3D studies of indentation by combined X-ray tomography and digital
volume correlation
Mahmoud Mostafavi
1,#
, Yelena Vertyagina
1
, Christina Reinhard
2
,
Robert Bradley
3
, Xia Jiang
1
, Marina Galano
1
and James Marrow
1,#
1
Department of Materials and
#
Oxford Martin School, University of Oxford, UK
2
Diamond Light Source, Harwell Science and Innovation Campus, Oxfordshire, UK
3
Manchester X-ray Imaging Centre, University of Manchester, UK
james.marrow@materials.ox.ac.uk
Keywords: Computed Tomography, Hardness, Indentation, Digital Image Correlation
Abstract Hardness testing obtains material properties from small specimens via measurement of
load-displacement response to an imposed indentation; it is a surface characterisation technique so,
except in optically transparent materials, there is no direct observation of the assumed damage and
deformation processes within the material. Three-dimensional digital image correlation (‘digital
volume correlation’) is applied to study deformation beneath indentations, mapping the relative
displacements between high-resolution synchrotron X-ray computed tomographs (0.9 µm voxel
size). Two classes of material are examined: ductile aluminium-silicon carbide composite (Al-SiC)
and brittle alumina (Al
2
O
3
). The measured displacements for Hertzian indentation in Al-SiC are in
good agreement with an elastic-plastic finite element simulation. In alumina, radial cracking is
observed beneath a Vickers indentation and the crack opening displacements are measured, in situ
under load, for the first time. Potential applications are discussed of this characterization technique,
which does not require resolution of microstructural features.
Introduction
Hardness testing has long been used to interrogate materials to understand their deformation and
fracture. The dimensions of an indentation (e.g. projected area and depth, sometimes with surface
profile analysis of pile-up/sink-in) are used to infer the processes of deformation that have occurred
underneath indenters of various size and shape [1-6]. With appropriate assumptions or
understanding of deformation processes, such as strain-hardening in ductile materials, hardness tests
on small samples can be used to evaluate the effects of subtle changes in microstructure on the
mechanical properties of engineering components (for instance, the effects of fast neutron
irradiation and thermal ageing on structural steels in structural nuclear steels [7, 8]). Fracture
behaviour can also be studied; there is a range of indentation methods for fracture toughness
measurements in brittle materials, although these are complicated by the various types of cracking
that develop depending on material properties [9, 10].
Hardness testing is a surface characterisation technique and, except in transparent materials, it
provides no direct observation of the assumed damage and deformation processes. Digital image
correlation is a highly precise displacement measurement method with many applications [11];
some of the authors have applied it to measure surface deformation in studies of cracking processes
[12], for instance. In appropriate microstructures, X-ray tomography can be combined with three-
dimensional digital image correlation (digital volume correlation or DVC) [13] to measure the
deformation within materials [14-16]. Such data may be used to validate models for deformation
and fracture behaviour. Sufficient contrast may be achieved from microstructural heterogeneities of
Authors' copy of a paper presented at Seventh International Conference on
MATERIALS STRUCTURE & MICROMECHANICS OF FRACTURE (Brno, 1-3 July 2013, http://
msmf.fme.vutbr.cz/msmf7/). The manuscript has been accepted publication in Key Engineering Materials.
the order of the voxel
1
size. The precision of displacement resolution increases with the multi-voxel
interrogation window size, enabling DVC to measure sub-voxel displacements [17].
DVC is applied here to high-resolution synchrotron X-ray computed tomography observations,
obtained in situ under load, to study indentation behaviour in two types of materials; aluminium-
silicon carbide composite (Al-SiC: 6061 Al alloy reinforced with 15wt% SiC particles of 500 nm
average size, extruded at 450 °C) and commercial purity alumina (Al
2
O
3
). The high brilliance and
penetration of synchrotron X-rays, compared to laboratory sources, allows sequential observations
of representative volumes of engineering materials [18-20]. The measured displacements in Al-SiC
are compared with predictive models for elastic-plastic indentation in a ductile material, whilst in
Al
2
O
3
the displacement field is used to detect cracking beneath the indentation in a brittle material
and to extract the crack opening displacements.
