Donald t. Reilly 75

8/6/2019 Donald t. Reilly 75

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J. Biomechanics, 1975, Vol. , pp. 93405. Pergamon Press. Printed inGreat Britain

THE ELASTIC AND ULTIMATE P ROP ER TIES OFCOMPACT BONE TISSUE?

DONALD T. REILLY and ALBERT H. BURSTEINBiomechanics Laboratory, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio44106, U.S.A.Abstract-The use of a transversely isotropic model is tested for the elastic behavior of bovine andhuman bone and the five independent constants of this model are determined. The accuracy of themodel is tested for eight cases by comparing the off-axis modulus predicted by a rotation of stiffnessmatrix with an experimentally determined off-axis modulus.Ultimate properties are presented for bovine and human bone for tension, compression, andtorsional loads. A Hankinson type failure criterion is proposed for off-axis ultimate stress and thispredicted value compared with experimental values for nine cases.

INTRODUCTIONThe elastic and ultimate characteristics of boneare a description of the response of the tissue, in amechanical sense, to stress incurred as it performsits biologic structural function. Those loadingcycles that do not cause permanent irreversiblechanges occur within the region of elastic tissuebehavior. The response of the tissue in this regionto external loads, that is, the relation betweendeformations and applied forces, is characterizedby the elastic constants. In order to describe themechanical response of bone in the elastic range,Lang (1970) postulated a five elastic constant ortransversely isotropic model for the elastic natureof bone and determined the constants using anultrasonic technique. A transversely isotropicmaterial exhibits one set of elastic properties in onedirection and a second set of elastic properties inthe two directions (or plane) perpendicular to thatdirection. Past work on the anisotropic mechanicalproperties of bone is difficult to compare consider-ing the varying methods of testing and is summar-ized in the review by Reilly and Burstein (1974a).Since very little plastic deformation was shown inall work previous to Burstein and others (1972), onecan directly criticize the ultimate values obtained.Indirectly, one could speculate that since the bonespecimen was not wet enough to undergo plasticdeformation, its elastic properties might also havebeen altered.

One recent paper by Pope and Outwater (1974)bears special mention since it concerns itself withthe topic of this paper yet contains seriousmisconceptions in its approach to the testing ofboth elastic and ultimate properties of bone.Mechanical properties of bovine bone were investi-

*Received 2 Au gust 1974.t This investigation was supported by an NSF GrantNo. GK-37023X; NIH Grants, Nos. Hd-00669-11; and AM16058-02; and NFWO Grant No. S. 2/S-A.O.G./JO.E6Z.

gated by three point bending of machined corticalspecimens. Analysis of the load-deformation of thespecimens was done using linearly elastic beamtheory. This brings up two problems. First, in thechoice of specimen dimensions the authors haveintroduced the problems of biaxiality. The only hintof specimen dimensions was given when theyclaimed that the biaxial stress problem was reducedby maintaining the thickness/width ratio (set) at


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394 D. T. REILLY etal.Outwater took some unfounded steps. They at-tempted to develop a formulation for the fracturestress as a function of the angle rotated from thelong axis of the bone. They substituted theirexperimental values for strength ,derived frombending tests into this formula which they clearlystated assumes uniaxial tension strengths. Thisformulation was further confused by their substitu-tion of an ultimate shear strength value for wethuman femur where a value for their tested bonetype (bovine tibia) was called for. Hence, whiletheir topic was an important one, their expositionwas unclear and unsound.

This study has five specific objectives:Elastic properties

1. To verify the existence of an isotropic plane,perpendicular to the long axis of the bone forbovine Haversian bone and therefore, the suitabil-ity of a transversely isotropic model for thishistological bone type.

2. To determine the five elastic constants for atransversely isotropic model for bovine and humanbone.

3. To test the accuracy of the transverselyisotropic model by predicting the elastic modulusexpected in off-axis tension and compression tests.Ultimate properties

4. To determine the ultimate properties ofhuman and bovine bone tissue and the variation ofthese properties in different directions.

5. To formulate a failure criterion for uniaxialtension or compression loads.

MATERIALSSpecimen preparationSpecimens were machined from the cortex of themiddle third of fresh frozen long human and bovinebones. Since mechanical properties were determined as afunction of direction, a reference system had to be chosenfor the orientation of the specimens in the cortex. TheCartesian coordinate system chosen (Fig. 1) takes thex-axis parallel to the long axis of the bone and a specimenmachined with its long axis in this direction was termed alongitudinal specimen. The y-axis was the circumferentialdirection while the z-axis was the endosteal-periostealdirection. Specimens with their long axis in the y directionare called transverse specimens and specimens with theirlong axis in the z direction are called radial specimens.Off-axis bovine specimens were also prepared with theirlong axis in the x-y plane and y-z plane at angles of 30and 60 to the x-axis and y-axis respectively. Bonespecimens from both human and bovine bones wereprepared by first thawing frozen bones following theprocedure outlined by Reilly and others (1974b). At therectangular parallelepiped stage of fabrication, the his-tological types of bovine bone was checked. Both ends ofthe 15 mm block were ground flat and stained withbrilliant green stain to highlight structure. If a mixed typeof histology was noted (at 30x) a final specimen was notmilled. As we have noted previously, the possibility of astrain concentration at interfaces between differenthistological bone types may prejudice the results.Therefore, bovine bone values are for completely

X-longitudinalY-TransverseZ-RadialFig. 1. Cartesian coordinate system chosen for corticalbone specimens.

