Beam Theory-Derived Strain Magnitudes
Strain due to bending.
The cross-sectional geometry of a beam has a significant effect on the distribution of strain throughout it. Under pure bending, normal strain within a cross-section increases with distance from the neutral axis of bending (Fig. 4A). In addition, sections closer to the fixed end of the beam will experience higher normal strains at equivalent distances from the neutral axis. This pattern was not seen in our beam model data (Table 2) because we modeled each section as having a unique second moment of area and hence different distances to the gauge locations from the neutral axis.
Variation in compressive strain magnitude at the maxillary, prefrontal, and jugal sites is a function of the magnitude of the perpendicular distance from the neutral axis and the second moment of area. Not surprisingly, strains in our solid beam model are significantly lower (1–2 orders of magnitude) than those typically seen in limb bones (Biewener, 1990) or mandibles (Dechow and Hylander, 2000), which are generally either flat or have medullary cavities, resulting in lower second moments of area and consequently higher strains.
Strain due to torsion.
In a beam with an elliptical cross-section loaded in pure torsion, the maximum shear stress and strain is expected to occur at a point on the surface of the beam that is closest to the twisting axis (Fig. 4B). Additionally, in cross-sectional view, shear stresses increase with the distance outward from the twisting axis (Hibbeler, 2000: p. 221). Unlike during bending, the distribution of shear stress will be the same at any cross-section in the beam, regardless of the distance between the cross-section and the fixed end of the beam.
Comparison of Models and In Vivo Data
As noted above, there were clear differences between the beam model, FE model, and the recorded in vivo strains, both with regard to the absolute strain magnitudes and the patterns of strain gradients across the skull. Low strains in the beam model were expected. There are several possible explanations why principal strain magnitudes were much higher in vivo than in the FEM. The most likely cause is that the scaling method used for constructing this model from a larger specimen (geometric similarity) resulted in thickened rostral bone relative to size for the scaled models, which would decrease principal strain values. In reality, it is expected that rostral bone thickness does not scale with isometry and therefore would be thinner than in the model. Two other possibilities for this discrepancy relate to the material properties of bone in the FEM. First, if an incorrect elastic modulus were used, this would not have much of an impact on stress magnitudes, but would greatly impact strain magnitudes (Daniel and McHenry, 2001). Second, during preliminary testing of the FEM, modeling the bone as isotropic resulted in lower stresses and presumably lower strains, and it is very likely that the crocodilian skull, like other vertebrate skulls, is generally orthotropic (Peterson and Dechow, 2003). Material testing of gauge sites will help to address these two issues.
Because in vivo and FEM loading is complex, it is difficult to determine why there was no similarity in rank-order patterns of strain magnitude between the three data sets. Incongruence between the models and the in vivo data may be attributable to a number of factors, such as local effects of muscles that were not accounted for in the FE or beam models, local effects of bite force, effects of sutures, or the presence of a complex, combined loading regime in vivo.
The strain orientations predicted by beam theory (Fig. 1) and the FEA and in vivo orientation values measured from the alligator rostum, jugals, and frontal bone (Figs. 5–7) allow evaluation of hypotheses that during midline (middle) biting, the snout can be modeled as a beam experiencing dorsally directed bending, and that during unilateral biting, the snout acts as an ellipsoid beam subjected to superimposed twisting and bending regimes.
During anterior midline bites, the strain orientations from both the FEM and in vivo strain gauges (Fig. 5) are indicative that the rostrum is acting as a cantilevered beam that is being bent dorsally concave (as in Fig. 1A). Although the FEM orientations are slightly more consistent with this hypothesis than in vivo orientations, this is not surprising considering that anterior midline bites during the in vivo experiments may have sometimes deviated slightly from the midline, creating torsional moments.
It is more difficult to assess whether unilateral biting causes the snout to act like a twisted beam because of the potentially confounding effects of the superimposed bending regime. However, if it is presumed that unilateral biting causes the working side to be twisted in a dorsal direction due to the dorsally directed bite force (Busbey, 1995; Preuschoft and Witzel, 2002), we should expect strain orientations to be approximately 45° during right side bites and −45° during left side bites (relative to the sagittal plane). The in vivo strain orientations generally confirm this hypothesis, especially when the animals bit at the right anterior, right anterior/middle, and left side bite point locations (Figs. 6 and 7).
FEM strain orientations confirmed the hypothesis of a combined loading in bending and torsion, although there was a noted asymmetry in the FEM results (Figs. 6 and 7). Orientations during right side bite FEM loadings were less consistent with this loading regime than during left side bites. A possible explanation for these unexpected results is that most of the strain gauge sites were located on the right side of the skull, and the FEM may have exhibited unusual behavior when the bite point too closely approached the gauge site. This deviation of strain orientation from the expectation of a twisted ellipsoid beam during unilateral biting was also found in the in vivo results when the bite point was located adjacent to the gauge (e.g., Fig. 6, right posterior bites; Fig. 6, left middle/posterior, left posterior bites), lending credence to this hypothesis. In these cases, ε1 is oriented in the sagittal plane and probably represents a local loading regime due to localized effects of bite force rather than a global one acting on the entire snout.
