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First published online June 15, 2007
Journal of Experimental Biology 210, 2231-2243 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.001578
Death by small forces: a fracture and fatigue analysis of wave-swept macroalgae
1 Hopkins Marine Station of Stanford University, Pacific Grove, CA 93950,
USA
2 Department of Mechanical Engineering, Stanford University, Stanford, CA
94305, USA
* Author for correspondence (e-mail: mach{at}stanford.edu)
Accepted 8 April 2007
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Summary |
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Key words: fracture mechanics, macroalgae, seaweed, fatigue, breakage
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Introduction |
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;
Denny and Gaylord, 2002) are
repeated for each of the more than 8000 waves impinging on shore each day.
Despite its harshness, the wave-swept environment is home to some of Earth's
most diverse and productive assemblages of organisms
(Smith and Kinsey, 1976;
Connell, 1978;
Leigh et al., 1987).
Accounting for such rich diversity in the midst of physical adversity is one
of the central goals of intertidal ecomechanics.
To that end, studies of macroalgal material properties and morphological
attributes have elucidated ways by which intertidal seaweeds withstand
wave-imposed forces (e.g. Carrington,
1990; Holbrook et al.,
1991; Denny and Gaylord,
2002; Pratt and Johnson,
2002; Kitzes and Denny,
2005; Martone,
2006). Although wave-induced flows potentially result in a variety
of hydrodynamic forces (such as lift, acceleration reaction and impingement
force), the bulk of hydrodynamic force for most seaweeds can be approximated
as drag, which imposes tension on algal thalli
(Gaylord et al., 1994;
Gaylord, 2000;
Gaylord et al., 2001). Most
studies have thus examined breakage of seaweeds through pull-to-break tensile
tests, which mimic imposition of drag force by a single wave. Strengths
measured in tensile tests are often much greater than predicted maximal
stresses encountered in the field (e.g.
Koehl and Alberte, 1988;
Gaylord et al., 1994;
Gaylord, 2000;
Johnson and Koehl, 1994;
Friedland and Denny, 1995;
Utter and Denny, 1996;
Denny et al., 1997;
Johnson, 2001;
Kitzes and Denny, 2005),
indicating that wave-swept macroalgae may be over-designed for resisting wave
forces. The consequent low risk of breakage and dislodgment may help explain
extant algal diversity (Denny,
2006). However, these conclusions must be viewed with skepticism.
Contrary to prediction, large fractions of many algal populations are broken
and dislodged each season (Seymour et al.,
1989; Dudgeon and Johnson,
1992; Dudgeon et al.,
1999; Johnson,
2001; Pratt and Johnson,
2002). How, then, can we reconcile our mechanical predictions with
field observations?
Several factors, such as herbivory, abrasion, senescence and fatigue, have
been suggested to increase breakage rates beyond those calculated for maximal
hydrodynamic forces (Friedland and Denny,
1995; Utter and Denny,
1996; Kitzes and Denny,
2005; Denny,
2006). The action of these phenomena could bring predicted rates
of breakage more in line with observations. As a first step towards evaluating
these factors, we explore the role of fatigue, asking a fundamental question:
does repeated loading of seaweeds lead to their breakage when single loadings
do not?
Repeated force imposition may break seaweeds through several scenarios.
First, repeated loading by waves may cause small fatigue cracks to form
(Koehl, 1984;
Koehl, 1986;
Kitzes and Denny, 2005). The
presence of such cracks – in addition to any cuts or nicks formed
through abrasion or herbivory – reduces breaking strength (the stress
that algae can resist before fracturing in two), thereby increasing the
probability of breakage by large forces (e.g.
Black, 1976;
Johnson and Mann, 1986;
Armstrong, 1987;
Biedka et al., 1987;
Denny et al., 1989;
Lowell et al., 1991;
DeWreede et al., 1992).
Furthermore, once cracks are present (regardless of their source), repeated
loading may cause them to grow until the alga breaks, even when algae are
subjected only to small forces. Our goal here is to characterize the effects
of cracks and the speeds with which they grow.
