Oliver-Pharr Method Research Articles

A Correction Function to Improve the Accuracy of Measuring Elastic Modulus by Instrumented Spherical Indentation

Abstract Instrumented indentation combined with the classic Oliver–Pharr method has been widely utilized to measure elastic modulus of various materials. However, the elastic modulus measured by instrumented spherical indentation (ISI) is not as accurate as that measured by instrumented sharp indentation, especially at large indentation depth. In this work, the effect of the maximum indentation depth on measurement of elastic modulus by ISI was deeply investigated through finite element simulations and experiments. It was found that errors in measured elastic moduli increase significantly due to the inaccurate estimation of contact radius and excessive increase in initial unloading stiffness as maximum indentation depth increases. A correction function was then proposed to correct the measured elastic modulus. After correction, the errors were effectively reduced to within ±5 % for most cases. This work contributes to discovery of the error source in the measurement of elastic modulus by ISI, thereby improving the measurement accuracy.

The mechanical properties of Al3Sc single crystal investigated by nanoindentation using Hertzian elasticity theory method and Oliver-Pharr method

The crystal structure, orientation, reduced modulus, and hardness of Al3Sc single crystal were studied using X-ray diffraction (XRD), electron backscatter diffraction (EBSD) and nanoindentation techniques, combined with Hertzian elastic theory method (Hertz) and Oliver-Pharr method (O&P). Millimeter-sized Al3Sc single crystals were prepared using quasi-equilibrium solidification of hypereutectic Al-15(wt %) Sc alloy. The XRD results showed that the samples were composed of L12-Al3Sc and Al phases. The EBSD and nanoindentation results showed that for five Al3Sc single crystals with different orientations, the reduced moduli measured by Hertz method were 146.6 ± 0.4–148.3 ± 0.2 GPa; while measured by O&P method they were 147 ± 2–150 ± 3 GPa. The corresponding hardnesses were varied between 3.8 ± 0.1 and 3.9 ± 0.1 GPa. The Poisson's ratio and elastic constants were determined through iterative calculation based on the relationship between the reduced modulus and Young’s modulus at various orientations. The polycrystal elastic properties, bulk modulus, shear modulus and Young's modulus, were determined. The elastic constants c11, c12, and c44 were 179.2 ± 0.4, 44.1 ± 0.1, and 68.3 ± 0.3 GPa, respectively; the Poisson's ratio was 0.196 ± 0.001. The Young's modulus of five Al3Sc single crystals oriented in different directions was in the narrow range 161–164 GPa. The anisotropy factor of Al3Sc crystal was 1.011 ± 0.002, confirming that it is elastically isotropic. These results provide fundamental information for the research and development of the Al3Sc intermetallic compound and related alloys in future.

A novel impact indentation technique with dynamic calibration method for measurement of dynamic mechanical properties

High strain rates and continuous real-time data acquisition are difficult to balance in existing impact indentation techniques. Meanwhile, the inaccuracy of the testing P-h curves limits the development of quantitative applications for dynamic testing of mechanical properties at micro region. This article develops a momentum exchange impact indentation testing method, which extends the testing range of strain rate to above 105 s−1, and at the same time avoids secondary impact. Impact indentation tests are conducted on single-crystal aluminum specimens with impact velocity from 76.9 to 176 mm/s. Based on calibrated data, the comparison of calculation methods for dynamic hardness is carried out. The dynamic elastic modulus is obtained by means of the Oliver-Pharr method, which increases from 71.9 to 89.4 GPa with increasing impact velocity. A multi-dimensional data analysis method is proposed based on the obtained experimental data to quantitatively study the damping and stiffness characteristics of materials. This testing method is particularly useful for quantitatively obtaining the micro region dynamic mechanical properties of solid materials at high strain rates.

