Toward an Instrumented Strength Microprobe – Origins of the Oliver-Pharr Method and Continued Advancements in Nanoindentation: Part 2

Abstract

Numerable advancements have afforded many benefits to nanoindenter system operators since the late 20th century, such as automation of measurements, enhanced load and displacement resolutions, and indentation with in-situ capabilities. Accordingly, the present chapter details how the Oliver-Pharr method of nanoindentation testing and analysis was adopted and relied upon as a framework that brought about widespread advancements in instrumented indentation testing. The present chapter introduces an emergent and theoretically consistent approach to assessing true stress–strain curves at a micromechanical scale using a flat-punch nanoindenter tip geometry and reliance upon Hollomon power-law plasticity and constitutive parameter fitting. Finally, a novel flat-punch nanoindentation testing method and approach to plasticity parameter analysis for metallic materials using nanoindentation systems can be implemented, bringing about an instrumented strength microprobe – a long sought-after tool.

1. Introduction

Similar to the work rendered and published by Bolshakov and Pharr in 1998, as discussed in Part 1, both Bolshakov and Pharr collaborated with Hay and Oliver in [1] to reformulate the Bolshakov-Pharr pile-up prediction ratio relation in terms of load–displacement curve slope-to-elastic contact stiffness ratio. Ultimately, Hay et al. found that $S_l/S$ maintains a one-to-one ratio with $h_f/h_{\mathit{max}}$ and $E/{\sigma }_y,$ too, while also maintaining the capacity for direct measurement during testing, regardless of $h_f$ or $h_{\mathit{max}}$ inspection capabilities at a given facility. Indeed, Figure 1 captures a few plots presented by Hay et al. when they formulated a pile-up constraint factor as a function of $S_l/S$ as part of their 1998 research effort detailed in [1].

Figure 1.

(a) relation between$S_l/S$ and $h_f/h_{\mathit{max}}$. (b) $S_l/S$ vs. depth data obtained from testing fused silica [1]. (c) used the normalized contact area vs.$S_l/S$ data to highlight that$S_l/S$ ratios obtained at finite depths resulted in Anano’s underestimation of actual contact area, which caused an overestimation of hardness as well as modulus. (d) dependency of the constraint factor as a function of $S_l/S$, wherein the constraint factor was found to be virtually independent of work-hardening when evaluated at large depths. Reproduced from [1].

Nevertheless, post-OP article publication developments and research indeed focused upon relations beyond predicting and correcting for pile-up and relating the tendency of a material to pile up during nanoindentation to specimen properties, such as modulus, yield strength, work-hardening rate, and the like. Consequently, consider additional sources of influence and other externality-driven considerations of importance when performing and recording load–displacement testing and analysis. The work of Feng and Ngan highlighted how creep and thermal drift were both likely to influence modulus values recorded via instrumented indentation testing (IIT) or depth-sensing indentation since the OP method assumes that materials experience purely elastic recovery – an assumption that diverges from the physical reality underpinning the influencing factors upon recording and analyzing unloading segments of relevant nanoindentation data. More to the point, Feng and Ngan carefully constructed a simple scheme of creep effect corrections while measuring the modulus of elasticity. Furthermore, Feng and Ngan concurrently derived a way to nullify the thermal drift effects when measuring the modulus of elasticity [2]. Feng and Ngan achieved said derivations through the consideration of Al, Cu (111), and Ni₃Al (111) in [2].

In summation, Feng and Ngan showed that creep-based influences upon the compliance associated with the contact of an indenter tip with a given specimen during depth-sensing indentation and contact at the load–displacement point of initial unloading could be mathematically resolved and experimentally addressed via careful consideration of the unloading rate and max load hold times too. With the aforementioned in mind, one ought to recall that Feng and Ngan were mainly focused on the modulus of elasticity value modification or evolution when thermal drift and creep were not adequately understood and accounted for during nanomechanical, micromechanical, or both forms of indentation testing. However, unlike accounting for creep, which maintained significant improvements in measurement accuracy, a simple nullification protocol presented by Feng and Ngan ensured that thermal drift effects could be readily overcome and accounted for.

Around the same period, influences and relationships between those above-discussed phenomena and the ascertained mechanical properties gained traction through other researchers’ perspectives. Indeed, in 1998, relations between the work of indentation, modulus of elasticity, and indentation hardness were considered by Cheng et al. [3]. Intriguingly, Cheng et al. resolved a proximal relation between the hardness-to-modulus proportionality and the proportionality between irreversible work of indentation and total work of indentation. As a result, Cheng et al. provided the nanomechanical research community with an alternative approach to estimating hardness and modulus via ITT systems equipped with conical or pyramidal indenter probes, as illustrated in part by Figure 2 [3].

Figure 2.

Relationship between the proportional mechanical properties, including specimen hardness, measurable elastic modulus values, and various forms of indentation work, i.e., total, irreversible, etc. (a,b) Present the relationship between hardness/reduced modulus vs. the ratio of irreversible work-to-total work of indentation loading and unloading using computational finite element analysis and experimental data for various materials. (c) Presents the ratio of irreversible work-to-total work of indentation loading and unloading vs. the initial yield stress/modulus of elasticity, while (d) presents the hardness/reduced modulus vs. the initial yield stress/modulus of elasticity. Reproduced from [3].

