Über die Autorin bzw. den Autor

Roland Ennos

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"The publication of this book is nothing if not auspicious. In Solid Biomechanics, Ennos brings to bear his unusually wide experience, from early work on insect flight to extensive recent research on plants. This is a book that we very much need."--Steven Vogel, professor emeritus, Duke University

"This accessible and clear book looks at how the structure and mechanical properties of tissues and organs of living organisms relate to their function. Discussing many kinds of tissues--plant, animal, and human--Solid Biomechanics will generate broad interest in the subject."--Rod Lakes, University of Wisconsin

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"The publication of this book is nothing if not auspicious. In Solid Biomechanics, Ennos brings to bear his unusually wide experience, from early work on insect flight to extensive recent research on plants. This is a book that we very much need."--Steven Vogel, professor emeritus, Duke University

"This accessible and clear book looks at how the structure and mechanical properties of tissues and organs of living organisms relate to their function. Discussing many kinds of tissues--plant, animal, and human--Solid Biomechanics will generate broad interest in the subject."--Rod Lakes, University of Wisconsin

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Solid Biomechanics

PRINCETON UNIVERSITY PRESS

Contents

Preface.....................................................................xiAcknowledgments.............................................................xiiiCHAPTER 1 The Properties of Materials.......................................3CHAPTER 2 Biological Rubbers................................................29CHAPTER 3 Complex Polymers..................................................42CHAPTER 4 Polymer Composites................................................59CHAPTER 5 Composites Incorporating Ceramics.................................83CHAPTER 6 Tensile Structures................................................95CHAPTER 7 Hydrostatic Skeletons.............................................111CHAPTER 8 Structures in Bending.............................................123CHAPTER 9 Structures in Compression.........................................147CHAPTER 10 Structures in Torsion............................................159CHAPTER 11 Joints and Levers................................................170CHAPTER 12 Attachments......................................................183CHAPTER 13 Interactions with the Mechanical Environment.....................198CHAPTER 14 Mechanical Interactions between Organisms........................206CHAPTER 15 The Future of Structural Biomechanics............................219Glossary....................................................................223References..................................................................231Index.......................................................................247

Chapter One

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 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 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 consequently be normalized by dividing it by its...