Fundamental relations used in nanoindentation: Critical examination based on experimental measurements

2002 ◽  
Vol 17 (9) ◽  
pp. 2227-2234 ◽  
Author(s):  
M. Martin ◽  
M. Troyon
Keyword(s):  
Elastic Modulus ◽  
Correction Factor ◽  
Contact Stiffness ◽  
Elastic Half Space ◽  
Critical Examination ◽  
Experimental Measurements ◽  
Contact Depth ◽  
Power Law Exponent ◽  
Fused Quartz ◽  
Measured Hardness

The fundamental relations used in the analysis of nanoindentation load–displacement data to determine elastic modulus and hardness are based on Sneddon's solution for indentation of an elastic half-space by rigid axisymmetric indenters. It has been recently emphasized that several features that have important implications for nanoindentation measurements are generally ignored. The first one concerns the measurement of the contact depth, which is actually determined by using a constant value ε = 0.75 for the geometry of a Berkovich indenter and for any kind of material, whereas the reality is that ε is a function of the power law exponent deduced from the analysis of the unloading curve. The second feature concerns the relation between contact stiffness, elastic modulus, and contact area, in which a correction factor γ larger than unity is usually ignored leading to a systematic overestimation of the area function and thus to errors in the measured hardness and modulus. Experimental measurements on fused quartz are presented that show the variation of ε with the geometry of the tip–sample contact; that is to say with the contact depth, as well as the existence of the correction factor γ, as predicted in some recent articles. Effects of both ε and γ on harness and modulus measurements are also shown.

A critical examination of the fundamental relations used in the analysis of nanoindentation data

1999 ◽  
Vol 14 (6) ◽  
pp. 2296-2305 ◽  
Author(s):  
Jack C. Hay ◽  
A. Bolshakov ◽  
G. M. Pharr
Keyword(s):  
Finite Element ◽  
Elastic Modulus ◽  
Half Space ◽  
Contact Stiffness ◽  
Elastic Half Space ◽  
Careful Examination ◽  
Critical Examination ◽  
First Order ◽  
Property Measurement ◽  
Measured Hardness

Methods for analyzing nanoindentation load-displacement data to determine hardness and elastic modulus are based on analytical solutions for the indentation of an elastic half-space by rigid axisymmetric indenters. Careful examination of Sneddon's solution for indentation by a rigid cone reveals several largely ignored features that have important implications for nanoindentation property measurement. Finite element and analytical results are presented that show corrections to Sneddon's equations are needed if accurate results are to be obtained. Without the corrections, the equations underestimate the load and contact stiffness in a manner that leads to errors in the measured hardness and modulus, with the magnitudes of the errors depending on the angle of the indenter and Poisson's ratio of the half-space. First order corrections are derived, and general implications for the interpretation of nanoindentation data are discussed.

On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation

1992 ◽  
Vol 7 (3) ◽  
pp. 613-617 ◽  
Author(s):  
G.M. Pharr ◽  
W.C. Oliver ◽  
F.R. Brotzen
Keyword(s):  
Elastic Modulus ◽  
Contact Area ◽  
Contact Stiffness ◽  
Elastic Half Space ◽  
Elastic Contact ◽  
Simple Relationship ◽  
Depth Sensing ◽  
Force Microscopy ◽  
Atomic Force ◽  
The Relationship

Results of Sneddon's analysis for the elastic contact between a rigid, axisymmetric punch and an elastic half space are used to show that a simple relationship exists among the contact stiffness, the contact area, and the elastic modulus that is not dependent on the geometry of the punch. The generality of the relationship has important implications for the measurement of mechanical properties using load and depth sensing indentation techniques and in the measurement of small contact areas such as those encountered in atomic force microscopy.

scholarly journals Inaccuracies in Sneddon's Solution for Elastic Indentation by a Rigid Cone and their Implications for Nanoindentation Data Analysis

1996 ◽  
Vol 436 ◽  
Author(s):  
A. Bolshakov ◽  
G. M. Pharr
Keyword(s):  
Finite Element ◽  
Elastic Modulus ◽  
Data Analysis ◽  
Half Space ◽  
Contact Stiffness ◽  
Elastic Half Space ◽  
Berkovich Indenter ◽  
First Order ◽  
Effective Angle ◽  
First Order Correction

AbstractMethods currently used for analyzing nanoindentation load-displacement data to determine a material's hardness and elastic modulus are based on Sneddon's solution for the indentation of an elastic half-space by a rigid axisymmetric indenter. Although this solution is widely used, no attempts have been made to determine how well it works for conditions of finite deformation, as is the case in most nanoindentation experiments with sharp indenters. Analytical and finite element results are presented which show that corrections to Sneddon's solution are needed if it is to be accurately applied to the case of deformation by a rigid cone. Failure to make the corrections results in an underestimation of the load and contact stiffness and an overestimation of the elastic modulus, with the magnitude of the errors depending on the angle of the indenter and Poisson's ratio of the half-space. For a rigid conical indenter with a half-included tip angle of 70.3°, i.e., the angle giving the same area-to-depth ratio as the Berkovich indenter used commonly in nanoindentation experiments, the underestimation of the load and contact stiffness and overestimation of the elastic modulus may be as large as 13%. It is shown that a simple first order correction can be achieved by redefining the effective angle of the indenter in terms of the elastic constants. Implications for the interpretation of nanoindentation data are discussed.

