The Anisotropic Hooke's Law for Cancellous Bone...

Journal of Elasticity53: 125–146, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

125

The Anisotropic Hooke’s Law forCancellous Bone and Wood

GUOYU YANG1, JESPER KABEL2, BERT VAN RIETBERGEN3,ANDERS ODGAARD2, RIK HUISKES4 and STEPHEN C. COWIN1?1The Center for Biomedical Engineering and The Department of Mechanical Engineering,The School of Engineering of The City College and The Graduate School of The City University ofNew York, New York, NY 10031, U.S.A.2Orthopaedic Research Laboratory, Aarhus University Hospital (ÅKH), Nørrebrogade 44,8000 Aarhus C, Denmark.3Institute for Biomedical Engineering, Swiss Federal Institute of Technology, Moussonstrasse 18,8044 Zurich, Switzerland.4Biomechanics Section, Institute of Orthopaedics, University of Nijmegen, P.O. Box 9101,6500 HB Nijmegen, The Netherlands.

Received 19 January 1999; accepted in final form 21 March 1999

Abstract. A method of data analysis for a set of elastic constant measurements is applied to databases for wood and cancellous bone. For these materials the identification of the type of elasticsymmetry is complicated by the variable composition of the material. The data analysis methodpermits the identification of the type of elastic symmetry to be accomplished independent of theexamination of the variable composition. This method of analysis may be applied to any set of elasticconstant measurements, but is illustrated here by application to hardwoods and softwoods, and toan extraordinary data base of cancellous bone elastic constants. The solid volume fraction or bulkdensity is the compositional variable for the elastic constants of these natural materials. The finalresults are the solid volume fraction dependent orthotropic Hooke’s law for cancellous bone and abulk density dependent one for hardwoods and softwoods.

1. Introduction

Many materials are anisotropic and inhomogenous due to the varying compositionof their constituents. The identification of the type of elastic symmetry is com-plicated by the variable composition of the material, which makes the analysisof the elastic constant measurement data difficult. A solution to this problem inwhich identification of the type of elastic symmetry and analysis of the variablecomposition are separated and analyzed independently was described in Cowinand Yang [4]. The method consists of averaging eigenbases, that is to say the bases

? Corresponding author: Fax (212) 787-3757; e-mail: [email protected] Mailing Address: 107 West 86th Street, Apartment 4F, New York, NY 10024,Work Address: Department of Mechanical Engineering, The City College, 138th Street and ConventAvenue, New York, NY 10031, U.S.A.

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126 GUOYU YANG ET AL.

composed of the orthogonal sets of eigenvectors of different measurements of theelasticity tensor, in order to construct an average eigenbasis for the entire data set.This is possible because the eigenbases, composed of eigenvectors, are independentof composition while the eigenvalues are not. The eigenvalues of all the anisotropicelastic coefficient matrices can then be transformed to the average eigenvectorbasis and regressed against their compositional parameters. This method treats theindividual measurement as a measurement of a tensor instead of as a collection ofindividual elastic constant or matrix element measurements, recognizing that themeasurements by different authors will reflect the systematic invariant tensorialproperties of a material, like eigenvectors and eigenvalues. This method for av-eraging different measurements of the anisotropic elastic constants for a specificmaterial has advantages over the traditional method of averaging the individualmatrix components of the elasticity or compliance matrices. Averaging invariantsremoves the effect of the reference coordinate system in the measurements, whilethe traditional method of averaging the components may induce errors due to thevarious reference coordinate systems and may distort the nature of the symmetry.This averaging process explicitly retains the orthonormality of the eigenvectorbasis.

An interesting result that emerged from the Cowin and Yang [4] analysis wasa method dealing with variable composition anisotropic elastic materials whoseelastic coefficients depend upon the particular composition of the material. In thecase of porous isotropic materials, for example, it is customary to regress theYoung’s modulus against the solid volume fraction and obtain expressions for theYoung’s modulusE as a function of the solid volume fractionφ; for exampleE = (constant)φn. The results of Cowin and Yang [4] provided a means to extendthis empirical method to anisotropic materials. In [4] this method was applied tofeldspar and it was discovered that the eigenvectors, but not the eigenvalues, wererelatively independent of material composition. That result is extended here to threenatural porous materials: cancellous bone, hardwood, and softwood. The previousresults and the present work establish this method of analysis as a valid approachto the construction of anisotropic stress-strain relations for other compositionallydependent materials.

This new method of analysis also identifies the type of elastic symmetry pos-sessed by the material. Noa priori assumption as to the type of elastic symmetryis made. The type of symmetry is identified from the character of the eigenvectorsthat are calculated. For example, in the present work the analysis shows that humancancellous bone has orthotropic elastic symmetry at the 95% confidence level.However, the data base we employ for wood incorporated the assumption of or-thotropic material symmetry so this feature is not illustrated by the analysis of thewood data.

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THE ANISOTROPIC HOOKE’S LAW FOR CANCELLOUS BONE AND WOOD 127

2. The Anisotropic Hooke’s Law: Notation

The elasticity tensor may be expressed either as a fourth-rank tensor, with com-ponentsCijkm, in a space of three dimensions or as a second-rank tensor, withcomponentsckm, in a space of six dimensions. The averaging processes are appliedhere to the six eigenvalues of the matrixc and to the eigenbasis ofc; or, equiva-lently, to the inverse ofc, the compliance tensors. The effect is to determine theaverage of the simultaneous invariants ofc and s. The second-rank tensorc in 6dimensions whose components areckm appears in a representation of the stress-strain relations due, in principle, to Kelvin [16, 17], but expressed by Rychlewski[15] and Mehrabadi and Cowin [12] in contemporary linear algebra notation. In thenext section the Kelvin formulation of the generalized Hooke’s law is summarized.In the next paragraph these notations are explained in greater detail.

