A. Dynamic hardness as function of indentation depth

A proportional relation exists between the impact load and impact velocity, v _in. Due to the downward direction of gravitational force on the pendulum indenter, increasing the indentation load from 10 mN to 100 mN causes a linear increase in v _in from 0.93 × 10⁶ ± 0.24 × 10⁵ nm/s to 2.5 × 10⁶ ± 1.0 × 10⁵ nm/s. Hence, increasing loads increase the mechanical energy of indentation and cause greater penetration, h _residual. H _Impact as a function of h _residual for the three tested materials is presented in Fig. 1. The H _Impact presented in Fig. 1 is corrected for pileup using Eq. (3). The percentage of pileup values with respect to h _residual $\hbox( {{\rm{or}}\;{{{h_{{\rm{pileup,}}n}}} \over {{h_{{\rm{residual}}}}}} \times 100\% } \hbox)$ were statistically similar and remained unchanged with increasing loads. The average pileup values for AA7075-T6, AA7075-Annealed, and Ni-200 were computed to be 3.5 ± 2.3, 7.1 ± 1.3, and 3.4 ± 2.1%. As presented in Fig. 1, the materials display decreasing hardness H _Impact with increasing h _residual that extends up to several micrometers. After correcting for pileup the load independent or true dynamic hardness, H _Impact,o of AA7075-T6, AA7075-Annealed, and Ni-200 were measured to be ∼1.50 ± 0.15 GPa, ∼1.03 ± 0.07 GPa, and ∼1.38 ± 0.08 GPa, respectively. The influence of various microstructures on H _Impact,o is beyond the scope of the current work.

FIG. 1.

Dynamic hardness, H _Impact as a function of residual depth, h _residual.

For comparing the indentation size effect of dynamic indentations with quasistatic indentations, H _Static versus h _residual were performed at quasistatic regime using the cube-corner indenter and corrected for pile-up in a similar fashion as H _Impact versus h _residual. The average percentage quasistatic indentation pileup values for AA7075-T6, AA7075-Annealed, and Ni-200 were computed to be 13.9 ± 1.7, 13.9 ± 1.3, and 9.1 ± 0.08%. The true or load independent quasistatic hardness H _Static,o for AA7075-T6, AA7075-Annealed, and Ni-200 were measured to be ∼1.39 ± 0.07 GPa, ∼0.74 ± 0.02 GPa, and ∼1.13 ± 0.03 GPa, respectively. The varying strain rate sensitivities for the three materials could be obtained from the difference between H _Impact,o and H _Static,o. However, the current section focuses on comparing the indentation size effects between H _Impact versus h _residual and H _Static versus h _residual. The indentation size effect is quantified using the characteristic indentation depth, h*. Figure 2 presents the h* for nano-impact indentations and quasistatic indentations for AA7075-T6, AA7075-Annealed, and Ni-200. A comparison of h* between dynamic and quasistatic indentations using cube-corner tip demonstrate that h* for the nano-impact indentations are several times higher than for the quasistatic nanoindentations. The h* for dynamic indentations extend well beyond the surface regime. Hence, surface effects that influence quasistatic nanoindentations become less significant for the dynamic indentation size effects. ^{Reference Menčík and Nemecek23} Indentation size effect in quasistatic nanoindentation hardness profile is widely explained by the Nix-Gao model, which uses geometrically necessary dislocation (GND) theories to explain this hardness reduction dependency. ^{Reference Nix and Gao24} Recently, Arreguin et al. demonstrated a significant deviation of H _Impact values from the linear Nix-Gao model, indicating departure from quasistatic indentation behavior. ^{Reference Arreguin-Zavala, Milligan, Davies, Goodes and Brochu25} This mismatch suggests a lack of significant contribution from GNDs ^{Reference Arreguin-Zavala, Milligan, Davies, Goodes and Brochu25} and deviation from dislocation motion by slip phenomenon during impact indentations, contributing to an enhanced decrease in hardness as function of h _residual. ^{Reference Zener and Hollomon8,Reference Armstrong, Arnold and Zerilli26,Reference Kun, Li-Ming and Shi-Sheng27}

FIG. 2.