Haversian or laminar type. Bones with a cortex too thin toproduce the 5 mm outside dimensions of the abovespecimens were cut into long rectangular parallelepipeds(square cross section of approx. 3 mm) and used fortorsion tests for the shear modulus.The nonlinear response of the material in the torsiontests necessitated round cross section specimens for thedetermination on the ultimate torsion strength (see sectionbelow on torsion testing). The 5 mm square cross sectionparallelepipeds were used for fabrication of thesespecimens. They were placed in a small lathe and areduced section was turned under a constant water flow(Fig. 2).Haversian bone with its osteons oriented in a preferreddirection has an axis of symmetry and therefore,histologically, an isotropic plan perpendicular to this axisof symmetry. We attempted to fit the transverselyisotropic material model (five independent constants) tothe inherent histological symmetry. Figure 3 shows anidealized block of Haversian bone with the chosenCartesian coordinates. The longitudinal direction is alongthe osteonal axis and according to the transverselyisotropic model, the elastic modulus (E) nd Poissonsratio (u) in the plane perpendicular to the longitudinaldirection should be the same regardless of specimenorientation. Therefore, to test the transversely isotropicmodel, specimens were cut in the isotropic plane (y-z) forbovine Haversian bone. (The required size of the

Fig. 2. Dimensions for the circular torsion specimen.


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The elastic and ultimate properties of compact bone tissue 395xt

,0steon

Fig: 3. Orientation of x-y-z axes with respect to theosteon systems.

specimens precluded this test in human cortical bone).The longitudinal specimens gave two more of theindependent constants E and Y. Tension and compres-sion tests at a strain rate of 0.02-0.05 see- wereperformed on longitudinal, transverse, radial and inter-mediate angle (30 and 60) specimens taken from theplane of isotropy and the x-y plane.

Testing machinesMETHODS

Uniaxial tension or compression testing was done on atesting machine incorporating a loading pendulum on ashaft to which a displacement cam was attached describedin Reilly and others (1974b). Torsion tests were run on atesting apparatus described by Burstein and Frankel(1971).

Extensometers were used for the determination of bothlongitudinal and transverse (Poissons) strain on longitudi-nal, transverse, and radial specimens. Only longitudinalstrain was measured for off-axis specimens. Strain wascalculated by dividing the change in dimension by the

When possible, a case consisted of specimens from asingle bone. If enough specimens from a single bone werenot available to warrant a statistically meaningful group,specimens from several bones were combined (noted inTables). For instance, the approximately twenty speci-mens needed for an off-axis type of test might come fromthree bones with cortical material randomly assigned tothe four directional (0. 30, 60, 90) groups.

Fig. 5. The Poissons extensometer shown in detail. Thedowel pins allow the extensometer to hang from the bonespecimen while recording change in width in only onedirection.

gage length of the extensometer. The extensometers wereinstrumented with 2000 Cl semiconductor type straingages in a four element bridge. They were designed to beclipped to the specimen gage length quickly and theirattachment did not entail any procedure which wouldchange the bone specimen surface (e.g. gluing or drying),and thus could be used on wet bone. The extensometerused for longitudinal strain measurement (strain in thedirection of the tensile or compression forces) wasdescribed by Reilly and others (1974b).The extensometer used for Poissons strain (v forlongitudinal loading and v for transverse loading) isshown in Fig. 4 on the left hand side. Here again the springsteel clip portion, instrumented with strain gages, affordeda light preload against the specimen. Two steel dowel pinswere used so that Poissons strain in the perpendiculardirection would not be constrained or load the extensome-ter. Any Poissons strain in the direction perpendicular tothat of interest would. only cause rotation and notdeflection of the extensometer due to jaw design shown inFig. 5.

