We provide an overview of the basic concepts of scaling and dimensional analysis, followed by a review of some of the recent work on applying these concepts to modeling instrumented indentation measurements. Specifically, we examine conical and pyramidal indentation in elastic-plastic solids with power-law work-hardening, in power-law creep solids, and in linear viscoelastic materials. We show that the scaling approach to indentation modeling provides new insights into several basic questions in instrumented indentation, including, what information is contained in the indentation load-displacement curves? How does hardness depend on the mechanical properties and indenter geometry? What are the factors determining piling-up and sinking-in of surface profiles around indents? Can stress-strain relationships be obtained from indentation load-displacement curves? How to measure time dependent mechanical properties from indentation? How to detect or confirm indentation size effects? The scaling approach also helps organize knowledge and provides a framework for bridging micro- and macroscales. We hope that this review will accomplish two purposes: (1) introducing the basic concepts of scaling and dimensional analysis to materials scientists and engineers, and (2) providing a better understanding of instrumented indentation measurements.

Scaling, dimensional analysis, and indentation measurements

In recent years, near-nano (submicron) and nanostructured materials have attracted increasingly more attention from the materials community. Nanocrystalline materials are characterized by a microstructural length or grain size of up to about 100 nm. Materials having grain size of ∼0.1 to 0.3 μm are classified as submicron materials. Nanocrystalline materials exhibit various shapes or forms, and possess unique chemical, physical or mechanical properties. When the grain size is below a critical value (∼10–20 nm), more than 50 vol.% of atoms is associated with grain boundaries or interfacial boundaries. In this respect, dislocation pile-ups cannot form, and the Hall–Petch relationship for conventional coarse-grained materials is no longer valid. Therefore, grain boundaries play a major role in the deformation of nanocrystalline materials. Nanocrystalline materials exhibit creep and super plasticity at lower temperatures than conventional micro-grained counterparts. Similarly, plastic deformation of nanocrystalline coatings is considered to be associated with grain boundary sliding assisted by grain boundary diffusion or rotation. In this review paper, current developments in fabrication, microstructure, physical and mechanical properties of nanocrystalline materials and coatings will be addressed. Particular attention is paid to the properties of transition metal nitride nanocrystalline films formed by ion beam assisted deposition process.

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Nanocrystalline materials and coatings

Using a variety of depth-sensing indentation techniques, the creep response of high-purity indium, from room temperature to 75 °C, was measured. The dependence of the hardness on the variables of indentation strain rate (stress exponent for creep (n)) and temperature (apparent activation energy for creep (Q)) and the existence of a steady-state behavior in an indentation test with a Berkovich indenter were investigated. It was shown for the first time that the indentation strain rate (-este-/h) could be held constant during an experiment using a Berkovich indenter, by maintaining the loading rate divided by the load (-este-/P) constant. The apparent activation energy for indentation creep was found to be 78 kJ/mol, in accord with the activation energy for self-diffusion in the material. Finally, by performing -este-/P change experiments, it was shown that a steady-state path independent of hardness could be reached in an indentation test with a geometrically similar indenter.

Indentation power-law creep of high-purity indium

A study of the indentation size effect (ISE) in aluminum and alpha brass is presented. The study employs rate effects to examine the fundamental mechanisms responsible for the ISE. These rate effects are characterized in terms of the rate sensitivity of the hardness, ∂ H/ ∂ ln e eff , where H is the hardness and e eff is an effective strain rate in the plastic volume beneath the indenter. ∂ H/ ∂ ln e eff can be measured using indentation creep, load relaxation, or rate change experiments. The activation volume V ∗ , calculated based on ∂ H/ ∂ ln e eff which can traditionally be used to compare rate sensitivity data from a hardness test to conventional uniaxial testing, is calculated. Using materials with different stacking fault energy and specimens with different levels of work hardening, we demonstrate how increasing the dislocation density affects V ∗ ; these effects may be taken as a kinetic signature of dislocation strengthening mechanisms. We noticed both H and ∂ H/ ∂ ln e eff (V ∗ ) exhibit an ISE. The course of V ∗ vs. H as a result of the ISE is consistent with the course of testing specimens with different level of work hardening. This result was observed in both materials. This suggests that a dislocation mechanism is responsible for the ISE. When the results are fitted to a strain gradient plasticity model, the data at deep indents (microhardness and large nanoindentation) exhibit a straight-line behavior closely identical to literature data. However, for shallow indents (nanoindentation data), the slope of the line severely changes, decreasing by a factor of 10, resulting in a “bilinear behavior”.

