Hardness across the multi-scales of structure and loading rate: a post-meeting response to the ‘pre-editorial’

There are many different concepts of hardness depending on the method used to measure it. For example, there is hardness measured by penetration, by scratching, by resilience, by machinability, by yield point or by magnetic or electric properties. However, it is indentation hardness that has become the most useful as it can provide a local measure of mechanical response. Since the pioneering work of Brinell in the early 1900s the concept of indentation hardness has been used for quality assurance or control of materials and more recent developments in instrumentation and analysis methods have greatly increased the utility of the technique. As time has passed the scales of deformation produced by indentation tests have become progressively smaller and it is now possible to assess the mechanical response of volumes of material with nanometre dimensions. In such circumstances there is a real need to understand what the hardness of the material means if the mechanical behaviour of the tested material is to be properly understood and there has been considerable progress in the last few years in relating the deformation processes which occur beneath an indenter to the measured hardness and bulk mechanical properties.

This special issue of MST contains a range of review papers and original articles which attempt to summarise the state of the art in hardness testing and analysis across a range of length scales. The historical origins of the test are reviewed by Walley¹ up to the publication of the seminal book by Tabor, ‘The hardness of metals’, in 1951, which marks the baseline for the understanding of indentation hardness. The development of instrumented indentation tests in the 1980s has enabled hardness to be measured at scales which are commensurate with the microstructure in polycrystalline metals and allow the details of deformation mechanisms to be identified when tests are performed in conjunction with suitable microscopy – indeed in situ indentation tests in the SEM or TEM are now preformed by a number of workers for this very purpose. This work is critical in the understanding of size effects which are routinely seen in the hardness of crystalline materials at submicrometre length scales.

At the same time, the test instrumentation methods available for doing hardness testing have expanded in line with the multiple uses now made of one or another versions of the hardness penetration methods, reaching the apparent zenith for nano-indentations where initial elastic loading and local plastic resistances can be measured at levels near to the theoretical limit of strength for essentially perfect material volumes: the ultimate indentation size effect. And for cases where the imposed indentation strain requirement exceeds that available from the initially unrestricted material deformation systems, there is even indentation fracture mechanics analysis to monitor the generation of crack propagation lengths and to specify the local material toughness.

John Knott asked a number of questions in the setting up of this issue which have been addressed in many of the papers in this issue:

(i)	Are standard instrumented indentation tests sufficiently sensitive to determine from the slopes of the force-displacement curves the anisotropy of a metal’s elastic constants with crystal orientation? For materials with a reasonable degree of anisotropy such as copper this is possible (e.g. Bull et al.2) and for very anisotropic materials such as zinc this is easily achievable. However, following the work of Vlassak and Nix,3 it should be realised that the material displacements during indentation are not always parallel to the loading axis and the measured variation in elastic modulus can be different from that measured in uniaxial tensile tests.
(ii)	Can we measure the activation of different slip systems in a material by using an asymmetric indenter and relate this to the Schmid factor? Microhardness anisotropy using a Knoop indenter has been widely studied and related to dislocation generation and propagation due to resolved shear stress variations around this indenter. The onset of plasticity can very conveniently be probed with instrumented indentation with strain bursts associated with formation of dislocation rosettes clearly visible in the load displacement curves for defect-free material. The distribution of statistically stored dislocations in the test volume is critical for this (e.g. Gao and co-workers4).
(iii)	Can dislocation interactions with obstacles be probed by instrumented indentation tests? The relationship between hardness and grain size follows Hall–Petch-like behaviour over a wide range of grain sizes (e.g. Armstrong and Elban5) but there are deviations at low grain size. The hardening effect of high angle (twin) boundaries can be probed and is greater than the low angle grain boundaries usually observed in copper (Salehinia and Bahr6). Solute effects are also visible in the early stages of plasticity in aluminium alloys (Wang and Conrad7). However, in all cases once a well-defined plastic zone is formed beneath the indenter these effects become subsumed in the apparently continuum behaviour of the high dislocation density region created. What is critical is that it is very important to prepare test samples correctly for the test – polished surfaces with high dislocation density are too heavily work hardened for most of these effects to be resolved.
(iv)	To what extent has the expanding spherical cavity model of Marsh8,9 for the hardness of glassy materials been developed further? The original analysis of Marsh was developed further by Johnson in his classic paper on the correlation of indentation experiments10 in which the indenter is encased in a hemispherical core (which equates to the highly deformed material) within which there is assumed to be a hydrostatic pressure, whereas outside the stresses and displacements have radial symmetry. By this approach the hardness of a material is governed by (E/Y) tanβ where β is the inclination angle of the indenter; Y is the stress associated with a representative plastic strain of 0·2tanβ. There has been some argument about the precise value of the representative strain (and where it occurs around the indentation), but there has been good agreement with experimental data in many systems. However, as Marsh observed, in many ceramic systems deformation may be discontinuous and associated with the formation of shear bands with intense deformation. This is a major contribution to the indentation size effect in ceramic materials but is mostly observed in the elastic–plastic transition range. Once a well defined plastic zone is produced individual shear bands cannot be seen. During the symposium the contact mechanics aspects of indentation in ceramic and glass systems were reviewed by Atkinson and co-workers,11 and Mokios and Aifantis12 examined gradient effects, coupling their gradient plasticity formulation with Johnson’s cavity model.
(v)	How does plasticity develop as the scale of the indentation increases with respect to the microstructure? There is a transition from single-crystal-like behaviour for small indents in the middle of a grain to polycrystalline behaviour with well defined Hall–Petch behaviour (e.g. in copper2). However, this is only clear if coatings or foils are tested where there is a single grain through the sample thickness. Bulk crystalline behaviour is observed when only a small number of grains are involved in the deformation (one grain and all its nearest neighbours in the case of copper, some 6–10 grains) but this is very dependent on grain shape and size.
(vi)	What is the effect of pre-yield microstrains on indentations? – the effect of surface roughness and polishing damage are an example of this and can be very significant. Existing defects are very easy to propagate and plastic deformation is easy, smooth and continuous. Creating new dislocations in dislocation-free material is more difficult and leads to discontinuous deformation (pop-ins, e.g. Nowak et al.13 on current spikes during nano-indentation of GaAs). Dislocations are surface nucleated at roughness, or within grain boundaries in indentation tests in the TEM.
(vii)	Do we see bulk deformation mechanisms in nanometre grain size material? It is surprising how bulk deformation mechanisms can be observed in fine grained materials – for instance, the Hall–Petch effect can be seen in materials with grain size below 100 nm, although deviations and inverse behaviour are observed in other cases. Dislocation generation and propagation has been observed in sub 100 nm grains in both TEM14 and atomistic simulations of indentation,15 although often these dislocations are generated in grain boundaries or at surfaces,16 run across the grain and disappear into the opposite boundary so are not visible after the test. Dislocation sources and indentation-induced grain boundary motion have also been seen operating in TEM studies.17

It was notable that indentation rate (and deformation rate) effects, a theme touched on only obliquely in the pre-editorial questions above, attracted significant interest in the symposium that is reflected in the contributions to this issue.^18–22

Steve Bull

Newcastle University ([email protected])