# Modelling and Experimental Validation of the Porosity Effect on the Behaviour of Nano-Crystalline Materials

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

## 3. Experimental Procedure

#### 3.1. Materials

#### 3.2. Compression Testing of the cW-Cu and W-Cu Samples

## 4. Numerical Model

#### 4.1. Porosity Classification

_{i}is the individual pore volume. The volume of the RVE is calculated by

_{pores}is the volume fraction of pores. Then, the mean radius of pores was calculated by

_{i}is the radius of each pore. Since the pores are considered spherical, the volume of the pores in the equivalent set of the porous model is calculated by:

#### 4.2. The RVE

- 1.
- Calculate number of pores in the domain by$${V}_{void}=poros\times {V}_{total}$$$${N}_{sp}=\frac{{V}_{void}}{\frac{4}{3}\pi {r}_{sp}^{3}}$$
_{total}is the total volume (volume of all grains + grain boundary), poros is the porosity percentage, V_{void}is the total porous empty volume, R_{sp}is the radius of the spherical porous hole and N_{sp}is the number of porous holes in the domain - 2.
- Get Nsp as the nearest whole integer number (NINT function in ANSYS APDL) (ANSYS Inc., Canonsburg, PA, USA) [27] and then recheck that total volume of all porous is not greater than Vvoid.
- 3.
- Get the properties of the surface area of GB
- Calculate the area of each surface of GB
- Calculate normal of each area
- Determine min/max location of key point (Kp) from the origin and its associated Kp number for each area (GB surface)

- 4.
- Determine parallel Surface
- d.
- Check whether GB surface lies on the boundary of a cube or not from its max/min centroid location
- e.
- Find all groups of GB surfaces (excluding boundary surfaces) who’s normal vectors are parallel to each other.$$\frac{{\overrightarrow{n}}_{area1}\cdot {\overrightarrow{n}}_{area2}}{\left|{n}_{area1}\right|\cdot \left|{n}_{area2}\right|}=1$$
- f.
- Calculate the distance between the centroid of each parallel surface and find the pair of surfaces which has minimum centroid distance.

- 5.
- Generate a random number
- g.
- For each pair, calculate distance from the origin of global coordinate (0,0,0) to the centroid of both the surfaces. In order to check which surface is close to the origin and which surfaces are far from the origin. The surface close to the origin is considered as Area 1 and the surface far from the origin is considered as Area 2.
- h.
- From the previously calculated min/max kp location, get the minimum and maximum location from the origin for each pair. This will define a small cubical space covered by a pair of surfaces in the GB and GI domain. Size of that cubical space will be xmin-xmax, ymin-ymax and zmin-zmax.
- i.
- Generate 500 (randsize) random numbers within the range of that cubical space for each pair.
- j.
- Find a minimum of one random number who’s XYZ location lies in between the pair of surfaces (Area 1 and Area 2)
- First, check that XYZ location of random number does not lie on the boundary of GB and GI domain
- Calculate the normal distance (DNrandmin) of the random number from Area 1 and similarly the normal distance (DNrandmax) from Area 2
- Recalculate the distance (Dcentmin) between the centroid of Area 1 and Area 2
- Check the XYZ location of the random number is greater than the location of Kp (xcent1, ycent1, zcent1) of Area 1 that is close to the origin (0,0,0) and the XYZ location is less than Kp (xcent2, ycent2, zcent2) of Area 1 that is far from the origin. By doing so, it is decided whether the random number lies inside the cubical space of area pair, not on the boundary of area pair.
- Considering GB as imported solid bodies, it is assumed that normal of all surfaces will be in an outward direction. Hence, the normal direction of both the surfaces of each pair will be in opposition to each other’s direction. Now check the location of the random number with respect to the normal direction of Area 1 and Area 2. If the normal vector from Area 1 to the XYZ location of the random number has the same direction as the normal vector of Area 1, then the previously calculated normal distance DNrandmin will be positive, otherwise, it will be negative. Therefore, check that normal distance DNrandmin of the random number is greater than 0 and DNrandmax is less than 0.
- Further, check that sum of the absolute value of the normal distance of a random number from Area 1 and Area 2 is not greater than the centroidal distance (Dcentmin). This confirms if the random number perfectly comes in between the pair of areas (Area 1 and Area 2)
- In order to make sure that random number does not lie very close to Area 1 or Area 2, check that the absolute value of DNrandmin is greater than tolerance TLdrand. Tolerance is nothing but 5% of Dcentmin. This is the optional condition.

- k.
- Repeat the same process. The total number of random numbers (Nsp) is found.

