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# On the uniqueness of measuring elastoplasticproperties from indentation

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Indentation is widely used to measure material mechanical properties such as hardness, elastic modulus, and fracture toughness (for brittle materials). Can one use indentation to extract material elastoplastic properties directly from the measured force-displacement curves? Or simply, is it possible to obtain material stress-strain curves from the corresponding indentation load-displacement curves? In an upcoming paper in JMPS titled "On the uniqueness of measuring elastoplastic properties from indentation: The indistinguishable mystical materials," Xi Chen and colleagues at Columbia University and National Defense Academy, Japan show the existence of "mystical materials", which have distinct elastoplastic properties yet they yield almost identical indentation behaviors, even when the indenter angle is varied in a large range. These mystical materials are, therefore, indistinguishable by many existing indentation analyses unless extreme (and often impractical) indenter angles are used. The authors have established explicit procedures of deriving these mystical materials. In many cases, for a given indenter angle range, a material would have infinite numbers of mystical siblings, and the existence maps of the mystical materials are also obtained. Furthermore, they propose two alternative techniques to effectively distinguish these mystical materials. The study in this paper addresses the important question of the uniqueness of indentation test, as well as providing useful guidelines to properly use the indentation technique to measure material elastoplastic properties.

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## Comments

## Thank you for interest in our work

Xiaodong,

Thank you for your interest in our work. As you probably agree, there are many unresolved issues in indentation mechanics and this field is far from saturated. The mystical material is just one such example, and your group has also done quite a few interesting works in indentation area. We first reported the idea of mystical materials in this post on iMechanica, and I am very pleased to see other iMechanicians followed up.

Xi

## Nanoindentation should find more applications in nanomechanics

I am so happy to see that your paper has stimulated more discussions on nanoindentation and its applications. In particular, I see the link between mechanics and materails science in your paper (personally I like such papers). I would like to see more papers on nanoindentation from your group and other groups in our community. Hopefully, one day, with the help of the advances in nanoindentation, one can measure almost all properties from one indentation load-displacement curve. I am looking forward to that day.

## Does viscoelastic material show similiar behavior?

Xi,

Fantastic paper. I am sure indentation community will be highly interested in this finding.

I am working on inverse FE analysis of indentation test for viscoelastic materials. I have thought about the uniqueness issue but not had a clue. In a simple viscoelastic model (say, a Kelvin model), there are three unknown parameters: E, viscosity eta and Poisson's ratio v.

Any comment on this?

Kaifeng

## what do you measure in an experiment?

Kaifeng,

Thank you. Can you please be a bit more specific regarding what do you measure in experiment, e.g. continuous force-displacement relationship or do you hold at the maximum load, etc.? The problem really depends on how many truly independent indentation variables you could measure, versus the unknown material properties.

Xi

## Indentation depth-force experiment

Xi,

I am using a typical Hysitron nano-indenter. So really the measured variables are load and displacement. To investigate the viscoelastic effect, a creep style load function is used (a hold time of about 10s at max. load). Will this help?

Kaifeng

## depends on what model of viscoelastic material you use

Kaifeng,

If you use Kelvin's model, it cannot be applied to stress relaxation at constant strain (i.e. holding). If you use other models the number of material parameters will be much more. From my instinction, it's going to be difficult to uniquely solve for all those material parameters with one simple test at constant loading rate and holding, probably you need measurements at other rates and cyclic loading, and/or use very different indenter geometries. Right now we do not have much experience on viscoelastic indentation problems. A more detailed investigation into such problem should be interesting, which our group may consider in the future.

Xi

## Viscoelastic indentation

Our group has substantial background and experience in viscoelastic indentation problems and there is a quite large and fast-growing literature on the subject. Some background is on my website here and elsewhere in iMechanica.

