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A phase field model for hydrogen-assisted fatigue
Alireza Golahmar, Philip K. Kristensen, Christian F. Niordson, Emilio Martínez-Pañeda
TL;DR
Hydrogen reduces metallic materials’ fatigue crack-growth resistance, yet phase-field fracture formulations for hydrogen-assisted static cracking had not been extended to fatigue. This paper presents a coupled phase-field model for deformation, diffusion, and fatigue damage. The model captures hydrogen-sensitive crack growth, recovers Paris-law behavior, maps loading-frequency effects, and yields Virtual S-N predictions in promising agreement with experiments.
Problem
Phase-field models for hydrogen-assisted static fracture had not been extended to fatigue, despite the need to predict fatigue crack growth affected by hydrogen and cyclic loading.
Source: §1 · p. 2Method
The paper introduces a coupled phase-field formulation for deformation, hydrogen diffusion, and fatigue damage, incorporating hydrogen-dependent fracture-energy degradation.
Source: §1 · p. 2Results
The model captures hydrogen sensitivity in fatigue crack-growth rates, recovers Paris-law behavior, maps loading-frequency sensitivity, and predicts S-N curves with promising agreement with experiments.
Source: §4 · pp. 23–24Takeaways & Limitations
The reported framework spans crack-growth and S-N predictions, including quantifying hydrogen’s influence on Paris-law behavior and on S-N curves.
Source: §4 · pp. 23–24The cracked-plate example prescribes constant hydrogen concentration at crack faces for simplicity; generalized Neumann-type or hydrostatic-stress-dependent Dirichlet conditions are described as more appropriate.
Source: §3.1 · p. 10
Abstract
from arXiv · showhide
We present a new theoretical and numerical phase field-based formulation for predicting hydrogen-assisted fatigue. The coupled deformation-diffusion-damage model presented enables predicting fatigue crack nucleation and growth for arbitrary loading patterns and specimen geometries. The role of hydrogen in increasing fatigue crack growth rates and decreasing the number of cycles to failure is investigated. Our numerical experiments enable mapping the three loading frequency regimes and naturally recover Paris law behaviour for various hydrogen concentrations. In addition, Virtual S-N curves are obtained for both notched and smooth samples, exhibiting a good agreement with experiments.
1. Introduction
Hydrogen can reduce metallic materials’ fatigue crack-growth resistance, while predicting its combined effects with cyclic fatigue remains challenging. This work introduces a phase-field model for hydrogen-assisted fatigue and reports frequency-sensitive crack-growth behavior, Paris-law recovery, and Virtual S-N curves in agreement with experiments.
1. Introduction
- Hydrogen-assisted fatigue is important because susceptible components in transport, construction, defence, and energy face alternating mechanical loads, while hydrogen reduces metallic materials’ fatigue crack-growth resistance.The concern is heightened for modern, high-strength alloys.Source: §1 · pp. 1–2
- Although phase-field models have advanced hydrogen-assisted static fracture, extending them to fatigue is challenging because hydrogen-affected growth depends on several material and loading factors.These include hydrogen content, embrittlement susceptibility, diffusivity, loading amplitude, and frequency.Source: §1 · p. 2
- The proposed first phase-field model for hydrogen-assisted fatigue couples deformation, diffusion, and fatigue damage through a variational framework.Its components include hydrogen- and fatigue-dependent degradation, crack-solid interface evolution, and evolving environmental boundary conditions.Source: §1 · p. 2
- The model captures fatigue crack-growth sensitivity to loading frequency and environment, recovers Paris-law behavior, and produces Virtual S-N curves in promising agreement with experiments.The S-N predictions cover smooth and notched samples.Source: §1 · p. 3
2. A phase field theory for hydrogen-assisted fatigue
The framework extends variational phase-field fracture to hydrogen-assisted fatigue by coupling crack evolution with hydrogen transport and cumulative fatigue damage. Its constitutive choices represent hydrogen-related fracture-energy degradation, loading-dependent fatigue degradation, and the evolving crack-environment interface.
