PSI - Issue 2_B

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

2539

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

5

where Σ vm eq is the macroscopic equivalent von Mises stress, Σ h is the macroscopic hydrostatic stress and θ is the angle in the deviatoric plane, taking values on the range 0 ≤ θ ≤ 2 π . The stress triaxiality T is defined on the usual form T = Σ h Σ vm eq (10) The unit cells are subjected to a wide range of stress states, corresponding to θ = 0 ◦ , 30 ◦ , . . . , 330 ◦ and T = 0 . 667 , 1 . 0 , 1 . 667 , 3 . 0, which results in a total of 48 simulations for each texture. A multi-point constraint (MPC) user subroutine was implemented in the implicit FE solver Abaqus / Standard to maintain the imposed stress state through out the numerical simulations. Details regarding the MPC subroutine and its application to unit cell analyses may be found elsewhere, see for instance Faleskog et al. (1998), Barsoum and Faleskog (2007), and Dæhli et al. (2016).

15 . 10

θ = 0 ◦ θ = 90 ◦ θ = 120 ◦ θ = 240 ◦

T = 1 . 0

11 . 57

f f 0

8 . 05

(a)

(b)

4 . 53

1 . 00

0 . 0

0 . 2

0 . 4

0 . 8

0 . 6

E vm eq

(e)

(c)

(d)

Fig. 3: Deformed configurations for the Goss texture at maximum equivalent stress for loading situations corresponding to a stress triaxiality of T = 1 . 0 for the deviatoric angles (a) 0 ◦ , (b) 90 ◦ , (c) 120 ◦ , and (d) 240 ◦ . Fringes of accumulated plastic strain are shown on the deformed configurations. The corresponding void growth plots are presented in (e). Fig. 3 shows fringes of accumulated plastic strain on deformed configurations of the unit cell with the Goss texture for various deviatoric angles θ and a stress triaxiality of T = 1 . 0. The deformed configurations correspond to the first frame after the macroscopic equivalent stress reaches its maximum ˙ Σ vm eq = 0, and consequently the macroscopic e ff ective deformation is in general di ff erent between the shown configurations. Fig. 3e shows that the void growth for the Goss texture is linked to the magnitude of the imposed stress, from which it is evident that the void evolution is a ff ected by the loading. Another interesting observation is that even though the loading states θ = 120 ◦ and θ = 240 ◦ are almost indistinguishable in terms of Σ vm eq , seen from the radius of the yield surface in Fig. 1b, the void growth rate is very di ff erent. Also, their void shapes evolve quite di ff erently. This indicates that the plastic flow direction, or correspondingly the normal to the yield surface, is important for the void evolution. However, the void shape is also a ff ected by the local field quantities in the proximity of the void, and it is thus di ffi cult to separate the various e ff ects. In general terms, the interplay between the plastic flow and the equivalent stress seems decisive for the macroscopic behaviour of the unit cell. 5. Calibration of porous plasticity model The constitutive relation given by Equation (6) was calibrated to the numerical unit cell calculations by means of a non-linear least-square error routine. Model parameters q i were varied in the optimization procedure. The stress and void volume fraction residuals to be minimized were defined as e σ = E max eq 0 | Σ GT eq − Σ UC eq | d E eq E max eq 0 1 2 Σ GT eq + Σ UC eq d E eq , e f = E max eq 0 | f GT − f UC | d E eq E max eq 0 1 2 f GT + f UC d E eq (11)