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ARTICLE IN PRESS
Journal of Biomechanics 40 (2007) 252–264
www.elsevier.com/locate/jbiomech
www.JBiomech.com
Anisotropic Poisson’s ratio and compression modulus of cortical
bone determined by speckle interferometry
R. Shahara,, P. Zaslanskyb, M. Barakb, A.A. Friesemc, J.D. Curreyd, S. Weinerb
a
Koret School of Veterinary Medicine, The Hebrew University of Jerusalem, P.O. Box 12, 76100 Rehorot, Israel
b
Department of Structural Biology, Weizmann Institute of Science, Israel
c
Department of Physics of Complex Systems, Weizmann Institute of Science, Israel
d
Department of Biology, University of York, UK
Accepted 16 January 2006
Abstract
Young’s modulus and Poisson’s ratios of 6 mm-sized cubes of equine cortical bone were measured in compression using a micromechanical loading device. Surface displacements were determined by electronic speckle pattern-correlation interferometry. This
method allows for non-destructive testing of very small samples in water. Analyses of standard materials showed that the method is
accurate and precise for determining both Young’s modulus and Poisson’s ratio. Material properties were determined concurrently
in three orthogonal anatomic directions (axial, radial and transverse). Young’s modulus values were found to be anisotropic and
consistent with values of equine cortical bone reported in the literature. Poisson’s ratios were also found to be anisotropic, but lower
than those previously reported. Poisson’s ratios for the radial–transverse and transverse–radial directions were 0:15 0:02, for the
axial–transverse and axial–radial directions 0:19 0:04, and for the transverse–axial and radial–axial direction 0:09 0:02
(mean7SD). Cubes located only millimetres apart had significantly different elastic properties, showing that significant spatial
variation occurs in equine cortical bone.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Bone; Mechanical properties; Interferometry; Poisson’s ratio; ESPI
1. Introduction
Cortical bone is anisotropic, with the elastic modulus
in the axial direction being significantly higher than in
the transverse and radial directions (Reilly and Burstein,
1975, Taylor et al., 2002, Dong and Guo, 2004, Iyo
et al., 2004). In fact the mechanical properties of bone
are affected by many aspects of its complex structure
(Weiner and Wagner, 1998), and in particular by the
mineral content (Currey, 2002).
Few studies have attempted to correlate structure
with function at the micron to millimeter meso-scale
Corresponding author. Tel.: +972 9 7433968;
fax: +972 9 7488994.
E-mail address: [email protected] (R. Shahar).
0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jbiomech.2006.01.021
(Zysset et al., 1999, Liu et al., 1999, Turner et al., 1999,
2000; Hengsberger et al., 2003; Enstrom, et al., 2001).
Since many structural differences between (and even
within) various types of cortical bone are found at that
length scale, the study of the mechanical properties
using millimeter-sized samples is of great importance.
However testing such samples is complicated, posing
many technical challenges. Various experimental methods have been reported for determining the elastic
properties of cortical bone, with sample sizes ranging
from several centimetres (Reilly et al., 1974, Reilly and
Burstein, 1975) to single osteons with dimensions of
hundreds of micrometres (Ascenzi and Bonucci, 1968).
Most methods were based on loading relatively bulky
bone samples in material testing machines, and recording load–displacement curves for loads such as tension,
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R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264
compression, torsion and bending. In tension and
compression experiments, the slope of the stress–strain
curve within the elastic region was used to estimate the
Young’s modulus of the bone. Most tests are performed
by applying a controlled deformation rate, and often
progress continuously until failure. Such experiments
are usually used to also provide information regarding
the yield point (the point at which the behaviour of the
sample ceases to be linearly elastic), load and work to
failure and ultimate strain. However, since these tests
are destructive, often only one estimate of modulus can
be obtained from each sample, thus experimental
precision cannot be determined. Furthermore, since
the use of large and bulky samples is required, local
micro-scale variations are ignored.
Several non-destructive methods are widely used for
obtaining mechanical properties of small samples.
Micro- and nano-indentation have been used to measure
the hardness and elastic constants of cortical bone at the
micro-structural level (Weiner et al., 1997, Rho et al.,
1997, Zysset et al., 1999, Silva et al., 2004, Hengsberger
et al., 2003; Bensamoun et al., 2004). These methods
allow estimation of hardness and Young’s modulus
from contact stiffness between the indenter tip and the
sample. They require a highly polished sample surface,
and the calculation assumes knowledge of the Poisson’s
ratios of the sample. When used to measure the elastic
modulus of anisotropic materials such as bone, the
modulus derived from the method is an average of the
anisotropic constants biased towards the modulus of the
direction of testing (Rho et al., 1997). Another approach
is based on the measurement of the speed at which
sound travels through bone (Yoon and Katz, 1976a,b,
Ashman et al., 1987, Rho et al., 1993). Although these
methods are non-destructive, they are also indirect, and
rely on the application of theories of composite
materials to the measurements in order to obtain
estimates of the elastic constants.