Characteristic indentation depth, h*, comparison for quasistatic and dynamic indentation hardness.

B. Strain rate profile and nonuniform loading

The deviation from the Nix-Gao model for H _Impact versus h _residual could be explained by the strain rates achieved by the nano-impact indentations. Strain rate, $\dot \varepsilon$ plays a key role in the behavior of net plastic flow underneath the indenter, as discussed by Tirupataiah et al. ^{Reference Tirupataiah and Sundararajan28} The primary characteristic of an impact loading that promotes heterogeneity, instability, and localization of strain is the nonuniformity in loading. ^{Reference Zener and Hollomon8} Unlike quasistatic indentations, the loading rate in nano-impact indentations decrease after coming into contact with the material, or at h = 0. A rapid decrease in $\dot \varepsilon$ with increase in indentation depth, h is observed. For macro-scale tests, the significantly higher $\dot \varepsilon$ at the early stages of an impact is a major factor in nonuniformity and the resulting instability. ^{Reference Zener and Hollomon8,Reference Rao and Varma29} Similarly, for nano-impact indentations the nonuniformity in loading is facilitated by the rapidly decreasing $\dot \varepsilon$ versus h curves, as shown in Fig. 3. The impact strain rate, ${\dot \varepsilon _{{\rm{Impact}}}}$ , is the $\dot \varepsilon$ at the point of impact. The ${\dot \varepsilon _{{\rm{Impact}}}}$ values for the loads ranging from 10 to 100 mN are statistically similar and have an average value of 1.34 × 10⁴ ± 0.1 × 10⁴ s⁻¹. The overall indentation strain rate is given by the relationship ${\dot \varepsilon _{{\rm{Average}}}} = {{{v_{{\rm{in}}}}} \over {{h_{{\rm{residual}}}}}}$ . ^{Reference Song, Shen, Su, Lai and Chiu6} On the other hand, the ${\dot \varepsilon _{{\rm{Average}}}}$ value of roughly 2.3 × 10² ± 0.8 × 10² s⁻¹ for loads 10–100 mN also lie in the impact regime, which ensures the required energy localization and consequent promotion of dynamic deformation mode. In essence, the high ${\dot \varepsilon _{{\rm{Impact}}}}$ and ${\dot \varepsilon _{{\rm{Average}}}}$ , and excessively high $\dot \varepsilon$ values at initial depths of penetration promote the dynamic nonuniformity and localization of deformation in a nano-impact indentation.

FIG. 3.

Strain rate profiles, $\dot \varepsilon$ , of AA7075-T6, AA7075-Annealed, and Ni-200 at 50 mN load.

C. F versus d and plastic wave generation during indentation

A representative resistance force versus indentation depth curve (F versus h) curve has been displayed in Fig. 4 using a 50 mN impact load on AA7075-T6, AA7075-Annealed and Ni-200. The wave like F versus h profile obtained in Fig. 4 suggests a dynamic deformation behavior during the nano-impact indentation process. Complete explanation of this wave-like deformation mechanism has been discussed in the literature. ^{Reference Johnson16,Reference Qiao, Yang and Bobaru17} Such wave-like formations have been discussed by Meyers to be the transmission of stress pulses exceeding the amplitude of elastic limit. ^{Reference Meyers30} Lawn et al. have reported the presence of fluid-like dynamic flow in metals. ^{Reference Lawn, Hockey and Wiederhorn14} The occurrence of such viscous plastic flow has been demonstrated for deformations with $\dot \varepsilon > 1000\;{{\rm{s}}^{ - 1}}$ . ^{Reference Cheval and Priester31–Reference Kumar and Kumble33} As seen from a representative profile of $\dot \varepsilon$ versus h in Fig. 3, a significant portion of a nano-impact deformation profile lies above $\dot \varepsilon$ of 1000 s⁻¹. This explains the wave generation in the F versus h profile shown in Fig. 4. Furthermore, the viscous flow phenomenon is reinforced by the presence of wave-like folds on the periphery of indentations, as illustrated in the representative images of 50 mN impact indentation load for AA7075-T6 and AA7075-Annealed in Fig. 5.

