A. Load versus depth curves

The effect of the angular misalignment on indentation load versus depth measurement was numerically investigated. The indenter was deliberately tilted over an angle of 0, 0.25°, 0.5°, 0.75°, and 1°, respectively, with respect to its perfectly aligned position. The corresponding load–displacement curves were recorded in Fig. 3, where an increasing angle of misalignment leads to a decrease load, especially in the earlier stage of the indentation due to the point contact and localized plastic deformation.

FIG. 3.

Load versus depth for various angular misalignment at maximum depth of 0.35 μm.

B. Young’s modulus and hardness

The mechanical properties, such as Young’s modulus E and hardness H, were calculated from the nanoindentation load-depth modeling data and listed in Table II, where the misalignment angle ranges from 0° to 1° and the maximum indentation depth from 0.3 μm to 0.4 μm. It can be seen that the influence of the misalignment angle and the maximum depth on Young’s modulus is less than ±0.5% and can be neglected. The reason is that the difference of the contact areas between a perfect alignment case and a misalignment angle of 1° case is less than 1‰ which can be neglected, and after fully loaded indentation, the unloading for both cases would follow the same linear elastic pathway, on which the Young’s moduli were calculated. Therefore, the effect of a small misalignment angle can be neglected. The contact area used in Table II is based on a perfect alignment indentation and the very small perturbation of Young’s modulus values extracted from modeling is within the simulation error. While the influence on hardness close to ±10%, and as the misalignment angle increases the error increases. The reason is that the hardness is very sensitive to the indentation depth, since AISI 1018 is a elastoplastic material with strain hardening. A larger indentation depth leads to a larger strain, a higher deformation load, and a greater hardness, thus, proper defining the indentation depth for a misalignment model is critical. After adjusting the maximum depth (listed as h ^r _max in the last column in Table II) by counting the center point on the bottom area of the flat-tip indenter as the contact point, the error induced by the angular misalignment on the hardness measurement was minimized, and the hardness versus the revised maximum depth are consistent whatever the misalignment angles are, as shown in Fig. 4.

TABLE II.

Nanoindentation data analysis for different misalignment angles and at various maximum depths (θ = 0°, 0.25°, 0.5°, 0.75°, and 1°, h _max = 0.3, 0.35, and 0.4 μm).

Note: ${h'_{\max }} = {h_{\max }} - a\,\sin \theta$ , where a is the radius of the cylindrical flat-tip indenter.

FIG. 4.

Hardness versus revised maximum depth for various misalignment angle (θ = 0°, 0.25°, 0.5°, 0.75°, and 1°).

C. Stress–strain development at the beginning of indentation

The stress–strain development in the beginning of the indentation process for different angular misalignments is shown in Figs. 5 and 6. It is observed that for the case of having an angular misalignment, even for a small load a local plastic deformation occurred near the edge of the indenter tip. As the misalignment angle increases, the model makes a steady hemispheric stress and strain distribution at a greater depth. Therefore, instead of an idealized plane contact at the very beginning of the indentation, with an angular misalignment, the stress and strain development process is starting from a point contact to a line contact and then plane contact, and a deeper depth is needed to form a steady hemispherical stress or strain distribution as the misalignment angle gets larger. In these cases, a modified depth, corresponding to the formation of a steady hemispheric stress and strain distribution, should be considered to find the yield strength.

FIG. 5.

von Mises stress development in the beginning of the indentation process for different angular misalignments (θ = 0°, 0.25°, and 1°).

FIG. 6.

von Mises strain development in the beginning of the indentation process for different angular misalignments (θ = 0°, 0.25°, and 1°).

D. Yield strength estimation

If a perfect elastic model is considered, ignoring the plasticity of the material, the indentation load P _e is proportional to the Young’s modulus of the specimen E and the ratio of the indentation depth to the indenter radius h/a. ^{Reference Hu, Lynne, Markondapatnaikuni and Delfanian5,Reference Hu, Lynne and Delfanian6} The comparison of the indentation load versus depth curves for the perfectly elastic model and the elastoplastic model is shown in Fig. 7. It shows that when the model is perfectly aligned (misalignment angle θ = 0°), the load of an elastoplastic material at the depth corresponding to the hemispherical stress–strain distribution contour model formed with σ _y0.1 and σ _y0.2 are 26% and 32%, respectively, lower than that of the perfectly elastic material model. A significant lower load can be observed from the model with angular misalignment than that from the perfectly elastic material and without angular misalignment. The pressure around the cylindrical indenter contact edge is infinite for the perfectly elastic material model while the pressure is finite value for an elastoplastic material model due to the plastic deformation, refer to Fig. 6.