Few studies describe experimental determination of
Poisson’s ratios of cortical bone. Reilly and Burstein
(1975) assumed transverse isotropy of fibrolamellar
bone, and used extensometers to measure strains in
two orthogonal directions concurrently. They found
Poisson’s ratio values which ranged between 0.29 and
0.63. Ashman et al. (1984) reported on the use of an
ultrasonic continuous wave technique, and found
Poisson’s ratio values which ranged between 0.27 and
0.45. Pithioux et al. (2002) also used an ultrasonic
method, and found Poisson’s ratios between 0.12 and
0.29. Despite this wide range of reported values
(0.12–0.63), many studies, especially finite element
analyses, often use values in the much narrower range
of 0.28–0.33.
Optical metrology techniques allow non-contact
measurement of displacements on surfaces of samples
subjected to static or dynamic mechanical loading.
253
Electronic speckle pattern-correlation interferometry
(ESPI) (Jones and Wykes, 1989, Rastogi, 2001) has
recently been used to determine sub-micron surface
displacements on the surface of millimetre-sized tooth
dentin samples loaded elastically in compression (Zaslansky et al., 2005). Displacements are directly determined from variations of laser light reflected from
samples immersed in water. Using this technique, it is
possible to perform quantitative analysis of strain on
compressed samples of mineralized biological tissues
such as bone and dentin, by loading them in a highprecision micro-mechanical loading device. Such measurements can be performed without damaging the
sample and hence anisotropic Young’s moduli and
Poisson’s ratios can be determined from multiple
measurements of each sample.
We report on measurements performed using a
commercial ESPI system (Q300—Ettemeyer, Ulm,
Germany) which has been combined with a custombuilt loading device allowing non-destructive compression tests of small cubes of cortical bone in three
orthogonal directions. Our set-up allows the measurement of in-plane and out-of-plane deformation fields on
surfaces of very small samples. Due to the high
sensitivity of our system, experiments can be conducted
non-destructively by repeatedly loading the sample
within its elastic region. Measurements may be performed on wet samples, thus satisfying a basic requirement for the study of biological specimens.
Measurements by this technique require load to be
applied in small increments, since large deformations
cause optical decorrelation that renders the measurements invalid.
While the ESPI technique allows for the determination of the elastic constants from the traditional
stress–strain curve, as is common in classical mechanical
testing methods, it also allows for the use of a
compliance-based method (‘Estimated best E’ from
Zaslansky et al., 2005). The compliance method is based
on the principle that each incremental loading step
within the elastic region can be considered as an
independent experiment in which the strain field created
within the sample and the incremental load causing it
are determined. Thus, the compliance of the sample is
determined repetitively and non-destructively.
An interesting feature of the ESPI method is its
inherent ability to concurrently measure strain along two
orthogonal directions on the sample surface. This allows
the derivation of Poisson’s ratios from the same
experimental data on exactly the same sample under
identical loading conditions. For each incremental
loading step, in addition to the axial strain of the
sample, the lateral strain is also determined. The
negative ratio of the latter to the former can be used
to estimate Poisson’s ratio. Since these loading steps can
be repeated at the discretion of the investigator, the
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254
statistical error associated with mechanical tests such as
these can be reduced.
This paper describes the results of optically determined measurements of strain of mechanically loaded
millimetre-sized samples of equine cortical bone. These
measurements were used to determine the anisotropic
variations in Young’s moduli and Poisson’s ratios.
2. Materials and methods
Six cubic bone samples of equine cortical bone were
obtained from the cadavers of two horses. They were
loaded in compression, and their surface displacements
were determined using ESPI following procedures
similar to those described in Zaslansky et al. (2005).
These data were used to determine three Young’s
moduli and three Poisson’s ratios related to the three
orthotropic directions (axial, radial and transverse).
2.1. Bone sample preparation
The right third metatarsal bones (MT3) were obtained
from a 4-year old-male Quarter horse and a 6-year-old
female Arabian horse. The cause of death of both horses
was unrelated to the musculoskeletal system. All
external soft tissue was meticulously removed, and 2cm thick slices were cut from the mid-diaphysis of each
bone using a hand saw (Fig. 1). Transverse sections of
200–300 mm thickness were then cut from each bone
slice, using a low-speed water-cooled diamond saw
(South Bay Technology Inc.) (Fig. 1). These sections
were ground and polished (Buehler Minimet Polisher)
and examined by reflected light microscopy. The area of
the cranial mid-diaphysis in the bones of both horses
consisted almost entirely of secondary remodelled
osteonal bone, with small areas of primary bone. Each
bone slice was further cut and used as a source of three 2
2 2 mm cubes. The cubes were cut such that their
faces were aligned with the anatomical axes of the bone:
proximo-distal (axial), antero-posterior (radial) and
cranio-caudal (transverse) orientations (Fig. 2). Care
was taken to note and mark the orientation of the cubes,
so that distinct axial, radial and transverse faces could
be identified. The cubes were then stored for 2-7 days on
water-saturated cotton swabs at 4 1C until testing.
2.2. Experimental set-up
The experimental set-up, shown schematically in Fig.
3, consisted of a mechanical tension–compression
device, a water chamber, an optical ESPI head capable
of determining surface displacements and a computer
which controlled the various components of the system
and analysed the data. Aside from the computer, the
entire set-up was enclosed in an acoustically insulated
box mounted on top of a floating optical table.
