## SUMMARY

Biomechanical analyses of intertidal and shallow subtidal seaweeds have elucidated ways in which these organisms avoid breakage in the presence of exceptional hydrodynamic forces imposed by pounding surf. However, comparison of algal material properties to maximum hydrodynamic forces predicts lower rates of breakage and dislodgment than are actually observed. Why the disparity between prediction and reality? Most previous research has measured algal material properties during a single application of force, equivalent to a single wave rushing past an alga. In contrast, intertidal macroalgae may experience more than 8000 waves a day. This repeated loading can cause cracks– introduced, for example, by herbivory or abrasion – to grow and eventually cause breakage, yet fatigue crack growth has not previously been taken into account. Here, we present methods from the engineering field of fracture mechanics that can be used to assess consequences of repeated force imposition for seaweeds. These techniques allow quantification of crack growth in wave-swept macroalgae, a first step towards considering macroalgal breakage in the realistic context of repeated force imposition. These analyses can also be applied to many other soft materials.

## Introduction

Wave-swept rocky shores are one of the most physically severe habitats on
the planet. At low tide, organisms in the intertidal zone are exposed to
terrestrial rigors, including substantial temperature fluctuations,
desiccation and increased insolation
(Denny and Wethey, 2001;
Tomanek and Helmuth, 2002).
At high tide, breaking waves are accompanied by water velocities that often
exceed 10 m s^{-1} and impose large hydrodynamic forces (e.g.
Dudgeon and Johnson, 1992;
Gaylord, 1999;
Gaylord, 2000;
Denny and Gaylord, 2002;
Denny, 2006). Nonetheless,
communities of organisms in this harsh environment are highly diverse and
productive (Smith and Kinsey,
1976; Connell,
1978; Leigh et al.,
1987). The unusual presence of dense and diverse assemblages of
organisms in a stressful environment, coupled with the experimental
tractability of the system (as a result of steep environmental gradients,
rapid turnover of organisms, and abundant sessile and slow-moving organisms),
has made wave-swept shores a test bed for ecomechanics. The connections
between community ecology (e.g. Paine,
1966; Paine, 1984;
Dayton, 1971;
Connell, 1978;
Menge, 1995;
Bertness and Leonard, 1997;
Harley and Helmuth, 2003),
physiology (e.g. Wolcott,
1973; Hofmann and Somero,
1995; Stillman and Somero,
1996; Somero,
2002; Stillman,
2002; Tomanek,
2002) and physical adaptations (e.g.
Koehl, 1986;
Carrington, 1990;
Blanchette, 1997;
Martone, 2006) of intertidal
and nearshore organisms have been explored for decades.

Even though physical and physiological intertidal stresses are repetitive in nature, associated with the flow and ebb of tides, most experiments have focused on acute lethal stresses and repercussions for competitive ecological interactions. Sublethal consequences of repeated desiccation, high and low temperatures, hydrodynamic forces and other environmental conditions have proven difficult to address (e.g. Koehl, 1984; Koehl, 1986; Davison and Pearson, 1996). Here we describe methods for quantifying the potentially lethal effects of repeated hydrodynamic forces.

## Hydrodynamic consequences for macroalgae

Although intertidal seaweeds occur in myriad forms, their morphologies
share some common elements. A macroalga attaches to the substratum
*via* a holdfast, from which one or several stem-like structures (often
called stipes) emerge. Each stipe supports one or more blades. Together
holdfast, stipe(s), and blade(s) constitute the thallus of the alga.

For seaweeds, hydrodynamic stresses imposed on thalli represent a
substantial facet of rocky shores' extreme physical environment. Subtidally,
water velocities reach several m s^{-1}
(Denny, 1988), while
magnitudes of water velocities increase manyfold intertidally (commonly to
10–20 m s^{-1}) as waves break and are funneled by substratum
topography (Denny et al.,
2003; Denny,
2006).

Intertidal macroalgae, as sessile organisms, cannot actively avoid the
violent water motion of the wave-swept environment. Instead, as water flows
past an intertidal seaweed, the water exerts force, primarily drag, on the
organism (Gaylord et al.,
1994; Gaylord,
2000; Boller and Carrington,
2006a). Intertidal macroalgae thus experience forces,
predominantly in tension, throughout their lengths with each passing wave. And
macroalgae endure substantial forces: drag forces imposed by water moving at
10 m s^{-1} are comparable to the forces that would be exerted by
winds traveling at 1050 km h^{-1}, nearly Mach 1, if air were
incompressible. Furthermore, intertidal seaweeds must endure these
hydrodynamic forces frequently; approximately 8600 waves break on shore each
day.

Many biomechanical studies have investigated the mechanical properties and morphological attributes that enable wave-swept macroalgae to survive drag forces imposed by breaking waves (e.g. Carrington, 1990; Holbrook et al., 1991; Denny and Gaylord, 2002; Pratt and Johnson, 2002; Kitzes and Denny, 2005; Martone, 2006). These studies have investigated algal material properties primarily in tensile tests, finding macroalgae highly extensible and generally compliant (the opposite of stiff), with low breaking strength, compared to other biomaterials (Hale, 2001; Denny and Gaylord, 2002). In addition, investigations have suggested the importance of algal flexibility, which is in part a consequence of the compliance of algal materials. Seaweeds align, deform and bundle with flow, thereby reconfiguring to reduce drag (Vogel, 1984; Koehl, 1986; Boller and Carrington, 2006b).

To date, studies of algal materials have evaluated their abilities to resist large wave forces through pull-to-break tests, in which samples are loaded in tension until they break. The force required for breakage, normalized as stress (applied bulk force per initial material cross-sectional area), is taken as the ultimate strength, or breaking stress, of the material. This strength is then compared to the stresses imposed by the largest waves to predict an alga's risk of breakage. These comparisons have repeatedly predicted low probabilities of breakage (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), leading to the suggestion that wave-swept algae are mechanically over-designed (Denny, 2006).

However, these predictions are at odds with reality: many seaweeds experience consistent, substantial seasonal breakage and dislodgment (Seymour et al., 1989; Dudgeon and Johnson, 1992; Dudgeon et al., 1999; Johnson, 2001; Pratt and Johnson, 2002), presumably due to wave-induced forces. For example, for two turf-like intertidal macroalgae, Dudgeon and Johnson (Dudgeon and Johnson, 1992) observed wintertime reduction in canopy cover reaching 13% for one species and 30% for another. In kelp forests, Seymour et al. (Seymour et al., 1989) documented mortality ranging from 2 to 94% over four winter seasons. And the sometimes meter-deep piles of seaweed washed up on beaches after storms stand testament to frequent breakage and dislodgment.

Failure in seaweeds assumes a variety of forms. For example, breakage of
blades or load-bearing midribs may occur primarily at distal or marginal
regions. This `tattering' reduces the sizes of algal thalli
(Black, 1976;
Blanchette, 1997;
Dudgeon et al., 1999) and
presumably lowers the risk of more catastrophic damage. Other seaweeds,
especially those with perennial holdfasts capable of regenerating stipes,
break primarily at the holdfast-stipe junction
(Carrington, 1990;
Hawes and Smith, 1995;
Shaughnessy et al., 1996;
Carrington et al., 2001;
Johnson, 2001). For instance,
when experimentally pulling a turf-like red macroalga, Carrington
(Carrington, 1990) found that
90% of thalli broke at the stipe-holdfast junction. Failure of this weak link
ensures survival of the holdfast and allows regeneration of stipes and blades.
Nonetheless, holdfast dislodgment, due to holdfast or substratum failure, does
occur frequently (Black, 1976;
Koehl, 1986;
Seymour et al., 1989;
Utter and Denny, 1996;
Gaylord and Denny, 1997). For
feather-boa kelp (*Egregia laevigata* Setchell) washed onto beaches,
Black (Black, 1976) documented
dislodgment due to holdfast or substratum failure for 35% of individuals, and
Koehl and Wainwright (Koehl and
Wainwright, 1977) determined holdfast detachment responsible for
3–55% of dislodged and broken individuals of a subtidal kelp,
*Nereocystis luetkeana* (Mertens) Postels & Ruprecht, with tangled
plants more likely to fail at the holdfast.

In sum, although wave-swept macroalgae appear over-designed on the basis of measured algal strengths and maximal wave-induced stresses, breakage nonetheless occurs commonly at various locations on macroalgal thalli.

To account for the discrepancy between predicted and observed algal
breakage rates, several external factors, aside from maximum water speeds,
have been invoked. Studies have suggested that stipe entanglement, low-tide
physiological stress, senescence, water-propelled projectiles, and damage from
herbivory or abrasion may increase breakage beyond rates predicted on the
basis of maximum water velocities alone
(Friedland and Denny, 1995;
Utter and Denny, 1996;
Kitzes and Denny, 2005;
Denny, 2006). Along these
lines, two studies, for two different kelp species, linked herbivorous damage
to breakage in approximately 30–50% of solitary individuals washed
ashore (Black, 1976;
Koehl and Wainwright, 1977),
and for the subtidal kelp *N. luetkeana*, Koehl and Wainwright
(Koehl and Wainwright, 1977)
observed breakage at abraded locations on thalli in approximately 40% of
solitary individuals cast ashore. In addition, various researchers have
speculated that *repetition* of wave-induced stress, not just the
maximum stresses, may contribute to algal breakage
(Koehl, 1986;
Hale, 2001;
Kitzes and Denny, 2005).
Experiencing in excess of 8000 waves per day, each with imposition of rapid
flow variation (Gaylord,
1999), intertidal macroalgae may be weakened by the repeated
loading of stresses too low to break them in pull-to-break tests.

In this primer, we focus on the potential role of repeated loads in mechanical failure of wave-swept algae. Repeated loading may act in concert with damage initiated by abrasion and herbivory to cause breakage and dislodgment by fatigue.

## The role of fatigue

Repeated stresses contribute to breakage in several ways. Through fatigue processes, wave-induced stresses below ultimate strength may cause formation of cracks, originating from existing material defects. Although the potential importance of fatigue crack initiation has been cited (Koehl, 1984; Koehl, 1986; Hale, 2001; Kitzes and Denny, 2005), fatigue has not been evaluated in macroalgae. Once a crack has formed in an alga through fatigue, herbivory or abrasion, it can locally amplify stress, thereby decreasing the alga's ultimate strength (where strength is calculated from bulk force applied to a specimen, disregarding local amplifications) and rendering the alga more susceptible to breakage by the imposition of a single large stress (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). Even if an alga containing a crack does not experience stress sufficient to break it in a single loading, repeated stresses below the alga's ultimate strength may cause a crack to grow to a length at which breakage occurs (Hale, 2001). In other words, repeated wave stresses that never reach a cracked alga's ultimate strength may cause fatigue crack growth to the point of complete fracture.

Most structural failures in human construction result from stresses well below the ultimate material strengths of building materials. Consequently, engineering theory includes a robust literature on crack formation through fatigue and on growth of cracks introduced by fatigue or other means. We focus specifically on fracture mechanics theory relevant to crack growth. Fatigue has been evaluated, but not with fracture mechanics methods, in biological materials ranging from bone to elastic proteins (e.g. Caler and Carter, 1989; Currey, 1998; Keaveny et al., 2001; Gosline et al., 2002). Failure in the presence of cracks has been assessed using fracture mechanics in biological materials such as bone, shell, horse hoof and grasses (e.g. Behiri and Bonfield, 1984; Bertram and Gosline, 1986; Vincent, 1991; Kasapi and Gosline, 1996; Kuhn-Spearing et al., 1996; Kasapi and Gosline, 1997; Currey, 1998; Taylor and Lee, 2003). However, these biological studies involving fracture mechanics have focused on the parameters relevant to propagation of cracks when materials fail catastrophically in response to single loadings. Although gradual crack extension may eventually cause complete fracture in conditions of repeated loading, few biological studies have examined incremental crack growth at sub-critical repetitively applied loads. Thus, studies to date do not address our central question: can repeated loading of seaweeds lead to their breakage?

Literature regarding fracture mechanics is almost exclusively written for specialized engineering audiences, and deciphering it, with the aim of applying it to biological situations, remains difficult for most biologists and even for many engineers. In response to the opacity of fracture mechanics literature, we provide here a coherent primer as a starting point for studies of fracture in organisms and as a strong basis for further investigation of the literature. To this end, we present a guide to relevant fracture mechanics techniques. We use consistent terminology for various fracture mechanics methods (a luxury often absent in the literature) and introduce relevant equations with intuitive explanations instead of extensive derivations. Interested readers are guided to cited literature for more detailed descriptions of equations' origins.

Although we use macroalgae as organisms of focus, presented techniques have been applied, at least in part, to biological materials such as bone and horse hoof (Behiri and Bonfield, 1984; Bertram and Gosline, 1986; Kasapi and Gosline, 1997; Currey, 1998) and are relevant to more extensible, softer biological materials such as cnidarian mesoglea, arterial wall, skin, tendon and muscle (Purslow, 1989). We discuss applied wave forces, but imposed stresses from any source can cause repeated-loading damage. The accompanying article (Mach et al., 2007) tests the feasibility of applying these techniques to several macroalgae.

We begin with two central parameters in linear elastic fracture mechanics
(LEFM), stress intensity factor and strain energy release rate, describing use
of these parameters as background for our presentation of techniques relevant
to flexible, extensible materials. [Readers interested in applying LEFM
techniques to botanical materials are referred to Farquhar and Zhao
(Farquhar and Zhao, 2006).] We
then discuss strain energy release rate as it has been applied to fracture and
incremental crack growth in highly extensible elastomeric materials. Finally,
we discuss another parameter, the *J*-integral, that has been effective
in characterizing fracture and fatigue in materials not well described by LEFM
and fracture mechanics of elastomers.

For each fracture mechanics approach, we describe the methods used to evaluate the lifetime of a material with a crack of a particular size. That is, presented parameters enable estimation not only of the force necessary to fracture a specimen in a single loading, but also of the number of smaller repeated loadings that would eventually lead to fracture through incremental crack growth. We hypothesize that, by quantifying the effects of repeated loadings in this manner, we will be better able to predict algal breakage on wave-swept shores.

## Cracks reduce strength

### The stress intensity factor (linear elastic fracture mechanics)

If you attempt to open a bag of peanuts by pulling on the bag in tension, you will likely have trouble tearing the plastic. Notch a side of the bag with scissors, and it will tear with ease. The same phenomenon occurs with seaweeds. Notches – in the form of cracks or discontinuities of any sort– reduce strength (calculated from bulk applied force) because they concentrate stress at their tips (e.g. Andrews, 1968; Shigley and Mischke, 2001). In other words, the material at a crack tip experiences local stresses that exceed the applied stress in the bulk of the specimen. In this fashion, failure may originate at the crack tip even when the bulk stress applied to the rest of the material is not sufficient to cause breakage. Once failure starts, the crack can propagate through the material. As the crack increases in length, it concentrates more stress at its tip, causing crack growth to accelerate and further decreasing the specimen's strength (Broek, 1982).

In the following sections, we consider several types of stress–strain behavior, depicted in Fig. 1, where strain is the ratio of change in length to original length as stress is applied to a material (engineering strain). Linear elastic stress–strain behavior refers to materials with linear relations between stress and strain that return to their original length when unloaded (Fig. 1A). Non-linear elastic materials also recover deformations upon unloading but display non-linear relations between stress and strain (Fig. 1B). Finally, elastic–plastic materials, upon loading, exhibit non-linear relations between stress and strain but additionally, upon unloading, leave irreversible deformation, termed plastic strain (or permanent set) (Fig. 1C). This plastic deformation exemplifies an inelastic strain.

First, we consider linear elastic fracture mechanics (LEFM). Although linear elastic material behavior may not characterize most seaweeds, LEFM provides basic fracture concepts and background information helpful in presenting other fracture mechanics approaches described here.

