**3. Molecular dynamic simulations**

### **3.1. Simulation procedure**

The reason for traditional engineering dimensioning rules becoming obsolete at small enough metric scales is that, with decreasing structural size, the number of surface close atoms as compared to number of bulk atoms increases and, at some point, no longer is negligible. Electron redistribution close to the surfaces will leave the surface close atoms in energy states deviating from those of bulk atoms. This influences the interatomic bonding forces and, thereby, the material response to mechanical loading as discussed by e.g., [1–3]. This effect is accentuated if the atoms are placed at, or close to, corners and edges of the structure, e.g., [4]. As a consequence, the material properties will vary with size for small enough structures, and

Also, the crystallographic orientation is of uttermost importance for the material properties, e.g., [5, 6]. The crystallographic orientation imposes anisotropy in the structure and also decides the surface topology which influences the mechanical properties. The orientation further sets the preferred slip plane directions and, thereby, influences the plasticity development. Here, the tensile response of single-crystal nanosized Cu beams, holding square-shaped, through-the-thickness voids, will be investigated with respect to elastic and plastic behavior and eventual size dependence in the mechanical response. The rational for this investigation is that even if defect-free structures might be intended at manufacturing, defects will always be present to some extent. If their presence influences the mechanical response of the struc-

To this end, 3D single-crystal nanosized Cu beams of different cross section sizes, holding throughthe-thickness voids of different sizes, will be investigated using the molecular dynamics free-ware large-scale atomic/molecular massively parallel simulator (LAMMPS), see [7]. The loading will be displacement controlled tension along the length direction, and two different crystallographic

The structures considered are single-crystal face centered cubic (fcc) Cu beams of length

*a*<sup>0</sup> = 3.615 Å denoting the lattice parameter for Cu. Each beam holds a symmetrically placed, square-shaped through-the-thickness void of width *w* and height *h*, cf. **Figure 1(a)** where a centrally placed coordinate system (*x, y, z*) is introduced. The sensitivity to voids will be

Two different crystallographic orientations are considered to determine its influence on the mechanical response. For the first orientation, referred to as the [100]-orientation, the coordinates (*x, y, z*) coincide with crystallographic directions according to: *x =* [100], *y* = [010] and *z* = [001], cf. **Figure 1(b)**. For the second, referred to as the [110]-orientation, *x* = [110], *y* = [−110]

A beam is built from the repetition of fcc Cu unit cells, and to mimic clamped end boundary conditions, all atoms within four unit cells at each end of the beam are restricted from movements in the *y*- and *z*-directions. During the relaxation step, one atom is fixed in all directions.

, *N* = 1, 2, 3, with

and square cross section with three different side lengths *s* = 6*Na*<sup>0</sup>

investigated by varying the width *w*, keeping the relative void height *h* = *s*/3 constant.

the effects become obvious below about 50–100 nm.

ture, the product functionality might be at risk.

orientations will be considered and compared.

**2. Geometry and boundary conditions**

*L* = 100*a*<sup>0</sup>

62 Molecular Dynamics

and *z* = [001], cf. **Figure 1(c)**.

For the simulations, the molecular dynamics free-ware LAMMPS has been employed and the atomic images are produced using OVITO, developed by [8].

In the simulations, an NVT-ensemble, with constant number of particles N, constant volume V and absolute temperature T, kept at a constant temperature of 0.01 K through a Nosé-Hoover thermostat as found in [9] is employed. Before load application, relaxation of the atomic ensemble in order to reach the equilibrium state is imposed during 5000 time steps, with each time step equal to 5 fs. This gives the relaxation time equal to 25 ps, which was found to be sufficiently long to reach equilibrium with good accuracy as judged from the variations in axial stress with time. Thereafter, a constant velocity of *vend* = *a*<sup>0</sup> /200/ps is imposed in the +*x*- and −*x*-directions at all atoms within the clamped ends using a constant time step of 5 fs.