Comparative Analysis and Error Assessment of Nanoindentation Evaluation Techniques for NafionTM117

AbstractAdvances in the application of polymers for electrochemical cells require an understanding of their viscous deformation mechanisms and their interaction with moisture. Nanoindentation offers a localized, microscale testing alternative to traditional tensile testing. However, the viscoelastic nature of the polymers, combined with their increased compliance, presents challenges in the analysis of nanoindentation results. In addition, the dependence on moisture results in significant scatter and low repeatability. This study combines nanoindentation and tensile testing as a verification method and compares different correction protocols for static nanoindentation to investigate the mechanical behavior of polymer electrolyte membranes. Comparisons of different indentation devices, analysis methods, and indentation protocols show a significant overestimation of Young’s modulus using the classical Oliver–Pharr method compared to values determined from tensile tests. Nanoindentation at different humidity levels revealed different mechanisms leading to a decrease in Young’s modulus and hardness with increasing humidity.

Berkovich indentation and the Oliver-Pharr method for shape memory alloys

Advances in instrumentation and interpretation of nanoindentation data with the Oliver-Pharr (O-P) method have fueled the reliable assessment of hardness and elastic modulus. However, shape memory alloys (SMAs) undergo a reversible phase transformation that complicates and challenges the conventional practice. This paper examines through finite element simulations the variation in nanoindentation response with SMA constitutive properties and the deformation regimes underneath a Berkovich indenter. Unique features of phase transformations – including tension-compression asymmetry, limited magnitude of transformation strain compared to plastic strain, and pressure-dependent transformation stress – inject fundamentally different patterns of elastic and inelastic deformation, compared to conventional elastic-plastic indentation. This can generate errors of up to 16% and 40% in hardness and modulus, respectively, when a conventional O-P approach for elastic-plastic materials is used. These errors are decreased if (1) contact stiffness is measured over a reduced (2% or less) portion of the initial unloading response and (2) new material-dependent expressions are used for geometric and correction factors in the O-P nanoindentation analysis. Approximate analytic expressions are obtained for the effective elastic modulus and martensite volume fraction in the inhomogeneous medium created by indentation. This new method is applied to Ni50.5Ti49.5 (at%) wire and for other SMAs, general guidance for selection of geometric and correction factors is provided.

Orientation-dependent hardness of Mo3Si studied by cube-corner nanoindentation

Mo3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{3}$$\\end{document}Si crystals having crystallographic out-of-plane orientations close to (001)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0\\,0\\,1)$$\\end{document}, (011)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0\\,1\\,1)$$\\end{document} and (123)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1\\,2\\,3)$$\\end{document} were studied by nanoindentation to better understand the hardness anisotropy. The Mo3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{3}$$\\end{document}Si precipitates were grown in a Mo-17.5Si-6B alloy and identified by electron backscatter diffraction. Pronounced material pile-up around the indents was observed, which causes inaccuracies in the projected contact area determined using the Oliver–Pharr method of around 30% compared to the real one. Based on the analysis of scanning electron micrographs, the projected contact areas and thus the hardness values were determined. To rationalize the observed pronounced orientation dependence of the hardness of Mo3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{3}$$\\end{document}Si, we suggest an easy-slip-cutting model accounting for the available slip systems as well as the upward flow of material along the surface of the cube-corner indenter tip.

Nanoindentation behaviour simulation of a polycrystalline NiTi shape memory alloy using the crystal plastic finite element method

Polycrystalline NiTi shape memory alloys (SMAs) are inevitably involved in plastic deformation under the action of an external force, thus significantly affecting the mechanical behaviour of materials. To deeply understand the plastic deformation mechanism of the NiTi SMA at 400 ℃, the representative volume element (RVE) model combined with the crystal plastic finite element method and the improved Voronoi algorithm were used to construct a three-dimensional polycrystalline spherical nanoindentation model for studying the contribution of different slip systems to the plastic deformation. The calculation results show that the constructed polycrystalline model can well match the experimental curve of the macroscopic stressstrain response curve of the polycrystalline NiTi SMA; the nanoindentation curve also confirms this result. Based on the constructed 3D polycrystalline nanoindentation model, three slip systems ({110}<100>, {010}<100>and {110}<111>) were introduced at the same time. According to the average cumulative shear strain calculated during the indentation process, the results show that the {110}<111> slip system provides the largest contribution to plastic deformation, and the {010}<100>slip mode provides the smallest contribution. The elastic modulus of the model was calculated using the Oliver-Pharr method, and the calculated results are consistent with the experimental results, which proves the validity of the model.