Fortunately, this inquiry into the potential role of algal fatigue can be
guided by a robust engineering literature on formation and growth of cracks
under repeated loading (e.g. Broek,
1982; Meguid,
1989; Janssen et al.,
2004). Here we apply fracture mechanics techniques to assess crack
growth in four intertidal and shallow subtidal macroalgae with flat-bladed
morphologies. In the field, these algae may break at blade, stipe, and
holdfast regions, but in this study, we focus on breakage of blades. We
evaluate the fracture toughness of these macroalgae, assessing their
resistance to complete fracture in the presence of cracks. We then measure
crack growth in conditions of cyclic repeated loading, ultimately determining
the number of imposed waves required for small cracks to grow to the point of
complete fracture. See the accompanying article
(Mach et al., 2007) for an
extended description of fracture mechanics parameters and techniques.
In this study, we focus our attention on the tensile stresses in algae with
flat-bladed morphologies, but fracture mechanics can also be applied to more
complex algal structures, such as holdfasts or stiff, large-diameter stipes,
which experience a variety of stresses. For such structures, finite element
computer models previously developed for elastomeric components can be used to
assess fatigue crack growth due to the combined effects of tensile, bending
and shear stresses (Busfield et al.,
2005).
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Materials and methods |
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Testing apparatus
All tests were performed using a hydraulically driven tensometer, the
driving arm (Parker Electrohydraulics, Elyria, OH, USA, PLA series) of which
has positional accuracy of 51 µm. A 0–10 V signal from a 16-bit
input/output board (National Instruments Corporation, Austin, TX, USA, model
AT-MIO-16X) regulated the position of the driving arm, controlled through an
interface written in LabVIEW (National Instruments Corporation).
Extension of test samples was determined from the position controller of
the tensometer's actuator, an appropriate method because samples were thin
blades tested at low stress with no observed slippage from sample grips.
Engineering strain, 
, in the bulk of a specimen was computed as change
in specimen length divided by initial length.
Force exerted on samples was measured using a waterproofed cantilever-style force transducer milled from an aluminum block. Four strain gauges (Measurements Group Inc., Raleigh, NC, USA, model CEA-13-062UW-350) were attached to the cantilever base in a full Wheatstone-bridge configuration. Force measurement accuracy was 0.005 N. Stress in the bulk of a specimen was defined as applied force divided by initial specimen cross-sectional area.
Samples were gripped by the tensometer as follows: thin strips of rubber were glued across algal sample ends with cyanoacrylate glue, and the tensometer grips were affixed to these rubber strips. Cyanoacrylate glue adhered well to all materials with no observed slippage. During loading, the rubber strips deformed a very small amount, but an analysis of this deformation showed it to have negligible effect on measurements of specimen extension.
Crack length in samples was measured using a telemicroscope (Questar Corporation, New Hope, PA, USA) fitted with an ocular Filar micrometer, the accuracy of which is 5 µm.
Single-edge-notch, pull-to-break tests
Critical strain energy release rate, TC, quantifies
strain energy release associated with crack propagation in a material [for an
extended description, see accompanying article
(Mach et al., 2007)].
TC was determined from pull-to-break tests of
single-edge-notch specimens of M. flaccida, U. expansa and P.
occidentalis. Specimens cut from blades varied in length from
approximately 7 to 11 cm and in width from approximately 0.9 to 1.7 cm.
Samples were either left un-notched or given a notch of length ranging from
0.4 to 2.5 mm, with notches introduced as small razor-blade cuts perpendicular
to one of a sample's long edges. M. flaccida and P.
occidentalis were pulled until failure at a strain rate of approximately
0.2 s-1, while U. expansa was extended to failure at a
strain rate of 0.1 s-1. P. occidentalis and U.
expansa samples were submerged in seawater during tearing, while M.
flaccida test pieces were wetted but not submerged.
For notched specimens, we made two calculations of critical strain energy
release rate (Rivlin and Thomas,
1953; Lake, 1983)
[see Eqn 9 in accompanying
article (Mach et al., 2007)].
First, we calculated TC,T:
 | (1) |
br is strain at breaking, and
a is crack length in the specimen, the length of the introduced
notch. The subscript `T' indicates total strain energy density, the
subscript `C' indicates critical strain energy release rate, and the
factor 
is
the appropriate value for k in eqn
9 in the accompanying article
(Mach et al., 2007).