Still, fundamental explorations and evaluations of said relations need not stop with the work of Cheng et al.; rather, Hay et al. and others also maintained an early critical eye when inspecting underlying relations between properties and measured nanoindentation responses in [4, 5, 6, 7].

Just as small-scale indentation-based and/or nanoindentation mechanical stress–strain curve evaluation captured the imagination of a subset of researchers cited in the previous subsection of the present manuscript, the period between 2003 and 2012 was also concerned with stress–strain behavioral insights via the use of nanoindentation systems. By 2003, Rodriguez and Gutierrez coupled an investigation of the ISE with a correlative analysis between tensile properties and nanoindentation-based behavior [8]. Rodriguez and Gutierrez invoked the use of the OP method of data analysis for performing sub-micron pyramidal nanoindentation while also invoking the strain gradient plasticity framework proposed by Nix and Gao for understanding the ISE affiliated with materials of various mesoscale mechanical tendencies such that the material flow stress could be compared with the bulk tensile properties of the very same specimens.

Therefore, when mathematically speaking, it stands to reason that plastic shear strain gradient-induced dislocations, or geometrically necessary dislocations, ${\rho }^G$, yields Eq. (1).

$${\rho }^G\approx \mathit{\delta \gamma }\frac{1}{\mathit{b\lambda }}$$E1

wherein $\delta$ is a constant, $\gamma$ is the plastic shear strain, $b$ is the Burgers vector, and $\lambda$is the localized deformation field length scale relative to indentation size. At the same time, ${\rho }^G$ and ${\rho }^S$, or the statistically stored dislocations, are cumulatively related to one another concerning the total dislocation density ${\rho }^T$ via Eq. (2), such that.

$${\rho }^T={\rho }^G+{\rho }^S$$E2

Rodriguez and Gutierrez also invoked Taylor’s relation, which expresses the mechanical flow stress $\sigma$ of a material with the total dislocation density, such that Eq. (3) yields.

$$\sigma =\mathit{\text{αMμb}}\sqrt{{\rho }^T}=\mathit{\text{αMμb}}\sqrt{{\rho }^G+{\rho }^S}$$E3

wherein$\alpha$ is a constant, $M$ is the Taylor factor, and $\mu$ is the elastic shear modulus.

Regrettably, Rodriguez and Gutierrez simply accepted the observation that the nanoindentation modulus of elasticity associated with the specimens studied, especially the steel-based subset of specimens, deviated from the macroscopic material’s modulus. Rodriguez and Gutierrez did so by viewing the matter of deviating elastic modulus values through the lens of indentation-induced pile-up or sink-in influences. Ideally, a study claiming to be exploring a linkage between tensile stress–strain behavior and ISE-informed indentation data would have anticipated the effects introduced by pile-up and/or sink-in, which could have been experimentally addressed in part via confocal microscopy-corrected contact area values (given confocal microscopy’s commercialization more than 20 years before the work of Rodriguez and Gutierrez). Rodriguez and Gutierrez also situated the then-debated direct relation between pile-up and material work-hardening.

Still, Rodriguez and Gutierrez correctly noted that grain-scale texture, surface roughness, and pile-up all influence nanoindentation data analysis, even going so far as to note that O&P, in 1992, did not correct for pile-up or sink-in effects. Of course, the lack of a method to account for pile-up or sink-in may limit O&P’s approach as of 1992. However, one ought to note that O&P went on to incorporate such a correction factor during post-1992 refinements to the originally formulated and presented OP method.

Having just remarked that O&P made refinements to their original approach to analyzing nanoindentation derived data, one may consider one of the most noteworthy articles concerned with such refinements, which O&P has unsurprisingly presented since 1992. Said article by O&P is cited herein as [9]. As such, in their 2004 article, which was also published in the Journal of Materials Research (that is, the same journal that initially published their 1992 manuscript), O&P noted that improved surface contact identification was achievable via dynamic nanoindentation in comparison with static nanoindentation, which was first described in their appendix to the original 1992 article. O&P also noted the promising capacity of dynamic or CSM-based nanomechanical and nanoindentation-based micromechanical characterization to unveil mechanical properties as a function of depth throughout the entire loading and unloading process. Thirdly, O&P detailed how improvements could be integrated into nanomechanical testing systems to ensure more accurate area function calibration and load frame compliance assessment [9]. Each of the three improvements and refinements may be considered noteworthy when presented one by one, let alone all together in one document.