The anisotropic form of Hooke’s law is often written in the indicial notation asTij = CijkmEkm, where theCijkm are the components of the elasticity tensor. Writ-ten as a linear transformation in six dimensions, Hooke’s law has the representationT = cE, or

T11

T22

T33

T23

T13

T12

=

c11 c12 c13 c14 c15 c16

c12 c22 c23 c24 c25 c26

c13 c23 c33 c34 c35 c36

c14 c24 c34 c44 c45 c46

c15 c25 c35 c45 c55 c56

c16 c26 c36 c46 c56 c66

E11

E22

E33

2E23

2E13

2E12

(1)

in the notation of Voigt [20]. The relationships of the components ofCijkm to thecomponents of the symmetric matrixc are easily derived from these definitions andappear many places in the literature. Introducing new notation, (1) may be rewrittenin the formT = cE where the shearing components of these 6-dimensional stressand strain vectors, denoted byT andE respectively, are multiplied by

√2, andc is

a new six-by-six matrix (Mehrabadi and Cowin [12]). The matrix form ofT = cEis given by

T11

T22

T33√2T23√2T13√2T12

=

c11 c12 c13√

2c14√

2c15√

2c16

c12 c22 c23√

2c24√

2c25√

2c26

c13 c23 c33√

2c34√

2c35√

2c36√2c14

√2c24

√2c34 2c44 2c45 2c46√

2c15√

2c25√

2c35 2c45 2c55 2c56√2c16

√2c26

√2c36 2c46 2c56 2c66

×

E11

E22

E33√2E23√2E13√2E12

(2)

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Table I. The elastic constants of hardwoods from Hearmon [4]. The repetition of a wood speciesin a second row (e.g., Quipo, Mahogany, Ash) indicates a second measurement. The units ofρ areg/cm3 and the units of the elastic constants are GPa= 1010 dynes/cm2.

Species ρ c11 c22 c33 c12 c13 c23 c44 c55 c66

Quipo 0.1 0.045 0.251 1.075 0.027 0.033 0.025 0.226 0.118 0.078

Quipo 0.2 0.159 0.427 3.446 0.069 0.131 0.178 0.430 0.280 0.144

White 0.38 0.547 1.192 10.041 0.399 0.360 0.555 1.442 1.344 0.022

Khaya 0.44 0.631 1.381 10.725 0.389 0.520 0.662 1.800 1.196 0.420

Mahogany 0.50 0.952 1.575 11.996 0.571 0.682 0.790 1.960 1.498 0.638

Mahogany 0.53 0.765 1.538 13.010 0.655 0.631 0.841 1.218 0.938 0.300

S. Germ 0.54 0.772 1.772 12.240 0.558 0.530 0.871 2.318 1.582 0.540

Maple 0.58 1.451 2.565 11.492 1.197 1.267 1.818 2.460 2.194 0.584

Walnut 0.59 0.927 1.760 12.432 0.707 0.936 1.312 1.922 1.400 0.460

Birch 0.62 0.898 1.623 1.7173 0.671 0.714 1.075 2.346 1.816 0.372

Y. Birch 0.64 1.084 1.697 15.288 0.777 0.883 1.191 2.120 1.942 0.480

Oak 0.67 1.350 2.983 16.958 1.007 1.005 1.463 2.380 1.532 0.784

Ash 0.68 1.135 2.142 16.958 0.827 0.917 1.427 2.684 1.784 0.540

Beech 0.74 1.659 3.301 15.437 1.279 1.433 2.142 3.216 2.112 0.912

Ash 0.80 1.439 2.439 17.000 1.037 1.485 1.968 1.720 1.218 0.500

The matrix c is called the matrix ofelastic coefficientsand its inverses, E =sT, s = c−1 is called thecompliancematrix. A chart relating these various nota-tions for the specific elastic coefficients is given in Table I of Cowin and Yang [4] orTable I of Cowin and Mehrabadi [3]. The 6-dimensional orthogonal transformationis represented byQ, which is a second-rank tensor in 6 dimensions that is directlyrelated to an associated orthogonal second-rank tensor in 3 dimensions (Mehrabadiand Cowin [12], Cowin and Mehrabadi [3]); thus the tensor transformation law forc or s to a new or primed coordinate system is

c′ = QcQT or s= QsQT . (3)

The eigenvalues and the eigenvectors ofc(s) are determined from the equation(s)

(c= 31)N = 0, ((s− (1/3)1N = 0)), (4)

where the vectorsN represent the normalized eigenvectors ofc (or s). Sincec (or s)is positive definite it has six positive eigenvalues. These eigenvalues are called theKelvin moduliand are denoted by3i, i = 1, . . . ,6, and are ordered (if possible)by the inequalities31 > · · · > 36 > 0. The eigenvalues ofsare the inverses of theeigenvalues ofc. The eigensystems for various anisotropic elastic symmetries aredescribed in Appendix A of Cowin and Yang [4]. The spectral representation of the

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matricesc ands in terms of the eigenvalues3i, i = 1, . . . ,6, and the normalizedeigenvectors is given by

c=6∑k=1

3kN(k) ⊗ N(k), s=6∑k=1

1

3k

N(k) ⊗ N(k). (5)

The symmetry of (linearly elastic) materials may be characterized by the num-ber and orientation of the planes of mirror, or reflective, symmetry (Cowin andMehrabadi [1, 4]). The material symmetry of wood is an appropriate example; ithas three perpendicular planes of mirror symmetry. One plane is generally perpen-dicular to the long axis of the tree trunk, another is perpendicular to the tangentto the growth ring, and the third is perpendicular to the radial direction associatedwith the growth rings. In colloquial speech, by the ‘grain of the wood’ we meanthe direction along the long axis of the wood fibers, fibers that are generally coin-cident with the long axis of the tree trunk. Here we define ‘wood grain’ as a setof three orthogonal, ordered directions, the first one of which coincides with thelocal ‘colloquial’ grain direction, which is locally the stiffest direction; the secondand third directions are directions orthogonal to each other in the plane perpendic-ular to the ‘colloquial’ grain direction and also represent directions of extrema instiffness in the local region of the wood. These directions are the directions tangentand perpendicular to the growth rings. The existence of these three perpendicularplanes of mirror symmetry, and no others, mean that wood has orthotropic materialsymmetry. We refer to the symmetry coordinate system for an orthotropic materialas the ‘grain’ coordinate system. In the grain coordinate system the compliancematrix may be expressed as

s=

(E1)−1 −ν12(E1)

−1 ν13(E1)−1 0 0 0

−ν21(E2)−1 (E2)

−1 −ν23(E2)−1 0 0 0

−ν31(E3)−1 −ν32(E3)

−1 (E3)−1 0 0 0

0 0 0 (2G23)−1 0 0

0 0 0 0 (2G13)−1 0

0 0 0 0 0 (2G12)−1

.

