# Multiscale Simulation of Surface Defects Influence Nanoindentation by a Quasi-Continuum Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{1}is 0.4032 nm, one atomic lattice spacing in [1 1 1] direction (d

_{0}) is 0.2328 nm, one atomic spacing in $[\overline{1}\text{}1\text{}0]$ direction (h

_{0}) is 0.1426 nm, Burgers vector $\overrightarrow{\mathrm{b}}$ is 0.285 nm, shear modulus μ is 33.14 GPa, Poisson ν is 0.319 and (1 1 1) surface energy γ

_{111}is 0.869 J/m

^{2}, which is comparable with the experimental values of 1.14–1.20 J/m

^{2}[29]. The elastic modulus predicted by this potential are

**C**

_{11}= 117.74 GPa,

**C**

_{12}= 62.06 GPa, and

**C**

_{44}= 36.67 GPa. The experimental values extrapolated to

**T**= 0 K are

**C**

_{11}= 118.0 GPa,

**C**

_{12}= 62.4 GPa, and

**C**

_{44}= 32.5 GPa [25]. Figure 1 shows the nanoindentation model used in the simulation and the corresponding schematic of local and non-local representative atoms with initial surface pit, and its unit cell model of Al in the selected directions. The rectangular indenter is set rigid with its width of 0.932 nm (four times the lattice constant of Al in [1 1 1] direction (d

_{0})). It is necessary to note that the indenter size is chosen refer to the ‘Nano-Indentation by a Square Punch’ simulation example in QC toturial document [30] and previous works [21,22]. The indenter shape is chosen rectangular in this simulation because the boundary of energy field (displacement field) and the distance between the pit and the indenter remain unchanged when driven down into the $(\overline{1}10)$ surface, which is exactly necessary to investigate the distance influence of the pit. The width D and depth H of surface pit are respectively 0.688 nm and 0.730 nm. We choose this pit size based on our previous works [21,22] and some necessary pre-simulations, which prove that such pit size is relatively moderate and proper, more sensitive to the distance effect. When the pit is too small, the influence of the pit on the nanohardness is not obvious; when the pit is too large, it is similar to that of a step, which is not exactly our focus. In the out-of-plane direction, the thickness of this model is equal to the minimal repeat distance with periodic boundary condition applied (namely 0.4938 nm for this model). The distance of adjacent boundary between the pit and indenter shows d in Figure 1. Fifteen different distances of adjacent boundary d have been simulated in this paper, which is respectively 1d

_{0}, 2d

_{0}, 3d

_{0}, 4d

_{0}, 5d

_{0}, 6d

_{0}, 7d

_{0}, 8d

_{0}, 9d

_{0}, 10d

_{0}, 11d

_{0},12d

_{0}, 13d

_{0}, 17d

_{0}, 21d

_{0}. These distances are selected in order to make a more comprehensive investigation.

**C**(

**C**=

**F**, where

^{T}F**F**is the deformation gradient) to the factor “epscr”, where the factor epscr is used to determine if a repatom must be made nonlocal due to significant variations in the deformation gradients around the repatom. Specifically, taking the eigenvalues of

**C**in two elements a and b to be ${\lambda}_{k}^{a}$ and ${\lambda}_{k}^{b}$ (k = 1, 2, 3), nonlocality is triggered if $\underset{a,b;k}{\mathrm{max}}\left|{\lambda}_{k}^{a}-{\lambda}_{k}^{b}\right|>epscr$.

## 3. Results and Discussion

#### 3.1. Nanohardness in the Case of no Surface Defect

#### 3.2. Nanohardness with Various Distances between Surface Defect and Indenter

_{0}, 2d

_{0}, 3d

_{0}, 4d

_{0}, 5d

_{0}, 6d

_{0}, 7d

_{0}, 8d

_{0}, 9d

_{0}, 10d

_{0}, 11d

_{0}, 12d

_{0}, 13d

_{0}, 17d

_{0}, 21d

_{0}, the nanohardness curve rises up in a wave pattern and finally tends towards the nanohardness value of no surface defect.

_{0}, d = 2d

_{0}, d = 3d

_{0}distance, do not match the wave pattern. To explain such special phenomenon, atomic structure and corresponding strain distribution of Al crystal are probed.

