Composite model of the shark's skeleton in bending: A novel architecture for biomimetic design of functional compression bias

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Composite model of the shark's skeleton in bending: A novel architecturefor biomimetic design of functional compression bias

Xiaoxi. Liu a, Mason N. Dean b,c, Adam P. Summers b,d, James C. Earthman a,!a Department of Chemical Engineering and Materials Science, 916 Engineering Tower, University of California, Irvine, CA 92697-2575, USAb Department of Ecology and Evolutionary Biology, 21 Steinhaus Hall, University of California, Irvine, CA 92697-2525, USAc Department of Biomaterials, Max Planck Institute of Colloids and Interfaces, 14424 Potsdam, Germanyd Friday Harbor Labs, University of Washington, Friday Harbor, WA 98250, USA

a b s t r a c ta r t i c l e i n f o

Article history:Received 3 August 2009Received in revised form 6 February 2010Accepted 7 May 2010Available online 24 May 2010

Keywords:Elasmobranch cartilageTessellated tissueEquivalent cross sectionFunctional compression bias

Much of the skeleton of sharks, skate and rays (Elasmobranchii) is characterized by a tessellated structure,composed of a shell of small, mineralized plates (tesserae) joined by intertesseral ligaments overlaying a softcartilage core. Although tessellated cartilage is a de!ning feature of this group of !shes and has beenmaintained for millions of years, the signi!cance of this skeletal tissue type — particularly from a mechanicalperspective — is unknown. A cross-sectional model, based on empirical material property and morphologicaldata, was developed in the present work to analyze the function of intertesseral joints in regulating the stressdistribution within the skeletal tissue during bending. The results indicate that this structure distributesmore stress to the tesserae loaded in compression when compared to those loaded in tension. A functionalbias towards compression has also been observed for bone, but with the formation of microcracks in theregion under greatest tension. The present model demonstrates how functional compression bias can also beachieved in tessellated cartilage structures but in the absence of microcracking. This behavior providespossible advantages including increasing the resistance to fatigue damage as well as mitigating the risk oftearing under excessive bending loads.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The typical adult chordate skeleton is made up of bones withcartilaginous elements serving as bearing surfaces, but the elasmo-branch (sharks, skates and rays) !shes are unusual in havingskeletons of only mineralized and unmineralized cartilage. Thisskeletal design is counterintuitive since cartilage, unlike bone, cannotself-heal [1]; and yet elasmobranch !shes lead particularly dynamiclifestyles. They are some of the fastest and largest animals in theoceans, and many feed on prey larger than themselves or as hard andtough as mollusk shell. We hypothesize that a key component of thehigh level of performance of their skeletons is the unique arrange-ment of tissue: much of the axial skeleton and the entire appendicularskeleton is made of tessellated cartilage, a composite of mineralizedhydroxyapatite blocks (tesserae) over a core of uncalci!ed cartilage(Fig. 1) [2–6].

Adjacent tesserae are adjoined by stout intertesseral ligamentsthat apparently allow for movement of tesserae relative to oneanother: either separation when the tesseral mat is loaded in tensionor abutting when under compression (Fig. 2). Both of these situations

can occur in bending of a skeletal structure (e.g. a beam of uncalci!edcartilage sheathed in tiles) where an applied nonaxial load produces amaximum tensile stress in the tesseral layer farthest from the axis ofbending and amaximum compressive stress in the contralateral layer.This differential loading would result in starkly different intertesseraljoint volume fractions from one side of the element to the other; theintertesseral joints must therefore in"uence the stress distribution oftessellated cartilage in bending. Since external surfaces are typicallymore susceptible to fatigue crack initiation, bending is particularlydetrimental because the particularly damaging maximum tensilestresses are localized to one of the surfaces of the skeletal element.Further, shear stresses are also produced in bending that areproportional, albeit smaller in amplitude, to the maximum tensile orcompression stresses. Hence, it is reasonable that tessellated cartilageloaded in bending (a particularly common mode for a swimming orbiting !sh) is in serious danger of fatigue and fracture. This risk isparticularly poignant for cartilage since, having essentially no vascularsystem, it cannot readily repair itself.

