PSI - Issue 61

Orhun Bulut et al. / Procedia Structural Integrity 61 (2024) 3–11

6 4

O. Bulut et al. / Structural Integrity Procedia 00 (2024) 000–000

Table 2. Crystal plasticity model parameters Parameters Basal < a >

Prismatic < a >

Pyramidal < a + c >

g 0 (MPa) g s (MPa) h 0 (MPa)

213 338 180

212 413 180

427

1127

180

˙ γ 0

0.0023

0.0023

0.0023

n

30

30

30

E total = E s ( ϕ, ∇ ϕ ) + E e ( ε e ,ϕ ) + E p ( ε p ,ϕ )

(6)

where E s , E e and E p represent the crack surface, elastic and plastic energy 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 (7) G c is the fracture toughness of the material while ˆ γ is referred to as the crack surface density function, providing a description of the crack surface over the domain in terms of the phase field parameter, as where l 0 is the length scale governing the di ff usiveness of the crack. Crack surface density function has a dependence on ∇ ϕ and hence incorporates non-local e ff ects. 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 (9) where g ( ϕ ) is the degradation function, which is the central coupling mechanism of the framework. ψ 0 is the unde graded bulk energy density given by: ˆ γ = ϕ 2 2 l 0 + l 0 2 |∇ ϕ | 2 (8)

1 2

ε e : C : ε e

(10)

ψ 0 =

A similar definition is employed for the plastic component of the energy functional: E p = g ( ϕ ) W p ( ε ) d Ω

(11)

where the definition of W p essentially represents the damage contribution as a result of plastic slip. To scale with the elastic damage, an energy representation of the plastic contribution is essential. The following definition is proposed: ˙ W p = α τ ( α ) ˙ γ ( α ) . (12) This is essentially a representation of the cumulative dissipated energy due to plastic slip on every slip system. One of the features of the current implementation of the phase field model in comparison to a brittle model is that the fracture toughness is several scales higher for a ductile material while damage accumulation is relatively slower. This translates to unrealistic material responses with early peaks and very gradual evolution of the phase field parameter. This necessitates additional controls to ensure a more accurate material response. An example of such a