Critical Examination of the Two-slope Method in Nanoindentation

2005 ◽  
Vol 20 (8) ◽  
pp. 2194-2198 ◽  
Author(s):  
M. Troyon ◽  
L. Huang
Keyword(s):  
Correction Factor ◽  
Contact Area ◽  
Contact Stiffness ◽  
Critical Examination ◽  
Slope Method ◽  
Loading And Unloading ◽  
Factor Α ◽  
Analytical Expressions ◽  
Good Agreement

In this paper, we derive corrected analytical expressions for calculating the hardness and modulus by the two-slope method. This method relies on the determination of the slopes of the loading and unloading curves rather than the indenter displacement as an input. These expressions take into account the correction factor α in the fundamental relations among contact stiffness, elastic modulus, and contact area, which is frequently forgotten or misused in the literature. It is shown that these corrected expressions allow measurements of the hardness and modulus in very good agreement with the commonly used technique based on the determination of the contact area. Additionally, the correction factor α can be easily determined if Young's modulus of the material is known.

Characterization of Surface Layer of Silicon Wafer by Using Nano Indenter

2007 ◽  
Author(s):  
Yu-Li Sun ◽  
Dun-Wen Zuo ◽  
Yong-Wei Zhu ◽  
Feng Xu ◽  
Min Wang
Keyword(s):  
Elastic Modulus ◽  
Silicon Wafer ◽  
Bulk Material ◽  
Contact Stiffness ◽  
Indentation Test ◽  
Oxide Coating ◽  
Stiffness Measurement ◽  
Contact Depth ◽  
Continuous Stiffness Measurement

Mechanical properties of the silicon wafer are evaluated by a nano indenter system with the continuous stiffness measurement (CSM) technique. Contact stiffness, hardness and elastic modulus of the silicon wafer are continuously measured during the loading in an indentation test. The results show that when the contact depth is between 20 and 32 nm, its contact stiffness is linear with the contact depth, and its hardness and elastic modulus keep constant at 10.2 GPa and 140.3 GPa respectively, which belong to the oxide coating of the silicon wafer. When the contact depth is between 32 and 60 nm, its contact stiffness is not linear with the contact depth, and the hardness and elastic modulus increase rapidly with the contact depth, because they are affected by the bulk material. When the contact depth is over 60 nm, the contact stiffness of the silicon wafer is linear with the contact depth again, and the hardness and elastic modulus keep constant at 12.5 GPa and 165.6 GPa respectively, which belong to the silicon wafer, the bulk material.

Correction factor for contact area in nanoindentation measurements

2005 ◽  
Vol 20 (3) ◽  
pp. 610-617 ◽  
Author(s):  
Michel Troyon ◽  
Liye Huang
Keyword(s):  
Elastic Modulus ◽  
Correction Factor ◽  
Contact Area ◽  
Basic Equation ◽  
Elastic Moduli ◽  
Contact Stiffness ◽  
Factor Α ◽  
Tip Radius ◽  
The Relationship ◽  
Indenter Shape

In the relationship between unloading contact stiffness, elastic modulus, and contact area, which is the fundamental basic equation for nanoindentation analysis, a multiplicative correction factor is generally needed. Sometimes this correction factor is called γ to take into account the elastic radial inward displacements, and sometimes it is called β to correct for the fact that the indenter shape is not a perfect cone. In reality, these two effects simultaneously coexist and thus it is proposed that this correction factor is α = βγ. From nanoindentation data measured on three materials of different elastic moduli with a sharp Berkovich indenter and a worn one, the tip of which was blunt, it is demonstrated that the correction factor α does not have a constant value for a given material and indenter type but depends on the indenter tip rounding and also on the deformation of the indenter during indentation. It seems that α increases with the tip radius and also with the elastic modulus of the measured materials.

An Experimental Evaluation of the Constant β Relating the Contact Stiffness to the Contact Area in Nanoindentation

2004 ◽  
Vol 841 ◽  
Author(s):  
Jeremy H. Strader ◽  
Sanghoon Shim ◽  
Hongbin Bei ◽  
W. C. Oliver ◽  
G. M. Pharr
Keyword(s):  
Mechanical Properties ◽  
Contact Area ◽  
Contact Stiffness ◽  
Experimental Measurements ◽  
Berkovich Indenter ◽  
Cube Corner ◽  
Fused Quartz ◽  
The Relationship ◽  
Theoretical Analyses ◽  
Scanning Electron

ABSTRACTMeasurements of mechanical properties by nanoindentation with triangular pyramidal indenters like the Berkovich rely heavily upon the relationship between the contact stiffness, S, the contact area, A, and the reduced elastic modulus, Er. This relationship is often written in the form S = 2βEr(A/π)1/2, where β is a constant that depends on the geometry of the indenter. Although the most common values for β used in experimental measurements are 1.000 and 1.034, various theoretical analyses have yielded values as small as 1.00 or as large as 1.2, depending on the assumptions made to model the deformation. Here, we explore the most appropriate value of β by performing careful experiments in fused quartz with thin gold coatings applied to the surface to reveal the actual contact area when observed in the scanning electron microscope. Experiments were performed not only with the Berkovich indenter, but with five other three-sided pyramidal indenters with centerline-to-face angles ranging from 35.3° (cube corner) to 85°. Results are discussed as they apply to obtaining accurate measurements of mechanical properties by nanoindentation.

