Solid Biomechanics 9781400840649, 9780691135502, 2011010837

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Solid Biomechanics ..................................................

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Solid Biomechanics

.................................................

Roland Ennos

................................................. PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

................................................. c 2012 by Princeton University Press Copyright
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Ennos, A. R. Solid biomechanics / Roland Ennos. p. cm. Includes bibliographical references and index. ISBN 978-0-691-13550-2 (hardcover : alk. paper) 1. Biomechanics. I. Title. QH513.E56 2012 571.4
3–dc22 2011010837 British Library Cataloging-in-Publication Data is available This book has been composed in Din PRO and Warnock PRO Printed on acid-free paper ∞ Typeset by S R Nova Pvt Ltd, Bangalore, India Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

To Yvonne .................................................

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CONTENTS .................................................

Preface Acknowledgments PART 1

Understanding Elasticity

CHAPTER 1 The Properties of Materials Forces: Dynamics and Statics Investigating the Mechanical Properties of Materials Determining Material Properties Loading, Unloading, and Energy Storage The Effect of Direction Changes in Shape during Axial Loading Shear Performing Material Tests Failure and Breaking Stress Concentrations and Notch Sensitivity Energy Changes and the Work of Fracture Measuring Work of Fracture Comparing the Properties of Materials PART 2

xi xiii

3 3 4 7 8 11 11 12 14 17 17 19 21 24

Biological Materials

CHAPTER 2 Biological Rubbers The Problem of Raw Materials Biological Polymers The Shape and Behavior of Random-Coil Chains The Structure and Mechanical Properties of Rubbers Biological Protein Rubbers Resilin Abductin Elastin

29 29 30 32 32 35 35 37 38

CHAPTER 3 Complex Polymers The Mechanics of Polymers Investigating Polymer Behavior A Typical Polymer: Sea Anemone Mesoglea Mucus and Gels

42 42 44 46 48

viii

CONTENTS

Making Protein Polymers Stiffer Silks

51 53

CHAPTER 4 Polymer Composites Combining Materials The Behavior of Soft Composites Natural Soft Composites Rigid Composites Keratinous Structures The Theory of Fillers and Discontinuous Composites Insect Cuticle The Plant Cell Wall Wood

59 59 59 62 66 68 74 75 79 80

CHAPTER 5 Composites Incorporating Ceramics The Advantages of Incorporating Minerals Spicule-Reinforced Connective Tissue Bone Tooth Ceramics Mollusk Shell

83 83 83 84 88 89

PART 3

Biological Structures

CHAPTER 6 Tensile Structures An Introduction to Structures Ropes and Strings Using Multiple Ropes Membranes, Skins, and Plates Resisting Out-of-Plane Forces Stresses in Pipes, Cylinders, and Spheres The Design of Arteries The Design of Lungs The Design of Swim Bladders The Design of Gas Vesicles

95 95 95 97 98 102 103 105 107 108 109

CHAPTER 7 Hydrostatic Skeletons The Advantages of Being Pressurized Cartilage The Hydrostatic Skeletons of Plants Cylindrical Pressure Vessels Pressure Vessels with Orthogonal Fibers Muscular Hydrostats Helically Wound Cylinders Helical Fibers to Control Growth and Shape Helical Fibers as Muscle Antagonists Fibers as Limits to Movement

111 111 111 112 113 113 115 115 116 119 121

CONTENTS

ix

CHAPTER 8 Structures in Bending The Complexity of Bending Simple Beam Theory The Four-Point Bending Test The Three-Point Bending Test The Consequences of Simple Beam Theory Fracture in Bending Shear in Beams The Consequences of Shear Biological Trusses Optimal Taper and the Scaling of Cantilever Beams

123 123 123 125 126 128 134 135 138 139 143

CHAPTER 9 Structures in Compression Material Failure in Compression Structural Failure in Compression The Buckling of Struts Buckling within Structures Cork

147 147 147 148 152 157

CHAPTER 10 Structures in Torsion Torsional Stresses and Strains Torsion Tests The Effect of Cross Section Designs That Resist Torsion Designs That Facilitate Torsion The Mechanics of Spiral Springs The Torsional Rigidity of Plates

159 159 160 162 162 163 165 166

CHAPTER 11 Joints and Levers Support and Flexibility Passive Movement in Plants Active Movement in Plants Hinges in Animals Moving Joints

170 170 170 171 172 175

PART 4

Mechanical Interactions

CHAPTER 12 Attachments Holding On Hooking On Attachments to Soft Substrates Attachments to Particulate Substrates Attachments to Hard, Flat Surfaces

183 183 183 184 185 189

CHAPTER 13 Interactions with the Mechanical Environment Optimizing Design for Strength

198 198

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CONTENTS

Factors of Safety How Optimization is Achieved CHAPTER 14 Mechanical Interactions between Organisms Biotic Interactions The Mechanics of Climbing Plants The Mechanics of Fungal Hyphae and Appressoria Plant Defenses against Fungi Food Processing by Animals Adaptations of Potential Food Other Biotic Interactions PART 5

198 201 206 206 206 209 210 210 212 215

Looking Forward

CHAPTER 15 The Future of Structural Biomechanics Successes Limitations and Future Developments New Frontiers for Biomechanics

219 219 219 222

Glossary References Index

223 231 247

PREFACE .................................................

Organisms, like everything else, have to obey the laws of physics, and biomechanics is the science that seeks to understand how the adaptations of animals and plants are constrained to their mechanical environment. It is a science that has a surprisingly long pedigree. Both Leonardo da Vinci and Galileo considered the mechanical designs of animals and plants even before Newton developed our understanding of mechanics itself. The early comparative anatomists, such as Baron Cuvier and Richard Owen, and the later functional morphologists were also interested in the mechanical design of the animals they studied. German and Swiss botanists of the late nineteenth century, such as Simon Schwendener and Wilhelm Pfeffer, similarly started to relate the form and anatomy of plants to the mechanical forces they had to withstand (Sachs, 1887; Haberlandt, 1914). Further impetus was provided to biomechanical study by D’Arcy Thompson’s 1917 book On Growth and Form, which sought to emphasize the role of physical laws as an important determinant of the form and structure of organisms. However, it was not until the 1940s that modern biomechanics research truly got going, with a particularly strong group in the Department of Zoology at Cambridge University, who began to investigate how animals move (Gray, 1953). Since then there has been a great deal of research on animal locomotion and many excellent books in this area (Alexander, 2003; Biewener, 2003). Terrestrial animals are mostly influenced by the force of gravity, so much of the research in this area has concentrated on the dynamics and energetics of movement and of collisions between bodies. Swimming and flying involve interactions of animals’ bodies with the surrounding water and air, whereas sessile organisms have to withstand the movements of these fluids. Most multicellular organisms also have to power flows of air and water to maintain their physiology; therefore, much has also been written on the subject of biological fluid dynamics (Vogel, 1994). In recent years there has been a great explosion in the study of the mechanics of biological materials (Currey and Vincent, 1980; Vincent, 1990b; Currey, 2002), both within biology departments and departments of materials science and engineering. There is great excitement at the prospect of applying this research to develop biomimetic materials and grow replacement body tissues. Research on the biomechanics of plants has also expanded rapidly (Niklas, 1992) with possible spin-offs for agriculture and biomimetics. However, despite the recent advances, there is no one place where a student can find an integrated account of structural biomechanics, although

xii

PREFACE

Gordon’s marvelous books on engineering (Gordon, 1968, 1978) did have a strong biomechanical bias, and Vogel included excellent chapters on materials and structures in his books on comparative biomechanics (Vogel, 1988, 2003). I seek to redress the balance and aim to draw the reader through all aspects of the field. The book begins with an introduction to the science of the mechanical properties of materials and shows how they can be determined. The first main section of the book then examines how organisms have developed such a wide range of structural materials using only the few building blocks that are available to cells. This is followed by the second section, which looks at how these materials are arranged into structures that are able to resist the many different sorts of loads to which they are subjected and how some structures facilitate controlled deformation and flexibility. The third section addresses an area of biomechanics that has recently been the subject of the fastest growing attention but that has been little commented on or reviewed: mechanical interactions. It examines how organisms interact with the mechanical environment around them: how they make attachments; how they respond to mechanical stress; and how they mechanically exploit or attempt to avoid being exploited by other organisms. Finally, I look into the future and make an attempt to identify areas in which structural biomechanics could, or should, provide new insights for biology, medicine, and engineering.

ACKNOWLEDGMENTS .................................................

This book sums up the fruits of the best part of thirty years, during which I have carried out research on and taught students about this most interdisciplinary area of biology. I have been singularly fortunate to link my childhood loves of animals and airplanes in this way. First, I was lucky enough to be taught physics at an excellent school where we studied the Nuffield Physics course, perhaps the best training anyone could have to carry out biomechanics research. Then I was fortunate to study natural sciences at Cambridge University; first, because it put paid to my then ambitions of becoming a physicist by the clever ruse of making the mathematics far too difficult for me to understand; second, because in the zoology course where I ended up, I was able to learn about animal locomotion from two excellent teachers and researchers in biomechanics: the late Ken Machin and Charlie Ellington. I then learned my trade as a biomechanics researcher from Robin Wootton in Exeter and John Currey in York, while also learning about botany from Alastair Fitter. I have for the last twenty years been happily working at the University of Manchester, where I have been free, through teaching a wide variety of students and learning more about the natural world from my colleagues, to carry out a good deal of collaborative research in many aspects of biomechanics. Throughout this time I have been lucky enough to meet and learn from so many of the great biomechanists at the enjoyable meetings of the Biomechanics Group of the Society for Experimental Biology. This has all helped to make it a lot of fun. There are many people to whom I am grateful for helping in various ways with this book. First, I would like to thank the biomechanics PhD students with whom I have worked and who have taught me so much, making me think and training me in all the latest techniques: Mitch Crook, Joanna Bradley, Guy Blackburn, Adrian Goodman, Sam Jackson, Jane Mickleborough, Bobby Mickovski, Karen Christensen-Dalsgaard, Laura Farran, Adam van Casteren, and Duncan Slater. I would like to thank my colleagues Jonathan Codd, Bill Sellers, Stephen Eichhorn, and Paul Mummery, with whom I have recently enjoyed learning about many new topics in biomechanics and setting up the MS course in biomechanics at the University of Manchester. My course on biological materials and structures in this program (together with an earlier undergraduate biomechanics course) is the basis of this book, which I undertook to write under the delusion that it would not therefore involve much more work. I would like to thank the students of these courses for their forbearance and feedback. Despite the work it has involved, I would still like to thank Princeton University Press for

xiv

ACKNOWLEDGMENTS

commissioning the book and the staff there for all their help in production. I would also like to thank Walter Federle for reading and commenting on part of the manuscript. Above all I would like to thank my partner Yvonne for her support, not only during the writing of this book but also during all the years of research and teaching before that.

CHAPTER 1 .................................................

The Properties of Materials

FORCES: DYNAMICS AND STATICS We all have some intuitive idea about the mechanics of the world around us, an idea built up largely from our own experience. However, a proper scientific understanding of mechanics has taken centuries to achieve. Isaac Newton was of course the founder of the science of mechanics; he was the first to describe and understand the ways in which moving bodies behave. Introducing the concepts of inertia and force, he showed that the behavior of moving bodies could be summed up in three laws of motion. 1) The law of inertia: An object in motion will remain in motion unless acted upon by a net force. The inertia of an object is its reluctance to change its motion. 2) The law of acceleration: The acceleration of a body is equal to the force applied to it divided by its mass, as summarized in the equation F = ma,

(1.1)

where F is the force; m, the mass; and a, the acceleration. 3) The law of reciprocal action: To every action there is an equal and opposite reaction. If one body pushes on another with a given force, the other will push back with the same force in the opposite direction. To summarize with a simple example: if I give a push to a ball that is initially at rest (fig. 1.1a), it will accelerate in that direction at a rate proportional to the force and inversely proportional to its mass. The great step forward in Newton’s scheme was that, together with the inverse square law of gravity, it showed that the force that keeps us down on earth is one and the same with the force that directs the motion of the planets. All this is a great help in understanding dynamic situations, such as billiard balls colliding, guns firing bullets, planets circling the sun, or frogs jumping. Unfortunately it is much less useful when it comes to examining what is happening in a range of no-less-common everyday situations. What is happening when a book is lying on a desk, when a light bulb is hanging from the ceiling, or when I am trying to pull a tree over? (See fig. 1.1b.) In all of these static situations, it is clear that there is no acceleration (at least until the tree does fall over), so the table or rope must be resisting gravity and the tree must be resisting the forces I am putting on it with equal and opposite

4

CHAPTER 1

(a)

F

(b)

m

a F

Figure 1.1. Forces on objects in dynamic and static situations. In dynamic situations, such as a pool ball being given a push with a cue (a), the force, F , results in the acceleration, a, of the ball. In static situations, such as a tree being pulled sideways with a rope (b), there is no acceleration.

reactions. But how do objects supply that reaction, seeing as they have no force-producing muscles to do so? The answer lies within the materials themselves. Robert Hooke (1635–1703) was the first to notice that when springs, and indeed many other structures and pieces of material, are loaded, they change shape, altering in length by an amount approximately proportional to the force applied, and that they spring back into their original shape after the load is removed (fig. 1.2a). This linear relationship between force and extension is known as Hooke’s law. What we now know is that all solids are made up of atoms. In crystalline materials, which include not only salt and diamonds but also metals, such as iron, the atoms are arranged in ordered rows and columns, joined by stiff interatomic bonds. If these sorts of materials are stretched or compressed, we are actually stretching or compressing the interatomic bonds (fig. 1.2b). They have an equilibrium length and strongly resist any such movement. In typically static situations, therefore, the applied force is not lost or dissipated or absorbed. Instead, it is opposed by the equal and opposite reaction force that results from the tendency of the material that has been deformed to return to its resting shape. No material is totally rigid; even blocks of the stiffest materials, such as metals and diamonds, deform when they are loaded. The reason that this deformation was such a hard discovery to make is that most structures are so rigid that their deflection is tiny; it is only when we use compliant structures such as springs or bend long thin beams that the deflection common to all structures is obvious. The greater the load that is applied, the more the structure is deflected, until failure occurs; we will then have exceeded the strength of our structure. In the case of the tree (fig. 1.1b), the trunk might break, or its roots pull out of the soil and the tree accelerate sideways and fall over. INVESTIGATING THE MECHANICAL PROPERTIES OF MATERIALS The science of elasticity seeks to understand the mechanical behavior of structures when they are loaded. It aims to predict just how much they

THE PROPERTIES OF MATERIALS

5

Force

(a)

Deflection

Interatomic force

(b)

tension

Interatomic distance

compression

Figure 1.2. When a tensile force is applied to a perfectly Hookean spring or material (a), it will stretch a distance proportional to the force applied. In the material this is usually because the bonds between the individual atoms behave like springs (b), stretching and compressing by a distance that at least at low loads is proportional to the force applied.

6

CHAPTER 1

(a)

Displacement

(b) σmax

Stress (σ)

σyield

εyield

εmax

Strain (ε) Figure 1.3. In a tensile test, an elongated piece of a material is gripped at both ends (a) and stretched. The sample is usually cut into a dumbbell shape so that failure does not occur around the clamps, where stresses can be concentrated. The result of such a test is a graph of stress against strain (b), which shows several important mechanical properties of the material. The shaded area under the graph is the amount of elastic energy the material can store.

should deflect under given loads and exactly when they should break. This will depend upon two things. The properties of the material are clearly important—a rod made of rubber will stretch much more easily than one made of steel. However, geometry will also affect the behavior: a long, thin length of rubber will stretch much more easily than a short fat one. To understand the behavior of materials, therefore, we need to be able separate the effects of geometry from those of the material properties. To see how this can be done, let us examine the simplest possible case: a tensile test (fig. 1.3a), in which a uniform rod of material, say a rubber band, is stretched.

The Concept of Stress If it takes a unit force to stretch a rubber band of a given cross-sectional area a given distance, it can readily be seen that it will take twice the force to give the same stretch to two rubber bands set side by side or to a single band of twice the thickness. Resistance to stretching is therefore directly proportional to the cross-sectional area of a sample. To determine the mechanical state of the rubber, the force applied to the sample must

THE PROPERTIES OF MATERIALS

7

consequently be normalized by dividing it by its cross-sectional area. Doing so gives a measurement of the force per unit area, or the intensity of the force, which is known as stress and which is usually represented by the symbol σ , so that σ = P /A,

(1.2)

where P is the applied load and A the cross-sectional area of the sample. Stress is expressed in SI units of newtons per square meter (N m−2 ) or pascals (Pa). Unfortunately, this unit is inconveniently small, so most stresses are given in kPa (N m−2 × 103 ), MPa (N m−2 × 106 ), or even GPa (N m−2 × 109 ).

The Concept of Strain If it takes a unit force to stretch a rubber band of a given length by a given distance, the same force applied to two rubber bands joined end to end or to a single band of twice the length will result in twice the stretch. Resistance to stretching is therefore inversely proportional to the length of a sample. To determine the change in shape of the rubber as a material in general, and not just of this sample, the deflection of the sample must consequently be normalized by dividing by its original length. This gives a measure of how much the material has stretched relative to its original length, which is known as strain and which is usually represented by the symbol, ε, so that ε = dL/L ,

(1.3)

where dL is the change in length and L the original length of the sample. Strain has no units because it is calculated by dividing one length by another. It is perhaps unfortunate that engineers have chosen to give the everyday words stress and strain such precise definitions in mechanics, since doing so can confuse communications between engineers and lay people who are used to the vaguer uses of these words. As we shall see, similar confusion can also be a problem with the terms used to describe the mechanical properties of materials.

DETERMINING MATERIAL PROPERTIES Many material properties can be determined from the results of a tensile test once the graph of force against displacement has been converted with equations 1.2 and 1.3 into one of stress versus strain. Figure 1.3b shows the stress-strain curve for a typical tough material, such as a metal. Like many, but by no means all, materials, this one obeys Hooke’s law, showing linear elastic behavior: the stress initially increases rapidly in direct proportion to the strain. Then the material reaches a yield point, after which the stress increases far more slowly, until finally failure occurs and the material breaks.

8

CHAPTER 1

The first important property that can be derived from the graphs is the stiffness of the material, also known as its Young’s modulus, which is represented by the symbol E . Stiffness is equal to the initial slope of the stress-strain curve and so is given mathematically by the expression E = dσ/dε

(1.4)

or by the original force-displacement curve E =

LdP . AdL

(1.5)

Stiff materials therefore have a high Young’s modulus. Compliance is the inverse of stiffness, so compliant materials have a low Young’s modulus. In many materials, the slope of the curve changes as the material is stretched. For such materials one can distinguish between the initial stiffness and the tangent stiffness, which is the slope at higher strains. The second important property that can be derived is the strength, or breaking stress, of the material; this is simply the maximum value of stress, σmax , along the y-axis. Breaking stress can alternatively be calculated from the original force-displacement curve using the formula σmax = Pmax /A.

(1.6)

Strong materials have a high breaking stress, whereas weak ones have a low breaking stress. The yield stress, σyield , can also be read off the graph, being the stress at which it stops obeying Hooke’s law and becomes more compliant; this is the point at which the slope of the graph falls. A third useful property of a material is its extensibility, or breaking strain, εmax , which is simply the maximum value of strain along the x-axis. Breaking strain can alternatively be calculated from the original force-displacement curve using the formula εmax = (L max − L)/L .

(1.7)

The yield strain can also be determined from this curve, being the strain at which the slope of the graph falls. A further material property that can be derived by examining the shape of the stress-strain curve is its susceptibility or resistance to breakage. A brittle material, such as glass, will not have a yield region but will break at the end of the straight portion (fig. 1.4), whereas a tough material, such as a metal, will continue taking on load at strains well above yield before finally breaking.

LOADING, UNLOADING, AND ENERGY STORAGE A final useful aspect of stress-strain graphs is that the area under the curve equals the energy, We , that is needed to stretch a unit volume of the material to a given strain. This factor is given in units of joules per cubic meter (J m−3 , which is dimensionally the same as N m−2 ). Under the linear part of the

THE PROPERTIES OF MATERIALS

9

tough

Stress

brittle

Strain Figure 1.4. Contrasting stress-strain graphs of brittle and tough materials. The tough material shows appreciable stretching after yield.

stress-strain curve, this energy equals half the stress times the strain, so We = σ ε/2. But strain equals stress divided by stiffness, so We = σ (σ/E )/2 = σ 2 /2E .

(1.8)

The elastic storage capability, Wc , of a material is the amount of energy under the curve up to the point at which yield occurs and is given by the equation 2 /2E . Wc = σyield

(1.9)

The amount of energy an elastic material can store, therefore, increases with its yield stress but decreases with its stiffness, because stiffer materials do not stretch as far for a given stress. So the materials that store most energy are ones that are strong but compliant. In a perfectly elastic material, all of this energy would be stored in the material and could be recovered if it were allowed to return to its original length. However, no materials are perfectly elastic; the percentage of energy released by a material, known as its resilience, is never 100% and falls dramatically in tough materials after yield, since yield usually involves irreversible damage to the sample. The resilience of a material can be readily

10

CHAPTER 1

Stress

(c)

Stress

(a)

Strain

Stress

(d)

Stress

(b)

Strain

Strain

Strain

Figure 1.5. The results of loading/unloading tests for (a) a perfectly elastic material, (b) a perfectly plastic material, (c) an elastic-plastic material, and (d) a viscoelastic material.

measured using a modified tensile test in which the sample is stretched to a point before yield occurs and then allowed to return to its rest length. The unloading curve will always be below the loading curve. The resilience is the percentage of the area under the unloading curve divided by the area under the loading curve; the percentage of energy that is lost is known as the hysteresis and is the remainder of 100% minus the resilience. Loading/unloading tests can be used to differentiate between different sorts of materials. In a perfectly elastic material (fig. 1.5a), the unloading curve follows the loading curve exactly, there is no hysteresis, and the material returns to its original shape after the test. In a perfectly plastic material, on the other hand (fig. 1.5b), the material will be permanently deformed by the load, and all the energy put into it will be dissipated. Tough materials often show elastic-plastic behavior (fig. 1.5c), acting elastically before and plastically after yield, in which case the sample will return only part of the way to its original shape and some energy will be dissipated in deforming it permanently. Finally, even before yield, materials often show

THE PROPERTIES OF MATERIALS

11

viscoelastic behavior (fig. 1.5d), in which energy is lost as they deform, just as it does in liquids, due to internal friction. The amount of energy lost and hence the shape of the loading/unloading curve will vary with the speed at which the test is carried out, as we shall see in Chapter 3, but unlike with elastic-plastic behavior, the material will eventually return to its original shape.

THE EFFECT OF DIRECTION Many engineering materials, such as metals, plastics, and concrete, are essentially homogenous and have the same material properties in all directions. These are said to be isotropic. Many other materials, on the other hand, particularly those with a complex internal structure (including many if not most biological materials), have very different mechanical properties in different directions. These materials are said to be anisotropic, and to fully characterize them, materials tests must be carried out in all three planes.

CHANGES IN SHAPE DURING AXIAL LOADING When a typical material sample is put into axial loading, that is, being stretched or compressed, it does not only get longer or shorter; it also gets narrower or thicker, necking or bulging under the load (fig. 1.6). As a consequence, in a tensile test the load will be spread over a smaller area, and so the actual stress in the sample will be greater than the stress given by dividing the load by the original area. The shape of the sample will also be elongated by more than the value given by dividing the change in length by the original area. In other words, both the stress and the strain will be underestimated. In most engineering materials, which deform by no more than 0.1–1% of their original length before they break, this is not a great problem. Engineers usually do not bother to try and calculate the true stress and true strain in their samples. Instead they use the convention of ignoring the change in shape and instead calculating what are known as engineering stress and engineering strain from the original dimensions of the sample. With such small changes in shape, the error would in any case be small. For many biological materials, on the other hand, strains can be far greater, reaching values up to 10, meaning stretches of 1000%! In these cases the differences between true stress and strain and engineering stress and strain can be very great indeed. However, because it is difficult to measure changes in shape during the course of materials tests, even biologists usually use engineering stress and strain, although, as we shall see, measuring the actual changes in shape can also provide other information about the material. The degree to which a material necks or bulges when stretched or compressed is given by its Poisson’s ratio, which is denoted by the symbol

12

CHAPTER 1

tension (b)

compression

(a) (c)

Figure 1.6. Shape changes during loading. If a specimen (a) is stretched (b), it will also tend to get narrower, whereas if it is compressed (c), it will tend to bulge outwards.

ν and calculated using the equation ν=−

lateral strain axial strain

(1.10)

For engineering materials that are isotropic, ν is usually between 0.25 and 0.33. The upper theoretical limit for ν is supposed to be 0.5, since at this value the volume of material will be unchanged as it is stretched; if the length increases by 1%, both the thickness and width will decrease by 0.5%, and the total volume will remain the same. If the lateral strain in both directions were greater than half the longitudinal strain, it would result in the volume decreasing when a material was stretched and increasing when compressed, which would seem to be physically improbable. Many biological materials behave in rather odd ways, however; being anisotropic they may have different Poisson’s ratios in different directions. As we shall see later in the book, some biological materials also have very high Poisson’s ratios, whereas others, such as cork, have values near zero; it is even possible to design materials with negative Poisson’s ratios, materials that expand laterally when stretched.

SHEAR We have seen how the axial stresses of tension and compression deform materials, but materials can also be deformed by a different kind of stress, shear stress. Shear stress acts parallel to a material’s surface (fig. 1.7),

THE PROPERTIES OF MATERIALS

13

(a)

(b)

F

F

γ F

F Figure 1.7. Shear stress deforms a square piece of material (a) into a rhombus shape (b) with shear strain γ .

tending to deform a rectangle into a parallelogram, a deformation known as shear strain.

Shear Stress Just as for axial stresses, the shear stress, which is denoted by τ , is a measurement of the intensity of the shear force and is therefore given by the expression τ = F /A,

(1.11)

where F is the shearing force that has been applied, and A is the area parallel to that force over which the force is applied. The units of shear stress are the same as those of axial stress: newtons per square meter (N m−2 ), or pascals (Pa). Note that if a unit of material is put into shear (fig. 1.7b), the righthand element being pushed upward, an equal and opposite shear force must act downward on the left-hand face for the element to be in equilibrium. However, if those were the only forces on the element, they would form a couple, spinning the material counterclockwise. Therefore two other shear stresses are set up, a stress on the top surface acting toward the right, and one on the lower surface acting toward the left.

Shear Strain Just as axial stresses cause axial strains, so shear stresses set up shear strains, which are the change in the angles within the elements, denoted by γ (fig. 1.7). Shear strains are expressed in radians, which, being ratios of the

14

CHAPTER 1

angular displacement relative to a portion of the full circumference, are dimensionless, just like axial strains.

Determining Material Properties in Shear The shear properties of a material can be determined by carrying out direct shear tests or torsion tests (see chapters 3 and 10), the results of which can be worked up just like the results of axial tests to give a graph of shear stress against shear strain. The most important shear property, the shear modulus, G, is determined similarly to Young’s modulus, using the equation G = dτ/dγ .

(1.12)

The Relationship between Axial Forces and Shear At first glance it seems as if axial and shear forces are quite different, unconnected forces. However, if we look at what happens during axial and shear loading, it becomes apparent that they are inextricably linked. In tensile and compressive tests, a square element at 45◦ to the loading will be sheared (fig. 1.8a–c), whereas in a shear test, a square element at 45◦ to the loading will be stretched in one direction and compressed in the other (fig. 1.8d,e). The amount of shear produced by a tensile test depends on the Poisson’s ratio of the material: materials with a larger ν will contract more laterally than those with a smaller ν, so the shear strain caused by a given tensile strain will be greater. For this reason materials with a high Poisson’s ratio will have a relatively lower shear modulus, G, compared with their Young’s modulus, E . It can be readily shown by a geometrical argument (Gere, 2004) that E and G are related by the expression G=

E 2(1 + ν)

(1.13)

so the shear modulus G is typically between 1/3 and 1/2 of the Young’s modulus of a material, depending on its Poisson’s ratio. Note that this expression is valid only for isotropic materials and so should not be used for biological materials, where it can prove highly misleading!

PERFORMING MATERIAL TESTS Many of the mechanical properties of a material can therefore be readily determined by carrying out one of two sorts of mechanical tests in which materials are put into axial loading: tensile tests and compressive tests. Both of these are most conveniently carried out in universal testing machines on specially prepared samples.

THE PROPERTIES OF MATERIALS

15

(b) (a) (c)

(d)

(e)

Figure 1.8. The relationship between axial and shear strains. If a square element at 45◦ (a) is stretched (b) or compressed (c), it will be sheared into a rhombus. Similarly if a square element at 45◦ (d) is sheared (e), it will be stretched and compressed into a rectangle.

Tensile Tests For a tensile test, the test piece typically has a “dumbbell” form (fig. 1.3a) with a relatively long, thin central portion and broad shoulders at each end. The sample is gripped firmly at its shoulders by two clamps: the lower one is mounted in the base of the machine; the upper one is attached via a load cell to a movable crosshead. To carry out the test, the crosshead is driven upward at a constant speed, while the force required to stretch the sample is measured by the load cell. This data is transferred to a computer that produces a readout of force versus deflection and, given the original dimensions of the sample, has the ability to calculate its material properties. It is assumed (fairly accurately) that all the stretching has occurred in the narrow central section of the sample. The widening at the ends ensures that the sample breaks in this central section and not at the clamps, where stresses can be concentrated. Tensile tests have three main pitfalls. The first is that with relatively thick samples of stiff materials, the rigidity of the sample may approach that of the testing machine. When a test is carried out in this situation, the machine itself will deform significantly, meaning that the readout overestimates the deflection of the sample and stiffness is underestimated. There are three

16

CHAPTER 1

solutions to this problem: you can use a longer, narrower sample to reduce its rigidity; you can attach an extensometer, which directly measures strain, to the sample; or you can attach an electronic strain gauge to the sample (Biewener, 1994). The second problem with tensile tests is the difficulty in producing the complex shape of the sample. Biological materials can be prepared by cutting around a machined template, but doing so can prove difficult, particularly for samples cut from small pieces of tissue. The third problem is that many biological materials, particularly the soft, wet, and slippery ones, can prove extremely difficult to clamp. In such cases biomechanists may resort to a range of techniques: using sandpaper to roughen the clamps; freezing the clamps to harden the material within the jaws; gluing the sample to the clamps using a cyanoacrylate glue that binds to water; or simply wrapping the sample around purpose-built attachments.

Compressive Tests Some of the problems of tensile tests can be overcome by carrying out compressive tests, in which a relatively thicker rod of material is squashed between two plates. The sample is much easier to machine because no expanded ends are needed, but the sample and plates must both be machined flat. Because the sample is relatively thicker, it will also be more rigid than a tensile sample, so it is much more likely that strain will have to be measured with an extensometer. Compressive tests usually give values of stiffness very similar to those of tensile tests, but as we shall see, materials often have very different tensile and compressive strengths. Therefore to fully characterize a material both tests may be needed.

Torsion Tests The shear properties of materials can be determined using the sorts of torsion tests we will examine more thoroughly in chapter 10.

Mechanical Testing with Homemade Equipment Not everyone has access to a materials testing machine or can afford to buy one. It may also be impossible to transport samples to the laboratory (for instance, if you want to investigate the properties of wood in a tropical rainforest). Finally, most commercially available testing machines are just not sensitive enough to measure the material properties of structures such as lengths of spider silk, which are very thin and compliant. For these situations, it is often necessary to construct purpose-built apparatus, which can work perfectly well. Nowadays, electronic force and displacement transducers are fairly inexpensive and data logging into laptop computers is fairly straightforward. However, in certain situations electronic equipment

THE PROPERTIES OF MATERIALS

17

may not be practicable or affordable, and good results may instead be achieved by purely mechanical means, either by measuring force with a spring gauge or by hanging weights on the end of a sample. Whichever way the forces are measured, though, it must be remembered that tests can be divided into two main types. For larger samples, displacement-controlled tests, in which the length of the sample is progressively altered while the force required to do this is measured, are recommended. Not only is one controlling the independent variable of the stress-strain curve, but these tests are also fairly safe, since when the material breaks the only energy released is that which is stored in the sample. In contrast, load-controlled tests, such as those in which weights are hung on the end of a sample, are very easy to perform, but failure of the sample can result in potentially damaging deflection of the clamps and of the mechanism that is applying the load.

FAILURE AND BREAKING As we have seen, it is relatively easy to explain how and why materials resist being deformed; one just has to consider the forces set up between their atoms. The fracture behavior of materials is more difficult to understand. It might be expected that the strength of a piece of material will be directly proportional both to the strength of its interatomic bonds and to its crosssectional area. Hence its breaking stress should be high and independent of the sample size. However, most materials have much lower breaking stresses than would be predicted from the strength of their chemical bonds, and larger pieces of material often have far lower breaking stresses than small ones. Brittle materials also tend to be much easier to break than tough ones, even if they have the same breaking stress. Throughout the last century, with the pioneering work of C. E. Inglis and A. A. Griffith (recounted very clearly by Gordon [1968]), it has been shown that to explain fracture, it is necessary to consider not only the overall stress in materials but also the distribution of stress within the sample and the changes in energy involved.

STRESS CONCENTRATIONS AND NOTCH SENSITIVITY Let us first examine the distribution of stress within a material that is being stretched in a tensile test. If the test piece used is perfectly smooth and free of internal flaws, the stresses will be evenly distributed throughout the material and the strength of the sample will equal the breaking stress of the material times its cross-sectional area. However, if there is a small scratch or ridge in the surface, or a flaw within the material, the stresses will have to divert around these obstructions, and stress concentrations will be set up at their sides (fig. 1.9a). The stress concentration factor will depend on the shape of these imperfections. For a circular hole or semicircular notch,

18

CHAPTER 1

(b)

Strength

(a)

tough

brittle

Relative notch length

Figure 1.9. The effect of stress concentrations on the strength of materials. In a brittle material, stress concentrations will form at the tip of cracks (a). Therefore the strength of a piece of brittle material will fall rapidly (b, dashed line) if a notch is introduced. In contrast, in tough materials the strength will fall only linearly (b, solid line) in proportion to the length of the notch.

it has been calculated that the stress at the sides will be three times the mean stress, whereas for an elliptical hole or semielliptical notch, the stress concentration, C, is given by the formula C = 1 + (2rpe /rpa ),

(1.14)

where rpe is the radius perpendicular to the force and rpa is the radius parallel to it. The longer the crack and the smaller the crack tip, therefore, the higher the stress concentration. Long, narrow cracks or holes oriented at right angles to the applied force will therefore increase stress far more than ones oriented parallel to it. If a brittle material with a notch cut in its side is stretched, the stress at its tip will increase more rapidly than in the material as a whole until the breaking stress of the material is reached and a crack opens up; this opening makes an even sharper notch, which quickly runs through the material. The strength of a piece of brittle material will therefore fall rapidly with the size of any flaws or notches at its surface, which is illustrated by a concave graph of strength against notch size, such as that shown in figure 1.9b. Such a material can be said to be notch sensitive. One reason why large pieces of

THE PROPERTIES OF MATERIALS

19

glass have a lower breaking stress than small ones is that they are more likely to have larger, sharper notches. In contrast, if a tough material with a notch cut into it is stretched, the material will yield rather than break at the tip of the notch, deforming markedly and so blunting the crack tip. As a consequence the strength of a piece of tough material will fall slowly and linearly with notch length (fig. 1.9b) so that the strength is proportional to the area of intact tissue at the end of the notch. Materials showing this sort of behavior are said to be notch insensitive. Tough materials therefore show low notch sensitivity, and brittle materials show high notch sensitivity.

ENERGY CHANGES AND THE WORK OF FRACTURE The argument above based on stress concentrations works well qualitatively, but it is less successful in quantitatively predicting and understanding the behavior of tough materials. Another, even more useful, way of thinking about what happens during failure is to look at the energy changes involved. When an object breaks, interatomic bonds are broken, creating two new surfaces; this process requires energy, and at first glance it appears difficult to understand where that energy might come from. Let us examine the situation shown in figure 1.10, in which a plate of material of thickness t and with a crack in it of length a is being stretched, producing an overall tensile stress in the plate, σ . Elastic energy is stored in the plate, but because the stresses are being diverted around the tip of the crack, small areas above and below the crack will be unstressed and will store no energy. Here it is assumed that these areas have the shape of a right triangle, but this is just a rough approximation. If the crack extends by a distance da, a greater volume of material will become unstressed. The amount of elastic energy, We , that this will release is equal to the energy stored per unit volume of material (which we have seen from equation 1.8 is σ 2 /2E ) times the extra volume, which from geometry can readily seen to be 2t a da. Extending the crack increases its surface area by the amount 2t da, since the crack has both an upper and lower surface, and if the surface energy (in J m−2 , the energy required to produce a unit area of new surface) of the material is g , the surface energy required to extend the crack, Ws , is 2tg da. For the crack to spontaneously extend, the energy released by unstressing the material around the crack must at least equal the surface energy required to extend it, so that We ≥ Ws so σ 2 /2E × 2ta da ≥ 2tg da. Therefore σ 2 ≥ 2E g /a.

20

CHAPTER 1

da

a

Figure 1.10. The effect of increasing crack length on the volume of unstressed material. If the crack extends a distance da, the material between the dashed and solid lines becomes unstressed, releasing energy.

More sophisticated and precise analysis gives a slightly different figure for σ of (1.15) σ ≥ (2E g /πa)0.5 At low stresses a crack cannot extend because not enough energy is released from the relaxation around the crack tip to open the crack. Once σ exceeds the critical value, however, the crack can and will extend, and as it lengthens it becomes more and more energetically favorable for it to do so. The crack will run very rapidly across the material and break it. Note that the longer the crack is initially, the lower the stress required to break the material. The critical crack length acrit can readily be obtained by rearranging 1.15 to give acrit = 2E g /π σ 2 .

(1.16)

It is greater for stiffer materials and ones with higher surface energy and decreases rapidly with the stress applied. In fact, the surface energy of most materials, the energy that is required to break the top layer of interatomic bonds, is very low, approximately 1 J m−2 , so for a brittle material even tiny scratches can make it much weaker.

THE PROPERTIES OF MATERIALS

21

In tough materials, the amount of energy needed to make new surfaces, or the work of fracture, is far higher, because yielding may involve a wide range of mechanisms that absorb energy; it may involve deforming the material near the crack tip plastically, as in metals, or creating a rough fracture surface with a much greater surface area, as in fiberglass and green wood. Many different fracture tests may be used to calculate the work of fracture, Wf , which is defined as the energy to produce the crack, e, divided by the crack’s area, A (not the total new area produced, which would be twice the area of the crack) giving the expression Wf = e/A.

(1.17)

Work of fracture, like surface energy, therefore has the units J m−2 . The critical crack length for a tough material is derived by substituting work of fracture into equation 1.16 to give the expression acrit = E Wf /π σ 2 .

(1.18)

As we shall see, many biological materials have particularly sophisticated toughening mechanisms. MEASURING WORK OF FRACTURE You might think it should be very easy to measure the work of fracture of a biological material. All you would need to do would be to perform a simple tensile test, and the work of fracture could be estimated from the area under the stress-strain curve up to the point of failure. Unfortunately things are not that simple. When a tensile piece is broken, some of the energy that was stored elastically may not be used to break the material but may instead be released explosively, making a snapping noise and flinging material about. The area under the stress-strain curve will therefore overestimate the work of fracture, and the error will be most severe when using long test specimens that store more energy. The problem could be minimized by using a very short test specimen, but machine compliance would then become a major difficulty. Instead, materials scientists have developed a range of tests to measure the work of fracture, although each of these has its own limitations. Controlled Cracking One method of overcoming the loss of stored elastic energy is to carry out a more controlled test in which the crack grows in a stable fashion. One way of doing this is to sequentially load and unload a test piece that is clamped asymmetrically in the testing machine. Examples of such methods include the compact tension test (Vincent, 1992) and the double cantilever beam (fig. 1.11a). The specimen may be sequentially loaded and unloaded several times, driving the crack across the specimen, between which actions it should return to its original shape. The work of fracture can be calculated

22

CHAPTER 1

(a)

(b)

(d)

(e)

(c)

Figure 1.11. Mechanical tests used to determine the work of fracture of materials: (a) a double cantilever test, (b) a Tattersall and Tappin notch test, (c) a trouser tear test, (d) a cutting test, and (e) an impact test.

by dividing the area between the loading and unloading curves by the area of new crack formed. One problem with this test is that it can prove difficult to drive the crack in the right direction, though this can be overcome by cutting notches and guide slots to weaken the material in the required direction. Another, more intractable problem is the difficulty in machining and clamping suitable samples. A similar test method is to carry out a Tattersall and Tappin notch test (fig. 1.11b; Tattersall and Tappin, 1966), very slowly bending a sample that is cut in such a way as to drive a crack gently through a beam.

Tearing and Peeling Another method, which is useful for thin flexible material samples, is to carry out a trouser tear test (Vincent, 1992), in which two legs of a thin specimen are pulled apart (fig. 1.11c). The work of fracture is the area under the force-deflection curve divided by the area of the new crack formed. A similar test can also be used to peel a narrow sliver of a material from the rest. Unfortunately, although they are easy to perform, these tests are only

THE PROPERTIES OF MATERIALS

23

really useful for materials that have preferred lines of failure, such as plant tissues, in which longitudinal fibers constrain the cracks to run between them; otherwise the tears are all too readily diverted in the wrong direction. Cutting Tests To overcome the problem of cracks moving in the wrong direction, several different kinds of cutting test have recently been devised. Materials may be cut through with a sharp blade (fig. 1.11d), whether it be that of an instrumented microtome (Atkins and Vincent, 1984), a guillotine (Atkins and Mai, 1979), an inclined razor blade (Ang et al., 2008), or nail clippers (Bonser et al., 2004). Alternatively, one can use the double blades of a sharp pair of scissors (Darvell et al., 1996). In all of these cases the fracture is constrained to run in the desired direction. All of these tests are straightforward to perform, and the work of fracture is found by dividing the energy needed to cut through the specimen by the area of the fracture surface produced, although in scissor and guillotine tests, the friction of the devices also must be taken into account. The tests can also detect particularly tough regions of the material, such as fibers or veins, since the force needed to cut through them is greater and allows their toughness to be calculated separately. However, because the cuts constrain the direction of the crack very precisely, these tests measure the minimum work of fracture of the material and cannot detect how much toughening is given to the material by the sorts of mechanisms that involve diverting the crack. Impact Tests A final series of tests to measure work of fracture involves specimens being struck by the impact of a moving pendulum and measuring the energy required to snap them (Vincent, 1992). Typically a notched bar or rod of material is mounted in one arm of the apparatus and is hit by the other swinging pendulum (fig. 1.11e). The specimen is broken transversely, the energy required being supplied by the kinetic energy of the pendulum, which consequently does not rise to so great a height after the impact as before. The work of fracture is the change in potential energy of the pendulum before and after the test divided by the cross-sectional area of the bar. This test works well for many stiffer materials, but since the precise conditions of loading are usually unknown, it is often hard to relate the results from this test to those of the other tests. In particular, the impacts tend to be very rapid, so less energy is used to break viscoelastic materials than in the other slower methods, and the work of fracture is consequently underestimated. Other Measurements of Work of Fracture The work of fracture of a material is usually regarded as being the same thing as its toughness. However, other, quite different definitions of toughness

24

CHAPTER 1

Table 1.1 Properties of Some Man-Made and Biological Materials.

......................................................

Material

Stiffness Tensile Extensibility Strain Resilience Work of (MPa) Strength energy (%) fracture (J m−2 ) (MPa) (MJ m−3 )

Steel

200,000

400

0.008

1

Glass Concrete Rubber Resilin Abductin Elastin Mucus Mesoglea Moth Silk Tendon Keratin Cuticle Unlignified plant cell wall Wood Bone Dentine Enamel Shell (nacre)

70,000 100,000 3 2 4 1.2 0.0002 0.001 4000 2000 3000 5000 3,000

170 5 7 4

0.002 0.00005 3 2

0.2 0.0001 10 4

2 0.0005

1 5

0.8

2000 100 300 60 100

0.3 0.1 0.2 0.01 0.05

200 3 0.3

15,000 2,000 3,000

4,000 17,000 15,000 50,000 30,000

40 200 50 35 50

0.01 0.006 0.003 0.0005 0.006

0.5 0.6 0.1 0.02 0.15

30,000 2,000 500 200 1600

100,000 –1,000,000 1–10 3–40 90 93 75–90 75 55 1,200 35 93

are sometimes presented in papers. Some authors (see, for instance, Gosline et al., 2002) present values for toughness with the units J m−3 . These values were actually derived from the area under the stress-strain graphs and so are more correctly measurements of the energy-storage capabilities of the materials. It is better, therefore, to use the term work of fracture.

COMPARING THE PROPERTIES OF MATERIALS All the complexity of the properties of materials and of materials testing means that it is surprisingly complicated to compare the properties of different materials. They can differ in their stiffness, their strength, their ability to store and release energy, and in their toughness. Some materials are also better at resisting tension, whereas others resist compression better; and some materials have the same properties in all directions, whereas with others, their performance depends on the direction in which they are stressed. A good way to get an instinctive idea of how to compare materials is to think of some everyday objects familiar even to children. Jell-O has very

THE PROPERTIES OF MATERIALS

25

low stiffness and strength and is also very brittle, making it easy for even very young children to cut up and eat. Cookies, in contrast, are stiff but not very strong, so a child can carry one around without breaking it yet still readily bite into it. Toffees are both stiff and strong, although they can be smashed to smithereens when hit by a hammer, showing that they are brittle. To eat them a child has to put them whole into their mouth, before softening them and dissolving them with their warm saliva. Pencils are stiff, strong, and extremely tough, so they can take a lot of punishment without breaking, although the brittle leads are easy to snap off. Rubber bands have low stiffness, but they are very stretchy, so they are reasonably strong and can store a lot of energy, which makes them ideal for use as slingshots. What about “grown-up” materials? Well, steel is one of the most popular materials in engineering because it is stiff, strong, and tough and so is ideal for taking loads and resisting impacts. This is why it is used to make the shells of cars. Glass is far less useful, because although it is almost as stiff and strong as steel, it is extremely brittle and so shatters on impact. Concrete is widely used to build walls and floors because it is stiff and extremely strong in compression, but because it cannot take tension, it cannot be used to make the roofs of buildings without being reinforced with steel. The properties of some important man-made materials and those of the natural materials we will encounter in the next few chapters are shown for ease of comparison in table 1.1. Perhaps the material that acts as the best single benchmark is rubber, since we can deform it fairly easily by hand and so get an intuitive feel for its properties. Most of its properties also, have small integer values for stiffness and strength (in MPa), strain energy (in MJ m−3 ), and maximum strain. Some materials, such as mucus and mesoglea, are more compliant than rubber, but most “rigid materials” are thousands of times stiffer and tens of times stronger but many times less extensible. As we shall see in the next few chapters, it is practically impossible to produce materials in which all the properties are maximized; in general, the stiffer and stronger a material is, the less it can be stretched. There is therefore no one “super” material that is ideal for all purposes, and the mechanical design of organisms consists of making materials that are suitable for a particular role and arranging them in the right way within the body. That topic is what most of the rest of this book is about.

