anisotropic poisson’s ratio and compression modulus of cortical bone

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Journal of Biomechanics 40 (2007) 252–264

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Anisotropic Poisson’s ratio and compression modulus of corticalbone determined by speckle interferometry

R. Shahara,�, P. Zaslanskyb, M. Barakb, A.A. Friesemc, J.D. Curreyd, S. Weinerb

aKoret School of Veterinary Medicine, The Hebrew University of Jerusalem, P.O. Box 12, 76100 Rehorot, IsraelbDepartment of Structural Biology, Weizmann Institute of Science, Israel

cDepartment of Physics of Complex Systems, Weizmann Institute of Science, IsraeldDepartment of Biology, University of York, UK

Accepted 16 January 2006

Abstract

Young’s modulus and Poisson’s ratios of 6mm-sized cubes of equine cortical bone were measured in compression using a micro-

mechanical 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 theaxial–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 modulusin the axial direction being significantly higher than inthe transverse and radial directions (Reilly and Burstein,1975, Taylor et al., 2002, Dong and Guo, 2004, Iyoet al., 2004). In fact the mechanical properties of boneare affected by many aspects of its complex structure(Weiner and Wagner, 1998), and in particular by themineral content (Currey, 2002).

Few studies have attempted to correlate structurewith function at the micron to millimeter meso-scale

e front matter r 2006 Elsevier Ltd. All rights reserved.

iomech.2006.01.021

ing author. Tel.: +972 9 7433968;

8994.

ess: [email protected] (R. Shahar).

(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 evenwithin) various types of cortical bone are found at thatlength scale, the study of the mechanical propertiesusing millimeter-sized samples is of great importance.However testing such samples is complicated, posingmany technical challenges. Various experimental meth-ods have been reported for determining the elasticproperties of cortical bone, with sample sizes rangingfrom several centimetres (Reilly et al., 1974, Reilly andBurstein, 1975) to single osteons with dimensions ofhundreds of micrometres (Ascenzi and Bonucci, 1968).Most methods were based on loading relatively bulkybone samples in material testing machines, and record-ing load–displacement curves for loads such as tension,

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compression, torsion and bending. In tension andcompression experiments, the slope of the stress–straincurve within the elastic region was used to estimate theYoung’s modulus of the bone. Most tests are performedby applying a controlled deformation rate, and oftenprogress continuously until failure. Such experimentsare usually used to also provide information regardingthe yield point (the point at which the behaviour of thesample ceases to be linearly elastic), load and work tofailure and ultimate strain. However, since these testsare destructive, often only one estimate of modulus canbe obtained from each sample, thus experimentalprecision cannot be determined. Furthermore, sincethe use of large and bulky samples is required, localmicro-scale variations are ignored.

Several non-destructive methods are widely used forobtaining mechanical properties of small samples.Micro- and nano-indentation have been used to measurethe hardness and elastic constants of cortical bone at themicro-structural level (Weiner et al., 1997, Rho et al.,1997, Zysset et al., 1999, Silva et al., 2004, Hengsbergeret al., 2003; Bensamoun et al., 2004). These methodsallow estimation of hardness and Young’s modulusfrom contact stiffness between the indenter tip and thesample. They require a highly polished sample surface,and the calculation assumes knowledge of the Poisson’sratios of the sample. When used to measure the elasticmodulus of anisotropic materials such as bone, themodulus derived from the method is an average of theanisotropic constants biased towards the modulus of thedirection of testing (Rho et al., 1997). Another approachis based on the measurement of the speed at whichsound travels through bone (Yoon and Katz, 1976a,b,Ashman et al., 1987, Rho et al., 1993). Although thesemethods are non-destructive, they are also indirect, andrely on the application of theories of compositematerials to the measurements in order to obtainestimates of the elastic constants.

Few studies describe experimental determination ofPoisson’s ratios of cortical bone. Reilly and Burstein(1975) assumed transverse isotropy of fibrolamellarbone, and used extensometers to measure strains intwo orthogonal directions concurrently. They foundPoisson’s ratio values which ranged between 0.29 and0.63. Ashman et al. (1984) reported on the use of anultrasonic continuous wave technique, and foundPoisson’s ratio values which ranged between 0.27 and0.45. Pithioux et al. (2002) also used an ultrasonicmethod, and found Poisson’s ratios between 0.12 and0.29. Despite this wide range of reported values(0.12–0.63), many studies, especially finite elementanalyses, often use values in the much narrower rangeof 0.28–0.33.

Optical metrology techniques allow non-contactmeasurement of displacements on surfaces of samplessubjected to static or dynamic mechanical loading.

