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# Mystical materials in indentation

As an indenter penetrates an elastoplastic material, the indentation load P can be measured as a continuous function of the indentation displacement δ, to obtain the so-called P-δ curve. A primary goal of the indentation analysis is to relate the material elastoplastic properties (such as the Young's modulus, yield stress, and work-hardening exponent) with the indentation response (i.e. the shape factors of the P-δ curve, including its curvature, unloading stiffness, loading work, unloading work, maximum penetration, residual penetration, maximum load, etc.). The sharp indenters (e.g. Berkovich or Vickers) are the most widely used in practice.

Some earlier studies have proposed that for a given sharp indenter angle, there is a one-to-one correspondence between the material elastoplastic properties and the indentation response. However, some recent studies found this is not true, that is, there are special sets of materials with different elastoplastic properties that could lead to the same P-δ curves for a given indenter angle. From dimensional analysis, most people in the indentation community tend to believe that for these special materials, if a distinct indenter angle is used, their P-δ curves must be sufficiently different such that their behaviors are distinguishable by using the dual (or plural) indenter method. Consequently, in recent papers the indentation community begins to claim that the dual (or plural) indenter method is "sufficient and necessary" to determine the material elastoplastic properties just from the P-δ curves obtained from dual (or plural) sharp indentation tests.

Nevertheless, the important topic of the uniqueness of indentation analysis is not yet systematically studied, at least to our knowledge. Are there sets of **mystical materials** with distinct elastoplastic properties, yet they always yield the same P-δ curves, even when different sharp indenters are used? In principle, one could build an enormous material database including every material with different elastoplastic properties, and perform numerical indentation tests to obtain their P-δ curves, and then do a reverse search to find such mystical materials. However, this is an impossible task to complete due to endless possible combinations. If we wish to derive such mystical material, one must find a concise way to explicitly relate the indentation responses with material properties, which is a nontrivial task considering the finite plastic deformation beneath the indenter tip, as well as the number of variables involved.

Recently, we have succeeded to explicitly derive the mystical materials. For the material examples listed below, which have different elastoplastic properties (according to their uniaxial stress-strain curves):

Their P-δ curves are identical when a Berkovich indenter tip (half apex angle 70.3 degrees) is used:

Moreover, their P-δ curves are identical when other indenter angles are used (63, 75, and 80 degrees), where the 63 and 75 indenters have cross-sectional areas that are half and twice of that of the Berkovich tip, respectively. Such identity also holds regardless of the maximum indentation load/depth one takes on an semi-infinite bulk specimen:

Therefore, it is impossible for the dual (or plural) indenter method to distinguish these mystical materials. In other words, the "well-established" dual (or plural) indenter method is proven to be not robust. In our paper which will be submitted for publication shortly, we will further show that one can explicitly derive infinite siblings which yield the same indentation response for plural indenters. The process is relatively straightforward.

Does this finding mean that there is no way to distinguish these mystical materials? We will propose alternative indentation techniques to distinguish them (which were previously established by our group), and we will explain why these methods may still work from a mechanics point of view. Meanwhile, we also note that if one uses extreme indenter angles in a dual indentation test (e.g. one very sharp and one very blut), the existence of mystical materials becomes difficult.

In my personal view, this work is a fundamental progress of indentation mechanics. The paper has been accepted for publication in J. Mech. Phys. Solids

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## Comments

## Comment on mystical materials in indentation

Dear Xi,

I support your point of view. In our previous work, the uniqueness of reverse analysis from conical indentation tests has been carried out and we concludes that the reverse analysis of a single load-displacement curve will yield non-unique combinations of elasto-plastic material properties. We have proposed reverse analysis algorithm that can uniquely recover the elasto-plastic material properties from the load–displacement curves of two conical indenters with different apex angles.

References: 1. Tho, K.K., Swaddiwudhipong, S., Liu, Z.S., Zeng, K. and Hua, J. (2004), “Uniqueness Of Reverse Analysis From Conical Indentation Tests”, Journal of Materials Research, Vol. 19(8), 2498-2502.2. Swaddiwudhipong, S., Tho, K.K., Liu, Z.S., and Zeng, K. (2005), “Material Characterization Based On Dual Indenters”, International Journal of Solids and Structures, Vol. 42 (1), 69–83.

Zishun

## Dual indenter method cannot promise unique solution

Dear Zishun,

Thank you for your interest in our work. I have noticed your paper which has derived special materials that have same loading/unloading curves for 1 particular indenter. However, the mystical materials I referred in the post are valid for a broad range of indenter angles, which means most existing dual indenter method cannot distinguish them. In the figures I have posted, those materials have distinct elastoplastic properties, yet they not only yield same loading-unloading force-displacement curves for one indenter angle, but also for any indenter angle between 63.14 and 80 degrees (thus, your IJSS paper which uses indenter angles 60 and 70.3 degrees, may not distinguish these mystical materials well; moreover, we could derive mystical materials that are valid between 60 and 70.3 degrees).

In almost all previous works which have claimed that the dual indenter method could uniquely recover material elastoplastic properties with 2 different conical indenters, only few materials were tested in the reverse analysis -- that's insufficient given the broad material space. Indeed, the existence range of mystical materials (which severely challenges the dual indenter method) is not very large and it would be very hard to find them by "trial and error" -- that's why these mystical materials were not found in the past.

We proposed an explicit method to derive them; we have also systematically discussed the existing condition (map) of the mystical materials, the general properties of mystical materials, as well as new methods to distinguish them by using two alternative techniques previously established by our group; more details will be available after we submit our paper this week.

I am pleased to see our work has attracted attention and I welcome discussion on this topic. Thank you.

Xi