Nix and Gao [J. Mech. Phys. Solids46, 411 (1998)] established an important relation between the microindentation hardness and indentation depth for axisymmetric indenters. We use the conventional theory of mechanism-based strain gradient plasticity [Huang et al., Int. J. Plast.20, 753 (2004)] established from the Taylor dislocation model [Taylor, Proc. R. Soc. London A145, 362 (1934); Taylor, J. Inst. Met.62, 307 (1938)] to study the Berkovich and other triangular pyramid indenters. The three-dimensional finite element analysis shows that the widely used equivalence of equal base area leads to significant errors, particularly in microindentation. A new equivalence of equal angle is proposed for triangular pyramid indenters, and it has been validated for a large range of indenter angles and indentation depths.

Indentation is widely used to extract material elastoplastic properties from measured force-displacement curves. Many previous studies argued or implied that such a measurement is unique and the whole material stress-strain curve can be measured. Here we show that first, for a given indenter geometry, the indentation test cannot effectively probe material plastic behavior beyond a critical strain, and thus the solution of the reverse analysis of the indentation force-displacement curve is nonunique beyond such a critical strain. Secondly, even within the critical strain, pairs of mystical materials can exist that have essentially identical indentation responses (with differences below the resolution of published indentation techniques) even when the indenter angle is varied over a large range. Thus, fundamental elastoplastic behaviors, such as the yield stress and work hardening properties (functions), cannot be uniquely determined from the force-displacement curves of indentation analyses (including both plural sharp indentation and deep spherical indentation). Explicit algorithms of deriving the mystical materials are established, and we qualitatively correlate the sharp and spherical indentation analyses through the use of critical strain. The theoretical study in this paper addresses important questions of the application range, limitations, and uniqueness of the indentation test, as well as providing useful guidelines to properly use the indentation technique to measure material constitutive properties.

A simple empirical method that extracts the elastic moduli of both thin films and the underlying substrates is proposed and validated by both new nanoindentation experiments and published data. Deconvolution of thin film’s elastic properties from the substrate is achieved by statistical estimation, where a simple function relating the elastic moduli of the thin film and substrate to the film-substrate composite modulus is used to fit the experimental data plotted against the logarithmic indentation depth normalized by film thickness. Experimental data from a wide range of soft and hard films on substrate were used to demonstrate the deconvolution and validate the method. The estimated elastic moduli of thin films and substrates agree well with their corresponding standard values or values obtained by other methods. The advantages of this method are discussed, and recommendations are made on how to design experiments to obtain reliable data for this method.

An attempt was made to predict the macroscopic plastic flow of a high-performance pipeline steel, consisting of dual constituent phases (soft ferrite and hard bainite), by performing nanoindentation experiments on each microphase with two spherical indenters that have different radii (550 nm and 3.3 μm). The procedure is based on the well known concepts of indentation stress-strain and constraint factor, which make it possible to relate indentation hardness to the plastic flow of the phases. Additional consideration of the indentation size effect for sphere and application of a simple “rule-of-mixture” led us to a reasonably successful estimation of the macroscopic plastic flow of the steel from the microphases properties, which was verified by comparing the predicted stress-strain curve with that directly measured from the conventional tensile test of a bulky sample.

A method is described for the creation of surfaces with cyclically reversible topographical form. Using spherical and cylindrical indenters applied to NiTi shape-memory alloys, an indentation-planarization technique is shown to result in a two-way shape memory effect that can drive flat-to-wavy surface transitions on changing temperature. First, it is shown that deep spherical indents, made in martensitic NiTi, exhibit pronounced two-way cyclic depth changes. After planarization, these two-way cyclic depth changes are converted to reversible surface protrusions, or “exdents.” Both indent depth changes and cyclic exdent amplitudes can be related to the existence of a subsurface deformation zone in which indentation has resulted in plastic strains beyond that which can be accomplished by martensite detwinning reactions. Cylindrical indentation leads to two-way displacements that are about twice as large as that for the spherical case. This is shown to be due to the larger deformation zone under cylindrical indents, as measured by incremental grinding experiments.