(6)

The orthotropic elastic coefficients, 1/E1, −ν12/E1, −ν13/E1, −ν21/E2, 1/E2,−ν23/E2, −ν31/E3, −ν32/E3, 1/E3, 1/(2G23), 1/(2G13) and 1/(2G12) may beconsidered either as the components of a fourth-rank tensor in a space of 3 di-mensions or as a second-rank tensor in a space of 6 dimensions (Mehrabadi andCowin [12], Cowin and Mehrabadi [3]). For cancellous bone we refer to the graincoordinate system as the trabecular grain to distinguish it from the wood grain. Bytrabecular grain we mean a set of three ordered orthogonal directions, the first one

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Figure 1. An illustration of the trabecular grain. By trabecular grain we mean a set of threeordered orthogonal directions, the first one of which lies along the local predominant tra-becular direction, which is locally the stiffest direction; the second and third directions aredirections orthogonal to each other in the plane perpendicular to the first direction and repres-ent directions of extrema in stiffness in the local region of the cancellous bone. The specimenis a 7 mm cube.

of which lies along the local predominant trabecular direction, which is locally thestiffest direction; the second and third directions are orthogonal to each other inthe plane perpendicular to the first direction and represent directions of extremain stiffness in the local region of the cancellous bone, see Figure 1. We know thatthese directions are orthogonal because it has been shown that cancellous bone hasorthotropic elastic symmetry (shown by the present data analysis to be at the 95%confidence level for 141 specimens), that is to say, three perpendicular planes (or,alternatively, axes) of mirror or reflective symmetry, exist in each local region ofthe bone tissue.

3. The Elastic Constant Data Bases

In this section we describe the source of the elastic constant data we analyze. Theresults for cancellous bone presented in this work are based upon an analysis ofa data base consisting of 141 human cancellous bone specimens. This data base,reported by van Rietbergen et al. [18, 19] and Kabel et al. [9], is superior to pre-vious data bases because the authors provide the entire set of anisotropic elasticconstants without ana priori assumption of a particular material symmetry andwithout an assumption of the direction in which the maximum Young’s modulusoccurs. This data base is unique in many different ways, the most important of

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which is the large number of specimens and its method of construction, but par-ticularly because it is not based entirely on measurements of real specimens. Thedata base of elastic constants of 141 human cancellous bone specimens employedhere was constructed by imaging real specimens, then computationally determiningtheir elastic constants. We believe that this cyberspace method of construction ismore accurate than the conventional mechanical testing procedures for evaluatingthe elastic constants of human cancellous bone. The determination of the elasticconstants of cancellous bone by conventional mechanical test procedures is verydifficult. The basic problem is that, due to the size of the human body it is difficultto obtain specimens of cancellous bone that are more than 5 mm cubes. The logicalway to test small cubes such as these is compression testing. However compressiontesting is highly inaccurate for cancellous bone because of (1) the frictional endeffects of the platens, (2) the near impossiblility of identifying,a priori, the graindirections in a bone specimen and thus to cut a specimen in the grain directions,(3) the stiffening effect of the platens on the bone near the platens and (4) theunpredictable inhomogeneity of the specimen. These and other difficulties in themechanical testing of cancellous bone are described in Keaveny [11].

The construction of the data base of elastic constants of 141 human cancellousbone specimens employed here was a breakthrough because it provided a relativelyinexpensive method of determining the full set of anisotropic elastic constantsfor a small specimen of cancellous bone by a combination of imaging the spe-cimen (Odgaard et al. [13], Kabel, et al. [9, 10]) and subsequent evaluation ofthe effective elastic constants using computational techniques based on the finiteelement method (van Rietbergen et al. [18, 19], Hollister et al. [8]). A sequenceof loadings was applied to the finite element models of the specimen and theresponses determined (van Rietbergen et al. [18, 19]). The sequence of loadingswas sufficient in number to determine all 21 elastic constants. Thus no materialsymmetry assumptions were made in the determination of the constants.

In this method the actual matrix material of the trabeculae comprising the bonespecimen is assumed to have an axial Young’s modulusEt . The value ofEt maybe fixed from a knowledge of the axial Young’s modulus for the tissue, or fromthe shear modulus about some axis, or by measuring the tissue modulusEt itself.For purposes of numerical calculationEt was taken to be 1 GPa (van Rietbergen etal. [18, 19], Odgaard et al. [13], Kabel et al. [9]). However, since these are linearFE-models, theFE-results can be scaled for any other modulus by multiplyingthe results with the new value ofEt (in GPa). The tissue modulusEt thus is a scalefactor that magnifies or reduces all the elastic constants. The inverse of the tissuemodulus 1/Et multiplies each component in the elastic compliance matrix. Thecancellous bone elastic constant results are presented here as multiples ofEt (cf.,e.g., Tables III and IV).

Once the image of the specimen was in the computer and a finite element meshwas generated, a sequence of loadings (van Rietbergen et al. [18, 19]) was ap-plied to the specimen and the responses determined. The sequence of loadings was

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Table II. The elastic constants of softwoods from Hearmon [4]. The repetition of a wood speciesin several rows (e.g., Spruce) indicates several measurements. The many measurements for Sprucereflect its use in early aircraft. The units ofρ are g/cm3 and the units of the elastic constants areGPa= 1010 dynes/cm2.

