211
J. Exp. Mar. Biol. Ecoi., 1989, Vol. 127, pp. 211-228
Elsevier
JEM 01240
Fracture mechanics and the survival of wave-swept
macroalgae
Mark Denny ‘, Virginia Brown ‘, Emily cannon
Alan Miller 3
‘, George
Kraemer2
and
Department of Biological Sciences, Stanford University, ‘Hopkins Marine Station, Pacific Grove, Cal$ornia,
U.S.A.; ‘Depa~ment of Biolqgy, Universityof Catgotxia at Los Angeles, Ca&$ornia, U.S.A.; 3Department of
Biology, Calgornia State Universir.v,Long Beach, Cali$omia, U.S.A.
(Received 16 August 1988; revision received 20 January 1989; accepted 15 February 1989)
Abstract:
Wave-swept macroalgae are constructed from materials which are brittle compared to most
biological structural materials. As a consequence, these plants are susceptible to breakage when they receive
a sharp-ended surface injury. Field tests indicate that injuries such as razor cuts can result in rapid mortality
even under benign surf conditions. However, because algal blade material is highly extensible, it allows for
a rounding of the apex of a surface flaw, resulting in a substantial lowering of the stress concentration in
the material. For all but the most sharply ended initial cracks, the rounding of the flaw is sufficient to make
the local stress concentration the limiting factor in fracture. in this fashion macroalgal blades manage largely
to avoid the dire consequences of being constructed from brittle materials.
Key words: Fracture mechanics; Intertidal ecology; Zridaea; Macroalga; Survivorship; Wave exposure
On exposed coastlines, breaking waves are typically accompanied by water velocities
of 5-10 m ’ s- ’ and water accelerations
in excess of 100 m * SC* (Denny, 1985, 1987,
1988; Denny et al., 1985). This harsh flow regime can impose potentially fatal hydrodynamic forces on intertidal organisms, and plants and animals exhibit a variety of
evolved “strategies” for coping with the exigencies of wave-induced flow (for reviews,
see Koehl, 1982, 1984, 1986; Denny, 1988). Of particular interest here are the mechanical characteristics of nearshore macroalgae. Intertidal and shallow subtidal macroalgae
are characterized by materials that have a low stiffness and a low strength relative to
many other biological support tissues (e.g., wood, shell, bone, etc.), but have a very high
extensibility (Table I). Further, the mo~holo~ of many macroalgae (narrow stipes and
long, flattened blades), coupled with the low stiffness of their materials, renders them
extremely flexible. Koehl (1982, 1984, 1986) has proposed that this combination of
extensibility and flexibility allows these plants to survive on wave-swept shores. When
a macroalga encounters wave-induced water velocities, it deforms in the direction of
Correspondenceaddress: M. Denny, Hopkins Marine Station, Department of Biologicaf Sciences,
Stanford University, Pacific Grove, CA 93950, U.S.A.
0022-0981/89/~03.50
0
1989 Efsevier Science Publishers B.V. (Biomedical Division)
M. DENNYET AL.
212
TABLE I
Typicalmechanicalpropertiesof biologicalstructuralmaterialsin comparisonto propertiesof macroalgal
stipesand blades.Algaldata from TableIII, other data from Denny(1988).
Material
Insect cuticle
Coral skeleton
Musselshell
Tensilestrength(MN. m - *)
9.5
40
56
Stiffness (MN. m - *)
Extensibility(%)
9500
60000
31000
l-2
0.03
0.18
1
1
Bone
190
18000
Wood
Algalstipes
and blades
115
12000
0.7-10
9.6-36
19-35
water motion (assuming a more streamlined shape) and “goes with the flow”, thereby
reducing the maximum relative velocity and acceleration with which it must cope. Both
of these attributes reduce the force imposed on the alga, presumably keeping it below
that which can be resisted by the relatively weak material (Koehl & Wainwright, 1977;
Koehl, 1982, 1984, 1986; Denny, 1988).