FIG. 4.

A typical F versus h curve h = 0 to h = h _max, shown for Ni-200 at 50 mN load.

FIG. 5.

Wave-like patterns in the pile-up regions of 50 mN indentation imprints, shown for (a) AA7075-T6 and (b) AA7075-Annealed.

D. Deformation related energy during impact

The quantification of energy dispersed toward creation and transmission of plastic waves is an important step toward evaluating the dynamic behavior of the nano-impact indentations. The assessment of energetics of deformation is primarily done using the plastic energy dissipated during indentation, ΔE _Plastic and the plastic work of deformation during indentation, W _Plastic. The components of the ΔE _Plastic and W _Plastic in the context of indentation size effect or the decreasing H _Impact values with respect to the increase indentation depths (or indentation sizes), to the best of our knowledge, has not been reported or discussed in the literature. ΔE _Plastic and W _Plastic is calculated using Eqs. (4) and (5), respectively:

(4) $$\displaylines{ \Delta {E_{{\rm{Plastic}}}} = \Delta KE = K \cdot {E_{{\rm{initial}}}} - K \cdot {E_{{\rm{final}}}} \cr = {1 \over 2}\left( {{m_{{\rm{effective}}}} \times v_{{\rm{in}}}^2} \right) - {1 \over 2}\left( {{m_{{\rm{effective}}}} \times v_{{\rm{out}}}^2} \right) \quad ,}}$$

(5) $${W_{{\rm{Plastic}}}} = \int_0^{{h_{{\rm{residual}}}}} {F \times {\rm{d}}x} \quad .$$

During quasistatic nanoindentation, the ΔE _Plastic and W _Plastic of deformation are equal. ^{Reference Andrews, Giannakopoulos, Plisson and Suresh34} However, due to energy dissipation in the form of plastic waves during dynamic indentations, ΔKE _Plastic is greater than ΔW _Plastic. ^{Reference Andrews, Giannakopoulos, Plisson and Suresh34} The energy related to the dynamic response of materials could be termed as ΔE _Dynamic. ΔE _Dynamic is calculated using Eq. (6):

(6) $$\Delta {E_{{\rm{Dynamic}}}} = \Delta K{E_{{\rm{Plastic}}}} - {W_{{\rm{Plastic}}}}\quad .$$

Increasing indentation loads cause an increase in energy of impact, leading to a greater overall dynamic response. Increasing ΔE _Dynamic values with increasing impact loads have been displayed in Fig. 6.

FIG. 6.

ΔE _Dynamic as function of increasing impact loads 10–100 mN for AA7075-T6, AA7075-Annealed, and Ni-200.

In Fig. 6, AA7075-Annealed displays the highest ΔE _Dynamic at almost every load. This might be attributed to the lower hardness values of AA7075-Annealed. The increased viscous flow or plastic waves has been related by Tabor to the softness in a material, explaining the higher ΔE _Dynamic in the relatively soft AA7075-Annealed. ^{Reference Tabor35} A significant percentage of ΔE _Plastic is converted to ΔE _Dynamic during the nano-impact indentations on the three materials. For example, at 100 mN the percentage of ΔKE _Plastic converted to ΔE _Dynamic; %(ΔE _Dynamic/ΔKE _Dynamic) for AA7075-T6, AA7075-Annealed, and Ni-200 are 60 ± 1, 44 ± 4%, and 41 ± 2% respectively. High values of %(ΔE _Dynamic/ΔKE _Dynamic) represent the energy dispersion toward plastic wave generation in the materials, further reinforcing the evidence of dynamic response induced in the materials by the nano-impact indentations.