Species ρ c11 c22 c33 c12 c13 c23 c44 c55 c66

Balsa 0.2 0.127 0.360 6.380 0.086 0.091 0.154 0.624 0.406 0.066

Spruce 0.39 0.572 1.030 11.950 0.262 0.365 0.506 1.498 1.442 0.078

Spruce 0.43 0.594 1.106 14.055 0.346 0.476 0.686 1.442 1.0 0.064

Spruce 0.44 0.443 0.775 16.286 0.192 0.321 0.442 1.234 1.52 0.072

Douglas Fir 0.45 0.929 1.173 16.095 0.409 0.539 0.539 1.767 1.766 0.176

Spruce 0.50 0.755 0.963 17.221 0.333 0.549 0.548 1.25 1.706 0.07

Pine 0.54 0.721 1.405 16.929 0.454 0.535 0.857 3.484 1.344 0.132

Douglas Fir 0.59 1.226 1.775 17.004 0.753 0.747 0.941 2.348 1.816 0.160

sufficient in number to determine all 21 elastic constants. Thus no material sym-metry assumptions were made in the determination of the constants. Quantitativestereological programs were used to determine the solid volume fractionφ of eachspecimen. These are the data employed in the analysis reported here.

The source of the elastic constant data on hardwood and softwood is easier todescribe; it was taken from Hearmon [5]. These data (Tables I and II) are relativelyunique because they include both the elastic constants and the apparent density ofthe specimens. Note that the data on cancellous bone are given as a function ofsolid volume fractionφ, while those for the hardwoods and softwoods are given asfunctions of bulk or apparent density,ρ. These two quantities are related byρ =γ φ whereγ is the density of the actual solid matrix material (γ is approximately1.9 g/cm3 for human bone).γ is considered to be a constant for these materials, soρ is proportional toφ.

4. Analysis of the Data

The data on the elastic constants of cancellous human bone and their volumefractions were analyzed in five steps. A detailed example for a particular humancancellous bone specimen is given in the Appendix. First, the eigenvalues andeigenvectors associated with the six-by-six matrix of elastic constant data weredetermined for each specimen. Second, the average of the 141 eigenbases wasdetermined (see the Appendix for details). Third, observing that the elements inthe average eigenvector basis associated with other than orthotropic symmetrywere near zero, the eigenvector basis was statistically tested for its closeness toorthotropic symmetry. It was found that the set of 141 specimens had orthotropicsymmetry at the 95% confidence level. Fourth, the data for each specimen were

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Figure 2. A plot of the data on the first eigenvalue for the human cancellous bone data setagainst the solid volume fraction. The vertical scale is dimensionless because the value ofthe first eigenvalue is divided byEt . The horizontal scale (solid volume fraction) is alsodimensionless.

referred to the common average eigenvector basis. The transformation law forsecond-rank Cartesian tensors was employed to transform the elasticity tensor fromits original coordinate system which reflected no symmetry to the common averageeigenvector basis where it reflected orthotropic symmetry except for a few smalldeviations. The elastic constants that were associated with other than elastic ortho-tropic symmetry were near zero and were neglected. The error associated with theneglect of these near-zero terms was less than 3.5% of the largest eigenvalue for thesixth specimen (Appendix) and the maximum error was 4.4% for specimen 36. Thisresult demonstrated that the common average eigenvector basis was indeed com-mon and almost independent of the volume fraction, paralleling the result notedby Cowin and Yang [4] that the eigenvectors for feldspar were independent of thecomposition of those materials. (The proof of that result in Cowin and Yang followsa different numerical argument, however.) Fifth, the six compositionally dependentinvariants were regressed against their volume fractions employing linear log-logrelationships. The rationale for the selection of log-log relationships is given byHodgskinson and Currey [6]. The results of this analysis for cancellous bone,the dependence of the eigenvalues upon the solid volume fraction, are shownin the second column of Table III. In Table III, under each of the eigenvalues,are the squared correlation coefficients(R2) for the correlation of that eigenvaluewith the corresponding set of eigenvalues of the original data. A plot of the dataon the first eigenvalue vs. solid volume fraction is presented in Figure 2. Then,using the spectral representation (5) of the matrix of elastic coefficients in termsof its eigenvalues and eigenvectors, where the eigenvalues are compositionallydependent and the eigenvectors are not, the strain-stress relations were constructed.These relations reflect the explicit dependence upon solid volume fractionφ. The

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Table III. Results of the linear log-log regression of the eigen-values against volume fractionφ for cancellous human bone andthe linear log-log regression of the eigenvalues against apparentdensityρ for hardwoods and softwoods.

Eigenvalue

(R2)Cancellous bone Hardwood Softwood

31

(R2)

1460Etφ1.72

(0.911)28.44ρ1.34GPa

(0.968)32.56ρ0.99GPa

(0.941)

32

(R2)

871Etφ1.89

(0.905)6.11ρ1.95GPa(0.936)

3.92ρ1.46GPa(0.874)

33

(R2)

493Etφ1.92

(0.868)0.86ρ0.75GPa(0.684)

1.79ρ1.74GPa(0.894)

34

(R2)

533Etφ2.03

(0.879)3.94ρ1.23GPa(0.893)

4.55ρ1.27GPa(0.704)

35

(R2)

633Etφ1.97

(0.902)3.14ρ1.37GPa(0.875)

4.01ρ1.36GPa(0.824)

36

(R2)

973Etφ1.98

(0.889)0.83ρ1.21GPa(0.453)

0.166ρ0.66GPa(0.271)

elastic constants obtained are given in the second column of Table IV as a functionof the solid volume fractionφ and a multiple ofEt . Et has the dimension of stressand the other numbers multiplying allφ’s raised to a power are dimensionless. Thesquared correlation coefficients(R2) for the orthotropic elastic coefficients are asfollows: for 1/E1, R