The multiple interacting constituents of tessellated cartilage tissueincrease the dif!culty of predicting itsmechanical behavior, particularlywhen it is subjected to multiaxial stresses such as those associated withbending. To address this complexity, we developed an idealizeddynamic numericalmodel to gain a better understanding of the bendingbehavior of tessellated cartilage as depicted schematically in Fig. 2.

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! Corresponding author. Tel.: +1 949 824 5018; fax: +1 949 824 2541.E-mail address: [email protected] (J.C. Earthman).

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The purpose of the present work was three-fold: 1) to develop a modelthat it is reasonably realistic in terms of geometry and mechanicalproperties but also provides insightful and easily interpretable predic-tions of bending behavior; 2) to predict the bending behavior of thetessellated structure for different elastic moduli of intertesseral joints(forwhichempirical data are lacking) todeterminehowthevalueof thisproperty affects the distribution of stresses; and 3) to gain a betterunderstanding of shark skeletal tissue under bending loads, which arecommon and often repeated under physiological loading conditions.The limited healing capacity of cartilage suggests that the tessellateddesign imparts some resistance to fatiguedamage, a typical failuremodefor man-made structures loaded in bending. The present model wastherefore also intended to yield qualitative insights as to how themorphology of tessellated cartilage tissue might affect its susceptibilityto damage from excessive or repeated bending loads.

2. Elastic moduli and composite geometry

Tessellated cartilage is comprised of three distinct tissue phases:unmineralized cartilage, mineralized cartilage, and !brous jointtissue. The elastic properties and dimensions used in the presentmodel were derived from representative measured values. Weimagined a simple cross-sectional slice of a tessellated beam, with asolid uncalci!ed cartilage block sandwiched between upper and lowertessellated layers. Each tissue component was treated as a linearelastic element in the present model. While this simpli!cation doesnot completely capture the nonlinear behavior of the soft tissuecomponents, it was useful for providing tractable and insightfuldescriptions of the role that the overall structure has in determining

the stress distribution in bending. The elastic modulus for mineralizedtessera, ET, has been found to be approximately equal to 3 GPa inrecent nanoindentation tests [7]. A representative elastic modulus ofapproximately 3 MPa was used for the uncalci!ed cartilage sand-wiched between the tessellated layers; this value is also supported bythis recent experimental work. There are no empirical data forintertesseral !bers. Depending on the elastin to collagen ratio andstrain rate, the elastic modulus of the joints between the tesserae canrealistically lie somewhere in the range of 3 MPa to 1 GPa [8].Accordingly, the present model was exercised with different valuesfor the joint modulus within this range.

The dimensions of each mineralized tessera in the outer layers ofthe tissue was approximated at 0.7 mm thick by 1 mm wide and thesize of the joints was set at 20 µm wide in the y direction and 0.7 mmbetween the tessera based on anatomical measurements [9–12].These estimates give an unloaded joint volume fraction, fJi, of 0.02within each of the tessellated layers. The uncalci!ed cartilage wasassigned a representative thickness of 10 mm. The dimensions used inthe present model are all within the typical ranges of values observedin cartilaginous !shes. Whereas these dimensions can vary withspecies and skeletal location [e.g. 4,11], the understanding gained inexecuting the present models should be qualitatively relevant formost tessellated tissues.

