First published online November 19, 2007
Journal of Experimental Biology 210, 4198-4212 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.010371
Burrowing in marine muds by crack propagation: kinematics and forces
Kelly M. Dorgan1,*,
Sanjay R. Arwade2 and
Peter A. Jumars1
1 Darling Marine Center, University of Maine, 193 Clark's Cove Road,
Walpole, ME 04573, USA
2 Department of Civil and Environmental Engineering, University of
Massachusetts, Amherst, 223 Marston Hall, 130 Natural Resources Road, Amherst,
MA 01003, USA
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Fig. 1. Diagram of experimental setup. Two light tables are shown as yellow blocks
on opposite sides of the aquarium (light yellow block in center) from the
cameras. Camera 1 recorded the lateral view of the worm and camera 2 recorded
the dorsal or ventral view. Schemes of the 2-D views of each camera are shown
with corresponding axes, and the orientation of the worm in the crack is shown
in the aquarium. Between the light table and the aquarium are a colored filter
and a right-handed, circular polarizing filter with a left-handed, circular
polarizing filter and another color filter between the aquarium and the
camera. The circular polarizing filter is shown here as a linear polarizing
filter with a quarter-wave retarder; actual filters combine the two
components. Filters on the far side of the cameras completely covered the
light tables with no other light passing through, and the filters on the
camera side were attached to the lenses. Cameras were run from separate
computers with LabView software. The defined coordinate system is used in all
relevant figures.
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Fig. 2. Calibration curve using three different test tubes (with water added) of
known mass. Test tubes have the same diameter (0.01135 m), but were cut to
different heights to reduce their mass, 0.52 g (*), 1.1 g (o), and
2.6 g (x). A linear relationship exists for the larger test tube (x), but does
not extend into the range of pixel areas around worms, indicated by the
horizontal dotted lines. We instead used a second-order polynomial fit through
the medium-sized test tube (o) that covered the range of observed pixel areas
around worms (r2=0.997, N=11).
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Fig. 3. Video frame of pharynx eversion and corresponding thresholded image (frames
from camera 1, the y–z plane). In the thresholded
image, the small upper patches of light are posterior of the pharynx and
result from body stress (B). The lower patches that join in the middle
indicate tensile stresses (T) at the crack tip. Tensile stresses are shown in
blue in the image of stress contours resulting from modeled stresses along the
crack tip (right frame). Central patches in the thresholded image indicate
compressive stress (C) from the force of pharynx eversion, and are the only
pixels included in force calculations. Compressive stress is shown in red in
the modeled image. Scale bar, 0.005 m.
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Fig. 4. Loading–unloading curves for (A) muddy sediment and (B) gelatin
showing linear elastic behavior for both materials. Force was measured as a
0.0254-m diameter cylindrical probe was lowered onto the surface of the
material using a Vitrodyne-V1000 Universal Tester. Both materials exhibit
linear elastic loading, although sediment shows a low resilience and a small
plastic deformation after the first loading, visible as an approximately
0.5x10–3 m shift to the right from the initial loading
curve to the second loading curve (also visible as a slight compression of the
surface sediment, not shown). Subsequent loadings show minimal
deformation.
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Fig. 5. (Ai) Plot of worm movement, crack extension and pharynx thickness over time
for a representative worm. The position of the anterior end of the worm's head
and the tip of the crack and the thickness of the pharynx (lateral view) were
recorded from video frames at 3.75 frames s–1. In one
burrowing cycle, the worm begins to move forward, extending the crack, then
everts its pharynx. Before the pharynx is fully everted, the worm's head and
the crack tip reach the most anterior point, then the anterior end of the
pharynx moves back as the pharynx everts completely. The worm then moves
laterally within the crack (not shown) with little anterior movement before
beginning to move forward again to repeat the cycle. (Aii) One pharynx
eversion shown in greater detail. (B) Sequence of images from one pharynx
eversion as indicated by corresponding labels on Aii. For each row, the left
image is a lateral view (from camera 1, y–z plane)
showing the stress fields, the center image is a thresholded copy of the left
image, and the right image is the corresponding image from the dorsal view
(from camera 2, x–z plane). Because the cameras were
not run from the same computer, the images from the two cameras are nearly,
but not perfectly, synchronous. Scale bar, 0.005 m.
