Early View publication on wileyonlinelibrary.com
(issue and page numbers not yet assigned;
citable using Digital Object Identifier – DOI)
ZAMM · Z. Angew. Math. Mech., 1 – 12 (2012) / DOI 10.1002/zamm.201100128
On the ability of nanoindentation to measure anisotropic elastic
constants of pyrolytic carbon
T. S. Gross1,∗ , N. Timoshchuk1 , I. I. Tsukrov1 , R. Piat2 , and B. Reznik3
1
Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824, USA
Institute of Engineering Mechanics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
3
Institute for Chemical Technology and Polymer Chemistry, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
2
Received 22 September 2011, revised 7 January 2012, accepted 25 April 2012
Published online 11 June 2012
Key words Nanoindentation, pyrolytic carbon, nanobuckling, elastic constants.
We used cube corner, Berkovich, cono-spherical, and Vickers indenters to measure the indentation modulus of highly
oriented bulk pyrolytic carbon both normal to and parallel to the plane of elastic isotropy. We compared the measurements
with elastic constants previously obtained using strain gage methods and ultrasound phase spectroscopy. While no method
currently exists to extract the anisotropic elastic constants from the indentation modulus, the method of Delafargue and
Ulm (DU) [17] was used to predict the indentation modulus from the known elastic constants. The indentation modulus
normal to the plane of isotropy was ∼ 20% higher than the DU predictions and was independent of indenter type. The
indentation modulus parallel to the plane of isotropy was 2–3 times lower than DU predictions, was depth dependent, and
was lowest for the cube corner indenter. We attribute the low indentation modulus to nanobuckling of the graphite-like
planes and the indenter type dependence to the impact of differing degree of transverse stress on the tendency toward
nanobuckling.
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Introduction
This work is part of a larger effort to create a multiscale model of carbon/carbon composites. The fidelity of the model would
be enhanced with anisotropic elastic property measurements of the pyrolytic carbon (PyroC) matrix with micrometer spatial
resolution because the properties are expected to be a function of distance from the fibers. For instance, bulk PyroC can
exhibit strong microstructural anisotropy [1, 2] and there are several reports documenting elastic anisotropy [3–5]. The
degree of microstructural anisotropy is governed by precursor gas and manufacturing conditions (temperature, pressure,
residence time, etc). In composites, Reznik [6] and Chen [7] showed that the microstructural texture of the PyroC coating
on a carbon fiber varies strongly with radial distance from the fiber and is also a function of processing conditions.
We have used nanoindentation to measure the spatial variation of the indentation modulus in carbon-carbon composites.
Figure 1 shows that the nanoindentation modulus is a function of radial distance from the fiber. The AFM image on the
right was obtained using the indenter as the probe in a scanning force microscope. The indentations were taken at 3 µm
intervals with a cube corner indenter at 2 mN force and the modulus values are in GPa.
While nanoindentation should be an ideal technique to determine the elastic constants with high spatial resolution,
pyrolytic carbon exhibits nearly reversible, elastic nanoindentation response and does not yield in a similar manner to the
materials for which the nanoindentation analysis methods were developed. To ensure localized measurements of elastic
response were representative of the properties, we performed experiments to correlate the nano and micro indentation
response with the anisotropic elastic response of a bulk pyrolytic carbon that had been previously characterized using
ultrasound phase spectroscopy [3] and standard strain gage methods [4]. We review other work on indentation of pyrolytic
carbon and analytical models of indentation in anisotropic materials in the following paragraphs.
Brief review of literature on nanoindentation of carbon
Diss et al. [8] performed micro and nano indentation tests on a variety of carbons including HOPG, glassy carbon, and pyrocarbon films as well as carbon/carbon composites. They found that most carbons exhibit a purely elastic microindentation
response at low loads (< 100 mN) and initiate cracks at higher loads. Since the response was purely elastic, they utilized
Sneddon’s equation for elastic loading of a purely isotropic material with a conical indenter to estimate the effective elastic
modulus from the loading-unloading curves. The effective (reduced) elastic modulus derived from the indentation data, was
compared with the results found in the literature.
∗
Corresponding author
E-mail: [email protected]
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
2
T. S. Gross et al.: Measure anisotropic elastic constants of pyrolytic carbon
27.8
28.5
28.7
12.8
12.9
15.2
11.8
13.5
12.3
13.2
5µm
50µm
(a)
17.7
19.2
(b)
Fig. 1 (online colour at: www.zamm-journal.org) Spatial variation of the indentation modulus in carbon-carbon composites. (a) Polarized light image of a polished cross section of a carbon-carbon composite with pyrolytic carbon matrix consisting of low- (first grey ring)
and high-textured (second darker ring) zones. Dashed box indicates the region for the AFM scan obtained with a cube corner indenter.
(b) AFM image indicating the indentation modulus (in GPa) as a function of radial position. Note the decay of the modulus values within
the high-textured matrix zone.