In coating/substrate bilayer systems, the indentation contact behavior transitionally varies from coatinglike to substratelike behaviors. Spatial confinement effects of the substrate induce very complicated plastic flows in the coating beneath the indenter, leading to a crucial difficulty that is not accounted for by any of the present quantitative analytical/theoretical predictions for the substrate-affected contact hardness. In this work, the author presents finite-element-based studies on the elastoplastic indentation contact mechanics of coating/substrate systems. The effect of the substrate is taken into account by introducing the spatially variable elastic modulus and the yield stress; this approach quantitatively describes the substrate-affected stress/strain field in the spatially localized area beneath the indenter. The elastoplastic constitutive relationship of the contact hardness for semi-infinite homogeneous bulks combined with these spatially variable material characteristics are successfully applied to analytically as well as quantitatively predict the substrate-affected contact hardness of bilayer composite systems having wide ranges of elastoplastic coating/substrate characteristics. The experimental procedures for determining the elastic/plastic parameters both of the coating and the substrate are also discussed, in which the importance of the experimental determination of substrate-affected indentation contact radius/area is emphasized.

In ductile metals, sliding contact induces plastic deformation resulting in subsurfaces, the mechanical properties of which are different from those of the bulk. This article describes a novel combination of nanomechanical test methods and analysis techniques to evaluate the mechanical behavior of the subsurfaces generated underneath a wear surface. In this methodology, nanoscratch techniques were first used to generate wear patterns as a function of load and number of cycles using a Hysitron TriboIndenter. Measurements were made on a (001) single crystal plane along two crystallographic directions, <001> and <011>. Nanoindentation was then used to measure mechanical properties in each wear pattern. The results on the (001) single crystal nickel plane showed that there was a strong increase in hardness with increasing applied load that was accompanied by a change in surface deformation. The amount of deformation underneath the wear patterns was examined from focused ion beam cross-sections of the wear patterns.

Nonlinear viscoelastic problems are in general not analytically solvable. However, it is shown here that, for any viscoelastic materials describable by a constitutive law with linear elastic and (in general) nonlinear viscous elements arranged in any network fashion, such as the Maxwell or standard linear solid arrangements, it is always possible to eliminate the viscous terms by replacing the displacement, strain, and stress fields of the problem by the jumps in rates of these fields. After the viscous terms are eliminated, the problem is reduced to a linear elastic problem defined on the same spatial domain and with the same elastic constant as in the original viscoelastic problem. Such a reduced elastic problem is analytically solvable in many practical cases, and the solution yields a relation between jumps in the load rate and the displacement rate, pertinent to the boundary conditions in the original problem. Such a relation can often be used as the basis for an experimental scheme to measure the elastic constants of materials. The material can be time- or strain-dependent, and the value of the elastic constant measured corresponds to the time instant or the strain value when the jump in load or displacement rate is implemented.

The storage and loss stiffnesses for the composite response of the sample, indenter, and load frame during dynamic nanoindentation are derived. In the first part of the analysis, no physical model is assigned to the composite system. It is shown that this case is equivalent to the conventional nanoindentation analysis. In the second part of the analysis, the sample is modeled as a standard linear solid in series with the indenter and load frame. The results for the storage and loss stiffnesses as computed by the two methods differ by at most ∼3% for the elastomeric system under consideration. Results for the storage and loss moduli are also similar. The relative merits and weaknesses of each analysis are discussed.

Despite active development over the past 15 years, contemporary nanoindentation methods still suffer serious drawbacks, particularly long thermal stabilization and thermal drift, which limit the duration of the measurements to only a short period of time. The presented work introduces a novel ultra nanoindentation method that uses loads from the μN range up to 50 mN, is capable of performing long-term stable measurements, and has negligible frame compliance. The method is based on a novel patented design, which uses an active top referencing system. Several materials were used to demonstrate the performance of the method. The measurements with hold at maximum load confirm extremely low levels of instrument thermal drift. The presented Ultra Nanoindentation Tester opens new possibilities for testing thin films and long-term testing, including creep of polymers at high resolution without the need of long thermal stabilization.