In any case, O&P continued their 2004 review of the progress made since their 1992 article by recognizing the fact that their method of nanoindentation testing and data analysis could be generalized to a greater degree and across more indenter geometries than previously thought. Importantly, O&P note that careful consideration must also be given to nanoindentation testing analysis of materials with reversible plasticity upon unloading since they assumed that only elastic displacements would be recoverable; yet, for those cases wherein materials partially unload plastically, O&P noted that FEM analysis had shown the effect to be virtually negligible for most monolithic materials. Of course, this remedied the previously noted concerns echoed by others in the field. Furthermore, in their original pedagogical formulation, O&P assumed that their expression for stiffness was sufficient; however, one significant improvement made by O&P by 2004 was the inclusion of a multiplicative $\beta$ term as a function of the physical processes that may affect the value of the term such that Eq. (4) yields.

$$S=\frac{\mathit{dP}}{\mathit{dh}}=\frac{\beta 2E_r}{\sqrt{\pi }}\sqrt{A}$$E4

In so far as non-Berkovich, spherical nanoindentation load vs. displacement relations were concerned, O&P also demonstrated that at shallow indentation depths relative to the radius of a spherical tip, the load vs. depth relation may be expressed, in the elastic deformation regime, as Eq. (5), wherein.

$$P=\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\sqrt{R}{E}_r{\left(h-h_f\right)}^{1.5}$$E5

such that differentiation of $P$ relative to $h$, in conjunction with Eq. (5), yields Eq. (6), wherein.

$$S=\frac{\mathit{dP}}{\mathit{dh}}=2\sqrt{R}{E}_r{\left(h-h_f\right)}^{0.5}$$E6

and Eq. (7) follows such that.

$$h_c=\frac{h_{\mathit{max}}+h_f}{2}$$E7

When proper substitution and algebraic manipulation are rendered, $R$ is expressed as Eq. (8),

$$R={\left(\frac{1}{R_1}+\frac{1}{R_2}\right)}^{-1}$$E8

while $R_1$ is the radius of the spherical indenter probe, and $R_2$ is the spherical hardness impression remaining after loading.

Interestingly, O&P also noted that the hardness obtained via spherical indenter tips was not necessarily equivalent to the hardness obtained using a Berkovich tip alongside their spherical nanoindentation contact mechanics analysis [9]. Work by Sousa et al. (see [10]) concurs with said hardness variability when a cono-spherical diamond indenter tip is dynamically applied to a material that is also tested via dynamic Berkovich nanoindentation according to OP methods in [10].

During the following year, Bei et al. examined the influence of using the contact area function, as determined via the OP calibration procedure, upon the nanomechanical phenomena known as pop-in events. The theoretical strengths are quantifiable from the analysis of said pop-in events [11]. Unlike sink-in or pile-up, which are related to either more or less material in contact with a given indenter tip during load vs. displacement cycling as part of indentation testing, pop-in events can be identified by abrupt bursts in depth at a given indentation load(s), which yield disjointed behavior within otherwise continuous load-depth data according to Bei et al.’s analysis. When Bei et al. presented their research, analysis, and findings, pop-in events were believed to signify the point of purely elastic to elastoplastic deformation during nanoindentation testing for both crystalline and amorphous material systems. Furthermore, pop-in events were believed to be connected with dislocation nucleation, although such associations have since been widely debated. Nevertheless, Bei et al. went on to detail the approach taken during their analysis, wherein finite element analysis was coupled with experimental load vs. displacement analysis to compare the effect of assuming a rounded spherical Berkovich tips’ apex and the apex geometry gleaned from calibration procedures that had been documented in the past by O&P.

More specifically, the Virtual Indenter FEA simulation package from MTS Corporation (Knoxville, TN, USA), at the time, was utilized alongside MTS’s nanoindentation testing system such that a diamond Berkovich indenter tip could be utilized in conjunction with a Cr₃Si single-crystal material system for their respective study. Interestingly, Bei et al. observed that the magnitude of the load at which a pop-in event occurred varied across crystallographic orientations and, after that, was rationalized through the lens of different resolved shear stress states and slip systems present in the material. Moreover, the findings by Bei et al. were consistent with their hypothesis that assuming a spherical apex, rather than a conical-spherical apex of a sharp pyramidal tip at sub 100 nm depths or so, could not provide an adequate geometrical description of the tip.

The influence of elastic anisotropy was not considered by Bei et al. as a potential influence upon indentation stress fields nor was the fact that sharp edges on Berkovich indenter tips were observable in residual deformation impressions considered. Such lack of consideration suggests that future work must address or overcome such problems, given that edge effects would undoubtedly change the stress field or stress state compared with a purely spherical geometry. Furthermore, because of the work of Bei et al. in 2005, the nanoindentation and nanomechanical characterization communities were made aware that simply assuming a spherical apex resulted in a maximum resolved shear stress that was overestimated by more than 41%. Lastly, Bei et al.’s experimental data correlated with the onset of the first pop-in event and dislocation nucleation once a material’s theoretical shear strength was surpassed.

Pedagogical consideration of nanoindentation testing and load vs. displacement data analysis methods, especially those defined by O&P and widely accepted by the general nanomechanical and material characterization communities, continued into the mid-2000s. Another staple of the early sub-micron mechanical property investigation community, A. C. Fischer-Cripps was well aware of the general issues that users encountered through 2006 while attempting to perform nanoindentation testing, thus leading Fischer-Cripps to document common sources of error associated with performing nanoindentation testing such that ITT users who wished to utilize nanoindentation techniques could more readily and confidently do so in Ref. to [12].

