# Microhardness, Indentation Size Effect and Real Hardness of Plastically Deformed Austenitic Hadfield Steel

^{*}

(This article belongs to the Section Manufacturing Processes and Systems)

## Abstract

**:**

^{2}). Instead, the P and d parameters obey Meyer’s power law

**(**P = A · d

^{n}) with n < 2. The plastically strained samples showed not only significant work hardening, but also different ISE significance, as compared to the non-deformed bulk steel. After extensive assessment of several theoretical models, including the Hays-Kendall model, Li-Bradt model, Bull model and Nix-Gao model, it was found that the real hardness can be determined by Vickers microhardness indentation and subsequent analysis using the Nix-Gao model. The newly developed method was subsequently utilised in two case studies to determine the real hardness properties of sliding worn surfaces and the subsurface hardness profile.

## 1. Introduction

^{2}

^{2}, kgf, mm

^{2}and mm, respectively. If the units of P and d are Newton (N) and micrometer (μm), respectively, Equation (2) is re-written as Equation (3), with H in the unit GPa [2].

_{0}, and the diagonal length d are kgf, kgf/mm

^{2}and mm, respectively. Li and Bradt modified the Hays–Kendall model by proposing a linear resistance to the diagonal length d [7]. Then, the Li–Bradt model is written, as shown in Equation (6). Bull and co-researchers set up a polynomial P-d relationship, as shown in Equation (7) [4]. Nix and Gao made a different approach to the explanation of the ISE by attributing it to the generation of a strain gradient and geometrically necessary dislocations in the indentation-induced deformation volume, to distinguish from the existing statistically distributed dislocations [8]. The Nix-Gao model is shown in Equation (8), where H and h are the measured hardness and indentation depth, H

_{0}stands for the real hardness and h

^{*}is a constant related to the indenter geometry, the elastic shear modulus and hardening property of the tested material [14]. In Vickers indentation, the Nix-Gao is also written as Equation (9) for the linear relationship between the indentation depth h and the indent diagonal length d.

## 2. Experimental Method and Data Analysis

#### 2.1. The Sample Materials

#### 2.2. Methods of Microhardness Testing and Characterisation

_{0.3}385.2, which showed a measured hardness of HV

_{0.3}381.8 ± 8.0 corresponding to the indent diagonal length of 0.03811 ± 0.00040 mm. Accordingly, the relative deviation of the hardness and diagonal length are −0.5% and 0.3%, respectively. The latter is less than 1.5% as specified by the standard ISO 6507-2. The repeatability and percent bias were calculated, according to the equations defined in the standard ISO 6507-2, to be 5.2% and −0.54%, respectively. These values are below the criteria of the repeatability (8.0%) and percent bias (7.01%) as specified by the standard ISO 6507-2, respectively.

_{α}radiation (wavelength λ = 0.1789 nm) was employed for the XRD analysis. The XRD scans were conducted using the θ−2θ scan mode, with a step size of 0.026° and a scanning speed of 0.0022°·s

^{−1}.

#### 2.3. Methods of Data Analyses

#### 2.3.1. Calculations of the Meyer Index n, Real Hardness H_{0}, and ISE Significance Coefficient η

_{0}. The calculations were based on Equations (5)–(8), and performed using the regression analysis function provided in MS Excel. Details of the calculation methods are summarized in Table 1. In case of the Hays–Kendall model, for example, the indentation loads {P} were plotted versus the square of indent diagonal length {d

^{2}}, followed by a linear regression to deduce the real hardness H

_{0}and the constant indentation resistance W, according to Equation (5). After that, an ISE significance coefficient was defined as η = $\frac{H-{H}_{0}}{{H}_{0}},$ according to the measured indentation hardness H and the real hardness H

_{0}. Thus, a positive or negative η value means the presence of ISE, whereas η = 0 stands for an ISE-free state. The calculations of the coefficient are also provided in Table 1.

#### 2.3.2. Prediction of Hardness Using the Theoretical Models

_{1}and H

_{2}, where H

_{1}is the hardness value determined from the load P and diagonal d using Equation (3), and H

_{2}is the hardness value determined by the related theoretical model. When both H

_{1}and H

_{2}vary with the diagonal d, there should be only one d value which satisfies both Equations (2) and (9), i.e., making H

_{2}= H

_{1}. Then, the projected hardness can be determined. Similarly, Equations (11)–(13) are developed to project the hardness values using the Li–Bradt model, Bull model and Nix-Gao model, respectively.