2.3. Mechanical testing device
The mechanical tension–compression device used for
loading the bone samples was custom-designed with
stainless-steel parts (SS 316) (see Fig. 4). The device
consists of a sealed chamber which includes a high-grade
glass window (BK-7 l/10 grade), allowing the use of
aqueous solutions as a medium in which sample testing
took place. The loading apparatus in the test chamber
consisted of an axial motion DC motor (PI M-235.5
DG, Physik Instrumente, GmbH, Germany) capable of
displacing a metal shaft in small sub-micron steps while
applying substantial force (4100 N). The metal shaft
acted as a movable upper anvil, pressing against samples
mounted on the stationary lower anvil which was
attached rigidly to the testing chamber base. The
movable metal shaft was fitted with an in-chamber
2 mm
2 mm
Radial
2 mm
Axial
Transverse
Fig. 2. The preparation of three 2 2 2 mm cubes from the slice
shown in Fig. 1, with the orthogonal anatomic orientations preserved.
Fig. 1. Removal of a 2-cm thick slice from the mid-diaphyseal area of the third metatarsal, and a thin transverse section.
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DC axial
motor
Sample
Laser
Water
Chamber
Glass
window
Trigger
Micro-step (PI) motor controller
and computer
ESPI System
Fig. 3. Schematic of the entire experimental system.
immersable load cell (AL311BN-6I, Sensotec, Honeywell, USA) capable of measuring loads up to 120 N in
tension or compression. Force measurements were
collected and stored on a computer using an A/D
converter (Omega DAQP-308 PCMCIA 16-bit Analogue I/O) and further analysed by custom-written
software (National Instruments Labview v 7.0 and
Matlab v 6.0).
2.4. Sample mounting
A thin lining layer of soft composite material (Z250,
3 M ESPE, St. Paul, MN USA) was placed over the
lower, stationary anvil in the test chamber. The face of
the bone cube which corresponded to the direction being
tested was then placed upon it. The cube was gently
pushed down onto the composite such that it was
mounted in the appropriate orientation (inset in Fig. 4).
Manual adjustment facilitated the alignment of the
cube. The composite material was then polymerized
using a hand-held light-cure device (LITEX 682,
Dentamerica CA, USA). A small amount of composite
was placed on the edge of the upper (travelling) anvil
which was then lowered until the composite was brought
into contact with the entire upper surface of the cube.
While the cube was held in the desired line of
compression (axial, radial or transverse) between the
two anvils, the top composite was cured. This mounting
procedure ensured that an intimate and stiff contact was
established between both anvils and the bone cube
through composite load–transfer layers. With each
sample mounted and fixed in place, the chamber was
sealed and filled with physiologic saline. Small pieces of
bone were added to the saline solution which was
refrigerated at 4 1C, for at least 24 h prior to initiation
of testing, in order to ensure mineral saturation of
the solution. The same solution was used for all
experiments.
2.5. Mechanical compression testing
A small compression preload of about 10 N was
applied at the beginning of each experiment, and the
sample was allowed to reach near-equilibrium force
readings while undergoing some initial stress-relaxation.
Once a stable reading of force was reached (approximately within 200–300 s), a series of 15 compression
increments was initiated. Each loading step consisted of
a small incremental load, which was produced by
forcing the upper anvil to move 2 mm downwards. Just
before and immediately after each such loading step,
force was determined by averaging 20,000 readings
(taken at 80 kHz), and surface deformation was
determined by laser speckle intensity measurements
(ESPI, Section 2.6). Each experiment was repeated 15
times, so as to collect a large and robust data set. Then,
at the end of 15 experiments, the test chamber was
emptied and the cube was dismounted. The cube was
then remounted in another (orthogonal) orientation,
and 15 load-deformation experiments repeated.
We were thus able to collect data from 15 repeated
experiments of force-deformation measurements for
each of the three orthogonal sample axes for each cube
(therefore each cube underwent 45 experiments). Every
experiment contained 15 incremental compression steps.
These measurements provided the database required to
determine the 3 Young’s moduli and 3 of the 6 Poisson’s
ratios of the cubes of cortical bone reported here. For 2
of the cubes we were also able to calculate the three
remaining Poisson’s ratios, (see Section 5).
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R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264
Fig. 4. Mechanical tension–compression device and loading chamber, with a bone cube in place. The first stage of sample mounting, showing the
bottom anvil, with a layer of composite and a bone cube.
2.6. Optical determination of surface displacements
Surface displacements of the compressed cube along
and across the loading direction were used to calculate
the longitudinal and lateral strains caused by each
incremental load. These displacements were determined
using two orthogonally aligned horizontal (X-axis) and
vertical (Y-axis) speckle pattern-correlation interferometers (see detailed description in Zaslansky et al., 2005).
Briefly, the surface of the sample is illuminated by laser
light from two symmetrical opposite angles relative to
the normal to the surface. The laser illumination creates
a speckled light interference field on the surface. By
imaging the speckle patterns onto a CCD detector array,
variations to the interference intensity patterns can be
detected and captured by a computer. Any sub-micron
displacement of sub-sections on the surface will affect
the optical path of the light propagating towards the
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CCD detector array, changing the detected interference
pattern. If the phase of the light of one of the beams is
modulated by means of a phase shifter, the phase
relations over the entire CCD field can be determined.
