A comparative study of indentation size effect models for different materials.

The phenomenon of indentation size effect (ISE) has received great attention in aerospace, nuclear power, microelectronics and medicine. Although researchers have proposed various ISE models, these models often involve different form and number of parameters that can make our wonder which is the best in existed ISE models. Herein, three types of ISE test data, namely, normal ISE, reverse ISE and transition of normal to reverse ISE, are used to evaluate the sixteen ISE models. The comparatively study indicates that Hou-Jennet(H-J), Nix-Gao-Feng (N-G-Fs), Nix-Gao-Hausild (N-G-H), Nix-Gao-Abu Al-Rub (N-G-A) and Nix-Gao-Qius (N-G-Qs) models can accurately predict the normal ISE. The reason for this is that the friction stress that is not related to dislocation activities or the indentation size effect of plastic zone has been introduced into these models. Therefore, these two factors should be considered in future ISE models. The sixteen ISE models are originally proposed to describe the normal ISE of different materials. However, to our surprise, some of these models are able to capture the reverse ISE and the transition of normal to reverse ISE of different materials. The determination coefficients (DC) of the sixteen ISE models are also determined for different materials. For reverse ISE, the highest DC value for Ni Carbide Silicon (NiCSi), TC4 titanium alloy (TC4) and Pulsed electro-deposited Ni (PED Ni) are given by the Exponential (EXP), Nix-Gao-Feng (N-G-Fs) and Nix-Gao-Abu Al-Rub (N-G-A) models, respectively. For the transition of normal to reverse ISE, the Nix-Gao-Yuan-Chen (N-G-YC), Nix-Gao-Feng (N-G-Fs), and Nix-Gao-Hausild (N-G-H) models produce the maximum DC for ZrO2 ceramic (ZrO2), Cu single crystals (Cu) and Y2O3-ZrO2 ceramic (Y2O3-ZrO2), respectively. Moreover, the mean DC of the Nix-Gao-Feng (N-G-Fs) model is the maximum among the sixteen ISE models, followed by the Nix-Gao-Hausild (N-G-H) model, but they cannot accurately predict the reverse ISE. Therefore, the Nix-Gao-Feng (N-G-Fs) and Nix-Gao-Hausild (N-G-H) models should be further modified to accurately predict the reverse ISE.

Nanoindentation Test of Ion-Irradiated Materials: Issues, Modeling and Challenges.

Exposure of metals to neutron irradiation results in an increase in the yield strength and a significant loss of ductility. Irradiation hardening is also closely related to the fracture toughness temperature shift or the ductile-to-brittle transition temperature (DBTT) shift in alloys with a body-centered cubic (bcc) crystal structure. Ion irradiation is an indispensable tool in the study of the radiation effects of materials for nuclear energy systems. Due to the shallow damage depth in ion-irradiated materials, the nanoindentation test is the most commonly used method for characterizing the changes in mechanical properties after ion irradiation. Issues that affect the analysis of irradiation hardening may arise due to changes in the surface morphology and mechanical properties, as well as the inherent complexities in nanoscale indentation. These issues, including changes in surface roughness, carbon contamination, the pile-up effect, and the indentation size effect, with corresponding measures, were reviewed. Modeling using the crystal plasticity finite element method of the nanoindentation of ion-irradiated materials was also reviewed. The challenges in extending the nanoindentation test to high temperatures and to multiscale simulation were addressed.

Microhardness Variation with Indentation Depth for Body-Centered Cubic Steels Pertinent to Grain Size and Ferrite Content.

For a micro-indentation hardness test with non-destructivity, the Nix-Gao model is widely used to describe tested hardness or microhardness variation with an indentation depth induced by indentation size effect, in which tested hardness approaches the macrohardness when the indentation depth is large enough. Based on an analysis of hardness measurements on 10 body-centered cubic steels with diverse microstructure, this paper proposes an analytical relation between microhardness to macrohardness ratio and the indentation depth by explicitly linking characteristic indentation depth (a data-fitting parameter) to grain size and ferrite volume fraction using two different methods. In addition, the normal distribution theory is incorporated to consider the inevitable scatter of identical measurements resulting from material heterogeneity and machining/testing errors. Results show that the proposed model, with 96% reliability, can effectively predict microhardness variation with the indentation depth and its scatter.

Irradiation-Hardening Model of TiZrHfNbMo0.1 Refractory High-Entropy Alloys.

In order to find more excellent structural materials resistant to radiation damage, high-entropy alloys (HEAs) have been developed due to their characteristics of limited point defect diffusion such as lattice distortion and slow diffusion. Specially, refractory high-entropy alloys (RHEAs) that can adapt to a high-temperature environment are badly needed. In this study, TiZrHfNbMo0.1 RHEAs are selected for irradiation and nanoindentation experiments. We combined the mechanistic model for the depth-dependent hardness of ion-irradiated metals and the introduction of the scale factor f to modify the irradiation-hardening model in order to better describe the nanoindentation indentation process in the irradiated layer. Finally, it can be found that, with the increase in irradiation dose, a more serious lattice distortion caused by a higher defect density limits the expansion of the plastic zone.

