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Recent work has shown that muddy sediments are elastic solids through which animals extend burrows by fracture, whereas non-cohesive granular sands fluidize around some burrowers. These different mechanical responses are reflected in the morphologies and behaviours of their respective inhabitants. However, Armandia brevis, a mud-burrowing opheliid polychaete, lacks an expansible anterior consistent with fracturing mud, and instead uses undulatory movements similar to those of sandfish lizards that fluidize desert sands. Here, we show that A. brevis neither fractures nor fluidizes sediments, but instead uses a third mechanism, plastically rearranging sediment grains to create a burrow. The curvature of the undulating body fits meander geometry used to describe rivers, and changes in curvature driven by muscle contraction are similar for swimming and burrowing worms, indicating that the same gait is used in both sediments and water. Large calculated friction forces for undulatory burrowers suggest that sediment mechanics affect undulatory and peristaltic burrowers differently; undulatory burrowing may be more effective for small worms that live in sediments not compacted or cohesive enough to extend burrows by fracture.
(a) Schematic dorsal view of forces resisting movement (solid line) and forces generating thrust (arrows) against lateral crack edges, calculated from linear elastic fracture mechanics (LEFM), with lengths proportional to calculated magnitudes. Thrust forces applied normal to the curved body must balance axial resistance from friction (Fr) and anterior fracture (Cr) and elastic (El) resistance to burrow (scale bar, 2 mm). (b) Modelled Armandia brevis with ideal meander shapes (A/l ¼ 0.1 to 0.5; black lines) with distribution of normal forces (arrows) necessary based on LEFM to overcome resistive forces along the body. (inset) Relative thrust (upper dashed lines) and resistance (upper solid lines) in the x-direction and in the y-direction (lower dashed and solid lines, respectively) are plotted as a function of increasing % body length over which T(s) ¼ 1 is applied. Modelled results for y-direction forces are shown only for A/l ¼ 0.1 but are representative of all values of A/l. (c) Schematic cross-section view of forces generating thrust against lateral crack edges. Maximum thrust (Th) is limited primarily by fracture resistance under LEFM, but plastic deformation, here illustrated on the ventral side, could increase lateral resistance through work to plastically deform sediment with additional elastic–plastic resistance resulting from the deformed geometry (hypothetical forces shown as dotted lines; oblique muscles (OM) shown connecting lateral and ventral grooves). (Online version in colour.)
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, 20122948, published 27 February 2013280 2013 Proc. R. Soc. B
Kelly M. Dorgan, Chris J. Law and Greg W. Rouse
muds
Meandering worms: mechanics of undulatory burrowing in
Supplementary data
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Research
Cite this article: Dorgan KM, Law CJ, Rouse
GW. 2013 Meandering worms: mechanics of
undulatory burrowing in muds. Proc R Soc B
280: 20122948.
http://dx.doi.org/10.1098/rspb.2012.2948
Received: 10 December 2012
Accepted: 5 February 2013
Subject Areas:
biomechanics, ecology, environmental science
Keywords:
burrowing, gait, biomechanics, kinematics,
polychaete, annelida
Author for correspondence:
Kelly M. Dorgan
e-mail: kdorgan@ucsd.edu
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rspb.2012.2948 or
via http://rspb.royalsocietypublishing.org.
Meandering worms: mechanics of
undulatory burrowing in muds
Kelly M. Dorgan, Chris J. Law and Greg W. Rouse
Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 920930202, USA
Recent work has shown that muddy sediments are elastic solids through
which animals extend burrows by fracture, whereas non-cohesive granular
sands fluidize around some burrowers. These different mechanical responses
are reflected in the morphologies and behaviours of their respective inhabi-
tants. However, Armandia brevis, a mud-burrowing opheliid polychaete,
lacks an expansible anterior consistent with fracturing mud, and instead
uses undulatory movements similar to those of sandfish lizards that fluidize
desert sands. Here, we show that A. brevis neither fractures nor fluidizes sedi-
ments, but instead uses a third mechanism, plastically rearranging sediment
grains to create a burrow. The curvature of the undulating body fits meander
geometry used to describe rivers, and changes in curvature driven by muscle
contraction are similar for swimming and burrowing worms, indicating that
the same gait is used in both sediments and water. Large calculated friction
forces for undulatory burrowers suggest that sediment mechanics affect undu-
latory and peristaltic burrowers differently; undulatory burrowing may be
more effective for small worms that live in sediments not compacted or
cohesive enough to extend burrows by fracture.
1. Introduction
Mechanical interactions between organisms and their environments are integral
to locomotion, but mechanical responses of soils and sediments to forces applied
by burrowing organisms are poorly understood. How morphologies and beha-
viours of infauna affect burrowing performance is important in understanding
the evolution of burrowing animals and the ecology of sediment communities.
Many worms with diverse morphologies and behaviours extend burrows
through muddy sediments by fracture, using eversible mouth parts and muscu-
lar expansions to apply dorsoventral forces to burrow walls that are amplified at
the burrow tip [1– 3]. Fracture of muddy sediments results not only from direct
forces applied by burrowers, but also from hydraulic pumping by infauna during
burrow irrigation [4].
Nematodes and some polychaetes, however, lack expansible anteriors used
for burrow extension by fracture, and instead move through muddy sediments
by undulation [5]. For these organisms, kinematics are similar to those of sand-
fish lizards, which use body undulations rather than limbs to generate thrust and
fluidize desert sands [6]. Fluidization of the medium is indicated by backward
slipping of the animal and bulk transport of suspended grains in the opposite
direction of locomotion. Mechanical responses differ, however, among burrowers
in non-cohesive granular sands: fishes such as sand lances and eels burrow in
saturated marine sands with no slipping [7,8]. Kinematics are similar to terrestrial
crawling, and burrowing results in small discrete movements of sand grains
rather than bulk transport. In both cases, movement involves displacement of dis-
crete grains against gravitational forces, in contrast to elastic muds, which are
held together by adhesion and cohesion of the intra-granular organic matrix [9].
These different responses of sands to undulatory burrowers are quantified
using wave efficiency,
h
¼v
x
/v
w
, the ratio of the animal’s forward velocity, v
x
,
to the velocity of the posteriorly travelling undulatory wave, v
w
. Wave efficiencies
of
h
0.5 for burrowing sandfish lizards are consistent with fluidization of gran-
ular sand [6], and of
h
1 for sand lances and eels are consistent with a solid
response of sand [7,8]. For swimming animals, wave efficiency varies
&2013 The Author(s) Published by the Royal Society. All rights reserved.
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considerably and depends on Reynolds number (Re)[10],with
high values of greater than 0.7 for swimming eels [11,12] down
to 0.23 for sperm [13]. For swimming leeches,
h
is proportional
to size and drops from 0.55 to 0.43 with a 10-fold increase in
viscosity [14]. Crawling animals have high
h
, greater than
0.8 compared with 0.190.29 for swimming nematodes [5],
approaching 1 on surfaces with sufficient friction in which
no slipping occurs.
Undulatory burrowing has not yet been quantitatively
explored in muddy sediments, which are elastic solids that
differ mechanically from non-cohesive sands [9]. Armandia
brevis (Moore, 1906), an opheliid polychaete that inhabits
muddy to sandy sediments, lacks both eversible mouthparts
and circular muscles needed for peristalsis. Rather, it has
bands of oblique muscles that act antagonistically to longitudi-
nal muscles, enabling lateral bending and undulatory
movements (figure 1). Armandia brevis also swims by undula-
tion, enabling direct comparison of wave efficiency while
burrowing and swimming. Wave efficiency values for burrow-
ing A. brevis close to 1 and higher than that of swimming
worms would indicate solid response of the medium, whereas
h
1 would indicate fluidization.
For A. brevis, swimming is a derived behaviour—apart
from this taxon and other members of its own subfamily
Ophelininae, no other opheliid swims [15]. Though dispersal
is one explanation, most swimming A. brevis are reproductive,
and spawning occurs only once before death [16]. That swim-
ming seems to be a secondary mode of locomotion raises the
question of whether swimming behaviours are distinct from
burrowing, i.e. whether the same gait is used in both media
with kinematic differences attributable to mechanics of solids
compared with fluids.
Discrete gaits are characterized by discontinuous changes
in movement patterns. Both sandfish lizards and sandlance
fishes substantially change their body shapes, increasing the
amplitude relative to wavelength (A/
l
), when transitioning
to burrowing from running and swimming, respectively
[6,7]. Larger A,
l
and frequency have been observed for nema-
tode worms swimming in water compared with media with
higher resistance, but positive linear relationships indicate con-
stant A/
l
and reveal no discrete transition indicative of gait
change [17,18]. Although the transition from fluid to elastic
mud is inherently discontinuous, similar body shape of
A. brevis, characterized by A/
l
, would be consistent with the
same gait used in both media. As body shape is determined
from discrete time points and is affected by external forces
[19], we also compared changes in body angle over a cycle
of undulatory movement as well as patterns of body curvature
to determine whether burrowing and swimming gaits are the
same. Whereas undulatory movements have been described as
sinusoidal [10] and a best-fit sine function has been used for
measuring amplitude, wavelength and wave speed [6], curva-
ture of the path of A. brevis appeared flatter and broader than a
sine function. Similar geometry was first described for paths of
meandering rivers, which can be approximated as a sine-
generated curve; the relationship between direction angle of
the path,
u
, and distance along the path, s, fits a sine curve
[20]. A sine fit to
u
sis a better approximation of snake
shape than an x–ysine fit, consistent with waves of alternating
muscle contraction travelling along the length of the animal
changing the body curvature [21]. Direct measurement of kin-
ematic parameters such as amplitude, wavelength, radius of
curvature and v
w
from profiles enables comparisons, but
lacks a mechanistic basis; fit to a meander curve would be con-
sistent with sinusoidal muscle contraction and relaxation,
whereas deviations indicate non-sinusoidal contraction or
non-uniform external forces.
We explored the burrowing and swimming behaviour of
A. brevis and combined experiments with theory to assess
three hypotheses for the mechanism of undulatory burrow
extension in muds: (i) A. brevis extends burrows by fracture
[1], (ii) A. brevis fluidizes muddy sediments, indicated by
h
1[6], and (iii) the muddy sediment deforms plastically
through grain rearrangement, indicated by
h
1 [7,8].