The interference patterns can be shown to correspond to
light phase differences on small surface sub-sections,
which are determined after computerized image filtering
and phase unwrapping (Huntley, 1998, Rastogi, 2001).
The phase relations were determined for small subareas on bone sample surfaces before and after each
load increment. By substraction, 2D matrices of phase
differences were obtained from the X and Y interferometers, of the form:
MATX ;Y ¼ Djði; jÞX ;Y
(1)
where i and j represent the rows and columns of the
matrix respectively, and corresponds to the phase
difference determined independently for each axis. Each
element in the two matrices corresponds to one sub-area
on the surface of the sample.
A scaling relation of the form:
l Djði; j ÞX ;Y
(2)
4p sin y
was used, in which l is the laser light wavelength
(l ¼ 780 nm in our system) and y is the angle between
the incident laser light and the normal to the sample
surface. Our system allows detection of surface displacements with magnitudes as small as l/30 (Jones and
Wykes, 1989). Thus, the discrete displacement values u
and v were determined, corresponding to the components of displacements of all sub-areas within the area of
interest along X and Y axes, respectively (Rastogi, 2001;
Jones and Wykes, 1989).
u; vði; j Þ ¼
2.7. Determination of surface strains
The uði; j Þ and vði; j Þ displacement matrices were
assumed to represent the displacements of the bulk of
the compressed bone samples. They were assumed to
vary linearly along the rows and columns of the
displacement fields in the Y- and X-directions, respectively. They could therefore be approximated as planes
by least-squares regression analysis that provided
continuous and differentiable displacement fields of
the form:
uði; j Þ ¼ ai þ bj þ c,
(3)
vði; j Þ ¼ di þ ej þ f .
(4)
For infinitesimal strains, the axial normal strain
(along the loading direction), the lateral normal strain
(perpendicular to the loading direction) and the shear
strain could then be determined according to
eii ði; j Þ ¼
@uði; j Þ
¼ a,
@i
(5)
@vði; j Þ
¼ e,
@j
1 @uði; j Þ @vði; j Þ
1
eij ði; j Þ ¼
þ
¼ ðb þ d Þ.
2
@j
@i
2
ejj ði; j Þ ¼
257
(6)
(7)
In our experiments, each detector on the CCD covered a
sub-area of approximately 12 mm 12 mm on the surface
of the sample. Due to the small dimensions of the
samples relative to the distance to the ESPI lens (2 mm
versus 225 mm, respectively), spherical aberrations
could be neglected. Therefore, the distance between
adjacent rows and columns could be assumed to be
12 mm. This number was then used as a scaling factor to
obtain estimates of strain from the slopes of the
regression planes.
3. Data analysis
Stress values were calculated as the ratio between the
force measurements recorded for each load increment
and the initial cross-sectional area of the corresponding
sample. These stresses and the optically determined
strains were used for the determination of Young’s
modulus and Poisson’s ratio along each of the bone
axes. In order for an experiment to be considered valid,
the lateral normal strain had to be larger than the shear
component by at least an order of magnitude. Otherwise
it was repeated entirely, including repositioning of the
sample.
3.1. Young’s modulus (E)
Two complementary methods were employed to
determine Young’s modulus (E) for each of the
orientations of the samples (axial modulus EA, radial
modulus ER, and transverse modulus ET):
3.1.1. Stress– strain curves
The cumulative stress was plotted against the
cumulative strain for the 15 increments in each of 15
repeated experiments, for all three compression orientations. The slopes of the stress–strain curves were then
estimated using linear least-squares regression analysis.
In all experiments a tight linear relationship between
stress and strain was found. A typical stress–strain curve
obtained from a 15-step compression experiment performed in the radial direction is shown in Fig. 5.
3.1.2. Compliance method
Individual compliance estimates were determined for
each of 15 incremental compression loads. Strain
increments for each loading step were determined by
least squares regression analysis of the surface displacement values, as described in Eqs. (3) and (4) above. The
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258
Sample Strain Stress curve, Radial compression
F
16
F
Stress [MPa]
14
12
10
8
Stress[MPa]= 11.6x103 Strain - 0.30
R2 = 0.996
6
4
2
0
0.00
2.00
4.00
6.00 8.00 10.00 12.00 14.00 16.00
6.00(strain(x105))
Middle
area
6 mm
Fig. 5. A typical stress–strain curve obtained from a compression
experiment on cube from the Arabian horse.
compliance estimate was then estimated by dividing the
strain increment by its corresponding stress increment.
The inverse of the median of these 15 values was
considered the estimate of Young’s modulus for the
experiment. In this manner 15 modulus estimates were
obtained for each direction (axial, radial and transverse). For each experiment, whole body rotation and
shear deformation were found to be much smaller than
axial deformation, and therefore could be ignored.
3.2. Poisson’s ratio (v)
In each experiment, longitudinal strains elong and
lateral (orthogonal to the direction of loading) strains
elat were obtained for each of the 15 incremental loads.