_{gb}, was 20% lower than the one of the grains, E

_{GI}. The hypothesis of the reduced value of the GB elastic constant was based on the results of ab initio calculations found in Reference [34] and experimental findings reported in Reference [35]. The yield stress of the GB phase was equal to 861 MPa because of the pile-up breakdown phenomenon. The reason behind this effect is based on the Hall–Petch effect.

_{cr}and v are the shear modulus, the Burger’s vector, the critical grain size and Poisson’s ratio respectively. Ιt was assumed that grain boundaries and grain interiors exhibit the elastic-plastic properties described by the following relationships:

_{0}is the elastic strain at the yield point, E is Young’s modulus and h is the work-hardening coefficient. The plasticity of grain boundaries was approached based on the Hill criterion [37]. The equivalent stress can be described as:

_{m}. Additionally, the consequential displacement at the surface of the RVE has been conducted by its length in the vertical direction to acquire the macroscopic deformation, e

_{m}. The elastic constant (Young’s modulus) and the yield strength of the nanocrystalline alloy can be numerically predicted. The periodic boundary conditions were applied so as to derive the homogenised behaviour of the RVE, employing the maximal plastic strain value as a boundary condition at each loading step. This led us to overpass the commonly known production issue of nanocrystalline materials at sufficient quantities in order to complete a considerable mechanical test campaign. Taking as granted the above-said proposed numerical procedure, a reduced number of specimens for the experimental tests is required for validation purposes.

## 5. Comparison of Numerical and Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Specimen in the testing machine. (

**a**) before experiment; (

**b**) after experiment; (

**c**) digital image correlation.

**Figure 10.**Comparison of the experimental results of cW-Cu specimens with the full dense and porous numerical results.

**Figure 11.**Comparison of the experimental results of W-Cu specimens with the full dense and porous numerical results.

Material | Diameter (mm) | Height (mm) | Mass (g) | Theoretical Density (g/cm^{3}) | Calculated Relative Density (%) |
---|---|---|---|---|---|

W-Cu | 13.08–13.11 | 25.1 | 45.12–46.53 | 14.95 | 89.04–92.25 |

cW-Cu | 13.08–13.14 | 24.72–25.16 | 44.92–46.14 | 88.84–92.04 |

**Table 2.**X-ray diffraction (XRD) results of the cW-Cu: cold-pressed sample, hot isostatic pressed specimens, average aged (36 h).

Step | Status | W | Cu |
---|---|---|---|

Composition (wt%) | 74% | 26% | |

1 | As-received powder size (nm) | 1300 ± 300 | 279 ± 62 |

2 | Cold pressed powder size (nm) | 1011 ± 225 | 74 ± 55 |

3 | HIP grain size (nm) | 113 ± 25 | 45 ± 25 |

4 | Grain size after 36 h annealing (nm) | 206 ± 70 | 247 ± 55 |

**Table 3.**XRD results of the W-Cu: milled powder, cold pressed and hot isostatic pressed specimens, average aged (36 h).

Step | Status | W | Cu |
---|---|---|---|

- | Composition (wt%) | 73.67% | 26.33% |

1 | Milled powder size (nm) | 30 ± 15 | 6 ± 5 |

2 | Cold pressed powder size (nm) | 16 ± 4 | 8 ± 3 |

3 | HIP grain size (nm) | 95 ± 60 | 31 ± 7 |

4 | Grain size after 36h annealing (nm) | 187 ± 65 | 108 ± 24 |

Features | Phases | Young Modulus (GPa) | Hall-Petch Parameters | Poisson Ratio | |
---|---|---|---|---|---|

σ_{0} (MPa) | k_{y} (MPa m^{½}) | ||||

Grain interiors | Cu | 120 | 40 | 0.11 | 0.336 |

W | 96 | 800 | 1 | 0.28 |

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**MDPI and ACS Style**

Bazios, P.; Tserpes, K.; Pantelakis, S.
Modelling and Experimental Validation of the Porosity Effect on the Behaviour of Nano-Crystalline Materials. *Metals* **2020**, *10*, 821.
https://doi.org/10.3390/met10060821

**AMA Style**

Bazios P, Tserpes K, Pantelakis S.
Modelling and Experimental Validation of the Porosity Effect on the Behaviour of Nano-Crystalline Materials. *Metals*. 2020; 10(6):821.
https://doi.org/10.3390/met10060821

**Chicago/Turabian Style**

Bazios, Panagiotis, Konstantinos Tserpes, and Spiros Pantelakis.
2020. "Modelling and Experimental Validation of the Porosity Effect on the Behaviour of Nano-Crystalline Materials" *Metals* 10, no. 6: 821.
https://doi.org/10.3390/met10060821