Free Body Analysis, Beam Mechanics, and Finite …rosslab.uchicago.edu/publications/Porro et al...

Free Body Analysis, Beam Mechanics, and FiniteElement Modeling of the Mandible of Alligatormississippiensis

Laura B. Porro,1* Casey M. Holliday,2 Fred Anapol,3 Lupita C. Ontiveros,4 Lolita T. Ontiveros,4

and Callum F. Ross1

1Department of Organismal Biology and Anatomy, University of Chicago, Chicago, Illinois 606372Department of Pathology and Anatomical Sciences, University of Missouri School of Medicine,Columbia, Missouri 652123Department of Anthropology, University of Wisconsin, Milwaukee, Wisconsin 532014Department of Biological Sciences, University of Wisconsin, Milwaukee, Wisconsin 53201

ABSTRACT The mechanical behavior of mammalianmandibles is well-studied, but a comprehensivebiomechanical analysis (incorporating detailed musclearchitecture, accurate material properties, and three-dimensional mechanical behavior) of an extant archo-saur mandible has never been carried out. This makes itunclear how closely models of extant and extinct archo-saur mandibles reflect reality and prevents comparisonsof structure–function relationships in mammalian andarchosaur mandibles. We tested hypotheses regardingthe mechanical behavior of the mandible of Alligatormississippiensis by analyzing reaction forces and bend-ing, shear, and torsional stress regimes in six models ofvarying complexity. Models included free body analysisusing basic lever arm mechanics, 2D and 3D beam mod-els, and three high-resolution finite element models ofthe Alligator mandible, incorporating, respectively, iso-tropic bone without sutures, anisotropic bone withsutures, and anisotropic bone with sutures and contactbetween the mandible and the pterygoid flange. Com-pared with the beam models, the Alligator finite elementmodels exhibited less spatial variability in dorsoventralbending and sagittal shear stress, as well as lower peakvalues for these stresses, suggesting that Alligator man-dibular morphology is in part designed to reduce thesestresses during biting. However, the Alligator modelsexhibited greater variability in the distribution of medio-lateral and torsional stresses than the beam models.Incorporating anisotropic bone material properties andsutures into the model reduced dorsoventral andtorsional stresses within the mandible, but led toelevated mediolateral stresses. These mediolateralstresses were mitigated by the addition of a pterygoid-mandibular contact, suggesting important contributionsfrom, and trade-offs between, material properties andexternal constraints in Alligator mandible design. Ourresults suggest that beam modeling does not accuratelyrepresent the mechanical behavior of the Alligatormandible, including important performance metrics suchas magnitude and orientation of reaction forces, andmediolateral bending and torsional stress distributions.J.Morphol. 000:000–000, 2011. � 2011 Wiley-Liss, Inc.

KEY WORDS: archosaurs; crocodilians; biomechanics;beam theory; bone stress

INTRODUCTION

In both aquatic and terrestrial vertebrates, thejaws are important in transmitting muscle force tothe external environment. Likewise, externalforces, such as those generated at the bite point orby struggling prey or opponents, are transferred tothe organism through the jaws. Thus, jaws areused by vertebrates to interact with their environ-ment, these interactions are expected to exertselective pressure on jaw morphology, and thismorphology may be optimized for performance inthese interactions. The morphology of the craniumreflects a compromise between multiple functions:feeding, protecting the brain, housing senseorgans, respiration, and communication. The man-dible, in contrast, has feeding as its primary func-tion, suggesting that morphological adaptations forapplying and resisting feeding forces should bemore obvious in the mandible than elsewhere inthe skull. The biomechanics of mammalian mandi-bles (particularly stem mammals, primates, andcarnivorans) are relatively well-studied. In con-trast, the mandibles of archosaurs (crocodilians,birds, and their fossil relatives) have received sub-stantially less attention, making it difficult to com-pare mandibular structure–function relationshipsbetween mammals and other vertebrates. Thisarticle begins to rectify this bias by presenting and

Additional Supporting Information may be found in the onlineversion of this article.

*Correspondence to: Laura Porro, Department of OrganismalBiology and Anatomy, University of Chicago, 1027 East 57th Street,Chicago, IL 60637. E-mail: [email protected]

Received 31 August 2010; Revised 24 January 2011;Accepted 10 February 2011

Published online inWiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/jmor.10957

JOURNAL OF MORPHOLOGY 000:000–000 (2011)

� 2011 WILEY-LISS, INC.

testing hypotheses regarding stress regimes in anarchosaur mandible during feeding using free bodyanalysis, beam theory, and finite element analysis(FEA).

In a series of articles, Hylander used free bodyanalysis and in vivo bone strain to predict loadingand stress regimes in primate mandibles during bit-ing and mastication (Hylander, 1977, 1979a,b, 1981,1984, 1985). (The term loading regime refers to aspecific combination of external forces; stress re-gime refers to a specific combination of internalstresses.) Hylander hypothesized that the workingside mandibular corpus experiences twisting aboutits long axis, shearing perpendicular to its long axis,and bending in sagittal planes, whereas the balanc-ing side corpus experiences sagittal bending andlong-axis twisting. Hylander also predicted that themandibular symphysis is dorsoventrally sheared,twisted about a transverse axis, and bent in a trans-verse plane due lateral transverse bending (i.e.,wishboning) of the corpora. Hylander’s workprompted subsequent studies relating variation inmammalian mandibular morphology to hypothe-sized loading regimes, although some aspects ofmandibular morphology are apparently not opti-mized to resist feeding forces (Daegling, 1993).

The degree to which these results apply tomandibles of nonmammals remains to be eval-uated because mandibular biomechanics of archo-saurs are poorly understood. Bock’s detailed freebody analysis of external forces acting on the birdbill (Bock, 1966), and his theoretical and photo-elastic studies of the internal stresses in themandible of the crow (Bock and Kummer, 1968),suggested to him that mandibular shape reflectsloading regime: under dorsoventral bending, themandible is argued to function like an I-beam,with the thickened dorsal and ventral corticesresisting tension and compression, respectively,and the lateral and medial cortices resistingshear. The region of the mandibular fenestra,present in most archosaurs, was argued to be anarea of low stress, making it possible for bone tobe absent. Bock and Kummer (1968) suggestedthat the mediolateral narrowness of the mandiblepermits flexibility and that ‘‘this flexibility isespecially important in birds which have anintramandibular hinge such as the pelicans, her-ons, gulls, and numerous other groups.’’ Birdsrequiring more rigid mandibles were suggested tohave rounder cross sections.

Similarly, free body analysis of the mandible ofCaiman crocodilus, including integration (sensuDullemeijer, 1974) of the effects of several differentkinds of loading patterns, was also said to supportthe hypothesis that the transverse cross-sectionalshape of the mandible is appropriate for resistingthese loading patterns. As in the crow, the mandib-ular fenestra was argued to be an area of lowstress, allowing bone to be absent in this area

(Dullemeijer, 1974; van Drongelen and Dulle-meijer, 1982).

These analyses suggest that archosaur mandibu-lar morphology reflects loading and stress regimes,providing support for attempts to reconstruct feed-ing behavior in fossil archosaurs. Lever armmechanics have been used to estimate the effi-ciency of the feeding apparatus, including themandible, and to differentiate between feedingstrategies, i.e., jaws adapted for powerful ratherthan fast jaw closure (Ostrom, 1961; Mazzettaet al., 1998; Desojo and Vizcaıno, 2009). Molnar(1998) used free body and space frame analyses topredict tensile and compressive stress trajectoriesin the mandible of Tyrannosaurus. His analysessuggested that the anterior portion of the mandi-ble (dentary) in Tyrannosaurus behaves as a canti-levered beam; posteriorly, tensile and compressivestresses were concentrated at the dorsal and ven-tral margins of the mandible. This was argued tobe consistent with mandibular cross-sectional mor-phology, in that the anterior mandible is a dorso-ventrally elongated ellipse, whereas the posteriormandible features thickened dorsal and ventralmargins supporting a thin vertical plate of bone.The great depth of the dentary in Tyrannosauruswas suggested to be an adaptation to resist sagit-tal bending. Therrien (2005) applied beam analysisto the mandibles of several theropod dinosaursand assumed that bite force applied at any pointalong the mandible would be proportional to thegeometry of that section. Furthermore, heassumed that dorsoventrally deep mandibular cor-pora were adapted to resist sagittal bending,whereas corpora that were mediolaterallyexpanded were adapted to resist torsional stresses.On the basis of mandibular morphology, he dividedtheropods into five feeding categories, includinglarge prey hunters, small prey specialists, andbone crushers. Using FEA, Rayfield (2001) exam-ined deformation, stress, and strain in the skull ofthe theropod Allosaurus; her work demonstratedthat the lower jaw functioned as a beam in three-point sagittal bending. However, because the lowerjaw was attached to the cranium at the quadrate-articular joint, each hemimandible was also bentin the mediolateral plane due to condylar reactionforce. A similar study was carried out on the iso-lated mandible of the theropod Carnotaurus (Maz-zetta et al., 2004). Bell et al. (2009) compared fi-nite element models (FEMs) of ceratopsian andhadrosaur mandibles that were constrained at thejaw joint and loaded at two different bite points;muscle forces were not applied. Their loading ofthe ceratopsian mandible resulted primarily insagittal bending stresses; in contrast, the hadro-saur mandible experienced high degrees of torsionin the postdentary bones.

These hypotheses regarding the biomechanicsof archosaur mandibles remain untested because

2 L.B. PORRO ET AL.

Journal of Morphology

in vivo data on mandible function are lacking andcomprehensive biomechanical analyses of extantarchosaur mandibles have yet to be performed; incontrast, several studies have examined themechanics of the crocodilian cranium (Daniel andMcHenry, 2001; Metzger et al., 2005; McHenryet al., 2006; Rayfield et al., 2007 Pierce et al.,2008; Rayfield and Milner, 2008). Furthermore,several features present in archosaurs (such aspatent mandibular sutures) are absent in mam-mals, and their effects on mandibular mechanicsare unknown.

HYPOTHESES

The aims of this study are: 1) to calculate theexternal forces acting on the mandible resultingfrom muscle architecture (cross-sectional area, pin-nation, and sarcomere length), muscle orientation,and insertion sites; 2) to estimate and comparereaction forces on the mandible using lever armmechanics, beam modeling, and FEA; 3) to esti-mate shear, bending, and torsional stresses in theAlligator mandible and their relative importanceusing beam modeling and FEA; and, 4) to comparereaction forces and stress regimes in a simplebeam FEM to a high-resolution FEM of the Alliga-tor mandible.

This study will address the following questions:1) Does mandibular morphology in this archosaurminimize stress while minimizing bony materialbetter than would a similarly loaded simple beam?2) How do the material properties of the archosaurmandible affect its mechanical behavior? 3) Howdoes the enlarged pterygoid flange of archosaursaffect mandibular stress patterns?

Bending Stresses

The mandibles of alligators and other archo-saurs must be subjected to bending stresses duringbiting because the mandible is much longer than itis deep; thus, we expect that bending stressesshould be larger than stresses due to sagittal (zy)or transverse (xz) shear. Bending stresses (r) in across section through a beam-like member aredirectly proportional to the magnitude of the exter-nal bending moment about that cross section (M),and inversely proportional to the second momentof area of the cross section (I). These stress magni-tudes vary in proportion to their distance (y) fromthe neutral axis of bending, and are either com-pressive or tensile.

The resultant bending moment acting on a crosssection is the sum of the individual bendingmoments acting on that section. These bendingmoments will vary along the length of the mandi-ble, as well as with variations in bite point.

If cross-sectional geometry (and therefore, sec-ond moment of area) varies along the Alligator

mandible to minimize stress (thus minimizing thelikelihood of failure or maximizing possible load-ing) while minimizing material, then stress due tobending will remain relatively constant. To quan-tify this effect in this article, bending stress mag-nitudes along the length of a simple prismaticbeam (in which cross-sectional geometry is con-stant along its length) is compared with bendingstress magnitudes along the length of the Alligatormandible. If Alligator mandible cross-sectionalproperties are optimized to maintain strengthwhile minimizing material, stress magnitudesshould vary less along the length of the Alligatormandible than they do along the simple, prismaticbeam. (We emphasize that we view optimization asa process that minimizes stress relative to theamount of bone material, not an endpoint charac-terized by specific values.)