3. Composite behavior of the tessellated layer

Our model approximates a skeletal cross section, represented by amonolithic block of uncalci!ed cartilage sandwiched between twoexternal tessellated layers (mats of tesserae adjoined by intertesseral!bers). The effective elastic modulus of each of the externaltessellated layers, Eext, can be approximated by the Reuss (isostress)description of composite behavior given by

Eext =fJEJ

+1!fJ

! "

ET

2

4

3

5!1

=ET

1 + ETEJ!1

! "fJ

!1"

where fJ is the volume fraction of intertesseral joint phase, EJ is theelastic modulus of the intertesseral joint phase, and ET is the elasticmodulus of the tesserae. We note that there are also joints runningparallel to the stress direction in a three-dimensional tessellatedstructure as illustrated in Fig. 3. The isostrain relation governing theircontribution to the composite modulus would be

Eext = fJoEJ + 1!fJo! "

ET : !2"

Considering this equation and the fact that the volume fraction ofparallel joints is approximately 0.02 andwould not change, it is safe toassume that the effect of parallel joints on the overall compositemodulus is negligible. By contrast, it can be seen that the value of EJ

Fig. 1. Uncalci!ed and mineralized phases of tessellated tissue, depicted as a schematicof the inset cross sectional BSE image.

Fig. 2. Schematic of the present model of a tessellated composite inspired by the tissue in Fig. 1 in both (a) unloaded and (b) bending con!gurations. The thickness of the tessera andcurvature in (b) are exaggerated for the purposes of illustration.

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could have a signi!cant effect on Eext in Eq. (1) if it is suf!cientlysmaller than ET, even for fJ values as low as 0.02. Accordingly, only theeffect of joints running perpendicular to the stress direction needs tobe considered in the present model. By not including the paralleljoints, we assume that all of the stress in these joints is transferred tothe tesserae and perpendicular joints in series. This assumption is notonly consistent with the low value of fJ in Eq. (2) but also the relativelylow modulus of the joint tissue, EJ.

Constraint on the joint by the much stiffer tesserae requires thatstraining the intertesseral joint in the plane of the tesserae isaccompanied by a change in its volume fraction within the tessellatedlayer. Consequently, the volume fraction of the joint is given by

fJ =lJ

lJ + lT! " !3"

where lJ is the length of the intertesseral joint (i.e. length of the !bersin Fig. 2) and lT is the length of the tesserae (Fig. 2). The true strain ofthe intertesseral joint is de!ned as

!J = "l

lJo

dll

= lnlJlJo

!!4"

where lJ and lJo are the !nal and initial joint widths respectively.Similarly, the strain in the tesserae is given by

!T = lnlTlTo

# $: !5"

Rearranging terms and substitution into Eq. (3) gives

fJ =lJo exp !J

! "

lJo exp !J! "

+ lTo exp !T! "= 1 +

lTolJo

exp"ET

! "EJ

!" #!1

!6"

where the applied stress !=EJ "J=ET "T acts on both the joint and thetesserae (isostress). Substituting Eq. (6) into Eq. (1), the modulus forthe external layers is then given by

Eext =ET

1 + ETEJ!1

! "1 + lTo

lJoexp "

ET! "

EJ

! "h i!1 !7"

3.1. Equivalent cross section method

The stress distribution in a composite beam (e.g. a tessellatedskeletal element) can be determined using a method that derives amonolithic equivalent cross section from the original compositecon!guration [13]. Consider a composite beam cross section consist-ing of materials with elastic moduli E1 and E2 as shown in Fig. 4. In thisillustration, the elastic modulus of the top section, E1, is lower thanthat for the section below, E2. From elastic beam theory, strain varieslinearly according to

! =yd#$d#

=y$

!8"

where " is the strain in the plane of the cartilage, y is the distance fromthe neutral axis normal to this plane, and # is the curvature of theneutral layer of the cartilage as de!ned in Fig. 5. Hooke's law then gives

"1 = E1! =E1y$

;"2 = E2! =E2y$

: !9"

A composite beam can be treated as a monolithic beam by creatingan equivalent cross section so that it deforms in a manner that isidentical to that for the composite beam. Once the correspondingstresses are determined for the equivalent cross section, they can thenbe transformed to those in the composite structure using a relativelysimple procedure. In order to illustrate this procedure, considerelemental forces on the sections illustrated in Fig. 4:

dF1 = "1dA =E1y$

dA;dF2 = "2dA =E2y$

dA: !10"