2. Material and methods
To address whether A. brevis extends burrows by fracture, we
combined experiments on worms in gelatin, an analogue for cohe-
sive elastic muds through which worms burrow by fracture [1– 3],
with theoretical predictions based on linear elastic fracture mech-
anics (LEFM). Because A. brevis lives in surface sediments [16], we
developed an additional analogue for weak surface muds com-
prising organic-mineral aggregates, fragments of concentrated
gelatin, in which kinematics were analysed to determine whether
fluidization or solid grain rearrangement occurs. Kinematics of
burrowing and swimming worms were compared to assess
whether A. brevis uses the same gait in the different media.
(a) Kinematics
Armandia brevis were collected from shallow subtidal sediments
in Mission Bay, San Diego, CA, and from sediments in the flow-
ing seawater tanks at Scripps Institution of Oceanography, La
Jolla, CA. For experiments to determine whether worms bur-
rowed by fracture, glass aquaria were filled with seawater
gelatin (28.35 g l
21
), and worms were placed in pre-made
cracks and filmed following methods of Che & Dorgan [3]. For
elastic aggregate burrowing experiments, concentrated seawater
gelatin (85 g l
21
) was chopped in a food processor to fragments
OM
LM
LM
c
g
OM
es
p
(a)
(b)(c)
Figure 1. (a) Morphology of Armandia brevis.(b) Ventral view of oblique
muscles (OM) shown using polarized light microscopy with segments distin-
guished by parapodia (p) and segmental eye spots (es). (c) Cross-section of
internal anatomy showing large bands of oblique (OM) and longitudinal
muscle (LM), gut ( g), and cuticle (c). (Online version in colour.)
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roughly 200– 500 mm diameter. Gelatin pieces were slowly added
to a plexiglass ant farm aquarium (7 71.5 cm) with seawater
and were gently mixed to remove air bubbles. Worms were
placed below the surface of the aggregates equidistant from the
two walls of the aquarium, and movements were recorded.
Only video segments in which the worms moved in a plane per-
pendicular to the camera angle (in focus within an approx. 1 cm
focal plane) and did not reach either wall were used for analysis.
Worms were filmed swimming in a Petri dish (10 2 cm) with a
Canon T2i DSLR camera at 60 fps. For both burrowing and
swimming, wall effects may occur, but eliminating or substan-
tially reducing wall effects was logistically difficult because of
visualization of burrowing worms and the speed of swimming
worms. Videos were first subsampled using QUICKTIME PRO v. 7
(swimming) or LABVIEW v. 7.1 (burrowing), and processed in
IMAGEJ v. 1.44 to obtain body outlines and head and tail positions
for each frame.
To characterize gait, first body shape was analysed by
measuring amplitude and wavelength. From outline coordinates
and head and tail positions, midlines were calculated and further
analysis done using custom MATLAB (R2010B) scripts. To measure
amplitude and wavelength, curvature of the path of travel
(smoothed centre of mass (COM) path) was removed by sequen-
tially rotating the midline for each frame and later frames about
the COM for that frame. Next, maximum deviations of midlines
from that straightened path and distances between peaks were cal-
culated. For consistency, midlines were shortened to total body
length, L
s
, of 95 per cent of the shortest midline in a sequence.
We compared fits with a meander curve and sine curve for
both burrowing and swimming worms. Shortened midlines
were converted from x–y to
u
scoordinates using a cartesian-
to-polar coordinate conversion for each segment dsof the
midline. Best-fit sine function in x–y coordinates was the sine
fit, in
u
scoordinates was the meander fit. Because both sine
and meander curves were fit to a line of points, residuals were
not randomly distributed and the autocorrelation of midline
points likely resulted in inflated R
2
, but subsampling did not
effectively remove the autocorrelation or change the R
2
.To
better assess the fit of the two models, we also compared the
body angle, sin(
u
), and curvature, d
u
/ds, profiles to those
predicted by a sine and meander fit (see the electronic supplemen-
tary materials for details). The sine and meander curves were fit to
individual frames rather than the path, for which both amplitude
and wavelength varied considerably, making modelling the path
as a single travelling sine wave infeasible.
Wave efficiency was calculated as
h
¼v
x
/v
w
, where the wave
velocity in
u
–s coordinates calculated from cross correlation of
subsequent frames was transformed to x–y coordinates by the
sinuosity, the ratio of the body length (before shortening) to
the shortest distance from head to tail, to obtain v
w
. Sinuosity
was inflated for swimming worms by yaw, seen as side-to-side
movement of the COM, so rather than using v
x
calculated from
the smoothed COM path, velocity was calculated along the
unsmoothed COM path (greater than v
x
for swimming, but
approx. v
x
for burrowing) and this corrected velocity was used
in wave efficiency calculations.
(b) Model
LEFM theory was used to calculate a force balance for burrow
extension by fracture for an undulating worm. Forces resisting
forward movement in linear elastic muds include cohesive frac-
ture resistance, elastic resistance to sediment deformation and
friction. The work to extend a burrow by fracture is the sum of
the work of fracture, W
Cr
¼G
c
(Dx)w
cr
, and the elastic work,
W
El
¼2s
w
w
w
(Dx)h[22]. G
c
is the fracture toughness (J m
22
),
Dxis the distance the crack extends (m), w
cr
is the crack width
(m), s
w
is the internal pressure of the worm (Pa), w
w
is the
width of the worm (m) and his the half-thickness measured
dorsoventrally (m). A factor of 2 is included because elastic dis-
placement occurs along both dorsal and ventral crack surfaces.
Thrust force, F
Cr
þF
El
, required to drive a wedge-shaped
worm forward a distance Dxmust balance the resistance:
ðFCr þFElÞDx½GcðDxÞwcr þ2
s
wwwðDxÞh¼0:ð2:1Þ
For peristaltic burrowing, friction is ignored because the normal
force on narrow moving segments is greatly reduced by nearby
dilated stationary segments, an assumption that does not apply
to more rigid-bodied undulatory burrowers. Here, we include a
friction force, F
Fr
¼2ms
w
w
w
L, with normal force based on elas-
ticity, which acts along L, the entire length of the worm (m). In a
crack-shaped burrow, friction acts on both dorsal and ventral
surfaces (requiring a factor of 2). In sediments with density
greater than that of the worm, overlying weight also contributes
to friction and F
El
, but in gelatin, this term is small and can
be ignored.
Rather than travelling in a straight line, undulatory move-
ment occurs in a two-dimensional plane. Resistive forces occur
along the body axis, s, and we calculate total external resistive
forces as
FresðsÞ¼FCr ðsÞþFElðsÞþFFr ðsÞ
¼Gcwcr þ2
s
wwwhþðL
0
2
ms
wwwds:ð2:2Þ
During undulation, thrust, F
Th
, is applied normal (n) to the body
axis, and here, we assume that these normal forces, F
Th
(n), are
limited by material resistance rather than the amount of muscu-
lar force the worm can generate. Assuming that burrow
extension follows LEFM and the burrow is a planar crack com-
pressing the worm dorsoventrally [1], material resistance to
these lateral forces is limited by the lateral fracture resistance.
Maximum lateral thrust force, therefore, depends on the lateral
work of fracture, elastic work and friction, the same components
as axial resistance but differing in geometry. For lateral resist-
ance, the crack length is the axial length of the worm, L, rather
than w
cr
:
FThðnÞ¼FCr ðnÞþFElðnÞþFFr ðnÞ
¼ðL
0
GcdsþðL
0
2
s
whdsþðL
0
2
ms
wwwds:ð2:3Þ
In reality, thrust force is not applied along the entire length of the
worm or at the maximum possible magnitude, so a scaling factor
TðsÞ[[0;1] is incorporated into equation (2.3). This thrust force,
F
Th
(n)T(s), results in an added (to F
res
) frictional resistance term
proportional to the thrust and distributed along Lon the outer
curved side of the body, F
Fr_added
(s)¼
m
F
Th
T(s).
Axial resistive forces and lateral thrust forces are balanced by
the curvature of the body (figure 2a). Converting to x–y coordi-
nates with the x-axis aligned with the s-axis at L¼0 (when the
head is oriented parallel to velocity, with
us
the body angle),
and assuming steady state,
FresðsÞcos
u
þFThðnÞTðsÞsin
u
¼0:ð2:4Þ
The internal pressure of the worm, s
w
, depends on the material
stiffness, E, and the elastic displacement, here h, the half-thick-
ness of the worm, as well as the crack geometry. For the
peristaltic burrowing polychaete Cirriformia moorei (Blake,
1996), this value was determined from measured displacements
along the length of the body and material stiffness using a
two-dimensional finite-element model [22]. Elastic modulus, E,
relates stress to strain in a material, but for burrowers, relating
displacement (h) to strain (1¼h/length scale) is confounded
by what length scale to use. We can calculate from C. moorei
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results a scale factor as a function of the geometry, f(g)(m
21
),
s
w¼E1Ehf ðgÞ:ð2:5Þ
Finite-element modelling shows that f(g) depends primarily on
the distance from the worm to the lateral crack edge and that
only a twofold increase in body thickness occurred as the crack
tip was extended from the lateral side of the worm out to a
large distance at which thickness reached an asymptote, with
s
w
and Eheld constant. These results suggest that differences
in f(g) between C. moorei and A. brevis do not exceed a factor of
2 and are probably much smaller (cf. fig. 7bin [2]).
Applying the assumption in equation (2.5), assuming that the
crack width is related to worm width as w
cr
¼aw
w
, and rearrang-
ing to obtain non-dimensional terms, the force balance (equation
(2.4)) becomes
ðL
0
TðsÞsinð
u
Þds
ww
¼
aGc=Eh2fðgÞþ2þ2
m
ðL
0
cosð
u
Þds=hþ
mc
ðL
0
TðsÞcos
u
ds=ww
Gc=Eh2fðgÞþ2þ2
m
ww=h;
ð2:6Þ
where
c
¼Gc
Eh2fðgÞþ2þ2
m
ww
h:ð2:7Þ
The right-handside of equation (2.6) is the ratio of non-dimensional
resistive forces to non-dimensional maximum thrust forces,
c
,and
the left-hand side indicates over what length of the body those
thrust forces must be applied for forces to balance.