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CHAPTER 2 .................................................

Biological Rubbers

THE PROBLEM OF RAW MATERIALS At first glance it appears extraordinary that organisms made up of cells could produce large structures that rival those of human technology. After all, cells are just simple bags of chemicals surrounded by a flexible double layer of lipid molecules that are held together by nothing more than hydrogen bonding. Typical lipid membranes are approximately 10−8 m thick, with a Young’s modulus of about 10 MPa and a breaking stress of 0.5 MPa (Lingard, 1977). We can calculate the mechanical properties of animal cells with the typical radius of 10 µm surrounded by such membranes using the analysis developed in chapter 7 for hydrostatic skeletons (Money et al., 1987). This calculation will show that such cells would have an effective Young’s modulus of only about 5 kPa and a compressive breaking stress of no more than 1 kPa. These properties would make cells tens of thousands times weaker than materials such as wood or steel. The cells would be far too weak to build into a multicellular organism, since a column of cells more than 10 cm thick would be crushed by its own weight! Some strength and stiffness can be provided by the cell’s internal skeleton, but it has to be able to move to allow cell division to take place; the internal skeleton would therefore have to continually break and reform chemical bonds to keep itself stiff, just as the bonds between actin and myosin in our muscle cells are broken and reformed to keep them tensed. Such a process would take up a huge amount of a cell’s energy. It is therefore far better for cells to produce extracellular tissues. It is these that for the most part have provided organisms with the stiffness, strength, and toughness that have enabled them to grow so large. Yet it must have been extremely difficult for organisms to evolve such tissues, because they had few raw materials available to them, and none of those had useful mechanical properties. Organisms cannot make pure metals, which have long been humans’ favorite technological materials. All that organisms have at their disposal are mineral salts, proteins, and sugars. These substances have no strength at all when they are held in solution within the cell and are far too brittle when dry; imagine trying to make useful structures out of salt, dried egg white, sugar cubes, or pasta! It is one of the miracles of life, therefore, that out of these unpromising raw materials nature has been able to craft a bewildering variety of structural materials. Many of them outperform the best products of human technology and reveal far greater sophistication. This achievement is all the more remarkable because

30

CHAPTER 2

(a)

R Cα

H

OH C

N H H (b)

O H

R

O

R

C

Cα

H Cα

H N

N C

Cα

N

O

R

H

H H

H

Figure 2.1. Proteins are composed of amino acids (a), each of which has a different side group R on its α carbon atom. These are joined at the amide link to form polymers (b), which are free to rotate each side of the α carbon.

skeletal materials are rarely fully formed inside the cell. Organisms have had to develop techniques to secrete the necessary materials through the cell membrane into the extracellular space, where they undergo a final selfassembly.

BIOLOGICAL POLYMERS The key to understanding the mechanical behavior of the majority of biological materials is to recognize that they are made up of polymers. Just like some man-made polymers, such as polystyrene, polyethylene, and nylon, the biological structural polymers—proteins and sugars—are made up of repeating units of molecules joined end to end by bonds that can rotate more or less freely. Unlike most man-made polymers, however, they can be far more complex and they are invariably found in association with water. Proteins Proteins are composed of amino acids joined end to end via the peptide bond or amide link to form a single long chain (fig. 2.1). However, although the single bonds each side of the central α carbon are free to rotate, the molecule may be held in a secondary structure that is largely the result of hydrogen bonding: bonds form between the positively charged hydrogen attached to the nitrogen (NH) of one peptide link and the negatively charged oxygen of the carboxyl group (C O) of another. The ability to form such bonds and the strength of this bonding will depend on the identity of the side groups, which can also interact to help proteins such as enzymes form a more complex, folded tertiary structure. Individual protein molecules may

BIOLOGICAL RUBBERS

31

(a)

CH2OH O

H H OH

OH

H H

HO H

OH

(b) OH

CH2OH O

O O

HO

OH

OH

CH2OH O

OH O

OH

OH OH

O OH

CH2OH

O OH

CH2OH

Figure 2.2. Polysaccharides are composed of hexoses, such as β-glucose (a), which are joined via one of the several –OH groups to form polymers, such as cellulose (b).

also be attached to their neighbors, at single points or along their entire length, by such bonding to form solid materials.

Polysaccharides Polysaccharides are composed of hexoses (fig. 2.2a) joined via an –O– bond. Since each hexose has several –OH groups, each of which may react with the –OH group of another hexose, polymerization may result in the formation of branched as well as single-chain molecules. In storage molecules such as starch and glycogen, the units are joined by α linkages, and the molecules can be well branched. This makes them easy to break down but not structurally useful. In contrast, in the structural molecules we will examine, such as cellulose (fig. 2.2b) and chitin, the units are joined by β linkages. These molecules are generally unbranched and hard to break down. Sugars, like proteins, may also be held in a secondary structure or attached to each other by hydrogen bonding and other molecular interactions.

Factors Affecting the Properties of Polymers The extent of hydrogen bonding and the resulting form of both types of polymer are influenced by two other factors: the presence of other modifying side groups in the amino acids and hexoses, and the presence of water

32

CHAPTER 2

and dissolved ions. As we shall see, the mechanical behavior of biological polymers is strongly dependent on their molecular structure. Therefore, their mechanical performance depends ultimately both on their molecular sequence, or primary structure, and on the aqueous environment in which they are held. In this and the following chapters we will examine the relationship between the structure and mechanical properties of rubbers, gels, silks, and fibrous composite materials and look at how these materials are constructed and used.

THE SHAPE AND BEHAVIOR OF RANDOM-COIL CHAINS If individual parts of the polymer molecules do not interact with each other or with other polymer molecules, they form random-coil chains, which writhe around because of the heat, somewhat like snakes in a snake-charmer’s basket. This movement of the long molecules is driven by the same molecular oscillations and collisions that cause the shaking movements of small particles that is known as Brownian motion. Each molecule may be free to move randomly, but the overall distance between the two ends of any single molecule is not random. It is very unlikely indeed that at any one time each alternate bond in a molecule is bent in the opposite direction to its neighbor, so that the molecule forms a straight line. It is also very unlikely that all the bonds will be bent in the same direction, so that the molecule is curled up into a tight circle. The most likely shape is one in between those extremes, with the distance, D, between the two ends of the molecule equal to the expression √ D = a N, (2.1) where a is the length of each link in the chain and N is the number of links in the chain. The situation is, in fact, analogous to the so-called drunkard’s walk describing the rate of diffusion of fluid molecules: the distance a molecule has diffused equals the mean free path between collisions multiplied by the square root of the number of collisions.

THE STRUCTURE AND MECHANICAL PROPERTIES OF RUBBERS Rubbers, or elastomers as they are also known, are polymers in which there is little interaction between the molecules over almost their entire lengths, but in which they are joined every now and then by strong, usually covalent bonds known as cross-links. Natural rubber, for instance, is composed of isoprene chains that are obtained from the latex, or sap, of the rubber tree Hevea brasiliensis. This liquid, which the tree produces as a defense when it is damaged, can be turned into a useful solid material by cross-linking the chains by sulphur-to-sulphur (S–S) bonds, which are introduced in the process of vulcanization.

BIOLOGICAL RUBBERS

33

(b)

(a)

Figure 2.3. Changes in the molecular structure of a rubber when it is stretched. The original disordered arrangement (a) is changed to one where the molecules are more aligned along the direction of stretch (b) and therefore more ordered.

In an unstressed piece of rubber, the molecules are free to move around √ between the cross-links, which are typically separated by a distance D=a n, where n is the number of links between cross-links (fig. 2.3a). Most of the chains between cross-links in a block of unstressed rubber will at any one time be in this intermediate state, and the rubber will have a more or less constant shape, despite the movements of the molecules within it. If the rubber is stretched, however, the molecules will be pulled in the direction of stretching and on average displaced into a more ordered arrangement (fig. 2.3b), one with greater distances between cross-links in the direction of stretch and shorter distances between cross-links at right angles to that direction. Such an ordered arrangement is vastly less likely to maintain itself, so the molecular movements tend to return the rubber to its original shape. The restoring force therefore depends on the resistance of nature to the decrease in entropy, or “muddled-upness,” that would be caused by this

34

CHAPTER 2

ordering of the molecules. The stiffness of a rubber is greater when the distance between the cross-links is shorter, and so when each chain is less free to move. Stiffness is also greater at higher temperatures, when the writhing of the chains is more powerful. This is in marked contrast to almost all other materials, which tend to get softer when heated. A major way of determining that a material is a rubber, and that it relies on entropy to provide its stiffness, therefore, is to see if it gets stiffer at higher temperatures. Theoretical analysis in fact shows that the shear modulus, G of a rubber is given by the expression G = ρ RT/M

(2.2)

where ρ is the density of polymer in the material, R is Boltzmann’s constant (8.3 J K−1 mol−1 ), T is absolute temperature, and M is the mean molecular mass between cross-links, which has SI units of kg mol−1 . Since the Poisson’s ratio of most rubbers is close to 0.5, the Young’s modulus of a rubber may be approximated by the equation E ∼ = 3ρ RT/M.

(2.3)

The stiffness of a rubber will increase with the volume fraction and density of the polymer chains it contains, which is important because many natural protein rubbers also contain large quantities of water. As a consequence of their polymeric structure and the entropic mechanism that causes the long mobile molecules to return to their resting length, rubbers have several unique properties. First, compared with most materials, in which length changes are prevented by resistance of their bonds to being stretched or bent, rubbers are extremely compliant. Both natural and man-made rubbers typically have a Young’s modulus on the order of 1 MPa, compared with the values for such crystalline materials as metals and glassy materials (including glass itself) of approximately 1 GPa, a difference of three orders of magnitude. Second, rubbers can stretch much further than other materials because of the large-scale deflections of their molecules; they typically have a breaking strain greater than 1, compared with 0.01 for most metals and glass. Third, because rubbers can be deformed so much, they can store relatively large amounts of energy, and most of that energy can be released when the rubber is allowed to return to its resting shape; rubbers typically have an energy storage of 1–5 kJ m−3 and a resilience of over 90%, so they can store and release much more energy than the equivalent mass of steel! These properties have proved extremely useful to humans, hence the economic importance of rubber. Rubber tires and springs make good use of the compliance of rubber to give vehicles a smoother ride and are much lighter than steel springs. Elastic straps and rubber bands make good use of the high breaking strain to hold up our clothing and clamp stationery together. More annoyingly, children’s slingshots make use of rubber’s high energy storage and resilience to cause mischief.

BIOLOGICAL RUBBERS

35

BIOLOGICAL PROTEIN RUBBERS Nature has not been slow to take advantage of the benefits of polymeric materials with these useful rubbery properties. Although rubber itself is not used as a mechanical material by rubber trees, three protein-based rubbers have evolved and are extensively used by three very different groups of animals. In some ways the rubbers are all very similar. Each has evolved with covalent cross-links that join the molecules together, and the strands between the cross-links contain a large percentage of the amino acid glycine. This amino acid lacks a side chain and is nonpolar, characteristics that prevent the formation of the sort of electrostatic bonds that would otherwise constrain the shape of the molecule. This freedom from constraint allows the molecules to form random-coil chains. Each of the rubbers also contains a high proportion of water, whose molecules separate the side chains of other, more polar amino acids and so plasticize the rubber. As we shall see in the case of elastin, water may also contribute toward the rubberiness of the material itself.

RESILIN Perhaps the best understood and characterized biological rubber is resilin, which was discovered by Weis-Fogh (1960) during his studies of the flight systems of desert locusts and dragonflies. Weis Fogh showed that this material was composed of protein chains joined by covalent cross-links between tyrosine residues to produce di- and trityrosine. He also found that it contained about 50% water in its natural state and was soft and rubbery. Given the material’s composition and behavior, Weis-Fogh and his colleagues subjected resilin to a classic series of investigations to determine whether it did in fact show rubbery elasticity, to measure its mechanical properties, and to determine its function in these insects. Using the dragonfly Aeschna, he first measured the mechanical properties of the elastic tendons that join a minor wing muscle to the wing. Despite the fact that they were only a millimeter long, he managed to stretch them with a purposebuilt testing apparatus and measure the force required. He showed that the stiffness of the tendon was approximately 1.8 MPa, and the breaking strain about 2, properties similar to those of man-made rubbers. The material also showed the high resilience of a typical rubber, releasing more than 90% of the energy needed to stretch it when it was vibrated at frequencies of 50 Hz (Jensen and Weis-Fogh, 1962). To demonstrate that this was due to rubbery elasticity, Weis-Fogh (1961) also carried out tests in which the resilin was stretched by different lengths at different temperatures and held at those lengths while the restoring force was measured. The material behaved in just the way a rubber should, the force required being proportional to the temperature, as predicted by equation 2.3. So why do these large insects have resilin in their flight systems? Manipulations of the wings of locusts showed that there are resilin pads in the wing

36

CHAPTER 2

(a) wing base

resilin (b) thorax wall adductor muscle

abductin

1mm Figure 2.4. Location and action of natural rubbers. (a) Resilin pads in the wing hinges of locusts (redrawn after Jensen and Weis-Fogh, 1962). The pads (clear material) stretch, storing energy at the top of the upstroke of its wingbeat (lower part of figure). (b) The abductin inner hinge of a bivalve stores energy as it is compressed when the animal closes its shell.

hinges (fig. 2.4a) and in the prealar arm which are strained when the wings are raised, making up a substantial fraction of the stiffness of the wing joint (Jensen and Weis-Fogh, 1962). Consequently they store energy at the top of the wingbeat, powering the downstroke and reducing the energy needed to flap the wings. This role therefore makes use of the high elastic storage and high resilience of the rubber. The elastic tendons of the dragonfly, on the other hand, are too small and are linked to such minor flight muscles to have such an important role, but they may act as shock absorbers.

The Distribution, Comparative Function, and Exploitation of Resilin Following this initial work, the distribution and function of resilin in insects and other arthropods have been quite extensively studied. This research is worth looking at in its own right, and also because it presents a good example of how initial discoveries may be exploited in biomechanics. One obvious stage is repetition. Scientists looked for other examples of the use of resilin by insects, helped by the fact that it exhibits UV fluorescence and so is easy

BIOLOGICAL RUBBERS

37

to detect. Essentially they repeated Weis-Fogh’s research, but in different organisms and on different structures. Resilin was found to be present in the sound-production organs of cicadas (Young and Bennet-Clark, 1995), where it helped maintain their vibrations, just as it helps locusts flap their wings. The energy-storage capability of resilin was also exploited in the catapult jumping apparatus of both fleas and froghoppers (Bennet-Clark and Lucey, 1967; Burrows, 2003). The jumping muscles of a flea do not directly straighten its legs. Instead, they slowly compress a resilin pad via a ratchet mechanism, storing energy. Only when the flea wants to jump is the ratchet released, allowing the pad to recoil, which it does in under a millisecond, straightening the insect’s legs, and flinging the insect into the air. Research has also revealed the use of resilin in structures where its ability to cope elastically with large deformations is more important than high energy storage or resilience; for instance, it has been found in the flexion lines of folding insect wings (Haas et al., 2000) and the complex stinging apparatus of bees (Hermann and Willer, 1985). Repeating other people’s work but on different organisms can show differences between and within species. Much can therefore be learned about the functions of materials or structures from comparative studies. For instance, investigations have shown that resilin does not appear to be present in the wing hinges of small insects such as beetles, flies, or bees, suggesting that they must use other structures to store the kinetic energy of their wings. Further research has shown that these insects store kinetic energy in their stiff flight muscles as well as in the cuticle of the thorax. Finally, research may attempt to exploit the discovery. The purpose of the science of biomimetics is to imitate the mechanical properties of biological materials and structures. Because resilin has a higher resilience than man-made rubbers and also because it has more evenly spaced crosslinks, there might be an advantage in copying it, attempting to cross-link man-made rubbers more evenly, or even in synthesizing artificial resilin (Elvin et al., 2005). However, the high resilience of resilin may instead be due to the water it contains and that acts as a plasticizer, preventing the chains from interacting between the cross-links. In an attempt to make a more efficient rubbery material, artificial resilin has recently been produced from recombinant pro-resilin molecules synthesized within Escherichia coli bacteria and subjected to photochemical cross-linking (Elvin et al., 2005). The resulting material does have a resilience of well over 90%, but it contains much more water than natural resilin and is nowhere near as stiff; it has a Young’s modulus of just 2.5 kPa, making it about 700 times more compliant.

ABDUCTIN Another natural protein rubber is abductin, which is found in the hinges between the two halves of bivalve shells. The outer hinge is composed of a fairly inextensible strap of protein; the abductin is found in a block, as the inner hinge, and its resistance to compression acts as an antagonist

38

CHAPTER 2

to the adductor muscle that closes the shell (fig. 2.4b). In many ways the structure and properties of abductin are similar to those of resilin. Like resilin, the abductin found in scallops acts as a true rubber (Alexander, 1966), with the protein chains separated and lubricated by water and joined by tyrosine cross-links, although in the case of abductin, these form 3,3-methylene bistyrosine. Experiments in which shells were closed while the force required to do this was monitored showed that the stiffness of scallop abductin is similar to that of resilin, with values ranging from 1 to 4 MPa (Kelly and Rice, 1967). Tests were also carried out to determine the resilience of abductin: the muscles were removed and the shells were set swinging, opening and closing with gradually reduced amplitude because of energy loss in the hinge. Such experiments produced an early value for the resilience of scallop hinges of 91% (Alexander, 1966). The hinges of scallops are also capable of deforming to a great extent, with compressive strains of about 0.45 when their shells are closed (Trueman, 1953), but because of the geometry of the hinges, the breaking strain of abductin is not known. Bivalve hinges have also been the subject of a range of comparative studies. Scallops are very unusual bivalves in that they actively swim by jet propulsion, rapidly opening and closing their shells and expelling a jet of water behind them. Maintaining the reciprocating movement requires energy, so there should be strong selection pressure for scallop abductin to have high resilience to reduce energy losses. Measured values of resilience of up to 95% have been recorded (Bowie et al., 1993). Such high values allow swimming scallops to greatly reduce the energy they use to swing their shells and reduce transport costs by up to 30% (DeMont, 1990). In contrast to scallops, high resilience in sessile bivalves might be a disadvantage, because it means that the adductor muscle will have to work hard continuously just to keep the shell closed. Studies have shown that the resilience of the hinges of sessile bivalves such as mussels is far lower, about 70% (Trueman, 1953); this value is lower because the composition of the protein is somewhat different, and also because the hinges contain more calcium carbonate than protein (Kahler et al., 1976). There has even been a study that compared the resilience of the abductin hinges of scallops from different climates (Denny and Miller, 2006). The resilience of rubbers tends to decrease at lower temperatures, for reasons we will examine in the next chapter, and so should pose a problem for Antarctic scallops. The authors, however, showed that the resilience of abductin in the Antarctic scallop, Adamussium colbecki, is 3% higher than that of its temperate relative at any given temperature, although how it achieves this enhanced performance is not known.

ELASTIN The rubber that has been subjected to the most study by far, undoubtedly because it is the one found in humans and other vertebrates, is elastin. At first it seemed likely that elastin obtains its stiffness from the same conventional entropic mechanisms displayed by resilin and abductin. However,

BIOLOGICAL RUBBERS

39

(a)

F

(b)

1.7nm

2.4nm 7.2nm

5.5nm Figure 2.5. The structure of elastin. Molecules are arranged in helixes (a), which in turn are held in larger triple helixes (b). (Reproduced by permission of Informa Healthcare Solutions from Urry, 1983.)

although x-ray diffraction studies showed little evidence of structure at the smallest atomic scale, elastin does differ in microscopic structure from both resilin and abductin: rather than being found in a single amorphous mass, it is split into ordered fibers on the order of 5 nm in diameter. Furthermore, the molecular structure of elastin also shows many repeated sequences that are five and six amino acids long (Urry, 1983; Urry and Parker, 2002). It now seems likely that elastin has a complex coiled-coil

40

CHAPTER 2

(a) 1.5

Stress (MPa)

1.0

0.5

0.0 0.0

0.5

1.0

Strain (b)

(c)

Figure 2.6. (a) The results of a tensile test on the ligamentum nuchae from the neck of a deer (b). The changes in its shape (c) of the ligament (dotted area) as the deer lowers its head to feed. (Redrawn after Dimery et al., 1985.)

BIOLOGICAL RUBBERS

41

structure (fig. 2.5). The amino acid sequence allows the formation of βturns, which enable the proteins to form tubes (fig. 2.5a), which themselves then coil up to form the fiber (fig. 2.5b). Water can penetrate into the tubes, altering its material properties. The flexibility is the result of the rotational freedom of the bonds in the β-turns that give the protein its rubberlike elasticity, which is fairly similar to that of resilin and abductin. It has recently been suggested that these other rubbers may also have a mechanism of elasticity somewhat like that of elastin and so are not conventional rubbers. Certainly, molecular evidence suggests that there are repeated sequences that might produce such a structure. The properties of elastin can be more difficult to study than those of the other rubbers, because it is rarely found on its own in the vertebrate body. The best example of nearly pure elastin is found in the ligamentum nuchae of ungulates. This ligament runs along the dorsal surface of their necks, from the neural spines of the thoracic vertebrae to the top of their skull, with branches to some or all of the neck vertebrae. Dimery et al. (1985) removed the ligamentum nuchae of a deer and stretched it by simply hanging weights from it. They found that the ligament had an initial Young’s modulus of approximately 1 MPa, but that it became stiffer at higher strains, up to a breaking stress of about 1.5 MPa and a breaking strain of 1 (fig. 2.6a). The high breaking strain allows ungulates to stretch their necks down to graze (fig. 2.6c), while the stiffness supplies most of the restoring force that the animals need to support the weight of their head and neck. One can get a feel for the softness and stretchiness of this ligament by cutting one out of an untrimmed neck of lamb, obtainable from a butcher shop. Similar ligaments are found in the necks of birds such as turkeys (Bennett and Alexander, 1987) and perform the same job, but in these birds they are split into many short lengths, which allow them to stretch by similar amounts even though they are positioned much closer to the centers of the vertebral joints. The most extensive use of elastin is in the walls of our large blood vessels, particularly the arteries. The elastin walls act to equalize the flow of blood and prevent unduly high peak pressures as the blood is pumped out of the heart by allowing the vessels to stretch in diameter by about 30%; not only does this allow us to take our pulses, but it also reduces the cost of pumping blood by about a third.

CHAPTER 3 .................................................

Complex Polymers

THE MECHANICS OF POLYMERS Our analysis of the mechanical behavior of rubbers in the last chapter assumed that the chain molecules were totally free to move about. As we saw, the protein rubbers that were evolved by nature have large amounts of glycine and other amino acids between their cross-links that do not interact with each other and so allow this free movement. But protein rubbers are unusual in this respect; most polymer units, especially in proteins and sugars, have nonuniform charge distributions over their length and so will tend to interact with each other. This will have two results. Individual points on different molecules may form temporary bonds between themselves, creating entanglements that mimic the permanent cross-links of rubbers. And different points on the same molecule may also form bonds with each other, reducing the flexibility of the molecule and giving it a more permanent shape. Both sorts of bond will have the effect of reducing the compliance of the polymer; it will take much longer, or will require more heat energy in the material, for chance vibrations due to the heat to break the bonds and so allow the material to change shape. The stiffness of the polymer will therefore increase as the length of time over which a force is applied decreases and as the temperature of the material is lowered.

The Behavior of a Typical Polymer A typical polymer shows a fairly complex pattern of behavior over time or with temperature (fig. 3.1). Over very short periods of time or at very low temperatures, no free rotation of the bonds is possible, so the stiffness of the material is dominated by the resistance of the polymer’s bonds to being bent. The stiffness and resilience of the material will therefore be high, but the material will also be brittle because fracture will involve only the breaking of surface bonds; there will be no deformation of the material below. Polymers in these conditions are said to exhibit glassy behavior. Over longer periods of time and at slightly higher temperatures, some rotation of the bonds is possible, so stiffness decreases. However, deforming the material requires energy that may not be recovered, so the resilience of the material decreases and its toughness increases. Polymers in these conditions are said to exhibit leathery behavior. Over still longer times or at higher temperatures, the molecules can move fairly freely, but that movement is

COMPLEX POLYMERS

109

43

glassy

E (Pa)

leathery

plateau

106

transition equilibrium

103 Log time

Figure 3.1. The changes in stiffness, E , over time of a typical polymer. Over time, its behavior changes from glassy through leathery and plateau to eventual equilibrium behavior. The dashed line indicates the behavior of polymers that lack permanent cross links.

impeded by the entanglements between the molecules. The material may reach a plateau in its mechanics, at which it exhibits rubbery behavior. Finally, over even longer times and higher temperatures, the entanglements have time to slip past each other. At first not all the entanglements slip, and those that remain still supply some stiffness to the polymer at the expense of decreased resilience. The material is said to be showing transition behavior. Finally, however, the material reaches its equilibrium behavior: if there are permanent cross-links between the molecules, the material behaves as a compliant and resilient rubber; if there are no cross-links, the polymer molecules can slip past each other and it behaves like a viscous liquid.

Factors Affecting Polymer Behavior Of course, all polymers are different and the magnitude of the changes in stiffness and the times and temperatures over which these changes occur will vary greatly; increased bond stiffness will shift the behavior toward the glassy end of the spectrum, whereas adding water will shift it toward the rubbery end. The complete spectrum of behavior is rarely seen in the course of everyday temperatures and time. However, one material, the so-called Potty Putty, or Silly Putty, which is sold in toy stores, does show it. If hit with a hammer, it smashes like glass; if rolled into a ball and dropped on the floor, it bounces like rubber; and if pulled slowly, it stretches and flows like a piece of soft toffee. No biological material shows quite this range

44

CHAPTER 3

of behavior. The biological rubbers we examined in the last chapter, for instance, have such free-moving molecules that they exhibit equilibrium rubbery behavior and consequently have high resilience. But we also saw in the last chapter that the resilience of the rubber abductin decreases at lower temperatures, suggesting that it is then working in the transition zone. There is also evidence that the resilience of elastin declines rapidly at higher frequencies; this precludes its use as a store of elastic energy to help reduce power consumption in the flapping flight of hummingbirds.

INVESTIGATING POLYMER BEHAVIOR There are, in fact, a range of other biological materials that make extensive use of the changes in behavior with time. These changes can be examined using two different sorts of transient mechanical tests—creep tests and stress relaxation tests—to examine the change of stiffness over time, whereas the change in resilience over different time periods can be investigated using dynamic tests.

Performing Transient Tests Transient tests involve applying an instantaneous change to the material and examining its response over time. In a creep test, a material is subjected to a sudden load and its resulting deflection is noted over time. There is an initial deflection, due to the instantaneous compliance, and the deflection then increases, more or less exponentially over time, to the equilibrium compliance. The results are best plotted as a graph of the logarithm of compliance, the inverse of Young’s modulus, against the logarithm of time. Doing so fits the results meaningfully onto a single graph and gives an idea of the time course of the exponential component (fig. 3.2a). In a stress relaxation experiment, the material is subjected to a sudden deflection, and the load required to maintain the deflection is recorded. There is an initial force due to instantaneous stiffness, which decreases, more or less exponentially, to the equilibrium stiffness. The results are best plotted as a graph of the logarithm of Young’s modulus against the logarithm of time (fig. 3.2b). In an ideal world, the ratio between the modulus at time zero and the modulus at an infinite time would be equal to the ratio of the glassy modulus to the equilibrium modulus of the material and so give the ultimate relative importance of stiffness and flow within the material. However, because force or extension cannot be applied literally instantaneously, in many materials with flexible molecules you are unlikely to be able to catch their glassy behavior. Conversely, in materials with stiff molecules you are unlikely to be able to carry on the experiments long enough to catch the equilibrium behavior. In most cases you will see just a part of the behavioral spectrum of the material. The slope of the graph of modulus against the

COMPLEX POLYMERS

45

(c)

Force

Displacement

(a)

Log time

Log time (d)

Force

Displacement

(b)

Log time

Log time

Figure 3.2. The two types of transient test. In creep tests (a), a force is applied instantaneously and the deflection is monitored to give a graph (b) of displacement versus the log of time. In stress relaxation tests (c), the sample is given an instantaneous deflection and the restoring force is monitored to give a graph (d) of force versus the log of time.

log of time, though, should provide information about the retardation time of the material—in other words, how long it takes the molecules to reorder themselves in response to force; a steeper slope represents a shorter retardation time and faster-moving molecules. In fact, few realworld materials show an exactly linear response with a single retardation time. In practice, a material will have a wide spectrum of retardation times: some parts of the material may relax faster than others and some parts may flow like a liquid. As a consequence, the analysis of creep and stressrelaxation experiments is quite complex and the results are hard to interpret (Vincent, 1992).

Performing Dynamic Tests The time-dependent behavior of materials can also be investigated by carrying out dynamic tests in which the material is subjected to sinusoidal cycles of stretching and relaxation while the restoring force is monitored. From the output, a graph of stress against strain can be plotted (fig. 3.3). For perfectly elastic and Hookean materials, the result will be an upwardly sloping straight line, but viscosity in the material will result in added resistance to the stretching that is proportional to the velocity. In an ideal fluid with no stiffness, the resistance would lag behind the displacement by 90◦ , and the graph would be a circle. In real viscoelastic materials, however,

46

CHAPTER 3

Stress

σ0

δ

ε0

Strain

Figure 3.3. The results of a dynamic test on a typical viscoelastic polymer. The greater the viscosity of the material relative to its stiffness, the greater is the angle δ.

the result will be somewhere in between the straight line and a circle, and so will give an ellipse (fig. 3.3). The energy loss per cycle is given by the area inside the ellipse, and the modulus of the material can be separated into the elastic component—the elastic modulus (G
)—and the viscous component—the viscous modulus (G

), which are related by the equation G

= G
tan δ,

(3.1)

where δ is the angle between the point at which strain is greatest, ε0 , and the point where stress is greatest, σ0 (fig. 3.3).

A TYPICAL POLYMER: SEA ANEMONE MESOGLEA Perhaps the best-studied example of time-dependent behavior seen in a biological polymer is that of the mesoglea, a layer of material in the body wall of sea anemones. The mesoglea is mostly composed of a polymeric

COMPLEX POLYMERS

47

2.0

Strain

1.5

1.0

Metridium 0.5

Anthopleura 0.0 1

10

103

100

104

105

Time(s) 1 min

1h

24 h

Figure 3.4. Stress-strain graph showing creep behavior of the mesoglea of two sea anemones: the calm water species Metridium (dashed line) and the rough water species Anthopleura (solid line). (Redrawn after Koehl, 1977).

sugar that is unusual in being constituted of neutral, rather than negatively charged, polysaccharide chains. Their neutrality allows the chains to move relatively freely and take on a random-coil conformation. Stress relaxation tests on the isolated mesoglea of the sheltered-water species Metridium senile (Gosline, 1971), in which it was stretched to strains of 0.2 to 0.3, showed that its instantaneous stiffness was approximately 20 kPa, but that it fell over the course of 28 h to just 0.2 kPa. Creep tests produced similar results (fig. 3.4; Alexander, 1962; Koehl, 1977). The material appeared to be in the transition zone between plateau and rubbery equilibrium behavior—it was stiff during short, initial periods, and then stretched during the course of anywhere from a few minutes to several hours to reach strains of more than 1, after which the stretching slowed down quickly. Following removal of the load, the material recovered, albeit slowly, to its original length, showing that there were permanent cross-links in the material that had given it its long-term rubbery behavior. This time-dependent behavior of the mesoglea is quite beneficial for sea anemones in their slow-motion lives: the high instantaneous stiffness helps them withstand buffeting by waves, whereas the compliance over long periods allows them to alter their posture over the tidal cycle. Koehl’s study (1977) was also a comparative one in that she examined not only the

48

CHAPTER 3

calm-water anemone Metridium but also Anthopleura xanthogrammica, a rough-water species that does not change its shape much. She found that the mesoglea of Anthopleura not only was stiffer (fig. 3.4) but also exhibited less creep, even after 28 h, which would allow it to resist the buffeting of the waves better and return more quickly to its resting shape. Therefore, the properties of the two anemones’ mesoglea were adapted to the mechanical conditions in which they live.

MUCUS AND GELS Despite the fact that the anemone mesoglea behaves like a rubber, the polysaccharide that gives it its mechanical properties makes up only about 2% of its weight; the mesoglea is mostly water. It could therefore be better regarded as an example of a group of biomaterials that are composed of small amounts of polymer that entrap much larger amounts of water and that include animal mucuses and plant and animal gels. The properties of such materials depend not only on the amounts of polymer but also on the interactions of the polymer molecules both between each other and with the water that surrounds them. In turn, of course, these interactions depend on the molecular structure of the polymers.

Mucus If the polymer molecules are not joined by permanent cross-links, they will form a mucus, which over long periods of time will flow like a fluid. Intuitively, the more polymer present and the larger the polymer molecules, the more they should increase the viscosity of the mucus. However, because their molecules can form temporary entanglements, the mechanical behavior of mucus can be complex. The best-studied mucus is probably the pedal mucus that covers the feet of slugs, snails, and other mollusks and that they use in their unique form of crawling locomotion. Slug mucus is composed of large glycoprotein molecules—essentially long protein chains with shorter, branched sugar molecules attached—held within 96–97% water. The properties of the mucus and the way in which it may be used was investigated in a classic series of papers by Denny and his coworkers (Denny and Gosline, 1980; Denny, 1984). Because the feet of mollusks seem to slide over the ground when they crawl, the mucus must be being sheared, so Denny investigated the shear behavior of the mucus using a specially designed cone and plate viscometer (fig. 3.5a). In the tests the plate was spun around as the torque acting on the plate was measured. The beauty of this apparatus is that, because the separation distance between the cone and the plate is proportional to the radius, all the mucus is sheared by the same angle. What Denny found was that at shear strains up to 5, the mucus behaved like a solid, albeit a very compliant one, with a shear modulus of

COMPLEX POLYMERS

49

(a)

Torque

(b)

Rotational velocity

Time

Time Figure 3.5. The shear properties of slug mucus in a rotating plate viscometer (a) showed that slug mucus behaves as a compliant solid (b) until a shear strain of approximately 5, after which it behaves like a viscous liquid. Short periods of rest allow the mucus to reform.

50

CHAPTER 3

approximately 100 Pa (fig. 3.5b). This behavior was probably the result of entanglements between the molecules. At higher strains, however, the entanglements started to break and the material “shear softened,” with the shear force decreasing by about 40%. At still higher strains, the material behaved like a viscous liquid, with the restoring force increasing with velocity. If the rotation was stopped, however, the material seemed to reform over a matter of a few seconds, turning back into a solid and regaining its stiffness. So how would this help slugs crawl? It seems that the mucus acts somewhat like a ratchet. As the slug crawls, a wave of muscle shortening and lengthening moves forward along its foot. The stationary mucus beneath the extended part of the foot anchors it, whereas the moving, shortening parts of the foot are allowed to slide; they are pulled along by shortening immediately in front of them. Mucus clearly helps slugs move, but it also makes them horribly messy to pick up, and it can prove almost impossible to remove the mucus from our fingers. We usually remove sticky substances by rubbing our fingers together and dislodging them. The mucus, though, merely flows like a liquid when we do this; it stays bound to the original finger and also forms a new film on the cleaning finger, so things just get worse! As well as facilitating movement, mollusk mucus can become a solid material that is used in dry conditions to attach the animals to the substrate. Different mollusks use different techniques to do this. Limpets stick themselves to their home rock at low tide by secreting special adhesive proteins into their pedal mucus. These proteins form cross-links between the glycoprotein molecules to form a glue (Smith, 2002), which we will examine further in chapter 12. In contrast, slugs pump salts into their mucus (Denny and Gosline, 1980), which seems to allow the molecules to line up with each other, forming a visible “fuzz” of fibers and increasing the stiffness of the mucus more than 40-fold.

Gels Many of the jellylike substances in the soft tissues of animals are, like mucus, also made of polymers that comprise proteins and sugars and are variously called such names as glycoproteins and mucopolysaccharides. These materials, which occur in structures such as the cornea, umbilical cord, and lung, differ from mucus in having permanent cross-links between the molecules and are often stiffened by fibers of elastin or collagen, as we shall see in chapter 4. Other common animal gels are pure polysaccharides, the most notable being the hyaluronic acids that constitute the vitreous humor of the eye and the gel within our cartilage (Vincent, 1990b). Crosslinks between these molecules are constantly being broken and reformed in these materials, giving them complex time-dependent behavior. The molecules are also charged, enabling them to attract and stabilize large quantities of water. As we shall see in chapter 7, this property makes them ideal to form the compressive central parts of hydrostatic structures.

COMPLEX POLYMERS

(a)

51

(b)

Figure 3.6. Plant gel molecules in solution (a) may be solidified and given their stiffness and strength by cations (closed circles) that link the polysaccharides and help them form cage structures (b) that trap large amounts of water.

Animal gels are less commonly made only of proteins—the gelatin that we use to bind other ingredients together in pies and puddings is composed of many short cross-linked strands of the protein collagen. The most common gels in nature are the polysaccharide gels of seaweeds and land plants, most notably agar, carrageenan, and pectin. These gels are used to glue cells together and buffer them from changes in the composition of the surrounding water. The cross-linking between the polysaccharide molecules is mediated by cations in the water around them, binding the negatively charged molecules together so that they form straight multiple lengths of molecules (Rees, 1977). The result is a cagelike structure (fig. 3.6) that gives the gels useful properties. Apart from holding large amounts of water, the structure gives them almost as high a stiffness (up to 0.1 MPa) as rubbers, but they have a much lower breaking strain, of 0.2–0.5, and are very brittle (Mitchell and Blanchard, 1974). The strength of gels also depends on the amount of cross-linking, so decreasing the cation concentration or raising pH tends to weaken them. Many of the plant gels are economically important. Agar gel is of course extensively used in microbiology as a convenient culture medium for microbes, and it is also used to make a wide range of Asian desserts; carrageenan provides texture to ice cream; and pectin is used to set jams and marmalades.

MAKING PROTEIN POLYMERS STIFFER As we have seen, gels can be stiffened in various ways: by incorporating more polymer material, by increasing the number of cross-links between the molecules, or by using cations to glue lengths of the molecules together

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(a)

0.51 nm

(b) 0.6 nm

Figure 3.7. The two main stable structural forms of proteins. (a) The α helix. (b) The β sheet.

to form a strong molecular cage. However, there is a limit to how much can be achieved in any of these ways. A better way to increase the strength of a material would instead be to use polymers that align themselves together automatically, even without using cations, to form rigid fibers or crystals. The hydrogen bonding between the active N–H and C O groups in proteins, which helps stabilize enzymes in their complex shapes, can also allow materials to form rigid, structurally useful, elongated shapes. The most stable state for many proteins is, in fact, to curl up into a helix with hydrogen bonds forming between one peptide link and the third following link. In this long, straight α helix (fig. 3.7a), the protein curls in a right-handed helix with quite a steep angle. As we have seen, in rubbers

COMPLEX POLYMERS

53

Table 3.1 The Helix-Forming Properties of Different Amino Acids.

......................................................

Helix breaker Glycine Serine Proline Asparagine Aspartic acid

Helix indifferent Lysine Tyrosine Threonine Arginine Cystine Phenylaalanine

Helix former Valine Glutamine Isoleucine Histidine Alanine Tryptophan Methionine Leucine Glutamic Acid

After Vincent, 1990.

the stability of this configuration is reduced by including large amounts of “helix breaker” amino acids, such as glycine, serine, and proline (table 3.1); conversely the α helix is stabilized by the presence of amino acids such as alanine and glutamine. Another stable configuration is the β sheet (fig. 3.7b), in which large numbers of adjacent protein molecules run parallel to each other and are held together by hydrogen bonds between the active groups to form stable monolayers or sheets of protein. This sheetlike structure can be stabilized by incorporating repeating sequences of glycine-alanine and glycine-serine, all of them hydrophobic molecules. Weaker van der Waal’s forces can then bind these pleated sheets together into a three-dimensional crystal. Many arthropods synthesize their very own construction material, incorporating lengths of protein that form either β sheets or α helixes that both stiffen and strengthen it, to produce one of the most remarkable of all biomaterials—silk.

SILKS Silks can best be described as fringed micelle materials (fig. 3.8) in which each protein molecule has regions with very contrasting structure. Just as in rubbers, there are stretches of protein containing large amounts of the helix-breaking amino acids, which prevent the strands from attaching to each other except via occasional covalent cross-links. These lengths form regions of rubbery material. In between, there are lengths that form crystals composed of β sheets or α helixes. This complex structure, in which the rubbery regions are stiffened by the rigid crystals, gives silks their unique mechanical behavior. When a strand of silk is first stretched, deformation can occur only by rearrangement of the short strands between the cross-links (fig. 3.8a). After a strain of a few percent, however, the cross bridges are broken and the long strands that this breakage releases are free to straighten up (fig. 3.8b). This breaking

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(b)

(a)

Figure 3.8. Changes in the fringed micelle silk structure (a) as it is stretched. Stretching initially rearranges molecules within the amorphous regions before cross links are broken (b), making further extension easier until the molecules become orientated parallel to the stretch.

reduces the stiffness of the silk until the strands eventually align along the length of the strand, and the silk becomes stiffer again, before finally snapping. The result is that silks have extraordinary mechanical properties (fig. 3.9; Denny, 1976). They are hundreds of times stiffer and stronger than rubbers, with an initial Young’s modulus of 1–10 GPa and breaking stress of approximately 1 GPa, yet they also have a moderately high breaking strain of 0.2–0.5. The consequence is that huge amounts of energy are required to break silk strands; estimates range from 50 to 150 MJ m−3 , or when tested at high strain rates, even higher, about 1000 MJ m−3 (Gosline et al., 1999). The breaking of the cross bridges seems to use a large amount of this energy, so silk has an extremely low resilience of 35% and a high hysteresis of about 65% (fig. 3.9). Of course it is not easy to make such a complex structure. The materials are synthesized and stored within silks glands, where the proteins take up

COMPLEX POLYMERS

55

600

Stress (MPa)

400

200

0 0.00

0.04

0.08

0.12

0.16

0.20

Strain Figure 3.9. Results of a tensile test on a typical insect silk, showing a high initial stiffness, an S-shaped stress-strain curve, a high breaking stress and strain, and high hysteresis.

tangled-coil shapes in solution. But how do arthropods prevent crystals forming within the silk glands and make crystals form when they spin their silk? The answer is that crystal formation depends not only on the presence of other components in the silk—such as water, small polymer molecules, and electrolytes—but also crucially on physical forces. As the silk is pulled out through the narrow funnels of the spinnerets, shear forces align the protein molecules parallel to the emerging strand, allowing adjacent molecules to interact and form the long thin crystal regions. The process may also be aided by loss of water and changes in pH.

Insect Silks Bees and other hymenopterans produce silks with α helixes (Denny, 1980), but silks that incorporate β sheets are far more common. These are produced by a wide variety of arthropods, including moths, lacewings,

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Figure 3.10. Schematic structure of the silk of the cultivated silk moth Bombyx mori illustrating the packing of adjacent β sheets. The small circles represent glycine residues, and the large ones represent alanine or serine.

and spiders. These silks all show broad similarities but can have markedly different molecular structures and mechanical properties. The silk produced by moths such as the cultivated silk moth, Bombyx mori, has crystals approximately 21 × 2 × 6 nm in dimensions, which contain approximately even amounts of the small glycine residues and large alanine or serine residues. As a result, its β sheets pack together well, one on top of the other, with large and small residues linking between others of the same type on the adjacent sheet (fig. 3.10). The function of the silk strands is to protect the pupa, because the large amounts of energy that are needed to bite or cut through the silk cocoon would make it costly for predators to penetrate. Lacewings produce silk for a quite different purpose: they lay their eggs on the ends of long silken rods. The silk they produce is extremely unusual: it has a crossed β silk structure in which the crystals are arranged at right angles to the strand. The crystals are consequently unfolded when the strand is stretched, rather like the polymer molecules in thermoplastics (Gordon, 1968), giving the silk a higher breaking strain but reduced stiffness.

Spider Silks It is, however, the silks of spiders that have been most thoroughly studied and that show the widest range of adaptations that optimize their mechanical performance in many different ways (Gosline et al., 1999). The draglines and frame silk of orb web spiders is fairly similar to the silk of moths but contains somewhat shorter crystalline regions. This composition makes the silk stiff, strong, and tough, which is ideal for making the frame of the web, whereas the high hysteresis is capable of absorbing enough energy to catch a falling spider or stop a flying insect. In addition to the proteins themselves, the silk also contains small molecules that attract water, causing

COMPLEX POLYMERS

57

1.2 1.0

frame silk

Stress (GPa)

0.8

0.6

0.4 viscid silk

0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Strain

Figure 3.11. Stress-strain graph of the mechanical behavior of the web silks of a typical ecribellate spider: strong, stiff frame silk (radial threads) and compliant viscid silk (spiral threads).

the frame to expand laterally and contract longitudinally, a process called supercontraction (Savage et al., 2004). This phenomenon keeps the frame taut even if it is attached to flexible vegetation, but the silk also absorbs so much water on damp autumn mornings that the web becomes a mass of droplets and is highly visible. Spiders use two quite different silks to capture their prey. The primitive cribellate spiders produce capture strands by an involved and timeconsuming process. To the surface of a normal silk strand they attach many nanoscale strands of gossamer silk by heckling, that is, using the combs on their legs (Blackedge and Hayashi, 2006). When these complex multistrands are stretched, the structure remains intact even after the main thread has been broken, and more energy is absorbed as, one by one, the nanofilaments also stretch and finally snap. The more advanced and much more diverse ecribellate spiders have developed an even more ingenious capture thread. The capture spiral of a typical orb web spider is made of viscid silk, which has very short crystalline regions but much glycoprotein glue. This thread readily absorbs water, producing a sticky strand in which great lengths of protein are held within water globules. This sort of silk has a stiffness much more like that of a rubber and can extend to several times its original length (fig. 3.11). The high breaking strain of viscid silk helps the strand trap insects because they are unable to break it (Gosline et al., 2002); no matter how much they struggle, the silk simply stretches with them.

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So impressive are the mechanical properties of spider silk that a great deal of research effort has been and continues to be put into trying to mimic it or to develop ways of farming it (Vollrath and Knight, 2001). Spiders are tricky to farm, however, so many attempts have been made to insert the genes for making the proteins into other organisms. So far the results have not been wholly successful, but such is the military interest in producing better body armor that efforts will no doubt continue.

CHAPTER 4 .................................................