Transverse and Sagittal Shear Stresses

The mandibles of alligators and other archo-saurs must also be subjected to sagittal (zy), coro-nal (xy), and transverse (xz) shear stresses becausethe muscle, bite, and joint reaction force vectorsare not coincident. Here, we concentrate on theshear stresses in sagittal and transverse planescaused by vertical and horizontal components ofbite, joint reaction and muscle forces. These shearstresses will vary along the length of the mandi-ble, depending on the position of the cross sectionrelative to the external shearing forces; they alsovary with changes in bite point. Because the ulti-mate strength of bone is greater in tension or com-pression than in shear (Reilly and Burstein, 1975),and because the Alligator mandible approximatesa long beam in its morphology, we hypothesizethat bending stresses will be larger than sagittalor transverse shear stresses. This predicts thatmandibular morphology is unlikely to minimizesagittal and transverse shear stresses. However, inareas of the mandible where shear stresses pre-dominate [e.g., between bite point and jaw elevatormuscles (during posterior biting) or at the symphy-sis], shear stress may be an important determi-nant of mandibular morphology; this could bereflected by morphological changes such asincreased dorsoventral mandibular height.

Shear stress in a beam section is directly propor-tional to the magnitude of the external shearingforces and inversely proportional to the cross-sec-tional area of the section. In hollow beams withwalls of uniform thickness shearing stresses areuniformly distributed, unless the section warpsunder load. In hollow beams with walls of nonuni-form thickness, shear flow can result in accumula-tions of high stresses in thin-walled areas (Hibb-eler, 2000). These considerations suggest thatthere may be significant differences in shear stressmagnitudes between a simple prismatic beam and

MECHANICS OF THE Alligator MANDIBLE 3


the Alligator mandible. If sagittal and transverseshear stresses are more uniformly distributed inthe Alligator mandible than in the prismaticbeam, this will support the hypothesis that Alliga-tor mandible morphology minimizes these shearstresses while minimizing material.

Torsional Shear Stresses

The Alligator mandible, and those of other arch-osaurs, must experience torsional stresses if theexternal forces acting on the mandible do not passthrough the centroidal axis of the mandible. In asolid beam with a circular cross section, the shearstress due to torsion (sT) at a point in the crosssection is a function of the external torque (T) act-ing on the beam and the distance of the point ofinterest from the twisting axis. sT is inverselyrelated to the polar moment of inertia (or torsionconstant), J, which reflects the resistance of thebeam to torsion (Roark, 1965; Roark and Young,1975; Hylander, 1979b; Turner and Burr, 1993).

The simple torsion formula cannot be applied tononcircular, hollow cross sections, such as mandi-bles, and other methods must be used (Daegling,1989, 2002, 2007a,b; Daegling and Hylander,1998). Various formulae can be used to account foreccentricity of the ellipse and wall thickness(Roark and Young, 1975), but all these formulaeare approximations (Daegling and Hylander, 1998;Daegling, 2007a; Ameen, 2008). Indeed, the applic-ability of elliptical models is not supported by em-pirical observation (Daegling, 1989) and even ellip-tical models with nonuniform wall thicknesses(Biknevicius and Ruff, 1992) ‘‘impose a geometricregularity to (mandibular) corpus shape that israrely observed in nature’’ (Daegling andHylander, 1998). Values of J can be calculatedfrom digitized images of cross sections, providingthe most direct estimate of resistance to torsion(Turner and Burr, 1993; Daegling, 2002); however,even these estimates fail to capture important

aspects of the behavior of a bony cross section whensubjected to a load, such as shear flow and warpingof the section (Hibbeler, 2000). FEA remains ‘‘thepreferred solution for modeling torsion (Hermann,1965)’’ (Turner and Burr, 1993; Daegling, 2002).These considerations suggest that in order to deter-mine whether the Alligator mandible functions toreduce torsional stress, comparisons between FEMsof the Alligator mandible and a simple beam of ellip-tical cross section would be useful.

MATERIAL AND METHODS

To calculate the stresses acting within the Alligator mandibleas accurately as possible, we estimated the external muscle forcesacting on the mandible, predicted the resulting external reactionforces using lever arm mechanics, and mapped these forces ontoa free body diagram of the mandible. External muscle forces andconstraints were applied to a hollow, elliptical beam with a con-stant cross section using 2D stress analysis software, our own cal-culations in Excel, as well as a 3D FEM of the simple beam. Todetermine the extent to which Alligator mandible morphologyacts to minimize shear and bending stresses, these forces andconstraints were also applied to a high-resolution FEM of the Al-ligator mandible, and the resulting stresses in the mandiblemodel were compared with those in the beams.

Alligator Mandible Free Body Diagram

A preserved head of Alligator mississipiensis was CT scanned(June 2005) at Stony Brook University Medical Center using aGE LightSpeed 16 CT scanner at 100 kV/70 mA. The individualhad been acquired when �1 m long and was kept in captivityfor 6 months. Basal skull length (anterior rostrum—occipitalcondyle) of the scanned specimen was 160 mm; mandibularlength (anterior tip—retroarticular process) was 189 mm. Atotal of 645 coronal slices (DICOM image format) was producedwith a slice spacing of 0.31 mm; scan resolution was 0.25 mm/pixel. Segmentation of the CT data, as well as construction andanalysis of the finite element model, were carried out at theUniversity of Chicago from 2009 to 2010. CT data were proc-essed using Amira 5.2.2 (Visage Imaging GmbH, Berlin, Ger-many). The scans were segmented to exclude soft tissues andseparate the bones, sutures, teeth, and tooth sockets of thelower jaw (detailed procedures for segmentation are describedby Lemberg et al., 2009); the cranium, including all bones andteeth, was segmented as a single part (Fig. 1A). Sutures andperiodontal ligament (PDL) were identified as low-density areas

Fig. 1. A: Three-dimensional model of Alligator created in Amira; varying colors denote mandibular bones, teeth, periodontum,and sutures that have been separated. B: Meshed finite element model in Strand7; muscle attachment sites (various colors) havebeen mapped onto the cranium and mandible (shown in light blue). Axes indicate global coordinate system of the finite elementmodel. C: Detail of tetrahedral mesh at bone-suture interface (medial view of right splenial-angular-coronoid contact); bone shownin blue, sutures in yellow.

4 L.B. PORRO ET AL.


between bones or between bones and teeth. Sutural morphologyand thickness varies throughout the mandible, although theyare continuous between bone surfaces (i.e., at no point do twoadjacent bones contact each other).The surface model generated in Amira was meshed in the fi-

nite element (FE) software package Strand7 2.4.1 (Strand7 PtyLtd, Sydney, Australia). Muscle attachment sites on the cra-nium and mandible (and the nodes closest to their centroids)were mapped onto the FEM (Fig. 1B) based on dissections ofthe jaw muscles of Alligator and other crocodilians (Hollidayand Witmer, 2007). Muscles accounted for in this study include:M. adductor mandibulae externus superficialis (mAMES), medi-alis (mAMEM), and profundus (mAMEP); M. adductor mandi-bulae internus [composed of M. pterygoideus dorsalis (mPTd)and ventralis (mPTv), and M. pseudotemporalis superficialis(mPSTs) and profundus (mPSTp)]; M. adductor mandibulae pos-terior (mAMP); M. depressor mandibulae (mDM); and M. intra-mandibularis (mIM). The small muscle comprising M. constric-tor internus dorsalis was excluded. (We prefer the term jaw ele-vator muscles to refer to the jaw ‘‘adductors,’’ but retain‘‘adductor’’ in the proper names of these muscles for continuitywith prior work.)The 3D model and the centroids of muscle attachment sites

were imported into the computer-aided design software Solid-Works 2006 (Dassault Systemes SolidWorks, Concord, MA) andthe orientation and moment arm of each muscle were deter-mined (see Discussion section below for calculation of muscleforce magnitude). Lever arm mechanics were used to predictthe magnitude and orientation of reaction forces at the jawjoint, bite point (working side), and symphysis (balancing side).Final images of the free body diagram (Fig. 2) were createdusing Adobe Illustrator and Photoshop CS (Adobe SystemsIncorporated, San Jose, CA), with the length of the force vectorsindicating their relative magnitude. For all models, the medio-lateral/transverse axis is set as the x axis; the dorsoventral asthe y axis; and the anteroposterior/longitudinal as the z axis(Fig. 1B). Mesh refinement is constant across all parts of themodel (bones, teeth, sutures, PDL; Fig. 1C).The cartilago transiliens presents a challenge in modeling

jaw elevator muscle forces in Alligator. The cartilago transiliensconsists of two cartilaginous discs in a bag of connective tissueonto which several muscles insert (Schumacher, 1973; VanDrongelen and Dullemeijier, 1982). The cartilago transiliens liesbetween the mandible (surangular-coronoid contact) and thethickened, cartilage-covered lateral edge of the pterygoid flange(torus transiliens; Iordansky, 1964, 1973). Some authors suggestthat the cartilago transiliens attaches to the coronoid bone ofthe lower jaw via a short tendon (Iordansky, 1973; Van Dronge-len and Dullemeijier, 1982); our dissections demonstrated thatthe cartilago transiliens is not attached to the upper jaw, andthere is only a weak connection between the cartilago transili-ens and the coronoid bone. This observation is supported byHolliday and Witmer (2007), who describe the cartilago transili-ens as an intramuscular fibrocartilage sesamoid joining mPSTsand mIM. Thus, we have resolved the forces of mPSTs andmIM into a single force (acting on the cranium at the origin ofmPSTs and acting on the mandible at the insertion of mIM)directed toward the location of the cartilago transiliens.

Muscle Force Calculations

The external muscle forces acting on the mandible were esti-mated from morphological measurements of the jaw elevatormuscles (Table 1). The force generated by individual jawmuscles of Alligator during biting is a function of muscle fiberrecruitment, frequency of motor unit action potentials, musclecontraction velocity, and physiological cross-sectional area(PCSA) taking into account sarcomere lengths at differentgapes. In this study, muscle fiber recruitment was assumed tobe maximal, action potential rate was assumed to be optimal,contraction velocity was assumed to be zero (i.e., static bites

were modeled), and sarcomere length at the selected gape (308)was estimated using modeled length–tension curves.

The physiological cross-sectional area of each muscle in afemale Alligator (mandibular length of 206 mm; 1.33-m totalbody length) was calculated from measures of muscle architec-ture following methods detailed elsewhere (Anapol and Barry,1996; Anapol and Gray, 2003; Anapol et al., 2004, 2008), butsummarized here.

In dissected specimens, muscle fiber length and pinnationangle from up to six fibers were calculated and averaged withineach muscle. Normalized fiber length was then calculated cor-recting for sarcomere contraction using an optimum sarcomerelength of 2.5 lm. Physiological cross-sectional area was thencalculated using the equation:

Fig. 2. Alligator-free body diagrams created in SolidWorks.A: Lateral view of balancing side. B: Lateral view of workingside. C: Dorsal view of mandible. D: Anterior view of workingside. E: Anterior view of balancing side. Muscle force vectors(solid arrows) and reaction force vectors (broken arrows) shown.Jtbal, balancing side joint reaction force; Jtwk, working side jointreaction force; Ptbal, balancing side pterygoid flange reactionforce; Ptwk, working side pterygoid flange reaction force; Symbal,balancing side symphyseal reaction force; Symwk, working sidesymphyseal reaction force; Bite, bite force [illustrated in (C) byan open circle]. Muscle abbreviations defined in text.