To compare the relative importance of fracture resistance, elas-
ticity and friction for A. brevis,weusevaluesforwormgeometry
and material properties to calculate approximate values for the
non-dimensional terms in equation (2.6). Measured for A. brevis,
L¼14 mm, w
w
¼0.7 mm, h¼0.35 mm; for gelatin, E¼7100 Pa
[3], G
c
¼0.4 J m
22
[3,23], calculated from C. moorei,f
1
(g)¼50 m
21
[22] and a¼2 [3], and we assume
m
¼0.3. For sinuosity of approxi-
mately 1.3, Ðcos(
u
)ds0.75L. Next, the length along the body over
which thrust must be applied for forces to balance and the corre-
sponding added friction were calculated numerically for simulated
worms with ratios of amplitude to wavelength, A/
l
, varying from
0.1 to 0.5. Worms were modelled as cosine-derived ideal meander
curves using custom MATLA B scripts.Forameandercurvederived
from a cosine curve, for which the head of the worm was oriented
at
u
, the balance of resistance, F
res
(s), and thrust, F
Th
(n), was calcu-
lated by converting from s–n to x–y coordinates, with the x-axis
oriented along the COM path. This force balance in the x-direction is
ðL
0
TðsÞsinð
u
Þds
ww
¼
aðGc=Eh2fðgÞÞcos
u
0þ2cos
u
0þ2
m
ðL
0
cosð
u
Þds=hþ
mc
ðL
0
TðsÞcos
u
ds=ww
Gc=Eh2fðgÞþ2þ2
m
ww=h
ð2:8Þ
and in the y-direction, perpendicular to the COM path,
ðL
0
TðsÞcosð
u
Þds
ww
¼
aðGc=Eh2fðgÞÞsin
u
0þ2sin
u
0þ2
m
ðL
0
sinð
u
Þds=hþ
mc
ðL
0
TðsÞsin
u
ds=ww
Gc=Eh2fðgÞþ2þ2
m
ww=h:
ð2:9Þ
0 0.002 0.004 0.006 0.008 0.010 0.012
−2
−1
0
1
2
×10−3 10
−3 m
−2
−1
0
1
2
−2
−1
0
1
2
−2
−1
0
1
2
−2
−1
0
1
2
0100
% body length
dimensionless force
A/l = 0.1
A/l = 0.15
A/l = 0.3
A/l = 0.2
A/l = 0.5
A/l = 0.1
A/l = 0.5
m
Fr
El
linear elastic
fracture
elastic–plastic
fracture
OM OM
plastic
elastic–plastic
Th
Fr
Cr+El
Fr
Cr+El
Cr
(a)(b)
(c)
Figure 2. (a) Schematic dorsal view of forces resisting movement (solid line) and forces generating thrust (arrows) against lateral crack edges, calculated from linear elastic fracture
mechanics (LEFM), with lengths proportional to calculated magnitudes. Thrust forces applied normal to the curved body must balance axial resistance from friction (Fr) and anterior
fracture (Cr) and elastic (El) resistance to burrow (scale bar, 2 mm). (b)ModelledArmandia brevis with ideal meander shapes (A/
l
¼0.1 to 0.5; black lines) with distribution of
normal forces (arrows) necessary based on LEFM to overcome resistive forces along the body. (inset) Relative thrust (upper dashed lines) and resistance (upper solid lines) in the
x-direction and in the y-direction (lower dashed and solid lines, respectively) are plotted as a function of increasing % body length over which T(s)¼1 is applied. Modelled results
for y-direction forces are shown only for A/
l
¼0.1 but are representative of all values of A/
l
.(c) Schematic cross-section view of forces generating thrust against lateral crack
edges. Maximum thrust (Th) is limited primarily by fracture resistance under LEFM, but plastic deformation, here illustrated on the ventral side, could increase lateral resistance
through work to plastically deform sediment with additional elasticplastic resistance resulting from the deformed geometry (hypothetical forces shown as dotted lines; oblique
muscles (OM) shown connecting lateral and ventral grooves). (Online version in colour.)
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Maximum possible thrust was applied (T(s)¼1) along an increas-
ing percentage of the body until thrust forces and resistive forces
balanced (left side of equation (2.8) exceeded the right side). We
chose this approach rather than using a constant T(s) along the
length of the worm both to minimize the added friction term,
F
Fr_added
(s), and to be consistent with qualitative observations of
focused forces applied by the related Ophelina acuminata (Orsted,
1843) moving in a pre-made crack in gelatin visualized using photo-
elastic stress analysis (K. M. Dorgan 2009, unpublished data). Force
was applied in the direction normal to the modelled midline and
opposite the direction of forward movement. The force balance in
the y-direction was used to determine at what position dsto
increase T(s). T(s) was incrementally increased at the position at
which
u
was closest to the ratio of the discrepancy in the y-direction
force balance (difference between right and left sides of equation
(2.9)) relative to the discrepancy in the x-direction force balance
(difference between right and left sides of equation (2.8)) until the
x-direction thrust force exceeded resistance.
3. Results and discussion
(a) Armandia brevis does not extend burrows by linear
elastic fracture
Worms placed in different orientations in pre-formed cracks of
varying widths in gelatin, an analogue for elastic muds, exhibited
undulatory movements with small-amplitude head wiggling,
but no worms extended the burrow even when curved with
the posterior braced against the crack edge (n.10). Shapes of
crawling snakes depend primarily on external forces, specifically
the ratio of lateral resistance to gravitational force (which deter-
mines the normal force upon which ventral friction depends): if
lateral resistance is low relative to friction, the body is more
curved [19]. Armandia brevis, oriented dorsoventrally compressed
in a crack, experienced low lateral resistance compared with
dorsoventral friction augmented by fracture resistance.
Modelling results showed that relative magnitudes of
dimensionless components of axial resistance (right side numer-
ator, equation (2.6)) are: fracture component, aG
c
/Eh
2
f
1
(g)¼18;
elastic work associated with burrow extension, 2; friction com-
ponent, 18; and added friction depends on T(s), and is
calculated numerically. Fracture resistance higher than elastic
resistance is consistent with calculations for C. moorei of a
work of fracture approximately 10elastic work [22]. Relative
magnitudes of dimensionless components of lateral thrust
(right side denominator, equation (2.6)) are: fracture component,
9; elastic work, 2; and friction, 1 (figure 2a). That friction plays a
substantial role in resisting axial movement but does little to
preventing lateral slipping is consistent with the elongate
shape of the worm.
For small A/
l
, maximum thrust forces along the entire
length of the worm were insufficient to balance resistive
forces; as A/
l
increased, thrust forces applied along decreas-
ing percentages of the body could balance resistance
(figure 2b). The relative thrust force (left side of equation
(2.8), dashed lines in figure 2binset) increased as T(s)
increased, reaching the resistive forces relative to maximum
thrust (right side of equation (2.8); solid lines) for A/
l
.0.1.
Added friction from thrust forces increased the resistance
forces (solid lines in figure 2binset) substantially, to approxi-
mately 2.5the resistance force with no added friction for
A/
l
¼0.1. As A/
l
increased, initial resistance force decreased
owing to a decrease in the x-component of friction, and the
added friction from thrust increased less steeply because as
u
increases, more of the added friction acts in the y-direction.
T(s) was increased at positions along the length of the worm to
maintain equilibrium in the y-direction—the resistance (right
side of equation (2.9), solid line in figure 2binset) was close
to zero and thrust (left side of equation (2.9), dashed line in
figure 2binset) fluctuated around 0. T(s) was more variable
for the largest A/
l
, 0.5, owing to the many positions at
which
u
p
/2, and indicates that forces smaller than maxi-
mal forces could be distributed along this region of the
body. Even at intermediate A/
l
, the substantial portion of
the body required to exert the maximum possible force indi-
cates that the LEFM model is only barely mechanically
feasible. Our model applies maximum forces; exceeding
these would result in lateral crack extension and reduced lat-
eral resistance. Applying smaller forces by reducing T(s),
however, would require force to be applied along a greater
percentage of the body, increasing the added friction, which
is already substantial at low-to-intermediate A/
l
(figure 2b).
Only at very high A/
l
does this mechanism seems mechani-
cally feasible, but at high A/
l
efficiency is low and
manoeuverability may be limited.
(b) Non-cohesive granular media exhibits solid response
to undulatory burrowing
Based on LEFM, lateral fracture resistance is only barely
sufficient for A. brevis to overcome anterior resistance, but
dorsoventral plastic deformation of sediments could increase
resistance to lateral slipping (figure 2c). In natural muds, frac-
ture toughness and stiffness are low in the top approximately
23 cm of sediments [24,25], corresponding to the depth dis-
tribution of A. brevis [16]. At the surface, fracture toughness
approaches zero, indicating that surface muds are non-cohe-
sive, high-porosity aggregates and that linear elastic fracture
occurs only below the surface layer. Armandia brevis was
able to burrow through an analogue of non-cohesive elastic
fragments of gelatin simulating surficial, unconsolidated
sediments comprising organic-mineral aggregates (see the
electronic supplementary materials, movie S1). Elastic– plastic
fracture with dorsoventral plastic deformation is, however,
theoretically feasible and, assuming a gradient in sediments
from surface aggregates to cohesive elastic mud, would increase
worms’ depth limit. Ventral and lateral grooves increase the
angle of contact between the worm and sediment, probably
increasing lateral resistance and facilitating elastic– plastic
fracture (figure 2c).
Worms burrowed through this analogue material with a
non-slipping undulatory wave (figure 3a), consistent with the
hypothesis of sediment exhibiting solid behaviour with plastic
reorganization of grains. By contrast, worms swimming through
a fluid medium clearly slip backwards (see the electronic sup-
plementary material, movie S2; figure 3b). Wave efficiencies
were significantly higher for burrowing than for swimming
worms (figure 4a). For burrowing worms,
h
¼1.00 +0.10
(mean +s.d.), and for swimming worms,
h
¼0.58 +0.11,
similar to values of 0.50– 0.58 for the related Ophelina sp. [26]
and of approximately 0.5 for burrowing sandfish lizards that
fluidize sands [6]. Calculating wave efficiency from smoothed
v
x
values rather than from velocity along the unsmoothed
COM path results in
h
¼0.48 +0.10 for swimming worms
and
h
¼0.97 +0.11 for burrowing worms. Faster swimming
worms slipped less, consistent with dependence on Re,whereas
no relationship was observed between
h
and velocity for
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burrowing worms (figure 4a). Velocity showed a strong depen-
dence on cycle frequency for both burrowing and swimming
(figure 4b). Swimming worms travelled the same distance per
cycle as burrowing worms (figure 4b), suggesting that inertia
may balance backward slipping.