DATA RECORDING AND REDUCTIONRecording of data from the load cell and extensometersfor the tension and compression tests were done on astorage oscilloscope. A photograph of the trace was thentaken.A stylized and an actual load deformation curve for atension test of a longitudinal specimen are shown in Fig. 6.Straight lines were fitted by eye to the two portions of thecurve and the following points used for reduction of thedata. The point of intersection of the two fitted straightlines were chosen as the yield point. This method wasconvenient and gave repeatable results for the yield pointwhich represented approximately the same value as a0.2% offset strain method. The stress at this point wasfound by dividing the load (on the vertical axis) by theoriginal cross sectional area of the specimen and ishereafter called the yield stress (u>).The slope of the straight line from the origin to the yieldpoint was used for determination of the elastic modulus(E in longitudinal direction, E in transverse direction)after appropriate conversion of load and area to stress anddeformation to strain.The slope of the second straight line fitted to the loadcurve again expressed as the ratio of stress to strain wasused for the determination of what was defined as thestrain hardening modulus (S), a term borrowed fromthe engineering description of metallic behavior.The ultimate stress (u..) was found from the highestload (P..) carried by the specimen before fracture.divided by the initial cross sectional area. Ultimate strain(E.,,) was calculated from the deformation undergone tothe fracture point (IL.,,) divided by the initial length of thespecimen gage section.Similarly a stylized and an actual load deformationcurve is shown for a compression test of a longitudinalspecimen in Fig. 7. The initial portion of the curve wasfitted by eye with a straight line whose slope representedthe elastic modulus (E or E) with appropriate transfor-mation to stress and strain.The ultimate compression stress (u,,J was found fromthe highest load supported by the specimen beforefracture divided by the initial cross sectional area. Manyof the human specimens tested had the type of curveshown in Fig. 7 for compression load-deformation and theproblem of ultimate strain determination arose since thecontinued horizontal deflection of the trace representedimpaction of the fracture pieces after the highest loadpoint. The ultimate compressive strain to fracture wastherefore defined as that strain calculated from thedisplacement undergone by the specimen when itsupported the highest load.The shear modulus was determined using a torsional


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3% D. T. REILLY et al.test. A straight line fitted to the initial slope of the torquedisplacement curve represents the product of the shearmodulus (G) and the torsional constant (.I).Thus:

$= GJ,where T = torque and 0 = angle of twist per unit length.For the circular cross sectional specimens, the value for _Iis simply the polar moment of inertia (&rr4, where r wasthe radius). The torsional constant for the nominallysquare cross section however was approximated by:

J = 16a3b $-StanhE >where a = half of smaller dimension of cross section andb = half of larger dimension of cross section, after Wang(1953).Determination of the ultimate shear strength in thetorsion test when the material behaves in a linearly elasticfashion is not difficult. Linear elastic response was foundto be the case for the torsion test when laminar bovinebone was used. Then the shear stress (which for therectangular cross section would be highest at the midpointof the longest side) was found using the series given byWang (1953):

T~x=~[1-~(~~(2n+I)lcoshkb)] nwhere r,,. = maximum shear stress, k,, = (2n + 1)/2a,T = torque at failure and .I, a and b, are defined above.The validity of the torsion test to determine shearstrength and the points of maximum shear stress havebeen discussed by Burstein and others (1973).Torsional tests performed on human and bovineHaversian bone specimens showed the material had anonlinear torque versus angular deformation curve.Determination of the shear modulus as described above,was taken from the initial slope of the curve. The ultimateshear stress, however, was found using the elegantsolution described by Nadai (1950). This solution holdsonly for the circular cross section and entails a graphicalinterpretation of the load-deformation curve but is anexact solution given as:

where a = radius of specimen, 0 = angle of twist/unitlength and M = torque at fracture. Then from a torquedeflection curve (Fig. 8) Nadai showed:&$(Me)=@+3M,

where D is taken from load deformation curveas in Fig. 8.In those tension or compression tests in whichPoissons ratiojvas found, the load deflection curves forboth longitudinal and transverse strain were displayed onthe same oscilloscope screen through a dual channel input(Fig. 9). The trace on the left hand side is the familiar load(vertical axis) vs deformation in the direction of the load(horizontal axis) while the trace on the right hand siderepresents, simultaneously, load (vertical axis) vs defor-mation perpendicular to the direction of the load(horizontal axis). Positive is to the right and hence thedeflection here requests a reduction in cross sectional areaunder a tensile load. A straight line was fitted by eye tothis curve and the ratio of the reduction in width to

Angle of twist/unit lengthFig. 8. Torsional load-deformation curve showing graphi-cal solution. After Nadai (1950), see text.

elongation at the same load below the yield point gavePoissons ratio.RESULTS

Elastic propertiesThe results of the experiment for the initialobjective of verifying the isotropic plane in bovineHaversian femora (y-z plane) are shown in Table1. A comparison of the constants in the transverseand radial direction with a t-test show no significantdifference (at the P < O-05 level) for both E and u.Also an analysis of variance for the data for thefour directions showed no significant added var-iance (at the P < 0.05 level) for the two elasticconstants. Thus, the values of the elastic constantsfor all directions in the y-z plane are not shown tobe different and the y-z plane is a plane of isotropyor has constant elastic properties regardless ofdirection of load in that plane.

The results of the experiments for the secondobjective, the determination of the five constants,E, Y, E, v and G are shown in Tables 2 (human)and 3 (bovine). The value given for the shearmodulus (G) in Case X represents an overall meanfor all torsional specimens of human bone.