However, the reduction in imposed force that is afforded by flexibility and extensibility
cannot alone explain the ability of wave-swept algae to survive in their physically
stressful environment. Consider the following. The presence of a sharp-ended flaw in
a material (e.g., a crack, surface nick, or grazing mark) acts to concentrate the applied
force per area (the stress) in the vicinity of the flaw (Fig. 1, see Boresi et al., 1980;
Gordon, 1976; Wainw~ght et al., 1976, for a discussion). As a consequence. the stress
caused by even a small force can be amplified locahy to exceed the breaking stress of
a material. For instance, an unflawed piece ofglass is quite strong - its breaking strength
of 1.1 . 10lo N * m - 2 exceeds that of steel (Gordon, 1976). Nonetheless, glass easily can
be broken. All one needs do is introduce a sharp-ended flaw with a diamond stylus, and
the resulting stress concentration ensures that a small applied force will locally exceed
the material’s breaking strength.
Although the presence of a local stress in excess of the breaking stress is necessary
for a material to fail, it is not in itself sufficient. As a crack propagates through a material,
new surface area is created in the vicinity of the crack tip as chemical bonds are broken.
When a material fails ~atastrophic~ly, the energy to create this surface area is drawn
from the potential energy stored elastically in the deformed material (the strain energy).
If the energy expended in creating new surface area (the work offracture, W) is greater
than the potential energy available from the deformed material, the material will not
spontaneously fracture even though the local stress exceeds the breaking stress. As a
consequence, materials with a high work of fracture are relatively immune to the action
of local stress concentrations and have a high “fracture toughness” (sensu W~nwri~t
et al., 1976). Conversely, materials with a low work of fracture are susceptible to the
action of local stress concentrations and are termed “brittle”. Classic brittle materials
MACROALGAL FRACTURE MECHANICS
213
FORCE
CROSS-SECTIONAL
AREA
CRACK
STRESS
Fig. 1. A schematic representation of manner in which stress trajectories are ahered by presence of a
sharp-ended crack, resulting in concentration of stress at crack tip.
such as glass and pottery have works of fracture of l-10 J * m - 2 (Gordon, 1976) and
are notoriously easy to fracture when flawed. Su~~singly, rubber, with a work of
fracture of x50 J .rn-* (Andrews, 1980), is also a brittle material. Its susceptibility to
catastrophic breakage is easily demonstrated - if a small flaw (a pin hole) is introduced
into an inflated balloon, the balloon pops.
In contrast, most biological structural materials have relatively high fracture
toughness. Mother of pear1 (the nacre layer of mo~usc~ shell) has a work of fracture
of 1650 J. m -2 for cracks propagating across the shell (Currey, 1977, 1980.) Bone has
an even larger work of fracture, z 10000 J * mm2 (Harris, 1980), and a similar value is
found for wood when cracked across the grain (Gordon, 1978). It was thus surprising
when Biedka et al. (1987) reported that the stipe material of Pterygophora cafzfornica, a
wave-swept macroalga, has a work offracture of only 824 J . m - ’ for fractures extending
across the stipe and an even lower W of 415 J - me2 for fractures extending along the
stipe. On the basis of these low works of fracture, it appears that the stipe of Pterygophora
is susceptible to breakage if there are any nicks or cracks present within the material.
This susceptibility is easily demonstrated both in Pterygophora and in other macroalgae.
The intact stipes of many kelps (e.g., Pterygophora, Laminaricz, Posteisia, Nereocystis,
~acr~ey~ti~) are difficult to break - they can be bent double or tied in knots without
fracturing. However, once a sharp-ended flaw is introduced into the stipe, the stipe is
easily snapped. Further, a cursory examination of wave-swept algae reveals that surface
214
M. DENNY ET AL.
flaws
are common: e.g., the grazing marks of sea urchins, gastropods or chitons, the
scrapes caused by abrasion, the holes left behind when spores are liberated. Two
questions are thereby raised: is a low work of fracture common among wave-swept
macroalgae? And, if so, how do these algae manage to survive despite the unavoidable
presence of surface flaws?
To answer these questions, we examine the fracture
mechanics of wave-swept macroalgae with a particular emphasis on Zridaeaflaccida
(Setchell et Gardner) Silva. Our findings suggest that the blades of macroalgae are
indeed characterized by materials having a low work of fracture, and that the presence
of sharp-ended surface flaws can result in breakage by ambient hydrodynamic forces.