PCSA ¼ MmuscleCosu

Lf 1:0564ð1Þ

with Mmuscle as muscle weight, y as pinnation angle, Lf as nor-malized fiber length, and 1.0564 as the specific gravity of themuscle. Maximum possible force production (Fpmax) was calcu-lated by multiplying PCSA by 25 N/cm2 (250 kN/m2), the spe-cific tension of mammalian muscle (Herzog, 2007):

Fpmax ðNÞ ¼ PCSA ðcm2Þ325 N=cm2 ð2Þ

Ideally, length–tension curves for the elevator muscles of Alliga-tor would be used to estimate maximum possible force genera-tion potential for each muscle at our selected gape of 308 (Fp30)(Nordstrom et al., 1974; Nordstrom and Yemm, 1974; Weijs andvan der Wielen-Drent, 1983). At present, length–tension curvesare not available for Alligator muscle; instead, we constructedlength–tension curves for the individual elevator muscles(Herzog, 2007) assuming an optimal sarcomere length (l0) of 2.5lm. Assuming a thick filament length of 1.6 lm (Walker andSchrodt, 1974), H-zone length of 0.2 lm and Z-disk length of 0.1lm, a thin filament length of 1.1 lm yields an l0 of 2.5 lm:

l0 ¼ ð231:1 lmÞ þ 0:1 lmþ 0:2 lm ¼ 2:5 lm ð3Þ

Using data obtained for frog (Gordon et al., 1966), the length–tension curve was anchored at a minimum sarcomere length of1.27 lm and the point of inflection on the ascending limb of thecurve was estimated to be at a sarcomere length of:

1:6 lmþ 0:1 lm ¼ 1:7 lm ð4Þ

The lower end of the plateau was set at a sarcomere length of:

ð231:1 lmÞ þ 0:1 lm ¼ 2:3 lm ð5Þ

Optimal sarcomere length (l0, Eq. 3) was at the right end of theplateau and the maximum sarcomere length is equal to:

ð231:1 lmÞ þ 1:6 lmþ 0:1 lm ¼ 3:9 lm ð6ÞStandardizing these by l0 yields relative sarcomere lengths (l/l0) of 0.508, 0.68, 0.92, 1.0, and 1.56 at these points and relativeforce magnitudes of 0, 84, 100, 100, and 0%, respectively(Fig. 3). Linear interpolation was then used to estimate theforce generated at a gape angle of 308 for the jaw elevatormuscles using estimates of muscle stretch derived from theAlligator FEM (Fig. 1B). This was done by comparing musclelength on the FEM with the jaws at minimum gape and rotatedopen to 308 (Table 2). The proportion of cross-sectional areaavailable at this gape was multiplied by muscle stress to yieldthe Fp30 for each muscle at 308 (Table 3):

Fp30 ¼ PCSA%Fpmax

100Fpmax ð7Þ

To estimate the force components acting in sagittal, coronal,and transverse planes, the muscle forces and their momentarms were resolved into components acting about the x, y, and zaxes (Table 4). The resulting forces were applied to the FEMs.Muscle contraction was assumed to be simultaneous and symmet-rical. For the beam models, forces acting on the right mandiblewere applied to models representing both the working side andbalancing side. In the Alligator FEMs, biting occurred on the rightdentary; thus, the left mandible is the balancing side. The modelscreated in Beam 2D (see below) were not subjected to anteroposte-rior muscle/reaction forces because this software package can onlymodel forces acting perpendicular to the beam.

For each modeling method, patterns of stress were calculatedfor five separate bite point locations [3rd (anterior), 7th (ante-rior–middle), 12th (middle), 14th (posterior-middle), and 19th(posterior) dentary teeth], and changes in stress due to varyingbite points were evaluated.

Calculation of Shear and Bending StressUsing Beam 2D

The stress analysis software Beam 2D (ORAND Systems,Mississauga, ON, Canada) was used to calculate reaction forces

TABLE 1. Muscle architecture data from Alligator mississippiensis

Muscle

Measures at minimum gape

Muscleweight (g)

Distal tendonlength (mm)

Proximaltendon

length (mm)Muscle bellylength (mm)

Fasciclelength (mm)

Length of 10sarcomeres (lm)

Percentsarcomere

contraction atminimum gapea

mAMES and mAMEM 3.70 0.00 0.00 30.00 20.80 19.77 79mAMEP 1.74 0.00 0.00 31.60 28.50 18.55 74mAMP 11.04 12.27 0.00 49.00 14.30 19.61 78mPSTs and mPSTp 2.81 0.00 5.18 34.00 15.93 19.57 78mPTd Ib 13.06 0.00 8.82 54.00 19.70 19.98 80mPTd IIb 18.92 0.00 11.87 75.00 20.97 20.07 80mPTv 35.41 0.00 18.18 77.40 46.75 18.84 75

MusclePinnation

angle yTendonpercentage

Adjustedfascicle

length (cm) Cos yPCSA (cm2)/force

(N)

Adjustedfascicle

length (mm)

Adjustedmuscle bellylength (mm)

mAMES and mAMEM 0.00 0.00 2.70 1.00 1.30/32.42 27.01 23.73mAMEP 0.00 0.00 3.87 1.00 0.43/10.63 38.74 23.45mAMP 16.64 39.59 1.84 0.95 5.45/136.20 18.38 38.44mPSTs and mPSTp 12.44 25.58 2.06 0.98 1.26/31.51 20.61 26.63mPTd Ib 13.04 29.87 2.46 1.00 5.02/125.51 24.59 43.17mPTd IIb 8.50 34.21 2.60 0.99 6.80/170.12 26.03 60.23mPTv 16.87 27.79 6.20 0.96 5.17/129.25 62.04 58.32

aNote that at minimum gape all of the muscles are contracted to �80% of their assumed optimal sarcomere length.bMuscle architecture data obtained for two separate portions of mPTd; combined PCSA of these portions and identical values (0.80)for percent sarcomere contraction at minimum gape used to determine force available at minimum gape.

6 L.B. PORRO ET AL.


and stress due to shear and bending along the mandible assum-ing a simple beam geometry. The beam was given a length of189 mm, corresponding to the length of the mandible (Fig. 4A–D). The cross-sectional geometry of the beam (a hollow, dorso-ventrally elongated ellipse) was constructed, based on measure-ments from CT scans of the right Alligator mandible under the19th dentary tooth (Table 5). This cross section was assumed tobe constant along the length of the beam. Young’s modulus wasset at 15 GPa, the approximate mean elastic modulus of Alliga-tor mandibular cortical bone from Zapata et al. (2010).The muscle forces acting on the mandible were applied along

the beam model at positions matching the z-coordinates of theircentroids in the Alligator FEM. The beam was modeled in bothlateral and dorsal views (Fig. 4A–D). For lateral views, the ver-tical (y) components of each muscle were applied to the beam.Additionally, for mPTd and mPTv, an anteriorly directedmoment generating a counter-clockwise (positive) torque aboutthe jaw joint was applied; this was done in consideration of thelarge cross-sectional area and anterior orientation of the ptery-goideus muscles. For dorsal views, the horizontal (x) componentof each muscle was applied to the beam.The balancing side beam was supported at the jaw joint and

symphysis in lateral view, generating vertical reaction forces atboth points (Fig. 4A). The constraint at the symphysis repre-sents the working side fixed at the bite point. The working sidebeam was supported at the jaw joint and bite point in lateralview, and a force equal but opposite the symphyseal reactionforce recorded on the balancing side beam model was applied tothe symphysis of the working side (Fig. 4B). This symphysealforce represents the forces exerted by the balancing side muscu-lature on the working side mandible. In dorsal view, the balanc-ing side was supported at the jaw joint and symphysis; theworking side was supported at the jaw joint, bite point, andsymphysis. A preliminary set of trials included an additionalsupport representing the pterygoid flange–mandibular contactfor both the balancing side and working side in dorsal view;

this support was subsequently removed to examine the effect ofits presence (see more below).

Calculation of Torques and TorsionalStresses

To calculate torques and torsional stresses (about the z axis),vertical and horizontal muscle and reaction forces (the latterextracted from Beam 2D) were used to calculate the externaltorques acting along the length of the beam in Excel. Maximumshear stress due to torsion (sT-max) in each cross section throughthe theoretical, hollow, elliptical beam was then calculatedusing Roark’s formula 13 (Young, 1989; Table 20):

sT�max ¼ T

2ptða� 12 tÞðb� 1

2 tÞð8Þ

where t is wall thickness, a is maximum height of the cross sec-tion and b is maximum width (see Table 5).

Finite Element Modeling—Beam Model

The cross section described above (Table 5) was used to createa 3D FEM of the beam (Fig. 4E). The beam FEM is 189 mm inlength, is composed of 64,071 solid hexahedral elements, andwas given the same elastic modulus (15 GPa) as the Beam 2Dmodel. Values for density (1772.6 kg/m3) and Poisson’s ratio(0.29) corresponding to the averages obtained for Alligator man-dibular cortical bone (Zapata et al., 2010) were applied. Mate-rial properties were assumed to be isotropic. Muscle forces (x, y,and z components) and constraints were placed at the samez-coordinates as in the Beam 2D models; each muscle (and, forthe working side, symphyseal) force was evenly distributed over

Fig. 3. Diagram illustrating methods for modeling length tension curves of Alligator muscles. Based on Herzog (2007). ‘‘Resting’’or optimal sarcomere length (l0) at the right end of the plateau was assumed to be 2.5 lm, thick filament length to be 1.6 lm(Walker and Schrodt, 1974), H-zone length to be 0.2 lm and Z-disk length to be 0.1 lm, yielding a thin filament length of 1.1 lm.The length–tension curve was anchored at a minimum sarcomere length of 1.27 lm (Gordon et al., 1966), yielding a point of inflec-tion on the ascending limb of the curve at a sarcomere length of 1.7 lm, the lower end of the plateau at a sarcomere length of 2.3lm and a maximum sarcomere length of 3.9 lm. Standardizing these by l0 yields relative sarcomere lengths (l/l0) of 0.508, 0.68,0.92, 1.0, and 1.56 at these points and relative force magnitudes of 0, 84, 100, 100, and 0% respectively.



five nodes. Three nodes, aligned mediolaterally, were con-strained at the jaw joint whereas one node was constrained atthe bite point (working side) or symphysis (balancing side).Thus, constraints in the beam FEM closely match those in theAlligator FEMs (see below). Additionally, muscle forces and con-straints were placed either on the dorsal or ventral surface ofthe beam, and either medial or lateral to the midline, based onthe position of muscle insertions, and the jaw joint, toothrow,and symphysis in the Alligator mandible. Such asymmetricalplacements allow these forces/constraints to generate torqueabout the z-axis of the beam. Five load cases were analyzed cor-responding to the five bite points on the working side, as wellone additional case representing the balancing side. The jawjoint and bite points were constrained in all six degrees of free-dom; the symphysis (in the balancing side model) was con-strained only against translation along the y axis.The models were solved using the linear static solver in

Strand7. Reaction forces at the jaw joint and bite point wererecorded and compared with results from Beam 2D and the Al-ligator mandible FEM. Bending stress [i.e., parallel to the long(z) axis of the beam] was recorded along longitudinal transectson the ventral sagittal midline of the beam, and along the mid-line of the lateral aspect of the beam. Additionally, shearstresses in zy planes were recorded along lateral and medialtransects in order to calculate torsional stress along the beam.Twenty coronal cross sections were taken along the length ofthe beam (one section every 10 mm). For each section, undereach load case, we obtained the total shear force acting on theplane in the transverse and sagittal directions; this was dividedby the cross-sectional area of the beam to obtain average shearstress in each section. Stresses recorded from the beam FEMwere compared with results from the Beam 2D models and Alli-gator FEMs.

Finite Element Modeling—Alligator Model

The FEM of the Alligator skull contains 1,001,484 linear tet-rahedral elements (cranium: 360,399; mandible: 641,085). Mus-cle forces were evenly distributed onto origins/insertionsmapped onto the skull (Fig. 1B) using the Visual Basic programBoneLoad (Grosse et al., 2007), which accounts for tensile, tan-gential, and normal traction loads due to muscles wrappedaround curved bone surfaces. This is an important considera-tion for Alligator because, for example, mPTv wraps around theventral margin of the mandible to insert on the lateral surfaceof the retroarticular process.Three nodes aligned mediolaterally across the joint surfaces

of each quadrate and articular were constrained against dis-placement (Fig. 4F), creating an axis around which the craniumand mandible rotated when muscle forces were applied. Theseconstraints prevented rigid body motion and generated reaction

forces at the jaw joints. The tip of a single dentary tooth wasconstrained to generate bite force; five load cases were analyzed(corresponding to the five bite point locations used in both theBeam 2D and beam FEM). For each case, the tip of the closestopposing maxillary/premaxillary tooth was also constrained.These constraints are similar to the ‘‘Strait method’’ used insome mammalian FEMs (Dumont et al., 2005; Ross et al., 2005;Strait et al., 2005). Constraining a single node at the bite pointresults in artificially high stress/strain values; however, thiseffect is highly localized (Grosse et al., 2007).

