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A generalised phase field model for fatigue crack growth in elastic-plastic solids with an efficient monolithic solver
Z. Khalil, A. Y. Elghazouli, E. Martínez-Pañeda
TL;DR
Fatigue-induced fracture motivates computational models that predict fatigue cracking and generalize across materials, geometries, and loading histories. The paper presents a generalized fatigue-damage formulation with an efficient solution approach, reports agreement with experimental data, and compares favorably with staggered approaches.
Problem
Fatigue-induced fracture is frequent, motivating computational models capable of predicting fatigue cracking and generalizing across materials, geometries, and loading histories.
Method
The paper presents a generalized formulation for fatigue damage using a crack surface density functional and critical fracture parameter.
Results
The model shows good agreement with experimental data, while the monolithic approach is more robust and efficient than staggered approaches; the staggered scheme requires over 25 times more load increments to match it.
Takeaways & Limitations
The presented formulation supports modeling fatigue crack growth across the paper’s investigated settings while offering an efficient alternative to staggered approaches.
Takeaways & Limitations
The approach comes at the cost of defining one additional parameter.
Abstract
from arXiv · showhide
We present a generalised phase field-based formulation for predicting fatigue crack growth in metals. The theoretical framework aims at covering a wide range of material behaviour. Different fatigue degradation functions are considered and their influence is benchmarked against experiments. The phase field constitutive theory accommodates the so-called AT1, AT2 and phase field-cohesive zone (PF-CZM) models. In regards to material deformation, both non-linear kinematic and isotropic hardening are considered, as well as the combination of the two. Moreover, a monolithic solution scheme based on quasi-Newton algorithms is presented and shown to significantly outperform staggered approaches. The potential of the computational framework is demonstrated by investigating several 2D and 3D boundary value problems of particular interest. Constitutive and numerical choices are compared and insight is gained into their differences and similarities. The framework enables predicting fatigue crack growth in arbitrary geometries and for materials exhibiting complex (cyclic) deformation and damage responses. The finite element code developed is made freely available at www.empaneda.com/codes.
1. Introduction
The paper develops a generalised phase field framework for fatigue crack growth in elastic-plastic metals, addressing the limited generality of empirical fatigue methods and existing phase field studies. It combines multiple fracture and hardening models with a quasi-Newton monolithic solver and evaluates them across 2D and 3D problems.
- Motivation and gap: Empirical stress- and strain-based fatigue approaches have limited applicability across arbitrary materials, geometries, and loading histories.Variational phase field models are presented as a computational alternative for high-, low-, and extremely low-cycle fatigue.
- Motivation and gap: Existing phase field fatigue studies were largely restricted to linear elastic materials, motivating formulations for elastic-plastic solids.Recent work addressed elastic-plastic fatigue, including low- and high-cycle regimes and ratcheting effects.
- Contributions: The proposed formulation models a general class of elastic-plastic materials with combined non-linear isotropic and kinematic hardening.The framework is designed for complex cyclic deformation and damage responses.
- Contributions: The constitutive theory accommodates AT1, AT2, and PF-CZM phase field fracture models rather than restricting analysis to one class.This enables comparison of brittle and quasi-brittle formulations within the same fatigue framework.
- Numerical strategy: A quasi-Newton monolithic solution scheme is reported as more robust and significantly more efficient than widely used staggered schemes.The comparison matters because fatigue predictions require many cycle-by-cycle computations.
- Validation and applications: The framework is tested on planar specimens, Compact Tension samples, an asymmetrically notched bar, and a three-dimensional pipe-to-pipe component.These studies compare constitutive and numerical choices and examine crack nucleation, growth, hardening, and damage.
2. Theory
The theory is formulated to support arbitrary constitutive choices for crack density, fracture driving force, degradation, and cyclic material response. It derives coupled force balances and then specialises them to deformation and fracture models.
- General formulation: The formulation is designed for arbitrary choices of crack density function, fracture driving force, degradation function, and cyclic material response.This provides the general constitutive basis for the fatigue framework.
- Balance laws: The theory derives force balances using the principle of virtual power before specifying constitutive choices for deformation and fracture.The derivation proceeds from kinematics to balance equations and then to constitutive specialisation.
2.1. Kinematics
The kinematic framework uses displacement, strain, and a smooth scalar phase field to represent diffuse cracks, while extending fracture energy to fatigue through a history-dependent degradation description.
- Kinematic variables: The primary kinematic variables are the displacement field u and the scalar damage phase field φ.The phase field takes φ = 0 for the intact phase and φ = 1 for fracture.