However, the great extensibility and low stiffness of the blade material provide a
mechanism by which fracture can be avoided in the case of most naturally occurring
flaws.
MATERIALS
AND ~~ETHODS
PRIMARY TEST SPECIES
I.jluccida is a macroscopic red alga living attached to rocks in the mid to low
intertidal zones of exposed shores north of Point Conception, California. The growth
form of the g~etophytes
and tetrasporoph~es consists of a flexible, elongate bIade
connected to a short stipe that is fumly attached by a holdfast to the rocky substratum.
An individual plant may have several blades growing from the holdfast region (Abbott
& Hollenberg, 1976).
TENSILE TESTS
Blades or whole plants of I.flaccida were collected at Hopkins Marine Station,
Pacific Grove, California (HMS), and stored in flowing seawater until used (never
> 2 days). Tensile tests were performed on dumbbell-shaped strips cut from the blades
{Fig. 2). Strips of cardpaper (2 x 4 cm) were glued using cyanoacrylate “super glue” to
the ends of the strip to facilitate clamping, and the sample was installed in a tensometer.
Two fiducial marks were made on the center section of the sample using typists’
Fiducia I Mark
Cardpaper
t
2 cm
4
A
8 cm ,--
Fig. 2. A schematic representation of a tensile-test sample.
MACROALGAL
FRACTURE
215
MECHANICS
correcting fluid. A constant rate of deformation was then imposed on the sample and
the resulting tension was measured by a force transducer attached to the stationary
clamp. The instantaneous voltage output of the transducer (proportional to force) was
measured and displayed by a digital voltmeter. This value, along with the deformations
of the material, were recorded simultaneously using two video cameras, a screen-splitter
(RCA model TC1470A), and a video cassette recorder. Corresponding values of force
and deformation were subsequently obtained from a frame-by-frame analysis of the
Each deformation is expressed as true strain, E:
video tape.
E = In
instantaneous distance between marks
original distance between marks
’
(1)
The instantaneous crosssectional area is calculated by assuming that the blade material
is isovolumetric - probably a reasonable assumption given the high water content of the
material. Thus at any extension:
instantaneous crosssectional area =
initial crosssectional area
(2)
eE
Each force was divided by the instantaneous crosssectional area of the center portion
of the test sample to yield a value for tensile stress (0). The stiffness of the material at
any extension is expressed as its modulus, E = da/de.
The tensile strength of four other intertidal algae present at HMS, Laminariu dentigeru
Kjellm., Mustocurpuspupillutus KUtzing, Mustoculpus jurdinii (J. Ag.) West, and an Ulvu
species, were measured in a similar fashion.
MEASUREMENT
OF WORK
OF FRACTURE
The work of fracture was measured for the blades of 10 species of nearshore macroalgae and one subtidal angiosperm (Table II). Specimens were collected from attached
plants at HMS, and were kept in flowing sea water until tested. All tests were conducted
within 48 h of the blade’s collection. A rectangular sample was cut from each blade and
a sharp-ended crack introduced with a razor blade; the crack’s length was generally
parallel to the axis of the blade. The prepared sample was installed in a tensometer for
a “trouser tear ” test as described by Biedka et al. (1987) (Fig. 3a). The two “legs” of
the sample were pulled in opposite directions at a known, constant velocity, tearing the
sample. The majority of cracks propagated parallel to the blade axis even if the initial
cracks were transverse. The force required to extend the crack was continuously
measured. This force (N), multiplied by the velocity with which the crack is extended
(m . s _ ‘), is equal to th e rate at which energy is expended in creating new crack surface
area(W). Note that the velocity of crack propagation is halfthe rate at which the clamps
of the tensometer move apart (Fig. 3a). The rate at which new surface area is created
216
M. DENNY ET AL.
0
15
30
45
60
f5
SECONDS
Fig. 3. Determination of work of fracture. (A) Schematic representation of a test sample undergoing a
“trouser tear”. Note that as crack extends by x, ends of “legs” move apart by 2 x. (B) A typical record of
force required to fracture f.flaccidu blade material.
by crack propagation is equal to the product of twice the thickness of the blade (there
are two sides to the crack) and the velocity with which the crack is extended. Thus:
energY(J.m-2)
=
rate of energy expenditure (J * s - ‘)
rate of surface area production (m” . s- ‘)
area
=
force of extension * crack propagation velocity
2 * blade thickness . crack propagation velocity ’
(3)
This area-specific energy is the work of fracture. As shown in Fig. 3b, the force required
to extend a crack typically fluctuates slightly, and the average force is used in calculating
the work of fracture.
