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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4

In the present parameter set, the strength of carbide σ cP = 2 σ 0 is smaller than that of the matrix σ cM = 4 σ 0 , and a micro-crack initiates from the carbide (stage 2 ). Subsequently, the crack grows into the grain and quickly crosses the grain boundary due to the interaction of dislocation pile-up between the interface of carbide-matrix and grain boundary. After the microcrack crossed over the grain boundary, the dissipation of plastic strain energy at the tips of the growing micro-crack leads to a limited macroscopic stress recovery (stage 3 ). The ferritic matrix might also arrest nucleated cracks. Further loading is necessary to drive the microcrack through the elastic-plastic second grain (stage 4 ). Finally, the crack propagates dynamically through the ligament of the second grain until complete separation of the material. To investigate the influences of carbide shape on micro-crack initiation, models with an ellipsoidal carbide and spherical carbide ( a = b ) are generated, both located at the grain boundary. Figure 5 shows the gradient plasticity model with micro-hard interface conditions predicts realistically that the micro-crack initiates from the ellipsoidal carbide at a relatively low amplitude of the macro strain E z = 0 . 04. In contrast, a spherical carbide is only fractured at high amplitude of macroscopic strain E z = 0.18. The simulation results imply that the shape ratio of carbide plays an importance role in micro-crack initiation. This factor e ff ect is not obtained with classical plasticity for which a micro-crack does neither initiate in the model with elliptical carbide nor with a spherical carbide, even at a relatively low cleavage strength of the matrix σ cM /σ 0 = 2 . 0.

4.2. Influence of cohesive strength on micro-crack mechanism

1

3

The influences of the strengths of the particle σ cP and of the matrix σ cM shall be investigated by means of sensitivity studies. Firstly, the strength of the matrix is fixed at σ cM /σ 0 = 4 . 0 and the strength of the particle σ cPM is varied (relative to yield stress σ 0 ). Corresponding macroscopic stress-strain curves are plotted in Fig. 6. It shows that the micro-crack does not initiate in a spherical carbide particle if the strength σ cP is greater than 6 times of the initial u z Σ z / σ 0 2

E z

4

σ I / σ 0 8.20 6.52

a )

1

3

2

4

4.84 3.16 1.50

1

3

-1. 80 -0.20

Σ z / σ 0

u z

2

E z

4

Grain 2

Grain 1

σ I / σ 0 8.20 6.52

b )

a )

1

3

2

4

Gradient plasticity u z Fig. 4: Influence of particle position: a) macroscopic stress-strain relation, b) micro-crack mechanism ( R g 1 / b = 2 . 3) 1 3 cracks arrested 5 E z = 0.04 1 2 Grain 1 -0.52 1 Grain 2 Grain 1 E z = 0.04

σ I / σ 0 53.6 44.3

3 σ I / σ 0

4.84 3.16 1.50

E E z = 0.04 1

E z = 0.04

24.7 20.4 16.2 11.6 7.30

3

Grain 2

-1. 80 -0.20

35.1 25.7 16.4

1

1

σ z / σ 0

3

-2. 20 7.20

-1. 45 2.92

Grain 2

Grain 1

Elliptical carbide E z = 0.04

Spherical carbide E z = 0.18

b )

E z

3

cracks arrested 5 E z = 0.04

3

Gradient plasticity

Grain 1 E z = 0.04

Grain 2

Grain 1

u z

1

E E z = 0.04 u z 1

2

E z = 0.04

-0.52

3

Fig. 5: Influence of carbide shape

3

Grain 2

u z

σ z / σ 0

σ z / σ 0