192:(optical or stylus) to obtain the full shape of the residual indent. With a spherical indenter (and a sample that is isotropic in the plane of the indented surface), the indent will exhibit radial symmetry and its shape can be captured in the form of a single profile (of depth against radial position). The details of this shape (for a given applied load) exhibit a high sensitivity to the stress-strain relationship of the sample. Also, it is easier to obtain than a load-displacement curve, partly because no measurements need to be made during loading. Finally, such profilometry has potential for the detection and characterization of sample anisotropy (whereas load-displacement curves carry no such information).
233:
detection and characterisation of sample anisotropy – see above. The figure gives an indication of the sensitivity of the profile to the stress-strain curve of the material. The term PIP thus encompasses the following features: 1) Obtaining stress-strain curves characteristic of the bulk of a material (by using relatively large spherical indenters and relatively deep penetration), 2) Experimental measurement of the residual indent profile and 3) Iterative FEM simulation of the indentation test, to obtain the stress-strain curve (captured in a constitutive equation) that gives the best fit between modelled and measured profiles.
201:“equivalent”, “effective” or “representative” values of the stress in the loaded part of the sample (from the applied load) and a corresponding set of values of the strain in the deformed region (from the displacement). The assumptions involved in carrying out such conversions are inevitably very crude, since (even for a spherical indenter) the fields of both stress and strain within the sample are highly complex and evolve throughout the process – the figure shows some typical plastic strain fields. Various empirical correction factors are commonly employed, with
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205:“training” procedures sometimes being applied to sets of load-displacement data and corresponding stress-strain curves, to help evaluate them. It’s also common for loading to be periodically interrupted, and data from partial unloading procedures to be used in the conversion. However, unsurprisingly, universal conversions of this type (applied to samples with unknown stress-strain curves) tend to be unreliable and it is now widely accepted that the procedure cannot be used with any confidence.
224:), followed by convergence on the best fit version (set of parameter values in the equation), giving optimal agreement between experimental and modelled outcomes (load-displacement plots or residual indent profiles). This procedure fully captures the complexity of the evolving stress and strain fields during indentation. While it is based on relatively intensive modelling computations, protocols have been developed in which the convergence is automated and rapid.
181:” - for which both the load (down to the mN range) and the displacement (commonly sub-micron) are very small. However, as noted above, if the deformed volume is small, then it’s not possible to obtain “bulk” properties. Moreover, even with relatively large loads and displacements, some kind of “compliance correction” may be required, to separate the response of the sample from displacements associated with the loading system.
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The strains created in the sample must therefore also range up to values of this order. This typically requires that the “penetration ratio” (penetration depth over indenter radius) should be at least about 10%. Finally, depending on the hardness of the metal, this in turn requires that the facility should have a relatively high load capability – usually of the order of several kN.
177:“instrumented” set-ups, in which the load is progressively ramped up and both load and penetration (displacement) are continuously monitored during indentation. A key experimental outcome is thus the load-displacement curve. Various types of equipment can be used to generate such curves. These include those designed to carry out so-called “
164:, in which the measured hardness tends to increase as the deformed volume becomes small, is at least partly due to a failure to interrogate a representative volume. The indenter, which is normally spherical, therefore needs to have a radius in the approximate range of several hundred microns up to a mm or two.
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Two main approaches have evolved for obtaining stress-strain relationships from experimental indentation outcomes (load-displacement curves or residual indent profiles). The simpler of the two involves direct “conversion” of the load-displacement curve. This is usually done by obtaining a series of
176:
The simplest indentation procedures, which have been in use for many decades, involve the application of a pre-determined load (often from a dead weight), followed by measurement of the lateral size of the residual indent (or possibly its depth). However, many indentation procedures are now based on
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A further requirement concerns the plastic strains generated in the sample. The indentation response must be sensitive to the plasticity characteristics of the material over the strain range of interest, which normally extends up to at least several % and commonly up to several tens of %.
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It has become clear that important advantages are offered by using the residual indent profile as the target outcome, rather than the load-displacement curve. These include easier measurement, greater sensitivity of the experimental outcome to the stress-strain relationship and potential for
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For tractable and user-friendly application, an integrated facility is needed, in which the procedures of indentation, profilometry and convergence on the optimal stress-strain curve are all under automated control
212:(a) FEM fields of equivalent plastic strain, after spherical indentation to a penetration ratio of about 20%, for as-extruded and annealed copper samples, and (b) corresponding measured and modelled indent profiles
140:, with far greater potential for mapping of spatial variations, this is an attractive concept (provided that the outcome is at least approximately as reliable as those of standard uniaxial tests).
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focused on this, in the form of (relatively crude) measurement of the “width” of the indent – commonly via simple optical microscopy. However, much richer information can be extracted by using a
136:, which gives numbers that are only semi-quantitative indicators of the resistance to plastic deformation). Since indentation is a much easier and more convenient procedure than conventional
312:
Roters, F (2010). "Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications".
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Huber, N (1999). "Determination of constitutive properties from spherical indentation data using neural networks. Part I: the case of pure kinematic hardening in plasticity laws".
595:
Tang, Y (2021). "Use of
Profilometry-based Indentation Plastometry to obtain Stress-Strain Curves from Small Superalloy Components made by Additive Manufacturing".
220:– FEM) modelling of the indentation procedure. This is first done with a trial stress-strain relationship (in the form of an analytical expression – often termed a
434:
Campbell, J (2019). "Comparison between stress-strain plots obtained from indentation plastometry, based on residual indent profiles, and from uniaxial testing".
152:(size-independent) properties brings in a requirement to deform a volume of material that is large enough to be representative of the bulk. This depends on the
216:
The other main approach is a more cumbersome one, although with much greater potential for obtaining reliable results. It involves iterative numerical (
39:
630:
Kim, J (2006). "Determination of
Tensile Properties by Instrumented Indentation Technique: Representative Stress and Strain Approach".
86:
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58:
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Lee, J (2009). "Reverse analysis of nano-indentation using different representative strains and residual indentation profiles".
729:"An Alternative Approach to Determine Material Characteristics Using Spherical Indentation and Neural Networks for Bulk Metals"
65:
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Zhan, X (2017). "Identification of plastic anisotropy using spherical indentation on different anisotropic yield criterions".
43:
657:
Hernot, X (2014). "Study of the concept of representative strain and constraint factor introduced by
Vickers indentation".
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Jeon, E (2009). "A Method for
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821:
Huang, F (2021). "Uncertainties in the representative indentation stress and strain using spherical nanoindentation".
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Yonezu, A (2009). "Estimation of the anisotropic plastic property using single spherical indentation - An FEM study".
469:
Bocciarelli, M (2005). "Parameter identification in anisotropic elastoplasticity by indentation and imprint mapping".
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The other main form of experimental outcome is the shape of the residual indent. As mentioned above, early types of
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Ostoja-Starzewski, M (2006). "Material
Spatial Randomness: From Statistical to Representative Volume Element".
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Bolzon, G (2004). "Material model calibration by indentation, imprint mapping and inverse analysis".
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Zambaldi, C (2010). "Plastic anisotropy of gamma-TiAl revealed by axisymmetric indentation".
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228:Profilometry-based indentation plastometry (PIP)
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