PSI - Issue 61

Berkehan Tatli et al. / Procedia Structural Integrity 61 (2024) 12–19 B. Tatli et al. / Structural Integrity Procedia 00 (2024) 000–000

15

4

Here, σ y , m represents the current yield strength, ε p eq , m denotes the accumulated plastic strain, and C m corresponds to the carbon content expressed as a weight percentage (wt.%). Material parameters that define the given constitutive relation are initial yield stress ( σ y 0 , m ), hardening modulus ( k m ), and hardening exponent ( n m ). Throughout this study, elastic constants ( E , ν ) and the hardening parameters of the martensite phase are adopted from the values presented in Pierman et al. (2014) and summarized in Table 2.

Table 2. J 2 plasticity parameters of the martensite phase. E (GPa) ν C m (wt.%)

C 0 (wt.%)

n m

a (GPa)

b (GPa)

q

210

0.3

0.3

0.7

120

33

360

1.45

2.2. Phase Field Fracture Model

Failure is modeled through the phase field fracture framework initially developed for brittle materials and extended to model many other failure phenomena, including ductile fracture. This framework is particularly useful for the current application since crack initiation and propagation can be simulated without any extra criteria. Moreover, since the crack is defined with an extra field variable, it can nucleate anywhere in the model and propagate through di ff erent phases of the material, simulating transgranular cracking. The coupling of crystal plasticity for the ferrite phase and J2 plasticity for the martensite phase with the phase field fracture model is performed through the addition of the plastic dissipation to the crack driving force. The total energy description is as follows, E total = E s ( ϕ, ∇ ϕ ) + E e ( ε e ,ϕ ) + E p ( ε p ,ϕ ) (5) where E s , E e , and E p represent the crack surface, elastic, and plastic energy components over the domain, respectively. ϕ is the phase field parameter, ε e is the elastic strain and ε p is the plastic strain. Crack surface energy is approximated through an integral over the entire domain, Ω , rather than the discrete crack surface, Γ , as, E s = G c d Γ ≈ G c ˆ γ d Ω (6) G c is the fracture toughness of the material while ˆ γ is referred to as the crack surface density function given by, where l 0 is the length scale governing the di ff usiveness of the crack. The elastic component of the functional is described as follows, E e = ψ e ( ϕ,ε ) d Ω= g ( ϕ ) ψ 0 ( ε ) d Ω (8) where g ( ϕ ) is the degradation function taken as g = (1 − ϕ ) 2 and ψ 0 is the undegraded bulk energy density given by, ψ 0 = ε e : C : ε e (9) where ε e is the elastic strain tensor and C is the elastic tangent modulus. A similar definition is adopted for the plastic counterpart of the energy functional, E p = g ( ϕ ) W p ( ϕ,ε ) d Ω (10) where the definition of W p essentially represents the damage contribution as a result of plastic deformation. The following definition is used to couple the crystal plasticity and phase field models, ˙ W CP p = α τ ( α ) ˙ γ ( α ) (11) ˆ γ = ϕ 2 2 l 0 + l 0 2 |∇ ϕ | 2 (7) 1 2