- Deformation: The deformation description adopts the standard decomposition of strains into elastic and plastic parts.
- Diffuse crack representation: A smooth continuous phase field represents discrete cracks diffusely, with the smearing controlled by the length scale ℓ.This auxiliary field implicitly tracks crack interfaces.
- Fracture energy: The diffuse crack representation approximates fracture energy over the crack surface using a crack surface density functional Υ and material toughness Gc.The phase field formulation regularises the discontinuous crack surface through these quantities.
- Fatigue extension: The rate-independent fracture description is extended to time- and history-dependent fatigue through a cumulative history variable and degradation of fracture energy.The fatigue degradation function depends on the fatigue history variable ¯ϑ.
2.2. Principle of virtual power. Balance of forces
The coupled deformation-fatigue system is derived through virtual power, introducing conventional mechanical fields together with phase-field micro-force variables and their associated balances and boundary conditions.
- Balance derivation: The principle of virtual power is applied to derive local force balances and natural boundary conditions for the coupled system.The derivation uses the Gauss divergence theorem and the fundamental lemma of calculus of variations.
- Mechanical fields: The mechanical description uses the symmetric Cauchy tensor σ, boundary traction T, and prescribed body force b.These fields describe conventional forces acting on the solid.
- Phase-field micro-forces: The damage problem introduces a scalar stress-like quantity ω conjugate to φ and a micro-stress vector ξ conjugate to ∇φ.No external traction is associated with the phase field.
- Kinematics: The body kinematics are defined by the displacement and phase fields u and φ, with corresponding rates ̇u and ̇φ.
2.3. Constitutive theory
The constitutive framework combines phase-field fracture, elastic-plastic deformation, fatigue degradation, and irreversible damage evolution. It accommodates AT1, AT2, and PF-CZM formulations while incorporating elastic and plastic energy contributions and nonlinear isotropic/kinematic hardening.
- Energy and degradation: The total potential energy combines strain energy with a fracture energy depending on the phase field and its gradient.The phase field degradation function reduces solid stiffness, while no damage-plasticity coupling term is defined.
- Damage irreversibility: The phase-field evolution is irreversible, with φ̇≥0 enforced through a history field based on the fracture driving force.The effective plastic work is assumed to increase monotonically, so the history field relates to the elastic fracture driving force.
- Fatigue damage: Fatigue damage is represented through a cumulative history variable, a threshold parameter, and asymptotic or logarithmic degradation functions.The logarithmic function includes κ as a material parameter controlling its slope, and the alternatives are assessed against experiments.
- Phase-field models: The framework accommodates AT1, AT2, and PF-CZM models through distinct degradation and dissipation-function choices.AT2 uses w(φ)=φ2, AT1 uses w(φ)=φ, and PF-CZM uses w(φ)=4φ; damage thresholds are defined for AT1 and PF-CZM but not AT2.
- Energy and degradation: The fracture driving force includes both elastic and plastic strain energy densities in the considered phase-field models.This choice is used across the AT1, AT2, and PF-CZM formulations.
- Elastic-plastic deformation: The deformation model combines nonlinear isotropic and kinematic hardening to represent cyclic plasticity phenomena.Isotropic hardening changes the yield-surface size, while kinematic hardening translates the yield surface in stress space.
3. Numerical implementation
The numerical implementation discretizes displacement and phase-field variables in finite elements and solves their coupled equations with a monolithic quasi-Newton scheme. The formulation is implemented in ABAQUS through a UELMAT subroutine, with BFGS preserving symmetric positive-definite stiffness approximations.
- Finite element discretisation: The finite element implementation discretizes nodal displacements and phase-field variables using shape functions and corresponding B-matrices.The discretized variables include nodal displacement components and the phase field at each node.
- Residuals and stiffness matrices: The weak forms produce coupled residuals and consistent tangent stiffness matrices for the displacement and phase-field problems.The tangent matrices are obtained by differentiating residuals with respect to incremental nodal variables and use the elastic-plastic material Jacobian.
- Residuals and stiffness matrices: A hybrid energy split applies the strain-energy decomposition only to the phase-field balance while degrading total strain energy in momentum balance.A small κ=1×10^-7 prevents ill-conditioning when φ=1.
- Solution strategy: The coupled displacement and phase-field solutions can be obtained monolithically or through a staggered sequential approach.Staggered schemes require sufficiently small load increments because accumulated cycle errors can cause deviations from equilibrium.
- Solution strategy: A quasi-Newton scheme using BFGS provides a robust and efficient monolithic implementation for fatigue crack-growth calculations.The approximated stiffness is updated after a prescribed number of iterations without convergence rather than after every iteration.