2 = 0.934; for 1/E2, R2 = 0.917; for 1/E3, R

2 = 0.879; for1/(2G23), R

2 = 0.870; for 1/(2G13), R2 = 0.887; for 1/(2G12), R

2 = 0.876; for−ν12/E1, R

2 = 0.740; for−ν13/E1, R2 = 0.841; and forν21/E2, R

2 = 0.666.The representations (Table IV) of the elastic constants for human cancellous

bone are quite interesting and contain revealing insights. The Young’s modulus ofhuman cancellous bone plotted as a function of direction at the volume fractionφ =0.35 is shown in Figure 3. This 3-dimensional closed, peanutshell shaped, surfacereflects the three orthogonal planes of mirror symmetry associated with orthotropy.For an isotropic material the surface shown in Figure 3 would be a sphere. Asthe solid volume fraction of cancellous bone changes, the overall average shapeof the surface in Figure 3 stays the same, but the intercept values of the surfacewith the three principal directions increase or decrease with the volume fraction;that is to say the plot in Figure 3 is simply enlarged or reduced. It is interesting tonote that the anisotropy ratios(E1/E2, E1/E3 andE2/E3) calculated from (3) aremildly decreasing with increasing volume fraction, thus the more dense material isassociated with less pronounced anisotropy.

The data on the hardwoods and softwoods were analyzed in a manner similarto that for cancellous bone. The dependence of the eigenvalues for the wood typesupon the apparent density is shown in the third and fourth columns of Table III.In Table III, under each of the eigenvalues, are the squared correlation coefficients

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Figure 3. The Young’s modulus of human cancellous bone of the 141 specimen averagerepresentation (Table IV) plotted as a function of direction at the highest volume fractionmeasured,φ = 0.35. The scales of the axes are dimensionless because the value of the Young’smodulus is divided byEt .

(R2) for the correlation of that eigenvalue with the corresponding set of eigenvaluesof the original data. The elastic constants obtained for the woods are given in thethird and fourth columns of Table IV as a function of the apparent density. Theresults for the woods are not as strong as the results for the cancellous bone. Thereason for this difference is the difference in the quality of the original data base.The data set on the cancellous bone is high quality data in that it contains all 21elastic constants and the solid volume fraction for 141 specimens.? The resultsconcerning these data are strong because of the large number of specimens; thecentral limit theorem states that the difference between the true mean and theestimated mean depends inversely upon the number of specimens. Thus the truemean and the estimated mean approach each other as the number of specimensincreases. The data on wood, given in Tables I and II, is based on about one-tenththe number of specimens. In addition, the assumption of orthotropic symmetry wasincorporated into the data base for wood, while the data base for human cancellousbone is independent of any assumption concerning elastic material symmetry.

5. Discussion

The superiority of this model for representing the compositional dependence ofthe elastic constants is established by comparing it with the results obtained usingother models. In the case of porous materials the customary model is to assumeelastic isotropy and to regress the Young’s modulus against solid volume fractionand obtain expressions for the Young’s modulusE as a function of the solid volumefractionφ; for exampleE = (constant)φn. This is done for cancellous bone and

? In the preparation of Cowin and Yang [4] an exhaustive search of published data bases on aniso-tropic elastic constants of materials was accomplished and no data base was found that approachedthe quality of this cancellous bone data base.

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Table IV. The functional dependence of the orthotropic elastic constants ofbone upon solid volume fraction,φ, and the functional dependence of theorthotropic elastic constants of the hardwoods and softwoods upon apparentdensity,ρ. Note thatρ = γ φ whereγ is the density of the actual solid matrixmaterial (γ is approximately 1.9 g/cm3 for human bone).γ is considered to bea constant for these materials, soρ is proportional toφ.

Elastic Constant Cancellous bone Hardwood Softwood

E1 1240Etφ1.80 1.307ρ0.89GPa 2.05ρ1.71 GPa

E2 885Etφ1.89 2.97ρ1.50GPa 3.14ρ1.59 GPa

E3 528.8Etφ1.92 27.63ρ1.41 GPa 32.01ρ1.01 GPa

2G23 533.3Etφ2.04 3.94ρ1.23GPa 4.56ρ1.27 GPa

2G13 633.3Etφ1.97 3.14ρ1.37GPa 4.10ρ1.36 GPa

2G12 972.6Etφ1.98 0.825ρ1.21GPa 0.166ρ0.66 GPa

ν23 0.256φ−0.09 0.024ρ−0.73 0.028ρ+0.18

ν32 0.153φ−0.05 0.227ρ−0.82 0.286ρ−0.40

ν13 0.316φ−0.19 0.016ρ−0.76 0.019ρ0.10

ν31 0.135φ−0.07 0.345ρ−0.23 0.300ρ−0.60

ν12 0.176φ−0.25 0.724ρ+0.90 0.269ρ−0.17

ν21 0.125φ−0.16 1.645ρ+1.50 0.412ρ−0.29

other materials, particularly sintered materials. We have not located a reference tothis type of model for wood. For cancellous bone, when the traditional model isemployed (Rice et al. [14], Cowin [2], Hodgskinson and Currey [6, 7]) the squaredcorrelation coefficients(R2) are in the range of 0.4 to 0.70; with the present modelthe squared correlation coefficient is 0.934 for the largest Young’s modulus, aboutone-third higher. However the reader should keep in mind that the squared correl-ation coefficients we report were not obtained directly from experimental data aswere the earlier ones. By reminding the reader of this fact we do not mean to implythat the method employed to obtain the elastic constants reported in van Rietbergenet al. [18, 19] and Kabel et al. [9] is any less accurate, only different. In fact, webelieve that the data reported by van Rietbergen et al. [18, 19] and Kabel et al.[9] are superior to previous data because they provide the entire set of anisotropicelastic constants without ana priori assumption of a particular material symmetryand without an assumption of the direction in which the maximum Young’s mod-ulus occurs. The data bases employed by Rice et al. [14] (see also Cowin [2])consisted generally of uniaxial compression tests of small cubes of cancellous bonein which only the Young’s modulus was reported. In these studies the direction ofthe maximum Young’s modulus was assumed or estimated.

