**4.1. Recorded strain levels**

**3.3. Centrosymmetry parameter**

64 Molecular Dynamics

for an fcc lattice. The vectors **R***<sup>i</sup>*

and for an atom, the *CSP* is defined according to

*CSP* = ∑

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,

<sup>|</sup>**R***<sup>i</sup>* <sup>+</sup> **<sup>R</sup>***<sup>i</sup>*+*<sup>N</sup>*/2|<sup>2</sup> (2)

. (a) *ε x* = 0 and (b) *ε x* = 0.1.

are the vector pairs of opposite nearest-neighbors to

*i*=1 *N*/2

and **R***i+N/*<sup>2</sup>

where also edges that have emerged through slip events attain high *CSP* values.

**Lattice structure** *CSP* Ideal fcc structure *CSP* < 3 Fault sites 3 ≤ *CSP* < 9 Surface atoms [100] 9 ≤ *CSP* < 21 Surface atoms [110] 9 ≤ *CSP* < 25 Edge and corner atoms [100] *CSP* ≥ 21 Edge and corner atoms [110] *CSP* ≥ 25

**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>

Here, *N* is the number of nearest neighbors in the lattice surrounding the atom, equal to 12

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, 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 strain at plastic initiation, *ε<sup>i</sup>* , the strain *εmc* at eventual closure of the mid-section of the void, the strain at eventual total void closure, *ε<sup>c</sup>* and the strain at rupture were determined through monitoring the instantaneous *CSP* values for all atoms during the loading process. If the beam ruptures due to necking as a result of void closure, this strain is denoted *ε<sup>f</sup>*<sup>1</sup> . In case the void does not close, but instead expands so that the ligaments above and below the void rupture independently, these strains are denoted *ε<sup>f</sup>*<sup>1</sup> and *ε<sup>f</sup>*<sup>2</sup> . Recorded strains are plotted for each beam size and orientation in **Figure 3**, and the values are given in **Table 2**.

### **4.2. Elastic response**

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>* , is, 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**, ε*i* tends to increase with beam cross section size as well as with decreasing ratio *w/h* for both orientations. It can also be noted that ε*<sup>i</sup>* for the solid beams is markedly higher than for the voided beams, and in all cases, the [110] orientation initiates first.

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

Another observation is that the initiation strain for the two thicker beams, with *s* = 12*a*<sup>0</sup> and *s* = 18*a*<sup>0</sup> , is relatively close in comparison with the thinnest with *s* = 6*a*<sup>0</sup> , which initiates at markedly 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 holding the smallest voids with *w/h* = 1 shows that, for both orientations, ε*<sup>i</sup>* /ε*isolid* ≈ 0.3 for *s* = 6*a*<sup>0</sup> and ε*<sup>i</sup>* /ε*isolid* ≈ 0.6 for *s* = 12*a*<sup>0</sup> and *s* = 18*a*<sup>0</sup> so that the void influence has decreased. Even so, the influence from a void, even if small, is always present.

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

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 except for *w/h* = 1. For the other ratios of *w/h*, still for *s* = 6*a*<sup>0</sup> , the voids close first at the middle of the void at strain *εcm,* thus forming two separate voids. Final void closure appears shortly after, at *ε<sup>c</sup>* , where after rupture occurs at *ε<sup>f</sup>*<sup>1</sup> . The different scenarios are shown in **Figure 5**.

**Figure 3.** Strains at plastic initiation, *ε<sup>i</sup>* , at closure of the mid-section of the void, ε*mc*, at void closure ε*<sup>c</sup>* and at ruptures *ε<sup>f</sup>*<sup>1</sup> and *ε<sup>f</sup>*<sup>2</sup> . (a), (c) and (e): [100] orientation. (b), (d) and (f): [110] orientation. Values from **Table 2**.

In the case *s* = 6*a*<sup>0</sup>

*εf*1 and *ε<sup>f</sup>*<sup>2</sup>

*w/h,* [110] *s* = 6*a*<sup>0</sup>

*w/h,* [100] *s* = 12*a*<sup>0</sup>

*w/h,* [110] *s* = 12*a*<sup>0</sup>

*w/h,* [100] *s* = 18*a*<sup>0</sup>

*w/h,* [110] *s* = 18*a*<sup>0</sup>

along the entire beam.

void rupture independently at *ε<sup>f</sup>*<sup>1</sup>

**Table 2.** Strains at plastic initiation, *ε<sup>i</sup>*

. Graphs in **Figure 3**.

and *w/h* = 1, the void expands and the two ligaments above and below the

, at closure of the mid-section of the void, ε*mc*, at void closure ε*<sup>c</sup>*

case are shown in **Figure 5(a)**–**(c)**, coded in the *CSP*. The state *ε<sup>x</sup>* = 0 is taken directly after relaxation. Activated {111} slip planes are seen to appear after plastic initiation and spread

