Marine Botany

Mazzaella flaccida (was Iridaea): Biomechanics

Stressed Out in 3-D

  • The iridescent cuticle of I. flaccida gives the thallus some surprising material properties. I performed trouser-tear tests on a number of blades to investigate differences in their the work of fracture and the effects of the cuticle and reproductive sori on crack propagation. The blade's mechanism of fracture serves to reduce stress concentrations and may be the source of the surprisingly high strength of the decidedly anisotropic carposporophyte blade. Variations in thallus strength between seasons, life phases, and levels of exposure correlate with expected patterns.

    Mechanisms of Fracture

    Using the trouser-tear machine in the Denny lab here at Hopkins Marine Station I measured the work of fracture for a variety of specimens of Iridaea. I cut a rectangle from the center of each thallus and slit this rectangle either longitudinally along the blade


    or laterally across the blade .

    I calibrated tensometer by attaching weights to the pulling apparatus. After I clamped the testing apparatus to the loose flaps of the thallus, the machine pulled the blade apart at a speed of 16 cm per minute.


    Instead of separating the blade cleanly along a plane perpendicular to the direction of tearing force (as shown above), all tests produced a diagonal crack surface as shown below.


    The diagonal was far more pronounced in the carposporangial blade than the gametophyte. Under the microscope it appeared that the diagonal crack surface was due to the heteromorphic nature of the thallus layers. The iridescent cuticle ripped at a different rate than the filamentous medulla.

    Strain Energy

    This mechanism of fracture forces the production of a greater surface area and increases the amount of strain energy which much be released during fracture to pay for separation. The area of the fracture surface is calculated as the product of the crack width and crack length7:

    crack surface area = 2 * crack width * crack length

    If the blade had ripped in one clean plane, the crack width would have equaled the blade thickness, t. In this case:

    area = 2 * t * crack length

    However, the diagonal nature of the crack forces calculation of the crack width from the Pythagorean relationship of blade thickness and crack height , i.e., the height of the diagonal tear:

    area = 2 * (t2 + h2)1/2 * crack length

    The length of this hypotenuse, the new crack width, will always be greater than the blade thickness, so the area produced by a diagonal crack will always exceed that of a simple planar crack.

    By forcing the production of greater surface area, this diagonal tearing would appear to inhibit crack propagation. The production of surface area requires the input of enough energy to supply the "free surface energy" associated with any surface. The energy used for crack propagation must be paid for with the strain energy released during the relaxation of the slit material. This seems intuitive-- you must pay for the energy you consume (by increasing surface area) by releasing stored strain energy. The energy required for fracture is referred to as the work of fracture7:

    Work of fracture

    = Energy / Area = J / m2

    = [ rate of energy expenditure (J/s) ] / [ rate of surface area production (m/s2) ]

    = [ force of extension (J/m) * crack propagation velocity (m/s) ] / [ 2 * crack width (m) * crack propagation velocity (m/s)) ]

    Work of fracture = F / (2 * crack width)

    Work of fracture is a constant property of a homogenous material8. By increasing the amount of surface area produced (through increasing the crack width) during fracture, Iridaea apparently raises the amount of force it can withstand given the blade's fixed work of fracture.

    Critical Crack Length

    Yet Denny et al. demonstrated that the release of strain energy does not control fracture in I. flaccida. 7 Fairly large cracks grow at stable rates in I. flaccida. If only strain energy is considered as the criterion for fracture, slits in the thallus appear to exceed the crack distance at which the onset of catastrophic failure is predicted. When the crack reaches a length at which the rate of energy release exceeds the rate of surface area production (and energy cost), the crack propagates spontaneously. The thallus fails catastrophically as the split separates at a higher rate than the constant rate of force application. The energy released through crack propagation increases as the square of crack length while the price of propagation varies linearly with this length. This critical crack length for catastrophic failure can be calculated:

    L (critical) = 2 * W *E / (2)

    Using a modulus of elasticity (E) of 107 N/m 2 and a work of fracture (W) of 278 J/m 2 these researchers calculated the critical crack length under a stress () of 4 * 106 N/m 2 . Although the critical crack length works out as .11 mm, the alga actually survives a crack of 2 mm under this level of stress. If the force the alga can withstand is not limited by the critical crack length, the release of strain energy cannot be the determining factor in failure.