Stress intensity factor is a parameter that, for linear elastic materials,
characterizes stress fields at very sharp crack tips. As an example, for a
sheet with an edge crack experiencing bending or axially applied stress
(Fig. 2A), the stress intensity
factor, *K*_{I} (measured in
), can be expressed as:
(1)
where σ is the bulk tensile stress applied to the sheet, computed as if
no crack were present; *a* is a measure of crack length; *w* is
the width of the specimen; and *f*(*a*/*w*) is a
dimensionless function of the crack geometry and sheet width. For derivation
and further description of Eqn 1,
see elsewhere (Broek, 1982;
Broek, 1989;
Atkins and Mai, 1985;
Saxena, 1998).
*f*(*a*/*w*), often theoretically derived, assumes
various forms (Saxena, 1998;
Anderson, 2005).
Eqn 1 can be applied to a variety
of specimen and crack geometries with appropriate relations for
*f*(*a*/*w*), as given in Saxena
(Saxena, 1998), Anderson
(Anderson, 2005) and other
sources. As a straightforward example, for a center-cracked sheet
(Fig. 2B) with dimensions much
larger than crack length, *f*(*a*/*w*)=1
(Broek, 1982;
Anderson, 2005), and:
(2)
The subscript `I' of *K*_{I} indicates that this parameter
refers to mode I loading, illustrated in
Fig. 3A. Although mode I
loading is depicted for a specimen with a single edge crack
(Fig. 3A), a sample with a
central crack (Fig. 2B), for
example, pulled in tension will also experience mode I, tensile-opening
loading. Although seaweeds may experience some mode II
(Fig. 3B) and mode III
(Fig. 3C) loading, many of the
imposed loads on seaweeds can be approximated as mode I, tensile-opening
loading. Accordingly, we predominantly address this first loading mode, not
giving analogous equations for other loading modes.

LEFM was originally developed for application to metals, in which concentrated stresses near crack tips cause plastic (permanent) deformations in tip vicinities. As long as plastic deformations are confined to a small zone around the crack tip, LEFM stress intensity factors, as well as strain energy release rate expressions described in the next section, can be applied to metals and other materials.

The critical value of stress intensity factor, *K*_{I}, at
which cracks advance is termed fracture toughness, *K*_{C}.
This critical value of stress intensity factor does depend on loading mode;
*K*_{C} here denotes fracture toughness for mode I loading.
*K*_{C} can be considered a material property in that it
characterizes strength in the presence of a crack. As with properties such as
ultimate tensile strength (breaking stress), fracture toughness typically
varies with factors like temperature and rate of load application. For a given
material, *K*_{C} is approximately constant for different
combinations of crack lengths and applied stresses, as well as for different
specimen geometries, such as the examples shown in
Fig. 2. *K*_{C}
is determined, using a relation such as Eqn
1, by measuring breaking stress for a material specimen of known
dimensions, geometry and crack length. Once determined for one combination of
specimen and crack geometry, *K*_{C} can be applied to assess
resistance to cracking for other geometries of the material.

With units of , stress
intensity factors may seem abstract. A comparison between applied stress and
stress intensity factor may thus be helpful. When a specimen with no crack is
loaded, stress applied to the material can be measured easily. If the loaded
sample breaks in two, the stress at failure is a measure of ultimate tensile
strength. When the material contains a crack, however, due to variation of
stresses within the sample, the applied stress at failure will no longer be
constant for the material. Instead, stress at failure will vary with size and
shape of the crack as well as with geometry of the test specimen, with
geometry determined by relative specimen dimensions and crack location.
Consequently, in the case of a cracked specimen, stress intensity factor,
*K*_{I}, instead of simply applied stress, can be used to
describe the physical state of the material. If the loaded sample breaks in
two, the pertinent parameter becomes not applied stress at fracture, but
stress intensity factor at failure, which is called fracture toughness,
*K*_{C}. Thus, this geometry-independent term (fracture
toughness) is the material property preferred for characterizing loading in
materials with cracks.

Once determined for one combination of specimen geometry and crack size
(and given linear elastic conditions with limited crack-tip plasticity),
fracture toughness can be used to assess the reduction in a material's
strength for different specimen geometries and crack sizes. For
tensile-opening, mode I loading (Fig.
3A), the strength of the material, σ_{C}, is reduced
by the presence of a crack according to:
(3)
where *a* is crack size and *f*(*a*/*w*) is
selected appropriately for the specimen and crack geometry of interest
(Broek, 1989;
Saxena, 1998). In other
words, given fracture toughness, strength of a specimen with known crack
length can be predicted.

LEFM, which includes stress intensity factors and
strain-energy-release-rate expressions described below, performs best for
materials such as glass and ceramics, which have little or no ability to
deform plastically and which have high moduli of elasticity (i.e. they are
stiff materials) and therefore experience relatively small bulk strains when
loaded to fracture. (Modulus of elasticity is the slope of a
stress–strain curve, with units of N m^{–2}.) For such
materials, strength reduction can be predicted reliably with
Eqn 3. For seaweeds, however,
large deformations act to round the crack tip and reduce stress
concentrations, thereby limiting the utility of linear elastic expressions in
predicting strength reduction in the presence of cracks
(Biedka et al., 1987;
Denny et al., 1989;
DeWreede et al., 1992).
Crack-tip rounding ameliorates the strength reduction predicted by
Eqn 3. Nonetheless, cracks of
various geometries have been demonstrated to increase the likelihood of
breakage in several macroalgae (Denny et
al., 1989; DeWreede et al.,
1992).

In summary, stress intensity factor *K*_{I} characterizes
the stress field at a crack tip for linear elastic behavior, and fracture
toughness *K*_{C} quantifies the critical value of this factor
at which a crack will propagate unstably to failure. Higher fracture toughness
values occur in materials more resistant to fracture in the presence of
cracks.

## Crack propagation

### Energy considerations (linear elastic fracture mechanics)

Crack propagation can also be examined in terms of energy
(Broek, 1982;
Broek, 1989). When a material
specimen is pulled, work is done on the sample. In this case, work is force
multiplied by specimen displacement, where displacement is specimen current
length minus specimen initial length. If a sample is pulled to an extension
and held, work is no longer done on the sample, but the sample still contains
energy – as evident, for example, in a stretched rubber band flying
across a room when released. This energy stored in the material is strain
energy, *U*.

Consider a laboratory sample of an elastic material held by grips and
pulled in tension to a fixed displacement. This fixed-grip condition can be
used to explain another important concept: `strain energy release rate'. Work
expended in extending the sample is stored as elastic strain energy,
*U*, and no further work is done once the grips are stationary.
Assuming no dissipative energy loss (e.g. through heat), the density of this
stored energy, the elastic strain energy density (J m^{–3}),
equals the area under the material's stress–strain curve at the fixed
strain imposed by the grips.

Now, imagine introducing a sharp slit (or crack) into this extended
fixed-grip specimen. When the crack extends incrementally, creating new crack
surface *dA*, strain energy in material around the crack will relax,
causing the elastic energy stored in the specimen to decrease by *dU*.
This decrease in stored energy as new crack surface forms is known as the
strain energy release rate, *G* (J m^{–2}), given by:
(4)
Note that energy release rate is defined with respect to crack area, not time,
unlike other common rates such as velocity. Crack surface area, *A*, is
not to be confused with crack length, *a*. Crack area *A* is
calculated as crack length multiplied by specimen thickness.

Some confusion in biological literature has arisen due to differing
definitions of *dA* (Biedka et al.,
1987; Denny et al.,
1989; Hale, 2001).
Sometimes new crack surface area created in crack extension is taken to
include surface area of both faces of the crack, while at other times it
includes surface area on only one face of the crack. Here, *dA* refers
to newly created surface area on one face of the crack, and we encourage use
of this convention to standardize measurements.

For a central crack in a sheet (Fig.
2B) with length and width much greater than crack size, evaluating
Eqn 4 for mode I loading gives
strain energy release rate as:
(5)
where σ is the bulk stress applied to the specimen, *a* is half
the length of the central crack, *E* is the elastic modulus of the
material, and subscript `I' again indicates mode I, tensile-opening loading
(Broek, 1982;
Perez, 2004;
Anderson, 2005).

Fracture testing usually involves pulling a cracked specimen while
recording force-*versus*-displacement data. In such cases, strain
energy release rate can be evaluated at displacements selected by the analyst,
applying an experimental procedure similar to that introduced in Appendix
A.

A crack will extend in a material when strain energy released in crack
growth (expressed as a rate, *dU*/*dA*) exceeds energy required
for the increase in crack surface area, *dV*/*dA.* The energy,
*V*, absorbed during crack extension includes energy to create new
surface as well as any energy dissipated through plastic deformation at the
crack tip. The per-area rate at which energy is required for creation of crack
surface, *dV*/*dA*, is often termed crack resistance,
*R*.

Crack extension occurs when strain energy release rate, *G*, reaches
a critical value *G*_{C} equal to *R*. This crack
advance may be stable or unstable. For example, when the driving force,
*G*, for crack extension increases with crack growth, but crack
resistance *R* remains constant, unstable growth occurs, which means
that, once it begins to elongate, a crack will grow to specimen fracture.
However, when *R* increases more than *G* with crack extension,
stable crack growth occurs, in which crack extension occurs but does not lead
to specimen fracture. In this scenario, a crack can advance a certain distance
(while *R*<*G*) and then stop (when
*R*>*G*), until higher loads are applied. Stable crack
advance occurs mainly in materials that produce large plastic deformations
with crack extension, such as thin plastic grocery bags, for which crack edges
ruffle significantly during tearing, indicating plastic deformations. In most
cases, *G*_{C}, the critical strain energy release rate,
corresponds to onset of unstable growth and fracture. Although we have
described *G* here in terms of stationary grips to explicate the
concept, *G*_{C} in practice is usually determined by pulling
specimens with initial cracks until unstable crack extension occurs.

We thus arrive at two different criteria for rapid crack propagation.
First, the stress intensity factor, *K*_{I}, must equal
fracture toughness, *K*_{C}. Second, the energy release rate,
*G*_{I}, must have reached its critical value,
*G*_{C}. For linear elastic materials, these criteria are
equivalent. From Eqn 2 and
Eqn 5, we can deduce that:
(6)
and at fracture,
(7)
These relations remain valid for different crack and specimen geometries.
Thus, for linear elastic materials, the reduction of material strength in the
presence of a crack can be assessed through either of these criteria. Knowing
*K*_{C} yields *G*_{C}, and *vice versa*.
*K*_{C} and *G*_{C} are properties of a given
material, so for a crack of length *a*, the stress σ required for
fracture in the presence of this crack can be derived either by the
stress-intensity-factor approach (Eqn
1 or Eqn 2) or by the
strain energy release rate approach (Eqn
4 or Eqn 5).

## Fatigue crack growth

### Techniques from linear elastic fracture mechanics

We now discuss a crucial point. As we have just noted, for a material with
a crack, applied stress must reach a critical value corresponding to
*K*_{C} or *G*_{C} for crack extension to occur.
In a given specimen, longer cracks require lower applied stresses to propagate
(Eqn 3). However, *for
repeatedly applied stresses resulting in sub-critical values of
K*_{I} *and G*_{I}*, crack growth can still
occur, in a very slow, incremental manner*. In conditions of repeated
loading, incremental fatigue crack growth at sub-critical
*K*_{I} and *G*_{I} can result in gradual growth
of a crack to a length at which it does rapidly propagate, fracturing the
material.

In other words, for a macroalga with a crack, wave forces causing stresses
less than the material's ultimate strength in the absence of cracks, and less
than the applied stress required for complete fracture, may still cause crack
growth (that is, small increases in crack length) with each force imposition.
At a certain point, the alga's crack may grow to a length at which applied
wave forces reach *G*_{C} and *K*_{C}, leading
to the fracture described in the previous sections. As a result, examining
algal fracture in a manner that considers only maximum wave forces may neglect
breakage that will occur due to incremental crack growth during smaller,
repeated loadings. The curious, important phenomenon of `sub-critical' crack
growth can be characterized (although not mechanistically explained) using the
following LEFM procedure, which allows prediction of a material's lifetime in
conditions of repeated loading. The physical mechanisms for incremental crack
growth have been documented for some engineering materials but not for
macroalgae. For metals, for example, when a crack opens in response to
sub-critical bulk stresses, localized plastic straining at the crack tip
causes the tip to blunt on a microscopic scale, which elongates the crack a
small amount. Upon removal of the bulk stress, the crack tip re-sharpens with
increased length, iteratively elongating with repeated loading
(Pook, 1983).

Predictions of specimen lifetimes proceed in two steps. First, baseline data are generated to describe the pattern of crack growth in a material. This baseline curve (Fig. 4) is then combined with real-world loading histories to predict time to failure.

#### Baseline curve

To create a baseline curve of crack growth, tests are conducted on samples
of a given material with different initial crack sizes or different imposed
stresses. For example, samples can be loaded with stress varying sinusoidally
from zero to a maximum value, with concurrent observation of increases in
crack length, *a*, as a function of cycle number, *N*. Note that
a cycle of imposed stress corresponds to the period spanning from maximum
imposed stress, to minimum imposed stress, and back to maximum imposed stress.
Repeated cyclic loading is often imposed on a specimen, with periodic
measurement of crack length, until the sample fractures. The magnitude of
stress range imposed (maximum stress minus minimum stress) generally exerts
the greatest influence on crack growth rates, as compared to loading
characteristics such as cycling frequency.

From a curve fitted to *a*-*versus*-*N* data, crack
growth rates (mm cycle^{-1}), *da/dN*, are calculated from the
curve's slope at different values of crack length, *a*. For each value
of crack length, a range of stress intensity factor,Δ
*K*_{I}, is computed from the range of applied stress,Δσ
(maximum stress minus minimum stress in a cycle), using a
relation such as Eqn 1. If the
minimum stress is zero, then Δ*K*_{I} equals the maximum
value of *K*_{I} applied in a cycle.

Crack growth rate values, *da/dN*, are then plotted against values
of stress intensity range, Δ*K*_{I}, on logarithmic axes,
where Δ*K*_{I} (for cyclic loading from zero to maximum
stress) equals the value of the stress intensity factor,
*K*_{I}, at the maximum imposed stress
(Fig. 4). Each material has a
characteristic log–log plot of *da/dN versus*Δ
*K*_{I}, which often has the shape depicted in
Fig. 4. Growth rate generally
increases with increasing crack length and with increasing applied stress. At
low Δ*K*_{I}, crack growth is extremely slow, and there
is sometimes a threshold value of Δ*K*_{I} below which no
crack growth occurs (Broek,
1982), shown as Δ*K*_{TH} in
Fig. 4. Similar baseline curves
can be generated for other loading modes
(Fig. 3) as well.

#### Lifetime

Once baseline data are generated, lifetime of a cracked material in
repeated loading conditions can be determined. Determination of lifetime
requires a loading history, a plot of stress applied to a material over time.
The loading history is analyzed to predict when fracture will occur, that is,
when *K*_{C} or *G*_{C} will be reached. There
are multiple approaches to this calculation. In one common LEFM approach,
crack growth is assumed to occur only during rising, tensile ranges of loading
(Nelson, 1977). In other
words, crack growth is assumed to occur only while applied stress stretches a
specimen beyond its initial length, not while specimen extension decreases in
tension and not while a specimen is loaded in compression. In the loading
history, each time that applied tensile stress increases from one value to
another and then drops, that increase (or range) of stress is considered
equivalent to a loading cycle used in generation of the baseline
*da/dN-versus*-Δ*K*_{I} curve
(Fig. 4).

For each successive rising tensile range in a loading history or in a
representative sequence of loading, crack growth for that cycle is added to
current crack length. The increment of crack growth for the cycle,
*da/dN*, corresponds, on the baseline data plot (i.e.
Fig. 4), to the stress
intensity range Δ*K*_{I} of the tensile loading. As
subsequent stress impositions are analyzed, crack length increases, and when
the stress intensity range reaches fracture toughness, fracture is predicted
to occur. In other words, the critical crack length corresponding to
*G*_{C} or *K*_{C} for the applied stress has
been reached, and material rupture is predicted, as long as resistance to
fracture, *R*, does not increase significantly with crack extension, as
described in the previous section. In this fashion, the number of loadings to
failure, or lifetime, of the cracked material is estimated.

In sum, to apply this procedure to a seaweed, one experimentally generates
a baseline curve describing crack growth in response to repeated loading in a
macroalga containing cracks, i.e. a
log–log*da/dN-versus*-Δ*K*_{I} curve
(Fig. 4). Each species, and
perhaps each population, requires a separate baseline curve characterizing its
crack growth behavior. Then a history of imposed wave forces is converted to
imposed wave stresses through consideration of a macroalga's cross-sectional
area. For each rising tensile imposition of stress in this wave stress
history, the calculated range of applied stress, Δσ, combined with
crack length, *a*, can be used to determineΔ
*K*_{I} for the loading (e.g.
Eqn 1). Then, for each rising
stress imposition, the corresponding crack growth is determined fromΔ
*K*_{I} for the stress imposition and from the
corresponding *da/dN* in the baseline data curve. When crack length is
sufficient for Δ*K*_{I} to equal *K*_{C},
breakage of the alga is predicted.

