Phase field cohesive zone modeling for fatigue crack propagation in quasi-brittle materials

TL;DR

This work addresses fatigue crack propagation in quasi-brittle materials such as concrete by extending the phase field cohesive zone model (PF-CZM) to include fatigue degradation within Griffith’s energy balance, complemented by a history field to enforce irreversibility. It develops a regularized PF-CZM framework with a geometric crack surface density, an energetic degradation function, and a fatigue accumulation mechanism, implemented in ABAQUS via UMAT/HETVAL and solved with a staggered scheme. The authors validate the approach against 2D and 3D problems across mode I and mixed-mode loadings, supported by experimental programs for calibration and evaluation of S–N curves and Paris-law behavior; findings show the model can reproduce fatigue crack growth, fatigue creep curves, and complex crack topologies, while highlighting limitations in low-cycle S–N predictions and hysteresis loops. The study’s significance lies in providing a physics-based, unified tool for fatigue analysis in quasi-brittle materials, with potential extensions to multi-scale coupling and plasticity to enhance predictive accuracy for design applications.

Abstract

The phase field method has gathered significant attention in the past decade due to its versatile applications in engineering contexts, including fatigue crack propagation modeling. Particularly, the phase field cohesive zone method (PF-CZM) has emerged as a promising approach for modeling fracture behavior in quasi-brittle materials, such as concrete. The present contribution expands the applicability of the PF-CZM to include the modeling of fatigue-induced crack propagation. This study critically examines the validity of the extended PF-CZM approach by evaluating its performance across various fatigue behaviours, encompassing hysteretic behavior, S-N curves, fatigue creep curves, and the Paris law. The experimental investigations and validation span a diverse spectrum of loading scenarios, encompassing pre- and post-peak cyclic loading, as well as low- and high-cyclic fatigue loading. The validation process incorporates 2D and 3D boundary value problems, considering mode I and mixed-modes fatigue crack propagation. The results obtained from this study show a wide range of validity, underscoring the remarkable potential of the proposed PF-CZM approach to accurately capture the propagation of fatigue cracks in concrete-like materials. Furthermore, the paper outlines recommendations to improve the predictive capabilities of the model concerning key fatigue characteristics.

Figures (15)

• Figure 1: Key characteristics of cyclic and fatigue behavior of concrete based on experimental observations, including the following abbreviations: force ($F$), crack mouth opening displacement (CMOD), upper level of the loading range ($S^\mathrm{max}$), number of cycles ($N$), number of cycles to fatigue failure ($N^\mathrm{f}$), crack length ($a$), and stress intensity factor ($K$)
• Figure 2: A cohesive zone-based phase field description of fracture and fatigue: a) cracking solid with a sharp crack and phase field regularization; b) geometrical function of the phase field cohesive zone model for varied coefficient $\xi$
• Figure 3: Implementation concept of the PF-CZM approach: a) flow chart of the subroutine for implementing a coupled deformation-phase field model utilizing the analogy to heat transfer, b) flowchart for the solution of the phase field at each integration point for a given increment using a staggered scheme
• Figure 4: Test setup and geometry of the notched three-point bending tests of concrete (mode I crack propagation)
• Figure 5: Monotonic and post-peak cyclic response of concrete under mode I crack propagation: a) experimental response (including three monotonic tests (LS1) and one cyclic test (LS2))