# Cyclic Indentation of Iron: A Comparison of Experimental and Atomistic Simulations

^{1}

^{2}

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## Abstract

**:**

_{a,p}on the cycle number. In both approaches, we obtain a power-law demonstrating h

_{a,p}with respect to the hardening exponent e. A detailed analysis of the atomistic simulation results shows that changes in the dislocation network under repeated indentation are responsible for this behavior.

## 1. Introduction

_{CHT}[11,12], which is described further in Section 2. In order to increase the comparability, we used the same method to analyze the outcome of the simulation and experiment. Additionally, the substrate material on which the cyclic indentation took place in the simulation was chosen to be as comparable as possible to that in the experiment (e.g., regarding the crystallographic structure of the indented surface). This comparison will allow us to obtain an understanding of the plastic processes occurring during indentations, and their dependence on cycle number. Furthermore, evaluation of the cyclic hysteresis behavior of the indentation in the simulation and experiment allow us to quantitatively demonstrate the similarities by calculating the key parameters describing cyclic indentation behavior, such as hardening-exponent

_{CHT}.

## 2. Materials and Experimental Method

_{CHT}was employed, as described by Kramer et al. [11] and developed at TU Kaiserslautern, Institute of Materials Science and Engineering. The results of the PHYBAL

_{CHT}method, which induces multiaxial compressive loadings, was proven to show a good correlation with uniaxial compression fatigue tests. This confirms that the hardening-exponent

_{CHT}(e

_{II}) obtained from PHYBAL

_{CHT}can be compared with the cyclic hardening behavior of materials in uniaxial fatigue loading conditions [12]. Further studies by Bambach et al. [14] have demonstrated a successful application of the PHYBAL

_{CHT}(e.g., to determine optimized heat treatment conditions of high strength steel for maximum damage tolerance during fatigue loadings). The present paper, however, focuses on understanding the local microstructural changes during cyclic indentation with help of MD simulations.

_{a,p}) was defined as the half-width of this hysteresis (see Figure 2a). The maximum indentation force chosen for this investigation was 500 mN, and therefore the hysteresis width was measured at 250 mN. Equation (1), a power-law function, was employed to describe the hysteresis width vs the number of cycles after the sixth cycle:

_{a,pII}is the half-width of the hysteresis after the sixth indentation cycle, a

_{II}is the coefficient of the h

_{a,p,II}-N curve, N is the number of indentation cycle, and e

_{II}is the hardening-exponent.

_{a,p}-N relation (green line in Figure 2b) was described by dividing it into two sections. Up to the fourth cycle, where the slope of the h

_{a,p}-N was not constant, the hysteresis width of the cyclic indentation was described by adding up the two power-law functions h

_{a,pI}(dark blue line in Figure 2b) and h

_{a,pII}(red line in Figure 2b). This change in the slope shows that the macroscopic plastic deformation processes were not completed until the fourth cycle. After the fourth cycle with the completion of macroplastic deformation, the slope of the h

_{a,p}-N became constant (i.e., the curve can be described with one power-law function h

_{a,pII}in order to evaluate the cyclic microplastic behavior of the material). The slope of the h

_{a,pII}-N after the fourth cycle is denoted as e

_{II}.

_{a,p}-N curve, occurred after the sixth cycle. As a result, the power-law Equation (1) provided a better fit to the experimental data after the sixth cycle. This variation in the load-depth curve was because of the different materials on which the cyclic indentation took place. In this work, we have used GOES, and the indentation took place in the middle of a large grain in contrast to [11,12], where fine-grained 18CrNiMo7-6 and 42CrMo4 were investigated.

## 3. Simulation Method

^{3}[17,18]. The crystal had a lateral extension of 65.7 nm and a depth of 42.8 nm; it contained 15,922,900 atoms.

_{disl}. The free software tools ParaView [21] and OVITO [22] were employed to visualize the atomistic configurations.

## 4. Experimental Results

_{max}was measured and plotted vs N.

_{a,p}to calculate the hardening-exponent e, was measured at 50% of the maximum indentation force based on the abovementioned PHYBAL

_{CHT}method.

_{II}between N = 6 and 10. For the investigated material with the abovementioned characteristics, e

_{II}was calculated to −0.57.

## 5. Simulation Results

#### 5.1. Plasticity

_{pl}of the farthest dislocation segment from the center of the contact area. Only dislocations that were adjacent to the indent pit are counted here, while emitted dislocation loops were discarded. Figure 10 shows that upon multiple indentations, the plastic zone increased in size; the effect amounted to around 10%. This was most pronounced in the first five cycles, before saturating. This result quantifies the size of the dislocation networks observed in the snapshots (see Figure 7).

_{dis}

_{l}have to be divided by the volume of the plastic zone. This results in

^{8}cm

^{−2}, since at these densities the probability of finding a dislocation line in the simulation volume is negligible. After the first cycle, it increased to 10

^{12}cm

^{−2}, which is in agreement with the literature [21].

^{12}cm

^{−2}) [27]. This reduced under subsequent indentation cycles, mainly because dislocation reactions simplify the network. Strain hardening (i.e., the process where subsequent cycles increase the dislocation density) cannot occur at these high densities. This feature may be in contrast with indentations possessing larger indenter radii; these have a correspondingly increased plastic zone and a reduced dislocation density, which may allow for strain-hardening reactions.