(c) Undulatory kinematics fit a meander curve
Meander curves showed good fit, significantly better than x–y
sine curve fits for both burrowing (12/15 worms) and
swimming (17/17 worms) when comparing frames for
individual worms (see figure 3c,d and electronic supplementary
material, table S1). R
2
was high, however, and more variable
across individuals for both meander (0.94 +0.03 for bur-
rowing; 0.99 +0.01 for swimming; mean +s.d.) and sine
(0.91 +0.03 for burrowing; 0.95 +0.03 for swimming) fits,
probably inflated by autocorrelation of midline points (see
figure 3c,dand electronic supplementary material, table S1).
More substantial differences between sine and meander fits
were found for body angle profiles, with R
2
for meander fits
510152025
4
6
8
10
12
14
16
18
20
22
24
46810 12 14 16 18 20 22 24
2
4
6
8
10
12
14
16
mm mm
mm
mm
(a)(b)
(c)
(d)
0 2 4 6 8101214161820
−2
0
2
mm
mm
*
12 14 16
−1
0
1
mm
mm
*
51015202530
−4
−2
0
2
4
mm
mm
10 12 14 16
−4
0
4
mm
mm
*
*
14 12 10 8 6 4 2 0
−1
0
1
position along body (mm)
*
76543210
−1
0
1
position along body (mm)
q (rad)
q (rad)
*
Figure 3. Midlines of A. brevis (a) burrowing (at 0.067 s intervals) and (b) swimming (at 0.017 s intervals), coloured sequentially from green to black with cor-
responding centre of mass as solid circles (dotted black line is smoothed COM path). Raw images superimposed are shown in insets (scale bar, 2 mm). For both
(c) burrowing and (d) swimming, a sine fit (solid black) through midlines (solid line with asterisk (*) at head) does not match curvature (residuals in dotted line) as
well as a meander fit to body angle,
u
, as a function of body position, s(black dotted line shows the converted sine fit). (Online version in colour.)
0
velocity (mm s−1)
f (cycles s−1)
10
20
30
40
50
60
70
2 4 6 8 10 12
0
velocity (mm s−1)
0.2
0.4
0.6
0.8
1.0
1.2
(a)(b)
10 20 30 40 50 60 70
wave efficiency
Figure 4. Parameters quantifying gait for burrowing (open circles) and swimming (fillled squares). (a) Wave efficiency is correlated with velocity for swimming
(R
2
¼0.52; p,0.01), but not for burrowing worms. (b) Distances travelled per cycle for burrowing (5.7 +2.0 mm cycle
21
, mean +95% CI; R
2
¼0.74) and
swimming (6.9 +2.6 mm cycle
21
;R
2
¼0.67) do not differ (combined data, solid line; 5.4 +0.8 mm cycle
21
,R
2
¼0.87).
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of 0.84 +0.06 and 0.96 +0.02, and for sine fits of 0.45 +0.08
and 0.47 +0.07 for burrowing and swimming, respectively
(see the electronic supplementary material, figure S1 and
table S1). For swimming worms, curvature profiles also fit a
meander curve significantly better than a sine curve (see the
electronic supplementary material, table S1). Midline profiles,
as well as body angle and curvature, of swimming worms
showed a better fit than burrowing worms (see the electronic
supplementary material, table S1). However, worms burrowed
with smaller
l
/Lthan when swimming, and we re-analysed the
meander fit of each half of the body length of burrowing worms
(because
l
/L0.5); this increased the fit for burrowing
worms, removing differences in fits of body shape and
curvature between burrowing and swimming worms and
substantially reducing the difference for body angle (see the
electronic supplementary material, table S1).
(d) The same gait is used for burrowing and swimming
Swimming worms undulated with significantly larger ampli-
tudes and wavelengths and at higher cycle frequencies ( f)
than burrowing worms (all ANOVA p,0.001; electronic sup-
plementary material, table S2 and figure S2). Higher ffor
swimming worms corresponds with higher velocities, and
there are no obvious discontinuities for burrowing and swim-
ming, with similar distances travelled per cycle in both media
(figure 4b). Moreover, body shapes, quantified by A/
l
,were
the same, 0.18 +0.03 (mean +s.d.) for burrowing and
0.19 +0.05 for swimming worms, similar to nematodes
[5,17] and burrowing sandfish [6]. Similar body shapes (A/
l
)
are consistent with the hypothesis that burrowing and swim-
ming worms use the same gait [17,18], but body shape
depends primarily on external forces [19]. For undulating
fishes, increasing amplitude from head to tail corresponds
with lags between waves of body curvature and muscle activity
[27], but constant amplitude along the length of A. brevis
enables use of change in curvature as a first-order approxi-
mation of muscle activity. The relationship between muscle
activation and body curvature is complex [28], and changes
in curvature are also influenced by internal elasticity and exter-
nal forces [19]; however, similar changes in curvature in the two
media are consistent with similar muscle activity patterns. As a
meander curve, the relationship between
u
at a fixed location on
the worm and time is sinusoidal, as is the relationship between
d
u
/dtand time. Amplitude, maximum d
u
/dtper cycle (see the
electronic supplementary material, figure S3), shows a strong
linear relationship with fwith no discontinuities that would
indicate a gait transition (figure 5). Linearity indicates that
the rate of body bending, presumably resulting from the rate
of muscle contraction, used in both modes of locomotion
is directly proportional to the cycle frequency, which is in
turn directly proportional to velocity (figure 4b): the pattern
of movement does not change with speed. More importantly,
this pattern of movement does not differ between burrowing
and swimming. Similar fit to a meander curve of body shape,
body angle and curvature (see the electronic supplementary
material, figure S1 and table S1) indicates that muscle contrac-
tion is similar and probably sinusoidal along the length of the
body for both swimming and burrowing worms. Larger Aand
l
for swimming worms can be attributed to low resistance of
water to internal elastic forces. For the nematode C. elegans,
crawling worms experience comparable external loads and
internal elastic forces, whereas for swimming worms external
forces are much smaller than internal elasticity [18].
4. Conclusions
Whereas transitions from swimming to burrowing [7,8] and
from running to burrowing [6] involve substantial changes in
locomotory behaviour, undulatory burrowing worms change
only the frequency of movement when transitioning to swim-
ming, a derived mode of locomotion in this group [15]. Use of
a burrowing undulatory gait for swimming could explain why
neither A. brevis, most nematodes [5], nor larval lampreys [29]
swim with undulatory waves increasing in amplitude, in con-
trast to most elongate undulatory swimmers, e.g. snakes [30],
eels [11] and amphioxus [31]. Muscle contraction patterns
and body stiffness contribute to this increasing amplitude
[30,31], which reduces yaw, probably resulting in greater effi-
ciency [5]. This side-to-side movement of swimming A. brevis
(figure 3b) may reduce swimming efficiency but does not
affect burrowing, their primary mode of locomotion.
Fit to a meander curve explicitly links the shapes of
elongate animals to the muscular activity that drives move-
ment. This sinusoidal change in angle along a path or over
time describes shapes of rivers [20], snakes [21], flagella
[32], A. brevis and probably other biological and abiotic
elongate patterns as well. In animal locomotion, deviations
from this meander model shape may indicate non-uniform
external forces or alterations of behaviour. For A. brevis, simi-
larity in meander fit supports our hypothesis that swimming
worms use the same muscle contraction patterns but at
higher frequency than burrowing worms.
Our finding that LEFM is an unlikely mechanism for
undulatory burrowing in muds is based on the size of
A. brevis and limited data for mechanical properties of
muds. Behaviours of burrowers using fracture depend on
0
f (c
y
cles s−1)
20
40
60
80
100
120
2 4 6 8 10 12
.
.
.
0.15 0.20 0.25 0.30
8
10
12
14
16
A/l
(dq/dt)/f
q
q
q
max(dq/dt) (rad s−1)
Figure 5. The amplitude of the sinusoidal relationship of d
u
/dtas a function of
time is strongly correlated with cycle frequency (R
2
¼0.95) with slopes not sig-
nificantly different for burrowing (open circles), swimming (closed squares), and
combined data (at 95% CI). For swimming worms (inset), some variability can
be explained by a positive correlation (R
2
¼0.61, p,0.05) between slopes cal-
culated for individual worms [max(d
u
/dt)/f(rad s
21
)/(cycles s
21
)] and A/
l
faster muscle contraction per cycle results in larger A/
l
. Sequential images of a
burrowing worm (0.067 s apart) show changing
u
(upper inset; scale bar, 2 mm).
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the ratio of work of fracture (/G
c
, fracture toughness) to elas-
tic work (/Eh, stiffness and body thickness) [23]. For
peristaltic burrowing, work of fracture is approximately
10elastic work [22], but for undulatory burrowers, this
large friction component that in elastic gels depends primar-
ily on stiffness, E, combines with elastic work, altering this
ratio. Compared with its behaviour with peristaltic bur-
rowers, the same material would seem much stiffer. For
smaller undulatory burrowers like the nematode C. elegans,
G
c
/Eh is much higher than for A. brevis, possibly explaining
their ability to burrow through gelatin [17] and agar. Simi-
larly, in soft sediments with higher G
c
/E than gelatin,
friction would be less important and fracture a more feasible
mechanism of undulatory burrowing. Friction increases not
only the apparent stiffness of muds, but also total work
to burrow, potentially exceeding the muscle capacity of
A. brevis in deeper sediments (a possible explanation for the
large longitudinal muscle bands, figure 1c). Friction is
probably high regardless of sediment mechanics: in weak
sediments with less elastic cohesion, the normal force
depends primarily on overlying weight, and friction would
increase with depth, similar to in elastic muds with increasing
E. Sediments with heterogeneous mixtures of sand and mud
are common habitats for burrowers and probably have mech-
anics that fall between those of sands and muds. The similar
granular responses of surface muds on small scales and of
sands to larger burrowers (figure 6) and the potential use
of elasticplastic fracture suggest interesting hypotheses
about burrowing in heterogeneous sandy muds, which may
involve a combination of elastic and granular mechanisms
that depend on burrower size, morphology and behaviour,
as well as small-scale differences in sediment mechanics.