Polymer Composites

COMBINING MATERIALS Despite their many merits, silks are materials that are literally thrown together at the last minute. The reinforcing crystals within the rubbery matrix are produced by momentarily changing the mechanical and chemical environment of their proteins; such changes cause the proteins to align and form fibers, but only imperfect and short ones. Much better control of fiber production could theoretically be achieved by manufacturing the fibrous and rubbery components from separate materials and then combining them. Such methods are, in fact, used extensively in human technologies and also throughout the natural world to produce composite materials. These materials are generally far more ordered than silks, contain much longer fibers, and have mechanical properties that make them far better for forming skeletons. Fortunately for biomechanists, natural composites have an internal structure similar to those that we have independently invented— such materials as reinforced concrete, car tires, fiberglass, and carbonfiber plastics. This means that we already have a fairly good understanding of the mechanical behavior of at least simple composite materials (Kelly and MacMillan, 1987). The most straightforward composites to understand are soft composites, which are made of a rubbery matrix reinforced by continuous fibers.

THE BEHAVIOR OF SOFT COMPOSITES A material that is composed of two quite different materials—such as large numbers of continuous crystalline fibers held within a rubbery matrix (fig. 4.1)—can have better properties than either material on its own. It can be stiffer and stronger than the matrix and tougher than both the matrix and the fibers.

Stiffness Consider first the simplest case of a matrix reinforced by parallel fibers (fig. 4.1). When the material is stretched along the direction of the fibers, both the fibers and the matrix have to be stretched in parallel and are subjected to equal strain. The stiffness, E V , of the composite material is

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Voigt

Reuss

Figure 4.1. The structure of a composite material reinforced with parallel continuous fibers. Properties parallel to the fibers (Voigt) are changed more readily than those at right angles (Reuss).

given according to the Voigt model of composites by the expression E V = E f Vf + E m (1 − Vf )

(4.1)

where E f and E m are the stiffnesses of the fiber and matrix, and Vf is the volume fraction of the fibers. The stiffness increases linearly with the volume fraction of fibers (fig. 4.2). In contrast, stretching the material at right angles to the fibers, the two materials are stretched in series and are subjected to equal stress. In this situation the stiffness, E R , of the material is given according to the Reuss model of composites by the more complicated expression E R = 1/[(Vf /E f ) + {(1 − Vf )/E m }].

(4.2)

At low fiber fractions the reinforcing effect of the fibers is much reduced in this direction (fig. 4.2), since the matrix can stretch unhindered by the fibers. Strength The effect of fiber reinforcement on the strength of a composite is rather more complicated because fibers usually break at a much lower strain than

Composite stiffness Ec

POLYMER COMPOSITES

61

Ef

Voigt Reuss Em 0

1 Fiber volume fraction

Figure 4.2. The effect of the volume fraction of fibers on the stiffness, E c , of a fibrous composite stretched parallel to (Voigt) or at right angles to (Reuss) the fibers.

the softer matrix. Adding the first few fibers to the matrix will actually reduce its strength parallel to the fibers, because these few fibers will break at a low load, leaving less matrix to withstand subsequent loads. The breaking stress, σ , will therefore decrease as the fraction of fibers increases (fig. 4.3) according to the expression σ = σm (1 − Vf ),

(4.3)

where σm is the breaking stress of the matrix. However, as the fraction of fibers increases, the strength of the material will eventually begin to increase again. The strength of the fibers combined with the stress in the matrix, σm
, at the breaking strain of the fibers will exceed the strength of the matrix. The material will break, therefore, when the fibers break, the strength being given by the expression σ = σf Vf + σm
(1 − Vf ).

(4.4)

The strength thereafter increases linearly up to a maximum of σf , when all the material is fiber (fig. 4.3). At right angles to the fibers, the fiber reinforcement will have no effect at all on strength, because the weaker matrix will always fail; breaking stress will equal σm (fig. 4.3).

Toughness Soft composites are tougher and more notch resistant than either the fibers or matrix on their own, for two main reasons. First, the softer matrix protects the brittle fibers from cracks simply by surrounding it. Second, and

CHAPTER 4

Composite strength σmax

62

σf Voigt

Reuss

σm

σ’m 0

1 Fiber volume fraction

Figure 4.3. The effect of the volume fraction of fibers on the strength, σmax , of a composite stretched parallel (Voigt) or at right angles (Reuss) to the fibers.

more important, the matrix is far too compliant to transmit cracks from one fiber to another, so tears cannot travel through the material. If a notched composite is stretched, the material in the crack will simply be pulled apart, blunting the crack tip, and the load will be held evenly between all the intact fibers (fig. 4.4).

NATURAL SOFT COMPOSITES Two examples of natural soft composites with continuous fibers have been subjected to extensive mechanical study: the rather bizarre slime of hagfish, and the much more important tendon.

Hagfish Slime When threatened by predators, hagfish, which are carrion-feeding jawless fish that look not unlike eels, exude substantial amounts of a transparent slime that seems to clog up the gills of their attackers (Lim et al., 2006). The material, which is produced in special slime glands, comes in two parts. There are protein threads, 1–3 µm wide and up to 20 cm long, which are coiled up inside cells; and mucins, which are held within vesicles (Fudge et al., 2005). On contact with seawater the threads unravel and the vesicles burst open, releasing their contents. The mucin binds to the threads, holding them close enough together so that it effectively traps huge amounts of water that cannot readily escape from between the threads. The result is

POLYMER COMPOSITES

63

(a)

(b)

Figure 4.4. Blunting of the tip of a crack (a) as a composite material with compliant matrix continues to be stretched (b).

a material that is energetically extremely cheap to produce, being composed of only 0.0020% threads, 0.0015% mucin, and 99.996% water! Yet over short periods, because the water cannot escape, the slime acts as a fiber-reinforced composite. Tensile tests on the slime (using purpose-built clamps to grip the sticky substance) show that the stiffness is only of the order of 40 Pa, with a breaking strain of between 1 and 2 (Fudge et al., 2005). The extremely low stiffness (and therefore high compliance) of the slime has three explanations. First, of course, the concentration of fibers is low, so there is little material to stiffen the slime. Second, the fibers themselves are very compliant. Tensile tests have shown that the fibers have an initial Young’s modulus of only around 6 MPa, not much more that of rubbers. This is because the fibers are composed of many short protein molecules, each of which has a stiff central region with an α-helix conformation bracketed by rubbery ends (Fudge et al., 2003). Since the compliant ends act in series with the rigid central regions, the whole fibers resemble more the Reuss model of a composite and are extremely flexible and stretchy. Third, calculations using the Voigt model to calculate the stiffness of the thread-mucus composite give estimates of Young’s modulus of about 120 Pa compared with the actual value of 40 Pa, so only approximately a third of the fibers in the slime seem to be taut enough to be actively stretched.

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x ray EM

x ray EM

x ray

MICROFIBRIL

TROPOCOLLAGEN

15Å 35Å

SUBFIBRIL

SEM OM

Evidence: x ray EM EM SEM SEM OM

TENDON FIBRIL

FASCICLE 35Å staining sites

640Å periodicity

reticular membrane waveform fibroblasts fascicular or crimp structure membrane

100–200Å 500–5000Å

50–300λ

100–500λ

SIZE SCALE

Figure 4.5. The hierarchical structure of tendon. (Reproduced by permission of Informa Healthcare Solutions from Kastelic et al., 1978.)

Of course, having no solid matrix, the slime is basically unstable, so it exhibits marked stress relaxation when stretched, as the water leaks away and as the threads shear past each other. Eventually the material becomes more and more dilute; it simply falls apart and dissolves in a matter of minutes, but by then the hagfish has long since effected its escape. Tendon Compared with hagfish slime, tendon is a far more ordered, sophisticated, and mechanically competent material. It is a highly hierarchical composite of the protein collagen within a matrix of other softer proteins and mucopolysaccharides. The collagen protein has a triple-helix structure; each tropocollagen molecule is formed by three α helixes joined together to form a shallowly wound spiral (fig. 4.5). This structure forms automatically because of the highly ordered sequence of amino acids in the collagen; it contains large numbers of the repeated sequences glycine-X-proline, and glycine-Xhydroxyproline, where X is another amino acid. The glycine allows extensive hydrogen bonding that stabilizes the triple helix. Each tropocollagen molecule is 280 nm long, but the molecules pack together automatically in a quarter-stagger every 67 nm, so that the microfibrils they form are in fact much longer. The material has several more layers of hierarchy (Kastelic et al., 1978). Several microfibrils are joined together within a subfibril (fig. 4.5), and several of these are bound to form fibrils 50–500 nm wide. The

POLYMER COMPOSITES

65

50

Stress (MPa)

40

30

20

10

0 0.00

0.01

0.02

0.03

Strain Figure 4.6. Results of a cyclic tensile test on a typical tendon. Note the compliant toe region of the curve and the high resilience.

fibrils themselves can extend the entire length of the tendon fascicle and are glued together via a mucopolysaccharide matrix, so a fascicle is essentially a reinforced composite with continuous fibers. The final complexity of the material, however, is that the fibrils are not straight but zigzag, at an angle of about 15◦ , to form what is known as “crimp,” every straight section being about 100 µm long (fig. 4.5). All of these characteristics give tendons unique and useful properties. Tensile tests (it is best to grip the ends using freezing clamps) show that the collagen endows tendon with a very high stiffness, approximately 1.5–3 GPa (Ker, 1981, 2006), a strength of 60–100 MPa, and a breaking strain about 0.03 (fig. 4.6). Because of their high breaking stress and reasonably high breaking strain, tendons can store quite large amounts of energy, about the same as rubbers, at 3 MJ m−3 ; they also have far higher resilience than silk, at approximately 93%. These properties allow tendons, which of course act in series with muscles in the vertebrate body, to be used in two main ways. The tendons (including our Achilles tendon) attached to large leg muscles, such as the plantaris and gastrocnemius, act as springs during running and jumping. They are relatively long and thin compared with their muscles, so they stretch a long way when the animals put their feet on the ground

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at the end of each step, storing energy, before releasing it again to power the new step (Alexander, 2003), like the springs in a pogo stick. These sorts of tendons therefore make use of the high-energy storage and resilience of tendon. The tendons joining the small muscles in our forearms that control our finger movements to our finger joints themselves are used in a very different way. These tendons, which you can feel running down the back of your knuckles, are relatively thick compared with their muscles and are essentially rigid (Ker et al., 1988). They transmit the contractions of the muscles directly to the fingers, like the control wires of airplanes, and make use of the high stiffness of tendon. The final useful properties of tendons are due to the soft matrix. It stops cracks running through the tendon, so damage to the outside of tendons rarely leads to rupture. Lastly, the matrix has low shear stiffness, allowing the crimp in the tendon to straighten and the tendon to lengthen readily when loads are first applied. The result is the characteristic high compliance of the toe region of the stress-strain curve (fig. 4.6). But what advantage does this confer? The answer seems to be that crimping allows tendons to bend readily around joints, rather like the ropes we will examine in chapter 6, because bending an unstressed tendon involves only shearing its matrix and not stretching or compressing its fibers. Unpublished results from Ker and his colleagues show that storing tendons in formaldehyde greatly stiffens the matrix; consequently, the toe region of the stress-strain curve disappears and the tendon becomes far harder to bend, even though its tensile properties are largely unchanged.

RIGID COMPOSITES There are only two downsides to the high compliance of the matrix in tendon. First, it makes tendons extremely compliant and weak in compression (fig. 4.7), so they could never be used as rigid skeletal elements. Second, the fibers have to be extremely long to prevent them from being pulled out of the matrix; this characteristic makes the manufacture of large pieces of material very tricky, bearing in mind that the material is secreted from thousands of individual cells. To overcome these two disadvantages the obvious answer would be to make the matrix out of a much stiffer, glassy material rather than a rubbery one. The former would resist compression much better and be able to transfer shear faster to the fibers, enabling much shorter fibers to be used. The disadvantage of a glassy matrix, though, is that cracks can run much more readily through it and could then be transmitted to the next fibers, so composites with a glassy matrix could become extremely brittle. Fortunately, if the fibers are not bonded too strongly to the matrix, cracks can be stopped at the interface between them by the Cook–Gordon mechanism (Gordon, 1968). As we have seen, a crack traveling through an isotropic solid sets up high tensile stresses at right angles to it, which tend to extend the crack. However,

POLYMER COMPOSITES

67

Stress (MPa)

0.1

0.2

compression

tension –0.001

0.001

0.002

Strain

Figure 4.7. Results of a dynamic test on tendon, showing its low compressive stiffness and strength.

it will also set up tensile stresses 1/5 of the magnitude in front of it parallel to the crack. Therefore, when a crack approaches an interface (fig. 4.8a), it will tend to pull it apart, and if the strength of the interface is less than 1/5 of the fiber strength, the interface will break before the crack reaches it (fig. 4.8b). This will blunt the crack tip and instead cause cracks to run up and down the fibers (fig. 4.8c). When the material is stretched, therefore, the fibers will tend to pull out before breaking, creating a very rough fracture surface and using up large amounts of energy. The composite will consequently be greatly toughened, giving it a work of fracture in the region of 1–10 kJ m−2 . One problem with fiber-reinforced composites is, of course, that the fibers strengthen and toughen the material only when it is stretched parallel to the fibers, so unidirectional composites would be highly anisotropic; they would be weak and brittle when stretched at right angles to the fibers. To obtain decent properties in all directions, fibers need to be oriented in all directions. It is not as easy to arrange them in this fashion as it might first seem. The obvious way would be to organize them randomly in space; however, just like random piles of sticks, fibers organized in this way do not pack closely together, so the volume fraction of fibers, and consequently the amount of reinforcement, would be low, and the material would still be weak. A somewhat better method, which is used in fiberglass, is to arrange the fibers in two-dimensional mats; doing so certainly increases the fiber fraction, but not by a great deal. The best method is to lay down many layers of material, in each of which the fibers are all aligned in the same orientation and packed closely together. The individual layers can then be oriented at different angles, like the grain of wood in the different sheets of plywood, to give more or less isotropic properties, at least in the plane of the sheets.

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(a)

(b)

(c)

Figure 4.8. The Cook-Gordon crack-stopping mechanism. As a crack travels through the matrix towards a fiber (a) it sets up a tensile stress parallel to its travel that tends to break the interface between the two materials (b). When the crack reaches the fibers it is thereby blunted and diverted (c). (Redrawn after Gordon, 1968.)

KERATINOUS STRUCTURES A good example of a natural rigid composite whose properties are dominated by its fibers is keratin, which forms the hair, horns, hooves claws, nails, and outer skin layer of mammals. Mammalian keratin is a composite that comprises two sorts of proteins. The fibrous component is made of lowsulphur proteins that contain large amounts of helix-forming amino acids, such as glutamine and leucine, and so has large sections that form α helixes. Like tendon, keratin is very hierarchical (fig. 4.9): the helixes wrap around each other to form coiled-coil protofibrils, and eleven of these are packed together to form a microfibril. In turn the microfibrils are arranged into macrofibrils, which are held in a matrix that is composed of the other major

POLYMER COMPOSITES

69

epicuticle exocuticle endocuticle nuclear a medulla remnant

high-S proteins

cuticle

low-S proteins

left-handed matrix cell right- coiled-coil intermediate membrane handed rope filament para cell ortho cell complex α-helix macrofibril cortex 1

2

7

200

2000

70000nm

Figure 4.9. The hierarchical structure of mammalian hair keratin. (Reproduced by permission of Elsevier Science from Marshall et al., 1991.)

component of the keratin: high-sulphur proteins (Fraser and MacRae, 1980; Hearle, 2000). These proteins contain large amounts of proline, serine, and cysteine and so form a disordered amorphous solid with plenty of S–S crosslinks that stiffen it and bind it to the nonfibrous parts of the low-sulphur proteins. Keratin is an unusual biomaterial for animals in that it is produced inside the cell, unlike that main structural tissues or extracellular matrix, which are formed outside. This characteristic has important implications. The arrangement of the fibers can be controlled readily, and they are usually closely aligned to the orientation of the cells; however, since the cells die after the macrofibrils are laid down, the material will also contain unwanted cell membranes. Keratin is laid down in layers outside the epidermis of the animal. A consequence of this is that because there are no living cells within the material, it cannot be remodeled, unlike materials such as tendon or bone, and therefore might tend to weaken over time. Fortunately, it is extremely resistant to decay.

Mammalian Keratins Tensile tests on mammalian keratinous structures show that they have unusual mechanical properties (fig. 4.10), with some similarities to silks (Vincent, 1992). Initially they are extremely stiff parallel to the fibers, with a Young’s modulus of 2–5 GPa. They then yield, however, at a strain of about 0.02, and extension occurs thereafter up to strains of

70

CHAPTER 4

250

Stress (MPa)

200

150

100

50

0 0.00

0.05

0.10

0.15

0.20

Strain Figure 4.10. Stress-strain graph of the mechanical behavior of a typical mammalian keratin, showing high initial stiffness, a yield region where the α helix unwinds, a high breaking stress, and high hysteresis.

0.2–0.3 without much increase in stress. Finally the stiffness increases again, and the material only breaks at stresses of 200–300 MPa. The reason for the yield and high breaking strain of mammalian keratin rests in the fiber structure. As the material is stretched, the α helix fibers are deformed irreversibly, being pulled almost straight, at which point they bind to their neighboring fibers to produce a twisted β sheet. Of course this molecular rearrangement uses up lots of energy, so keratins are extremely tough, with works of fracture on the order of 10–15 kJ m−2 . Because the α-to-β transition is irreversible, the resilience of the material at high strains is also low, and the return part of the stress-strain curve is much lower than the rising part (fig. 4.10). This is ideal in structures, such as horns and hooves, that must be able to absorb large impacts. The only disadvantage is that, because the transition is irreversible, once horns have been highly stressed, they lose some of their ability to absorb energy. Fortunately, animals keep on growing their keratin structures, so horns and hooves are gradually replaced as the old material wears away. Studies using x-ray diffraction and polarized light microscopy have shown that the fibers within keratinous structures are oriented in ways that

POLYMER COMPOSITES

71

(a)

tubule

direction of fractures

(b)

Figure 4.11. Control of fracture in mammalian keratin structures. In horses’ hooves (a) the fibers (dashed line) and tubules (solid lines) divert cracks downward or forward (arrows). In human fingernails (b), fibers in the thick central layers divert cracks transversely.

optimize their mechanical behavior. In long thin structures, such as horns and claws, that are loaded mostly in bending, the fibers are aligned parallel to the long axis (Earland et al., 1962), parallel to the stresses set up when they are bent (see chapter 8). In horses’ hooves, in contrast, the majority of the tissue is in flat, platelike cells oriented at right angles to direction of growth (fig. 4.11a; Bertram and Gosline, 1986; Kasapi and Gosline, 1997). There are also hollow tubes running posterodistally through the hoof, with layers

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of curved cells wrapped around their outside to form tubules. Together these two sorts of cells prevent cracks from running up through the hoof to its base (fig. 4.11a). A similarly sophisticated arrangement is seen in our own fingernails, which have a sandwich structure (Farren et al., 2004). The fracture properties of the nail are dominated by the thick intermediate layer in which the fibers are laid down in needlelike cells parallel to the half moon and free edge of the nail (fig. 4.11b). This configuration makes the nail twice as tough in the proximal direction as in the transverse direction, so any cracks in the nail are diverted laterally, protecting the quick and making the nail self-trimming. However, if all of the nail were arranged in this way, it would crack all too easily. To prevent this, the nail also has thin dorsal and ventral layers composed of pancake-like cells in which the fibers are oriented in all directions parallel to the nail’s surface. These layers provide the nail with bending rigidity and also wrap around the lateral edges of the nail, preventing cracks from forming. The only downside to this arrangement is that since there is no intermediate layer at the edges of the nail, cracks can run in any direction there. Tears can therefore run down the corner of our nails into our skin, which is surprisingly painful. Avian Keratins Compared with mammalian keratin, the keratins found in the scales and claws of reptiles and in the feathers and beaks of birds have been much less studied; however, what work has been carried out has demonstrated that they are quite different. X-ray crystallography has shown that the fibrous part of the structure is in the form of β sheets, rather than α helixes (Fraser et al., 1971). They form in that way because the molecules contain segments with large amounts of serine and glycine, just as in silks, which pack together readily. This difference in structure has several consequences. Avian keratins are approximately twice as stiff as mammalian keratins (fig. 4.12), but since they are already held in the β-sheet configuration, they do not undergo a conformational change and have no obvious yield region; consequently, they break at strains of only 0.06–0.15. They also have higher resilience and lower hysteresis, although they still have similar toughness as mammalian keratins. Just as in mammalian keratin, the fibers are arranged in such a way to optimize performance. In the shafts of feathers they are arrayed parallel to the long axis, particularly near the base of the rachis, which enables them to efficiently resist the stresses set up by bending when the feathers are loaded in flight (Cameron et al., 2003). There is also a thin outer layer in which the fibers are helically wound, preventing the shaft from splitting (Earland et al., 1962; Purslow and Vincent, 1978). The Effect of Hydration Since they are laid down on the outside of animals’ bodies, keratins can be subjected to large changes in humidity, which can dramatically alter

POLYMER COMPOSITES

73

250 0% RH

Stress (MPa)

200

50% RH

150

100

100% RH 50

0 0.00

0.05

0.10

0.15

Strain Figure 4.12. The results of tensile tests on bird keratin at 0, 50, and 100% relative humidity (RH). Note the lack of a yield region, the biphasic behavior of the material, and its lower breaking strain. Hydration reduces both its stiffness and strength. (Redrawn after Taylor et al., 2004.)

their mechanical properties. Water can readily penetrate into the matrix and plasticize its proteins, making it rubbery rather than glassy. The classic composite theory given earlier in this chapter would predict that wetting would reduce the stiffness and strength across the fibers but would have little effect parallel to them, because the bulk of the stiffness is given by the more rigid fibers. In fact, however, hydration reduces the stiffness and strength of keratinous structures in all directions (fig. 4.13), even parallel to the fibers (Bertram and Gosline, 1987; Kitchener and Vincent, 1987; Vincent, 1990b), so it is clear that the simple continuous composite model is inappropriate for keratins. In fact, hydration weakens keratins because the fibers within it are not continuous, as the Voigt model assumes, but relatively short; the material can deform and break between the ends of the fibers. Whatever the causes of the changes due to hydration, however, they provide another mechanism by which the properties of keratinous structures can be optimized. Structures such as hooves, feathers, and nails are usually held at intermediate humidities of approximately 60%, at which their

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6

0% RH 53% RH

75% RH

Stress (MPa)

4

100% RH

2

0 0.00

0.05

0.10

0.15

0.20

0.25

Strain Figure 4.13. The effect of relative humidity (RH) on the mechanical properties of a typical mammalian keratin, horse hoof. (Redrawn after Bertram and Gosline, 1987.)

toughness is maximized (Bertram and Gosline, 1987; Taylor et al., 2004; Farran et al., 2008), whereas at the base where they are laid down, they are far wetter and softer, reducing stress concentrations at the join with soft living tissues. The effect of hydration also explains why it is so much easier to cut your fingernails after a bath and why soaking your feet helps you cut even the thickest toenails.

The Effect of Melanin In contrast to adding water, adding melanin to keratins makes them stiffer and harder. It acts as a filler, reducing the volume of matrix where deformation can take place. Birds tend to have melanin in their beaks (Bonser and Witter, 1993) and in their outermost wing feathers, where the increase in hardness reduces wear (Bonser, 1995). This is why so many seabirds that have mainly white plumage still tend to have black tips to their wings.

THE THEORY OF FILLERS AND DISCONTINUOUS COMPOSITES The effect of incorporating fillers and discontinuous fibers in composites has, in fact, been widely investigated because of our own development of carbon-filled rubbers and of a whole host of composites with short reinforcing fibers.

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75

Fillers cannot strengthen a matrix but they do stiffen and harden it (Mullins, 1980). Rigid spherical particles occupying a volume fraction Vp of a composite will raise the stiffness in all directions (Guth, 1945) according to the equation E = E m (1 + 2.5Vp + 14.1Vp2 ).

(4.5)

The effects of elongated particles and short fibers are more complex because they not only reduce the volume of the matrix but also reinforce it, although not as much as continuous fibers. For fibers to reinforce a matrix, they have first to be loaded by shear between the matrix and the ends of the fibers (fig. 4.14a), so the longer and thinner they are, the greater the percentage of the length of the fibers that will be fully loaded. Therefore, the efficiency of reinforcement rises with the aspect ratio L/D of the fibers, reaching the Voigt value for infinite aspect ratios (Kelly and MacMillan, 1987). When a composite reinforced by discontinuous fibers is stretched (fig. 4.14b), the matrix will also be loaded in shear and tension around the ends of the fibers and will tend to deform there. This complex pattern of loading has several consequences. First, the stiffness of the composite will be reduced below that of a continuous composite. Second, the properties of the composite will change as the tensile stress increases (fig. 4.15) to produce biphasic behavior (Harris, 1980). The matrix will start to yield in shear and tension, so the material will show a “yield” region: the stiffness will decrease as the fibers slip through the matrix until failure eventually occurs. The plastic deformation of the matrix means that the material will absorb large amounts of energy, but some of the deformation will be irreversible, and when the material is released, it will not return to its resting length. Such biphasic behavior is certainly seen in feather keratin (fig. 4.12; Taylor et al., 2004), where the changes in the matrix are not masked by the conformational changes seen in the fibers of mammalian keratin. But it is also demonstrated by two discontinuous composites that use sugars rather than proteins as fibers: insect cuticle and plant cell wall.

INSECT CUTICLE Insect cuticle is a composite of fibers of the straight-chain sugar chitin within a protein matrix (Andersen, 1977; Vincent and Wegst, 2004). In chitin, the monosaccharide units are held together by β1-4 links, making the polymeric chains very straight and so able to crystallize readily. The fibers are in fact composed of 19 parallel chains held together in a hexagonal prism approximately 2.8 nm wide and 300 nm long. They are probably the stiffest of all biological fibers, with a Young’s modulus of about 150 GPa, and they bind readily via hydrogen bonds to the protein component, which can itself be held in a rather ordered silklike β arrangement. Unlike keratin, cuticle is an extracellular tissue and, although like keratin it covers the surface of the body, it is used to construct not just a protective

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(a)

Stress

fiber tension

interface shear

(b)

tension shear

Figure 4.14. Deformations within a short-fibered composite material when it is stretched. The fibers are loaded from their ends via shear at the interface with the matrix (a), so only the central region is fully stressed in tension. Looking at the entire material (b), it can be seen that the matrix is stressed in shear on the sides of the fibers and in tension between their ends.

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77

Stress

yield

Strain Figure 4.15. Stress-strain graph of the biphasic behavior of a typical short-fibered composite. As the material is stretched, the matrix yields, resulting in a decrease in its stiffness. It then fails to return to its resting length when the load is removed.

covering but the actual exoskeleton of insects. Of course, to make an animal that can move about and eat, such an exoskeleton must have very different properties in different regions. Insects make their jointed skeletons by varying both the thickness of cuticle and its mechanical properties; these can be altered both by changing not only the chitin and protein content but also the extent to which the cuticle is tanned, or sclerotized, a stiffening process similar to the one we use to make leather. By these mechanisms, insects can make a wide range of structures, from the softest of membranes to the hardest of mandibles. Newly laid down cuticle and the cuticle that is used to make the membranes at the joints between the body segments is unsclerotized, typically having a water content of 40–75%, with the remainder being composed of equal amounts of chitin and protein. The softest of all cuticles so far studied is the intersegmental membrane of the locust abdomen (Vincent, 1975). Not only is this cuticle very wet, which plasticizes the protein matrix, but the chitin fibers are arrayed parallel to the edge of the abdominal segments. This arrangement makes it extremely easy to pull the segments apart, since doing so stretches the material at right angles to the fibers. The stretchiness of the material is put to good use by the female locust, which digs its abdomen into the desert sand using terminal appendages to lay its eggs. Mimicking the reciprocating action of its appendages, Vincent was able to show that the membrane showed stress softening (fig. 4.16), giving it an effective Young’s modulus of just 1 kPa and a breaking strain of over 10. This allows the insect to stretch its abdomen more than 12 cm into the cooler, damper sand, greatly improving the survival chances of its eggs. The softness of untanned

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Stress (kPa)

10

5

0 0

5

10 Strain

Figure 4.16. The mechanical behavior of a locust intersegmental membrane when stretched intermittently. (Redrawn after Vincent, 1975.)

cuticle is also vital for another reason: it allows insects to grow. One major problem of having an exoskeleton is that this rigid covering prevents an animal from growing continuously. To get bigger, an insect periodically has to molt, splitting its skin and dragging itself out of its original covering. The softness of the new, untanned cuticle then allows the insect to expand its body by pumping itself up with air and forcing body fluids into its wings. Finally, the cuticle is tanned to leave a new, larger insect. In most areas of an insect’s body the outer layer, or exocuticle, is tanned shortly after molting to produce the rigid elements of its skeleton; just as we soak animal skins with tannins from tree bark to tan them, insects pump dihydroxyphenols into the cuticle, stiffening it dramatically. The actual mechanism by which tanning stiffens the cuticle has long been a subject of debate (Vincent and Wegst, 2004). Initially it was thought that the dihydroxyphenols formed cross-links between the proteins, stiffening the matrix, just as the sulphur cross-links introduced by vulcanization stiffen natural rubber. However, this mechanism alone this could not raise the stiffness of the matrix much above 10 MPa. Instead, it now seems likely that the main action of the dihydroxyphenols is to expel water, after which the proteins can form β sheets and become stiffer. Certainly, drying out untanned cuticle has a stiffening effect very similar to that of tanning, and some tanned cuticles, notably that of the abdomen of the blood-sucking bug Rhodnius, can be softened by acid-induced rewetting (Reynolds, 1975). A final effect of the dihydroxyphenols is that they can polymerize to form a melanin filler that can harden the cuticle; the hardest regions of an insect’s body, its wing hinges and mandibles, are consequently very dark in color or even black.

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79

Figure 4.17. The helicoidal arrangement of fibers in the cuticle of many insects, which has many layers of unidirectional fibers, each of which is rotated slightly relative to the layer beneath.

The properties of different insect cuticles are controlled not only by the degree of tanning but also by the orientation of the chitin fibers. One of the stiffest cuticles tested is the one that makes up the apodemes of the locust hind leg (Ker, 1977). These structures act just like the tendons of vertebrates but contain no collagen; instead they have a chitin content of 17%, with the fibers arranged longitudinally, as in mammalian tendon. This gives them a longitudinal stiffness of around 11 GPa. Since this cuticle is highly tanned and has a stiff matrix, these apodemes cannot bend and are far more brittle than the tendons of mammals; this brittleness is not much of a problem because the apodemes are held inside the exoskeleton and so are protected from damage. Unlike apodeme cuticle, the body cuticle of most insects is arranged in such a way as to be much more isotropic. The elytra, or wing cases, of many beetles, for instance, have a plywood-like arrangement, with alternating layers in which the fibers are arranged at right angles. Even more common is the helicoidal configuration (fig. 4.17), in which the fiber orientation changes gradually throughout the thickness of the cuticle. Both arrangements produce cuticle that is relatively stiff, with a Young’s modulus of 2–5 GPa, but also tough, with a work of fracture of 2–3 kJ m−2 (Vincent and Wegst, 2004). The layers of similarly oriented fibers are so thin in many beetles that they can also result in the production of physical colors, making the insects iridescent (Parker et al., 1998)). THE PLANT CELL WALL Plants are very different anatomically from animals in that rather than producing blocks of skeletal material, each cell has its own skeleton: the

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plant cell wall. As we shall see, this architecture means that plant structural design is heavily influenced by the arrangement of the cells. Nevertheless, the design of plant cell walls shares many similarities with that of insect cuticle; both plant cells and insects have their own exoskeleton. Like cuticle, the plant cell wall is a composite of fibers of a straight-chain sugar, in this case cellulose, within a matrix made of branched hemicellulose molecules. The cellulose fibrils are somewhat larger than those of chitin, being around 3.5 nm in diameter, and have a stiffness of about 130 GPa. The mechanical properties of cell-wall material is strikingly similar to that of cuticle. It has biphasic behavior in tension, with a linear initial region (Kohler and Spatz, 2002) followed by yield of the matrix, after which the stiffness is markedly reduced. Typically the Young’s modulus of cell wall is 2–5 GPa, with strength of 100 MPa and a breaking strain of 0.05–0.1. The toughness, as determined from cutting tests, is similar to that of cuticle at 2–5 kJ m−2 (Lucas et al., 1997). The high compliance of the matrix allows developing cells to stretch their cell wall using turgor pressure, just as insects stretch their skeletons at molt. As we shall see in chapter 7, this cell expansion is controlled by the orientation of the cellulose fibrils. Another similarity of cell walls to insect cuticle is in the way that many of them are stiffened and strengthened by the process of lignification, which is very similar to the process of sclerotization. During lignification, the complex polyphenol molecule lignin is incorporated into the cell wall, and it seems to act just as dihydroxyphenols do in insect cuticle: it expels water from the matrix, resulting in cell-wall material that is stiffer and stronger but that has a lower breaking strain, only around 0.01–0.02 (Kohler and Spatz, 2002). Young, growing plant tissues are unlignified and have to be stiffened by turgor pressure (see chapter 7), which is why the tips of flower stems are the quickest to wilt, whereas in older plant tissue, the lignified cell walls provide permanent support. A final similarity of plant cell walls to cuticle is in the way that the angle of cellulose fibers controls the mechanical properties of the tissue. The stems of herbaceous plants are given rigidity by two different tissues that are located around their perimeter. Young stems may be stiffened by unlignified collenchyma cells, whereas older ones are stiffened by lignified sclerenchyma cells. Both are long, thin cells oriented parallel to the stem, and in both, their cell walls are stiffened by cellulose microfibrils that run more or less up and down the walls, just like the chitin fibrils in insect tendon. This reinforcement maximizes the cells’ stiffness and strength.

WOOD In one tissue—wood—plant cell walls are arranged into a design that endows them with mechanical properties that far outdo those of cuticle. The individual long, narrow cells that make up the trunks of trees are mostly oriented parallel to the trunk, just like the fiber cells in herbaceous plants. However, the cell walls of the former have a more complicated structure:

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81

S3

S2

S1

primary wall

Figure 4.18. The fiber orientation within the tracheid cells of wood. In the thickest layer (S2 ), the fibers wind at approximately 20◦ to the long axis of the cell.

in the 60–80% of their thickness that makes up the S2 layer (fig. 4.18), the fibers are oriented 20◦ to the long axis of the cell, winding helically up it. This configuration slightly reduces the wood’s stiffness below what could be achieved if the fibrils were parallel to the long axis. The reason for this unusual arrangement, though, was determined by Jeronimidis (1980). He showed that as the xylem cells in the wood are stretched, the cell walls buckle inward, splitting parallel to the cellulose fibrils, and the helical lengths of cell wall then unwind and straighten, like springs. This produces a very rough fracture surface that takes large amounts of energy to form. The stress-strain curve of wood consequently has a long post-yield region, and the process makes wood extremely tough, with a work of fracture of about 30–50 kJ m−2 , ten times higher than that of cell walls of herbaceous plants or cuticle and weight for weight as high as steel. This toughness is important in a tissue that is found in such large pieces, since large cracks and notches can occur in trees; the toughness of wood is also one of the main reasons why we find it so useful as a construction material: no matter how many notches or holes we make in it, it is hardly weakened.

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(a)

(b) R tracheid orientation T L ray orientation

rays Figure 4.19. The structure of a typical tree trunk. The tracheid cells are oriented longitudinally, making it strongest and toughest in this (L) direction, but the rays also strengthen the wood radially (R). Wood is weakest in the tangential direction (T).

Of course the energy-absorbing mechanism of wood can only work if the cell walls can buckle inward; in very dense wood such as ebony, the cell walls are too thick and the lumen is too narrow, so the wood breaks across more readily, with the consequence that dense woods such as ebony are far brittler than light woods (Lucas et al., 1997). It is also much easier to break wood at right angles to the cells, since failure mainly involves splitting the cells apart. Wood is therefore much weaker and far less tough across the grain, but this is generally not a problem for trees because the bending of the branches sets up forces that are mainly parallel to the grain, as we shall see in chapter 8. In any case, wood is also strengthened radially by the presence of rays (fig. 4.19; Burgert and Eckstein, 2001), which prevent the outer growth rings from peeling off the tree. The presence of rays makes the radial strength of wood about 50% greater than the tangential strength; this is why it is easiest to split wood or chop logs radially into pie-shaped pieces, since these cuts avoid crossing any rays.

CHAPTER 5 .................................................

Composites Incorporating Ceramics

THE ADVANTAGES OF INCORPORATING MINERALS Fibrous composites composed solely of polymers are excellent materials, but they have two disadvantages. First, even when reinforced by polymer fibers, they are not as stiff or hard as truly crystalline materials. Second, they are metabolically expensive to produce. Both disadvantages would be overcome if organisms could make use of minerals such as calcium carbonate, calcium phosphate, or silica, which form extremely stiff crystals and which are very common in the natural environment. Some arthropods do, in fact, incorporate metal salts into their cuticle, where they act as fillers, increasing the cuticle’s hardness. Many insects have heavy metals, such as zinc, manganese, or iron, in their mandibles to harden the cutting edge (Robertson et al., 1984; Schofield et al., 2002), while many crustaceans reinforce larger areas of their cuticle by incorporating calcium salts. For instance, some mantis shrimps have extremely hard hammer-like front claws, which are impregnated with calcium salts; they use these to deliver devastating blows to their prey while male shrimps use the claws in macho trials of strength with their rivals (Currey et al., 1982). It is somewhat more difficult to use minerals to replace, rather than to add to, reinforcing fibers, because minerals have the disadvantage of being denser than polymers and extremely brittle. Nevertheless, many groups of animals have evolved a range of biological ceramics that combine salt crystals with a soft protein matrix. The factors that affect the mechanical properties of such materials can readily be predicted from the composite material theory we examined in chapter 4. However, they are best demonstrated by considering the results of a study of one of the simplest of biological ceramics: spicule-reinforced connective tissue (Koehl, 1982).

SPICULE-REINFORCED CONNECTIVE TISSUE Many sponges (though not the bath sponge) and relatives of corals called sea pens contain small pieces of calcium carbonate or silica in their collagenous body walls. These structures, which are known as spicules, vary greatly in size and shape and constitute a highly variable volume fraction of the wall, anywhere from 2 to 35%. To separate the effects of the number, size, shape, and orientation of spicules on the mechanical behavior of the body wall, Koehl (1982) isolated spicules and used them to make a wide range of model

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materials. The spicules were added to long thin strips of (raspberry) jelly, and the strips were then subjected to tensile tests. Koehl found, as might be expected, that the stiffness of the strips rose with the volume fraction of spicules, because this reduced the space for the jelly matrix and amplified the strain within it. Less intuitively, the stiffening effect actually rose with the surface area of the spicules, so a smaller volume fraction of small spicules, which have a greater ratio of surface area to volume, is as effective as a greater volume fraction of large spicules. The reason is that with a larger surface area stresses can be transferred more efficiently to the spicules. Finally, long thin spicules were more effective than short fat ones because they acted more like the reinforcing fibers in fibrous composites, stiffening the material parallel to their long axis. The stiffness of the spicules themselves was not important because almost all the deformation occurred in the matrix, which also protected the spicules from damage. The take-home message of this research is clear: to get a truly stiff ceramic, the largest possible numbers of tiny needlelike crystals should be incorporated into a matrix. The structure and mechanical behavior of all other biological ceramics—bone, dentin, enamel, and shell—can all be understood in this context.

BONE The main skeletal tissue of vertebrates is bone, which is an array of collagen in which are embedded crystals of hydroxyapatite (Ca10 (PO4 )6 (OH)2 ; Currey, 2002). The size of the crystals in bone may vary, but they are usually about 4 nm thick, 10–20 nm wide, and 20–50 nm long, and at least initially tend to be laid down in the spaces between tropocollagen molecules, which are 67 nm apart. This basic structure is found in the simplest, most ordered of bonelike materials: ossified tendon. Limb bones are rather more complex and have a hierarchical structure, like so many of the fibrous composites we have already seen (fig. 5.1). The crystals aggregate to form fibrils 0.1–3 µm in diameter (fig. 5.1a), and these can then be arranged in one of two very different ways. In woven bone (fig. 5.1c), the fibrils are arrayed fairly randomly. In contrast, in the less highly mineralized lamellar bone (fig. 5.1b), they are arranged like plywood in sheets 2–6 µm thick; in each sheet the fibrils are oriented in the same direction, but adjacent sheets have fibrils at different orientations. The two forms of bone can be found separately, or they may combine to form laminar bone, but they seem to have separate functions and are usually laid down in quite different situations. Woven and laminar bone can be deposited rapidly in young animals and in repairing fracture calluses but seem to have inferior mechanical properties. Lamellar bone is usually found in mature bones. The cells that create the bone, the osteocytes, remain within it as the bone grows, and in large animals the primary bone is also remodeled to produce Haversian systems, or secondary osteons (fig. 5.1d). Many osteoclasts, which are one kind

COMPOSITES INCORPORATING CERAMICS

85

(a)

0.1µm

(b) lamellar

(c) woven

10µm

10µm (d) Haversian

0.5mm Figure 5.1. The hierarchical structure of bones. Platelets are aggregated into fibrils (a), which may be arranged into plywood-like sheets to form laminar bone (b) or randomly to form woven bone (c). Laminar bone can be remodelled around blood vessels to form Haversian systems (d). (Redrawn after Currey, 1970.)

of osteocyte, travel through the bone, dissolving away cylinders of tissue, while other osteocytes, the osteoblasts, follow and fill in the cavity again by producing layers of new bone tissue; secondary osteons are arranged in concentric cylinders, like the tissue of a leek, and leave only a tiny blood vessel at the center of the cylinder. Because of the small size and stiffness of the crystals and its high mineral content (from 50 to 80%), bone has a high stiffness, between 5 and 30 GPa, which is somewhat more than the fibrous composites such as insect cuticle or plant cell wall. Long bones are usually about twice as stiff parallel to the long axis as they are perpendicular to it, because of the preferred orientation of the fibrils. Bone is also extremely strong, particularly in compression, with a breaking stress of around 200 MPa. It is weaker than wood in tension, however, because of the brittleness of the crystals, but tensile tests show that

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200

failure 150

Stress (MPa)

yield

100

50

0 0.00

0.01

0.02

0.03

Strain Figure 5.2. Results of a tensile test on typical bone. Note the long post-yield region that toughens the bone.

it has a long post-yield region, so that failure occurs at strains of around 0.03 (fig. 5.2).Cracks are stopped in bone at the interfaces between crystals and the matrix, just as they are stopped at the interface between the fibers and the matrix in polymer composites. Cracking also occurs between adjacent lamellae, so bone has quite a reasonable work of fracture of 2–3 kJ m−2 . The properties of bones are strongly influenced by their mineral content. Stiffness increases with mineral content (Katz, 1971), as might be expected, though not according to any straightforward model such as the Voigt or Reuss model (fig. 5.3). The other properties also vary with mineral content, but in a more complex way. Currey (1979a) compared the properties of three very different bones: the femur of a cow, which has a conventional mechanical function; the inner-ear bone, or bulla, of a whale, which is never mechanically stressed; and the antler of a deer, which receives impact loads in fights. The femur, with its intermediate mineral content (table 5.1) was the strongest of the bones and had reasonable stiffness and toughness. In

COMPOSITES INCORPORATING CERAMICS

87

120

enamel

E (GPa)

80

Voigt 40

Reuss bone

0 0.0

0.5

1.0

Volume fraction of mineral Figure 5.3. The influence of mineralization on the Young’s modulus of bone and enamel. The stiffness follows neither the Voigt nor the Reuss model. (Redrawn after Katz, 1971.) Table 5.1 Properties of Three Contrasting Bones.

......................................................

Antler Femur Bulla

% Mineral 59 67 86

E (GPa) 7.4 13.5 31.3

σ max (MPa) 179 247 33

Work of fracture (Jm) 6,200−2 1,700 200

From Currey, 1979a.

contrast, the bulla with its very high mineral content was very stiff, which would aid its sound-conducting role, but it was very weak and brittle. Finally, the antler, with its low mineral content, was not very stiff or strong, but because it had such a high breaking strain, it was by far the toughest, ideal for absorbing the energy of impacts. Bone also changes in its mineral content and mechanical properties throughout ontogeny (Currey, 1979b); its mineral content rises, resulting in an increase in stiffness but a drop in toughness (fig. 5.4). These changes

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50

25 sheep

E (GPa)

humans 15

30

10

20

5

10

1mo

1y

5y

20y

Work of fracture (kJm–2)

40

20

50y

Figure 5.4. Changes in the Young’s modulus (dashed lines) and work of fracture (solid lines) of the bones of sheep and humans with age. (Redrawn after Currey, 1979b.)

are clearly adaptive, since the stiff bones of adult animals increase their locomotor efficiency. In contrast, young animals have less need to move about efficiently, but they play more and are more prone to accidents, so the high toughness of their bones helps them avoid fractures. A final aspect of bones was until recently something of a mystery. It has long been known that Haversian bone has inferior mechanical properties compared with primary bone, so the question arises, why do large animals do so much bone remodeling? The answer now appears to be related to the fracture properties of bone. As it is stretched past its yield point, bone undergoes an increasing amount of microcracking (Zioupos et al., 1994), which can be picked up with a microphone and which shows up under the microscope as whitening of the bone. This process absorbs energy and helps give bone much of its toughness, but over time it can weaken the bone; microcracking results in painful conditions, such as shin splints, and can even eventually lead to fatigue fracture. There is now good evidence that bone is reabsorbed and relaid preferentially in areas with larger amounts of microcracking (Bentolila et al., 1997). Remodeling therefore effectively repairs our bones and maintains their strength and toughness. TOOTH CERAMICS Apart from bone, mammals also lay down hydroxyapatite in their teeth (which are in fact modified scales), forming two rather different tissues, dentin and enamel. Dentin provides the main structure of the tooth, whereas enamel provides a hard surface and cutting edge.