Similarly, Marx et al. [9] used nanoindentation to measure the near-surface mechanical properties of a carbon-carbon
composite and two chemical vapor deposited (CVD) carbons one being isotropic (ICVD) and the other being anisotropic
(ACVD). They also noted that the indentations were almost completely elastic and showed that the indentations fit well
to the elastic formula for a conical indenter. However, the effects of anisotropy were not considered as only one value of
elastic modulus was given for anisotropic CVD carbon. They used standard methods for isotropic materials to estimate
the modulus [10]. The difference in modulus’ values of ICVD (30.69 ± 0.97 GPa) and ACVD (29.92 ± 0.94 GPa) is not
statistically significant.
Hofmann et al. [11] used nanoindentation methods to measure the hardness and modulus of pyrolytic carbon layers on a
graphite rod formed in a fluidized bed reactor at 1350◦ C from a mixture of propane and nitrogen at atmospheric pressure.
The indentations were made normal to the growth direction on metallographically prepared sections. They observed a
smooth deposit that had moduli of 24.4 ± 0.8 GPa and a rough deposit that had a modulus of 8.8 ± 1.9 GPa. The hardness
values were 3.6 ± 0.2 GPa for the smooth deposits and 0.9 ± 0.2 GPa for the rough deposits. They noted that the indents
were almost completely reversible. They used standard methods for isotropic materials to estimate the modulus [10].
López-Honorato et al. [12] used nanoindentation to measure the modulus and hardness on polished cross sections of
PyroC deposited on 0.5 mm alumina spheres in a fluidized bed in a CVD coater at atmospheric pressure with acetylene
and a mixture of acetylene/propylene in the range of 1250–1450◦C. All measurements were transverse to the growth
direction of the coating. The structure observed by TEM consisted of well ordered, small graphite-like domains at low
deposition temperatures. The domain structure progressed to longer, less regular domains at the higher temperatures with
nearly amorphous character in between the long domains. They used standard elastic plastic methods for isotropic materials
to estimate the modulus [10]. They found that the nanoindentation-derived reduced modulus decreased from 34.5 GPa to
13 GPa with increasing deposition temperature and the hardness decreased from 4.8–1.5 GPa with increasing temperature.
Ozcan et al. [13] studied the structure and nanoindentation behavior of two different C/C composites. One was fabricated
by chemical vapor phase infiltration (CVI) and the other was fabricated by a combination of charred resin with CVI. Optical
activity and high resolution transmission electron microscopy were used to show that the CVI matrix was highly oriented
and the charred resin/CVI matrix was nearly isotropic. They reported indentation modulus values between 12–35 GPa
depending on orientation of the fiber and matrix and material type and heat treatment. Their loading-unloading curves
appeared to be primarily reversible with very small amounts of residual indentation considering the 300–400 nm depth of
indentation. They attribute the discrepancy between indentation modulus and macroscopic tensile modulus of the fiber to
kinking of the graphite-like planes.
Richter et al. [20] performed careful nanoindentation studies using a cube corner indenter on diamond, highly oriented
pyrolytic graphite (HOPG), and fullerene films. Like others, they found that indentations were almost completely reversible.
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.zamm-journal.org
ZAMM · Z. Angew. Math. Mech. (2012) / www.zamm-journal.org
3
They estimated the modulus of HOPG to be 10.5 GPa normal to the plane of the rings. They also observed load drops when
the planes cracked.
Guellali et al. [14] performed three-point bending tests to measure the elastic modulus for high-temperature for the same
PyroC used in this work subjected to different heat treatments. The calculated in-plane elastic modulus was found to be on
the order of 17.8–18.5 GPa both for the as-deposited PyroC and for the samples that were heat treated up to 2500◦ C/2 hr.
There was an approximately 10% increase in Young’s modulus for the sample which was heat treated up to 2900◦C/2 hr
(∼ 21 GPa). Guellali et al. attribute this phenomenon to the fact that heat treatment leads to further graphitization (i.e.
PyroC become more highly-textured) and a higher fraction of σ-bonding in the plane of the graphene-like sheets.
Gebert et al. [3] characterized the microstructure of PyroC (from the same sample studied in this paper) and performed
ultrasound phase spectroscopy to measure the elastic constants. They used polarized light microscopy and high-resolution
transmission electron microscopy to show that this particular pyrolytic carbon material is highly-textured and does not
exhibit a textural gradient in the direction of deposition. The independent elastic constants were estimated from the velocity
of longitudinal and shear waves. While some of the stiffness components C12 and C13 have relatively large error (17% and
46% respectively), the engineering elastic modulus values were estimated and exhibited transverse isotropy with 12.8 GPa
parallel to the growth direction (out-of-plane modulus), and 27.1 GPa transverse to the growth direction (in-plane modulus).
We recently used strain gage methods to measure the engineering modulus and Poisson’s ratios on the material used
in this study in both tension and compression. The in-plane elastic modulus exhibited tension-compression anisotropy
(30.2 GPa in tension and 18.8 GPa in compression) and the out-of-plane modulus was 5.2 GPa in compression [4].