Muddy sediments are ubiquitous and are inhabited by
diverse animals, many of which, such as A. brevis,aresmall,
live close to the sediment– water interface, and exhibit undula-
tory or non-peristaltic movements. Reduction of friction by
alternating expansion and contraction during peristalsis
suggests higher efficiency than undulatory burrowing in com-
pacted sediments. The limited distance over which forces can
be applied during peristalsis, however, may be insufficient to
overcome fracture resistance or even to anchor small worms
in less consolidated sediments. Mechanics indicate that undu-
latory burrowing is more effective in these weak surface
sediments and that these differences are greater for small
worms—sediments that are too tough for small peristaltic bur-
rowers to crack [3] exert smaller normal forces and less
frictional resistance on small undulatory burrowers. These
different mechanisms of burrowing in muds—plastic defor-
mation or elasticplastic fracture for undulatory burrowers
versus elastic fracture for peristaltic burrowers (figure 6)—
suggest habitat partitioning and different functional roles of
infauna based on sediment mechanics and body size.
This project was funded by NSF OCE grant no. 1029160. We thank
C. Hermans, P. Jumars, M. Koehl and M. Pruett for helpful discus-
sions; T. Tucker and A. Francoeur for helping with data analysis;
and S. Woodin and an anonymous reviewer for constructive
feed-back on the manuscript.
grain size
z
z
0
Armandia brevis swimming
sand fluidization
plastic grain rearrangement
plastic grain rearrangement
Armandia
brevis
e.g. sandfish lizard
e.g., sand lances, eels
burrow extension by fracture
depth, sediment compaction
granular
solid
elastic solid
fluid
mud sand
Figure 6. Scheme of different mechanisms of burrowing in idealized muds (left) and sands (right). Dotted line indicates a later time, and the differing mechanics of
the media are indicated. Plastic grain rearrangement by Armandia brevis (upper left) is consistent with descriptions of kinematics of non-slipping burrowing in sands
[7,8] (lower right), although in muds, aggregate grains have lower density and are deformable, suggesting that friction may be more important and gravity less
important than in sands. This solid granular response differs from sand fluidization (upper right), which has thus far only been observed in dry terrestrial sands by
lizards considerably larger than A. brevis [6]. Depth or sediment compaction is hypothesized to distinguish between the two mechanisms in each sediment type, but
more research is needed on the mechanical responses of natural sediments to burrowing behaviours on spatial scales corresponding to burrower morphologies. Many
natural sediments are heterogeneous mixtures of sand and mud and probably exhibit properties of both media. Lower left panel of burrow extension by fracture
adapted from Dorgan et al. [1] with permission.
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References
1. Dorgan KM, Jumars PA, Johnson BD, Boudreau B,
Landis E. 2005 Burrow extension by crack
propagation. Nature 433, 425. (doi:10.1038/
nature03232)
2. Dorgan KM, Arwade S, Jumars PA. 2007 Burrowing
in marine muds by crack propagation: kinematics
and forces. J. Exp. Biol. 210, 41984212. (doi:10.
1242/jeb.010371)
3. Che J, Dorgan KM. 2010 It’s tough to be small:
dependence of burrowing kinematics on body size.
J. Exp. Biol. 213, 1241 1250. (doi:10.1242/
jeb.038661)
4. Volkenborn N, Polerecky L, Wethey D. 2010
Oscillatory porewater bioadvection in marine
sediments induced by hydraulic activities of
Arenicola marina.Limnol. Oceanogr. 55,
1231 1247. (doi:10.4319/lo.2010.55.3.1231)
5. Gray J, Lissmann HW. 1964 The locomotion of
nematodes. J. Exp. Biol. 41, 135– 154.
6. Maladen RD, Ding Y, Li C, Goldman DI. 2009
Undulatory swimming in sand: subsurface
locomotion of the sandfish lizard.
Science 325, 314318. (doi:10.1126/science.
1172490)
7. Gidmark NJ, Strother J, Horton J, Summers AP,
Brainerd EL. 2011 Locomotory transition from water
to sand and its effects on undulatory kinematics in
sand lances (Ammodytidae). J. Exp. Biol. 214,
657664. (doi:10.1242/jeb.047068)
8. Herrel A, Choi H, Dumont E, de Schepper N,
Vanhooydonck B, Aerts P, Adriaens D. 2011
Burrowing and subsurface locomotion in
anguilliform fish: behavioral specializations
and mechanical constraints. J. Exp. Biol. 214,
1379 1385. (doi:10.1242/jeb.051185)
9. Dorgan KM, Jumars PA, Johnson BD, Boudreau
BP. 2006 Macrofaunal burrowing: the medium
is the message. Oceanogr. Mar. Biol. 44,
85 121.
10. Taylor G. 1952 Analysis of the swimming of long
and narrow animals. Proc. R. Soc. Lond. A
214, 158183. (doi:10.1098/rspa.1952.0159)
11. D’Aout K, Aerts P. 1999 A kinematic comparison of
forward and backward swimming in the eel
Anguilla anguilla.J. Exp. Biol. 202, 1511 1521.
12. Gillis GB. 1998 Environmental effects on undulatory
locomotion in the American eel Anguilla rostrata:
kinematics in water and on land. J. Exp. Biol. 201,
949 961.
13. Gray J, Hancock G. 1955 The propulsion of sea-
urchin spermatozoa. J. Exp. Biol. 32, 802– 814.
14. Jordan CE. 1998 Scale effects in the kinematics and
dynamics of swimming leeches. Can. J. Zool. 76,
18691877. (doi:10.1139/z98-131)
15. Paul C, Halanych KM, Tiedemann R, Bleidorn C.
2010 Molecules reject an opheliid affinity for
Travisia (Annelida). Syst. Biodivers. 8, 507 512.
(doi:10.1080/14772000.2010.517810)
16. Hermans CO. 1978 Metamorphosis in the opheliid
polychaete Armandia brevis.InSettlement and
metamorphosis of marine invertebrate larvae
(eds F-S Chia, ME Rice), pp. 113 126. Amsterdam,
The Netherlands: Elsevier.
17. Berri S, Boyle JH, Tassieri M, Hope IA, Cohen N. 2009
Forward locomotion of the nematode C. elegans is
achieved through modulation of a single gait. HFSP J.
3, 186193. (doi:10.2976/1.3082260)
18. Fang-YenC,WyartM,XieJ,KawaiR,KodgerT,Chen
S, Wen Q, Samuel A. 2010 Biomechanical analysis of
gait adaptation in the nematode Caenorhabditis
elegans.Proc. Natl Acad. Sci. USA 107,20323
20 328. (doi:10.1073/pnas.1003016107)
19. Guo ZV, Mahadevan L. 2008 Limbless undulatory
propulsion on land. Proc. Natl. Acad. Sci. USA 105,
31793184. (doi:10.1073/pnas.0705442105)
20. Langbein W, Leopold LB. 1966 River meanders-
theory of minimum variance. Washington, DC:
United States Government Printing Office.
21. Hirose S. 1993 Biologically inspired robots.
New York, NY: Oxford University Press.
22. Dorgan KM, Lefebvre S, Stillman J, Koehl MAR.
2011 Energetics of burrowing by the cirratulid
polychaete Cirriformia moorei.J. Exp. Biol. 214,
22022214. (doi:10.1242/jeb.054700)
23. Dorgan KM, Arwade S, Jumars PA. 2008 Worms as
wedges: effects of sediment mechanics on
burrowing behavior. J. Mar. Res. 66, 219– 254.
(doi:10.1357/002224008785837130)
24. Johnson BD, Barry MA, Boudreau BP, Jumars PA,
Dorgan KM. 2011 In situ tensile fracture toughness
of surficial cohesive marine sediments. Geo Mar.
Lett. 32, 39 48. (doi:10.1007/s00367-011-0243-1)
25. Barry MA, Johnson BD, Boudreau BP. 2012 A new
instrument for high-resolution in situ assessment of
Young’s modulus in shallow cohesive sediments.
Geo Mar. Lett. 32, 349 357. (doi:10.1007/s00367-
012-0277-z)
26. Clark RB, Hermans CO. 1976 Kinetics of swimming in
some smooth-bodied polychaetes. J. Zool. 178,
147– 159. (doi:10.1111/j.1469-7998.1976.tb06004.x)
27. Katz SL, Shadwick RE. 1998 Curvature of swimming
fish midlines as an index of muscle strain suggests
swimming muscle produces net positive work.
J. Theor. Biol. 193, 243 256. (doi:10.1006/jtbi.
1998.0696)
28. McMillen T, Holmes P. 2006 An elastic rod model
for anguilliform swimming. J. Math. Biol. 53,
843886. (doi:10.1007/s00285-006-0036-8)
29. Ayers J. 1989 Recovery of oscillator function
following spinal regeneration in the sea lamprey.
In Cellular and neuronal oscillators (ed. JW Jacklet),
pp. 371405. New York, NY: Marcel Dekker.
30. Jayne BC. 1988 Muscular mechanisms of snake
locomotion: an electromyographic study of lateral
undulation of the florida banded water snake
(Nerodia fasciata) and the yellow rat snake (Elaphe
obsoleta). J. Morphol. 197, 159 181. (doi:10.1002/
jmor.1051970204)
31. Webb J. 1973 The role of the notochord in forward
and reverse swimming and burrowing in the
amphioxus Branchiostoma lanceolatum.J. Zool.
Lond. 170, 325 338. (doi:10.1111/j.1469-7998.