Experiment
The specimens, an Al-SiC cylinder (3 mm in diameter, 4 mm high) and an Al
2
O
3
rectangular prism
(3 × 3 mm square, 4 mm high), were indented using a 5 mm radius Al
2
O
3
ball (i.e. Hertzian) and a
‘Vickers’ square pyramidal diamond respectively, by a loading stage that had been modified to
accommodate the indenters. High resolution computed synchrotron X-ray tomography was
performed at the Diamond Light Source, Joint Engineering, Environmental and Processing
beamline (I12 – JEEP), using radiographic projections obtained at an X-ray beam energy of 53 keV
with a nominal voxel size of 0.9 µm. The exposure time was 2 seconds per radiograph for both
materials, with projections at increments of 0.04 degrees over 180° rotation. Reference tomographs
were recorded under a small pre-load (nominally 10 N), applied to reduce rigid body movement
between successive scans. The maximum applied load on the Al-SiC sample was 500 N
(indentation depth 47 µm, measured from radiographs), which dropped to 480 N during the
tomography scan, and that on the alumina was 360 N (indentation depth 35 µm), which dropped to
330 N during the tomography scan.
A combined Fourier-wavelet ring artefact removal algorithm [21] was used to suppress ring
artefacts in the reconstructed tomographs. These arise from instrument features such as defective
pixels in the scintillator; if not adequately supressed they can significantly increase noise in the
DVC analysis as they do not displace with the material. Example xy-plane slices of the tomographs
are shown in Figure 1; these are effectively maps of the differences in X-ray attenuation in the
sample due to its microstructure. The fine porosity in the Al
2
O
3
is well resolved, but the contrast in
the Al-SiC composite is low as both phases have very similar X-ray attenuation: Al (density 2.7 g
cm
-3
) has an attenuation coefficient (including coherent scattering) of 0.334 cm
2
g
-1
at 53 keV,
compared to 0.331 for SiC (density 3.1 g cm
-3
). The calculated absorption contrast difference is
only 13% [22]; this could be enhanced if energies below the absorption edge in SiC around 2 keV
were used, but this is not practical. However, there is a degree of phase contrast from
microstructure interfaces due to the sample-camera distance and coherent X-rays [23]. Artefacts
remain in the tomographs; low contrast bands that are a characteristic of the wavelet algorithm are
observed in the Al-SiC composite and radial streaks emerge from the region below the indentation
in Al
2
O
3
. These may be edge artefacts [24], from the strong difference in attenuation between
Al
2
O
3
and open regions of radial cracks that are aligned with the projection axis of some
radiographs. Cracking, expected at the applied indentation load, is not observable in the
tomographic image.
1
A voxel is the three-dimensional equivalent of a pixel.
a)
b)
Figure 1: Examples of reconstructed tomography slices in the xy plane; a) Al
2
O
3
and b) Al-SiC composite
(inset top right: scanning electron microscope image of Al-SiC microstructure).
a)#
b) #
Figure 2: Visualisations of maximum principal strain a) Al
2
O
3
and b) Al-SiC composite. (the sample field of
view is approximately 3 mm in each case)
Analysis
The DVC analyses were carried out using the Davis Strain Master 8.1 software [25], correlating
a loaded 3D dataset (i.e. tomograph) against its reference to map the relative displacements. Each
dataset measured 4016×4008×2672 voxels (160 GB as 32 bit data), cropped to 3504×3504×2000
voxels and converted to 8 bit data (reduced to 24 GB) for Al-SiC and 2600×2600×2000 voxels
(13GB at 8 bit) for Al
2
O
3
. Vertical (z) rigid body movements between datasets were corrected by
visual matching of image slices in an xy plane close to the indented surface. For Al
2
O
3
, the
following image correlation parameters were judged to be optimal; 256×256×256 voxel
interrogation window, 50% overlap and 2 passes, followed by 128×128×128 interrogation window,
75% overlap and 3 passes. For Al-SiC it was necessary to use a 256×256×256 interrogation
window, 50% overlap, 2 passes and followed by 64×64×64 interrogation window, 50% overlap and
2 passes. Reducing the final interrogation window size increases the displacement map spatial
resolution, though excessive noise arises with smaller window size. Overlapping interrogation
windows may improve the displacement map spatial resolution in smoothly changing fields,
allowing the use of larger interrogation windows to reduce measurement noise. Increasing the
number of passes may also reduce noise, with a diminishing effect with increasing passes. The
deformation may be visualised as a nominal strain, obtained from the displacement gradient;
examples are given in Figure 2. These show that deformation has occurred under the indentor in the
Al-SiC and Al
2
O
3
, but the images are not very suitable for quantitative analysis.