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Fig. 6. Franc2d lateral model, a 2-D model of the y–z plane
of the 3-D experiments. (A) The displaced finite element mesh from model 2
(378 Pa pharynx stress, 92 Pa, linearly decreasing to 60 Pa, body stress) is
shown as solid lines, and the original geometry with the crack is shown by
dotted lines. (B) The displaced mesh shows the shape of the worm in gelatin
and the head region (boxed area in A) enlarged below. The surface of the
displaced mesh is slightly raised, a result of the displacements along the
crack walls. Scale bar, 5 mm; magnification factor, 1.
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Fig. 8. Worm thickness for different models and real worms (black). All five models
have an applied pharynx stress of 378 Pa. Model 1 (blue dotted line) has
constant body stress of 92 Pa, model 2 (green dotted line) has body stress
linearly decreasing from 92 to 60 Pa, model 3 (red dotted line) has constant
body stress corrected for the lateral constraints to 64 Pa, model 4 (cyan
dotted line) has body stress corrected for the lateral constraints to 64
linearly decreasing to 42 Pa. In models 1–4, pharynx stress is applied
from the crack tip to 0.00725 m, resulting in a pharynx displacement equal to
the average pharynx length observed, 0.00667 m. Model 5 (green broken line)
has the same stresses as model 2, but the pharynx stress extends to 0.00839 m,
resulting in pharynx displacement equal to the maximum pharynx length
observed, 0.00779 m (Table
3).
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Fig. 9. Worm thickness for model 2 in gelatin and in natural sediment with
calculated stresses. The solid line is the worm thickness in gelatin (green
dotted line in Fig. 8), with
pharynx stress of 378 Pa and body stress 92 Pa, linearly decreasing to 60 Pa.
The stresses were increased in natural sediments by
Esed/Egel to a pharynx stress of 5305
Pa and body stress 1291 Pa, linearly decreasing to 842 Pa (dotted line).
Because the change in Poisson's ratio was not considered, the calculated
stresses resulted in higher displacements. Stresses were reduced by the ratio
of the displacements in gelatin to the displacements from the first stresses
applied in sediment, to pharynx stress 4408 Pa and body stress 1073 Pa,
linearly decreasing to 700 Pa (dash-dotted line).
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Fig. A1. (A) Pixel area resulting from stresses applied to the surface of gelatin by
test tubes and flat-bottomed cylinders of different diameters
(r2=0.11; N=44). (B) Pixel area as a function of
force instead of stress for the data in A (r2=0.94;
N=44). (C) Results of finite element model of calibration showing
thresholded pixel area as a function of stress (500, 750 and 1000 Pa) for
three different modeled radii (0.005, 0.010 and 0.015 m)
(r2=0.16; N=9). (D) Pixel area as a function of
force instead of stress for the data in C (r2=0.996;
N=9).
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Fig. A2. Balloon pressure transducer setup. The balloon is stretched to a 2-D disk
using a wire and is glued to the end of a tube (grey line), which is connected
to a Luer lock adapter (thicker grey line). The adapter is connected to a
syringe filled with water and a pressure transducer. The balloon, syringe and
pressure transducer were held in the horizontal x–y
plane, as shown in the top-view scheme, to minimize variations in pressure
(P) with water height.
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Fig. A3. (A) Calibration curve comparing weight exerted by a test tube on the
surface to pressure exerted by a balloon inflated in a crack in the gelatin.
The areas of the primary compression fringes on each side of the balloon were
measured separately (* and +), and regressions are shown as dotted
lines (r2=0.65 and 0.62; N=13). The regression
through the test-tube data (x; solid line; r2=0.998;
N=6) was extended through the ranges of the balloon data. (B) Stress
contours from franc2d models of stress on the surface (Bi), along the crack
starting at the tip (Bii), and along the crack starting 0.002 m up from the
tip (Biii). The models are 2-D representations of the y–z plane
in the 3-D experiments. Stress (500 Pa) is exerted along 0.0089 m in each
model and is indicated by vectors. In each image, red and white colors
indicate compressive stress; blue is tensile stress. Images were thresholded
to the light red/white boundary (scale bar=0.005 m).
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© The Company of Biologists Ltd 2007
max) and half-length of crack
(a) indicated. Scale bar, 1 mm; magnification factor, 1. (B) Change
in maximum displacement in the anterior model as the crack is extended
a at each tip. The dotted line at 0.0047 m indicates the
measured distance from the worm's body to the lateral edge of the crack (data
not shown).