EVALUATION OF DAMAGE AND PLASTIC PROPERTIES BY
MICROINDENTATION AND INVERSE METHOD
Bruno Guelorgeta, Manuel Françoisa, Jian Lub
Université de technologie de Troyes, ICD-CNRS FRE 2848, LASMIS, 12 rue Marie Curie,
B.P. 2060, 10010 Troyes Cedex, France
b
Department of Mechanical Engineering, Hong Kong Polytechnic University,
Hung Hom Kowloon, Hong Kong
a
ABSTRACT
Berkovich microindentation was performed on a longitudinal section of a semi-hard copper sheet, which was broken in a
tensile machine. The shorter the distance between the measured point and the fracture, the higher the hardness, due to the
work hardening. Young's modulus decreases by 60 % within a distance of a few hundreds of micrometers from the fracture
and is constant beyond. The corresponding evaluated damage is 0.6. Yield stress and strain hardening coefficient were
evaluated, in the localized neck, through an inverse method applied to spherical tip indentation tests. When the distance to the
fracture is higher than 30 µm, the yield stress is growing up when approaching the fracture, because of strain hardening. When
the distance is lower than 30 µm, damage becomes predominant and yield stress decreases. No clear tendency can be drawn
with strain hardening coefficient evolution.
Introduction
Indentation has been widely used for characterizing material properties such as Young's modulus, hardness or yield stress.
Tabor and Stilwell proposed a method for determining hardness through spherical and conical indentation [1-3]. The first
authors concerned with determining the Young's modulus through indentation were Bulychev and al. [4]. Later on, Doerner
and Nix [5] proposed a method which was improved by Oliver and Pharr [6].
For Lemaitre and Chaboche [7], damage D can be evaluated through Young’s modulus variation. In case of damage, the
measured Young's modulus E’ is lower than E, the nominal one:
E ' = (1 − D ) E
(1)
Indentation tests have already been used to identify damage law of a tensile test specimen [8-9], but it was based on
microhardness measurement associated with a phenomenological model. The present authors propose to evaluate damage
directly through Young’s modulus variation as measured by microindentation [10] performed with a Berkovich tip.
Extracting the plastic properties of a material from indentation test is a hot question. This needs the use of an inverse method.
Microindentation with a spherical tip, coupled with an inverse method, was used to determine the yield stress and strain
hardening coefficient evolutions in the localized neck of the cross section of a semi-hard copper sheet tensile specimen.
Specimen, methodology and results
The specimen was cut in a 0.8 mm thick semi-hard copper sheet. It was first tested by a tensile machine till fracture.
Localization first appeared in the centre of the specimen and a crack was initiated there and propagated to the edges. The
specimen was then cut and carefully polished along the longitudinal section, in order to avoid surface hardening. The indenter
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used was a NanoIndenter XP , from MTS .
Indents were performed with a Berkovich tip, on the longitudinal section, at different distances from the fracture (Fig. 1).
Attention was paid to avoid border effects (free surface effects). The indentation depth was 500 nm, so the width of the imprint
was 3.5 µm. Normally, the minimum distance between two indentation tests should be three times larger than the imprint
width, i.e. in the present study 10,5 µm. That is the reason why no point was closer than 10 µm from the border (from the free
surface).
Variations of Young's modulus E and hardness H have been measured versus the distance d to the fracture. Results are
reported in Fig. 2. No error bar was estimated at distances equal to 10 and 20 µm from the fracture, due to the lack of space to
perform enough tests. For comparison, data obtained (five tests average) in undamaged zone (called "E bulk" and "H bulk")
are plotted in Fig. 2 at an arbitrary distance to the fracture equal to 80 000 µm. The NanoIndenter XP was equipped with the
Continuous Stiffness Measurement option (CSM) [6,11], which allows to measure continuously Young's modulus and
hardness during the loading process. The indents, the modulus and hardness values plotted in Fig. 2 are the averages of
these values during the loading process between 300 and 450 nm. Damage was determined using Eq. 1, results are plotted in
Fig. 3.
Figure 1. 3D view of the sample.
