PSI - Issue 13

Per Hansson / Procedia Structural Integrity 13 (2018) 837–842 Per Hansson/ Structural Integrity Procedia 00 (2018) 000 – 000

3

839

1.2 Molecular dynamics For the simulations the molecular dynamics free-ware LAMMPS ( http://lammps.sandia.gov ) has been employed and the atomic images are produced using OVITO, developed by Stukowski (2010). The interaction between the Cu atoms is described by an EAM-potential, giving the potential energy of an atom. It consists of one pair-wise repulsive part and one N-body attractive part with a cut-off radii, cf. Holian and Ravelo (1995). The potential energy, E i , of atom i is thus given by Eq. (1): (1) where, r ij is the distance between atoms i and j ,   is a pair-wise potential function,   is the contribution to the electron charge density from atom j at the location of atom i , and f is an embedding function that represents the energy required to place atom i into the electron cloud. For the present study the potential file named Cu_u3.eam, provided by LAMMPS and developed by Foiles et al. (1986), has been used. For the simulations, a NVT-ensemble held at a constant temperature of 300K by a Nosé-Hoover thermostat as found in Ellad and Miller (2011) is generated. Initially the atomic ensemble constituting the beam is relaxed to its equilibrium state for 5000 time steps. Thereafter the load is effectuated by applying the constant velocity v 0 , to the atoms within four unit cells at each end of the beam, thus mimicking clamped ends, whereas the atoms in between these end cells are free to move without constraints. The results are evaluated using the Centro-Symmetry Parameter CSP according to Kelchner et al. (1998), as being a measure of the instantaneous lattice disorder, i.e. the instantaneous plasticity. The CSP for an atom is defined according to Eq. (2), where the N is the number of nearest neighbors in the surrounding lattice, equal to 12 for an fcc material. R i and R i+N/ 2 are the vectors corresponding to pairs of opposite nearest-neighbors in the lattice. The value of the CSP signals whether an atom is part of a perfect lattice, a local defect, or part of a free surface. Commonly used CSP values in fcc lattices are: ideal fcc structure CSP <3, partial dislocation 321. 1 ( ) ( ) 2 ij ij i j i j i E f r r              

N

/ 2

1 i CSP R R      i

2

(2)

i N

/ 2

2. Results and discussion 2.1 Monotonic loading at 0.01K and 300K

The first investigation was to compare strain at plastic initiation, eventual defect closure and strain at rupture for monotonic loading at temperatures T =0.01K and T =300K for all three geometries. The results at T =0.01K are taken from previous work by Ahadi et al (2016) for the same geometries and the results are given in Table 1. By comparing the results in Table 1 it can be seen that both the strain at plastic initiation and at rupture decreases with increasing temperature. This effect is strongest in the case of solid beams and for the case of strain at plastic initiation with a void. The latter case is the result of partial closure of the void at T =300K, not seen at 0.01K, leading to earlier plastic initiation. This can be explained by the increased movements of the atoms, increasing dislocation mobility and decreasing stress for dislocation nucleation with increasing temperature.

Table 1. Strain at plastic initiation, ε xi , at defect closure, ε xc , and at rupture, ε xf .

T =0.01K, solid

T =0.01K, edge

T =0.01K, void

T =300 K , solid

T =300 K , edge

T =300 K , void

ε xi ε xc ε xf

0.0920

0.0375

0.035

0.0725

0.0350

0.0200 0.0250

-

-

-

-

-

0.270

0.110

0.103

0.135

0.0975

0.100

3.2 Stress-time curves during fatigue loading In this subsection the axial stress in the x -direction of the beams, σ x , is presented as function of simulation time t for the three different geometries for different values of maximum displacement, δ xmax according to Fig. 2, for