The negative ratio between elat and elong was determined,
and Poisson’s ratio was estimated as the median value of
these 15 separate estimates. Since each experiment was
repeated 15 times, the reported result was obtained as
the average of 15 medians. For each cube, 3 different
combinations of lateral/longitudinal load orientation
pairs (out of 6 possible combinations) were performed.
4. Method validation
Three different approaches were used to validate our
methodology. First, the method described above was
used to determine the elastic properties of a small
2 2 2 mm cube made of an isotropic material: Acetal
polyether imide (Ultem-1000s, General Electric Plastics), whose elastic constants are precisely known. Since
our experimental procedure is very different from
industrial ISO and ASTM testing recommendations,
these results allowed us to validate the application of
our novel methodology to the testing of very small
samples in water and without contact.
Second, an experiment was designed to determine the
effect of the non-slip conditions occurring at the anvilsample interface on the determined elastic constants.
Bottom
area
2 mm
(a)
F
(b)
Fig. 6. Schematic of the parallelepiped bone sample, showing the
central and lower areas of interest.
To this end a 2 2 6 mm rectangular parallelepiped
sample of equine cortical bone was compressed along its
long axis to allow calculation of Young’s modulus and
Poisson’s ratio in an area distant from the area of
applied loads (centre of the sample), and in an area close
to loaded edge (see Fig. 6). The results were compared
with those obtained in cube experiments using samples
from the same horse and an identical load orientation.
Third, a finite element model was created to simulate
a cube compression experiment. In this model the lower
nodes of the cube were fixed, and the topmost nodes
(area of load application) were fully restrained except
along the loading direction. The modelled bone material
was assigned orthotropic properties, using Young’s
moduli determined here, and shear moduli reported by
Ashman et al. (1987). The model was analysed with two
different Poisson’s ratios: 0.3 and 0.1. The model was
created with Nastran software, NFW version 2002, and
consisted of 12,800 8-node brick elements, with 13,671
nodes. The composite layer above and below the cube
was modelled as isotropic, with values for Young’s
modulus (11 GPa) and Poisson’s ratio (0.30) obtained
from the manufacturer (3 M). The results were examined
to determine the level at which the lateral constraints
ceased to affect the lateral strain.
5. Results
Table 1 shows the calculated Young’s moduli
and Poisson’s ratio results based on 10 compression
experiments conducted on a 2 2 2 mm cube of
Ultem-1000s. The measured Young’s moduli were very
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259
Table 1
Results of a compression experiment performed on 2 2 2 mm cubes made of Ultems
Ultem-1000s
Young’s modulus ET
(GPa) reported by
the manufacturer
Young’s modulus ET
(GPa) by the
compliance method
(in brackets standard
deviation, n ¼ 15)
Young’s modulus ET
(GPa) by the stressstrain curve method
(in brackets standard
deviation, n ¼ 15)
Poisson’s ratio
reported by the
manufacturer
Poisson’s ratio (in
brackets standard
deviation, n ¼ 15)
3.2
3.42 ( 0.01)
3.43 (0.01)
0.36
0.38 (0.07)
close to the value supplied by the manufacturer
(3.42–3.43 GPa and 3.2 GPa, respectively). The measured and manufacturer-supplied Poisson’s ratios were
also very similar (0.38 and 0.36, respectively). These
results indicate that our methodology is accurate (the
means are in excellent agreement) and precise (small
standard deviations).
Table 2 shows the results of Young’s moduli obtained
in a typical set of 15 experiments compressing a Quarterhorse bone cube in the radial direction. For each
experiment the Young’s modulus was obtained from
the slope of stress–strain curves (S–S) as well as from the
inverse of the median compliance. Also shown are the
mean and standard deviations for these values. Clearly
the results from both methods are very similar, both
with respect to the mean values and their errors.
Table 3 summarizes the results obtained from
compression tests of all cubes of cortical bone used in
this study: 3 from the 4-year-old male Quarter horse,
and 3 from the 6-year-old female Arabian horse. Results
of Young’s moduli determined by both stress–strain
curves and compliance-median methods for the three
anatomic orientations (axial, radial and transverse) are
given, as well as three Poisson’s ratios (for the
combination of orientations measured). Only 3 of 6
Poisson’s ratios were experimentally determined. In
materials of orthotropic symmetry, the 6 different
Poisson’s ratios are not independent (Cowin and Van
Buskirk, 1986). It can be shown that Poisson’s ratios
of orthotropic materials must satisfy the following
relationships:
uAT =E A ¼ uTA =E T ,
(8)
uAR =E A ¼ uRA =E R ,
(9)
uRT =E R ¼ uTR =E T ,
(10)
where EA is Young’s modulus in the axial direction, ER
is Young’s modulus in the radial direction, ET is
Young’s modulus in the transverse direction, and uij
are the 6 different Poisson’s ratios, where ij denotes the
respective axial, radial or transverse direction-combinations (Cowin and Van Buskirk, 1986). The three
Poisson’s ratios not determined experimentally were
calculated according to Eq (8)–(10) for one cube of
each horse.
