| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
Physiology and Biochemistry |
University of Maryland, Baltimore County, Mechanical Engineering, 1000 Hilltop Circle, Baltimore, Maryland 21250 USA
Received for publication April 2, 2006. Accepted for publication July 18, 2006.
ABSTRACT
In the field of fracture mechanics, an analytical framework has been established for understanding the mechanical failure of any structure made of inherently flawed materials. In the context of botany, this includes an extraordinarily wide variety of turgid and/or woody structures made of cellulose-based tissues, the diverse soils penetrated by their roots, and a multitude of plant-based commodities and foodstuffs. The goal of this article is to provide an overview of the theory of engineering fracture mechanics and to identify some special characteristics of wood and other plant-based materials that require further development in this area.
Key Words: crack • fracture mechanics • fracture toughness • stress intensity factor
One might think material (or biomaterial) failure could always be avoided by ensuring that the maximum stress was kept well below the ultimate strength. On the contrary, experience has shown that engineering structures designed on this basis can and do fail with predictable regularity. From one perspective, the brittle-like failure of ductile materials subject to time-varying stress arises from deviatoric strain energy cycling (Liu and Ross, 1996
). However, at a slightly larger size scale, there is another equally important factor. Real materials are invariably flawed by incipient cracks, including those created by molecular defects, those induced by sharp internal forms, and those due to in-service damage (Anderson, 1995
). In consequence, a large irregular old structure is more likely than a small smooth new structure to contain a flaw large enough to grow into a destructive runaway crack. Moreover, the rate at which this crack grows is accelerated by increased stress and as its size increases. In general, damage-tolerant structures of any kind must be strong enough to slow the inexorable process of crack growth.
As described later, most of the early work on fracture mechanics was motivated by interest in the cracking of relatively stiff isotropic materials subjected to quasistatic loading. However, at least in principle, most of the resulting concepts can be generalized to plant tissues especially including wood. While engineering structures made of natural wood are characteristically flawed by the presence of knots and splits, the work required to fracture a wooden structure can be very high. In part, this is because cellulose is a remarkably strong and light material whose capacity to absorb strain energy is comparable or greater than that of many steels and aluminums (Jeronimidis, 1980
). In fact, its specific tensile strength (i.e., strength per unit weight) is typically 5–10 times higher than that of a hot-rolled structural steel. At the same time, the wood cell wall in particular and plant tissue in general is only partly comprised of cellulose (Niklas, 1992
) and is better viewed as a composite material at multiple size scales.
The effects of plant tissue composition and organization on fracture toughness have always been of great practical interest in the wood products industry (Hamad, 1998
; Heyden and Gustafsson, 1998
; Smith et al., 2004
). The precise functional roles of hemicellulose and lignin are still being debated (Kohler and Spatz, 2002
). What is clear is that the total fracture energy of most woods is disproportionately high in comparison to the intrinsic toughness of the cell wall material. In essence, crack propagation through wood is influenced by material that is relatively remote from the crack plane itself. A wood cell can be idealized as a long narrow tube whose diameter is <5 µm. At the microstructural level, its secondary cell wall, denoted S2, is reinforced primarily by spirally wrapped cellulose microfibrils inclined about 15° from the long axis. As a result and in contrast to polycrystalline metals, this special microfibrillar morphology serves to couple normal and shear behaviors (Lucas et al., 1993). Thus, when a wood cell and its fellows are stretched along their long axis, the lignin matrix within which the microfibrils are embedded begins to fracture. This selective failure process allows the hollow cell to buckle inward, to elongate, to narrow, and to pull itself away from the surrounding cells. Each of these processes helps generate new surface area and thereby dissipates energy, even though the cellulose fibrils can continue to carry tensile load. Of primary importance here is the plastic buckling of the helically reinforced cell wall into its lumen. Experiments suggest that buckling accounts for most of the fracture toughness of wood, whereas all other effects together account for less than 10% of the total in softwood species (Lucas et al., 1997
). The balance is somewhat different in hardwoods, where additional toughening arises from the presence of large water-conducting vessels (i.e., 50–500 µm diameter), which can help arrest crack growth.