Effects of dynamic indentation on the mechanical response of materials

2008 ◽  
Vol 23 (6) ◽  
pp. 1604-1613 ◽  
Author(s):  
M.J. Cordill ◽  
N.R. Moody ◽  
W.W. Gerberich
Keyword(s):  
Thin Films ◽  
Mechanical Properties ◽  
Mechanical Response ◽  
Length Scale ◽  
Polycrystalline Nickel ◽  
Dynamic Indentation ◽  
Crystalline Materials ◽  
Fused Quartz ◽  
Hardness Values ◽  
Measured Hardness

Dynamic indentation techniques are often used to determine mechanical properties as a function of depth by continuously measuring the stiffness of a material. The dynamics are used by superimposing an oscillation on top of the monotonic loading. Of interest was how the oscillation affects the measured mechanical properties when compared to a quasi-static indent run at the same loading conditions as a dynamic. Single crystals of nickel and NaCl as well as a polycrystalline nickel sample and amorphous fused quartz and polycarbonate have all been studied. With respect to dynamic oscillations, the result is a decrease of the load at the same displacement and thus lower measured hardness values of the ductile crystalline materials. It has also been found that the first 100 nm of displacement are the most affected by the oscillating tip, an important length scale for testing thin films, nanopillars, and nanoparticles.

Interaction Model for “Shelled Particles” and Small-Strain Modulus of Granular Materials

2018 ◽  
Vol 85 (10) ◽  
Author(s):  
Shun-hua Zhou ◽  
Peijun Guo ◽  
Dieter F. E. Stolle
Keyword(s):  
Elastic Modulus ◽  
Contact Stiffness ◽  
Scaling Law ◽  
Interaction Model ◽  
Contact Problems ◽  
Hertzian Contact ◽  
Spherical Particles ◽  
Thin Coating ◽  
Normal Contact ◽  
Laboratory Test Results

The elastic modulus of a granular assembly composed of spherical particles in Hertzian contact usually has a scaling law with the mean effective pressure p as K∼G∼p1/3. Laboratory test results, however, reveal that the value of the exponent is generally around 1/2 for most sands and gravels, but it is much higher for reclaimed asphalt concrete composed of particles coated by a thin layer of asphalt binder and even approaching unity for aggregates consisting of crushed stone. By assuming that a particle is coated with a thin soft deteriorated layer, an energy-based simple approach is proposed for thin-coating contact problems. Based on the features of the surface layer, the normal contact stiffness between two spheres varies with the contact force following kn∼Fnm and m∈[1/3,  1], with m=1/3 for Hertzian contact, m=1/2 soft thin-coating contact, m=2/3 for incompressible soft thin-coating, and m=1 for spheres with rough surfaces. Correspondingly, the elastic modulus of a random granular packing is proportional to pm with m∈[1/3,  1].

Numerical study on the dynamic characteristics of water-lubricated rubber bearing under asperity contact

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Guo Xiang ◽  
Yijia Wang ◽  
Cheng Wang ◽  
Zhongliang Lv
Keyword(s):  
Elastic Modulus ◽  
Contact Stiffness ◽  
Water Film ◽  
Radial Clearance ◽  
Asperity Contact ◽  
Stiffness Coefficient ◽  
Content Type ◽  
Damping Coefficients ◽  
Rubber Bearing ◽  
Stiffness And Damping

Purpose In this study, the dynamic characteristics of the water-lubricated rubber bearing considering asperity contact are numerically studied, including water-film stiffness and damping coefficients and plastic-elastic contact stiffness coefficient. Design/methodology/approach The Kogut-Etsion elastic-plastic contact model is applied to calculate the contact stiffness coefficient at the bearing-bush interface and the perturbed method is used to calculate the stiffness and damping coefficients of water-film. In addition, the rubber deformation is determined by the finite element method (FEM) during the simulation. Parametric studies are conducted to assess the effects of the radial clearance, rubber thickness and elastic modulus on the dynamic characteristic of water-lubricated rubber bearing. Findings Numerical results indicate that stiffness and damping coefficients of water film and the contact stiffness of asperity are increased with the decreasing of the radial clearance and the dynamic coefficients are less sensitive to the rubber thickness compared with the elastic modulus of rubber. Furthermore, due to the existed groove, a sudden change of the water film direct stiffness and damping coefficients is observed when the eccentricity ratio ranges from 0.6 to 1.0. Originality/value It is expected this study can provide more information to establish a dynamic equation of water-lubricated rubber bearings exposed to mixed lubrication conditions.

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