#### 5.2. Load-Depth Curves

_{a,p}as a function of the number of indentation cycles N. We calculate 2h

_{a,p}from the load-depth curves of Figure 13 as an average of the widths determined at 40%, 50%, and 60% of the maximum load. The double-logarithmic plot, Figure 15, reveals that the curve can be roughly approximated by a power-law, Equation (1). We note that in simulation both the maximum indentation depth, Figure 14, and the width of the hysteresis loop, Figure 15, show a tendency of saturating with cycle number, while in the experiment they appeared to continue changing, see Figure 5 and Figure 6. The stabilizing trend was caused by the smaller plastic volume in the simulation, where dislocation reactions ceded after around five cycles (see the discussion above), while in the larger plastic zone in the experiment, reactions continued beyond 10 cycles.

_{a,p}only changed between 0.62 and 3.29 Å. Compared to the nearest-neighbor distance in Fe, 2.47 Å, these changes are of atomic dimensions or below. Note that the determination of the exact power exponent in Equation (1) is subjected to some arbitrariness, see Figure 16, where two fits are provided. However, similar power-laws were found in the experiment [11], albeit with smaller exponents.

## 6. Conclusions

- Under subsequent indentations: (i) the dislocation network widens slightly, since the stress gradients exerted by the indenter move the farthest shear loops further out; (ii) the network simplifies in the vicinity of the indent pit, since the highly non-equilibrium dislocation structure generated in the first indentation is subject to dislocation reactions when dislocations move in the stress fields of the subsequent indentation cycles.
- Both effects lower the dislocation density. We thus observe in the atomistic simulations effectively a softening under cyclic indentation.
- As a consequence, the indentation depth increases slightly under constant load.
- As the changes in the dislocation network become more and more negligible, so do the changes in the indentation depth and hysteresis width.

_{a,p}can be associated with the freezing-in of dislocation reactions. While dislocations still moved under the time-varying stress fields exerted by the indenter, their motion became reversible in the sense that the network topology did not change, and energy was only dissipated by dislocation mobility. In larger-scale experiments, dislocation immobilization by pinning and formation of junctions would have a similar effect. Since the hysteresis is associated with the plastic work done under indentation, consequently, the hysteresis widths decrease.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic grain structure of the Goss texture longitudinal to the rolling direction (

**a**) and the cyclic hardness indentation (

**b**) of investigated electrical steel sheet.

**Figure 4.**Hysteresis curve formed after the second indentation cycle, and plastic indentation depth amplitude h

_{a,p}at 250 mN.

**Figure 5.**Dependence of maximum indentation depth (at F

_{max}) on the indentation cycle number in the experiment.

**Figure 6.**Plastic indentation depth amplitude h

_{a,p}vs indentation cycles N and the power-law describing the h

_{a,pII}with respect to N for N ≥ 6.

**Figure 7.**Snapshots of the dislocation network after (top) the first and (bottom) tenth indentations. Yellow: deformed surface and other unidentified defects. Dislocations are colored according to their Burgers vector b: blue ½ <111>, red <100>.

**Figure 8.**Dependence of the total dislocation length during cyclic indentation at full penetration and after the retraction of the indenter on cycle number.

**Figure 9.**Change of the dislocation structure during the third cycle. Colors denote dislocation type as in Figure 7. The white circle highlights the region where the most changes in dislocation structure occurred during this cycle.

**Figure 12.**Comparison of the (

**right**) hydrostatic pressure and (

**left**) dislocation in the specimen at the end of the 10th cycle. Positive pressures are compressive, negative pressures are tensile. Dislocations are colored as in Figure 7. The figure shows a cross-sectional view on the indented specimen; the thickness of the slab in the left-hand image is 2 nm.

**Figure 15.**Dependence of the hysteresis width 2h

_{a,p}on cycle number and fits a power-law, Equation (1). Two fits are provided to demonstrate the limits of the power-law exponents obtained.

**Figure 16.**Comparison of experimental and simulated hysteresis widths as a function of cycle number N.

Chemical Composition | C | Mn | Si | Fe |
---|---|---|---|---|

In wt% | 0.007 | 0.080 | 3.230 | Bal. |

In at% | 0.03 | 0.76 | 6.22 | Bal. |

Description | Value |
---|---|

Elastic modulus | 110 ± 10 GPa |

Yield stress | 320 ± 5 MPa |

Tensile strength | 325 ± 2 MPa |

Hardness | 180 ± 5 HV0.1 |

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**MDPI and ACS Style**

Deldar, S.; Alabd Alhafez, I.; Smaga, M.; Beck, T.; Urbassek, H.M.
Cyclic Indentation of Iron: A Comparison of Experimental and Atomistic Simulations. *Metals* **2019**, *9*, 541.
https://doi.org/10.3390/met9050541

**AMA Style**

Deldar S, Alabd Alhafez I, Smaga M, Beck T, Urbassek HM.
Cyclic Indentation of Iron: A Comparison of Experimental and Atomistic Simulations. *Metals*. 2019; 9(5):541.
https://doi.org/10.3390/met9050541

**Chicago/Turabian Style**

Deldar, Shayan, Iyad Alabd Alhafez, Marek Smaga, Tilmann Beck, and Herbert M. Urbassek.
2019. "Cyclic Indentation of Iron: A Comparison of Experimental and Atomistic Simulations" *Metals* 9, no. 5: 541.
https://doi.org/10.3390/met9050541