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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Table 1: Material properties for the cell model (Kroon and Faleskog, 2005; Giang et al., 2018)

Material

E /σ 0

N

/ b 0.4

Γ 0 / ( σ 0 b )[-]

ν

σ c /σ 0 2 . . . 24 2 . . . 24

Ferrite (the matrix)

333 666

0.3 0.3

0.1

1.05

Carbide

-

-

0.525

surface r = R 0 is assumed to remain cylindrical but without resulting radial loading, compare e. g. Steglich and Brocks (1998). Regarding the plasticity model, micro-free boundary conditions are applied at the surface of the cell. Micro hard interface conditions are applied at the grain boundaries and at the carbide-ferrite interface in order to account for the dislocation pile-up. Dynamic simulations with quasi-static loading are performed to deal with potentially unstable propagation of microcracks, cf. Giang et al. (2018).

3.2. Cohesive zone model

The cohesive zone model developed by Roth et al. (2014) is used for two types of potential failure (carbide cracking and cleavage in the ferrite). Under loading, the traction-separation relation follows an exponential curve

exp 1 −

δ δ 0

δ δ 0

(2)

t = σ c

.

However, beyond the cohesive strength σ c as maximum transmittable stress at δ = δ 0 , the unloading path towards the origin deviates from Eq. (2). This behaviour represents dissipation of energy during the separation process as illustrated in Fig. 3. The work of separation Γ 0 = exp(1) δ 0 σ c corresponds to the area under the loading curve. The cohesive zone is implemented in Abaqus / Standard via the UEL interface as a user-defined element (Roth et al., 2014). Subscripts () M , () P , () PM are appended to the cohesive parameters σ c and Γ 0 to indicate whether respective values refer to cleavage of the matrix, breakage of the particle or of the interface between particle and matrix, respectively. The exact experimental determination of the cohesive parameters is di ffi cult. Here, the values Γ 0PM = Γ 0M = 2 Γ 0P from literature (Kroon and Faleskog, 2005; Li and Zhou, 2013) and σ cPM = σ cP (Giang et al., 2018) are adapted. The influence of the cohesive strengths σ cM and σ cP and σ cPM shall be investigated within a parameter study. The nonclassical parameters h and are adopted from (Giang et al., 2018) as well. The used material properties are given in Table 1. 4. Results and discussion The distance of a carbide to a grain boundary is highly relevant for the microcrack initiation. In the present model, this distance is reflected by R g 1 , compare Fig. 2b. Figure 4 shows the predictions of the model for various values of R g 1 (relative to b ). The results in Fig. 4a show that the limit load, i. e. the macroscopic strength, is lowest if the particle is located at the grain boundary. That means that a carbide particle at the grain boundary is easy to break while a particle far away from the grain boundary requires higher loadings to break. The reason is that when a carbide is located at or near a grain boundary, the dislocations pile-ups at the interface of carbide-matrix and at the grain boundary interact with each other. In turns, the stress concentration around carbide increases and triggers cracking of the particle. If the carbide is far away from the grain boundary, the mechanism of micro-crack initiation in cell model includes three distinct stages as can be seen in Fig. 4b: firstly, local stresses increase, both, on interface of carbide-matrix and at the grain boundary due to dislocation pile-up (stage 1 ). When the principal stress exceeds the (cohesive) strength σ cP of the carbide or σ cM of the matrix, it triggers cracking of the carbide or cleavage of the matrix, respectively. 3.3. Material parameters 4.1. Influence of carbide position and carbide shape on micro-crack initiation