In order for the equivalent cross section to have the same bendingrigidity as the composite, it is necessary to use a larger cross sectionwith modulus E1 to replace the material with modulus E2 if E1bE2.Therefore, the external force on the E2 material is altered to

dF2 =nE1! "y$

dA =E1y$

nb1dy! " where n =E2E1

: !11"

Thus, the equivalent cross section width of the beam is

becs = b2 =E2E1

b1 !12"

in the section where the modulus is E2, and becs=b1 in the sectionwith modulus E1.

The stress in the equivalent cross section in bending is then givenby

"ecs =Mzy!Iz"ecs

!13"

whereMz is the applied bending moment and (Iz)ecs is the moment ofinertia for the equivalent cross section area, Aecs. The equivalent crosssection moment of inertia is given by

Iz! "ecs = "Aecs

y2dA = "yy2becsdy: !14"

Fig. 3. Top view of the tessellated layer under tension in the present composite bendingmodel. The low joint volume fraction of 0.02 in Eq. (2) predicts that the joints parallel tothe applied stress would not have a signi!cant effect on the overall modulus of thestructure and, thus, they can be omitted in the model. This assumption implies that thestress in the parallel joints is transferred to the tesserae and joints in series, which isalso consistent with the relatively low modulus of the joint phase.

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As shown in Fig. 4, !=!ecs at locations in which the elasticmodulus is E1. Stresses at locations that correspond to the greaterelastic modulus, E2, are transformed according to

" =E2E1

"ecs: !15"

4. Equivalent cross section for calci!ed cartilage

The calci!ed cartilage structure shown in Fig. 2 is a typical sectionthrough an elasmobranch skeletal element, and contains two layers ofgreater modulus, one at the top and one at the bottom. However,because of the intertesseral joints, the modulus of each of these layersdepends on the stress and loading regime (tension or compression)imposed in each. Substituting the relation for Eext in Eq. (7) for E2 inEq. (15) leads to the following equivalent cross section width for theexternal tessellated layers in Fig. 2:

bext =EextEu

b1 =ETb1

Eu 1 + ETEJ!1

! "fJ

h i !16"

where Eu is the modulus of the uncalci!ed cartilage sandwichedbetween the two external tessellated layers, and b1 is the actual crosssection size of the composite structure. For convenience, b1 wasassigned a value of unity in the present model.

The transformation from the composite cross section to equivalentcross section is illustrated schematically in Fig. 6. Once bext isdetermined for both tessellated layers, (Iz)ecs can be calculated usingEq. (14).We note that the equivalent cross section of the tensile side issmaller than the compressive side when loaded due to the increase involume of the soft tissue joints in tension and the decrease in joint

volume in compression. In addition, the equivalent external layershave the shape of isosceles trapezoids as opposed to rectangles due tothe gradient in stress, and therefore elastic modulus, in these regions.We also note that the shape of the present equivalent cross sectionvaries with changes in bending moment as intertesseral jointscompress or stretch from their resting states. In particular the widthof the regions corresponding to the tessellated layers changesigni!cantly as the bending moment increases or decreases.

The present model begins its determination of the stressdistribution from the unloaded state. Substituting the unloadedjoint volume fraction, 0.02, into Eq. (7) gives (Eext)o=131 MPa and(bext)o is then equal to 44 mm for both external layers. According toEq. (14), the initial unloaded value for the equivalent second momentof inertia is 1.84!10!9 m4. Starting with these values, the stressdistribution for the composite cartilage was then determined for anincrementally small increase in the bending moment. Based onEqs. (13) and (15), this incremental change in stress with a smallchange in bending moment is given by