    Stress Concentrations

    But there is another condition which must be met in a material before failure will occur. The first condition, as discussed above, involves strain energy: you must release enough strain energy to pay for increasing crack surface area. Yet the Denny lab determined that this was not the factor controlling failure. The second condition for failure concerns the concentration of stress at the crack tip. The local stress at the crack tip must exceed the breaking stress of the material. This seems intuitive; in order to break bonds, you must provide a force greater than that which holds them together. Apparently this is the limiting factor in the fracture of the thallus. If stress concentrations control fracture, we would expect a tough alga to develop mechanisms of reducing the concentration of stress at the crack tip.

    Avoiding Stress Concentrations in 2-D

    The Denny lab identified two mechanisms of avoiding stress concentrations in 2 dimensions. 7 Like any flexible material, the tip of a crack in I. flaccida becomes rounded as the stretching force increases (See A). In contrast, a rigid material like paper simply splits when force is applied perpendicular to the axis of a crack (See the high concentration of stress at the crack tip in B). Recall that stress is simply force divided by the area on which it acts. Rounding of a crack tip increases the area on which a force must act to produce a stress. This is merely a passive mechanism of resisting failure, common to many flexible materials--just try slitting and ripping a plastic bag to prove this to yourself. More interesting was the Denny lab's discovery of the growth of a pit at the tip of a crack within 48-72 hours after being slit by a razor (See C). In the  such a slit would easily result from the scraping of the thallus on a bed of barnacles. Development of a pit ensures that forces aiming to further rip the thallus must act on a larger surface area. Recall that stress equals force divided by area, so the larger the surface on which the force acts, the less stress experienced by the blade. A pit in a blade helps distribute the force around the edges of the pit even better than would a mere rounding of the crack tip.

    Avoiding Stress Concentrations in 3-D

    My research points to a third mechanism of reducing stress concentrations, this one in 3-dimensions. Recall that the heterogeneity of cuticle and medullary fibers forced the diagonal propagation of a crack through the thallus. Instead of acting only on the plane perpendicular to the direction of crack propagation, the force of separation had to act on a tilted plane. This 3-D tear process compels the separation force to act on a larger surface in a less favorably oriented direction. Imagine trying to rip a sponge lengthwise if the initial crack is oriented parallel to the direction of force application. It would be much easier if the crack were oriented perpendicular to your pulling axis. By ripping diagonally, I. flaccida achieves a situation half-way between these extremes. A second look at Figure 10 will help to clarify the meaning of diagonal in this context. Note that "diagonal" does not imply that the length of the crack is tilted diagonally in the plane of thallus; rather, the width of the crack extends at a diagonal from this plane across the thickness of the blade.

    Especially notable in my tests was the much greater crack height (more diagonal crack) which developed in female gametophyte's blade (remember, the female gametophyte Becomes the carposporophte) when a crack hit a carpospore. In the carposporangial blade of I. flaccida the cuticle appeared play an important role in preventing catastrophic failure of the blade. Each peak on the graph of applied force corresponded to the stalling of a crack at cystocarp . Here a stress concentration developed and when the applied force finally reached a critical threshold the thallus ripped suddenly, potentially catastrophically. However, at this point the height of the crack increased dramatically as the differential ripping of the cuticle and medulla produced a diagonal crack surface. This apparently stalled the progress of the crack enough to return to the crack to a state of equilibrium ripping and at this point the height of the crack returned to a more typical value.

    In contrast, the nonreproductive gametophyte had a uniform tear rate.

    A Tetrasporophyte Adapted to Avoid Stress?

    The mechanism of the carposporangial blade fracture has some interesting implications for Iridaean adaptations to environmental stress. Although I did not perform tests on the tetrasporangial blade of I. flaccida, its smooth margins would seem to help avoid fracture after nicking. This thallus has scattered sori just like the carposporangial blade. These tetrasporangial sori also create inhomogeneities in the fabric of the blade which should also concentrate stress when the blade becomes torn. I would predict a similar variation in the degree of diagonal tearing for the tetrasporangial blade as I observed in the carposporangial plant.

