Introduction and synopsis

The honeycomb of the bee, with its regular array of prismatic hexagonal cells, epitomizes a cellular solid in two dimensions. Here we use the word ‘honeycomb’ in a broader sense to describe any array of identical prismatic cells which nest together to fill a plane. The cells are usually hexagonal in section, as they are in the bee's honeycomb, but they can also be triangular, or square, or rhombic. Examples were shown in Fig. 2.3.

Man-made polymer, metal and ceramic honeycombs are now available as standard products. They are used in a variety of applications: polymer and metal ones for the cores of sandwich panels in everything from cheap doors to advanced aerospace components; metal ones for energy-absorbing applications (the feet of the Apollo 11 landing module used crushable aluminium honeycombs as shock absorbers); and ceramic ones for high-temperature processing (as catalyst carriers and heat exchangers, for example). And many natural materials – wood is one – can be idealized and analysed as honeycombs (as we do in Chapter 10). If such materials are to be used in load-bearing structures an understanding of their mechanics is important.

There is a second good reason for studying honeycombs: it is that the results shed light on the mechanics of the much more complex three-dimensional foams. Analysing foams is a difficult business: the cell walls form an intricate three-dimensional network which distorts during deformation in ways which are hard to identify.

Introduction and synopsis

Structural members made up of two stiff, strong skins separated by a lightweight core are known as sandwich panels. The separation of the skins by the core increases the moment of inertia of the panel with little increase in weight, producing an efficient structure for resisting bending and buckling loads. Because of this, sandwich panels are often used in applications where weight-saving is critical: in aircraft, in portable structures, and in sports equipment. Figure 9.1 shows the rotor blade from a helicopter and a flooring panel from a Boeing 747; Fig. 9.2 shows a prefabricated housing wall panel; and Fig. 9.3 shows sections from a downhill ski and from the hull of a racing yacht. In these examples, the skins or face materials are, typically, aluminium or fibre-reinforced composites; the cores are aluminium or paper-resin honeycombs, polymeric foams or balsa wood, all of which have a cellular structure.

Nature, too, makes use of sandwich designs. Sections through the skull of a human and the wing of a bird both clearly show a low-density core separating the solid faces (Fig. 9.4). Plants, too, can have a sandwich structure: that of the iris leaf, described in more detail below, is shown in Fig. 9.22(b) while that of a stalk was shown earlier, in Fig. 2.6(h). In all of these examples, the faces and core are made from the same material with the faces almost fully dense and the core a foam.

The mechanical behaviour of a sandwich panel depends on the properties of the face and core materials and on its geometry.

Introduction and synopsis

The structure of cells has fascinated natural philosophers for at least 300 years. Hooke examined their shape, Kelvin analysed their packing, and Darwin speculated on their origin and function. The subject is important to us here because the properties of cellular solids depend directly on the shape and structure of the cells. Our aim is to characterize their size, shape and topology: that is, the connectivity of the cell walls and of the pore space, and the geometric classes into which these fall.

The single most important structural characteristic of a cellular solid is its relative density, ρ*/ρs (the density, ρ*, of the foam divided by that of the solid of which it is made, ρs). The fraction of pore space in the foam is its porosity; it is simply (1 — ρ*/ρs). Generally speaking, cellular solids have relative densities which are less than about 0.3; most are much less – as low as 0.003. At first sight one might suppose that the cell size, too, should be an important parameter, and sometimes it is; but (as later chapters show) most mechanical and thermal properties depend only weakly on cell size. Cell shape matters much more; when the cells are equiaxed the properties are isotropic, but when the cells are even slightly elongated or flattened the properties depend on direction, often strongly so. And there are important topological distinctions, too.

Introduction and synopsis

The word ‘cell’ derives from the Latin cella: a small compartment, an enclosed space. Our interest is in clusters of cells – to the Romans, cellarium, to us (less elegantly) cellular solids. By this we mean an assembly of cells with solid edges or faces, packed together so that they fill space. Such materials are common in nature: wood, cork, sponge and coral are examples (cellulose is from the Latin diminutive cellula: full of little cells).

Man has made use of these natural cellular materials for centuries: the pyramids of Egypt have yielded wooden artefacts at least 5000 years old, and cork was used for bungs in wine bottles in Roman times (Horace, 27 BC). More recently man has made his own cellular solids. At the simplest level there are the honeycomb-like materials, made up of parallel, prismatic cells, which are used for lightweight structural components. More familiar are the polymeric foams used in everything from disposable coffee cups to the crash padding of an aircraft cockpit. Techniques now exist for foaming not only polymers, but metals, ceramics and glasses as well. These newer foams are increasingly used structurally – for insulation, as cushioning, and in systems for absorbing the kinetic energy from impacts. Their uses exploit the unique combination of properties offered by cellular solids, properties which, ultimately, derive from the cellular structure.