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Dentin Dentin is a tissue very much like well-mineralized bone in composition (Waters, 1980; Currey, 2002), with approximately 70% mineral content, 20% protein, and 10% water, but it differs from bone in certain respects. It is laid down from the outside in: the dentin gradually fills up the pulp cavity at the center of a tooth, and contains no cells: only tubules reaching inward radially to the pulp. The crystals are also rather different from those of bone, being hexagonal prisms 3 nm in diameter and 64 nm long. Nevertheless, the mechanical properties of dentin are basically similar to those of wellmineralized bone, with stiffness in the region of 15 GPa, a strength of 50 MPa, and a fairly low work of fracture of 500 J m−2 . The low work of fracture is not too much of a problem in a tissue that is loaded mostly in compression and that seldom receives mechanical impacts. In larger, more prominent teeth, such as the tusks of narwhals, which can be used for fighting, the degree of mineralization is lower and the work of fracture, at over 30 kJ m−2 , is far higher (Brear et al., 1993). Enamel Enamel is quite different from both dentin and bone. It is far more heavily mineralized, having a mineral content of around 97% (Waters, 1980; Currey, 2002). The high percentage of minerals makes enamel extremely stiff, with a Young’s modulus of 30–80 GPa, a property that makes it ideal for cutting up food, even though its breaking stress, at about 35 MPa, is not very high. Such a high mineral content is also liable to make enamel extremely brittle. This problem is overcome to some extent by the structure of the material. The crystals, which are large plates about 25 nm×100 nm×500 nm, are arranged in a complex pattern within keyhole-shaped prisms that are approximately 4 µm in diameter (fig. 5.5b). The crystal orientation prevents cracks from traveling across the prisms. The prisms are themselves also arranged in a complex way in the tooth (Waters, 1980). Near the surface of the tooth, they are closely packed together perpendicular to the surface (fig. 5.5a), which maximizes the hardness and minimizes the chances of decay. Beneath this, though, the prisms are redirected so that they cross each other at right angles in a basketwork-like mesh that deflects and effectively stops cracks. Consequently, the incidence of fracture is much reduced even though the work of fracture is only around 200 J m−2 . As we shall see in chapter 14, the shape of teeth, and the precise role of the dentin and enamel, is shaped by the diet of the animal and by the mechanical properties of the food. MOLLUSK SHELL The other important ceramics in nature are the shell materials produced by mollusks, which are made using the mineral calcium carbonate in the

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(a)

(b)

2µm

Figure 5.5. The hierarchical structure of enamel. The tooth is made up of interlinked keyhole-shaped prisms (a), which are themselves composed of a complex pattern of crystals (b). (Redrawn after Waters, 1980.)

(a)

(b)

100µm

(c)

5µm

50µm

Figure 5.6. Arrangement of crystals within different shell materials: (a) prismatic; (b) nacre, or mother of pearl; and (c) crossed lamellar.

form of either aragonite or calcite (Currey, 1980). Mollusk shell contains even less protein matrix than enamel, usually between 0.5 and 4%, so one might expect shell to be hopelessly brittle. In fact, a range of sophisticated arrangements of the mineral means that the shells of mollusks can be pretty tough and act as an excellent defense against a range of predators. One type of shell with excellent mechanical properties is mother of pearl, or nacre, which is found in the shells of a wide range of both gastropod and bivalve mollusks but which has been best studied in mussels. Nacre is composed of aragonite tiles 0.3–0.5 µm thick that are arranged in layers parallel to the surface of the shell (fig. 5.6b). The rows are staggered like bricks, so any crack running through the shell has to constantly be diverted around the edges of the tiles. This arrangement toughens the material, giving a work of fracture of about 1.6 kJ m−2 across the shell (Currey, 1977), which is similar to that of bone, although nacre is much easier to break in the plane of the shell, when one is merely splitting apart the layers of tiles.

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Another mechanically competent arrangement seen in shells of mollusks such as razor shells and cone shells (Currey, 1980) is the crossed lamellar structure (fig. 5.6c). As in lamellar bone the structure is separated into lamellae, here 20 µm thick, within which the aragonite needles are all parallel but at right angles to those in the adjacent lamella. This plywoodlike structure helps divert cracks just like the crossed prisms of enamel (Currey and Kohn, 1976), once again creating a rougher fracture surface and toughening the material. Not all shells have equally impressive mechanical properties. In oysters some parts of the shell have a foliated structure, in which the needles are arranged in overlapping bundles, whereas other parts of the shell resemble miniature piles of rubble (Currey, 1980). This rubblelike material seems to have extremely poor mechanical properties, but it can be laid down extremely quickly, an important trait for a sessile organism whose best defense against boring predators is to make its shell thicker than the length of its predators’ mouthparts. In contrast to nacre, which is deposited in tiles parallel to the surface, prismatic tissue is laid down in columns 10–200 µm wide at right angles to the surface of many shells (fig. 5.6a). The alignment of the edges of the prisms makes this tissue vulnerable to cracking, but like the surface prisms of tooth enamel, it presents a smooth, hard surface to the world and strongly resists chemical attack. In many shells, different types of structures can be combined. In razorshells, thick layers of crossed lamellar tissue are separated by thin prismatic layers. Studies using Raman spectroscopy have shown that that the prismatic layers are held in precompression (Eichhorn et al., 2005); if the crossed lamellar tissue were removed they would expand laterally. When such a shell is subjected to crushing tests, the cracks that run through the crossed lamellae are deflected when they reach the boundary with the prismatic layer, because it expands laterally as the crack hits it. The layers then separate, a process known as delamination, absorbing more energy and further toughening the shell.

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CHAPTER 6 .................................................

Tensile Structures

AN INTRODUCTION TO STRUCTURES Having examined the wide range of materials that organisms produce, the next few chapters examines how they are arranged into larger-scale structures such as plant stems, blood vessels, and arms. In some ways, of course, the distinction between materials and structures is an artificial one. Few biological materials are homogenous, and a fibrous composite could equally be regarded as a structure. Conversely, cartilage, which will be introduced as a structure in chapter 8, is usually regarded as a material. Nevertheless, structural design can be distinguished from materials design because it involves control of geometry—of the shape, size, and arrangement of the components that make up a structure. In contrast, as we saw in chapter 1, geometry is excluded from materials tests by the process of normalization. Much of the following chapters is therefore devoted to understanding how and why organisms arrange the materials of which they are composed.

ROPES AND STRINGS We saw in chapter 4 that organisms are capable of making materials, such as tendon, cuticle, and wood, that show preferential resistance to forces in one direction. In particular, a material reinforced with unidirectionally oriented microfibrils or crystals will be much stiffer and stronger against tensile forces in that direction than against all others. When organisms design complete structures to resist tensile forces in just one direction, therefore, one of the things they do is to incorporate tissue reinforced by fibers or crystals with the preferred orientation. The insect apodeme we discussed in chapter 4, for instance, is many times stiffer and stronger in tension along its axis than laterally. However, there are two problems with having a solid rod of material reinforced along its axis. First, it is vulnerable to surface cracking, which can weaken it drastically if the material from which it is made is brittle. Second, because of its axial stiffness, such a rod will also be resistant to bending and so will not be able to act as a flexible string. These are not totally disastrous problems. Insect apodemes work fairly well as longitudinal springs, but they do so only because they are protected within the exoskeleton and do not have to bend around joints. In vertebrate tendons, these problems have been overcome to some extent by incorporating the stiff collagen fibers within a soft matrix; doing so blunts the tips

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(a)

(b)

Figure 6.1. Methods of constructing structures to resist tension in one direction. Rigid material may either be concentrated into a central strand (a) or separated into many helically wound strands (b).

of cracks, reducing the tendon’s vulnerability and making it far easier to bend. The crimp in the collagen fibers also allows easy bending and prevents kinking; the only disadvantage is the low initial stiffness of the tendon. To make a stiff structure that is easy to bend, resistant to cracks, and has a high initial modulus, organisms use either one or both of the two following designs. The first design is to arrange all the stiff reinforced tensile material into a single narrow central strand (fig. 6.1a), which can bend readily, while its surface is protected from cracks by a soft outer casing. This design is common in man-made technology, where wires are protected by a plastic coating, and it is even more common in nature. The underground organs of plants are an excellent example. As we shall see in chapter 12, the distal roots of plants are subjected only to tensile forces and are consequently strengthened by a central stele of lignified tissue that is surrounded by a soft parenchymatous cortex (Ennos, 2000; Evert and Eichhorn, 2006). This arrangement is generally stated to be the defining characteristic of root anatomy, but it is also seen in underground structures that are anatomically part of the shoot system, such as rhizomes and tubers (Haberlandt, 1914). Many above-ground structures that have to resist mostly tensile forces also exhibit similar anatomy: for instance, the stems of waterweeds such as Myriophyllum spp. and Ranunculus fluitans (Usherwood et al., 1997) and some tendrils and fruit stalks. Tensile cores are also seen in

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spider silks, which have a rigid fibroin center and a softer senicin coating (Denny, 1980). The second design is to separate the tensile material into many separate elements, adding a higher level of hierarchy to that of the microfibrils within the matrix of the material. This separation helps prevent cracks from running across the tensile element. However, to allow bending to occur without the elements on the concave surface buckling, the fibers also need to wind around each other in a multiple helix (fig. 6.1b). When such a structure is bent, the fibers can rearrange themselves, shearing past each other rather than being stretched, compressed, or buckled. People have used this technique for millennia: ropes and strings are made by simultaneously stretching and coiling together natural fibers, such as hemp or cotton, or nowadays man-made fibers, such as nylon or Kevlar. Metal ropes can also be made by doing the same with thin metal wires. Fibers can also be bound together using the alternative method of braiding, as when people interlock their own hair into flexible braids. A major advantage of both of these arrangements is that they help the structure resist the weakening effects of tensile failure in single fibers. When ropes are stretched, the tension pulls all the helical fibers inward, making them grip each other more tightly. This gripping will induce friction between a broken fiber and adjacent ones, resulting in rapid transfer of tension from the adjacent intact fibers into the broken fiber on either side of the fracture. Therefore even if many fibers in a rope are broken, as long as they are broken in different places, it will retain almost all of its original strength. This is the key to the effectiveness of spinning thread: twisting the relatively short fibers together as they are stretched holds them together. With cotton, the friction between adjacent fibers increases with the humidity of the air around them, further strengthening the cotton thread. This is why cotton spinning was most economically carried out in the rainy northwest of England, where the spinning mills were further cooled and humidified by flooding their flat roofs with water. In nature, the most common ropelike structures are the stems of woody climbers or lianas; they show what is called anomalous thickening, in which woody fibrous strands are separated by soft parenchyma tissue (Haberlandt, 1914) and twist up the length of the stem. These stems are certainly strong enough in tension to support swinging apes as well as their own weight and are flexible enough to bend out of the way without breaking if the trees and branches that support them collapse.

USING MULTIPLE ROPES Just as it is advantageous for tensile structures to be separated into many thin strands, it is often advantageous for organisms to use many ropes, rather than just a single one, to resist tension; each element can be kept narrow and, therefore, flexible. There is also one further advantage. The size and cost of attachments at each end of a rope should theoretically increase more rapidly

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than the force it has to withstand (Gordon, 1980); the area of the attachment should increase in line with the force, but its thickness must also increase to maintain the same shape. Its volume should therefore rise by the power of 3/2 in relation to the force it has to resist. The greater cost and difficulty in making attachments to the deck was probably one reason why square-rigged sailing ships had such complex rigging. There is one potential disadvantage with this arrangement, though: the ropes must be moderately extensible or the loads will not be well distributed between the elements. The organisms that most clearly demonstrate such a multistrand attachment mechanism are mussels, which hold onto rocks using many byssus threads (the “beards” on mussels). One difficulty in using so many threads is that, because waves can strike from any angle, force can easily be applied very unevenly between the threads; some might be at their breaking point while others are unstressed. This uneven distribution is overcome to a large extent because the threads have not only a stiff collagen-like distal region but also a very compliant elastin-like proximal region (Carrington, 2002). Therefore heavily loaded threads can stretch readily, allowing others to be loaded and to take up at least some of the force.

MEMBRANES, SKINS, AND PLATES Structures that have to resist stretching in two dimensions are even more common in nature than biological ropes. Most organisms have skins and internal membranes separating and dividing their organs, and almost all organisms are composed of cells that are surrounded by a membrane. Many organisms also possess specialized platelike regions that increase their surface areas for particular tasks; good examples are external ears, the wings of insects and bats, and most important of all, the leaves of plants. These structures have to be able to withstand forces in all directions within the plane without tearing. There is therefore little point in a membrane being reinforced by fibers oriented preferentially in just one direction; it would simply tear between the fibers.

Macroscopic Crack Stoppers Of all membranes, plates have the most problems in resisting tearing because, unless they are perfectly circular, stresses will be concentrated at irregularities around their rim, just as they are concentrated at the tips of cracks. To prevent cracks from developing around their edges, leaves such as those of holly have reinforced margins. Other leaves have veins running around the bottom of notches or have lignified cells known as sclereids reinforcing their edges (Haberlandt, 1914), just as stitching reinforces the ends of buttonholes. Cracks can be prevented from running through the middle of plates by reinforcing the membrane itself; the veins of both insect wings and leaves do this job (Wootton, 1981; Lucas et al., 1991), acting just

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(c) (b)

15

(a)

Stress (MPa)

10

5

0 0.0

0.2

0.4 0.6 Strain

0.8

1.0

Figure 6.2. The supposed structure and mechanical behavior of skin (a). As the skin is stretched (b), the random collagen fibers become oriented along the direction of tension. The result is a large lateral contraction (b) and a J-shaped stress-strain curve (c).

like the stitching in ripstop nylon. A final technique is to prevent cracks from developing at the edges of a membrane in the first place. We do this by folding the fabric in our clothes at the ends to produce hems, something that is also seen in the trailing edges of the hindwings of locusts. Many seaweeds, on the other hand, have frilly edges to their fronds, which stops them ever being loaded in tension.

Mammalian Skin At first glance, the skin of mammals does not seem to be well adapted at all to resist tearing because it does not seem to contain any of the macroscopic reinforcing mechanisms that are seen in leaves and insect wings. Yet our skin is remarkably resistant to fracture, and even when it is punctured, it seldom tears as a woven fabric would. The secret of the tear resistance of mammalian skin appears to lie in the arrangement of material within the dermis, the layer of skin that lies beneath the keratinized epidermis. The dermis is composed largely of compliant elastin, but it is reinforced by collagen fibers that, rather like rubber molecules, are partly folded and randomly oriented (fig. 6.2a). When skin is stretched, the fibers initially add little resistance to the stiffness of the elastin matrix, so the skin is at first very compliant. However, as stretching proceeds, the collagen fibers gradually

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straighten and reorient themselves parallel to the stretching force (fig. 6.2b), providing greatly increased reinforcement and increasing the stiffness of the skin. This gives skin its characteristic J-shaped stress-strain curve (fig. 6.2c; Veronda and Westman, 1970). Because of the reorientation of the collagen, the skin also exhibits a high Poisson’s ratio, contracting laterally rapidly as it is stretched (fig. 6.2b). There are two main advantages to this design. First, skin is readily held in prestrain, so that it can cover our bodies smoothly without us having to put a lot of effort to change its shape as we move. The importance of this characteristic is emphasized by the difficulty divers have in moving about when wearing their thick rubber wet suits. Gordon (1978) also pointed out how the high Poisson’s ratio of skin allows it to hug our bodies snugly as we move. In this respect skin behaves just like knitted fabrics which provide a closer and more flexible fit than woven cloth. Rib stitch has particularly good properties in this respect and so is used in parts of garments where a close fit is most important, such as the ends of sleeves, sock cuffs, and turtle necks. The second advantage of the design is that it makes skin very hard to tear; although the mechanisms involved are still unclear, several factors could be involved. One is that when a cut is made in skin, its high initial compliance allows the tear to widen dramatically, blunting the crack tip and reducing the stress concentration. Another is that the area under the J-shaped stressstrain curve is very small, so there is little stored elastic energy available to drive the crack forward. Third, the low initial stiffness of skin may make it much harder for this energy to be transferred to the crack tip. Finally, the fibrous nature of skin also gives it a reasonably high work of fracture.

Bat Skin and Bias-Cut Fabric Bats are perhaps the mammals that have the most mechanically important skin—their wing membrane—which is composed of two thin layers of epidermis with a thin dermis between (Swartz et al., 1996). The membrane has to combine three apparently mutually exclusive qualities: it has to show high toughness, to resist tearing; it has to be stiff when the wing is held out, to resist aerodynamic forces; yet it must also be able to contract when the bat folds its wings, so that it does not get tangled up. The solution adopted by bats is to reinforce the wings with fibers (fig. 6.3a), as in insect wings or leaves, although in the case of the bat wing, the fibers are composed of an elastin core with a collagen sheath (fig. 6.3b). I am unaware of the significance of this rather unusual arrangement, in which stiffer material surrounds a more compliant core. The orientation of the fibers is easier to understand, however. Their stiffening efficiency is maximized by having them oriented more or less orthogonally to each other when the wing is stretched out, running parallel to and at right angles to the supporting bones. The orthogonal arrangement of the fibers, however, also gives the wing its ability to contract when the wing is folded. The fibers shear easily relative

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(a)

101

(b)

Figure 6.3. (a) The orthogonal arrangement of reinforcing fibers in the skin of the wings of bats. The fibers stiffen and strengthen the wing membrane, while the arrangement makes it easy to shear and contract when the wing is folded up. The fibers (b) are composed of an elastin core (large open circles) with a collagen sheath (closed circles). (Reproduced by permission of Elsevier Science from Swartz et al., 1996.)

to each other, the wing membrane in between folding into tiny corrugations that run from corner to corner of the fibers, greatly reducing the wing area. Tensile tests have shown that the stiffness of the folded wing membrane is many times higher parallel to these corrugations than at right angles to them (Swartz et al., 1996). A somewhat similar use of orthogonally oriented fibers is seen in bias-cut fabric (Gordon, 1978), in which the warp and weft of the fabric are oriented 45◦ to the horizontal. This orientation allows the warp and weft of the fabric to shear past each other and the fabric to stretch, making it more drapable and flexible. Bias-cut material is used in skirts and dresses, ties, binding, and trims, among other applications. The same arrangement is used to more mechanical and less aesthetic effect by cable grips, which are socklike structures composed of helically arranged fibers, used by engineers to hold onto the ends of ropes and cables. The grip is readily pulled over the ends of the rope, like a sock onto your foot. When you pull on the grip, the fibers shear, orienting them more parallel to the long axis of the grip and making them grip tighter onto the rope. The design is the same as that seen in the bamboo finger traps, which are sold as toys. Incidentally, there have been suggestions (Langer, 1978) that human skin also contains a rhomboidal pattern of macroscopic fibers, just like bat-wing skin and cable grips. Langer suggested that these fibers are responsible for the way in which skin gapes when punctured. Having perforated corpses all over with circular stab wounds using an ice pick, he found that the

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σc σc P t

L

2R

Figure 6.4. The circumferential stresses, σC , set up within a membrane of thickness t billowing up into a semicircular vault of radius R under a pressure P .

skin gaped preferentially in certain directions at different points around the body; he claimed that they did so parallel to the long axis of the rhombus. Langer’s discovery has had practical benefits for surgery. Surgeons make cuts preferentially parallel to Langer’s lines, because they hold stitches and heal better than cuts in other orientations. Despite the practical importance of this idea, however, the subject seems not to have been studied seriously since Langer.

RESISTING OUT-OF-PLANE FORCES The main function of the many membranes is to resist out-of-plane forces— which in the case of bat wings is the lift produced by the wing. Because membranes have negligible bending stiffness, the only way they can resist out-of-plane forces is to billow up; their tensile strength then allows them to resist the stretching stresses set up within them. To calculate the stress that will be set up, let us examine the simple case of a rectangular membrane of thickness t that is supported by booms of length L along two sides and that billows up into a semicircular vault shape of radius R when acted on by a pressure, P , from beneath (fig. 6.4). In equilibrium, the upward force, 2RLP, which is applied to the membrane by the air, must be equaled by the downward force, 2Ltσ , provided by the tension in the membrane, where σ is the tensile stress. Therefore 2R L P = 2Ltσ , so σ = R P /t.

(6.1)

Consequently, the stress in the membrane is proportional to the pressure and inversely proportional to the thickness of the membrane, as one would expect. But surprisingly, the stress is also proportional to the radius of curvature of the membrane; the flatter the membrane the larger the stress. This is a potential problem for flying bats, since for economical flight they ideally should have as flat a wing membrane as possible. However, a flatter membrane would require having a stiffer or thicker wing membrane. The

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(a)

103

(b)

actinofibrils 1mm

Figure 6.5. The design of the wings of pterosaurs. The outer wing was supported by a single finger (a), but the membrane contained keratinous actinofibrils (b) that may have added some support or control. (Redrawn from Wellnhofer, 1991.)

bat would also have to have stronger wing bones and stronger muscles to pull the wing forward enough to keep the membrane taut. Bats overcome this dilemma by separating their wings into several smaller areas using their fingers and hind legs (fig. 6.3). This allows them to hold their membranes out sufficiently flat without having to make too heavy an investment in wing structures. As Gordon (1978) pointed out, a similar design is employed in Chinese junks, in which the large sail is separated into many easily billowing areas by a series of lightweight bamboo booms. So what about the wings of pterosaurs, which were supported by just a single finger bone along their leading edge (fig. 6.5a)? It was previously thought that the wing membranes of these huge flying animals must have been held extremely taut, like the sails of modern racing yachts, to achieve a reasonable aerodynamic wing shape without undue billowing (Gordon, 1978). More recently, however, it has been found that the membrane of their wings was reinforced by keratinous actinofibrils (Wellnhofer, 1991), which ran in approximately the same orientation as the feathers of birds, from the leading edge backward and outward to the trailing edge (fig. 6.5). It is possible that the fibers provided additional support for the membrane, like the additional booms of Chinese junks. However, because they had a radius of only 0.1 mm, they were probably not rigid enough to do this (Bennett, 2000). Clearly there is room here for some model studies to see just what their function actually was.

STRESSES IN PIPES, CYLINDERS, AND SPHERES The mechanics of a pipe that is pressurized from the inside is just the same as that of the billowing membrane of the bat wing. Since a pipe can be regarded as two semicircular membranes stuck together, the pressure inside the pipe will set up circumferential stresses, σC , in the walls that are given by the

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P t

σL 2R Figure 6.6. The longitudinal stresses, σL , set up within a cylindrical pressure vessel of wall thickness t and radius R by an internal pressure P .

equation σC = R P /t.

(6.2)

If two end walls are stuck on the ends of the pipe, it will become a cylindrical pressure vessel. The circumferential stresses in its walls will be identical to those in the pipe, but longitudinal stresses will also be set up. Looking at a diagram (fig. 6.6), it is clear that the forces on the end walls set up by the pressure within, π R 2 P , must be equal to the longitudinal restoring force set up in the walls, which is equal to the wall cross-sectional area, 2πRt, times the stress within it, σL , and so given by the expression 2πRtσL . Therefore 2πRtσL = πR 2 P , and rearranging, the longitudinal stress in the walls is given by the equation σL = RP/2t.

(6.3)

Finally, the end walls can be replaced by hemispheres, and the side walls can be made infinitesimally short. This will produce a spherical pressure vessel, in which the stresses are uniform and are the same as the longitudinal stress in a cylindrical vessel σ = RP/2t.

(6.4)

The equations that describe the stresses in the walls of pressure vessels are known as the Laplace equations and can be summarized for those nonmathematically inclined in three sentences: 1) Stresses in the walls of pressure vessels rise with the pressure difference between inside and outside, and with the radius of the vessel, and fall with the thickness of the walls. 2) In a sphere, stresses are equal in all directions. 3) In a cylinder, the circumferential stresses are twice as great as the longitudinal stresses.

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As we shall see, these equations have profound implications on the design of a wide range of biological structures, but meanwhile they can illuminate an important aspect of a student’s life in the kitchen: the cooking of sausages. A sausage can be regarded as a typical cylindrical pressure vessel, in which the meat (and other less salubrious ingredients) are held within a protein skin. When it is cooked, the filling heats up and tends to expand, putting the skin under tensile stresses. Since the circumferential stress in such a vessel is twice the longitudinal stress, the skin will fail circumferentially and the sausage will split along its length. Of course, it is possible to cook sausages without splitting them, especially high-quality sausages with more meat and less filling. The Laplace equations tell us that narrower sausages and ones with thicker skins should be more resistant to splitting than wider ones with thin skins. If you want to avoid split-sausage misery, stick to the narrower sausages.

THE DESIGN OF ARTERIES We have already seen in chapter 2 that arteries, which are essentially pipes linking the heart to the capillaries, must have very compliant walls to equalize the flow of blood from the beating heart and reduce peak blood pressures. They achieve their high compliance by being largely composed of the rubbery material elastin. However, if artery walls were composed entirely of elastin, there would be a problem. If one region of the artery wall were slightly less rigid than the others, it would bulge outward more under the peak blood pressure. However, because the wall stress in a pipe is proportional to its radius, this means that the more extended region would be put under greater stress than the rest of the wall and bulge outward still further (fig. 6.7). This positive feedback could thereby cause the development of a permanently stretched and weakened bulge, or aneurysm. Long balloons develop just such bulges when we attempt to blow them up, bulging in the middle long before the tip of the balloon is expanded. Aneurysms are much rarer in our arteries than in balloons, which is fortunate because such bulges are very dangerous. Not only can they disrupt the flow of blood, but more worryingly they can eventually split, causing massive internal bleeding; if the blood vessel is in our brains this can cause a hemorrhagic stroke, whereas if it is in the aorta, it is invariably fatal. We are protected from aneurysms because many narrow layers of randomly oriented collagen molecules are incorporated into the tunica media of our artery walls (Bergel, 1961). Just as happens in skin, this collagen has little reinforcing effect on unstressed tissue, but as the artery expands under high pressure pulses, the collagen molecules are straightened and become oriented more circumferentially. As a consequence, stretched arteries resist further stretching more strongly; the relationship can be graphed as a J-shaped pressure-radius curve (fig. 6.8). The restoring stress

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(a)

(b)

(c)

Figure 6.7. The development of an aneurysm in a blood vessel (a). A slight bulge in the blood vessel under internal pressure (b) causes an increase in circumferential stress, further stretching it (c).

50

Pressure (kPa)

40

30

20

10

0 0

1 2 Relative radius

3

Figure 6.8. The J-shaped pressure-radius curve of a typical blood vessel. (Redrawn from Bergel, 1961.)

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rises exponentially with the radius of the artery and so stabilizes it against aneurysm formation. The system is perhaps most well developed in the aorta of whales (Shadwick, 1999). These mammals have a particularly large aortic bulb that acts to equalize blood flow during diving, when the whale’s heart beats particularly slowly. The collagen in the bulb is oriented more radially in this structure than elsewhere in the vascular system, making it even more extensible yet still highly resistant to aneurysms.

THE DESIGN OF LUNGS Another organ system in our body also has to undergo large changes in shape and so is vulnerable to the consequences of the Laplace equation— our lungs. To increase the surface area for gas exchange, mammalian lungs are divided into large numbers of linked air bags called alveoli; we expand them when we breathe by lowering our diaphragm, reducing the pressure inside our thorax. This is equivalent to increasing the pressure inside each alveolus. It is relatively straightforward to inflate and deflate a single pressure vessel such as a balloon. However, if you try to do the same with two pressure vessels—for instance, if you try to simultaneously blow up two or more balloons—the Laplace equation creates difficulties. If one balloon has slightly weaker walls, it will stretch more than the other; it will consequently develop higher stresses in its walls, which will cause it to expand yet further, whereas the stronger-walled balloon has lower wall stresses and remains deflated. The result is that only one of the balloons will inflate. In our lungs the result could be disastrous: only one of the many thousands of alveoli would inflate until it burst, whereas all the rest of them would remain collapsed. To prevent this happening it, is clear that a J-shaped relationship between pressure and radius is needed, like the one we saw for our arteries. This is not as easy to achieve as it sounds. The walls of our alveoli are so thin that there is little chance of incorporating randomly oriented collagen fibers, and in any case this would make the alveoli too stiff to inflate. Instead, much of the resistance of alveoli to being expanded is due not to the elasticity of their walls but to the surface tension of the fluid that coats them (Clements, 1962), and this has a major disadvantage. The surface tension of water, at 70 mN m−1 is high, which would oppose expansion strongly; even worse, it is a constant whatever the area. Lungs coated in pure water would therefore be impossible to inflate. Studies of lung mechanics have investigated the relative importance and volume-dependence of tissue elasticity and surface tension by inflating the lungs either with air (which will be resisted by both forces) or with saline solution (which will be resisted only by tissue elasticity) (Clements, 1962). The results show that surface tension is the most important component (fig. 6.9), but that, far from remaining constant, it falls as the lung volume decreases, thus stabilizing alveolar expansion. The reason the surface tension falls with

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Pressure (kPa)

2

1

0 0.0

0.1 Volume (l)

0.2

Figure 6.9. The pressure needed to inflate and deflate lungs with air (solid lines) and with saline water (dashed lines). The difference between the two curves is the pressure needed to overcome the surface tension of the alveolar fluid. (Redrawn from Clements, 1962.)

surface area is because of the surfactant molecules that the lung secretes into the fluid. As the lung deflates they crowd together on the surface and consequently reduce surface tension more than when the lung is expanded. The only downside to their action appears to be the high hysteresis, which increases the energetic cost of breathing. The importance of the surfactant to the operation of our lungs can be seen in premature babies. They often find it hard to breathe, a condition called infant distress syndrome, because they have not had time to manufacture a sufficient amount of surfactant; since their lungs cannot be expanded sufficiently, they have to be supplied with oxygen.

THE DESIGN OF SWIM BLADDERS If the alveoli of our lungs have the problem that they can collapse catastrophically, the reverse is true of the swim bladders of bony fish (Alexander, 1967). The swim bladder is a flexible structure that can be blown up or contracted by a fish to control its buoyancy and so help it float in the water. The fish secretes gas into the bladder to blow it up and provide more buoyancy, then either absorbs the gas or ejects it via a duct into its gut to contract the bladder and reduce buoyancy. The problem is that the buoyancy provided by such a flexible structure depends on the hydrostatic pressure acting on the outside of the fish. If the fish moves to a greater depth,

TENSILE STRUCTURES

(a)

109

(b)

36º

2nm

50µm Figure 6.10. Gas vesicles in cyanobacteria are torpedo-shaped floats (a) with walls made from brick-shaped proteins (b) oriented at 36◦ to the long axis of the cylinder. (Redrawn from Walsby, 1994.)

the hydrostatic pressure on the bladder will increase, causing the bladder to shrink. This results in reduced buoyancy, so the fish will tend to sink further. The opposite happens if the fish rises in the water; the pressure falls and the bladder expands, increasing the fish’s buoyancy. The result is that the fish’s position in the water is highly unstable. It therefore has to regulate its swim bladder actively as it changes position in the water. For shoals of fast-moving pelagic fish, such as mackerel, this is just not possible, and they consequently have to do without swim bladders. They have to swim continually to avoid sinking in the water, although they do reduce their density to some extent using lightweight fish oils. Unfortunately for mackerel, these omega oils are supposed to confer health benefits to us when we eat their flesh, so in recent years we have become keener than ever to catch them!

THE DESIGN OF GAS VESICLES A final example of a structure that has to cope with the consequences of Laplace’s equations, although it has to withstand compression rather than tension, is the gas vesicle. Gas vesicles are tiny torpedo-shaped buoyancy devices (fig. 6.10a) 10–100 nm in diameter that are synthesized within

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photosynthetic bacteria such as cyanobacteria and halobacteria (Walsby, 1994). Like submarines, gas vesicles contain only low-pressure air and hence have to withstand the crushing pressure caused by the osmotic pressure, or turgor, of the cells and the hydrostatic pressure of the water outside them. To do this they have walls that are composed of two layers of bricklike proteins that are laid down at an angle of approximately 36◦ to the long axis of the vesicle (fig. 6.10b). Both of these aspects are important to their mechanical function. First, because the wall thickness of the vesicles is constant, the Laplace equations show that the stresses in the walls will rise with the diameter of a gas vesicle. Therefore, although wide gas vesicles provide more buoyancy using less wall material than narrower ones, they are more vulnerable to being crushed. Second, the Laplace equations show that in a cylindrical pressure vessel, circumferential stresses are twice as great as longitudinal ones. The maximum stresses therefore occur at an angle of 54.7◦ to the long axis of the cylinder. At 36◦ to the long axis of the vesicle, the proteins are optimally arranged almost at right angles to this maximum stress, just as the horizontal bricks in a wall are arranged at right angles to gravity. As we shall see in the next chapter, this fact that the maximum stresses act at 54.7◦ to the long axis of a cylinder is even more important to organisms with hydrostatic skeletons. Gas vesicles are, in fact, used to regulate the height of bacteria in water columns and constitute a fascinating example of the interactions of mechanics with biology. Lake-dwelling cyanobacteria constantly synthesize gas vesicles, which therefore gradually increase their buoyancy. The cyanobacteria then rise to the surface of the lake, where they are ideally positioned to maximize photosynthesis. However, as they photosynthesize, the sugars they produce increase the turgor pressure of the cyanobacteria, causing the gas vesicles to collapse and the cyanobacteria to sink again. The result is a continual cycle of rising and falling that may also help improve the access of the cyanobacteria to nutrients, which tend to be more common some way below the surface of the water. The mechanics of gas vesicles also influences the adaptations of cyanobactaria to different environments. Cyanobacteria dwelling in shallow lakes tend to have large gas vesicles, which are cheap to construct. In contrast, those living in deeper lakes have to make do with smaller, less cost-efficient vesicles, because the vesicles have to be able to withstand the higher hydrostatic pressures at the greater depths to which the water currents could take them; wider vesicles would collapse at such depths, stranding the cells at the bottom. In the sea, the depth of the water is so great and the currents so strong that bacteria could easily be sunk to depths at which even small vesicles would collapse. Marine cyanobacteria therefore do without gas vesicles; buoyancy is in any case less of a problem in sea water, which is denser than fresh water.

CHAPTER 7 .................................................

Hydrostatic Skeletons

THE ADVANTAGES OF BEING PRESSURIZED We have just seen the problems that structures such as arteries and alveoli have in coping with the stresses set up within them by internal pressures. But there can also be advantages for pressure vessels in being stressed this way: because internal pressure puts the walls into tension, it protects them from buckling. Consequently one can produce a rigid, yet cheap, structure in which the walls have to withstand only tension and in which all the compression is taken by the fluid within. In human technology the most successful pressurized structures use air as the fluid. In pneumatic tires, which were essential for the development of bicycles and automobiles, the rubber tire holds the air, which helps withstand the weight of the vehicle. The two elements, air and rubber, also combine to give some suspension, producing a smooth ride as the wheel rolls over bumps, and the arrangement allows a large area of rubber to contact the ground, which, as we shall see in chapter 14, improves grip. Various inflatable boats have also proved successful, partly because of their buoyancy, while many nonrigid airships (blimps) are stiffened by the pressure of helium within their lightweight skins. In nature, in contrast, there are very few examples of pneumatic structures. Male frigate birds, like several other bird species, have brightly colored throat pouches that can be temporarily expanded for display purposes, and some seaweeds have air-filled floats. However, organisms make much more use of hydrostatic structures, because water is easily obtained, can be readily pressurized osmotically, and can be contained in many separate individual cells as well as by single larger structures. In fact, as we shall see, compartmentalization can be an advantage in hydrostatic structures. CARTILAGE One of the most important osmotically stiffened structures is cartilage, which is often regarded as a material but which is far from homogenous in structure. It is composed of 75% water, 16% collagen, and 7% proteoglycans, all of which are arranged in a highly organized way. There are also cells called chondrocytes that manufacture the other elements and which compose 2% of the volume. Within the structure, the proteoglycans osmotically attract the water, forming a soft jelly, which pressurizes the balloon-like structure up to 0.2 MPa or 2 atmospheres (Vincent, 1992). The collagen provides both a skin to resist the tension and internal bracing to maintain

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cartilage

calcified layer bone

Figure 7.1. The structure of articular cartilage, showing the orientation of collagen fibers (arrows) within the structure.

the structure’s shape. In the articular cartilage in our joints, for instance (fig. 7.1), the collagen is preferentially oriented depending on its position. The outer layer, which provides a tensile skin and a low-friction articulating surface, has fibers that are oriented parallel to the surface. In the center, fibers are oriented at 45◦ to the surface, crisscrossing the cartilage to resist shear forces. Finally, close to the bone, fibers are oriented at right angles to the bone surface where they provide a firm attachment. The result is a material that is far more compliant than bone or even tendon, with a Young’s modulus of 1–10 MPa and a breaking strain of approximately 0.25. Cartilage therefore makes excellent compliant hinges and also acts as a shock absorber in our backbone. There is just one disadvantage. If cartilage is continually compressed, as it is in our backbone, water gradually seeps away resulting in creep of the material; consequently we can be 1–2 cm shorter at the end of the day than at the beginning.

THE HYDROSTATIC SKELETONS OF PLANTS The most important hydrostatic skeleton of all is the plant cell, in which the rigid walls are stressed by osmotically generated internal pressures, or turgor, of up to 1 MPa. The difficulty in understanding the mechanics of plants, however, is that plant cells do not act on their own but rather in concert with millions of other cells. The splitting of the plant body into separate cells prevents large-scale movement of water within it; this limits local deformations and helps prevent buckling. The importance of this can be seen if one examines the mechanical behavior of single-celled seaweeds, such as the coenocytic green algae, the so-called sea grapes, of the genus Caulerpa. These amazing algae, which can reach lengths of up to a meter,

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have the contents of their single huge multinucleate cell stabilized by gels, which prevents them from leaking when they are wounded. Their stipes are also stabilized by struts that cross the cell, like the struts within the hollow bones of birds (see chapter 8). Despite this, however, the stipes are still extremely prone to kinking when bent, since the gel inside is easily squashed. The weakness of Caulaerpa stipes contrasts with the strength of the stipes of multicellular brown seaweeds, such as the sea palm Postelsia, which are made up of millions of separate cells. Unlike Caulerpa, Postelsia is capable of standing upright out of the water (Holbrook et al., 1991), despite having similar rubbery cell walls. Multicellularity was, in fact, probably essential to provide plants with enough support to stand upright on dry land. From the point of view of a biomechanist, however, it makes the mechanical behavior of even the simplest blocks of plant tissue—lumps of parenchyma such as potatoes— extremely complex. The simplest plant organs, such as primary plant stems, are even more complex, being hydrostatic structures at more than one level. They are made up of a central mass of closely packed thin-walled cells, which botanists refer to as parenchyma (each cell of which is, of course, itself a hydrostatic structure). These are in turn held in compression by an outer skin, or epidermis, whose cells have relatively thicker cell walls. This arrangement makes a large-scale hydrostatic structure, so that the parenchyma cells are put into compression and pack together into 14-sided shapes called orthotetrakaidecadedrons (Vogel, 1988). Inserting longitudinal fibers composed of thick-walled cells around the perimeter of plant stems further stiffens them but adds yet another level of complexity! Therefore, although the structural design of plant stems can appear straightforward, it is very hard indeed to fully analyze or model.

CYLINDRICAL PRESSURE VESSELS The majority of hydrostatic structures in biology are completely or partly cylindrical (Wainwright, 1988) with their outer tensile skin reinforced by fibers. The fibers may be oriented in two very different patterns (Vogel, 1988): they may be orthogonally arranged, with longitudinal and circumferential fibers (fig. 7.2a); or they may be arranged helically in either one or two directions (fig. 7.2b). These two arrangements give quite different mechanical properties to these hydrostatic structures and are intimately related to how they are used.

PRESSURE VESSELS WITH ORTHOGONAL FIBERS Hydrostatic structures reinforced with orthogonal fibers have high bending rigidity, because the longitudinal fibers are arranged parallel to the forces set up by bending. They are also highly resistant to compression. However, they are very prone to kinking and have a low torsional resistance, since

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(a)

(b)

Figure 7.2. The two main arrangements of fibers to support cylindrical pressure vessels: (a) a cylinder with orthogonal fibers, and (b) a cylinder with helically wound fibers.

(a)

(b)

Figure 7.3. The arrangement of fibers in the corpus cavernosum of the penis of the nine- banded armadillo: (a) unaroused, and (b) aroused. (Redrawn from Kelly, 1997.)

when they are twisted, the fibers can readily shear past each other. Maybe because of the kinking problem such structures are relatively rare, but orthogonal fibers are found in the structure that gives the mammalian penis its rigidity—the corpus cavernosum. The mechanics of this structure has been studied in the nine-banded armadillo (Kelly, 1997), which is extremely well endowed, having a penis about two-thirds as long as its body. Kelly has shown that the tunica albuginea, which composes the surface of the corpus cavernosum, is made up of two layers of orthogonally oriented collagen fibers, although at rest the organ is limp and the collagen fibers are crimped (fig. 7.3a). When the armadillo is aroused, blood flow into the corpus cavernosum is increased, and outflow is prevented by valves. The volume increases, therefore, and the fibers straighten, expanding the corpus

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cavernosum and inflating the penis (fig. 7.3b). The structure is not actually cylindrical, but it is held in shape by walls between the various parts; the walls probably keep fluid evenly distributed through the organ, acting like the cell walls in a plant stem, and so helps prevent kinking. Once the mammalian penis is extended, the pressure inside the corpus cavernosum is further increased, at least in dogs, by muscles around the base of the penis that increase the pressures to over 1 MPa or 10 atmospheres (Purohit and Beckett, 1976). It is likely that the mechanics of the human penis is similar to that of the armadillo and dog, but it has not, as far as I know, been studied.

MUSCULAR HYDROSTATS In many tubular structures in animals, such as annelid worms, orthogonally arranged muscle fibers alter their shape. Longitudinal muscles shorten the animal, while circumferential muscles can act as antagonists, narrowing the animal and therefore, since its volume is constant, causing it to lengthen. Worms bend by contracting their longitudinal muscles down one side while keeping their circumferential muscles active. Some animal structures, the muscular hydrostats, such as the tongues of lizards, the legs of cephalopods, and the trunks of elephants (Kier and Smith, 1985), are composed almost entirely of muscle. In muscular hydrostats the arrangement of the circumferential muscles gives them seemingly amazing properties. First, their contraction extends the organ, so it seems as if it contains a muscle that actually pushes. Second, if the organ is narrow enough, even a small contraction of the circumferential muscles results in a large extension of the organ, so the muscles act at a very high velocity advantage. This allows the tongues of lizards and frogs to be fired rapidly out of their mouths to catch flying insects (Van Leeuwen et al., 2000).

HELICALLY WOUND CYLINDERS Hydrostatic structures reinforced by helical fibers (fig. 7.2b) have a very low bending rigidity, in contrast to those with orthogonal fibers, because the fibers can readily shear past each other when the structure is bent. The helical winding, however, helps prevent kinking of the structure. This is why garden hoses so often have helical reinforcement; they can be moved readily around the garden and rolled up, but the winding is supposed to stop them flattening out and blocking water flow. Unfortunately this arrangement is not always wholly successful, possibly because the plastic matrix is too stiff to allow the fibers to rearrange. Helically wound cylinders are also extremely difficult to twist because the fibers are oriented to resist the shear this sets up in the walls. However, if the membrane between the fibers is fairly compliant, they cannot control the length of the cylinder, so its shape is critically dependent on the winding angle of the fibers.

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The volume of fluid enclosed by a helically wound cylinder is maximized when the fibers are oriented at an angle of 54.7◦ to the long axis of the cylinder. The length along a cylinder, H, that is enclosed by a fiber of length L, oriented at an angle θ to the long axis, is found readily by geometry: H = L cos θ. Meanwhile the circumference, C, that such a length can enclose is Lsin θ , which gives the radius, R, for the cylinder of R = L sin θ/2π. The total volume of a cylinder is π R 2 H, so the volume, V, enclosed by the fiber is given by V = L 3 (sin2 θ cos θ )/4π,

(7.1)

which is maximized when θ = 54.7◦ . The greatest volume is therefore enclosed when reinforcing fibers are oriented at 54.7◦ to the long axis of the cylinder, and the volume decreases at both smaller and larger angles (fig. 7.4). Since the circumferential stress is twice the longitudinal stress, as we saw in the last chapter, stresses are also maximized at 54.7◦ , so fibers at this angle are also optimally oriented to strengthen the walls of the pressure vessel. Because of the relationship between fiber angle and volume, the mechanical properties of a helically wound pressure vessel depend crucially on the orientation of the fibers relative to 54.7◦ . If the fibers are wound at a small angle to the long axis, the volume enclosed will be low. Therefore applying pressure inside will cause the fiber angle to increase, and the cylinder to shorten and fatten until the fiber angle reaches 54.7◦ . Similarly, if the angle of the fibers is large, the volume of the cylinder will also be low, but applying internal pressure will cause it to lengthen and thin until the fibers once again reach 54.7◦ . It is only when the fibers are oriented at exactly 54.7◦ that the cylinder has maximum volume; pressurizing the cylinder will merely stiffen it. In fact, very few organisms use hydrostatic skeletons in which the fibers are oriented at the magical 54.7◦ . Helical fibers are instead laid down at very different angles and used in three main ways: as a way of controlling growth; as muscle antagonists; and as a means of limiting movement.

HELICAL FIBERS TO CONTROL GROWTH AND SHAPE Plant Cells It has been accepted for some time that helical fibers control the expansion of plant cells (Taiz, 1984), which is an important aspect of plant growth. The expansion of plant cells is powered by osmotic pressure within the cells—or turgor, as it is known to biologists—but the control of expansion is mediated

HYDROSTATIC SKELETONS

117

(a) L

L

D πD

θ

Relative volume

(b)

0

10

20

30 40 50 60 Fiber angle (degrees)

70

80

90

Figure 7.4. The effect of fiber angle on the volume of the cylindrical pressure vessel it encloses (a). Volume enclosed (b) is maximized at a fiber angle of 54.7◦ .

by changes in the mechanical properties of the cell wall. Proteins called expansins loosen the cellulose microfibrils within the cell wall matrix, allowing them to shear past each other. In a typical young cell that is produced in the extension region of a root or shoot (or primary meristem as it is known to botanists), the cellulose fibers are laid down at a large angle to the long axis (fig. 7.5a). Loosening the fibers will therefore allow internal pressure to extend the cell, the fiber angle will decrease toward the maximum volume angle of 54.7◦ , and the structure will elongate. In contrast, if a plant structure is subjected to mechanical stimulation, such as if a root is growing through strong soil or if a stem is shaken, the fibers are laid down at a much smaller angle to the long axis (fig. 7.5b). Therefore, when the fibers are loosened,

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(a)

(b)

Figure 7.5. The control of cell growth in plant cells. If fibers are laid down at a high angle to the long axis (a), turgor will cause them to extend. If fibers are laid down at a high angle (b), turgor instead causes lateral cell expansion.

internal pressure will cause the cells to expand laterally as the fiber angle increases toward 54.7◦ , and the stem or root will grow thicker rather than longer.

Vertebrate Notochords Another case of a cylindrical organ whose expansion is controlled by helical fibers is that of the notochord of vertebrate embryos, whose behavior has been studied by Koehl et al. (2000). When it is first produced, the notochord is a short, curved, cylindrical structure that has to straighten as the embryo grows. Because these structures are so tiny, Koehl et al.

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investigated the mechanics of the notochord using Koehl’s favorite physical modeling technique—making models with polyurethane walls reinforced with helical nylon fibers and pressurizing them with compressed air. In all cases, the models were straightened by the internal pressure. In other respects the models behaved just as predicted by the mathematical analysis shown above: the ones with large fiber angles extended when pressurized, whereas those with small angles contracted. In the notochords of a real specimen—the clawed frog Xenopus laevis—the fibers were laid down at around 54◦ , which allowed them to straighten without changing length.