Three FEMs of the Alligator mandible were analyzed: twomodels differ in their material properties, and are referred to assimple and suture. Simple used the same homogeneous and iso-tropic material properties as the beam FEM. Suture reflects ourcurrent best estimate of the material properties of all the com-ponents of the Alligator mandible (Table 6). Anisotropic bonematerial properties are from an individual of similar size to thespecimen used to construct the FEM (Zapata et al., 2010).There is no published information on the material properties ofcrocodilian teeth, sutures, or PDL, so the teeth were assignedthe material properties of bovine dentine (Gilmore et al., 1969),which has been applied to other archosaur FEMs (Rayfieldet al., 2001). Density and Poisson’s ratio for sutures were aver-ages of values reported in the literature (Currey, 2002; Kupcziket al., 2007). Elastic modulus for mammalian cranial suturesreported in the literature varies considerably, ranging from 1.16MPa to 12.1 GPa (Margulies and Thibault, 2000; Radhak-rishnan and Mao, 2004; Kupczik et al., 2007). Preliminary trialsusing the Alligator FEM found that sutural elastic modulusstrongly influenced strain magnitude and orientation in thesuture and the immediate surrounding bone. For our suturemodel we assigned sutures an elastic modulus of 90 MPa, whichis approximately two orders of magnitude lower than the lowestelastic modulus we used for Alligator bone and falls within therange of values recorded in other taxa. Mammalian PDL is farmore compliant than sutures (Rees and Jacobsen, 1997; Yoshidaet al., 2001). Unlike mammals, in which the teeth are attachedto the alveolar bone socket via PDL (gomphosis), most reptileshave teeth ankylosed directly to the jawbone. Crocodilian PDLappears to be structurally intermediate between mineral-freemammalian PDL and mineralized attachment in lizards (McIn-tosh et al., 2002). For the suture model, PDL was assigned thesame material properties as sutures.

The elongate pterygoid flanges of crocodilians have been sug-gested to act as braces resisting medial translation (inverse

TABLE 3. Calculation of Fp available at 308 gape in Alligatormississippiensis

Musclel/l0 at

min. gapea l/l0 at 308InterpolatedFp (N) at 308

% of totalmuscle forceb

mAMESc 0.79 1.10 13.30 2.04mAMEMc 0.79 1.10 13.30 2.04mAMEP 0.74 0.88 10.36 1.58mAMP 0.78 1.04 125.59 19.32mPSTsd 0.78 1.15 11.61 1.78mIMe — — 33.18 5.10mPSTpd 0.78 1.15 11.61 1.78mPTd 0.80 0.90 291.17 44.79mPTv 0.75 0.86 124.03 19.07mDMe — — 15.90 2.44

aEquivalent to % contraction of sarcomeres at minimum gape.bPercentage each muscle contributes to total muscle force (650N, the sum of Fp at 308 for all muscles) produced on one side ofthe head.cmAMES and mAMEM were considered a single muscle massin Table 1; Fp was divided equally between these muscles.dmPSTs and mPSTp were considered a single muscle mass inTable 1; Fp was divided equally between these muscles.eData on muscle architecture were not collected for mDM ormIM. Fp for these muscles was extrapolated using data fromBusbey (1989) and Cleuren et al. (1995).

TABLE 2. Muscle length (centroid of origin to centroid ofattachment) measured from the FE model at minimum gape

and at 308 gape

Muscle

Musclelength atminimumgape (mm)

Muscle lengthat 308 (mm) % Change

Right mAMES 21.52 32.68 51.89Right mAMEM 28.22 39.26 39.12Right mAMEP 79.48 94.41 18.78Right mAMP 41.24 52.13 26.42Right mPSTs 39.80 55.87 40.37Right mIM 24.66 24.66 0.00Right mPSTp 38.88 56.75 45.97Right mPTd 49.62 54.86 10.31Right mPTv 47.37 52.20 11.40Right mDM 24.77 21.00 215.24

8 L.B. PORRO ET AL.


wish-boning) of the mandibles during jaw closure (Busbey,1995; Iordansky, 1964). The third model, Pterygoid (PT) flange,evaluates this hypothesis by using the same material propertiesas the suture model, but differing in constraining a single nodeon the medial surface of each mandible, opposite its contactwith the pterygoid flange at 308 of gape, to restrict medialmovement (along the x axis) of each mandibular ramus.

TABLE

4.Resultantandresolved

muscle

forces,mom

entarm

s,andmom

ents

forea

chsideof

theAlligatormandible

Muscle

Total

force(N

)X

force(N

)Y

force(N

)Z

force(N

)

Total

mom

ent

arm

(m)

Xmom

ent

arm

(m)

Ymom

ent

arm

(m)

Zmom

ent

arm

(m)

Total

mom

ent

(Nm)

Mom

ent

abou

tX

(Nm)

Mom

ent

abou

tY

(Nm)

Mom

ent

abou

tZ

(Nm)

mAMES

13.30

8.19

10.33

21.74

0.0242

0.0005

0.0044

0.0237

0.3212

0.2530

0.1989

20.0584

mAMEM

13.30

10.66

6.92

23.93

0.0313

0.0091

0.0028

0.0298

0.4159

0.2379

0.3537

20.1215

mAMEP

10.36

7.42

6.38

23.40

0.0411

0.0120

0.0068

0.0388

0.4260

0.2843

0.3312

20.1344

mAMP

125.59

55.68

111.79

213.67

0.0238

0.0069

0.0007

0.0227

2.9865

2.5611

1.3602

0.8705

mPSTs

11.61

20.66

10.83

24.13

0.0405

0.0035

0.0142

0.0374

0.4704

0.4635

20.1571

20.1588

mMI

33.18

21.89

30.94

211

.81

0.0405

0.0035

0.0142

0.0374

1.3444

1.3247

20.4491

20.4537

mPSTp

11.61

20.24

8.56

7.84

0.0335

0.0005

0.0226

0.0247

0.3888

0.3887

0.1937

0.1938

mPTd

291.17

186.17

70.96

212.35

0.0135

0.0034

0.0130

0.0014

3.9424

2.9361

21.0321

2.6839

mPTv

124.03

95.86

9.53

78.13

0.0155

0.0050

0.0146

0.0005

1.9200

1.1530

20.6204

1.4900

mDM

15.90

5.17

9.38

11.75

0.0106

0.0063

0.0051

0.0068

0.1679

20.1280

20.1187

20.0864

TotalResultant(N

)X

Resultant(N

)Y

Resultant(N

)Z

Resultant(N

)532.0

366.27

275.63

271.39

Signvalues

given

apply

totheright(w

orking)sideramuswithin

thecoordinate

system

discu

ssed

inthetext—

xaxis

ismed

iolateral(positiveis

toward

stheleft),yaxis

isdor-

soven

tral(positiveis

dorsa

llydirected),andzaxis

isanteroposterior

(positiveis

anteriorly

directed).

Fig. 4. Illustrations of models discussed in this article. A–D:Models created in Beam 2D; black arrows represent muscle/symphyseal forces, bent arrows represent torques from mPT,gray triangles represent supports. E, F: Finite element models.A: Beam representing the balancing side in lateral view, withsupport at symphysis. B: Beam representing the working sidein lateral view, with anterior bite point. C: Beam representingthe balancing side in dorsal view, with supports placed on themedial aspect of the beam. D: Beam representing the workingside in dorsal view during an anterior bite, with supportsplaced on the medial aspect of the beam. E: Three-dimensionalbeam FEM during anterior bite on the working side, with sim-plified muscle vectors (black arrows) and constraints (gray tri-angles). F: Finite element model of Alligator mandible duringanterior bite with simplified constraints (gray triangles) anddetailed applied muscle forces (black areas). E and F shown inthe same coordinate system.



Results from the Alligator mandible FEMs were extractedand analyzed in the same manner as for the beam FEM; how-ever, only bones were included in transects (i.e., sutures, peri-odontal ligament and teeth were not sampled). Because the bal-ancing side is a left mandible in the Alligator FEMs (they aremodeled as right mandibles in the beam models), plots of medio-lateral (zx) shear stress and torsional stress for the balancingside of all Alligator FEMs were inverted for easier comparisonwith results from beam models.

Hypothesis Testing

To determine whether bending stress magnitudes vary lessalong the length of the Alligator mandible than they do alongthe simple beam, the following estimates of maximum compres-sive and tensile stress due to bending (rmax) were calculatedand compared:

� Maximum bending stress (rmax) along a theoretical, hollow, el-liptical beam using

rmax ¼ My

Ið10Þ

� the magnitude of rmax along a ventral transect of the beamFEM;

� the magnitude of rmax along a ventral transect of the Alliga-tor mandible FEM without and with sutures, and a pterygoidflange contact.

To quantify the effect of Alligator mandibular geometry onsagittal and transverse shear stresses, the following esti-mates of maximum shear stress (smax) were calculated andcompared in sagittal and transverse planes:

� smax in each cross section through the theoretical, hollow,elliptical beam using:

smax ¼ Fs

Að11Þ

where Fs is shearing force (the sum of all shearing forces actingto one side of the cross section) and A is the area of the crosssection;

� smax in each cross section through the beam FEM;� smax in each cross section through the Alligator mandibleFEM without and with sutures, and a pterygoid flange contact.

To determine whether torsional stress in the Alligator mandi-ble is less variable than that in a simple beam, the followingestimates of maximum shear stress due to torsion (sT) were cal-culated and compared:

� sT along the theoretical, hollow, elliptical beam using Roark’sformula 13 from (Young, 1989) [our Eq. (8)];

� sT in the beam FEM;� sT in the Alligator mandible FEM without and with sutures,and a pterygoid flange contact.

RESULTSExternal Forces and Free Body Analysis

At minimum gape, all of the measured Alligatorjaw elevator muscles are contracted to �80% oftheir optimal sarcomere length, whereas at 308 ofgape mAMP is on the plateau, three (mPST andmAMES/mAMEM) are on their descending (right)limb, and three (mPTd, mPTv, and mAMEP) areon the ascending limbs of their length–tensioncurves (Fig. 5). M. pterygoideus dorsalis shows thegreatest change in length with change in gape,mPTv shows the least, and neither reaches theplateau of its length–tension curve at 308 of gape.

The resultant muscle force on the lower jaw (Ta-ble 4) is directed upward and medially at approxi-mately 2538 relative to the y-axis in anterior view,and forward at approximately 2458 from the y-axis in right lateral view (by convention, negativevalues are clockwise). Muscle forces acting on theworking side and balancing side are the samesince symmetrical muscle activity was assumed. Inlateral view (Fig. 2A,B) the mandible is subjectedto muscle forces with orientations that range fromantero-dorsal (mPTv, mPTd, mPSTp) to postero-dorsal (mAME, mAMP, mPSTs). Most muscles actbetween the jaw joint and the posterior end of thetoothrow; mPTv and mPTd attach to the mandibleposterior and ventral to the jaw joint, while mDMattaches posterior to and level with the jaw joint.Lever arm mechanics were used to predict verticalreaction forces at the jaw joints, biting tooth, andsymphysis (Table 7). As expected, bite force is cal-culated to increase with more posterior bites; as aresult, reaction force on the working side jaw jointis predicted to decrease with more posterior bites.Reaction force at the balancing side jaw joint iscalculated to always be higher than that on theworking side and is predicted to be unchanged bychanges in bite point on the working side. As aresult, total reaction force at the balancing sidejaw joint should be more vertically oriented andlarger in magnitude than reaction force on theworking side.

TABLE 6. Material properties assigned to suture version of theAlligator mandible FEM

Material OrientationDensity(kg/m3)

Elasticmodulus(GPa)

Shearmodulus(GPa)

Poisson’sratio

Bone X 1662.8 8.1Y 1662.8 9.26Z 1662.8 19.71XY 1662.8 3.17 0.38XZ 1662.8 4.45 0.08YZ 1662.8 5.51 0.15

Teeth All 2,076 21 N/Aa 0.31Sutures

and PDLAll 1,200 0.09 N/Aa 0.3

aShear modulus is automatically calculated in Strand 7 for iso-tropic materials.

TABLE 5. Cross-sectional geometry of beam used in Beam2Dand finite element model, taken from CT scans under 19th

dentary tooth

Maximum externalheight (a) (mm)

19.15

Maximum externalwidth (b) (mm)

14.48

Average wallthickness (t) (mm)

2.94

Cross-sectionalarea (mm2)

129.32

Torsional inertiaconstant (J) (m4)

5.77865 E 209

Cross section shown at right.

