Modified Griffith's Criterion: Hydrogen Brittle Fracture

• The paper demonstrates that hydrogen significantly reduces the fracture toughness in Fe-based alloys by modifying the classical Griffith's criterion through surface energy reduction and trap effects.
• It employs atomistic simulations and continuum phase-field models to quantify hydrogen-induced changes, revealing measurable drops in stress intensity factors even at low concentrations.
• The study provides a practical predictive framework to inform the design and assessment of high-strength steels prone to hydrogen embrittlement.

Hydrogen embrittlement remains a critical limitation in the use of high-strength steels and other iron-based alloys. Conventional fracture mechanics, centered on Griffith's criterion, provides a foundational metric for crack initiation in brittle solids but does not capture the dramatic reductions in fracture toughness induced by even dilute concentrations of hydrogen. Recent advances, leveraging atomistic simulations and continuum phase-field models, have established a rigorously modified Griffith's criterion that reflects hydrogen-induced changes in surface energy, hydrogen trapping, and rapid diffusion, thus offering a predictive framework for hydrogen-assisted brittle fracture in structural metals (Egorov et al., 14 Dec 2025, Kristensen et al., 2020).

1. Classical Griffith's Criterion and Its Limitations

Griffith's criterion posits that brittle fracture occurs when the elastic energy release rate $G$ equals or exceeds a critical value $G_c$, traditionally written as $G_c^0 = 2\gamma_s^0$, where $\gamma_s^0$ is the surface energy of the pristine material. In terms of the mode I stress-intensity factor $K_I$, the critical condition is

$K_I \geq K_{Ic,0} = \sqrt{E' \cdot 2\gamma_s^0},$

where $E' = E/(1-\nu^2)$ for plane strain, with $E$ the Young's modulus and $\nu$ the Poisson's ratio. This formulation assumes a purely brittle response and neglects environmental effects such as hydrogen adsorption, trapping, and surface energy modification. These omissions render $G_c^0$ and $K_{Ic,0}$ inadequate when hydrogen is present, leading to systematic overestimates of fracture toughness in practical settings where hydrogen embrittlement is operational (Egorov et al., 14 Dec 2025).

2. Mechanistic Basis for Hydrogen-Induced Modification

Atomistic simulations using density-functional-theory-accurate machine-learned interatomic potentials for iron-hydrogen systems have revealed that dilute hydrogen concentrations (0.05–200 appm) can fundamentally alter crack-tip processes. In the absence of hydrogen, bcc Fe exhibits ductile blunting mediated by dislocation emission. However, hydrogen dramatically accelerates the transition to cleavage by rapidly diffusing to the crack tip, adsorbing onto nascent surfaces, and reducing the local surface energy $\gamma_s(\Theta)$ (Egorov et al., 14 Dec 2025). This drop in $\gamma_s$ lowers $G_c$ and correspondingly $K_{Ic}$, shifting the ductile-to-brittle balance toward brittle fracture via decohesion.

Table 1 summarizes key energetic quantities.

Parameter	Pure Fe	H-Saturated Fe
Surface energy, $\gamma_s$ (J/m$^2$)	$\approx$ 2.4	$\approx$ 1.3
$K_{Ic}$ (MPa$\sqrt{\mathrm{m}}$)	1.16 (500 K)	0.76–0.96 (0.05–200 appm H)

The process is essentially athermal on simulation timescales, as hydrogen diffusion and trapping are much faster than crack propagation, ensuring hydrogen is always available at the crack tip (Egorov et al., 14 Dec 2025).

3. Mathematical Formulation of the Modified Griffith Criterion

Hydrogen-Modified Critical Condition

The presence of hydrogen leads to a modified criterion, expressed in energy-release-rate and stress-intensity forms: $G \geq G_c(\theta, H),$

$$K_I \geq K_{Ic}(\theta, H) = K_G(\theta, H) + \Delta K_{\mathrm{trap}},$$

with

$K_G(\theta, H) = \sqrt{2 \gamma_s(\Theta) E'},$

where $\gamma_s(\Theta)$, the surface energy as a function of hydrogen coverage $\Theta$, interpolates between the clean and fully saturated states: $$\gamma_s(\Theta) = (1-\Theta) \gamma_s^0 + \Theta \gamma_s^{\mathrm{sat}}.$$ Here, $\gamma_s^0 \approx 2.4$ J/m$^2$ for pure Fe and $\gamma_s^{\mathrm{sat}} \approx 1.3$ J/m$^2$ for H-saturated {110} Fe. $\Delta K_{\mathrm{trap}} \approx 0.15$ MPa$\sqrt{\mathrm{m}}$ represents the lattice trapping resistance at 0 K.