Electronic speckle pattern-correlation interferometry(ESPI) (Jones and Wykes, 1989, Rastogi, 2001) hasrecently been used to determine sub-micron surfacedisplacements on the surface of millimetre-sized toothdentin samples loaded elastically in compression (Za-slansky et al., 2005). Displacements are directly deter-mined from variations of laser light reflected fromsamples immersed in water. Using this technique, it ispossible to perform quantitative analysis of strain oncompressed samples of mineralized biological tissuessuch as bone and dentin, by loading them in a high-precision micro-mechanical loading device. Such mea-surements can be performed without damaging thesample and hence anisotropic Young’s moduli andPoisson’s ratios can be determined from multiplemeasurements of each sample.

We report on measurements performed using acommercial ESPI system (Q300—Ettemeyer, Ulm,Germany) which has been combined with a custom-built loading device allowing non-destructive compres-sion tests of small cubes of cortical bone in threeorthogonal directions. Our set-up allows the measure-ment of in-plane and out-of-plane deformation fields onsurfaces of very small samples. Due to the highsensitivity of our system, experiments can be conductednon-destructively by repeatedly loading the samplewithin its elastic region. Measurements may be per-formed on wet samples, thus satisfying a basic require-ment for the study of biological specimens.Measurements by this technique require load to beapplied in small increments, since large deformationscause optical decorrelation that renders the measure-ments invalid.

While the ESPI technique allows for the determina-tion of the elastic constants from the traditionalstress–strain curve, as is common in classical mechanicaltesting methods, it also allows for the use of acompliance-based method (‘Estimated best E’ fromZaslansky et al., 2005). The compliance method is basedon the principle that each incremental loading stepwithin the elastic region can be considered as anindependent experiment in which the strain field createdwithin the sample and the incremental load causing itare determined. Thus, the compliance of the sample isdetermined repetitively and non-destructively.

An interesting feature of the ESPI method is itsinherent ability to concurrently measure strain along twoorthogonal directions on the sample surface. This allowsthe derivation of Poisson’s ratios from the sameexperimental data on exactly the same sample underidentical loading conditions. For each incrementalloading step, in addition to the axial strain of thesample, the lateral strain is also determined. Thenegative ratio of the latter to the former can be usedto estimate Poisson’s ratio. Since these loading steps canbe repeated at the discretion of the investigator, the

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statistical error associated with mechanical tests such asthese can be reduced.

This paper describes the results of optically deter-mined measurements of strain of mechanically loadedmillimetre-sized samples of equine cortical bone. Thesemeasurements were used to determine the anisotropicvariations in Young’s moduli and Poisson’s ratios.

Axial

Transverse

Radial

2 mm

2 mm

2 mm

Fig. 2. The preparation of three 2� 2� 2mm cubes from the slice

shown in Fig. 1, with the orthogonal anatomic orientations preserved.

2. Materials and methods

Six cubic bone samples of equine cortical bone wereobtained from the cadavers of two horses. They wereloaded in compression, and their surface displacementswere determined using ESPI following proceduressimilar to those described in Zaslansky et al. (2005).These data were used to determine three Young’smoduli and three Poisson’s ratios related to the threeorthotropic directions (axial, radial and transverse).

2.1. Bone sample preparation

The right third metatarsal bones (MT3) were obtainedfrom a 4-year old-male Quarter horse and a 6-year-oldfemale Arabian horse. The cause of death of both horseswas unrelated to the musculoskeletal system. Allexternal soft tissue was meticulously removed, and 2-cm thick slices were cut from the mid-diaphysis of eachbone using a hand saw (Fig. 1). Transverse sections of200–300 mm thickness were then cut from each boneslice, using a low-speed water-cooled diamond saw(South Bay Technology Inc.) (Fig. 1). These sectionswere ground and polished (Buehler Minimet Polisher)and examined by reflected light microscopy. The area ofthe cranial mid-diaphysis in the bones of both horsesconsisted almost entirely of secondary remodelledosteonal bone, with small areas of primary bone. Eachbone slice was further cut and used as a source of three 2� 2 � 2mm cubes. The cubes were cut such that theirfaces were aligned with the anatomical axes of the bone:proximo-distal (axial), antero-posterior (radial) andcranio-caudal (transverse) orientations (Fig. 2). Carewas taken to note and mark the orientation of the cubes,so that distinct axial, radial and transverse faces could

Fig. 1. Removal of a 2-cm thick slice from the mid-diaphyseal a

be identified. The cubes were then stored for 2-7 days onwater-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–compressiondevice, a water chamber, an optical ESPI head capableof determining surface displacements and a computerwhich controlled the various components of the systemand analysed the data. Aside from the computer, theentire set-up was enclosed in an acoustically insulatedbox mounted on top of a floating optical table.