FRACTURE
Samples ofl. jlaccida were collected and mounted as described above for tensile tests.
Cracks of various types and sizes were introduced into the samples with a razor blade
MACROALGAL
FRACTURE
MECHANICS
217
or the sharpened end of a section of hollow tubing. Sample width, blade thickness, and
the aspect ratio of the crack were measured with vernier calipers. The aspect ratio is
the ratio of crack length pe~endicul~ to the direction of applied force, X. to rn~~urn
crack width measured parallel to applied force, Y (Fig. 4). After the introduction of a
crack, each sample was tested in tension as described above. Crack aspect ratio at the
moment of failure (final aspect ratio) was determined from the video tape of the
experiment.
Fig. 4. Definition of parameters used to describe cracks in algal blades.
A similar, but less extensive, set of experiments was carried out with the blades of
L. dentigera.
FIELD EXPERIMENTS
The effect of flaws was tested in the field on a series of individual ~.~ucci~u plants
living in a typical surge channel at HMS. One blade on each plant was injured as
described below and an adjacent blade of similar size was left undisturbed as a control.
All other blades, if any, on the plant were cut off at their stipes. 19 blades had one
2-4 mm sharp-ended slit cut into their edge, perpendicular to the edge z */3-5’2 of the
distance between the holdfast and the free end of the blade; a few received up to four
such injuries. Five blades received 2 mm-diameter semicircular notches instead of
sharp-ended slits; another five had 5-7 mm circular holes punched into the interior of
218
M. DENNY ET AL.
the blade. The injuries were made at low tide in early July 1987 and the plants were
monitored every 24 h for up to 9 days. At the end of this period, blades with slits that
had not fractured were brought into the laboratory to test the tensile strength of the blade
at the injury site as described above.
In a similar set of expe~ments, two adjacent a~egations of a brown alga, P~~~e~~~
~~l~aef~~~s (Rupr~ht~, the sea palm, were chosen at an exposed site (Garrapata State
Beach) on the open coast south of Carmel, Catifornia. All individuals (z 30) in one
aggregation received a l-mm deep slit in their stipe near the base, inflicted with a razor
blade. The individu~s in the other a~e~ation were left uninjured as a control.
The blade material of I. flaccida is typical of many wave-swept algae (Table II). Its
breaking stress averages 4.0 * lo6 N. m - 2 (SD = 0.88 1106 N * m -‘, n = 3), and the
modulus at 10% extension is E 1.10’ N -rne2, rising with further extension to a value
of 3.6 *IO’ N-m -2 just before breaking (Fig. 5). Average breaking strain was 0.25
(SD = 0.026, n = 3), equivalent to an extension of 29%.
Works of fracture for 11 species of nearshore algae and one subtidal angiosperm are
Typical mechanical properties of marine algal materials. B, blade material; S, stipe material,
Alga
Rhodophyta
iriduea fraccida
~astacurp~s pap~l~~~s
Tissue
Strength (MN+m-“)
Tensile stiffness (MN r m - *)
+,rlt
B
B
4.0
2.7
36
9.#
0.25
0.30
s
4.6
28.5
0.22
0.23
0.29
(gametophyte phase)
Mastacarpus jardinii
Potphyra laciniata’
~areap~yl~ edulis’
Phaeoph_~ta
Ascophyllum nodosum ’
L)uTviliaeaantarcticaZ
Eiserzia arborea4
Fticus serra tus ’
Fucus vesictdosas’
Laminariu dentigera
Laminarib digitata 1
Laminate saccharina
Lession~a nigrescen?
~ereocystis feutkearta2
Postelsa pubnaeformfs~
Chiorophyta
U&a sp.