Table 2
Experimental results of Young’s moduli for a typical set of 15
experiments for a Quarter horse cube
Experiment #
S–S curve
Compliance
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Mean
Standard deviation
9.6
10.0
10.2
10.0
10.0
10.0
10.1
10.5
10.6
10.2
9.9
10.0
10.0
10.0
9.8
10.1
0.22
9.7
9.9
10.1
9.8
9.9
9.9
9.7
10.2
10.4
10.2
9.8
9.8
9.6
10.0
9.7
9.9
0.23
Compression was in the radial direction. Moduli are shown both by
the compliance method and by the stress–strain (S–S) method.
By rearranging Eqs. (8)–(10) it can be seen that the
ratio of orthogonal moduli is equal to the reciprocal
ratio of Poisson’s ratios:
E i uji
¼ ,
E j uij
(11)
where i and j are two orthogonal anatomic directions.
Table 4 presents a comparison of the ratio of reciprocal
Poisson’s ratios and the ratio of the corresponding
Young’s moduli, in those experiments were the appropriate Poisson’s values were both measured. It can be
seen that the two ratios behave roughly as predicted by
Eq. (11).
The measured Young’s moduli values clearly demonstrate the well known axial anisotropy of secondary
osteonal bone. Interestingly, there are significant differences in the modulus values between the 6 cubes. In both
horses, one-way analysis of variance performed on the
results obtained for the axial Young’s modulus of the
three cubes revealed that they are significantly different
(po0.00001).
Poisson’s ratios were found to be in the lower range of
values reported to date for cortical bone. For strains
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260
Table 3
Summary of results for all 6 cubes
Horse
Cube
EA
ER
ET
Quarter horse
Cube 1
23.5
(0.3)
23.4
(0.6)
10.0
(0.5)
9.9
(0.7)
11.4
(0.7)
11.3
(0.5)
Cube 2
19.3
(0.8)
19.4
(0.6)
9.9
(0.2)
10.0
(0.3)
Cube 3
21.7
(0.7)
21.6
(0.5)
Cube A
Arabian horse
uRA
uAR
uTA
uAT
uRT
uTR
0.060
0.208
(0.044)
—
—
0.122
(0.063)
0.141
(0.031)
8.8
(0.3)
9.1
(0.3)
0.059
(0.028)
0.132
0.078
(0.027)
0.128
(0.060)
—
—
9.3
(0.4)
9.6
(0.5)
9.5
(0.4)
9.6
(0.5)
0.074
0.165
(0.050)
0.111
(0.046)
0.138
0.166
(0.046)
17.4
(0.4)
17.5
(0.5)
13.9
(0.4)
13.8
(0.5)
11.3
(0.5)
10.9
(0.5)
0.122
(0.018)
0.196
(0.017)
0.070
(0.017)
0.108
—
—
Cube B
22.1
(1.1)
21.7
(0.8)
9.9
(0.7)
9.9
(0.4)
11.1
(0.6)
11.3
(0.5)
—
—
0.098
0.192
(0.023)
0.150
(0.015)
0.156
(0.020)
Cube C
22.6
(0.5)
22.8
(0.7)
11.5
(0.3)
11.8
(0.2)
11.1
(0.3)
11.2
(0.3)
0.102
(0.010)
0.208
0.172
(0.015)
0.166
0.124
0.244
(0.015)
0.174
E represents Young’s modulus, and u represents Poisson’s ratio. The subscripts A, R, and T represent the axial, radial and transverse directions,
respectively. In each cell are shown the Young’s moduli obtained from the stress–strain curve above, with its standard deviation in brackets, the
Young’s modulus obtained by the compliance method below, and its standard deviation in brackets. All means and standard deviations are based on
15 observations. Poisson’s ratios are given either as measured, or in underlined italics when calculated based on Eqs. (8)–(10).
along the axis of compression a reliable signal to noise
(S/N) ratio was fairly easy to obtain requiring only
moderate filtering of the v(i,j) displacement fields (local
weighted averaging with each point averaged with its 4
nearest neighbors at a ratio of 100:40). The S/N ratio for
the orthogonal transverse strain values was found to be
smaller since the signal was weaker while the noise
remained the same, and resulted in larger standard
deviations of the estimation of Poisson’s ratio.
In addition to the similarity found between the
measured and manufacturer-supplied values of Poisson’s
ratios for the synthetic material we tested, we decided to
investigate the possibility that the low values of measured
Poisson’s ratios of the bone samples might have been
caused by the effect of constrained sample edges on the
measured strains. Specifically, we assessed the possibility
that these constraints limit the lateral expansion of the
sample and yield erroneously low Poisson’s ratios.