E. Characterization of indentation imprint and strain-field

The nano-impact indentations featuring: (a) impact regime strain rates, (b) sharp cube-corner indenter tip, and (c) dynamic stress-wave creation provide an ideal condition for localization of strain. The nonuniformity of impact loading, illustrated by $\dot \varepsilon$ profiles in Fig. 3, causes instability due to heterogeneity in dislocation distribution ^{Reference Armstrong and Walley36} and hence strain. ^{Reference Zener and Hollomon8} The instability induces stress wave generation, as indicated by Figs. 5–7. Previous studies have discussed the generation of extra dislocations by the waves that result in increased flow stress and localized strain. ^{Reference Arreguin-Zavala, Milligan, Davies, Goodes and Brochu25,Reference Armstrong and Walley36} Localized strain during high strain rate deformation has been reported to result in adiabatic softening. ^{Reference Zener and Hollomon8,Reference Rajaraman, Jonnalagadda and Ghosh37,Reference Chou, Robertson and Rainey38} The dynamic instability has been discussed to induce abrupt generation and release of extremely localized thermal energy. ^{Reference Armstrong, Coffey and Elban39,Reference Coffey and Armstrong40} If the rate of heat generation is higher than the rate of heat conduction to the bulk, a thermally induced local softening or adiabatic softening is possible. ^{Reference Rajaraman, Jonnalagadda and Ghosh37,Reference Hodowany, Ravichandran, Rosakis and Rosakis41} Macro-scale indentations have widely shown to cause adiabatic softening, or ${{{\rm{d}}H} \over {{\rm{d}}\varepsilon}} \lt 0$ , at high ɛ and $\dot \varepsilon$ . ^{Reference Sundararajan and Tirupataiah15,Reference Tabor35} Similarly, the decreasing H _Impact as a function of h _residual, or ${{{\rm{d}}{H_{{\rm{Impact}}}}} \over {{\rm{d}}{h_{{\rm{residual}}}}}}$ , are in agreement with the thermally induced softening witnessed in several macro-scale impact tests. ^{Reference Zener and Hollomon8,Reference Lawn, Hockey and Wiederhorn14,Reference Sundararajan and Tirupataiah15,Reference Tirupataiah and Sundararajan28,Reference Tabor35,Reference Johnson and Cook42} However, strain localization is a pre-requisite for the dynamic softening phenomenon. Strain localization due to nano-impact indentations is assessed by characterizing the extent of strain fields around the indents for a range of residual depths. The strain affected zones of the cube-corner indenter at dynamic strain rate regime are compared to the affected zones at quasistatic strain rate regime at comparable indentation depths to assess the role of strain rate on the strain localization phenomenon.

FIG. 7.

Band contrast image of quasistatic indent on Ni-200 (a) and the GROD contour map, indicating extent of strain effect, l _maximum (b).

