# Indentation Measurement in Thin Plates under Bending Using 3D Digital Image Correlation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0},Y

_{0},Z

_{0}). Every facet presents a different gray scale of intensity. During the deformation, some algorithms of correlation [29] track the speckle contained into a facet. Each facet at the reference image is searched in the deformed image P’(X’

_{0},Y’

_{0},Z’

_{0}). The result is a displacement vector located in the center of each virtual facet.

## 2. Materials and Methods

#### 2.1. Experimental Methodology

^{®}. The script takes the output data from VIC-3D (displacements in matrix structures) and a loop cycle links the minimum displacement with the corresponding load, as both systems were synchronized. The measured out-of-plane displacement values were referred to a reference image, corresponding to zero indentation to obtain the indentation value for each loading state.

_{T}is the energy applied to the specimen, E

_{c}is the involved energy due to contact, E

_{b}is the involved energy due to bending and E

_{sys}is the elastic energy absorbed by the loading frame during the test. Both energies terms E

_{c}and E

_{b}, have an elastic component that is recovered during the unloading path, (E

_{c},

_{elastic}and E

_{b},

_{elastic}) and a plastic component that is absorbed by the specimen, which is responsible for the remaining permanent damage after the load is removed (E

_{c},

_{plast}and E

_{b,plast}). Thus, the energy balance is presented as:

_{sys}was evaluated by comparing the displacements of the machine actuator provided by the LVDT and those measured using DIC for the clamp frame holding the specimen.

#### 2.2. Experimental Validation of the Proposed Methodology

^{2}, of the measurement differences between 3D-DIC (values in the considered region of analysis) and the LVDT was calculated for each vertical displacement step according to the following expressions [42].

_{i}is the relative error in the measurement. Values for the mean error, µ, and standard deviation of the error, s, at each displacement step were evaluated. Table 2 shows the average mean and the standard deviation for the different displacements steps.

^{−3}mm and the average standard deviation lower than 1·10

^{−3}mm.

^{−3}mm, was placed and vertically aligned with the center of the sphere, in the non-contact side of the specimen. Figure 5 shows the results obtained for four of these indentation experiments using the dial indicator and DIC technique simultaneously. Error bars show a deviation percentage of ±6% respect to the average value. It can be observed that in all the cases, experimental results are within the error bars which indicate that repeatability of the results. During the unloading path, values were also inside of the 6%; however, more scatter in the results was found. Figure 5b shows the same test conditions as in Figure 5a, but in this case 3D-DIC was employed. Figure 5c overlays the results of both techniques. In all the cases, differences are smaller than 6%, using this value as a reference from the previously calculated dispersion bands of 3D-DIC. These differences in the indentation results could be associated to small deviations in the position of the dial gauge or to a slight loss of perpendicularity of the gauge when large deflections are experienced due to bending.

## 3. Results and Discussion

## 4. Conclusions

- Results from the performed experiments made it possible to infer the experimental contact law for 2 mm, 3 mm, 4 mm, 5 mm and 6 mm thick specimens made of AL 1050 H14 and an energy balance was performed to quantify the amount of energy to generate contact damage and bending deformation.
- The recovered elastic energy during the unloading has been also quantified. Higher contact damage was observed as the thickness increases while bending effect is minimized when thickness increases.
- The adopted experimental methodology makes potentially possible the contact behavior evaluation at different loading rates as well as obtaining the real contact law and a complete energy balance during the test. Then, the proposed method could be used to characterize the contact behavior during impact events or creep effects.
- Additional experimentation could highlight some uses as a method to measure the contact laws and harness in complex geometries or scenarios.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic of the setup to perform the experiments; (

**b**) laboratory-adopted setup for the proposed experimental methodology.

**Figure 2.**(

**a**) Schematic illustration showing the indentation measurement principle during a contact experiment which includes bending. (

**b**) Real test illustrating the thickness reduction at the contact area and the indentation measurement.

**Figure 5.**(

**a**) Tests results for the dial indicator setup; (

**b**) tests results for 3D-DIC setup; (

**c**) results comparison using 3D-DIC and the dial indicator.

**Figure 6.**Out-of-plane displacements at the region of analysis when a 2 mm thick specimen is analysed at different loads: (

**a**) 0 mm and 0 N; (

**b**) −0.05 mm and 320 N; (

**c**) −0.096 mm and 611 N; (

**d**) −0.111 mm and 731 N.

**Figure 8.**Experimental contact force for: (

**a**) 3 mm; (

**b**) 4 mm; (

**c**) 5 mm and (

**d**) 6 mm thick specimens.

**Figure 9.**(

**a**) Normalized contact load as a function of the indentation/thickness ratio for different specimen thicknesses, (

**b**) normalized contact load as a function of the maximum deflection/thickness ratio for different specimen thicknesses.