Analytical estimates of the effect of elastic anisotropy on indentation modulus
For homogeneous and isotropic materials, microindenter and nanoindenter load-displacement curves are analyzed to determine the effective elastic modulus and hardness using some variation of the method described by Oliver and Pharr [10].
They estimate the modulus using the slope of the unloading curve at maximum displacement, S = dP/dh, where P is the
applied load and h is the displacement, and a contact area function. The contact depth hc , is the actual depth of penetration
into the sample not including elastic deformation of the sample or the indenter system
Pmax
(1)
S
with the suggested shape parameter value of ε of 0.72 for conical indenters and 0.75 for paraboloids of revolution (spherical
indenters) [2]. The contact depth is used in the tip area function, A(hc ) to estimate the reduced Young’s modulus Er (also
called the indentation modulus, M).
√
S π
Er ≡ M = p
.
(2)
2 A(hc )
hc = hmax − ε
The tip area function, A(hc ), is an estimate of an area of contact between a tip and a substrate as a function of depth. The
use of the area function in the Oliver and Pharr method was validated by comparing scanning electron microscope images
of plastic indentations with the fit of the area function on fused quartz and several other materials that exhibit elasto-plastic
deformation that is homogeneous on the scale of the indentation. There is no currently accepted method to estimate the
area of contact for a purely elastic contact. Since standard methods assume plastic deformation, the A(hc ) estimated using
the software included with most commercial nanoindenters is likely to be an overestimate the area of contact for pyrolytic
carbon which exhibits mostly reversible indents.
Several investigators have proposed extensions of the Oliver-Pharr method to determine the elastic properties of elastically anisotropic materials. Vlassak and Nix [15] estimated the contact stiffness for a flat triangular punch and a parabola
of revolution punch acting on an elastic half space of a material with elastic anisotropy. The material was assumed to
have three fold and four fold symmetry and they used an assumed pressure distribution on the indentation surface. They
measured the nanoindentation response for metals with low and high elastic anisotropy ratios and obtained reasonable
agreement with their predictions. All of the materials they tested exhibited plastic deformation. Swadener and Pharr [16]
extended this analysis to predict indentation response for elastically deforming materials with arbitrary symmetry. They
demonstrated good agreement with experimental measurements. Their analysis also assumes a pressure distribution under
the indenter.
Delafargue and Ulm [17] proposed explicit approximations for the indentation modulus of a conical indenter in transversely isotropic materials based on an assumed pressure distribution. The coordinate system correlation to the material
in this study is shown in Fig. 2. The indentation modulus M1 (and M2 ) in the directions orthogonal to the solid’s axis of
symmetry was simplified to the following form
p
(3)
M1 = M12 M13 ,
www.zamm-journal.org
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
4
T. S. Gross et al.: Measure anisotropic elastic constants of pyrolytic carbon
Fig. 2 Coordinate system correlation to deposition direction. The plane of isotropy is the 1–2
plane.
where M13 is the indentation modulus obtained by indenting in an isotropic solid and given by
M13 =
c211 − c212
c11
and M12 is the indentation modulus in the direction along the solid’s axis of symmetry and given by
r
c11
.
M12 = M3
c33
(4)
(5)
The constants Cij represent the components of the stiffness tensor of a transversely isotropic material. The indentation
modulus M3 in the direction along the solid’s axis of symmetry simplifies to the following expression
s
−1
1
2
c231 − c213
M3 = 2
+
.
(6)
c11
c44
c31 + c13
The authors showed that Eqs. (1)–(4) also exhibit good correlation with experimental results. The explicit approximation
greatly enhances the utility of their solution.
2
Experimental
2.1 Material
The highly textured pyrolytic carbon material discussed in this paper (referred to as PyroC) was provided by Schunk
Kohlenstofftecknik GmbH, Heuchelheim, Germany. It was manufactured in the form of 100 mm diameter disk and thickness of 1.5 mm. This is the same material characterized in Gebert et al. [3] and Guellali et al. [14].
Small straight sections of the material were cut from the PyroC disk using a Buehler Isomet low speed saw and glued
to 15 mm diameter steel AFM discs. The samples were mounted so that the surface was either parallel to or transverse to
the growth plane. The growth plane is the plane of isotropy and indentations normal to the growth plane are also normal to
the plane of isotropy. Indentations on the transverse sections are defined as parallel to the plane of isotropy. The prepared
samples were ground using SiC 1200 grit paper and the final polish was performed with 0.02 µm colloidal silica solution.
The average surface roughness was estimated to be 10 nm for sections where the surface is transverse to the plane of
isotropy and 40 nm for sections that are parallel to the plane of isotropy.