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;
Kinloch and Young, 1983;
Kadir and Thomas, 1984;
Seldén, 1995) that
Eqn 1 can be applied to materials
that dissipate energy by assuming that the stored energy density available for
crack extension is the recoverable energy density under the retraction curve,
Wo,S, (Fig.
1) instead of the input energy density under the extension curve,
Wo,T. We thus calculated critical strain energy release
rate from Wo,S as well:
 | (2) |
The total strain energy density, Wo,T, absorbed by a
material before fracture was measured directly as area under the material's
stress–strain curve during extension to breaking.
Wo,S, on the other hand, was calculated for
Eqn 2 using estimates of
R, the resilience at breaking strain, because retraction curves
cannot be measured for samples extended to fracture:
 | (3) |
).
Each species' resilience as a function of strain was estimated by cycling
un-notched test pieces to increasing extensions
(Fig. 2). Resilience as a
function of strain, determined from these cyclic-test measurements
(Fig. 2B), was then used to
estimate resilience at breaking strain for each single-edge-notch,
pull-to-break test.
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,
and the slope of the plot yielded critical energy release rate for total or
stored strain energy density.
Many samples left un-notched broke at the grips due to tissue damage or
stress concentrations introduced by the grips. These samples were removed from
analysis. When an un-notched specimen – that is, a specimen without an
intentionally introduced crack – broke at a location away from the
grips, fracture was assumed to originate at a small, naturally occurring crack
of unknown length ao. Given that critical energy release
rate TC,S is known from the single-edge-notch tests
described above, ao was estimated for an un-notched
specimen by treating the assumed initial crack as an edge crack and solving
for its length:
 | (4) |
Finally, for each species, critical strain energy release rate was
expressed as:
 |
 | (5) |
, is found as a function of applied stress, ,
from representative pull-to-break stress-strain curves for each species; total
strain energy density as a function of applied stress,
Wo,T,, is also calculated from representative
stress-strain curves; and stored strain energy density as a function of
stress, Wo,S,, is determined by applying
Eqn 3. The subscript `'
indicates variables determined as fitted polynomial functions of applied
stress. Breaking stress was determined for given crack lengths, a, by
finding the values of stress, , that gave and
Wo,T, (or and
Wo,S,) necessary to balance
Eqn 5.
Crack propagation: single-edge-notch, repeated-loading tests
Single-edge-notch tests involving repeated cycling were performed on
rectangular test pieces with lengths of 6.6–13.0 cm and widths of
1.4–3.1 cm. Sample sizes were constrained by the extent of flat and
undamaged areas in blades of our test species. After samples were cut from
blades, tissue adjacent to sites of future cracks was sectioned, and its
thickness was determined using a compound microscope. Cross-sectional area of
the test piece was calculated as the product of this approximately uniform
thickness and the sample's width. All test pieces were immersed in cooled
seawater during testing. For all tests, samples were cycled between zero
strain and a fixed maximum cyclic strain. Cycling occurred at 1.0 Hz, with
strain oscillating sinusoidally. Maximum imposed bulk strains ranged from
0.073 to 0.180, and maximum strain rates ranged from 0.23 s-1 to
0.57 s-1.
Testing protocols were adapted from Seldén
(Seldén, 1995), who
describes standard methods for studying cyclic crack growth in rubber. Each
test consisted of several phases. First, un-notched test pieces were
conditioned by repeated cycling from zero strain to the test's maximum strain.
When subjected to cyclic stretching, numerous elastomers and biological soft
tissues experience the `stress-softening' Mullins effect
(Emery et al., 1997;
Edsberg et al., 1999;
Mars and Fatemi, 2004;
Franceschini et al., 2006;
Dorfmann et al., 2007), in
which stress needed to reach a certain strain drops with each repeated cycle
of stretching (e.g. Fig. 3).
Conditioning prior to cyclic crack growth testing of rubber-like materials was
suggested by Seldén (Seldén,
1995) to reduce the Mullins effect
(Mullins, 1969), and
experiment-appropriate conditioning of soft biological tissues has been found
to reduce specimen- and test-related variability and to stabilize material
behavior (Carew et al., 2004).