2.3. Mechanical testing device

The mechanical tension–compression device used forloading the bone samples was custom-designed withstainless-steel parts (SS 316) (see Fig. 4). The deviceconsists of a sealed chamber which includes a high-gradeglass window (BK-7 l/10 grade), allowing the use ofaqueous solutions as a medium in which sample testingtook place. The loading apparatus in the test chamberconsisted of an axial motion DC motor (PI M-235.5DG, Physik Instrumente, GmbH, Germany) capable ofdisplacing a metal shaft in small sub-micron steps whileapplying substantial force (4100N). The metal shaftacted as a movable upper anvil, pressing against samplesmounted on the stationary lower anvil which wasattached rigidly to the testing chamber base. Themovable metal shaft was fitted with an in-chamber

rea of the third metatarsal, and a thin transverse section.

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Glass window

DC axial motor

Water Chamber

Micro-step (PI) motor controller and computer ESPI System

Trigger

Sample Laser

Fig. 3. Schematic of the entire experimental system.

R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264 255

immersable load cell (AL311BN-6I, Sensotec, Honey-well, USA) capable of measuring loads up to 120N intension or compression. Force measurements werecollected and stored on a computer using an A/Dconverter (Omega DAQP-308 PCMCIA 16-bit Analo-gue I/O) and further analysed by custom-writtensoftware (National Instruments Labview v 7.0 andMatlab v 6.0).

2.4. Sample mounting

A thin lining layer of soft composite material (Z250,3M ESPE, St. Paul, MN USA) was placed over thelower, stationary anvil in the test chamber. The face ofthe bone cube which corresponded to the direction beingtested was then placed upon it. The cube was gentlypushed down onto the composite such that it wasmounted in the appropriate orientation (inset in Fig. 4).Manual adjustment facilitated the alignment of thecube. The composite material was then polymerizedusing a hand-held light-cure device (LITEX 682,Dentamerica CA, USA). A small amount of compositewas placed on the edge of the upper (travelling) anvilwhich was then lowered until the composite was broughtinto contact with the entire upper surface of the cube.While the cube was held in the desired line ofcompression (axial, radial or transverse) between thetwo anvils, the top composite was cured. This mountingprocedure ensured that an intimate and stiff contact wasestablished between both anvils and the bone cubethrough composite load–transfer layers. With eachsample mounted and fixed in place, the chamber wassealed and filled with physiologic saline. Small pieces ofbone were added to the saline solution which wasrefrigerated at 4 1C, for at least 24 h prior to initiation

of testing, in order to ensure mineral saturation ofthe solution. The same solution was used for allexperiments.

2.5. Mechanical compression testing

A small compression preload of about 10N wasapplied at the beginning of each experiment, and thesample was allowed to reach near-equilibrium forcereadings while undergoing some initial stress-relaxation.Once a stable reading of force was reached (approxi-mately within 200–300 s), a series of 15 compressionincrements was initiated. Each loading step consisted ofa small incremental load, which was produced byforcing the upper anvil to move 2 mm downwards. Justbefore and immediately after each such loading step,force was determined by averaging 20,000 readings(taken at 80 kHz), and surface deformation wasdetermined by laser speckle intensity measurements(ESPI, Section 2.6). Each experiment was repeated 15times, so as to collect a large and robust data set. Then,at the end of 15 experiments, the test chamber wasemptied and the cube was dismounted. The cube wasthen remounted in another (orthogonal) orientation,and 15 load-deformation experiments repeated.

We were thus able to collect data from 15 repeatedexperiments of force-deformation measurements foreach of the three orthogonal sample axes for each cube(therefore each cube underwent 45 experiments). Everyexperiment contained 15 incremental compression steps.These measurements provided the database required todetermine the 3 Young’s moduli and 3 of the 6 Poisson’sratios of the cubes of cortical bone reported here. For 2of the cubes we were also able to calculate the threeremaining Poisson’s ratios, (see Section 5).

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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.

R. Shahar et al. / Journal of Biomechanics 40 (2007) 252–264256

2.6. Optical determination of surface displacements

Surface displacements of the compressed cube alongand across the loading direction were used to calculatethe longitudinal and lateral strains caused by eachincremental load. These displacements were determinedusing two orthogonally aligned horizontal (X-axis) andvertical (Y-axis) speckle pattern-correlation interferom-eters (see detailed description in Zaslansky et al., 2005).