## 3. Results

#### 3.1. Effect of Straining on The Microstructure of the Austenitic Hadfield Steel

#### 3.2. The Vickers Microhardness Property Determined by Indentation

#### 3.3. The Real Hardness and ISE Significance Determined Using the Theoretical Models

^{2}≥ 0.999. The values of Meyer index n are not equal to, but less than, 2, suggesting positive ISE. The constant A also varies between the three samples, and the rail top shows the maximum value because of its highest hardness as shown in Figure 4.

^{2}suggests that the four models have good feasibility in analysing the real hardness and the ISE phenomenon. The real hardness of the three samples is listed in Table 3, in which the results show good consistency. Compared to the low hardness of the bulk steel, the tensile sample reached a high hardness of 443 kgf/mm

^{2}. The rail top gained a hardness of 697 kgf/mm

^{2}, which can be considered as the maximum achievable hardening when the associated embrittlement became sufficient to trigger spalling and delamination failures. Considering the high relevance factor R

^{2,}as shown in Figure 6, an attempt was made to undertake the same modelling analyses using the hardness data obtained in the microhardness range, i.e., from 0.01 to 0.2 kgf. The determined real hardness values are shown in Table 4. Comparing Table 3 and Table 4, the real hardness of the two strained samples shows little change, despite a slight increase in the relative deviation. The results suggest that the real indentation hardness can be measured with reasonably high accuracy by using the indentation loads not exceeding 0.2 kgf.

#### 3.4. Compatibility of the Theoretically Projected Hardness to the Measured Hardness

#### 3.5. Case Studies: Initial Applications of the Established Method

#### 3.5.1. The Microhardness Properties of the Worn Surfaces of Steels Having Different Microstructure

_{N-G}, determined using the Nix-Gao model, and the microhardness H

_{0.2}of the six samples. The real hardness H

_{N-G}differs from the microhardness H

_{0.2}in all cases, due to the influence of the ISE. Comparing to the bulk steel, the wear-induced hardening of the austenitic Hadfield steel is the most significant, followed by the pearlitic rail steel. The 300M steel also exhibits a certain scale of hardening, suggesting that the applied sliding wear was able to make further hardening to the martensite microstructure.

^{2}. The samples, however, show quite different values of Meyer’s index n. The Hadfield steel worn surface shows a higher index of 1.81 than the bulk steel (n = 1.69), suggesting decreased ISE significance of the worn surface. The worn surfaces of the pearlitic and martensitic steels show a decreased index as compared to the bulk samples, suggesting increased ISE significance of the worn surfaces.

#### 3.5.2. The Subsurface Hardness Profile of the Worn Hadfield Steel Turnout

^{2}. In contrast, the microhardness values measured at the highest indentation load of 0.2 kgf are the lowest. The real hardness at each depth is even lower than the values of the microhardness.

^{2}, following by decreasing values, with increasing depth. Within the depth of 1.5–2.0 mm, the real hardness is higher than 400 kgf/mm

^{2}, being lower than the real hardness of the rail top (689 kgf/mm

^{2}, Figure 8c) and the worn surface (550 kgf/mm

^{2}, Figure 10b), but higher than or equivalent to the hardness obtained on the strained tensile bar (443 kgf/mm

^{2}, Figure 8b).

## 4. Discussion

#### 4.1. Selection of Theoretical Models to Determine the ISE-Independent Real Hardness

_{0}of micro-scale volumes where macro-scale indentation is impossible. Comparing between the real hardness and the conventional Vickers microhardness, a drawback of the latter is obvious. The microhardness property has been shown to depend strongly on the applied independent load (Figure 4, Figure 10a and Figure 12). The real hardness, on the other hand, approached the hardness measured at macro-scale loads, which makes it independent to the indentation load (Table 1, Table 2, Table 3 and Table 4). Because of the ISE and its dependence on indentation load and on material characteristics, large uncertainties are expected in a comparative study of microhardness properties of different materials, or when the properties are obtained at different indentation conditions. These uncertainties have been overcome by measuring the real hardness, as is demonstrated in the two cases (Figure 10 and Figure 12).

