Scaling, dimensional analysis, and indentation measurements

Abstract

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 macro-scales. 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.

Introduction

The perception of hardness is perhaps as ancient as human existence. It is the natural consequence of our ability to sense the surrounding world through physical contact. In fact, we develop an intuitive appreciation of “hard” and “soft” as infants before we learn to talk or walk. However, it is only fairly recent (about 100 years or so) that quantitative scales of hardness and measurement methods were developed for materials property characterization. The first modern technique for measuring the hardness of metals was due to Brinell [1], who was concerned about the consistency of steels produced by the company he was employed by. He took steel plates of each batch, placed a hard steel ball between them and squeezed them together in a vice. The size of the dents was chosen to represent the hardness of his steel. Today, a standard Brinell test consists of pressing a hard steel ball normally onto the surface of the metal under investigation. The Brinell hardness is given by the load divided by the surface area of the indentation. Other standard hardness tests have been developed and are routinely used, including the Vickers, Berkovich, Knoop, and Rockwell tests [2], [3], [4]. Typically, an indenter of a given geometry and material type, such as a sphere, a cone, or a pyramid made of steel or diamond, is pushed into the solid. Upon unloading, a permanent depression in the surface of material is examined with a microscope. The hardness value is usually defined as the ratio of the indentation load and either the surface or projected area of residual indents, though the depth of indentation may also be used as a measure of hardness (i.e., in the Rockwell indentation test). These standard hardness tests are vital in nearly all areas of materials science and engineering.

Since the early 1970s, instrumented micro- and nano-indentation techniques have been developed and are now widely available [5], [6], [7], [8], [9], [10], [11]. In instrumented indentation experiments, load and indenter displacement are recorded simultaneously during the entire loading and unloading process. These instruments also allow the precise control of either load or displacement during indentation experiments. Instrumented indentation can be accurately made using forces as small as a few micro Newtons over depths in the nano-meter range (Fig. 1). Thus came the term “nanoindentation”. Together with improved understanding of indentation mechanisms and advances in analysis methods, such as that due to Doerner and Nix [12] and Oliver and Pharr [13], instrumented indentation has become an indispensable tool for probing mechanical properties at small length scales.

Since instrumented indentation can also be performed at the load range typical of conventional Vicker’s and Rockwell hardness tests, it is one of the few experimental techniques that can be performed at both micro- and macro-scales, allowing the investigation of materials behavior across length scales from nano- to millimeters. Recently, the field of “multiscale” modeling has attracted significant attention [14], [15], [16], [17], [18]. Theoretical models are being developed for describing materials behavior using quantum mechanics for a few atoms, molecular dynamics for hundreds of millions of atoms, dislocation dynamics for interacting dislocations, and continuum mechanics at macroscopic scales. Because it offers an opportunity for comparison with experiments, modeling indentation from small number of atoms to continuum has become popular in recent publications on multiscale modeling [14], [18], [19], [20], [21].

Indentation measurements have been applied to a wide range of materials, ranging from metals, ceramics, polymers, and composites. The application areas include microelectronics, optoelectronics, coatings for low-emission window glasses, and tribological coatings [22], [23], [24], [25], [26], [27], [28]. Presently, there is also a growing interest in probing biological materials [29], [30], [31], [32], [33], [34], [35] and food products [36], [37]. Instrumented indentation measurements have also been used in the study of the deformation and fracture of rocks for a better understanding of rock mechanics related to geological evolution of the planets [38], [39].

A review of every aspects of indentation is beyond the scope of the present paper, though there are several reviews [40], [41], [42], [43], [44], [45], [46], bibliographies on indentation modeling [47], [48], books [2], [3], [49], [50], special journal volumes [24], [27], [28], and topical conference proceedings [22], [23], [24], [25], [26] dedicated to the broad areas of indentation hardness measurements. Instead, this review will focus on a scaling approach to indentation modeling [51], [52], [53], [54], [55]. This approach helps understand several basic questions in instrumented indentation measurements, including:

• What information is contained in the indentation load–displacement curves?

• What is hardness? How does hardness depend on mechanical properties and indenter geometry?

• 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?

We show that the scaling approach to indentation modeling provides some new insights into these issues. It also helps organize knowledge and provide a framework for bridging micro- and macro-scales.

We begin with a tutorial on dimensional analysis. Although dimensional analysis is routinely used in certain branches of science and engineering, most materials scientists and engineers are not exposed to this powerful analysis tool. The basic idea of dimensional analysis is that the physical laws do not depend on the arbitrariness in the choice of units of physical quantities. This concept often allows the number of arguments in functions describing physical phenomena to be reduced, thus making them simpler to obtain either from calculations or experiments. This review shows that dimensional analysis is very valuable to the analysis of a range of indentation problems.

Since indentation measurements have been applied to a great variety of materials, ranging from metals, ceramics, polymers, to biomaterials, it is necessary to model indentation using general though simplified descriptions for the mechanical properties of solids, including the power-law work-hardening, power-law creep, and linear viscoelastic effects. Dimensional analysis is then used to explore various aspects of indentation in these model systems with the aid of finite element calculations when necessary. The focus is on indentation using self-similar conical and pyramidal indenters with some discussions on spherical indenters.

This review is intended to serve 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.