PSI - Issue 2_B

L.E.B Dæhli et al. / Procedia Structural Integrity 2 (2016) 2535–2542 L.E.B. Dæhli et al. / Structural Integrity Procedia 00 (2016) 000–000

2540

6

4.70

2.10

θ = 0 ◦ θ = 60 ◦ θ = 120 ◦ θ = 180 ◦ θ = 240 ◦ θ = 300 ◦ Unit cell Gurson

T = 1 . 0

3.78

1.58

f f 0

Σ vm eq σ 0

2.85

1.05

θ = 0 ◦ θ = 60 ◦ θ = 120 ◦ θ = 180 ◦ θ = 240 ◦ θ = 300 ◦ Unit cell Gurson

1.92

0.52

T = 1 . 0

1.00

0.00

0.000

0.125

0.250 E vm eq

0.375

0.500

0.000

0.125

0.250 E vm eq

0.375

0.500

(a)

(b)

4.50

2.10

θ = 30 ◦ θ = 90 ◦ θ = 150 ◦ θ = 210 ◦ θ = 270 ◦ θ = 330 ◦ Unit cell Gurson

T = 1 . 0

3.62

1.58

f f 0

Σ vm eq σ 0

2.75

1.05

θ = 30 ◦ θ = 90 ◦ θ = 150 ◦ θ = 210 ◦ θ = 270 ◦ θ = 330 ◦ Unit cell Gurson

1.88

0.52

T = 1 . 0

1.00

0.00

0.000

0.150

0.300 E vm eq

0.450

0.600

0.000

0.150

0.300 E vm eq

0.450

0.600

(c)

(d)

Fig. 4: Response of unit cell and the homogenized material model in terms of (a) and (c) equivalent von Mises stress, and (b) and (d) void volume fraction against the equivalent strain. All curves shown are for the Cube texture. where the superscripts GT and UC denote Gurson-Tvergaard and unit cell, respectively, and |◦| denotes the magnitude. Note that the limit of the definite integral is the equivalent strain at maximum stress from the respective unit cell analyses E max eq = E UC eq ˙ Σ vm eq = 0 with the equivalent strain being defined by E eq = E i j E i j (12) (13) was adopted in the optimization process. In the present work, the residuals were given equal weight w σ = w f because the calibrated q i -values were only slightly a ff ected by the residual weights. Table 2: Parameters retrieved from the optimization procedure. 2 3 where E i j are the macroscopic strain deviator components. A weighted residual on the form e = w σ e σ + w f e f

Texture

q 1

q 2

Cube Goss

1 . 912 1 . 282

0 . 791 0 . 842

The resulting parameters of the optimization procedure can be found in Table 2. For comparative reasons, we note that the calibrated parameters in the study by Steglich et al. (2010) were q 1 = 1 . 22 and q 2 = 1 . 16. An obvious reason