In practical terms, a Kelvin model is almost never used for this type of problem. Similarly, a Maxwell (series spring and dashpot) is also not used in practice for solids being indented. Even in relatively short experimental time-frames, more complicated behaviour is seen and the simple models, although good teaching tools for fundamental physics of viscoelasticity, fall away as practically useless.

The simplest useful solid model is a "Standard Linear Solid" with three parameters (two spring constants and a dashpot coefficient) although for many real experiments more complicated models are required and the functions are sums of several empirical exponential decay terms that do not map easily onto the simple spring-and-dashpot ideas.

In practice, it is actually quite easy to solve for 5- or 7- parameter viscoelastic functions under indentation creep or relaxation conditions, because you see behavior that has very clearly delineated short-time and longer-time responses. It is absolutely true, as stated above, that it becomes more difficult with tests at constant rate although I have looked at this issue some (Phil. Mag. 2006). However, it's generally true, not just in indentation conditions, that you find there are very good reasons experimentally for the emphasis on constant load (stress) or constant displacement (strain) tests for really probing the viscoelastic behavior most sensitively, and constant rate tests are better for emphasis on elastic response.

(If you have more specific questions just ask; I probably have about 100 papers here in my files including 10 of my own on this subject -- much has been said on the subject of viscoelastic indentation in recent years.)

## How to determine which model to choose?

Michelle, while our group has published over two dozens of papers on indentation mechanics, we have not yet touched the viscoelastic topic. I have a simple question for you, which model (e.g. how many springs and dashpots, or how many parameters) is more widely used in practice? For metals we usually use the power-law hardening model but for viscoelastic materials, if you get an experimental curve, how do you determine which model (and how many parameters) should be used to describe its indentation behavior?

This is also related with uniqueness of your approach: with 5-7 parameters (or sometimes you do not even know how many parameters should you use to describe the viscoelastic model), how can you be certain that your numerical solution is unique?

## viscoelastic model selection

There are really no hard-and-fast rules for viscoelastic model selection, but there are some guidelines that you can keep in mind. Most empirical approaches that use exponential decay functions for the creep or relaxation function can be considered as reasonably accounting for behavior at one time constant per decade. They are multiple time constant functions that are not usefully related back to individual spring or dashpot coefficients in most circumstances.

In terms of how many time-constants are needed for any particular set of experimental data: You have two real time-scales in the experiment itself: (1) the time it takes to reach the peak load or displacement (assuming a hold-type test for creep or relaxation with a finite ramp- or rise-time, since real experiments do not involve magical machines with Heaviside step-function capability) and (2) the length of the holding time during which you are measuring time-dependent responses. These two time-scales will guide you on what the base times are in your experiment and thus how many time constants you can reasonably expect to "experience" with any particular experiment. So, for example, if you do an experiment that lasts a few minutes, with a ramping time of a few seconds, around 2-3 time constants is about right (a few minutes divided by a few seconds is 60 which is near 10^2). For an experiment a few hours long, with a ramp time of a few seconds, you're closer to 10^4 and 4-5 time constants would be a good guess. These are good "rule-of-thumb" guidelines but of course the thing to do is to let the experiment guide you. If the function you choose does not fit the data, then you need to rethink your function!!!

In terms of the uniqueness question, it almost takes a little bit of re-thinking what one is trying to do with viscoelastic materials analysis. These types of summed exponential functions we use are mathematically convenient and their primary usefulness is to describe the observed response and perhaps, even better, be able to predict the response under different applied loading conditions (with the caveat that you are limited to predictions in time-scales that are similar to the ones you have been observing in your experiment; there is little to no good in extrapolation beyond the experimental time-frame).

So really, the idea here is not to uniquely identify "THE" constitutive response of the viscoelastic material with a single test--these are not necessarily material property measurements in the traditional sense.

It's always key in these situations to stop and sort out what you are doing, and why. You hand me a lump of a viscoelastic material. You say that you want me to measure it's mechanical properties by indentation. Really, in order to establish the best-practice mechanism for dealing with this lump of material, I need more information before I start!

