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PALAIOS, 2010, v. 25, p. 356–360
Research Article
DOI: 10.2110/palo.2009.p09-164r
FOSSIL VERTEBRATE TRACKS AS PALEOPENETROMETERS: CONFOUNDING EFFECTS OF
FOOT MORPHOLOGY
P. L. FALKINGHAM,
1
* L. MARGETTS,
1,2
and P. L. MANNING
1,3
1
School of Earth, Atmospheric and Environmental Science, University of Manchester, Williamson Building, Oxford Road, Manchester, M13 9PL, UK;
2
Research
Computing, University of Manchester, Devonshire House, Oxford Road, Manchester, M13 9PL, UK;
3
Department of Earth and Environmental Science, University of
Pennsylvania, 254-b Hayden Hall, 240 South 33rd Street, Philadelphia, Pennsylvania 19104-6316, USA
e-mail: peter.falkingham
@
manchester.ac.uk
ABSTRACT
The depth to which a vertebrate track is indented can provide a wealth of
information, being a direct result of the weight, duty factor, and limb
kinematics of the animal as well as media (5 substrate or sediment)
consistency. In order to recreate the formation of the track and elucidate
media consistency at the time of track formation, such factors as animal
mass, duty factor, and foot morphology must be taken into consideration.
This study uses Finite Element Analysis and physical modeling to
demonstrate for the first time that the shape of the foot is an important
factor that influences the depth to which the sediment is penetrated. In
cohesive sediment, less compact morphology allows more sediment to
move vertically upwards at the edges of the foot, dissipating force at the
surface, and retarding transmission of load vertically down into the
sediment. The reverse of this effect is seen in noncohesive sediment. Foot
morphology, therefore, has a direct impact on preservation potential, both
of surface tracks and undertracks, that is irrespective of the pressure
exerted on the sediment surface by the foot and independent of mass and
duty factor.
INTRODUCTION
In this paper, the effect of foot morphology on track depth is
investigated by using Finite Element Analysis (FEA) to indent a series
of abstract foot geometries. As a measure of foot morphology, a metric
derived from circumference, or edge length, has been used, where edge
length can be defined as the boundary between foot and sediment when
seen in plan view. FEA has previously been applied to the study of
track formation to show that interdigital webbing may arise as an effect
of media deformation, rather than as the impression of true webbing
(Falkingham et al., 2009), and to illustrate subsurface deformation and
undertrack formation beneath vertical and heel-toe cycle loads
(Margetts et al., 2005, 2006; Falkingham et al., 2007, 2008).
While vertebrate ichnotaxa may be difficult to constrain to specific
environmental or media (5 substrate) conditions, we are provided with
tracks produced in multiple media with relatively consistent loading
conditions (e.g., pressure, force vectors, pedal morphology, etc),
resulting in track morphologies that may vary to a considerable degree
due entirely to media consistency (Manning, 2004; Mila`n and Bromley,
2006, 2008; Dı
´
az-Martı
´
nez et al., 2009). Even if two different animals
that share foot morphology and limb kinematics are separated
temporally and spatially, there may be enough consistency in loading
conditions that differences in the tracks—magnitude of displacement
rims, radial cracks, track depth—can be used to infer media conditions
at the time of track formation in different strata. Much of this
sedimentary variation may be linked to water content (e.g., Platt and
Hasiotis, 2006), which is in turn controlled by environmental factors
(e.g., Hasiotis, 2007). In a cohesive medium, water content directly
determines both the shear strength of the material (through creating
cohesion between particles) and the Poisson ratio (compressibility).
Extremes of moisture content prevent track formation either because
the medium is too loose, or too liquid (Laporte and Behrensmeyer,
1980; Platt and Hasiotis, 2006).
Given that track depth is a function of the force applied through the
sole of the foot over a given medium, there is therefore the potential to
use tracks as paleopenetrometers (Lockley, 1987; Allen, 1989; Nadon
and Issler, 1997; Nadon, 2001), whereby the depth of a track may be
used to gauge the medium consistency at the time of track formation,
and subsequently used to refine paleoenvironmental interpretations
(Lockley, 1986; Nadon and Issler, 1997; Nadon, 2001; Platt and
Hasiotis, 2006). In order to do so with confidence, other confounding
factors influencing the depth of a track must be understood and taken
into account.