Anisotropic Deformation Behavior and Indentation Size Effect of Monocrystalline BaF2 Using Nanoindentation.

In this study, our objective is to investigate the anisotropic deformation behavior and the indentation size effect (ISE) of monocrystalline barium fluoride (BaF2) using nanoindentation experiments with a diamond Berkovich indenter. BaF2 is known for its anisotropy, which results in significant variations in its mechanical properties. This anisotropy poses challenges in achieving high processing quality in ultra-precision machining. Through our experiments, we observed numerous pop-in events in the load-displacement curves, indicating the occurrence of plastic deformation in BaF2 crystals, specifically in the (100), (110), and (111) orientations; these pop-in events were observed as the indentation depth increased to 56.9 nm, 58.2 nm, and 57.8 nm, respectively. The hardness-displacement and elastic modulus-displacement curves were obtained from the tests exhibiting the ISE. The nanoindentation hardness of BaF2 is found to be highly dependent on its crystallographic orientation. Similarly, for BaF2 in the (100) orientation, the range is from 2.43 ± 0.74 and 1.24 ± 0.12 GPa. For BaF2 in the (110) orientation, the values range from 2.15 ± 0.66 to 1.18 ± 0.15 GPa. For BaF2 in the (111) orientation, the values range from 2.12 ± 0.53 GPa to 1.19 ± 0.12 GPa. These results highlight the significant influence of crystallographic orientation on the mechanical properties of BaF2. To better understand the ISE, we employed several models including Meyer's law, the Nix-Gao model, the proportional specimen resistance (PSR) model, and the modified PSR (mPSR) model, and compared them with our experimental results. Among these models, the mPSR model demonstrated the best level of correlation (R2>0.9999) with the experimental measurements, providing a reliable description of the ISE observed in BaF2. Our reports provide valuable insights into the anisotropic mechanical characteristics of BaF2 materials and serve as a theoretical guide for the ultra-precision machining of BaF2.

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

Microhardness testing is a widely used method for measuring the hardness property of small-scale materials. However, pronounced indentation size effect (ISE) causes uncertainties when the method is used to estimate the real hardness. In this paper, three austenitic Hadfield steel samples of different plastic straining conditions were subjected to Vickers microhardness testing, using a range of loads from 10 to 1000 g. The obtained results reveal that the origin of ISE is derived from the fact, that the indentation load P and the resultant indent diagonal d do not obey Kick's law (P = A · d2). Instead, the P and d parameters obey Meyer's power law (P = A · dn) 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.

Indentation Size Effect in CoCrFeMnNi HEA Prepared by Various Techniques.

High entropy alloys (HEAs) are materials of great application potential and which have been extensively studied during the last two decades. As the number of possible element combinations is enormous, model materials representing certain groups of HEAs are used for the description of microstructure, properties, and deformation mechanisms. In this study, the microstructure and mechanical properties of the so-called Cantor alloy composed of Co, Cr, Fe, Mn, and Ni in equiatomic ratios prepared by various techniques (casting, melt-spinning, spark plasma sintering) were examined. The research focused on the indentation measurements, namely, the indentation size effect describing the evolution of the hardness with penetration depth. It was found that the standard Nix-Gao model can be used for this type of alloy at higher penetration depths and its parameters correlate well with microstructural observations. The Nix-Gao model deviates from the measured data at the submicrometer range and the applied modification affords additional information on the deformation mechanism.

Effect of Indentation Load on Nanomechanical Properties Measured in a Multiphase Boride Layer.

The study investigated the dependence of the indentation load on nanomechanical properties for a gas-borided layer produced on Inconel 600-alloy. During the measurements, the indentation load range from 10 mN to 500 mN was used. Three types of tested areas, differing in the concentration of chromium, were examined. The increase in chromium concentration was accompanied by an increase in indentation hardness and Young's modulus. Simultaneously, the increase in the indentation load resulted in a decrease in the indentation hardness and Young's modulus, for each type of the tested area. The presence of the indentation size effect was analyzed using four models: Meyer's law, Hays and Kendall model, Li and Bradt model, Nix and Gao model. For all tested areas, good agreement with the Meyer's law was obtained. However, areas with a higher chromium concentration were more susceptible to indentation size effect (ISE). The proportional specimen resistance (PSR) model was used to describe the plastic-elastic behavior of the tested materials, as well as to detect the presence of ISE. It was found that the increase in chromium concentration in the tested area was accompanied by a greater tendency to elastic deformation during nanoindentation.