The power of this procedure is that breakage of seaweeds can be examined in a manner that considers each force imposition (each wave) that seaweeds experience. It thus estimates the lifetime of a cracked alga as number of waves required for a crack to grow to failure.

In general, crack growth in engineering materials involves substantial variability (Broek, 1982), and differences between predicted and actual growth result from variability in material cracking and fracture behavior, as well as from idealizations and simplifications in prediction methods. Similar variability likely occurs for macroalgae.

LEFM may not effectively characterize algal fracture. Because LEFM performs best for materials displaying brittle fracture (which seaweeds often do not, compared to engineering materials), alternative methods should be explored, and two such approaches are described below. Even if other methods are found superior for application to seaweeds, LEFM might be well applied to some plant materials such as leaves and wood (Farquhar and Zhao, 2006) or to shells to predict cycles to failure during predator loadings (e.g. Boulding and LaBarbera, 1986; LaBarbera and Merz, 1992) or wave force impositions.

## Crack growth in macroalgae

### Fracture mechanics of elastomeric materials

Macroalgae generally exhibit high extensibility and non-linear stress–strain curves (e.g. Fig. 5), factors that potentially limit the utility of LEFM techniques in analyzing and predicting fracture in seaweeds. However, these characteristics of seaweeds (along with their incompressibility) are similar to the properties of rubber and other elastomers, and a common modified approach to fracture mechanics designed for elastomers is likely applicable to seaweeds.

Based on energy considerations, Rivlin and Thomas
(Rivlin and Thomas, 1953)
pioneered the fracture mechanics of rubber-like materials. They demonstrated
that Eqn 4 for strain energy
release rate can be applied to such materials. Their approach does not assume
linear stress–strain behavior, but does presume elasticity. This
presumption is often approximately true in regions far from crack tips (i.e.
not at stress concentrations) in rubber or macroalgal specimens. Furthermore,
their methods can be applied when bulk strains are large, even 100% or more,
which involves a doubling of specimen length during loading
(Lindley, 1972). To be
consistent with their nomenclature, and that of subsequent researchers in
fracture mechanics of elastomers, when discussing this approach we denote
strain energy release rate as *T* instead of *G*.

Strain energy release rate, *T*, may be found experimentally, as for
linear elastic materials, by loading specimens with initial cracks until
cracks extend unstably (Appendix A). Critical values of *T*_{C}
defined by these loadings are analogous to *G*_{C} for linear
elastic materials. Experimental results demonstrate that
*T*_{C} is approximately constant for different specimen and
crack geometries and therefore can be considered a material property
characterizing resistance to fracture
(Rivlin and Thomas, 1953;
Thomas, 1994). Thomas
(Thomas, 1955;
Thomas, 1994) also showed
that *T* can be related to *W*, the strain energy density around
the surface of a crack tip of diameter *d*:
(8)
where θ is the angle shown in Fig.
6 and *W*(θ) indicates that *W* is a function
of θ. Thomas determined this relation by considering a specimen's change
in energy with an increment of crack extension, which is dominated by elastic
strain energy relaxed in a small zone ahead of the crack tip
(Thomas, 1994). Derivation of
Eqn 8 assumes elastic, including
non-linear elastic, behavior and strains less that 200%
(Thomas, 1955). This equation
indicates that blunting of a crack tip, which increases crack-tip diameter
*d*, can thus be expected to increase values of
*T*_{C}.

Fortunately, relatively simple analytical expressions for *T* have
been derived for a number of important specimen types. These expressions also
assume elastic stress–strain behavior and permit substantial bulk
strains. We briefly describe relevant equations without detailing
corresponding derivations. We refer the reader to cited sources for
derivations.

#### Single-edge-crack specimens

For rubber specimens each with a single edge crack (e.g.
Fig. 2A), pulled in tension,
energy release rate is given by:
(9)
where *W*_{o} is strain energy density (that is, energy per
unit volume) present in the bulk of the specimen and *k* is a parameter
related to specimen extension, with extension expressed as an extension ratioλ
(Rivlin and Thomas,
1953; Lake, 1983).
Extension ratio is simply a ratio of a specimen's current length to its
initial length. That is, an extension ratio of 2 corresponds to strain of 1,
or a doubling in length. Eqn 9
assumes an incompressible elastic material and crack sizes small compared to
specimen width. Greensmith (Greensmith,
1963) found experimentally that *k* is approximately π
at λ=1, then drops to approximately 1.6 at λ=3. Numerical
analysis confirmed these results (Lindley,
1972). Variation of *k* with λ can be adequately
approximated by the simple relation:
(10)
(Atkins and Mai, 1985;
Lake, 1995;
Seldén, 1995).

To determine critical energy release rate, *T*_{C}, a
single-edge-crack specimen with crack length *a* is stretched until it
breaks, and force and extension at fracture are measured.
*W*_{o} in Eqn 9
can be found from the stress–strain curve of a specimen without a crack;
*W*_{o} is the area under that stress–strain curve up to
the bulk stress at which fracture occurred in the cracked specimen.

#### Trouser-tear specimens

Another common method for determining *T*_{C} of rubber-like
materials involves trouser-tear specimens
(Fig. 7). A trouser-tear
specimen consists of a rectangular sheet cut along its long axis to form a
pants-shaped test piece. The `legs' are pulled in opposite directions to
create tearing action (Fig.
3C). Greensmith and Thomas
(Greensmith and Thomas, 1955)
note the convenience of this test piece, for which *T*_{C} and
rate of tear propagation are independent of crack length.

Ahagon et al. (Ahagon et al., 1975) indicate that crack growth in rubber trouser-tear specimens may actually occur on inclined planes such that tensile stresses applied to the legs act in a normal direction to the planes of cracking, which results in mode I cracking (Fig. 3A). On the other hand, Mai and Cotterell (Mai and Cotterell, 1984) and Joe and Kim (Joe and Kim, 1990) note that trouser-tear testing of rubber may involve a mixture of mode I and mode III cracking. For thin sheets of biological materials, the mode or modes of cracking in trouser-tear testing are unclear at this time.

For trouser-tear tests, critical energy release rate can be found from:
(11)
where λ is extension ratio in the legs during tearing, *F* is
force applied to the legs during tearing, *b* is initial thickness of
the test piece, *W*_{o} is strain energy density in the legs
during tearing, and *C* is initial cross-sectional area of both legs
combined, the cross-sectional area of the `body' of the test piece
(Rivlin and Thomas, 1953;
Greensmith and Thomas, 1955;
Lake, 1983). Often, extension
of legs and strain energy stored in legs are assumed negligible relative to
energy associated with crack extension
(Rivlin and Thomas, 1953;
Greensmith and Thomas, 1955;
Seldén, 1995), in
which case:
(12)
Thus, for trouser-tear specimens, critical values of *T*_{C}
can be found by monitoring force required to propagate a crack. Another
approach for finding *T*_{C} with this specimen type involves
finding the net energy, Λ, expended in loading, tearing, and unloading
of a specimen, obtained from the area under a force–displacement plot
(e.g. Fig. 8). Then, critical
energy release rate is given (Purslow,
1983) by:
(13)
where Δ*ab* is the crack extension surface area, taken as
distance traveled by a crack between its initial and final lengths,Δ
*a*, multiplied by thickness of a specimen, *b*.
Fluctuations in force with crack extension of the kind illustrated in
Fig. 8 are typical of
variations observed for macroalgae as well as other pliant biological tissues
(Purslow, 1989).

Biedka et al. (Biedka et al., 1987) and Denny et al. (Denny et al., 1989) determined critical strain energy release rates for seaweeds from trouser-tear tests using formulations similar to Eqn 12 and Eqn 13 except that they referenced fracture energy to two times the fracture surface area. They termed the measured property `work of fracture', even though they measured critical energy release rate. Multiplying their works of fracture by two (and again by two for Denny et al.'s values to account for a spurious factor introduced in their calculations) yields critical strain energy release rates for seaweeds comparable to calculations from Eqn 12 and Eqn 13.

#### Center-crack specimens

For another specimen geometry, a center-cracked specimen
(Fig. 2B), strain energy
release rate for tensile-opening loading
(Fig. 3A) is given by:
(14)
(Seldén, 1995;
Yeoh, 2002). This relation
resembles the formulation for single-edge-notch specimens
(Eqn 9 and
Eqn 10), with crack length
*a* as defined in Fig.
2B. The parameter *k* (here
) varies, strictly speaking,
for center-crack (Eqn 14) and
edge-crack (Eqn 9 and
Eqn 10) specimens because
deformation of an edge crack is less constrained, as discussed, for example,
by Sanford (Sanford, 2003).
However, the difference in *k* for these two specimen types is small
compared to other sources of variability and is often ignored.

In the linear elastic case, Eqn
14 is equivalent to the LEFM expression for *G*
(Eqn 5). Under elastic
conditions, strain energy density, *W*_{o}, is area under a
stress–strain curve. Under linear elastic conditions, this area under
the curve, and thus strain energy density, equalsσ
^{2}/2*E*. In addition, for linear elastic conditions,
specimen extensions are usually small compared to elastomer extensions, so
that λ≅1. Substitution of these values yields
Therefore, in the linear elastic case for a center-cracked specimen,
*T* reduces to *G* in Eqn
5, demonstrating consistency of the approaches. Equivalency of
*T*, which assumes elastic stress–strain behavior and permits
substantial bulk strains, and *G*, which assumes linear elastic
behavior, holds true for other crack and specimen geometries.

#### Effects of viscoelasticity

Many elastomers, as well as macroalgae, display some degree of viscoelastic behavior, a combination of elastic and time-dependent viscous stress–strain behavior. As such, they violate the assumption of elasticity inherent in the analyses so far.

Viscoelastic behavior is characterized, for example, by stresses relaxing if material is moderately stretched and held fixed or by inelastic (creep) strains developing if material experiences constant load over time. Under constant-amplitude, cyclic loading, viscoelastic behavior appears in loops formed by stress–strain curves for repeated loading and unloading cycles (e.g. Fig. 9, as compared to elastic behavior shown in Fig. 1A,B). Viscoelastic loading–unloading loops displayed by seaweeds (Fig. 9A) resemble loops exhibited by rubbers (Fig. 9B). Often, for elastomers and macroalgae, loop width decreases with repeated cycles and tends towards much smaller values for lower specimen extension (e.g. Fig. 9). In addition, a residual inelastic (plastic) strain may remain after the first cycle, but additional increments of residual strains often become negligible (Fig. 9). Similar stress–strain behavior has been observed in other plant tissue (Spatz et al., 1999) and in muscle of soft-bodied arthropods (Dorfmann et al., 2007). This viscoelastic behavior will be most pronounced in crack-tip regions where stresses and strains are much higher than in the bulk of the specimen.

In spite of such complexities in material behavior, range of energy release
rate, Δ*T*, has been used successfully to correlate crack growth
rate under zero-to-tension repeated loading in rubber
(Lake, 1995;
Seldén, 1995;
Mars and Fatemi, 2003;
Schubel et al., 2004;
Busfield et al., 2005), much
as Δ*K* has been used successfully in linear elastic analyses.
Since the value of *T* at the maximum point of a load cycle isΔ
*T* when the minimum point is zero (no extension), *T*
will be used here without Δ.

#### Characterizing crack growth rate

As with LEFM analyses, crack growth per cycle of loading, *da/dN*,
may be evaluated in a nonlinearly elastic material over a range of strain
energy release rates, *T* (Atkins
and Mai, 1985; Lake,
1995; Seldén,
1995). Then, the relationship between *da/dN* and
*T* is determined. The cyclic crack growth per cycle is represented as
some simple function of *T*, a relationship often maintained over a
wide range of crack growth rates:
(15)

When *f*(*T*) is known, incorporating effects of specimen
shape and applied forces, this equation can be used to predict crack growth
rate behavior for a given material, analogous to data presented in
Fig. 4. For specimens each with
a single-edge crack (Fig. 2A),
cycled in tension, energy release rate is described by
Eqn 9. For this equation, for a
cyclically loaded sample, *W*_{o} is taken as the strain energy
density at maximum extension, λ_{max}. *W*_{o}
is often measured directly from stress–strain plots for un-notched
samples, assuming that regions far from a crack behave as if no crack were
present (Atkins and Mai, 1985).
Likewise, *k* is calculated for λ_{max}. Calculated
*T* is plotted *versus da/dN*, as done forΔ
*K*_{I} data (Fig.
4).

For crack extension in regions of intermediate-to-high strain energy
release rates, crack growth rate per cycle commonly follows an empirically
determined power-law form (Lake,
1995; Seldén,
1995):
(16)
Experimentally observed for a variety of rubbers, this relationship may aptly
describe algal crack growth as well because of the resemblance between seaweed
and rubber material behavior. B and β are constants fitted to
*T-versus*-*da/dN* data. Once determined from tests using one
set of crack sizes, *T* values, and particular specimen and crack
geometry, these constants can be used in Eqn
16, for the same material, but for other crack sizes, *T*
values and geometries.

#### Predicting lifetime

Eqn 16 can be integrated to
determine the number of loading cycles, *N*, required for a crack to
grow from length *a*_{1} to length *a*_{2}.
Consider the case of a single-edge-cracked specimen
(Fig. 2A) that experiences
cyclic loading with constant maximum extension. From
Eqn 9, *T* in
Eqn 16 is set equal to
*2kW*_{o}*a*. Eqn
16 then becomes
For these loading conditions, *2kW*_{o} assumes a constant
maximum value because *k* and *W*_{o} are proportional
to extension. One can then integrate
yielding (Lake, 1995;
Seldén, 1995):
(17)
From this equation, the number of loading cycles required for an increment of
crack growth can be determined. β often has a value of 2 to 6
(Lake, 1995). If, in addition,
*a*_{2}*a*_{1},
the second term in parentheses in Eqn
17, (1/*a*_{2}^{β–1}), is
negligibly small and can be dropped:
(18)
In this way, the lifetime, in number of loading cycles, can be determined for
a specimen with a small introduced crack.

This equation allows for powerful predictions [see accompanying article
(Mach et al., 2007)]. Once
baseline crack-growth behavior of an alga has been evaluated with
Eqn 16,
Eqn 18 can be used to estimate
the number of waves of a certain magnitude required to break an edge-cracked
alga through incremental crack growth. That is, for a flat-bladed alga with an
edge crack, Eqn 9 can be used
with stress–strain curves to determine the wave force required to
fracture the alga in a single wave, once critical values of *T* have
been measured. Eqn 18, in
contrast, enables prediction of the wave force required to break an
edge-cracked alga in, for example, 100, 1000 or 10 000 waves, thereby
estimating lifetime of the notched alga in different wave conditions. Wave
force can then be correlated with offshore wave height, given various
assumptions about wave breaking (Gaylord,
1999; Denny et al.,
2003; Helmuth and Denny,
2003; Denny,
2006), allowing predictions of the frequency with which notched
algae experience waves sufficient to break them in these 100, 1000 or 10 000
wave loadings.

*J*-integral and elastic–plastic fracture

An important advance in the field of fracture mechanics was development of
the *J*-integral (Rice,
1968a; Rice,
1968b), a line integral that evaluates the stress–strain
field along a contour surrounding a crack tip
(Fig. 10A). The
*J*-integral is given in Appendix B. *J* and associated
techniques have been applied successfully to assess fracture in the presence
of cracks and to evaluate incremental crack growth even in specimens that
experience substantial plastic deformation at crack tips or dissipative
viscoelastic processes. This more flexible approach may have advantages for
application to seaweeds as well as other biological materials (e.g.
Bertram and Gosline, 1986).

*J* can be thought of as an energy-related parameter, the integral
of two terms that contain strain energy density (or a product with units of
strain energy density, J m^{–3}). Rice
(Rice, 1968a;
Rice, 1968b) derived
*J* for non-linear elastic stress–strain behavior, and the
integral is independent of the contour selected. The *J*-integral also
characterizes intensity of strains in the crack-tip region, analogous in that
respect to the stress intensity factor for linear elastic behavior.
Computational and experimental methods for evaluating the integral are given
in texts such as Kanninen and Popelar
(Kanninen and Popelar, 1985),
Saxena (Saxena, 1998) and
Anderson (Anderson, 2005).

