PSI - Issue 42

Tuncay Yalçinkaya et al. / Procedia Structural Integrity 42 (2022) 1651–1659 Tuncay Yalc¸inkaya et al. / Structural Integrity Procedia 00 (2019) 000–000

1653

3

to obtain the Schmid’s resolved shear stress, τ α , which drives the crystallographic slip. In the current rate-dependent crystal plasticity formulation, the slip rates are directly related to the immediate resolved shear stress. If the resolved shear stress is higher than the current slip resistance, g α in a slip system α , the slip is activated. This means that all the slip systems are potentially active, yet, the amount of the plastic slip depends on the value of the projected stress. The classical power law relation is used for the flow equation governing the evolution of the rate of plastic slips,

g α 1 / m

0 τ α

sign( τ α ) ,

˙ γ α = ˙ γ

(4)

where ˙ γ 0 is a reference slip rate, g α is the slip resistance on the slip system α which governs the hardening of the single crystal, m is the rate sensitivity parameter. Strain hardening is related to the evolution of the slip resistance on the slip systems through ˙ g α = β h αβ ˙ γ β , (5) with h αβ and h αα representing respectively the latent hardening matrix and the self-hardening rate ( α β , and no summation on α ) and, for which a simple form is used here

h αα = h g s − g 0 , h αβ = q αβ h αα ( α β ) . 0 sech 2 h 0 γ

(6)

where g 0 is the initial slip resistance and g s is the saturation value of slip resistance. The above equation sets are implemented as a user material subroutine in ABAQUS software based on the framework of Huang where the in cremental stress update is obtained implicitly (see Huang (1991)). { 112 } 111 slip family is used in crystal plasticity calculations of ferrite phase (see e.g. Yalc¸inkaya et al. (2008) and Yalc¸inkaya et al. (2009) for BCC crystal plasticity). Cubic elastic parameters are taken from Woo et al. (2012) as C 11 = 231 . 4 GPa, C 12 = 134 . 7 GPa and C 44 = 116 . 4 GPa. The reference slip rate ˙ γ 0 is used as 0 . 001 and rate sensitivity exponent m is determined to be 25. The ratio of latent hardening with respect to the self hardening in q αβ matrix is taken to be unity. The hardening parameters for ferrite phase are identified through a polycrystalline pure ferrite RVE (see e.g. Yalc¸inkaya et al. (2019)) and tabulated in Table 1.

Slip Systems { 112 } 111

g s (MPa)

g 0 (GPa)

h 0 (MPa)

252

98

75

Table 1: Crystal plasticity parameters for ferrite phase.

The martensite phase deforms according to the classical J 2 plasticity theory with isotropic hardening as presented in Pierman et al. (2014): σ y , m = σ y 0 , m + k m 1 − exp ( − ε P n m ) , σ y 0 , m = 300 + 1000 C 1 / 3 m , k m = 1 n m   a + bC m 1 + Cm C 0 q   , (7) where ε P is the accumulated plastic strain, σ y , m , C m , and k m are respectively the current yield strength, carbon content which is assumed to be constant and equal to 0.3 wt%, and hardening modulus of the martensite phase. Remaining hardening parameters are also taken as the values given in Pierman et al. (2014) as shown in Table 2. Moreover, the Young’s modulus and Poission’s ratio for martensite are taken to be E = 210 GPa and v = 0 . 3, respectively.