10 L.B. PORRO ET AL.


The medially directed components of all the jawmuscles (except mIM) are evident in dorsal view(Fig. 2C). Laterally directed reaction forces aregenerated at the jaw joints, at the contact with thepterygoid flange (Busbey, 1995), and at the bitepoint on the working side. Because mPTd andmPTv are large (Table 4) compared with theremaining elevator muscles, are strongly inclinedmedially, and attach posterior to the jaw joint, thelever arm calculations predict that the majority ofthe laterally directed reaction force will occur atthe jaw joints, with only small reaction forces pre-dicted at the bite point and pterygoid flange-man-dibular contact. The symphysis is calculated to bein mediolaterally oriented tension.

The jaw elevators and mDM attach to the dorsalor ventral margins of the mandible, so their medi-ally directed components generate torques aboutthe long (z) axis (Fig. 2D,E). Because mPTd,mPTv, and mAMP attach to the ventral margin ofthe mandible, these powerful muscles are calcu-lated to produce a positive (counterclockwise in an-terior view) torque on the right side mandible(clockwise or negative torque on the left side).

The pterygoid flange does not restrict anteropos-terior motion of the mandible. Furthermore, it was

assumed that the bite point does not experienceanteroposterior reaction forces. Thus, a longitudi-nal reaction force equal but opposite to that gener-ated by the jaw muscles is calculated to occur atthe jaw joints (Table 7).

Beam Analyses

In describing our results, shearing and bendingstresses are described as occurring within anatom-ical planes. Specifically, dorsoventrally directedmuscle, symphyseal, bite and joint reaction forcesproduce shearing and bending stresses within sag-ittal (yz) planes. Mediolaterally directed muscle,bite, and joint reaction forces produce shearingand bending stresses within transverse planes(xz). Torsion is described as occurring around thelong axis of the Alligator mandible and the long (z)axis of the beam models, generating shear stressesboth in planes tangent to the surface of the crosssection, as well as between adjacent cross sections.

Reaction forces. In the Beam 2D model, verti-cal reaction forces generated at the bite points andbalancing side jaw joint are very similar to thosepredicted by lever arm mechanics and vertical

Fig. 5. Length tension curves calculated for each of the muscles modeled in this study. Thethick lines indicate the range of the length–tension curve explored by each muscle from mini-mum gape (left end of the thick line) to 308 of gape. Muscle abbreviations described in text.



TABLE 7. Reaction forces generated by each model and in other studies

Model Bite position

Bite force (N) Jaw joint (N)Total biteforce/totaljoint rxnforce

Joint rxnforce angle(8, anterior

view)

Joint rxnforce angle(8, lateralview)X Y Z X Y Z

Leverarm mech.

Balancing — — — 2364.5 2218.2 2271.4 — 259.09 251.2

Anterior 0 2119.6 0 2364.5 2156.0 2271.4 0.25 266.83 260.11Anterior-middle 0 2135.1 0 2364.5 2140.5 2271.4 0.28 268.92 262.63Middle 0 2154.2 0 2364.5 2121.5 2271.4 0.33 271.57 265.89Posterior2middle 0 2176.9 0 2364.5 298.7 2271.4 0.38 274.84 270.01Posterior 0 2214.1 0 2364.5 261.5 2271.4 0.47 280.42 277.23

Beam 2D Balancing — — — 2369.2 2219.7 — — 259.24 —Anterior 32.06 2118.6 — 2371.0 2212.9 — 0.29 260.15 —Anterior2middle 13.5 2135.0 — 2371.5 2196.5 — 0.32 262.12 —Middle 10.4 2156.7 — 2372.2 2174.8 — 0.38 264.84 —Posterior-middle 9.7 2186.7 — 2373.1 2144.8 — 0.47 268.79 —Posterior 10.43 2238.4 — 2374.8 293.1 — 0.62 276.05 —

BeamFEM (XYZ)

Balancing — — — 2366.2 2215.8 2271.4 — 259.49 251.51Anterior 5.2 2124.1 2202.3 2371.4 2207.3 269 0.55 260.83 218.41Anterior2middle 5.7 2142 2170.7 2372 2189.5 2100.6 0.52 263.01 227.96Middle 6.4 2164.3 2140.5 2372.7 2167.1 2130.8 0.50 265.85 238.05Posterior2middle 7.3 2195.1 2111.7 2373.6 2136.4 2159.5 0.52 269.94 249.46Posterior 9 2249.6 280.4 2375.3 281.8 2190.8 0.61 277.70 266.79

AlligatorFE model(simple)

Balancing, anterior — — — 2283.9 2300.9 2128.8 — 243.33 223.17Balancing, ant-mid — — — 2284.8 2299.6 2135.4 — 243.55 224.32Balancing, middle — — — 2287.6 2297.9 2144.7 — 243.99 225.91Balancing, post-mid — — — 2289.0 2297.1 2148.7 — 244.21 226.59Balancing, post — — — 2293.6 2295.7 2158.7 — 244.80 228.22Anterior 25.5 283.8 238.1 2278.0 2304.9 2192.6 0.21 242.36 232.28Anterior-middle 31.1 297.7 215.0 2284.1 2292.8 2209.0 0.23 244.14 235.52Middle 29.1 2116.8 23.5 2286.5 2275.3 2211.1 0.27 246.14 237.48Posterior-middle 25.3 2131.6 216.3 2285.3 2261.0 2194.4 0.31 247.55 236.68Posterior 2.2 2170.6 246.1 2270.6 2221.7 2154.6 0.46 250.67 234.89

AlligatorFE model(suture)


AlligatorFE model(PT flange)


Ericksonet al., 2003(in vivo biteforce data)

Middle (11thmaxillary tooth)

— 2230a — — — — — — —

Sinclair andAlexander, 1987

Posterior — 274b 0 — 2103 248 0.65 — 225

Cleuren et al., 1995 Posterior — 293c 278 — 2136 263 0.81 — 265

Sign values given apply to the right side mandible within the coordinate system discussed in the text—x axis is mediolateral (posi-tive is toward the left), y axis is dorsoventral (positive is dorsally directed), and z axis is anteroposterior (positive is anteriorlydirected). Joint reaction force angles are relative to the y axis for a right side mandible; by convention, negative values are clock-wise from the y axis.aEstimated total bite force for 1 meter (total body length) Alligator from regression curve based on bite forces obtained in vivo dur-ing a ‘‘high-force snapping bite.’’bEstimated bite force (assumed vertical) for a �0.64 m Caiman. Bite force obtained for working side (37 N) doubled to account forthe contribution of the balancing side jaw elevator muscles.cEstimated bite force (assumed 21308 from tooth row) for 0.65 m Caiman. Bite force obtained for the working side doubled toaccount for the contribution of the balancing side jaw elevator muscles.

reaction forces on the working side jaw joint areonly slightly higher (Table 7). The beam FEMexperiences higher vertical and lower horizontalreaction force at the biting tooth, and lower verti-cal reaction force at the working side jaw jointthan the Beam 2D model at each bite position.Vertical reaction force on the balancing side jawjoint and lateral reaction forces at both jaw jointsare similar in both the beam FEM and Beam 2D(Table 7).

In dorsal view, laterally directed reaction forcein the Beam 2D model is similar for the balancingside and working side jaw joints, and is slightlyhigher than predicted by lever arm mechanics. Apreliminary set of Beam 2D trials included a sup-port at the pterygoid-mandibular contact; reactionforces generated at this support were very low (<1N) and tensile. Thus, this support was removedfrom most FEM models; however, it was includedin the PT flange Alligator FEM. The beam FEMpredicts a large posteriorly directed reaction forceat the biting tooth, which is highest in magnitudeduring anterior bites and decreases at posteriorbite positions. Posteriorly directed reaction forceon the working side jaw joint is highest in magni-tude during posterior biting. Posteriorly directedforce on the balancing side jaw joint is identical tothat predicted by lever arm mechanics.

Internal stresses—Bending stress in sagit-tal planes. Peak dorsoventral bending stress ishigher than any other stress experienced by thebeams. In both beams, muscle forces cause thebeam (between the jaw joint and anterior muscleinsertions) to deform dorsally (negative bending),resulting in compression along the ventral surface(Fig. 6) and tension dorsally. This negative bend-ing stress peaks dorsal and ventral to the positioncorresponding to the area of insertion of severallarge muscles and the mandibular fenestra in themandible. The entire balancing side experiencesnegative bending (Fig. 6H), with bending stressgradually decreasing to zero at the symphysis. Incontrast, there is an inflection point in the workingside beams (Fig. 6C–G) reflecting positive bendingof the anterior mandible due to bite reaction force;this inflection point is located posterior to the bitepoint, and bending stress due to bite reaction forcepeaks at the bite point itself and decreases to zeroat the symphysis. As the bite point moves anteri-orly, the bending regime approaches that exhibitedby the balancing side. Except during posterior bit-ing, peak bending stress is a result of negativebending produced by muscle forces, and is higheston the balancing side and during anterior biting.Peak dorsoventral bending stress during posteriorbiting is a result of positive bending about the bitepoint. Peak bending stress is higher in the beamFEM than in the Beam 2D models but onlybecause of irregularities due to localized highstress near the muscle insertions (Fig. 6).

Internal stresses—Shear stress in sagittalplanes. Sagittal shear stresses are lower than thetorsional shear stresses discussed below and morethan an order of magnitude lower than dorsoven-tral bending stresses discussed above. The effectsof variation in bite point on the patterns of shearstress in the beams are illustrated in Figure 7.The highest shear stresses generated in sagittalplanes occur between the posterior bite point andthe muscle insertions. As the bite point movesanteriorly, the magnitude of the shear stresses act-

Fig. 6. Dosoventral bending stress in the beam models. A:Lateral view of Alligator working side mandible with muscleforce vectors (solid arrows) and reaction force vectors (brokenarrows). B: Vertical forces (black arrows) and supports (blacktriangles) applied to Beam 2D model representing working side;black arrow at symphysis represents contribution from balanc-ing side muscle force. C–H: Compressive bending stresses(MPa) and location relative to neutral axis indicated in graybox (Beam 2D estimate). Black line indicates stress magnitudeand type (positive is tensile, negative is compressive) recordedalong the ventral transect of the beam FEM. C–G: Workingside, with location of bite point indicated by black triangle. H:Balancing side. Key indicates scale and conventions for positiveand negative bending. Numbers to right of figures indicate max-imum bending stress estimated using Beam 2D (gray) andbeam FEM (black).



ing in the beam between the bite point and muscleinsertions decreases. Sagittal shear stress in thebeam between the muscle insertions and the jawjoint peaks on the balancing side and during ante-rior bites, and decreases as the bite point movesposteriorly. Peak sagittal shear stress magnitudesare higher in the beam FEM than in the Beam 2Dcalculations for each load case but only marginally(Fig. 7).

Internal stresses—Bending and shearstress in transverse planes. Bending and shearstresses in transverse planes predicted by Beam2D are similar for the balancing side and working

side (regardless of bite point) because horizontalreaction forces at the jaw joints are similar acrossall trials and horizontal reaction forces at the sym-physis and bite point are very small. Powerfulmuscles pull medially on the beam both posterior(mPTd and mPTv) and anterior (mAMP andmAME) to the jaw joint. As a result, both bendingand shear stresses peak at the jaw joint (Fig. 8)and the beam is laterally bowed (i.e., experiencespositive bending), placing the pterygoid flange con-tact in ‘‘tension.’’ The mandible experiences medial(negative) bending anterior to the insertion ofmAMP, decreasing to zero at the symphysis.Mediolateral bending stresses are lower than dor-soventral bending stresses; however, shear in thetransverse plane is higher than that recorded insagittal planes in Beam 2D.

Fig. 7. Sagittal shear stress in the beam models. A: Lateralview of Alligator working side mandible with muscle force vec-tors (solid arrows) and reaction force vectors (broken arrows).B: Vertical forces (black arrows) and supports (black triangles)applied to Beam 2D model representing working side; blackarrow at symphysis represents contribution from balancing sidemuscle force. C–H: Shear stresses (MPa) and location relativeto the neutral axis indicated in gray box (Beam 2D estimate).Black line indicates stress magnitude recorded in xy sections ofthe beam FEM. Location of bite point on the working side trials(C–G) is indicated by a small black triangle. H: Balancingside. The standard conventions are shown at the bottom of thefigure.