. Snapshots of the events during deformation for this

Effects of Voids in Tensile Single-Crystal Cu Nanobeams http://dx.doi.org/10.5772/intechopen.74169 67

and at ruptures

and *ε<sup>f</sup>*<sup>2</sup>

*w/h,* **[100]** *s* **= 6***a***<sup>0</sup>** *ε<sup>i</sup> εcm ε<sup>c</sup> ε<sup>f</sup>***<sup>1</sup>** *ε<sup>f</sup>***<sup>2</sup>** 0.0353 — — 0.0978 0.1087 0.0136 0.0707 0.0734 0.1495 — 0.0082 0.0217 0.0245 0.1033 — 0.0108 0.0109 0.0162 0.1223 —

 0.0109 — 0.0299 0.0951 — 0.0082 — 0.0217 0.1196 — 0.0082 — 0.0163 0.1223 — 0.0082 — 0.0162 0.1223 —

 0.0571 — — 0.2011 0.2310 0.0408 — — 0.2826 0.3043 0.0326 — — 0.2609 0.2609 0.0300 — — 0.1984 0.2609

 0.0353 — — 0.1440 0.1821 0.0299 0.2065 — 0.2636 0.2880 0.0245 0.2120 — 0.2962 0.2962 0.0245 0.2418 — 0.2826 0.3261

 0.0625 — — 0.3777 0.4293 0.0571 — — 0.2853 0.3125 0.0462 — — 0.2717 0.3451 0.0435 — — 0.2690 0.3206

 0.0380 — — 0.6603 — 0.0326 — — 0.7470 — 0.0326 — — 0.3451 0.3750 0.0299 — — 0.4620 0.4701


**Table 2.** Strains at plastic initiation, *ε<sup>i</sup>* , at closure of the mid-section of the void, ε*mc*, at void closure ε*<sup>c</sup>* and at ruptures *εf*1 and *ε<sup>f</sup>*<sup>2</sup> . Graphs in **Figure 3**.

In the case *s* = 6*a*<sup>0</sup> and *w/h* = 1, the void expands and the two ligaments above and below the void rupture independently at *ε<sup>f</sup>*<sup>1</sup> and *ε<sup>f</sup>*<sup>2</sup> . Snapshots of the events during deformation for this case are shown in **Figure 5(a)**–**(c)**, coded in the *CSP*. The state *ε<sup>x</sup>* = 0 is taken directly after relaxation. Activated {111} slip planes are seen to appear after plastic initiation and spread along the entire beam.

**Figure 3.** Strains at plastic initiation, *ε<sup>i</sup>*

and *ε<sup>f</sup>*<sup>2</sup>

66 Molecular Dynamics

, at closure of the mid-section of the void, ε*mc*, at void closure ε*<sup>c</sup>*

. (a), (c) and (e): [100] orientation. (b), (d) and (f): [110] orientation. Values from **Table 2**.

and at ruptures *ε<sup>f</sup>*<sup>1</sup>

directly after relaxation, at *ε<sup>x</sup>* = 0, is included. As seen, the plasticity in this case is localized to the void vicinity, similar to the case of void closure for the [100] orientation and *s* = 6*a*<sup>0</sup>

A comparison between the corresponding relaxed states for the [100]- and the [110] orientations, at *ε<sup>x</sup>* = 0, shows the impact of crystallographic orientation. The [110] orientation creates a more roughened surface as compared to the [100] orientation. Comparing **Figures 5(a)** and **6(a)**, with the thinnest ligaments surrounding the voids and thus the weakest cross sections, it is seen that the deformation of the void in the relaxed state is much more obvious for the [110] orientation. The void seems to bulge due to the lower constraints on the atoms in combination

tern. In all cases, the void is expanding, so that two ligaments rupture individually above and

voids first close at the center, thus forming two separate voids. One of these will grow until the

and *ε<sup>f</sup>*<sup>2</sup>

, **Figure 3(d)**, displays a different deformation pat-

. But for all cases apart from *w/h* = 1, the initial

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

. This is illustrated in **Figure 6(d)**–**(f)**

, *w/h* = 1, and (d)–(f): *s* = 12*a*0, *w/h* = 2. [110] orientation.

with the orientations of the preferred {111} slip planes for the [110] orientation.

and *ε<sup>f</sup>*<sup>2</sup>

Increasing the cross section size to *s* = 12*a*<sup>0</sup>

ligaments above and below it rupture at strains *ε<sup>f</sup>*<sup>1</sup>

**Figure 6.** Events during deformation of beams with (a)–(c): *s* = 6*a*<sup>0</sup>

Coded in the *CSP*. The states *ε<sup>x</sup>* = 0 are taken directly after relaxation.

below the growing void at strains *ε<sup>f</sup>*<sup>1</sup>

**Figure 5(d)**–**(f)**.

for *w/h* = 2.

in

69

**Figure 4.** Strains at plastic initiation *ε<sup>i</sup>* . (a) [100] orientation and (b) [110] orientation.

**Figure 5.** Events during deformation of beams with *s* = 6*a*<sup>0</sup> and (a)–(c) *w/h* = 1 and (d)–(f) *w/h* = 3. [100] orientation. Coded in the *CSP*. The states *ε<sup>x</sup>* = 0 are taken directly after relaxation.