Around the same time that Fischer-Cripps, O&P, and Bei et al. published their mid-2000s research, Troyon and Huang laid the additional groundwork for a correction factor, which was expressed by the $\beta$ term presented in Eq. (4) by O&P in 2004; concurrent effects were appropriately accounted for during nanoindentation data analysis [13]. In doing so, Troyon and Huang detailed how radially inward and elastic displacements as well as indenter shapes deviations from a perfect cone geometry can concurrently be linked with one another in place of $\beta$, in the case of O&P in [6], or $\gamma$, in the case of work performed by Hay et al. in [4], alone. Soon after that, Troyon and Huang continued their research into the matter of a comprehensive multiplicative correction factor for mathematically detailing relations between contact area, modulus of elasticity, and unloading contact stiffness in another article too (see [14]). Moreover, [14] suggests that the Troyon and Huang methods should undergo continued analysis and extended applications; however, there exists no generalized advantage of their method with that of the OP method.

Like decades prior, the advancements made between 2013 and the present point centered upon pedagogical matters, the continued applicability of nanoindentation-based analysis, the development of a deeper understanding of plasticity and the mechanisms of materials, and more. Accordingly, we may first consider the work of Siu and Ngan, which was published in 2013 and suggested that the dynamic/CSM-based measurement method artificially induced sample strength modifications during oscillatory nanoindentation testing [15]. If true, such findings would generally question the integrity of CSM-based nanoindentation testing and the CSM-dependent advancements made within materials characterization as a field of study. Siu and Ngan made their case by coupling nanoindentation with microscopy (EBSD and TEM), ex-situ, while a Berkovich indenter tip geometry was utilized, CSM frequencies were varied, and a ductile, commercially pure, Al test specimen was used. Tentatively ignoring the veracity of the approach taken by Siu and Ngan for the time being and the correctness of their philosophical claims, the authors showed that CSM-induced errors could be decoupled from sample strain rate sensitivities. Interestingly, the work by Siu and Ngan resulted in the procurement of methodological findings and material deformation mechanism-based findings.

Stated otherwise, Siu and Ngan went beyond the realm of demonstration in 2013 and into the arena of physical mechanisms surrounding material plasticity. Siu and Ngan suggested that their unexpected observation of material softening could be interpreted through the lens of variable pressure during oscillatory cycling, enabling stress relief due to elastic recovery during the unloading half cycle, wherein the material was in an elastoplastic state. Moreover, Siu and Ngan argued that those mentioned above, in turn, induced dislocation motion reversals, which resulted in dipole dislocation annihilation, decreased dislocation density, and dislocation motion-induced subgrain formation. Finally, although Siu and Ngan noted that the OP method of hardness determination appeared to be flawed when CSM capability was enabled, Siu and Ngan also conceded that the influence of nanometric oscillations upon nanometer length scale deformation of metals was unknown at the time [15].

Nevertheless, Al was not the only material or work concerned with related or similar matters of relevance following their Al-based 2013 article. In fact, during the same year, Siu and Ngan extended such oscillation-induced strength modification effects to Cu and Mo, too [16]. Furthermore, motivated by their original findings detailed in [15], Siu and Ngan stated that oscillation-induced softening of Al was intrinsically intertwined with enhanced annihilation of dislocations and formation of sub-grains due to the simultaneous imposition of oscillatory stresses [16]. Accordingly, such an interpretation and distillation of their earlier findings were consistent with their respective motivation underlying the use of Cu and Mo to go beyond proof-of-concept demonstration and toward the realm of generalized phenomena for a particular class of metallic materials.

While Siu and Ngan continued to build upon their 2013 research elsewhere [17, 18], other researchers garnered a maintained interest in the topic [19, 20, 21]. Additional oscillatory-induced and CSM measurement methods incurred errors studied by varying nanoindentation research and development community members. Inspired by the 2006 publication by Durst et al. [22], as well as the work of Cordill et al. in 2009 [23], alongside Pharr et al.’s more recent account of Durst et al.’s findings in [24] via modulus-to-hardness material ratios in addition to unloading curve assumptions and their effect on evaluated stiffness values, and Vachhani et al.’s reported outcome departures during dynamic nanoindentation testing as a function of harmonic frequency in [25], Merle et al. set out to advance a generalized understanding surrounding Vachhani et al.’s findings in [26].

Consequently, Merle et al. found that caution must be exercised appropriately when evaluating materials with high modulus-to-hardness ratios because selected harmonic parameters cause notable contact stiffness underestimation, which directly influences the resultant modulus of elasticity recorded when high loading rates are employed [26]. In doing so, Merle et al. were able to identify the culprit responsible for such oscillation-induced behavior in terms of a biased phenomenon associated with lock-in amplifier signal processing. Furthermore, Merle et al. illustrated how phase angle signal data could indicate the occurrence of said oscillatory artifacts during dynamic nanoindentation testing.