Investigation of indentation size effect and R-curve behaviour of Li2O-SiO2 and Li2O-2SiO2 glass ceramics.

Indentation size effect (ISE) and R-curve behaviour of Li2O-SiO2 and Li2O-2SiO2 glass ceramics are investigated using micro-indentation and indentation-strength (IS) techniques, respectively. Vickers micro-indentations were applied on both materials at the load of 0.10-19.6 N to determine the load influence on the measured hardness. For the IS-measured fracture toughness, the load ranged from 1.96 to 19.6 N. The hardness decreased with increasing load by 20% and 18% on Li2O-SiO2 and Li2O-2SiO2 glass ceramics, respectively, indicating the ISE behaviour on both materials. The fracture toughness increased with the load by 27% and 59% on Li2O-SiO2 and Li2O-2SiO2 glass ceramics, respectively, signifying the R-curve behaviour. The ISE behaviour of both materials was analysed using the Meyer's, Hays-Kendall (HK), proportional specimen resistance (PSR), Nix-Gao (NG), modified PSR (MPSR) and elastic plastic deformation (EPD) models while the R-curve behaviour was analysed by the fractional power law. The Meyer's index of both materials was less than 2, strongly confirming the ISE existence. The HK, PSR and NG models were only suitable to determine intrinsic Vickers hardness for Li2O-2SiO2 glass ceramic while the MPSR and EPD models were successful for both materials. The fractional power law gave higher R-curve steepness for Li2O-2SiO2 than Li2O-SiO2 glass ceramics. Also, material and brittleness indices predicted, respectively, higher quasi-plasticity and better machinability for Li2O-2SiO2 than Li2O-SiO2 glass ceramics indicating superior performance in the former to the latter. Finally, this study presents a new significant insight into the micro-mechanisms of fracture tolerance behaviour of these glass ceramics which is critical to their functional performance as structural ceramics.

Exploring the origins of the indentation size effect at submicron scales.

The origin of the indentation size effect has been extensively researched over the last three decades, following the establishment of nanoindentation as a broadly used small-scale mechanical testing technique that enables hardness measurements at submicrometer scales. However, a mechanistic understanding of the indentation size effect based on direct experimental observations at the dislocation level remains limited due to difficulties in observing and quantifying the dislocation structures that form underneath indents using conventional microscopy techniques. Here, we employ precession electron beam diffraction microscopy to "look beneath the surface," revealing the dislocation characteristics (e.g., distribution and total length) as a function of indentation depth for a single crystal of nickel. At smaller depths, individual dislocation lines can be resolved, and the dislocation distribution is quite diffuse. The indentation size effect deviates from the Nix-Gao model and is controlled by dislocation source starvation, as the dislocations are very mobile and glide away from the indented zone, leaving behind a relatively low dislocation density in the plastically deformed volume. At larger depths, dislocations become highly entangled and self-arrange to form subgrain boundaries. In this depth range, the Nix-Gao model provides a rational description because the entanglements and subgrain boundaries effectively confine dislocation movement to a small hemispherical volume beneath the contact impression, leading to dislocation interaction hardening. The work highlights the critical role of dislocation structural development in the small-scale mechanistic transition in indentation size effect and its importance in understanding the plastic deformation of materials at the submicron scale.

Indentation Induced Mechanical Behavior of Spark Plasma Sintered WC-Co Cemented Carbides Alloyed with Cr3C2, TaC-NbC, TiC, and VC.

The focus of this paper is on examining the mechanical behavior of spark plasma sintered WC-Co based composites doped with Cr3C2, TaC-NbC, TiC, and VC, as well as defining some parameters characterizing deformation and fracture processes during hardness measurement. The calculated microhardness of WC-Co cemented carbides for all the studied compositions is found to be higher than the results obtained during hardness testing. Therefore, the ratio of the experimental and calculated values of microhardness is shown to be an approximate indication of WC-Co cemented carbide sensitivity to damage processes during indentation. Some parameters characterizing the microstructure-microhardness relationship are defined, and the nanomechanical properties of WC-Co cemented carbide phases are examined in order to separate the deformation and fracture processes during the indentation process. Strain gradient linear function parameters are calculated for 10-cycle nanoindentation. It was found that the nanoindentation curve after 10 cycles shows anomalous behavior of the WC grains, which indicates their fracture processes.

The Influence of the Third Element on Nano-Mechanical Properties of Iron Borides FeB and Fe2B Formed in Fe-B-X (X = C, Cr, Mn, V, W, Mn + V) Alloys.