Fig. 8. Mediolateral stresses and moments in beam models.A: Dorsal view of Alligator working side mandible with muscleforce vectors (solid arrows) and reaction force vectors (brokenarrows). B: Horizontal muscle forces (black arrows) and sup-ports (black triangles) applied to working side Beam 2D model.All stresses and moments were similar regardless of bite point;only those calculated for the posterior bite are shown. C: Calcu-lated bending moments in the beam model shown in (B). Stand-ard conventions for positive and negative moments are illus-trated at left. D: Compressive and tensile bending stresses forBeam 2D model indicated in gray box; tensile stresses for beamFEM (lateral transect) indicated by black line. E: Shear forces(N) and shear stress (in N/m2) for Beam 2D model indicated ingray box; tensile shear stresses for beam FEM indicated byblack line. Standard conventions shown at left.



The working side beam FEMs experience strongbut localized positive (lateral) bending at the bitepoint, whereas the balancing side experiences weaknegative (medial) bending at the symphysis. Peakmediolateral bending stress is highest at the jawjoint, and was slightly lower than that recorded inBeam 2D. Transverse shear stress predicted bythe beam FEM is identical on the working side andbalancing side (regardless of bite point) and peaksat the jaw joint (Figs. 8 and 13). Peak transverseshear stress in the beam FEM is lower in magni-tude than that predicted by Beam 2D.

Torsional stresses. Figure 9 illustrates esti-mated torsional stresses along the working sideand balancing side beams. Torsional stressesderived from Excel calculations predict that theworking side beam below and adjacent to the jawjoint experiences strong counterclockwise (positive)torque, whereas the portion between the muscleinsertions and the bite point experiences clockwise(negative) torque. Calculations for the balancingside suggest that the entire beam experiences posi-tive torque. These results support predictionsderived from the free body diagram that thepowerful jaw elevators (mAMP, mPT) tend torotate the ventral margin of the lower jaws medi-ally relative to the tooth rows. Twisting of themandible on both sides must be resisted at the jawjoint and symphysis, and at the biting tooth of theworking side. The working side and balancing sideof the beam FEM model also experience counter-clockwise torque between the jaw joint and the an-terior muscle insertions due to the actions of mPTand mAMP (see videos in Supporting Information);however, torsional stress rapidly decreases anteri-orly. Additionally, the beam FEM also experiencesclockwise torque posterior to the jaw joint due tothe action of mDM. For all beam models, torsionalstress is highest near the jaw joint and is higheron the balancing side than the working side.

Simple Alligator FEMReaction forces. Vertical bite forces in the sim-

ple Alligator FEM are more than 30% lower thanat corresponding bite positions in the beam models(Table 7). Vertical reaction force at the workingside jaw joint is greater in magnitude at all bitepositions in the simple Alligator FEM than in thebeam models. Vertical reaction force at the balanc-ing side jaw joint is more than 80 N higher thanthat predicted by the beam models and remainsnearly constant with changes in bite position onthe working side. The simple Alligator FEM pre-dicts higher medially directed forces at the bitepoint than the beam models (Table 7). Laterallydirected reaction forces at the jaw joints are lowerin than those generated in the beam models.

Posteriorly directed reaction forces at the bitepoint of the simple Alligator FEM are much lowerthan those in the beam FEM, whereas posteriorlydirected forces on the working side jaw joint aregenerally higher, peaking during middle bites.Unlike lever arm mechanics or the beam models,in which the balancing side and working side weremodeled independently, the two sides of the man-dible were modeled together in the AlligatorFEMs, and changes in bite point on the workingside cause changes in the mechanical behavior ofthe balancing side. Posteriorly directed reactionforce on the balancing side jaw joint of the simpleAlligator FEM is much lower than those predicted

Fig. 9. Torsional stresses in beam models. A: Lateral view ofAlligator working side mandible with muscle force vectors (solidarrows) and reaction force vectors (broken arrows). B: Verticalforces (black arrows) and supports (black triangles) applied toBeam 2D model representing working side; black arrow at sym-physis represents contribution from balancing side muscle force.Standard conventions for positive and negative torques [as wellas scale bar for (C–H)] shown in key below (B). The direction oftorque is that of the mandible anterior to the reference section(RS), indicated by a dashed line. C–H: Torsional shear stresses(MPa) along the beam indicated in gray box (Beam 2D esti-mate) and black line (beam FEM). Location of bite point on theworking side trials (C–G) is indicated by a small black triangle.H: Balancing side mandible.



by lever arm mechanics or beam models, andincreases during posterior bites.

Internal stresses—Bending stress in sagit-tal planes. Like the beams, the balancing side ofthe simple Alligator FEM undergoes negativebending (dorsal deformation) along most of itslength (Fig. 10F–J). Peak dorsoventral bendingstress on the balancing side is highest during pos-terior biting (on the working side), occurs midwayalong the length of the mandible (Fig. 10F), and issubstantially lower on the balancing side of thesimple Alligator FEM than in the beam FEM.

As in the beam models, the posterior workingside mandible of the simple Alligator FEM experi-ences negative bending, whereas the anterior man-dible undergoes positive bending (Fig. 10A–E).During anterior to posterior-middle bites, peakdorsoventral bending stress occurs near applica-tion of muscles forces and is lower than thatrecorded in the beam FEM. During posterior bites,peak bending stress occurs near the bite point.Dorsoventral bending stress varies less along thesimple Alligator FEM than along the beam FEM;this is particularly evident during anterior bitesand on the balancing side, but is true at all bitepositions (Fig. 10).

Internal stresses—Shear stress in sagittalplanes. Plots of sagittal shear stress in the simpleAlligator FEM resemble those generated by thebeam models (Fig. 11), except that shear stressdecreases abruptly at the symphysis (on both theworking side and balancing side). Peak sagittalshear stress is lower on the working side (buthigher on the balancing side) of the simple Alliga-tor FEM than the beam FEM. Overall, peak dorso-ventral shear stress is very low compared withbending stress.

Internal stresses—Bending and shearstress in transverse planes. Mediolateral bend-ing stress is more variable in the simple AlligatorFEM than along the beam models (Fig. 12). Theworking side and balancing side of the simple Alli-gator FEM experience positive (lateral) bending atthe jaw joint, negative (medial) bending in thearea of the elevator muscle insertions, negativebending at the bite point (working side) and posi-tive bending at the symphysis. In both sides, peakbending stress is due to medial bending and ishighest during posterior bites; on the balancingside, peak mediolateral bending stress occurs inthe area of the muscle insertions while on theworking side it occurs at the bite point. Peakmediolateral bending stresses are higher on thebalancing side than on the working side, arehigher in the simple Alligator FEM than in thebeam FEM, and are nearly as high as peakstresses generated by dorsoventral bending.

The working side simple Alligator FEM experi-ences transverse shear stresses similar to those inthe beam FEM (Fig. 13). The balancing side of the

simple Alligator FEM experiences positive trans-verse shear anterior to the muscle insertions.Transverse shear stresses are generally lower inthe simple Alligator FEM than in the beam FEM.

Torsional stresses. The working side of thesimple Alligator FEM experiences positive (coun-terclockwise) torsion from the jaw joint to the an-terior muscle insertions, indicating that the poster-oventral margin of the working side is beinginverted (Fig. 14). Torsional stress decreasesbetween the muscle insertions and bite point; themandible between the bite point and the symphy-sis experiences high positive torques, again indi-cating that the tooth row is being everted. This isevident in deformation videos (see Supporting In-formation and Discussion).

In contrast, the balancing side of the simplemodel exhibits predominantly negative (clockwise)torsional stress, indicating the ventral margin ofthe lower jaw is being everted. Again, videos of de-formation confirm this. The only exception to thispattern is the posterior balancing side during ante-rior bites, which experiences positive torsion andis inverted. Peak torsional stress on the balancingside occurs near the jaw joint and at the symphy-sis, and is higher than peak torsional stresses onthe working side.

On both the working side and balancing side,torsional stress peaks posterior to the symphysis(unlike the beam FEM in which torsional stresspeaks at the jaw joint and decreases to zero anteri-orly), is highest during posterior bites, and is gener-ally higher than those recorded in the beam FEM(but lower than values recorded in Beam 2D models).

Suture Alligator FEMReaction forces. Vertical bite forces are lower

in the suture Alligator FEM than in the simple Al-ligator FEM (Table 7), whereas vertical reactionforces at the working side jaw joint are higher.Medial reaction force at the bite point and lateralreaction force at the working side jaw joint arelower than those recorded in the simple AlligatorFEM, whereas lateral reaction force on the balanc-ing side jaw joint was higher (Table 7). Like thesimple Alligator FEM, the suture Alligator modelexperiences posteriorly directed force at the jawjoints; these are higher on the balancing side andlower on the working side than those recorded inthe simple Alligator FEM.

Internal stresses—Bending stress in sagit-tal planes. Plots of dorsoventral bending stressfor the suture Alligator FEM are similar to thoseof the simple Alligator FEM and the beam FEM(Fig. 10); however, peak dorsoventral bendingstress magnitudes are lower in both the workingand balancing side of the suture model than in thesimple Alligator or beam FEMs, particularly dur-ing posterior bites.



Fig. 10. Dorsoventral bending stresses in the FEMs. A–J: Bending (zz) stresses taken along the ventral transect of the fourmodels (color identified in legend) on the working side (A–E) and the balancing side (F–J). Closed triangles indicate bite point onworking side; open triangles are used in plots of the balancing side. Maximum dorsoventral bending stress displayed next to eachplot. K: Color contour plot of the simple Alligator FEM in ventral view, showing zz stress during posterior biting. L: Color contourplot of the suture Alligator FEM in ventral view, showing zz stress during posterior biting. ‘‘WS’’ denotes working side, ‘‘BS’’denotes balancing side.



Internal stresses—Shear stress in sagittalplanes. Plots of sagittal shear stress for the sutureAlligator FEM strongly resemble those from thebeam and simple Alligator FEMs (Fig. 11). Peaksagittal shear stresses are generally lower withthe addition of sutures (compared with the solid

model), particularly anterior to the bite point (onthe working side) and in the anterior portion ofthe balancing side.

Internal stresses—Bending and shearstress in transverse planes. Plots of mediolat-eral bending stress in both the working side and

Fig. 11. Sagittal shear stresses in the FEMs. A–J: Shear (zy) stresses taken within transverse cross sections of the four models(color identified in legend) on the working side (A–E) and the balancing side (F–J). Closed triangles indicate bite point on workingside; open triangles are used in plots of the balancing side. Maximum zy shear stress within sections sampled displayed next toeach plot. K: Color contour plot of the simple Alligator FEM in oblique view, showing zy stress within cross sections of the model.



balancing side suture Alligator mandibles are simi-lar those generated by the simple Alligator FEM(Fig. 12), although peak mediolateral bendingstresses are higher in the suture model than in thesimple model on the working side. Plots of trans-

verse shear stress in the suture model resemblethose generated by the simple model (Fig. 13),although transverse shear stress is lower in theanterior balancing side mandible of the suture Al-ligator FEM.

Fig. 12. Mediolateral bending stress in the FEMs. A–J: Bending (zz) stresses taken along the lateral transect of the four models(color identified in legend) on the working side (A–E) and the balancing side (F–J). Closed triangles indicate bite point on workingside; open triangles are used in plots of the balancing side. Maximum mediolateral bending stress displayed next to each plot. K–N: Color contour plots of the working (K, L) and balancing (M, N) sides of the sutures (K, M) and PTflange (L, N) Alligator FEMsin lateral view during a posterior bite, showing zz stress.



Torsional stresses. Torsional stress plots (Fig.14) of the working side mandible of the suture Alli-gator FEM differ from those of the simple model inthat stress magnitudes are substantially lower inthe anterior mandible, particularly, during middleto posterior bites. The balancing side suture Alli-gator model also experiences lower negative tor-sional stresses in the anterior mandible than thesimple model. Additionally, the posterior balancingside of the suture model experiences positive tor-que at all bite points, suggesting inversion of theventral margin of the posterior mandible relativeto the tooth row, whereas the anterior mandiblecontinues to undergo negative torsion. This is con-firmed by deformation videos (see Supporting In-

formation). Torsional stresses on both sides peakduring middle bites.

