PSI - Issue 13

N.A. Giang et al. / Procedia Structural Integrity 13 (2018) 45–50 Giang N.A. / Structural Integrity Procedia 00 (2018) 000–000

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2. Scalar gradient plasticity

Within implicit gradient-enriched theory by Peerlings (2007), non-local scalar variable ε p is introduced as coun terpart to the equivalent plastic strain ε p . The nonlocal equivalent plastic strain ε p is introduced implicitly via the Helmholtz-type PDE ε p − ε p = 2 ∇ 2 ε p . (1) Therein, ∇ 2 refers to the Laplace operator whose prefactor introduces as an intrinsic material length scale. Equa tion (1) can be interpreted that a di ff erence between ε p and ε p is a source of gradients of ε , and vice versa. In this sense, a gradient-enriched formulation of J 2 plasticity can be constructed by making the flow stress σ f dependent on ε p : σ f = σ y ( ε p ) − 3 2 h · ε p − ε p . A Ramberg-Osgood power law is employed for the flow curve σ y ( ε p ), which is thus given in implicit form as σ y σ 0 = σ y σ 0 + E σ 0 ε p N . Furthermore, PDE (1) requires to formulate suitable boundary conditions. This can be a micro-hard boundary ε p = 0 or a micro-free boundary n · ∇ ε p = 0, respectively. Based on the formal similarity of Eq. (1) and the stationary heat equation, Seupel et al. (2018) developed a technique to implement the Helmholtz PDE (1) into the commercial FEM code Abaqus / Standard via built-in thermomechanical elements. This technique is employed here as well. For details on the FEM implementation, the reader is refered to (Giang et al., 2018).

3. Micromechanical Model

3.1. Cell model

σ c t

Uz

Uz

Uz

Dissipated energy

z

Microhard �ε p = 0

Microhard �ε p = 0

2H 0

Γ 0

r

2a

Grain-1

Grain-2

Exponential law

2R 0 2b

Carbide particle

δ 0

R g1

a )

b )

Symmetric crack plane

b

Uz

δ

Grain-1

Fig. 2: Axisymmetric unit cell model under uniaxial loading: a) carbide at grain boundary (GB), b) carbide near GB

Fig. 3: Traction-separation law of cohe sive zone model

A microcrack in ferritic steel might initiate from a broken carbide at a grain boundary (GB) or from a debonded particle. Subsequently, the micro-crack can propagate into the ferritic matrix and finally it might also cross some GBs and attain a macroscopic size. In order to model the micro-crack initiation, an axisymmetric unit cell model under uniaxial macroscopic loading is employed. Two types of cell model are employed to investigate the initiation of cleavage micro-cracks: A cell model with a carbide located at the GB and one with a carbide arranged near GB as shown schematically in Fig. 2. The cell encompasses an ellipsoidal carbide particle embedded within ferrite grains. When two axes of elliptical shape are equal, the spherical shape of carbide model is obtained. According to Shibanuma et al. (2016), the shape ratio of elliptical carbide can be chosen a = 3 . 5 b whereby a and b are major axis and minor axis, respectively. Potential fracture of the ferrite by cleavage or of the carbide in both cells is modelled by cohesive zones. Thereby, it is assumed that cleavage of ferrite or of the carbide appears only along the plane of symmetry z = 0. Therefore, only a quarter of the cell needs to be incorporated in the finite element analysis. The dimensions of the cell model are chosen as H 0 = R 0 = 4 . 64 b , which is calculated from the volume fraction of carbide. The latter is chosen as V p = 1 . 4%. A uniform vertical displacement U z is applied at the top surface z = H 0 . The outer