Stomatal Guard Cells A final example of a cylindrical structure whose shape is (presumably) controlled by the orientation of helical fibers is that of stomata in plant leaves. It is well known that the opening of stomata is controlled by altering the hydraulic pressure of the sausage-shaped guard cells on either side; increasing the internal pressure in the two guard cells causes them to bow outward, opening the stoma (Haberlandt, 1914; Willmer and Fricker, 1996). It has been suggested that this action is the result of the “radial” orientation of the cellulose fibers in the guard cells and the relative thicknesses of the inner and outer walls. However, the literature reveals there is still a basic ignorance of the mechanics, and even of the structure, of the cell walls. This situation is unfortunate considering the great importance of stomata and the huge number of papers botanists have published on them. Presumably the “radial” fibers are actually circumferential to the guard cells and prevent lateral expansion of the guard cells, but structural studies and modeling of stomatal mechanics are still urgently needed, a century after Haberlandt’s book.

HELICAL FIBERS AS MUSCLE ANTAGONISTS We have already seen that orthogonally oriented muscles in a hydrostatic skeleton can be used as antagonists to one another to lengthen, shorten, and bend the cylinder. However, the problem with using muscles in this way is that they require energy to generate the necessary forces, and the two sets of muscles have to be stimulated in a coordinated manner. If one could make a hydrostatic skeleton that required the activation of only one set of muscles, it would therefore require less energy and be easier to control.

Nematodes A good example of helical fibers as muscle antagonists is seen in nematode worms. These simple creatures (phylum Nematoda) possess only longitudinal muscles, which are antagonized by collagen fibers in the cuticle that are

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oriented at 75◦ to the long axis of the worm (Harris and Crofton, 1957). The longitudinal muscles of the worm are, in fact, under constant lowintensity contraction, which would tend to shorten the worm. However, such shortening would result in it getting wider, which is resisted by the large-angle fibers, so the result is that the worm is pressurized instead. This pressurization produces a rigid structure that can be bent backward and forward by asymmetrical contraction of the longitudinal muscles, first on one side and then on the other, allowing the worm to swim through the water. Unlike with the notochords Koehl et al. (2000) examined, a mathematical model produced by Alexander (1987) suggests that the nematode should be more stable when bent, rather than straight; certainly at rest nematodes are often seen curved in an exaggerated S-shape, flipping suddenly from side to side when they move. The only problem with the design of nematodes is that the internal pressurization makes it difficult for them to feed. They have to have specialized needlelike mouthparts with a special valve arrangement to allow them to suck up their food and prevent them from vomiting.

The Squid Mantle The opposite arrangement, in which mostly longitudinally oriented fibers act as antagonists to circumferential muscles, is seen in the mantle of the squid (Ward and Wainwright, 1972), which is used by the animal for jet propulsion. The collagen fibers in the outer tunic of the mantle are arranged at an angle of approximately 25◦ to its long axis. Therefore when the squid contracts its powerful circumferential muscles, the fibers strongly withstand the lengthening of the mantle that would otherwise occur. The consequence is a reduction in the volume enclosed by the mantle, so the water within it is forcibly expelled through the squid’s siphon, propelling the squid rapidly (if inefficiently) in the opposite direction.

Shark Skin We have already seen that helical fibers seem to affect the stability of hydrostatic structures in bending but in different ways. The notochords Koehl et al. (2000) examined seem to be straightened by internal pressure, whereas nematodes seem to be bent by it. Many swimming vertebrates, such as sharks and whales, also have crossed arrays of helical fibers in their skin, which may enable the skin to act as a hydrostatic skeleton, reducing the need for more extensive ossification of its cartilaginous skeleton. It has been suggested (Wainwright et al., 1978) that the skin can act as an antagonist to the swimming muscles of sharks, tending to restore the body to a straight position and perhaps helping to reduce the energy needed for locomotion. It has indeed been found that during swimming the bodies of sharks can become pressurized up to 200 kPa, but the mechanics of the situation, like the pattern of muscle fibers, seems to be extremely complex. Moreover, the

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121

limp

stiff

Relative volume

stiff

0

10

20

30

40

50

60

70

80

90

Fiber angle (degrees) Figure 7.6. Graph showing how limits to the changes in shape of limp worms depends on their initial volume. As the worms stretch or contract, their fibers enclose less volume and become more circular in cross section. (Redrawn from Clark and Cowey, 1958.)

fiber angles in shark skin also varies around the body. The helical fibers also confer another benefit to sharks; they greatly increase the body’s torsional rigidity. This stiffening prevents their bodies from being twisted by the swimming forces produced by their uneven heterocercal tails, which have a much large dorsal lobe than ventral lobe.

FIBERS AS LIMITS TO MOVEMENT A final use of helical fibers—to limit shape changes—is seen in two groups of worms: the ribbon worms (phylum Nermertea) and flatworms (phylum Turbellaria). As their common names suggest, these worms have a far smaller body volume than their helical fibers could contain, so at rest, with their fibers at an angle close to 54.7◦ , they are limp and fairly flat. Both groups change their shape using antagonistic longitudinal and circumferential muscles, which shorten and lengthen them, respectively. However, as the worms lengthen or shorten, the helical fibers change their orientation, reducing the volume that they can contain. As the worms get shorter or

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longer, they become more and more circular in cross section until a limit is reached and they can change shape no further. As figure 7.6 shows, the smaller the initial volume, and the flatter they are at rest, the further can they stretch and contract. For most worms the limits to movement due to the helical fibers pose no problem. However, this may be more of a problem for a group of annelid worms, the leeches (subclass Hirudinea). Leeches only rarely encounter their prey, so when they do, they engorge as much blood as possible. A feeding leech takes in blood until it has expanded to form an immobile cylinder, presumably limited by the length of fibers that are now at the 54.7◦ angle. It then falls off the victim, but by this stage it is, of course, unable to move and therefore extremely vulnerable to predators (or to the revenge of the victim). Only when the water in the blood has been excreted can the leech once again move about.

CHAPTER 8 .................................................

Structures in Bending

THE COMPLEXITY OF BENDING A large array of biological structures have to resist bending—from the stems, branches, leaves, and roots of plants; to the legs, wings, necks, and tails of animals; and to the fruiting bodies of fungi. Yet compared with the relative simplicity of forces within structures that are subjected to tension, bending induces a far more complex pattern. The stresses set up in a beam by a lateral force are far from intuitively obvious, and they depend enormously on where and how the beam is supported, as well as on its size, shape, and orientation. To get a grip on the subject of bending, it is therefore best to look first at the simplest possible case and analyze it using what is known as simple beam theory.

SIMPLE BEAM THEORY When a uniform rod is held at both ends and bent (fig. 8.1a), it is put into what is known as pure bending. What happens to each piece of material depends on where it is within the rod. Pure geometry means that the material toward the outside of the arc of the rod will be stretched along its long axis, whereas material toward the inside of the arc will be compressed along the long axis. The size of the strain imposed will therefore rise linearly with the distance on either side of the line running through the center of area, or the neutral axis of the rod. Only material directly on the neutral axis will remain unstrained. The resistance to bending of a beam, or its flexural rigidity, is therefore due not to the resistance of any element within it to being bent, but to the resistance of the material to axial strains—to being stretched or compressed. The rigidity will depend, therefore, on the tensile and compressive stiffness of the material, E . However, the material that is positioned further away from the neutral axis will be more effective and provide more resistance to bending than that close to it, for two reasons. First, material further from the neutral axis will be stretched or compressed more for a given curvature and so will resist with a greater stress. Second, the restoring force it produces, and which tends to straighten the beam, will act further from the neutral axis and so will generate a greater restoring moment around it. Therefore, the effectiveness of a piece of material is proportional to the square of its distance from the neutral axis.

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Tension

(a)

Neutral axis Compression

(b)

w

dy

ymax

y

Figure 8.1. A bending moment (arrows) produces axial strains along a beam (a), stretching material on the outside and compressing material on the inside of the neutral axis (dashed line). The further the material is from the neutral axis (b), the more it is strained and the greater its contribution to bending rigidity.

Mathematically, the flexural rigidity of a beam is given by the expression Flexural Rigidity = EI,

(8.1)

where E is the Young’s modulus of the material, and I is the second moment of area of the beam, which can be found by summation or

STRUCTURES IN BENDING

R

πR4 4

125

Ri

Ro

π (R 4 −R 4) 4 o i

W

WD3 12

R1 D

R2

π(R13R2) 4

Figure 8.2. The second moments of area of some regular cross sections when bent in the vertical plane.

integration over the cross section with the expression
ymax I = wy 2 dy,

(8.2)

ymin

where w is the width of the rod a distance y from the neutral axis (fig. 8.1b). The second moments of area of regular cross sections are straightforward to calculate using integral calculus and are shown in figure 8.2. I has units m4 , so the flexural rigidity, EI, has units N m2 . A simple formula then links the curvature of the rod, c, to the moment, M, applied to it c = M/E I,

(8.3)

and the maximum longitudinal stress, σmax , at the outside of the beam is given by the expression σmax = Mymax /I,

(8.4)

where ymax is the maximum distance of the chord from the neutral axis. Unfortunately, it is actually quite difficult to measure both an applied moment and the curvature it induces, so other, more practical methods are usually used to determine the flexural rigidity and strength of a beam and the mechanical properties of the material of which it is composed. THE FOUR-POINT BENDING TEST The method that most closely mimics the bending of the rod at its ends and that puts a beam into pure bending is to support the beam at both ends and push it down with a rig that meets the rod at two points which are inside the two supports and equidistant from them (fig. 8.3a). This arrangement is known as four-point bending. The flexural rigidity, EI, and flexural strength, Mmax , of the rod can then be readily calculated from the force-deflection curve of the rig using the following formulas: EI =

dF a (3L 2 − 4a2 ), dx 48

(8.5)

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(a)

(b)

Figure 8.3. Two methods for determining the mechanical properties of beams. In four-point bending (a), the beam is bent by pushing down at two points equidistant from the outer supports. In three-point bending (b), the beam is pushed by only a single central probe.

where dF /dx is the initial slope of the force displacement curve of the rig, L is the distance between the lower supports, and a is the distance between inner and outer probes. And Mmax = F max a/2,

(8.6)

where F max is the maximum force. Unfortunately, four-point bending tests can be rather tricky to carry out. They depend on accurate placement of the inner points, and a major problem is that to get this accurate placement, the probes need to be fairly sharp-edged. The probes will therefore tend to dent the beam, particularly if, like the stem of a herbaceous plant, it is composed of soft material, and so the results often tend to underestimate flexural stiffness and strength. To overcome this problem, the ends of the test piece may be stiffened by encasing them in resin (Corning and Biewener, 1998) or using loops of string or rope instead of contact points.

THE THREE-POINT BENDING TEST A much more frequently used technique, which overcomes some of the disadvantages of four-point bending, is to put the beam into three-point bending (fig. 8.3b), by either hanging a weight off the middle of the beam or pushing down on it using a single probe attached to the crosshead of a mechanical testing machine. The technique has the great advantage that it is very simple to perform; moreover, the supports and central probe are far apart, and both can therefore be fairly blunt, reducing denting. The threepoint bending test gives excellent values for flexural rigidity and flexural

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strength of a beam, at least for beams that are many times as long as they are wide. A typical three-point bending test will produce a plot of force against displacement, just like a tensile test, and this can then be used to calculate the properties of both the beam and of the material of which it is composed.

Calculating Structural Properties The rigidity of the beam EI, which has units N m2 , is calculated using the equation EI =

dF L 3 , dx 48

(8.7)

where d F /dx is the initial slope of the force displacement curve of the central probe and L is the distance between the supports, while the bending strength, Mmax , of the beam, which has units N m, is given by Mmax = F max L/4,

(8.8)

where F max is the maximum force.

Calculating Material Properties If the geometry (and hence the I ), of the beam is known, the average stiffness, or flexural modulus, E B (with units N m−2 or Pa), of the beam can also be readily calculated from equation 8.7 by dividing the rigidity by I . Note that the flexural modulus may not be the same as the Young’s modulus calculated from a tensile test, E T , mainly because the flexural modulus is more dependent on the material further from the neutral axis, but also because half the beam will be loaded in compression. The ratio E B /E T is a useful measure of how material is arranged within a solid beam (Usherwood et al., 1997). In their study of the stems of terrestrial and aquatic buttercups, they found that the terrestrial forms, which had to resist bending to stand up, had higher values of E B /E T (around 2.5) than the aquatic forms (around 0.8), which only had to resist tension from flowing water. The terrestrial buttercups had their stiff fibrous material arranged closer to the outside of their stems. Finally, the breaking stress of the material, σmax (also with units N m−2 or Pa), can be calculated using the equation σmax = Mmax ymax /I.

(8.9)

Note that calculations of flexural rigidity and flexural stiffness assume that the material obeys Hooke’s law. However, they also rely on two more assumptions: that the movement of the probe and the depth of the beam are both small relative to the length of the beam. Calculations of breaking stresses and strains also assume that failure has occurred within the material

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itself. As we shall see, in many beams failure occurs because of the buckling of the structure rather than the failure of the material.

THE CONSEQUENCES OF SIMPLE BEAM THEORY The Optimal Numbers of Beams Because the effectiveness of a piece of material depends on the square of its distance from the neutral axis, the flexural rigidity and strength of a beam are both strongly dependent on its radius. The rigidity is proportional to the fourth power of its radius, and the strength to the third power of radius, whereas the mass per unit length is proportional only to the radius squared. The consequence is that it is more efficient to resist bending using a single thick beam than many narrow ones placed side by side. This is one reason why trees and other plants support their crowns with just a single stem or trunk. It also explains why there has been a consistent evolutionary trend in ungulates to lose their outer toes (Simpson, 1950). In horses the outer toes have been totally lost, leaving just the single central toe with its single hoof. In even-toed ungulates, two hooves remain but the two central toes are fused together, effectively giving a single, more or less cylindrical cannon bone.

The Optimal Cross Sections of Beams A second consequence of the squared term is that the bending rigidity and strength of beams depends on their cross-sectional shape and orientation as well as the cross-sectional area; beams with more material placed further from the neutral axis resist bending more strongly. A rectangular beam, such as a ruler or plank, is more resistant to bending parallel to the long axis of its cross section than to its short axis, hence the vertical orientation of the joists beneath our floorboards. Expanding the top and bottom of the beam will further increase the flexural rigidity per unit mass of material. This is why engineers are so fond of using I-beam girders (fig. 8.4a), or RSJs (rolled steel joists), as they are known in the building trade. In biology the best example of beams that have to resist bending in just a single plane are the lateral roots of trees; when the tree sways in the wind, these roots are either bent upward or downward depending on whether they are on the leeward or windward side of the trunk, and they respond, as we shall see in chapter 13, by laying down wood on their top and bottom surfaces to produce a figure-eight cross section (fig. 8.4b). To resist bending in all directions, one could combine several I beams to produce a beam with a cross section like a carriage wheel. In fact, though, because the outer flanges are joined, there is no real need to have the central spokes, and so the optimal shape to maximize flexural rigidity is a hollow tube. Both people and nature have thoroughly exploited tubular construction, and most large man-made structures, such as bridges, oil rigs,

STRUCTURES IN BENDING

(a)

129

(b)

Figure 8.4. Beams that efficiently resist bending forces in the vertical plane: (a) a man-made I beam, and (b) the lateral root of a tree.

and even bicycles, are made by welding together a number of thin-walled metal tubes. Tubes are even more common in nature: many plants have hollow stems, for instance; the legs of arthropods are hollow tubes; and even the long bones of vertebrates are usually hollow. The optimal degree of hollowness depends on what is inside the tube. If the tube is filled with a fluid, as is the case for the marrow-filled bones of mammals, then increased hollowness will mean that it will contain a greater weight of marrow. The optimal shape of marrow-filled bones will therefore be relatively thick walled. Mammals indeed have thick-walled long bones (Currey and Alexander, 1985). If the hollow tube is filled with air, then it will be better to be thinner walled. Optimal stiffness is achieved by having infinitely thin walls, but those would have two main disadvantages. A thin-walled bone would not be so strong or resistant to impacts. It is not surprising, therefore, that the wing bones of birds such as crows, that lead a rather rough-and-tumble life are rather thicker walled than those of soaring birds, such as albatrosses. The record for the thinnest-walled bones seems to have been held by the humeri of the large pterosaurs, which have a wall thickness of no more than 1/15 the radius. Presumably these creatures lived a very careful life.

Problems with Local Buckling Getting rid of the central spokes of the cross section of a tube will certainly optimize stiffness, but it does have disadvantages. The webbing in an I beam has several functions, one of the most important of which is to keep the upper and lower flanges a constant distance apart. As we have seen, when a beam is bent, the material on the convex edge is stretched and on the concave side it is compressed. If the central webbing were not there, the outer flange could move inward rather than stretching, and the inner flange could bulge outward (fig. 8.5a). The two flanges would each act as weak independent beams and would come together, and the entire beam would bend readily about the narrow hinge that formed. In a tubular beam the

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(a)

(b)

Figure 8.5. A hollow tube will buckle rather than break when it is bent because the outer wall will move inward and the inner wall move outward, reducing the changes in length (a) and so ovalizing the cross section of the tube (b).

problem is not quite so acute, but with no material in the center of the cross section to keep the two sides apart, the tube will tend to ovalize when it is bent (fig. 8.5b; Spatz et al., 1990). The second moment of area of the tube will decrease, and eventually the walls will collapse inward, resulting in the local buckling that you can see when you bend a drinking straw or piece of rubber tubing. The ease with which local buckling happens will depend on the thickness of walls and on the properties of the walls themselves. Obviously, thinnerwalled tubes are more prone to local buckling. Young and Budynas (2001) give the following equation for the buckling moment, M, of a thin-walled cylinder: M=

k Ert 2 , (1 − ν 2 )

(8.10)

where r is the outer radius of the cylinder, t is its wall thickness, and E and ν are the Young’s modulus and the Poisson’s ratio of the material. The numerical factor k can vary somewhat depending on the material,

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(a)

131

(b)

Figure 8.6. Ovalization and buckling of tubes may be prevented by stiffening the outer walls with stringers (a) or maintaining the cross section by incorporating bulkheads (b).

but usually k ≈ 1. Note that the strength of the rod therefore depends on the stiffness of the material rather than on its strength. Unfortunately this equation is only true for cylinders of wall thicknesses less than 1/10 the radius and that are made of isotropic materials. It has been applied fairly successfully to predict the bending strength of thin-walled feather shafts (Corning and Biewener, 1998). The predicted values for bending strength, though, were 20% higher than the actual values. The likely cause of the difference was that the keratin within feather shafts is anisotropic. It is reinforced mainly by longitudinally oriented fibrils, so the transverse stiffness will be lower than the longitudinal stiffness; the shaft will therefore be more prone to flattening and be weaker than equation 8.10 predicts. The problem of local buckling is even more acute in hollow plant stems, which are reinforced by longitudinally oriented fibers in a soft parenchyma matrix and which therefore have a far lower transverse stiffness than longitudinal stiffness. This makes their cross sections extremely prone to being flattened. Bend the flower stalk of a daffodil or dandelion, or the chives on your salad, and you will be able to see how readily they flatten out, allowing them to buckle. The analysis of this situation is rather more complex and involves finding the result that minimizes energy; however, it has been admirably tackled by Spatz and his coworkers (Spatz et al., 1990). They have also investigated the mechanical design of a number of hollowstemmed grasses and horsetails (Speck et al., 1998).

Methods to Prevent Local Buckling Developing thicker walls is not the only way to prevent local buckling in hollow beams; many different design solutions have been developed both by humans and nature. Longitudinal stringers (fig. 8.6a) locally increase the flexural rigidity of the wall, reducing flattening, whereas bulkheads (fig. 8.6b) maintain the circular cross section perfectly wherever they are positioned. Each can be used on its own (bulkheads can be found, for

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instance, at the nodes of bamboo and horsetails [Niklas, 1989] or together (as in the spines of hedgehogs [Vincent and Owers, 1986]), the stems of umbellifers such as cow parsley, and the petioles of bananas [Ennos et al., 2000]). It is noteworthy that in most biological situations, the bulkheads do not act in compression, but rather in tension, like the spokes of a bicycle wheel. In bamboos, for instance, the nodal septa are very thin and tear parallel to the bending moment. (Kappel et al., 2004) showing they have failed due to lateral tension. Certainly neither these thin nodes nor the even thinner bulkheads of bananas, made as they are from a single layer of starshaped cells (Ennos et al., 2000), are suitable to resist compression. Rushes show an interesting variability in the mechanical reinforcement of their hollow stems. Some, such as the jointed rush (Juncus articulatus),have sections held open by bulkheads. In others, such as the soft rush (Juncus effuses),the section is kept open by a lightweight pith. The pith is composed, like the bulkheads of bananas, of star-shaped cells that make up what is known to botanists as stellate parenchyma; this pith is relatively resistant to lateral compression but may actually function by preventing tension. Animals may also resist ovalization with spokelike structures. The shafts of bird feathers are filled with a keratin foam (Purslow and Vincent, 1978), while on a larger scale, the struts, or trabeculae, within the hollow wing and leg bones of flying birds, and indeed at the ends of our own long bones, also prevent flattening of the bone, although it is uncertain whether they act more in tension or in compression. Even if no lightweight filling is available, organisms may make use of energetically inexpensive filling materials, such as water, to resist compression. Many herbaceous plants, for instance, have solid stems, whose centers are filled with parenchyma tissue, which is composed of large, thin-walled cells filled with water, while the reinforcing fibers that provide the flexural rigidity are located near the edge. Of course, the orientation of these fibers is crucial in their effectiveness: fibers oriented longitudinally, parallel to the long axis of the stem and so to the forces set up in bending, will stiffen a rod much more than ones that spiral around it (fig. 6.1b). In self-supporting herbs, therefore, the fibers are almost invariably parallel to the long axis of the stem. Spiral fibers are seen only in woody lianas, which are subjected mainly to tension (see chapter 6), and in the trunks of trees subjected to high wind speeds. Such trees develop a spiral grain (Kubler, 1991), making them far more flexible and enabling them to flex with the wind without breaking.

Other Effects of Transverse Stresses Transverse stresses are not confined to hollow tubes. Whenever a solid beam is bent, the longitudinal forces set up within it will also produce transverse stresses (Ennos and van Casteren, 2010). In initially straight rods, or in initially curved rods that are bent more, they will set up transverse compressive stresses, just as in hollow beams. In contrast, if an initially curved beam is bent in such a way as to straighten it, transverse tensile

STRUCTURES IN BENDING

(a)

133

(b)

(c)

Figure 8.7. Mechanisms of bending failure in wood. Branches made of light wood buckle inward when bent (a) because of its low transverse compressive strength, whereas curved branches split when straightened (b) because of the low transverse tensile strength (redrawn from Ennos and van Casteren, 2010). Similar splitting occurs at the junction between the trunk and lateral roots (c).

stresses will be set up (Mattheck and Kubler, 1995). These stresses are much smaller than the longitudinal stresses, except where the curvature is very high, so they can be ignored in beams made of isotropic material. However, in highly anisotropic materials, such as wood, they can become a real problem. Low-density wood has extremely low transverse compressive strength because the thin-walled cells of which it is composed can be readily crushed. Consequently, when the branches of a willow are bent, they do not snap but instead buckle inward (fig. 8.7a; Ennos and van Casteren, 2010), a property that helpfully allows us to weave willow into baskets. Conversely, all varieties of wood have low transverse tensile strength, because cracks can readily run between the longitudinally oriented wood cells, or tracheids, and the radially oriented rays (see chapter 4). This causes particular problems for trees that have undergone gravitropic and phototropic growth responses that have bent their trunks or branches upward (see chapter 9), forming what Claus Mattheck calls hazard beams (Mattheck and Kubler, 1995). When gravity bends these beams downward, they are extremely prone to splitting exactly down their middle (fig. 8.7b). Trees experience an apparently even more serious problem at the junctions between their lateral roots and trunk. When the trunk is blown laterally by the wind, high lateral tensile forces will be set up at the highly curved root-trunk junction, which tends to split, causing what is known as root delamination (fig. 8.7c; Mattheck and Kubler, 1995). Fortunately trees can prevent delamination in two ways. First, they

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(a)

(b)

Figure 8.8. The fracture of straight branches of dense wood. Straight branches break halfway across before splitting either in both directions (a); tapered branches only split distally (b). (From Ennos and van Casteren, 2010.)

can increase the rate of cambial growth at the junction of the root and trunk, producing a buttress and reducing the rate of curvature. Second, they can incorporate more and stronger rays at the root-trunk junction, which increases the lateral strength of the wood (Mattheck and Kubler, 1995).

FRACTURE IN BENDING We saw in chapter 1 that when long thin structures are loaded in tension, they tend to break transversely across their short axis, more or less in a straight line; the traveling crack causes an increasingly large stress concentration at its tip that drives it forward. For many brittle materials, the same thing happens in bending: bend a piece of glass or a biscuit and it breaks straight across. However, for tough materials, and particularly for anisotropic ones, the process of fracture in bending can be rather different (Ennos and van Casteren, 2010). In the living twigs and branches of trees, in which transverse tensile strength down the center line is extremely low, the crack can get diverted when it reaches halfway across (fig. 8.8a). The crack will then run longitudinally up and down the stick, leaving one half intact and causing the characteristic greenstick fracture. Tapered branches may split only distally before eventual fracture of the final half of the stick (fig. 8.8b). Tearing through the second half of a broken cylindrical twig can be extremely difficult, and one often has to resort to twisting it in order to fully break it. Early humans must have found it extremely hard to break off living branches to make his tools, but orangutans often make a virtue out of

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necessity. They construct the frame of their sleeping nests using branches that are still half attached to the tree, and so ensure that they can pass a worry-free night. The so-called greenstick fractures of bones, in which the bone breaks only halfway before splitting lengthwise, just like branches, is common in children, but the reasons do not seem to have been investigated. The conventional wisdom appears to be that children’s bones are more flexible and so are more likely to suffer from displacement-limited failure. However, it may be that, like living twigs, children’s bones are more anisotropic because they have undergone less remodeling and so contain less Haversian bone. They are therefore more prone to longitudinal splitting (Liu et al., 1999; Ennos and van Casteren, 2010).

SHEAR IN BEAMS We have seen that the central webbing in a beam has a crucial role in keeping the outer edges equidistant and preventing the section either collapsing inward or splitting apart, but it also has another, even more crucial task in beams. The cantilever beam shown in figure 8.9a, in which the upper and lower flanges are joined only by vertical rods, will be extremely easy to bend. If a vertical load is applied, the two flanges will flex readily, like two separate narrow beams, and each cell will shear into a more or less rhomboidal shape (fig. 8.9b). A major function of the webbing is to resist this shear, which rises to a maximum at the neutral axis (Gere, 2004). The webbing need not in fact be complete, and it could just as well be replaced by alternating compression struts and tension wires to produce an open truss (fig. 8.9c). After all, as we have seen in chapter 1, shear is just alternating tension and compression. To understand why the material in a cantilever beam has to resist shear whereas the material in the beam that was bent at both ends does not (fig. 8.1), we have to examine the patterns of stresses that are set up in beams by applied forces. If a perpendicular force is applied to a beam, this sets up shear within it on either side of the force. The shear in a beam is equal to the integral over distance of the lateral forces applied to it (Gere, 2004), so a point force results in constant shear. In turn the shear sets up a bending moment that is the integral over distance of the shear force (Gere, 2004). Consider a cantilever beam that is point loaded at its end (fig. 8.10). In equilibrium the forces on the beam must balance so there will be an equal and opposite reaction force at its base (fig. 8.10a). The point force will also cause a constant shear to be set up all the way along the beam (fig. 8.10b). Finally, this in turn will set up a bending moment in the beam. This will be the integral over distance of the constant shear force and so will increase linearly to the base of the cantilever (fig. 8.10c). What about a beam in three-point bending? Well, this is equivalent to two inverted cantilevers held together at their center. The downward

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(a)

(b)

(c)

Figure 8.9. A cantilever truss without cross-bracing (a) will be easy to bend, since each cell can readily shear (b). Cross-bracing (or a solid web) resists the shear, greatly stiffening the beam.

force at the center point is of course balanced by upward forces of half its magnitude at the two supports (fig. 8.11a). As in the cantilever there will be a constant shear all the way along the beam (fig. 8.11b), and the bending moment will rise to a maximum at its center (fig. 8.11c). Therefore a beam

STRUCTURES IN BENDING

137

(a)

force

(b)

shear 0

(c) bending moment 0

Figure 8.10. Force (a), shear (b), and bending moment (c) in a cantilever beam loaded at its end. Shear is the integral of applied force, and bending moment is the integral of shear.

in three-point bending is subjected to shear, as well as to a pure bending moment, and the shear should increase its flexural rigidity. Fortunately, if the beam is relatively narrow, the shear forces will be low relative to the bending moment. This is why beams used in a three-point bending test should be more than 15 times as long as they are wide, and even longer for anisotropic material (Spatz et al., 1996). In four-point bending the situation is quite different. The upward force at each outer support equals the downward force applied at the two inner loading points (fig. 8.11d). Therefore, although the two outer regions are put into shear, the main central region is under no shear stress (fig. 8.11e). The bending moment will consequently rise from zero at the outer supports to a maximum at the inner loading points and will remain constant throughout the central length (fig. 8.11f). Under constant pure bending, the beam will therefore exhibit a constant curvature like we saw for the beam bent at both ends in figure 8.1.

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(a)

(d)

force

(b)

(e) 0

(c)

0

shear

0

(f) 0 bending moment

Figure 8.11. Force, shear, and bending moment in beams held in three-point bending (a, b, c) and four-point bending (d, e, f). Shear is the integral of applied force and bending moment the integral of shear.

THE CONSEQUENCES OF SHEAR As we have seen, shear rigidity is essential for a beam to resist bending adequately. Structures that look like beams but that lack shear rigidity, such as paperback books and packs of playing cards, are therefore extremely flexible in bending; the leaves and cards simply slip past each other when a bending load is applied. I know of no equivalent structures in nature, but one natural structure that has limited shear resistance, the tail fins of fish, does exist and has very unusual mechanical properties. Tail fins are stiffened by rays that are split into two longitudinally orientated half rays (fig. 8.12.). These are firmly joined at their tips but are separated toward their base by a jellylike squidge (McCutchen, 1970; Geerlink and Videler, 1987) that has little shear stiffness. The bases of the rays are held on either side of a sternpost by tendons, and a strap prevents them separating. This arrangement means that relative longitudinal movements of the half rays are possible, making the structure far more flexible than a proper beam would be. If one half ray is pulled forward by its basal tendon, this causes the entire fin to be bent toward that side (fig. 8.12b). Similar curvature can be caused by holding the bases of the rays at the same position and applying a lateral load to the middle of the fin. The fish therefore has the ability to change the shape of its tail fin during its power strokes to optimize its hydrodynamic performance. Some apparently rigid beams can also fail as a result of the low shear strength of the material of which they are made. Hollow trees, for instance,

STRUCTURES IN BENDING

139

sternpost

(a)

half ray

squidge tendon

strap

(b)

Figure 8.12. The mechanism of action of the tail fins of fish. If one of the half rays is pulled forward (b), the fin is bent toward that side. (Redrawn from McCutchen, 1970.)

are rather prone to mechanical failure. This has usually been attributed to a reduction in their bending strength. However, recent research suggests that failure is instead due to shear, because the trunk splits along its length right down its center, where the rays cannot strengthen it (Mattheck et al., 2006). Hollowing out a tree raises the shear stresses by a much greater factor than it does the longitudinal stresses. Once split, the trunk acts as two separate half tubes; it will be weaker and the cross sections will also be prone to flattening, so the trunk will ultimately fail in bending. In fact, much larger shear stresses are set up toward the base of tree trunks as a result of the anchorage forces set up by their roots (fig. 8.13; Mattheck et al., 2006), and it is probably these that are responsible for the initial splitting of the trunk.

BIOLOGICAL TRUSSES As we have seen, beams do not need to be made of continuous material to resist bending; trusses consisting of alternating tension and compression members are even more effective and are commonly seen in engineering, from the horizontal trusses in railway bridges and the arms of cranes to the vertical trusses of the Eiffel Tower and electricity pylons. They are just as common in nature, although they are usually hidden beneath the skin. Cantilever beams made of bones, ligaments, and tendons are common in the necks and tails of large animals, such as the ungulates we saw in chapter 2 (fig. 2.6b,c) and, even more impressively, in dinosaurs. In these cases the bones usually resist the compressive forces, and the ligaments and tendons, the tensile forces. The skeleton of a large bipedal dinosaur, such as

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wind

Figure 8.13. Shear stresses in trees. High up the trunk there is shear due to the force of the wind, but much greater shear stresses in the opposite direction are caused by the large anchorage forces around the roots.

a hadrosaur, with its neck and tail cantilevered out and balancing on either side of the legs, is in fact very reminiscent of one of the main cantilevers of the Forth rail bridge. The hip region in such dinosaurs was undoubtedly subjected to the greatest bending moments and shows the most pronounced adaptations for rigidity: the hip bones and vertebrae were crisscrossed by ossified ligaments. Similar trusses can be seen in many platelike bones, which look solid from the outside. In cross section it can be seen, for instance, that the ischium of a horse is a cleverly arranged hollow truss with solid laminar bone at the top and bottom surfaces, and narrow trabeculae running at 45 degrees to the surface across the hollow center (fig. 8.14). These trabeculae resist the shear stresses that are set up by muscles inserted at the end of the bone. Sandwich construction is also seen in the leaves of plants such as irises (Gibson et al., 1988) and the giant reed Typha (Rowlatt and Morshead, 1992). In many cases in biology, however, thick, hollow plates are far too heavy or energetically expensive to produce, and at first glance it would seem to be difficult to make thin platelike structures, such as leaves or insect wings, rigid enough to support themselves. Many plants support their leaves using a fairly thick midrib, with more fibrous material incorporated toward the upper surface and more parenchyma tissue toward the bottom to resist compression (Vogel, 1988). Midribs are not the only way to achieve flexural rigidity, though. Plates can be stiffened by making them corrugated or U shaped in cross section, dramatically increasing their second moment of area and allowing the surface to act as its own truss. The leaves of hornbeams, for instance, are corrugated on either side of the midrib, this arrangement having the advantage that the lamina is opened out and unfurls from the bud as the midrib lengthens. The leaves of traveler’s and European palms are even more impressive corrugated structures and open out even more spectacularly from the crown.

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141

Figure 8.14. The plate truss of a horse’s ischium. Bending forces (arrow) produced by the muscle are withstood by the upper and lower flanges of compact bone (stippled), while trabeculae criss-cross the central region along the lines of stress. (Redrawn from Currey, 2002.)

The wings of insects, too, are supported by corrugation of the wing membrane, especially at their leading edge (Wootton, 1981). The corrugation can be maintained by angle brackets or thickenings of the veins (fig. 8.15), while the tops and bottoms of the creases are reinforced, against compressive forces in particular, by the expansion of the membrane to form tubular longitudinal veins. Cross veins also prevent the membrane from crumpling. At first glance it looks as if the cross veins of dragonfly wings, being at right angles to the longitudinal veins, are poorly oriented to resist the shear in the structure. However, as Newman and Wootton (1986) pointed out, the membrane itself is a structural member, acting as a stressed skin to reduce bending, just like the fabric or metal skins covering the structures of aircraft and ships. Plates with many corrugations are equally resistant to upward and downward bending. However, beams with just a single V- or U-shaped corrugation behave very differently when loaded from above and below. This is because the cross-sectional shapes of beams are altered as a result of the longitudinal deflections within the beam. Just as an axially loaded rod will get thinner when stretched and thicker when compressed, so the convex side of a bent rod (which is in axial tension) will contract laterally, and the concave side (which is in axial compression) will expand laterally. As a consequence, a flat plank will deform to form a saddle shape when bent (fig. 8.16a,b). The transverse radius of curvature, c T , is given by the expression c T = νc L ,

(8.11)

where c L is the longitudinal curvature and ν is the Poisson’s ratio of the plank (Gere, 2004). Therefore when a U-shaped beam is bent downward, the sides will curl upward and inward even more, increasing the second moment of area of the

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Figure 8.15. The V-shaped leading edge of the wing of a damselfly is stabilized by angle-bracket cross veins and stiffened by a stressed skin membrane. Arrows show stress direction in the wing membrane. (Redrawn from Wootton, 1990.)

(a)

(b)

Figure 8.16. Changes in the cross-sectional shape of flat beams when they are bent. A flat plank (a) will curve into a saddle shape (b).

beam and increasing its flexural rigidity (fig. 8.17a). In contrast, when bent upward, the sides will deflect downward and outward, flattening the section and reducing the second moment of area (fig. 8.17b). As the beam is bent further, the bending will become more concentrated at this point, until a flat hinge point is formed along which the beam bends readily. The asymmetric resistance of U-shaped beams to flexing is once again extensively exploited by both humans and nature. The measuring tapes used by builders have a U-shaped section, which allows them to be held outward a meter or two from the hand. They can be readily bent upward, however, and flatten out as they are wound up inside their drum. In nature, the leaves of grasses are often U-shaped, which helps support them against gravity while allowing

STRUCTURES IN BENDING

(a)

143

(b)

Figure 8.17. Changes in the shape of cambered beams when they are bent. In a U-shaped beam the sides will move inward, stiffening it when it is bent downward (a), and move outward, flattening it and weakening it when bent upward (b). (Redrawn from Wootton, 1981.)

them to flex upward in the wind. Conversely the wings of butterflies (Wootton, 1990) and the basal veins of in the wings of some flies (Ennos, 1989) are arched in the other direction, helping them support the body against aerodynamic forces from below while allowing them to flex downward at the end of their downstroke. As we shall see in chapter 10, the channel shape of butterfly wings also affects their torsional properties and alters how they are used in flight. A final method of increasing the flexural stability in platelike structures is seen in the vertical leaves of some grasses and daffodils, most notably the giant reed Typha. Instead of remaining in one plane, these leaves are twisted along their long axis (fig. 8.18), so that they cannot easily bend along a single axis (Schulgasser and Witztum, 2004). Analysis suggests that this arrangement increases the maximum height that such leaves can reach by up to 26%.

OPTIMAL TAPER AND THE SCALING OF CANTILEVER BEAMS The trunks and branches of trees both act as cantilever beams. It is of course easy to see this in branches, since they are bent down both by the leaves and fruit that they support and also by their own weight. Tree trunks also act as cantilevers, though, because they are subjected to large lateral forces as the wind blows against the crown of the tree. In such cantilever beams we have seen that the bending moment increases toward the base. The precise way in which it rises, however, will depend on how the beam is loaded.

The Optimal Taper of Tree Trunks For the simplest case in which the beam is loaded by a point force at its tip, the bending moment rises linearly from zero at the tip to a maximum at the base of the cantilever. An optimally tapered beam should maintain a

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Figure 8.18. Twisting of the upright leaves of grasses and other monocots stabilizes them against gravity.

constant maximum stress at its surface all the way along its length, avoiding laying down too much material in one place and having a weak spot in another. In this case, since the bending moment rises linearly with the distance from the tip and the breaking strength of a beam is proportional to the cube of its radius (Mattheck, 1991), the optimal taper follows the expression D ∝ L 1/3 ,

(8.12)

where D is diameter and L is the distance from the tip. This is precisely the shape that is often seen in trees that have a small windblown crown (fig. 8.19a) and are hence loaded essentially by a point force at their tip. In contrast, if the lateral load rises linearly toward the base, as it might do in a windblown tree that has a deep conical crown (fig. 8.19b), then the bending moment should rise with L to the third power (Mattheck, 1991) and the optimal taper follows the expression D ∝ L.

(8.13)

In other words, the optimal shape is for the trunk to be conical. This shape is indeed seen in trees with conical crowns, such as swamp cypresses and dawn or giant redwoods. If a tree is loaded in some intermediate way, such as with a constant force all the way up its length, the optimal taper should lie somewhere between these extremes. The Optimal Taper of Branches A branch or other horizontal cantilever that is loaded primarily by its own weight has quite different patterns of force on it from the trunk. The loading

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(a)

(b)

L

D

Figure 8.19. Optimal shapes of the trunks of trees to resist wind forces. In a tree with a small crown (a), the diameter, D, should increase with the cube root of the distance from the crown, L, whereas in a conifer with a conical crown (b), the trunk should also be conical.

per unit length due to its weight will rise with its cross-sectional area—in other words, with the square of its diameter—so both the strength of the beam and the force applied to it depends on how it tapers. To maintain a constant stress or strain all along its length, the optimal taper follows the expression D ∝ L 2,

(8.14)

an extreme shape in which branches get much thicker as they get longer. However, in this case the sag is concentrated toward the tip of the branch. For a branch to show a constant rate of sagging all along its length, the optimal shape is less extreme, following the expression D ∝ L 3/2 .

(8.15)

Such a branch is said to show elastic similarity (McMahon, 1975).

The Scaling of Tree Trunks This final pattern of taper is also relevant to the stability of self-loaded columns, such as bare tree trunks or the stalks of grass. If a vertical column gets too tall and thin, it will become unstable, start to sag, lean over, and finally collapse under bending forces caused by its own weight. This is what makes the Indian rope trick so apparently impressive; flexible ropes simply

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should not be able to stand upright. The maximum height, h, a self-loaded column can reach is given by Greenhill’s formula h = 0.887 r 2/3 (E /ρ)1/3 ,

(8.16)

where r is the radius, E the Young’s modulus, and ρ the density of the material of which the column is made (Greenhill, 1881). The diameter of a tree trunk should therefore rise with its height to the power of 3/2 to ensure that the tree does not collapse under its own weight. A great deal of effort has been spent by foresters and plant biomechanists in examining the scaling of trees. Just by looking at them, it seems obvious that taller trees tend to be relatively thicker than shorter ones, but what exactly the exponent in the scaling equation is or should be remains unclear. In a seminal paper McMahon (1975) examined the diameters and heights of the largest trees of each species and the pattern of taper in their branches. McMahon showed that both the scaling of the trunk and taper of the branches roughly follow the elastic similarity model: diameter is proportional to length to the power of approximately 3/2. However, the branches of trees do not sag appreciably in the way emphasized by McMahon, and the heights of trees is well below the maximum height they could attain according to Greenhill’s equation; no branch or tree trunk fails under its own weight! The main forces on trees are instead caused by the wind and vary extremely erratically depending on many factors: the size of the tree and its crown shape; the tree’s position, both geographically and relative to other trees; and on the time of year. Moreover, tree trunks also have an important hydraulic role in supplying the leaves with water from the roots. Therefore the results of the many scaling studies that have since been carried out on trees are inconclusive. Indeed there is no real reason to suppose that there should be any one optimal scaling pattern within or between trees (Bertram, 1989). I would recommend that the young biomechanist avoid getting involved in scaling investigations, and the arguments to which they almost inevitably lead, until much later in their scientific careers. Study individual organisms and look at how mechanics influences their biology instead!

CHAPTER 9 .................................................

Structures in Compression

MATERIAL FAILURE IN COMPRESSION We saw in chapter 1 that most materials have similar stiffness in compression as in tension. However, their fracture properties are often very different, due to the nature of the two loads. In compression the bonds between the atoms and molecules are being shortened rather than stretched, and any cracks that are formed tend to close up rather than extend. For these reasons many solid materials, especially brittle ones, have higher compressive strengths than tensile strengths. In fact, when they are compressed, many materials fail because of the shear that is set up by the compressive stress and that is maximum at 45◦ to it. The two pieces on either side of a diagonal line of failure either slide past each other (fig. 9.1a), or a more complex pattern of failure can occur in more brittle materials in which two wedges of material break off laterally (fig. 9.1b). The former type of fracture can be seen in the part of our body most susceptible to being crushed, our lumbar vertebrae, whereas when leg bones are broken in bending, wedges of material often break off on the side that is compressed. STRUCTURAL FAILURE IN COMPRESSION Shear failure is the most common form of failure for blocks and relatively short columns of material, but if a long narrow strut, such as a ruler, is compressed end to end, something quite different happens: it tends to bend away from the forces, bulging outward in what is known as Euler buckling, which was named after the eighteenth-century mathematician who first analyzed the process. What happens is that because no strut is ever exactly straight, the compressive force will always act to one side of its center (fig. 9.2a), setting up a bending moment on the strut. This force will tend to bend it further outward (fig. 9.2b), so increasing the bending moment. If the force is large enough, the process will continue, and the compressive force will overcome the restoring moment provided by the rigidity of the strut, which will bulge further and further outward and then collapse. Two things are worth pointing out about this process. First, the force required is proportional not to the compressive strength of the strut, nor even to its flexural strength, but rather to its flexural rigidity, EI. Structural failure will occur due to the change in shape of the strut long before material failure occurs. Second, the force required falls with the square of the length (fig. 9.2c). This happens because for a given angle of bulge, the length of

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(a)

(b)

Figure 9.1. Patterns of failure of short columns in compression. The structure can fail in pure shear (a), or following shear failure, two wedges of material can break off (b). (Redrawn from Gordon, 1978.)

the moment arm supplied by the compressive force is directly proportional to the strut’s length, whereas the curvature in the strut, and hence the restoring moment it can supply, is inversely proportional to its length. The compressive strength, PE , is therefore given by the formula PE = kπ 2 E I /L 2 ,

(9.1)

where L is the length of the strut. The constant k, and hence the force required to cause Euler buckling, will depend on how the ends of the strut are held (fig. 9.3). In a pin-jointed strut that bows in the middle (fig. 9.3a), k = 1. If the rod is held firmly at its ends (fig. 9.3c), however, a lot more bending has to occur, and the force required is consequently quadrupled. In contrast, a rod held firmly at its base but entirely free to move at its tip requires only a quarter of the force needed to buckle the pin-jointed strut (fig. 9.3d). Note that putting two free-ended struts end to end at their base would form a pin-jointed rod of twice the length, just as we saw in the last chapter that a beam in three-point bending is equivalent to two cantilevers joined end to end. THE BUCKLING OF STRUTS There are several consequences of these mechanical properties of struts. First, to prevent compressive forces from buckling a strut, it is best to make

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149

compression

compression

(a)

(b)

moment arm

(c)

Failure load

Crushing strength

Length Figure 9.2. The cause of Euler buckling in long struts. Slightly noncentric compressive forces will set up bending moments about the strut (a), which cause it to bow outward, further (b) increasing the moment arm of the compressive force, and causing still further bowing. The force required decreases with the square of its length (c).

