PSI - Issue 42

António Mourão et al. / Procedia Structural Integrity 42 (2022) 1744–1751 António Mourão / Structural Integrity Procedia 00 (2019) 000 – 000

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a)

Metallographic diagram.

b)

Grain size distribution.

Fig. 2. Várzeas bridge material, a) metallographic diagram and b) grain size distribution.

3. Computations 3.1. Crystal plasticity model

The crystal plasticity model was derived in a finite strain framework by the classical multiplication decomposition of the deformation gradient components in to elastic and plastic parts is considered as follows, = (1) where is the elastic component of the deformation gradient and denotes the plastic component of the deformation gradient. The rate of change of is related to the slipping rate of the slip system by, ˙ ( ) −1 = ∑ =1 ˙ ( ) ( ) ⊗ ( ) (2) where ˙ ( ) is the slip rate of the α -th slip system, N is the total number of the slip systems, ( ) and ( ) are the slip direction and norm of the α -th slip plane, respectively. The velocity gradient is defined as, = = = ˙ −1 = + (3) where D is the symmetric part and Ω is the antisymmetric spin tensor of the velocity gradient L . The plastic component of the velocity gradient is given by, = ˙ ( ) −1 ( ) −1 = ∑ =1 ˙ ( ) ( ) ⊗ ( ) ( ) −1 = ∑ =1 ˙ ( ) ∗( ) ⊗ ∗( ) (4) where ∗( ) = ( ) and ∗( ) = ( ) ( ) are the slip direction and norm in the deformed configuration. Assuming that the elastic deformations of the crystal are not affected by the sliding deformation, the constitutive relationship can be written as ˆ = ℂ: − ∑ =1 (ℂ: ( ) + ( ) ⋅ − ⋅ ( ) ) ˙ ( ) (5) where ˆ is the Jaumann rate of Cauchy stress, C is the elastic moduli tensor. ( ) and ( ) are determined by ( ) = 1 2 ( ∗( ) ⊗ ∗( ) + ∗( ) ⊗ ∗( ) ) (6)