**3.2. Interatomic potential**

To calculate the atomic interactions, an embedded atom method (EAM) potential according to [10, 11] is employed. The potential has one pair-wise repulsive and one N-body attractive part and the potential energy *Ei* of atom *i* is given by:

$$E\_i = f\left(\sum\_{\mu l} \rho\{r^{\mu}\}\right) + \frac{1}{2} \sum\_{\mu l} \phi\{r^{\mu}\} \tag{1}$$

where *rij* is the distance between atoms *i* and *j*, *φ* is a pair-wise potential function, *ρ* 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 Cu\_u3.eam, provided by LAMMPS and developed by [12], is used.

#### **3.3. Centrosymmetry parameter**

The results are evaluated and illustrated using the centrosymmetry parameter, *CSP,* as defined by [13]. The *CSP* is a measure of the deviation from a perfect lattice configuration, and for an atom, the *CSP* is defined according to

$$\text{CSP} = \sum\_{i=1}^{N/2} \left| \mathbf{R}\_i + \mathbf{R}\_{i \leftrightarrow \text{N/2}} \right|^2 \tag{2}$$

**4. Results and discussion**

Beams of geometry according to **Figure 1**, holding voids with aspect ratios *w/h* = 1, 2, 3, 4, are loaded under displacement controlled tension until final rupture. After an initially elastic phase, plasticity will appear through slip along close-packed {111} planes. For each case, the

monitoring the instantaneous *CSP* values for all atoms during the loading process. If the beam

does not close, but instead expands so that the ligaments above and below the void rupture

In **Figure 4**, all strains at plastic initiation, taken from **Table 2**, have been merged for comparison; in **Figure 4(a)**, for the [100] orientation and in **Figure 4(b)**, for the [110] orientation. From simulations of solid beams, it was observed that the strain at plastic initiation, ε*<sup>i</sup>*

in practice, independent on cross section size for each orientation as also concluded in e.g., [14], where pure metric scaling effects were investigated for solid single-crystal Cu beams. In [14], it was found that ε*<sup>i</sup>* ≈ 0.094 for the [100] orientation and ε*<sup>i</sup>* ≈ 0.068 for the [110] orientation, so that the [110] orientation yields first, with the ratio between initiation strains about 0.7. The values for solid beams are included in **Figure 4** as circles at *w/h* = 0. As seen in **Figure 4**,

tends to increase with beam cross section size as well as with decreasing ratio *w/h* for

The higher initiation strain for solid beams is expected since a void acts as a local stress raiser,

edly lower strains for both orientations. This indicates that a limiting value is approached with increasing *s*. A comparison of the initiation strains between solid beams, ε*isolid*, and beams

Starting with the [100] orientation, with recorded strains in **Table 2** and plotted in **Figure 3(a)**, **(c)** and **(e)**, it is seen that for the smallest cross section, *s* = 6*a*<sup>0</sup> **Figure 3(a)**, the void closes in all cases

the void at strain *εcm,* thus forming two separate voids. Final void closure appears shortly after,

Another observation is that the initiation strain for the two thicker beams, with *s* = 12*a*<sup>0</sup>

and *ε<sup>f</sup>*<sup>2</sup>

ruptures due to necking as a result of void closure, this strain is denoted *ε<sup>f</sup>*<sup>1</sup>

beam size and orientation in **Figure 3**, and the values are given in **Table 2**.

for the voided beams, and in all cases, the [110] orientation initiates first.

, is relatively close in comparison with the thinnest with *s* = 6*a*<sup>0</sup>

holding the smallest voids with *w/h* = 1 shows that, for both orientations, ε*<sup>i</sup>*

**4.3. Atomic arrangements during plastic deformation for the [100] orientation**

and *s* = 18*a*<sup>0</sup>

influence from a void, even if small, is always present.

except for *w/h* = 1. For the other ratios of *w/h*, still for *s* = 6*a*<sup>0</sup>

, where after rupture occurs at *ε<sup>f</sup>*<sup>1</sup>

, the strain *εmc* at eventual closure of the mid-section of the void,

and the strain at rupture were determined through

Effects of Voids in Tensile Single-Crystal Cu Nanobeams http://dx.doi.org/10.5772/intechopen.74169

. Recorded strains are plotted for each

for the solid beams is markedly higher than

so that the void influence has decreased. Even so, the

. The different scenarios are shown in **Figure 5**.

. In case the void

, is,

65

and

, which initiates at mark-

, the voids close first at the middle of

/ε*isolid* ≈ 0.3 for *s* = 6*a*<sup>0</sup>

**4.1. Recorded strain levels**

strain at plastic initiation, *ε<sup>i</sup>*

**4.2. Elastic response**

weakening the structure.

/ε*isolid* ≈ 0.6 for *s* = 12*a*<sup>0</sup>

ε*i*

*s* = 18*a*<sup>0</sup>

and ε*<sup>i</sup>*

at *ε<sup>c</sup>*

the strain at eventual total void closure, *ε<sup>c</sup>*

independently, these strains are denoted *ε<sup>f</sup>*<sup>1</sup>

both orientations. It can also be noted that ε*<sup>i</sup>*

Here, *N* is the number of nearest neighbors in the lattice surrounding the atom, equal to 12 for an fcc lattice. The vectors **R***<sup>i</sup>* and **R***i+N/*<sup>2</sup> are the vector pairs of opposite nearest-neighbors to the atom. For a perfect lattice, the *CSP,* through the definition (Eq. (2)), becomes *CSP* < 3. Since the *CSP* of an atom is a measure of the positions of the atoms nearest neighbor pairs, both crystallographic orientation and structure geometry are of importance. For the present crystallographic orientations and beam geometries, *CSP* values for atoms located at surfaces, edges and corners are shown in **Table 1**. In the present investigation, the *CSP* reached values up to 60 for atoms situated at or close to corners and edges. Values in the interval between 9 and 21 are found for atoms affected by local defects such as voids, partial dislocations or stacking faults.

As an illustration of the placements of atoms with *CSP* > 21, a beam with orientation [100] is shown at two different strain levels in **Figure 2**. In the figure, each individual atom is shown as a filled circle, with color according to the *CSP* value. In **Figure 2**, all atoms with *CSP* ≤ 21 are colored red; the rest, found at edges and at corners, have their *CSP* in the interval 21 < *CSP* ≤ 60. **Figure 2(a)** shows the situation directly after relaxation, at zero axial load, with high *CSP* values along the edges of the beam, and **Figure 2(b)** at an axial strain of *ε<sup>x</sup>* = 0.1, where also edges that have emerged through slip events attain high *CSP* values.


**Table 1.** *CSP* values for present geometries and orientations.

**Figure 2.** Red atoms: *CSP* ≤ 21, nonred atoms: 21 < *CSP* ≤ 60. [100]-orientation and *s* = 6*a*<sup>0</sup> . (a) *ε x* = 0 and (b) *ε x* = 0.1.