2.2 Experiments
The indentation tests on PyroC were performed on the micro and nano length scales. The microindentation measurements
R
H100C dynamic microindenter. The instrument was equipped with the optical microwere performed on a FisherScope
scope and the indenter head. The optical microscope allowed the user to observe the surface of a specimen to navigate to
the desired location of indentation. The load range of the instrument is 0.4–1000 mN with the load resolution of 0.2 µN. A
Vickers indenter tip (square pyramid with apex angle of 138◦) was used throughout all experiments. The tip was calibrated
on a sample of steel with known material properties. The penetration depth ranged from 2–8 µm.
R
R
nanoindenter. The Hysitron
three-plate capacitive
The nanoindentation was performed using a Hysitron TriboScope
R
force/displacement transducer was attached to Veeco Dimension SPM. The range of the applied loads of this instrument
is 25 µN to 15 mN with the load resolution of approximately 1 nN and stated displacement resolution of about 0.0004 nm
although the displacement drift is on the order of 0.1 nm/sec. Depending on the applied load and the indenter tip, the
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.zamm-journal.org
ZAMM · Z. Angew. Math. Mech. (2012) / www.zamm-journal.org
5
displacement range of nanoindentation experiments was 50–1000 nm with most indents being in 100–350 nm range. A
cube corner, Berkovich, and cono-spherical indenter (60◦ included angle, 3 µm radius) were used to perform the indents.
Each tip was calibrated on a fused quartz substrate over the expected depth of indentation.
We estimated the indentation modulus using the Oliver-Pharr method as implemented by Hysitron’s Triboscope software
version 8.3 and using finite element methods described in the following section. There is no currently accepted method to
estimate the area of contact for a purely elastic contact. Since standard methods assume plastic deformation, the A(hc )
estimated using the software included with most commercial nanoindenters is likely to be an overestimate the area of
contact.
Diss et al. [8] noted that the Oliver-Pharr method might not apply to PyroC since it exhibits elastic response. They used
the Sneddon solution for a conical indenter in an elastically isotropic material to estimate the modulus from the maximum
load and displacement. We analyzed our data by fitting the loading curves to the Sneddon solution for a conical and a
spherical indenter for the Berkovich and cube corner tip and for the spherical indenter for the cono-spherical tip.
E
2 · tan(θ) 2
·
· h (conical indenter)
1 − ν2
π
√
4E
P (h) =
· R · h3/2 (spherical indenter)
2
3(1 − ν )
P (h) =
(7)
(8)
Finite element methods
We used a simple two-dimensional axisymmetric finite element model shown in Fig. 3 to estimate the indentation response
both normal and parallel to the plane of isotropy for all three indenters using the elastic properties measured by strain
gages. We realize that this simplified model does not accurately represent the behavior of indentations parallel to the plane
of isotropy because the in-plane properties for that orientation are not isotropic. However, the results do yield insight into
the material behavior. The method we describe does not assume a pressure distribution and should therefore more accurately
capture the effects of in-plane modulus on the indentation modulus than the methods described in [15, 16].
The substrate was modeled as a deformable body with either isotropic or transversely isotropic properties using MSC
Marc Mentat software. The size of the substrate was chosen to be ten times larger than the maximum depth of an indent.
Substrate
Indenter
(not to scale)
depth
Axis of
symmetry
Bottom
fixed
Fig. 3 (online colour at: www.zammjournal.org) 2D axisymmetric FE model of
indentation.
width
www.zamm-journal.org
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
6
T. S. Gross et al.: Measure anisotropic elastic constants of pyrolytic carbon
The finite element mesh was created using quadrilateral elements with finer elements near the location of an indent and
coarse elements farther away.
The indenter was modeled as a rigid body with a specified radius of curvature and a constant velocity of penetration.
The tip radius was set at 200 nm for the Berkovich and cube corner indenters and 3000 nm for the cono-spherical with
appropriate included angles for each indenter. Boundary conditions were defined so that the bottom of the substrate was
fixed in all directions.
The indentation modulus was determined from the actual area of contact and the stiffness determined from the derivative
of integrated force in the indentation direction. We validated the model by comparing to the Hertz solution for a 150 nm
spherical indenter in an isotropic elastic material and the indentation modulus using Eq. (2) was within 1% of the predicted
value.
3
Results and discussion
3.1 Microindentation
Five tests were performed normal and five test parallel to the plane of isotropy of PyroC with the Vickers indenter. The
applied loads were 150, 300, 600, 800, and 1000 mN. The penetration depth ranged from 2 µm to 8 µm. All the tests
produced consistent data as shown in Figs. 4a and 4b.
800
800
Load (mN)
1000
Load (mN)
1000
600
400
200
0
0
(a)
600
400
200
2
4
6
Displacement (Pm)
8
(b)
0
0
2
4
6
Displacement (Pm)
8
Fig. 4 Bulk PyroC, (a) normal to the plane of isotropy, (b) in the plane of isotropy.
The loading and unloading curves exhibited nearly elastic behavior with some hysteresis. The hysteresis was observed
to be larger for deeper indents than for shallower indents. The reduced Young’s moduli in the directions normal and parallel
to the plane of isotropy were measured to be 10.0 ± 0.32 GPa and 9.97 ± 0.2 GPa, respectively. These estimates were made
using the Oliver-Pharr method.