_{0}, 2d

_{0}and 3d

_{0}. It shows that when the distance d equals 1d

_{0}and 2d

_{0}, there appears a notch phenomenon at the left side of surface pit, which directly induces serious damage to the structure of materials and great strain concentration (as shown in Figure 5A–D). When the distance d equals 3d

_{0}, there is no notch (as shown in Figure 5E,F). Consequently, when the distance d equals 1d

_{0}and 2d

_{0}, the nanohardness is greatly reduced. That is to say that the first three atoms in nanohardness curve as shown in Figure 4 will not match the wave pattern associated with a cycle of three atoms.

#### 3.3. Formula Modification of Necessary Load for Elastic-To-Plastic Transition

_{cr}is the critical load value at the onset of dislocation emission, k is the slope of elastic stage in the load-displacement curve, h is the depth of dislocation dipole when it is emitted, a is the half width of indenter, γ

_{111}is the energy of (1 1 1) surface of Al crystal.

_{0}and d = 2d

_{0}is not taken into account because of notch.

## 4. Conclusions

- The pitted surface plays a great role in the emission of dislocation that it causes significant reduction on nanohardness, compared with defect-free situation.
- As the distance between the pit and indenter increases, nanohardness increases in a wave pattern associated with a cycle of three atoms, which is closely related to periodic atoms arrangement on {1 1 1} atomic close-packed planes of face-centered cubic metal; when the adjacent distance between the pit and indenter is more than 16 atomic spacing, there is almost no effect on nanohardness.
- The theoretical formula for necessary load of the elastic–plastic transition of Al film has been effectively modified to accommodate the effect of the initial surface pit. This modified formula closely fits the decreasing trend of nanohardness as the distance between the pit and indenter increases, and such trend agrees well with the experimental results of surface step with various distances. Such modification may contribute to the investigation of material property with surface defects, particularly in mircochips and MEMS.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The schematic illustration of nanoindentation model with a pit defect and its unit cell model of Al in the selected directions, where the unusual shapes in local region are not finite elements, they are just the schematic of its specific region that one corresponding representative atom belongs to.

**Figure 5.**Von Mises strain distribution of notch propagation. (

**A**) d = 1d

_{0}at the load step of the indenter 0.38 nm; (

**B**) d = 1d

_{0}at the load step of the indenter 0.4 nm; (

**C**) d = 2d

_{0}at the load step of the indenter 0.44 nm; (

**D**) d = 2d

_{0}at the load step of the indenter 0.46 nm; (

**E**) d = 3d

_{0}at the load step of the indenter 0.46 nm; (

**F**) d = 3d

_{0}at the load step of the indenter 0.48 nm.

**Figure 6.**The comparison of the necessary load for elastic-to-plastic transition of Al thin film with various distances between the pit and the indenter calculated by the theoretical formula before and after modification.

Distance (d_{0}) | QC Data (N/m) | Theory Load (N/m) | Data Difference (N/m) |
---|---|---|---|

3 | 14.28 | 18.02 | 3.75 |

4 | 14.46 | 17.29 | 2.83 |

5 | 14.48 | 17.88 | 3.39 |

6 | 14.24 | 17.41 | 3.15 |

7 | 14.86 | 17.96 | 3.14 |

8 | 14.85 | 17.65 | 2.83 |

9 | 14.38 | 17.92 | 3.07 |

10 | 14.87 | 17.87 | 3.49 |

11 | 14.86 | 18.03 | 3.16 |

12 | 14.49 | 17.56 | 2.70 |

13 | 14.70 | 17.99 | 3.50 |

17 | 15.06 | 18.04 | 3.34 |

21 | 15.09 | 18.17 | 3.11 |

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**MDPI and ACS Style**

Zhang, Z.; Ni, Y.; Zhang, J.; Wang, C.; Jiang, K.; Ren, X.
Multiscale Simulation of Surface Defects Influence Nanoindentation by a Quasi-Continuum Method. *Crystals* **2018**, *8*, 291.
https://doi.org/10.3390/cryst8070291

**AMA Style**

Zhang Z, Ni Y, Zhang J, Wang C, Jiang K, Ren X.
Multiscale Simulation of Surface Defects Influence Nanoindentation by a Quasi-Continuum Method. *Crystals*. 2018; 8(7):291.
https://doi.org/10.3390/cryst8070291

**Chicago/Turabian Style**

Zhang, Zhongli, Yushan Ni, Jinming Zhang, Can Wang, Kun Jiang, and Xuedi Ren.
2018. "Multiscale Simulation of Surface Defects Influence Nanoindentation by a Quasi-Continuum Method" *Crystals* 8, no. 7: 291.
https://doi.org/10.3390/cryst8070291