**Figure 10.**(

**a**) Normalized evolution of the applied energy due to contact versus normalized load for different specimen thicknesses. (

**b**) Normalized evolution of the applied energy due to bending versus normalized load for different specimen thicknesses.

**Figure 11.**Applied energy percentage due to contact and bending as a function of the specimen thickness.

**Figure 12.**Percentage of elastic recovered energy due to contact and bending as a function of the specimen thickness.

**Figure 13.**Percentage of absorbed energy due to contact and bending as a function of the specimen thickness.

**Table 1.**Aluminum 1050 H14 properties [34].

AL 1050-H-14 | E (GPa) | ν | Density (kg/m ^{3}) | Yield Stress (MPa) | Hardness (HB) |
---|---|---|---|---|---|

69 | 0.33 | 2700 | 105 | 35 |

**Table 2.**Results of the mean and the variance of the measurement differences between 3D DIC and LVDT for each displacement step.

Step (mm) 0. | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
---|---|---|---|---|---|---|

Mean error (mm) and (%) | 0.97 × 10^{−3}(0.19%) | 1.85 × 10^{−3}(0.19%) | 1.37 × 10^{−3}(0.09%) | 0.73 × 10^{−3}(0.04%) | 0.75 × 10^{−3}(0.03%) | 1.36 × 10^{−3}(0.05%) |

Standard deviation, s, (mm) and (%) | 0.69 × 10^{−3}(0.14%) | 0.61 × 10^{−3}(0.06%) | 0.51 × 10^{−3}(0.03%) | 0.32 × 10^{−3}(0.02%) | 0.32 × 10^{−3}(0.01%) | 0.29 × 10^{−3}(0.01%) |

Thickness (mm) | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|

Total applied energy (J) | 0.80 | 2.06 | 4.21 | 5.53 | 7.31 |

Total applied energy: contact (%) | 4.80 | 8.32 | 12.09 | 14.49 | 16.50 |

Total applied energy: bending (%) | 95.20 | 91.26 | 86.40 | 83.84 | 81.60 |

Total applied energy: system (%) | 0.00 | 0.42 | 1.51 | 1.67 | 1.90 |

Recovered energy (elastic) (J) | 0.38 | 0.80 | 1.50 | 1.90 | 2.42 |

Elastic energy: contact (%) | 3.12 | 5.26 | 7.99 | 10.94 | 13.48 |

Elastic energy: bending (%) | 96.88 | 94.74 | 89.33 | 86.84 | 83.50 |

Energy to generate permanent damage (plastic) (J) | 0.42 | 1.26 | 2.71 | 3.63 | 4.89 |

Energy to generate permanent damage: contact (%) | 6.27 | 10.26 | 14.35 | 16.38 | 18.01 |

Energy to generate permanent damage: bending (%) | 93.83 | 89.06 | 84.78 | 82.28 | 80.61 |

Energy to generate permanent damage: structure (%) | 0.00 | 0.68 | 0.87 | 1.34 | 1.38 |

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**MDPI and ACS Style**

Almazán-Lázaro, J.-A.; López-Alba, E.; Rubio-García, L.; Díaz-Garrido, F.-A. Indentation Measurement in Thin Plates under Bending Using 3D Digital Image Correlation. *Appl. Sci.* **2021**, *11*, 2706.
https://doi.org/10.3390/app11062706

**AMA Style**

Almazán-Lázaro J-A, López-Alba E, Rubio-García L, Díaz-Garrido F-A. Indentation Measurement in Thin Plates under Bending Using 3D Digital Image Correlation. *Applied Sciences*. 2021; 11(6):2706.
https://doi.org/10.3390/app11062706

**Chicago/Turabian Style**

Almazán-Lázaro, Juan-Antonio, Elías López-Alba, Luis Rubio-García, and Francisco-Alberto Díaz-Garrido. 2021. "Indentation Measurement in Thin Plates under Bending Using 3D Digital Image Correlation" *Applied Sciences* 11, no. 6: 2706.
https://doi.org/10.3390/app11062706