GROD function has been previously used in literature to characterize process induced plastic strain in metals. ^{Reference Kumar and Pollock43} GROD function uses crystal rotation at each point with respect to the average orientation of the grain to generate a contour map, describing strain field induced by the indentation. The contour maps were utilized to measure the extent of strain affected zones of the indent imprints. The contour maps displayed a general reduction in GROD angles with increasing distance from the edges of the indentation imprints, until the deviation angles became negligible. From the GROD maps, the maximum length of strain effect, l _maximum, is measured perpendicular to the edge of the triangular indentation imprint. An illustration of the color contour based on GROD angles has been presented in Fig. 7, a representative case shown for a quasistatic indent on Ni-200. For the quasistatic indent, the GROD angles decrease from ∼0.6° to ∼0° over a maximum length of ∼17 μm. The presence of strain gradient in quasistatic indentations using cube-corner tips has also been detailed elsewhere. ^{Reference Rester, Motz and Pippan44} On the other hand, a GROD contour map of a nano-impact indent on Ni-200 shows negligible strain field around the indent, indicated by the negligible GROD contouring. Instead, in the case of nano-impact indents on Ni-200, the maximum strain seems to be concentrated in a narrow band in the extreme vicinity of the imprint edges, generating very low l _maximum values ranging between 0.5 and 2 μm. The strain affected area is associated to significant crystallographic distortions caused by the impact indentations, potentially causing lack of crystal orientation indexing and therefore lack of signal in such areas. AA7075-T6 and AA7075-Annealed do not display such excessive localization of strain as Ni-200. Dynamic indentations on the two aluminum alloys display some degree of strain field. In comparison to quasistatic indentations on AA7075-T6 and AA7075-Annealed however the l _maximum is much lower. Table I presents the ratio of maximum length of strain affected region to the average vertex size of the indentation imprints, l _maximum/l _{avg.indent vertex} for AA7075-T6, AA7075-Annealed, and Ni-200. Varying h _residual values are obtained by the nano-impact indentation loads of 1 mN, 10 mN, 50 mN, and 100 mN. In Ni 200, l _maximum/l _{avg.indent vertex} for nano-impact indentations for the given h _residual values range between 0.056 ± 0.012 and 0.11 ± 0.012. For AA7075-T6 the l _maximum/l _{avg.indent vertex} ranges between 0.13 ± 0.02 and 0.37 ± 0.03. Based on Table I, the l _maximum/l _{avg.indent vertex} at quasistatic and dynamic indentations for AA7075-T6, AA7075-Annealed, and Ni-200 at similar h _residual values could be compared. For AA7075-T6, at h _residual of 7.2 ± 0.4 and 6.9 ± 0.13 μm for quasistatic and dynamic indentation respectively, l _maximum/l _{avg.indent vertex} for quasistatic indents is significantly larger than the dynamic indents. Similarly, quasistatic and dynamic indents at h _residual of 8.1 ± 0.1 and 8 ± 0.34 for AA7075-Annealed, and h _residual of 7.7 ± 0.20 and 8 ± 0.4 for Ni-200 generate l _maximum/l _{avg.indent vertex} ratio for quasistatic indents that far exceeds the ratio for dynamic indents, demonstrating the effect of strain rates on the strain localization process.

TABLE I.

h _residual and the corresponding l _maximum/l _{avg.indent vertex} for dynamic and quasistatic indentations on AA7075-T6, AA7075-Annealed, and Ni-200.

The GROD contour maps could be complemented using line misorientation angle mapping measured normal to the imprint edges, yielding 2-dimensional intra-granular misorientation gradients. The magnitude of intra-granular misorientation angle represents the degree of distortion due to plastic strain in a grain. It was previously discussed that the dynamic indentations for Ni-200 generate a narrow band of heavily deformed region in the imprint vicinity, displaying negligible strain field and hindering line misorientation measurements. Quasistatic indentations for Ni-200, on the other hand, display a gradual decrease in misorientation with increasing distance from the imprint. The dynamic indents on Ni-200 displayed a relatively higher localization of strain compared to AA7075-T6 and AA7075-Annealed. The comparison of intra-granular strain affected regions generated by quasistatic and dynamic indentations is more easily performed for AA7075-T6. Figure 8 presents the misorientation angle versus distance from the edge of the indent imprint for quasistatic and dynamic indentations on AA7075-T6. For the dynamic indentations, negligible reduction in misorientation angles are observed throughout the length of the affected regions. The misorientations appear to be similar throughout the strain affected length until dropping to very low or negligible values beyond the affected length. Similar to Ni-200, the strain affected region of quasistatic indentations on AA7075-T6 display a gradual decrease of misorientation angles starting with maximum values at imprint vicinity, as shown in Fig. 8 for the same line distance as that of the dynamic indentation. Comparison of quasistatic and dynamic indentations in AA7075-Annealed follows similar trend as AA7075-T6. Overall, the considerable degree of strain localization by dynamic indentations, compared to quasi-static indentations, suggests the adiabatic nature of deformation toward the resultant softening effect seen for H _Impact versus h _residual in Fig. 1.

FIG. 8.

Line misorientation angle mapping from unaffected region to edge of indent imprint, angles measured relative to the starting point, for quasistatic (a) and dynamic (b) indentations for AA7075-T6.