For low-toughness, glasslike materials, brittle fracture is the dominant failure mechanism, and linear elastic fracture mechanics (LEFM) is generally viewed as the most appropriate failure theory. In this case, the critical stress for fracture has a linear dependence on the so-called stress intensity factor defined later. In contrast, for high toughness materials, ductile flow leading to plastic collapse is the usual cause of failure, and fracture mechanics may not even be applicable. In between these two extremes lie the ductile engineering metals and the cellulose-based plant tissues. For the former, fatigue-induced cracking is the primary cause of failure but nonlinear fracture mechanics theory is required. While various approaches have been validated for polycrystalline metals, their direct relevance to fibrous plant tissues is largely unproven and must therefore be assessed on a case-by-case basis.
In fracture mechanics theory, the stress field close to a crack is often assumed to have the same shape regardless of the larger geometry of the body. In reality, the spatial variation of the stress is affected by material heterogeneity and directionality and by the absolute size of the crack relative to that of the body. For example, the stress field at the root of a notch in a serrated leaf will be distorted in nonlinear proportion to the prevailing direction of fibrous reinforcement, and by the close proximity of the leaf margins (Fig. 1). In practice, an incipient crack in a thin leaf can be easily deflected from its original path by a vein or other tough structure. From an experimental standpoint, this particular challenge is best overcome using a scissor-like cutting instrument to direct the crack in the desired way (Atkins and Mai, 1979
). Even then, care must be taken to prevent artefacts caused either by sticky latex or slippery sap from distorting the measured work of fracture. Observe that there are three distinct ways that an imposed force P can drive a crack into the leaf from the serrated notch. In mode I, P produces tensile stress acting normal to the plane of the crack and tends to open the incipient crack. In mode II, P produces shear stress acting parallel to the plane of the crack and perpendicular to its front and tends to slide the two crack faces relative to one another. In mode III, P produces shear stress acting parallel to the plane of the crack and parallel to its front and tends to tear the crack front open.
|
). Close to the crack tip, the stress is invariably dominated by a term proportional to 1/(r)0.5, where r is the distance from the crack tip. Any mode of loading produces a 1/(r)0.5 stress singularity at the crack tip, but the exact nature of the r and dependence varies with the loading conditions (Anderson, 1995
). In general, the stress ahead of a crack in an isotropic linear elastic material can be expressed as
|
| (1) |
) depend on alone, but are different for each stress component and between modes. For instance, one can show that the stress component 
xx due to mode I loading is
|
| (2) |
|
| (3) |
) depend on alone but are different for each direction and between the modes. For example, the x displacement due to mode I loading is
|
| (4) |
= 3 – 4
for plane strain or = (3 – )/(1 + ) for plane stress, with G being the shear modulus and being the Poisson's ratio. Most commonly, the three constants KI, KII, and KIII appearing in Eqs. 1–4 are used to quantify the strength of the crack tip singularity associated with each of the three modes of loading, respectively. In mixed mode loading, each of these so-called stress intensity factors contributes to the total stress (whose non-zero components are 22, 12, and 32)
|
| (5) |
). For example, Patton-Mallory and Cramar (1987)
used finite element methods to obtain stress intensity factors for various wooden timbers. In many practical situations, mode I dominates, and KI can be rescaled by a correction factor close to unity to capture the lumped effect of mixed mode loading. Once KI (and possibly the other two stress intensity factors) are known, LEFM assumes that all components of stress, strain, and displacement can be expressed in terms of r and only. The use of a single parameter (i.e., the stress intensity factor) to describe crack tip conditions is one of the key concepts in fracture mechanics.