#### 4.2. The Origin of ISE

^{2}. In fact, Kick’s law is a special case of Meyer’s power law, in which the index n is assumed to be 2, whereas in most cases n ≠ 2 has been experimentally confirmed.

^{2}. This is true especially when the applied indentation loads fell into the microhardness range.

#### 4.3. Effect of Mechanical Straining on the ISE and Real Hardness of Hadfield Steels

_{0.01}945 and HV

_{1}718, corresponding to the indentation loads of 0.01 and 1 kgf, respectively, Accordingly, the strain-hardening ratio against the non-deformed bulk steel is 2.4 and 3.3, respectively, whereas the average strain-hardening ratio calculated from the microhardness range of 0.01–0.2 kgf is 2.6. Meanwhile, another strained sample, the tensile bar, shows strain-hardening ratios different from the rail top sample. These differences suggest that the experimentally measured strain-hardening severity depends both on the material, i.e., the plastic deformation scale, and on the applied indentation load. The method reported in this paper brings about the feasibility of quantifying the hardness and strain-hardening properties by using microhardness measurements.

#### 4.4. Effect of Microstructure on the ISE Properties

## 5. Conclusions

- The real hardness of small-scale materials can be determined by Vickers microhardness indentation and subsequent analysis using the Nix-Gao model. The origin of ISE derives from the mismatch between the experimentally determined P ~ d relationship and Kick’s law (P = A · d
^{2}). Within the limitation of the present experimental activity, the results suggest that Kick’s law should be replaced by Meyer’s power law (P = A · d^{n}) with n < 2. - Because of the ISE, indentation hardness measured under small loads does not present the real hardness property and the real strain-hardening ratio. The plastically strained samples exhibited not only strong work hardening, but also different ISE significance, as compared to the non-deformed bulk steel. The bulk steel retained the lowest A and n values, obtaining the lowest hardness and the strongest ISE significance. The strained samples showed increased an A value, with the n approaching 2, indicating higher hardness and lower ISE significance, respectively. The sample experienced the extreme plastic straining showed the highest A value, reaching a real hardness of 689 kgf/mm
^{2}. - When the indentation loads were in macro scale, the four theoretical models show good precision in calculating the real hardness and in predicting the Vickers hardness values at various loads. When the indentation loads were in micro scale, the Nix-Gao model outperformed the other theoretical models in these calculations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Photographic illustration of the hardness tester employed in the experiments: part (

**A**) is the main body of the tester; parts (

**B**,

**C**) are the monitors of the control panel. The insert 1 in Part (

**A**) shows details of the indenter, a sample under indentation and the optical lens for observation; The inserts 2 shows the measurement of the two diagonal lengths, d

_{1}and d

_{2}; and the insert 3 shows the precise positioning of an indent tip point.

**Figure 2.**The microstructure of Hadfield steel under various strained conditions: (

**a**) the bulk steel; (

**b**) the 47%-elongated tensile bar; and (

**c**) the worn top of turnout.

**Figure 3.**The X-ray diffraction curves of the Hadfield steel under various strained conditions, showing different broadening of the austenite diffraction peaks.

**Figure 4.**The microhardness properties of the three Hadfield steel samples plotted versus the applied indentation load.

**Figure 5.**Meyer’s P~d profiles of the Hadfield steel, showing effect of plastic strain on the power law relationship.

**Figure 6.**Linear regression plots to calculate the real hardness of the three Hadfield steel samples using the theoretical models: (

**a**) using the Hays–Kendall model; (

**b**) using the Li–Bradt model; (

**c**) using the Bull model; and (

**d**) using the Nix-Gao model.

**Figure 7.**The ISE significance coefficient η of the three Hadfield steel samples determined using the theoretical models: (

**a**) using the Hays−Kendall model; (

**b**) using the Li−Bradt model; (

**c**) using the Li−Bradt model; and (

**d**) using the Nix−Gao model.

**Figure 8.**Comparison between the measured hardness and the hardness projected using the theoretical models: (

**a**) the bulk steel; (

**b**) the 47%−elongated tensile bar; and (

**c**) the rail top.

**Figure 9.**SEM micrographs showing the worn surfaces: (

**a**) the M18 austenitic Hadfield steel; (

**b**) the pearlitic rail steel; and (

**c**) the martensitic 300 M steel.

**Figure 10.**The hardness and ISE properties of the three different microstructure steels and their worn surfaces: (

**a**) the measured microhardness plotted versus indentation load; (

**b**) the real hardness (H

_{N-G}determined from the Nix-Gao model) as compared to the microhardness at the indentation load 0.2 kgf (H