In this study, the influence of alloying elements on the mechanical properties of iron borides FeB and Fe2B formed in Fe-B-X (X = C, Cr, Mn, V, W, Mn + V) alloys were evaluated using instrumented indentation measurement. The microstructural characterization of the alloys was performed by means of X-ray diffraction and scanning electron microscope equipped with an energy dispersive X-ray analyzer. The fraction of the phases present in the alloys was determined either by the lever rule or by image analysis. The hardest and stiffest FeB formed in Fe-B-X (X = C, Cr, Mn) alloys was observed in the Fe-B-Cr alloys, where indentation hardness of HIT = 26.9 ± 1.4 GPa and indentation modulus of EIT = 486 ± 22 GPa were determined. The highest hardness of Fe2B was determined in the presence of tungsten as an alloying element, HIT = 20.8 ± 0.9 GPa. The lowest indentation hardness is measured in manganese alloyed FeB and Fe2B. In both FeB and Fe2B, an indentation size effect was observed, showing a decrease of hardness with increasing indentation depth.

Round Robin into Best Practices for the Determination of Indentation Size Effects.

The paper presents a statistical study of nanoindentation results obtained in seven European laboratories which have joined a round robin exercise to assess methods for the evaluation of indentation size effects. The study focuses on the characterization of ferritic/martensitic steels T91 and Eurofer97, envisaged as structural materials for nuclear fission and fusion applications, respectively. Depth-controlled single cycle measurements at various final indentation depths, force-controlled single cycle and force-controlled progressive multi-cycle measurements using Berkovich indenters at room temperature have been combined to calculate the indentation hardness and the elastic modulus as a function of depth applying the Oliver and Pharr method. Intra- and inter-laboratory variabilities have been evaluated. Elastic modulus corrections have been applied to the hardness data to compensate for materials related systematic errors, like pile-up behaviour, which is not accounted for by the Oliver and Pharr theory, and other sources of instrumental or methodological bias. The correction modifies the statistical hardness profiles and allows determining more reliable indentation size effects.

Indentation size effect of Y-TZP dental ceramics.

OBJECTIVE: The purpose of this study was to investigate and analyze the indentation size effect (ISE) in Vickers hardness of monolithic yttria partially stabilized zirconia (Y-TZP) dental ceramics without and with the addition of dental dye A3. The ISE is analyzed using the Mayer law, a proportional specimen resistance (PSR) model and a modified proportional specimen resistance (MPSR) model. METHODS: Two samples of Y-TZP dental ceramics, trade names BruxZir (provided by Glidewell Laboratories, CA, USA), were investigated. The first sample was polished Y-TZP and the second sample was polished Y-TZP with the addition of dental dye A3, by VITA Classical Shade Guide. The Vickers hardness was measured under the following loads: 0.49N, 0.98N, 1.96N, 4.90, 9.81N and 29.42N. Thirty indentations were made on each sample, under each load. Relationships between the applied load, F, and the resulting indentation size, d, have been analyzed by the Mayer law, the PSR model and the MPSR model. RESULTS: The Meyer index (n) for both Y-TZP dental ceramics is less than 2, which indicates that hardness is dependent on test loads. The PSR model and the MPSR model were used to calculate "true" Vickers hardness or load-independent hardness. SIGNIFICANCE: All applied mathematical models are suitable for the data analysis, which is confirmed with high correlation coefficients, but the best correlation between measured values and mathematical models was achieved with the MPSR model with a correlation coefficient of 0.9999.

Effect of surface roughness in the determination of the mechanical properties of material using nanoindentation test.

A quantitative model is proposed for the estimation of macro-hardness using nanoindentation tests. It decreases the effect of errors related to the non-reproducibility of the nanoindentation test on calculations of macro-hardness by taking into account the indentation size effect and the surface roughness. The most innovative feature of this model is the simultaneous statistical treatment of all the nanoindentation loading curves. The curve treatment mainly corrects errors in the zero depth determination by correlating their positions through the use of a relative reference. First, the experimental loading curves are described using the Bernhardt law. The fitted curves are then shifted, in order to simultaneously reduce the gaps between them that result from the scatter in the experimental curves. A set of shift depths, Δhc , is therefore identified. The proposed approach is applied to a large set of TiAl6V4 titanium-based samples with different roughness levels, polished by eleven silicon carbide sandpapers from grit paper 80 to 4,000. The result reveals that the scatter degree of the indentation curves is higher when the surface is rougher. The standard deviation of the shift Δhc is linearly connected to the standard deviation of the surface roughness, if the roughness is high-pass filtered in the scale of the indenter (15 µm). Using the proposed method, the estimated macro-hardness for eleven studied TiAl6V4 samples is in the range of 3.5-4.1 GPa, with the smallest deviation around 0.01 GPa, which is more accurate than the one given by the Nanoindentation MTS™ system, which uses an average value (around 4.3 ± 0.5 GPa). Moreover, the calculated Young's modulus of the material is around 136 ± 20 GPa, which is similar to the modulus in literature.