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(a)

π2

EI L2

(b)

4π2

EI L2

(c)

(d)

π2 4π2

EI 4L2

EI L2

Figure 9.3. The forces required to cause Euler buckling for a uniform beam held in different ways: (a) pin-jointed at both ends, (b) pin-jointed at both ends and translation prevented at the center point, (c) held firmly at both ends, and (d) held firmly at the base but free to move at one end.

it as short and as thick as possible. That is why nails are best made relatively thick compared with their length: so they do not buckle when they are hammered into a piece of wood. Similarly the stingers of most bees and wasps are fairly short and thick. In some cases, though, long thin struts are absolutely necessary. An excellent example is the ovipositor of ichneumon wasps. These wasps lay their eggs in caterpillars and beetle larvae that live deep inside the branches of trees, and to reach them the wasps have to drill down with their long ovipositors through more than 2 cm of wood. The ovipositor must therefore be long and rigid enough to withstand Euler buckling when drilling is just starting, yet thin enough not to require too much force to drill. To reconcile these mutually incompatible pressures, the insects hold their ovipositor between their hind legs when they are drilling, approximately at its midpoint, preventing it bowing outward except above and below this point. This halves its effective length and so quadruples the downward force the insect can apply without the ovipositor buckling (fig. 9.3b).

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151

Figure 9.4. Euler buckling on the radially oriented veins in the hind-wing vane of locusts allows the wing to develop camber as the wing is drawn forward. (Redrawn from Wootton, 1990.)

Insects can also allow Euler buckling to occur for their own benefit. The hind wings of locusts, walking sticks, and praying mantises have a trailing edge fan that is supported by a series of veins that radiate from the wing base (fig. 9.4). When the insect is at rest, the fan is folded up, but in flight the hindwing is drawn forward, tending to straighten the trailing edge. Eventually the taut trailing edge is drawn forward and the longitudinal veins buckle upward, developing a strong camber in the wing membrane and producing what Wootton (1990, 1995) calls the “umbrella effect.” Another consequence of the mechanics of Euler buckling is the optimal cross-sectional shape of struts. The strength of a strut depends only on its flexural rigidity, whereas with a beam loaded in bending, its strength will depend on its breaking strength. In a strut, material positioned well away from the neutral axis is therefore particularly effective at increasing its strength, because it increases the rigidity much more than does material further in. For this reason, the optimal cross-sectional shape for a solid strut is not circular, but triangular, since the material at the apexes of the triangle are further away from its center than the material at the surface of a circle of the same cross-sectional area. Triangular struts occur in nature in the form of the solid stems of sedges (see fig. 10.5; Ennos, 1993a). However, just as in bending, it is far better for struts to be hollow rather than solid. As in conventional bending, the disadvantage of hollow tubes is that they are prone to local buckling, although in axially compressed struts it is more likely to take the form of Brazier buckling (Brazier, 1927) rather than ovalization; the walls themselves undergo a form of Euler buckling (fig. 9.5), the force PB required being given by the equation PB = kπ t 2 E ,

(9.2)

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Figure 9.5. Brazier buckling, showing the crumpling of the outer shell when a thinwalled tube is compressed.

where t is the wall thickness and k a value between 0.5 and 0.8. The shorter a strut and the higher the Euler buckling force it can withstand, therefore, the relatively thicker walled it must be to prevent Brazier buckling. Just like ovalization, Brazier buckling can be resisted by incorporating stringers and bulkheads. Two keratinous structures that employ extremely sophisticated stringers and bulkheads—porcupine quills and hedgehog spines—are both well suited to resist Brazier buckling (Vincent and Owers, 1986) but are used in different ways and hence have subtly different mechanical designs. Porcupines have long, arrow-tipped quills that are used defensively. The quills need to be as stiff as possible to allow the porcupine to back into an attacker and apply enough force for the sharp tip to penetrate the adversary’s skin. A line of weakness then allows the tip to break off. The uniformly short spines of hedgehogs, in contrast, seem to be designed not only for protection but also for absorbing energy. Hedgehogs are good climbers, but they do fall from walls and trees. Landing on their spines could be dangerous since if the spines could resist too great an axial force, they could be driven into the hedgehog’s body, impaling it. To prevent this, the spines are slightly bent so that Euler buckling is promoted and the force required to buckle the spines is reduced; the spines then absorb large amounts of energy as they bend, cushioning the animal’s fall. The base of a spine, meanwhile, like the base of a porcupine quill, inserts into a broad mushroom-like base that reduces the stresses on the hedgehog’s body. BUCKLING WITHIN STRUCTURES We have seen that buckling is a problem for long thin structures, but it can also be a problem in structures or materials that contain long thin structures within them. In biological materials, many structures contain long thin fibers or rods that are prone to Euler buckling and tubes with thin walls that are prone to Brazier buckling.

STRUCTURES IN COMPRESSION

(a)

153

(b)

tension

compression Figure 9.6. Prestressing of cell walls in herbaceous plants (a) in tension, prevents them from being compressed, even when a bending moment is applied to the stem (b).

Prestressing in Herbaceous Plants The most important thin-walled tubes in biology are undoubtedly plant cells, whose walls, as we saw in chapter 7, are protected from buckling by turgor pressure within the cell. Turgor pressure also helps prevent buckling of the epidermis that stiffens young plant stems. And as we have seen, herbaceous plants are also often stiffened around their periphery by the inclusion of reinforcing collenchyma and sclerenchyma fibers. Although these types of cells are thick walled, the fibers themselves are long and narrow and so are predisposed to Euler buckling. The fibers are protected from collapse by being held in pretension, which is provided by the turgor of the thin-walled parenchyma cells inside the plant’s pith. Turgor holds all the cell walls of the plant in tension (fig. 9.6a; Vincent and Jeronimidis, 1991), preventing them from being put into compression even when the stem is bent (fig. 9.6b). The entire plant structure is strikingly reminiscent of that of a prestressed concrete pillar, in which the concrete is held in precompression and the metal rods in tension. Prestressing concrete is usually regarded as a method that protects concrete from being loaded in tension, but equally it could be considered a way of preventing the narrow reinforcing rods from being loaded in compression!

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(a)

(b)

wind

tension compression

Figure 9.7. Prestressing in the trunks of trees puts the outside of the trunk into pretension (a) and the inside into compression, reducing compressive stresses on the leeward side of the trunk when the wind subjects the tree to bending (b).

Prestressing in Trees Wood is another plant material that is composed of large numbers of narrow thin-walled cells that are stuck together. However, in the case of wood, the cells are dead, so there are no cell contents that can pretension the cell walls. As a consequence, wood has only half the yield strength in compression along the grain as in tension and readily forms compression creases when crushed. If a tree is bent by the wind, the leeward side would fail well before the windward side. This is not a problem if the wind loading is all from the same direction, because the crease will merely allow the wood on the leeward side to densify and it will not break. However, it would be disastrous if the load were then reversed, for instance if the tree were swaying in the wind, because the crease would then form the start of a tension crack that could run catastrophically across the trunk. Fortunately for trees, they can overcome this problem using a quite different form of prestressing from that used in herbaceous plants. Trees possess a ring of generative tissue called the vascular cambium, just below their bark, that allows them to grow in diameter by laying down new layers of wood cells toward the interior of the tree. Once laid down the new cells try to shrink longitudinally but are prevented from doing so because they are firmly attached to the layer of cells toward the interior. The outer cells are thus held in pretension and tend to compress the interior cells. The result is a pattern of longitudinal prestress in the trunk, in which the inside is held in compression and the outside in tension (fig. 9.7a; Jeronimidis, 1980). This alters the pattern of longitudinal stress that results when a tree is bent in a storm. On the windward side, the bending moment puts the wood under even higher tensile stresses (fig. 9.7b), but the cells can cope with this.

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155

The advantage for the tree is that on the leeward side the tensile prestress reduces the compressive stresses. As a result the tree can bend almost twice as far in the wind without breaking and resist almost twice the wind load. Shipwrights used the prestressing of tree trunks to their own advantage: they made masts and booms as far as possible from whole tree trunks and boughs, which proved to be resilient and strong. Prestressing has just one disadvantage for the tree. It puts the dead heartwood of the tree into compression, which makes it vulnerable to compressive failure. Trees overcome this problem to a large extent by packing the heartwood cells with gums and resins that inhibit both buckling and rotting. Consequently heartwood is stronger in compression than the outer sapwood. Nevertheless, over time the heartwood of trees can rot, making the cells weaker and more brittle. This leads to the formation of a hollow trunk that, as we saw in the last chapter, is much more vulnerable to splitting. Prestressing also has implications for our use of trees. On the positive side, it does make it easier for us to cut trees down. As we cut through the outer layers with our axes or saws, the tissue will contract, freeing the cutting tool and preventing it getting stuck in the kerf. By the time we have cut right through the trunk, however, some disadvantages emerge. With the bottom of the trunk released from its constraints, it is free to respond to the prestress. The outer trunk contracts longitudinally and the inner trunk expands, and this can cause the trunk to bend outward, splitting along its length and resulting in the formation of the “shakes” seen in the trunks of trees such as poplars and the tropical Koompassia, which make the timber useless. Even worse for us, the splitting can occur very rapidly indeed, releasing large amounts of energy. The trunk of a large Eucalyptus tree can spring outward as much as 1–2 m, killing the lumberjack who is felling it. Even in trees that do not split, sawyers have to pay great attention to the problem of prestress to enable them to cut useful pieces of timber that neither bend nor split.

Reaction Wood Trees also make use of prestressing to control their own shape—by altering the amount of prestress on each side of their branches and trunks, largely by changing the angle at which the cellulose microfibrils are laid down in the S2 layer of the cells. Decreasing the angle at which the fibrils are laid down increases their longitudinal shrinkage and hence the pretension they cause. One function of asymmetric prestress is to keep branches growing straight. If a branch grew by laying down unstressed wood, the bending load on the branch would increase, and it would have to sag downward before the new wood could counteract the effect of its own weight. Over time the tree would start to “weep.” Trees prevent this bending by laying down wood with a slightly lower fiber angle on the top of the branch than on the bottom, prestressing the wood so that it counteracts its own weight, even without sagging.

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(a)

(b) tension wood

compression wood

Figure 9.8. The role of reaction wood in reorienting branches. In conifers (a), compression wood along the bottom of a branch pushes it upward, whereas in angiosperm trees (b), tension wood along the top pulls it upward.

Even greater asymmetries can be employed if the trunk of a tree is displaced and it has to grow upward again, or if the leading shoot dies and a branch has to grow upward to replace it. Conifers lay down “compression wood” along the bottom of the branch (fig. 9.8). The cells differ in two respects from those in normal wood. First, their fibers are oriented at approximately 45◦ to their long axis. Second, the cells are circular, not square. As a consequence they are free to twist, and when the matrix swells, they tend to stretch, not shorten (Burgert et al., 2007). The stretching of the compression wood pushes the branch upward (fig. 9.8a). In contrast, the broad-leaved trees belonging to the flowering-plant group, or Angiosperms, do the same thing by laying down “tension wood” along the top of the branch. This also has a characteristic structure, with an inner, celluloserich G layer in its cell walls (fig. 4.18). Goswami et al. (2008) suggest that it is the swelling of this layer that causes the outer layers of the cell wall to contract longitudinally. The contraction of the tension wood therefore pulls the branch upward (fig. 9.8b). Both techniques work well, but tension wood performs its task more efficiently, because its cell walls are stiffer than those of compression wood. Recent theoretical research (Almeras and Fournier, 2009) suggests that ultimately the narrowness of tree trunks is not constrained by the need to resist bending forces themselves but rather by the ability of prestressing to keep the trunk growing straight upward. If so, this might help explain why conifers are so rare in the windless tropics. Because they have a less efficient form of reaction wood, they have to be thicker to keep their trunks growing upward; such bulk would be a particular disadvantage in an area where trees have little lateral wind forces to resist since it would mean conifers could not grow upward so quickly. There are only two problems with reaction wood. First, the bent trunk or branch forms a hazard beam (see chapter 8), which is prone to splitting. Second, the wood in bent areas has highly asymmetric prestress, so it is prone to warp when cut up, making it useless for timber.

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157

50µm Figure 9.9. The structure of cork. It has a honeycomb structure, with the hexagonal prisms of the hollow cells oriented radially to the trunk, but the radial walls are corrugated, reducing their Euler buckling strength. (Redrawn from Gibson et al., 1981.)

CORK A final disadvantage of wood as a construction material for trees is that it is even weaker in compression across the grain than along it; the cells, which are closely packed tubes held side to side are readily crushed by lateral blows. As we saw in chapter 8, this vulnerability can also lead to branches made of light wood buckling when they bend. Fortunately, trees are protected from lateral blows by their outer covering of bark, which contains a clever system for absorbing shocks when put into compression—cork. Like wood, cork is composed of closely packed, hollow cells, although in the case of cork, the cells are filled with air and are oriented with their long axis pointing radially relative to the trunk (fig. 9.9). Cork cells are, in fact, hexagonal prisms, at first glance not unlike a honeycomb in structure. However, whereas honeycombs are designed to be rigid and have straight longitudinal walls, those of cork cells are designed to be collapsible: they are corrugated, facilitating Euler buckling at a fraction of the force that would be required to buckle straight walls. Even when initially stressed in compression, therefore, cork shows low stiffness, with a Young’s modulus of around 20 MPa. As each layer of cells starts to buckle, moreover, the walls become even more corrugated, and they become even easier to deform. The stress-strain curve of cork therefore flattens out as the layers of cells get squashed like a concertina (fig. 9.10). The cork compresses by more than 60% of its original thickness before the walls jam together, the material densifies, and the stiffness rises again. The result is a material that requires only a low force to flatten but has a very high energy absorption capacity of

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15

Stress (MPa)

10

5

0 0.0

0.2

0.4 0.6 Strain

0.8

1.0

Figure 9.10. The stress-strain behavior of cork in compression. It has a low Young’s modulus, a high compressive strain until the walls eventually densify, and high hysteresis. (Redrawn from Gibson et al., 1981.)

about 1–2 MJ m−3 . Cork also has very low resilience, because the walls are disrupted by their large changes in shape (fig. 9.10). Cork therefore protects the tree efficiently from blows and as a result of its waterproof properties has also proved extremely useful for humans. Cork acts as an excellent shock absorber and seal, making it ideal for car gaskets. Of course cork also makes an excellent bottle stopper, partly because of its low compressive stiffness, but also because it has a low Poisson’s ratio. As the cells are compressed, they collapse into themselves and the cork does not bulge outward. This is why it is so much easier to force a cork into a bottle than to force a rubber stopper, which bulges outward as it is pushed into the neck. Finally, cork has even successfully been used to make flooring that is both soft and antislip, and before the advent of refrigerator magnets, it used to be invaluable in the kitchen, where corkboards were readily pierced by drawing pins bearing messages and postcards.

CHAPTER 10 .................................................

Structures in Torsion

TORSIONAL STRESSES AND STRAINS We saw in chapter 8 that loading a rod or other beam with forces that are oriented at right angles to it, but applied along its axis, puts it into bending. When such forces are applied off-axis, however, they put it into torsion and tend to make the beam twist along its length. When you are twisting a beam, you are putting the outer regions into shear, stretching the outer surface at 45◦ to the long axis in one direction, and compressing it at 45◦ to the long axis in the other (fig. 10.1). The torsional resistance of a beam is therefore due to the shear stiffness, G, of the material of which it is composed. The beam will twist in response to the torsional moment, and as it continues to do so, another effect occurs. The more highly sheared outer layers of the beam tend to be stretched, and when the material is deflected enough, this stretching starts to apply a compressive force to the inner material. That is why twisting a wet cloth is so effective at removing water: the inner regions are compressed so much that the water they hold is squeezed out of them. In many ways, the manner in which the shape of a beam affects its torsional rigidity is strikingly similar to how it affects its flexural rigidity. Material further from the center of area of the beam is much more effective at resisting torsion for the same two reasons: it will be sheared more for a given twist angle; and since it is further from the center, a given shear force will set up a greater restoring moment. Therefore the effectiveness of a ring of material is proportional to the square of its distance from the center of the area. Mathematically, the torsional rigidity of a beam is given by the expression Torsional rigidity = G J ,

(10.1)

where G is the shear modulus of the material and J (or in some books and papers, K ) is the polar second moment of area of the beam, which can be found by summation or integration over the cross section of the expression
rmax r 2 dA, (10.2) J = 0

where r is the distance of each ring from the center of area. For a circular rod, for instance, J = πr 4 /2. Like the second moment of area, the polar second moment of area has units m4 , and like flexural rigidity, the torsional rigidity has units N m2 .

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compression

shear

Figure 10.1. The pattern of stresses set up when a beam is twisted or put into torsion.

A simple formula, then links the torsional deflection of a beam, θ (given in radians), to the torsional moment, M, applied to it: θ = ML/G J ,

(10.3)

where L is the length of the beam.

TORSION TESTS If you can grip the two ends of a beam, it is fairly straightforward to carry out a mechanical test to measure its torsional rigidity and strength. The easiest way nowadays is to use an electronic torsion balance. It grips one end of the sample while the other end is held in another clamp. Either end can be rotated (using a protractor to measure the angle) as the restoring moment produced by the beam is measured by the balance (fig. 10.2a). If you do not have such a balance, or if you are dealing with very flexible structures, results just as good can be obtained by mechanical means. Within elastic limits, metals show Hookean behavior in torsion, the restoring moment they produce being directly proportional to their angular deflection. Therefore the torque balance can be replaced by a metal torsion spring. For relatively rigid beams, such as plant stems or bones, the spring can be helical (Ennos et al., 2000), while for very flexible structures, such as insect legs and wings, a long thin metal wire can be used (Ennos, 1988). If the end of the spring or wire is rotated, a torsion moment is set up and causes a smaller angular deflection of the free end of the beam (fig. 10.2a), which can be measured optically, ideally by attaching the ends to a balanced crossbeam. The relative angular deflections of the metal wire and the beam

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161

torsion

(a)

(b)

wire sample

cross bar

sample

torque meter Figure 10.2. Methods to investigate the torsional behavior of a structure: (a) the torsion balance, and (b) the torsion pendulum.

depends on their relative torsional rigidity; the torsional moment applied to the beam is proportional to the difference between the angular deflection of the two ends of the metal wire and to the torsional rigidity of the spring, R: G J /L = (θ1 − θ2 )R.

(10.4)

The torsional rigidity of the wire can in turn be measured by using it as a torsion pendulum (fig. 10.2b). If the crossbar is set swinging back and forth, the more rigid the wire, the higher the oscillation frequency, f , so the rigidity of the wire can be found using the equation R = 4π 2 f 2 H,

(10.5)

where H is the moment of inertia of the crossbar. If you only require the torsional rigidity of your beam and not its strength, you can simply use the entire apparatus as a torsion pendulum, measuring the rigidity of the beam and the wire and then subtracting the rigidity of the wire. A final alternative is to hang a crossbar directly beneath your beam. You could then twist it using a pair of strings that pull in opposite directions and are operated via a set of pulleys (Vogel, 2003), or use the beam and crossbar as a simple torsion pendulum apparatus. As with so many tests in biomechanics, the main difficulty can be in holding the two ends of your beam without weakening them. Gluing the ends or encasing them in a solid such as epoxy resin is often the best answer to this problem.

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R

J

I J

πR4

0. 5

Ri

Ro

W

π 4 4 (R R ) 2 o i

βDW3

0. 5

D2 12βW2

R1 D

R2

π(R13R23) R12 R22 R12 R22 4R22

Figure 10.3. Polar second moments of area J for regular cross sections. For the rectangular cross section, D is the length of the long side, and the factor β rises from a value of 0.141 when D/W = 1 to 0.333 when D/W = ∞.

THE EFFECT OF CROSS SECTION The polar second moments of area of beams are more difficult to calculate than the flexural second moments of area. The reason is that achieving the most effective torsional resistance requires a complete ring of material; projections of the cross section outside the central circular region are far less effective at resisting torsion than the central region itself. In one way, however, the effect of shape is very similar for torsion as for bending: tubes are far more effective at resisting torsion weight for weight than solid rods, although as in bending, they are vulnerable to collapsing inward. Several values of I for a regular cross section are given in figure 10.3. Also included in this figure are comparisons of their relative resistance against bending and torsion. These are quantified by the ratio of flexural to polar second moment of area, I /J , which gives “twist-to-bend” ratios for solid sections (Vogel, 1992); these rise from 0.5 for a solid cylinder to higher values for other shapes.

DESIGNS THAT RESIST TORSION Most structures in nature, like most of those designed by humans, are constructed to avoid torsion and to limit torsional deflections in their beams. The long bones of most animals, for instance, seldom have projections that could act as long levers to apply dangerous twisting forces; when our limbs are twisted, it is our ligaments that tend to fail first. This is fortunate because our bones are rather susceptible to torsional fractures, failing in tension along a helical path running at approximately 45◦ to their long axis and resulting in a complex fracture that is difficult to set and that heals poorly. Of course skiers do attach long levers—skis—to the ends of their legs that

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163

could easily result in terrible torsional injuries. Fortunately the invention of quick-release bindings makes skiing at least moderately safe. Another good example of natural structures that avoid torsion are the feathers of birds, which have to resist forces imposed on them by the air as they fly. The outer primary feathers, which can be used as independent airfoils, have their shafts positioned toward the leading edges of the feathers, at the quarter-chord position, where lift is centered. The lift therefore acts more or less through the center of the shaft and so does not subject it to appreciable torsion. The hollow section of the feather shaft will in any case enable it to strongly resist any torsional moments that are applied. In contrast to the primary feathers, secondary feathers overlap each other to form a single large wing airfoil. They are therefore more likely to be subjected to approximately even aerodynamic forces across their chord, and their shafts are located near the midpoints of the feathers, so once again the torsional moments will be small. In a few situations, however, torsion is inevitable, the most obvious case being the leading edge bones in the wings of bird and bats (Swartz et al., 1992); aerodynamic forces will always act behind the leading edge, and the resulting lift force produced by the wing will twist it nose down. We have already seen that the leading edge bones of birds are hollow, which certainly helps them to resist torsion, and this is also true for bats. One would also expect the hydroxyapatite crystals in these bones to be oriented obliquely around the long axis of the bone to further strengthen them against torsion, but whether this is the case has not, to my knowledge, been investigated.

DESIGNS THAT FACILITATE TORSION We saw earlier that hollow tubes are better than rods at resisting twisting; however, the integrity of the cross section also has a major effect on torsional rigidity. If a hollow tube is split along its length, for instance, its torsional rigidity will be dramatically reduced, because the split sides are now free to shear past each other (fig. 10.4). The split tube will, in fact, behave more like a flat plate of similar cross section—J will have the value 2π Rt 3 /3—and the split tube will be very easy to twist. If the wall thickness of a tube is a quarter of its radius, for example, splitting the tube will reduce its rigidity by a factor of approximately 32! Structures with longitudinal reinforcing fibers separated by a more flexible matrix will also be much easier to twist than ones with a complete ring of rigid material (Ennos, 1997), because the fibers can readily slide past each other. The relative ability of a structural beam to resist bending and torsion therefore depends not only on its external geometry but also on its internal structure. It is best quantified by another twist-to-bend ratio first discussed by Vogel (1992): Twist to Bend Ratio = E I /G J , in other words, the flexural rigidity divided by the torsional rigidity.

(10.6)

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Figure 10.4. Splitting a tube along its length will greatly reduce its torsional rigidity since the free edges can slide past each other.

Vogel pointed out that the leaf stalks of many species of plants are much easier to twist than to bend, having ratios of EI/GJ of 3–7. Most of these structures have noncircular cross sections, so some of this facility is the result of their geometry and so of a high I /J ratio. However, geometry can account for only a fraction of the effect, because values of I /J for the sorts of shapes Vogel encountered were only about 1.4–1.6. Most of the effect resulted from the low shear modulus of the stalks, which gave a value of E /G of approximately 3, probably because the stalks are strengthened by isolated fibers. The low torsional rigidity has advantages for leaf stalks such as those of aspen: they can support the leaf while still being able to move from side to side and twist in the wind, reducing the thickness of the boundary layer of air and so speeding up photosynthesis (Vogel, 1992). Twisting and bending combined also allow tree leaves to flex downwind (see chapter 11), reducing the drag of the crown. The high EI/GJ value of 7 for the stems of daffodils also allows their flower heads to twist downwind in high winds, protecting them from damage (Vogel, 2007). Even more impressive are the stems of sedges (Ennos, 1993a), with EI/GJ values of 30–80, and the petioles of bananas (Ennos et al., 2000), with values of 60–100. In both of these cases the shape of the beam is far from circular (fig. 10.5a), but the torsion is no doubt once again facilitated much more by the isolation of the longitudinal lignified fibers. In both of these structures, which have a built-in downward curvature, their high torsional flexibility allows them to twist away from the wind, reducing drag and, in the case of the sedge, reducing the risk of self-pollination (fig. 10.5b). The effect of isolating fibers in reducing torsional compliance is not only confined to large-scale structures such as plant stems. It can also be seen

STRUCTURES IN TORSION

(a)

165

aerenchyma

parenchyma lignified material

1 mm

(b) wind

Figure 10.5. The noncircular shape and isolated fibers within the stems of sedges (a, transverse section) allows these downward-curving beams to reconfigure away from the wind by twisting (b). (From Ennos, 1993.)

in composite materials that are reinforced by isolated microfibrils, such as tendon and keratin. Both horn (Kitchener and Vincent, 1987) and human fingernails (Farran et al., 2009) have a very low shear stiffness, particularly when the matrix is softened at high humidity levels; values of E /G can therefore rise to as high as 30. Bone can also have high values of up to 20 (Spatz et al., 1996).

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THE MECHANICS OF SPIRAL SPRINGS Although in general both man-made and natural structures avoid torsional forces, there are some structures that are designed to be loaded largely in torsion: helical springs. When a helical spring is stretched, the wire of which it is made is twisted, allowing it to unwind and lengthen (Spring Research Association, 1974). The longitudinal stiffness of a spring as an entire structure is far lower than the tensile stiffness of all but the finest wires and is given by the equation dF/dl = 4G J /πnD 3 ,

(10.7)

where J is the polar second moment of area of the wire, n is the number of coils in the spring, and D is the diameter of the spring. For a spring made from a homogenous cylindrical wire of diameter, d, the stiffness is therefore given by the equation dF/dl = Gd 4 /8nD 3 .

(10.8)

Helical springs are used by humans largely in suspensions, to reduce shocks. The same is true in nature, where the tendrils of climbing plants, such as white bryony and cucumbers, are coiled into springs (although the two ends have to be coiled in opposite directions (fig. 14.1c)). This pulls the plant upward and provides a suspension, preventing the vine from being damaged when its host plant is moved by the wind. In helical tendrils the optimal design is for the strengthening tissue to be laid down in a ring around the outside of the tendril, to maximize its torsional resistance and hence reduce the compliance of the spring. One would therefore expect tendrils to have low values of EI/GJ. I know of no investigation that has examined the mechanics of such tendrils, but it should be possible to measure EI/GJ because when a helical spring is twisted, the wire of which it is composed is bent (Brown, 1981). The torsional stiffness of a spring is given by the expression dM/dθ = 2E I /nD,

(10.9)

where I is the second moment of area of the wire, n is the number of coils, and D the diameter of the spring. For a homogenous wire of diameter, d, the torsional stiffness is therefore given by the equation dM/dθ = E d 4 /10.2nD.

(10.10)

THE TORSIONAL RIGIDITY OF PLATES We have seen that the torsional modulus, J , of thin plates is very low. However, in a relatively wide plate structure there is another way in which the torsional rigidity can be increased. Twisting a cantilevered plate also involves bending the front and back in opposite directions (fig. 10.6), so if the front and back are folded into beams with high flexural rigidity, they will

STRUCTURES IN TORSION

167

Figure 10.6. The differential bending rigidity of two beams on either side of a structure can give it torsional rigidity because both beams have to be bent to twist it.

provide torsional rigidity due to their differential bending stiffness. The wings of early aircraft were stiffened in torsion in this way by incorporating rigid spars (Gordon, 1978). At first glance you would think that the corrugated wings of insects would be hard to twist because of the differential bending rigidity caused by their corrugations. This rigidity would be a problem because these insects have to reverse both the twist and camber of their wings between the two wing strokes so they can act like propeller blades during both strokes (fig. 10.7a,b). However, the pattern of corrugation in such wings is cleverly arranged to reduce torsional rigidity and to allow coupled control of torsion and camber (Ennos, 1988a). Instead of being separately attached to the thorax, the rear corrugations of the wings of dragonflies and many other insects diverge directly from the corrugated leading-edge spar (fig. 10.7c). As the wing swings down during a stroke, the force of air behind the leading-edge spar causes it to twist, raising the spars behind it, cambering the wing automatically (fig. 10.7c), and forming it into a propeller shape. The wing then reverses both its twist and camber on the reverse stroke, the process being aided by the inertia of the wing (Ennos, 1988b). A similar pattern of rigid supports radiating from a leading-edge spar is seen in the arrangement of the primary feathers of hummingbirds. It is possible that when hovering, hummingbirds could change the shape of their wings, like insects, by controlling the orientation of their primaries. The wings of butterflies show some corrugation, but they are basically broad cambered plates with a convex dorsal surface. They are therefore unable to twist at their base readily. However, they also show useful passive deformations of their wings because of the unique torsional properties of cambered plates. In chapter 8 we saw that cambered plates resist bending more from the concave side than from the convex, but they also resist

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(a)

(b)

(c)

Figure 10.7. In hovering flight, insects reverse both the torsion and camber of the wings between strokes (a,b; redrawn from Ennos, 1988). These deformations are allowed by the high torsional compliance of the corrugated wing and the diverging corrugations (c) that cause the wing to automatically camber.

torsion asymmetrically (Wootton, 1993; Ennos, 1995). Such structures are far easier to twist by forces applied to their convex side than to the concave, because when pushed on the convex side, the cross section of the opposing diagonal flattens out and allows the plate to flex along it (fig. 10.8b). Forces applied to the concave side, in contrast, result in a

STRUCTURES IN TORSION

(a)

169

(b)

Figure 10.8. Asymmetric behavior of a cambered plate in torsion. When pushed from the convex side (a) the camber increases, strongly resisting further torsion. When pushed from the convex side (b), the section flattens out, allowing it to twist by flexing along an oblique line. (Redrawn from Ennos, 1995.)

deepening of the section along the opposing diagonal (fig. 10.8a), preventing such flexion. (This contrasting torsional behavior is readily investigated using builders’ flexible measuring tapes.) Butterflies in flight move their wings forward and downward during the downstroke, and because of the wings’ torsional stiffness, they remain untwisted. As the butterflies flap their wings upward and backward during the upstroke, in contrast, the forces on the convex side cause the wings to twist readily, supinating the outer half of the wing and allowing it to provide useful lift and thrust (Wootton, 1993).

CHAPTER 11 .................................................

Joints and Levers

SUPPORT AND FLEXIBILITY The last few chapters have shown that most biological structures are built to be as strong and rigid as possible for a given investment in structural material. Such construction enables them to economically withstand the forces to which they are subjected without breaking or unduly deforming. However, organisms also need to be able to change shape, either to flex away from forces in their environment, or, in animals especially, to allow them to move. Such changes in shape can be achieved in two contrasting ways. The structure can be compliant throughout, perhaps with some regions being more flexible to localize and control the deformation, or it can be composed of rigid elements interspersed with totally mobile joints. As Vogel (1999) pointed out, most man-made mechanisms incorporate joints, whereas in nature more use is made of controlled flexibility; however, there is overlap since joints also occur in nature and some man-made mechanisms employ flexibility.

PASSIVE MOVEMENT IN PLANTS Plants are almost by definition immobile organisms, yet they often have to be flexible enough to undergo large-scale deformation: trees bend before the wind and water plants flex downstream to produce a more streamlined shape that reduces the forces on the plant’s body. One way of increasing compliance is to use noncircular cross sections. We have already seen in chapter 10 that in certain downwardly curved plant stems or leaf stalks, the low torsional rigidity allows them to twist away from the wind. The U-shaped or vertically flattened petioles of the leaves of trees also allows them to twist and bend away from the wind (Vogel, 1992), enabling the leaf to point away from it. Vogel has also shown, however, that the leaves of many trees are themselves well adapted to curl up into streamlined tubes when placed in strong winds (Vogel, 1989). In lobed leaves, such as those of the tulip tree Liriodendron and of sycamores and maples, the lobes curl upward and wrap around each other (fig. 11.1a), whereas in pinnate leaves, such as those of the honey locust or ash, the leaflets fold up together (fig. 11.1b). In neither case has the structural adaptations been investigated; however, superficial examination suggests that the stalks of the leaflets in pinnate leaves have a channel-like form that allows them to twist and bend

JOINTS AND LEVERS

(a)

171

(b)

5 ms–1

15 ms–1

20 ms–1

Figure 11.1. Reconfiguration in the wind of the leaves of (a) the lobed leaf of the tulip tree Liriodendron and (b) the pinnate leaf of the honey locust. (Redrawn after Vogel, 1989.)

upward, whereas the lateral veins of lobed leaves seem to readily hinge or bend upward near their base. This is clearly a system that would repay further investigation. An alternative way of increasing compliance in the long structures of plants is to reduce their diameter, so that even though they are circular in cross section, they can readily bend away from the wind. This design is evident in the branches of trees, whose wood is denser than that of the trunk (King and Loucks, 1978). Because its woody material is located closer to its neutral axis, a branch of dense wood will bend more readily than an equivalent mass of light wood, so the dense branch would appear to be less efficient at holding up leaves. However, such a branch can bend much further before it breaks, enabling the crown of the tree to reconfigure better in the wind while the rigid trunk holds the branches up to the light (Bertram, 1989). It seems to be no accident that high-density wood is more common in long-lived trees and those living in more exposed areas. However, denser wood also seems to be better able to maintain water conduction in drought and also to resist rotting, so it is hard to say how important each factor is in the choice of wood density.

ACTIVE MOVEMENT IN PLANTS Many plants also make their long organs compliant enough to be bent via their own actions by concentrating their rigid material into a narrow hinge region. The leaves of many peas and beans are moved up and down in a diurnal rhythm and flexed toward the sun by means of the pulvinus, a short region at the distal end of the leaf stalk. In the pulvinus the lignified material

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(a)

(b)

Figure 11.2. Capture movements of the Venus flytrap Dionaea. When set (a), the trap has its bottom hinge curved downward and the walls are curved outward and downward. The hinge is flexed upward when the trap is sprung, causing the walls to move to their other bistable state (b), with the walls curved inward and upward, trapping the fly.

is localized to a narrow central region (Haberlandt, 1914). The leaf is then moved by unilaterally weakening the cell walls in the surrounding tissue; the properties of the cell walls are changed by altering the concentrations of potassium and chloride ions (Satter, 1981). The turgor within them therefore expands cells on one side of the pulvinus, bending the leaf toward the other side. A similar motor drives the much faster and more dramatic movements of the leaves of the sensitive plant Mimosa and the Venus flytrap Dionaea (Forterre et al., 2005). In the Venus flytrap, the lower hinge, which joins the two plates of the trap, is straightened by loosening of the cell walls, which allows the tissue beneath the lignified rigid element to expand. The ingenious aspect in the design of the Venus flytrap is the magnification of this movement by the curvature of the walls. These have a dome shape, curving both outward and downward when the trap is set (fig. 11.2a). When the hinge is straightened, the walls initially resist inward movement, but only up to a point, when the curvature suddenly flips over from one bistable state to the other, like the popping of the lid of a tin; the walls rapidly pivot and curve inward (fig. 11.2b) to bring the spikes together and capture the unfortunate insect.

HINGES IN ANIMALS Some animals use similar techniques as plants to make hinges. We have already seen in chapter 10, for instance, how cambered butterfly wings are twisted and flexed during the upstroke. Many insect wings also contain more specific hinges that allow transverse flexion and cambering. A number of flies’ wings have cross sections of an inverted U shape in the main veins at their base, allowing the wing to flex downward at the end of the downstroke, before being flicked rapidly upward at the start of the upstroke (Ennos, 1989a). Deformation is often further localized by incorporating flexion lines

JOINTS AND LEVERS

173

(b) (a)

cartilage

membrane

synovial fluid

synovial membrane

Figure 11.3. Joints in insects (a) are usually made of two tiny peg-and-socket joints on either side of a limb (a). In contrast the typical synovial joints of vertebrates are much larger. They have weak but lubricated cartilage, and the expanded hollow ends of the bone are supported by trabeculae. (Redrawn from Currey, 1970.)

of untanned tissue across the wing (Ennos, 1989a). The cross veins of many insect wings are also converted into flexible hinges by being reinforced only by isolated rings of tanned tissue (Wootton, 1981) that are strikingly like the design of the hoses of vacuum cleaners. Similar arrangements are used to make the abdomens of insects mobile, with soft, thin arthrodial membrane joining the thicker sclerotized plates.

Joints in Arthropods The most impressive joints in insects, however, are the articulating hinges in the legs of arthropods. Most of these are composed of two peg-and-socket joints located on either side of the limb (fig. 11.3a); the tiny articulating areas are often hardened by inclusion of metal salts. To some extent it is the resulting smoothness of the surfaces that reduces friction in these joints, making the legs free to move. However, the small size of the pegs and sockets is even more important: any frictional forces that occur will act so close to the center of rotation that they will hardly impede movement at all. The only problem with the peg-and-socket arrangement is that it allows movement only in one plane. Free movement in two planes can be achieved, as in our hips and shoulders, by a ball-and-socket joint. Many insects have such joints between the thorax and coax at the top of their legs, but they can also achieve free movement by placing two hinge joints at right angles to each other

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close together at the base of the leg. This arrangement closely parallels the structure of the universal joint which was invented in the mid-seventeenth century by Robert Hooke. It may be no coincidence that Hooke was also a renowned early microscopist and would have been among the first people to notice the arrangement of the joints in insects such as the flea that he illustrated in his treatise Micrographia (Hooke, 1665).

Joints in Vertebrates The joints of vertebrates are quite different from those of arthropods, primarily because of the difference in the mechanical properties of the articulating material: cartilage is far weaker than either cuticle or bone. As a result, the abutting surfaces in vertebrate joints must be much larger, a requirement that has two main consequences. First, the long bones of vertebrates have to expand at their ends (fig. 11.3b). The second consequence of the large articulating surface is that vertebrate joints have to be very well lubricated to ease movement and to reduce wear on the soft cartilage. The lubrication of vertebrate joints is certainly improved by the synovial fluid, which is held within the joint cavity, but exactly how it works is still under debate. The concept of weeping lubrication, as water is forced out of the cartilage between two touching surfaces, has been suggested, as has the role of electrostatic repulsion between the two surfaces. Whatever the mechanism, the coefficient of friction of vertebrate joints, µ, the ratio of friction force to the normal force applied (see chapter 12) is low: between 0.01 and 0.1. Vertebrate joints, whether hinges or ball and socket joints, involve two surfaces that can be easily separated from each other; they therefore have to be held together by collagen-rich ligaments. These are of necessity joined to the bones on either side of the joint, well away from the center of movement. Therefore they greatly affect the freedom of movement of the joint; how they limit the rotation depends on their precise positioning (Alexander and Bennett, 1987). In a simple hinge joint, for instance, only ligaments that are inserted at the center of rotation of the joint will hold the joint together firmly while allowing free movement (fig. 11.4a). If the ligaments are inserted instead on one side or other, or on the near side of the center of rotation, the ligament will usually be limp, only becoming taut and limiting movement in one (fig. 11.4b) or both (fig. 11.4c) directions. Even more intriguingly, if the ligament attaches at the far end of the center of rotation (fig. 11.4d), it will act as a bistable, like a light switch—taut at the center of movement and free on either side. Such a bistable operation seems to act in the pasterns of horses, enabling the hoof to be flicked rapidly from one orientation to the other and giving the horse a precise “action.” Perhaps the best-known joint ligaments in the human body, because of their frequent mention in sports bulletins, are the cruciate ligaments in our knees. These ligaments hold our knees together and are located at the center of our knee joints between the inner and outer cartilages (fig. 11.5). The

JOINTS AND LEVERS

175

(a)

(c)

(b)

(d)

Figure 11.4. Different arrangements of ligaments that limit and control the rotation of vertebrate joints. Attachment at the center of rotation (a) allows free movement. Ligaments attached to the sides of the center of rotation (b) limit movement in one direction, and attachments on the near side (c) limit movement in both directions. Attachment at the far side (d) produces a bistable joint (Redrawn from Alexander and Bennett, 1987.)

crossed ligaments act with the shape of the knee joint, which has a nearly flat lower surface and a curved upper surface, to control its movements; they force the top and bottom surfaces to roll rather than slide over each other. Our knee joints are not, therefore, simple hinge joints but rolling joints. This prevents them suffering from sliding wear, which would be particularly severe in a large habitual biped such as us. The only problem with these ligaments is their vulnerability to impacts from the side during contact sports such as football and rugby; such impacts stretch and break the ligaments, making the knee joint unstable. Fortunately sportspeople with this injury can nowadays be helped by operations that transplant ligament tissue from elsewhere in the body or replace it by artificial ligaments. In contrast to most joint ligaments, the ligament in the arch of our foot has a very different function. It acts as a spring, limiting the flattening of the arch of the foot and storing energy (Ker et al., 1987).

MOVING JOINTS Moving a joint of an animal ultimately relies on the force and power produced by muscles, but the transmission systems in animals, like those used in human technology, can be very diverse. It has been suggested (Ellis, 1944; Parry and Brown, 1959) that spiders extend their legs using a hydraulic

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(a)

(b)

(c)

femur

tibia

Figure 11.5. Action of the human cruciate ligaments. They control the rotation of the joint, forcing the surfaces to roll rather than slide over each other. (Redrawn from Barnett et al., 1961.) (a)

(b) blood pressure membrane

Figure 11.6. Hydraulic pressure (arrow) straightens the legs of spiders.

system like that used in dump trucks. Blood is pressurized by muscles that compress the cephalothorax, and it can be let into the legs using a complex system of valves. Because the joints of spiders’ legs are on the outside (fig. 11.6), the rising blood pressure causes the joint to extend and the leg to straighten. The legs can then be flexed using a conventional muscle system. The system seems to work fairly well; jumping spiders, for example, can jump several centimeters onto their prey. However, this system is not seen in other arthropods, perhaps because the valve system is too complex or because the high blood pressures required interfere with the actions of other organs. Certainly, spiders seem to limit locomotion to bursts of a few seconds with long rests in between. The most common transmission system in animals, especially vertebrates, is of course mechanical: the muscles attach to bones on either side of a joint, either directly or via tendons. The muscle has to act to one side of the joint to move it, and the joints of vertebrates often have projections to increase that distance. Depending on the function of the limbs, the geometry

JOINTS AND LEVERS

(a)

177

(b)

Figure 11.7. Contrasting mechanical advantages in the forelegs of (a) a horse and (b) an armadillo. Compared with the length of the leg, the muscles (which act along the dotted line) have a far larger moment arm around the shoulder in the armadillo. (Redrawn from Currey, 1970.)

can be altered to change the mechanical advantage of the joint system. To obtain high-velocity motion, the muscle should act close to the center of the joint, whereas to produce large forces, the muscles should act further away. In formal terms, the velocity mechanical advantage MAv of a muscle system is given by the equation MAv = L/ h,

(11.1)

where L is the distance to the end of the limb and h is the perpendicular distance of the muscle from the joint. The force advantage MAf is simply the inverse of this: MAf = h/L .

(11.2)

A comparison of the skeletons of running and digging animals is indicative of the way in which the system can be adapted to the mode of life. A horse has long legs, a narrow scapula, and a short elbow (fig. 11.7a), which will result in a high velocity advantage for its leg muscles, whereas the digging armadillo has short legs, an expanded scapula, and a long elbow (fig. 11.7b), which will give a much higher force advantage. Unfortunately, in many cases there is a problem with the properties of muscles and the geometry of the skeleton. Vertebrate muscles produce their maximum work when they contract by 25% of their length (fig. 11.8b). In our legs, however, our knee bones do not stick out far enough for our legstraightening muscle, the quadriceps, to contract by 25% of its length. One solution to overcome this mismatch would be to have much knobbier knees,

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(a)

(b)

(c)

(d)

Figure 11.8. Anatomy and operation of (a, b) parallel-fibered and (c, d) pennate muscles. When the fibers contract, parallel-fibered muscles (b) shorten substantially and bulge outward, producing relatively low forces; pennate muscles (d) shorten less but produce higher forces and do not bulge.

but this would be inconvenient. Another would be to have a very short, fat quadriceps joined to the knee by a long tendon, but this arrangement would result in awkward bulging thighs and would demand an extra investment in the tendon. The solution that has been adopted in such cases is a pennate muscle. These muscles are composed of sets of short muscle fibers oriented at an angle to the long axis of the muscle. In double pennate muscles they are joined to a central tendon sheath, like the barbs on the quill of a feather pen (fig. 11.8c), hence the name. Because the individual muscle fibers are so short, even when they contract by the full 25% of their length, the whole muscle will contract by far less (fig. 11.8d), but their contraction will produce a much greater force than that produced by an equivalent conventional muscle. For the conventional muscle (fig. 11.8b), the force produced, F , is given by the expression F = σ A,

(11.3)

where σ is the stress produced by the muscle, and A is its cross-sectional area. For a muscle of volume V and length L that equation is equivalent to

JOINTS AND LEVERS

179

the expression F = σ V/L .

(11.4)

The length change, dL meanwhile is given by the expression dL = εL ,

(11.5)

where ε is the shortening strain. For the pennate muscle, in contrast, with much shorter fibers of length λ acting at an angle α to the long axis of the muscle, the force transmitted to the tendon sheath is given by the expression F = σ V/λ

(11.6)

and the force parallel to the sheet is F = σ V cos α/λ.

(11.7)

The force is therefore multiplied by the factor L cos α/λ. However, the contraction distance (fig. 11.8d) is reduced by the same factor and is given by the expression dL = ελ cos α.

(11.8)

Pennate muscles are fairly common in vertebrates; for instance, both our quadriceps and our gastrocnemius muscles in our legs are pennate. However, pennate muscles are even more common in arthropods, for two reasons. First, arthropod muscles are located inside the exoskeleton, and so both their diameter and their moment arm about the hinge are extremely limited. Increasing the force advantage by using pennate muscles is therefore essential and is most clearly seen in the hind legs of locusts and other grasshoppers, where the pinnate pattern of the muscles is visible in the patterning of the hind tibia. A second advantage of pennate muscles in arthropods is that unlike conventional muscles they do not bulge as they shorten, so the skeleton can be totally packed with muscle. This can certainly be seen in crab claws: the pennate muscle fills the claw, attaching on either side of a flat apodeme sheet, and the muscle can contract to give the creature a powerful nip.

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CHAPTER 12 .................................................

Attachments

HOLDING ON We have now seen how biological structures are arranged to maximize their resistance to the forces to which they are subjected in their environment and also how they incorporate compliant regions or joints to allow deformation and movement. Another equally vital aspect of the mechanics of organisms has until recently, however, been relatively less well studied: their ability to form attachments. Organisms must be able to make temporary or permanent attachments, not only between different organs in their own bodies but also to other objects or to the substrate around them, whether it is soft, particulate, or hard. As we shall learn, these requirements have been extremely important in shaping the bodies of both plants and animals.