The physics underlying the fracture of brittle materials were first explained by Griffith (1920)
, and most of the early theory applied to botanical fracture (e.g., Porter, 1964
) can be traced back to the metals fracture community. In both cases, simple elasticity theory indicates (incorrectly) that the stress at the tip of a sharp crack approaches infinity, seeming to imply that a cracked structure must fail if subjected to any level of external load. Instead, Griffith discerned the basic relationship between flaw size and fracture stress by equating the extra energy required to create a new crack surface to the loss of elastic strain energy (associated with relaxation of the local stress field) due to crack propagation. Griffith showed that there is a critical crack length LG below which a crack cannot increase in length at a given level of load. Rather, further crack growth requires a distortion of the stress field, which can only be accomplished by supplying an amount of energy equal to the work of fracture. In theory, a brittle material having no cracks longer than some critical length LG is safe from fracture at the associated load level. The safe length depends on the ratio of the work of fracture to that of the strain energy stored in the material. Because this length is inversely proportional to the material's resilience, a brittle material with a large capacity to absorb strain energy is also more likely to crack. In any event, the goal of linear elastic fracture mechanics is to predict the critical load that will cause a crack to grow. In the case of dynamic fracture, the rate and direction of growth are also of interest. LEFM is not accurate very close to the crack tip and within the small process zone where the material is undergoing irreversible damage. It is also inaccurate very far from the crack tip where the shape of the stress field is more strongly influenced by the body's geometry and the boundary conditions. However, there is an intermediate zone within which the idealized stress field is often sufficiently accurate to be useful. This is called the region of K dominance (Fig. 2).
|
; Hiller and Jeronimidis, 1996
). The work of fracture usually decreases with increasing moisture content (Niklas, 1992
; Williamson and Lucas, 1995
), but this is not the case for thin sections of wet wood (Lucas et al., 1990
). Similarly, in contrast to metals, temperate zone woods are more resistant to cracking (across the grain) at lower temperatures (Vincent, 1990
). Again, one must recognize that most of fracture mechanics theory is based on idealizations of manmade material behavior, which may or may not be appropriate within the botanical world. As a guideline, LEFM is most likely to be useful if the length and width of the specimen of interest are both at least 25 times the process zone radius and if the specimen thickness is at least as large as the process zone radius. The process zone radius in mode I loading is roughly rp = 2.5(KI/S*), where S* is the yield stress Sy for a ductile material, or the ultimate stress St for a brittle material.
In North America, the stress intensity factor KI is most often measured using one of the five American Society for Testing and Materials (ASTM E1820–01) standard specimen geometries, which are (1) the compact tension specimen, (2) the disk-shaped compact specimen, (3) the single-edge notch bend specimen, (4) the arc-shaped specimen, and (5) the middle tension specimen. Similar testing methods are recommended by the British Standard (BS 7448) and by the European Structural Integrity Society (ESIS P1–92). In practice, special cutting techniques are often required to prepare plant-based materials without splitting or otherwise damaging their surface structures. The fracture toughness of plant-based foodstuffs can also be assessed using a wedge penetration test that approximates tooth action (Khan and Vincent, 1993
). Regardless of the specific test configuration, the stress intensity factor(s) can then be expressed in terms of the applied load, the crack length, and the other relevant features of the specimen geometry. It is often difficult to detect the exact onset of crack growth. Instead, the usual alternative approach is to plot the applied load as a function of the measured crack tip opening displacement (CTOD). The response is assumed to depart from a straight line relation as the crack begins to grow. By convention, the intersection of the plotted P vs. CTOD curve with a secant line whose slope is 5% less than the initial linear slope is taken as the fracture load (Fig. 3).