1973.tb01381.x)
32. Silvester NR, Holwill MEJ. 1972 An analysis of
hypothetical flagellar waveforms. J. Theor. Biol. 35,
505523. (doi:10.1016/0022-5193(72)90148-8)
rspb.royalsocietypublishing.org Proc R Soc B 280: 20122948
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... Burrowing by undulation is used by several organisms that feature flexible, elongated bodies (Maladen et al., 2009;Dorgan et al., 2013;Gray and Lissmann, 1964;Gidmark et al., 2011;Ozkan-Aydin et al., 2021a;Herrel et al., 2011). With rapid undulation of their bodies, organisms like lizards (Maladen et al., 2009; Sharpe et al., 2015), snakes (Sharpe et al., 2015), worms (Ozkan-Aydin et al., 2021a), and fish (Atkinson et al., 1987), can push against and relocate environmental media around their bodies to generate propulsion. ...
... With rapid undulation of their bodies, organisms like lizards (Maladen et al., 2009; Sharpe et al., 2015), snakes (Sharpe et al., 2015), worms (Ozkan-Aydin et al., 2021a), and fish (Atkinson et al., 1987), can push against and relocate environmental media around their bodies to generate propulsion. Burrowing with undulation is possible in cohesionless and cohesive media, like dry sand or muddy ground (Maladen et al., 2009;Dorgan et al., 2013;Atkinson and Pullin, 1996). Animals typically match their undulation amplitude, wavelength, and frequency to the intended granular material (Rieser et al., 2024;Pierce et al., 2024). ...
... Burrowing methods are divided into submerging mechanisms (vertical axis) and subterranean locomotion mechanisms (horizontal axis). Information for burrowing methods are from the following sources: Dorgan et al. (2013), Che and Dorgan (2010a), Trueman (1975) Summary of the discussed grand challenges soft burrowing robots face in granular media. In this illustration, the robots are in an environment with cohesionless granular media, but the represented challenges apply for cohesive media as well. ...
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Robotic burrowing holds promise for applications in agriculture, resource extraction, and infrastructure development, but current approaches are ineffective, inefficient, or cause significant environmental disruption. In contrast, natural burrowers penetrate substrates with minimal disturbance, providing biomechanical principles that could inspire more efficient and sustainable mechanisms. A notable feature of many natural burrowers is their reliance on soft body compositions, raising the question of whether softness contributes to their burrowing success. This review explores the role of soft materials in biological burrowing and their implications for robotic design. We examine the mechanisms that soft-bodied organisms and soft robots employ for submerging and subterranean locomotion, focusing on how softness enhances efficiency and adaptability in granular media. We analyze the gaps between the capabilities of natural burrowers and soft robotic burrowers, identify grand challenges, and propose opportunities to enhance robotic burrowing performance. By bridging biological principles with engineering innovation, this review aims to inform the development of next-generation burrowing robots capable of operating with the efficiency and efficacy seen in nature.
... These counterintuitive observations have in the past been explained qualitatively by suggesting that the gel-like network provides much higher resistance in a direction normal to the swimmer than in the tangential direction (Magariyama and Kudo 2002;Nakamura et al. 2006). More recently, a series of experimental studies of locomotion in materials with a yield stress (Harman et al. 2012;Kudrolli and Ramirez 2019;Dorgan et al. 2013;Nazari et al. 2023) have observed the same feature: helical and undulatory swimmers are able to move much faster, relative to the speed of their swimming waveform, than in a Newtonian fluid, and, in particular, seem able to 'cut' through the ambient material, travelling almost along their axis. We return to consider these observations-which are rationalised by the modelling presented here-in the "Summary and discussion" section. ...
... Some similar qualitative agreement with theoretical predictions has been observed for undulation. Dorgan et al. (2013) studied undulatory motion of Armandia brevis worms in gelatin and showed with very clear overlain pictures that the worms burrowed essentially along their axis in this material; the same experiments in water showed a large amount of drift and much less forward motion. They also presented more quantitative measurements: they reported a scaled swimming speed W * = W s /c ≈ 0.97 in gelatin, roughly double the value reported in water, and measured the amplitude of the worms to be a/λ ≈ 0.18, again broadly consistent with the theoretical predictions in "Undulatory swimming" section for swimming at the wave speed (a/λ 0.12) or with maximum efficiency (a/λ ≈ 0.15). ...
... 5 In a separate study, Kudrolli and Ramirez (2019) investigated the locomotion of different worms (blackworms and earthworms) and found that they both moved with a mixed peristaltic and undulatory gait. They observed the same qualitative feature as Dorgan et al. (2013): overlain snapshots of the worms' locations reveal that they move almost along their axis when travelling through sediments (here a packing of hydrogel grains, which presumably has an effective bulk yield stress, as well as other additional rheological complexity). In contrast, the same worms in (Newtonian) water are unable to follow their axis and travel forwards at an appreciably lower speed. ...
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Locomotion at small scales in the absence of inertia is a classical and enduring research topic. Here, recent developments in the theory of such locomotion through a viscoplastic ambient fluid are reviewed and explored. The specific focus here applies to motion of cylindrical filamentary bodies that are long and thin, for which an asymptotic slender-body theory can be exploited. Details of this theory are summarised and then applied to describe different swimming waveforms: undulation, peristalsis, and helical motion. It is shown that, in general, strong force anisotropy close to the limit of axial cylindrical motion has a significant effect on locomotion in viscoplastic media, allowing for highly efficient motion in which the swimmer is able to ‘cut’ through the material following very closely the path of its own axis. Some qualitative comparison with experiments is presented, and future extensions and research directions are reviewed. Graphical abstract Deformation fields around cylinders moving at different angles to their axis through a yield stress fluid, showing (a) a low yield stress and (b) a high yield stress
... Biological organisms exhibit diverse features to optimize survival and navigate through disordered habitats [1][2][3][4][5][6][7][8]. Active filaments, from actin filaments in the cytoskeleton to cyanobacteria and earthworms, move through complex, crowded environments. ...
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We investigate the locomotion of thin, living T. Tubifex worms, which display active polymerlike behavior, within quasi-2D arrays of cylindrical pillars, examining varying spatial arrangements and densities. These active worms spread in crowded environments, with a dynamics dependent on both the concentration and arrangement of obstacles. In contrast to passive polymers, our results reveal that in disordered configurations, increasing the pillar density enhances the long-time diffusion of our active polymer-like worms, while we observe the opposite trend in ordered pillar arrays. We found that in disordered media, living worms reptate through available curvilinear tubes, whereas they become trapped within pores of ordered media. Intriguingly, we show that reducing the worm's activity significantly boosts its spread, enabling passive sorting of worms by activity level. Our experimental observations are corroborated through simulations of the tangentially driven polymer model with matched persistence length predicting the same trends.
... Despite this high resistance encountered in soil, various organisms have evolved to burrow effectively and efficiently using diverse locomotion strategies. These strategies include body undulation, as observed in sandfish lizards (Scincus scincus), worms (Armandia brevis), and the burrowing eel (Pisodonophis boro) (Maladen et al., 2009(Maladen et al., , 2011Herrel et al., 2011;Dorgan et al., 2013); dual-anchor and fluidization techniques utilized by razor clams (Trueman et al., 1966;Trueman, 1967Trueman, , 1966; peristaltic crawling typical of earthworms (Quillin, 2000;Calderón et al., 2016); and circumnutation movements seen in plant roots (Migliaccio et 2013; Taylor et al., 2021). The overarching principle behind these strategies is to either minimize penetration resistance or to enhance anchorage or propulsion force during the burrowing process, showcasing nature's varied approaches to overcoming the challenges of soil navigation. ...
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Soil-dwelling organisms have evolved diverse strategies for efficient subterranean movement. For example, the seeds of Erodium cicutarium and Pelargonium species employ continuous rotational motion for self-burial, while the angled worm lizard Agamodon angeliceps tunnels by oscillating its head around its trunk's axis. These rotational movements significantly reduce penetration resistance. This study presents comprehensive experiments investigating the effects of various factors on rotational penetration forces and energy consumption. Results reveal that force reduction follow an approximately hyperbolic decay with the tangential-to-axial velocity ratio (u). Penetrator geometry, particularly roundness and conical tip shape, is found to significantly influence reduction at low velocity ratios, whereas relative density and material type exhibit moderate impact. Reduction is also observed to increase with interfacial friction angle but decreases with confining pressure and depth. Energy consumption analysis shows that while penetration force-related energy decreases with u, total energy consumption increases due to rotational torque. For self-burrowing robot designs, lower velocity ratios are recommended to balance penetration force reduction and energy efficiency effectively.
... Neoichnological studies on the physicochemical controlling factors of well-known ichnofauna have led to the solidification of ichnology as a standard line of evidence for paleodepositional interpretations, with perhaps the most famous being the development of the Seilacherian Ichnofacies (Seilacher, 1967). Expanding these studies into the microscopic realm would lead to a better understanding of: 1) the distribution of various meiofaunal organisms in relation to redox boundaries; 2) how burrowing strategies change with pore water chemistry and sediment mechanics (e.g., Dorgan et al., 2005Dorgan et al., , 2007Dorgan et al., , 2013Dorgan et al., , 2016Dorgan, 2015); and 3) the preservation potential of these structures (e.g., . Concerning reservoir analysis, future investigations of microbioturbation within organic-rich mudstones may evaluate how microscopic burrows alter the reservoir properties of unconventional reservoirs. ...
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Conventionally, geologists have regarded mudstones as deposits formed through suspension settling in environments located at the terminus of sediment transport pathways, with the sediment sourced from a mix of detrital inputs into the basin and in situ production within the basin. However, mudstones are sedimentologically enigmatic as they are characterized by intricate small-scale features. Analysing mudstones with the typical techniques used for coarse grained siliciclastics does a disservice to the intricacies of these deposits. Grains, pores, and depositional fabrics within these rocks are not visible in hand sample, and often not even at the petrographic scale. Study of these features, at appropriate scales, can generate valuable insights into the physical and chemical conditions of their deposition. Along with analytical techniques, the conventionally held interpretations of these rocks are out of date. New insights into the origins and composition of grain components reveal significant variability, indicating these deposits are much more complex than traditionally understood. As a result, historical nomenclature and interpretation paradigms have undergone significant revision. However, there is still more research needed to fully address the challenges of mudstone description, classification and interpretation. This paper presents digestible discussions of changes in mudstone paradigms, the most effective practices consistent with modern understandings of mudstones, and considers areas that merit further consideration. Ideas presented herein are aimed at all those interested in mudstones, but is primarily meant for those new to the challenge of conducting mudstone analyses. Herein we recognize several preferred practices that have gained consensus in the literature, these include: (1) clearly defining common historical terms such as ‘clay’, ‘silt’, ‘bed’, and ‘shale’ depending on modern chosen usage; (2) outlining the transportational (i.e., functional) grain size of the deposit, as many constituents may be transported as amalgamated clasts; (3) clearly defining if reported mudstone composition is based on transported or apparent grain size (i.e., individual grain measurements); (4) thin section preparation methods and their integration with other complementary analytical techniques. As well, we discuss: (1) the use of both petrographic trace fossil analysis and microfacies analysis; (2) complex depositional mechanisms, beyond suspension settling, that lead to the accumulation of fine-grained deposits; and, (3) the interaction of several variables involved in accumulating organic-rich deposits. Ultimately, when embarking on mudstone analysis, one must first decide what question they are trying to answer. This will dictate the approach used, and if the focus is on the intricacies of grain size, composition, or depositional fabric.