*J* has been found useful in analyzing resistance to crack extension
in materials with extensive plastic deformation emanating from crack tips. The
critical value of *J* at which onset of crack extension occurs,
*J*_{C}, can be considered a material property. Like fracture
toughness *K*_{C}, *J*_{C} is in principle
independent of specimen and crack geometry as well as crack size. Over the
years, *J* has been applied successfully as a fracture parameter for
metals and plastics (Kim et al.,
1989; Bose and Landes,
2003; Wainstein et al.,
2004).

*J* can be interpreted graphically
(Fig. 11). Suppose test
specimens with two different crack lengths, *a* and (*a+da*),
are pulled to a fixed displacement. The area between the curves for two
different crack lengths (Fig.
11) represents the change in stored energy, *dU*, that
occurs for crack extension *da. dU* is (*Jtda*), where
*t* is specimen thickness and *dA=tda*
(Broek, 1982). For this fixed
displacement example,
(19)
The *J*-integral, despite its apparent complexity (Appendix B), is
equivalent to *G* and *T*, given certain assumed material
behaviors. For example, if the path (contour) for the *J*-integral is
taken around the boundaries of an edge-cracked specimen
(Fig. 10B), evaluation of
*J* for rubber-like materials produces results equivalent to
Eqn 9
(Oh, 1976). In this scenario,
*J*_{I}=*T*_{I}=*2kW*_{o}*a*.
Furthermore, calculating a *J*-integral for a rounded crack tip
(Fig. 6) with the contour taken
around the surface of the crack tip yields an expression equivalent to
Eqn 8 for *T*. Also, if
linear elastic behavior is considered, *J*_{I} can be shown to
equal (*K*_{I})^{2}/*E*, which by
Eqn 6 is *G*_{I}
(Rice, 1968a;
Rice, 1968b).

Note that energy release rate usually involves energy released from a
specimen to `feed' a growing crack in elastic material. If large amounts of
plastic deformation occur when a cracked specimen is loaded, much energy
absorbed by the specimen is not recovered upon unloading or crack advance
(Anderson, 2005). In such a
situation, Eqn 19 relates
*J* to the difference in energy absorbed by identical specimens with
two different crack sizes.

Although derived for non-linear elastic behavior
(Fig. 1B), *J* can be
applied to the loading portion of an elastic–plastic stress–strain
curve (e.g. Fig. 1C). The
*J*-integral is not defined for unloading. Nevertheless,Δ
*J* has been successfully correlated with crack growth rate,
*da*/*dN*, for repeated cycles of loading and unloading
(Dowling and Begley, 1976)
even when gross amounts of plasticity accompany crack growth. With such
correlations, *J* can be used, as described for *G* and
*T*, to predict lifetime of materials with cracks, including
seaweeds.

Furthermore, for some elastomeric materials and certain specimen designs,
energy dissipates during specimen deformation and does not contribute to
cracking processes. This dissipated energy should be separated from energy
that contributes to cracking in determining fracture resistance. *J*
may provide a means of partitioning energy in the crack-tip region from energy
dissipated in the bulk of a specimen (Lee
and Donovan, 1985).

Because it accommodates non-linear stress–strain curves and extensive
plastic deformation at crack tips during loading, *J* may be another
fracture parameter that could be fruitfully applied to macroalgal fracture
processes.

## Conclusions

Macroalgae frequently incur cracks due to herbivory, abrasion and fatigue. The fracture mechanics methods outlined here allow assessment of material strength reduction in the presence of cracks and of the effects of stresses below a material's ultimate strength. In repeated loadings imposed by breaking waves, cracks in macroalgal materials likely grow even when individual forces are not sufficient to cause complete fracture. These methods suggest a first avenue for investigating seaweed breakage in the realistic context of repeated wave force imposition.

Furthermore, the methods presented from LEFM, fracture mechanics of elastomers and elastic–plastic fracture mechanics enable prediction of the lifetime for breakage of other biological materials with cracks or flaws in the presence of isolated large loads or of repeated loadings. Although incremental crack growth at sub-critical loads has been largely ignored for many biological materials, such fatigue crack growth may contribute importantly to ecologically, evolutionarily and physiologically relevant breakage in organisms ranging from seaweeds to terrestrial plants to animals.

## Appendix A

Critical strain energy release rate, *T*_{C}, may be found
experimentally through the following procedure. Specimens with introduced
cracks of different lengths *a* (m) are pulled until unstable tearing
occurs. For the various tested specimens, load (N), applied to a specimen
until it tears completely, is plotted against specimen displacement δ
(m), the difference between specimen length at a given time and initial
specimen length (Fig.
A1A).

For a selected value of specimen displacement, δ^{*}, a plot
of stored strain energy *U* (J) *versus* initial crack length is
constructed (Fig. A1B). Given
elastic material behavior, stored strain energy is the area under the
load–displacement curve, in this case between zero displacement andδ
^{*} (Fig.
A1A). If unstable tearing occurs before a specimen reaches the
selected displacement, the specimen's load–deformation curve in
Fig. A1A is extrapolated to
estimate stored energy.

Then, a third plot (Fig.
A1C) of specimen displacement at tearing *versus* initial
crack length is generated from the load–displacement plots in
Fig. A1A. From this plot, for
the displacement selected for Fig.
A1B, δ^{*}, one determines initial crack length,
*a*^{*}, for which tearing would have occurred at the given
displacement from a line fitted to the data points. At this value of
*a*^{*}, the tangent to the
*U*-*versus*-*a* curve
(Fig. A1B) is determined. This
tangent (or slope) yields *dU*/*da*, which can be converted to
critical strain energy release rate, *T*_{C}, or–
*dU*/*dA*, by multiplying the tangent by–
1/specimen thickness. Any selected value of specimen displacement for
construction of the second plot (Fig.
A1B) should yield approximately the same value of
*T*_{C}.

## Appendix B

The *J*-integral (Rice,
1968a; Rice,
1968b) is given by the line integral:
(A1)
where Γ is a path-independent, counterclockwise contour surrounding a
crack tip, *W* is strain energy density, and **P** is a stress
vector acting on an element of path length *ds*
(Fig. A2A). **P** is
defined according to the outward-direction, unit-vector normal to Γ,
**n** (Fig. A2A; see
Eqn A2 and
Eqn A3 below). In
Fig. A2A, **u** denotes a
vector quantifying displacement of the material at the same location
(*ds*), while (∂**u***/*∂*x*) is a
displacement gradient (see Eqn
A4 and Eqn A5
below). Although the stress (traction) vector is usually notated with
`**T**', here we use `**P**' to avoid confusion with strain energy
release rate *T*.

To explain these terms and illustrate evaluation of the integral,
two-dimensional stress will be considered. Two-dimensional stress occurs, for
example, in a stretched thin sheet of material. It is described by three
stress components, σ_{x}, σ_{y}, andσ
_{xy}, acting on a small element of material
(Fig. A2B). Theσ
_{x} and σ_{y} components elongate (or compress)
material, while the τ_{xy} component shears material. Also for
illustration, a rectangular contour Γ around a crack tip will be
considered, depicted in Fig.
A2C.

The traction vector **P** can be expressed as
*P*_{x}**i**+*P*_{y}**j**, where **i**
and **j** are unit vectors in the *x* and *y* directions,
respectively. *P*_{x} and *P*_{y} can be found
from:
(A2)
(A3)
where *n*_{x} and *n*_{y} are components of the
outward unit vector **n** normal to a segment. For instance, along the
segment 1–2 in Fig. A2C,
(*n*_{x}, *n*_{y})=(0, –1), so that
*P*_{x}=–τ_{xy} and
*P*_{y}=–σ_{y}. Along 2–3,
(*n*_{x}, *n*_{y})=(1, 0), so that
*P*_{x}=σ_{x} and
*P*_{y}=τ_{xy}. Along 3–4,
(*n*_{x}, *n*_{y})=(0, 1) so that
*P*_{x}=τ_{xy} and
*P*_{y=}σ_{y}. Along 4–5 and 6–1,
(*n*_{x}, *n*_{y})=(–1, 0), yielding
*P*_{x}=–σ_{x} and
*P*_{y}=–τ_{xy}.

Deformation of an object is commonly represented by a displacement vector
that describes the change in coordinates of a point in the object, from
(*x*_{1}, *y*_{1}) to (*x*_{2},
*y*_{2}). The vector is given by:
(A4)
where *u*=*x*_{2}–*x*_{1} and
*v*=*y*_{2}–*y*_{1}
(Boresi, 2000). The vector can
vary in magnitude and direction from one location to another in an object.
Differentiation of Eqn A4 leads
to:
(A5)

Forming the scalar product
**P**·(∂**u**/∂*x*) along segment 1–2
yields
along segment 3–4, the product is the same, except multiplied by–
1 throughout. Along segment 2–3, the product is
along segments 4–5 and 6–1, the product is the same, except again
multiplied by –1 throughout.

Each segment will contribute to the *J*-integral as indicated in
Table A1. Note that along this
rectilinear path *ds* becomes either *dx* or *dy*,
depending on the segment.

The strain energy density term along a segment can be evaluated for
two-dimensional stress from
(A6)
where ϵ_{x}, ϵ_{y}, and γ_{xy} are
normal (ϵ) and shear (γ) strain components present along a segment.
Integration is carried out from the initial state (no strains) to the final
state (maximum strains reached).

Several approaches exist for evaluating the terms in the
*J*-integral when significant plastic straining is present. For
example, the displacement terms ∂*u*/∂*x* and∂
*v*/∂*x*, as well as∂
*u/*∂*y* and ∂*v*/∂*y*, can
be found by optical methods such as Moire interferometry
(Dadkhah and Kobayashi, 1990),
digital image correlation (Sutton et al.,
1991), and electronic speckle pattern interferometry
(Moore and Tyrer, 1994).
Corresponding strains can then be computed fromϵ
_{x}=∂*u*/∂*x*,ϵ
_{y}=∂*v*/∂*y* andγ
_{xy}=½[(∂*u*/∂*y*)+(∂*v*/∂*x*)].
From a material's stress–strain curve, stress componentsσ
_{x}, σ_{y} and τ_{xy} can be
computed from these strain components using relations between stresses and
strains available from the theory of plasticity
(Sutton et al., 1996;
Chakrabarty, 2006). If a
contour is taken far enough from a crack-tip region to make plastic straining
negligible, simpler linear elastic stress–strain relations can then be
used (Kawahara and Brandon,
1983). Determination of the variation of displacements, strains
and stresses along the segments of the contour provides input to the
evaluation of the terms in the line integrals in
Table A1. The variation of a
given term (e.g. *W*) along a segment can be fitted by a mathematical
function of *x* or *y* to facilitate integration
(Read, 1983).

The *J*-integral may also be determined using commercially available
finite element programs that compute the terms involved in the integral from
loads applied to a given specimen geometry, without the need for experimental
data other than a stress–strain curve.

**List of symbols and abbreviations**

Equation in which each symbol is first used is given (if symbol is used in an equation).

- A
- crack surface area, Eqn 4
- a
- measure of crack length, Eqn 1
*a*_{1}- initial crack length, Eqn 17
*a*_{2}- final crack length, Eqn 17
- B
- fitted constant, Eqn 16
- b
- thickness, Eqn 11
- C
- cross-sectional area of trouser-tear test piece, Eqn 11
- c
- leg width of trouser-tear test piece
- d
- crack-tip diameter, Eqn 8
- ds
- contour element path length, Eqn A1
- da/dN
- crack growth rate, Eqn 15
- E
- modulus of elasticity, Eqn 5
- f(a/w)
- dimensionless function of the crack geometry and sheet width, Eqn 1
- F
- force, Eqn 11
- G
- strain energy release rate, Eqn 4
*G*_{C}- critical strain energy release rate, Eqn 7
*G*_{I}- strain energy release rate (mode I loading), Eqn 5
- J
*J*-integral, Eqn 19*J*_{C}- critical value of
*J* *J*_{I}*J*for mode I loading*K*_{C}- critical stress intensity factor, fracture toughness, Eqn 3
*K*_{I}- stress intensity factor (mode I loading), Eqn 1
- k
- specimen extension parameter, Eqn 9
- LEFM
- linear elastic fracture mechanics
- N
- cycle number, Eqn 17
- n
- normal vector
- P
- traction vector, Eqn A1
- R
- crack resistance
- t
- thickness
- T
- strain energy release rate, Eqn 8
*T*_{C}- critical strain energy release rate, Eqn 11
*T*_{I}- strain energy release rate (mode I loading), Eqn 9
- U
- strain energy, Eqn 4
- u
- displacement vector, Eqn A1
- V
- energy absorbed during crack extension
- W
- strain energy density in crack-tip region, Eqn 8
*W*_{o}- strain energy density in bulk of specimen, Eqn 9
- w
- width, Eqn 1
- β
- fitted constant, Eqn 16
- Γ
- contour surrounding crack tip, Eqn A1
- ΔJ
- range of
*J*-integral - Δ
*K*_{I} - range of stress intensity factor (mode I loading)

- Δ
*K*_{TH} - threshold range of stress intensity factor
- ΔT
- range of strain energy release rate
- Δσ
- range of applied stress
- δ
- displacement
- λmax
- maximum extension ratio
- ϵ
- normal strain component, Eqn A6
- τxy
- shear stress component, Eqn A2
- γ
- shear strain component, Eqn A6
- σ
- applied stress, Eqn 1
- σC
- strength of specimen with crack, Eqn 3
- θ
- crack-tip angle, Eqn 8
- λ
- extension ratio, Eqn 10
- Λ
- energy released in crack extension (trouser-tear test), Eqn 13

## ACKNOWLEDGEMENTS

This paper was inspired and motivated by work begun by B. Hale. The manuscript benefited from the comments 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 project.

- © The Company of Biologists Limited 2007

## References

## SUMMARY

Biomechanical analyses of intertidal and shallow subtidal seaweeds have elucidated ways in which these organisms avoid breakage in the presence of exceptional hydrodynamic forces imposed by pounding surf. However, comparison of algal material properties to maximum hydrodynamic forces predicts lower rates of breakage and dislodgment than are actually observed. Why the disparity between prediction and reality? Most previous research has measured algal material properties during a single application of force, equivalent to a single wave rushing past an alga. In contrast, intertidal macroalgae may experience more than 8000 waves a day. This repeated loading can cause cracks– introduced, for example, by herbivory or abrasion – to grow and eventually cause breakage, yet fatigue crack growth has not previously been taken into account. Here, we present methods from the engineering field of fracture mechanics that can be used to assess consequences of repeated force imposition for seaweeds. These techniques allow quantification of crack growth in wave-swept macroalgae, a first step towards considering macroalgal breakage in the realistic context of repeated force imposition. These analyses can also be applied to many other soft materials.

- fracture mechanics
- breakage
- fatigue
- intertidal
- macroalgae
- seaweed
- biomechanics

## Introduction

Wave-swept rocky shores are one of the most physically severe habitats on
the planet. At low tide, organisms in the intertidal zone are exposed to
terrestrial rigors, including substantial temperature fluctuations,
desiccation and increased insolation
(Denny and Wethey, 2001;
Tomanek and Helmuth, 2002).
At high tide, breaking waves are accompanied by water velocities that often
exceed 10 m s^{-1} and impose large hydrodynamic forces (e.g.
Dudgeon and Johnson, 1992;
Gaylord, 1999;
Gaylord, 2000;
Denny and Gaylord, 2002;
Denny, 2006). Nonetheless,
communities of organisms in this harsh environment are highly diverse and
productive (Smith and Kinsey,
1976; Connell,
1978; Leigh et al.,
1987). The unusual presence of dense and diverse assemblages of
organisms in a stressful environment, coupled with the experimental
tractability of the system (as a result of steep environmental gradients,
rapid turnover of organisms, and abundant sessile and slow-moving organisms),
has made wave-swept shores a test bed for ecomechanics. The connections
between community ecology (e.g. Paine,
1966; Paine, 1984;
Dayton, 1971;
Connell, 1978;
Menge, 1995;
Bertness and Leonard, 1997;
Harley and Helmuth, 2003),
physiology (e.g. Wolcott,
1973; Hofmann and Somero,
1995; Stillman and Somero,
1996; Somero,
2002; Stillman,
2002; Tomanek,
2002) and physical adaptations (e.g.
Koehl, 1986;
Carrington, 1990;
Blanchette, 1997;
Martone, 2006) of intertidal
and nearshore organisms have been explored for decades.