Examples of the displacement fields in the indented Al-SiC and Al
2
O
3
are presented in Figure 3,
illustrating the relative vertical displacements (V
z
) in the xy plane. White patches are regions where
displacement vectors were removed due to poor correlation. Due to higher levels of noise in the
low contrast Al-SiC data, a relatively stringent criterion was applied, removing displacement
vectors with poor correlation (correlation coefficient < 0.6); less stringent filtering was applied to
the Al
2
O
3
. Consequently Figure 3a shows more white patches than Figure 3b. The image artefacts
described earlier may be responsible for some regions of poor correlation.
As presented, both datasets are affected by a small rigid body rotation between the reference and
loaded tomographs. This dominates the relative displacement field and thus the deformation arising
from the indentation is difficult to observe. The DVC analysis software can be used to correct for
this by first calculating the relative rotations between tomographs using a large interrogation
window, which increases precision and reduces local effects, then adjusting one tomography dataset
by interpolation to remove this rotation. Correlation with smaller interrogation windows is then
applied to the interpolated dataset. The authors have implemented a more computationally efficient
method, based on [26], which considers the relative displacement field from the original
tomography data and thus does not require interpolation. Applying this method, the Al-SiC rotated
by -0.26°, 0.58° and 3.6° about the x, y and z-axes respectively when loaded (clockwise rotation
about the axis is positive), while the Al
2
O
3
rotated by -0.54°, -0.33° and 0.34° about the x, y and z-
axes respectively. These rotations are attributed to compliance of the loading rig; no evidence was
found for flexure of the sample. The corrective rigid body rotation gives a displacement field
aligned with the original sample orientation. The vertical displacements below the indentations,
after correction for the rigid body rotations, are shown in Figure 4; the effects of the indentations
are now well visualised. The compressive strains in the xz plane below the indentions, along with
the displacement vectors are illustrated in Figure 5. The displacement measurement noise,
estimated as the standard error after rotation correction from regions of negligible deformation that
are remote from the indentations, was 1.8 µm for the Al-SiC and 0.3 µm for the Al
2
O
3
.
a)#
b)#
Figure 3: Original vertical (V
z
) displacement field in a) Al-SiC in the xy plane at 213 voxels (0.192 mm)
below the indenter b) Al
2
O
3
in the xy plane at 96 voxels (0.087 mm) below the indenter.
a)#
b)#
Figure 4: Vertical (V
z
) displacement field in xy plane below the indenter after rigid body rotation correction
in a) Al-SiC at 0.192 mm (213 voxels), b) Al
2
O
3
0.087 mm (96 voxels).
a)
b)
c)
d)
Figure 5: Displacement vectors and compressive strain distribution, ε
zz
, below the indenter in the xz plane at
the centre of the indented region. Data before rotation correction (left) and after (right) are compared. Data
are shown for Al-SiC (a & b) and Al
2
O
3
(c & d). (Every fifth vector is shown, magnified for clarity) – The
white patches show the areas where the displacement vectors could not be reliably found. An example of the
V
z
displacement field in the xy plane is inset in c) at the position of the dashed line. Note: z=0 corresponds to
the centre of an image correlation window adjacent to the original surface; 64 voxels (58 µm) and 32 voxels
(29 µm) below the non-indented surface for Al
2
O
3
and Al-SiC respectively.
A 3D finite element simulation of the Hertzian indentation was carried out (ABAQUS v 6.10 [27]),
with 1200 rigid elements to simulate the indenter and 18000 solid eight-node brick elements for the
Al-SiC. Finite sliding, frictionless contact was modelled and nonlinear elastic (Ramberg-Osgood)
material properties defined: elastic modulus 102 GPa; Poisson ratio 0.27; yield stress (0.2% proof
stress) 230 MPa and hardening exponent 9. These values were obtained by curve-fitting to tensile
data [28], and agree with published results for a similar material [29]. Nonlinear geometry was