Generally, stress softening diminishes after a sufficient number of
constant-amplitude conditioning cycles, and stable stress–strain
behavior ensues. If force–extension curves become uniform during
conditioning, further changes in the force–extension curves can be
assumed due to the soon-to-be introduced crack and not due to further stress
relaxation.
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Even after 2000–5000 conditioning cycles, algal blades demonstrated a
slight, but measurable, decrease in stress that we considered in subsequent
calculations. For each un-notched test piece during conditioning, maximum
stored strain energy density per cycle, Wo,S, was measured
as the area under the test piece's stress–strain curve during return
from maximum strain to zero strain. To account for the continued relaxation
observed in test specimens, Wo,S was determined as a
function of cycle number, N, during conditioning cycles for each
sample. Wo,S decreased as a logarithmic function of
N:
 | (6) |
After conditioning cycles were complete, enough seawater was drained from the tank to make a single edge notch of 0.5–3 mm perpendicular to a long edge of the test specimen. The initial length of the cut was determined by straining the sample slowly at 0.005 s-1 until the crack opened wide enough to be visible. A wet glass coverslip was placed on the back of the algal blade to ensure the region around the crack was flat and perpendicular to the telemicroscope, which was then used to measure crack length. Following this measurement, the coverslip was removed, seawater was replaced, and the notched sample was again cycled at 1.0 Hz between zero and maximum strain. After 500–1000 cycles, the tank was drained enough to expose the sample to air, and crack length was again measured. This procedure was repeated until the crack had grown in length 10% beyond the introduced razor cut, at which point fracturing at the crack tip was assumed independent of any effects of the initial cutting. When 10% growth was observed, the number of cycles needed for the crack to grow an additional 10% of its current length was estimated (200–25 000 cycles). These cycles were applied, and crack length was then remeasured. This process was iterated until the crack stopped growing or the sample broke in two.
For each crack-length measurement, crack growth rate, da/dN, was
estimated as increase in crack length between measurements, 
a,
divided by the number of cycles between measurements, N. Also
for each measurement, strain energy release rate, TS, was
calculated as was done for rubber (e.g.
Seldén, 1995):
 | (7) |
max the maximum
strain imposed on the sample in each cycle [see
eqn 9 in accompanying article
(Mach et al., 2007)]. Again,
the subscript `S' indicates that T is calculated from stored
strain energy density. This equation resembles
Eqn 2, used for pull-to-break
tests, with the substitution of an average measure of crack length.
Several precautions minimized the time required for measurement of crack
length, during which loading stopped. First, crack length was measured at a
relatively infrequent rate, only after the crack was estimated to have grown
by 10%, as was done for rubber
(Seldén, 1995). And
second, tank draining and subsequent measurement of crack length were
completed as rapidly as possible.
Predictions of lifetime
Fatigue lifetime, as a function of maximum cyclic stress, was calculated
for P. occidentalis, U. expansa, and M. flaccida using a
crack-growth-based approach. An empirically determined power-law function was
used to describe the relationship between energy release rate,
TS, and crack growth rate, da/dN, during
repeated-loading tests (Lake,
1995; Seldén,
1995) [see eqn 16 in accompanying article
(Mach et al., 2007)]:
 | (8) |
;
Seldén, 1995) [see eqn
17 in accompanying article (Mach et al.,
2007)]:
 | (9) |
 | (10) |
cyclic, and stored strain energy density,
Wo,S,cyclic, were determined as functions of
cyclically applied stress by subjecting rectangular test pieces to 25 cycles
between zero strain and a given maximum, beginning at maximum strain of 0.10.
Maximum strain was increased by 5% after each 25-cycle test until the sample
broke. For the last cycle at each maximum strain level, maximum stress and
maximum stored strain energy density were determined. Polynomials were then
fit to these data to find maximum cyclic strain and stored strain energy
density as functions of maximum cyclic stress. The number of cycles required
for an assumed, naturally occurring crack in a sample to grow to failure could
then be estimated from Eqn
9. For each species, several values of B and ß were used in Eqn 9 to indicate the range of fatigue lifetimes predicted from crack growth data. B and ß were first calculated from all experimental data for each species. For M. flaccida, fatigue lifetimes were also determined for `upper-bound' values of B and ß, and for U. expansa, B and ß determined for a subset of the species' data were used to exclude effects of an outlying crack-growth-rate point. Finally, fatigue lifetimes were calculated for each species using B and ß values calculated from the combined data of all species.