... Here, we consider the resistive forces based on a viscous response of a Newtonian medium, as opposed to models based on Coulomb friction [23], such that the local force depends linearly on the velocity of the body section. This approach is known to be widely applicable across all scales in highly dissipative systems [19,20,[24][25][26][27], with great accuracy for small deformations, that may decrease when non-local effects play an important role. ...
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Dissipative environments are ubiquitous in nature, from microscopic swimmers in low-Reynolds-number fluids to macroscopic animals in frictional media. In this study, we consider a mathematical model of a slender elastic locomotor with an internal rhythmic neural pattern generator to examine various undulatory locomotion such as Caenorhabditis elegans swimming and crawling behaviours. By using local mechanical load as mechanosensory feedback, we have found that undulatory locomotion robustly emerges in different rheological media. This progressive behaviour is then characterized as a global attractor through dynamical systems analysis with a Poincaré section. Furthermore, by controlling the mechanosensation, we were able to design the dynamical systems to manoeuvre with progressive, reverse and turning motions as well as apparently random, complex behaviours, reminiscent of those experimentally observed in C. elegans. The mechanisms found in this study, together with our dynamical systems methodology, are useful for deciphering complex animal adaptive behaviours and designing robots capable of locomotion in a wide range of dissipative environments.
... Different organisms have developed adaptations that aid their needs in different soil environments. Some animals use undulating body motions to burrow, such as the polychaetes (Dorgan et al 2013), oigochaetes (Kudrolli and Ramirez 2019), sandfish lizards (Maladen et al 2009), and sand lances (Gidmark et al 2011). Razor clams use a dual anchor strategy (Trueman 1967, Winter andHosoi 2011) and marine and earth worms use peristalsis (Dorgan 2018). ...
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Probes that penetrate soil are used in fields such as geotechnical engineering, agriculture, and ecology to classify soils and characterize their properties in situ. Conventional tools such as the Cone Penetration Test (CPT) often face challenges due to the lack of reaction force needed to penetrate stiff or dense soil layers, necessitating the use of large drill rigs. This paper investigates more efficient means of penetrating soil by taking inspiration from a plant-root motion known as circumnutation. Experimental penetration tests on sands are performed with circumnutation-inspired (CI) probes that advance at a constant vertical velocity ( v) while simultaneously rotating at a constant angular velocity ( ω). These probes have bent tips with a given bent angle ( α) and bent length ( L1). The variation of the mobilized vertical force ( Fz), torque ( Tz.), and the mechanical work components with the ratio of tangential to vertical velocity (ωR/ν, where R is the distance of the tip of the probe from the vertical axis of rotation) is investigated along with the effects of probe geometry, vertical velocity, and soil relative density ( DR). The results show that the soil penetration resistance does not vary with v, but it increases as α, L1, and DR are increased. Fz decays exponentially with increasing ωR/v, Tz initially increases and then plateaus, while total work ( WT) shows little magnitude changes initially but later increases monotonically. The mechanisms leading to these trends are identified as the changes in the probe projected areas and mobilized normal stresses due to differences in probe geometry and the effects of ωR/v on the resultant force direction and soil disturbance. The results show that CI penetration within a specific range of ωR/v leads to small increases in WT (i.e., ⩽25%), yet mobilizes Fz magnitudes that are 50%–80% lower than that mobilized during non-rotational penetration (i.e., CPT). This indicates that CI penetration can be adopted for in situ characterization or sensor placement with smaller vertical forces, allowing for use of lighter rigs.
Article
Bio-inspired probes have emerged as a promising solution for in-situ site characterisation, particularly in challenging terrains and extraterrestrial exploration. This study presents a viable and computationally efficient Material Point Method (MPM) framework for studying Bio-Inspired Cone Pressuremeter (BICP) probe mechanism. With its inherent advantage of particle and continuum frameworks, MPM allows seamless simulation of multi-staged BICP probe propulsion that involves large deformation. A novel implementation strategy was developed for this study to simulate the complex movement of the BICP probe in three sequential stages, including penetration, pressuremeter module expansion, and tip advancement. Sensitivity analysis was conducted to achieve an objective solution and determine the optimum mesh size and mass scaling factor for the BICP probe within the realms of current state-of-the-art MPM formulation. Furthermore, investigations were performed on the established MPM framework to study the influence of probe geometry, material state, and layered soil strata. The findings reveal that in probes with longer pressuremeter modules, larger zone of stress relaxation was observed around the cone tip during module expansion stage than their shorter or double-module counterparts. Meanwhile, the BICP probe’s response during all stages in different material states corroborates its sensitivity to the soil’s mechanical properties. Although the layered strata significantly influenced the BICP probe’s response during the penetration and module expansion stages, it had minimal impact during the tip advancement stage.
Article
This paper presents a novel locomotion soft robot that exploits the properties of auxetic structures to achieve bio-inspired undulatory locomotion. To reduce the dependence on computation and control strategies, we propose to develop a soft structure using a combination of positive Poisson’s ratio lattice structure and negative Poisson’s ratio lattice structure that creates a dorsoventral undulating wave pattern under compressive load. This is combined with a laterally undulating gait pattern exhibited by giant salamanders. The soft structure is actuated with nylon cables attached to servo motors, mimicking muscles. We use finite element analysis (FEA) methods to accurately model the soft structure’s deflection pattern, which is then used to create a control strategy for the robot. We develop a mathematical model and a subsequent gait pattern based on the sequential actuation of the nylon cables. The gait was experimentally tested and further improved with closed-loop error compensation. The research proves linear locomotion is possible through the proposed design with the lowest computational requirements.
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The extended phenotype of helical burrowing behavior in animals has evolved independently many times since the Cambrian explosion (~540 million years ago [MYA]). A number of hypotheses have been proposed to explain the evolution of helical burrowing in certain taxa, but no study has searched for a general explanation encompassing all taxa. We reviewed helical burrowing in both extant and extinct animals and from the trace fossil record and compiled 10 hypotheses for why animals construct helical burrows, including our own ideas. Of these, six are post‐construction hypotheses—benefits to the creator or offspring, realized after burrow construction—and four are construction hypotheses reflecting direct benefits to the creator during construction. We examine the fit of these hypotheses to a total of 21 extant taxa and ichnotaxa representing 59–184 possible species. Only two hypotheses, antipredator and biomechanical advantage, cannot be rejected for any species (possible in 100% of taxa), but six of the hypotheses cannot be rejected for most species (possible in 86%–100% of taxa): microclimate buffer, reduced falling sediment (soil), anticrowding, and vertical patch. Four of these six are construction hypotheses, raising the possibility that helical burrowing may have evolved without providing post‐construction benefits. Our analysis shows that increased drainage, deposit feeding, microbial farming, and offspring escape cannot explain helical burrowing behavior in the majority of taxa (5%–48%). Overall, the evidence does not support a general explanation for the evolution of helical burrowing in animals. The function and evolution of the helix as an extended phenotype seems to provide different advantages for different taxa in different environments under different physicochemical controls (some traces/tracemakers are discussed in more detail due to their association with body fossils and well‐constrained physicochemical parameters). Although direct tests of many of the hypotheses would be difficult, we nevertheless offer ways to test some of the hypotheses for selected taxa.
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Burrowing by benthic infauna mixes both sediment grains and interstitial fluids, affecting sedimentary redox conditions and determining fates of organic matter and pollutants. Explicit, quantitative analyses of material properties of sediments, however, have been applied only recently to understand mechanisms of burrowing. Muds are elastic solids that fracture under small tensile forces exerted by burrowers, and are dominated by adhesive forces between sediment grains and the surrounding mucopolymeric gel and (or) by cohesion of this gel. By contrast, in clean sands behaving as granular materials, gravity is a much more significant force holding grains together than is adhesion or cohesion. Burrowers in muds have diverse structures that act as wedges to propagate cracks and elongate their burrows. In sands, increased rugosity on a small, and lique-faction on a larger scale, facilitate displacement of the grains that carry compressive forces along distinct force chains or arches. The classic dual-anchor system described for burrowers is reinter-preted as having several additional functions. The characteristic dilations or expansions function primarily as wedges that exert lateral tensile forces to propagate cracks forward, secondarily as double O-ring seals holding fluid pressure in the advancing burrow (maintaining tensile stresses needed to open a crack), and thirdly as anchors (to pull the shell along in bivalves in particular). Burrowing bivalves are wedges. In the case of burrowing gammarid amphipods, the dorsal exo-skeleton mirrors the shape of half a sedimentary bubble and constitutes a wedge. A great many anatomical features of burrowers can now be understood analogously. The identification of the mechanisms of burrowing by crack propagation suggests that a substantial revision of the previously described feeding guilds of polychaetes is required.
