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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3

Fig. 1. Experimental results for crack paths

G c

l 0 ∆ ϕ + g ′ ( W e

ϕ l 0 −

p )

+ W

= 0

(2)

where G c represents the fracture toughness of the material, l 0 is the length scale and ∆ ϕ is the Laplacian of the phase field. Elastic and plastic contributions are computed as

1 2

= σ eq ˙ ε p

W e

σ : ˙ ε e dt , W p

eq dt

(3)

=

where ˙ ε e is the rate of elastic strain tensor, ˙ ε p eq is the equivalent plastic strain rate and σ eq is the von Mises equivalent stress. A quadratic degradation function is used in the current model, g = (1 − ϕ ) 2 . This formulation, however, results in a gradual degradation of material as the plastic strain builds up and leads to an unphysical material response. As a solution to this, a threshold or a critical value has been introduced to the ductile phase field fracture models (Ambati et al., 2015; Borden et al., 2016; Li et al., 2022; Waseem et al., 2023) which act as a limiting factor for the crack initiation. With this addition, models can predict the sudden decrease in stress carrying capacity during the last phase of ductile fracture. In the current model, W p is replaced with < W p − W p c > where <.> is the Macaulay operator and W p c is the critical plastic energy. The addition of stress state e ff ects to the phase field framework is done through scaling the accumulation of plastic contribution to the crack driving force, as

=

p eq

σ eq ˙ ε

W p

dt

(4)

ε f

where ε f is the failure strain in the classical uncoupled ductile damage models. In this work MMC model formulation is used for ε f in the PF model. In addition to the phase field fracture model, ductile failure is simulated with uncoupled ductile failure criteria and element deletion to demonstrate the mesh dependency of the solution. JC and MMC failure criteria are used in this study, and the damage is assumed to accumulate linearly with the following relation: D = ¯ ε p 0 d ¯ ε p ε f . (5)