Turning our attention to advancements made in so far as the utility of the work of indentation is concerned, per modernized understanding of nanoindentation as a field of study, the 2013 publication by Jha et al. is considered next [27]. Jha et al. executed extensive nanoindentation modeling via finite element analysis methods to probe elasticity and elastoplastic behavior to garner insights into the total work of indentation and the elastic work of indentation from load–displacement data. Jha et al. found that the aforementioned work parameters could characterize the mechanical response of materials under indentation [27]. Based on the work of Jha et al., one may consider the effort as a means of better appreciating the concept of the work of indentation (which was already discussed in the previous subsections) and the agreement shared between the contact depths obtained for Berkovich and spherical indenter tips coupled with the OP method and that obtained via elastic energy constant-based contact depth determination methods in [27].

In much the same way the nanoindentation research and development concerned with the notion and concept of the work of indentation extended into the current decades’ advancements, nanoindentation stress–strain curve evaluation was undoubtedly a focal point of research activity to date too. Such sentiment holds in so far as the influence of CSM measurement parameters is considered to a greater extent at the same time too. However, by keeping with the notable research published in 2013 concerning said matters of inquiry, we may first consider the work of Bobzin et al. [28]. In 2013, Bobzin et al. attempted to address the standing need for prospective nanoindentation-based mechanical flow curve derivation methods via iterative comparative analysis of FEA computations. They experimentally obtained load vs. displacement curves coupled with adaptive plastic behavior model parameter identification. Interestingly, Bobzin et al. echoes less-utilized stress–strain nanoindentation analysis by Juliano et al. [29], such that reference stress and reference strain values are measurable from load-depth data and curves via Eq. (9) and Eq. (10), respectively. Thus, the following mathematical relation for reference stress was invoked by Bobzin et al., such that.

$$\sigma =\frac{0.9P}{\pi a^2}\left(\frac{h_t}{h_c}-1\right)$$E9

while the following mathematical relation for reference strain was also provided by Bobzin et al., such that.

$$\epsilon =\frac{\sigma }{1-v^2}\left(\frac{4}{2.7P\left(h_t{h}_c^{-1}-1\right)}{\left(2h_t-2h_c\frac{1}{{\left[2h_c{\left(a^2+h_c^2\right)}^{-1}\right]}^{\frac{1}{3}}}\right)}^{\frac{3}{2}}-\frac{1-v_i^2}{E_i}\right)$$E10

and linked together via the Johnson-Cook model, as given in the following expression, depicted herein as Eq. (11), such that.

$$\sigma =A+B{\epsilon }^n$$E11

Just a few years after the work of Jha et al., Bobzin et al., Siu and Ngan, and Merle et al., which are detailed above, nanoindentation stress vs. strain evaluation methods, other than Bobzin et al.’s FEA-supported protocol, were applied to the mechanical characterization and property evaluation tasks surrounding alloyed Ti64 materials in [30]. Though the spherical nanoindentation-derived effective stress vs. strain-curve methodologies and protocols were established before the work of Weaver and Kalidindi, which is currently under consideration, Weaver and Kalidindi clearly and concisely noted their intended objectives and motivation behind their 2016 manuscript in its’ introductory section. Weaver and Kalidindi provided the materials development and design research and engineering communities with a referable case study that ideally captured the high-throughput nature and feasibility of such material characterization in so far as metal systems were concerned, including Ti64, which maintains microstructural complexity concerning site-specific, microstructural, and local length-scale dependencies of measured material response too [30].

Moreover, the protocols above of relevance, which include Kalidindi and Pathak’s approach, as detailed in the previous section of the present literature review, were relied upon throughout most of the analysis and mechanical characterization performed in Weaver and Kalidindi’s 2016 study. Building upon the research presented in 2016, as well as the advancements and protocol developments by Kalidindi and close colleagues across Drexel University of the Georgia Institute of Technology, as well as others, Weaver, and others from the same network relating Kalidindi, Pathak, and Weaver with one another, continued to extend the range of spherical nanoindentation stress vs. strain exploration and data analysis. Intriguingly, Weaver et al. invoked CSM spherical nanoindentation stress–strain analysis to probe and measure the evolution of mechanical properties of He, W, and (He+W) ion-irradiated tungsten at a granular level in [31].

Nevertheless, an unexpected observation of significance for those utilizing OP analysis modalities was the secondary finding by Weaver et al. that ion-irradiated hardness values obtained via OP and Berkovich indentation and Weaver et al.’s spherical dynamic nanoindentation approach can be directly compared with one another. Still, Weaver et al. are not the only modern researchers concerned with spherical indentation and indentation-based stress–strain characterization and data analysis. Instead, Xiao et al. in 2019 unveiled a mechanistic model for spherical nanoindentation stress vs. strain relations, which intertwined three principal deformation mechanisms of relevance [32]. More to the point, said principal deformation mechanisms invoked by Xiao et al. included the indentation size effect and, therefore, the matter of geometrically necessary dislocations, followed by irradiation hardening and were concluded in terms of strain-softening affected by the removal of defects. The performance of Xiao et al.’s model for non-irradiated reference materials is compared to experimentally measured values and the model’s predictions of indentation stress versus indentation strain. Additionally, this performance is compared to an alternative model previously reported in the literature before Xiao et al.’s publication.