If a material behaves in a transversely isotropicmanner, the elastic modulus for specimens cut inthe x-y plane at an angle C#Jo the x-axis (Fig. 10)can be found for uniaxial load by the equation:

Therefore by determining the five elastic constants,we could calculate the off-axis moduli. FiguresII-18 represent the results of these experiments.The solid line represents the values for E(4)calculated using the above expression and the meanvalues for the elastic constants. The dotted linerepresents the E(4) calculated using constants onestandard deviation from the mean. Experimentalpoints are shown on the graphs with one standarddeviation in each direction. The values and numberof specimens for each experimental point are givenin the Appendix. Both tensile and compressivedeterminations are shown.


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The elastic and ultimate properties of compact bone tissue 397Table 1. Bovine. Haversian femur

ELASTIC PROPERTIESTension in Isotropic (y-z) PlaneE

Direction n ix10gN/m2) "Transverse 4 12.0(4.62) .45(.250)3o" 7 LL.3(3.19) .47(.106)60' 4 10.6(1.69) .34(.071)Radial 6 10.0(0.50) .44(.040)

ENTRIES IN TABLES ARE GIVEN AS: NEAN(St.Dev.)n - number of specimensE = elastic modulus for transverse or radial specimens (in y-z plane)E' = elastic modulus for longitudinal specimens (in x direction)G' = shear modulus" = Poisson's ratio for transverse or radial specimensY' = Poisson's ratio for longitudinal specimensS = strain hardening modulus'Jult = ultimate stress=Y = yield stressEUlt = strain at fracture

Table 2. Human femurELASTIC PROPERTIES

TENSION E E' C'Case (x10gN/m2) n u n (x10gN/m2) n Y' n (xLOgN/m2) nI* 10.1(2.35) 6 .62(.257) 5 17.9(3.92) 15 .40(.157) 13IIf 14.1(3.31) 7 .45(.173) 7 18.3(4.57) 12 .51(.183) 12III (right)+ 13.2(2.87) 7 .51(.264) 7 15.6(2.74) 5 .35(.203) 5 3.71(.467) 6IV (Left)+ 13.4(3.34) 5 .59(.x22) 5 17.9(3.22) 6 .29(.025) 6 3.10(.379) 4XU 3.28(.383) 166

COMPRESSIONv* 11.7(101) 5 .63(.197) 5 lS.Z(O.85) 4 .38(.154) 4

* Composite of bones aged 41 female, 49, 61, 71 males** 21 year male+ 23 year male+t Specimens from 15 male, 5 female - ages 21 - 86

Table 3. Bovine femurELASTIC PROPERTIES

(xlO'N/m )2 E'Y n (x10gN/m2) n G'MVRRSIAN n Y' n (xlO'N/m*) nTensioncase VI 10.40.64) 5 .51(.236) 5 23.1(3.18) 3 .29(.079) 3Compression 3.57(.250) 22case VII lO.l(l.78) 8 .51(.115) 8 22.3(4.61) 5 .40(.209) 5

LAMINARTensioncase VIII 11.0(.173) 25 .63(.226) 6 26.5(5.37) 6 .41(.2m) 10 5.09(.389) 6


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Fig. 4. Compression test set up. Both longitudinal and Poissons strain extensometers are shown inplace.

(Facingp. 396)


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Tension

Fig. 6. Actual and idealized tension load-deformationcurves. The lower drawing shows how data points were

interpreted from photographic data recordings.

Compression

Fig. 7. Actual and idealized compression load-deformation curves.


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Fig. 9. Load-deformation curves for tension tests with Poissons strain recorded. See text.

Fig. 19. Load-defiection curve from a compression test on a transverse specimen from a human fe


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398 D. T. REILLY et l.

Fig. 10. Off-axis specimen orientation to coordinatesystems.

Fig. 11. Case I. Off-axis modulus determined from atension test on human femoral specimens.

Angle 9, deg

Fig. 12. Case II. Off-axis modulus determined from a Fig 15. Case V. Off-axis modulus determined from atension test on human femoral specimens. compression test on human femoral specimens.

Ultimate propertiesTable 4 shows the ultimate properties for tensile

tests in the longitudinal and transverse directionsfor specimens from ihe same bone. Included also inthese table are the values for the yield stress (cry)and strain hardening modulus (S) which relates

0 30 60 90Angle P, de g

Fig. 13. Case III. Off-axis modulus determined from atension test on human femoral specimens.

i I I I I , ,0 30 60 90Angle P, deg

Fig. 14. Case IV. Off-axis modulus determined from atension test on human femoral specimens.

