Deformation-Diffusion-Fracture Phase Field Model

• Deformation–diffusion–fracture phase field model is a variational framework that couples mechanical deformation, solute diffusion, and fracture evolution using a continuous field variable.
• The approach integrates multiple physics by combining mechanical, chemical, and fracture energy terms, with stress, strain, and concentration dependencies driving crack evolution.
• It is applied to simulate phenomena like hydrogen-assisted cracking, lithium-induced battery degradation, and hydraulic fracturing, with robust finite element numerical implementations.

A deformation–diffusion–fracture phase field-based model is a variational, continuum-scale, multi-physics modeling framework that simultaneously describes mechanical deformation, mass transport (diffusion), and fracture evolution in solids using the phase-field methodology. This approach regularizes sharp cracks with a continuous field variable and couples the evolution of cracks to stress, strain, and local concentration of diffusible species. The methodology is particularly suited to situations where degradation, embrittlement, or fracture is triggered or accelerated by the transport and accumulation of solutes—exemplified by hydrogen-assisted cracking in metals, lithium-induced stress in Li-ion battery electrodes, and hydraulic or thermal fracturing in porous media. The phase field methodology enables robust numerical simulation and facilitates the incorporation of atomistically-informed constitutive laws and multi-physics coupling.

1. Variational Framework and Energy Functional

The core of deformation–diffusion–fracture phase field modeling is the definition of a total free energy (or pseudo-energy) functional that encompasses mechanical, chemical, and fracture energies, as well as coupling terms between these subsystems.

A generic functional in the isothermal, small-strain regime with a single diffusive species can be written as: $$\mathcal F[u, d, c] = \int_\Omega \Big\{ g(d)\,\psi_+(\varepsilon(u), c) + \psi_-(\varepsilon(u), c) + f(c) \Big\} \, dV + \int_\Omega W_{\text{fracture}}[d, \nabla d, c]\,dV$$ where $u$ is the displacement vector, $d \in [0,1]$ is the phase-field variable (%%%%2%%%%: intact, $d=0$: fully broken), $c$ is concentration, $\psi_+$ and $\psi_-$ are tensile and compressive portions of the strain energy, $f(c)$ is the free energy of mixing or chemical storage, and $W_{\text{fracture}}$ is the regularized crack surface energy, often of Ambrosio–Tortorelli (AT1 or AT2) form: $$W_{\text{fracture}} = \frac{G_c(c)}{4c_n}\left[\frac{(1-d)^n}{\ell} + \ell |\nabla d|^2\right]$$ with fracture toughness $G_c$ possibly a function of $c$.

Specific models introduce additional terms reflecting chemo-mechanical coupling (e.g., elastic energy depending on $c$), strain gradient plasticity, poroelastic contributions, or temperature effects as needed in the physical system of interest (Ahmadi, 2020, Martínez-Pañeda et al., 2018, Liu et al., 2024, Kristensen et al., 2020, Kristensen et al., 2020).

2. Governing Equations and Coupled Field Evolution

The functional's stationarity yields a system of coupled partial differential equations for displacements, concentration, and crack evolution:

Mechanical equilibrium:

$$\nabla \cdot \sigma + b = 0, \quad \sigma = g(d)\,\sigma_+ + \sigma_- + \text{(coupling terms)}$$

Stress may be modified by chemical strains (e.g., from intercalation in batteries), hydrostatic swelling, or fluid/thermal terms in poroelastic and thermo-mechanical models (Ahmadi, 2020, Liu et al., 2024).

Mass transport (diffusion):

$$\frac{\partial c}{\partial t} = \nabla \cdot (D(c)\, \nabla \mu), \quad \mu = \frac{\partial f}{\partial c} + \frac{\partial \psi}{\partial c} + \text{(stress/strain terms)}$$

Fluxes often include stress-assisted terms, e.g., for hydrogen, $$J \sim - D\,\nabla c + (D\,c\,V_H/RT) \nabla \sigma_H$$ (Martínez-Pañeda et al., 2018, Kristensen et al., 2020, Kristensen et al., 2020).

Phase field (fracture) evolution:

$$G_c(c)\left(\frac{d}{\ell} - \ell\,\Delta d\right) - 2 (1-d)\psi_+(\varepsilon(u), c) = 0$$

or corresponding forms for AT1/AT2, typically with irreversibility enforced (e.g., $d(x,t) \leq d(x, t + \Delta t)$) (Liu et al., 2024). In "hybrid" models for battery materials, $\psi_+$ restricts crack driving forces to tension, preventing spurious fracture under compression (Ahmadi, 2020).

Additional equations govern fluid flow (Darcy), energy balance (heat), or plastic evolution (gradient plasticity) as required (Liu et al., 2024, Kristensen et al., 2020).

3. Constitutive Modeling and Coupling Strategies

Crucial model choices include:

• Degradation function $g(d)$: Quartic, quadratic, or more general forms with residual stiffness to regularize numerics; $g(d)$ is typically designed such that $g(1)=1$ (intact), $g(0)\approx 0$ (fully broken) (Ahmadi, 2020, Martínez-Pañeda et al., 2018).
• Energy decomposition: Split of the strain energy into positive (tension) and negative (compression) parts, typically by volumetric–deviatoric methods, ensuring that only tensile energy drives fracture (Kristensen et al., 2020, Liu et al., 2024).
• Concentration-dependent elasticity and toughness: For battery and hydrogen-embrittlement problems, both Young's modulus and $G_c$ may decrease with local $c$ according to rule-of-mixtures or atomistically-informed degradation laws. Hydrogen toughness degradation employs empirical or DFT-calibrated laws: $G_c(c) = G_c^0 (1 - \chi \theta)$, with $\theta$ a Langmuir–McLean coverage (Martínez-Pañeda et al., 2018, Kristensen et al., 2020, Kristensen et al., 2020).
• Poromechanics and thermal effects: Poroelastic moduli, Biot coefficient, effective thermal expansion, and porosity can be made dependent on both phase field and strain, enabling accurate representation of fractured, heated, fluid-filled rocks or geomaterials (Liu et al., 2024).
• Plasticity: Coupling to gradient or rate-dependent plasticity is achieved via higher-order energetic and dissipative hardening terms, critical for ductile-to-brittle transitions in metals (Kristensen et al., 2020).