3.2 Nanoindentation
Nanoindentation of PyroC was performed using a cube corner indenter with a ∼ 200 nm radius of curvature, a Berkovich
indenter with a ∼ 200 nm radius of curvature, and a cono-spherical tip with 3 µm radius of curvature where the radius
R
of curvature was determined from Imagemet
software’s (Version 3.3.2) implementation of the blind tip reconstruction
method on AFM images obtained by each indenter. The radius of curvature for the cono-spherical tip was also measured to
be 3 µm by SEM.
Discontinuities were observed in the load-penetration curves only for indentations normal to the plane of isotropy (1-2
plane) for both the cube corner and conical indenters at high loads (∼ 400 µN for the cube corner tip and ∼ 5000 µN for
the cono-spherical indenter). They were not observed for Berkovich indenters at the force limit of the indentation system,
15 mN. These discontinuities have been observed by Barsoum et al. [19] for microindentation and have been attributed to
the propagation of kink bands. We did not utilize data from curves with discontinuities in the indentation modulus estimates.
Figure 5 shows typical load penetration curves for all three indenters at approximately 200 nm depth of penetration
normal to and parallel to the plane of isotropy. The material exhibited mostly elastic behavior with distinct loading and
unloading curves. There was small residual penetration in some cases as was also observed by Diss et al. [8] and by Ozcan
et al. [13]. The cube corner tip penetrated the most for a given load and the cono-spherical indenter penetrated the least. We
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.zamm-journal.org
ZAMM · Z. Angew. Math. Mech. (2012) / www.zamm-journal.org
7
1200
1200
Berk
800
Cone
600
Cube
400
200
0
0
Berk
1000
Load (PN)
Load (PN)
1000
800
Cone
600
400
Cube
200
50
100
150
200
0
0
250
Displacement (nm)
50
100
150
200
250
Displacement (nm)
Fig. 5 Force-displacement curves for indentations normal to the plane of isotropy (left) and parallel to the plane of
isotropy (right).
did not observe any residual indentation on the surface in topography images obtained in AFM mode using the indenter to
image the surface immediately after indentation or by observation with a scanning electron microscope.
We fit the load-displacement (P-h) curves for 200–250 nm deep indentations to a power law relationship and found that
the exponent varied between 1.2 and 1.65. This is consistent with the equation for a spherical indenter. We used Eq. (8) to
solve for E using a Poisson’s ratio of 0.25 and a tip radius of 200 nm for the Berkovich and cube corner indenters and a
tip radius of 3000 nm for the cono-spherical indenter. The results are summarized in Table 1. There does not appear to be
a consistent trend and the results are not consistent with the macroscopic results. We did not investigate this approach any
further as it is not a commonly used approach and it is difficult to calibrate.
Table 1
E3 (GPa)
E1 = E2 (GPa)
Cube corner
7.9
4.3
Berkovich
20.6
17.3
Cono-spherical
4.2
3.4
Cube
Cone
Berk
20
Indentation modulus (GPa)
Indentation modulus (GPa)
Figure 6 shows the indentation modulus for all three indenters normal and parallel to the plane of isotropy. The Delafargue and Ulm predictions of indentation modulus using the elastic constants determined from the ultrasonic phase spectroscopy [3] and strain gage methods [4] are indicated as dashed lines. We used the Saint-Venant’s principle to approximate
the shear modulus based on in-plane and out-of-plane Young’s moduli.
15
10
DU
5
0
0
Normaltothe
planeofisotropy
200
400
hmax (nm)
600
20
Cube
Cone
Berk
DU
15
10
5
0
0
Transversetothe
planeofisotropy
200
400
600
hmax (nm)
Fig. 6 Indentation modulus normal to the plane of isotropy (left) and parallel to the plane of isotropy (right). The dashed
lines labeled DU indicates the values predicted from the predictions of Delafargue and Ulm [17] using the elastic properties
measured using strain gages.
www.zamm-journal.org
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
8
T. S. Gross et al.: Measure anisotropic elastic constants of pyrolytic carbon
The values of indentation modulus normal to the plane of isotropy are roughly the same (∼ 12–13 GPa) for all three
indenters and, within the scatter of the data, do not depend on the depth of the indentation. We attribute the scatter to the
∼ 40 nm surface roughness. The predictions of Delafargue and Ulm (9.6 GPa) are less than the observations. As mentioned
earlier, we believe that the standard methods for estimating the area function overestimate the actual area of contact so the
indentation modulus values may be underestimates. This would result in a larger difference between the predictions and the
measurements.
The values of the indentation modulus parallel to the plane of isotropy exhibit much less scatter but are strongly dependent on the indenter type and well below the predictions of Delafargue and Ulm. The values also depend upon indentation
depth. The indentation modulus values for the cube corner and Berkovich indenters stabilize for indentation depths between 200-300 nm but the values for the cono-spherical indenter continue to decrease with increasing indentation depth.