Even though physical and physiological intertidal stresses are repetitive in nature, associated with the flow and ebb of tides, most experiments have focused on acute lethal stresses and repercussions for competitive ecological interactions. Sublethal consequences of repeated desiccation, high and low temperatures, hydrodynamic forces and other environmental conditions have proven difficult to address (e.g. Koehl, 1984; Koehl, 1986; Davison and Pearson, 1996). Here we describe methods for quantifying the potentially lethal effects of repeated hydrodynamic forces.

## Hydrodynamic consequences for macroalgae

Although intertidal seaweeds occur in myriad forms, their morphologies
share some common elements. A macroalga attaches to the substratum
*via* a holdfast, from which one or several stem-like structures (often
called stipes) emerge. Each stipe supports one or more blades. Together
holdfast, stipe(s), and blade(s) constitute the thallus of the alga.

For seaweeds, hydrodynamic stresses imposed on thalli represent a
substantial facet of rocky shores' extreme physical environment. Subtidally,
water velocities reach several m s^{-1}
(Denny, 1988), while
magnitudes of water velocities increase manyfold intertidally (commonly to
10–20 m s^{-1}) as waves break and are funneled by substratum
topography (Denny et al.,
2003; Denny,
2006).

Intertidal macroalgae, as sessile organisms, cannot actively avoid the
violent water motion of the wave-swept environment. Instead, as water flows
past an intertidal seaweed, the water exerts force, primarily drag, on the
organism (Gaylord et al.,
1994; Gaylord,
2000; Boller and Carrington,
2006a). Intertidal macroalgae thus experience forces,
predominantly in tension, throughout their lengths with each passing wave. And
macroalgae endure substantial forces: drag forces imposed by water moving at
10 m s^{-1} are comparable to the forces that would be exerted by
winds traveling at 1050 km h^{-1}, nearly Mach 1, if air were
incompressible. Furthermore, intertidal seaweeds must endure these
hydrodynamic forces frequently; approximately 8600 waves break on shore each
day.

Many biomechanical studies have investigated the mechanical properties and morphological attributes that enable wave-swept macroalgae to survive drag forces imposed by breaking waves (e.g. Carrington, 1990; Holbrook et al., 1991; Denny and Gaylord, 2002; Pratt and Johnson, 2002; Kitzes and Denny, 2005; Martone, 2006). These studies have investigated algal material properties primarily in tensile tests, finding macroalgae highly extensible and generally compliant (the opposite of stiff), with low breaking strength, compared to other biomaterials (Hale, 2001; Denny and Gaylord, 2002). In addition, investigations have suggested the importance of algal flexibility, which is in part a consequence of the compliance of algal materials. Seaweeds align, deform and bundle with flow, thereby reconfiguring to reduce drag (Vogel, 1984; Koehl, 1986; Boller and Carrington, 2006b).

To date, studies of algal materials have evaluated their abilities to resist large wave forces through pull-to-break tests, in which samples are loaded in tension until they break. The force required for breakage, normalized as stress (applied bulk force per initial material cross-sectional area), is taken as the ultimate strength, or breaking stress, of the material. This strength is then compared to the stresses imposed by the largest waves to predict an alga's risk of breakage. These comparisons have repeatedly predicted low probabilities of breakage (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), leading to the suggestion that wave-swept algae are mechanically over-designed (Denny, 2006).

However, these predictions are at odds with reality: many seaweeds experience consistent, substantial seasonal breakage and dislodgment (Seymour et al., 1989; Dudgeon and Johnson, 1992; Dudgeon et al., 1999; Johnson, 2001; Pratt and Johnson, 2002), presumably due to wave-induced forces. For example, for two turf-like intertidal macroalgae, Dudgeon and Johnson (Dudgeon and Johnson, 1992) observed wintertime reduction in canopy cover reaching 13% for one species and 30% for another. In kelp forests, Seymour et al. (Seymour et al., 1989) documented mortality ranging from 2 to 94% over four winter seasons. And the sometimes meter-deep piles of seaweed washed up on beaches after storms stand testament to frequent breakage and dislodgment.

Failure in seaweeds assumes a variety of forms. For example, breakage of
blades or load-bearing midribs may occur primarily at distal or marginal
regions. This `tattering' reduces the sizes of algal thalli
(Black, 1976;
Blanchette, 1997;
Dudgeon et al., 1999) and
presumably lowers the risk of more catastrophic damage. Other seaweeds,
especially those with perennial holdfasts capable of regenerating stipes,
break primarily at the holdfast-stipe junction
(Carrington, 1990;
Hawes and Smith, 1995;
Shaughnessy et al., 1996;
Carrington et al., 2001;
Johnson, 2001). For instance,
when experimentally pulling a turf-like red macroalga, Carrington
(Carrington, 1990) found that
90% of thalli broke at the stipe-holdfast junction. Failure of this weak link
ensures survival of the holdfast and allows regeneration of stipes and blades.
Nonetheless, holdfast dislodgment, due to holdfast or substratum failure, does
occur frequently (Black, 1976;
Koehl, 1986;
Seymour et al., 1989;
Utter and Denny, 1996;
Gaylord and Denny, 1997). For
feather-boa kelp (*Egregia laevigata* Setchell) washed onto beaches,
Black (Black, 1976) documented
dislodgment due to holdfast or substratum failure for 35% of individuals, and
Koehl and Wainwright (Koehl and
Wainwright, 1977) determined holdfast detachment responsible for
3–55% of dislodged and broken individuals of a subtidal kelp,
*Nereocystis luetkeana* (Mertens) Postels & Ruprecht, with tangled
plants more likely to fail at the holdfast.

In sum, although wave-swept macroalgae appear over-designed on the basis of measured algal strengths and maximal wave-induced stresses, breakage nonetheless occurs commonly at various locations on macroalgal thalli.

To account for the discrepancy between predicted and observed algal
breakage rates, several external factors, aside from maximum water speeds,
have been invoked. Studies have suggested that stipe entanglement, low-tide
physiological stress, senescence, water-propelled projectiles, and damage from
herbivory or abrasion may increase breakage beyond rates predicted on the
basis of maximum water velocities alone
(Friedland and Denny, 1995;
Utter and Denny, 1996;
Kitzes and Denny, 2005;
Denny, 2006). Along these
lines, two studies, for two different kelp species, linked herbivorous damage
to breakage in approximately 30–50% of solitary individuals washed
ashore (Black, 1976;
Koehl and Wainwright, 1977),
and for the subtidal kelp *N. luetkeana*, Koehl and Wainwright
(Koehl and Wainwright, 1977)
observed breakage at abraded locations on thalli in approximately 40% of
solitary individuals cast ashore. In addition, various researchers have
speculated that *repetition* of wave-induced stress, not just the
maximum stresses, may contribute to algal breakage
(Koehl, 1986;
Hale, 2001;
Kitzes and Denny, 2005).
Experiencing in excess of 8000 waves per day, each with imposition of rapid
flow variation (Gaylord,
1999), intertidal macroalgae may be weakened by the repeated
loading of stresses too low to break them in pull-to-break tests.

In this primer, we focus on the potential role of repeated loads in mechanical failure of wave-swept algae. Repeated loading may act in concert with damage initiated by abrasion and herbivory to cause breakage and dislodgment by fatigue.

## The role of fatigue

Repeated stresses contribute to breakage in several ways. Through fatigue processes, wave-induced stresses below ultimate strength may cause formation of cracks, originating from existing material defects. Although the potential importance of fatigue crack initiation has been cited (Koehl, 1984; Koehl, 1986; Hale, 2001; Kitzes and Denny, 2005), fatigue has not been evaluated in macroalgae. Once a crack has formed in an alga through fatigue, herbivory or abrasion, it can locally amplify stress, thereby decreasing the alga's ultimate strength (where strength is calculated from bulk force applied to a specimen, disregarding local amplifications) and rendering the alga more susceptible to breakage by the imposition of a single large stress (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). Even if an alga containing a crack does not experience stress sufficient to break it in a single loading, repeated stresses below the alga's ultimate strength may cause a crack to grow to a length at which breakage occurs (Hale, 2001). In other words, repeated wave stresses that never reach a cracked alga's ultimate strength may cause fatigue crack growth to the point of complete fracture.

Most structural failures in human construction result from stresses well below the ultimate material strengths of building materials. Consequently, engineering theory includes a robust literature on crack formation through fatigue and on growth of cracks introduced by fatigue or other means. We focus specifically on fracture mechanics theory relevant to crack growth. Fatigue has been evaluated, but not with fracture mechanics methods, in biological materials ranging from bone to elastic proteins (e.g. Caler and Carter, 1989; Currey, 1998; Keaveny et al., 2001; Gosline et al., 2002). Failure in the presence of cracks has been assessed using fracture mechanics in biological materials such as bone, shell, horse hoof and grasses (e.g. Behiri and Bonfield, 1984; Bertram and Gosline, 1986; Vincent, 1991; Kasapi and Gosline, 1996; Kuhn-Spearing et al., 1996; Kasapi and Gosline, 1997; Currey, 1998; Taylor and Lee, 2003). However, these biological studies involving fracture mechanics have focused on the parameters relevant to propagation of cracks when materials fail catastrophically in response to single loadings. Although gradual crack extension may eventually cause complete fracture in conditions of repeated loading, few biological studies have examined incremental crack growth at sub-critical repetitively applied loads. Thus, studies to date do not address our central question: can repeated loading of seaweeds lead to their breakage?

Literature regarding fracture mechanics is almost exclusively written for specialized engineering audiences, and deciphering it, with the aim of applying it to biological situations, remains difficult for most biologists and even for many engineers. In response to the opacity of fracture mechanics literature, we provide here a coherent primer as a starting point for studies of fracture in organisms and as a strong basis for further investigation of the literature. To this end, we present a guide to relevant fracture mechanics techniques. We use consistent terminology for various fracture mechanics methods (a luxury often absent in the literature) and introduce relevant equations with intuitive explanations instead of extensive derivations. Interested readers are guided to cited literature for more detailed descriptions of equations' origins.

Although we use macroalgae as organisms of focus, presented techniques have been applied, at least in part, to biological materials such as bone and horse hoof (Behiri and Bonfield, 1984; Bertram and Gosline, 1986; Kasapi and Gosline, 1997; Currey, 1998) and are relevant to more extensible, softer biological materials such as cnidarian mesoglea, arterial wall, skin, tendon and muscle (Purslow, 1989). We discuss applied wave forces, but imposed stresses from any source can cause repeated-loading damage. The accompanying article (Mach et al., 2007) tests the feasibility of applying these techniques to several macroalgae.

We begin with two central parameters in linear elastic fracture mechanics
(LEFM), stress intensity factor and strain energy release rate, describing use
of these parameters as background for our presentation of techniques relevant
to flexible, extensible materials. [Readers interested in applying LEFM
techniques to botanical materials are referred to Farquhar and Zhao
(Farquhar and Zhao, 2006).] We
then discuss strain energy release rate as it has been applied to fracture and
incremental crack growth in highly extensible elastomeric materials. Finally,
we discuss another parameter, the *J*-integral, that has been effective
in characterizing fracture and fatigue in materials not well described by LEFM
and fracture mechanics of elastomers.

For each fracture mechanics approach, we describe the methods used to evaluate the lifetime of a material with a crack of a particular size. That is, presented parameters enable estimation not only of the force necessary to fracture a specimen in a single loading, but also of the number of smaller repeated loadings that would eventually lead to fracture through incremental crack growth. We hypothesize that, by quantifying the effects of repeated loadings in this manner, we will be better able to predict algal breakage on wave-swept shores.

## Cracks reduce strength

### The stress intensity factor (linear elastic fracture mechanics)

If you attempt to open a bag of peanuts by pulling on the bag in tension, you will likely have trouble tearing the plastic. Notch a side of the bag with scissors, and it will tear with ease. The same phenomenon occurs with seaweeds. Notches – in the form of cracks or discontinuities of any sort– reduce strength (calculated from bulk applied force) because they concentrate stress at their tips (e.g. Andrews, 1968; Shigley and Mischke, 2001). In other words, the material at a crack tip experiences local stresses that exceed the applied stress in the bulk of the specimen. In this fashion, failure may originate at the crack tip even when the bulk stress applied to the rest of the material is not sufficient to cause breakage. Once failure starts, the crack can propagate through the material. As the crack increases in length, it concentrates more stress at its tip, causing crack growth to accelerate and further decreasing the specimen's strength (Broek, 1982).

In the following sections, we consider several types of stress–strain behavior, depicted in Fig. 1, where strain is the ratio of change in length to original length as stress is applied to a material (engineering strain). Linear elastic stress–strain behavior refers to materials with linear relations between stress and strain that return to their original length when unloaded (Fig. 1A). Non-linear elastic materials also recover deformations upon unloading but display non-linear relations between stress and strain (Fig. 1B). Finally, elastic–plastic materials, upon loading, exhibit non-linear relations between stress and strain but additionally, upon unloading, leave irreversible deformation, termed plastic strain (or permanent set) (Fig. 1C). This plastic deformation exemplifies an inelastic strain.

First, we consider linear elastic fracture mechanics (LEFM). Although linear elastic material behavior may not characterize most seaweeds, LEFM provides basic fracture concepts and background information helpful in presenting other fracture mechanics approaches described here.

Stress intensity factor is a parameter that, for linear elastic materials,
characterizes stress fields at very sharp crack tips. As an example, for a
sheet with an edge crack experiencing bending or axially applied stress
(Fig. 2A), the stress intensity
factor, *K*_{I} (measured in
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\mathrm{Pa}\sqrt{\mathrm{m}}\) \end{document}), can be expressed as:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ K_{\mathrm{I}}={\sigma}\sqrt{{\pi}a}f\left(\frac{a}{w}\right),\] \end{document}(1)
where σ is the bulk tensile stress applied to the sheet, computed as if
no crack were present; *a* is a measure of crack length; *w* is
the width of the specimen; and *f*(*a*/*w*) is a
dimensionless function of the crack geometry and sheet width. For derivation
and further description of Eqn 1,
see elsewhere (Broek, 1982;
Broek, 1989;
Atkins and Mai, 1985;
Saxena, 1998).
*f*(*a*/*w*), often theoretically derived, assumes
various forms (Saxena, 1998;
Anderson, 2005).
Eqn 1 can be applied to a variety
of specimen and crack geometries with appropriate relations for
*f*(*a*/*w*), as given in Saxena
(Saxena, 1998), Anderson
(Anderson, 2005) and other
sources. As a straightforward example, for a center-cracked sheet
(Fig. 2B) with dimensions much
larger than crack length, *f*(*a*/*w*)=1
(Broek, 1982;
Anderson, 2005), and:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ K_{\mathrm{I}}={\sigma}\sqrt{{\pi}a}{\ }.\] \end{document}(2)
The subscript `I' of *K*_{I} indicates that this parameter
refers to mode I loading, illustrated in
Fig. 3A. Although mode I
loading is depicted for a specimen with a single edge crack
(Fig. 3A), a sample with a
central crack (Fig. 2B), for
example, pulled in tension will also experience mode I, tensile-opening
loading. Although seaweeds may experience some mode II
(Fig. 3B) and mode III
(Fig. 3C) loading, many of the
imposed loads on seaweeds can be approximated as mode I, tensile-opening
loading. Accordingly, we predominantly address this first loading mode, not
giving analogous equations for other loading modes.

LEFM was originally developed for application to metals, in which concentrated stresses near crack tips cause plastic (permanent) deformations in tip vicinities. As long as plastic deformations are confined to a small zone around the crack tip, LEFM stress intensity factors, as well as strain energy release rate expressions described in the next section, can be applied to metals and other materials.

The critical value of stress intensity factor, *K*_{I}, at
which cracks advance is termed fracture toughness, *K*_{C}.
This critical value of stress intensity factor does depend on loading mode;
*K*_{C} here denotes fracture toughness for mode I loading.
*K*_{C} can be considered a material property in that it
characterizes strength in the presence of a crack. As with properties such as
ultimate tensile strength (breaking stress), fracture toughness typically
varies with factors like temperature and rate of load application. For a given
material, *K*_{C} is approximately constant for different
combinations of crack lengths and applied stresses, as well as for different
specimen geometries, such as the examples shown in
Fig. 2. *K*_{C}
is determined, using a relation such as Eqn
1, by measuring breaking stress for a material specimen of known
dimensions, geometry and crack length. Once determined for one combination of
specimen and crack geometry, *K*_{C} can be applied to assess
resistance to cracking for other geometries of the material.

