PSI - Issue 13

L. Laiarinandrasana et al. / Procedia Structural Integrity 13 (2018) 1751–1755 L. Laiarinandrasana et al. / Structural Integrity Procedia 00 (2018) 000–000

1754

4

3

142

5

140

10

92

10

98

Void height FE σ zz

Void height FE σ zz

4

120

8

82

2

132

8

68

3

100

6

72

h v ( µ m)

h v ( µ m)

h v ( µ m)

σ zz (MPa) h v ( µ m)

σ zz (MPa)

σ zz (MPa)

σ zz (MPa)

2

80

4

62

1

122

6

38

1

60

2

52

Void height FE σ zz

Void height FE σ zz

0

112

0

40

0

42

4

-1 -0.5 0 0.5 1 8

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

-1.2 -0.8 -0.4 0

r/R

r/R

r (mm)

2t/T (-)

6

30

6

60

10

32

10

20

8

8

22

10

4

10

4

30

6

6

φ r , φ θ ( µ m)

φ r , φ θ ( µ m)

φ r , φ θ ( µ m)

σ rr , σ θθ (MPa) φ r , φ θ ( µ m)

Void φ r Void φ θ FE σ rr FE σ θθ

12

0

σ rr , σ θθ (MPa)

σ rr , σ θθ (MPa)

σ rr , σ θθ (MPa)

Void φ r Void φ θ FE σ rr FE σ θθ

Void φ r Void φ θ FE σ rr FE σ θθ

4

4

2

-10

2

0

Void φ r Void φ θ FE σ rr FE σ θθ

2

-10

2

2

0

-30

0

-30

0 -1.2 -0.8 -0.4 0 -8

0

-20

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

-1 -0.5 0 0.5 1

r/R

r/R

r (mm)

2t/T (-)

a)

b)

c)

Fig. 4. Void height (h v ) and diameter ( φ ) vs. principal stress through the 3 directions of the coordinates (z being the load direction). From top to bottom, porosity contour maps, void height vs. maximum principal stress σ zz , void diameters vs. transverse stresses: a) NT4 round bar; b) NT045 round bar; c) CT specimen.

r and θ corresponded to the crack propagation and the thickness directions respectively. h v and φ were the plotted through r and through the normalized thickness coordinate (2t / T). The main results are illustrated in the bottom rows of diagrams in figs. 4a,b,c. The symbols (open squares and circles) represent the void characteristic lengths corresponding to the first Y-axis. As mentioned earlier voids were cylindrical showing here a 3D shape with a “unique” height but with an elliptical basis, thus two diameters: φ r , φ θ . For NT4 and NT045 notched round bars (figs. 4a,b), it was observed that the circumferential diameter φ θ was the major axis, that is, greater than φ r . The diagram relative to the CT specimen (fig. 4c, bottom left) indicates the particular case where the ellipse major axis φ r followed the crack propagation direction.

3.3. Finite Element analysis

A porous-visco-plastic constitutive model already implemented in the FE code Zset was used under finite strain formulation (Selles et al. (2017)). The material parameters were optimized thanks to the macroscopic data ( δ c ) in fig. 2 as well as the local distribution of V f . The obtained set of parameters allowed a good agreement between the simulations (lines) and the experimental data (symbols) in fig. 2. Moreover, the porosity contour maps in fig. 4 (top row) were in accordance with those of the measured V f in fig. 3 (bottom row). Following then Laiarinandrasana et al. (2016), the principal stresses ( σ rr , σ θθ , σ zz ) were computed and compared with the void characteristic lengths, according to their specific directions. As shown in fig. 4 diagrams, a fair agreement was obtained between the void characteristic lengths and the corresponding FE stress components. It is to be noted that these stress space distributions were obtained at the onset of the tertiary stage, that is at the end of the secondary creep stage. The discussion on the time singularities of the stress is out of the scope of this work.