METHODS
Two experiments consisting of FEA simulations were carried out to
investigate the effects of varying pedal complexity on penetration of the
medium. FEA provides a means for investigating stress and strain
within a continuous medium under load, and is now a tried and tested
technique in paleontology. See Rayfield (2007) for a comprehensive
review of the method, and Falkingham et al. (2007, 2008, 2009) and
Margetts et al. (2005, 2006) for details of the method as applied to track
simulations.
The first experiment involved generating geometrically abstract
shapes to act as indenters, which maintained a consistent surface area,
but differed in complexity. The same pressure was then applied to the
surface of each indenter. By using geometrically abstract shapes,
complete control over shape complexity could be achieved, and FEA
meshes consisting of relatively few elements used, facilitating rapid
analysis (Rayfield, 2007).
These analyses were undertaken using a program written by us using
ParaFEM (www.parafem.org.uk, checked March 2010), a freely
available parallel finite element library. The program was validated
using a series of geotechnical engineering test problems, such as the
bearing capacity of a smooth flexible footing (Smith and Griffiths,
2004). Results of these validation examples were compared with
empirical solutions and with analyses carried out using Abaqus/CAE
version 6.8-2 (http://www.simulia.com/, checked March 2010).
A second experiment was carried out using physical modeling.
Indenters of equivalent shape as those used in experiment 1 were cut
from wood and used to indent natural media.
Measuring shape.—In order to draw comparisons between indenters
of differing morphology, a metric was required. As a measure of shape,
edge length—the circumference of the indenter—was used as this varies
with shape for a given sized indenter. Using absolute circumference or a
* Corresponding author.
Copyright
g
2010, SEPM (Society for Sedimentary Geology) 0883-1351/10/0025-0356/$3.00
ratio of edge length to surface area, however, provides a function that
varies with size; a small square has a higher circumference to surface
area ratio than a larger square. To take account of this, edge length was
normalized using equation 1.
e’~ e
=
4ðÞ
2
.
A ð1Þ
Where e9 is the normalized edge length, e is edge length, or
circumference, of the indenter, and A is the surface area of the
indenter. A square will always have an e9 value of 1, regardless of size,
whilst less compact morphologies will have a higher value, but one that
will remain constant as size varies.
Experiment 1
An FEA mesh representing a volume of elastic–perfectly plastic soil
was created using 20-node hexahedral elements and given the properties
of a stiff mud (Young’s Modulus 5 100,000 kPa, Poisson Ratio 5 0.4,
Shear Strength 5 100 kN/m
2
(Leach, 1994)). On the surface of this
mesh, an indenter was created, and given a Young’s Modulus and Shear
Strength sufficiently high as to make the indenter nondeformable
relative to the medium. This is a technique used by FEA users in
geotechnical engineering to model rigid indenters (Potts and Zdravko-
vic
´
, 1999, 2001). The elements used to define the indenter were arranged
in seven different configurations, forming seven indenters, each with a
surface area of nine square units, but with edge lengths ranging from 12
units (the minimum possible for nine elements) to 20 units (the
maximum possible) (Fig. 1). Corresponding values of e9 ranged from 1
to 2.78. Two variations were created each for edge lengths of 14 and 20
units, one (Figs. 1C, F) more complex than the other (Figs. 1B, G). A
uniform pressure of 10 units per unit area was applied to the surface of
each indenter to provide a vertical load.
A second series of indenters were generated to explore the effects of
indenter size on track depth, and each had a surface area of 16 square
units (using 16 elements), providing a greater range of e9 (1 to 4.52)
(Fig. 2). The same pressure was applied as for the above scenario, and
the soil properties remained constant. The parameters of each indenter
are summarized in Table 1.
There are many more possible indenter shapes that would retain
constant surface area over a range of edge lengths, but it is not feasible
to attempt to model them all here. The indenters used herein represent
most of the extreme forms of complexity and simplicity.
In order to avoid effects of low-resolution meshes, a series of analyses
were run on consecutively higher resolution meshes until the difference
in final result became negligible. Final mesh sizes were on the order of
400,000 elements. Figure 3 shows how meshes were refined.
Experiment 2
For this experiment, indenters matching those used in experiment 1
(Fig. 1) were made from wood. These indenters were used to indent a
soft mud with shear strength of ,5–10 kN/m
2
as measured in situ with a
penetrometer. A consistent pressure of 3 kN/m
2
was slowly applied
through each indenter using the penetrometer. Subsequent displace-
ment was then measured. The above procedure was repeated for dry,
fine-grained sand. These experiments were carried out numerous times
and recorded depths were averaged for each indenter.
FIGURE 1—Indenter shapes used in experiment 1 (surface area 5 9 units
2
) in plan
view. Indenters are subsequently referred to as 1A–G.