Briefly, the surface of the sample is illuminated by laserlight from two symmetrical opposite angles relative tothe normal to the surface. The laser illumination createsa speckled light interference field on the surface. Byimaging the speckle patterns onto a CCD detector array,variations to the interference intensity patterns can bedetected and captured by a computer. Any sub-microndisplacement of sub-sections on the surface will affectthe optical path of the light propagating towards the


CCD detector array, changing the detected interferencepattern. If the phase of the light of one of the beams ismodulated by means of a phase shifter, the phaserelations over the entire CCD field can be determined.The interference patterns can be shown to correspond tolight phase differences on small surface sub-sections,which are determined after computerized image filteringand phase unwrapping (Huntley, 1998, Rastogi, 2001).

The phase relations were determined for small sub-areas on bone sample surfaces before and after eachload increment. By substraction, 2D matrices of phasedifferences were obtained from the X and Y inter-ferometers, of the form:

MATX ;Y ¼ Djði; jÞX ;Y (1)

where i and j represent the rows and columns of thematrix respectively, and corresponds to the phasedifference determined independently for each axis. Eachelement in the two matrices corresponds to one sub-areaon the surface of the sample.

A scaling relation of the form:

u; v i; jð Þ ¼lDj i; jð ÞX ;Y4p sin y

(2)

was used, in which l is the laser light wavelength(l ¼ 780 nm in our system) and y is the angle betweenthe incident laser light and the normal to the samplesurface. Our system allows detection of surface dis-placements with magnitudes as small as l/30 (Jones andWykes, 1989). Thus, the discrete displacement values u

and v were determined, corresponding to the compo-nents of displacements of all sub-areas within the area ofinterest along X and Y axes, respectively (Rastogi, 2001;Jones and Wykes, 1989).

2.7. Determination of surface strains

The u i; jð Þ and v i; jð Þ displacement matrices wereassumed to represent the displacements of the bulk ofthe compressed bone samples. They were assumed tovary linearly along the rows and columns of thedisplacement fields in the Y- and X-directions, respec-tively. They could therefore be approximated as planesby least-squares regression analysis that providedcontinuous and differentiable displacement fields ofthe form:

u i; jð Þ ¼ ai þ bj þ c, (3)

v i; jð 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 shearstrain could then be determined according to

eii i; jð Þ ¼@u i; jð Þ

@i¼ a, (5)

ejj i; jð Þ ¼@v i; jð Þ

@j¼ e, (6)

eij i; jð Þ ¼1

2

@u i; jð Þ

@jþ@v i; jð Þ

@i

� �¼

1

2bþ dð Þ. (7)

In our experiments, each detector on the CCD covered asub-area of approximately 12 mm� 12 mm on the surfaceof the sample. Due to the small dimensions of thesamples relative to the distance to the ESPI lens (2mmversus 225mm, respectively), spherical aberrationscould be neglected. Therefore, the distance betweenadjacent rows and columns could be assumed to be12 mm. This number was then used as a scaling factor toobtain estimates of strain from the slopes of theregression planes.

3. Data analysis

Stress values were calculated as the ratio between theforce measurements recorded for each load incrementand the initial cross-sectional area of the correspondingsample. These stresses and the optically determinedstrains were used for the determination of Young’smodulus and Poisson’s ratio along each of the boneaxes. In order for an experiment to be considered valid,the lateral normal strain had to be larger than the shearcomponent by at least an order of magnitude. Otherwiseit was repeated entirely, including repositioning of thesample.

3.1. Young’s modulus (E)

Two complementary methods were employed todetermine Young’s modulus (E) for each of theorientations of the samples (axial modulus EA, radialmodulus ER, and transverse modulus ET):

3.1.1. Stress– strain curves

The cumulative stress was plotted against thecumulative strain for the 15 increments in each of 15repeated experiments, for all three compression orienta-tions. The slopes of the stress–strain curves were thenestimated using linear least-squares regression analysis.In all experiments a tight linear relationship betweenstress and strain was found. A typical stress–strain curveobtained from a 15-step compression experiment per-formed in the radial direction is shown in Fig. 5.

3.1.2. Compliance method

Individual compliance estimates were determined foreach of 15 incremental compression loads. Strainincrements for each loading step were determined byleast squares regression analysis of the surface displace-ment values, as described in Eqs. (3) and (4) above. The

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Sample Strain Stress curve, Radial compression

0

2

4

6

8

10

12

14

16

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.006.00(strain(x105))

Stre

ss [

MPa

]

Stress[MPa]= 11.6x103 Strain - 0.30

R2 = 0.996

Fig. 5. A typical stress–strain curve obtained from a compression

experiment on cube from the Arabian horse.