Figure 2. Young’s modulus and hardness versus distance to the fracture. Error bars are equal to +/- one standard deviation,
estimated on 3 to 6 indents (no error bar for d = 10 and 20 µm).
One inverse method had to be used to extract the plastic properties of the material through indentation. Having experimentally
tested five of them [12], one was found more reliable, the Cao and Lu’s sphere method [13]. Indents were performed with a
spherical tip (radius = 8.1 µm), at different distances from the fracture. The indentation depth was 810 nm. Using the inverse
method from Cao and Lu [13], yield stress σy and strain hardening coefficient n were determined at several distances from the
fracture. Results are plotted on Fig. 4 and 5. For comparison, data obtained by indentation in the undamaged zone (called
"Bulk") and data obtained by tensile test (called "Ref.") are plotted at an arbitrary distance to the fracture equal to 80 000 µm.
This method consists in extracting a set of ten couples of representative strain and representative stress (εr, σr) for ten different
depths hi/R = 0.01, 0.02… 0.1, where R is the tip radius. According to the method, one should use two of these couples to fit a
behaviour law. It has been found from experimental results that it is better to use more than two points, up to seven for depths
hi/R = 0.04, 0.05… 0.1, and not to take into account depths corresponding to hi/R = 0.01, 0.02 and 0.03 (i.e depths equal to 81,
162 and 243 nm). At such depths, small defects may not be negligible anymore. These defects could be (i) in the homogeneity
Figure 3. Damage versus distance to the fracture.
Figure 4. Yield stress versus distance to the fracture. Error bars are equal to +/- one standard deviation, estimated on 2 to 6
indents.
Figure 5. Strain hardening coefficient versus distance to the fracture. Error bars are equal to +/- one standard deviation,
estimated on 2 to 6 indents
of the specimen (small surface hardening due to polishing…), (ii) of the tip shape, (iii) due to size effect or (iv) due to
roughness.
Discussion
The two curves of Fig. 2 have inverse behaviours. When approaching the fracture, the closer the measuring point, the higher
the hardness. Such kind of results have already been published [10,14]. It is due to the higher dislocations density in the
specimen, because of the plastic strain, especially in the neighboring of the fracture.
The variation of the modulus versus the distance to the fracture is different: between 150 µm and 20 mm, the modulus is more
or less constant and equal to the bulk value; for distances lower than approximately 150 µm, the measured modulus becomes
lower and lower, which is interpreted as an increasing damage, up to 60 % (Fig. 3). This value is very high and could be
oversestimated, but is quite difficult to check by adding other tests, because of the lack of space. Aluminium specimens were
tested and gave similar tendencies. Copper specimens tested with hydraulic bulge test also gave similar results. A discussion
may arise about the fact that the loaded volume is mainly submitted to a compression load while damage usually has
significant effect in tension, however it is beyond the scope of the present paper.
It can be seen in Fig. 4 that when the distance to the fracture is higher than 30 µm, the yield stress is growing up when
approaching the fracture. This can be attributed to strain hardening: the closer to the fracture, the higher the plastic strain, so
the higher the yield stress. But when the distance is lower than 30 µm, damage becomes predominant and yield stress
decreases. In Fig. 5, no clear tendency can be drawn with strain hardening coefficient evolution. As already outlined elsewhere
[12], for the moment this inverse method is much more appropriated to catch yield stress than strain hardening coefficient.
Conclusion
Berkovich microindentation tests were done on a longitudinal section of a broken tensile specimen. Variations of Young's
modulus and hardness were measured in the localized neck of the specimen. Due to the increasing dislocations density in the
neighboring of the fracture, the hardness increases when approaching the fracture. On the contrary, due to the increasing
damage in the few last hundreds of micrometers, the modulus decreases when approaching the fracture. The variation of
measured modulus was used as a way of determination of local damage. Moreover, using an inverse method, yield stress
evolution in the neck was successfully determined experimentally: the closer the measuring point, the higher the yield stress,
until damage become predominant (i.e. for distance to the fracture lower than 30 µm). Strain hardening coefficient results are
not as reliable as those of yield stress.
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