PSI - Issue 21

Tuncay Yalçinkaya et al. / Procedia Structural Integrity 21 (2019) 61–72 T. Yalc¸inkay et l. / Structur l Integrity Procedia 00 (2019) 000–000

63

(a)

(b)

(c)

(d)

Fig. 1: Artificially generated dual-phase steel microstructures that belong to DP1 (a), DP2 (b), DP3 (c), and DP4 (d).

The parameters for the ferrite phase are taken from Lai et al. (2016) and presented below in Table 2, where σ y , f is the current yield stress, σ y 0 , f is the initial yield stress, α f and β are parameters that are related to the average ferrite grain size. θ f is the initial, θ IV is the stage-IV hardening rate and it is taken to be 100 MPa for all the steels investigated in this study. Finally, σ tr y and ε tr P respectively represent the flow stress and the plastic strain. Table 2: Parameter set used for ferrite flow curves(Lai et al. (2016)).

Steel

σ y 0 , f ( MPa )

α f ( GPa )

β ( GPa )

θ f ( MPa )

DP1 DP2 DP3 DP4

250 279 300 307

4.9

11 13 17 20

4895 5980 8925

6

8.9

10.3

10260

The flow behavior of the martensitic phase is governed by the phenomenological equations and parameter sets given by Pierman et al. (2014): σ y , m = σ y 0 , m + k m (1 − exp ( − ε P n m )) (4) where σ y , m is the current yield stress, ε P is the accumulated plastic strain, and σ y 0 , m , k m , n m are material parameters. C m is the martensite carbon content in wt%, whose e ff ect on strain hardening is given below σ y 0 , m = 300 + 1000 C 1 / 3 m . (5) The hardening modulus k m reads

1 n m   a +

q  

bC m 1 + ( C m

k m =

(6)

C 0 )

with a = 33 GPa, b = 36 GPa, C 0 = 0.7, q = 1.45, n m = 120, C m = 0.3 wt%. For the second numerical approach, the crystal plasticity constitutive model is assigned to ferrite phase (see Huang (1991)) while martensite is still governed by the J2 plasticity with isotropic hardening. The plastic slip rate in each slip system, ˙ γ ( α ) , is obtained through a classical power law relation,

0 � � � � � � τ ( α )

g ( α ) � � � � � � 1 / m h αβ � � � ˙ γ β � � �

˙ γ ( α ) = ˙ γ

sign( τ ( α ) )

(7)

where, τ ( α ) is the resolved Schmid stress on the slip systems, ˙ γ

( α ) is the slip resistance on

0 is the initial slip rate, and g

each slip system, which governs the hardening of the material and evolves according to

n ∑ β = 1

˙ g ( α ) =

(8)