S
S
s
s
3.7
0.7
S
S
S
B
S
S
S
S
S
B
0.25
12.5-24.1
4.0
5.1
10
4.6
0.23
1.2
3.6
1.3
20
9.6
2.6
’ Delf (1932); ’ Koch1 (X986); 3 Holbrook et aL (in prep.); 4 Charters et al. (1969).
0.26
0.18
MACROALGAL
FRACTURE
219
MECHANICS
6
5
A
W
0
-
4-
N
E
2
3
z
k!
i7l
2
v
0
0
0.04
0.08
0. I2
0. I 6
TRUE
0.20
0.24
0.28
STRAIN
Fig. 5. Typical stress-strain
curves of I.flaccidu blade material. Data points are from three separate
tests and solid curve is least squares fit to data. For strains > 0.04 c~(N m 2, = 1.28 10s s2.54.
shown in Table III. All species tested have a work of fracture much lower than biological
structural materials such as wood (Table III). In this respect, the materials from which
macroalgal blades are constructed are brittle.
-a l-
l
1
0
0
0.2
0.4
FINAL
l
_&
e
-I
s,
0.6
ASPECT
0.8
I@
1.0
RATIO
Fig. 6. Breaking stress decreases with increasing final aspect ratio. Solid line is a least-squares
experimental
data. Breaking stress (MN. m - ‘) = 1.022. final aspect ratio-“.9*4(r*
= 0.705).
tit to
220
M. DENNY ET AL.
TABLE
III
Work of fracture for marine macroalgae and other materiais. B, blade material; S, stipe material. With
exception of h4aslocarpq all algae identified according to Abbott L Hollenberg (1976).
Tissue
Species
Rhodophyta
Iridaea jlaccida
Gastroclonium coulteri
M~i~a~~
papillams
(g~etophyte
W (J.m-*)
SD
n
B
B
B
139
84
234
46
21
32
22
I
10
B
B
208
63
9
204
67
9
phase)
Prionitis IanceoIa~a
Rhodymenia pacifica
Phaeophyta
Cystoseira osmundacea
Egregib menziesii
Fucus distiehus
Laminaria dentigera
B
B
B
B
(longitudinal)
(~ansverse~
Petvetia ~~t~~~ta
Pferygophora ca&omica 1
B
S
(longitudinal)
(transverse)
99
37
8
139
143
35
47
8
10
605
185
171
197
7
45
10
415
824
106
388
10
12
17
9
I
Tracheophyta
Grass’ (with the grain)
Wood3 (across the grain)
30
10000
Zosfera marina
(with the grain)
Man-made
B
64
mate~als
Cement, brick, stone3
Glass, pottery’
Nylon, polyethylene’
Mild steel3
Rubber3
3-40
l-10
1000
105-106
20-100
’ Biedka et al. (1987); ’ Vincent (1982); 3 Gordon (1978).
The stress required to break a sample of I.~ffccida blade material decreases as the
final aspect ratio of the crack increases (Fig. 6). In other words, the more sharp-ended
the flaw, the more susceptible the material is to breakage.
The survivorship of I. flaccida blades is shown in Fig. 7. Within 3 days 63 % of those
blades receiving a sharp-ended 2-4 mm edge slit had fractured. None of the blades with
either semicircular edge injuries or circular internal holes fractured in the 9-day
experiment. All controls remained intact throughout the experiment. Wave heights were
monitored during this experiment using a subsurface pressure transducer (operated by
the Monterey Bay Aquarium) at an offshore site x 700 m away form the study site. Surf
conditions were moderate during this experiment (Table IV). During winter storms,
significant wave heights can be two to three times the maxima encountered during the
period of this experiment (Table IV), suggesting that plants present in the winter may
MACROALGAL FRACTURE MECHANICS
0
1
3
2
4
221
9
DAYS
Fig. 7. Survivorship of I.flaccidu
TABLE
in surf-zone.
IV
Significant wave heights at HMS for days during which field studies were carried out and, for comparison,
significant waves heights from stormy days in following winter. Significant wake height is four times SD of
surface elevation, approximately equal to average of highest 33% of waves present during measurement
period (see Kinsman, 1965, or Denny, 1988, for a discussion).