HOOKING ON Perhaps the easiest things to attach to are objects that have an obvious shape: particularly small or highly curved objects that can be readily gripped or ones with many hairs or asperities into which things can get entangled. Hooks and anchors are ideally suited to grip these sorts of structures. The best examples are seen in the hooked seeds of plants such as wood avens (Geum urbanum), cleavers (Gallium aparine), and cocklebur (Xanthium strumarium), whose numerous hooks readily become entangled in the hair of passing animals or human clothing (Nachtigall, 1974). In a textbook case of biomimetics, copying the action of these seeds gave rise to the invention of Velcro (Vogel, 1999). The eggs of tardigrades are in a similar manner covered in hairs that end in anchors which help to lodge them in the moss in which they are laid. The mechanics of such “probabilistic fasteners” have been investigated by Gorb and Popov (2002) and Williams et al. (2007). An alternative way of attaching to a hairy substrate is to use a comblike arrangement: the hairs become gripped between the teeth of the comb. This technique is found in fleas, which have a spacing between the teeth on their leg that is directly proportional (if rather larger) than the diameter of the host species’ hairs (Nachtigall, 1974). This allows the hairs to fit into, and become jammed in, the comb when the host scratches itself in its attempts to remove the flea. Making interlocking attachments is even easier between two parts of the same organism (Nachtigall, 1974). The two valves of radiolarians, for instance, are neatly interlocked by a series of hook-and-eye connectors.

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shaft

barb

distal barbule

proximal barbule

Figure 12.1. The attachment mechanism between the barbs of feather vanes. The barbs and barbules both diverge at low angles, so pushing the barbs distally loosens the barbule hooks and releasing them will let them reattach.

Many other interlocking organs of animals have analogues in human technology. The wings of some species of cicada are held at rest using snap fasteners, in which a projection on the wing snaps snugly into a small hole in the thorax. Structures found in the skeletons of echinoderms and in the anogenital plates of spider mites are strikingly similar to zippers. In the vanes of feathers, hooked distal barbules grip the grooves within the proximal barbules of the next barb, holding the vane together. The sophistication in this arrangement, though, lies in the orientations of the barb and barbules, which allow the birds to readily repair a broken vane (Ennos et al., 1995); both the barbs and barbules diverge at a low angle from their attachments, so that when the bird pushes the barbs distally with its beak, the hooked distal barbules are moved into the grooves of the distal barbules (fig. 12.1) and grip onto them when released. Another possibility is to tie a structure to another stronger one. The tendrils of plants wrap themselves around the stems and branches of selfsupporting plants in this way, while “knotting” is used by many birds— weaver birds use their beaks to tie their nests to branches using reef knots, while the pendulous tit ties an even more complex knot (Nachtigall, 1974).

ATTACHMENTS TO SOFT SUBSTRATES Attaching to soft substrates, such as flesh, is more difficult than attaching to solid structures because the substrates deform when gripped, making firm

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mechanical attachment impossible. Instead, the best way to attach seems to be to penetrate the soft tissue with a point and then to grip it using curved hooks. The ideal design of such an attachment mechanism resembles the head of an arrow or harpoon, and such arrangements are common in nature, for instance, in the heads of tapeworms and acanthocephalan worms and in the stingers of bees (Nachtigall, 1974). A more temporary attachment is achieved by the arrowheaded tongues of insectivorous birds, which are ideal for spearing their food, such as soft-bodied caterpillars; and some of the defensive hairs of plants are also shaped like harpoons. Of course these attachments often appear to be just too secure; honeybees are notoriously unable to pull out their stingers which, fatally for the bee, break off their body and stick in their victims. Temporary attachment is better achieved using curved hooks whose orientation can be controlled. Some parasitic worms attach to their hosts with a rosette of such hooks, but the archetypal curved hooks are the claws found in the feet of animals as different as arthropods, birds of prey, carnivorous dinosaurs, and cats. The action of claws is so intuitively obvious that it has until recently received little research effort, although Dai et al. (2002) have examined how the claws of beetles utilize friction to grip rough surfaces. Research has been stimulated by the suggestion that the huge central claw of the small predatory dinosaur Deinonychus was used to disembowel its prey. Tests in which a model dinosaur leg with a reconstructed claw was kicked into a pig’s carcass showed that the supposed inner cutting edge of the claw was far too blunt to tear through flesh (Manning et al., 2006). Instead, the tip of the claw punctured the skin and the curvature of the claw caused it to dig into the flesh. The claws, like those of other carnivores, must have acted more like crampons, holding onto the prey while the razor sharp teeth did the cutting (see chapter 14).

ATTACHMENTS TO PARTICULATE SUBSTRATES Failure of the substrate is a particular problem for organisms, such as land plants, that need to attach to particulate substrates such as soil, sand, or silt. These substrates are far too weak in tension to allow plants merely to stick themselves to the surface, like the holdfasts of seaweeds do on rocky shores; the top surface of particles would simply separate from those beneath them. Instead, plants have developed roots that, like worms, burrow down into the substrate and provide the plant both with the anchorage it requires and a supply of water and nutrients.

Resisting Pullout The situation of a worm or single root being pulled out of the soil is, in fact, analogous to a fiber being pulled out of the matrix of a composite material (chapter 4). Tension will be transferred from the root to the soil

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(a)

(b) tension

shear

tension

shear

air

soil

Figure 12.2. Shear and tensile stresses around the surface of a root that is being pulled out of the soil. In unstiffened flexible roots (a), shear stress is concentrated at the top of the root at the soil surface, whereas in woody roots (b), it is concentrated at the tip, deep in the soil.

via shear, but exactly what happens will depend on the relative material properties of the root and soil (Ennos, 1989b, 2000). If the root is less stiff than the soil in which it is embedded, it will tend to stretch more than the soil surrounding it and the shear stress will be concentrated around the top of the root (fig. 12.2a); if the root is stiffer than the soil, its tension will be transferred all the way down its length to its tip, where shear will be concentrated (fig. 12.2b). In fact, nonwoody roots are usually less stiff than soil; they behave more or less elastically, with a Young’s modulus of about 10 MPa, and have a breaking stress of around 1 MPa and breaking strain of around 10%. In contrast, agricultural soils exhibit rather curious elastic-plastic behavior in shear. They have a high initial stiffness, in the region of 100–1000 MPa, but yield at stresses of only 1–100 kPa. They do not break, though, but continue to resist shear stress due to friction even at high strains. As a consequence, when nonwoody roots are pulled from the soil the events are quite complex. Because the roots are less stiff than the soil, shear stress will be concentrated around the top of the root, and the soil (or the bond between the root and the soil) will yield first at this point and its shear resistance (or friction between root and soil) will transfer tension in the root into the soil. The greater the force applied, the further down into the soil will the root be stretched and the soil yield (Ennos, 1990, 2000).

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(a)

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(b)

(c)

(d)

Figure 12.3. Anchorage systems of herbaceous plants. Climbing and procumbent plants are anchored against uprooting by a fibrous root system (a). Some small herbs are anchored against overturning by rigid elements in the shoot system, such as a rosette of lateral branches (b) or multiple stems (c). More commonly anchorage is provided by a tap root (d).

The system will eventually fail in one of two ways. Short roots will pull out of the soil, but longer roots will break before the soil strength around the root is fully mobilized. Because of the relative material properties of roots and soils, a typical unstrengthened root will break even if it is very short, a matter of several millimeters, and at a force of at most 0.1–0.3 N. This result has important implications. A plant cannot increase its anchorage merely by lengthening its roots; they must also be strengthened, and doing so will represent a real energetic cost of anchorage for the plant. The cost can be minimized in two ways: first, by strengthening the root only toward its base, where it is under most tension; second, by having many narrow roots rather than a single thick one, because these will have a greater surface area and transfer tension into the soil nearer the surface. Plants that merely have to resist being pulled out of the ground as opposed to supporting themselves—climbers and short procumbent plants—therefore tend to have fibrous root systems with roots whose bases are strengthened by thick lignified steles (fig. 12.3a). Even so, the vast majority of the root length in such plants—the flexible distal roots—could never be stressed by forces from above and have no role in anchorage; only the basal parts of the root system anchor plants. For this reason it is unlikely that root hairs could have an important role in anchorage except in young seedlings. Indeed, studies by Bailey and others (2002) showed that mutant plants of the weed Arabidopsis that have no root hairs had just as great resistance to pullout as normal plants. Root hairs probably have the even more important role of enabling the roots to burrow through the soil; they can anchor the root just behind its tip, helping it to push through the soil. Much research has already been carried out on the resistance of soil to root growth (Bengough and Mullins, 1990), but more research looking at the comparative ability of normal plants and root-hair mutants to grow through soil should help verify the role of root hairs.

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Resisting Lateral Forces Although fibrous root systems are good at resisting uprooting by vertical forces, tall, self-supporting plants more often have to resist falling over under their own weight or being pushed sideways by the wind. Such forces will cause the root system to rotate in the soil, and a simple fibrous system would be useless at preventing this; each root would just bend. To resist falling over, a process known to botanists as lodging or windthrow, a more energetically expensive anchorage system is required. It must contain at least one rigid element at the base of the stem to act as a lever (Ennos, 2000). One method used by annual herbs, such as Arabidopsis, is a rosette of side shoots for the plant to lean on (fig. 12.3b.) or the presence of several stems, each of which runs horizontally outward at its base before curving upward (fig. 12.3c). However, most plants anchor themselves via rigid roots, and many small plants have tap roots for this purpose (fig. 12.3d; Goodman et al., 2001). These can anchor the plant directly like the point of a stake, because lateral movement is resisted by the compressive strength of the soil, and they can also act as insertions for lateral roots that act like guy ropes. Taproot systems work well for small plants, such as herbs and young trees, but as plants get bigger, their stems tend to get relatively thicker, and a single thick tap root would start to become too long and too energetically expensive (Ennos, 1993b). Instead, large trees usually develop anchorage systems dominated by thick lateral roots and sinker roots. When these trees are blown over by the wind, the plant rotates about the leeward side, and a ball or plate of roots and soil is levered out of the ground (fig. 12.4a). Most of the anchorage resistance is supplied by the sinker roots, with smaller contributions from the bending resistance of the lateral roots at the hinge and the weight of the root-soil plate (Crook and Ennos, 1996). Of course the roots of trees are composed of wood, which, though rather more flexible than stem wood, has a stiffness of 1–2 GPa, similar to or greater than that of soil. Such roots will transmit forces deeper into the soil than the roots of herbs, and they are often anchored toward their ends by rosettes of short roots (Mickovski et al., 2007). The relative importance of different parts of the root system depends on their precise shape, which in turn depends on soil conditions. Taproots tend to be largest in trees growing in sandy soils, in which deep rooting is likely to improve anchorage the most compared with other soils. At the other extreme, in tropical soils lateral roots are particularly shallow, since nutrients are only found in a narrow layer of leaf litter. Many tropical trees anchor themselves with widely spaced sinker roots, and the narrow lateral roots are linked to the trunk with spectacular root buttresses (fig. 13.3d) these act essentially as angle brackets, transmitting forces smoothly from the trunk down into the sinkers (Crook et al., 1997). The importance of the sinker roots for trees can be seen by considering the case of Sitka spruce (Picea sitchensis) trees, which in the United Kingdom tend to be grown on waterlogged peat soils. The waterlogging kills the sinker roots, so these trees have to rely on the plate of lateral roots alone for anchorage (Coutts, 1986).

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(a)

(b)

Figure 12.4. Anchorage systems of trees (a) and cereals (b). Rotation in trees occurs around a leeward hinge and the root-soil plate is pulled out of the soil. In cereals the hinge is on the windward side and the root–soil cone is pushed into the soil.

Despite making an investment into their root systems of up to 50% of their dry mass, they are extremely vulnerable to windthrow and must be harvested well before they are mature. Monocots, which include grasses and palms, do not have secondary thickening and so cannot produce a thick taproot to anchor themselves. Instead, these plants are anchored by thick adventitious roots that grow out of the stem later in the life of the plant. The roots may be either underground, as in the coronal roots of wheat, or emerge above the soil, as in the prop roots of maize. In either case the roots act in both tension and compression, like the feet of a tripod; failure in these narrower root systems occurs around a windward hinge (fig. 12.4b), the inverted cone of roots being pushed into the soil (Crook and Ennos, 1993), which fails in compression rather than tension.

ATTACHMENTS TO HARD, FLAT SURFACES In many ways, attaching to a hard, flat surface seems more difficult than gripping mechanically to an object or anchoring into a soft or particulate substrate. The attachment must withstand three different forces with no obvious mechanical way of doing so (fig. 12.5). It must withstand shear forces that would pull it off sideways; it must withstand tensile forces that would pull it off vertically; and it must withstand peel forces that would

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(a)

(b)

(c)

stress concentration glue

Figure 12.5. Attachments on hard flat surfaces must withstand three different forces: (a) shear, (b) tension, and (c) peel.

Figure 12.6. The optimal tapered shape of an attachment organ to minimize the chances of peel.

pry it off gradually. Of these the most worrying is peel, because stresses will become concentrated around the edge of the attachment, and failure will therefore occur at a much lower force. This is why adhesive tape or plasters hold objects together so well yet can be readily peeled off. If a firm attachment is needed, it is best to try and avoid peel forces by locating tensile and shear forces toward the center of the attachment zone. A good way of achieving this is for the attachment organ to be thickest at its middle and to taper toward the edges (fig. 12.6), a shape that is seen in such structures as the holdfasts of seaweeds and byssus pads of mussels.

Glued Joints To make a strong permanent attachment, the best technique appears to be to glue the two surfaces together. A glued joint has three components (fig. 12.7): the two surfaces and the intermediate layer of glue. To stick the surfaces together, the glue must be introduced as a liquid and must

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top surface

adhesion cohesion

glue

bottom surface

Figure 12.7. The components of a typical glued attachment. Failure must be resisted by the adhesion of the glue to the two surfaces and by the cohesion of the glue itself.

be capable of completely wetting (or with some glues, dissolving) both of the surfaces. It must then solidify, after which the strength of the joint will depend on the adhesion of the glue to each surface and on the strength, or cohesion, of the glue itself. We have already seen in chapter 3 that the mucus of mollusks such as limpets can act as glue when the mucus molecules are cross-linked by adhesive proteins (Smith, 2002). Mucilage is also released by the root hairs of plants (Ennos, 1989b), sticking them to the soil particles that surround them and improving the anchorage of this root region. The two best-studied biological glues are produced by mollusks and arthropods. The byssus threads of mussels, whose mechanical behavior was examined in chapter 6, are glued to the rocky substrate via byssus pads, which are about 3 mm in diameter and highly tapered to avoid peel. The glue is composed of several proteins, one of which (MfP1) acts as a coating and another of which (MfP3) is the adhesive itself (Lin et al., 2007). The proteins appear to be cross-linked to form a solid attachment. Even when they act through a tapered attachment organ, however, glued joints are rarely as strong as the cohesion of their glue, because differences in the stiffness of different parts of the joint almost always result in stress concentrations that weaken it. For the byssus pads, the attachment strength is approximately 0.1 MPa, well below the strength of the threads themselves, which explains why the attachment pads have to be so large. Barnacles, which despite looking superficially like limpets are actually highly modified crustaceans, also attach themselves to rocks using glue; in their case it is a cement that resembles the proteins found in arthropod cuticle (Walker and Yule, 1987). Barnacles themselves are conical in shape and rigid, so they are largely immune to peel forces, and the protein glue is strong, setting within two hours of its production. The setting process resembles tanning in cuticle, involving the injection of quinones and probably dehydration of the protein. The result is a bond that has a breaking stress of around 1 MPa.

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high pressure

low pressure

seal

Figure 12.8. Operation of a sucker. After attachment and sealing of the outer edge, the sucker is pulled back (arrow) to reduce the pressure beneath the sucker.

Suckers For temporary attachments glues are clearly inappropriate, and one of the commonest temporary attachment mechanisms is the use of suckers. To attach in this way, the edges of the sucker are placed flat onto the surface, usually with a seal of mucilage or other liquid, and the center of the sucker is then raised to reduce the pressure beneath it (fig. 12.8). The attachment force is provided by atmospheric pressure acting on the outer surface of the sucker. The scheme works well, but the attachment force is limited at best to atmospheric pressure, which is 0.1 MPa at sea level, although it increases beneath the water surface at a rate of 0.01 MPa m−1 . The most economical method of producing suction over a long period of time is to use elastic recoil of the sucker to raise the center, as in the toy suckers used in children’s arrows. This technique in animals therefore uses no muscular energy and is the method favored by male diving beetles, which use suckers on the inside of their front legs to hold onto females as they mate. The only problem with using elastic recoil is that it makes the suckers very difficult to detach. Male water beetles either slide their suckers off the edge of the female or waggle them around frantically to remove them (Nachtigall, 1974). Better control of the attachment, although at a much higher metabolic cost, is achieved by using muscles to raise the center of the sucker, a method found in animals as diverse as midge larvae, leeches, certain mollusks (called chitons), and octopus (Nachtigall, 1974). Octopuses are masters of sucker attachment. They seem to be able to use the suction cups on their tentacles both actively and passively (Kier and Smith, 2002), and there is some evidence that they can generate negative pressures beneath their suckers (Smith, 1991, 1996); in such cases the water acts like a solid in the confined space beneath the sucker, as it does in the xylem of trees, and is held in tension.

Liquid Film In the air, it is not necessary to use suction to join two wet surfaces; a thin liquid film will effectively glue them together by itself (fig. 12.9). Wet microscope slides, for example, can be very hard to separate in air, but they

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(a)

193

(b)

Figure 12.9. The mechanisms involved in wet adhesion. Surface tension forces (a) are proportional to the perimeter length, Laplace pressure forces are proportional to contact area, and Stefan adhesion (b) is proportional to shear rate (note the different lengths of the arrows).

will fall apart easily when dropped into water. Two forces are involved in this wet adhesion: capillary force and Stefan adhesion (Federle et al., 2002). The capillary force in turn has two components: Laplace pressure and surface tension. The force of surface tension is proportional to the length of the perimeter (fig. 12.9a). In contrast, Laplace pressure (which is also caused by the surface tension of the water) increases with the contact area and falls with the separation of the plates. Stefan adhesion is a result of the viscosity of the water. If the two plates are forced apart, the meniscus of the fluid will move inward (fig. 12.9b), so the water molecules will shear past each other. Stefan adhesion, is therefore proportional to the viscosity of the fluid and to the velocity of separation and inversely proportional to the separation of the plates. Theoretically, wet adhesion could resist almost infinite stresses, but in practice it is limited to approximately 1 MPa by the cohesive strength of water. Perhaps the best studied of the animals that use a liquid film for attachment are tree frogs, which adhere to smooth leaves using soft expanded toe pads (Hanna and Barnes, 1991). These pads secrete a mucus that keeps the toes moist and allows them to produce an adhesive force of around 1 kPa; adhesion is possibly limited by the large size of the toes and hence the difficulty in keeping them all closely pressed to the substrate. Using fluidcovered toes has two potential disadvantages for active creatures such as tree frogs. First, they have to overcome the pads’ adhesion to raise their feet to allow them to walk. They do this by peeling each toe off the surface as they take a step. The toes come off automatically when the frogs walk forward, because the toe is tilted and the surface is peeled off from the back to the front. When the frog walks backward, in contrast, special muscles have to be used to raise the tips of the toes off the leaf first. A second problem is that because the toes are attached to the leaf by a film of liquid, the frogs are all too apt to slide about like a bald car tire hydroplaning on a wet road. Experiments in which the closeness of the toe pad to a glass plate has been measured using interference reflection microscopy have shown that each

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cell on the frog’s toe is pressed extremely closely to the surface (Federle et al., 2006). Excess mucus seem to be channeled away from the surface through deep grooves that run between each cell and act like the treads of wet-weather car tires. This adaptation allows tree frogs to clamber up steep slopes even on very smooth surfaces, although because of the relative weakness of the attachment mechanism, they are not capable of walking upside down. Insects also use wet adhesion to complement the attachment provided by the hooks at the end of their feet, enabling them to attach even to smooth substrates. Ants, cockroaches, and stick insects all hold on by means of soft, fluid-covered smooth pads. The smooth pads deform enough to allow close contact with large surface irregularities, while the fluid fills the gaps between the pad and small depressions in the surface (Drechsler and Federle, 2006). Identification of the fluid is clearly difficult since it comes in such tiny amounts, but experiments suggest that it is a water-in-oil emulsion (Federle et al., 2002); it acts somewhat like a solid when held still, increasing the pads’ grip. The system is much more effective than that in frogs, providing a maximum adhesion of about 0.1 MPa. Such a force enables these insects to walk upside down even on smooth glass. The pads peel off the surface, just like those of frogs, to allow them to walk. Flies and beetles have quite different attachment organs: pads of long, thin hairs, each around a micron in diameter and each covered in its own fluid film (Bullock et al., 2008). Compared with smooth pads, this arrangement should theoretically have three advantages. First, since the long hairs are fairly flexible, they can each attach readily even onto rough surfaces. Second, less secretion is required. Third, since each hair has its own meniscus, there should be a far greater perimeter of fluid and hence greater attachment due to surface tension. In fact, however, the adhesion of hairy pads seems to be roughly similar to that of the smooth pads, possibly because it is Laplace pressure that provides the dominant attachment force. The insects can attach with forces equal to several times their own body weight yet peel off the surface readily, one hair at a time, although they do leave telltale fly prints behind them.

Dry Adhesion Although wet adhesion works well, it requires a constant production of fluid, and this fluid also limits the attachment strength because of its low cohesion. A potentially much stronger and cheaper bond could theoretically be achieved merely by abutting an attachment organ very close to the substrate. Van der Waal’s forces between the atoms of the two surfaces would thereby be set up and could take tensile loads of 20–200 MPa. Of course most solid objects do not stick to surfaces, and this is because of the great roughness of objects at the atomic scale; the surfaces approach each other only at very few points, so the contact area, and hence dry adhesion, between the two objects is usually negligible. One way to increase

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the contact area would be to make the attachment organ so soft and plastic that it could be pushed down firmly onto the surface and deform until it made close contact throughout its area. This seems to be the way in which Elmer’s-Tack (or in the United Kingdom, Blu-Tack) and modeling clays work. The problem they face, however, is the difficulty in removing them from the surface again and their habit of falling apart due to their softness. A much better method of forming a large temporary contact area is to make the attachment organ out of extremely fine hairs, which are so compliant that large numbers can approach and contact the surface. Animals that use this technique include spiders (Niederegger and Gorb, 2006) and geckos. Studies have long pointed out the fineness of the hairs on the toe pads of geckos, and it used to seem as if no decade would go without its paper in Nature pointing out this fact. Recent studies have, however, highlighted the great sophistication of the design of gecko feet (Autumn and Gravish, 2008). The hairs are not simple; if they were, the van der Waal’s forces would stick them to each other as well as to the surface, and they would end up in a confused tangle! Instead, the hairs are highly branched with flat spatulate ends to the short terminal twigs. This arrangement provides a large attachment area, while the rigid trunk and branches of the hairs keeps their tips separated, like the leaves of a tree, so that their full surface area can be pressed onto the surface. The attachment adhesion at the spatula level can be as high as 1 MPa, although complete arrays have rather lower values of 50 kPa (Gravish et al., 2008). This allows even large Tokay geckos to support their own weight while hanging from a single toe. Another aspect of the design of the hairs allows the gecko to readily remove its feet from the surface without having to peel them off. In their unstressed state the hairs curl inward and downward so that the spatulate ends are at right angles to the toe and hence do not have much attraction to surfaces (Autumn and Gravish, 2008). Geckos attach themselves by planting their toes on the surface and pulling their feet inward toward their bodies, thereby flattening the spatulate ends onto the surface. To detach itself, all the gecko needs to do is to stop pulling the toes inward; the hairs will curl up, peeling the spatulate ends off automatically. This process also allows the toes to shed debris that would otherwise reduce its efficiency. So exciting is the dry attachment mechanism of the gecko that there is great interest in trying to mimic it to produce reusable tape or other attachment devices. These attempts have been successful to some extent, but none has reached anywhere near the sophistication of the gecko design.

Friction, Traction, and Gripping The contact area between two solid surfaces that are touching is usually tiny, so that adhesion between them is negligible. However, if they are pushed together by a normal force, it does help to stop them shearing past each other, and any relative movement will be resisted by friction. The friction

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(a)

(b)

Figure 12.10. Friction of materials on a hard surface. In glassy materials (a), friction is low because of the small contact area, whereas in rubbers (b), the normal force increases the contact area, resulting in high friction.

force, F is given by the expression F = µN,

(12.1)

where N is the normal force and µ is the coefficient of friction. In most glassy solids, the friction results from the interlocking of projections on the two surfaces and so tends to be higher the rougher the surfaces are, and lower the smoother they are. The traction that an animal can exert on a surface will therefore depend largely on how rough the surfaces are. The friction between keratin structures, such as claws and hooves, and smooth hard surfaces, such as stone or tiles, is relatively low at 0.1–0.3 (Bonser et al., 2003), which is why horses, cattle, and dogs all slip rather easily on them. Fortunately in nature these animals mostly walk on soil with which they have far greater friction, both because it is rougher and because wet soils give under their weight, allowing the foot to push against the side of the depression that they make. Low friction also prevents hoofed animals from walking safely on naturally smooth surfaces such as tree trunks. Climbing animals that run along large branches, such as squirrels and cats, use sharp claws to dig into and grip the surface. However, this mechanism for gripping is not used by animals, such as primates and koala bears, that hold onto small-diameter branches and twigs. They grip using soft fingerpads, each of which is backed by a hard nail and is covered in ridges to produce fingerprints. Fingerpads grip far better than keratin hooves, particularly on smooth surfaces, because skin behaves like rubber and its low stiffness allows a large area of contact with the surface (fig. 12.10b). The resulting attractive

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van der Waal’s forces give high coefficients of friction, in the region of 1–2 against acrylic glass (Warman and Ennos, 2009). There is a mystery here, however. It is usually assumed that we have fingerprints to increase the friction of our fingerpads and so improve grip, but fingerprints reduce the contact area by a factor of about a third (Warman and Ennos, 2009). They would therefore decrease friction against smooth dry surfaces! It is possible that the prints improve friction against rough surfaces, or remove water like the grooves on car tires, so improving friction in the wet. They may even help prevent blistering. Once again it is clear that more research is needed in this area. A final point to make is that in some situations, low friction is an advantage, for instance, for snakes that need to glide smoothly over the ground or for sand lizards, which move through sand. Recent studies have shown that the scales of these reptiles have particularly low coefficients of friction, due either to the low adhesion of the keratin to other surfaces (Baumgartner et al., 2007) or to the microrelief of the scales (Berthe et al., 2009).

CHAPTER 13 .................................................

Interactions with the Mechanical Environment

OPTIMIZING DESIGN FOR STRENGTH A major theme of this book has been how well materials and structures seem to be “designed” in terms both of their material properties and of their overall shape for their mechanical function. The molecular sequences of proteins and sugars are arranged in such a way as to produce polymers with impressive mechanical performance, and proteins and sugars can be arranged together and embedded with crystals to produce useful biological composites. Similarly, materials are organized within structures to efficiently resist the loads to which organisms are subjected and to allow the necessary deformations and movements. Such apparent optimization of the design of organisms is, of course, the result of evolution by natural selection. However, another aspect of structural design, which until fairly recently was less often studied, is whether biological structures are optimally designed in terms of their overall strength or rigidity. If a bone were made too thin, it would break or bend when loaded even if it did have an optimal tubular shape. Conversely, if it was too thick, it would be too heavy and would cost too much to make, even though it was tubular. The questions that need to be posed, therefore, are, do biological structures have the correct strength and rigidity and, if so, how is that achieved?

FACTORS OF SAFETY In order to calculate if structures are strong enough to perform their mechanical role, one needs to know two things: the strength of the structure and the maximum load it is likely to encounter. These can then be combined to produce what engineers call the factor of safety (Alexander, 1981), which is given by the equation Factor of safety = Strength/Maximum applied load.

(13.1)

The structure should be safe if the factor of safety is greater than 1, but the greater the factor of safety, the more “over-designed,” heavier, and more costly the structure. From the time that factors of safety were first developed by engineers, they have proved to be a useful tool in designing safe structures. Using mathematical and, more recently, finite element models of the structures they are designing, engineers calculate the maximum stresses to which each

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element is likely to be subjected by the predicted loads, such as gravity and the wind. They then compare those stresses with the breaking stresses of the materials from which their structure is built. Even so, engineers still tend to use high factors of safety—well over 10 for structures such as bridges—to ensure that failure cannot occur and to minimize deflections. In biology, calculating factors of safety is a slightly different and more difficult process because the structures already exist. Strength can be measured by mechanically testing the structures to determine the failure loads, but it is almost impossible to measure the maximum applied load; instead it has to be estimated in some way. We can measure the strains on organisms in life using strain gauges or other techniques, or model the forces on them in the computer using finite element analysis. However, we are never certain that we have measured the maximum possible applied load. As Alexander (1981) pointed out, it is also not just a question of ensuring that the factor of safety exceeds 1. The optimal factor of safety of a biological structure will, in fact, depend on several things. First, one has to consider the variability inherent in nature. The loads experienced by organisms are likely to be unpredictable, and the strengths of natural materials may also vary. The greater the variability, the greater is the likelihood that the loads on the structure will exceed its strength, and so the likelihood that it will break. Therefore the higher the factor of safety will have to be to reduce the incidence of failure. Second, one has to consider the consequences of mechanical failure. The more severe the consequences of failure, the higher the optimal factor of safety will be.

Factors of Safety in Animals When Alexander (1981) examined a range of animal structures, he found that they had very different factors of safety, which he could relate to their functions and loading regimes. In general, the long bones of both running and flying vertebrates have quite high safety factors, with values from 2 to 8. This range reflects the unpredictability of the loading of bones, which tend to break in accidents involving falls or collisions, and the fact that bone fractures are often fatal. One also has to consider the problem of fatigue fracture, which will reduce the strength of frequently loaded bones (Currey, 2002). Animals can recover from breaks—in one study, more than a third of gibbons had healed fractures (Schultz, 1939), as did 3–6% of the birds studied by Brandwood et al. (1986)—but more often, they will result in a herbivore being eaten or a carnivore starving to death. In contrast to bones, the leg tendons of vertebrates tended to have much lower factors of safety—from just over 1 to 1.6. This reflects the fact that the loading of tendons, which are subjected to tension by muscles that produce a known maximum stress, are much more predictable. Moreover, since many tendons store energy in locomotion and the amount of energy stored increases with the distance they stretch, a high factor of safety would decrease the deflection and energy storage of a tendon. There will

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consequently be strong selection pressure to reduce the factor of safety to a minimum. In contrast to leg tendons, Alexander found that the opercula of fish and the tendons of our fingers have high factors of safety, about 8–10. These high factors of safety are explicable because both structures have to be extremely rigid. The operculum has to power water movements over the gills, so it must not bend when moved in and out. Our finger tendons also have to be extremely stiff in tension to transmit the movements from our muscles in the upper forearm to the fingers (Ker et al., 1988). Both of these structures are designed for optimal rigidity rather than optimal strength, and both cases show that designing structures for adequate stiffness rather than adequate strength is inevitably more costly (Gordon, 1978). Even within one organ, different parts can have very different factors of safety. Biewener and Dial (1994) found factors of safety between 1.9 (for torsion) and 3.5 (for compression) in the humerus of pigeons. In contrast, the factors of safety of the shafts of their flight feathers were much higher, in the range of 6–12 (Corning and Biewener, 1998). The humerus appears to be designed for strength, whereas feathers, which are made of the more compliant material keratin, are probably designed for optimal stiffness. In the light of these studies, the results of a study on the wing feathers of early birds, such as Archaeopteryx (Nudds and Dyke, 2010), are very interesting. Nudds and Dyke calculated safety factors for the feather shafts using estimates of the mass of these extinct creatures and by assuming that their feathers were made of keratin with similar properties to modern birds. They found that because the shafts of the feathers of Archaeopteryx were so narrow—about a third of that of similarly sized modern birds—they would have had very low factors of safety, even if it was assumed that the shafts were solid. This suggests that Archaeopteryx flew more like a glider than a powerful flapping bird. One study, that of Brandwood (1985) on spider dragline silk, calculated a factor of safety of less than 1. Brandwood looked at the situation of a falling spider and asked whether its dragline would be capable of halting its fall. Because this is a dynamic situation, Brandwood analyzed this process by considering whether the dragline could absorb the energy of the falling spider and found that it was capable of storing only around 80% of it. Of course spiders are held safely by their dragline when they do fall, so this result required some explanation! The resolution of this apparent paradox seems to be that the extra energy is absorbed by making new dragline by drawing silk through the spinnerets of the spider.

Factors of Safety in Plants The concept of the factor of safety has also been applied to trees. Much research, for example, has been carried out on the factor of safety of trees against collapsing under their own weight. Studies such as those by McMahon (1975) have shown that large trees are approximately 3–4 times

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as thick as they need to be to prevent Euler buckling of the trunk, so a factor of safety of 3–4 is often quoted. But calculations using the fact that bending rigidity is proportional to the fourth power of the diameter would yield a factor of safety more like 100, meaning that trees are vastly overdesigned. In fact, trees are far more likely to be damaged by the force of the wind, and a much more realistic investigation of their factor of safety was carried out by Mattheck (1993). He cut windows of different sizes and shapes into tree trunks and used finite element analysis to determine how much these would magnify stresses. He then surveyed how the presence of windows affected the incidence of failure of the trees in the next few seasons. Using this technique Mattheck found a factor of safety of about 4.5. In apparent contradiction to these findings, however, Mattheck et al. (2006) also found that trees started to fail when their trunks had been hollowed out to the extent that their wall thickness was 30% of the radius. Because material at the center of a beam is not very effective at resisting bending, this hollowness would have raised bending stresses by only about 30%. This discrepancy alerted Mattheck to the fact that (as we saw in chapter 8) the failure of hollow trees is due to a quite different stress: shear.

HOW OPTIMIZATION IS ACHIEVED The evidence above strongly suggests that the strength and/or rigidity of biological structures is optimized, but how is this achieved? It is generally accepted that most aspects of apparent design in nature are produced by the process of evolution by natural selection (fig. 13.1a). The genetic blueprint, or genotype, of an organism affects its morphology, or phenotype, and this in turn has an effect on the survival and reproductive success of the organism. Over many generations and turns of the feedback loop, the optimal design will be ever more closely approached. At least for structural design, however, there is excellent evidence that the phenotype of an organism is also affected by the mechanical environment to which it is subjected. Therefore there is a further shorter-term feedback loop called dynamic optimization (fig. 13.1b). The structure detects the stresses or strains to which it is subjected and responds either by laying down more material or removing some. This process should give an optimal structure in a much shorter time and acts to perfect each structure separately, so optimizing the whole organism more efficiently. The feedback loop itself can then be tuned over many generations by natural selection (fig. 13.1b). There is good evidence of dynamic optimization in two very different structures: the bones of vertebrates and the long organs of plants.

Dynamic Optimization in Bone Bones are capable of detecting and responding to mechanical stress (Currey, 2002). High stresses cause an increase in fibroblast activity, resulting in the

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(a) genetic blueprint

body

survival of the fittest

(b)

survival of the fittest genetic blueprint

body

stress measured

remodelling and redesign Figure 13.1. The two mechanisms by which the structures of organisms may be optimized: (a) evolution by natural selection, and (b) dynamic optimization, modified by natural selection.

laying down of more bone material, whereas low stresses cause an increase in fibroclast activity, resulting in the reabsorption of bone. It is well known that these processes happen both at the level of the entire skeleton and at a more local level: the bones of astronauts lose mass when they are in space, and tennis players’ racket arms are thicker than their throwing arms. What is perhaps less well known is that this effect is very well localized indeed; for example, broken bones that have healed crooked gradually straighten. Classic experiments in which the ulna of pigs’ and sheep’s forelegs were removed—increasing stresses on the two edges of the radius (fig. 13.2)— showed that more bone was laid down on highly stressed edges of the radius, changing its shape over a period of three months. This preferential distribution reduced the peak stresses on it back to normal levels (Goodship et al., 1979). In contrast, the unstressed ends of the ulna were reabsorbed. Many other experiments have shown that it is the size and shape of bones, rather than the properties of the bone material itself, that are affected by mechanical stimulation (Currey, 2002). The healing and development of bones can be simulated in a computer, by combining finite element analysis with a simple growth rule (Mattheck, 1998). A finite element model of a bone is made and is then loaded as it would be in nature. The model calculates the induced surface stress in the bone and then lays down more material in highly stressed areas and removes it from lowly stressed areas. The process is repeated many times in a series of feedback loops until equal stresses occur all around the bone. The pattern of growth and the final shape of the bone are often strikingly similar to that in real bones. The surprising corollary of this observation is that the shapes of our bones may not be genetically programmed themselves but instead may be the result of stress. In early development, the muscle movements of fetuses may drive the formation of bone shape around a number of centers of ossification.

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(a)

(b)

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Figure 13.2. Changes in the shape of the distal foreleg bones of a pig (a) when part of the ulna is removed (b). More bone is laid down along the front and rear edges of the radius (c), while the stumps of the ulna are reabsorbed. (Redrawn from Goodship et al., 1979.)

The phenomenon of stress-induced bone remodeling is therefore well established, but the mechanism by which bones detect mechanical stress are still unclear (Currey, 2002). One theory is that stresses set up piezoelectric fields in the bone that are detected by the cells; another is that changing stresses cause fluid flow within the bones and that this flow will cause electrical changes that can be picked up by the cells. As Currey describes, much effort is being invested in determining which of these theories is correct—or whether cells in the bone detect damage or strain directly. Obviously this is a very important area of research with great potential for fracture repair and for helping tissue engineers produce artificial bone.

Dynamic Optimization in Plants The effect of mechanical stimulation on plants has been known for even longer than its effect on bone. As far back as the early nineteenth century, it was known that plants respond to wind-induced shaking in two ways. First, there is a reduction in primary growth, so stressed plants are shorter than unstressed ones. Second, there is an increase in secondary thickening, so stressed plants are thicker than unstressed ones. Together, these effects are known as thigmomorphogenesis (Jaffe and Forbes, 1993). The effect on

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(a)

(b)

(c)

(d)

sinker roots Figure 13.3. Computer simulation of adaptive growth in trees to produce buttresses. Material is laid down faster in the highly stressed area at the top of the junction between the root and trunk (lines close together) resulting in gradual change in shape (b, c) and eventual production of a buttress (d). (Redrawn from Mattheck, 1998.)

primary growth appears to be on the plant as a whole, and some ingenious experiments by Coutand and Moulia (2000), in which different-sized tomato plants were carefully bent, have shown that the plants seem to detect strain rather than stress; the effect is proportional to the integral of strain over the entire plant. In contrast, the effect on the secondary growth of plants is more complex. Shaken plants generally grow thicker than unshaken ones, but the wood they produce tends to have a higher microfibrillar angle (Telewski, 1995), giving it a lower stiffness but a higher breaking strain. Some trees even respond by laying down their wood cells at an angle to the long axis of their trunks, producing what is called spiral grain (Kubler, 1991). All of these effects will make a tree stronger but better able to reconfigure in the wind, and so it will be more resistant to wind damage. As in bone, there is overwhelming evidence that the effect on the secondary growth of plants is a local one: strain is detected at each point in the cambium and more wood is laid down in highly stressed areas. If a tree is staked, for instance, it only thickens above the stake where it is stressed, while trees allowed to sway in only one plane thicken preferentially in that direction to form an elliptical trunk and grow thicker roots in that plane (Mattheck, 1991). As Mattheck points out, the simple growth rule of laying down more material in highly stressed areas should result in equal stresses around the tree, a result he calls the constant stress hypothesis. He suggests that this hypothesis explains many aspects of tree morphology: the rapid healing around the sides of wounds; the expansion of the bases of branches; the figure-eight cross section of roots (see chapter 8); even the growth of the bizarre buttress roots of tropical rainforest trees (fig. 13.3). Mattheck (1991)

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has simulated each of these patterns in a computer by combining finite element analysis with the simple growth rule that more wood should be laid down in highly stressed areas. Using another developmental program, he has even simulated the development of greater transverse strength that occurs around the root forks of trees (Mattheck, 1997). We saw in chapter 8 that this increase in strength is caused by the development of greater numbers of rays in this region, helping prevent delamination. Mattheck showed that this process could be produced by the simple growth rule that more rays will be laid down where transverse stresses are higher. Such is the persuasiveness of Mattheck’s ideas that it is tempting to think that tree development can be simulated by his rules alone. However, when these ideas are subjected to experimental investigation, things are not quite so clear-cut. In our tests on buttress formation, for instance (Crook et al., 1997), although stresses in rainforest trees were highest at the top of the join between the buttress and the trunk—just as Mattheck predicted—the same was also true of species of trees, both tropical and temperate, that did not form buttresses (Crook and Ennos, 1996). It is clear, therefore, that the process is more complex than Mattheck suggests; trees may differ, for example, in their response to stress. As in bone, the precise mechanism by which strains are detected remains unclear. The growth regulator ethylene has long been known to be involved in the process of thigmomorphogenesis (Jaffe and Forbes, 1993). More recently, several “touch” genes have been identified by molecular biologists, and the expression of one of them, PtaZFP2, has been shown to correlate with diameter growth (Coutand et al., 2009). Once again this is an extremely active and important area of investigation, but the complexity of the whole system has been barely touched on.

CHAPTER 14 .................................................

Mechanical Interactions between Organisms

BIOTIC INTERACTIONS Understandably, most of the research that has been carried out on the mechanics of biological structures has examined how they interact with the physical environment around them: how animals withstand the forces to which they subject themselves when they move, or how plants withstand the forces to which they are subjected by gravity and the wind. In recent years, however, researchers have also started to look at other equally important aspects of biomechanics that are related even more closely to how organisms survive: their mechanical interactions with other organisms. All organisms need energy to survive, so many of these biotic interactions have to do with how different organisms obtain their food. Of course different organisms do this in very different ways. Most plants make their own food by photosynthesis, using light as their energy source. Consequently the most important mechanical interaction between plants is the way that climbing plants and epiphytes parasitize the mechanical structures of self-supporting plants to reach the light. Fungi, in contrast, burrow through other organisms, digesting them extracellularly, so they must show mechanical adaptations for penetration. Finally, animals ingest other organisms and digest them within an internal tube or gut, so they must have mechanical adaptations to catch food and break it up into ingestible and digestible pieces. No plant or animal wants to be dug into or eaten, so both plants and animals have developed defenses, some of them mechanical, to protect themselves against fungal diseases, herbivores, or carnivores. Such interactions can result in so-called evolutionary arms races between the organisms. In the last few years one of the great successes of biomechanics research has been the gradual discovery of a wide range of sophisticated mechanical adaptations in organisms that have been driven by host-disease, plant-herbivore, and predator-prey interactions.

THE MECHANICS OF CLIMBING PLANTS Many aspects of the design of climbing plants are very obvious and were admirably described by nineteenth-century German botanists, such as Haberlandt (1914), long before the advent of biomechanics as a separate discipline. The simplest and most primitive method for plants to acquire support is generally regarded as scrambling, whereby narrow-stemmed

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(a)

acanthophylls (b) i

ii

supporting stem

expanding stipule

(c)

helical tendril

Figure 14.1. Mechanical adaptations of some climbing plants: (a) the cirrate leaf of the climbing palm Desmoncus polyacanthos with robust backward-pointing acanthophylls at the tip (redrawn from Isnard and Rowe, 2008a); (b) tensioning of the twining stem of Dioscoria bulbifera. The expanding stipule at the base of a leaf pushes the stem away from the support (redrawn from Isnard et al., 2009); and (c) the helical tendril of a bryony Bryonia dioica plant, showing the opposite winding at the two ends.

plants lean on the more sturdy self-supporting plants that are growing around them. To aid them, some scrambling plants, such as cleavers and hops, are covered in downward-pointing hairs, whereas scrambling roses use downward-pointing thorns. Many scrambling shrubs also use their own petioles and branches to provide support (Menard et al., 2009); once they touch a branch of a neighboring plant, they bend downward, securing their grip and levering their own stem upward. Climbing rattan palms have attachment organs called cirri and flagella (Isnard and Rowe, 2008a), which are armed with a series of hooks that are larger and stronger toward the base (fig. 14.1a). These attachment organs act as ratchets; any movement of the host plant that lowers its branches temporarily will allow lower and lower hooks to engage, pulling the rattan higher when the host returns to its original position. One of the problems with obtaining support from another, stronger plant, is that its movements could break or uproot a scrambler, so some of the latter have specific adaptations to prevent this happening. In hops, the lower hairs on the stem are anvil shaped, pointing upward as well as downward and preventing the lower stem from being levered out of the ground. In cleavers, in contrast, the lower stem and basal root are both extremely stretchy (Goodman, 2005), being able to extend by more than 15%

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of their original length before they break, so even quite large movements of the shoot system will not break or uproot the plant. Twining plants have long been regarded as more sophisticated than scramblers, because they coil helically up the stems of their hosts. Until very recently, though, the mechanism by which twiners attached to stems was unclear. Measurements of the internal force provided by twining stems, made by growing them up flexible plastic water-filled tubes (Silk and Hubbard, 1991), showed that twining stems develop a strong inward force on the support, which sets up a strong frictional contact. It is this that prevents the helix from collapsing downward. However, what produces this inward force remained a mystery. Undoubtedly, some of the force is produced by the resistance of the twining stem to being bent or twisted off; indeed, many herbaceous twiners have stems with an outer ring of lignified tissue that maximize their flexural and torsional rigidity. This would not be enough by itself, though. Recent research has shown that the inward force is provided by the stretching resistance of the stem (Isnard et al. 2009). After a region of twining stem has wrapped itself around its support and stopped extending, it pushes itself away from its support, either by swelling of the stipules at the base of each leaf (fig. 14.1b) or by simple secondary growth. This puts the twining stem in tension, massively increasing the inward force it develops. The growth in thickness of the supporting plant will also contribute to this tension. The tendril climbers are generally regarded as being the most sophisticated of the climbers, relying for support as they do on tendrils (usually modified petioles), which wave around until they touch a support, after which they wrap themselves around it. Both of these processes rely on asymmetric growth of the tendril, which is usually effected by controlled acidification of the cell walls. In some plants, such as white bryony (Bryonia dioica), a member of the cucumber family, the tendrils also coil up into a helix after they have grasped a support (fig. 14.1c), pulling the climber upward and producing a well-sprung attachment like that of the suspension system of a car. There is a final problem that many climbing plants must overcome. In order to grow tall enough to reach supports, the young stems have to be relatively rigid. The best way of achieving this is to have a stem with an outer ring of rigid tissue; however, later in the life of the climber, this rigidity would prevent the climber from flexing and following the growth of its host. Climbers have different ways of overcoming this difficulty. In rattans, the young stems are stiffened by being clasped by the bases of leaves (Isnard and Rowe, 2008b), just as grass stems are supported by their leaf sheaths. Later on in life, these leaves are shed, allowing the stem to flex freely. Something rather more sophisticated occurs in dicotyledonous climbers such as Aristolochia (Masselter and Speck, 2008). The young primary stem of these plants is stiffened by a ring of lignified cortical tissue. Later on, secondary growth occurring inside this ring produces such great circumferential stresses in the outer cylinder that it splits longitudinally, bursts open, and is shed. This arrangement has the advantage that the new

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secondary tissue can be made both compliant, which allows the stem to flex easily, and extremely permeable to water, so that it allows a large crown to develop, even at the end of a long narrow stem. Of course none of this is any benefit to the host plant, so many trees have defenses against climbers. They may shed their bark, for instance (Putz, 1984), while many pioneer trees of the tropical genera Macaranga and Cecropia co-opt ants that not only eat herbivores but also chew the growing points of climbing plants that attempt to attach. The trees provide shelter within their hollow twigs, helping the ants by providing weakened areas, or prostomata, through which the ants can readily drill (Brouat et al., 2001). The trees also protect their ant partners from ants of other species by providing slippery wax barriers that only the host species can cross (Federle et al. 1997).