To relate our results to flow conditions in the field, we calculated
approximate water velocities corresponding to experimentally applied stresses
for a M. flaccida frond. Of species in this study, M.
flaccida is the only one for which drag coefficients are available, and
it is generally exposed to the greatest wave forces in the field. Drag
coefficients from Bell (Bell,
1992) were used to determine drag force and thus stress as a
function of wave-imposed water velocities. As we show below, M.
flaccida is predicted to accumulate fatigue damage for stresses imposed
by water velocities greater than approximately 8 m s-1.
This estimated water-velocity threshold for fatigue damage was used to approximate time (rather than number of cycles) to failure. Based on continuous wave force measurements at Hopkins Marine Station (M. L. Boller and M. J. O'Donnell, personal communication), we found that 5% of waves are associated with intertidal water velocities greater than 8 m s-1 on a day of average offshore significant wave height (1.0 m). Given a wave period of 10 s, we then estimated that water velocities exceed 8 m s-1 every 200 s, on average. Thus, a cycle of loading that causes fatigue damage will occur every 200 s. This approximation likely overestimates the rate of imposition of water velocities over 8 m s-1 because some small wave-induced forces, below the sensitivity limit of measurement devices, were not recorded and because intertidal seaweeds are not exposed to waves for the entirety of each day.
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Fig. 5 depicts calculated breaking stresses as functions of crack size (Eqn 5) along with measured breaking stresses determined in single-edge-notch, pull-to-break tests. Breaking stress showed the most predictable correlation with crack length for M. flaccida specimens and demonstrated poor correlation with crack length for P. occidentalis. Because of the high variability in critical strain energy release rate for P. occidentalis, we show predicted lower bounds of breaking stress for different crack lengths (Fig. 5B), using the consistent lower-bound values of TC,S and TC,T in Eqn 5.
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Crack propagation in single-edge-notch, repeated-loading tests
In single-edge-notch tests involving repeated cycling, maximum stored
strain energy density, Wo,S, in each cycle decreased
during conditioning cycles, before crack introduction; a representative
example is depicted in Fig. 6.
The majority of decrease in strain energy density occurred during the first 50
cycles, but further decrease continued throughout the conditioning period and
was considered for each sample with Eqn
6. U. expansa, the only green alga in the study, showed
greater decrease in strain energy density than the red algae M.
flaccida and P. occidentalis.
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Predictions of lifetime
Crack-growth power-law functions (Eqn
8) were determined from plots of T versus da/dN
(Fig. 8A). B and ß for the
different species' power-law functions are given in
Table 3, and crack growth rates
predicted from these power-law functions are shown in
Fig. 8B. As depicted in
Fig. 8A, noticeable variation
occurred in measures of crack growth rate. One source of this variation was
branching of the crack tip, which caused cracks to propagate more slowly. For
M. flaccida and M. splendens, for which the variation was
most substantial, an upper bound on crack growth rate for each species was
thus determined. The upper bound for M. flaccida is shown as the
broken line in Fig. 8A. In
determining power-law functions to describe crack growth rates
(Eqn 8), points along the upper
bound of plots of T versus da/dN, were arbitrarily selected for
M. flaccida and splendens, to supplement power-law functions
determined for these species overall. No r2 values are
given for the upper-bound M. splendens and flaccida
relations in Table 3 since
regression points were arbitrarily selected. In addition, two power-law
functions were determined for U. expansa, with an outlying data point
that suggests an exceptionally low initial crack growth rate excluded for the
second relation.
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Fig. 9 depicts cycles to
failure in terms of each repeatedly applied stress's percent of breaking
strength, as calculated from Eqn
9. (Breaking strength is estimated stress at fracture for
un-notched specimens in pull-to-break tests.) Values of B and ß for
Eqn 9 are given in
Table 3, and polynomials for
Eqn 9 that relate maximum
strain, cyclic, and maximum stored strain energy
density, Wo,S,cyclic, to maximum cyclic stress are
listed in Table 4.
