PSI - Issue 23

Solveig Melin et al. / Procedia Structural Integrity 23 (2019) 137–142 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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To detect, identify dislocations, and to calculate the dislocation length a Dislocation extraction algorithm, DXA, developed by Stukowski et al. (2012) have been used. This analysis tool is incorporated in OVITO. Using this tool the length of different types of dislocations; i.e. Perfect, Shockley, Stair-rod, Hirth, Frank and Other dislocations, can be identified. In this paper mobile dislocations are defined as the sum of Perfect and Shockley dislocations, immobile as the sum of Stair-rod, Hirth and Frank dislocations. Also the sum of all dislocations is detemined. The stress tensor for an atom is calculated as a time averaged value over a number of time steps. In LAMMPS the per atom stress, σ ij , for atom α is calculated from eq. (1):

= − 1 ( +∑ )

(1)

where V α is the atomic volume, M α is the atomic mass, v αβ are the force and distance between atoms α and β , respectively. For simplification it is assumed that the atomic volume of all atoms in the beam is the same, calculated as the total volume of the beam divided by the number of atoms. To calculate the average axial stress in the loading direction the stresses in the x -direction is summarized over all atoms and averaged over a number of time steps. In this paper the stress is calculated every 0.5ps and the dislocation density every 25ps. i and v j are the atom velocity components and F i αβ and r j

3. Results and discussion 3.1. Single crystal beams

To determine the influence on the mechanical properties from a grain boundary, first simulations of single crystal Cu beams with three different crystallographic orientations, corresponding to the x -direction in the [100], [110] or [111] directions were performed. In Fig. 2a the axial stress vs. axial strain and in Fig. 2b the dislocation density vs. axial strain, are seen for the three different orientations. The dislocation density is calculated as the sum of all dislocation lengths divided by the volume of the beam. As seen, the highest stiffness and the highest stress at plastic initiation were obtained for the [111] beam. This is because the resolved shear stress along the preferred slip planes in this case is low. The lowest stiffness was for the [100] direction and the lowest stress at plastic initiation was found for the [110] direction. The stress at plastic initiation was defined as the stress at which the first dislocation in the beam was formed, found visually using OVITO. Hardening of the beams occurred after some plastic deformation, especially clear in the [110] direction at high strains and for the [100] direction for lower strains. It was found that this point for all cases corresponds to the first maximum in the corresponding stress-strain curve. Regarding the dislocation density it was found that, in the beginning of plastic deformation, most dislocations were formed in the [111] beam whereas in the later stages most dislocations were formed in the [110] beam.

Fig. 2. a) Stress vs. strain and b) dislocation density vs. strain for single crystal beams. Blue is the [100], red the [110] and green the [111] orientation.