This depth dependence is much greater than can be attributed to errors in area function determination and tip irregularities
(typically less than 100 nm) and can therefore be considered to be representative of depth dependent elastic properties. We
attribute this depth dependence to the increasing activation of a nanobuckling mechanism. Both Diss et al. [8] and by Ozcan
et al. [13] also point out this possibility. Ozcan et al. suggest that relative in-plane motion results in permanent deformation
and use this to explain the small residual indentation.
Riter [18] proposed that the hexane rings in graphite can buckle into boat modes and puckering modes. We suggest
that longer wavelength modes that consist of aligned boat modes are also feasible. Loidl et al. [5] used a 100 nm diameter
synchrotron radiation beam to measure spatial dependence of the graphite plane spacing and the deviation of the graphite
planes from the axis of PAN fiber loaded in bending. They inferred the strain from the graphite plane spacing and the
deviation from the axis from the increase in azimuthal width of the peak associated with the graphite plane spacing. They
showed that the neutral axis of the fiber was shifted and used that information to estimate a compressive and tensile elastic
modulus. They found that the width of the graphite plane reflection was much greater in compression which is consistent
with nanobuckling.
We proposed that the low values of indentation modulus when indenting in the plane of isotropy can be attributed to
activation of an elastic nanobuckling deformation mechanism when the material is compressively deformed in the nominal
plane of the graphite-like sheets. The difference between the indentation modulus obtained by different indenters can be
attributed to two factors. First, the magnitude of the compressive stress in the indentation direction is greater for sharp
indenters. This should enhance the tendency toward nanobuckling. Second, a transverse tensile stress exists when the
transverse elastic modulus is less than the modulus in the direction of indentation. The magnitude of this transverse tensile
stress is greater for sharper indenters and will therefore increase the contribution of the nanobuckling mechanism to the
indentation modulus. The following finite element results using the simplified model compare the stress fields for indents
normal to and in the plane of isotropy using the elastic properties in Table 2.
Table 2 Elastic properties used for the finite element model of indentation normal to and parallel to the plane of isotropy.
E3
E1 = E2
G12
G32 = G31
ν32
ν21
ν13
Normal to plane of isotropy
5.20 GPa
18.8 GPa
7.64 GPa
3.00 GPa
0.27
0.23
0.99
Parallel to plane of isotropy
18.8 GPa
5.20 GPa
2.11 GPa
10.0 GPa
0.99
0.23
0.27
Figure 7 and Fig. 8 show normal and transverse stress contours for a 116 nm deep indent for all three indenters and for
the case where the lateral modulus is stiffer (Fig. 7) and softer (Fig. 8) than the modulus in the direction of indentation.
The second two rows of the figures show the transverse compressive stress and transverse tensile stress separately to clearly
illustrate the region experiencing tensile stress.
First, the extent of the compressive stress field in the indentation direction is much greater for the soft transverse indentations. Second, there is a significant difference in the extent of the stress field between the different indenters for the
soft transverse direction but very little difference in extent for the stiff transverse indentations. The maximum compressive
stress is greatest for the cube indenter although the values of the stresses are quite high. Clearly these stresses will not be
this high if the elastic nanobuckling mechanism is activated.
The other important difference is the presence of a transverse tensile stress component for the soft transverse indentations which is not observed for the stiff transverse orientation. These would tend to enhance the operation of an elastic
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.zamm-journal.org
ZAMM · Z. Angew. Math. Mech. (2012) / www.zamm-journal.org
Cube Corner
2-5
Axial
0.5-2
Transverse tensile
Transverse compressive
0-0.5
1-6
tensile 0.5-1
0-0.5
Berkovich
9
Cono-spherical
2-3.5
0.5-2
0.5-2
0-0.5
1-4
0.5-1
tensile
0-0.5
1-1.5
tensile 0.5-1
0-0.5
0-0.5
1.4µm
0-.05
compressive
0-.05
compressive
0-.05
compressive
Units are in GPa
Fig. 7 (online colour at: www.zamm-journal.org) Stress contours for three different indenters where the indentation is
normal to the plane of isotropy. The first row shows the stresses in the direction of indentation. The second two rows show
the compressive and tensile stress fields transverse to the direction of indentation. Note that only the surface experiences
tensile stress when the transverse modulus is higher than the modulus in the direction of indentation.
nanobuckling mechanism. The magnitude of the tensile stress field is greatest for the cube corner indenter and lies immediately under the indenter. For the cube corner indenter, the region of transverse tensile stress is nearly coincident with the
region of high compressive stress. However for the Berkovich indenter, the region of transverse tensile stress and the region
of compressive stress in the direction of indentation are not coincident. The cono-spherical indenter exhibits intermediate
spatial overlap of the region of high compressive stress in the indentation direction with the region of high transverse tensile
stress.