_{0.2}); (

**c**) Meyer’s power profiles; and (

**d**) the ISE significance coefficient η.

**Figure 11.**Cross-sectional microstructure of the worn Hadfield steel turnout: (

**a**) an optical image showing deformation bands and cracks from the rail top at a certain depth; and (

**b**) a SEM image showing different orientations of mechanical twins beside a grain boundary.

**Table 1.**The regression analysis to calculate the real hardness

**H**and ISE significance coefficient

_{0}**η**.

Models and Equations | $\mathbf{Regression}:\phantom{\rule{0ex}{0ex}}\mathit{Y}={\mathit{a}}_{0}+{\mathit{a}}_{1}\xb7\mathit{X}$ | H_{0} | ISE Parameter | η | ||
---|---|---|---|---|---|---|

Y | X | |||||

Hays-Kendall | P = W + $\frac{{H}_{0}}{1.8544}$ · d^{2} | P | d^{2} | 1.8544 · ${a}_{1}$ | W = ${a}_{0}$ | $\frac{W}{P-W}$ |

Li-Bradt | P = a · d + $\frac{{H}_{0}}{1.8544}$ · d^{2}$\frac{P}{d}$ = a + $\frac{{H}_{0}}{1.8544}\xb7d$ | $\frac{P}{d}$ | d | 1.8544 · a_{1} | a = ${a}_{0}$ | $\frac{1.8544\xb7a}{d\xb7{H}_{0}}$ |

Bull | P = W + a · d + $\frac{{H}_{0}}{1.8544}$ · d^{2}(Y = a _{0} + a_{1} · X + a_{2} · X^{2}) | P | d | 1.8544 · a_{2} | W = ${a}_{0}$ a = ${a}_{1}$ | $\frac{1.8544}{d\xb7{H}_{0}}$ · (a_{1} + $\frac{{a}_{0}}{d}$) |

Nix-Gao | H = H_{0} · $\sqrt{1+\frac{{d}^{*}}{d}}$${H}^{2}={H}_{0}^{2}$ +${H}_{0}^{2}$·$\frac{{d}^{*}}{d}$ | ${H}^{2}$ | ${d}^{-1}$ | $\sqrt{{a}_{0}}$ | ${d}^{*}$ = $\frac{{a}_{1}}{{a}_{0}}$ | $\sqrt{1+\frac{{d}^{*}}{d}}$ − 1 |

**Table 2.**The average hardness and deviation of the three Hadfield steel samples measured at various loads.

Sample | Indentation Load, kgf | |||||||
---|---|---|---|---|---|---|---|---|

0.01 | 0.025 | 0.05 | 0.1 | 0.2 | 0.3 | 0.5 | 1.0 | |

Rail Top | 945 ± 57 | 835 ± 39 | 808 ± 28 | 770 ± 48 | 744 ± 32 | 751 ± 29 | 712 ± 43 | 718 ± 34 |

Tensile Strain 47% | 565 ± 102 | 535 ± 76 | 520 ± 64 | 495 ± 62 | 473 ± 31 | 443 ± 37 | 457 ± 44 | 452 ± 44 |

Bulk Steel | 386 ± 26 | 366 ± 28 | 305 ± 23 | 281 ± 5 | 265 ± 16 | 233 ± 15 | 227 ± 9 | 220 ± 6 |

**Table 3.**Determination of the real hardness (H

_{0}, in kgf/mm

^{2}) of the strained Hadfield steel using the theoretical models.

Theoretical Models | Bulk Steel | Tensile Bar | Rail Top |
---|---|---|---|

Hays-Kendall | 217 ± 3 | 449 ± 2 | 713 ± 4 |

Li-Bradt | 201 ± 5 | 436 ± 3 | 693 ± 7 |

Bull | 210 ± 13 | 443 ± 7 | 689 ± 7 |

Nix-Gao | 210 ± 13 | 443 ± 7 | 689 ± 7 |

Average | 210 | 443 | 697 |

deviation | 3.1% | 1.2% | 1.9% |

**Table 4.**Determination of the real hardness of the strained Hadfield steel using indentation loads from 0.01 to 0.2 kgf.