0 30 60 90Angle P> deg

stress and strain after the yield stress is reachedThe average ratio for the ultimate tensile stress inthe longitudinal direction to that in the transversedirection is 2.6 for the four individuals studied. Theultimate strain undergone by a specimen was on theaverage of four times higher in the longitudinal


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The elastic and ultimate properties of compact bone tissue 399Table 4. Human femur, tension

ULTMATE PROPERTIES

BONE specimen Oultorientation n (x106N/m2) EUltA+ Longitudinal .038(.0054) 0.637CO.1526) 113.(8.7)Transverse : .008(.0013)III (right)+

Longitudinal 5 .02n(.0071) 0.842(0.4133) 116(11.7)Transverse 7 .uo7(.0015)IV (Left)+t Longitudinal 137Transverse ; 6231"

11.6) .030(.0059) 0.862CO.1774) 117C7.3)8.9) .007(.0016)Longitudinal 3 .022(.0044) 1.069(0.0403) lOB(4.5)Tr.ClnSVerSe 3 .006(.0013)

Mean Value Longitudinal 21 135(15.6) .031(.0072) 0.812(.1470) 114C3.1)Transverse 20 53C10.7) .007(.0014)

+ 31 year male+t 23 year male* 63 year male

4 1 I I . . *0 30 60 90Angle , de9

Fig. 16. Case VI. Off-axis modulus determined from atension test on bovine Haversian specimens.30 1

Angle P, degFig. 17. Case VII. Off-axis modulus determined from acompression test on bovine Haversian specimens.direction than in the transverse. This would makethe average energy absorbed by the bone tissue inthe longitudinal direction greater than that of thetransverse by one order of magnitude. It is quiteobvious how this type of loading configuration

4 , . . , 10 30 M) 90

Angle p, deg

Fig. 18. Case VIII. Off-axis modulus determined from atension test on bovine laminar specimens.would result in very low energy absorptionconsidering the low stress levels and absence of thelong plastic phase.

The compression tests of human specimens inthe longitudinal and transverse directions did notproduce such great differences. The ultimate stressin the longitudinal direction (Table 5) was approxi-mately one and one-half times as great as theultimate stress needed for fracture in the transversedirection. The strains to failure in the transversecompression tests, however, were larger than thoseof the longitudinal tests. With impaction offragments and therefore an inability in determininga precise failure strain (as could be done in thetension tests), the comparison of ultimate strain andenergy for the two compression tests is moredifficult than in the tension test. However, whatevercriterion is used for the ultimate strain, the energyabsorbed in the two compression tests would notshow the order of magnitude difference of thetension tests. The transverse compression tests forsome of the Haversian type bone specimens(human A and bovine 3) gave an interesting


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4 0 0 D. T. REILLY et al.Table 5. Human femur, compression

LILTIMATB ROPERTIES

BONE SpecimenOrientation nOult

(x106N/m2) 'ultA* Longitudinal 6Transverse 3 203C27.6) .019(.0034)151C10.7) .087(.0248)22**

1131+1132i+Mean Value

Longitudinal 3Transverse 5Longitudinal 7Longitudinal 4LongitudinalTransverse

198C12.7) .018(.0038)lH(13.6) .028(.0029)206(10.0) .019(.0029)211C14.4) .018(.0007)205C17.3)131C20.7)

* 31 Year male** 52 Year male+ 21 year male++ 22 year male

Table 6. Bovine, tensionULTIMATE PROPERTIES

Specime OUlC s OYTYPE/BONE orientation n h106N/m2) 'ult (x10gN/m2) (x106N/m2)Haversian BoneFemur 3 Longitudinal .016(.0048) 0.340(.2725) 14OC6.8)Tra"SVerSe : %I ;:1"; .009(.0011)Tibia 1 Longitudinal 7 .022(.0064) 0.665CO.2179) 141C19.7)Transverse 5 1:;;18":;j .009(.0?37)Femur 9 H Tra"SVerSe .007(.0042)

Radial .007(.0018)

Lamlnar BoneFemur 4 Longitudinal .033(.0049) 0.767CO.2571) 156C7.9)Transverse .007(.0012)Femur 9 L Transverse .008(.0016)Radial : .002(.0006)

load-deformation curve. Figure 19 is typical of thecurves obtained from these transverse compressiontests. They were similar in form to the curvesobtained from longitudinal tension tests. Sinceshear cracks and microbuckling were seen in thefailed specimens, the mechanism producing theplastic deformation in transverse compression istherefore different than the tension yieldmechanism.

The bovine Haversian specimens used fortension tests (Table 6) showed an approximatestrength ratio for the principal directions oflongitudinal, transverse, and radial of 3 : 1: 0.7. Thistable also shows that for the laminar bonespecimens, the ratios of tensile strength for theprincipal directions was about 3 : 1: O-4. The differ-ence in the ratios for the two different histologicaltypes of bone tissue is due to that very histologicdifference. The Haversian type showed no differ-ence for the ultimate strain in the transverse andradial direction since in this plane whether thetensile load is transverse or radial, it is pullingperpendicular to the axis of the osteons. The

tension tests of the radial specimens of the laminartype bone all fractured through the region ofvascular network, which for these specimens isperpendicular to the load direction and produces areduced load carrying area. The ultimate strains forthe laminar type radial specimens hence show afour fold decrease from the transverse ultimatestrain. Energy to fracture of the Haversian typespecimens is three times higher in the longitudinaldirection than the transverse direction and about50% higher in the transverse direction than in theradial direction due to the lower radial ultimatestress. The laminar bovine bone showed about a2:3 ratio of longitudinal to transverse energyabsorbed to fracture and the transverse energy tofracture was an order of magnitude greater thanthat in the radial direction due to the vascularnetwork strain concentration.Compression of bovine Haversian specimensshowed an approximate ratio of strengths in theprincipal directions of 1.6 : 1: 1 (Table 7).

