Toward the correlation of indentation hardness in micro- and nano-scale: understanding of indentation edge behaviors in Fe–Cr alloys

Abstract

Nanoindentation hardness tests are used to measure indentation hardness at the micro- and nanoscales and further to predict Vickers hardness on larger scales. Hence, the relationship between Vickers and nanoindentation hardness has gained considerable research interest. Here we introduce the concept of Meyer hardness as a mean contact pressure correctly into the two hardness in order to explain their linear correlation. The Vickers hardness is converted to the Vickers Meyer hardness (HVM) as defined by a load divided by the projected contact area; the Nix–Gao model is used to calculate the bulk-equivalent nanoindentation hardness (H₀) from depth-dependent nanoindentation hardness. A linear relationship HVM = 0.86 × H₀ is observed in the Fe–Cr alloys with a wide range of elastic modulus to hardness ratios and is suggested to be a universal relationship for various metallic materials. A tangent method can distinguish the indentation edge behaviors such as pile-up, sink-in, and the pseudo-pile-up phenomenon. A novel pile-up correction estimates the real contact area in nanoindentation tests, and actual residual contact area is measured to correct the pile-up around Vickers imprints. The corrected HVM and corrected H₀ show almost same values after both pile-up corrections.

Appendices

Appendix

Appendix 1: Pile-up Area Calculation

In Fig. 2, $l$ and $h$ are the horizontal length and height, respectively.The details for the pile-up area calculation are

$$A_{c} + A_{p} = \frac{\sqrt 3 }{4} \times W^{2} + 3 \times \frac{1}{2} \times W \times a, A_{c + P} = \frac{\sqrt 3 }{4} \times \left( {W + 2\sqrt 3 l} \right)^{2}$$

Let $A_{c} + A_{p}$ be equal to $A_{c + P}$; then,

$$3Wl + 3\sqrt 3 l^{2} = 3 \times \frac{1}{2} \times W \times a$$

Here, $A_{c} + A_{p}$ and $A_{c + P}$ represent the residual projected areas calculated step-by-step and one-time, respectively. In Fig. 2, $A_{c}$ represents the central triangle (blue plus gray), which includes the conventional contact area and the pile-up around the corners; $A_{p}$ represents the total of three red triangles outside.

The high-order minimum can be neglected because $l$ is considerably smaller than the length of the edges $W$. Thus, we have

$$l = \frac{1}{2} \times a = \frac{1}{2} \times \left( {l_{{p\left( {edge} \right)}} - l_{{p\left( {corner} \right)}} } \right)$$

$$l_{p} = \frac{1}{2} \times \left( {l_{{p\left( {edge} \right)}} + l_{{p\left( {corner} \right)}} } \right)$$

Similarly,

$$h_{p} = \frac{1}{2} \times \left( {h_{{p\left( {edge} \right)}} + h_{{p\left( {corner} \right)}} } \right)$$

where $h_{p}$ represents the equivalent pile-up height.

The following $h_{total}$ will be used for the actual projected area.

$$h_{total} = h_{c} + h_{p} = h_{c} + \frac{1}{2} \times \left( {h_{{p\left( {edge} \right)}} + h_{{p\left( {corner} \right)}} } \right)$$

APPENDIX 2: Residual Nanoindentation Measurement

The twelve selected specimens were used to measure the pile-up height; two nanoindentation imprints were used for confocal laser microscope measurements. (Fig. 9) For each triangle nanoindentation, ten parallel line analyses were set perpendicular to each edge of one imprint, and the highest point of each line was used to determine the maximum pile-up height of an edge ($h_{{p\left( {edge} \right)}}$) as shown in Fig. 10. The maximum pile-up height was used as the pile-up height for the tangent method.

In addition, three-line analyses were set along the traces left by the edges of the triangle imprint to obtain the pile-up height of the corners ($h_{{p\left( {corner} \right)}}$) as shown in Table 1.

Figure 9

CLM measurements on the nanoindentation imprint for the pile-up around the edge (a) and the pile-up around the corner (b). Red lines represent the line analysis in the CLM measurements

Figure 10

An example of pile-up height analysis of 1000 h Fe-30Cr alloy obtained by confocal laser microscope

See Figs 9 and 10 Table 1

Appendix 3: Indentation Elastic Modulus in Nanoindentation Tests

The elastic modulus is obtained using the nanoindentation test. Figure 11 shows an example on 1000h Fe-50Cr.

Figure 11

Elastic modulus obtained via nanoindentation test on 1000 h Fe-50Cr

See Fig. 11

Appendix 4: Hardness in Nanoindentation Tests

The raw data of the nanoindentation tests are used for the bulk-equivalent nanoindentation hardness using the Nix–Gao model, as shown in Fig. 12.

See Fig. 12.

Figure 12

(a-c) Nix–Gao model for Fe–Cr alloys aged for different hours. (d) Indentation depth of Vickers imprints

Appendix 5: Ratio of ${\varvec{E}}_{{{\varvec{IT}}}} /{\varvec{HVM}}_{0.1}$ in Aged Fe-Cr Alloys

Based on the HV (H) and average elastic modulus (E) in the nanoindentation test, the $E_{IT} /HVM_{0.1}$ ratio is shown in Fig. 13. The specimens marked by dashed lines were selected to analyze the pile-up height using a confocal laser microscope. Modulus.

See Fig. 13.

Figure 13

Ratio of E_IT/HVM_0.1 in Fe–Cr alloys

Appendix 6: ImageJ Analysis

See Fig. 

Figure 14

Image area analysis from the residual surface area measured by the diagonals (left) to the residual projected area (right)

14 and Table 2.

Appendix 7: Corrected Nanoindentation Hardness

After the pile-up analysis, the corrected bulk-equivalent nanoindentation hardness H₀ was calculated using the novel correction method. The average bulk equivalent nanoindentation hardness values of the two nanoindentations used for the pile-up analysis before and after the correction are shown in Table 3. Here, only specimens showing pile-up behavior are included.

See Table 3.