FIG. 7.

Indentation load versus depth comparison for a perfectly elastic and elastoplastic material model at the beginning state for measuring yield strength.

Table III lists the indentation data obtained from modeling. With an appropriate depth modification, the yield strength at 0.1% offset and 0.2% offset were evaluated from the modeling. Fig. 8 shows the relationships between the misalignment angle θ and the relative depth h′/a for forming a hemispherical stress–strain distribution or modifier coefficient k for estimating yield strength.

TABLE III.

Indentation modeling data for extracting yield strength.

Note: The 0.1% and 0.2% offset yield strengths are ${\sigma _{y0.1}} = {{{P_{0.1}}} \over {{k_{0.1}}A}}$ and ${\sigma _{y0.2}} = {{{P_{0.2}}} \over {{k_{0.2}}A}}$ , respectively, where A is the indenter contacting area, A = πa ², ^{Reference Hu, Lynne, Markondapatnaikuni and Delfanian5,Reference Hu, Lynne and Delfanian6} k _0.1 and k _0.2 are the modifiers determined from the calibration test or modeling, and P _0.1 and P _0.2 are the indentation loads at the depth h′_0.1 and h′_0.2, respectively. h′_0.1 and h′_0.2 are the depths corresponding to the hemispherical stress–strain distribution contour model formed with angular misalignment for σ _y0.1 and σ _y0.2, respectively.

FIG. 8.

Misalignment angle versus ratio of depth to radius h′/a or modifier coefficient k.

E. Misalignment angle measurement from tests

The existence of angular misalignment on the cylindrical flat-tip indentation test creates significant errors for finding the yield strength of the material from the load–displacement graph. Indentation tests done by Hu et al. ^{Reference Hu, Lynne and Delfanian6} were used as the experimental results for the current work. The indentation profiles obtained from the tests were utilized to analyze the angular measurement. Figure 9 shows the surface profiles of a specimen scanned from KEYENCE VK-9710 Color 3D Violet Laser Scanning Microscope (Keyence Corp. of America, Itasca, Illinois). Profile lines (blue wavy line on the left bottom and pink wavy line on the upper right) were measured horizontally and vertically across the upper left indentation such that the maximum to minimum height of the indentation along these lines can be measured. Based on the profile lines, the laser microscope allows for calculating the angle of the specimen surface which was used to estimate the misalignment angles of the indentation. The detailed profile (surface height) measurements are shown in Fig. 10, in which the indentation region and the entire profile line for both horizontal and vertical lines are included. Table IV lists the misalignment angles measured from the indentation profiles of the specimens. Due to the difference of the testing and surface conditions from each indentation, the maximum misalignment angles measured range from 0.295° to 0.706° with a mean value of 0.450°.

FIG. 9.

Selected profile lines from specimen surface for surface measurement.

FIG. 10.

Profile measurement from a typical indentation for estimating angular misalignment: (a) horizontal and (b) vertical.

TABLE IV.

Calculated maximum misalignment angles from the indentation profile measurements.

F. Nanoindentation test data analysis

Table V summarizes the indentation data with measured misalignment angles. By considering the misalignment angle, the corrected relative depths h′_0.1/a and h′_0.2/a and the modifiers k _0.1 and k _0.2 can be interpreted from Table III or Fig. 8. Then the depths h′_0.1 and h′_0.2 corresponding to the formation of the hemispherical stress–strain contours of σ _y0.1 and σ _y0.2 can be calculated, where the practical cylindrical indenter tip diameter 2a takes 9.318 μm. Based on h′_0.1 and h′_0.2, the loads P _0.1 and P _0.2 from the load–depth curves can be read. Finally, the 0.1% offset and 0.2% offset yield strengths σ _y0.1 and σ _y0.2 can be calculated. The mean value of the yield strengths σ _y0.1 and σ _y0.2 are 481 and 503 MPa, respectively, with the standard deviations of 32.0 MPa and 28.4 MPa, respectively, and errors of 3.9% and 2.0%, respectively, comparing to the actual yield strengths of σ _y0.1 = 500.4 MPa and σ _y0.2 = 513.2 MPa. Possible error sources include testing specimen preparation (surface roughness) and the misalignment angle measurement.

