[Paper Review] A generalised phase field model for fatigue crack growth in elastic-plastic solids with an efficient monolithic solver

Zeyad Khalil, A.Y. Elghazouli+1 more|arXiv (Cornell University)|Oct 20, 2021

Numerical methods in engineeringEngineering84 references143 citations

TL;DR

This paper presents a generalized phase field model for fatigue crack growth in elastic-plastic metals, integrating AT1, AT2, and phase field-cohesive zone models with nonlinear isotropic and kinematic hardening. It introduces a monolithic quasi-Newton solver that significantly improves computational efficiency over staggered schemes, enabling accurate prediction of crack initiation and propagation in complex 2D and 3D geometries under cyclic loading.

ABSTRACT

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.

Motivation & Objective

• To develop a unified computational framework for predicting fatigue crack growth in elastic-plastic metals across high-, low-, and ultra-low-cycle fatigue regimes.
• To incorporate both brittle (AT1, AT2) and quasi-brittle (PF-CZM) phase field fracture models within a single formulation.
• To enable accurate simulation of cyclic deformation using combined nonlinear isotropic and kinematic hardening laws.
• To overcome the computational inefficiency of staggered solution schemes by introducing a monolithic quasi-Newton solver.
• To validate the model against experimental data and demonstrate its predictive capability on complex 3D engineering components.

Proposed method

• Formulates a generalized phase field fracture model using a scalar damage field to regularize crack topology.
• Integrates fatigue degradation functions (asymptotic and logarithmic) that depend on elastic and plastic strain energy densities.
• Employs a variational framework based on the Ambrosio-Tortorelli functional for AT1 and AT2 models, and a cohesive zone-inspired formulation for PF-CZM.
• Models cyclic plasticity using combined nonlinear isotropic and kinematic hardening with Bauschinger effect.
• Solves the coupled system of equations using a monolithic quasi-Newton algorithm to improve convergence and reduce computational cost.
• Implements the framework in a finite element code using 10-node tetrahedral elements and adaptive mesh refinement near crack regions.

Experimental results

Research questions

• RQ1 How do different fatigue degradation functions (asymptotic vs. logarithmic) affect the prediction of crack growth under cyclic loading?
• RQ2 What is the influence of phase field fracture model choice (AT1, AT2, PF-CZM) on fatigue crack growth rates and failure prediction?
• RQ3 How does the inclusion of kinematic hardening and the Bauschinger effect affect damage accumulation and crack growth in cyclic loading?
• RQ4 To what extent does a monolithic quasi-Newton solver outperform traditional staggered schemes in terms of efficiency and robustness for cycle-by-cycle fatigue simulations?
• RQ5 Can the framework predict crack nucleation and propagation in complex 3D geometries without initial defects?

Key findings

• The logarithmic degradation function provides better agreement with experimental data on carbon steel than the asymptotic function, though it requires an additional material parameter.
• The PF-CZM model predicts higher crack growth rates with increasing material strength σc due to its σc-dependent degradation function, unlike AT1/AT2 models.
• The monolithic quasi-Newton solver reduced the number of load increments required to reach convergence by more than 25 times compared to staggered schemes.
• Neglecting kinematic hardening leads to a significant underestimation of damage, particularly in the early cycles, due to the absence of the Bauschinger effect.
• The model successfully predicted crack nucleation and propagation in a 3D pipe-to-pipe connection without initial defects, demonstrating its capability for real-world engineering applications.
• The framework accurately captured the evolution of force-displacement and force-cycle responses, showing strong sensitivity to hardening mechanisms in the early stages of fatigue.