G.M. Pharr

An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments

The indentation load-displacement behavior of six materials tested with a Berkovich indenter has been carefully documented to establish an improved method for determining hardness and elastic modulus from indentation load-displacement data. The materials included fused silica, soda–lime glass, and single crystals of aluminum, tungsten, quartz, and sapphire. It is shown that the load–displacement curves during unloading in these materials are not linear, even in the initial stages, thereby suggesting that the flat punch approximation used so often in the analysis of unloading data is not entirely adequate. An analysis technique is presented that accounts for the curvature in the unloading data and provides a physically justifiable procedure for determining the depth which should be used in conjunction with the indenter shape function to establish the contact area at peak load. The hardnesses and elastic moduli of the six materials are computed using the analysis procedure and compared with values determined by independent means to assess the accuracy of the method. The results show that with good technique, moduli can be measured to within 5%.

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

Results of Sneddons analysis for the elastic contact between a rigid, axisymmetric punch and an elastic half space are used to show that a simple relationship exists between 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.

Measurement of mechanical properties by ultra-low load indentation

Abstract Ultra-low load indentation, also known as nanoindentation, is a widely used tool for measuring the mechanical properties of thin films and small volumes of material. One of the great advantages of the technique is its ability to probe a surface and map its properties on a spatially resolved basis, sometimes with a resolution of better than 1 μm. In this paper, techniques for measuring mechanical properties by ultra-low load indentation techniques are reviewed and discussed. Emphasis is given to the use of sharp indenters and how they can be used to measure elastic modulus, hardness, and fracture toughness. These fundamental mechanical properties characterize the three primary modes of deformation in solids—elasticity, plasticity, and fracture.

Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation

An experimental investigation was undertaken to measure the intrinsic elastic properties of several of the microstructural components of human vertebral trabecular bone and tibial cortical bone by the nanoindentation method. Specimens from two thoracic vertebrae (T-12) and two tibiae were obtained from frozen, unembalmed human male cadavers aged 57 and 61 years. After drying and mounting in epoxy resin nanoindentation tests were conducted to measure Youngs modulus and the hardness of individual trabeculae in the vertebrae and single osteons, and interstitial lamellae in the tibiae. Measurements on the vertebral trabeculae were made in the transverse direction, and the average Youngs modulus was found to be 13.5 +/- 2.0 GPa. The tibial specimens were tested in the longitudinal direction, yielding moduli of 22.5 +/- 1.3 GPa for the osteons and 25.8 +/- 0.7 GPa for the interstitial lamellae. Analysis of variance showed that the differences in the measured moduli are statistically significant. Hardness differences among the various microstructural components were also observed.

Influences of pileup on the measurement of mechanical properties by load and depth sensing indentation techniques

Finite element simulation of conical indentation of a wide variety of elastic-plastic materials has been used to investigate the influences of pile-up on the accuracy with which hardness and elastic modulus can be measured by load and depth-sensing indentation techniques. The key parameter in the investigation is the contact area, which can be determined from the finite element results either by applying standard analysis procedures to the simulated indentation load-displacement data, as would be done in an experiment, or more directly, by examination of the contact profiles in the finite element mesh. Depending on the pileup behavior of the material, these two areas may be very different. When pileup is large, the areas deduced from analyses of the load-displacement curves underestimate the true contact areas by as much as 60{percent}. This, in turn, leads to overestimations of the hardness and elastic modulus. The conditions under which the errors are significant are identified, and it is shown how parameters measured from the indentation load-displacement data can be used to identify when pileup is an important factor. {copyright} {ital 1998 Materials Research Society.}

The correlation of the indentation size effect measured with indenters of various shapes

Experimental results are presented which show that the indentation size effect for pyramidal and spherical indenters can be correlated. For a pyramidal indenter, the hardness measured in crystalline materials usually increases with decreasing depth of penetration, which is known as the indentation size effect. Spherical indentation also shows an indentation size effect. However, for a spherical indenter, hardness is not affected by depth, but increases with decreasing sphere radius. The correlation for pyramidal and spherical indenter shapes is based on geometrically necessary dislocations and work-hardening. The Nix and Gao indentation size effect model (J. Mech. Phys. Solids 46 (1998) 411) for conical indenters is extended to indenters of various shapes and compared to the experimental results.