F F

6 mm

2 mm

Middle area

Bottom area

F(a) (b)

Fig. 6. Schematic of the parallelepiped bone sample, showing the

central and lower areas of interest.


compliance estimate was then estimated by dividing thestrain increment by its corresponding stress increment.The inverse of the median of these 15 values wasconsidered the estimate of Young’s modulus for theexperiment. In this manner 15 modulus estimates wereobtained for each direction (axial, radial and trans-verse). For each experiment, whole body rotation andshear deformation were found to be much smaller thanaxial deformation, and therefore could be ignored.

3.2. Poisson’s ratio (v)

In each experiment, longitudinal strains elong andlateral (orthogonal to the direction of loading) strainselat 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 ofthese 15 separate estimates. Since each experiment wasrepeated 15 times, the reported result was obtained asthe average of 15 medians. For each cube, 3 differentcombinations of lateral/longitudinal load orientationpairs (out of 6 possible combinations) were performed.

4. Method validation

Three different approaches were used to validate ourmethodology. First, the method described above wasused to determine the elastic properties of a small2� 2� 2mm cube made of an isotropic material: Acetalpolyether imide (Ultem-1000s, General Electric Plas-tics), whose elastic constants are precisely known. Sinceour experimental procedure is very different fromindustrial ISO and ASTM testing recommendations,these results allowed us to validate the application ofour novel methodology to the testing of very smallsamples in water and without contact.

Second, an experiment was designed to determine theeffect of the non-slip conditions occurring at the anvil-sample interface on the determined elastic constants.

To this end a 2� 2� 6mm rectangular parallelepipedsample of equine cortical bone was compressed along itslong axis to allow calculation of Young’s modulus andPoisson’s ratio in an area distant from the area ofapplied loads (centre of the sample), and in an area closeto loaded edge (see Fig. 6). The results were comparedwith those obtained in cube experiments using samplesfrom the same horse and an identical load orientation.

Third, a finite element model was created to simulatea cube compression experiment. In this model the lowernodes of the cube were fixed, and the topmost nodes(area of load application) were fully restrained exceptalong the loading direction. The modelled bone materialwas assigned orthotropic properties, using Young’smoduli determined here, and shear moduli reported byAshman et al. (1987). The model was analysed with twodifferent Poisson’s ratios: 0.3 and 0.1. The model wascreated with Nastran software, NFW version 2002, andconsisted of 12,800 8-node brick elements, with 13,671nodes. The composite layer above and below the cubewas modelled as isotropic, with values for Young’smodulus (11GPa) and Poisson’s ratio (0.30) obtainedfrom the manufacturer (3M). The results were examinedto determine the level at which the lateral constraintsceased to affect the lateral strain.

5. Results

Table 1 shows the calculated Young’s moduliand Poisson’s ratio results based on 10 compressionexperiments conducted on a 2� 2� 2mm cube ofUltem-1000s. The measured Young’s moduli were very

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Table 1

Results of a compression experiment performed on 2� 2� 2mm cubes made of Ultems

Young’s modulus ET

(GPa) reported by

the manufacturer


(GPa) by the

compliance method

(in brackets standard

deviation, n ¼ 15)


(GPa) by the stress-

strain 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)

Ultem-1000s 3.2 3.42 ( 0.01) 3.43 (0.01) 0.36 0.38 (0.07)

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 9.6 9.7

2 10.0 9.9

3 10.2 10.1

4 10.0 9.8

5 10.0 9.9

6 10.0 9.9

7 10.1 9.7

8 10.5 10.2

9 10.6 10.4

10 10.2 10.2

11 9.9 9.8

12 10.0 9.8

13 10.0 9.6

14 10.0 10.0

15 9.8 9.7

Mean 10.1 9.9

Standard deviation 0.22 0.23

Compression was in the radial direction. Moduli are shown both by

the compliance method and by the stress–strain (S–S) method.


close to the value supplied by the manufacturer(3.42–3.43GPa and 3.2GPa, respectively). The mea-sured and manufacturer-supplied Poisson’s ratios werealso very similar (0.38 and 0.36, respectively). Theseresults indicate that our methodology is accurate (themeans are in excellent agreement) and precise (smallstandard deviations).

Table 2 shows the results of Young’s moduli obtainedin a typical set of 15 experiments compressing a Quarter-horse bone cube in the radial direction. For eachexperiment the Young’s modulus was obtained fromthe slope of stress–strain curves (S–S) as well as from theinverse of the median compliance. Also shown are themean and standard deviations for these values. Clearlythe results from both methods are very similar, bothwith respect to the mean values and their errors.