Mechanical Analysis and Constitutive Modeling of Nonlinear Behavior of Silver-based Conductive Ink

Stretchable electronic devices are progressively deployed in many applications of mechanical, electrical, and bio-medical engineering. These circuits are made of stretchable and flexible substrate, as well as conductive ink, and various electronic components. The choice of components and layouts for the substrate and conductive ink can regulate the stretchability of stretchable circuits. On top of that, the substance utilized to create the conductive ink must have high electrical conductivity and strong adherence to the substrate in order to produce a high-quality stretchable printed circuit. Thus, this study focused on the development of stretchable conductive ink using silver powder as a conductive filler and PDMS-OH as a binder. The mechanical properties of the synthesized ink were investigated via simple uniaxial tensile testing method and nanoindentation technique, respectively. Accordingly, the modulus of elasticity, tensile stress and yield stress of the ink were obtained as 5.72 MPa, 1.195 MPa, and 0.86 MPa, congruently at 137% strain before undergoing failure. The experimental stress-strain data was then employed on the elastic-plastic constitutive model to investigate the elastomeric properties of the ink as it is an alternative method of lengthy and expensive procedures of validating different polymers. Moreover, the hardness and reduced modulus of the ink were evaluated by nanoindentation method using 5 mN maximum load with 0.5 mN/s loading/unloading rate and 2 secs holding time. Consequently, the hardness and reduced modulus values were obtained as 1.45 MPa and 34.53 MPa, respectively. These values were further validated by Oliver-Pharr method, and were in a good agreement.

Investigation of nanomechanical properties of Ni-Mn based Heusler alloys using nanoindentation technique

Studies on mechanical properties of Ni-Mn based Heusler alloys are lacking which limits their use for practical applications. In the present work, nanomechanical analysis of ternary Ni-Mn-Sn and Ni-Mn-Al Heusler alloys were performed using nanoindentation test. Dense pellet specimens of selected 15 hr mechanical alloyed ball milled powder samples were prepared by consolidation, sintering and polishing processes. Load-displacement curves were obtained from a typical nanoindentation test by penetrating the indenter tip into the prepared pellet specimens surface with a prescribed loading and unloading data using Oliver-Pharr method of analysis to determine elastic modulus and hardness. The average elastic modulus and harness of Ni-Mn-Al alloy found more than Ni-Mn-Sn alloy. Also, the effects of sintering temperatures on the densities of pellet specimens were observed which showed that higher sintering temperatures are required to achieve the homogeneity and reduce the porosity of specimen.

Indentation creep behavior of Fe–8Ni–xZr oxide dispersion strengthened alloys

Abstract This study was conducted to understand the creep behavior of two oxide dispersion strengthened alloys containing Zr as the alloying addition by performing indentation creep tests at room temperature. The oxide dispersion strengthened alloys were Fe–8Ni–xZr (x = 1 and 4 at.%, i.e., Zr-1 and Zr-4 alloys, respectively), which had been previously fabricated by mechanical alloying; followed by consolidation via equal channel angular extrusion at 1000 °C. The indentation tests were conducted under a maximum load of 100 mN with the loading rates at 300 and 400 mN min−1. The hardness was calculated by the Oliver–Pharr method, and the creep properties, such as the creep displacement, creep strain rate, creep stress, and stress exponent n, were determined. The results showed that the Zr-4 alloy was harder than the Zr-1 alloy. However, the creep resistance of the Zr-1 alloy was better than that of the Zr-4 alloy. It was further demonstrated that both the hardness and creep resistance depended on the loading rate. Moreover, a possible creep mechanism was proposed. Although the tests were performed at room temperature, they can provide insight into the effect of an oxide dispersion strengthened alloys microstructure on creep at higher temperatures.