In this study, nucleation of dislocations in magnesium oxide (MgO) during nanoindentation with a spherical indenter is investigated. For flat and defect-free surfaces prepared by chemo/mechanical polishing, reversible load–displacement curves have been obtained for load as high as 300 mN, whereas on a cleaved MgO surface, pop-in and plastic deformation occur at 10 mN with the same indenter. Furthermore, these reversible curves deviate from the Hertz contact theory. Indented areas have then been characterized by atomic force microscopy and nanoetching. In some cases, few slip lines are observed for reversible indentation tests. However, the slip lines position indicate that the nucleation process of the corresponding dislocations is different from that involved during a pop-in phenomenon.

The constraint factor, C (given by the hardness-yield strength ratio H/Y in the fully plastic regime of indentation), in metallic glasses, is greater than three, a reflection of the sensitivity of their plastic flow to pressure. Furthermore, C increases with increasing temperature. In this work, we examine if this is true in amorphous polymers as well, through experiments on amorphous poly(methyl methacrylate) (PMMA). Uniaxial compression as well as spherical indentation tests were conducted in the 248–348 K range to construct H/Y versus indentation strain plots at each temperature and obtain the C-values. Results show that C increases with temperature in PMMA as well. Good correlation between the loss factors, measured using a dynamic mechanical analyzer, and C, suggest that the enhanced sensitivity to pressure is possibly due to β-relaxation. We offer possible mechanistic reasons for the observed trends in amorphous materials in terms of relaxation processes.

Decreasing scales effectively increase nearly all important mechanical properties of at least some “brittle” materials below 100 nm. With an emphasis on silicon nanopillars, nanowires, and nanospheres, it is shown that strength, ductility, and toughness all increase roughly with the inverse radius of the appropriate dimension. This is shown experimentally as well as on a mechanistic basis using a proposed dislocation shielding model. Theoretically, this collects a reasonable array of semiconductors and ceramics onto the same field using fundamental physical parameters. This gives proportionality between fracture toughness and the other mechanical properties. Additionally, this leads to a fundamental concept of work per unit fracture area, which predicts the critical event for brittle fracture. In semibrittle materials such as silicon, this can occur at room temperature when the scale is sufficiently small. When the local stress associated with dislocation nucleation increases to that sufficient to break bonds, an instability occurs resulting in fracture.

In the present work, a previously developed neural network approach for analyzing spherical indentation experiments is applied to prestressed specimens to determine the effect of residual stresses on the identified stress–strain curves. Within this scope, a comparison to other measurement errors has been made, which are caused by surface preparation and anisotropy of the material. To validate the experimental and analysis approach, the effect of compressive and tensile prestresses was also simulated using a three-dimensional finite element model. The material investigated is a rolled 2024 T351, which is widely used for manufacturing airplanes. It is shown that the existing neural network approach is able to determine the stress–strain behavior in agreement with that obtained from tensile tests. The method is robust against most error sources, such as surface roughness, coarse grain structure, and anisotropy, if a sufficient number of experiments are available. The most important influencing factor can be the residual stress causing errors up to 20% in the identified stress–strain curves.

The mechanical behavior of Ti-based metallic glass has been investigated by means of indentation experiments at different loading rates. Contrary to many crystalline materials, an increase of the loading rate causes a reduction of hardness, i.e., a mechanical softening. This effect is ascribed to deformation-induced creation of excess free volume, which is more pronounced for higher strain rates. The decrease of hardness is accompanied with an increase of the contact stiffness and a reduction of the reduced elastic modulus. Finite element simulations reveal that the mechanical response of this material can be described using the Mohr-Coulomb yield criterion. The changes in the nanoindentation curves with the increase of loading rate are well reproduced by decreasing the value of the Mohr-Coulomb cohesive stress. This result is consistent with the presumed enhancement of free volume.