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Fig. 10 depicts loading cycles to failure as a function of repeatedly applied stress for M. flaccida, the only species with known drag coefficients.
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Crack propagation rates in Figs 9 and 10 are maximum rates that assume cracks propagate as single sharp cracks. Power-law functions describing crack growth (Eqn 8, Table 3) were calculated for samples that did not display substantial bifurcation of growing cracks or peeling apart of blade cell layers.
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) and
bone (Currey, 1998), materials
for which fracture mechanics has been used successfully to characterize
structural integrity. Only variation in Porphyra occidentalis
TC exceeds published values for other biological and soft
materials. Calculations of TC using stored strain energy density (Table 1) are likely more accurate than calculations from total strain energy density because stored strain energy density excludes energy dissipated within a specimen, which is included in total strain energy density but does not contribute to crack elongation.
Cracks were introduced using razor blades, and specimen fracture in pull-to-break tests proceeded directly from the resulting sharp crack tips. Sharpness of these crack tips likely resulted in fracture at lower stresses than if the crack tips had more natural, somewhat rounded geometries. The initially high crack growth rates observed in repeatedly loaded, single-edge-notch test specimens also indicate the greater ease with which sharp, razor-introduced cracks propagate, but these initial high growth rates were excluded from analyses.
Biological implications
Variability in calculated critical strain energy release rates may indicate
that different parts of algal blades vary in their susceptibility to crack
propagation. This variability may help explain tattering in marginal regions
of thalli, which avoids more catastrophic breakage (e.g.
Blanchette, 1997), or may
affect dispersal if reproductive portions of blades are more prone to tearing.
Furthermore, holes found in blades, especially common in U. expansa,
may slow the overall course of crack growth by providing gaps that must be
bridged or bypassed by growing cracks. TC values were
lowest for U. expansa, indicating that blades of this species may be
the most susceptible to fracture. It is worth noting that healthy U.
expansa, in addition to growing on rocks and docks, is often found
floating freely in protected areas (Abbot
and Hollenberg, 1976). Thus, a tear propagating across an entire
blade may be less lethal for U. expansa than for P.
occidentalis and M. flaccida.
Critical strain energy release rate, breaking stress and strain energy density measurements throughout this study were least variable for M. flaccida, perhaps due to the cellular composition of its blades or to wave action it experiences in the field. M. flaccida blades are thicker than P. occidentalis and U. expansa blades, with more medullary tissue, which may behave as a more homogenous material than the closely packed cell layers of P. occidentalis and U. expansa. However, it is worth noting that cell size in all studied species is small (cell diameters of approximately 0.01 mm) compared to lengths of introduced cracks. In addition, of the three species, M. flaccida is exposed to the greatest wave-induced forces in the field, which may select over evolutionary time or among blades in a population for more uniformly robust material performance.
Critical energy release rates measured in this study are similar to values
measured for other algae and biological materials
(Biedka et al., 1987;
Denny et al., 1989)
(Table 5).
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Fatigue lifetime predictions
Our results suggest that fatigue may play an important role in breakage of
marine algae. For all species studied, imposed stresses well below breaking
strength are predicted to cause fracture within hours or days (Figs
9 and
10). The potential importance
of fatigue processes in wave-swept macroalgae has long been suggested, and
these results provide the first evidence that fatigue crack growth is indeed
relevant to seaweeds.
Previous comparisons of algal pull-to-break strengths and maximal field stresses have predicted that, contrary to observation, seaweeds should rarely break. Perhaps rectifying these discrepancies between prediction and reality, this study indicates that wave-induced stresses may break seaweeds commonly, not in single pull-to-break loadings, but due to the accumulated effects of repeated stressing.