|
f = (2E*
s/
a)1/2, where E* = E for plane stress and E* = E/(1 – 2) for plane strain when E is the Young's modulus, s is the surface energy, and a is the half crack length. In practice, the accuracy of this prediction is very good for glass but unrealistically low for the ductile metals of greater interest to engineers. As a general rule, Griffith-based theory will also substantially underestimate the fracture strength of the biological materials of primary interest to botanists. For example, natural rubber can be classified as a brittle material (in the sense that it does not yield prior to fracture) but is so deformable that a sharp notch is completely blunted long before the breaking stress is reached. Deformability increases the energy required to open a crack, and leads naturally to the traditional definition of fracture toughness R, whose SI units are J/m3 (i.e., energy per unit volume) Somewhat confusingly, within the fracture mechanics community, the stress intensity factor K, whose usual SI units are instead MPa · m0.5 (i.e., energy per square root of volume) has also come to be called fracture toughness K. For isotropic materials, the maximum possible value of K is the critical stress intensity factor Kc, which is proportional to the square root of the product of the elastic modulus E and the work of fracture W (or equivalently, R). For anisotropic materials, the relationship between R and Kc is algebraically more complex. For engineering metals, 20 < Kc < 200 MPa · m0.5 (i.e., energy per squam0.5 (Anderson, 1995
). In contrast, most engineering polymers and plant tissues lie within the range of 0.1 < Kc < 10 MPa · m0.5 (Niklas, 1992
). Various experimental approaches are used; of particular interest, Samarasinghe and Kulasiri (2004)
have used the digital image correlation technique to measure stress intensity factors in wood. As a general rule, any material whose yield strength is close to its ultimate strength will have a relatively low fracture toughness. In a fibrous material, fracture toughness across the grain direction can easily be 20 times greater than the fracture toughness parallel to the grain. However, woods that have experienced cyclically reversed load may be markedly reduced in toughness by cell wall damage during the compressive portion of each cycle.
In a brittle material such as glass, crack initiation rapidly progresses into an unstable catastrophic fracture. In contrast, in a ductile metal such as aluminum, slow, stable crack growth can continue for a very long time (i.e., several decades) prior to ultimate failure. And in a fibrous cellular material such as wood, cell wall buckling introduces a completely new mechanism for arresting crack growth. Thus, there are profound dissimiliarities in the ability of these three different types of material to resist crack extension. In the ductile metal, inelastic deformation within the crack wake serves to moderate and reduce the crack tip stress. In contrast, in a plant tissue, the crack wake may still be held together by a so-called bridging zone reinforced by unbroken fibers (e.g., consider elm wood). In practice, the effects of inelasticity and bridging can exert a very strong effect on apparent fracture strength once the crack length begins to increase. The key advance in understanding the influence of inelastic behavior (e.g., plasticity, viscoelasticity, poroelasticity) on fracture toughness is attributed to Irwin (1948
, 1957
). Irwin and his colleagues at the Naval Research Laboratory in Washington, D.C. were among the first to recognize that certain limitations of Griffith's theory can be addressed by accounting for the energy dissipated by plastic flow near the crack tip. Although their resulting theory was derived for metals, a similar approach could be used to account for different types of energy dissipation (e.g., by internal pore fluid motion, crack meandering, cell crushing). The parameter of central interest is the strain energy release rate 
, which is defined as the rate of change in potential energy per unit crack length in the region around a crack. In the simplest case, a crack in a linear isotropic elastic material is assumed to grow when reaches some critical value c. As anticipated by Irwin in the early 1960s,
one can show that the stress intensity factor K has a simple relationship to .
Over the next decade, various methods were proposed to account for the development of a plastic yield zone around the crack tip and for its effect on fracture toughness (Fig. 4). Of particular note, Wells (1963)
observed that the crack faces were moved apart by plastic deformation and proposed that this effect could be quantified using a parameter known as the crack tip opening displacement (CTOD). This parameter has proved especially useful for understanding the fracture properties of welded structures, such as those used for off-shore oil production. Independently, Rice and Rosengren (1968)
and Hutchinson (1968)
both identified a different parameter used to characterize dissipative material behavior ahead of a crack that is far from any edges. In a series of papers, they showed that a nonlinear energy release rate could be quantified by using a line integral evaluated along an arbitrary contour surrounding the crack. This so-called J integral is now understood as a nonlinear stress intensity parameter. The J parameter has proved especially useful for characterizing the tough steels used in nuclear power plants. Shih and Hutchinson (1976)
later demonstrated an underlying equivalency between the J integral and the CTOD methods. In materials (or plant tissues) that undergo large-scale yielding, the stress fields around the crack tip may no longer be unique and may in fact be path dependent. In this situation, at least one additional parameter is required to characterize the crack tip conditions. Research on this topic is ongoing, and there is no consensus on the best approach.
