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

14

3

2. Methodology

2.1. Constitutive Modelling

The ferrite phase is characterized using a rate-dependent crystal plasticity framework that e ff ectively links mi croscale crystallographic slip to macroscopic deformation measures. An example application of this framework can be seen in studies such as Acar et al. (2021) and Gu¨nay et al. (2023). Within this framework, the primary contributors to macroscopic deformation are identified as elastic lattice distortion and crystallographic slip, which originates from dislocation motion on active slip systems. In the context of the finite strain formulation utilized in this study, the defor mation gradient is multiplicatively decomposed into its elastic and plastic components. The velocity gradient tensor is used to calculate the rate of plastic deformation on each slip system, which is related to the shear stress acting on the slip system. The flow equation governing the evolution of the rate of plastic slips is described using the classical power law relation, α is the slip resistance on the slip system α which governs the hardening of the single crystal and n represents 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 (2) where h αβ represents the latent hardening matrix, h αα stands for the self-hardening rate (with α β , and there is no summation over α ). A simplified form is used for their evolution, h αα = h 0 sech 2 h 0 γ g s − g 0 , h αβ = q αβ h αα ( α β ) . (3) Here, g 0 denotes the initial slip resistance, g s represents the saturation value of slip resistance, h 0 is the initial hardening modulus, and the term q αβ includes latent hardening coe ffi cients. The equations mentioned above are integrated into ABAQUS software as a user material subroutine following a framework that utilizes an implicit approach for incre mental stress updates. In the crystal plasticity calculations for the ferrite phase, the slip family 112 ⟨ 111 ⟩ is considered active. The material parameters for the ferrite phase are determined with CPFEM simulations of the polycrystalline RVEs by utilizing the experimental data presented in Lai et al. (2015) to fit the flow curves obtained from the simu lations, see Yalcinkaya et al. (2019) for more details of the parameter identification study. The elastic constants, C 11 , C 12 , C 44 , are adopted from Woo et al. (2012). Table 1 provides a summary of the elastic constants and parameters for the crystal plasticity model. ˙ γ α = ˙ γ 0 τ α g α n sign( τ α ) (1) where ˙ γ 0 stands for the reference slip rate, g

Table 1. Crystal plasticity parameters of the ferrite phase. C 11 (GPa) C 12 (GPa) C 44 (GPa)

1 )

˙ γ 0 (1 / s −

Slip Systems { 112 }⟨ 111 ⟩

g s (MPa)

g 0 (GPa)

h 0 (MPa)

n

231.4

134.7

116.4

252

98

75

0.001

10

Constitutive behavior of the martensite phase is defined by the conventional J 2 plasticity theory with isotropic hard ening as proposed in Pierman et al. (2014), σ y , m = σ y 0 , m + k m 1 − exp − ε p eq , m n m , σ y 0 , m = 300 + 1000 C 1 / 3 m ,

(4)

) q

1 n m

bC m

a +

k m

.

=

Cm C 0

1 + (