# Unambiguous Identification of Crystal Plasticity Parameters from Spherical Indentation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Crystal Plasticity Finite Element Method

#### 2.2. Optimization

## 3. Results and Discussion

#### 3.1. Parameter Optimization

**Case 1:**fitting only LD curve,**Case 2:**fitting only surface topography,**Case 3:**fitting LD curve and surface topography.

**Population 1**and

**Population 2**) were examined. Thus, each case was run for two initial populations, thus resulting in six EA runs in total. Figure 4 shows the normalized difference (difference divided by the difference in the first generation) for each EA run. All the simulations converged maximally after generation 6, with the exception of

**Case 3**in

**Population 1**, where significant difference drop appeared in the 10th generation. However, it was checked that for that particular case additional five generations (results not shown) did not result in any further improvement.

**Case 2**) did not result in correct fit: the level of force was either too low (

**Population 1**) or too high (

**Population 2**). This shows that surface information alone is not sufficient to correctly reproduce LD curve. This highlights the fact that various choices of initial parameters can lead to very similar surface topographies.

**Case 1**(i.e., fitting only LD curves) results in somewhat different surface topography. In order to see it better, another figure was prepared. Figure 7 shows the maps of point-wise differences between the solution obtained in each EA run vs. the reference simulation. In addition, Table 3 shows the minimum, maximum, and mean values of the differences. The differences are now clearly visible. The largest differences are present in the

**Case 1**(fitting only LD). This shows that fitting LD curve is not sufficient to achieve excellent surface topography fit, which is a counterbalance to the previously observed fact that fitting only surface topography is not sufficient to obtain the correct LD curve.

**Case 3**regardless of population. Second best values were found by

**Case 1**, while

**Case 2**gave the worst answer. Concerning ${\theta}_{1}$, a good approximation was obtained with

**Case 2**and

**Case 3**when starting from

**Population 1**, but when starting from

**Population 2**, none of the EA runs resulted in correct value. Moreover, in the latter case, the values filled the investigated range almost completely. In order to understand the efficiency of various EA simulations in determination of correct parameters, a sensitivity analysis was performed.

#### 3.2. Sensitivity Analysis

**Case 1**that considered only LD curves, was able to find a close guess for ${\tau}_{0}$, ${\theta}_{0}$ and ${\tau}_{1}$. Its prediction of ${\theta}_{1}$ was however the worst. The correct determination of ${\tau}_{0}$, ${\theta}_{0}$, and ${\tau}_{1}$ is understandable in view of Figure 9a–c. On the other hand, it is not surprising that the value of ${\theta}_{1}$ was not determined correctly (remembering that LD curve was insensitive to this parameter alone). It was already outlined that

**Case 2**(considering only surface topography) provided the worst parameter calibration. Its prediction of ${\theta}_{1}$ is however better than the one provided by

**Case 1**. Although the surface topography was relatively insensitive to the value of this parameter (cf. Figure 10d), still the variability of surface topography with respect to ${\theta}_{1}$ was greater than the variability of LD curve in that case. Thus, it seems that this variability enabled

**Case 2**, to perform better in finding ${\theta}_{1}$, in particular when starting from

**Population 1**. Finally,

**Case 3**was able to find reasonably good values of ${\tau}_{0}$, ${\theta}_{0}$, and ${\tau}_{1}$ regardless of the initial population, which is understandable considering that both LD curves and surface topographies were sensitive to these parameters. As LD curves are insensitive to ${\theta}_{1}$ and surface topographies are almost insensitive to this parameter, it seems also reasonable that determination of its value when using both information for fitting heavily depended on the initial population.

#### 3.3. Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The graphical explanation of fitness evaluation based on surface topography. (

**a**) Regular grid of points. (

**b**) Point-by-point differences between two surfaces projected onto the regular grid.

**Figure 4.**Plot of the normalized difference vs. generation in each of the EA runs stating from

**Population 1**(

**a**) and

**Population 2**(

**b**).

**Figure 5.**The load–displacement curves obtained in each of the EA runs stating from population

**Population 1**(

**a**) and

**Population 2**(

**b**).

**Figure 6.**The surface topographies in the reference solution and each of the EA runs. Heights are given in μm.

**Figure 7.**The difference in surface topographies between each of the EA runs and the reference solution. The legends present the colors related to a difference value in μm.

**Figure 8.**The values of the parameters obtained at each generation of each EA run. The ranges of parameters are shown in red and the reference parameter value in light orange (thick line).

**Figure 9.**Sensitivity of LD curves to (

**a**) ${\tau}_{0}$, (

**b**) ${\theta}_{0}$, (

**c**)${\tau}_{1}$, and (

**d**) ${\theta}_{1}$ (other parameters were kept fixed).

**Figure 10.**Sensitivity of surface topographies to (

**a**) ${\tau}_{0}$, (

**b**) ${\theta}_{0}$, (

**c**)${\tau}_{1}$, and (

**d**) ${\theta}_{1}$ (other parameters were kept fixed). A difference with respect to the reference simulation is shown. Size of the map is 2.4 μm × 2.4 μm. The legend presents the colors related to a difference value in μm.

**Figure 11.**Sensitivity of LD curves and surface topographies to ${\theta}_{1}$ in a wider range (other parameters were kept fixed). A difference with respect to the reference simulation is shown. Size of the map is 2.4 μm × 2.4 μm. The legend presents the colors related to a difference value in μm.

**Figure 12.**The sensitivity of load–displacement curves with varying Coulomb friction coefficient (FL denotes the frictionless case).

${\mathit{\tau}}_{0}$ [MPa] | ${\mathit{\theta}}_{0}$ [MPa] | ${\mathit{\tau}}_{1}$ [MPa] | ${\mathit{\theta}}_{1}$ [MPa] |
---|---|---|---|

8 | 240 | 142 | 7.5 |

${\mathit{\tau}}_{0}$ [MPa] | ${\mathit{\theta}}_{0}$ [MPa] | ${\mathit{\tau}}_{1}$ [MPa] | ${\mathit{\theta}}_{1}$ [MPa] |
---|---|---|---|

5–15 | 100–400 | 50–250 | 5–10 |

**Table 3.**The minimum, maximum, and mean values (in μm) of the differences between surface topographies obtained in each of the EA run and the reference solution.

Minimum | Maximum | Mean | |||||||
---|---|---|---|---|---|---|---|---|---|

Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | |

Population 1 | −7.275 | −0.101 | −0.049 | 0.671 | 0.073 | 0.074 | −0.700 | −0.044 | 0.008 |

Population 2 | −4.466 | −0.154 | −0.023 | 1.042 | 0.755 | 0.185 | −0.126 | 0.154 | 0.017 |

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**MDPI and ACS Style**

Frydrych, K.; Papanikolaou, S.
Unambiguous Identification of Crystal Plasticity Parameters from Spherical Indentation. *Crystals* **2022**, *12*, 1341.
https://doi.org/10.3390/cryst12101341

**AMA Style**

Frydrych K, Papanikolaou S.
Unambiguous Identification of Crystal Plasticity Parameters from Spherical Indentation. *Crystals*. 2022; 12(10):1341.
https://doi.org/10.3390/cryst12101341

**Chicago/Turabian Style**

Frydrych, Karol, and Stefanos Papanikolaou.
2022. "Unambiguous Identification of Crystal Plasticity Parameters from Spherical Indentation" *Crystals* 12, no. 10: 1341.
https://doi.org/10.3390/cryst12101341