## Further references on identification of material parameters

Kaifeng,

I guess the following articles of N. Huber may help you

1. Tyulyukovskiy E,

Huber NNeural networks for tip correction of spherical indentation curves from bulk metals and thin metal films

JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 55 (2): 391-418 FEB 2007

2.

Huber N, Tyulyukovskiy E, Kraft OOn the analysis of the stress-strain behaviour of thin metal films on substrates using nanoindentation

PHILOSOPHICAL MAGAZINE 86 (33-35): 5505-5519 NOV-DEC 2006

3. Tyulyukovskiy E,

Huber NIdentification of viscoplastic material parameters from spherical indentation data: Part I. Neural networks

JOURNAL OF MATERIALS RESEARCH 21 (3): 664-676 MAR 2006

4. Klotzer D, Ullner C, Tyulyukovskiy E, et al.

Identification of viscoplastic material parameters from spherical indentation data: Part II. Experimental validation of the method

JOURNAL OF MATERIALS RESEARCH 21 (3): 677-684 MAR 2006

5.

Huber N, Tyulyukovskiy EA new loading history for identification of viscoplastic properties by spherical indentation

JOURNAL OF MATERIALS RESEARCH 19 (1): 101-113 JAN 2004

6.

Huber N, Nix WD, Gao HIdentification of elastic-plastic material parameters from pyramidal indentation of thin films

PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 458 (2023): 1593-1620 JUL 8 2002

V. Hegadekatte, University of Karlsruhe, Germany

## It's difficult to obtain unique solution in his paper

Vishwanath,

For the first paper (JMPS) you mentioned, if my understanding is correct and you only indent to h/R up to 0.08 (according to 4.1 of the Huber JMPS paper), it will be difficult to distinguish the P-h curves from spherical indentation on bulk material. As we showed in Fig.12 of our JMPS paper discussed above, to effectively distinguish mystical materials by spherical indentation, we need a deeper penetration h/R=0.3. Of course one may argue tiny differences may occur at very small small h/R but in my view those are too sensitive to small perturbation in real experiments. In addition, we suspect shallow spherical indentation (e.g. Huber's JMPS paper) will not work well, because that corresponds to a small range of effective sharp indenter angles, where we showed we could derive mystical materials explicitly. If you are interested, we could always find even better counter examples.

I also note that in the Huber papers you listed (I only looked at those without viscoplastic effects so far), uniqueness of inverse analysis was never discussed.

## Your comments

Xi Chen,

I will forward your comments to Prof. Huber.

V. Hegadekatte, University of Karlsruhe, Germany

## Stress Heterogeneity

My guess is that the force measured during indentation is the net result of a wide range of stresses underneath the indenter, and we can have an infinite number of variation in the consitutive relation that can give the same net force. My experience with large strain viscoelastic materials is that the portion of the stress-strain relationship at large strains is particularly difficult to get, probably because only a small region of the material actually achieves these stresses.

Having said that, other scientists have suggested using data from as many tests as possible (e.g. two indenters), which is not inconvenient for elasto-plastic materials but a little more pain for rate and strain dependent as well heterogeneous materials.

## the point is two indenters do not work well

That's the main finding of our paper discussed here. Many people believed the solution from multiple indenters should be unique based on dimensional analysis, but it is not -- of course we did not discuss the effect of strain rate, heterogeneous materials, and non-power law materials. Even for the simplest homogeneous, isotropic, power law materials and quasi-static loading, dual indenter method does not work well and therefore it cannot be used in practice when these ignored factors may be important. Similarly, spherical indentation and film indentation are also affected and the solution may not be unique if the experiment is not carefull carried out, see our paper for more discussions.

## nano-indentation test

I have benefited from this thread of discussion on indentation test.

I put a link in my teaching “Materials of Engineering Laboratory” (http://imechanica.org/node/1061#comment-2531) to this blog.