Nanoindentation of embedded particles

We address the effect of elastic inhomogeneity on elastic modulus and hardness determinations made by depth-sensing indentations performed on individual particles embedded within a matrix of different elastic modulus. Finite element simulations and nanoindentation experiments are used to quantify the consequences of particle/matrix elastic inhomogeneity and we propose an adaptation of the Oliver–Pharr method that gives access to particle properties knowing those of the matrix. The method is suitable for any combination of matrix and particle elastic modulus and for any type of indenter, provided that the area of the tested particles along the surface of the sample is measured and that a large number of particles are probed. Further conditions for the implementation of the method are that testing conditions be such (i) that permanent deformation of the matrix is avoided, and (ii) that permanent deformation in each probed particle under the indenter is not affected by the matrix.Graphical abstract

SURFACE PROPERTIES OF FLUOROCARBON COATINGS PRODUCED BY LOW-FREQUENCY PLASMATRON AT ATMOSPHERIC PRESSURE

This paper presents the results of studies conducted on coatings based on thin fluorocarbon films obtained by plasma-enhanced chemical vapor deposition using low-frequency plasmatron and low-temperature plasma at atmospheric pressure. The possibility of forming thin fluorocarbon layers by supplying a cyclohexane/carbon tetrafluoride gas mixture to substrates made of polymeric materials (polyethylene terephthalate and polystyrene) has been demonstrated and the main technological modes of deposition process have been established. The absorption spectra of the obtained coatings were studied, the influence of the gas discharge energy contribution on the concentration of fluorocarbon compounds was established, and the band gap was calculated using the Tauc method. The surface relief of the obtained structures was considered using atomic force microscopy and the root-mean-square deviation of the surface roughness was calculated, which reached a maximum of 36 &amp;#177; 3 nm. Using the Oliver-Pharr method, the nanohardness and Young's modulus of elasticity of the obtained coatings were calculated, which amounted to 0.447 &amp;#177; 0.025 and 6.10 &amp;#177; 0.39 GPa, respectively.

A simple method for the determination of contact area for nanoindentation tests

<p indent="0mm">The contact area is a key parameter for the mechanical property characterization via the nanoindentation technique. In the conventional Oliver-Pharr (OP) method, the contact area was calculated based on an empirical area function precalibrated with a standard sample. However, the procedure for precalibrating the area function is rather complex and concerns about its reliability still exist. In this paper, the efficiency of the precalibrated OP area function was evaluated based on the expression derived by Malzbender et al. [J Mater Res, 2000, 15: 1209–1212] to describe the nanoindentation loading behavior. It was shown for the first time that, for a relatively small contact depth, the loading segment of the measured load-displacement cannot be described sufficiently using the mechanical properties determined via the conventional OP method, implying that the method may not yield reliable results for small indentations. To overcome these limitations while using the OP area function, based on the consideration of a self-consistent description for the measured load-displacement curve, a simple method was proposed in the present study to directly determine the contact area without precalibration. The two parameters involved in this new method for calculating the contact area, the contact depth <italic>h</italic>_c, and the so-called “truncation distance of a rounded indenter tip” <italic>h</italic>_d, can be obtained directly and easily by analyzing the unloading and the loading segments of the same load-displacement curve, respectively. Compared with the conventional OP method, the main advantage of this method is that the contact area can be determined directly using the data measured with the test material, and the results may reflect the material responses to nanoindentation directly and accurately. The efficiency and reliability of this newly proposed method were confirmed by the nanoindentation data measured on typical materials.