With units of \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\mathrm{Pa}\sqrt{\mathrm{m}}\) \end{document}, stress
intensity factors may seem abstract. A comparison between applied stress and
stress intensity factor may thus be helpful. When a specimen with no crack is
loaded, stress applied to the material can be measured easily. If the loaded
sample breaks in two, the stress at failure is a measure of ultimate tensile
strength. When the material contains a crack, however, due to variation of
stresses within the sample, the applied stress at failure will no longer be
constant for the material. Instead, stress at failure will vary with size and
shape of the crack as well as with geometry of the test specimen, with
geometry determined by relative specimen dimensions and crack location.
Consequently, in the case of a cracked specimen, stress intensity factor,
*K*_{I}, instead of simply applied stress, can be used to
describe the physical state of the material. If the loaded sample breaks in
two, the pertinent parameter becomes not applied stress at fracture, but
stress intensity factor at failure, which is called fracture toughness,
*K*_{C}. Thus, this geometry-independent term (fracture
toughness) is the material property preferred for characterizing loading in
materials with cracks.

Once determined for one combination of specimen geometry and crack size
(and given linear elastic conditions with limited crack-tip plasticity),
fracture toughness can be used to assess the reduction in a material's
strength for different specimen geometries and crack sizes. For
tensile-opening, mode I loading (Fig.
3A), the strength of the material, σ_{C}, is reduced
by the presence of a crack according to:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ {\sigma}_{\mathrm{C}}=\frac{K_{\mathrm{C}}}{\sqrt{{\pi}a}f\left(\frac{a}{w}\right)},\] \end{document}(3)
where *a* is crack size and *f*(*a*/*w*) is
selected appropriately for the specimen and crack geometry of interest
(Broek, 1989;
Saxena, 1998). In other
words, given fracture toughness, strength of a specimen with known crack
length can be predicted.

LEFM, which includes stress intensity factors and
strain-energy-release-rate expressions described below, performs best for
materials such as glass and ceramics, which have little or no ability to
deform plastically and which have high moduli of elasticity (i.e. they are
stiff materials) and therefore experience relatively small bulk strains when
loaded to fracture. (Modulus of elasticity is the slope of a
stress–strain curve, with units of N m^{–2}.) For such
materials, strength reduction can be predicted reliably with
Eqn 3. For seaweeds, however,
large deformations act to round the crack tip and reduce stress
concentrations, thereby limiting the utility of linear elastic expressions in
predicting strength reduction in the presence of cracks
(Biedka et al., 1987;
Denny et al., 1989;
DeWreede et al., 1992).
Crack-tip rounding ameliorates the strength reduction predicted by
Eqn 3. Nonetheless, cracks of
various geometries have been demonstrated to increase the likelihood of
breakage in several macroalgae (Denny et
al., 1989; DeWreede et al.,
1992).

In summary, stress intensity factor *K*_{I} characterizes
the stress field at a crack tip for linear elastic behavior, and fracture
toughness *K*_{C} quantifies the critical value of this factor
at which a crack will propagate unstably to failure. Higher fracture toughness
values occur in materials more resistant to fracture in the presence of
cracks.

## Crack propagation

### Energy considerations (linear elastic fracture mechanics)

Crack propagation can also be examined in terms of energy
(Broek, 1982;
Broek, 1989). When a material
specimen is pulled, work is done on the sample. In this case, work is force
multiplied by specimen displacement, where displacement is specimen current
length minus specimen initial length. If a sample is pulled to an extension
and held, work is no longer done on the sample, but the sample still contains
energy – as evident, for example, in a stretched rubber band flying
across a room when released. This energy stored in the material is strain
energy, *U*.

Consider a laboratory sample of an elastic material held by grips and
pulled in tension to a fixed displacement. This fixed-grip condition can be
used to explain another important concept: `strain energy release rate'. Work
expended in extending the sample is stored as elastic strain energy,
*U*, and no further work is done once the grips are stationary.
Assuming no dissipative energy loss (e.g. through heat), the density of this
stored energy, the elastic strain energy density (J m^{–3}),
equals the area under the material's stress–strain curve at the fixed
strain imposed by the grips.

Now, imagine introducing a sharp slit (or crack) into this extended
fixed-grip specimen. When the crack extends incrementally, creating new crack
surface *dA*, strain energy in material around the crack will relax,
causing the elastic energy stored in the specimen to decrease by *dU*.
This decrease in stored energy as new crack surface forms is known as the
strain energy release rate, *G* (J m^{–2}), given by:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ G=-\frac{dU}{dA}.\] \end{document}(4)
Note that energy release rate is defined with respect to crack area, not time,
unlike other common rates such as velocity. Crack surface area, *A*, is
not to be confused with crack length, *a*. Crack area *A* is
calculated as crack length multiplied by specimen thickness.

Some confusion in biological literature has arisen due to differing
definitions of *dA* (Biedka et al.,
1987; Denny et al.,
1989; Hale, 2001).
Sometimes new crack surface area created in crack extension is taken to
include surface area of both faces of the crack, while at other times it
includes surface area on only one face of the crack. Here, *dA* refers
to newly created surface area on one face of the crack, and we encourage use
of this convention to standardize measurements.

For a central crack in a sheet (Fig.
2B) with length and width much greater than crack size, evaluating
Eqn 4 for mode I loading gives
strain energy release rate as:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ G_{\mathrm{I}}=\frac{{\pi}{\sigma}^{2}a}{E},\] \end{document}(5)
where σ is the bulk stress applied to the specimen, *a* is half
the length of the central crack, *E* is the elastic modulus of the
material, and subscript `I' again indicates mode I, tensile-opening loading
(Broek, 1982;
Perez, 2004;
Anderson, 2005).

Fracture testing usually involves pulling a cracked specimen while
recording force-*versus*-displacement data. In such cases, strain
energy release rate can be evaluated at displacements selected by the analyst,
applying an experimental procedure similar to that introduced in Appendix
A.

A crack will extend in a material when strain energy released in crack
growth (expressed as a rate, *dU*/*dA*) exceeds energy required
for the increase in crack surface area, *dV*/*dA.* The energy,
*V*, absorbed during crack extension includes energy to create new
surface as well as any energy dissipated through plastic deformation at the
crack tip. The per-area rate at which energy is required for creation of crack
surface, *dV*/*dA*, is often termed crack resistance,
*R*.

Crack extension occurs when strain energy release rate, *G*, reaches
a critical value *G*_{C} equal to *R*. This crack
advance may be stable or unstable. For example, when the driving force,
*G*, for crack extension increases with crack growth, but crack
resistance *R* remains constant, unstable growth occurs, which means
that, once it begins to elongate, a crack will grow to specimen fracture.
However, when *R* increases more than *G* with crack extension,
stable crack growth occurs, in which crack extension occurs but does not lead
to specimen fracture. In this scenario, a crack can advance a certain distance
(while *R*<*G*) and then stop (when
*R*>*G*), until higher loads are applied. Stable crack
advance occurs mainly in materials that produce large plastic deformations
with crack extension, such as thin plastic grocery bags, for which crack edges
ruffle significantly during tearing, indicating plastic deformations. In most
cases, *G*_{C}, the critical strain energy release rate,
corresponds to onset of unstable growth and fracture. Although we have
described *G* here in terms of stationary grips to explicate the
concept, *G*_{C} in practice is usually determined by pulling
specimens with initial cracks until unstable crack extension occurs.

We thus arrive at two different criteria for rapid crack propagation.
First, the stress intensity factor, *K*_{I}, must equal
fracture toughness, *K*_{C}. Second, the energy release rate,
*G*_{I}, must have reached its critical value,
*G*_{C}. For linear elastic materials, these criteria are
equivalent. From Eqn 2 and
Eqn 5, we can deduce that:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ K_{\mathrm{I}}=\sqrt{G_{\mathrm{I}}E}\] \end{document}(6)
and at fracture,
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ K_{\mathrm{C}}=\sqrt{G_{\mathrm{C}}E}{\ }.\] \end{document}(7)
These relations remain valid for different crack and specimen geometries.
Thus, for linear elastic materials, the reduction of material strength in the
presence of a crack can be assessed through either of these criteria. Knowing
*K*_{C} yields *G*_{C}, and *vice versa*.
*K*_{C} and *G*_{C} are properties of a given
material, so for a crack of length *a*, the stress σ required for
fracture in the presence of this crack can be derived either by the
stress-intensity-factor approach (Eqn
1 or Eqn 2) or by the
strain energy release rate approach (Eqn
4 or Eqn 5).

## Fatigue crack growth

### Techniques from linear elastic fracture mechanics

We now discuss a crucial point. As we have just noted, for a material with
a crack, applied stress must reach a critical value corresponding to
*K*_{C} or *G*_{C} for crack extension to occur.
In a given specimen, longer cracks require lower applied stresses to propagate
(Eqn 3). However, *for
repeatedly applied stresses resulting in sub-critical values of
K*_{I} *and G*_{I}*, crack growth can still
occur, in a very slow, incremental manner*. In conditions of repeated
loading, incremental fatigue crack growth at sub-critical
*K*_{I} and *G*_{I} can result in gradual growth
of a crack to a length at which it does rapidly propagate, fracturing the
material.

In other words, for a macroalga with a crack, wave forces causing stresses
less than the material's ultimate strength in the absence of cracks, and less
than the applied stress required for complete fracture, may still cause crack
growth (that is, small increases in crack length) with each force imposition.
At a certain point, the alga's crack may grow to a length at which applied
wave forces reach *G*_{C} and *K*_{C}, leading
to the fracture described in the previous sections. As a result, examining
algal fracture in a manner that considers only maximum wave forces may neglect
breakage that will occur due to incremental crack growth during smaller,
repeated loadings. The curious, important phenomenon of `sub-critical' crack
growth can be characterized (although not mechanistically explained) using the
following LEFM procedure, which allows prediction of a material's lifetime in
conditions of repeated loading. The physical mechanisms for incremental crack
growth have been documented for some engineering materials but not for
macroalgae. For metals, for example, when a crack opens in response to
sub-critical bulk stresses, localized plastic straining at the crack tip
causes the tip to blunt on a microscopic scale, which elongates the crack a
small amount. Upon removal of the bulk stress, the crack tip re-sharpens with
increased length, iteratively elongating with repeated loading
(Pook, 1983).

Predictions of specimen lifetimes proceed in two steps. First, baseline data are generated to describe the pattern of crack growth in a material. This baseline curve (Fig. 4) is then combined with real-world loading histories to predict time to failure.

#### Baseline curve

To create a baseline curve of crack growth, tests are conducted on samples
of a given material with different initial crack sizes or different imposed
stresses. For example, samples can be loaded with stress varying sinusoidally
from zero to a maximum value, with concurrent observation of increases in
crack length, *a*, as a function of cycle number, *N*. Note that
a cycle of imposed stress corresponds to the period spanning from maximum
imposed stress, to minimum imposed stress, and back to maximum imposed stress.
Repeated cyclic loading is often imposed on a specimen, with periodic
measurement of crack length, until the sample fractures. The magnitude of
stress range imposed (maximum stress minus minimum stress) generally exerts
the greatest influence on crack growth rates, as compared to loading
characteristics such as cycling frequency.

From a curve fitted to *a*-*versus*-*N* data, crack
growth rates (mm cycle^{-1}), *da/dN*, are calculated from the
curve's slope at different values of crack length, *a*. For each value
of crack length, a range of stress intensity factor,Δ
*K*_{I}, is computed from the range of applied stress,Δσ
(maximum stress minus minimum stress in a cycle), using a
relation such as Eqn 1. If the
minimum stress is zero, then Δ*K*_{I} equals the maximum
value of *K*_{I} applied in a cycle.

Crack growth rate values, *da/dN*, are then plotted against values
of stress intensity range, Δ*K*_{I}, on logarithmic axes,
where Δ*K*_{I} (for cyclic loading from zero to maximum
stress) equals the value of the stress intensity factor,
*K*_{I}, at the maximum imposed stress
(Fig. 4). Each material has a
characteristic log–log plot of *da/dN versus*Δ
*K*_{I}, which often has the shape depicted in
Fig. 4. Growth rate generally
increases with increasing crack length and with increasing applied stress. At
low Δ*K*_{I}, crack growth is extremely slow, and there
is sometimes a threshold value of Δ*K*_{I} below which no
crack growth occurs (Broek,
1982), shown as Δ*K*_{TH} in
Fig. 4. Similar baseline curves
can be generated for other loading modes
(Fig. 3) as well.

#### Lifetime

Once baseline data are generated, lifetime of a cracked material in
repeated loading conditions can be determined. Determination of lifetime
requires a loading history, a plot of stress applied to a material over time.
The loading history is analyzed to predict when fracture will occur, that is,
when *K*_{C} or *G*_{C} will be reached. There
are multiple approaches to this calculation. In one common LEFM approach,
crack growth is assumed to occur only during rising, tensile ranges of loading
(Nelson, 1977). In other
words, crack growth is assumed to occur only while applied stress stretches a
specimen beyond its initial length, not while specimen extension decreases in
tension and not while a specimen is loaded in compression. In the loading
history, each time that applied tensile stress increases from one value to
another and then drops, that increase (or range) of stress is considered
equivalent to a loading cycle used in generation of the baseline
*da/dN-versus*-Δ*K*_{I} curve
(Fig. 4).

For each successive rising tensile range in a loading history or in a
representative sequence of loading, crack growth for that cycle is added to
current crack length. The increment of crack growth for the cycle,
*da/dN*, corresponds, on the baseline data plot (i.e.
Fig. 4), to the stress
intensity range Δ*K*_{I} of the tensile loading. As
subsequent stress impositions are analyzed, crack length increases, and when
the stress intensity range reaches fracture toughness, fracture is predicted
to occur. In other words, the critical crack length corresponding to
*G*_{C} or *K*_{C} for the applied stress has
been reached, and material rupture is predicted, as long as resistance to
fracture, *R*, does not increase significantly with crack extension, as
described in the previous section. In this fashion, the number of loadings to
failure, or lifetime, of the cracked material is estimated.

In sum, to apply this procedure to a seaweed, one experimentally generates
a baseline curve describing crack growth in response to repeated loading in a
macroalga containing cracks, i.e. a
log–log*da/dN-versus*-Δ*K*_{I} curve
(Fig. 4). Each species, and
perhaps each population, requires a separate baseline curve characterizing its
crack growth behavior. Then a history of imposed wave forces is converted to
imposed wave stresses through consideration of a macroalga's cross-sectional
area. For each rising tensile imposition of stress in this wave stress
history, the calculated range of applied stress, Δσ, combined with
crack length, *a*, can be used to determineΔ
*K*_{I} for the loading (e.g.
Eqn 1). Then, for each rising
stress imposition, the corresponding crack growth is determined fromΔ
*K*_{I} for the stress imposition and from the
corresponding *da/dN* in the baseline data curve. When crack length is
sufficient for Δ*K*_{I} to equal *K*_{C},
breakage of the alga is predicted.

The power of this procedure is that breakage of seaweeds can be examined in a manner that considers each force imposition (each wave) that seaweeds experience. It thus estimates the lifetime of a cracked alga as number of waves required for a crack to grow to failure.

In general, crack growth in engineering materials involves substantial variability (Broek, 1982), and differences between predicted and actual growth result from variability in material cracking and fracture behavior, as well as from idealizations and simplifications in prediction methods. Similar variability likely occurs for macroalgae.

LEFM may not effectively characterize algal fracture. Because LEFM performs best for materials displaying brittle fracture (which seaweeds often do not, compared to engineering materials), alternative methods should be explored, and two such approaches are described below. Even if other methods are found superior for application to seaweeds, LEFM might be well applied to some plant materials such as leaves and wood (Farquhar and Zhao, 2006) or to shells to predict cycles to failure during predator loadings (e.g. Boulding and LaBarbera, 1986; LaBarbera and Merz, 1992) or wave force impositions.