TABLE V.

0.1% and 0.2% offset yield strength estimation by indentation testing data.

G. Dimensional analysis in nanoindentation

Dimensional analysis, or physical similarity analysis, is very important, since the dimension, i.e., the diameter (2a), of the cylindrical flat-tip indenter matters the indentation load versus depth measurements. In the above experimental and modeling data analysis, what size of the indenter diameter should be properly selected? How can the experimental data from the indentation tests using a cylindrical flat-tip indenter of a 9.318 μm in diameter be compared with the modeling data by a nominal diameter of 10 μm? Furthermore, is it possible that the aforementioned data and analysis, and even the conclusive statement can be applied to the general nondimensional parametric analysis? The answer is YES, ^{Reference Hu, Lynne, Markondapatnaikuni and Delfanian5,Reference Baker, Westine and Dodge25,Reference Zohuri26} based on the dimensional analysis and similarity principle on a replica model, i.e., a model of a dynamic system or structure of exactly the same geometry and materials as a prototype, but scaled in size alone. In the aforementioned analysis of a quasistatic mechanical model, nondimensional parameters, such as the ratio of the indentation depth to the indenter radius h/a, have been adopted, referring to Figs. 5, 6 and 8 and Tables III and V, so that the modeling data, especial the yield strength data, can be compared with the experimental data, even though the indenter diameter by the modeling is slightly larger than that by the testing. In addition, since the grain size of a low carbon steel, e.g., AISI 1018 in this study, is around a couple of micrometers in diameters, ^{Reference Davis27} and an indenter diameter of 10 μm can cover dozens of such grains underneath the contact area and plastic deformation involves hundreds of such grains during the indentation testing. Therefore, the effect of the grain orientations can be neglected.

H. Comparison of the misalignment angle effect on indentation by different types of indenters

As discussed in the Introduction, a more or less misalignment angle always exists due to the structural tolerance of the testing device and the imperfection of the specimen preparation. ^{Reference Tadmor, Miller and Phillips15–Reference Paoli and Volinsky23} For a spherical indenter, since it has a rotational symmetry, i.e., its shape looks the same after some rotation by a partial turn, thus, there is no influence on indentation measurements if a finite misalignment angle exists. For a pyramidal sharp-tip indenter, such as Berkovich, conical, or Vickers, the angular misalignment results in increases in the indentation load, contact area and contact stiffness at the same penetration depth. This results in a underestimation of indentation projected contact area, and in turn leads to an overestimation of the mechanical properties measured by nanoindentation. Research indicated that the influence of the angular misalignment on the load versus depth measurement of an indentation test can be ignored when the angle is less than 1.2°. However, when the angle is larger than 1.2°, the misalignment angle needs to be determined and the measurement compensation needs to be estimated. Simulation shows that a 5° misalignment angle results in about 25% contact area higher than untitled indentation. The misalignment angle can be assessed from measurements of the sides of the indented impression if a good image is available. A simple but effective method is based on the ratio of the long to short side of the indented impression. ^{Reference Oliver and Pharr16} But practically, the accuracy is questionable. For an advanced indentation device, the tilt correction functionality could be integrated into the controlling software for measuring the tilt and adjust the data. ^{Reference Srivastava17} For a cylindrical flat-tip indenter with a constant contact area, from this study, the influence of the angular misalignment on Young’s modulus assessment can be ignored when the angle is less than 1° or even a little bit larger. However, other properties assessed are sensitive to the misalignment angle and need to be adjusted accordingly, such as the hardness to be adjusted by considering an average or effective indentation depth instead of the contact depth, and yield strength—one of the advantages of the property measurement by a flat-tip indenter—to be adjusted by considering a misalignment angle related correction factor. The tilt angle assessment is easier and more accurate based on the indented impression.