Pterygoid Flange Alligator FEMReaction forces. Vertical bite and jaw joint

reaction forces in the PT flange Alligator FEM, aswell as medial forces on the biting tooth, are simi-lar to those in the suture model (Table 7); however,laterally directed reaction force on both jaw jointsdecrease substantially compared to the suturemodel. Unlike the suture model, the balancing sidejaw joint of the PT flange model does not experi-ence high lateral or posteriorly directed forces dur-ing biting on the posterior tooth. Laterally directed

Fig. 13. Mediolateral shear stress in the FEMs. A–J: Shear (zx) stresses taken within transverse cross sections of the four mod-els (color identified in legend) on the working side (A–E) and the balancing side (F–J). Closed triangles indicate bite point on work-ing side; open triangles are used in plots of the balancing side. Maximum zx shear stress within sections sampled displayed to theright of each plot.



forces occur at the contact with the pterygoid flangeon the working side and balancing side mandibles atall bite positions; the magnitude of these forcesrange from 30 to 70 N. This force is greatest on thebalancing side during posterior bites.

Internal stresses—Bending stress in sagit-tal planes. The PT flange Alligator modelstrongly resembles the suture model in the distri-bution of dorsoventral bending stresses (Fig. 10)and in peak dorsoventral bending stress magni-tudes.

Internal stresses—Shear stress in sagittalplanes. Plots of sagittal shear in the PT flangeFEM are virtually identical to those generated bythe suture model (Fig. 11), as are peak sagittalshear stresses.

Internal stresses—Bending and shearstress in transverse planes. Unlike the simpleand suture Alligator FEMs, which experiencemedial bending in the area of the muscle inser-tions, the PT flange model experiences lateralbending in this region (Fig. 12) due to the presenceof the pterygoid flange. Mediolateral bendingstresses are lower in both the working side andbalancing side of the PT flange model comparedwith the suture model, particularly during poste-rior bites. Peak mediolateral bending stress in thesuture model increases as the bite point movesposteriorly; in contrast, in the PT flange model itpeaks during anterior bites.

Unlike the simple and suture Alligator models,peak transverse shear stress in the PT flange

Fig. 14. Torsional shear stress in the FEMs. A–J: Torsional stresses taken along the longaxis of the four models (color identified in legend) on the working side (A–E) and the balancingside (F–J). Closed triangles indicate bite point on working side; open triangles are used in plotsof the balancing side. Maximum torsional stress displayed next to each plot.



model does not change with changes in bite posi-tion (Fig. 13). Transverse shear stresses are lowerthan those in the simple and suture models, partic-ularly, during posterior bites. Transverse shearstress occurs at the contact with the pterygoidflange.

Torsional stresses. Plots of torsional stressesin the PT flange Alligator model resemble thosegenerated by the suture model (Fig. 14); however,peak torsional stresses are somewhat higher inboth sides.

DISCUSSION

The aims of this study were several fold. Dataon muscle architecture (cross-sectional area, pin-nation, sarcomere length) and muscle orientationwere obtained and used to estimate the externalmuscle forces acting on the Alligator mississip-piensis mandible at 308 of gape. These data werethen used to estimate, using beam modeling andFEA, the bite and joint reaction forces acting on,and the shear, bending and torsional stresses act-ing in the Alligator mandible, as well as its gen-eral patterns of deformation. By comparing thereaction forces and stress regimes in the simplebeam models with the high-resolution FEMs of theAlligator mandible, we sought to address a num-ber of questions. Does mandibular morphology inthis archosaur minimize stress while minimizingbony material better than would a similarly loadedsimple beam? How do the material properties ofthe archosaur mandible affect its mechanicalbehavior? How does the enlarged pterygoid flangeof archosaurs affect mandibular stress patterns?

Muscle architecture and Length–TensionCurves

We found sarcomere lengths at minimum gaperanging between 1.88 and 2.01 lm; Busbey (1989)reported sarcomere lengths of 1.99–2.38 lm in Al-ligator elevator muscles at 08 gape. Failure to cor-rect muscle fascicle lengths for sarcomere lengthat muscle fixation can result in errors in calcula-tion of muscle PCSA. In this study, a resting, oroptimal, sarcomere length of 2.5 lm was assumed,a value larger than that measured in frog (2.0–2.2)but similar to that estimated for the upper end ofthe plateaus for cat and lower end of the plateauregion of humans (Herzog, 2007). The length–ten-sion curves presented here will need to be recalcu-lated as thin filament lengths and/or length ten-sion curves for A. mississippiensis become avail-able. These changes should not significantly alterthe relative PCSA magnitudes of the muscle mod-eled here, although they might result in changesin absolute stress and strain magnitudes.

In Alligator, some jaw elevator muscles (mAME,mPST) reach the inflection point of the ascending

limb of their length–tension curves at minimumgape, suggesting that these muscle groups aremore effective (in terms of optimal sarcomerelength) at low gape angles (see also Busbey, 1989).In contrast, mPTd and mPTv are below the pla-teau even at gape angles of 308, suggesting thesemuscles are most effective at wide gapes.

Iordansky (1964) observed that mPST andmAME are enlarged relative to mPT in long-snouted crocodilians, which feed primarily on fish;the lines of actions of these vertically orientedmuscles make them mechanically advantageous atsmall gape angles. In contrast, the pterygoidmuscles are well developed in short-snouted croco-dilians with generalized diets (including large ani-mals), and their highly oblique lines of actionmake them mechanically advantageous at highgape angles needed to capture large prey. Addi-tionally, EMG studies have demonstrated that ac-tivity in mPT ceases shortly after closing, whereasmAME remains active during holding and crush-ing at minimum gape (Busbey, 1989). Thus, opti-mal sarcomere length, mechanical advantage, andmuscle activity patterns all suggest that mAMEand mPST are adapted for closing the jaws atsmall gapes, while the pterygoids are better suitedfor closing the jaws at large gapes (Stern, 1974).

External Reaction Forces—Orientations andMagnitudes

Bite force and joint reaction force are importantperformance metrics. Previous studies have usedreaction forces as measures of jaw efficiency and toinfer changes in feeding behavior or diet amongdifferent taxa (Van der Meij and Bout, 2004; Wroeet al., 2005; Christiansen and Wroe, 2007) or dur-ing ontogeny (Thomason et al., 1990; Herrel andO’Reilly, 2006). For these reasons, it is critical tounderstand how differences in modeling methodand model complexity affect both the absolute andrelative magnitudes of the reaction forces gener-ated.

Some general patterns are observed in all mod-els of the Alligator mandible. Both jaw joints arein compression in all models at all bite points,with joint reaction force always directed ventrally,laterally, and posteriorly (Table 7). Additionally,vertical bite force increases as the bite point movesposteriorly, whereas vertical force at the workingside jaw joint decreases. The similarity of theseresults across models is due to the fact that therelative magnitudes of joint and bite reactionforces are determined by the geometry, magnitude,and orientation of the external forces acting on themandible and its overall external geometry, not bythe internal shape of the mandible or its materialproperties. Similarly, all models experience later-ally directed forces at the jaw joints, probablybecause the muscle forces were all applied near



the joint, or its equivalent position in the beammodels. Thus, modeling technique might beexpected to have little influence on the orienta-tions and relative magnitudes of external reactionforces.

However, despite general similarities betweenmodels, substantial differences in reaction forcesare observed between the Alligator FEMs and thebeam models. Bite forces in the beam models (andthose predicted using lever arm mechanics) arehigher than those generated in the AlligatorFEMs, while vertical reaction forces at the jawjoints are lower. This suggests that the morphologyof the Alligator mandible makes it less efficient attransferring force from the elevator muscles to thebite point than a simple beam. Additionally, thebeam models and Alligator FEMs experience sub-stantial medial (and in the FEMs, posterior) bitereaction force components, suggesting that thecommon assumption made in lever arm analysis ofliving and fossil taxa (i.e., that bite force is verti-cal) is probably incorrect. Reaction force on thebalancing side jaw joint was assumed to remainconstant in the lever arm calculations and beammodels, but varies with bite position in the Alliga-tor FEMs; thus, beam models that consider thebalancing side and working side separately do notaccurately predict reaction forces experienced bythe balancing side, particularly longitudinal reac-tion forces. Finally, the beam models predicted low,medially directed reaction forces at the pterygoid-mandibular contact; however, the PT flange modeldemonstrates that high laterally directed forcesoccur at this contact.

These differences between the results of beamand FE modeling suggest that the use of beammodels to estimate bite and joint reaction forcemagnitudes and orientations in fossil archosaursmay yield inaccurate results. In addition, differen-ces in reaction forces between the three AlligatorFEMs suggest that failure to accurately model ma-terial properties may also yield inaccurate esti-mates of bite and joint reaction forces, even if ge-ometry is precisely modeled using FEA. The addi-tion of anisotropy/sutures/PDL to our simplemandible model resulted in lower bite forces in thesutures FEM. Furthermore, when the pterygoidflange–mandible contact is added to the model, lat-erally directed reaction forces at the jaw joints arereduced, resulting in more vertically oriented jointreaction forces in anterior view than the other Al-ligator FEMs.

Cleuren et al. (1995) suggested that the orienta-tion of joint reaction forces in Caiman would bealigned with the quadrate, which forms the jawjoint; the reaction forces experienced by the Alliga-tor FEMs (but not the beam models) provide somesupport for this observation. The quadrate of thisspecimen is oriented 2388 from the y-axis (in ante-rior view) and 2478 from the y-axis (in lateral

view). The reaction forces experienced by the Alli-gator FEMs average 2438 in anterior view, and2318 in lateral view. Thus, joint reaction forcespredicted by FEA more closely align with thequadrate in anterior view than lateral view.

Erickson et al. (2003) measured changes in biteforce during ontogeny in Alligator; their work pre-dicts a bite force of �230 N in an animal with atotal body length of 1 meter, the approximate sizeof the animal we used to build the FEM. This issimilar to bite reaction forces predicted by ourbeam models (118.6–249.6 N) and lever armmechanics (119.6–214.1 N; Table 7), althoughhigher than maximum bite forces obtained fromthe Alligator FEMs (170.6 N). Other studies haveused lever arm mechanics (Sinclair and Alexander,1987; Cleuren et al., 1995) to estimate bite andjoint reaction forces in Caiman; their results pre-dict lower bite forces (for smaller animals) buthigher bite/joint force ratios than our AlligatorFEMs.

The discrepancy between bite force values calcu-lated by our Alligator FEMs and those predictedfrom in vivo measurements (Erickson et al., 2003)may have a number of causes. One possibility isthat the individual from which the muscles weretaken (a wild female from Rockefeller Wildlife Ref-uge, 1.3-m body length, 0.21-m head length) mayhave had unusually small muscles, leading us tounderestimate force produced by the elevatormuscles. Additionally, muscle PCSA was multipliedby a specific muscle tension of 25 N/cm2 in thisstudy; other workers have used muscle stresses of>30 N/cm2 when modeling reptilian muscles(Johnston and Gleeson, 1984; Sinclair andAlexander, 1987). Although increasing muscle ten-sion would not change relative muscle forces orthe overall mechanical behavior of our models, itwould result in a 20% increase in reaction forcemagnitude, and presumably increase stress magni-tudes and model deformation. These possibilitieswill be explored further in future sensitivity analy-ses by increasing muscle force. In addition, ourestimates of material properties of sutures andPDL (which were not based on data from Alliga-tor) may be inaccurate since the simple AlligatorFEM, which does not incorporate sutures or PDL,experiences higher bite reaction forces than thesuture and PT flange models. At present, we areconfident that the relative magnitudes of the cal-culated muscle forces are correct, as well as thedeformation regime and relative bite and jointreaction forces.

Internal Forces—Bending, Shearing, andTorsion

Our hypothesis, that bending stresses would behigher in the Alligator mandible than stresses dueto shear or torsion is corroborated. In all models,



peak bending stresses are an order of magnitudehigher than peak shear stresses. Peak stress dueto torsion is always lower than peak bendingstress and higher than peak shear stress. It is im-portant to note that high shear stresses at the jawjoint and in the area of the jaw elevator muscles,as well as high torsional stress at the symphysis,suggest that shear and torsion may be importantdeterminants of morphology in these areas.