Date
Significant wave height (m)
Average
~xperi~enfal period
7 July 1987
8 July
9 July
10 July
11 July
I2 July
13 July
14 July
15 July
1987
1987
1987
1987
1987
1987
1987
1987
0.55
0.65
Maximum
0.25
0.29
0.99
1.0s
1.09
0.76
0.50
0.45
0.36
0.37
0.57
1.40
1.11
1.28
2.65
2.51
2.63
0.62
0.39
0.3 1
0.27
0.24
Winter sforms
2 December 1987
15 January 1988
18 January 1988
222
M. DENNY ETAL.
be even more likely to be broken. All of the injured P. pa~~aef~~~s had fractured by
the follo~ng day and none of the control plants had fractured.
DISCUSSION
Measurement of the work of fracture for a variety of macroalgae indicates that they
are brittle; it takes very little energy to propagate a flaw in these plants. Further, our
field experiments demonstrate that the hydrod~~c
forces to which two algal species
are naturally exposed are sufficient to cause fracturing when a sharp-ended flaw is
present in the material, even during relatively benign surf conditions. In light of these
results, how is it possible that macroalgae survive on wave-swept shores?
To answer this question, we re-examine the two separate criteria that must be met
before a material will fracture. First, the local stress must exceed the breaking strength
of the material. In most stiff, inextensible materials (e.g., shell, bone, glass) this condition
is easily met. A sharp-ended flaw, no matter how small, serves to amplify greatly the
local stress, often many thous~d fold (Gordon, 1976; Wright
et al., 1976; Boresi
et al., 1980). Thus, when a sharp-ended flaw is present, any small applied force will
cause the local stress to exceed the breaking strength.
The second condition for fracture concerns the availability of energy for creating new
crack surface area. The energy available to spontaneously create new crack area is a
function of the length of the crack (Gordon, 1976; Wainwright et al., 1976). If the crack
is too short, the potential energy (strain energy) released by an incremental len~ening
of the crack is less than the energy required to create the new crack surface area. Under
these conditions the crack is not self-propagating. Above some critical crack length,
however, the potential energy released during crack lengthening and available to the
fracture process is sufficient to create new surface area. Thus, when the crack exceeds
the critical length, the crack can propagate spontaneously, and the material fails
catastrophically. The critical crack length, L, is a function of the properties of the
material and of the applied stress:
L=2
WE/(ncr”),
(4)
where W is the work of fracture, E is the tensile modulus of the material, and rr is the
average applied stress (Gordon, 1976). Only if both these criteria are met (local
exceedance of the breaking stress and availability of sufficient strain energy) will the
material fracture.
These criteria provide clues to the survivability of macroalgae. First, consider the
critical crack length. A typical sample of I. ji’accida blade material has a modulus of
s 10’ N. mm2 and a work of fracture (as measured by the trouser tear test) of
278 J . m - *. Our tensile tests show that a sample with a 2 mm long, rounded crack can
withstand a tensile stress of 4. lo6 N * rnm2. Inserting these values for W, E, and stress
into Eqn. 4 we see that the predicted critical crack length is 0.11 mm, much less than
MACROALGAL
FRACTURE
MECHANICS
223
the actual crack length of 2 mm. The fact that an intact sample can have a crack much
larger than the critical crack predicted on the basis of stored elastic energy suggests that
stress concentration rather than strain energy is the limiting factor in the fracture of the
material.
This suggestion is reinforced by a comparison of the time course of fracture in stiff
materials with that observed in algal blade materials. A tensile test on a sample of a
typical stiff material containing a crack proceeds as follows. During the initial stages
of extension, the crack gradually lengthens as the stress increases, but it maintains its
sharp-ended morphology. During this time, the energy to create new surface area is
provided by the action of the applied tensile force as the sample is extended. But because
the crack is below critical length, the potential energy available for crack propagation
(strain energy from the sample itself) is not sufficient to support crack propagation and
the growth of the crack is not self-sustaining. During this stage of the test, if the extension
of the sample as a whole is stopped, the growth of the crack stops. If extension is
resumed, at some point the crack reaches its critical length and the sample fractures
catastrophic~ly.