The relative invariance of principal strain orientations in the frontal gauge location, in both the FEM and the in vivo strain experiments, is notable. In all cases, this orientation is indicative of dorsal bending in the region between the orbits. We suggest that this similarity may indicate that the dorsal roof of the braincase is part of a functionally discrete region, distinct from more rostral areas of the cranium, and subject to different loading conditions. However, further testing is needed to confirm this hypothesis.
Beams (Greaves, 1985; Thomason, 1991; Weishampel, 1993; Covey and Greaves, 1994; Busbey, 1995) and finite-element analyses (Daniel and McHenry, 2001; Strait et al., 2003; Rayfield, 2005, this issue) have often been used to represent loading regimes within the vertebrate cranium theoretically. The results of this study indicate that neither simple beam theory nor a finite-element model is able to provide a completely accurate prediction of the nature of in vivo strain recorded from the alligator cranium. Specifically, the models were not able to represent absolute in vivo strain magnitudes or strain gradients accurately.
The poor correspondence between the Alligator FEA and in vivo strain data stands in stark contrast to the close correspondence in the studies of the macaque skull published elsewhere in this volume (Ross et al., 2005, this issue; Strait et al., 2005, this issue). These differing degrees of correspondence might be attributable to differences in modeling procedures. Most notably, the Alligator model used in this study was of relatively low resolution and so did not include the detailed geometry of the macaque model. Similarly, the material properties and muscle forces used in the macaque model are arguably more realistic than those used in the Alligator model.
We hypothesize that one significant difference between the two models lies in the relative importance of sutures in the biomechanical functioning of the two skulls. Numerous experimental studies have shown that sutures typically exhibit strain magnitudes that are an order of magnitude higher than those in the bones that they connect, and strains can be reduced or reoriented across sutures (Jaslow, 1990; Jaslow and Biewener, 1995; Rafferty and Herring, 1999; Herring and Teng, 2000; Metzger and Ross, 2001; Rafferty et al., 2003; Lieberman et al., 2004). The principal strain values recorded from the alligators in vivo are on average greater than those recorded from any other vertebrate cranial bones that have been extensively sampled (Hylander, 1979; Hylander and Johnson, 1992; Herring et al., 1996; Ross and Hylander, 1996; Hylander and Johnson, 1997; Herring and Teng, 2000; Ravosa et al., 2000; Ross, 2001; Thomason et al., 2001; Lieberman et al., 2004; Ross and Metzger, 2004). If sutural strain increases as a function of bone strain, the very high bone strain magnitudes recorded in this study predict extremely high sutural strains, suggesting that sutural morphology might be of great importance in the functioning of the Alligator skull, as it appears to have been in dinosaurs (Rayfield, 2005, this issue) and at least some mammals (Herring and Teng, 2000).
The results of this study should serve as a caveat against the exclusive use of FEA or beam modeling techniques to demonstrate loading patterns in a complex skeletal structure such as the cranium, which is subjected to unpredictable and highly dynamic forces. However, even using relatively low-resolution finite-element and beam models, we can reproduce the basic strain orientations that were seen in vivo, indicating that both of these modeling techniques have the utility of serving as a first-pass approximation of the in vivo loading conditions. While beam and FE analyses are suitable first-pass estimations for strain orientations, we advocate that whenever possible, hypotheses related to loading in the vertebrate cranium should be supported with either in vivo or in vitro strain data.
Loading Regimes in Crocodilian Skull
Although the principal strain orientations recorded in vivo are not precisely those predicted by either the beam models or the FEA, the strain orientation data strongly suggest that the snout is bent upward and twisted during biting. If this hypothesis is correct, then, as suggested by Busbey (1995), the cross-sectional profile of the Alligator rostrum is not optimized for resisting the loading regimes to which it is subjected during feeding. This is even more remarkable, considering the high principal strain magnitudes recorded from the Alligator skull in vivo (Ross and Metzger, 2004) compared to the relatively low strain magnitudes recorded from the Alligator postcranium (Blob and Biewener, 1999). Whether the cranial/postcranial differences reflect differing optimality criteria in the cranial versus the postcranial skeleton, differing material properties, or differing success in eliciting vigorous locomotor versus biting behavior (Ross and Metzger, 2005), it is clear that explanations for the cross-sectional geometry of the alligator snout must invoke a function other than dissipating feeding forces. One possibility is Busbey's (1995) suggestion that the platyrostral geometry of the crocodilian snout is optimized for lateral snapping movements used in the capture of prey in an aquatic environment. The consequent decreased ability of the snout to resist dorsoventral bending is compensated for by the evolution of a hard palate and decreased resistance to torsion is compensated for by scarf joints at the sutures (Busbey, 1995). If this hypothesis is correct, then the Alligator snout is similar to various parts of the primate skull in that the function of dissipating feeding forces appears to have exerted little constraint on the geometry of the cranium (Hylander et al., 1991; Ravosa et al., 2000; Ross, 2001).