The shear strength of bone tissue as determinedfrom the torsional tests are listed in Table 8 for both


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The elastic and ultimate properties of compact bone tissue 401Table 7. Bovine Haversian femur compression

ULTIMATE PROPERTIES

BONEFemur 3

Femur 9

Specimen OultOrientation n (x106N/m2) CultLongitudinal a 272( 3.3) .016(.0015)Transverse 146(31.8) .032(.0010)Tral-tSVZXSe 4 196(11.4) .053(.0197)Radial 5 190(18.0) .072(.0137)

Table 8. Human and bovine shear strengthULTIMATE PROPERTIES

TYPEHuman(femur)

BoneIv*41+

ma Longitudinal "ultn (x106N/m2) Tmax48 ::I 2:$ 1.92.2

BovineHaversian Cl 3 67( 5.5) 1.7(femur) C2 4 72(12.0) 1.8BOVilWPlexiform Dl 66( 7.8) 2.7(femur) DZ 6" 62( 6.0) 2.9

* 23 year male+ 56 year male

human and bovine bone. In the table, the shearstrength is compared to the longitudinal ultimatetensile strength found for specimens from the samebone. For human and Haversian bovine bone, theshear strength was found to be approximatelyone-half the ultimate tensile strength, but in thelaminar type bovine bone, it was one-third as great.This finding can also, we feel, be attributed to theplane in which the laminar type bone has itsvascular network. A fracture of this type wasdescribed by Burstein and others (1973).

An attempt was made to describe the ultimatestrength of bone specimens under uniaxial loadingat orientations between the principal directionswith an empirical strength criterion. A simple andconvenient type of criterion was introduced byHankinson (1921) to describe the ultimate compres-sive strength of wood. His criterion relates anoff-axis strength to the strengths determined in thetwo principal directions as follows:

S(9O)S(O)S(4) = [S(90) COS #J+ S(0) sin $1where S(C#J)= the off-axis ultimate stress at someangle of rotation, S(0) = longitudinal ultimate stress(tensile or compressive), S(90) = transverse ulti-mate stress (tensile or compressive), I#I= angle ofrotation from principal direction and n = anynumber.If the principal ultimate tensile stresses are used,this criterion is for the off-axis ultimate tensilestress and principal ultimate compressive stressvalues allow it to be used for off-axis ultimatecompressive stress. The ultimate stresses were

found for the off-axis specimens used for theoff-axis modulus experiments and are shown inFigs. 20-27. The solid line represents the predictedoff-axis ultimate stress using the mean values andn = 2 in the Hankinson formula. Again the dottedlines are calculated using the mean values with onestandard deviation added or subtracted. Actualexperimental values are included in the Appendix.Figure 28 is the ultimate stress found for thespecimens used in the isotropic plane elasticconstant experiment, and shows that the Hankin-son criterion holds for y-z plane also. This casewas run since the transverse ultimate stress wasfound to differ significantly (P


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402 D. T. REILLY et al.

36 66Angle, deg

9b

Fig. 21. Case II. Off-axis strength in tension for human Fig. 24. Case V. Off-axis strength in compression forfemoral specimens. human femoral specimens.

15c

E20; IOC

\ casem\\1 111\ \\ \ \1 \,. -___\ ---0 3; &J

Angle, deg9b

Fig. 22. Case III. Off-axis strength in tension for human Fig. 25. Case VI. Off-axis strength in tension for bovinefemoral specimens. Haversian femoral specimens.

0 - 60 90Angle, deg

Fig. 23. Case IV. Off-axis strength in tension for human Fig. 26. Case VII. Off-axis strength in compression forfemoral specimens. bovine Haversian femoral specimens.

CONCLUSIONS or three dimensional investigations. The results ofMany studies dealing with the mechanical be- this study with regard to Haversian bovine bone

havior of bone have assumed linear elastic proper- support the transversely isotropic model of Langties. We no longer feel such a model is valid for two (1970) who found value for the five elastic

30 60 90Angle, deg

casem

0 30 60 904m$e, deg

Angle, deg


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The elastic and ultimate properties of compact bone tissueG = 3.6 x lON/m*v = 0.36v= 0.51

40322

813.

For human femoral compact bone, our overallmean values were:

0-0Angle, deg

E = 17.0 x lON/m* $0)E= 11-5x lON/m* (31)

G = 3.28 x lo9 N/m2 (166)u = 0.46 (147)v = 0.58 (26).