Table 3 summarizes the results obtained fromcompression tests of all cubes of cortical bone used inthis study: 3 from the 4-year-old male Quarter horse,and 3 from the 6-year-old female Arabian horse. Resultsof Young’s moduli determined by both stress–straincurves and compliance-median methods for the threeanatomic orientations (axial, radial and transverse) aregiven, as well as three Poisson’s ratios (for thecombination of orientations measured). Only 3 of 6Poisson’s ratios were experimentally determined. Inmaterials of orthotropic symmetry, the 6 differentPoisson’s ratios are not independent (Cowin and VanBuskirk, 1986). It can be shown that Poisson’s ratiosof orthotropic materials must satisfy the followingrelationships:

uAT=EA ¼ uTA=ET, (8)

uAR=EA ¼ uRA=ER, (9)

uRT=ER ¼ uTR=ET, (10)

where EA is Young’s modulus in the axial direction, ER

is Young’s modulus in the radial direction, ET isYoung’s modulus in the transverse direction, and uijare the 6 different Poisson’s ratios, where ij denotes therespective axial, radial or transverse direction-combina-tions (Cowin and Van Buskirk, 1986). The threePoisson’s ratios not determined experimentally werecalculated according to Eq (8)–(10) for one cube ofeach horse.

By rearranging Eqs. (8)–(10) it can be seen that theratio of orthogonal moduli is equal to the reciprocalratio of Poisson’s ratios:

Ei

Ej

¼uji

uij

, (11)

where i and j are two orthogonal anatomic directions.Table 4 presents a comparison of the ratio of reciprocalPoisson’s ratios and the ratio of the correspondingYoung’s moduli, in those experiments were the appro-priate Poisson’s values were both measured. It can beseen that the two ratios behave roughly as predicted byEq. (11).

The measured Young’s moduli values clearly demon-strate the well known axial anisotropy of secondaryosteonal bone. Interestingly, there are significant differ-ences in the modulus values between the 6 cubes. In bothhorses, one-way analysis of variance performed on theresults obtained for the axial Young’s modulus of thethree cubes revealed that they are significantly different(po0.00001).

Poisson’s ratios were found to be in the lower range ofvalues reported to date for cortical bone. For strains

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Table 3

Summary of results for all 6 cubes

Horse Cube EA ER ET uRA uAR uTA uAT uRT uTR

Quarter horse Cube 1 23.5 10.0 11.4 0.060 0.208 — — 0.122 0.141

(0.3) (0.5) (0.7) (0.044) (0.063) (0.031)

23.4 9.9 11.3

(0.6) (0.7) (0.5)

Cube 2 19.3 9.9 8.8 0.059 0.132 0.078 0.128 — —

(0.8) (0.2) (0.3) (0.028) (0.027) (0.060)

19.4 10.0 9.1

(0.6) (0.3) (0.3)

Cube 3 21.7 9.3 9.5 0.074 0.165 0.111 0.138 0.166 0.174

(0.7) (0.4) (0.4) (0.050) (0.046) (0.046)

21.6 9.6 9.6

(0.5) (0.5) (0.5)

Arabian horse Cube A 17.4 13.9 11.3 0.122 0.196 0.070 0.108 — —

(0.4) (0.4) (0.5) (0.018) (0.017) (0.017)

17.5 13.8 10.9

(0.5) (0.5) (0.5)

Cube B 22.1 9.9 11.1 — — 0.098 0.192 0.150 0.156

(1.1) (0.7) (0.6) (0.023) (0.015) (0.020)

21.7 9.9 11.3

(0.8) (0.4) (0.5)

Cube C 22.6 11.5 11.1 0.124 0.244 0.102 0.208 0.172 0.166

(0.5) (0.3) (0.3) (0.015) (0.010) (0.015)

22.8 11.8 11.2

(0.7) (0.2) (0.3)

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).

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 uTR/uRT ¼ 1.16 ER/ET ¼ 1.14

Quarter horse cube 2 uTA/uAT ¼ 0.60 ET/ER ¼ 0.46

Arabian horse cube A uRA/uAR ¼ 0.62 ET/ER ¼ 0.80

Arabian horse cube B uRT/uTR ¼ 1.06 ET/ER ¼ 1.20


along the axis of compression a reliable signal to noise(S/N) ratio was fairly easy to obtain requiring onlymoderate filtering of the v(i,j) displacement fields (localweighted averaging with each point averaged with its 4nearest neighbors at a ratio of 100:40). The S/N ratio forthe orthogonal transverse strain values was found to besmaller since the signal was weaker while the noiseremained the same, and resulted in larger standarddeviations of the estimation of Poisson’s ratio.

