PSI - Issue 68

I. Yucel et al. / Procedia Structural Integrity 68 (2025) 1287–1293 Yucel et al. / Procedia Structural Integrity 00 (2024) 000–000

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refined. Thus, such methods demonstrate high mesh dependency in the simulations of the crack paths and in load displacement curves of the materials as demonstrated in Pijaudier-Cabot and Bazˇant (1987), Huespe et al. (2009), Tuhami et al. (2022). This raises the need for an appropriately sized and designed mesh structure in the region of ex pected crack initiation and propagation (Besson et al., 2001). As a solution, non-local methods have been developed. The non-local formulation of damage can stem from the coupling of the damage variable with the non-local strain through an integral formulation (Pijaudier-Cabot and Bazˇant, 1987). Di ff erential formulations or gradient forms have also been used in the implementation of non-locality to fracture models (Geers et al., 1998). Another way to address mesh dependency is proposed in Huespe et al. (2009). A weak discontinuity band of finite thickness is introduced through additional functions once the constituent equations lose their ellipticity, and the problem becomes ill-posed. Plastic deformation is allowed to happen inside the band, and the method proves to overcome mesh refinement prob lems. Phase field (PF) fracture framework, essentially a non-local damage model, has been applied in the context of ductile fracture recently. Initially developed for brittle fracture, this method is extended to cover many failure mech anisms such as dynamic crack growth (Borden et al., 2012), fatigue fracture (Carrara et al., 2020; Waseem et al., 2024), corrosion cracking (Cui et al., 2021) and hydrogen assisted failures (Cui et al., 2024). In PF method, cracks are defined as non-local di ff use entities, and the model has shown great success in the simulation of the formation and propagation of complex crack patterns as well as the branching and merging of cracks. The extension of this model to ductile fracture includes the consideration of total energy as the sum of elastic, plastic and fracture energy contri butions (Ambati et al., 2015; Borden et al., 2016; Li et al., 2022). How plasticity is coupled, the inclusion of various forms of degradation functions, and the introduction of stress-state dependency are key aspects that di ff erentiate the various variants of phase field models for ductile failure. This study utilizes the phase field fracture framework developed in Waseem et al. (2023) for ductile fracture sim ulations of Inconel 718 to address the mesh dependency issues in the uncoupled local modeling approach. The phase field model has been coupled with the MMC model to account for stress-state on ductile failure predictions with the parameters calibrated in Erdogan et al. (2023). Di ff erent mesh configurations for in-plane shear (ISS) and plane strain tension (PST) specimens are created with varying element sizes and orientations in the region of expected crack propagation. The results for the crack paths and load-displacement curves are compared with the experimental ones. The analyses are performed using the finite element software Abaqus. Uncoupled ductile failure simulations utilize Abaqus / Explicit solver while the phase field fracture model is performed with Abaqus / Standard, and the models are implemented through user subroutines for both methods.

2. Method

The phase field fracture and the classical uncoupled ductile fracture models are used to perform ductile failure simulations in this work. The Johnson-Cook (JC) and the modified Mohr-Coulomb (MMC) failure models are both employed, whose parameters have been calibrated in Erdogan et al. (2023) for Inconel 718. The work is focused on the two tensile test specimens, plane strain tension (PST) and in-plane shear (ISS), as shown in Fig. 1, together with the experimental crack patterns. The details of the specimen geometries and experimental work can be obtained from the aforementioned article.

2.1. Ductile phase field fracture model

The ductile phase field fracture model originates from the addition of plastic contribution to the brittle formulation presented and implemented for Abaqus in Navidtehrani et al. (2021). The formulation of the phase field fracture model can be seen in detail in Waseem et al. (2023). The plastic contribution to the total energy, E p , is formulated as E p ( ϕ,ε p ) = g ( ϕ ) W p ( ε p ) (1)

where ϕ is the phase field parameter, g is the degradation function which is identical for elastic and plastic parts of the energy and W p is the crack driving energy due to plastic deformations. This leads to the following strong form for the phase field problem: