## SUMMARY

Wave-swept macroalgae are subjected to large hydrodynamic forces as each wave breaks on shore, loads that are repeated thousands of times per day. Previous studies have shown that macroalgae can easily withstand isolated impositions of maximal field forces. Nonetheless, macroalgae break frequently. Here we investigate the possibility that repeated loading by sub-lethal forces can eventually cause fracture by fatigue. We determine fracture toughness, in the form of critical strain energy release rate, for several flat-bladed macroalgae, thereby assessing their resistance to complete fracture in the presence of cracks. Critical energy release rates are evaluated through single-edge-notch, pull-to-break tests and single-edge-notch, repeated-loading tests. Crack growth at sub-critical energy release rates is measured in repeated-loading tests, providing a first assessment of algal breakage under conditions of repeated loading. We then estimate the number of imposed waves required for un-notched algal blades to reach the point of complete fracture. We find that, if not checked by repair, fatigue crack growth from repeated sub-lethal stresses may completely fracture individuals within days. Our results suggest that fatigue may play an important role in macroalgal breakage.

## Introduction

As sessile organisms, intertidal macroalgae experience the full brunt of
wave-induced water velocities, which in breaking waves commonly exceed 10 m
s^{-1}. Hydrodynamic forces thereby imposed on seaweeds
(Gaylord, 2000;
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).

## Materials and methods

### Specimens

*Mazzaella flaccida* (Setchell & Gardner) Fredericq was
collected from intertidal rocks at Carmel River Beach in Carmel by the Sea,
CA, USA, and Hopkins Marine Station in Pacific Grove, CA, USA. *Porphyra
occidentalis* Setchell & Hus and *Mazzaella splendens*
(Setchell & Gardner) Fredericq were collected at subtidal depths of
3–9 m at Carmel River Beach. *Ulva expansa* (Setchell) Setchell& Gardner was collected from floating docks in Monterey, CA, USA. The
flat-bladed morphologies of these macroalgae facilitated observation of crack
growth in conditions of repeated loading.

### 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, ϵ, 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, *T*_{C}, quantifies
strain energy release associated with crack propagation in a material [for an
extended description, see accompanying article
(Mach et al., 2007)].
*T*_{C} 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 *T*_{C,T}:
(1)
*W*_{o,T} is the total strain energy density absorbed by
samples before fracture, ϵ_{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).

Eqn 1 is derived assuming
elastic stress–strain behavior with no energy dissipation. However,
algal blades exhibit some internal energy dissipation, as demonstrated by the
hysteresis loops shown in Fig.
1. It has been suggested
(Ahagon et al., 1975;
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,
*W*_{o,S}, (Fig.
1) instead of the input energy density under the extension curve,
*W*_{o,T}. We thus calculated critical strain energy release
rate from *W*_{o,S} as well:
(2)
The subscript `S' indicates *stored* strain energy density. When no
energy is dissipated, the extension and retraction curves follow the same
path, making *W*_{o,S} equivalent to
*W*_{o,T}.

The total strain energy density, *W*_{o,T}, absorbed by a
material before fracture was measured directly as area under the material's
stress–strain curve during extension to breaking.
*W*_{o,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)
Resilience is the ratio of area under a retraction curve (hatched area in
Fig. 1A) to total area under
the extension curve (hysteresis loss area plus hatched area in
Fig. 1A)
(Wainwright et al., 1976).
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.

For calculations of both critical energy release rates,
*T*_{C,T} and *T*_{C,S}, the inverse of crack
length, *a*, was plotted against
,
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 *a*_{o}. Given that critical energy release
rate *T*_{C,S} is known from the single-edge-notch tests
described above, *a*_{o} was estimated for an un-notched
specimen by treating the assumed initial crack as an edge crack and solving
for its length:
(4)
Here, average *T*_{C,S} for a species and estimated stored
strain energy density, *W*_{o,S}, for each specimen at breakage
were used.

Finally, for each species, critical strain energy release rate was
expressed as:
and
(5)
where *T*_{C,T} and *T*_{C,S} are average
critical strain energy release rates for each species; strain,ϵ
_{σ}, is found as a function of applied stress, σ,
from representative pull-to-break stress-strain curves for each species; total
strain energy density as a function of applied stress,
*W*_{o,T,σ,} is also calculated from representative
stress-strain curves; and stored strain energy density as a function of
stress, *W*_{o,S,σ}, is determined by applying
Eqn 3. The subscript `σ'
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, σ, that gave ϵ_{σ} and
*W*_{o,T,σ} (or ϵ_{σ} and
*W*_{o,S,σ}) 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.

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, *W*_{o,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, *W*_{o,S} was determined as a
function of cycle number, *N*, during conditioning cycles for each
sample. *W*_{o,S} decreased as a logarithmic function of
*N*:
(6)
where C_{1} and C_{2} are constants fitted to experimental
data. A linear regression of log(*N*) and strain energy density,
*W*_{o,S}, was calculated to determine C_{1} and
C_{2} for each sample, with the first 100 conditioning cycles excluded
from the regression analysis to best estimate continued decreases in strain
energy density after the initial steep drop. This equation was used for
subsequent cycles to determine cycle-specific stored strain energy density for
each sample.

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, Δ*a*,
divided by the number of cycles between measurements, Δ*N*. Also
for each measurement, strain energy release rate, *T*_{S}, was
calculated as was done for rubber (e.g.
Seldén, 1995):
(7)
with *W*_{o,S} determined as a function of cycle number from
Eqn 6, *a*_{1} the
crack length at the beginning of the interval, *a*_{2} the
crack length at the end of the interval, and ϵ_{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,
*T*_{S}, 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)
where B and β are constants measured for a species. Rearranging and then
integrating this equation yields predicted fatigue lifetime,
*N*_{f}, the number of cycles until fracture of an un-notched
sample (Lake, 1995;
Seldén, 1995) [see eqn
17 in accompanying article (Mach et al.,
2007)]:
(9)
where *a*_{o} is again the effective size of cracks assumed to
occur naturally in the test piece, and *a*_{C} is critical
crack length, the crack size at which complete fracture of the sample occurs,
given as:
(10)
where average *T*_{C,S} for each species was determined from
Eqn 2. Values of strain,ϵ
_{cyclicσ}, and stored strain energy density,
*W*_{o,S,cyclicσ}, 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 β were used in
Eqn 9 to indicate the range of
fatigue lifetimes predicted from crack growth data. B and β 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 β, and for *U. expansa*, B and β 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 β 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.

## Results

### Single-edge-notch, pull-to-break tests

A representative example of critical-strain-energy-release-rate calculation
is shown in Fig. 4, and values
for all species are listed in Table
1. Resilience as a function of strain
(Eqn 2 and
Eqn 3) is given in
Table 2. Initial effective
crack length, *a*_{o}, was calculated from
Eqn 4 as 0.25 mm for
*Mazzaella flaccida* and 0.31 mm for *Ulva expansa*. For
*Porphyra occidentalis, a*_{o} was not determined owing to the
large variation in values of *T*_{C,S} and to the difficulty of
producing fracture away from the grips in un-notched specimens. For use in
subsequent calculations, *a*_{o} for *P. occidentalis*
was set to 0.28 mm, the average *a*_{o} for *M.
flaccida* and *U. expansa* combined.

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 *T*_{C,S} and *T*_{C,T} in
Eqn 5.

For blades with notches less than 0.5 mm in length, *U. expansa*
blades are expected to break at the lowest stresses, and *P.
occidentalis* blades are predicted to break at the highest stresses. Few
data points are given for un-notched blades (on the ordinate) due the
difficulty of producing fracture away from the grips. Actual breaking stresses
for un-notched blades are probably greater than values shown here, especially
for *P. occidentalis* and *U. expansa*, because stronger
specimens were more likely to fail at the grips first.

### Crack propagation in single-edge-notch, repeated-loading tests

In single-edge-notch tests involving repeated cycling, maximum stored
strain energy density, *W*_{o,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*.

Growth of an introduced crack in a *M. flaccida* sample, occurring
over more than 60 000 loading cycles, is depicted in
Fig. 7. This representative
plot demonstrates that crack growth rate, *da/dN*, varies over the
course of crack growth. When the crack was first introduced, the sharpness of
the crack tip resulted in rapid initial crack growth. Once the initial crack
tip blunted, crack growth slowed. When crack length increased by 10% from its
initial value, crack growth rate was assumed independent of initial effects of
razor-blade nicking, and further data collected were used in crack growth
analyses. Crack growth became increasingly rapid as specimen fracture was
approached.

Measured crack growth rates for all cycled specimens are plotted as
functions of energy release rate in Fig.
8A. Critical energy release rates, *T*_{C,S}, were
estimated from these plots (Table
1). In addition, the strain energy release rate below which crack
growth did not occur, *T*_{o}, was estimated as 100 J
m^{–2} for *M. flaccida*, 35 J m^{–2} for
*P. occidentalis*, and 70 J m^{–2} for *U.
expansa*. Stable crack growth was extremely difficult to produce in *U.
expansa* blades; most cracks did not grow or else fractured after only a
few cycles following razor cutting.

### Predictions of lifetime

Crack-growth power-law functions (Eqn
8) were determined from plots of *T versus da/dN*
(Fig. 8A). B and β 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 *r*^{2} 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.

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 β for
Eqn 9 are given in
Table 3, and polynomials for
Eqn 9 that relate maximum
strain, ϵ_{cyclicσ}, and maximum stored strain energy
density, *W*_{o,S,cyclicσ}, to maximum cyclic stress are
listed in Table 4.

Fig. 10 depicts loading
cycles to failure as a function of repeatedly applied stress for *M.
flaccida*, the only species with known drag coefficients.

For Figs 9 and
10, curves are broken for
cyclically applied loading stresses that correspond to energy release rates
below the fatigue threshold, *T*_{o}, for unnotched specimens.
The repeatedly applied stress below which cracks of length
*a*_{o} will not propagate was estimated as 0.7 MN
m^{–2} for *U. expansa*, 0.7 MN m^{–2} for
*M. flaccida*, and 0.3 MN m^{–2} for *P. occidentalis.
U. expansa* blades are predicted to display no crack growth at stresses
less than 75% of pull-to-break strength
(Fig. 9C), while *P.
occidentalis*, at the other extreme, is predicted to fail eventually at
stresses only 10% of its pull-to-break strength
(Fig. 9B). *M. flaccida*
is predicted to accumulate fatigue damage due to stresses imposed by water
velocities greater than approximately 8 m s^{-1}
(Fig. 10). Figs
9 and
10 indicate the number of
cycles required for failure by fatigue cracking: *M. flaccida* may fail
in fewer than 10 000 loadings, *P. occidentalis* may fail in fewer than
30 000 loadings, and *U. expansa* may fail in fewer than 3000 loadings.
The secondary ordinates in Figs
9 and
10 represent lower-bound
estimates of time to failure and indicate that all three species may fail by
fatigue in hours or days.

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.

## Discussion

### Critical energy release rate calculations

For measurements of critical strain energy release rate
(Table 1), coefficients of
variation for *Mazzaella flaccida* and *Ulva expansa
T*_{C} values are consistent with *T*_{C}
coefficients of variation determined for rubber
(Nakajima and Liu, 1993) and
bone (Currey, 1998), materials
for which fracture mechanics has been used successfully to characterize
structural integrity. Only variation in *Porphyra occidentalis
T*_{C} exceeds published values for other biological and soft
materials.

Calculations of *T*_{C} 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. *T*_{C} 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).

### 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).

- a
- measure of crack length, Eqn 1
*a*_{C}- critical crack length, Eqn 9
*a*_{o}- initial (effective) crack length, Eqn 4
*a*_{1}- initial crack length, Eqn 7
*a*_{2}- final crack length, Eqn 7
- B
- fitted constant, Eqn 8
- C
_{1} - fitted constant, Eqn 6
- C
_{2} - fitted constant, Eqn 6
- da/dN
- crack growth rate, Eqn 8
- k
- specimen extension parameter
- N
- cycle number, Eqn 6
*N*_{f}- fatigue lifetime, Eqn 9
- R
- resilience, Eqn 3
- SEN
- single-edge-notch
- T
- strain energy release rate
*T*_{C}- critical strain energy release rate
*T*_{C,S}- critical strain energy release rate calculated from
*W*_{o,S}, Eqn 2 *T*_{C,T}- critical strain energy release rate calculated from
*W*_{o,T}, Eqn 1 *T*_{o}- threshold strain energy release rate
*T*_{S}- strain energy release rate calculated from
*W*_{o,S}, Eqn 7 *W*_{o}- strain energy density
*W*_{o,S}- stored (recoverable) strain energy density, Eqn 2

*W*_{o,S,cyclicσ}*W*_{o,S}as a function of maximum cyclically applied stress, Eqn 9*W*_{o,S,σ}*W*_{o,S}as a function of applied stress, Eqn 5*W*_{o,T}- total absorbed strain energy density, Eqn 1
*W*_{o,T,σ}*W*_{o,T}as a function of applied stress, Eqn 5- β
- fitted constant, Eqn 8
- ϵ
- strain
- ϵbr
- breaking strain, Eqn 1
- ϵcyclicσ
- strain as a function of maximum cyclically applied stress, Eqn 9
- ϵmax
- maximum strain in a cycle, Eqn 7
- ϵσ
- strain as a function of applied stress, Eqn 5
- σ
- applied stress in bulk of specimen
- σmax
- maximum stress in a cycle

## ACKNOWLEDGEMENTS

This manuscript benefited from the suggestions and insights of M. Boller, J. Gosline, B. Grone, L. Hunt, J. Mach, P. Martone, K. Miklasz, L. Miller, and two anonymous reviewers. NSF grants OCE 9633070 and OCE 9985946 to M. Denny supported this research.

- © The Company of Biologists Limited 2007