QUICK SEARCH:   [advanced] Author: Keyword(s):
Home     Help     Feedback     Subscriptions     Archive     Search     Table of Contents

First published online June 15, 2007
Journal of Experimental Biology 210, 2213-2230 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.001560

This Article Summary Full Text Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Related articles in JEB Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Mach, K. J. Articles by Denny, M. W. Search for Related Content PubMed PubMed Citation Articles by Mach, K. J. Articles by Denny, M. W.

Techniques for predicting the lifetimes of wave-swept macroalgae: a primer on fracture mechanics and crack growth

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

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 (4K): [in this window] [in a new window]   Fig. 1. Schematic showing different types of stress–strain behavior: (A) linear elastic, (B) non-linear elastic and (C) elastic–plastic with unloading.

View larger version (4K): [in this window] [in a new window]   Fig. 2. (A) Edge-cracked sheet experiencing bulk tensile stress , with crack length a and width w. (B) Center-cracked sheet with crack length denoted as 2a for reasons related to mathematical derivation of the corresponding stress intensity factor.

View larger version (5K): [in this window] [in a new window]   Fig. 3. Three loading modes of cracked specimens: (A) mode I: tensile opening (cleavage); (B) mode II: in-plane shearing; and (C) mode III: anti-plane shearing (tearing).

View larger version (4K): [in this window] [in a new window]   Fig. 4. A log–log plot illustrating patterns of crack growth for conditions of repeated loading. Crack growth rate (mm cycle-1), da/dN, indicates the increase in crack length for each cycle, N, of sinusoidally varying applied stress. Stress intensity range , KI indicates the variation in stress intensity factor during each cycle of loading. KTH indicates threshold stress intensity factor range, and KC denotes fracture toughness.

View larger version (5K): [in this window] [in a new window]   Fig. 5. Representative stress–strain curves for pull-to-break tests of three macroalgae (Hale, 2001).

View larger version (1K): [in this window] [in a new window]   Fig. 6. Schematic of a rounded crack tip with diameter d and angle .

View larger version (4K): [in this window] [in a new window]   Fig. 7. A trouser test piece after a crack has extended from an initial incision. The path of future crack growth is shown as a broken line. Force F is applied to both legs. Each leg has width c, and legs and body have thickness b. Cross-sectional area of the test piece, C, equals bx2c.

View larger version (12K): [in this window] [in a new window]   Fig. 8. Force applied to a trouser-tear test piece is plotted against distance of test piece's extension. Arrow 1 indicates the specimen's initial extension, arrow 2 indicates tearing of the test piece at average force F, and arrow 3 indicates final retraction of the specimen as applied force is removed. The stippled area in (A) depicts energy released in crack extension. In (B), hatched area 1, under the initial extension curve (arrow 1), indicates strain energy in legs before crack growth. Cross-hatched area 3, under the retraction curve (arrow 3), indicates strain energy stored in legs at the end of the test. Because legs are longer at the end of the test due to crack extension, final stored strain energy (area 3) is greater than initial stored strain energy (area 1).

View larger version (19K): [in this window] [in a new window]   Fig. 9. (A) Stress–strain curves of a red macroalga, Mazzaella flaccida (Setchell & Gardner) Fredericq, for two cycles of stretching (Hale, 2001) showing a sizeable loading–unloading loop on the first cycle, followed by a much smaller loop on the second cycle. A small amount of residual strain remains after the first cycle. (B) Stress–strain curves, adapted from Dorfmann and Ogden (Dorfmann and Ogden, 2004), of a rubber compound for several cycles of stretching, showing similar stress–strain behavior, plus some reduction in maximum stress levels with cycling (stress softening). Curves are shown for maximum strains of 0.5, 1.0 and 1.5. Plot, copyright 2003, is reprinted with permission from Elsevier.

View larger version (4K): [in this window] [in a new window]   Fig. 10. (A) Diagram showing an integration path (contour) taken in a counterclockwise direction around a crack tip. (B) An edge-cracked specimen stretched and held with fixed displacement, showing a contour taken around the boundaries of the specimen.

View larger version (5K): [in this window] [in a new window]   Fig. 11. Schematic showing change in load–displacement curves of specimens with two different crack lengths (a and a+da) but the same specimen displacement.

View larger version (12K): [in this window] [in a new window]   Fig. A1. Plots used in experimental determination of critical strain energy release rate. (A) Load–displacement plots for specimens with various initial crack lengths. Solid circles indicate points at which unstable tearing occurred. Lower curves are for specimens with longer initial cracks. * represents a displacement chosen for generation of plot (B). Area 4, A4 (hatched), indicates strain energy present in specimen 4 when it is pulled to displacement *; specimen 4's load–displacement curve must be extrapolated. breaking,1 indicates the displacement at which specimen 1 tore unstably, which is used for plot (C). (B) A plot, derived from (A), of strain energy versus initial crack length at a selected value of specimen displacement, *. For example, strain energy A4 is plotted against the length of specimen 4's initial crack, as depicted by the open circle 4. (C) A plot, derived from (A), of displacement at tearing versus initial crack length. For example, breaking,1 is plotted against specimen 1's initial crack length, as indicated by grey circle 1. From this plot, initial crack length, a*, corresponding to *, is determined. At this value of a*, the tangent to plot (B) is found, which gives (dU/da).

View larger version (6K): [in this window] [in a new window]   Fig. A2. (A) Counterclockwise contour around a crack tip showing an element of path length ds, with unit vector n normal to the path and with stress and displacement vectors, P and u, respectively, also shown. Crack length is a. (B) An element of material experiencing two-dimensional stress. Not all three stress components need be active; often only one or two components are active. (C) Rectangular path surrounding a crack tip, used to illustrate evaluation of a J-integral.