%"ext =Eext %Mz yEu Iz! "ecs

!17"

in the external tessellated layers while

%"u =%Mz yIz! "ecs

!18"

holds for the internal uncalci!ed cartilage. Taken together, Eqs. (17)and (18) can be used to determine the change in stress at verticalpositions, designated by y, through the entire composite cross sectionfor an incremental increase in the bending moment. The total stress ata position y can then be determined from

"ext =1Eu

# Eext %Mz yIz! "ecs

!19"

and

"u = #%Mz yIz! "ecs

: !20"

We note that (Iz)ecs is not constant during loading as a result of thevariation in bext. Accordingly, the value of y for a given position in thecomposite also changes during bending as a result of a shift in theneutral axis as (Iz)ecs changes. Eqs. (19) and (20) were solvednumerically by incrementally increasing the bending moment by1!10!5 Nm from 0 to 0.4 Nm. This increment in bending momentwas suf!ciently small to avoid signi!cant error. For each iteration, thecalculated stress distribution was used in Eq. (7) to update themodulus in each tessellated layer. New values of bext and then (Iz)ecswere then also determined using the updated Eext. These values were

Fig. 4. Illustration of an equivalent cross section for a composite beam with a stiffer lower component, E2NE1. Strain, ", equivalent cross section stress, !ecs, and actual stress, !, arealso shown as a function of distance, y.

Fig. 5. Geometric conventions for a beam in bending. The neutral axis in this !gurehappens to be in the same location as the central axis.

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then used in the subsequent iteration to calculate the stressdistribution for the next incremental increase in bending moment.The predictions were limited to bending moments that correspond toa maximum applied stress of approximately 300 MPa in thetessellated layer under compression, which is approximately anorder of magnitude greater than the ultimate compressive strengthsmeasured for vertebral calci!ed cartilage from elasmobranch species[14]. This is a reasonable limit due to the fact that stresses aretransferred from the uncalci!ed cartilage to the stiffer tessellatedcartilage and that the thickness of the uncalci!ed regions isapproximately 10 times greater than that for the tessellated layers.

5. Results

The composite elastic modulus calculated using Eq. (7) is plottedin Fig. 7 for different values of joint modulus, EJ, and as a function ofstress. These results demonstrate a pronounced and abrupt change incomposite modulus under compressive stresses for the lowest jointmoduli. This trend is due to the change in volume fraction of jointtissue that can decrease to very low values for applied compressivestresses and substantially increase under tension. This difference injoint volume gives rise to a very high modulus in compression and arelatively low modulus in tension. As EJ decreases, the compressiveand tensile composite moduli of the tessellated layer are increasinglydominated by the mechanical behavior of the separate hard and softtissues, respectively, resulting in strikingly different compositemoduli under different loading conditions. Conversely, the moduli

in compression and tension approach the same value as EJ increases invalue.

Predicted stress distribution for applied bending moments of 0.01,0.04 and 0.4 Nm are plotted in Fig. 8. Several features are worth notingin this !gure. First, the magnitudes of the maximum stresses in thetessellated layers becomes more different as the bending momentincreases, with the compressive layer exhibiting much largermaximum stresses. From the lowest to the highest bending moments,the maximum compressive stresses increase over 200-fold, whereasmaximum tensile stresses only increase by about a factor of 10.Second, the stresses in the uncalci!ed cartilage (central plots) arerelatively low in amplitude, remaining less than 0.03 MPa incompression and 8 MPa in tension (red) even though the stressamplitude in the tessellated layer under compression (blue) exceeds300 MPa. By comparison, the stress in the tessellated layer undertension reaches a value slightly less than 14 MPa. Finally, the neutralaxis in the tissue (where the stress transitions from tension tocompression) migrates, with increasing bending moment, toward thecompression side of the tissue until it enters the tessellated layerthere.