We propose that the combination of high compressive stress causing a greater degree of nanobuckling and the high
transverse tensile stress enhancing that nanobuckling is the reason that the cube corner indenter has the lowest indentation
modulus. The cono-spherical indenter has a lower indentation modulus than the Berkovich indenter because of the volume
of the stressed zone is much larger.
We attribute the depth dependence of the indentation modulus to the fact that the ratio of the transverse force to the
normal force is continually changing as the spherical portion of the indenter penetrates into the sample. It remains constant
after the indenter penetrates deeper than the spherical portion. Figure 9 below shows the angle of the force of the indenter
measured from the indentation direction as a function of depth. The Berkovich indenter saturates fairly quickly because it
does not take long for the indenter to penetrate past the spherical region at the tip. We were unable to run the simulations for
the cube corner indenter to the point where the angle saturates but the behavior should be similar to the Berkovich indenter
but at a greater depth. The angle of the force continually increases with increasing depth for the cono-spherical indenter
thereby increasing the impact of the soft transverse elastic modulus.
www.zamm-journal.org
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
10
T. S. Gross et al.: Measure anisotropic elastic constants of pyrolytic carbon
Cube Corner
Axial
3.5-18
2-3.5
0.5-2
Berkovich
Cono-spherical
3.5-13
2-3.5
3.5-4
2-3.5
0.5-2
0-0.5
0-0.5
0.5-2
Transverse tensile
Transverse compressive
0-0.5
1-5
0.5-1
tensile
1-3
0.5-1
1-1.2
0.5-1
tensile
tensile
0-0.5
0-0.5
0-0.5
1.4µm
.15-.6
.1-.15
.05-.1
.05.05-.1 .1
0-.05
compressive
compressive
.05-.1
compressive
Units are in GPa
Fig. 8 (online colour at: www.zamm-journal.org) Stress contours for three different indenters where the indentation is
in the plane of isotropy. The first row shows the stresses in the direction of indentation. The second two rows show the
compressive and tensile stress fields transverse to the direction of indentation. Note that the transverse tensile stress is very
high in magnitude for the cube corner indenter and directly under the indenter where the compressive stress is highest.
The region experiencing compressive stress under the indenter is separated from the region experiencing transverse tensile
stress.
4
Summary
The goal of this effort was to determine if it is possible to use nanoindentation methods to provide micrometer scale
anisotropic elastic property variations to those interested in modeling the composite performance of pyrolytic carboncoated carbon fiber composites. We proposed to resolve the question by comparing nanoindentation measurements to bulk
properties on a pyrolytic carbon sample for which the anisotropic elastic properties were characterized by two different
methods.
We observed that the indentations were almost completely reversible which was also observed by others [5, 8, 13].
We used the model predictions of Delafargue and Ulm [17] to estimate the expected indentation modulus from the elastic
constants determined by bulk measurements using two different methods [3,4]. While the measurements and estimates were
in reasonable agreement for the indentations normal to the plane of isotropy, the measured indentation modulus values for
indentations parallel to the plane of isotropy were factors of 2–3 below the predicted values. Furthermore, the indentation
modulus values were different for indenters with different included angles and tip radius of curvature and depended on
depth of indentation.
We proposed that the disparity between the measurements and predictions can be explained by the activation of an elastic nanobuckling mechanism of the graphene-like structures when subjected to stress in the plane of the sheets rather than
failure of the method to accurately detect the variation in properties. Furthermore, we utilized simplified 2D axisymmetric
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.zamm-journal.org
NormalForce
Vectors
ResultantForce
Vector
ɽ
Force angle relative to the indentation axis (deg)
IndentationAxis
ZAMM · Z. Angew. Math. Mech. (2012) / www.zamm-journal.org
11
35
Cube norm
Cube trans
30
25
Berk norm
Berk trans
20
Cone trans
15
Cone norm
10
5
0
0
100
200
300
400
500
600
Displacement (nm)
Fig. 9 Plot of angle of resolved force of the indenter tip on the sample for cube corner, Berkovich, and 3 µm radius of
curvature cono-spherical indenter. Note that the angle of the force is greatest for the cube corner indenter which would
tend to enhance the tendency toward nanobuckling under the tip due to the decrease in lateral constraint.
finite element models to show that the magnitude of compressive stresses were greatest for the cube corner indenter that exhibited the lowest indentation modulus and that the extent of the compressive stress zone was greatest for the cono-spherical
indenter which exhibited the next lowest indentation modulus. The models also predicted the existence of transverse tensile
stresses which increases the driving force for nanobuckling when the transverse elastic modulus was lower than the modulus in the direction of indentation. However, those stresses will change when the nanobuckling mechanism is activated and
can only be viewed as a driving force.
While the Delafargue and Ulm model is moderately successful estimating the indentation modulus from known elastic
properties normal to the plane of isotropy, it is not immediately obvious how to extract elastic properties from indentation
modulus measurements.
Acknowledgements The authors gratefully acknowledge the financial support of the National Science Foundation (NSF) and German
Science Foundation (DFG) through the grant DMR-0806906 Materials World Network: Multi-Scale Study of Chemical Vapor Infiltrated
Carbon/Carbon Composites. Romana Piat also acknowledges the support of DFG through the project PI 785/1-1 (Heisenberg fellowship).