Beyond the work of Xiao et al. cited above, additional developments of relevance include that detailed in 2018 by Jin et al., wherein Jin et al. claimed to have formulated a model for the quantifiable linkage of incipient irradiation damage via nanoindentation pop-in phenomena analysis in [33]. After that, in 2019, the near-surface nanoindentation response of ion-irradiated FCC metals was studied by way of strain gradient plasticity mechanisms, which were modified to account for irradiation effects [34]. At the same time, Xiao et al. extended their 2019 single-crystal irradiation studies into the realm of polycrystalline steel specimens [35].

On the other hand, just a couple of years prior to the work of Xiao et al. in 2019, Kumar et al. attempted to develop a nanoindentation-based means of evaluating argon-ion irradiation-induced hardening of a ferritic and martensitic dual-phase steel in [36]. As a result, Kumar et al. found changes in nanoindentation hardness of dual-phase steel because irradiation follows a power-law relation dependent upon irradiation dosage [36].

Returning to the realm of spherical nanoindentation stress vs. strain curve analysis through Weaver, Pathak, and Kalidindi, one may note [37], wherein Khosravani et al. focused on spherical nanoindentation as a means of characterizing two hierarchical and martensitic FeNiC steel systems. Interestingly, Khosravani et al. found that pop-in phenomena could be linked to dislocation and lath boundary interaction, followed by dislocation transmission through the boundary during nanoindentation loading. Beyond the critical examination that Khosravani et al. provided regarding how the measured properties may be related to uniaxial tensile test counterparts, Khosravani et al. also quantified the mechanical behavior of lath martensite phases across length scales and various indenter tip radii. Khosravani et al. reportedly observed minor indentation size effects across variable tip radii and nanometric length-scale plasticity and strength domination by nanostructured defects.

Unsurprisingly, steel continues to capture the attention of nanoindentation researchers and material science and engineering community members to date. For example, in 2020, Massar et al. applied dynamic Berkovich nanoindentation hardness and modulus of elasticity measurements, according to the OP method, for recycled battlefield scrap steel powders and cold spray-processed material consolidations in [38]. In addition to Massar et al., continued consideration of the use of nanoindentation to characterize steels in the 2010s unveils the work of Pham and Kim, which was published in 2015, wherein the authors identified the modulus of elasticity and nanoindentation hardness values associated with an SM490 steel weld zone via statistical data analysis [39]. Furthermore, Yang et al. also utilized statistical data analysis to quantify the transformation kinetics of bainite phase formation within an austempered steel [40]. Still, nanoindentation for such steel specimen characterization is also discussed elsewhere, including the following references of note: [41, 42, 43, 44, 45], to name a few.

One ought to also consider the recent work of Ruiz-Moreno et al., published in 2020, as another notable nanoindentation-related research article [46]. Ruiz-Moreno et al. performed nanomechanical characterization over a range of temperatures. Ruiz-Moreno et al. demonstrated how indentation hardness of a P91 system could be measured under ambient conditions and at elevated temperatures (873.15°K) for transient property inspection via quasi-static and dynamic nanoindentation testing, as shown in Figure 3. In addition to the range of transient properties explored at elevated temperatures and ambient temperature, Ruiz-Moreno et al. ventured into pedagogical consideration surrounding assessing spherical nanoindentation stress vs. strain curves. Conditions, Ruiz-Moreno et al. noted that consequences existed when adhering to the school of thought surrounding and underpinning the work of Weaver, Pathak, Kalidindi, and the like [46].

Figure 3.

Cyclic or quasi-static nanoindentation load-depth curves are presented in (a), while a dynamic nanoindentation load vs. depth curve is presented via (b). Indentation stress–strain curves as a function of testing temperature are presented in (e,f). Additional details shown in (c,d,g,h) can be understood through consulting [46], which is the reference the present figure is adopted and reproduced from.

2. Actualization of a frustrum instrumented strength microprobe

As was discussed above, many have endeavored to use a spherical nanoindenter tip probe to measure small-scale stress–strain curves with limited or variable degrees of success [47]. However, spherical nanoindentation includes several practical difficulties because the uncertainty in the contact area is generally more significant for spheres than for other indenter shapes (see the variation in an as-manufactured cono-spherical tip geometry obtained from a leading tip supplier for more nuanced appreciation, as shown in Figures 4–7, which are adopted from [10].

Figure 4.

Renderings of the imperfect tip geometry obtained when a cono-spherical tip was purchased from Micro-star technologies: (a–c) have been normalized in the x, y, and z directions. (a) Degree of deviation of the actual tip geometry from an ideally spherical tip at the apex of the indenter probe. Notice the scale bar in microns to the right of (a), which topographically signifies the distance from the point of Cartesian coordinate origin parallel to an x-y plane that is orthogonal to the z-axis. Reproduced from [10].

Figure 5.