- Solution strategy: The BFGS update retains symmetry and positive definiteness when the initial stiffness approximation has those properties.The implementation uses an approximated stiffness matrix satisfying the update relation based on changes in solution and residual vectors.
- ABAQUS implementation: The generalized model is implemented in ABAQUS through a UELMAT subroutine that defines element residuals and stiffness matrices.ABAQUS assembles global matrices and solves the system, while the material library supplies undamaged stresses and the elastic-plastic Jacobian.
4. Results
The framework is evaluated through uniaxial cyclic tests, Compact Tension crack-growth simulations, constitutive comparisons, and large-scale three-dimensional failure problems. Results show sensitivity to constitutive and phase-field choices, while monolithic quasi-Newton schemes provide robust, efficient cycle-by-cycle predictions.
- Uniaxial cyclic loading: The combined non-linear isotropic/kinematic hardening model provides good agreement with cyclic experiments, although differences remain at low fatigue lives.The model calibration requires combined hardening to attain a good fit with the experimental data.
- Uniaxial cyclic loading: Smaller phase-field length scales produce longer fatigue lives, while increasing Gc increases cycles to failure without noticeably changing the curve slope.The length-scale effect is linked to AT2 strength and the fatigue threshold; the Gc effect is most evident at low Gc values.
- Uniaxial cyclic loading: The logarithmic degradation function provides flexibility for matching experimental fatigue responses, and larger κ values increase sensitivity to strain amplitude.Higher fatigue thresholds also produce longer fatigue lives for the values considered.
- Compact-Tension crack growth: AT1 and AT2 predict similar crack-growth rates, but AT1 delays initiation and produces more cycles to failure because its strength is 935 MPa versus 496 MPa for AT2.The comparison uses the same Gc and ℓ values.
- Compact-Tension crack growth: PF-CZM initiation is largely insensitive to σc, whereas lower strength values produce larger fatigue lives; AT2 and PF-CZM predictions coincide for σc = 500 MPa versus σc = 496 MPa.The PF-CZM model nevertheless predicts different fatigue crack-growth rates as strength changes.
- Constitutive comparisons: Kinematic hardening changes early-cycle force and damage responses through the Bauschinger effect, while damage reduces differences between constitutive models later.Combined and non-linear kinematic hardening produce faster damage rates, whereas purely isotropic hardening underestimates damage.
- Three-dimensional applications: The framework predicts fatigue-damage crack nucleation, growth, and complete failure in three-dimensional settings with arbitrary geometries and dimensions.The demonstrations include large-scale structural failure and several two- and three-dimensional boundary value problems.
5. Conclusions
The paper presents a unified phase field framework for fatigue crack initiation and growth in elastic-plastic solids, combining multiple fracture models, fatigue degradation functions, cyclic hardening descriptions, and a monolithic solver. Case studies show agreement with carbon-steel experiments, distinct predictions across phase field models, improved computational efficiency, and applicability to complex geometries and cyclic responses.
- Generalised formulation: The formulation unifies AT1, AT2, and PF-CZM phase field models with asymptotic and logarithmic fatigue degradation functions driven by elastic and plastic fields.Material cyclic response is represented using combined nonlinear kinematic/isotropic hardening for a broad class of elastic-plastic materials.
- Numerical framework: The finite element implementation solves the coupled equations monolithically with a quasi-Newton algorithm and evaluates several 2D and 3D boundary-value problems.The studies examine the roles of phase field choices and the interaction between fatigue damage and cyclic hardening.
- Constitutive comparisons: The model agrees well with carbon-steel experiments, while the logarithmic degradation function better captures damage scaling at the cost of one additional parameter.This added flexibility improves matching to the reported experimental behaviour.
- Constitutive comparisons: Fatigue crack predictions differ between AT1/AT2 and PF-CZM models, with PF-CZM predicting higher growth rates as material strength σc increases because its degradation function depends on σc.The comparison shows that constitutive model selection affects both crack initiation and propagation predictions.
- Numerical framework: The quasi-Newton monolithic scheme is more robust and efficient than staggered approaches, which require over 25 times more load increments to match the monolithic result.The computational difference is especially relevant for cycle-by-cycle fatigue predictions.
- Cyclic deformation and applicability: Neglecting kinematic hardening in a crack-driving force containing elastic and plastic contributions is non-conservative and can significantly underestimate fatigue crack growth.The framework therefore captures interactions between damage and cyclic deformation effects in arbitrary dimensions and geometries.