# Numerical Simulation of the Depth-Sensing Indentation Test with Knoop Indenter

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Aspects

## 3. Numerical Simulation and Materials

#### 3.1. Indenters

#### 3.2. Finite Element Mesh

#### 3.3. Materials

## 4. Results

#### 4.1. Indentation Geometry and Equivalent Plastic Strain Distributions

#### 4.2. Indentation Contact Area and Young’s Modulus

#### 4.3. Flat Indenter

#### 4.4. Correlation with Experimental Results

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Knoop indenter modeled with Bezier surfaces: (

**a**) general view; (

**b**) detail of the indenter tip imperfection.

**Figure 3.**Evolution of the tangents of ${\theta}_{1}$ and ${\theta}_{2}$ as function of the R-ratio values.

**Figure 4.**Finite element mesh used in the numerical simulations: (

**a**) global view; (

**b**) detail of the central region where indentation occurs.

**Figure 5.**Surface indentation profiles obtained along the two diagonals, the shorter, m (z-axis) and the longer, L (x-axis), respectively: (

**a**,

**c**) obtained at maximum load; (

**b**,

**d**) after unloading.

**Figure 6.**Surface indentation profiles at maximum load, obtained from the results of Figure 5a,c, where z is multiplied by R = 7.11, in order to easily compare the profiles along the two diagonals, the longer, L, and the shorter, m: (

**a**) Material M1; (

**b**) Material 2; (

**c**) Material M3.

**Figure 7.**Equivalent plastic strain distributions obtained at maximum load in the numerical simulations using the Knoop and Vickers indenters: (

**a**,

**b**) Material M1; (

**c**,

**d**) Material M3.

**Figure 8.**Normalized contact area results obtained in the numerical simulation of the materials with the values of the strain hardening parameter, yield stress and Young’s modulus shown in Table 2, using the Knoop indenter. (

**a**) Contact area ${A}_{{h}_{\mathrm{c}}}/{A}_{\mathrm{REF}}$; (

**b**) Contact area ${A}_{\mathrm{FE}}/{A}_{\mathrm{REF}}$.

**Figure 9.**Normalized Young’s modulus results obtained in the numerical simulation of the materials with the values of the strain hardening parameter, yield stress and Young’s modulus shown in Table 2, using the Knoop indenter: (

**a**) Young’s modulus ${E}_{{h}_{\mathrm{c}}}/{E}_{\mathrm{REF}}$; (

**b**) Young’s modulus ${E}_{\mathrm{FE}}/{E}_{\mathrm{REF}}$.

**Figure 10.**Evolution of the ratio $P/{S}^{2}$ as a function of $H/{E}_{\mathrm{r}}^{2}$, obtained in the numerical simulations of all materials in Table 2, using the Knoop and Vickers indenters.

**Figure 11.**Evolution of load as a function of the elastic indentation depth obtained in the numerical simulations using flat indenters with different ratio R: (

**a**) R = 1; (

**b**) R = 4; (

**c**) R = 7.11.

**Figure 12.**Evolution of $\beta $ as a function of the R-ratio obtained in numerical simulations using the five flat and pyramidal indenters with different values of the R-ratio.

R = L/m | θ_{1} | θ_{2} | Area Function |
---|---|---|---|

1 (Vickers) | 74.0546 | 74.0546 | $A=24.5000{h}^{2}+0.5600h+0.0032$ |

2.5 | 69.1723 | 81.3478 | $A=34.5500{h}^{2}+0.6650h+0.0032$ |

4 | 67.0462 | 83.9559 | $A=44.6000{h}^{2}+0.7556h+0.0032$ |

5.5 | 65.8369 | 85.3366 | $A=54.6500{h}^{2}+0.8364h+0.0032$ |

7.11 (Knoop) | 64.8379 | 86.2199 | $A=65.4377{h}^{2}+0.9152h+0.0032$ |

Materials | Studied Cases | n | ${\mathit{\sigma}}_{\mathbf{y}}$ (GPa) | E (GPa) |
---|---|---|---|---|