The local hydrogen coverage $\Theta$ is given by

$$\Theta = \frac{N_{H, \mathrm{surf}}}{N_{\mathrm{sites}}},$$

where $N_{H, \mathrm{surf}}$ is the number of H atoms at the freshly created crack surfaces and $N_{\mathrm{sites}}$ the available binding sites. Oriani’s equilibrium relates the local trap occupancy to bulk hydrogen concentration $\theta_l$ and elastic binding energy $\Delta E_b$: $$\frac{\theta_c}{1-\theta_c} = \frac{\theta_l}{1-\theta_l} \exp\left[ -\frac{\Delta E_b}{k_B T} \right].$$ $\Theta \approx \theta_c$ for low $\theta_l$.

This framework reduces $G_c$ linearly with increasing hydrogen coverage, explicitly capturing the embrittlement mechanism (Egorov et al., 14 Dec 2025).

4. Application and Quantitative Predictions

Implementing the criterion involves:

1. Measuring or prescribing bulk hydrogen concentration $\theta_l$ and temperature $T$.
2. Computing trap occupancy $\theta_c$ at the crack tip using Oriani’s model.
3. Estimating hydrogen coverage $\Theta \approx \theta_c$ for representative low concentrations.
4. Determining reduced surface energy via $\gamma_s(\Theta)$ using known DFT/ML-potential data.
5. Calculating $K_G$ and $K_{Ic}$ using the above relations.
6. Comparing $K_{Ic}$ to applied $K_I$ in the material/structure of interest.

For example, at $T=500$ K and $\theta_l = 5$ appm, $\gamma_s(\Theta) \approx 2.07$ J/m$^2$, $K_G \approx 0.72$ MPa$\sqrt{\mathrm{m}}$, and $K_{Ic} \approx 0.87$ MPa$\sqrt{\mathrm{m}}$. This matches atomistic simulation results ($0.86 \pm 0.02$ MPa$\sqrt{\mathrm{m}}$), validating the criterion and confirming the dominant role of surface energy reduction over lattice trapping in practical regimes (Egorov et al., 14 Dec 2025).

5. Continuum Models and Phase-Field Perspectives

Complementary continuum approaches, such as the hydrogen-sensitive phase-field model of Martínez-Pañeda et al., generalize the Griffith-type criterion to elasto-plastic solids with stress-driven diffusion and strain-gradient effects (Kristensen et al., 2020). The total free energy incorporates hydrogen-modified fracture energy,

$G_c(c) = G_{c0}[1 - x\theta(c)],$

with $\theta(c)$ governed by a Langmuir-McLean isotherm,

$\theta(c) = \frac{c}{c + e^{-\Delta G^0 / RT}},$

where $x$ is a DFT-calibrated damage coefficient. In the sharp crack limit, the criterion for brittle advance is

$G \geq G_c(c),$

leading to a hydrogen-reduced fracture threshold,

$K_{th}(c) = K_{th}^0 \sqrt{1 - x\theta(c)}.$

This model replicates observed reductions of $K_{th}$ in ultra-high strength steels under increasing hydrogen charging, confirming the generality of the hydrogen-modified Griffith framework (Kristensen et al., 2020).

6. MD and Experimental Validation

Large-scale molecular dynamics (MD) using density-functional-theory-accurate machine-learning potentials for Fe-H predicts $K_{Ic}$ reductions (from 1.16 down to 0.76 MPa$\sqrt{\mathrm{m}}$) as hydrogen concentrations increase from 0 to 200 appm at 500 K (Egorov et al., 14 Dec 2025). These atomistic results are in quantitative accord with predictions from the modified Griffith criterion, lying between theoretical bounds defined by lattice trapping.

Experimental phase-field modeling for ultra-high strength AerMet100 steel yields $K_{th}$ drops from $\sim$30 (inert) to $\sim$10 MPa$\sqrt{\mathrm{m}}$ (high H) as a function of applied potential, accurately traced by the predicted $K_{th}$ trajectory with appropriate parameterization of $G_c(c)$ and the hydrogen isotherm (Kristensen et al., 2020).

7. Practical Implications and Usage

The modified Griffith criterion provides a robust framework for predicting hydrogen-assisted brittle fracture in Fe-based alloys. The approach offers a pathway for mapping out embrittlement “safety envelopes” by varying hydrogen concentration, temperature, and alloy elastic parameters. Accurate surface energy models, lattice trapping contributions, and hydrogen site occupancy are central to implementation. This enables material design, structural assessment, and alloy selection in critical applications where hydrogen embrittlement is a persistent threat (Egorov et al., 14 Dec 2025, Kristensen et al., 2020). The formalism is extendable to a variety of microstructural, thermodynamic, and loading scenarios, forming a foundational basis for ongoing research in environmentally assisted fracture.