Theoretical Models | Bulk Steel | Tensile Bar | Rail Top |
---|---|---|---|

Hays-Kendall | 258 ± 4 | 467 ± 6 | 734 ± 6 |

Li-Bradt | 232 ± 8 | 448 ± 8 | 696 ± 5 |

Bull | 226 ± 9 | 414 ± 6 | 680 ± 9 |

Nix-Gao | 234 ± 21 | 459 ± 10 | 682 ± 9 |

Average | 238 | 447 | 698 |

deviation | 5.6% | 5.2% | 3.6% |

**Table 5.**The absolute difference (ΔH) between the projected and measured hardness in the load range 0.02–0.2 kgf. (ΔH = H

_{measured}− H

_{projected}for H

_{projected}< H

_{measured}; ΔH = H

_{projected}− H

_{measured}for H

_{projected}> H

_{measured}).

Models | Bulk Steel | Rail Top | Tensile Bar | |
---|---|---|---|---|

Hays-Kendall | Range | 19–313 | 3–104 | 9–56 |

mean ± dev | 96 ± 145 | 32 ± 48 | 23 ± 22 | |

Li-Bradt | Range | 2–106 | 1–15 | 2–41 |

mean ± dev | 29 ± 43 | 7 ± 6 | 14 ± 16 | |

Nix-Gao | Range | 5–30 | 1–14 | 0–17 |

mean ± dev | 14 ± 10 | 6 ± 5 | 9 ± 6 | |

Bull | Range | 4–150 | 3–41 | 2–75 |

mean ± dev | 39 ± 63 | 20 ± 14 | 22 ± 30 |

Properties | Rail Top | Tensile Bar | Bulk Steel | |
---|---|---|---|---|

Meyer’s constant A and index n | A | 0.00057 | 0.00036 | 0.00033 |

n | 1.891 | 1.896 | 1.761 | |

ISE significance coefficient η | Mean value | 0.14 | 0.11 | 0.36 |

Range | 0.04–0.37 | 0.02–0.27 | 0.05–0.84 | |

Hardness H | H_{0} | 689 | 443 | 210 |

HV_{0.01} | 945 | 565 | 386 | |

HV_{1} | 718 | 452 | 220 | |

Strain-hardening ratio ($\frac{H-{H}_{bulk}}{{H}_{bulk}}$ · 100%) | By H_{N-G} | 3.3 | 2.1 | 1.0 |

By HV_{0.01} | 2.4 | 1.5 | 1.0 | |

By HV_{0.01–0.2} | 2.6 | 1.6 | 1.0 | |

By HV_{1} | 3.3 | 2.1 | 1.0 |

Steels | Meyer’s Index n | ISE Significance Coefficient η | ||
---|---|---|---|---|

Bulk | Worn | Bulk | Worn | |

Austenitic Mn18 | 1.69 | 1.81 | 0.63 | 0.29 |

Pearlitic rail | 1.92 | 1.78 | 0.10 | 0.31 |

Martensitic 300M | 1.89 | 1.77 | 0.13 | 0.42 |

Mean | 1.83 | 1.79 | 0.29 | 0.34 |

Deviation | 0.13 | 0.02 | 0.30 | 0.07 |

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**MDPI and ACS Style**

Luo, Q.; Kitchen, M.
Microhardness, Indentation Size Effect and Real Hardness of Plastically Deformed Austenitic Hadfield Steel. *Materials* **2023**, *16*, 1117.
https://doi.org/10.3390/ma16031117

**AMA Style**

Luo Q, Kitchen M.
Microhardness, Indentation Size Effect and Real Hardness of Plastically Deformed Austenitic Hadfield Steel. *Materials*. 2023; 16(3):1117.
https://doi.org/10.3390/ma16031117

**Chicago/Turabian Style**

Luo, Quanshun, and Matthew Kitchen.
2023. "Microhardness, Indentation Size Effect and Real Hardness of Plastically Deformed Austenitic Hadfield Steel" *Materials* 16, no. 3: 1117.
https://doi.org/10.3390/ma16031117