The results presented here validate the method of analysis developed in Cowinand Yang [4], in which the identification of the type of elastic symmetry is accom-plished independent of the examination of the variable composition of the material,

230245.tex; 30/06/1999; 11:40; p.12


as a valid approach to the construction of anisotropic stress-strain relations for otheranisotropic and compositionally variable materials.

Appendix: Analysis of the Data Base of Elastic Constant Measurements

The purpose of this appendix is to present a step-by-step summary of the numer-ical analysis performed on the human cancellous bone data base by describingthe analysis performed on one vertebral specimen. The specimen selected was thesixth specimen. The specimen was imaged, a finite element model of the trabecularstructure was constructed (van Rietbergen et al. [18, 19]), and, by analysis of theresponse of the model to various loading situations, the matrix of elastic constants(i.e., the elasticity matrix) for that particular specimen was determined. The actualspecimen is shown in Figure 1. For the sixth specimen this compliance matrix isgiven in the Voigt notation in the original coordinate system by:

s= 1

Et

0.141 −0.041 −0.035 −0.018 0.027 0.076

−0.041 0.219 −0.033 0.05 −0.0093 0.05

−0.035 −0.033 0.129 0.026 −0.0058 −0.013

−0.018 0.05 0.026 0.429 0.113 −0.036

0.027 −0.0093 −0.0058 0.113 0.309 0.016

0.076 0.05 −0.013 −0.036 0.016 0.408

, (A1)

whereEt is the tissue modulus (see Section 3) and is equal to 1 GPa. The elasticitymatrix in the Voigt notation is obtained from (A1) by inversion, thus

c=s−1=Et

9.86 2.773 3.147 −0.115 −0.581 −2.072

2.773 5.952 2.346 −0.926 0.382 −1.267

3.147 2.346 9.35 −0.851 0.315 −0.662

−0.115 −0.926 −0.851 2.805 −1.08 0.399

−0.581 0.382 0.315 −1.08 3.712 −0.168

−2.072 −1.267 −0.662 0.399 −0.168 3.016

. (A2)

The double index Voigt notation employed in these two equations does not producea tensor and the analysis to be performed requires tensors. The elasticity matrix inthe Voigt notation,c, may be converted to the second-rank tensor notation (in sixdimensions)c by multiplying the three-by-three matrices in the upper right andlower left hand corners of the six-by-six matrixc by

√2 and the three-by-three

matrix in the lower right hand corner of the six-by-six matrixc by 2, as may beseen from a comparison of equations (1) and (2), thus

230245.tex; 30/06/1999; 11:40; p.13


c= Et

9.86 2.773 3.147 −0.162 −0.821 −2.93

2.773 5.952 2.346 −1.309 0.54 −1.791

3.147 2.346 9.35 −1.204 0.445 −0.936

−0.162 −1.309 −1.204 5.61 −2.159 0.798

−0.821 0.54 0.445 −2.159 7.425 −0.335

−2.93 −1.791 −0.936 0.798 −0.335 6.032

. (A3)

This elasticity matrix is the matrix of components of a second-rank tensor in sixdimensions rather than the elasticity matrix in the Voigt notation and follows theconversion rules outlined above. The elasticity matrix was then sorted so thatc11 >

c22 > c33. This was accomplished by a sequence of coordinate transformations thatpermuted the axis labels, e.g. 1− > 2,2− > 3,3− > 1, etc. For the sixth specimenthe permutation was 1− > 1,2− > 3,3− > 2, thus

c= Et

9.86 3.147 2.773 −0.162 −2.93 −0.821

3.147 9.35 2.346 −1.204 −0.936 0.445

2.773 2.346 5.952 −1.309 −1.791 0.54

−0.162 −1.204 −1.309 5.61 0.798 −2.159

−2.93 −0.936 −1.791 0.798 6.032 −0.335

−0.821 0.445 0.54 −2.159 −0.335 7.425

. (A4)

Step1. Find the eigenvalues and eigenvectors:

Using standard contemporary mathematical analysis programs such as Math-Cad, MatLab, Maple, Mathematica or MacSyma, the eigenvalues and eigenvectorsof (A4) are calculated:31 = 15.75, 32 = 6.958, 33 = 3.69, 34 = 4.515, 35 = 3.922,

36 = 9.396,

N(1) =

0.64

0.542

0.396

−0.162

−0.333

0.047

, N(2) =

−0.278

0.773

−0.117

−0.11

0.5

−0.223

, N(3) =

−0.238

0.0068

0.602

0.712

−0.039

0.268

, (A5)

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N(4) =

0.472

0.036

−0.511

0.398

0.242

0.546

, N(5) =

0.313

−0.287

0.441

−0.216

0.76

−0.019

, N(6) =

−0.368

0.157

0.106

−0.499

0.026

0.761

.

After this sorting operation the eigenvectors for this specimen were similar in struc-ture to the eigenvectors calculated for other specimens. By similar we mean thateach set of eigenvectors followed a similar pattern. The typical pattern was thatthe first three components of the first eigenvector are all positive, the second com-ponent of the second eigenvector and the third component of the third eigenvectorare positive and greater in magnitude than any of the other components of thateigenvector. For the fourth, fifth and sixth eigenvectors the fourth, fifth and sixthcomponents, respectively, are nearer to the value+1 than any other component ofthat eigenvector. In a few cases these pattern guidelines were not satisfied and somejudgment was needed in the arrangement of the eigenvectors.

Step2. Find the average eigenvectors:

The nominal average (NA) of the eigenvectorsNNAk associated with a particu-

lar eigenmode, is the sum of all 141 eigenvectors associated with that particulareigenmode divided by 141,

NNAk ≡

1

141

141∑Y=1

NYk .