In contrast, a tensile test on an algal sample proceeds as follows. During the initial
phase of extension the crack does not lengthen at all, instead its shape changes - its
aspect ratio decreases as the initially sharp-ended flaw becomes rounded. The crack
continues to change shape (but not to lengthen) until a critical stress is reached, at which
point the material catastrophically fails. We propose that failure occurs at this time not
because the potential energy available for creating crack area suddenly becomes sufficient (as calculated above, the crack is already much in excess of its critical length and
therefore should have sufficient available potential energy), but rather because the local
stress becomes equal to the breaking stress. The fact that the crack does not extend
slowly prior to failure implies that up to the time of failure the local stress was less than
the breaking stress. The fact that crack propagation proceeds spontaneously as soon
as it begins is evidence in support of our supposition that the crack is already above
its critical length when the stress becomes suflicient.
In light of the strong possibility that the absence of local stress concentrations governs
fracture in I. flaccida, we can examine the crack-rounding behavior of algal material as
a mechanism for avoiding fracture. First we calculate the apparent stress concentrating
effect of cracks of different aspect ratios. The apparent stress concentration factor, k,
is equal to the mean breaking stress of the unflawed material (4. lo6 N. m - 2 for
1.flaccida) divided by the stress at which the flawed sample breaks (Fig. 6). Measured
values for k are plotted in Fig. 8 as a function of final aspect ratio; k increases with
increasing aspect ratio. In other words, the more sharp-ended the crack, the higher is
the stress concentration. This is in qualitative accordance with theory. For example,
Neuber (1958, as cited in Boresi et al., 1978) asserts that the stress concentration factor
of a rounded crack in one edge of a flat plate is:
k = 1 + [2X/R],
224
M. DENNY ET AL.
where X is the length of the crack perpendicular to the direction of applied force as
before, and R is the radius of curvature of the crack (Fig. 4). If we take the crack
width (Y in Fig. 4) divided by two as an estimate of R, we predict that:
k = 1 t [4X/q.
(6)
These predictions are compared to the measured stress concentrations
provide a reasonable fit to the data for final aspect ratios < ~0.7.
in Fig. 8 and
8r
1
8
7-
’
2
6-
l
0
0. I
I
I
I
I
0.3
0.5
0.7
0.9
1
1.1
FINAL ASPECT RATIO
Fig. 8. Measured stress-concentration factor, k, increases with increasing final aspect ratio. Solid line is a
least-squares fit to experimental data, k = 0.723 + 6.41 . final aspect ratio’.56. Dashed line is stress concentration factor predicted from Eqn. 6.
Neuber (1958) notes that Eqn. 5 is intended to apply only when X is much less than
the width of the blade. As X approaches the blade width, the k predicted by a more exact
theory is higher than that predicted by Eqn. 6, which perhaps can account for the
discrepancy evident in Fig. 7 at high aspect ratios. More precise predictions of k are
given by Neuber (1958) and Roark & Young (1975), but the simple predictions of Eqn.
6 seem sufficient for most purposes.
It is useful to compare the aspect ratio at breaking to the aspect ratio initially
introduced (Fig. 9). Because of the practical difficulty of measuring the aspect ratio of
a razor slit, these data have not been plotted. It is evident that the final aspect ratio is
smaller than the initial aspect ratio due to the ability of the material to deform. A time
sequence of the change in crack shape for a typical test is shown in Fig. 10. The final
aspect ratio for a razor slit is 0.7-l.
MACROALGAL FRACTURE MECHANICS
225
l
!NfTlAl
ASPECT
RATIO
Fig. 9. Final aspect ratio as a function of initial aspect ratio. Solid line is a least-squares fit to data; tinal
aspect ratio = 0.075 + 0.177*initial aspect ratio (r2 = 0.63).
A
l-r
“-?i-
Fig. 10. Aspect ratio of a slit decreases as sample is extended. Time sequence moves from A-D. (A) Initial
flaw before extension. (D) Flaw immediately prior to breakage.
226
M. DENNY ET AL.
The idea that fracture in algal material is controlled by local stress concentrations
explains several aspects of our field experiments. Those injured plants which fractured
during our experiments were those that received initial injuries with high aspect ratios,
implying that their final aspect ratios were 20.7-1 and resulting in a stress concentration factor of 5-7. Those plants which received semicircular or circular injuries did
not break. These injuries would result in a final aspect ratio of ~0.3 and a stress
concentration factor not substantially different from 1 (Fig. 9). Thus, we would not
expect these injured plants to be at substantially greater risk than the controls.