The spherical indentation strength of a lead zirconate titanate (PZT) piezoelectric ceramic was investigated under poled and unpoled conditions and with different electrical boundary conditions (arising through the use of insulating or conducting indenters). Experimental results show that the indentation strength of the poled PZT is higher than that of the unpoled PZT. The strength of a poled PZT under a conducting indenter is higher than that under an insulating indenter. Poling direction (with respect to the direction of indentation loading) did not significantly affect the strength of material. Complementary finite element analysis (FEA) of spherical indentation of an elastic, linearly coupled piezoelectric half-space is conducted for rationalizing the experimental observations. Simulations show marked dependency of the contact stress on the boundary conditions. In particular, contact stress redistribution in the coupled problem leads to a change in the fracture initiation, from Hertzian cracking in the unpoled material to subsurface damage initiation in poled PZT. These observations help explain the experimental ranking of strength the PZT in different material conditions or under different boundary conditions.

A novel optimization approach, capable of extracting the mechanical properties of an elasto-plastic material from indentation data, is proposed. Theoretical verification is performed on two simulated configurations. The first is based on the analysis of the load–displacement data and the topography of the residual imprint of a single conical indenter. The second is based on the load–displacement data obtained from two conical indenters with different semi-angles. In both cases, a semi-analytical approach [e.g., Dao et al., Acta Mater.49, 3899 (2001) and Bucaille et al., Acta Mater.51, 1663 (2003)] is used to estimate Young’s modulus, yield stress, and strain hardening coefficient from the load–displacement data. An inverse finite element model, based on a commercial solver and a newly developed optimization algorithm based on a robust stochastic methodology, uses these approximate values as starting values to identify parameters with high accuracy. Both configurations use multiple data sets to extract the elastic-plastic material properties; this allows the mechanical properties of materials to be determined in a robust way.

Molecular dynamics simulations were performed to gain fundamental insight into crystal plasticity, and its size effects in nanowires deformed by spherical indentation. This work focused on <111>-oriented single-crystal, defect-free Ni nanowires of cylindrical shape with diameters of 12 and 30 nm. The indentation of thin films was also comparatively studied to characterize the influence of free surfaces in the emission and absorption of lattice dislocations in single-crystal Ni. All of the simulations were conducted at 300 K by using a virtual spherical indenter of 18 nm in diameter with a displacement rate of 1 m·s−1. No significant effect of sample size was observed on the elastic response and mean contact pressure at yield point in both thin films and nanowires. In the plastic regime, a constant hardness of 21 GPa was found in thin films for penetration depths larger than 0.8 nm, irrespective of variations in film thickness. The major finding of this work is that the hardness of the nanowires decreases as the sample diameter decreases, causing important softening effects in the smaller nanowire during indentation. The interactions of prismatic loops and dislocations, which are emitted beneath the contact tip, with free boundaries are shown to be the main factor for the size dependence of hardness in single-crystal Ni nanowires during indentation.

Indentation tests were performed, using a flat punch probe, on silicone gels to induce failure under compression. The silicone gels were formed from networks of vinyl-terminated polydimethylsiloxane (PDMS) with molecular weights of 800 and 28,000 g/mol and a sol fraction of trimethylsiloxy-terminated PDMS with molecular weights ranging from 1250 to 139,000 g/mol. Cone cracks were observed in samples that fractured from defects at the sample surface, but failure more commonly originated from the corners of the indenter. Ring cracks were observed for the most highly compliant samples that fractured at indentation depths approaching the overall thickness of the sample. In these cases we generally observed a delayed fracture response, with a time delay that increased with increasing sol fraction and decreased with increasing indentation load.

The indentation size effect, in which the contact pressure increases as the size of contact decreases, has been observed for many years for both spherical and pointed indenters. The concept of strain gradient plasticity has been used to describe this phenomenon, but it is often necessary to introduce a material length scale l* to fit experimental data. Here we present a theory based on the concept of dislocation slip distance, which naturally generates the scaling and incorporates the material parameters that influence the size effect. We compare this model with experimental data for a range of ceramics and tungsten metal and show that the yield strain and Burgers vector are the important material parameters in the indentation size effect.