Our estimates of fatigue lifetimes, based on numbers of cycles required for crack growth to fracture in un-notched blades, require several qualifications. First, and most importantly, fatigue lifetimes as estimated here assume that initial, `effective' cracks in un-notched samples grow similarly to longer cracks observed in this study. However, crack growth may deviate from the power-law functions used here when cracks are very small. In addition, the fatigue lifetime curves in Fig. 9 predict that any blade containing a propagating crack subjected to repeated loading will resist cycling indefinitely or fracture within 30 000 cycles. However, some specimens in the single-edge-notch, repeated-loading tests fractured after many more than 30 000 cycles (e.g. Fig. 7). But in other single-edge-notch, repeated-loading tests, cracks slowed or stopped propagating, perhaps due to cracks encountering deformities in blades or due to single propagating cracks branching into two or more cracks.
Fracture mechanics estimates of lifetime are statistical predictions that cannot predict the exact fates of individual plants. Variability among individuals stemmed from unknown prior loading histories and from the variable nature of fracture and fatigue processes. Although blade samples used in this study had no macroscopic damage, they were collected from the field, where they had undergone unknown numbers of wave force loadings and had potentially accumulated some level of fatigue damage prior to testing. Fracture parameters and lifetime calculations were determined accordingly for algae of average fatigue damage, not for specimens with known pre-experiment loading histories. Thus, predictions made with techniques outlined here are applicable as averages for given populations. Nonetheless, our primary conclusion stands: fatigue crack growth may cause failure of wave-swept algae under conditions in which imposition of a single stress would not cause breakage.
Our predictions of fatigue lifetimes do not consider biological repair. Although a certain number of cycles at high stress may cause blade fracture (Figs 9 and 10), if these high-stress cycles are separated by low-stress cycles that do not cause crack propagation, algae may have time to repair fractured tissues or round crack tips in a way that mitigates subsequent crack growth. In addition, the energy release rate below which crack growth does not occur may be effectively increased through tissue repair.
In this study, standard loading protocols involving sinusoidal variation in strain were used to explore application of fracture mechanics to macroalgal crack growth. Subsequent studies could evaluate several aspects of loading protocol that may be important in comparing field and laboratory crack growth. First, the rate of strain energy input from waves or laboratory loading – with rate calculated per unit time or per loading cycle – may influence crack growth rates. Second, different strain rates and waveforms during each loading may change crack growth rates. Third, viscoelastic changes as well as biological repair, potentially acting over different time scales, may occur during loading stoppages, possibly altering the course of crack growth.
We emphasize, however, that standard laboratory loading protocols have
successfully characterized naturally variable fatigue processes in rubbers
even though typical loading regimes are invariably more complex for these
materials (e.g. Lake, 1995;
Seldén, 1995).
Simplified laboratory testing of engineering materials is adequate to predict
field performance, and indeed for seaweeds, more elaborate testing methods may
not improve accuracy of predictions, in light of inherent variability in
fatigue processes. Most field loadings on algal blades in this study can be
modeled as drag forces imposed in tension, and the sinusoidal, tensile cyclic
loadings we have used to study crack growth approximate field stresses as has
been done for engineering materials.
For future studies, we note that algal blades represent particularly good model organisms for investigating crack growth and repair because blades are often naturally planar, easily kept alive in seawater, and frequently only one or two cells thick, all of which make crack tips and new tissue growth easily observable.
Conclusion
Although subsequent experiments should probe additional aspects of fracture
and fatigue, the results reported here demonstrate that failure through
fatigue crack growth may be an important component of life for wave-swept
macroalgae, with breakage resulting from repeated imposition of small
stresses. Fatigue processes may have similar consequences for any biological
structure subjected to repeated loads.
List of symbols and abbreviations
The equation in which each symbol is first used is given (if it is used in an equation).
br
cyclic
max
max
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Acknowledgments |
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References |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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at fracture, where Wo is calculated as total
absorbed strain energy density, Wo,T, and estimated stored
strain energy density, Wo,S. Solid circles indicate data
points calculated from total strain energy density; solid diamonds indicate
data points calculated from stored strain energy density.
TC is the slope of the regression line for each data set:
regression line (broken line) for data points calculated from total strain
energy density yields an estimate of TC,T, and regression
line (solid line) for data points calculated from estimated stored strain
energy density yields an estimate of TC,S, as defined in
the text. Open circles and diamonds indicate data points for un-notched
samples, for which effective crack length was estimated.