The results for all six eigenmodes are:

NNA(1) =

0.791

0.396

0.237

0.0017

0.031

0.0067

, NNA

(2) =

−0.365

0.703

0.144

0.026

−0.035

−0.032

, NNA

(3) =

−0.137

−0.238

0.759

0.011

0.033

−0.0006

, (A6)

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NNA(4) =

0.0018

−0.0064

−0.015

0.797

0.014

−0.0064

, NNA

(5) =

−0.039

0.028

−0.05

−0.029

0.753

−0.003

, NNA

(6) =

−0.032

0.02

0.0048

−0.026

0.0008

0.773

.

The nominal averageNNAk of eigenvectors given above is not, in general, an or-

thonormal basis. To obtain the average orthonormal eigenbasisNAVGk nearest in a

least squares sense, to the eigenbases of all individual 141 specimens, Cowin andYang [4] (Equation 16) show that the nominal average must be multiplied by atensor which is the inverse square root of the tensor or open product ofNNA

k withitself:

NAVGk ≡

6∑q=1

NNAq ⊗ NNA

q

−1/2

NNAk ;

thus from (A6) theNAVGk are given by

NAVG(1) =

0.867

0.417

0.269

0.0035

0.04

0.015

, NAVG

(2) =

−0.468

0.863

0.181

0.022

−0.044

−0.041

, NAVG

(3) =

−0.16

−0.282

0.944

0.014

0.053

−0.0027

, (A7)

NAVG(4) =

0.0078

−0.015

−0.02

0.999

0.028

0.021

, NAVG

(5) =

−0.047

0.036

−0.052

−0.028

0.996

−0.0042

, NAVG

(6) =

−0.033

0.029

0.0061

−0.021

0.0014

0.999

.

Step3: Determine the type of elastic symmetry of these bone specimens:

There are eight possible types of linear elastic material symmetry (Cowin andMehrabadi [3]). To find the type of elastic symmetry characteristic of these 141

230245.tex; 30/06/1999; 11:40; p.16


specimens we compared the eight eigenbases of these symmetries with the averageeigenbasis (A7). The result of this comparison of the eight candidate eigenbasesof these symmetries given in Mehrabadi and Cowin [12] shows that the elasticmaterial symmetry with the closest eigenbasis is orthotropy. Two of the seven otherelastic material symmetries (monoclinic and triclinic) are less than orthotropicand therefore include orthotropic as a special case; the symmetry of a materialis designated by the greatest symmetry it exhibits. The other five elastic mater-ial symmetries (cubic, tetragonal, trigonal, transverse isotropy (hexagonal ) andisotropy) are greater than orthotropy. These five symmetries may be shown to beinconsistent with the data by comparing their eigenbases (Mehrabadi and Cowin[12]) with (A7). Alternatively they may be shown to be inconsistent with the databy comparing properties of the set of orthotropic elastic constants obtained with theproperties of the elastic constants of these lesser symmetries. A material symmetrygreater than orthotropy and close to orthotropy is transverse isotropy; in order foran orthotropic material to be transversely isotropic about, say the 3-axis, it wouldbe necessary for the orthotropic symmetry coefficients to satisfy the following fourrelationships:E1 = E12, ν13 = ν23, G13 = G23 andG12 = E1/(2(1+ ν12)). Itis easy to see that the elastic constants presented in (3) or (4) do not satisfy theserelationships about the 3-axis (nor the corresponding relationships about the 1- orthe 2-axis) and therefore there is no justification for the reduction to the greatersymmetry of transverse isotropy. Cubic, tetragonal, trigonal, and isotropy may alsobe excluded by parallel arguments to this argument.

The eigenbasis for orthotropic symmetry has the form (Mehrabadi and Cowin[12]);

NORTH(1) =

N1(1)

N2(1)

N3(1)

0

0

0

, NORTH

(2) =

N1(2)

N2(2)

N3(2)

0

0

0

, NORTH

(3) =

N1(3)

N2(3)

N3(3)

0

0

0

, (A8)

NORTH(4) =

0

0

0

1

0

0

, NORTH

(5) =

0

0

0

0

1

0

, NORTH

(6) =

0

0

0

0

0

1

,

where

(N1(m))

2+ (N2(m))

2+ (N3(m))

2 = 1 for m = 1,2,3.

230245.tex; 30/06/1999; 11:40; p.17


Comparison of (A7) with (A8) shows that, in order for the average eigenbasis (A7)to be orthotropic we may neglect components in (A7) whose values are very small,and approximate 0.999 by 1, to obtain the following estimate of the average (AVGEST):

NAVG EST(1) =

0.867

0.417

0.269

0

0

0

, NAVG EST

(2) =

−0.468

0.863

0.181

0

0

0

, NAVG EST

(3) =

−0.16

−0.282

0.944

0

0

0

,

(A9)

NAVG EST(4) =

0

0

0

1

0

0

, NAVG EST

(5) =

0

0

0

0

1

0

, NAVG EST

(6) =

0

0

0

0

0

1

.

The neglect of these small terms to obtain the eigenbasis for orthotropic symmetryis justified by use of a one-sided studentz test, with known standard deviation, atthe 95% confidence level. This test was applied to rotational difference measuresdescribed in Cowin and Yang [4]. This result is of considerable significance be-cause the type of elastic symmetry was not assumeda priori; the type of elasticsymmetry was obtained by matching the average eigenbasis obtained from the datato the typical form of the eigenbasis for orthotropic elastic symmetry. This demon-strates that these 141 specimens of human cancellous bone have elastic orthotropicsymmetry at the 95% confidence level.

