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First published online June 15, 2007
Journal of Experimental Biology 210, 2231-2243 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.001578

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Death by small forces: a fracture and fatigue analysis of wave-swept macroalgae

Katharine J. Mach^1,*, Benjamin B. Hale¹, Mark W. Denny¹ and Drew V. Nelson²

¹ Hopkins Marine Station of Stanford University, Pacific Grove, CA 93950, USA
² Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA

View larger version (9K): [in this window] [in a new window]   Fig. 1. (A) Example stress–strain curves for extension and retraction. Internal energy dissipation occurs as indicated by the hysteresis loss, the difference between input energy density and recoverable (stored) strain energy density. Input energy density is found as the area under the extension curve, while stored strain energy density is the hatched area under the retraction curve. (B) Loading of a M. flaccida specimen to maximum strain of approximately 0.35 demonstrates hysteretic loss.

View larger version (12K): [in this window] [in a new window]   Fig. 2. (A) Schematic of stress–strain behavior observed when a specimen is cycled to increasingly larger strains. The tips of the extension–retraction loops trace a path that closely approximates the curve from a pull-to-break test. In this example, the pull-to-break curve is `r-shaped'. Resilience at a strain level is taken as the ratio of area under the corresponding retraction curve to total area under the broken pull-to-break curve. For instance, for the strain level indicated by `Strain 1', resilience is the ratio of the hatched area to the total area under the pull-to-break curve up to `Strain 1'. (B) Experimental cycling behavior for M. flaccida. For clarity, only the retraction portions of loops (return curves) are shown. The bold line represents the reconstructed pull-to-break extension curve, which is gently `J-shaped'. Resilience (R) at four strain levels is shown.

View larger version (16K): [in this window] [in a new window]   Fig. 3. (A) Conditioning of a particle-reinforced rubber specimen stretched to an extension ratio of 3. (Extension ratio is current specimen length divided by initial length.) The stress–strain curves, upon extension and retraction, exhibit `stress softening', which is called the Mullins effect. This softening diminishes as the number of loading cycles increases. Plot is reprinted from Dorfmann and Ogden (Dorfmann and Ogden, 2004), copyright 2003, with permission from Elsevier. (B) Stress–strain curves of a brown alga, Egregia menziesii (Turner) Areschoug, for two cycles of extension and retraction. These curves also exhibit stress softening, with lower maximum stress reached in the second cycle. In addition, for both A and B, viscoelastic loading–unloading loops decrease in size with cycling.

View larger version (7K): [in this window] [in a new window]   Fig. 4. An example of calculating critical strain energy release rate, TC, for M. flaccida in single-edge-notch, pull-to-break tests. The inverse of crack length, a, introduced into each sample is plotted against at fracture, where Wo is calculated as total absorbed strain energy density, Wo,T, and estimated stored strain energy density, Wo,S. Solid circles indicate data points calculated from total strain energy density; solid diamonds indicate data points calculated from stored strain energy density. TC is the slope of the regression line for each data set: regression line (broken line) for data points calculated from total strain energy density yields an estimate of TC,T, and regression line (solid line) for data points calculated from estimated stored strain energy density yields an estimate of TC,S, as defined in the text. Open circles and diamonds indicate data points for un-notched samples, for which effective crack length was estimated.

View larger version (11K): [in this window] [in a new window]   Fig. 5. Measured and predicted breaking stresses as a function of crack length, a, for (A) M. flaccida, (B) P. occidentalis and (C) U. expansa. In each plot, data points indicate breaking stress measured as a function of crack length in single-edge-notch, pull-to-break tests. Data points on ordinate (open circles) are for un-notched test pieces. Solid line indicates predicted breaking stress as determined from Eqn 5 using critical strain energy release rate calculated from estimated stored strain energy density. Broken line indicates predicted breaking stress based on critical strain energy release rate calculated from total strain energy density. Curve equations are as follows, with breaking stress in units of MPa and crack length, a, in m: (A) solid line: breaking stress=3510a–0.81, r2=0.933; broken line: breaking stress=9530a–0.66, r2=0.935; (B) solid line: breaking stress=2123a–0.90, r2=0.0033; broken line: breaking stress=8668a–0.69, r2=0.0030; (C) solid line: breaking stress=17650a–0.58, r2=0.712; broken line: breaking stress=28790a–0.51, r2=0.724.

View larger version (9K): [in this window] [in a new window]   Fig. 6. Reduction in maximum strain energy density per cycle during the conditioning period for three algal species used in this study. Wo,S/(Wo,S)in is the ratio of maximum stored strain energy density in a given cycle to maximum stored strain energy density of the initial cycle for that specimen. The abscissa is on a log scale. Solid lines represent regressions calculated excluding data from the first 100 cycles.

View larger version (10K): [in this window] [in a new window]   Fig. 7. An example of crack growth in a single-edge-notch specimen for cyclic repeated loading at 1 Hz to 0.166 strain. Approximate energy release rate as a function of crack size, calculated as if change in strain energy density with cycling (Eqn 6) were negligible, is shown on the secondary ordinate. Only data points for which crack growth was equal to or greater than 10% of previously used value (filled points) were used in analysis of crack growth rate.

View larger version (18K): [in this window] [in a new window]   Fig. 8. Empirical (A) and calculated (B) fatigue crack growth rates (m cycle-1), da/dN, plotted against energy release rate (J m–2), TS, in single-edge-notch, repeatedly loaded test pieces. Axes in plots are logarithmic. Data for test specimens that did not demonstrate crack growth are not shown. In both A and B, diamonds represent M. flaccida, squares, M. splendens, triangles, U. expansa, and crosses, P. occidentalis. Symbols used in A are given in the plot. For B, the black line with no symbols represents the power-law series for all data collected for all species. Again in B, solid symbols (diamonds, squares, triangles, and crosses) represent power-law series calculated from all data for each species. Open diamonds (M. flaccida) and squares (M. splendens) are for relations determined from the upper bounds of crack growth rates for the two species. Open triangles are for a relation for U. expansa that excludes the lowest crack growth rate point for U. expansa in A.

View larger version (20K): [in this window] [in a new window]   Fig. 9. Fatigue lifetime curves depicting predicted cycles to fracture as a function of percent of breaking strength. Blue curves are calculated from parameters for all data for a given species (`All' in Table 3), green curves are determined from parameters for all algal data combined (`Overall' in Table 3), red curve is for the upper-bound parameters for M. flaccida (`Upper Bound' in Table 3), and gray curve is for U. expansa parameters determined excluding an outlying data point (`Excluding Outlier' in Table 3). Broken portions of curves are below measured fatigue thresholds, To, for which no crack growth is predicted. An approximate time to failure is given on the secondary ordinate. As described in the text, time to failure is calculated assuming that each wave has a period of 10 s and that approximately 1 in 20 waves imposes stresses great enough to cause fatigue damage.

View larger version (9K): [in this window] [in a new window]   Fig. 10. Fatigue lifetime curves depicting predicted cycles to fracture as a function of cyclic stress for M. flaccida. Broken portions of curves are below measured fatigue threshold, To, for which no crack growth is predicted. Color-coding of curves is as described for Fig. 9. Approximate water velocities corresponding to applied stresses are given on the secondary abscissa. Estimated time to failure, determined as described in the text, is given on the secondary ordinate.