PSI - Issue 42

Can Erdoğan et al. / Procedia Structural Integrity 42 (2022) 1643 – 1650 Erdog˘an et al. / Structural Integrity Procedia 00 (2019) 000–000

1645

3

2. Methods

2.1. Material

AISI 4340 steel in HRC 16 hardness condition is considered in the numerical analysis. This is a medium carbon high strength steel alloy frequently used in gears and shafts in the aerospace and automotive industries. The material is modelled as isotropic elasto-plastic, and the constitutive model is governed by the classical von Mises plasticity framework. The yield stress of the material is described by the Johnson-Cook plasticity as σ y = A + B ( ¯ ε p ) n 1 + Cln ˙¯ ε p ˙ ε 0 1 − ˆ θ (1) where σ y is the yield stress, ¯ ε p is the equivalent plastic strain, ˙ ε 0 is the reference strain rate and A , B , C and n are material specific constants. ˆ θ is given by the following relation ˆ θ =   0 , if θ < θ trans ( θ − θ trans ) / ( θ melt − θ trans ) , if θ trans ≤ θ ≤ θ melt 1 , if θ > θ melt (2) where θ trans is the transition temperature and θ melt is the melting temperature of the material. Inelastic heat fraction ( χ ) and specific heat ( C p ) parameters are used for adiabatic heating e ff ects. The material parameters are adopted from Ghazali et al. (2020) and Johnson and Cook (1985) as shown in Table 1.

Table 1: Material parameters.

E (GPa)

A (MPa)

B (MPa)

n

C

˙ ε 0

θ trans (K)

θ melt (K)

C p (J / kg K)

ν

χ

208

0.29

250.9

624.5

0.2474

0.0014

1

298

1793

0.9

477

2.2. Damage Model

Modified Mohr-Coulomb model (Bai and Wierzbicki, 2010) is a widely used stress triaxiality and Lode angle dependent damage model used for the FE simulations of ductile failure of metals. The model is extended to include strain rate and temperature e ff ects. The failure criteria in extended form is defined as ε f =   ˆ A ˆ C 2   ˆ C 3 + √ 3 2 − √ 3 ˆ C 4 − ˆ C 3 sec ¯ θπ 6 − 1     ×    1 + ˆ C 2 1 3 cos ¯ θπ 6 + ˆ C 1 T + 1 3 sin ¯ θπ 6    − 1 / n × (3)

1 + ˆ C 5 ln

˙¯ ε p ˙ ε 0 ×

1 + ˆ C 6 ˆ θ

where T is the stress triaxiality, ¯ θ is the Lode angle parameter and ε f is the strain value at failure. T and ¯ θ are the two dimensionless parameters that define the stress state and expressed as

2

3 J 2

3 / 2

6 θ L π

J 3

σ m σ eq

, ¯ θ = 1 −

, and cos(3 θ L ) =

(4)

T =

where the mean stress is σ m = tr ( σ ) / 3 = ( σ 11 + σ 22 + σ 33 ) / 3, σ eq is the von-Mises equivalent stress and θ L is the Lode angle. J 2 and J 3 are defined as the second and the third deviatoric stress invariants, respectively. The MMC model parameters are tabulated in Table 2. Damage evolution rule is expressed with the following integral