In addition to the similarity found between themeasured and manufacturer-supplied values of Poisson’sratios for the synthetic material we tested, we decided toinvestigate the possibility that the low values of measuredPoisson’s ratios of the bone samples might have beencaused by the effect of constrained sample edges on themeasured strains. Specifically, we assessed the possibilitythat these constraints limit the lateral expansion of thesample and yield erroneously low Poisson’s ratios.

We examined the effect of constrained sample edgesexperimentally by comparing the Poisson ratios mea-sured in 2 different areas of a parallelepiped bonesample. Table 5 shows the results obtained forcompression of the 2� 2� 6mm rectangular parallele-piped along its long axis (see Fig. 6). The sample was

divided into 3 areas of interest: proximal third, centralthird and distal third. Results are presented for both thecentral and distal 2� 2mm areas and are quite similar,falling within the range of variation shown to exist inthis bone within millimeter-range zones. These resultsshow that the end constrains only affect the measure-ments slightly.

We also tested edge effects and the validity of themeasured Poisson’s ratios using finite element analysis

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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) 10.7 (0.48) 10.9 (0.41) 0.142 (0.03)

Bottom area (n ¼ 15) 9.1 (0.57) 9.1 (0.79) 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� 2mm bone cubes withPoisson’s ratio of 0.1 and 0.3, respectively, and with0.5mm thick composite layers between the cube andboth loading anvils. Boundary conditions simulating theno-slip contact between the anvils and the sample wereset. The distributions of the computed strains along acentral 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 inFigs. 7a and b, respectively. It can be seen that thecentral area, coincident with the area on the sample’ssurface from which experimental data were obtained,exhibits a consistent strain in both the direction of loadapplication and the direction orthogonal to it. Theaverage absolute values of the ratio between the lateraland axial strains are 0.303 and 0.107 for simulationsusing Poisson’s ratios of 0.3 and 0.1, respectively. Theseresults suggest that the direct Poisson ratio measure-ments obtained by our experiments represent the trueproperties of the tested material, and are not biased bythe no-slip conditions between the anvils and thecomposite layer.

6. Discussion

This study shows that Young’s moduli and Poisson’sratios can be determined by direct observations ofsurface displacements in millimeter-sized samples ofsecondary osteonal cortical bone loaded in compressionunder water. All bone samples showed much highermodulus in the axial direction (EA) than in the radialand transverse directions (ER and ET), as expected.Furthermore, the radial and transverse moduli werequite similar. This is consistent with values obtained byothers using macroscopic specimens (Reilly and Bur-stein, 1975, Taylor et al., 2002, Dong and Guo, 2004,Iyo et al., 2004).

We calculated Young’s moduli from our experimentaldata by two independent methods (the stress–strain andcompliance methods). As can be seen in Table 3, themethods yield nearly identical results. Yet an importantdifference exists between these two methods. The S–Smethod uses a broad range of stresses and strains,

whereas the compliance method is based on measure-ments made with very small stress increments(o1MPa). Our results show that minute deformationcan be reliably determined with our ESPI-based experi-mental system. Low stress increments result in verysmall strain increments, thus allowing experimentalapproximation of the local derivatives of the stress–-strain curve. Furthermore, our ability to measureaccurately very small strains allows us to conduct elasticexperiments in which damage does not occur in thesample. Additionally, with the S–S curve slope methodonly one modulus estimate is obtained for eachexperiment, whereas with the compliance method alarge number of statistically independent measurementsare obtained. This has the advantage of providingmultiple results for each sample, allowing improvedstatistics. Furthermore, these features might haveapplication for studying other properties, such as rate-and load-dependant variations of the elastic constants.

We found significant local variations in Young’smoduli to occur within a range of 1–2mm of the equinecortical bone we tested. Axial Young’s moduli variedbetween 17.4 and 23.6GPa, whereas radial and trans-verse Young’s moduli varied between 8.8 and 13.9GPa.Similar variation was previously reported by Ashman etal. (1984). We have shown that the variations found insmall adjacent cubes do not arise from lack of precisionor lack of accuracy of our method. We suggest that inboth horses, structural differences in the meso-scale,such as regions of secondary osteons versus interstitialbone, variable pore distributions, and/or local differ-ences in mineral content have a major impact onlocal stiffness. In fact, when we remounted someof our samples and repeated the experiments, weobtained extremely similar results (data not shown).The broad range of results is however similar to resultspreviously reported in the literature (Currey, 2002,Cowin, 2001).

