COMPUTATIONAL MODELING OF INDENTATION - Nanoindentation with Biological Applications - page 163

α =° ), Alkorta et al . 76 have searched and
found a set of distinct materials with almost identical P − curves.
Consequently, more information must be sought to obtain the
constitutive relationship from the sharp indentation Ph

For Berkovich indenter (

70.3

curves. It is
widely assumed that the dual (or plural) indenter method provides such
needed independent information. 1,49,50,53-56,69,73,76,77,79 In principle (based
on the dimensional analysis, Eq. 6-4 ), unique loading and unloading
curves can be obtained for a given set of material properties ( E , ν

−

σ , n )
and two (or more) sharp indenters. That is, although special sets of
materials (with distinct elastoplastic properties) have been found to yield
indistinguishable Ph

,

y

curves when a particular α is used, these
materials are unlikely to again yield nearly identical Ph

−

curves when
another α is chosen. It is believed by some researchers that with the
additional independent equations established from other α s, there are
sufficient equations needed to uniquely solve the material elastoplastic
properties from the reverse analysis. 53,73
Such a “well-established” theorem was recently challenged by Chen
et al . 65 who have established step-by-step algorithms to explicitly derive
sets of mystical materials exist, which, despite of their different
elastoplastic properties, yield indistinguishable indentation load-depth
curves for not only one particular indenter angle, but also
indistinguishable P − curves other selections of α ; in some cases,
mystical materials cannot be distinguished effectively when α is varied
from 60 o to 80 o (covering most indenter angles used in practice), and the
resulting shape factors of the Ph

−

curves are within several percent of
difference. When any power-law material is given, its mystical siblings
can be derived explicitly (for a fixed Poisson's ratio) and when the
Poisson's ratio is allowed to vary, more mystical groups can be derived
with larger material property differences. It is further shown 65 that due to
their similarities to plural indentation, shallow spherical indentation also
cannot effectively distinguish these mystical materials.
The uniqueness problem (or finding of the mystical materials) is
analogous to a multi-variable problem ( Fig. 6-6 ), where the indentation
shape factors (the general “z-axis”) are related with the material
elastoplastic properties (the general “x-axis”) and indenter angle ranges
(the general “y-axis”) through a multi-dimensional “surface”. On such

−