture in ductile metals and agree promisingly with AerMet100 experiments.
- Model and implementation: The framework couples stress-assisted hydrogen diffusion, strain-gradient plasticity, and a chemo-mechanical phase field for hydrogen embrittlement.It includes energetic and dissipative plastic length scales, a phase field length scale, and thermodynamically consistent governing equations.
- Ductile-to-brittle transition: The hydrogen-dependent fracture-energy law rationalises quasi-cleavage and the transition from microvoid cracking to brittle fracture in ductile steels.The model connects this mechanism to atomistic calculations of surface-energy reduction with hydrogen coverage.
- Model and implementation: The model is implemented with displacements, plastic strains, hydrogen concentration, and the phase field as primary variables in a mixed finite element formulation.The coupled problem is solved using implicit time integration with suitable chemical and mechanical boundary conditions.
- Crack-tip fields: Plastic strain gradients predict much higher crack-tip stresses and hydrogen concentrations than conventional plasticity, providing a physical basis for atomic-scale decohesion.The calculations assess stationary cracks and connect gradient hardening with elevated crack-tip fields.
- Crack growth resistance: Increasing the plastic length scales relative to the plastic-zone size reduces R-curve steepness, whereas decreasing the phase field length scale increases fracture resistance.These results demonstrate sensitivity to the plastic and fracture length scales.
- Ductile-to-brittle transition: Brittle fracture occurs when the cohesive strength is on the order of 10σy or larger, and is promoted by sufficiently large Lp/R0.Without hydrogen, brittleness requires large Lp/R0; hydrogen reduces fracture energy and permits brittle fracture in ductile metals with initially small Lp/R0.
- Experimental comparison: Predictions for mode-I fracture in AerMet100 show promising agreement with experiments across a wide range of applied potentials.This comparison provides a quantitative benchmark for the model's predictive capability.
Appendix A. Details of numerical implementation
The appendix provides explicit expressions for the matrix operators and stiffness-matrix components used in the numerical implementation.
- Numerical implementation: Explicit matrix operators and stiffness-matrix components are provided for the numerical implementation.These expressions are used in Section 3.2.
Appendix A.1. Matrix operators
The matrix-operator appendix defines the plane-strain nodal deformation approximation and the associated shape-function and gradient discretizations.
- Matrix operators: Under plane-strain conditions, nodal solutions are introduced for the deformation problem.The formulation then specifies shape-function matrices and discretized gradient quantities.
- Matrix operators: Shape-function matrices and gradient quantities provide the discretization operators for the numerical fields.These operators support the finite element representation of the deformation problem.
Appendix A.2. Stiffness matrix components
The stiffness-matrix appendix constructs the element Jacobian by differentiating residuals with respect to nodal variables and includes deformation, plasticity, phase-field, and hydrogen-transport contributions.
- Stiffness construction: The stiffness matrix is constructed by differentiating the residuals with respect to the nodal variables.This yields the displacement-field entries and the coupling terms among the coupled fields.
- Stiffness construction: Separate stiffness contributions are specified for the plastic strain field and crack phase field.The appendix gives the corresponding field-specific matrix components.
- Hydrogen transport: Hydrogen mass transport contributes a diffusivity matrix and a concentration capacity matrix to the finite element system.The hydrogen concentration time derivative is discretized analogously to the concentration field.