We examined the effect of constrained sample edges
experimentally by comparing the Poisson ratios measured in 2 different areas of a parallelepiped bone
sample. Table 5 shows the results obtained for
compression of the 2 2 6 mm rectangular parallelepiped along its long axis (see Fig. 6). The sample was
Table 4
Comparison of ratio of reciprocal Poisson’s ratios and the ratio of the
corresponding Young’s moduli, in those experiments were the
appropriate Poisson’s values were measured
Bone cube
Ratio of measured
reciprocal
Poisson’s ratios
Ratio of
corresponding
measured Young’s
moduli
Quarter horse cube 1
Quarter horse cube 2
Arabian horse cube A
Arabian horse cube B
uTR/uRT ¼ 1.16
uTA/uAT ¼ 0.60
uRA/uAR ¼ 0.62
uRT/uTR ¼ 1.06
ER/ET ¼ 1.14
ET/ER ¼ 0.46
ET/ER ¼ 0.80
ET/ER ¼ 1.20
divided into 3 areas of interest: proximal third, central
third and distal third. Results are presented for both the
central and distal 2 2 mm areas and are quite similar,
falling within the range of variation shown to exist in
this bone within millimeter-range zones. These results
show that the end constrains only affect the measurements slightly.
We also tested edge effects and the validity of the
measured Poisson’s ratios using finite element analysis
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261
Table 5
Results of compression of a rectangular parallelepiped along its long axis (transverse direction)
Region of interest
Young’s modulus ET (Gpa) by
the compliance method (in
brackets standard deviation,
n ¼ 15)
Young’s modulus ET (Gpa) by
the S–S curve method (in
brackets standard deviation,
n ¼ 15)
Poisson’s ratio uTR (in brackets
standard deviation, n ¼ 15)
Central area (n ¼ 15)
Bottom area (n ¼ 15)
10.7 (0.48)
9.1 (0.57)
10.9 (0.41)
9.1 (0.79)
0.142 (0.03)
0.120 (0.02)
The displacements of the transverse-radial surface were measured. The measurements were performed both in the central area and the bottom area
(see Fig. 6).
simulating compression of 2 2 mm bone cubes with
Poisson’s ratio of 0.1 and 0.3, respectively, and with
0.5 mm thick composite layers between the cube and
both loading anvils. Boundary conditions simulating the
no-slip contact between the anvils and the sample were
set. The distributions of the computed strains along a
central longitudinal line of nodes, in the compression (y-)
direction and lateral (x-, orthogonal to the compression)
direction for Poisson’s ratios of 0.3 and 0.1 are shown in
Figs. 7a and b, respectively. It can be seen that the
central area, coincident with the area on the sample’s
surface from which experimental data were obtained,
exhibits a consistent strain in both the direction of load
application and the direction orthogonal to it. The
average absolute values of the ratio between the lateral
and axial strains are 0.303 and 0.107 for simulations
using Poisson’s ratios of 0.3 and 0.1, respectively. These
results suggest that the direct Poisson ratio measurements obtained by our experiments represent the true
properties of the tested material, and are not biased by
the no-slip conditions between the anvils and the
composite layer.
6. Discussion
This study shows that Young’s moduli and Poisson’s
ratios can be determined by direct observations of
surface displacements in millimeter-sized samples of
secondary osteonal cortical bone loaded in compression
under water. All bone samples showed much higher
modulus in the axial direction (EA) than in the radial
and transverse directions (ER and ET), as expected.
Furthermore, the radial and transverse moduli were
quite similar. This is consistent with values obtained by
others using macroscopic specimens (Reilly and Burstein, 1975, Taylor et al., 2002, Dong and Guo, 2004,
Iyo et al., 2004).
We calculated Young’s moduli from our experimental
data by two independent methods (the stress–strain and
compliance methods). As can be seen in Table 3, the
methods yield nearly identical results. Yet an important
difference exists between these two methods. The S–S
method uses a broad range of stresses and strains,
whereas the compliance method is based on measurements made with very small stress increments
(o1 MPa). Our results show that minute deformation
can be reliably determined with our ESPI-based experimental system. Low stress increments result in very
small strain increments, thus allowing experimental
approximation of the local derivatives of the stress–strain curve. Furthermore, our ability to measure
accurately very small strains allows us to conduct elastic
experiments in which damage does not occur in the
sample. Additionally, with the S–S curve slope method
only one modulus estimate is obtained for each
experiment, whereas with the compliance method a
large number of statistically independent measurements
are obtained. This has the advantage of providing
multiple results for each sample, allowing improved
statistics. Furthermore, these features might have
application for studying other properties, such as rateand load-dependant variations of the elastic constants.
We found significant local variations in Young’s
moduli to occur within a range of 1–2 mm of the equine
cortical bone we tested. Axial Young’s moduli varied
between 17.4 and 23.6 GPa, whereas radial and transverse Young’s moduli varied between 8.8 and 13.9 GPa.
Similar variation was previously reported by Ashman et
al. (1984). We have shown that the variations found in
small adjacent cubes do not arise from lack of precision
or lack of accuracy of our method. We suggest that in
both horses, structural differences in the meso-scale,
such as regions of secondary osteons versus interstitial
bone, variable pore distributions, and/or local differences in mineral content have a major impact on
local stiffness. In fact, when we remounted some
of our samples and repeated the experiments, we
obtained extremely similar results (data not shown).
The broad range of results is however similar to results
previously reported in the literature (Currey, 2002,
Cowin, 2001).