## Crack growth in macroalgae

### Fracture mechanics of elastomeric materials

Macroalgae generally exhibit high extensibility and non-linear stress–strain curves (e.g. Fig. 5), factors that potentially limit the utility of LEFM techniques in analyzing and predicting fracture in seaweeds. However, these characteristics of seaweeds (along with their incompressibility) are similar to the properties of rubber and other elastomers, and a common modified approach to fracture mechanics designed for elastomers is likely applicable to seaweeds.

Based on energy considerations, Rivlin and Thomas
(Rivlin and Thomas, 1953)
pioneered the fracture mechanics of rubber-like materials. They demonstrated
that Eqn 4 for strain energy
release rate can be applied to such materials. Their approach does not assume
linear stress–strain behavior, but does presume elasticity. This
presumption is often approximately true in regions far from crack tips (i.e.
not at stress concentrations) in rubber or macroalgal specimens. Furthermore,
their methods can be applied when bulk strains are large, even 100% or more,
which involves a doubling of specimen length during loading
(Lindley, 1972). To be
consistent with their nomenclature, and that of subsequent researchers in
fracture mechanics of elastomers, when discussing this approach we denote
strain energy release rate as *T* instead of *G*.

Strain energy release rate, *T*, may be found experimentally, as for
linear elastic materials, by loading specimens with initial cracks until
cracks extend unstably (Appendix A). Critical values of *T*_{C}
defined by these loadings are analogous to *G*_{C} for linear
elastic materials. Experimental results demonstrate that
*T*_{C} is approximately constant for different specimen and
crack geometries and therefore can be considered a material property
characterizing resistance to fracture
(Rivlin and Thomas, 1953;
Thomas, 1994). Thomas
(Thomas, 1955;
Thomas, 1994) also showed
that *T* can be related to *W*, the strain energy density around
the surface of a crack tip of diameter *d*:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ T=d{{\int}_{0}^{\frac{{\pi}}{2}}}W({\theta})\mathrm{cos}{\theta}{\ }d{\theta},\] \end{document}(8)
where θ is the angle shown in Fig.
6 and *W*(θ) indicates that *W* is a function
of θ. Thomas determined this relation by considering a specimen's change
in energy with an increment of crack extension, which is dominated by elastic
strain energy relaxed in a small zone ahead of the crack tip
(Thomas, 1994). Derivation of
Eqn 8 assumes elastic, including
non-linear elastic, behavior and strains less that 200%
(Thomas, 1955). This equation
indicates that blunting of a crack tip, which increases crack-tip diameter
*d*, can thus be expected to increase values of
*T*_{C}.

Fortunately, relatively simple analytical expressions for *T* have
been derived for a number of important specimen types. These expressions also
assume elastic stress–strain behavior and permit substantial bulk
strains. We briefly describe relevant equations without detailing
corresponding derivations. We refer the reader to cited sources for
derivations.

#### Single-edge-crack specimens

For rubber specimens each with a single edge crack (e.g.
Fig. 2A), pulled in tension,
energy release rate is given by:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ T_{\mathrm{I}}=2kW_{\mathrm{o}}a,\] \end{document}(9)
where *W*_{o} is strain energy density (that is, energy per
unit volume) present in the bulk of the specimen and *k* is a parameter
related to specimen extension, with extension expressed as an extension ratioλ
(Rivlin and Thomas,
1953; Lake, 1983).
Extension ratio is simply a ratio of a specimen's current length to its
initial length. That is, an extension ratio of 2 corresponds to strain of 1,
or a doubling in length. Eqn 9
assumes an incompressible elastic material and crack sizes small compared to
specimen width. Greensmith (Greensmith,
1963) found experimentally that *k* is approximately π
at λ=1, then drops to approximately 1.6 at λ=3. Numerical
analysis confirmed these results (Lindley,
1972). Variation of *k* with λ can be adequately
approximated by the simple relation:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ k=\frac{{\pi}}{\sqrt{{\lambda}}}.\] \end{document}(10)
(Atkins and Mai, 1985;
Lake, 1995;
Seldén, 1995).

To determine critical energy release rate, *T*_{C}, a
single-edge-crack specimen with crack length *a* is stretched until it
breaks, and force and extension at fracture are measured.
*W*_{o} in Eqn 9
can be found from the stress–strain curve of a specimen without a crack;
*W*_{o} is the area under that stress–strain curve up to
the bulk stress at which fracture occurred in the cracked specimen.

#### Trouser-tear specimens

Another common method for determining *T*_{C} of rubber-like
materials involves trouser-tear specimens
(Fig. 7). A trouser-tear
specimen consists of a rectangular sheet cut along its long axis to form a
pants-shaped test piece. The `legs' are pulled in opposite directions to
create tearing action (Fig.
3C). Greensmith and Thomas
(Greensmith and Thomas, 1955)
note the convenience of this test piece, for which *T*_{C} and
rate of tear propagation are independent of crack length.

Ahagon et al. (Ahagon et al., 1975) indicate that crack growth in rubber trouser-tear specimens may actually occur on inclined planes such that tensile stresses applied to the legs act in a normal direction to the planes of cracking, which results in mode I cracking (Fig. 3A). On the other hand, Mai and Cotterell (Mai and Cotterell, 1984) and Joe and Kim (Joe and Kim, 1990) note that trouser-tear testing of rubber may involve a mixture of mode I and mode III cracking. For thin sheets of biological materials, the mode or modes of cracking in trouser-tear testing are unclear at this time.

For trouser-tear tests, critical energy release rate can be found from:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ T_{\mathrm{C}}=\frac{2{\lambda}F}{b}-\frac{W_{\mathrm{o}}C}{b},\] \end{document}(11)
where λ is extension ratio in the legs during tearing, *F* is
force applied to the legs during tearing, *b* is initial thickness of
the test piece, *W*_{o} is strain energy density in the legs
during tearing, and *C* is initial cross-sectional area of both legs
combined, the cross-sectional area of the `body' of the test piece
(Rivlin and Thomas, 1953;
Greensmith and Thomas, 1955;
Lake, 1983). Often, extension
of legs and strain energy stored in legs are assumed negligible relative to
energy associated with crack extension
(Rivlin and Thomas, 1953;
Greensmith and Thomas, 1955;
Seldén, 1995), in
which case:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ T_{\mathrm{C}}=\frac{2F}{b}.\] \end{document}(12)
Thus, for trouser-tear specimens, critical values of *T*_{C}
can be found by monitoring force required to propagate a crack. Another
approach for finding *T*_{C} with this specimen type involves
finding the net energy, Λ, expended in loading, tearing, and unloading
of a specimen, obtained from the area under a force–displacement plot
(e.g. Fig. 8). Then, critical
energy release rate is given (Purslow,
1983) by:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ T_{\mathrm{C}}=\frac{{\Lambda}}{{\Delta}ab}.\] \end{document}(13)
where Δ*ab* is the crack extension surface area, taken as
distance traveled by a crack between its initial and final lengths,Δ
*a*, multiplied by thickness of a specimen, *b*.
Fluctuations in force with crack extension of the kind illustrated in
Fig. 8 are typical of
variations observed for macroalgae as well as other pliant biological tissues
(Purslow, 1989).

Biedka et al. (Biedka et al., 1987) and Denny et al. (Denny et al., 1989) determined critical strain energy release rates for seaweeds from trouser-tear tests using formulations similar to Eqn 12 and Eqn 13 except that they referenced fracture energy to two times the fracture surface area. They termed the measured property `work of fracture', even though they measured critical energy release rate. Multiplying their works of fracture by two (and again by two for Denny et al.'s values to account for a spurious factor introduced in their calculations) yields critical strain energy release rates for seaweeds comparable to calculations from Eqn 12 and Eqn 13.

#### Center-crack specimens

For another specimen geometry, a center-cracked specimen
(Fig. 2B), strain energy
release rate for tensile-opening loading
(Fig. 3A) is given by:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ T_{\mathrm{I}}=\frac{{\pi}W_{\mathrm{o}}(2a)}{\sqrt{{\lambda}}}.\] \end{document}(14)
(Seldén, 1995;
Yeoh, 2002). This relation
resembles the formulation for single-edge-notch specimens
(Eqn 9 and
Eqn 10), with crack length
*a* as defined in Fig.
2B. The parameter *k* (here
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(k={\pi}{/}\sqrt{{\lambda}}\) \end{document}) varies, strictly speaking,
for center-crack (Eqn 14) and
edge-crack (Eqn 9 and
Eqn 10) specimens because
deformation of an edge crack is less constrained, as discussed, for example,
by Sanford (Sanford, 2003).
However, the difference in *k* for these two specimen types is small
compared to other sources of variability and is often ignored.

In the linear elastic case, Eqn
14 is equivalent to the LEFM expression for *G*
(Eqn 5). Under elastic
conditions, strain energy density, *W*_{o}, is area under a
stress–strain curve. Under linear elastic conditions, this area under
the curve, and thus strain energy density, equalsσ
^{2}/2*E*. In addition, for linear elastic conditions,
specimen extensions are usually small compared to elastomer extensions, so
that λ≅1. Substitution of these values yields
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[T_{\mathrm{I}}=\frac{{\pi}\frac{{\sigma}^{2}}{2E}(2a)}{\sqrt{1}}=\frac{{\pi}{\sigma}^{2}a}{E}=G_{\mathrm{I}}.\] \end{document}
Therefore, in the linear elastic case for a center-cracked specimen,
*T* reduces to *G* in Eqn
5, demonstrating consistency of the approaches. Equivalency of
*T*, which assumes elastic stress–strain behavior and permits
substantial bulk strains, and *G*, which assumes linear elastic
behavior, holds true for other crack and specimen geometries.

#### Effects of viscoelasticity

Many elastomers, as well as macroalgae, display some degree of viscoelastic behavior, a combination of elastic and time-dependent viscous stress–strain behavior. As such, they violate the assumption of elasticity inherent in the analyses so far.

Viscoelastic behavior is characterized, for example, by stresses relaxing if material is moderately stretched and held fixed or by inelastic (creep) strains developing if material experiences constant load over time. Under constant-amplitude, cyclic loading, viscoelastic behavior appears in loops formed by stress–strain curves for repeated loading and unloading cycles (e.g. Fig. 9, as compared to elastic behavior shown in Fig. 1A,B). Viscoelastic loading–unloading loops displayed by seaweeds (Fig. 9A) resemble loops exhibited by rubbers (Fig. 9B). Often, for elastomers and macroalgae, loop width decreases with repeated cycles and tends towards much smaller values for lower specimen extension (e.g. Fig. 9). In addition, a residual inelastic (plastic) strain may remain after the first cycle, but additional increments of residual strains often become negligible (Fig. 9). Similar stress–strain behavior has been observed in other plant tissue (Spatz et al., 1999) and in muscle of soft-bodied arthropods (Dorfmann et al., 2007). This viscoelastic behavior will be most pronounced in crack-tip regions where stresses and strains are much higher than in the bulk of the specimen.

In spite of such complexities in material behavior, range of energy release
rate, Δ*T*, has been used successfully to correlate crack growth
rate under zero-to-tension repeated loading in rubber
(Lake, 1995;
Seldén, 1995;
Mars and Fatemi, 2003;
Schubel et al., 2004;
Busfield et al., 2005), much
as Δ*K* has been used successfully in linear elastic analyses.
Since the value of *T* at the maximum point of a load cycle isΔ
*T* when the minimum point is zero (no extension), *T*
will be used here without Δ.

#### Characterizing crack growth rate

As with LEFM analyses, crack growth per cycle of loading, *da/dN*,
may be evaluated in a nonlinearly elastic material over a range of strain
energy release rates, *T* (Atkins
and Mai, 1985; Lake,
1995; Seldén,
1995). Then, the relationship between *da/dN* and
*T* is determined. The cyclic crack growth per cycle is represented as
some simple function of *T*, a relationship often maintained over a
wide range of crack growth rates:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ \frac{da}{dN}=f(T).\] \end{document}(15)

When *f*(*T*) is known, incorporating effects of specimen
shape and applied forces, this equation can be used to predict crack growth
rate behavior for a given material, analogous to data presented in
Fig. 4. For specimens each with
a single-edge crack (Fig. 2A),
cycled in tension, energy release rate is described by
Eqn 9. For this equation, for a
cyclically loaded sample, *W*_{o} is taken as the strain energy
density at maximum extension, λ_{max}. *W*_{o}
is often measured directly from stress–strain plots for un-notched
samples, assuming that regions far from a crack behave as if no crack were
present (Atkins and Mai, 1985).
Likewise, *k* is calculated for λ_{max}. Calculated
*T* is plotted *versus da/dN*, as done forΔ
*K*_{I} data (Fig.
4).

For crack extension in regions of intermediate-to-high strain energy
release rates, crack growth rate per cycle commonly follows an empirically
determined power-law form (Lake,
1995; Seldén,
1995):
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ \frac{da}{dN}=\mathrm{B}T^{{\beta}}.\] \end{document}(16)
Experimentally observed for a variety of rubbers, this relationship may aptly
describe algal crack growth as well because of the resemblance between seaweed
and rubber material behavior. B and β are constants fitted to
*T-versus*-*da/dN* data. Once determined from tests using one
set of crack sizes, *T* values, and particular specimen and crack
geometry, these constants can be used in Eqn
16, for the same material, but for other crack sizes, *T*
values and geometries.

#### Predicting lifetime

Eqn 16 can be integrated to
determine the number of loading cycles, *N*, required for a crack to
grow from length *a*_{1} to length *a*_{2}.
Consider the case of a single-edge-cracked specimen
(Fig. 2A) that experiences
cyclic loading with constant maximum extension. From
Eqn 9, *T* in
Eqn 16 is set equal to
*2kW*_{o}*a*. Eqn
16 then becomes
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\frac{da}{dN}=B(2kW_{\mathrm{o}}a)^{{\beta}},{\ }\mathrm{or}{\ }\frac{da}{a^{{\beta}}}=B(2kW_{\mathrm{o}})^{{\beta}}dN.\] \end{document}
For these loading conditions, *2kW*_{o} assumes a constant
maximum value because *k* and *W*_{o} are proportional
to extension. One can then integrate
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[{{\int}_{0}^{N}}dN=\frac{1}{B(2kW_{\mathrm{o}})^{{\beta}}}{{\int}_{a_{1}}^{a_{2}}}\frac{da}{a^{{\beta}}},\] \end{document}
yielding (Lake, 1995;
Seldén, 1995):
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ N=\frac{1}{({\beta}-1)\mathrm{B}(2kW_{\mathrm{o}})^{{\beta}}}\left(\frac{1}{a_{1}^{{\beta}-1}}-\frac{1}{a_{2}^{{\beta}-1}}\right).\] \end{document}(17)
From this equation, the number of loading cycles required for an increment of
crack growth can be determined. β often has a value of 2 to 6
(Lake, 1995). If, in addition,
*a*_{2}*a*_{1},
the second term in parentheses in Eqn
17, (1/*a*_{2}^{β–1}), is
negligibly small and can be dropped:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ N=\frac{1}{({\beta}-1)\mathrm{B}(2kW_{\mathrm{o}})^{{\beta}}}\left(\frac{1}{a_{1}^{{\beta}-1}}\right)\] \end{document}(18)
In this way, the lifetime, in number of loading cycles, can be determined for
a specimen with a small introduced crack.

This equation allows for powerful predictions [see accompanying article
(Mach et al., 2007)]. Once
baseline crack-growth behavior of an alga has been evaluated with
Eqn 16,
Eqn 18 can be used to estimate
the number of waves of a certain magnitude required to break an edge-cracked
alga through incremental crack growth. That is, for a flat-bladed alga with an
edge crack, Eqn 9 can be used
with stress–strain curves to determine the wave force required to
fracture the alga in a single wave, once critical values of *T* have
been measured. Eqn 18, in
contrast, enables prediction of the wave force required to break an
edge-cracked alga in, for example, 100, 1000 or 10 000 waves, thereby
estimating lifetime of the notched alga in different wave conditions. Wave
force can then be correlated with offshore wave height, given various
assumptions about wave breaking (Gaylord,
1999; Denny et al.,
2003; Helmuth and Denny,
2003; Denny,
2006), allowing predictions of the frequency with which notched
algae experience waves sufficient to break them in these 100, 1000 or 10 000
wave loadings.