Fig. 27. Case VIII. Off-axis strength in tension for bovinelaminar femoral specimens.

?2h

60 --.-\aseX-.50 .I-%-.

. .-. -L_.\ --

40 . . ..-._ -_-

30-

20-

IO-

r 1 ,0 30 60 90

The values for E and E are associated with astandard deviation of approx. 15-20% of themodulus if determined in tension and 7-10% ifdetermined in compression. Standard deviationsfor the Poissons ratios were generally in the rangeof 30% of the ratio. These large ranges are mostlikely due to the inhomogeneous nature of the bonematerial (in porosity, mineralization, and osteonaldirection), and specimen size (presumably a largercross-sectional area would produce a better averag-ing of the local inhomogeneities). The high valuesfor the Poissons ratio, with many individualspecimens having a ratio greater than 0.5, attests tothe fact that the material is really a structure witha high degree of kinematic deformation. Standarddeviations for the shear modulus was approx. 10%of the values observed.

Angle, degFig. 28. Case IX. Off-axis strength in tension for bovineHaversian femoral specimens.constants as follows:

E = 22.0 x IO N/mE= 11.3 x lON/m*

G = 5.4 x 10 N/m2v = O-48v = 0.40.

His specimens were from the phalanx but he didnot, however, report the histological type of boneor give standard deviations. Our mean values forbovine Haversian femoral compact bone were:

nE = 22.6 x 10 N/m* 8E = 10.2 x 10 N/m* 13

The validity of the transversely isotropic modelwas tested by the attempt to predict the modulifound in uniaxial tests at angles of 30 and 60 to thelong axis of the bone using the five constants andmatrix rotation. While some predicted valuescorrespond very well with the experimental off-axisstiffnesses, others (notably Cases III and IV) donot. This is particularly disconcerting consideringthe fact that Case III represents bone specimenstaken all from one bone and Case IV is thecontralateral side. The discrepancy may very wellbe due to the method of obtaining the specimensfrom a single bone. When several bones were usedas the source of specimens for the off-axis tests, anattempt was made to obtain at least one of eachorientation specimen from each bone used. When asingle bone was used to obtain all orientations ofspecimens, sufficient numbers could be obtainedonly if a specimen of one orientation was takenfrom a rough slab of bone. That is, a rough piece ofcortical bone would most profitably yield all 30specimens or all transverse specimens. Thus,specimens of a given orientation would be takenfrom a specific portion of the bone, and thequestion of the different properties of differentportions of bones may play a role in producing theunexpected property variation with respect to


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404 D. T. REILLY et al.orientation. Better correspondence with predictedvalues is seen in those cases where groups of bonesare used for the different orientation specimens andprobably better sampling is had of different corticalpositions and their moduli. The maximal stiffnessfound in those two cases for the 30 direction arenot impossible theoretically since an appropriatevariation in the five elastic constants may predictthis value.

Using the constants determined in this investiga-tion, an experimentor can then model a threedimensional structure of cortical bone (plate,cylinder or shell) under complex loading configura-tions and predict load-deformation characteristicsin the elastic range.

The overall values for the ultimate strengths ofthe human bone specimens are listed below:Longitudinal Tension (+I~ 133 x 10N/m2

Compression u.1, = 193 x 10 N/m2Transverse Tension cult = 5 1 x lo6 N/m*

Compression uult = 133 X lo6 N/m*Torsion rma. = 68 x ION/m*.(axis inx-direction)

These values are associated with a standarddeviation of approx. 7-10% for the ultimate normalstresses and 5% for the shear stresses. All of theabove data are given with no consideration to theage of the donor bone or the site of provenance ofthe specimen. They are averages of specimens froma population over the age span of 19-80yr.The failure loads predicted with the Hankinsontype formulation although in excellent agreementwith experimental values, represent a first step inthe development of a failure criteria for corticalboneBone tissue may be regarded as a plasticmaterial (3.1% elongation) in its longitudinaldirection, but does not exhibit this behavior in itstransverse direction (0.7% elongation).Acknowledgement-The authors thank the Tissue Bank inNMRI National Naval Medical Center, Bethesda, Mary-land, for their cooperation and effort in obtainingspecimens for this study. In particular, we thank RobertW. Bright, LCDR, MC, USNR, for his effort and

Also, we would like to thank Marc Martens, M.D., ofthe Catholic University of Louvain.REFERENCES

