PSI - Issue 42

1748 António Mourão et al. / Procedia Structural Integrity 42 (2022) 1744–1751 António Mourão/ Structural Integrity Procedia 00 (2019) 000 – 000 ( ) = 1 2 ( ∗( ) ⊗ ∗( ) − ∗( ) ⊗ ∗( ) ) (7) Based on Schmid's law, the slipping rate ˙ ( ) on the α -th slip system is corresponding to the Schmid stress ( ) . The flow rule is suggested using a power law as: ˙ ( ) = ˙ ⟨ | ( ) − ( ) |− ( ) ⟩ ( ( ) − ( ) ) (8) where ( ) is the back stress, ˙ is the reference strain rate, K and n are the material constants, and ( ) is the isotropic hardening term. The rate of back stress ˙ ( ) was proposed as: ˙ ( ) = 1 ˙ ( ) − 2 | ˙ ( ) | ( ) (9) The evolution of isotropic hardening variable ( ) is suggested as: ˙ ( ) = ∑ ℎ (1 − ( ) )| ˙ ( ) | (10) The internal state variable ˙ ( ) , which represents the dislocation hardening, is defined by: ˙ ( ) = (1 − ( ) )| ˙ ( ) | (11) where c 1 , c 2 , b , Q are material constants. 3.2. FE modelling Given the previously mentioned microstructure a two-dimensional CPFE model of a representative volume element (RVE) with 100 grains randomly generated polygons called Voronoi tessellations commonly used in mesoscale modelling was generated using ABAQUS software, which can be seen in Figure 3, with both height and width of 320 μm and following the grain size distribution as presented in Figure 2. 5

a)

Generated RVE model in ABAQUS.

b) Boundary conditions and the inverse pole map of the RVE model

Fig. 3. RVE model.