Artificial intelligence based analysis of nanoindentation load–displacement data using a genetic algorithm

We developed an automated tool, Nanoindentation Neo package for the analysis of nanoindentation load–displacement curves using a Genetic Algorithm (GA) applied to the Oliver-Pharr method (Oliver et al.,1992). For some materials, such as polycrystalline isotropic graphites, Least Squares Fitting (LSF) of the unload curve can produce unrealistic fit parameters. These graphites exhibit sharply peaked unloading curves not easily fit using the LSF, which tends to overestimate the indenter tip geometry parameter. To tackle this problem, we extended our general materials characterization tool Neo for EXAFS analysis (Terry et al., 2021) to fit nanoindentation data. Nanoindentation Neo automatically processes and analyzes nanoindentation data with minimal user input while producing meaningful fit parameters. GA, a robust metaheuristic method, begins with a population of temporary solutions using model parameters called chromosomes; from these we evaluate a fitness value for each solution, and select the best solutions to mix with random solutions producing the next generation. A mutation operator then modifies existing solutions by random perturbations, and the optimal solution is selected. We tested the GA method using Silica and Al reference standards. We fit samples of graphite and a high entropy alloy (HEA) consisting of BCC and FCC phases.

Nanoindentation and Atomic Force Microscopy Derived Mechanical and Microgeometrical Properties of Tooth Root Cementum

In the present research, nanoindentation, atomic-force microscopy and optical microscopy were used to study the mechanical and microgeometrical parameters of tooth tissues. A nanoindentation test unit equipped with Berkovich indenter was used to determine the values of the reduced Young’s modulus and indentation hardness and both nanoindentation and atomic force microscopy using a diamond probe on a silicon cantilever were used to study microgeometrical parameters of human tooth root cementum. Three areas of cementum were studied: the cervical region near the dentine–enamel junction, the second third of the tooth root, and the apex of the tooth root. The interpretation of the results was carried out using the Oliver–Pharr method. It was established, that the mechanical properties of cementum increase from the cervical region to the central part of the root, then decrease again towards the apex of the tooth root. On the contrary, the microgeometrical characteristics of cementum practically do not demonstrate any change in the same direction. A decrease in the roughness parameters in the direction from cellular cementum to dentine was observed. Additionally, a decrease in the reduced Young’s modulus and indentation hardness of dentine in the cervical area compared to dentine in the crown part of the tooth was found using nanoindentation. The investigation of the dentine–cementum junction with high resolution revealed the interspaced collagen fiber bridges and epithelial rests of Malassez, whose sizes were studied. The parameters of the topographic features of the cementum in the vicinity of the lacunae of cementocytes were also established.

Comparison of Oliver-Pharr and Work of Indentation Approach to Determine the Mechanical Properties of Melt-Spun Al-12%Wt.Si-0.5%Sb Alloy

In this research, the hardness and reduced modulus of Al-%wt.12-%wt.0.5Sb melt-spun alloy were evaluated by using depth sensing indentation and atomic force microscopy techniques. We considered two approaches, Oliver-Pharr and Work of Indentation, to analyse the load-displacement curves. The ratio of final depth to maximum depth was found to be higher than the reported critical value of 0.70, which mean that pile-up was dominant in the melt-spun. A pile-up around the deformed surface was observed from atomic force microscope, which is consistent with the aferomentioned result. The hardness calculated by Oliver-Pharr method was higher than that calculated by Work of Indentation Approach. According to the results, Work of Indentation Approach was more reliable than the Oliver-Pharr approach because of reducing pile-up affect.

Mechanical Properties of Macromolecular Separators for Lithium-Ion Batteries Based on Nanoindentation Experiment.

