PSI - Issue 35

Varvara Romanova et al. / Procedia Structural Integrity 35 (2022) 196–202 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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2. CPFEM mechanical simulations The grain-scale mechanical analysis for the simulated microstructural model consisting of 3514 grains (Fig. 1a) is performed using CPFE calculations. The constitutive behavior of grains is described within the crystal plasticity theory formulated in terms of small strains. The generalized Hooke’s law is formulated w ith respect to a crystal frame whose axes for face-centered cubic crystals coincide with the [100], [010] and [001] crystal directions. Thus, for all grains the Hooke’s law is written in the same form irrespective of the grain orientation relative to the s pecimen frame ( ) p ij ijkl kl kl C    = − , (1) where ijkl C are the elastic constants, ij  is the stress rate tensor, and the difference in parentheses is equal to the elastic strain. The strain rates ij  are kinematically related to the velocities i u as ( ) , , ij i j j i u u  = + . (2) where ij  are the orientation tensor components calculated for an  slip system through its normal and slip direction vectors, i m and i s , ( ) ( ) ( ) ij i j j i m s m s    = + . (4) The slip system  is activated when the resolved shear stress ( )   reaches the critical value ( ) crss   . Then the slip rate ( )   follows a power law ( ) ( ) ( ) ( ) 0 ( ) sgn crss           = , (5) where 0  is the reference slip rate, the same for all slip systems, and the resolved shear stress is calculated as ( ) ( ) ( ) i ij j s m      = . (6) The critical resolved shear stress (CRSS) ( ) crss   is determined from the equation ( ) ( ) ( ) 1/2 0 p crss eq kD f      − = + + , (7) where the first term of the right-hand sum is the initial CRSS value. The second term is the Hall-Petch relation for the grain boundary strengthening where the grain size D calculated individually for each grain. The third term is an empirical strain hardening function chosen to fit the experimental stress-strain curve in the form ( ) ( ) 1 2 1 exp( / ) p p eq eq f a a   = − − . (8) where 1 a and 2 a are the constants obtained by fitting the experimental stress-strain curve for an AlSi10Mg alloy. In the simulations, the p eq  value is calculated individually for each finite element and, thus, the CRSS values being the same for all elements in the initial stage become inhomogeneous as plastic deformation develops. The model parameters and material constants used in the simulations are 1111 108 C = GPa, 1122 61 C = GPa, 2323 28 C = GPa, 0 35  = MPa, 1/2 0.07 k MPa cm =  , 1 113 a = MPa and 2 0.0164 a = . The micromechanical model was incorporated in ABAQUS/Explicit to simulate its quasistatic loading in terms of dynamic problem. The solution to the equations of motion using explicit schemes proved their high efficiency in view of computational demands. The equation of motion in a strong form 1 2 1 2 In the framework of crystal plasticity, the plastic strain rates p ij  are calculated as a summary slip in the active <111> {110} slip systems as 12 ( ) ( ) 1 1 2 p ij ij      = =   , (3)