PSI - Issue 13

20.4 16.2 11.6 7.30

35.1 25.7 16.4

N.A. Giang et al. / Procedia Structural Integrity 13 (2018) 45–50 Giang N.A. / Structural Integrity Procedia 00 (2018) 000–000 yield strength of the matrix σ 0 . In contrast, the micro-crack initiates from an ellipsoidal carbide even it has a high strength σ cP /σ 0 ≥ 24. E z σ z / σ 0 1 Elliptical carbide Circular carbide 1 3 3 3 -1. 45 2.92 -2. 2 7.20 E z = 0.04 E z = 0.18 49 5

u z

u z

σ z / σ 0

σ z / σ 0

E z σ I / σ 0 38.6 31.6

E z

a

a ) b ) Fig. 6: Influence of strength of carbide on micro-crack initiation: a) spherical carbide, b) ellipsoidal carbide 4 a u z

24.6 17.5 10.4

σ z / σ 0

-3. 65 6.92

It is well-known that the ratio σ cM /σ 0 varies with temperature due to dependency of the yield stress σ 0 on temper ature. Figure 7 shows that this ratio has a very strong influence on the macroscopic stress-strain behaviour. While the

Circular carbide E z = 0.15

E z

a )

σ I / σ 0 38.6 31.6

σ I / σ 0 35.7 28.8

u z

a

a

cr

4

u z

24.6 17.5 10.4

22.0 15.2 8.23

a

σ z / σ 0

σ z / σ 0

a

-3. 65 6.92

-5. 43 1.42

Spherical carbide E z = 0.15

Elliptical carbide E z = 0.13

E z

E z

a b ) Fig. 7: Influence of strength of matrix on micro-crack initiation: a) spherical carbide, b) ellipsoidal carbide a )

σ I / σ 0 35.7 28.8

u z

3 crack arrest

3

22.0 15.2 8.23

Elliptical carbide E z = 0.13 For a strength of the matrix σ cM /σ 0 < 16 . 0, three stages are again observed: 1 a micro-crack originates from carbide particle due to high stress concentration induced by dislocation pile-up; 2 the micro-crack grows further into the matrix, which leads the softening of the cell structure and finally 3 the micro-crack completely propagates through the grain. However, if the ratio σ cM /σ 0 ≥ 16, both cell models, with ellipsoidal and with spherical carbide, exhibit an arrest of the micro-crack in the matrix (stage a in Fig. 7). Independent of the particular value of the strength σ cPM of the interface, it can be concluded that much higher values of the strength σ cPM of the ferritic matrix can be used with the gradient theory than with conventional theory. Such higher values are more realistic since they lie closer to the theoretical strength of the material. The present study employed a gradient-enhanced theory to model the plastic deformations of ferrite within a unit cell model of the microstructure. The di ff erent potential failure mechanisms around a carbide particle were modelled by cohesive zones. The results show that the shape and strength of the carbide particle directly influence the micro crack mechanism. For a spherical carbide, no micro-crack is observed in the cell model if carbide strength σ cP /σ 0 ≥ 6 . 0. However, with an ellipsoidal carbide the strength of carbide needs to reach values σ cP /σ 0 ≈ 24 . 0 in order to prevent initiation of a microcrack. However, once a microcrack initiated, the influence of the relative strength σ cM /σ 0 of the ferritic matrix, which is directly related to the temperature, on potential crack arrest in the neighbouring ferrite grain is found to be almost independent of the shape of the carbide. A relative strength σ cM /σ 0 ≥ 16 . 0 is necessary to 1 strain at which the carbide breaks is only moderately dependent on σ cM /σ 0 , the further macroscopic strain to ultimate failure increases considerably with σ cM /σ 0 , i. e. with increasing temperature. 3 3 -5. 43 1.42 E z E z σ z / σ 0 b ) a b ) 5. Conclusions

σ z / σ 0