FIGURE 2—Indenter shapes with surface area 5 16 units
2
, viewed in plan view.
Subsequently referred to as indenters 2A–I. See table for details of edge lengths.
TABLE 1—Details of indenter surface area, edge length, edge length to surface area
ratio, and e9.
Indenter
Surface area
(units
2
)
Edge length
(units)
Edge-to-surface-area
ratio
Normalized edge
length (e9)
1A 9 12 1.33 1
1B 9 14 1.56 1.36
1C 9 14 1.56 1.36
1D 9 16 1.78 1.78
1E 9 18 2 2.25
1F 9 20 2.22 2.78
1G 9 20 2.22 2.78
2A 16 16 1 1
2B 16 18 1.125 1.26
2C 16 20 1.25 1.56
2D 16 22 1.375 1.89
2E 16 24 1.5 2.25
2F 16 26 1.625 2.64
2G 16 30 1.875 3.52
2H 16 32 2 4
2I 16 34 2.125 4.52
FIGURE 3—Increasing mesh resolution. Left; element size 5 1 unit
3
, Middle;
elements with dimensions 50% smaller (volume 25% of original), and Right; smallest
elements used (0.25 units
3
). Each image is shown to the same scale.
PALAIOS
EFFECT OF FOOT MORPHOLOGY ON TRACK DEPTH
357
RESULTS
Experiment 1
Maximum displacement was plotted against edge length (Fig. 4). The
maximum depth to which the indenters displaced the sediment
decreased as complexity (given by e9) increased. The most complex
shape (Fig. 1F), however, does not follow the pattern, instead indenting
to a greater depth than the simple indenter of edge length 20 (Fig. 1G).
The data show an overall decrease in maximum vertical displacement
corresponding to an increase in e9.
Experiment 2
The values for depth of indentation in the mud and dry sand are
shown in Figure 5. The indenters in mud showed slight reduction in
depth with increasing e9; it can be seen that indenters 1G and 1F (e95
2.78) both indented to a lesser degree than did indenter 1A (e951),
showing an extreme of only 50% of the depth of indenter 1A. The sand
showed the reverse trend seen in the experiments with mud and in the
FEA simulations; an increase in e9 produced a greater depth of
indentation. There is still the dichotomy between indenters 1B and 1C,
and between 1F and 1G.
DISCUSSION
The results from experiment 1 show a general trend for decreasing
displacement as edge length (e9) increases (Figs. 4, 6). Experiment 2
shows this trend in cohesive media, but that the reverse is true in
noncohesive sand. This is consistent with soil mechanics theory; a
noncohesive sand will displace to the greatest extent at the edges of an
indenter because values of Young’s Modulus vary with confining
pressure (Craig, 2004). Sediment grains are able to move past each
other, and as a result grains located between protrusions of indenters
are not pulled downwards by cohesion, unlike in muds and clays.
Penetration in cohesive media decreased by nearly 20% when
normalized edge length was increased from shortest to longest (most
compact shape to least compact). When displacement is normalized to a
percentage of the depth indented by the most compact form, it can be
seen that size of indenter does not significantly affect the pattern
(Fig. 6).
The larger indenters (surface area 5 16 units
2
) penetrated to greater
depths than the smaller set of indenters (Fig. 4). Even though pressure
remains constant, indenter size affects penetration depth independently
of shape. This is consistent with geotechnical theory, which shows a
relationship between footing size and bearing capacity of a soil (Zhu et
al., 2001; Kumar and Khatri, 2008).
The implication for vertebrate paleoichnology is that two dissimilar
foot morphologies may indent to very different track depths, even when
the same pressure is applied. A sauropod track may be deeper than a
theropod track, for instance, not due to the weight of the animal, which
when distributed over the surface area of the foot creates an equal
pressure to the smaller animal, but due to the morphology and
geometry of the foot being larger, and more compact in shape.
The mechanism by which normalized edge length affects track depth
can be explained through soil mechanics. As the load is applied, the
FIGURE 4—Graph plotting maximum vertical displacement beneath indenters of
varying edge length, for indenters consisting of 9 or 16 elements, with subsequent
surface area of 9 or 16 units
2
(diamonds and squares, respectively).
FIGURE 5—Mean depths of indentation for indenters 1A–F in mud (diamonds) and
dry sand (squares). Increasing normalized edge length results in a decrease in depth
indented in cohesive mud, but an increase in noncohesive sand.