An example of a sequence of events during void closure is seen in **Figure 5(d)**–**(f)** for *s* = 6*a*<sup>0</sup> and *w/h* = 3. As seen, the void closes first at the center of the void, forming two separate voids which both close shortly after formation. This leads to that the initially voided but now healed cross-sectional part necks, and final rupture occurs at the strain *ε<sup>f</sup>*<sup>1</sup> , cf. **Figure 3(a)**. Also, here the {111} slip planes are activated, but localization of the plasticity to the formerly voided region, followed by rupture, occurs before the plasticity has reached the beam ends. Instead, elastic regions remain away from the necking region.

For the larger beam cross sections, *s* = 12*a*<sup>0</sup> and *s* = 18*a*<sup>0</sup> , no void closure occurs. Instead, the voids expand and the ligaments above and below the void rupture independently at *ε<sup>f</sup>*<sup>1</sup> and *ε<sup>f</sup>*<sup>2</sup> . The series of events are thus similar to what is seen in **Figure 5(a)**–**(c)**.

## **4.4. Atomic arrangements during plastic deformation for the [110] orientation**

For the [110] orientation and *s* = 6*a*<sup>0</sup> , **Figure 3(b)**, total closure of the voids occurs for all *w/h* where after the now solid center cross section ruptures at strain *ε<sup>f</sup>*<sup>1</sup> . The voids are filled, atom plane by atom plane, from one side of the void to the other like a zipper, through slip along {111} planes. The case *s* = 6*a*<sup>0</sup> , *w/h* = 1 is illustrated in **Figure 5(a)**–**(c)**, where the configuration directly after relaxation, at *ε<sup>x</sup>* = 0, is included. As seen, the plasticity in this case is localized to the void vicinity, similar to the case of void closure for the [100] orientation and *s* = 6*a*<sup>0</sup> in **Figure 5(d)**–**(f)**.

A comparison between the corresponding relaxed states for the [100]- and the [110] orientations, at *ε<sup>x</sup>* = 0, shows the impact of crystallographic orientation. The [110] orientation creates a more roughened surface as compared to the [100] orientation. Comparing **Figures 5(a)** and **6(a)**, with the thinnest ligaments surrounding the voids and thus the weakest cross sections, it is seen that the deformation of the void in the relaxed state is much more obvious for the [110] orientation. The void seems to bulge due to the lower constraints on the atoms in combination with the orientations of the preferred {111} slip planes for the [110] orientation.

Increasing the cross section size to *s* = 12*a*<sup>0</sup> , **Figure 3(d)**, displays a different deformation pattern. In all cases, the void is expanding, so that two ligaments rupture individually above and below the growing void at strains *ε<sup>f</sup>*<sup>1</sup> and *ε<sup>f</sup>*<sup>2</sup> . But for all cases apart from *w/h* = 1, the initial voids first close at the center, thus forming two separate voids. One of these will grow until the ligaments above and below it rupture at strains *ε<sup>f</sup>*<sup>1</sup> and *ε<sup>f</sup>*<sup>2</sup> . This is illustrated in **Figure 6(d)**–**(f)** for *w/h* = 2.

An example of a sequence of events during void closure is seen in **Figure 5(d)**–**(f)** for *s* = 6*a*<sup>0</sup> and *w/h* = 3. As seen, the void closes first at the center of the void, forming two separate voids which both close shortly after formation. This leads to that the initially voided but now healed

. (a) [100] orientation and (b) [110] orientation.

the {111} slip planes are activated, but localization of the plasticity to the formerly voided region, followed by rupture, occurs before the plasticity has reached the beam ends. Instead,

plane by atom plane, from one side of the void to the other like a zipper, through slip along

voids expand and the ligaments above and below the void rupture independently at *ε<sup>f</sup>*<sup>1</sup>

**4.4. Atomic arrangements during plastic deformation for the [110] orientation**

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

, cf. **Figure 3(a)**. Also, here

. The voids are filled, atom

 and *ε<sup>f</sup>*<sup>2</sup> .