The suture and PT flange Alligator FEMs gener-ally experience lower, more evenly distributed dor-soventral bending and shear stress than the sim-ple model, suggesting that the combination of ani-sotropic bone [stiffer along the long axis of themandible (Zapata et al., 2010)], sutures, and PDLreduces dorsoventral bending stress in Alligatormandibles. However, the suture model exhibitshigher mediolateral stresses (specifically mediolat-eral bending stress on the working side and trans-verse shear stress on the balancing side); thesemediolateral stresses are reduced in the PT flangemodel. Anisotropy/sutures/PDL also result in lowerpeak torsional stress in the suture model comparedto the simple model. Thus, our results indicate: 1)that sutures/anisotropy/PDL in the Alligator man-dible reduce dorsoventral bending, sagittal shearstresses, and torsional stress; and 2) that the pter-ygoid–mandibular contact reduces high mediolat-eral stresses in the sutured Alligator mandible.

Model Deformation

The impact of sutures, bone anisotropy, PDL,and the pterygoid flange on the behavior of the Al-ligator mandible are evident when comparing over-all deformation regimes (see Supporting Informa-tion). In the absence of sutures/anisotropy/PDL(simple model), bending of the mandible was pri-marily dorsoventral; in particular, the balancingside of the simple model is bent strongly upwardand the tooth row is inverted (i.e., it experiencesnegative torsion). In all Alligator models, the ante-rior working side is bent dorsoventrally duringposterior bites; dorsoventral bending is far lesspronounced on the working side during anteriorbites. In the suture model, which lacks a ptery-goid-mandibular contact, mediolateral bending ofthe mandibles (particularly the balancing side dur-ing posterior bites) is more pronounced, resultingin higher mediolateral bending stress and trans-verse shear stresses in this model. Furthermore,the addition of sutures/anisotropy/PDL results indecreased torsional stress and the posterior bal-ancing side experiences positive torsion, whereasthe anterior balancing side continues to undergonegative torsion. The addition of the pterygoidflange to the model, yielding our ‘‘best guess’’hypothesis regarding mandibular deformation inAlligator, results in decreased dorsoventral andmediolateral bending; instead the primary defor-

mation of this model is twisting of the mandibles,with the ventral margin of the working side andposteroventral margin of the balancing side beinginverted relative to the tooth rows. In all models,bending is more pronounced during posterior bites,whereas twisting about the long axis is the pre-dominant deformation regime during anteriorbites.

Although these verbal descriptors accuratelydescribe deformation of the Alligator mandible, weemphasize that such descriptors are necessarilyimprecise. Rather, as noted elsewhere (Chalket al., 2010; Ross et al., 2011), the finite elementmodel, including its specific geometry and materialproperties, and the resulting loading, stress, anddeformation regimes, is our hypothesis regardingthe mechanical behavior of the Alligator mandibleduring biting. The model itself should be referredto for further details.

Form and Function of the Alligator Mandible

The lower magnitude and less variable dorsoven-tral bending and sagittal shear stresses in the Alli-gator mandible FEMs suggest that, compared witha simple beam, the mandible of Alligator is betteroptimized to reduce these stresses during biting.We hypothesize that these results primarily reflectimproved resistance to dorsoventral bending, andthat this is an important design criterion in Alliga-tor mandibles. Dorsoventral shear stress magni-tudes are very low in all FEMs, suggesting thatthe improved resistance of the mandible to dorso-ventral shearing forces is likely to be a secondaryeffect of changes in design to resist dorsoventralbending. Similarly, the low values of mediolateralshearing and torsional stresses suggest that theseparameters are unlikely to exert a significanteffect on Alligator mandible design.

The material properties of the Alligator mandi-ble appear to also reflect design criteria, especiallyin interaction with the pterygoid flange constraint.Alligator mandibular bone is stiffer along the longaxis of the mandible (Zapata et al., 2010). Ouranalyses show that incorporation of this aniso-tropic bone (as well as sutures and PDL) in the Al-ligator FEM reduces dorsoventral bending and tor-sional stresses. However, this comes at theexpense of increased mediolateral bending andshear stresses, and lowered bite reaction force, dueto strong mediolateral deformation of the mandible(compared with a homogeneous, isotropic mandi-ble).

The presence of the pterygoid flange amelioratesthe increased mediolateral stresses but does noth-ing to improve the poor efficiency suggested by thelow bite to joint reaction force ratios in Alligator.The results presented here reveal that in the Alli-gator FEM, the addition of a contact between themandible and pterygoid flange reduces: mediolat-



eral deformation of the mandible, reaction forcesat the balancing side jaw joint, mediolateral bend-ing stress in the working side mandible and trans-verse shear stress in the balancing side mandible.Lack of compressive reaction forces at the contactin the beam 2D model and lack of strong mediolat-eral deformation in simple Alligator FEM, how-ever, suggest this function of the pterygoid flangemay be necessitated by increased mediolateral de-formation due to anisotropic bone and/or suturesfound in Alligator mandibles. Future sensitivityanalyses will parse out the relative importance ofthese two input parameters.

Finally, our results demonstrate that the sym-physis is an area of high torsional stress in theAlligator mandible compared with the beam mod-els. Computed tomography imaging reveals thatthe symphysis (dentary–dentary contact) in thesubadult Alligator is strongly interdigitated in thecoronal plane; it is possible that the morphology ofthis contact (as well as its material properties) isimportant in mitigating torsional stresses.

Limitations of the Model

The FE model of the Alligator mandible pre-sented here incorporates the most detailed geome-try, accurate bone material properties, and com-prehensive information on muscle architecture andcontractile properties of any model of an archosaurskull yet published. Results derived from thismodel are still dependent on a number of assump-tions. Our model only tests the mechanical behav-ior of the mandible during simple, unilateralbiting; however, living crocodilians load their man-dibles in a number of ways: they can lift or throwprey out of the water, shake the head, and engagein ‘‘twist-feeding.’’ These behaviors would undoubt-edly induce higher torsional and mediolateralforces on the mandible. Because we cannot at pres-ent validate the mechanical behavior of the mandi-ble resulting from such activities, and becausesuch activities result in more complicated loadingregimes better suited for dynamic analyses, wechose not to load our model in such a manner. Theloading regime applied to the model assumes thatall muscles (and all portions of individual muscles)fire simultaneously and maximally; incompleteEMG data (Busbey, 1989; Cleuren et al., 1995)demonstrated that the jaw muscles of crocodiliansdo not behave in this manner. The model pre-sented in this article is tested at only one gapeangle (308); future analyses will investigatechanges in mechanical behavior with changes ingape angle in Alligator. Likewise, our model is of asubadult Alligator; a study in progress examineschanges in mandibular mechanical behaviorwithin an ontogenetic series of Alligator. Finally,information on the material properties of crocodi-lian teeth, sutures and PDL, as well as informa-

tion on the histological and material properties ofthe cartilago transiliens, is not currently available;as these data become available, we hope to refineour model accordingly.

CONCLUSIONS

Lever arm calculations and beam models havebeen used extensively to evaluate the mechanicalbehavior and morphology of archosaur crania andmandibles (Sinclair and Alexander, 1987; Busbey,1995; Cleuren et al., 1995; Mazzetta et al., 1998;Henderson, 2002; Therrien, 2005; Desojo and Viz-caıno, 2009; Tanoue et al., 2009). The results pre-sented here suggest that these calculations provideapproximate estimates of the general orientationof bite or joint reaction forces, and the general dis-tribution of dorsoventral bending stresses, andsagittal and transverse shear stresses in the man-dible. However, they perform poorly in predictingabsolute or relative reaction force magnitudes, pre-cise reaction force orientation, and mediolateralbending and torsional stress distribution withinthe mandible. Although the beam models experi-enced dorsoventral bending and sagittal and trans-verse shear stress patterns similar to the AlligatorFEMs, stress magnitudes were substantially differ-ent. One of the most notable differences betweenthe beam models and the Alligator FEMs was thebehavior of the balancing side, which varied withchanges in bite position in the Alligator FEMs,suggesting that the behavior of the balancing sideis poorly represented in simple beam models inwhich the working side and balancing side aremodeled separately. Overall, beam models do notaccurately represent the mechanical behavior ofthe archosaur mandible, despite its superficial re-semblance to a beam.

It has been suggested that the morphology ofthe Alligator mandible reduces stress during bitingwhile minimizing bony material (van Drongelenand Dullemeijer, 1982); we tested this hypothesisby evaluating whether stress is more evenly dis-tributed in the Alligator mandible than in a struc-ture with uniform geometry and material proper-ties throughout: i.e., a simple beam. Comparedwith the beam models, dorsoventral stresses weremore evenly distributed in the Alligator FEMs,corroborating van Drongelen, and Dullemeijer’sanalyses. However, the Alligator mandible doesnot appear optimized to reduce mediolateral andtorsional stresses, compared with a simple beam.

Most studies using FEA assume isotropic mate-rial properties for bone. Additionally, while severalstudies have focused on the mechanical signifi-cance of cranial sutures in Alligator and otherarchosaurs (Busbey, 1989; Rayfield, 2004, 2005),the sutures of the mandible have received far lessattention. Results from the Alligator FEMs demon-



strate that bone anisotropy/sutures/PDL reducedorsoventral stresses and torsional stresses, butmake the mandible susceptible to mediolateral defor-mation. These results suggest that sutures/anisot-ropy/PDL must be considered when attempting toaddress questions about mechanical behavior in themandibles of extinct archosaurs; this is especially im-portant in modeling archosaurs, such as many thero-pod dinosaurs, which possess patent mandibularsutures or a possible intramandibular joint.

The elongate pterygoid flanges of crocodilianshave been hypothesized to act as braces againstmedial deformation of the mandible in platyrostralforms, due to more medially inclined jaw elevators(Busbey, 1995; Iordansky, 1964). Our results cor-roborate this hypothesis. The development of thepterygoid flange varies among crocodilians andother archosaurs. Differences in cranial morphol-ogy among taxa that possess elongate pterygoidflanges (such as platyrostral crocodilians and orei-nirostral ornithopod dinosaurs) suggest that thepresence of this feature may be related to mandib-ular morphology, sutural morphology, or jaw mech-anism.

Both similarities and differences were observedbetween the mechanical behavior of our Alligatormodel and the behavior of mammalian (specificallyprimate) mandibles (see Introduction). Both jawjoints are in compression in the Alligator modeldespite changes in bite point; this contrasts withmammals, in which recruitment levels of workingside and balancing side muscles must be modu-lated to prevent the working side jaw joint frombeing put under tension during posterior bites(Greaves, 1978, 1982; Spencer, 1995, 1998, 1999;Thomason et al., 1990). This difference is attribut-able to the more anterior position of the Alligatortoothrow relative to mammals.

Our results demonstrated that bending stresswas higher than either shear or torsional stress inthe Alligator mandible. As the relative importanceof bending, shear, and torsional stresses in primatemandibles has not yet been quantified, resultsfrom the Alligator model cannot be compared withprimates at this time. However, we predict thatbending stresses are more important in Alligatormandibles than in the relatively shorter mandiblesof most primates.

Like the deformation regimes hypothesized byHylander (1977, 1979a,b, 1981, 1984) to occur inprimate mandibles, the PT flange Alligator modelexperiences twisting about its long axis and dorso-ventral bending, with bending being predominantduring posterior bites and twisting being larger inmagnitude during anterior bites. Unlike primatemandibles, which experience lateral transversebending of the corpora (i.e., ‘‘wishboning’’) duringmastication due to the action of the masseter, themandibles of Alligator experience medial trans-verse bending, being pulled inward by the power-

ful, medially inclined pterygoideus muscles. Thisdeformation is strongly reduced by the addition ofthe pterygoid flange contact. Finally, our resultsdemonstrated that the mandibular symphysis ofAlligator experiences dorsoventral shearing, twist-ing about the transverse axis (leading to elevatedtorsional stress) and bending in the transverseplane (due to ‘‘inverse wishboning’’); similar defor-mation is reported in the primate mandiblealthough transverse bending is due to lateral‘‘wishboning.’’ We note that these are preliminarycomparisons, based on a single archosaur taxon,and much work is still required to better under-stand the similarities and differences of the load-ing environment between mammalian and reptil-ian mandibles.

ACKNOWLEDGMENTS

The authors thank Hauke Bartsch (Visage Imag-ing) and Christian Wietholt (Department of Radiol-ogy, University of Chicago) for assistance with Amira.Anne Delvaux (Beaufort Analysis) provided technicalsupport for Strand7. Ruth Elsey at the RockefellerWildlife Refuge generously provided specimens forstudy. David Reed, Justin Lemberg, and KeithMetzger as well as two anonymous reviewers pro-vided valuable comments on the article.

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