Neutral axis migration is plotted for multiple values of the jointmodulus in Fig. 9 as a function of bending moment. It can be seen inthis !gure that thismigration ismore pronounced for the lower valuesof EJ and occurs at lower bending moments. For EJ=300 and1000 MPa, neutral axis migration is negligible. These results demon-strate how stress redistribution becomes less prominent as jointmodulus increases. We note that for EJ=3MPa the neutral axis shiftsinto the tessellated layer under compression at a relatively lowbending moment of about 0.04 Nm. Comparison with Fig. 8 indicatesthat this migration into the tessellated layer corresponds to theformation of a discontinuity in stress distribution at the interfacebetween the uncalci!ed cartilage and the tessellated layer.

Maximum tensile stress in the upper tessellated layer is plotted asa function of bending moment in Fig. 10 (a). With increasing jointstiffness, stress increases more rapidly in the tensile tessellated layerand reaches a larger maximum stress. As before, changes in jointstiffness have less of an effect at EJ$100 MPa. It can be seen that thetensile stress reaches about 57 MPa for EJN100 MPa and an appliedbending moment of 0.4 Nm. By comparison, the maximum tensilestress for EJ=3 MPa is less than 14 MPa for a bending moment of0.4 Nm as also shown in Fig. 8. Thus, reducing EJ from 1000 to 3 MParesults in a 75% decrease in maximum tensile stress.

Maximum compressive (most negative) stress in the lowertessellated layer is plotted against bending moment for differentvalues of EJ in Fig. 10 (b). The amplitude of this stress exceeds 300 MPaunder a bending moment of 0.4 Nm for EJ=3MPa indicating anextreme bias towards compression compared with the correspondingmaximum tensile stress in Fig. 10. As noted above, changes in joint

Fig. 6. Schematic of the transformation from a tessellated composite cross section to its equivalent cross section in bending with tension at the top and compression at the bottom.The shaded areas correspond to the tessellated layers shown in Fig. 2.

Fig. 7. Composite elastic modulus as a function of applied stress for layers made up oftesserae and joints as predicted by Eq. (7). Results are shown for joint elastic modulus,EJ, values ranging from 3 to 1000 MPa. The elastic modulus of the calci!ed tesserae wasassigned a value of 3 GPa.

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stiffness above EJ$100 MPa have negligible effect on predictedstresses. Whereas compressive stresses are much higher for cartilagecomplexes with lower joint stiffness values, for EJ$100 MPa, stressesin the tessellated layers are more symmetrically balanced such thatboth maximum compressive and tensile stresses approximate anabsolute value of 57 MPa under a bending moment of 0.4 MPa(Figs. 10 and 11).

Maximum tensile stress in the uncalci!ed cartilage layer is plottedversus bending moment in Fig. 11 (a). As shown in Fig. 8, thismaximum stress exists at the interface with the upper tessellatedlayer. The greatest stress at a bending moment of 0.4 Nm is exhibitedfor EJ=3 MPa while lower values are obtained for higher values of EJ .Stress versus bending moment at the center of the uncalci!edcartilage (central axis) is plotted in Fig. 11 (b) to examine thelikelihood of the development of fatigue damage there. This centralaxis corresponds to cartilage tissue that is exactly in the middle

between the two tesseral mats and furthest away from the !brousperichondrium that wraps skeletal elements (Fig. 1). This region istherefore likely to have the lowest access to nutrients that diffuse infrom the surrounding vascularized tissue and as a result would havethe lowest capacity for repairing any fatigue damage that developsunder repeated bending. Fig. 11 (b) shows that the only signi!cantly!nite stresses are tensile by nature as in the cases of EJ=3, 10, and30 MPa. The greatest tensile stresses were predicted for EJ=3 MPabut reach a value slightly less than 4.2 MPa for a bending moment of0.4 Nm.