    The tetrasporangial thallus of intertidal I. flaccida has an important quality lacking in the tetrasporangial blade of subtidal I. cordata. Recall that the fully diploid blades of I. flaccida have smooth margins uninterrupted by sori; this feature provides the most reliable mechanism for distinguishing it from I. cordata. Without these sori in the margins of the thallus, I. flaccida avoids placing nasty stress concentrators in a critical region of the blade. On the wave-swept shores favored by I. flaccida the thallus must often get raked across rough barnacles or nicked by debris. Smaller nicks might penetrate only the smooth margin of the blade and the lack of stress concentrators here would help to prevent the propagation of such a nick into the blade. I. cordata lives further down in the tidal zones, often subtidally, and might be helped less by such an adaptation. Of course, I have not even had time to test a tetrasporangial I. flaccida blade, let alone any type of I. cordata thallus.

    I am curious why it would be the tetrasporophyte which would show the stress-avoidance adaptation of smooth margins when its sori are actually smaller (and probably less of a problem) than those of the carposporangial blade, which does not have smooth margins. Carposporangial blades in general appear to be thicker and sturdier than tetrasporangial blades, so perhaps the advantage of the smooth margins is most pronounced for the more delicate tetrasporangial blade. Again, however, I need to measure tetrasporangial work of fracture before I can compare the blade strength of all three Iridaean life history phases.

    Several Comparisons of Works of Fracture

    The data which I did obtain does allows some comparison of blade strength between various life history phases, seasons, locations, and directions of crack propagation . Unfortunately, I had far too little time to do statistically accurate experiments so I stress that my information is generally based on only one test of thallus strength, not an average of multiple trials.

    Seasonality

    Blade strength shows a strong seasonality, with the blades almost twice as strong in the winter. While the Denny lab measured a summertime work of fracture of 278 J/m2 for the gametophyte stage growing at Hopkins Marine Station, I measured a winter value of 490 J/m2 for a blade also cut longitudinally. This information makes sense either selectively or adaptively. Those blades which survived the harsher wave thrashing in the winter may be better adapted to survive stress. Alternately, they may simply represent the strongest fraction of a genetically identical population, and these specimens alone survived until I went collecting in mid-February.

    Degree of Exposure

    Work of fracture also correlates with exposure. For the gametophyte winter stage, cut longitudinally, the work of fracture of 644 J/m2 from Pt. Pinos far exceeds the figure of 490 J/m2 obtained for a similar, but fairly sheltered Hopkins blade.

    The Carposporophyte

    Most surprising to me was the huge difference in strength between the gametophyte and carposporophyte at Hopkins. The fully reproductive part of the carposporophyte figured in at 1301 J/m2 , tremendously strong when compared with the 490 J/m2 figure for a similarly cut and located gametophyte. The less developed part of the carposporangial blade shows a less pronounced but still remarkably greater work of fracture (700 J/m2) than the gametophyte. I would speculate that the carposporophyte is stronger because it is an older blade. Recall that I am referring to the carposporangial blade of the female gametophyte as the carposporophyte, so the carposporangial blades I tested were all older gametophytes. Perhaps the gametophyte would be happier if it were stronger, but it needs time to mature into a more sturdy blade.

    Direction of the Initial Slit

    Blades cut horizontally were consistently weaker than blades cut longitudinally. At first this startled me because when I see a ripped thallus in the intertidal the tear generally extends horizontally across the blade, and I would think that the plant would want to have the highest work of fracture in this direction in order to stay intact. Yet upon further consideration I realized that it is no surprise that the blade rips horizontally if it is weakest in this direction.

    Strength From Base to Tip

    My last test addressed variation in strength along the blade of the less reproductive carposporangial thallus. From each of two thalli I cut two rectangles of material from different heights on the blade and compared their work of fracture. Surprisingly it appears easier to rip a thallus closer to the stipe than the tip of the blade because the work of fracture is lower in lower regions.

    Yet why would an alga be weaker towards its stem, and risk total loss of a thallus rather than simple pruning of its tip? At first I regarded this data as so irrational that it did not deserve examination without further experimentation. Yet again I changed my mind when I observed blades in the field. In the carposporophyte, the blade tends to tatter near the base first, as my strength tests would predict. This makes sense because the carpospores near the base develop before the ones towards the tips of the blades and would be ready for release sooner. Tattering of the blade is a common mechanism for assisting spore release, so again my data seem to have real meaning. Whatever biological necessity forces the blade to first become mature towards its base also requires that thallus fabric be weaker in this region in order to assist spore release.

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