High tensile strength and toughness play an important role in improving the mechanical performance of separator films, such as resistance to external force, improving service life, etc. In this study, a nanoindentation experiment is performed to investigate the mechanical properties of two types of separators for LIBs based on the grid nanoindentation method. During the indentation experiment, the “sink-in” phenomenon is observed around the indenter when plastic deformation of the specimen occurs. The “sink-in” area of the polyethylene (PE) separator is larger than that of the polypropylene/polyethylene/polypropylene (PP/PE/PP) separator, i.e., the plastic area of the PE separator is larger than that of the PP/PE/PP separator. In order to select a suitable method to evaluate the hardness and elastic modulus of these separators for LIBs, three theoretical methods, including the Oliver–Pharr method, the indentation work method, and the fitting curve method, are used for analysis and comparison in this study. The results obtained by the fitting curve method are more reasonable and accurate, which not only avoids the problem of the large contact area obtained by the Oliver–Pharr method, but also avoids the influence caused by the large fitting data of the displacement–force curve and the inaccuracy of using the maximum displacement obtained by the indentation method. In addition, the obstruction ability of the PP/PE/PP separator to locally resist external load pressed into its surface and to resist micro particles, such as fine metal powder, that can enter the lithium-ion battery during the manufacturing process is greater than that of the PE separator. This research provides guidance for studying the mechanical properties and exploring the estimation method of macromolecular separators for LIBs.

Measuring the elastic modulus of soft biomaterials using nanoindentation

The measurement of the elastic modulus of soft biomaterials via nanoindentation relies on the accurate determination of the zero-point of the tip-sample interaction on which the depth of penetration into the sample is based. Non-cantilever based nanoindentation systems were originally designed for hard materials, and therefore monitoring the zero-point contact presents a significant challenge for the characterisation of very soft biomaterials. This study investigates the ability of non-cantilever based nanoindentation to differentiate between hydrogels with elastic moduli on the order of single kiloPascals (kPa) using a bespoke soft contact protocol and low flexural stiffness of instrument. Polyethylene glycol (PEG) hydrogels were fabricated as a model system with a range of elastic moduli by varying the polymer concentration and degree of crosslinking. Elastic modulus values were calculated using the Oliver-Pharr method, Hertzian contact model, as well as a viscoelastic model to account for the time-dependent behaviour of the gels. The stiffness measurements were validated by measuring cantilever beams with the equivalent flexural stiffness to that of the PEG hydrogels being tested. The results demonstrated a high repeatability of the measurements, enabling differentiation between hydrogels with elastic moduli in the single kPa to hundreds of kPa range.

Atomistic Insights into the Phase Transformation of Single-Crystal Silicon during Nanoindentation.

The influence of the indenter angle on the deformation mechanisms of single-crystal Si was analyzed via molecular dynamics simulations of the nanoindentation process. Three different types of diamond conical indenters with semi-angles of 45°, 60°, and 70° were used. The load–indentation depth curves were obtained by varying the indenter angles, and the structural phase transformations of single-crystal Si were observed from an atomistic view. In addition, the hardness and elastic modulus with varying indenter angles were evaluated based on the Oliver–Pharr method and Sneddon’s solution. The simulation results showed that the indenter angle had a significant effect on the load–indentation depth curves, which resulted from the strong dependence of the elastic and plastic deformation ratios on the indenter angle during indentations.

Determination of mechanical properties of Bi12TiO20 crystals by nanoindentation

Bi12TiO20 (BTO) single crystal was grown by Czochralski method and investigated mechanically by nanoindentation measurements. X-ray diffraction pattern of the crystal presented one intensive peak around 37.95° associated with (330) plane of cubic crystalline structure. Nanoindentation experiments were performed at various loads between 5 and 100 mN. Hardness and Young's modulus of the crystal were determined by Oliver-Pharr method. The hardness-load dependency exhibited behavior of indentation size effect. True hardness value of BTO crystal was revealed as 4.4 GPa. Young's modulus decreased with increase of load and load-independent Young's modulus was found around 93 GPa at high loads. The load-dependent elastic and plastic deformation components were calculated and it was observed that the dominant component in BTO single crystal is plastic deformation at the applied loads. The present paper reports for the first time the mechanical characteristics of the BTO single crystal by carrying out nanoindentation experiments.