Maximum compressive stress in the uncalci!ed layer, plottedversus bendingmoment, is illustrated in Fig. 11 (c). The greatest stressamplitude at a bending moment of 0.4 Nm is exhibited forEJ=100 MPa while lower values are obtained for higher values of EJ.The absolute values of these stresses are about an order of magnitudeless than those for maximum tension shown in Fig. 11 (a). Larger

Fig. 8. Stress distributions throughout a calci!ed cartilage composite loaded in bending predicted for EJ=3 MPawhere compression is in blue and tension in red. The applied bendingmoments for each stress distribution are (a) 0.01, (b) 0.04, and (c) 0.4 Nm. The shaded areas correspond to tessellated layers that lie on either side of the thicker uncalci!ed cartilageas illustrated in Fig. 2. Each distribution corresponding to a given bending moment is split into three different horizontal scales to better illustrate the stresses within the tessellatedlayer under compression (left), the uncalci!ed cartilage (center), and the tessellated layer under tension (right).

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differences between maximum compression and tension stresseswere predicted for the lower EJ values. For EJ=3 and 10 MPa,compressive stresses in the uncalci!ed cartilage are eliminated withincreasing bending moment as a result of the migration of the neutralaxis into the lower tessellated layer. Despite this absence ofcompressive stress, the maximum tensile stress within the uncalci!edlayer remains extremely low compared to the compressive stressesthat develop in the adjacent tessellated mat. While the tessellatedstructure assures low tensile stresses, it can result in the broaddistribution of tension throughout the structure due to the migrationof the neutral axis. The possibility exists that this widely distributed

low-level tension is advantageous. For example, it could be hypoth-esized that the resulting hydrostatic tension throughout most or all ofthe uncalci!ed core layer could increase the presence of nutrients thatdiffuse from the vascularized perichondrium under compression.More work is needed to explore this possible advantage.

6. Discussion

The present model predicts that signi!cantly more stress will beborne by the tessellated layer on the compressive side compared tothe core and the tensile tessellated layer for a bending skeletalelement with joint tissue stiffness less than 100 MPa (EJb0.01ET). Thisresult is consistent with the fact that the tesseral mat more closelybehaves as a homogenous layer when the joints stiffness approachesthat for the tesserae. The predicted functional bias towards compres-sion also becomes more pronounced when the composite bears alarger bendingmoment. Based on a comparison of Figs. 10 (a) and (b),the maximum compressive stress could be over two orders ofmagnitude greater than the maximum tensile stress in the tessellated

Fig. 9. Neutral axis position within the cartilage complex as a function of appliedbending moment predicted for different values of the joint modulus, EJ. The shadedregions in this plot correspond to the tessellated layers.

Fig. 10. Maximum stress in the tessellated layers under (a) tension in bending and(b) compression in the corresponding tessellated layer as a function of bendingmoment for different values of the joint modulus, EJ.

Fig. 11. Model results for (a) maximum tensile, (b) central axis, and (c) maximumcompressive stress in the uncalci!ed layer for different values of the joint modulus, EJ.

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layers. We note that the resistance to fatigue damage of othercomposite materials has been found to be much greater undercompressive loading than it is under tensile loading [15,16]. Thus, themodel suggests that tessellated cartilage can reduce the risk of fatiguefailure by stress redistribution to a loading regime (compression) thatis less damaging for the tissue.

The present model of elasmobranch calci!ed cartilage in bendingprovides qualitative insights on the bene!cial functions of thesecomposite tissues. However, certain assumptions should be taken intoconsideration when analyzing the model's predictions. First, themoduli used in the model are estimates based on of the few valuesthat have been measured so far for elasmobranch tissues. Second,strain rate dependencies — which may play a large part in cartilagebiomechanics — are not addressed, though the effect of strain rateincrease could be roughly equivalent to an effective decrease in jointmodulus. Third, the size and spacing of the tesserae in the model areassumed to be constant though there is some variation in thesedimensions in actual tissues. We expect that the spatial inhomoge-neity of the tesserae would mean that the sharp discontinuitiespredicted for EJ=3MPa would not be observed in practice for a bulkspecimen. Rather, the actual variation in joint thickness and tesseraesize would make these anomalies more diffuse. However, stress anddisplacement discontinuities in certain individual joints could be justas pronounced locally depending on their dimensions and propertiesas well as those of the adjacent tesserae.