4. Numerical Implementation

Solution strategies vary, but finite element discretization with implicit (backward-Euler) time integration is common. The fields are either solved monolithically (all unknowns in a global vector, e.g., $Q=\{u, c, d\}$) (Ahmadi, 2020), or in a staggered (alternating minimization) framework, solving sequentially for mechanics, diffusion, and phase field at each time step (Martínez-Pañeda et al., 2018, Kristensen et al., 2020, Liu et al., 2024, Kristensen et al., 2020).

Some key implementation aspects:

• Irreversibility is enforced by projection or history field methods.
• FE discretization: 4- or 8-node quadrilaterals (2D), bricks, tetrahedra, etc. Equal-order interpolation for all fields is conventional.
• Coupling: Block-diagonal structure or monolithic Newton–Raphson with consistent tangents.
• Stabilization: Small residual stiffness and artificial diffusion (e.g., for advection-dominated heat flow) are used to ensure numerical stability (Liu et al., 2024, Kristensen et al., 2020).

Open-source and user-element codes, such as OpenGeoSys and ABAQUS UEL subroutines, are employed for computational experiments (Liu et al., 2024, Martínez-Pañeda et al., 2018, Kristensen et al., 2020).

5. Illustrative Applications and Model Validation

Major application domains include:

• Hydrogen-assisted cracking in high-strength steels (Martínez-Pañeda et al., 2018, Kristensen et al., 2020, Kristensen et al., 2020):
  • Models capture reduction in $G_c$, crack growth thresholds ($K_{\text{th}}$), transition from ductile to cleavage fracture, R-curve degradation, and environmental time-scale effects.
  • Experiments for AISI 4135, AerMet100, and duplex-SS demonstrate accurate threshold predictions and embrittlement effects.
• Lithium-induced fracture in Li-ion battery electrodes (Ahmadi, 2020):
  • Models incorporate lithium concentration effects on both modulus and fracture resistance; two-way stress–diffusion coupling; numerics reproduce nanowire crack evolution under various lithiation scenarios.
• Hydro-thermal-poroelastic fracture in geological media (Liu et al., 2024):
  • Models verified against benchmarks (Terzaghi's and thermal consolidation, KGD hydraulic fracture) capture fluid-driven and thermally activated crack evolution, improved porosity update accuracy, and interface interactions.
• Large-scale engineering integrity assessment (Kristensen et al., 2020):
  • Applications in bolt failure, pipeline defect propagation, and “digital twin” paradigms for real-world asset management, leveraging direct mapping of pre-existing defects onto the phase-field variable.

Typical validation includes analytic solution recovery in canonical problems, quantitative agreement with experimental load–displacement, R-curve, and $K_{\text{th}}$ data, and mesh/time-convergence studies.

6. Model Parameters, Calibration, and Practical Considerations

Parameter identification is context-dependent. Key parameters usually include elastic moduli ($E$, $\nu$), fracture energy ($G_c^0$), phase-field regularization length ($\ell$), diffusion coefficient ($D$), partial molar volume (for chemo-mechanical coupling), toughness degradation coefficients ($\chi$ from DFT), porosity evolution parameters, and in plasticity-augmented models, hardening/exponent and gradient length scales ($L_e$, $L_d$) (Martínez-Pañeda et al., 2018, Kristensen et al., 2020, Liu et al., 2024). Experimental data, atomistic simulations, and literature correlations inform parameter selection.

Boundary and initial conditions are selected to match the physical system under study: displacement/traction for mechanics, Dirichlet/Neumann for concentration and phase field, and, in poromechanical settings, pressure and temperature as appropriate.

7. Significance and Prospects

Deformation–diffusion–fracture phase field models provide a unified, flexible, and physically-grounded computational approach for simulating multi-physics crack evolution in solids exposed to aggressive environments or complex loading histories. The phase field methodology is key in enabling robust simulation of coupled field evolution, complex crack topologies (e.g., branching, merging, arrest), and realistic structural integrity assessment across scales. Continuing developments include improved multi-scale calibration, extension to finite deformation, incorporation of discrete heterogeneity, enhanced treatment of plasticity/viscoplasticity, non-isothermal effects, and real-time simulation capacities for “digital twin” applications (Kristensen et al., 2020, Liu et al., 2024, Kristensen et al., 2020).

Key references include phase-field models for hydrogen embrittlement in steels (Martínez-Pañeda et al., 2018, Kristensen et al., 2020, Kristensen et al., 2020), lithium diffusion–fracture coupling in batteries (Ahmadi, 2020), and thermo-poro-mechanical fracture in geomaterials (Liu et al., 2024). These works demonstrate the versatility and rigorous predictive capacity of the deformation–diffusion–fracture phase field framework across a wide range of material systems and engineering applications.