Nanoindentation and the indentation size effect: Kinetics of deformation and strain gradient plasticity

INSTRUMENTED INDENTATION TESTING (IIT), also known as depth-sensing indentation, continuous-recording indentation, ultra-low-load indentation, and nanoindentation, is a relatively new form of mechanical testing that significantly expands on the capabilities of traditional hardness testing. Developed largely over the past two decades, IIT employs high-resolution instrumentation to continuously control and monitor the loads and displacements of an indenter as it is driven into and withdrawn from a material (Ref 1–13). Depending on the details of the specific testing system, loads as small as 1 nN can be applied, and displacements of 0.1 nm (1 A) can be measured. Mechanical properties are derived from the indentation load-displacement data obtained in simple tests. The advantages of IIT are numerous, as indentation load-displacement data contain a wealth of information, and techniques have been developed for characterizing a variety of mechanical properties. The technique most frequently employed measures the hardness, but it also gives the elastic modulus (Young’s modulus) from the same data (Ref 8, 11). Although not as well-developed, methods have also been devised for evaluating the yield stress and strain-hardening characteristic of metals (Ref 14–16); parameters characteristic of damping and internal friction in polymers, such as the storage and loss modulus (Ref 17, 18); and the activation energy and stress exponent for creep (Ref 19–25). IIT has even been used to estimate the fracture toughness of brittle materials using optical measurement of the lengths of cracks that have formed at the corners of hardness impressions made with special sharp indenters (Ref 13, 26, 27). In fact, almost any material property that can be measured in a uniaxial tension or compression test can conceivably be measured, or at least estimated, using IIT. An equally important advantage of IIT results because load-displacement data can be used to determine mechanical properties without having to image the hardness impressions. This facilitates property measurement at very small scales. Mechanical properties are routinely measured from submicron indentations, and with careful technique, properties have even been determined from indentations only a few nanometers deep. Because of this, IIT has become a primary tool for examining thin films, coatings, and materials with surfaces modified by techniques such as ion implantation and laser heat treatment. Many IIT testing systems are equipped with automated specimen manipulation stages. In these systems, the spatial distribution of the near-surface mechanical properties can be mapped on a point-to-point basis along the surface in a fully automated way. Lateral spatial resolutions of about a micron have been achieved. An example of small indentations located at specific points in an electronic microcircuit is shown in Fig. 1. The purpose of this article is to provide a practical reference guide for instrumented indentation testing. Emphasis is placed on the better-developed measurement techniques and the procedures and calibrations required to obtain accurate and meaningful measurements.

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Instrumented Indentation Testing

The hardness, H , and rate sensitivity of the hardness, m = ∂ ln H /∂ ln ∊ eff | x p , where ∊ eff is an effective strain rate and x p the plastic depth, are measured in molybdenum at room and low temperature (160 and 170 K) using as-received and annealed specimens. Based on these measurements it is found that H separates into two components: one depending on indentation rate and temperature, and the other depending on the starting state of the material. An activation volume is defined, v ∗ = 9 kT/mH , which falls within the range of values derived from other experimental techniques. The values of m obtained from indentation creep, indentation load relaxation, and indentation rate-change experiments agree closely with each other provided a consistent analysis is used. The results of these experiments suggest that the rate- and temperature-dependence of the hardness can be used to discriminate between strengthening mechanisms at low temperature.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0884291400075531

K.B. Yoder

Papers

Division of the hardness of molybdenum into rate-dependent and rate-independent components