(a) location of the extracted tip profile, obtained through the very tip of the probe’s apex. (b) a circle with a nominal radius of 2.71 μm fit to the presented profile. The circle in (b) is positioned such that it is coincident with the tip of the apex and parallel to the z-axis. Notice the deviation from the imposed circle in (b) by the profile between approximately 1.5 and 3.5 μm on the horizontal axis. (c-d) deviation of the true surface from an ideal sphere of a 2.71 μm radius. (c-d) is normalized in the x- and y-directions with vertical distortion in the z-direction to capture the deviation via colored contour plotting. Reproduced from [10].

Figure 6.

Residual indentation imprint on a single crystal of commercially pure Al was obtained using 3D confocal microscopy-based analysis. The upper graphical rendering illustrates the true surface profile at the spherical region in the blue line versus the ideal surface profile for a spherical tip with a 2.71 μm radius in the black line. The plot at the bottom of this figure captures the surface profile deviation from a 2.71 μm nominal radius circle. Reproduced from [10].

Figure 7.

Even though the maximum indentation depth associated with a nanoindenter tip of 2.71 μm is 794 nm, the max cut-off depth before the transition from spherical shape to a conical geometry was set at 542 nm to showcase the spherical geometric profiles readily. Therefore, at 542 nm away from the apex of the perfectly spherical tip, the ideal spherical geometry is presented in (a). (b) Unveils the true surface profile associated with the tip from Micro-star technologies up to 794 nm from the apex. For comparison with (a), (c) captures the actual surface cropped to the same vertical dimension as the ideal surface, highlighting the deviation. Renderings are normalized in the x, y, and z directions. Reproduced from [10].

Furthermore, as the contact area grows during spherical nanoindentation testing, the volume of tested material also grows, gradually and continuously incorporating virgin material into the test. Thus, both the material and strain change concurrently. However, with a flat punch, the contact area is well known and fixed as the area of the punch face. Because the contact area is fixed, the volume of the tested material is roughly constant throughout the entirety of the test. Thus, the present section demonstrates how such fixed flat-punch indentation testing conditions or parameters enable microscopic true stress–strain relationships of a wide range of metallurgical materials to be probed using Hay et al.’s emergent protocol (a protocol that has ultimately brought the nanoindentation community one step closer to realizing the long sought-after Instrumented Strength Microprobe).

The microscopic stress–strain relationships were measured by nanoindentation using a new method [48]. This method required Young’s modulus as input, and it returned the yield point, true stress–strain ordered pairs beyond the point of complete contact, and coefficients $K$ and $n$ of the best power-law fit to the post-yield behavior Eq. (12):

$$\sigma =K{\epsilon }^n$$E12

With a KLA iMicro nanoindenter, a 90-degree frustum, having a flat circular end with a diameter of around 10 μm, can be pressed into a given target material at a prescribed strain rate and to a depth of 2 μm while continuously measuring load, $P$; penetration depth, $h$; and elastic contact stiffness, $S$, with the latter being measured by a small, superimposed oscillation [49, 50]. The strain is calculated as Eq. (13) for each point acquired during loading beyond the point of complete contact:

$$\epsilon =\left(\frac{2}{\pi }\right)\frac{h}{a}$$E13

wherein $a$ is the contact radius, calculated as the radius of the indenter tip at a distance $h$ from the frustum face along the indenter axis, or the following relation, shown in Eq. (14):

$$a=a+\mathit{\text{htan}}\psi$$E14

with $\psi$ representing the half-included angle of the cone. For each point acquired during loading beyond the point of complete contact, the stress $\sigma$ was calculated in proportion to the mean pressure of the contact $p_m$, as Eq. (15) below:

$$\sigma =\xi p_m$$E15

with the constant proportionality $\xi$ calculated as a linear function of the parameter $S^{\ast }$ according to Eq. (16) as follows,

$$\xi =0.3969\left(S^{\ast }\right)+0.3218$$E16

where $S^{\ast }$ is calculated as $S_L/S$ or the local slope of the unloading curve $\mathit{dP}/\mathit{dh}$ divided by the elastic contact stiffness at the same point. Finally, the expression for strain rate is given as Eq. (17):

$${\epsilon }^{\prime }\equiv \left(\frac{2}{\pi }\right)\frac{h^{\prime }}{a}$$E17

Logically, the target strain rate ${\epsilon }^{\prime }$ is prescribed by a given user, and the system calculates and imposes the necessary penetration rate $h^{\prime }$ to achieve the target strain rate. In any case, the point of complete contact between the punch face and test surface was determined as the point where measured contact stiffness first exceeded the expected value based on the known reduced modulus for the material $E_r$ [51, 52], which is given in Eq. (18):

$$S>2E_{r}a$$E18

To determine the yield point, the stress–strain ordered pairs acquired beyond the point of complete contact fit the power-law form of Eq. (12); the yield point for each test was determined as the point of intersection between this power-law fit and the linear part of the stress–strain curve (as determined by the input Young’s modulus). Thus, by the end of each test, the whole true stress–strain curve was available by automatically patching together these three segments into a whole: (i) from the origin to the yield point: the linear part (generated by the input Young’s modulus); (ii) from the yield point to the point of complete contact: an extrapolation of the power-law fit to data from segment (iii); and (iii) from the point of complete contact to the end of loading: measured stress–strain as calculated by Eq. (13) and Eq. (14), respectively.