We confirmed the accuracy of our measurements andvalidity of the results by performing compressionexperiments on small cubes of Ultems (see alsoZaslansky et al., 2005). Measurements made using aprotocol similar to that used for bone yielded values forboth Young’s modulus and Poisson’s ratio which were

ARTICLE IN PRESS

x- and y-strains along a central line of the cube in the direction of compression

-0.00012

-0.0001

-0.00008

-0.00006

-0.00004

-0.00002

0

0.00002

0.00004

position of element (mm)

stra

in x- strain

y- strain

0 0.5 1 1.5 2 2.5 3

(a)

(b)

x- and y-strains along a central line of the cube in the direction of

-0.00012

-0.0001

-0.00008

-0.00006

-0.00004

-0.00002

0

0.00002

0.00004

0.00006

0 0.5 1.5 2 2.5 3

position of element (mm)

stra

in

x - strain

y - strain

1

compression

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 ofcortical bone are however lower than those usuallyreported. In fact, previous studies reported a very largerange of values of Poisson’s ratios in bone, from 0.12 to0.63 (Pithioux et al, 2002; Reilly and Burstein, 1975).The values found in our series of experiments fall withina narrower range. The low values obtained by ourexperiments for Poisson’s ratios in the RA and TAdirections (0.07–0.124) are particularly striking.

In a material with orthotropic symmetry the Poisson’sratio values should be orientation-dependent. As can beseen from Eq. (11), the relationship between thedifferent Poisson’s ratios is related to the associated

Young’s moduli. Table 3 shows that when both uRA anduAR or uTA and uAT were measured in the same bonecube, they were different. Table 4 demonstrates that theratio between these reciprocal Poisson’s ratios and theratio between the corresponding Young’s moduli aresimilar. On the other hand when uRT and uTR were bothmeasured 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 areindeed representative values for Poisson’s ratios ofsecondary osteonal bone.

The large differences between many of our measure-ments of Poisson’s ratio and those often used in theliterature led us to investigate the possibility thatthe non-slip condition between the anvils of the


micro-mechanical loading device and the bone samplescould cause the measured lateral strain to be mislead-ingly low. Several validation studies were conducted toevaluate this scenario.

A compression experiment conducted on a6� 2� 2mm rectangular parallelepiped bone samplealong its long axis yielded similar results for Young’smodulus and Poisson’s ratio when data were obtainedfrom the central 2� 2 region (far from the confinededges) and from a 2� 2 region near the loaded edge.Furthermore, analysis of a finite element model ofcompression of a cube, with boundary conditionssimulating the non-slip edge effect arising in theexperiments of this study showed that in the entireregion of interest on the face of the cube the ratio ofstrain across the direction of loading to the strain at thesame point along the direction of loading yields the truePoisson’s ratio. We therefore conclude that the values ofPoisson’s ratio reported here are accurate.

Osteonal bone is the dominant bone type of adultequine bone (Mason et al., 1995); however it is notknown how this bone behaves in the meso-scale whenloaded under compression. Individual osteons havedifferent mineral contents, and indeed mineral contentvaries continuously throughout their structure. Mineralcontent also varies between Haversian and inter-osteonal areas (Currey, 2002). Hence different stiffnessvalues characterize different sites within the bone, and itis not clear how the load is distributed among these sites.We plan to further modify our system in order toachieve even greater resolution that will allow themeasurement of displacements in the meso-scale of50–500 mm. It will then be possible to determine localdisplacement variations in regions such as newly formedosteons with low mineral content and regions of older,more mineralized osteons. The variation of localdisplacements within individual regions (and the result-ing strain variations) can then be compared.

The results reported here demonstrate that themethod described can yield quantitative measurementsof surface displacements of small cubes of bone loadedin compression while in a water environment. Further-more, these results can be used to calculate the strains,and through them values for Young’s moduli andPoisson’s ratios. Measurements are performed withoutcontact with the sample, and can easily be repeatedmany times on the same sample, since the method isnon-destructive, allowing truly elastic measurements.

In conclusion, we described the determination ofYoung’s moduli and Poisson’s ratios using a novel, non-contact optical method. Small bone samples were testednon-destructively in an aqueous environment, andmeasurements of displacements were performed in threeorthogonal directions. This study showed that signifi-cant variation occurs locally in Young’s moduli andPoisson’s ratios, and the measured Poisson’s ratios of

cortical bone were found to be lower than previouslyreported.

Acknowledgements

S.W. is the incumbent of the Dr. Walter and Dr.Trude Burchardt Professorial Chair of StructuralBiology. A.A.F is the incumbent of the Peter andCarola Kleeman Professorial Chair of Optical Sciences.Support for this research was provided from Grant RO1DE006954 from the National Institute of Dental andCraniofacial Research, and from the Women’s HealthResearch Center to Dr. Stephen Weiner.

We wish to thank Benjamin Sharon, David Leibovitz,Gershon Elazar and Yosef Shopen for excellenttechnical assistance.

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