Although these tests have been conducted on the blade material of Z.flaccida, this
material is probably typical of macroalgal blade material in general (Tables II, III). Thus
we expect that the blades of many wave-swept algae will exhibit fracture mechanics
similar to those described here. This is confirmed for L. dentigera blade material, where
an initially semicircular edge crack has a k = 1.16 and a razor slit has a k = 2.23.
In some cases the stipe material of macroalgae is considerably stiffer than that of
f.Jlaccida blade material (Table III). If these materials have a work of fracture and a
breaking strength similar to those measured here (Tables II, III), the critical crack length
under equivalent applied stress will be larger (Eqn. 4), but, because they are stiffer, their
crack-rounding ability may be substantially reduced. Consequently, these stipe materials
may be at a greater risk of fracture in the presence of surface injuries. This is true for
L. dentigera where the stipe material is 2.3 times as stiff in tension as the blade material
(Table II). A razor slit in a blade of L. dentigera fractures on average at a stress 52%
of that required to break a rounded flaw (n = 5), indicating that the slit is effectively
rounded. In contrast, a razor slit in the stipe material fractures on average at a stress
only 18 y0 of that required to break a rounded flaw (n = 3), indicating that the flaw
remains sharp-ended until breakage occurs. Casual observations of the blades and
stipes of L. dentigera at HMS suggest that surface flaws are much more common on
blades than on stipes, and we propose that this difference may be due to fracture
mechanics. Unable to effectively blunt surface cracks, injured stipes are quickly broken
and therefore are not often observed. In contrast, the crack-rounding property of blade
material allows blades to survive in the presence of multiple injuries.
Aside from the safety provided by rounding of the crack, plants may be able to affect
actively the fracture process. In our field tests, experimental I. jlaccida that received slit
injuries but survived for 9 days exhibited morphological changes that may contribute
to further survival. Seven of the surviving slits had increased their aspect ratio - the slits
had become wedge shaped (Fig. 1lb). The remaining slits, in addition to increasing their
aspect ratio, had formed a l-2 mm diameter “pit” at the apex of the slit (Fig. 1lc). Pits
formed as early as 48 h after initial injury; most pits formed 72 h after injury.
The ability of these morphological changes to aid in resisting fracture was tested using
tensile tests as described earlier. Blade samples from the field plants containing slits with
and without pits were compared to controls with freshly cut slits. A one-way ANOVA
indicates a significant difference in the breaking stresses among the three treatments
(F = 30, P -C 0.001). A multiple comparison analysis (Tukey-Kramer; Sokal & Rohlf,
MACROALGAL FRACTURE MECHANICS
221
Fig. 11. Response to introduction of razor slits in I.flaccidu. (A) Initial flaw. (B) After 24 h, aspect ratio
of flaw has decreased. (C) A pit may form at tip of flaw after 48-72 h.
1981) found that the average breaking strength of the slits with pits
(X = 19.0. 105N~m-2, SD = 4.2. 10’ N .mP2, n = 8) was significantly greater than
that of freshly cut slits (X = 8.2. 10’ N. rne2, SD = 2.6 * 10’ N * rne2, n = 20; P < 0.01).
The aged slits also have a significantly greater average breaking stress than the freshly
n=4; P-cO.05). Aged slits
cut slits (X = 13.0. 10’ N.rne2, s~=5.1*10~N+rn-~,
with pits are significantly stronger that aged slits without pits (P < 0.05). These preliminary data indicate that Z.fluccida may have mechanisms for moderating the effects of
sharp-ended flaws. If the blade can survive for 48-72 h (for instance, if surf conditions
are particularly benign in the period immediately following injury), the sharp end of the
flaw may effectively be blunted, and the stress concentration reduced.
ACKNOWLEDGEMENTS
This research was supported by NSF grant OCE 83-14591 to M. Denny and a Myers
Trust Fund research grant to E. Carrington. E. Carrington was supported by an ARCS
fellowship, and V. Brown and G. Kraemer received partial support from the Friends
of Hopkins Marine Station.
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