References
[1] V. De Pauw, B. Reznik, A. Kalhöfer, D. Gerthsen, Z. J. Hu, and K. J. Hüttinger, Texture and nanostructure of pyrocarbon layers
deposited on planar substrates in a hot-walled reactor, Carbon 41, 71–77 (2003).
[2] X. Bourrat, B. Trouvat, G. Limousin, G. Vignoles, and F. Doux, Pyrocarbon anisotropy as measured by electron diffraction and
polarized light, J. Mater. Res. 15, 92–101 (2000).
[3] J.-M. Gebert, B. Reznik, R. Piat, B. Viering, K. Weidenmann, A. Wanner, and O. Deutschmann, Elastic constants of high-texture
pyrolytic carbon measured by ultrasound phase spectroscopy, Carbon 48, 3647–3650 (2010).
[4] T. Gross, K. Nguyen, M. Buck, N. Timoshchuk, I. Tsukrov, B. Reznik, R. Piat, and T. Böhlke, Tension-compression anisotropy
of in-plane elastic modulus for pyrolytic carbon, Carbon 49, 2145–2147 (2011).
[5] D. Loidl, O. Paris, M. Burghammer, C. Riekel, and H. Peterlik, Direct observation of nanocrystallite buckling in carbon fibers
under bending load, Phys. Rev. Lett. 95, 225501/1-4 (2005).
[6] B. Reznik, D. Gerthsen, W. Zhang, and K. J. Hüttinger, Texture changes in the matrix of an infiltrated carbon fiber felt studied by
polarized light microscopy and selected area electron diffraction, Carbon 41, 376–380 (2003).
[7] T. Chen, B. Reznik, D. Gerthsen, W. Zhang, and K. J. Hüttinger, Microscopical study of carbon/carbon composites obtained by
chemical vapor infiltration of 0◦ /0◦ /90◦ /90◦ carbon fiber preforms, Carbon 43, 3088–3098 (2005).
[8] P. Diss, J. Lamon, L. Carpentier, J. L. Loubet, and P. Kapsa, Sharp indentation behavior of carbon/carbon composites and varieties
of carbon, Carbon 40, 2567–2579 (2002).
[9] D. Marx and L. Riester, Mechanical properties of carbon-carbon composite components determined using nanoindentation, Carbon 37, 1679–1688 (1999).
www.zamm-journal.org
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
12
T. S. Gross et al.: Measure anisotropic elastic constants of pyrolytic carbon
[10] W. Oliver and G. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing
indentation experiments, J. Mater. Res. 7, 1564–1583 (1992).
[11] G. Hofmann, M. Wiedenmeier, M. Freund, A. Beavan, J. Hay, and G. Pharr, An investigation of the relationship between position
within coater and pyrolytic carbon characteristics using nanoindentation, Carbon 38, 645-653 (2000).
[12] E. López-Honorato, P. J. Meadows, P. Xiao, and T. J. Abram, Structure and mechanical properties of pyrolytic carbon produced
by fluidized bed vapor deposition, Nucl. Eng. Design 238, 3121–3128 (2008).
[13] S. Ozcan, J. Tezcan, and P. Filip, Microstructure and elastic properties of individual components of C/C composites, Carbon 47,
3403–3414 (2009).
[14] M. Guellali, R. Oberacker, and M. J. Hoffmann, Influence of heat treatment on microstructure and properties of highly textured
pyrocarbons deposited during CVD at about 1100◦ C and above 2000◦ C, Compos. Sci. Tech. 68, 1122–1130 (2008).
[15] J. Vlassak and W. D. Nix, Measuring the elastic properties of anisotropic materials by means of indentation experiments, J. Mech.
Phys. Solids 42, 1223–1245 (1994).
[16] J. Swadener and G. Pharr, Indentation of elastically anisotropic half-spaces by cones and parabola of revolution, Philos. Mag. A,
Phys. Condens. Matter Struct. Defects Mech. Prop. 81, 447–466 (2001).
[17] A. Delafargue and F. J. Ulm, Explicit approximations of the indentation modulus of elastically orthotropic solids for conical
indenters, Int. J. Solids. Struct. 41, 7351–7360 (2004).
[18] J. Riter, Interpretation of diamond and graphite compressibility data using molecular force constants, J. Chem. Phys. 52, 5008–
5010 (1970).
[19] M. W. Barsoum, A. Murugaiah, S. R. Kalindindi, T. Zhen, and Y. Gogotsi, Kink bands, nonlinear elasticity and nanoindentations
in graphite, Carbon 42, 1435–1445 (2004).
[20] A. Richter, R. Ries, R. Smith, M. Henkel, and B. Wolf, Nanoindentation of diamond, graphite, and fullerene films, Diam. Relat.
Mater. 9, 170–184 (2000).
c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.zamm-journal.org