That said, the relationship for strain that is defined in Eq. (13) was derived by considering elastic contact between a frustum and a test surface, with the load and depth related through the reduced modulus $E_r$, as Eq. (19) [51].

$$P\equiv 2E_r\mathit{ah}$$E19

Dividing both sides of Eq. (19) by the contact area gives an expression for mean pressure, Eq. (20):

$$\frac{P}{{\mathit{\pi a}}^2}=p_m=\left(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.\right)E_r\left(\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)$$E20

If we define the strain as in Eq. (13), then Eq. (20) becomes Eq. (21), such that.

$$p_m=E_r{\epsilon }_i$$E21

which is analogous in form to the stress–strain relation used to comprehend the elastic part of a uniaxial tension or compression test. Because one definition of strain should befit both elastic and plastic phases of the test, we used the definition of strain expressed by Eq. (13) for all testing, and by extension, the definition of strain rate expressed by Eq. (17).

The linear function for inferring the constant of proportionality $\xi$ from $S^{\ast }$ Eq. (16) was determined using extensive (90+) FEM simulations of flat-punch indentations into materials with systematically varied degrees of plasticity. The extent of plasticity for each finite-element simulation was captured by a single value of the parameter $S^{\ast }\equiv S_L/S$ [1, 53]. For each simulation, $S_L$ was determined as the slope of the loading curve prior to peak penetration; the contact stiffness $S$ was determined as the slope of the simulated load-depth relation at the onset of unloading. (Experimentally, $S_L$ and $S$ are available throughout loading, but $S$ is only available at the onset of unloading.) For fully elastic simulations, $S^{\ast }$ had a value of unity because the loading and unloading curves coincided. For the most plastic simulations, $S^{\ast }$ approached zero as the unloading curve was nearly vertical. Thus, the domain for $S^{\ast }$ is zero (fully plastic) to unity (fully elastic).

To determine the precise functional form of $\xi =f\left(S^{\ast }\right)$, each finite-element simulation was analyzed as follows: (i) the parameter’s value $S^{\ast }$ was calculated; (ii) strain was calculated according to Eq. (13); (iii) the true stress value $\sigma$ was calculated from the input stress–strain relation as the stress at the strain calculated in step (ii); and (iv) the value of $\xi$ was calculated by dividing the true stress (step iii) by the mean indentation pressure $p_m$. Finally, using the results of all simulations, it was found that $\xi$ depended linearly on $S^{\ast }$, so long as $S^{\ast }<0.8$; Eq. (16) is the best fit for this linear relation. We note here that the intercept of Eq. (16), which indicates the value of $\xi$ for the case of full plasticity ($S^{\ast }\to 0$), is very close to the scaling factor of 1/3 determined by David Tabor [54] and further verified by many others through finite-element analysis [55] and experiments [56].

Figure 8 presents the true stress–strain curves obtained for multiple flat-punch indents via the method described herein and applied to commercially pure Ti. The average yield stress obtained was 270 ± 50 MPa, and individual test data from twelve measurements using this method agreed with 240 MPa as measured via tensile testing while also being well within the upper and lower bounds of yield strength for Ti reported within the relevant literature. While the scatter within the data was high, the microstructure of the pure Ti system resulted in relatively large grains, which would allow for a single test location to be constrained to single as well as randomly oriented grains; further information about Hay’s original work and the KLA-patented flat-punch stress vs. strain analysis technique can be found in [57], while the present authors have detailed recent applications and extensions of the method in [58].

Figure 8.

True stress–strain for a Ti metal system characterized using the method formulated further herein and Hay’s patented and emergent flat-punch nanoindentation technique. Reproduced from [57].

3. Conclusions

Building off the discussion provided in Part 1, the present chapter continues the consideration of advancements within the theory and practice of small-volume instrumented indentation testing and nanoindentation testing and analysis enabled via the advent of the Oliver-Pharr method. Advancements presented within the literature following Oliver and Pharr’s 1992 research article (and discussed herein) focused upon contact area estimation, pile-up phenomena, stiffness, creep-correction, thermal drift, relations between measured properties and work of indentation terms, load–displacement curve, pop-in phenomena, the application of strain gradient plasticity, and more. Thereafter, consideration of prior work presenting spherical nanoindentation testing as a means of assessing indentation stress–strain curves of metallic materials (in particular) was discussed, followed by transient material property assessment with high-temperature nanoindenter systems and the effects of continuous stiffness measurement testing procedures and their influence upon recorded results. In conclusion, the present chapter is finalized through the introduction of an emergent and theoretically consistent approach to assessing true stress–strain curves at a micromechanical scale using a flat-punch nanoindenter tip geometry and reliance upon Hollomon power-law plasticity and constitutive parameter fitting, therefore detailing the long sought-after ability to utilize nanoindentation testing as an instrumented strength microprobe.