, no void closure occurs. Instead, the

and (a)–(c) *w/h* = 1 and (d)–(f) *w/h* = 3. [100] orientation. Coded

, **Figure 3(b)**, total closure of the voids occurs for all *w/h*

, *w/h* = 1 is illustrated in **Figure 5(a)**–**(c)**, where the configuration

cross-sectional part necks, and final rupture occurs at the strain *ε<sup>f</sup>*<sup>1</sup>

The series of events are thus similar to what is seen in **Figure 5(a)**–**(c)**.

where after the now solid center cross section ruptures at strain *ε<sup>f</sup>*<sup>1</sup>

elastic regions remain away from the necking region.

For the larger beam cross sections, *s* = 12*a*<sup>0</sup>

**Figure 5.** Events during deformation of beams with *s* = 6*a*<sup>0</sup>

in the *CSP*. The states *ε<sup>x</sup>* = 0 are taken directly after relaxation.

For the [110] orientation and *s* = 6*a*<sup>0</sup>

{111} planes. The case *s* = 6*a*<sup>0</sup>

**Figure 4.** Strains at plastic initiation *ε<sup>i</sup>*

68 Molecular Dynamics

**Figure 6.** Events during deformation of beams with (a)–(c): *s* = 6*a*<sup>0</sup> , *w/h* = 1, and (d)–(f): *s* = 12*a*0, *w/h* = 2. [110] orientation. Coded in the *CSP*. The states *ε<sup>x</sup>* = 0 are taken directly after relaxation.

influence of *w/h* on the failure strain can be drawn. In some cases, the strain increases, and in

Effects of Voids in Tensile Single-Crystal Cu Nanobeams http://dx.doi.org/10.5772/intechopen.74169 71

It was concluded that geometrical features such as beam size and crystallographic orientation played a crucial role for the mechanical behavior. Plasticity develops through slip along closed packed {111} planes, and the [110] orientation always initiates plasticity first. Further, the strain at plastic initiation increases with beam cross section size as well as with decreasing

Studying the deformation pattern, it was found that the plasticity developed and the void deformed in different ways depending on cross section size, void aspect ratio and crystal orientation. As regards the events that lead to final rupture of the beams, different scenarios were observed.

In some cases, the void elongated and the two beam ligaments, above and below the void, eventually necked and ruptured independently. In such cases, the plasticity, through slip along {111} planes before the last ligament rupture, tended to extend away from the regions

In the cases where closure of the voids occurred, the strain at closure decreased with increasing *w/h*. Also, it was observed that the strain at failure was relatively independent of crystal-

Sometimes the void first closes at the center, forming two separate voids. Then, two scenarios are possible. One is that the two voids both eventually close, followed by necking and rupture of the now healed cross section. In these cases, the plasticity localizes to the vicinity of the neck and leaves regions away from the neck elastic. The other possible scenario is that one of the created voids start to elongate and the ligaments above and below this void neck and

There were also cases where failure did not occur in the vicinity of the void; instead rupture

[1] Hommel M, Kraft O. Deformation behavior of thin cupper films on deformable substrates. Acta Materialia. 2001;**49**:3935-3947. DOI: 10.1016/S1359-6454(01)00293-2

[2] Schweiger R, Dehm G, Kraft O. Cyclic deformation of polychrystalline Cu films. Philo-

sophical Magazine. 2003;**83**:693-710. DOI: 10.1080/0141861021000056690

occurred near one beam end after that the plasticity had spread over the entire beam.

lographic orientation and that it increased with increasing cross section size.

some cases, it decreases with increasing value of *w/h*.

near the void and could sometimes reach the beam ends.

ratio *w/h* for both orientations.

rupture independently.

**Author details**

**References**

Aylin Ahadi\*, Per Hansson and Solveig Melin

\*Address all correspondence to: aylin.ahadi@mek.lth.se

Division of Mechanics, LTH, Lund University, Lund, Sweden

**Figure 7.** Events during deformation of beams with s = 18a0 and *w/h* = 1. [110] orientation. Coded in the *CSP*. At (a) ε<sup>x</sup> = 0, taken directly after relaxation, (b) ε<sup>x</sup> = 0.405 and (c) ε<sup>x</sup> = 0.649.

For *s* = 18*a*<sup>0</sup> in the [110] orientation with curves in **Figure 3(f)**, lastly, another phenomenon appears for *w/h* = 1 and *w/h* = 2. For these cases, initiation of plasticity as well as rupture occurs away from the void as illustrated in **Figure 7** for *s* = 18*a*<sup>0</sup> and *w/h* = 1. For the larger *w/h*, *w/h* = 3 and *w/h* = 4, the voids elongate and the ligaments rupture individually over the void similar to the events in **Figure 5(a)**–**(c)**.