|
showed that a power law of the form da/dN = c(
K)m could be used to relate the change in crack length a per load cycle N to the stress intensity range K (Fig. 5). The two constants c and m must be found empirically; m is typically between 2 and 4. When reasonable estimates of initial crack length and critical stress intensity are available, an engineering structure (e.g., the aircraft) can remain in service even as its subcritical cracks are growing in a slow, stable manner. However, in the final stage III, one or more cracks reaches the critical length associated with the imposed stress intensity. Once this has occurred, sudden catastrophic fracture can occur at any time. In practice, most fatigue tests are performed using sinusoidal mode I loading. Interestingly, crack growth rate within stage II is only weakly sensitive to the mean value of the stress intensity factor.
|
were the first to define a precise relationship between fracture toughness, imposed stress, and flaw size needed to use the J integral as a design tool. When a structure is sized using strength of materials analysis, a limit is placed on the maximum allowable stress relative to the yield (or ultimate) strength. In contrast, if a structure is sized using fracture mechanics analysis, three interrelated properties are considered. In various forms, these are a material property (e.g., the critical stress intensity factor or energy release rate), a kinetic property (e.g., the current stress intensity factor or energy release rate), and a geometric property (the flaw size or crack tip opening displacement). As indicated, the energy criterion approach considers crack extension to occur whenever the energy available for crack growth is sufficient to overcome the material's resistance to fracture. In ductile metals, this resistance is dominated by plasticity. However, in plant tissues, it may instead arise from a chemical phase change, pore water motion, or intrinsic viscoelasticity. As indicated, fracture toughness is usually but not always measured in mode I (see Lucas and Pereira, 1990
or Hiller and Jeronimidis, 1996
). When required, various empirical approaches can be used to construct mixed mode fracture criteria. For example, to arrive at the criterion of maximum hoop stress (Anderson, 1995
), one must postulate (with some justification) that a crack subjected to critical mixed mode loading will branch at the angle for which the opening stress is greatest, thereby relieving the local mode II stress intensity factor. In mode I, for a through-thickness crack of length 2a spanning a much wider plate subjected to uniaxial tensile opening stress , the energy release rate is = 2a/E*, where E* = E for plane stress and E* = E/(1 – 2) for plane strain, with the Young's Modulus E and the Poisson's Ratio defined in accordance with convention. At the instant of fracture, = c implying that the critical stress f is inversely proportional to the square root of a. The energy release rate is viewed as the driving force for fracture, and the critical value c is viewed as a global measure of intrinsic resistance to fracture, which is independent of the cracked body's exact geometry. The often preferred perspective offered by the stress intensity approach is that the stress throughout the K zone around a through-thickness crack tip (subject to mode I opening load) is proportional to KI = 2(a)0.5, where is the nominal crack opening stress and a is half of the crack length. In this case, KI is viewed as the driving force for fracture and the critical stress intensity KIc is viewed as a size-independent material property. For linear elastic materials, these two alternative perspectives on the mechanisms of fracture are essentially equivalent.
Certain defining characteristics of botanical structures limit the applicability of existing fracture mechanics theory. In particular, the mechanical behavior of plant tissues is often more sensitive to water content and temperature, is much more deformable, and is usually anisotropic, in contrast to the engineering metals that have motivated many of the key developments in fracture mechanics. Added to this, the toughness of cellulose-based materials can be degraded by fungal decay (Sexton et al, 1993
) or increased by a hardening response to environmental stress over relatively short time scales. Each of these differences pose analytical challenges that have not yet been addressed.