Without strain hardening | 5 | ≈0 | 0.2, 2, 6, 10 and 20 | 70 |

5 | 200 | |||

5 | 400 | |||

With strain hardening | 5 | 0.15 | 70 | |

5 | 200 | |||

5 | 400 | |||

5 | 0.30 | 70 | ||

5 | 200 | |||

5 | 400 |

**Table 3.**Mechanical Properties of The Materials Used in the Study of The Knoop Indentation Geometry.

Material | ${\mathit{\sigma}}_{\mathbf{y}}$ (GPa) | n | E (GPa) | ν | ${\mathit{h}}_{\mathbf{f}}/{\mathit{h}}_{\mathbf{max}}$ |
---|---|---|---|---|---|

M1 | 0.2 | 0.01 | 200 | 0.3 | 0.97 |

M2 | 6 | 0.3 | 0.40 | ||

M3 | 20 | 400 | 0.25 |

**Table 4.**Values obtained for the correction factor $\beta $ in the numerical simulations with flat indenters.

R = L/m | E (GPa) | Average Values of $\mathit{\beta}$ | ||||
---|---|---|---|---|---|---|

30 | 200 | 400 | 600 | 800 | ||

$\mathit{\beta}$ | ||||||

1.00 | 1.055 | 1.054 | 1.053 | 1.054 | 1.054 | 1.054 |

2.50 | 1.125 | 1.123 | 1.124 | 1.125 | 1.124 | 1.124 |

4.00 | 1.215 | 1.214 | 1.214 | 1.214 | 1.215 | 1.214 |

5.50 | 1.269 | 1.266 | 1.267 | 1.267 | 1.266 | 1.267 |

7.11 | 1.374 | 1.372 | 1.371 | 1.371 | 1.372 | 1.372 |

Materials | ${\mathit{E}}_{\mathbf{nom}}$ (GPa) [19] | ${\mathit{E}}_{\mathbf{G}}$ (GPa) [19] | Error (%) | ${\mathit{E}}_{\mathbf{M}}$ (GPa) [19] | Error (%) | $\frac{\sqrt{\mathit{\pi}}}{2\mathit{\beta}}\frac{1}{{\mathit{E}}_{\mathit{r}}}{(\mathsf{\mu}\mathbf{m}/\mathbf{N}}^{2})$ [19] | $\mathit{E}$ (GPa) | Error (%) |
---|---|---|---|---|---|---|---|---|

Si_{3}N_{4} | 317 ± 4 | 316.5 ± 4.24 | −0.16 | 300 ± 20.0 | −5.36 | 2.548 ± 0.029 | 302.8 ± 4.40 | −4.48 |

Ceramic-glass | 82 ± 2 | 85.0 ± 0.36 | 3.66 | 85 ± 4.0 | 3.66 | 7.930 ± 0.031 | 82.1 ± 0.35 | 0.15 |

Alumina | 385 ± 6 | 386.0 ± 7.75 | 0.26 | 380 ± 18.5 | −1.30 | 2.233 ± 0.032 | 359.4 ± 6.90 | −6.64 |

$\beta $-TCP | 130 ± 2 | 129.0 ± 0.85 | −0.77 | 142 ± 14.0 | 9.23 | 5.568 ± 0.032 | 120.7 ± 0.70 | −7.12 |

Fused silica | 68 ± 1 | 65.0 ± 0.30 | −3.00 | 70 ± 4.0 | 2.94 | 9.221 ± 0.034 | 69.9 ± 0.25 | 2.80 |

Average of the absolute value of the error | 1.57 | 4.50 | 4.27 |

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**MDPI and ACS Style**

Simões, M.I.; Antunes, J.M.; Fernandes, J.V.; Sakharova, N.A.
Numerical Simulation of the Depth-Sensing Indentation Test with Knoop Indenter. *Metals* **2018**, *8*, 885.
https://doi.org/10.3390/met8110885

**AMA Style**

Simões MI, Antunes JM, Fernandes JV, Sakharova NA.
Numerical Simulation of the Depth-Sensing Indentation Test with Knoop Indenter. *Metals*. 2018; 8(11):885.
https://doi.org/10.3390/met8110885

**Chicago/Turabian Style**

Simões, Maria I., Jorge M. Antunes, José V. Fernandes, and Nataliya A. Sakharova.
2018. "Numerical Simulation of the Depth-Sensing Indentation Test with Knoop Indenter" *Metals* 8, no. 11: 885.
https://doi.org/10.3390/met8110885