2. A phase field theory for hydrogen-assisted fatigue
- The phase field represents cracks as a smooth evolving interface, enabling a regularized Griffith formulation in which crack growth requires stored energy to overcome fracture resistance.The phase field is 0 in intact material and 1 inside the crack; the regularized energy converges to the Griffith energy as the length scale tends to zero.Source: §2 · pp. 3–4
2.1. Hydrogen degradation function
- Hydrogen reduces fracture resistance through a concentration-dependent degradation law inspired by atomistic surface-energy trends and hydrogen coverage at decohering interfaces.The stated choices assume interface decohesion, but the framework can accommodate other mechanisms through alternative degradation functions.Source: §2 · p. 4, §2.1 · p. 5
2.2. Fatigue degradation function
- Fatigue damage degrades the fracture response through a cumulative history variable and threshold; the history variable grows only during loading and is defined from degraded strain energy.The study generally uses an asymptotic fatigue degradation function in its numerical experiments unless otherwise stated.Source: §2.2 · p. 6
2.3. Coupled deformation-diffusion-fracture problem
- The model couples displacement, diffusion, and fracture: hydrostatic stress lowers hydrogen chemical potential, increasing solubility and attracting hydrogen toward high-volumetric-strain regions such as crack tips.An effective diffusion coefficient can represent the slowing effect of microstructural trapping sites.Source: §2.3 · pp. 6–7
2.4. Numerical implementation
- The finite-element implementation enforces damage irreversibility, prevents compressive damage, solves the coupled equations with a staggered approach, and updates moving chemical boundaries with a penalty method.The staggered solution converges to the monolithic result as the residual norm is controlled.Source: §2.4 · p. 7
3. Results
The results show that hydrogen exposure is associated with faster fatigue crack growth and shorter fatigue lives, while the model reproduces experimental S-N trends for smooth samples. Boundary-layer simulations also capture the interplay between loading frequency and hydrogen diffusion, including Paris-law behavior.
3.1. Cracked square plate subjected to fatigue in a hydrogenous environment
- In an inert environment, the model agrees well with prior predictions; omitting the tension-compression split predicts higher crack-growth rates.The validation compares no split, volumetric/deviatoric, and spectral decompositions.Source: §3.1 · p. 9, Figure 2 · p. 10
- In the cracked-plate simulations, fatigue crack growth rates increase as environmental hydrogen concentration rises.The trend is reported for environmental concentrations of 0.1, 0.5, and 1 wt ppm.Source: §3.1 · p. 10, Figure 3 · p. 11
3.2. Boundary layer model
- At a stationary crack, hydrogen concentration rises with applied load, peaks near the crack tip, and remains constant at Kmin = 0 for R = 0.The reported profiles compare the maximum, mean, and minimum stages of a low-frequency cycle.Source: §3.2 · p. 13, Figure 5 · p. 13
- At high loading frequency, hydrogen concentration near the crack tip oscillates less and can settle below low-frequency maxima; the authors suggest beneficial diffusion-slowing traps as a design strategy.At f = 10^3 Hz, the concentration flattens about 5% below the low-frequency maximum for the material properties and location considered.Source: §3.2 · p. 14, Figure 6 · p. 15
- In the Paris regime, the Paris coefficient C increases with hydrogen content, while the exponent is less environment-sensitive and is approximately 3.2.The model produces Paris-law behavior as an output rather than requiring it as an input.Source: §3.2 · p. 17, Figure 8 · p. 17
- The model captures a transition between fast tests, where hydrogen diffusion to the fracture process zone is limited, and slow tests, where hydrogen accumulates in high-σH regions.The paper maps the transition between these limiting frequency regimes.Source: §3.2 · pp. 17–18, Figure 9 · p. 18
3.3. Notched cylindrical bar
- For notched bars, increasing hydrogen concentration shortens fatigue life, while the S-N curve slope appears relatively insensitive to hydrogen content.The reported environments contain 0.1, 0.5, and 1 wt ppm hydrogen.Source: §3.3 · p. 19, Figure 11 · p. 20
3.4. Comparison with experimental S-N curves
- Hydrogen reduces cycles to failure by almost an order of magnitude in JIS-SM490B, and predicted Virtual S-N curves agree well with smooth-sample experiments.Agreement is less satisfactory at low stress amplitudes, particularly without hydrogen; the authors identify adding a fatigue endurance limit as a potential improvement.Source: Figure 13 · p. 22, §3.4 · p. 23
4. Conclusions
The coupled phase-field formulation links deformation, diffusion, and fatigue crack growth to model hydrogen-assisted fatigue. Its findings include parameter-free S-N predictions with promising experimental agreement, frequency-dependent crack-tip hydrogen behavior, and recovered Paris-law behavior.
4. Conclusions
- The model couples deformation, diffusion, and fatigue crack growth, incorporating stress-dependent solubility and an evolving environment-diffusion interface.Source: §4 · p. 23
- Crack-tip hydrogen distribution is sensitive to loading frequency and material diffusivity; at sufficiently high frequency, concentration stops oscillating and rises to saturation, even at R = 0.Source: §4 · p. 23
- The model captures fatigue crack-growth-rate sensitivity to hydrogen and recovers Paris-law behavior, allowing quantification of hydrogen’s influence on Paris-law parameters.Source: §4 · p. 23
- Mapping crack-growth-rate sensitivity to loading frequency reveals two experimentally observed limiting states and predicts a smooth transition between them.Source: §4 · p. 23
- Parameter-free predictions of hydrogen’s impact on S-N curves show promising agreement with experiments, with Virtual S-N curves obtained for notched and smooth samples.The framework is presented as a platform for addressing the challenge of predicting hydrogen-assisted fatigue failures.Source: §4 · p. 24