There have been a number of investigations [17–19] aimed atunderstanding the bending behavior of bone that are relevant to thepresent work. For example, Burstein and coworkers [17] !rst includedpost-yield straining in tension in their analysis to explain whyexperimental measurements of bending strength were generallygreater than those predicted by linear elastic bending models andultimate tensile strength data. The results of the nonlinear model theydeveloped indicate that stresses redistribute to the compressed side ofa bone specimen due to the fact that the yield stress in tension issubstantially lower in amplitude than the yield stress in compression.They further showed that his stress redistribution is accompanied by amigration of the neutral axis towards the compressive side of thespecimen. Currey [18] later pointed out that the bend strength forbone is consistently underestimated by the yield strength in tensionfor a large number of bone types and species as a result of thisasymmetric yielding behavior. Ebacher and colleagues [19] subse-quently investigated the bending strength of human long bones usingboth optical and backscattered electron (BSE) microscopy on cadaverspecimens tested in bending. Their results indicated very differentinelastic deformation mechanisms in tension and compression.Speci!cally, yielding in tension is facilitated by diffuse formation ofmicrocracks transverse to the tensile stress while that under muchhigher stress amplitudes in compression was accompanied bysomewhat larger shear cracks that were inclined to the compressivestress. In consideration of the strain redistribution from tension tocompression, it was concluded that the bending strength of boneprimarily depends on its compressive strength [19], and is thereforenot limited to its weaker strength in tension.

The similarity between the stress redistribution in bendingpredicted in the present work for tessellated cartilage and thatobserved earlier for bone is worth noting. In the case of bone,migration of the neutral axis towards the side under compression alsooccurs as a result of asymmetric yielding due to the formation ofmicrocracks under tension [17–19]. We surmise that this yieldingbehavior is sustainable under moderate loading amplitudes for tworeasons. First, the observed neutral axis migration keeps the tensilestresses low. Second, bone can repair the microcrack damagerelatively quickly by remodeling processes [20]. A bene!cial migrationof the neutral axis also occurs in tessellated cartilage but as a result ofan asymmetric and repeatable contribution of the joint ligaments that

does not require the formation of microcracks. This difference infacilitating a bias towards compressive stresses appears to be essentialbecause calci!ed cartilage tissue, unlike vascularized bone, cannotreadily repair microcrack damage.

7. Conclusion

A numerical model was used to gain insight into the functionalcharacteristics of tessellated cartilage found in elasmobranch species.The results indicate that tessellation serves to manage bending loadsin a way that can increase resistance to damage by distributing thehighest stresses to the tissues and loading regimes best able to bearthem. The stresses produced in the external tessellated layers can bewell over an order of magnitude greater than those in the internaluncalci!ed cartilage. The model also demonstrates how the externaltessellated layers can produce much lower tensile stresses comparedto the corresponding compressive stresses in bending. This functionalbias towards compression was also observed for bone in bending butwith the formation of microcracks in the regions under greatesttension. By contrast, the same functional bias can be achieved in atessellated cartilage structure if the modulus of the soft tissue joints isless than about 1% of the modulus of the tesserae, a reasonable rangefor the soft tissue ligaments. The predicted functional bias towardscompression is accompanied by a migration of the neutral axis fromthe midpoint of the tissue so that more of the uncalci!ed tissue isperiodically under relatively low tensile stresses. The tessellatedstructure modeled here could provide guidance for manufacturedbiomimetic materials in which a low modulus core is overlaid with ajointed or tiled high modulus layer. We show that soft tissue joints inthis outer layer are a key feature to this design, one that allows forneutral axis migration and the associated shifting of stresses to theless damaging compressive regime.

Acknowledgement

This work was supported by the National Science Foundation(Award No. IOB-0616322). Fellowship support from the Departmentof Chemical Engineering and Materials Science at UC Irvine is alsogratefully acknowledged.

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