Burstein, A. H., Currey, J. D., Frankel, V. H. and Reilly,D. T. (1973) The ultimate prop&ties of bone tissue: theeffects of yielding. J. Biomechanics 5, 35-44.Burstein, A. H. and Frankel, V. H. (1971) A standard testfor laboratory animal bone, J. Biomechanics 4,1.55-1.58.Burstein, A. H., Reilly, D. T. and Frankel, V. H. (1973)Failure characteristics of bone and bone tissue.Perspectives in Biomedical Engineering (Edited byKenedi, R. M.). University Park Press, Baltimore,Maryland.Hankinson (1921) Investigation of crushing strength ofspruce at varying angles of grain. U.S. Air ServiceInformation Circular No. 259.Lang, S. B. (1970) Ultrasonic method for measuringelastic coefficients of bone and results on fresh anddried bovine bone. IEE, Trans. Bio. Engr 101-105,April.McElhaney, J. H. and Byars, E. F. (1965) Dynamicresponse of biological materials. ASME 65-WA/HUF-9.Nadai, A. (1950) Theory of Flow and Fracture of Solids,Vol. 1. McGraw-Hill, New York.Pope, M. H. and Outwater, J. 0. (1974) Mechanicalproperties of bone as a function of position andorientation, J. Biomechanics 7, 61-66.Reilly, D. T. and Burstein, A. H. (1974a) The mechanicalproperties of bone tissue: A review. J. Bone ht. Surg.56-A, 4.Reilly, D. T., Burstein, A. H. and Frankel, V. H. j1974b)The elastic modulus for bone. J. Biomechanics 7,271-275.Wang, C. (1953) Applied Elasticity. McGraw-Hill, NewYork.Weiss, V. (1960) Current views and theories on fracture,crack initiation and propagation., 7th SagamoreOrdnance Materials Research Conference.

assistance.

NOMENCLATUREelastic modulus in isotropic planeelastic modulus perpendicular to isotropic planeoff-axis elastic modulusindependent shear moduluspolar moment of inertiaelongation at fractureload sustained at fracturestrain hardening modulusoff-axis ultimate stresstorqueultimate strainPoissons ratio in isotropic planesecond independent Poissons ration (see text)ultimate stressyield stressultimate shear stressangle of twist per unit length.


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The elastic and ultimate properties of compact bone tissueAPPENDIX

405

HUMANFF.MURE@ Oult

n (x10yN/m2) (x106N/m2) EUltTensionCase I L

3o"60'T

case II L3060'T

case III L3o"60'T

case IV L30D60'T

Compressioncase v L

3060'T

1295

1244755566555

13 18.2(0.85) 187(28.8)5 14.6(0.67) 173(13.8)7 11.3(0.94) 133(15.0)5 11.7(1.01) 132(11.4)

17.9(3.92) 128(13.1)14.3(2.08) 95(11.6)10.5(1.48) 49( 3.7)lO.l(Z.35) 50( 1.0)18.3(4.57) 133(18.6)13.2(3.65) 107( 4.0)14.9(2.80) 80( 5.2)14.1(3.31) 59( 6.6)15.6(2.74) 133122.4)18.7(7.83) 105( 3.4)15.1(2.54) 62( 5.7)13.2(2.87) 57( 3.6)17.9(3.22) 137(11.6)21.9(4.93) lOO(12.9)13.3(3.20) 55( 4.5)13.4(3.34) 62( 8.9)

.023(.006

.014(.003

.006(.002:

.008(.0011

.032).0073

.031(.0104

.009(.0014

.007(.0019)

.026(.0071)

.021(.0064)

.007(.0030)

.007(0015)

.030(.0059)

.020(.0059)

.006(.0008)

.007(.0016)

.026(.0004)

.028(.0052)

.031(.0010)

.028(.0029)

BOVINE HAVERSIAN FEMUREm OuLt

n (x10yN/m2) (x106N/m2) EUltTensionCase VI L 3

3o" 460' 6T 5

CompressionCase VII L 5

3o" 1460' 2T 4

BOVINE LAMINAR FEMURTensiollCase VIII L 6

3o" 17,60' 6

23.1(3.18)16.7(4.54)12.8(1.57)10.4(1.64)

22.3(4.62) 254(25.5)15.1(1.50) lqO(12.6)10.6(0.23) 148( 2.6)lO.l(l.78) 146(21.8)

26.5(5.37) L67( 8.8)18.0(1.69) lll( 7.9)15.2(1.90) 68(12.6)

144( 6.2)90( 7.5)60( 4.1)46( 7.1)

.016(.0048)

.011(.0036)

.008(.0014)

.009(.0011)

.016(.0015)

.025(.0038)

.032(.0006)

.031(.0016)

.033(.0049)

.012(.0023)

.007(.0023)

.007(.0012)25 lL.O(l.73) 52( 7.7)BOVINE HAVERSIAN FEMUR, Tension in Isotropic PlaneCase IX E OUlt

n (x10yN/m2) (x106N/m2) 'ult YTransverse (y) 4 13.0(4.62) 54(5.8) .007(.0042) .45(.250)

3o" 7 11.3(3.19) 53(8.7) .009(.0034) .47(.106)60 4 10.6(1.69) 49(9.1) .008(.0043) .34(.071)

Radial (z) 6 10.0(0.50) 39(4.7) .007(.0018) .44(.040)

Documents

Donald t. Reilly 75