*J*-integral and elastic–plastic fracture

An important advance in the field of fracture mechanics was development of
the *J*-integral (Rice,
1968a; Rice,
1968b), a line integral that evaluates the stress–strain
field along a contour surrounding a crack tip
(Fig. 10A). The
*J*-integral is given in Appendix B. *J* and associated
techniques have been applied successfully to assess fracture in the presence
of cracks and to evaluate incremental crack growth even in specimens that
experience substantial plastic deformation at crack tips or dissipative
viscoelastic processes. This more flexible approach may have advantages for
application to seaweeds as well as other biological materials (e.g.
Bertram and Gosline, 1986).

*J* can be thought of as an energy-related parameter, the integral
of two terms that contain strain energy density (or a product with units of
strain energy density, J m^{–3}). Rice
(Rice, 1968a;
Rice, 1968b) derived
*J* for non-linear elastic stress–strain behavior, and the
integral is independent of the contour selected. The *J*-integral also
characterizes intensity of strains in the crack-tip region, analogous in that
respect to the stress intensity factor for linear elastic behavior.
Computational and experimental methods for evaluating the integral are given
in texts such as Kanninen and Popelar
(Kanninen and Popelar, 1985),
Saxena (Saxena, 1998) and
Anderson (Anderson, 2005).

*J* has been found useful in analyzing resistance to crack extension
in materials with extensive plastic deformation emanating from crack tips. The
critical value of *J* at which onset of crack extension occurs,
*J*_{C}, can be considered a material property. Like fracture
toughness *K*_{C}, *J*_{C} is in principle
independent of specimen and crack geometry as well as crack size. Over the
years, *J* has been applied successfully as a fracture parameter for
metals and plastics (Kim et al.,
1989; Bose and Landes,
2003; Wainstein et al.,
2004).

*J* can be interpreted graphically
(Fig. 11). Suppose test
specimens with two different crack lengths, *a* and (*a+da*),
are pulled to a fixed displacement. The area between the curves for two
different crack lengths (Fig.
11) represents the change in stored energy, *dU*, that
occurs for crack extension *da. dU* is (*Jtda*), where
*t* is specimen thickness and *dA=tda*
(Broek, 1982). For this fixed
displacement example,
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ J=-\frac{dU}{dA}.\] \end{document}(19)
The *J*-integral, despite its apparent complexity (Appendix B), is
equivalent to *G* and *T*, given certain assumed material
behaviors. For example, if the path (contour) for the *J*-integral is
taken around the boundaries of an edge-cracked specimen
(Fig. 10B), evaluation of
*J* for rubber-like materials produces results equivalent to
Eqn 9
(Oh, 1976). In this scenario,
*J*_{I}=*T*_{I}=*2kW*_{o}*a*.
Furthermore, calculating a *J*-integral for a rounded crack tip
(Fig. 6) with the contour taken
around the surface of the crack tip yields an expression equivalent to
Eqn 8 for *T*. Also, if
linear elastic behavior is considered, *J*_{I} can be shown to
equal (*K*_{I})^{2}/*E*, which by
Eqn 6 is *G*_{I}
(Rice, 1968a;
Rice, 1968b).

Note that energy release rate usually involves energy released from a
specimen to `feed' a growing crack in elastic material. If large amounts of
plastic deformation occur when a cracked specimen is loaded, much energy
absorbed by the specimen is not recovered upon unloading or crack advance
(Anderson, 2005). In such a
situation, Eqn 19 relates
*J* to the difference in energy absorbed by identical specimens with
two different crack sizes.

Although derived for non-linear elastic behavior
(Fig. 1B), *J* can be
applied to the loading portion of an elastic–plastic stress–strain
curve (e.g. Fig. 1C). The
*J*-integral is not defined for unloading. Nevertheless,Δ
*J* has been successfully correlated with crack growth rate,
*da*/*dN*, for repeated cycles of loading and unloading
(Dowling and Begley, 1976)
even when gross amounts of plasticity accompany crack growth. With such
correlations, *J* can be used, as described for *G* and
*T*, to predict lifetime of materials with cracks, including
seaweeds.

Furthermore, for some elastomeric materials and certain specimen designs,
energy dissipates during specimen deformation and does not contribute to
cracking processes. This dissipated energy should be separated from energy
that contributes to cracking in determining fracture resistance. *J*
may provide a means of partitioning energy in the crack-tip region from energy
dissipated in the bulk of a specimen (Lee
and Donovan, 1985).

Because it accommodates non-linear stress–strain curves and extensive
plastic deformation at crack tips during loading, *J* may be another
fracture parameter that could be fruitfully applied to macroalgal fracture
processes.

## Conclusions

Macroalgae frequently incur cracks due to herbivory, abrasion and fatigue. The fracture mechanics methods outlined here allow assessment of material strength reduction in the presence of cracks and of the effects of stresses below a material's ultimate strength. In repeated loadings imposed by breaking waves, cracks in macroalgal materials likely grow even when individual forces are not sufficient to cause complete fracture. These methods suggest a first avenue for investigating seaweed breakage in the realistic context of repeated wave force imposition.

Furthermore, the methods presented from LEFM, fracture mechanics of elastomers and elastic–plastic fracture mechanics enable prediction of the lifetime for breakage of other biological materials with cracks or flaws in the presence of isolated large loads or of repeated loadings. Although incremental crack growth at sub-critical loads has been largely ignored for many biological materials, such fatigue crack growth may contribute importantly to ecologically, evolutionarily and physiologically relevant breakage in organisms ranging from seaweeds to terrestrial plants to animals.

## Appendix A

Critical strain energy release rate, *T*_{C}, may be found
experimentally through the following procedure. Specimens with introduced
cracks of different lengths *a* (m) are pulled until unstable tearing
occurs. For the various tested specimens, load (N), applied to a specimen
until it tears completely, is plotted against specimen displacement δ
(m), the difference between specimen length at a given time and initial
specimen length (Fig.
A1A).

For a selected value of specimen displacement, δ^{*}, a plot
of stored strain energy *U* (J) *versus* initial crack length is
constructed (Fig. A1B). Given
elastic material behavior, stored strain energy is the area under the
load–displacement curve, in this case between zero displacement andδ
^{*} (Fig.
A1A). If unstable tearing occurs before a specimen reaches the
selected displacement, the specimen's load–deformation curve in
Fig. A1A is extrapolated to
estimate stored energy.

Then, a third plot (Fig.
A1C) of specimen displacement at tearing *versus* initial
crack length is generated from the load–displacement plots in
Fig. A1A. From this plot, for
the displacement selected for Fig.
A1B, δ^{*}, one determines initial crack length,
*a*^{*}, for which tearing would have occurred at the given
displacement from a line fitted to the data points. At this value of
*a*^{*}, the tangent to the
*U*-*versus*-*a* curve
(Fig. A1B) is determined. This
tangent (or slope) yields *dU*/*da*, which can be converted to
critical strain energy release rate, *T*_{C}, or–
*dU*/*dA*, by multiplying the tangent by–
1/specimen thickness. Any selected value of specimen displacement for
construction of the second plot (Fig.
A1B) should yield approximately the same value of
*T*_{C}.

## Appendix B

The *J*-integral (Rice,
1968a; Rice,
1968b) is given by the line integral:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ J={{\int}_{{\Gamma}}}\left(Wdy-\mathbf{P}{\cdot}\frac{{\partial}\mathbf{u}}{{\partial}x}ds\right),\] \end{document}(A1)
where Γ is a path-independent, counterclockwise contour surrounding a
crack tip, *W* is strain energy density, and **P** is a stress
vector acting on an element of path length *ds*
(Fig. A2A). **P** is
defined according to the outward-direction, unit-vector normal to Γ,
**n** (Fig. A2A; see
Eqn A2 and
Eqn A3 below). In
Fig. A2A, **u** denotes a
vector quantifying displacement of the material at the same location
(*ds*), while (∂**u***/*∂*x*) is a
displacement gradient (see Eqn
A4 and Eqn A5
below). Although the stress (traction) vector is usually notated with
`**T**', here we use `**P**' to avoid confusion with strain energy
release rate *T*.

To explain these terms and illustrate evaluation of the integral,
two-dimensional stress will be considered. Two-dimensional stress occurs, for
example, in a stretched thin sheet of material. It is described by three
stress components, σ_{x}, σ_{y}, andσ
_{xy}, acting on a small element of material
(Fig. A2B). Theσ
_{x} and σ_{y} components elongate (or compress)
material, while the τ_{xy} component shears material. Also for
illustration, a rectangular contour Γ around a crack tip will be
considered, depicted in Fig.
A2C.

The traction vector **P** can be expressed as
*P*_{x}**i**+*P*_{y}**j**, where **i**
and **j** are unit vectors in the *x* and *y* directions,
respectively. *P*_{x} and *P*_{y} can be found
from:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ P_{\mathrm{x}}=n_{\mathrm{x}}{\sigma}_{\mathrm{x}}+n_{\mathrm{y}}{\tau}_{\mathrm{xy}},\] \end{document}(A2)
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ P_{\mathrm{y}}=n_{\mathrm{x}}{\tau}_{\mathrm{xy}}+n_{\mathrm{y}}{\sigma}_{\mathrm{y}},\] \end{document}(A3)
where *n*_{x} and *n*_{y} are components of the
outward unit vector **n** normal to a segment. For instance, along the
segment 1–2 in Fig. A2C,
(*n*_{x}, *n*_{y})=(0, –1), so that
*P*_{x}=–τ_{xy} and
*P*_{y}=–σ_{y}. Along 2–3,
(*n*_{x}, *n*_{y})=(1, 0), so that
*P*_{x}=σ_{x} and
*P*_{y}=τ_{xy}. Along 3–4,
(*n*_{x}, *n*_{y})=(0, 1) so that
*P*_{x}=τ_{xy} and
*P*_{y=}σ_{y}. Along 4–5 and 6–1,
(*n*_{x}, *n*_{y})=(–1, 0), yielding
*P*_{x}=–σ_{x} and
*P*_{y}=–τ_{xy}.

Deformation of an object is commonly represented by a displacement vector
that describes the change in coordinates of a point in the object, from
(*x*_{1}, *y*_{1}) to (*x*_{2},
*y*_{2}). The vector is given by:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ \mathbf{u}=u\mathbf{i}+v\mathbf{j},\] \end{document}(A4)
where *u*=*x*_{2}–*x*_{1} and
*v*=*y*_{2}–*y*_{1}
(Boresi, 2000). The vector can
vary in magnitude and direction from one location to another in an object.
Differentiation of Eqn A4 leads
to:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ \frac{{\partial}\mathbf{u}}{{\partial}x}=\frac{{\partial}u}{{\partial}x}\mathbf{i}+\frac{{\partial}v}{{\partial}x}\mathbf{j}.\] \end{document}(A5)

Forming the scalar product
**P**·(∂**u**/∂*x*) along segment 1–2
yields
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[-{\tau}_{\mathrm{xy}}\frac{{\partial}u}{{\partial}x}-{\sigma}_{\mathrm{y}}\frac{{\partial}v}{{\partial}x};\] \end{document}
along segment 3–4, the product is the same, except multiplied by–
1 throughout. Along segment 2–3, the product is
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[{\sigma}_{\mathrm{x}}\frac{{\partial}u}{{\partial}x}+{\tau}_{\mathrm{xy}}\frac{{\partial}v}{{\partial}x};\] \end{document}
along segments 4–5 and 6–1, the product is the same, except again
multiplied by –1 throughout.

Each segment will contribute to the *J*-integral as indicated in
Table A1. Note that along this
rectilinear path *ds* becomes either *dx* or *dy*,
depending on the segment.

The strain energy density term along a segment can be evaluated for
two-dimensional stress from
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ W={{\int}}({\sigma}_{\mathrm{x}}d{\epsilon}_{\mathrm{x}}+{\sigma}_{\mathrm{y}}d{\epsilon}_{\mathrm{y}}+{\tau}_{\mathrm{xy}}d{\gamma}_{\mathrm{xy}}),\] \end{document}(A6)
where ϵ_{x}, ϵ_{y}, and γ_{xy} are
normal (ϵ) and shear (γ) strain components present along a segment.
Integration is carried out from the initial state (no strains) to the final
state (maximum strains reached).

Several approaches exist for evaluating the terms in the
*J*-integral when significant plastic straining is present. For
example, the displacement terms ∂*u*/∂*x* and∂
*v*/∂*x*, as well as∂
*u/*∂*y* and ∂*v*/∂*y*, can
be found by optical methods such as Moire interferometry
(Dadkhah and Kobayashi, 1990),
digital image correlation (Sutton et al.,
1991), and electronic speckle pattern interferometry
(Moore and Tyrer, 1994).
Corresponding strains can then be computed fromϵ
_{x}=∂*u*/∂*x*,ϵ
_{y}=∂*v*/∂*y* andγ
_{xy}=½[(∂*u*/∂*y*)+(∂*v*/∂*x*)].
From a material's stress–strain curve, stress componentsσ
_{x}, σ_{y} and τ_{xy} can be
computed from these strain components using relations between stresses and
strains available from the theory of plasticity
(Sutton et al., 1996;
Chakrabarty, 2006). If a
contour is taken far enough from a crack-tip region to make plastic straining
negligible, simpler linear elastic stress–strain relations can then be
used (Kawahara and Brandon,
1983). Determination of the variation of displacements, strains
and stresses along the segments of the contour provides input to the
evaluation of the terms in the line integrals in
Table A1. The variation of a
given term (e.g. *W*) along a segment can be fitted by a mathematical
function of *x* or *y* to facilitate integration
(Read, 1983).

The *J*-integral may also be determined using commercially available
finite element programs that compute the terms involved in the integral from
loads applied to a given specimen geometry, without the need for experimental
data other than a stress–strain curve.

**List of symbols and abbreviations**

Equation in which each symbol is first used is given (if symbol is used in an equation).

- A
- crack surface area, Eqn 4
- a
- measure of crack length, Eqn 1
*a*_{1}- initial crack length, Eqn 17
*a*_{2}- final crack length, Eqn 17
- B
- fitted constant, Eqn 16
- b
- thickness, Eqn 11
- C
- cross-sectional area of trouser-tear test piece, Eqn 11
- c
- leg width of trouser-tear test piece
- d
- crack-tip diameter, Eqn 8
- ds
- contour element path length, Eqn A1
- da/dN
- crack growth rate, Eqn 15
- E
- modulus of elasticity, Eqn 5
- f(a/w)
- dimensionless function of the crack geometry and sheet width, Eqn 1
- F
- force, Eqn 11
- G
- strain energy release rate, Eqn 4
*G*_{C}- critical strain energy release rate, Eqn 7
*G*_{I}- strain energy release rate (mode I loading), Eqn 5
- J
*J*-integral, Eqn 19*J*_{C}- critical value of
*J* *J*_{I}*J*for mode I loading*K*_{C}- critical stress intensity factor, fracture toughness, Eqn 3
*K*_{I}- stress intensity factor (mode I loading), Eqn 1
- k
- specimen extension parameter, Eqn 9
- LEFM
- linear elastic fracture mechanics
- N
- cycle number, Eqn 17
- n
- normal vector
- P
- traction vector, Eqn A1
- R
- crack resistance
- t
- thickness
- T
- strain energy release rate, Eqn 8
*T*_{C}- critical strain energy release rate, Eqn 11
*T*_{I}- strain energy release rate (mode I loading), Eqn 9
- U
- strain energy, Eqn 4
- u
- displacement vector, Eqn A1
- V
- energy absorbed during crack extension
- W
- strain energy density in crack-tip region, Eqn 8
*W*_{o}- strain energy density in bulk of specimen, Eqn 9
- w
- width, Eqn 1
- β
- fitted constant, Eqn 16
- Γ
- contour surrounding crack tip, Eqn A1
- ΔJ
- range of
*J*-integral - Δ
*K*_{I} - range of stress intensity factor (mode I loading)

- Δ
*K*_{TH} - threshold range of stress intensity factor
- ΔT
- range of strain energy release rate
- Δσ
- range of applied stress
- δ
- displacement
- λmax
- maximum extension ratio
- ϵ
- normal strain component, Eqn A6
- τxy
- shear stress component, Eqn A2
- γ
- shear strain component, Eqn A6
- σ
- applied stress, Eqn 1
- σC
- strength of specimen with crack, Eqn 3
- θ
- crack-tip angle, Eqn 8
- λ
- extension ratio, Eqn 10
- Λ
- energy released in crack extension (trouser-tear test), Eqn 13

## ACKNOWLEDGEMENTS

This paper was inspired and motivated by work begun by B. Hale. The manuscript benefited from the comments 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 project.