FIGURE 6—Data for normalized edge length against maximum displacement
(normalized as a percentage of the most compact indenter) as recorded from FEA
simulations of geometric shapes. Trend lines are shown for both sets of data, and
show a close similarity (diamonds 5 surface area of 9 units
2
, squares 5 surface area
of 16 units
2
).
FIGURE 7—Vectors of displacement beneath the corner of a loading template.
Sediment is forced vertically down beneath the center of the indenter, but moves
outwards and upwards at the edge of the indenter, according to Prandtl theory. A
higher edge length to surface area ratio provides more opportunity for sediment to
move upwards and laterally, reducing energy transmitted down.
358
FALKINGHAM ET AL.
PALAIOS
medium is displaced. At the surface, beside the indenter, the path of
least resistance allows the sediment to move upwards (Fig. 7). Directly
beneath the indenter, sediment can only move vertically down, creating
a ‘dead zone’ (Allen, 1989, 1997; Manning, 2004). As such, an indenter
with a high edge to surface area ratio provides relatively more
opportunity for sediment to move upwards around the indenter. The
result is that energy is lost at the surface, rather than transmitted
vertically, and shallower tracks are produced. This is in agreement with
Jackson et al. (2009) who noted that it was the widest parts of indenters
that transmitted displacement most deeply.
There are exceptions to this, however, where an increase in edge
length can lead to indentation to a greater degree. For example,
indenter 1F, despite having a normalized edge length of 2.78, indented
further than indenter 1G with equal edge length (Fig. 4), this is because
the medium was unable to move upwards in the small gaps present in
indenter 1F due to cohesion. The stiffness of the medium prevented
easy movement, and instead the areas between protrusions were forced
down, essentially decreasing the effective e9 for indenter 1F. By creating
a more complex shape (with more corners), the effects of a stiff,
cohesive medium mean that effective e9 is reduced. This is exaggerated
in a low-resolution finite element mesh where only a single element is
present between indenting elements (e.g., in indenter 1F) and is unable
to deform to an extent allowing it to pass between the protrusions of the
indenter. Such a scenario highlights the importance of choosing the
correct FEA mesh resolution.
In order to use vertebrate tracks as paleopenetrometers, estimates of
mass and speed must be used in conjunction with observed or implied
pedal morphology and geometry. It is not enough to say that two
tracks, made by animals of similar size with similar-sized feet represent
comparable indenters; pedal morphology must also be constant.
Investigating the effects of pedal morphology also brings insight to
advantageous pedal forms. These experiments indicate that an animal
with a given mass may be provided with an advantage towards reducing
the depth to which its feet sink in soft media, either through an increase of
the surface area of the foot, which subsequently reduces pressure, or by an
increase in the edge length of the foot. Such an advantage may be linked
to the morphology of the feet of wading birds. Many wading birds possess
long, slender toes with no interdigital webbing (Brown et al., 1987;
Paulson, 1992). Such animals traverse soft media regularly. Increasing
surface area of the foot directly would be disadvantageous towards
moving the foot through water. By increasing edge to surface area ratio
and employing the effect described here, however, a low surface area can
be maintained whilst the effect of sinking into soft media may be reduced.
CONCLUSIONS
Tracks made by two animals of comparable size (mass and pedal
surface area) in similar media conditions may nevertheless be of
differing depth. The complexity of the foot morphology, as measured
using the normalized edge length e9, is one cause of this variation in
depth. Cohesion of the medium means that areas not directly in contact
with the indenter are still displaced down by neighboring medium,
essentially decreasing effective pressure. The effects of morphology are
reversed in noncohesive media, where increasing relative edge length
results in greater depth of tracks. Neoichnological and laboratory
experiments and observations must, therefore, be used comparatively
only with similar media if meaningful comparisons are to be drawn.
Size also has an independent effect on total displacement; larger
indenters penetrate the medium to a greater extent, when morphology
and pressure are kept constant.
ACKNOWLEDGMENTS
PLF was funded by Natural Environment Research Council (NERC,
award NER/S/A/2006/14033). HPCx project e46 funded through
Engineering and Physical Sciences Research Council (EPSRC, grant
EPF055595-1). We also acknowledge support from Louise Lever
for assisting with the FEA visualization, James Jepson and Karl
Bates for comments on an early draft, and Research Computing
Services at the University of Manchester for providing free access to the
local HPC system Horace. We also thank Stephen T. Hasiotis and two
anonymous reviewers whose comments helped to improve the
manuscript.
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ACCEPTED FEBRUARY 20, 2010
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