Measurement of Thin Film Mechanical Properties Using Nanoindentation

One of the simplest ways to measure the mechanical properties of a thin film is to deform it on a very small scale. Because indentation testing with a sharp indenter is one convenient means to accomplish this, nanoindentation, or indentation testing at the nanometer scale, has become one of the most widely used techniques for measuring the mechanical properties of thin films. Other reasons for the popularity of nanoindentation stem from the ease with which a wide variety of mechanical properties can be measured without removing the film from its substrate and the ability to probe a surface at numerous points and spatially map its mechanical properties. The utility of the mapping capability is illustrated in Figure 1, which shows several small indentations made at selected points in a microelectronic device. The hardness and modulus of the device were determined at each point. In addition to microelectronics, nanoindentation has also proved useful in the study of optical coatings, hard coatings, and materials with surfaces modified by ion implantation and laser treatment.

The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques.

Acoustic microscopy (30-60 microm resolution) and nanoindentation (1-5 microm resolution) are techniques that can be used to evaluate the elastic properties of human bone at a microstructural level. The goals of the current study were (1) to measure and compare the Youngs moduli of trabecular and cortical bone tissues from a common human donor, and (2) to compare the Youngs moduli of bone tissue measured using acoustic microscopy to those measured using nanoindentation. The Youngs modulus of cortical bone in the longitudinal direction was about 40% greater than (p<0.01) the Youngs modulus in the transverse direction. The Youngs modulus of trabecular bone tissue was slightly higher than the transverse Youngs modulus of cortical bone, but substantially lower than the longitudinal Youngs modulus of cortical bone. These findings were consistent for both measurement methods and suggest that elasticity of trabecular tissue is within the range of that of cortical bone tissue. The calculation of Youngs modulus using nanoindentation assumes that the material is elastically isotropic. The current results, i.e., the average anisotropy ratio (E(L)/E(T)) for cortical bone determined by nanoindentation was similar to that determined by the acoustic microscope, suggest that this assumption does not limit nanoindentation as a technique for measurement of Youngs modulus in anisotropic bone.

Influences of stress on the measurement of mechanical properties using nanoindentation: Part I. Experimental studies in an aluminum alloy

The influence of applied stress on the measurement of hardness and elastic modulus using nanoindentation methods has been experimentally investigated using special specimens of aluminum alloy 8009 to which controlled stresses could be applied by bending. When analyzed according to standard methods, the nanoindentation data reveal changes in hardness with stress similar to those observed in conventional hardness tests. However, the same analysis shows that the elastic modulus changes with stress by as much as 10%, thus suggesting that the analysis procedure is somehow deficient. Comparison of the real indentation contact areas measured optically to those determined from the nanoindentation data shows that the apparent stress dependence of the modulus results from an underestimation of the contact area by the nanoindentation analysis procedures.

Influences of stress on the measurement of mechanical properties using nanoindentation: Part II. Finite element simulations

The finite element method has been used to study the behavior of aluminum alloy 8009 during elastic-plastic indentation to establish how the indentation process is influenced by applied or residual stress. The study was motivated by the experiments of the preceding paper which show that nanoindentation data analysis procedures underestimate indentation contact areas and therefore overestimate hardness and elastic modulus in stressed specimens. The NIKE2D finite element code was used to simulate indentation contact by a rigid, conical indenter in a cylindrical specimen to which biaxial stresses were applied as boundary conditions. Indentation load-displacement curves were generated and analyzed according to standard methods for determining hardness and elastic modulus. The simulations show that the properties measured in this way are inaccurate because pileup is not accounted for in the contact area determination. When the proper contact area is used, the hardness and elastic modulus are not significantly affected by the applied stress. {copyright} {ital 1996 Materials Research Society.}