THE MECHANICS OF FUNGAL HYPHAE AND APPRESSORIA Several plant species are parasitic on other plants, using their roots or stems to burrow into the xylem or phloem of their hosts. This would no doubt repay biomechanical study, but I am unaware of any research in this area. Fungi are even more adept at burrowing into their hosts, be they plants or animals. The growth of fungal hyphae has been fairly well studied and appears to be rather like that of plant cells and root hairs. It is driven by turgor and controlled by the chitin skeleton of the hyphal wall (Money, 1998). Such a pattern of growth, powered by internal pressures of around 1 MPa, is perfectly adequate to drive the growth of hyphae through soil. However, these pressures would not be enough to break through most biological materials, particularly plant cell walls. Many fungi therefore produce specialized infective organs called appressoria (Howard et al., 1991); studies of these structures have shown that they can develop pressures of up to 80 MPa. This is possible because the appressorium is small with a thick and melanized wall. But how the infection of a plant cell occurs remains unclear. The appressorium, which is approximately hemispherical, is glued firmly to the cell wall with a strong ring of tissue around the perimeter of the join that prevents the appressorium from peeling off. The actual infection occurs via an infection peg, which is driven down through the cell wall at a special weakening of the appressorial cell wall. Whether even the 80 MPa pressure within the appressorium is enough to penetrate the plant cell wall is uncertain. It may be that there is some additional digestion of the cell wall at this point or that the actin-rich walls of the peg may also help develop force. Of course, chemical digestion of tissue is also extremely important in the processes of infection and exploitation in fungal infections of plants. In wood, different groups of rot fungi—brown, white, and soft rots— attack different components of the cell wall, producing different patterns of weakening, with the consequence that wood infected by different fungi breaks down in very characteristic ways (Schwarze et al., 2000).

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PLANT DEFENSES AGAINST FUNGI Plants do not benefit from being consumed by a fungus, so we should expect them to develop mechanical, as well as chemical, defenses against fungal diseases. Trees block off diseased areas with plugs of resin that must act at least partially via a mechanical role. Plants also use more active defenses. Many trees produce chitinases that break down the cell walls of the fungus, but these are produced at lower levels in the ant plants (Heil et al., 2000); presumably the ants protect the plant from fungi, but the plant may have to reduce the levels of chitinases to prevent them attacking the chitin within the ants themselves. In some herbaceous plants, such as cucumbers and grasses, plants that have been supplied with silicon tend to be more resistant to diseases such as mildew than ones that lack this element. It has been suggested that a plaque of silica essentially forms a glass wall that helps block off infections. However, some studies suggest that the presence of silica at infection wounds may be incidental and occurs simply because this is where transpiration water evaporates (Fauteaux et al., 2005). Several authors therefore suggest that silica may act indirectly to mobilize chemical defenses. More research is clearly also needed in this area. FOOD PROCESSING BY ANIMALS We saw in chapter 12 how the claws and feet of animals can help them grip their food, whether it be plant or animal. But one of the main problems faced by animals is how they can mechanically process their food: how they kill they prey and cut it up into pieces small enough that it can be swallowed and digested as quickly and as efficiently as possible. Of course most animals do this with their teeth, and much has been written about the functional morphology and relative merits of incisors, canines, and molars. Unfortunately most of that work was done without any reference to the science of fracture mechanics; it is only in the last two decades, with the research of people such as Lucas (2004), that a real understanding of tooth design and food processing has started to emerge. In turn, this has led to further research on the adaptations of food that protect it from being eaten, studies which are revolutionizing our understanding of animal-plant and predator-prey interactions. Teeth The best method to break up a food particle depends strongly on its mechanical properties. A material that is compliant but tough—such as skin, muscle, or plant parenchyma—is best cut up using one or more sharp blades that concentrate stress and so force a crack through the food particle (fig. 14.2a; Lucas, 2004). Hence, most mammals have sharp incisor teeth at the front of the mouth, and insectivores possess sharp-zigzag molars with notches in between. But why are sharp teeth so often serrated or notched?

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(b)

Figure 14.2. Methods of breaking up food particles with different mechanical properties. Soft, tough particles are best cut up using sharp teeth (a) to concentrate stresses. In contrast, stiff, brittle foods are best broken by loading them in bending between teeth with blunt cusps (b). (Redrawn from Lucas, 2004.)

One problem with cutting with a straight blade is that it can cause large deformations of the food particle, consuming excess energy, and the food particle can also slip from between the teeth. It has recently been shown that cutting with inclined blades reduces the energy needed to cut both plant and animal material, while notches in the blades traps flesh and reduces the energy needed to cut through it (Anderson, 2009). In contrast to tough materials, stiff, brittle materials would rapidly blunt a sharp-bladed incisor or carnassial molar. To break up brittle materials, they should ideally be loaded in bending by squeezing them between alternating blunt cusps (fig. 14.2b; Lucas, 2004). This puts one side in tension, where a crack will be induced and will run through material, breaking it up. This explains the design of blunt multicusped molar teeth, which occur in the back of the mouth, where greater closing forces can be induced, and which carnivores such as hyenas use to break up bones. Material such as wood, which is both stiff and tough, will be resistant to being broken up by both incisors and molars, one reason woody tissue is rarely eaten and why it acts as such a good defense in nuts. Once cut up into manageable pieces in the mouth, food is often further broken up into smaller and smaller particles by flat molar teeth, which crush them with compressive loads, like grains of wheat being crushed between milling stones (Lowrison, 1974). Large particles of brittle material fail at relatively low loads because irregularities in their shape concentrate stress and start cracks that run through them. Break-up gets more difficult, however, as the particles become smaller and smaller because they eventually are smaller than the irregularities that would reduce their breaking stress (see chapter 1). Gastroliths Birds have no teeth, and crocodiles lack molars with which to break up their food. Many of these creatures, particularly seed-eating birds, actively swallow stones, or gastroliths, which they use to grind up their food in their

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muscular gizzards. There has been little research on these stones, although milling theory (Lowrison, 1974) and initial experiments we have performed in our laboratory, suggest that the milling stones need to be larger than the ingested particles to effectively “nip” the latter and prevent them from slipping away from the stones. As break-up proceeds, however, smaller stones, which have a greater surface area per unit volume, should become more effective. This line of reasoning suggests that birds should ingest a range of sizes of gastroliths to optimize their digestion but should include particles larger than their food items. Once again this problem would repay further investigation, based on a sound understanding of fracture mechanics.

Food Processing in Humans The mechanical properties of foods seems not only to affect how they can be processed but also how they are experienced in terms of texture and even taste. Such notch-resistant foods as hard apples seem crisper and tastier, whereas softer apples seem mealier and lacking in taste (Vincent, 2004). For these reasons much research on food has been put into quantifying and optimizing the physical characteristics of foods, often it seems—as in the case of tomatoes—at the expense of giving them real taste. But perhaps the most surprising and intriguing research about human food processing investigated why we humans chew our food so much. Experiments by Prinz and Lucas (1997) showed that chewing our food allows us to stick it together more strongly into a solid bolus, adding saliva and pressing the food into the roof of our mouth with our tongue. This process makes it safer for us to swallow and is particularly important for humans, because with our high voice box we are extremely prone to food traveling down our respiratory tract and choking us. Large food particles have a low surface area, so even with saliva between them, they will not stick together. As we chew our food and break it into smaller particles, however, the surface area increases, and the pieces stick together better, like damp sand or clay, to produce a solid lump of material that is safe to swallow. There is a limit to the advantages of chewing, however. As time goes on, more saliva gets introduced to the bolus, reducing the cohesion between the particles, just as wetting clay eventually turns it into a liquid, and the bolus falls apart. In tests in which subjects were given diced carrot or brazil nuts to eat, the predicted cohesive strength of the bolus rose to a maximum at about 20–30 chews—just the number at which it would normally be swallowed.

ADAPTATIONS OF POTENTIAL FOOD Adaptations of Fruit Some plant organs—specifically, fruit—are designed to be eaten, since this ensures that the seeds they contain are dispersed and provided with fertilizer

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by the droppings of the herbivores. This process is why fruits contain a reward in the form of a sugary pulp. Most fruits do two things, however, to ensure that they are eaten only when the seeds within are ripe. The flesh is softened and loaded with sugars only when the seeds are ready; and at this point the peel is also softened and its softness signaled by a change in color or the production of scent. Fruit also exhibits finer aspects of mechanical design, because the cell walls are often oriented radially outward from the seeds (Khan and Vincent, 1990), permitting cracks to run more easily between the seed and the outside of the fruit and allowing the seed to be released.

Defenses of Seeds In contrast to fruit, for all other plant organs, most notably seeds, it is a distinct disadvantage to be eaten. Seeds in particular exhibit a wide range of defense mechanisms. Many seeds are, of course, highly poisonous, but mechanical defenses are just as common, if less well studied. Most nuts have strengthened woody shells made up of small, thick-walled, highly lignified cells. As a result, the shells are extremely hard and strong in compression, although since, like dense wood, they cannot utilize the woody toughening mechanism (chapter 4), they are fairly brittle. Nuts are therefore vulnerable to percussive fracture: they may be broken open by being hit with stones by primates or dropped from a height by crows. A major problem with having a thick woody shell is that it makes it difficult for the seed to germinate, because to do so the radicle must penetrate outward through the shell. In a comprehensive study of a single tropical species, Mezettia leptoda, Lucas et al. (1991) showed that its nut allows germination by incorporating a weak line of brittle tissue along one side and a plug at each end of the line to initiate cracks. As the authors point out, the fact that this line is along the long axis of the nut makes it hard for orangutans to exploit this weakness to crack open the nut for themselves; their gape is not large enough to allow the orangutan to insert its canines into the two ends of the crack and lever open the nut. Similar adaptations that allow germination can be found in many other nut species. Such modifications are well worth studying in more detail and provide excellent subjects for projects in biomechanics courses. There is also a lower limit to the size of nuts. Small seeds cannot use thick shells, since the pressures they would then require to split open such a shell would be too high. (This is yet another consequence of the Laplace equations we first met in chapter 6!) Small seeds instead use different techniques to prevent themselves from being crushed. Seeds, such as apple pips and linseeds, are coated with a gel that makes them extremely slippery; this should help prevent them being nipped between teeth or gastroliths, because they would simply slip from between the crushing surfaces. Initial tests in our laboratory suggest that removing the slippery coating makes them more vulnerable to being crushed, but again, more research in this area is required to test such ideas.

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Defenses of Leaves Next to fruit and seeds, the leaves of plants are the most attractive parts of plants to herbivores, being filled with the products of photosynthesis— sugars and starch—and the photosynthetic cells that contain large amounts of proteins, such as Rubisco which is needed for photosythesis. In some cases these cells are protected by thick walls, as in the bundle sheath cells of certain grasses that use the C4 photosynthesis pathway. The nonphotosynthetic cells can also be defensive structures. The midribs and veins of most leaves have thick-walled, lignified cells and make the entire leaf less profitable and palatable to eat. This is a particularly effective defense against large herbivores (Lucas et al., 2000). Small caterpillars, on the other hand, simply eat the material between the veins, skeletonizing the leaf (Choong, 1996). Grasses are probably the most successful of all plants, covering more than 30% of dry land and comprising more than 5,000 species. This success is at least partly related to the sophistication of their mechanical defenses. First, the leaves grow from their base, having a basal meristem that ensures that even if the top of the leaf is eaten, it can continue to grow. Second, the leaves are reinforced by large numbers of woody sclerenchyma fibers that run longitudinally along the leaf. This arrangement makes the leaves notch resistant (Vincent, 1982) since lateral cracks are stopped when they reach a new fiber, just as in a composite material. Consequently, herbivores find it more difficult to break leaves off. Small herbivores, such as caterpillars, locusts, and voles, have to snip through the entire leaf, which takes time and energy, whereas large ones, such as sheep and cows, have to grasp the leaf and pull it upward with great force. The third mechanical defense of grasses, whose action is just becoming understood, is the incorporation of silica bodies, effectively small lumps of glass, into the epidermis of the leaf. It has long been suggested that the silica makes the grasses much more abrasive (Walker et al., 1978), helping to wear down the teeth of large grazing animals. Indeed the coincident rise of the grasses with the evolution of the long hypsodont teeth in horses (Simpson, 1951) has long been regarded as a classic case of coevolution. Despite this observation, there was no strong evidence that herbivores avoid eating grasses because of their silica content. Recent research, though has shown that high levels of silica can protect grasses against small chewing herbivores (Massey et al., 2006). They avoid high-silica grasses, and when forced to eat them, they cannot digest them as efficiently as low-silica grasses. Further experiments explain how the silica defends the grasses. Wear tests show that silica does indeed make the grasses more abrasive (Massey et al., 2007), and high-silica grasses wear down the mouthparts of insects so much that they cannot digest the grasses as well, even if then fed low-silica grass (Massey and Hartley, 2009). The silica bodies also directly protect the photosynthetic cells, rather like the wooden props in mines; less chlorophyll (and sugar) is released when high-silica gasses are subjected to grinding action (Hunt et al., 2008), and more is retained in the cells, meaning

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less-efficient feeding. There is even some evidence that large herbivores, such as sheep, can detect and avoid high-silica grasses (Massey et al., 2009).

OTHER BIOTIC INTERACTIONS Of course, feeding and defending oneself from being eaten are not the only interactions between organisms. There are many instances of symbiotic interactions that may have mechanical aspects: the pollination of flowers by insects, for instance. Recently it has been shown that the conical epidermal cells on flowers help bees hold onto them (Whitney et al., 2009). There are plenty of other symbiotic interactions and examples of unusual ways of life, such as carnivory in plants, that are ripe for study; for example, the pitcher plants of the genus Nepenthes have several different ways of making their traps more slippery (Bauer and Federle, 2009). There is room for more research on all sorts of biotic interactions. At last, instead of being a separate isolated enterprise, biomechanics research is becoming an integral part of studies in ecology, evolution, and animal behavior.

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CHAPTER 15 .................................................

The Future of Structural Biomechanics

SUCCESSES Throughout this book we have seen that mechanical studies have been extremely successful in helping understand the form and function both of biological materials and structures. Having begun by looking mostly at animals, biomechanics research has more recently elucidated the design of plants, fungi, and even bacteria in substantial detail. Such work has not only helped us understand how organisms are adapted to their mechanical environment but has also shed light on the biotic interactions between them. Finally, we are also learning how to copy biological materials and structures and so turn the adaptations of creatures into engineering solutions.

LIMITATIONS AND FUTURE DEVELOPMENTS Until recently, however, structural biomechanics research has been held back by several technical limitations. Two of the main difficulties have been the heterogeneity of biological materials and the small size of many organisms. These factors have often made it hard to determine the organisms’ mechanical properties, since it is impossible to machine and test tensile specimens that are so small. This problem is compounded by the fact that so many biological materials are anisotropic, so to fully characterize a material, tests would have to be carried out in all three orthogonal directions. A further difficulty has been caused by the complex shapes of so many organs; it is just not possible to measure the local stresses around them by performing simple mechanical tests, nor has it been it possible to accurately calculate the stresses within them using the sort of straightforward analyses that were described in this book. Fortunately the march of technology, driven largely by the needs of materials scientists and physicists, has started to help us overcome these difficulties and given biomechanics a wide new range of tools to use.

Small-Scale Mechanical Testing Local stiffness and strength can now be investigated with microindentation (Bonser and Witter, 1993), and even more effectively with nanoindentation, testing machines (Akhtar et al, 2009a; Farran, Ennos, and Eichhorn, 2009). Unfortunately both of these techniques need the samples to be embedded

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in resin and ground smooth, so they are not suitable for live cellular tissues. Fortunately, two other techniques can be used instead. The atomic force microscope can provide quantitative morphological data in high resolution (Graham et al., 2010), while scanning acoustic microscopy can determine the stiffness and density of tissue components both on the surface and within a sample using ultrahigh-frequency sound vibrations (Akhtar et al., 2009b). Its resolution can reach the micron level. These techniques have the disadvantage that they cannot measure anisotropy, but ultrasound can be used at megahertz frequencies to measure anisotropic constants. Just as stiffnesses measured at high strain rates tend to be higher, due to viscoelasticity, so stiffness moduli measured at high frequencies are also higher.

Measuring Stresses and Strains Until recently most of the techniques to measure the stresses and strains were designed to work at a fairly large scale. Strain gauges are capable of measuring surface stresses in all three directions, but they are not capable of measuring at less than the millimeter level, the electrical circuitry they come with can be expensive, and they cannot be used for compliant materials, since attaching strain gauges would greatly alter their properties. Nowadays there are several techniques to measure surface strain. Laser speckle interferometry (which was invented by A. E. Ennos, among others) uses interference between scattered laser light before and after deflection to calculate displacements as small as the wavelength of light (5 × 10−7 m) (Jones and Wykes, 1989); it is therefore suitable for the study mainly of stiff materials, such as composites and ceramics. From the deflections, the strain can now be calculated by computer analysis. To study deflections on the surface of skin and other more compliant materials, a different technique is needed: image correlation (Quinta da Fonseca et al., 2005). This optical method tracks the movement of points on the surface and uses their displacements to calculate the strain pattern. Finally, the loading state within materials can be investigated using Raman spectroscopy (Eichhorn et al., 2005). This technique measures the changes in frequency of oscillation of the bonds within molecules and therefore gives information about how strained they are.

Dealing with Complex Shapes In the past it was virtually impossible to exactly characterize the shape of small structures because the techniques to examine them, such as light microscopy and scanning electron microscopy, just looked at their surface. Nowadays, there are several new techniques that can reconstruct the threedimensional shapes of small objects. For transparent cells, one can use confocal microscopy, which scans through the cell using a focused light

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beam and uses the reflection to reconstruct a two-dimensional or threedimensional image. More usefully, recent advances in x-ray technology means that there are now a wide range of x-ray computer tomography devices that can scan and reconstruct the three-dimensional structure of a huge size range of objects, from dinosaur bones right down to the microand nanoscales of cells. Once the three-dimensional structures are obtained, it is now possible to do several things with them. One is to make a model of the desired structure using fast prototyping technology. An approach that is far more widely used, however, is to make a finite element model in the computer for finite element analysis. In this technique a complete structure is split into many hundreds of tiny geometrical pieces, a load is applied, and a computer determines the pattern of stress around the object. We have already seen in chapter 13 how finite element models have been used to determine the stresses around idealized bones and trees (Mattheck, 1998) and then used to simulate the pattern of growth. However, they can also be used to calculate the stresses and strains around real structures (Rayfield, 2004) and hence predict the way the structure will deflect (Herbert et al., 2000).

Using the New Techniques There is no doubt that these new techniques are extremely useful. In particular, they have proved the ideal way of investigating the mechanics of extinct animals (Rayfield, 2004; Manning et al., 2009), allowing paleontologists to calculate just how strong a bite dinosaurs could have made and how strong their claws were. They can also be useful in evolutionary studies, for instance in understanding the pattern of shape changes in skulls and other bones (Moazen et al., 2009). Above all, they have been widely embraced by applied biomechanics for investigating the stresses around human feet and the effect of prostheses and implants. However, therein lies a weakness from the perspective of comparative biomechanics. These techniques have been developed and are most suitable for engineers, whose main aim is usually to characterize the performance of structures that they have designed. Apart from the adaptive growth technique of Mattheck, such methods do not readily shed light on exactly why the structures get stressed and deflect in the ways that they do which is of more interest to biologists. Of course, finite element simulations can be performed on abstracted models with most of the characteristics of real structures, or one can alter single features of real structures and examine the effect. However, even then the results do not necessarily imply an understanding of what is going on. Finally, if finite element models do not precisely predict the mechanical behavior, then it is hard to see what advantage they provide (Wootton et al., 2003). Therefore, although it is possible to study a wide variety of structures and obtain useful information about them, simply making and studying models that are more and more accurate does not give any new insights into the basic principles underlying their design. As an old timer, I would therefore

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like to point out that there is still mileage in the traditional methods. Simple mathematical analysis, the use of analogies with human technology, and above all making simple mechanical models in the spirit of origami, Erector Sets, and other modeling toys should still have their place even in this modern age!

NEW FRONTIERS FOR BIOMECHANICS Biomechanists not only have new techniques to help them, they also can move into new areas of research that could put them in the biological mainstream. One reason that biomechanists are considered quirky outsiders by most other biologists is that most of their research has been carried out on extracellular tissues. In contrast, the great majority of modern biological researchers seek to understand the internal workings of cells. Of course, there has been some excellent biomechanical research looking at the mechanism for mechanical detection in plants and bone (Coutand et al., 2009), and biomechanists have long been investigating the properties of muscle, which relies on interactions between the intracellular filaments actin and myosin. However, all cells, not just muscle cells, contain a complex cytoskeleton. This structure not only helps control their shape and mechanical function but also responds to the cell’s mechanical environment. Physicists have already started to become involved with aspects of research in this area, using such techniques as atomic force microscopy and scanning acoustic microscopy to probe the structure and mechanics of cells. They are also introducing cell biologists to ideas and theory with which biomechanists are more than familiar and which are described in this book. One example is the recent research on development, that is beginning to examine how the forces generated by filaments control morphogenesis and wound healing (Hutson et al., 2003). Developing and wounded Drosophila embryos use both internal “purse-string” filaments and adhesion-mediated “zipping” to close up holes in their epithelium. Understanding these processes is straightforward since it involves the simple analysis of stresses in the walls of pressure vessels that was examined in chapter 7. Physicists have also recently become involved in understanding the mechanical design of microtubules (Hunyadi et al., 2007). They suggest that the helical wall structure of these tiny tubular structures helps them resist catastrophic flattening, the sort of mechanical reasoning that was introduced in chapters 8 and 9. The world of cells is a physical world just as much as it is a chemical world. As biomechanists, we must be more prepared to learn enough of the language of cell and molecular biologists to understand them and to teach ourselves how to talk about biomechanics in as simple a way as possible, so we can form true interdisciplinary collaborations. The future of structural biomechanics at all scales and both inside and outside cells is a bright one.

GLOSSARY .................................................

α helix

stable right-handed helical arrangement of amino acids in a protein polymer

adhesion

attraction of two adjacent surfaces to each other

alveoli

tiny linked air sacs in lungs

amino acid

protein monomer

aneurysm

weakened area in a blood vessel that tends to bulge under blood pressure

anisotropic

having different mechanical properties in different directions

appressorium

infective organ of a parasitic organism such as a fungus

axial loading

stretching or compressing a material

β sheet

stable pleated-sheet arrangement of protein polymers

basal meristem

low, growing point where the leaves of grasses are produced

biomimetics

field of study that seeks to produce useful engineering solutions by copying nature

bistable

stable in two positions but not in the intermediate state, like a light switch

Brazier buckling

local buckling in the walls of hollow tubes when they are crushed

Breaking strain

strain at which a material breaks

breaking stress

strength of a material

brittle

susceptible to being broken by impact

Brownian motion

movement of small particles or polymer molecules due to bombardment by surrounding atoms

bulkhead

transverse reinforcement of a tube that helps it resist local buckling

capillary force

attachment caused by surface tension of a liquid

cartilage

osmotically pressurized hydrostatic structure that is reinforced with collagen

cephalothorax

front body segment of a spider

ceramics

inorganic salt crystals used to reinforce and harden biological materials

coefficient of friction

relative resistance of two surfaces to sliding over each other

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GLOSSARY

cohesion

tensile strength of a glue or liquid

collenchyma

stiffening material used in young plants; composed of long thick-walled unlignified cells

comparative study

biological investigation in which related organisms are compared

compliance

ease with which a material may be deformed; the opposite of stiffness

composite material

solid in which stiff fibers are held within a softer matrix

compression

force tending to shorten a structure

compressive test

mechanical test in which a short, squat piece of material is shortened

cork

material in tree bark that provides insulation and mechanical protection

creep

delayed deformation of a material following the application of a load

creep test

transient test in which a load is instantaneously applied and the deformation of the sample is monitored over time

crossed lamellar

plywood-like arrangement of material in some shells

crosshead

moving part of a universal mechanical testing machine

crystalline material

solid, such as a metal or salt, that is composed of an ordered array of atoms

cuticle

exoskeletal material of insects and other arthropods; composed of chitin fibers held within a protein matrix

cyanobacterium

photosynthetic bacterium; also known as a blue-green alga

delamination

separation of layers of material in fracture that absorbs energy

dentin

principal structural material of teeth

differential bending stiffness

torsional resistance of a beam conferred to it by its resistance to bending

displacementcontrolled test

mechanical test in which the sample is progressively deformed and the force required to do so is monitored

dynamic

mechanical situation involving the acceleration of objects

dynamic optimization

process by which the strength of structures can be tailored to the actual forces it has to withstand

dynamic test

mechanical test involving sinusoidal changes in displacement

elastic

type of behavior in which a material returns all its stored energy after being deformed and so has 100% resilience

elastic-plastic

type of behavior in which a material behaves elastically before and plastically after yield

elastic similarity

pattern of scaling in which diameter increase with length to the power of 3/2

elastic storage capability

amount of energy a material can store and return

GLOSSARY

225

elasticity

the behavior of solid materials in response to mechanical forces

elastomer

rubber; flexible material composed of freely moving chains of molecules

enamel

outer, hard, cutting layer of teeth

engineering strain engineering stress

strain calculated using the initial dimensions of the sample stress calculated using the initial dimensions of the sample

entropy

degree of disorder in a system

equilibrium

liquid or rubbery behavior of polymers at very high temperatures and over very long periods of time

Euler buckling

collapse of a strut as it bows outward when compressed end to end

extensibility

amount a material can deform before breaking

extensometer

component that can be attached to a sample in a universal mechanical testing machine to measure the sample’s deformation

factor of safety

relative strength of a structure compared with the load it has to withstand

filler

small particle that acts to stiffen and harden a composite material

flexural modulus

stiffness of a material calculated from the results of a bending test

flexural rigidity

resistance of a beam to being bent

foliated

arranged in overlapping fiber bundles as in some mollusk shells

four-point bending

flexural test that puts the central region of a beam into pure bending

fringed micelle

arrangement of polymeric material in which ordered crystalline regions are interspersed with disordered amorphous regions

gastrolith

stone in the crop of a bird that helps break up its food

gas vesicle

air-filled float of a cyanobacterium

gel

cross-linked polymer that traps large quantities of water to produce soft, brittle solids

glassy

stiff, brittle behavior of polymers at low temperatures and over short periods of time

greenstick fracture

fracture mode of branches and children’s bones in which the beam splits along its center line

halobacterium

salt-dwelling bacterium that uses a primitive sort of photosynthesis

hardness

resistance of a material to indentation

Haversian system

concentric cylinder of remodeled bone surrounding a blood vessel (syn., secondary osteon)

helicoidal

arrangement of cuticle in some arthropods, in which fiber orientation gradually changes through its thickness

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GLOSSARY

hexose

one type of sugar monomer

holdfast

attachment organ of a seaweed

Hooke’s law

observation that the deflection of a material or structure is directly proportional to the force applied, at least initially

hydrogen bonding

attraction between strongly positive and negatively charged parts of molecules, such as the hydrogen and oxygen in water and other organic molecules

hydrostatic structure

structure in which the compressive forces are withstood by water

hypha

threadlike body of a fungus

hypsodont

describes the high-crowned teeth of grazing mammals

hysteresis

percentage of elastic energy absorbed by a material, 100% minus the resilience

initial stiffness

resistance of a material to the first deformation

isotropic

having identical material properties in all directions

lamellar bone

bone in which the fibrils are arranged in plywood-like sheets

laminar bone

bone that is a combination of woven and lamellar bone

Laplace equations

equations governing the stresses in the walls of pressure vessels

Laplace pressure

component of capillary force due to surface tension and proportional to the contact area of the liquid

leathery

tough, viscoelastic behavior of polymers at intermediate temperatures and periods of time

lignification

process by which plant cell wall is stiffened and water-proofed

load cell

component of a universal mechanical testing machine that measures the force needed to deform the sample

load-controlled test

mechanical test in which the sample is progressively loaded and its deformation is monitored

local buckling

failure of a tube due to the flattening and collapse of its cross section when it is bent

material

substance that is arranged in space to make up a structure

mechanical advantage moment

relative movement and force produced by a lever system

mucus

liquid glycoprotein polymer that shows complex shear softening behavior

muscular hydrostat

structure, such as the trunk of an elephant, that uses muscles to provide support as well as to move

nacre

tilelike shell arrangement of mollusks such as mussels; also known as mother-of-pearl

nematode

primitive type of worm that is a common soil-dweller and parasite

force acting at a perpendicular distance from a point that tends to cause bending or torsion

GLOSSARY

227

neutral axis

line along the centerline of a beam where bending does not stretch or compress it

notch sensitivity

degree to which the strength of a material is affected by cracks

notochord

strengthening structure in the primitive and embryonic vertebrate backbone

osteocyte

bone-forming cell

parenchyma

soft packing material of plants made up of large closely packed cells with thin unlignified cell walls

pelagic

fast-moving wandering animal, such as a shoaling fish or sea bird

pennate muscle

muscle with short fibers oriented at an angle to its long axis

plastic

type of behavior in which a material fails to return to its original shape after being deformed and so has 0% resilience

plateau

rubbery behavior of polymers at intermediate temperatures and periods of time

Poisson’s ratio

degree to which a material necks or bulges when being stretched or compressed; minus transverse strain divided by axial strain

polar second moment of area

distribution of area around the center of a beam that governs its resistance to torsion

polymer

material made of long chains of molecules joined together

polysaccharides

long-chain polymers of sugars

prestress

stress held within structures even before they are subjected to external loads

primary structure

molecular sequence or order of the amino acids and hexoses in biological polymers

prismatic

arranged in parallel columns like the crystals in some mollusk shells

protein

biological polymer composed of large numbers of amino acids joined end to end

pure bending

bending that occurs when a beam is subjected only to axial forces, not shear

random-coil chain rays

shape of polymer that is free to writhe around

reaction wood

wood that holds high levels of prestress; used to reorient the trunk and branches

resilience

percentage of the stored elastic energy a material can return

retardation time

time taken for molecules to reorder following application of a force

radially oriented lines of cells in wood that increase its radial strength

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GLOSSARY

Reuss model

model used to calculate the properties of a composite material reinforced by continuous fibers oriented perpendicular to the applied force

rubber

flexible material composed of freely moving chains of molecules joined only at occasional cross bridges

sclerenchyma

stiffening material used in mature plants; composed of long, thick-walled, lignified cells

sclerotization

process by which insect cuticle is tanned

second moment of area

distribution of area around the neutral axis of a beam that governs its resistance to bending

secondary osteon

concentric cylinder of remodeled bone surrounding a blood vessel (syn., Haversian system)

secondary structure

shape in which proteins are held due to hydrogen bonds between different parts of the molecule

shear

force acting parallel to the surface of a piece of material that tends to deform a square into a rhombus

shear modulus

stiffness of a material in shear loading; the shear stress that would be required to give a shear strain of 1 radian

shear strain

angular deformation of a material under shear load

shear stress

intensity of shear load

shell

outer protective layer of mollusks

silk

protein polymer produced by insects and spiders in which short fibrous regions are interspersed with disordered rubbery lengths

simple beam theory

analysis used to predict the flexural behavior of long thin rods

spicule

piece of ceramic reinforcing material in the body wall of a sponge or sea pen

static

involving forces in equilibrium, so there is no acceleration

Stefan adhesion

attachment due to the viscosity of fluid under a plate

stiffness

resistance of a material to deformation

stipe

stem of a seaweed

stomata

openings in the leaves of plants that let carbon dioxide for photosynthesis and release oxygen to the air

strain

relative deformation of a material under load

strain gauge

electronic component that can be attached to a structure to measure the local strain

strength

force at which a structure will break, or stress at which a material will break

stress

intensity of force, or force per unit area, on a material

stress concentration

local increase in stress around a hole, defect, or region of change in material properties

GLOSSARY

229

stress relaxation

delayed decrease in the force needed to maintain the shape of a material following the application of a deformation

stress relaxation test

transient test in which a deformation is instantaneously applied and the force required to maintain it is monitored over time

stringer

longitudinal thickening of the walls of a tube that help it resist local buckling

structure

solid shape composed of an arrangement of materials

strut

beam whose role is to resist compression

sucker

structure using atmospheric pressure to attach to a surface

surface tension

resistance of fluids to developing new surface area

tangent stiffness

resistance of a material to being deformed at a given point during a test

tensile test

mechanical test in which a narrow strip of material is stretched

tension

force stretching a structure

tertiary structure

large-scale folded shape of proteins such as enzymes

thigmomorphogenesis

the process by which plants detect stresses and alter their growth to optimize their structural design

three-point bending

flexural test that loads a beam into maximum bending at its center

torsion

twisting

torsion pendulum

mechanical apparatus used to measure the torsional rigidity of a beam

torsional rigidity

resistance of a beam to being twisted

tough

resistant to being broken by impact

toughness

resistance of a material to fracture

trabecula

small rod of bone that support the thin cortex

transient test

mechanical test involving sudden application of a force or displacement

transition

toffee-like behavior of polymers at high temperatures and over long periods of time

true strain

strain calculated using the instantaneous dimensions of the sample

true stress

stress calculated using the instantaneous dimensions of the sample

truss

beam made out of alternative compression struts and tension wires

turgor

internal pressure of plant cells, fungi, and bacteria that is produced by the osmotic potential of their contents

twist-to-bend ratio

relative resistance of a beam to bending compared with torsion

230

GLOSSARY

universal joint

joint made up of two hinges at right angles that allows movement in two planes but not rotation

van der Waal’s force

attraction between molecules due to instantaneous changes in charge distribution

viscoelastic

type of behavior in which a material returns to its original shape after being deformed but loses energy by internal friction, meaning its resilience is less than 100%

Voigt model

model used to calculate the properties of a composite material reinforced by continuous fibers oriented parallel to the applied force

work of fracture

energy needed to produce a crack of unit area

woven bone

weak bone in which the fibrils are arranged randomly

yield

point at which a material undergoes an irreversible failure and becomes less stiff

yield strain

strain at which a material undergoes irreversible failure

yield stress

stress at which a material undergoes irreversible failure

Young’s modulus

stiffness of a material in axial loading; the stress that would be required to give an axial strain of 1

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INDEX .................................................

abductin, 37–38; mechanical properties of, 24 adhesion: dry, 194–195; of glue, 191; wet, 192–194 agar, 51 algae, 112–113 aneurysms, 105–107 animal cells: properties of, 29 anisotropy, 11 annelid worms, 115, 122 apodemes of insects, 79, 95 arteries: aneurysms in, 105–107; collagen in, 105–107; elastin in, 41; mechanical design of, 105–107 atomic force microscope, 220, 222 bambered plates: bending of, 141–143; torsion of 167–169 bamboo, 132 banana stalks, 132, 164 barageenan, 51 barnacles, 191 bartilage, 50, 111–112; structure of, 111–112 bats: bones of, 193; skin of, 100–101; wings of, 102–103 bending tests: four point, 125–126, 136–138; three point, 126–128, 135–138 biomimetics, 37 birds: keratin structures in, 72–74; ligaments in the necks of, 41 bivalves: abductin hinges in, 37–38 bone: changes with ontogeny, 87–88; effect of mineral content on, 87; greenstick fracture in, 133–135; hierarchical nature of, 84–85; mechanical properties of, 85–87; structure of, 84–85; twist-to-bend ratio of, 165 bones: factors of safety in, 199–200; hollowness of, 129; optimization in, 201–203; torsional failure of, 162; trabeculae within, 132, 140 branches: greenstick fracture in, 133–135; reconfiguration in, 171; scaling of, 144–145

brownian motion, 31 buckling: Brazier, 151–152; Euler, 147–151; local, 129–132; in willow, 133 buttercups, 127 buttress roots, 204–205 byssus threads, 98 cell mechanics, 29, 222 cellulose, 80 centin, 89; mechanical properties of, 24 cephalopod mollusks: legs of, 115; suckers of, 192 ceramic composites, 83–91 chitin, 75 cifferential bending stiffness, 166 claws, 68, 71, 185; structure of, 71 climbing plants, 206–209 collagen: in arteries, 105–107; in bone, 84–85; in cartilage, 111–112; composition and structure of, 64–65; in the mammalian penis, 114; in skin, 99–100; in tendon 64–65 composite materials; 59–82; biphasic behavior in, 75, 77; discontinuous fibers in, 75–77; effects of fillers, 74–75; stiffness of, 59–60; strength of, 60–62; structure of, 59; theory of, 59–63, 74–75; toughness of, 61–63, 67, 68; work of fracture of, 67 compression: material failure in, 147, 148; structural failure in, 147–158 compressive tests, 16 confocal microscopy, 220–221 cork, 157–158 corrugated beams, 140–141 creep tests, 44–45; on sea anemone mesoglea, 46–47 cuticle, 75–79; intersegmental membrane, 77–78; mechanical properties of, 24, 77–79; incorporation of metal salts within, 83; organization of layers in, 79; tanning of, 77–78; structure of, 75–77 cyanobacetria, 109–110 dinosaurs: claws of, 185; trusses in, 139–140 dynamic mechanical tests, 45–46

248

elastin: in arteries, 105; in ligaments, 40–41; mechanical properties of, 24; in skin, 109–110; structure of, 38–41 elephants: trunks of, 115 enamel, 89–90; mechanical properties of, 24 energy: storage within materials, 8–11 factors of safety, 198–201; in animals, 199–200; in plants, 200–201 feathers: buckling of, 131; composition of, 72; factors of safety in, 200; foam within, 132; melanin within, 74, structure of, 184; torsion of, 162–163 fibrous composites. See composite materials fingerpads, 196–197 finite element analysis, 221 fish: swim bladders in, 108–109; tails of, 138–139 flatworms, 121–122 flexural modulus, 127 flexural rigidity, 123–127 flexural strength, 123–127 food processing, 210–212 fracture mechanics, 17–21; investigating, 21–23 friction, 195–197; of fingerpads, 196–197; of keratin, 196, 197; in vertebrate joints, 174 fruit, 212–213 fungi, 209–210 gastroliths, 211–212 gas vesicles, 109–110 geckos: attachment in, 195 gels, 50–51 glycoproteins, 48, 50 grasses: leaves of, 142–143; mechanical defenses of, 214–215; silica in, 214; stems of, 131 greenstick fracture, 133–135 growth: of notochords, 118–119; of plant cells, 116–118 hagfish slime, 62–64 halobacteria, 109–110 hedgehog spines, 152 hemicellulose, 80 hinges: in insect wings, 172–173; in leaves, 171–172; of Venus flytrap, 172 Hooke’s law, 4 hooks, 183–185, 207 hooves, 68, 71–72, 128; reduction in number of, 128; structure of 71–72 horsetails: buckling of, 131 humans: skin of, 101–102; food processing by, 212 hydrostatic structures, 50, 111–121; gels within, 50

INDEX

image correlation, 220 insects: adhesion in, 194; apodemes of, 79, 95; cuticle of, 75, 77–79; resilin in, 35–37; silks of, 55–56; stingers of, 150, 185; wings of, 98–99, 102–103, 141–143, 151, 166–169, 172–173 isotropy: definition of, 11 joints: of arthropods, 173–174; of spiders, 175–176; of vertebrates, 174–175 keratin structures, 68–74: avian; 72–74; biphasic behavior of, 75; effects of hydration on, 72–74; effects of melanin within, 74; friction of, 196; mammalian, 69–72; mechanical properties of, 24, 69–71, 72–74; shear stiffness of, 165; structure of, 68–69 Langer’s lines, 101–102 Laplace equations, 102, 104–105, 213 laser speckle interferometry, 220 leaves: corrugation in, 140; edge reinforcement of, 98, of grasses, 142–143; mechanical defenses of, 214–215; of trees, 170–171 leeches, 122 lianas, 97 ligaments: around vertebrate joints, 174–175; cruciate, 174–176 ligamentum nuchae, 40–41 lizards: tongues of, 115 lungs, mechanical design of, 107–108 material properties: ways of investigating, 4–11; time-dependence of, 42–44 mechanical defenses: of leaves, 214–215; of nuts, 213; of seeds, 213 mechanical properties of common materials, 24 melanin: within bird keratins, 74 membranes: lipid, 29 mesoglea, 46–48; mechanical properties of, 24 microindentation, 219 mollusks: abductin hinges in bivalves, 37–38; mucus produced by, 48–50; shells of, 89–91 mucopolysaccharides, 50, 65 mucus, 48–50; mechanical properties of, 24 muscles: mechanical advantage of, 176–177; pennate, 178–179 muscular hydrostats, 115 mussels: beards of, 98; byssus pads of, 190; shells of, 90

INDEX

nails, 68, 72; structure of, 72 nanoindentation, 219 nematode worms, 119–120 nemertean worms, 121–122 Newton’s laws of motion, 3 notch sensitivity, 18–19 notochord, 118–119 nuts, 213 octopus: legs of, 115; suckers of, 192 optimization, 201–205; in bone, 201–203; in plants, 203–204; in trees, 204–205 parenchyma; 97, 113; stellate, 132 pectin, 51 penis, 114–115 plant cells: arrangement of, 113, mechanics of, 112–113; growth of, 116–118 plant cell wall, 79–82; arrangement in wood, 80–81; biphasic behavior of, 80; lignification of, 80; mechanical properties of, 24, 80; structure of, 80 Poisson’s ratio: definition of, 11–12; relationship to shear and axial stiffness, 14–15; and shape of beams, 141–142 polar second moment of area, 159; of regular cross sections, 162 polysaccharides, 31 polymers, 30–32, 42–44, 51–53; investigating the properties of, 44–46; time dependent mechanical behavior of, 42–44; random coil 32; structure of, 31–32 porcupine quills, 152 pressure vessels: cylindrical, 113–121; fiber arrangement in, 113–114; stresses in the walls of, 102, 103–104 prestressing: in herbaceous plants, 153; in trees, 154–156 protein polymers, 30–31, 51–53; hydrogen bonding within, 51–53 protein structure: α helix, 52–53, 55, 64, 68–70; β sheet, 53, 55–56, 72; effects of amino acids on, 52–53 pterosaurs: bones of, 129; wings of, 103 Raman spectroscopy, 220 rays, 82 reaction wood, 155–156 reconfiguration: of leaves, 170–171; of branches, 171 resilin, 35–37; mechanical properties of, 24 roots: anchorage by, 185–189, 191; delamination of, 133–134; mechanical properties of, 185, 188; structure of, 96, 128–129

249

ropes, 96–98 rubbers, 32–41; structure and properties of, 32–35; biological, 35–41 rushes: stems of, 132 sausages, 105 scaling: of attachments, 98; of trees, 145–146 scanning acoustic microscope, 220, 222 seaweeds: arrangement of cells in, 113; fronds of, 99; gels within, 51; holdfasts of, 190 second moment of area, 124–125; of regular cross sections, 125 sedges, 151, 164–165 seeds, 213 sharks, 120–121 shear: in beams, 135–139; definition of, 11–12; modulus, 14; in plant cell growth, 116–118; relationship to axial forces, 14; of slug mucus, 48–50; strain, 13–14; due to torsion, 159–160 shear modulus, 14; of mucus, 48–49; of rubbers, 34 shells of mollusks, 89–91; mechanical properties of, 24 silica defenses: against fungi, 210; against herbivores, 214–215 silks, 53–58; of insects, 55–56; mechanical properties of, 24, 54–55; of spiders 56–58, 97, 200; structure of, 53–54 simple beam theory, 123–125 skin: 99–102; of bats, 100–101; of humans, 101–102; of sharks, 120–121; tear-resistance of, 99–100 slime: of hagfish, 62–64 slugs: mucus produced by, 48–50 snakes, 197 soil: mechanical properties of, 185 spicules, 84–84 spiders: attachment in, 195; silk of, 56–58, 200 spiral grain, 132 spiral springs, 165–166 sponges, 83–84 squid, 120 stiffness: definition of, 8 stomata, 119 strain: definition of, 7; gauges, 220 stress: definition of, 6–7; in walls of pressure vessels, 102, 103–104; transverse, 132–133 stress relaxation tests, 44–45 suckers, 192 surface tension: in lungs, 107–108 swim bladders, 108–109

250

teeth: abrasion of, 214; food breakdown by, 210–211; structure and composition of, 88–89 tendons, 64–67; compression of, 66–67; crimp in, 65, 66; factors of safety in, 199–200; mechanical properties of, 24, 65; structure of, 64–65; uses as control wires, 66; uses as springs, 65–66 tendrils, 208 tensile tests, 4–6, 15–6 thigmomorphogenesis, 204 torsional rigidity, 160 torsion tests, 16, 160–161 toughness: definition of, 23–24 trabeculae, 132 transient mechanical tests, 44–45 tree frogs, 193–194 trees: anchorage of, 188–189; branch shape of, 144–145; buttresses in, 204–205; factors of safety in, 200–201; lateral roots of, 128–129; optimization in, 204–205; prestressing in, 154–156; reaction wood in, 155–156; scaling in, 145–146; shear failure in, 138–139; single trunks of, 128; trunk shape of, 143–145; wood of, 80–82 trusses, 135, 136, 139–140 turgor, 80 twist-to-bend ratio, 163–165

INDEX

ungulates: toe reduction in, 128; neck trusses of, 139 Venus flytrap, 172 viscoelasticity: definition of, 10–11 viscometer, 48–49 viscosity: of polymers, 46 waterweeds, 96 whales: arteries in, 107 wings: of bats, 102–103; of insects, 98–99, 141–143, 151, 166–169, 172–173; of pterosaurs, 103; torsion in bones of, 163 wood: arrangement of cells in, 82; cell structure of, 80–81; compressive failure in, 154–156; mechanical properties of, 24, 81–82; spiral grain in, 132; toughness of, 81 work of fracture: definition of, 19–21; measurement of, 21–23 x-ray tomography, 221 xylem, 80–82 yield: definition of, 8 Young’s modulus: definition of, 8