Step4. Transformation of the matrix of elasticity tensor components to the canon-ical coordinate system for orthotropic symmetry:

The matrix of elasticity tensor components (A4) for the sixth specimen willbe transformed to the average eigenbasisNAVG

k given by (A7) using the tensortransformation rule for Cartesian second-rank tensors, a transformation rule thatemploys the orthogonal transformation from the original basis used for speci-men six to the basisNAVG

k given by (A7). The tensor transformation rule isc′ =

230245.tex; 30/06/1999; 11:40; p.18


QcQT , where the orthogonal tensorQ is obtained from the open or tensorproduct of the basesNk, given by (A5), andNAVG

k , given by (A7), thus

Q =

0.725 0.088 −0.131 0.458 0.355 −0.338

0.116 0.884 0.307 0.065 −0.296 0.136

0.272 −0.078 0.643 −0.523 0.471 0.101

−0.275 −0.202 0.648 0.429 −0.176 −0.498

−0.552 0.329 −0.079 0.223 0.73 0.0006

0.083 −0.235 0.219 0.528 0.031 0.781

, (A10)

andc′ is given by

c′ = Et

13.477 3.034 2.533 −0.006 0.472 0.032

3.034 8.228 1.867 0.064 0.088 0.125

2.533 1.867 4.668 0.0041 0.093 0.058

−0.006 0.064 0.0041 4.521 0.024 −0.104

0.472 0.088 0.093 0.024 3.946 0.033

0.032 0.125 0.058 −0.104 0.033 9.395

. (A11)

Neglecting the elements with small values in the three-by-three matrices in theupper right and lower left hand corners of the six-by-six matrix (A11) as well as theoff-diagonal elements in the three-by-three matrix in the lower right hand cornerof the six-by-six matrix (A11), the transformed elastic matrix in the canonicalcoordinate system for orthotropic symmetry is given by

c′EST= Et

13.477 3.034 2.533 0 0 0

3.034 8.228 1.867 0 0 0

2.533 1.867 4.668 0 0 0

0 0 0 4.521 0 0

0 0 0 0 3.946 0

0 0 0 0 0 9.395

. (A12)

This result could also have been obtained by calculating the values of the eigenval-ues referred to theNAVG EST

k eigenbasis and then employing the spectral representa-

tion of c′EST in terms of its eigenvalues and eigenvectors,c′EST=6∑k=1

3′kNAVG ESTk ⊗

NAVG ESTk , to obtain the result (A12).

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In order to evaluate the accuracy of the representation (A12) for (A11) weinvestigate the differenceD between usingNAVG EST

k and NAVGk for the reference

eigenbases:

D =6∑k=1

3′kNAVG ESTk ⊗ NAVG EST

k −6∑k=1

3kNAVGk ⊗ NAVG

k . (A13)

The criterion we employ to evaluate the accuracy of the representation (A12) for(A11) is to say that (A12) is a good representation for (A11) if the largest compon-ent of D divided by the largest eigenvalue of (A11) is small. The value ofD forspecimen six is given by:

D=

−0.02 0.017 −0.0066 0.006 −0.472 −0.032

0.017 −0.014 0.004 −0.064 −0.088 −0.125

−0.0066 0.004 −0.012 −0.0041 −0.093 −0.058

0.006 −0.064 −0.0041 −0.0063 −0.024 0.104

−0.472 −0.088 −0.093 −0.024 −0.024 −0.033

−0.032 −0.125 −0.058 0.104 −0.033 0.0012

. (A14)

The largest number in this matrix is 0.472. If we divide this number by the valueof the maximum eigenvalue, 15.75, the estimate of error is 3%. Employing thesame method we found that the maximum error for all 141 specimens was 4.4%and it occurred in the data for specimen 36. The error for the data from most ofthe specimens was less than 3.5%. The calculation of error has presented a prob-lem because it can be done in many different ways. For example, the estimate ofmaximum error for this specimen is 12.8% if we consider the error relative to thesmallest, rather than the largest, eigenvalue. Thus, a statement about error onlyhas meaning in relation to the method of calculation. It appears reasonable to usto base our estimate of error on the largest eigenvalue because that number willlikely be the most significant in most situations. A disadvantage of this methodof error calculation, pointed out by a referee, is that the error will be different forthe same matrix if it applied to the inverse if the matrix. In this case the errorestimate will be based on the smallest rather the largest of the eigenvalues of theoriginal matrix. Another method of calculating the error was employed by vanRietbergen et al. [18]. That method uses a matrix norm which includes all theerror from each matrix component. Using van Rietbergen’s method we find thatthe maximum error is 47.5% and it occurred for specimen 39 although, for all but5 of the 141 specimens, the error was less than 25%. The van Rietbergen methodappears to make the error appear larger than it is due to the use of a matrix norm.

Step5. Analysis of the eigenvalue dependence upon volume fraction and the finalrepresentation:

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The six compositionally dependent invariants were regressed against their volumefractions using linear log-log relationships. The results of this analysis, the depend-ence of the eigenvalues upon the solid volume fraction, are shown in Table III.Then using the spectral representation of the matrix of elastic coefficients in termsof its eigenvalues and eigenvectors, (5), where the eigenvalues are compositionallydependent and the eigenvectors are not, the strain-stress relations were constructed.These relations reflect the explicit dependence upon volume fraction. The squaredcorrelation coefficients(R2) for the orthotropic elastic coefficients are as follows:for 1/E1,R2 = 0.934; for 1/E2,R2 = 0.917; for 1/E3,R2 = 0.879; for 1/(2G23),R2 = 0.870; for 1/(2G13), R2 = 0.887; for 1/(2G12), R2 = 0.876; for−ν12/E1,R2 = 0.740; for−ν13/E1, R2 = 0.841; and forν21/E2, R2 = 0.666.

Acknowledgments

This work was supported by NSF Grant No. CMS-9401518, by grant number665319 from the PSC-CUNY Research Award Program of the City University ofNew York, by the Netherlands Foundation for Research (NWO/Medical Sciences)and the Netherlands Computing Facilities Foundation (NCF). The authors thankSusannah Fritton for comments on an earlier version of this paper.

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