On a concluding note, soil fracture plays a key role in determining plant vigor and survival. The fracture strength of the soil substrate in which a plant is growing moderates root establishment and development rates and limits the anchorage required to resist wind forces. Cracking is a familiar feature of surface soils and allows water, oxygen, and nutrients to enter the underlying strata. Soil fracture contributes to the healthy rhizosphere, and the creation of artificial cracks via cultivation is a hallmark of purposeful agriculture. Cracking due to weathering stresses tends to improve soil structure and establishes drainage pathways. The degree of soil cohesion is moderated by many factors, but the foremost among these is water content. As its water content increases, a soil can change from an elastic solid to a plastic solid to a liquid consistency. In the solid form, soil cracking is usually due to environmentally induced reductions in moisture content (the channels created by earthworms are a notable exception). Factors of relevance include the mutual interrelationship between clay mineralogy, platelet orientation, salinity, root structure, and biological exudates with soil pore structure. It is generally understood that the capacity of a soil to support plant growth is linked to its ability to maintain a robust pore structure in the face of environmental disturbance. Complicating the situation, many of the organisms that live in soil are able to heal or reinforce cracks, for example, by moderating soil moisture content, producing hydrophobic exudates, or using their roots or hyphae to bind soil particles together. In this case, the assumption of monotonic crack growth at the heart of the existing theory of fracture mechanics is no longer valid. This parting observation may set the stage for further research on the analysis of fracture in botany.
FOOTNOTES
2 Author for correspondence (farquhar{at}umbc.edu
) 
LITERATURE CITED
Anderson T. L.. 1995. Fracture mechanics, fundamentals and applications, 2nd ed CRC Press, New York, New York, USA.
Atkins A. G. Mai Y.-W.. 1979. On the guillotining of materials. Journal of Materials Science 14: 2747-2754.[CrossRef][ISI]
Atkins A. G. Mai Y.-W.. 1985. Elastic and plastic fracture: metals, polymers, ceramics, composites and biological materials Wiley, Chichester, UK.
Gibson L. J. Ashby M. F.. 1997. Cellular solids: structure and properties Cambridge University Press, Cambridge, UK.
Griffith A. A.. 1920. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London, A 221: 163-198.[CrossRef]
Hamad W. Y.. 1998. On the mechanisms of cumulative damage and fracture in native cellulose fibres. Journal of Materials Science Letters 17: 433-436.[CrossRef][ISI]
Heyden S. Gustafsson P. J.. 1998. Simulation of fracture in a cellulose fibre network. Journal of Pulp and Paper Science 24: 160-165.[ISI]
Hiller S. Jeronimidis G.. 1996. Fracture in potato tuber parenchyma. Journal of Materials Science 13: 2779-2796.
Hutchinson J. W.. 1968. Singular behavior at the end of a tensile crack tip in a hardening material. Journal of the Mechanics and Physics of Solids 16: 13-31.[Medline]
Jeronimidis G.. 1980. The fracture behavior of wood and the relations between toughness and morphology. Proceedings of the Royal Society of London, B 208: 447-460.
Irwin G. R.. 1948. Fracture dynamics, In G. R. Irwin and J. A. Kies [eds.], Fracturing of metals, 147–166 American Society of Metals, Cleveland, Ohio, USA.
Irwin G. R.. 1957. Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics 24: 361-364.
Irwin G. R.. 1960. Plastic zone near a crack and fracture toughness. Proceedings of Seventh Sagamore Ordnance Materials Conference, Syracuse University, New York 4: 63.
Khan A. A. Vincent J. F. V.. 1993. Anisotropy in the fracture properties of apple flesh as investigated by crack-opening tests. Journal of Materials Science 28: 45-51.