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Slender-bodied organisms swimming with whole-body undulations exhibit what appears to be a high degree of kinematic parameter conservation, which is independent of body size. However, organisms of very different sizes swim in fundamentally different physical realms, owing to the relative scaling of viscous and inertial fluid stresses as a function of size and speed. In light of the size-dependent fluid forces, the kinematic constancy suggests three hypotheses: (1) swimming organisms adopt a single "ideal" swimming mode requiring the modification of muscle forces or motor patterns through ontogeny, (2) swimming kinematics are determined predominantly by the passive mechanical interaction of the body and the fluid, resulting in a single swimming mode independent of absolute body size, or (3) while undulatory swimming kinematics may be similar between organisms, there are important size-dependent kinematic differences. In this study, I address this issue by examining the swimming kinematics and dynamics of the medicinal leech Hirudo medicinalis L. as a function of body size. Over a 5-fold increase in body length, the relative amplitude of body undulations during swimming did not change; however, swimming speed, propulsive wave speed, and propulsive wave frequency all decreased, while propulsive wave number increased slightly, strongly supporting hypothesis 2. To determine the source of the observed size-dependent swimming kinematics, I manipulated the dynamic viscosity of the organism's fluid environment to alter the constraints placed on swimming behavior by the physical surroundings. In the elevated-viscosity treatment, all kinematic parameters changed in the opposite direction to that predicted by hypothesis 2, rejecting both the idea that swimming kinematics are simply determined by passive mechanical interactions and that leeches have a target swimming mode under active control.
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This paper describes a new, miniature, instrumented flat dilatometer (mIDMT) designed to assess variations in nearly continuous compressive stress–strain behaviour with depth in shallow cohesive sediments. The instrument was tested both in situ in the Bay of Fundy, Nova Scotia, Canada, and in cored samples from Willapa Bay, Washington, USA. Comparisons between probe and laboratory uniaxial assessments for other elastic materials—gelatine and foam rubber specifically—show strong agreement over the range of strains induced in the experiments. Observed values of Young's modulus (E) for the gelatine and ethylene-vinyl acetate foam ranged from 6–343 kPa. Sediment stress–strain curves were distinctly linear for the over-consolidated fine-grained sediments of the Minas Basin, and values of E were found to increase with depth from near zero to 500–1,300 kPa at 20 cm depth. At the Willapa site, the sandy tidal flat sediments also behave elastically but E tended to increase more strongly with depth than for sediments from the Minas Basin. Young's modulus was inversely correlated to porosity at all sites tested, and linearly related to shear strength in the Minas Basin. The newly designed instrument has much finer resolution than for other, similar methods of determining E in situ, and it provides data at a resolution sufficient to assess small-scale processes such as gas bubble growth and infaunal locomotion, for which elastic constants are needed for modelling and prediction.
Chapter
So ubiquitous are curves in rivers and so common are smooth and regular meander forms that they have attracted the interest of investigators from many disciplines. Also, investigations of the physical characteristics of glaciers and oceans have led to the recognition that analogous forms occur in melt-water channels developed on glaciers and even in the currents of the Gulf Stream. The striking similarity in physical form of curves in these various settings is the result of certain geometric proportions apparently common to all, that is, a nearly constant ratio of radius of curvature to meander length and of radius of curvature to channel width (Leopold and Wolman, 1960, p. 774). This leads to visual similarity regardless of scale.
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Historically, the study of swimming eels (genus Anguilla) has been integral to our understanding of the mechanics and muscle activity patterns used by fish to propel themselves in the aquatic environment. However, no quantitative kinematic analysis has been reported for these animals. Additionally, eels are known to make transient terrestrial excursions, and in the past it has been presumed (but never tested) that the patterns of undulatory movement used terrestrially are similar to those used during swimming. In this study, high-speed video was used to characterize the kinematic patterns of undulatory locomotion in water and on land in the American eel Anguilla rostrata. During swimming, eels show a nonlinear increase in the amplitude of lateral undulations along their bodies, reaching an average maximum of 0.08L, where L is total length, at the tip of the tail. However, in contrast to previous observations, the most anterior regions of their bodies do not undergo significant undulation. In addition, a temporal lag (typically 10-15% of an undulatory cycle) exists between maximal flexion and displacement at any given longitudinal position. Swimming speed does not have a consistent effect on this lag or on the stride length (distance moved per tailbeat) of the animal. Speed does have subtle (although statistically insignificant) effects on the patterns of undulatory amplitude and intervertebral flexion along the body. On land, eels also use lateral undulations to propel themselves; however, their entire bodies are typically bent into waves, and the undulatory amplitude at all body positions is significantly greater than during swimming at equivalent speeds. The temporal lag between flexion and displacement seen during swimming is not present during terrestrial locomotion. While eels cannot move forwards as quickly on land as they do in water, they do increase locomotor speed with increasing tailbeat frequency. The clear kinematic distinctions present between aquatic and terrestrial locomotor sequences suggest that eels might be using different axial muscle activity patterns to locomote in the different environments.
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We employed real-time pressure recording and high temporal resolution two-dimensional oxygen imaging to characterize the porewater bioadvection related to hydraulic activities of Arenicola marina, a widespread representative of benthic macrofauna. Behavior-specific positive and negative pressure oscillations and hydraulic pulses resulted in bidirectional porewater flow and highly dynamic redox oscillations on the scale of minutes. Pumping of water by the worm into its blind-ending burrow pressurized the sediment and caused sediment oxygenation at depth and the exit of anoxic porewater into the overlying water. The sediment volume that was affected by bioadvective transport of oxygen and the porewater flow patterns varied strongly among sediment types. In low-permeability sediments, localized plumes of anoxic porewater ascended from the sediment, presumably through sedimentary cracks, while porewater flowed evenly through highly permeable sediments. Hydraulic behaviors that moved water out through the open tail shaft caused a reduction of porewater pressures below the hydrostatic baseline which resulted in the collapse of plumes and enhanced oxygen penetration into the surficial sediments. Porewater bioadvection and the related perfusing and oscillatory phenomena will affect a variety of biogeochemical and ecological processes, including organic matter mineralization, benthic recruitment, and prey localization. We suggest that bidirectional porewater bioadvection and the associated transient geochemical conditions are prevalent features of biogenically active sediments.
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The movement of any short length of the tail of a spermatozoon of Psammechinus miliaris and the characteristic changes which it undergoes during each cycle of its displacement through the water can be described in terms of the form and speed of propagation of the bending waves which travel along the tail (Gray, 1953, 1955); the form of the wave depends on the maximum extent of bending reached by each element and on the difference in phase between adjacent elements. The object of this paper is to consider the forces exerted on the tail as it moves relative to the surrounding medium and to relate the propulsive speed of the whole spermatozoon to the form and speed of propagation of the bending waves generated by the tail. The mathematical theory of the propulsive properties of thin undulating filaments has recently been considered by Taylor (1951, 1952) and by Hancock (1953); the present study utilizes and extends their findings but approaches the problem from a somewhat different angle. resistance, and consequently the transverse displacement (Vy) elicits reactions tangential and normal to the surface of the element. The latter force (δNy) has a component(δNysinθ) acting forward along the axis (xx ′) of propulsion; it is this component which counteracts the retarding effect of all the forces acting tangentially to the surface.
Article
Slender-bodied organisms swimming with whole-body undulations exhibit what appears to be a high degree of kinematic parameter conservation, which is independent of body size. However, organisms of very different sizes swim in fundamentally different physical realms, owing to the relative scaling of viscous and inertial fluid stresses as a function of size and speed. In Light of the size-dependent fluid forces, the kinematic constancy suggests three hypotheses: (1) swimming organisms adopt a single "ideal" swimming mode requiring the modification of muscle forces or motor patterns through ontogeny, (2) swimming kinematics are determined predominantly by the passive mechanical interaction of the body and the fluid, resulting in a single swimming mode independent of absolute body size, or (3) while undulatory swimming kinematics may be similar between organisms, there are important size-dependent kinematic differences. In this study, I address this issue by examining the swimming kinematics and dynamics of the medicinal leech Hirudo medicinalis L. as a function of body size. Over a 5-fold increase in body length, the relative amplitude of body undulations during swimming did not change; however, swimming speed, propulsive wave speed, and propulsive wave frequency all decreased, while propulsive wave number increased slightly, strongly supporting hypothesis 2. To determine the source of the observed size-dependent swimming kinematics, I manipulated the dynamic viscosity of the organism's fluid environment to alter the constraints placed on swimming behavior by the physical surroundings. In the elevated-viscosity treatment, all kinematic parameters changed in the opposite direction to that predicted by hypothesis 2, rejecting both the idea that swimming kinematics are simply determined by passive mechanical interactions and that leeches have a target swimming mode under active control.
Article
The swimming of long animals like snakes, eels and marine worms is idealized by considering the equilibrium of a flexible cylinder immersed in water when waves of bending of constant amplitude travel down it at constant speed. The force of each element of the cylinder is assumed to be the same as that which would act on a corresponding element of a long straight cylinder moving at the same speed and inclination to the direction of motion. Relevant aerodynamic data for smooth cylinders are first generalized to make them applicable over a wide range of speed and cylinder diameter. The formulae so obtained are applied to the idealized animal and a connexion established between B/lambda , V / U and R1. Here B and lambda are the amplitude and wave-length, V the velocity attained when the wave is propagated with velocity U, R1 is the Reynolds number Udrho /mu , where d is the diameter of the cylinder, rho and mu are the density and viscosity of water. The results of calculation are compared with James Gray's photographs of a swimming snake and a leech. The amplitude of the waves which produce the greatest forward speed for a given output of energy is calculated and found, in the case of the snake, to be very close to that revealed by photographs. Similar calculations using force formulae applicable to rough cylinders yield results which differ from those for smooth ones in that when the roughness is sufficiently great and has a certain directional character propulsion can be achieved by a wave of bending which is propagated forward instead of backward. Gray's photographs of a marine worm show that this remarkable method of propulsion does in fact occur in the animal world.
Article
The phylogenetic position of the polychaete genus Travisia within Annelida is a matter of ongoing debate. Travisia is usually placed within Opheliidae, but morphological similarities with Scalibregmatidae, such as a rugose epidermis, are obvious. To further examine placement of this enigmatic group, we examined 28 annelid species from a range of families, but with special emphasis on Scalibregmatidae and Opheliidae. Our data set consisted of four genes: 16S rDNA, 18S rDNA, 28S rDNA and Histone 3. By combining genes and conducting Maximum Likelihood, Maximum Parsimony and Bayesian analysis, our results strongly support a sister-group relationship of Travisia and Scalibregmatidae. None of the phylogenetic analyses clustered Travisia with or within Opheliidae and such placements are also significantly rejected by hypothesis testing. Moreover, we obtained new insights on relationships within Opheliidae and Scalibregmatidae. Within Opheliidae, the traditional classification into Opheliinae and Ophelininae received strong support.