We confirmed the accuracy of our measurements and
validity of the results by performing compression
experiments on small cubes of Ultems (see also
Zaslansky et al., 2005). Measurements made using a
protocol similar to that used for bone yielded values for
both Young’s modulus and Poisson’s ratio which were
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262
x- and y-strains along a central line of the cube in the direction of
compression
0.00006
0.00004
0.00002
0
strain
0
0.5
1
1.5
2
2.5
3
-0.00002
-0.00004
x - strain
-0.00006
y - strain
-0.00008
-0.0001
-0.00012
position of element (mm)
(a)
x- and y-strains along a central line of the cube in the direction of
compression
0.00004
0.00002
0
0
0.5
1
1.5
2
2.5
3
strain
-0.00002
x- strain
-0.00004
y- strain
-0.00006
-0.00008
-0.0001
-0.00012
(b)
position of element (mm)
Fig. 7. Finite element analysis results, showing the longitudinal (y) and lateral (x) strains along a central line of the cube in the direction of
compression for bone cubes with Poisson’s ratio of 0.3 (a) and 0.1 (b).
very similar to the values reported by the manufacturers,
as can be seen in Table 1.
The results obtained here for Poisson’s ratios of
cortical bone are however lower than those usually
reported. In fact, previous studies reported a very large
range of values of Poisson’s ratios in bone, from 0.12 to
0.63 (Pithioux et al, 2002; Reilly and Burstein, 1975).
The values found in our series of experiments fall within
a narrower range. The low values obtained by our
experiments for Poisson’s ratios in the RA and TA
directions (0.07–0.124) are particularly striking.
In a material with orthotropic symmetry the Poisson’s
ratio values should be orientation-dependent. As can be
seen from Eq. (11), the relationship between the
different Poisson’s ratios is related to the associated
Young’s moduli. Table 3 shows that when both uRA and
uAR or uTA and uAT were measured in the same bone
cube, they were different. Table 4 demonstrates that the
ratio between these reciprocal Poisson’s ratios and the
ratio between the corresponding Young’s moduli are
similar. On the other hand when uRT and uTR were both
measured in the same bone cube they were quite similar,
as were the corresponding radial and transverse moduli.
These findings strongly support the notion that these are
indeed representative values for Poisson’s ratios of
secondary osteonal bone.
The large differences between many of our measurements of Poisson’s ratio and those often used in the
literature led us to investigate the possibility that
the non-slip condition between the anvils of the
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micro-mechanical loading device and the bone samples
could cause the measured lateral strain to be misleadingly low. Several validation studies were conducted to
evaluate this scenario.
A compression experiment conducted on a
6 2 2 mm rectangular parallelepiped bone sample
along its long axis yielded similar results for Young’s
modulus and Poisson’s ratio when data were obtained
from the central 2 2 region (far from the confined
edges) and from a 2 2 region near the loaded edge.
Furthermore, analysis of a finite element model of
compression of a cube, with boundary conditions
simulating the non-slip edge effect arising in the
experiments of this study showed that in the entire
region of interest on the face of the cube the ratio of
strain across the direction of loading to the strain at the
same point along the direction of loading yields the true
Poisson’s ratio. We therefore conclude that the values of
Poisson’s ratio reported here are accurate.
Osteonal bone is the dominant bone type of adult
equine bone (Mason et al., 1995); however it is not
known how this bone behaves in the meso-scale when
loaded under compression. Individual osteons have
different mineral contents, and indeed mineral content
varies continuously throughout their structure. Mineral
content also varies between Haversian and interosteonal areas (Currey, 2002). Hence different stiffness
values characterize different sites within the bone, and it
is not clear how the load is distributed among these sites.
We plan to further modify our system in order to
achieve even greater resolution that will allow the
measurement of displacements in the meso-scale of
50–500 mm. It will then be possible to determine local
displacement variations in regions such as newly formed
osteons with low mineral content and regions of older,
more mineralized osteons. The variation of local
displacements within individual regions (and the resulting strain variations) can then be compared.
The results reported here demonstrate that the
method described can yield quantitative measurements
of surface displacements of small cubes of bone loaded
in compression while in a water environment. Furthermore, these results can be used to calculate the strains,
and through them values for Young’s moduli and
Poisson’s ratios. Measurements are performed without
contact with the sample, and can easily be repeated
many times on the same sample, since the method is
non-destructive, allowing truly elastic measurements.
In conclusion, we described the determination of
Young’s moduli and Poisson’s ratios using a novel, noncontact optical method. Small bone samples were tested
non-destructively in an aqueous environment, and
measurements of displacements were performed in three
orthogonal directions. This study showed that significant variation occurs locally in Young’s moduli and
Poisson’s ratios, and the measured Poisson’s ratios of
263
cortical bone were found to be lower than previously
reported.
Acknowledgements
S.W. is the incumbent of the Dr. Walter and Dr.
Trude Burchardt Professorial Chair of Structural
Biology. A.A.F is the incumbent of the Peter and
Carola Kleeman Professorial Chair of Optical Sciences.
Support for this research was provided from Grant RO1
DE006954 from the National Institute of Dental and
Craniofacial Research, and from the Women’s Health
Research Center to Dr. Stephen Weiner.
We wish to thank Benjamin Sharon, David Leibovitz,
Gershon Elazar and Yosef Shopen for excellent
technical assistance.
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