Kohler L. Spatz H.-C.. 2002. Micromechanics of plant tissues beyond the linear-elastic range. Planta 215: 33-40.[CrossRef][ISI][Medline]
Liu J. Y. Ross R. J.. 1996. Energy criterion for fatigue strength of wood structural members. Journal of Engineering Materials and Technology 118: 375-378.[CrossRef]
Lucas P. W. Pereira B.. 1990. Estimation of the fracture toughness of leaves. Functional Ecology 4: 819-822.[CrossRef][ISI]
Lucas P. W. Tan H. T. W. Cheng P. Y.. 1997. The toughness of secondary cell wall and wood tissue. Philosophical Transactions of the Royal Society of London, B, Biological Sciences 352: 341-352.[CrossRef]
Matas A. J. Cobb E. D. Paolillo Jr. D. J. Niklas K. J.. 2004. Crack resistance in cherry tomato fruit correlates with cuticular membrane thickness. HortScience 39: 1354-1358.[ISI]
Niklas K. J.. 1992. Plant biomechanics, an engineering approach to plant form and function University of Chicago Press, Chicago, Illinois, USA.
Paris P. C. Gomez M. P. Anderson W. E.. 1961. A rational analytic theory of fatigue. Trend in Engineering 13: 9-14.
Patton-Mallory M. Cramer S. M.. 1987. Fracture mechanics: a tool for predicting wood component strength. Forest Products Journal 37: 39-47.
Porter A. W.. 1964. On the mechanics of fracture in wood. Forest Products Journal XIV 325-331.
Rice J. R.. 1968. A path independent integral and the approximate analysis of strain concentration of notches and cracks. Journal of Applied Mechanics 35: 379-386.[ISI]
Rice J. R. Rosengren G. F.. 1968. Plain strain deformation near a crack tip in a power law hardening material. Journal of Mechanics and Physics of Solids 16: 1-12.
Samarasinghe S. Kulasiri D.. 2004. Stress intensity factor of wood from crack-tip displacement fields obtained from digital image processing. Silva Fennica 38: 267-278.[ISI]
Sexton C. M. Corden M. E. Morrell J. T.. 1993. Assessing fungal decay by small scale toughness tests. Wood Fibre Science 25: 375-383.
Shih C. F. Hutchinson J. W.. 1976. Fully plastic solutions and large scale yielding estimates for plane stress crack problems. Journal of Engineering Materials and Technology 98: 289-295.
Smith I. Landis E. Gong M.. 2004. Fracture and fatigue in wood Wiley, Chichester, UK.
Vincent J. F. V.. 1990a. Fracture in plants. Advances in Botanical Research 17: 235-282.
Vincent J. F. V.. 1990b. Structural biomaterials Princeton University Press, Princeton, New Jersey, USA.
Vincent J. F. V. Saunders D. E. J. Beyts P.. 2002. The use of stress intensity factor to quantify ‘hardness' and ‘crunchiness' objectively. Journal of Texture Studies 33: 149-159.[CrossRef][ISI]
Vincent J. F. V.. 2004. Application of fracture mechanics to the texture of food. Engineering Failure Analysis 11: 695-704.[CrossRef][ISI]
Wells A. A.. 1961. Unstable crack propagation in metals: cleavage and fast fracture. Proceedings of the crack propagation symposium, vol. 1 210-230 College of Aeronautics and Royal Aeronautical Society, Cranfield, UK.
Wells A. A.. 1963. Application of fracture mechanics at and beyond general yielding. British Welding Journal 10: 563-570.
Williams M. L.. 1957. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics 24: 109-114.
Williamson L. Lucas P.. 1995. Effect of moisture content on the mechanical properties of a seed shell. Journal of Materials Science 30: 162-166.[CrossRef][ISI]
This article has been cited by other articles:
|
|
 |
|
  K. J. Mach, D. V. Nelson, and M. W. Denny Techniques for predicting the lifetimes of wave-swept macroalgae: a primer on fracture mechanics and crack growth J. Exp. Biol., July 1, 2007; 210(13): 2213 - 2230. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 K. J. Niklas, H.-C. Spatz, and J. Vincent Plant biomechanics: an overview and prospectus Am. J. Botany, October 1, 2006; 93(10): 1369 - 1378. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