The formal description of elastic anisotropy

For some purposes the linear-elastic response of a foam can be thought of as roughly isotropic: its moduli are the same for all directions of loading. Then it is completely characterized by just two moduli (any two of Young's modulus, E*, the shear modulus, G*, the bulk modulus, K*, and Poisson's ratio, ν*, Chapter 5). But most man-made foams are anisotropic (Chapter 6): the Young's modulus in the rise direction is often twice as great as that in the other two perpendicular directions and the shear moduli and Poisson's ratios, too, depend on the direction of loading. Natural cellular solids are more anisotropic: the moduli of wood can differ by a factor of 10 along the grain and across it. And honeycombs are more anisotropic still, with moduli normal to the plane of the honeycomb which can be hundreds of times greater than those in-plane. As the symmetry of the material decreases, more moduli are required to describe the elastic response completely. It is helpful to know how many are needed, what they describe, and how they relate to each other.

Introduction and synopsis

Foams can be made out of almost anything: metals, plastics, ceramics, glasses, and even composites. Their properties depend on two separate sets of parameters. There are those which describe the geometric structure of the foam – the size and shape of the cells, the way in which matter is distributed between the cell edges and faces, and the relative density or porosity; these are described in Chapter 2. And there are the parameters which describe the intrinsic properties of the material of which the cell walls are made; those we describe here.

The cell wall properties which appear most commonly in this book are the density, ρs, the Young's modulus, Es, the plastic yield strength, σys, the fracture strength, σfs, the thermal conductivity, λs, the thermal expansion coefficient, αs, and the specific heat Cps. Throughout, the subscript ‘s’ indicates a property of the solid cell wall material while a superscripted ‘*’ refers to a property of the foam itself. Thus, E*/Es means ‘the foam modulus divided by that of the cell wall material’; this is also referred to as ‘the relative modulus’.

It is helpful to start with an overview of the properties of solid materials, which have values which lie in certain characteristic ranges. A perspective on these is given by material property charts (Ashby, 1992), of which Figs. 3.1 and 3.2 are examples. The first shows Young's modulus, Es, plotted against the density, ρs. Metals lie near the top right: they have high moduli and high density.

Introduction and synopsis

Superficially, bones look fairly solid. But looks are deceptive. Most bones are an elaborate construction, made up of an outer shell of dense compact bone, enclosing a core of porous cellular, cancellous, or trabecular bone (trabecula means ‘little beam’ in Latin). Examples are shown in Fig. II.I: cross-sections of a femur, a tibia, and a vertebra. In some instances (as at joints between vertebrae or at the ends of the long bones) this configuration minimizes the weight of bone while still providing a large bearing area, a design which reduces the bearing stresses at the joint. In others (as in the vault of the skull or the iliac crest) it forms a low weight sandwich shell like those analysed in Chapter 9. In either case the presence of the cancellous bone reduces the weight while still meeting its primary mechanical function.

An understanding of the mechanical behaviour of cancellous bone has relevance for several biomedical applications. In elderly patients with osteoporosis the mass of bone in the body decreases over time to such an extent that fractures can occur under loads that, in healthy people, would be considered normal. Such fractures are common in the vertebrae, hip and wrist, and are due in part to a reduction in the amount of cancellous bone in these areas. The degree of bone loss in a patient can be measured using non-invasive techniques, so an understanding of the relationship between bone density and strength helps in predicting when the risk of a fracture has become high.

Introduction and synopsis

In Chapter 5 we described the uniaxial behaviour of isotropic foams. Expressions for their elastic moduli and compressive strengths, and their resistance to tensile fracture and fatigue, were derived assuming that the temperature and rate of loading corresponded to those for which the cell wall properties are tabulated (typically, T = 20°C and ɛ = 10−3/s). In practice, foams are often anisotropic. And in many engineering applications, they are loaded in more than one direction and at high rates or different temperatures.

Military, aerospace, automative and packaging applications often require a knowledge of foam properties at high rates of deformation and at temperatures other than room temperature. Here the modulus and failure-mechanism diagrams of Chapter 3 become useful. The moduli and strength of rigid foams (metals as well as polymers) decrease linearly as the temperature rises. Increasing the strain-rate does not affect their moduli but increases their strength. Semirigid foams (those used at a temperature close to the glass temperature Tg of the base polymer) are more complicated: both their moduli and strength increase with strain-rate, sometimes dramatically. Elastomeric foams are different again: their moduli increase slightly with increasing temperature, but are almost independent of strain-rate.

These effects are directly related to the temperature and strain-rate dependence of the cell-wall properties (Chapter 3): they are inherent properties of the material of which the foam is made.

Introduction and synopsis

Wood is the most ancient, but still the most widely used, structural material in the world – indeed – the word ‘material’ itself derives from the Latin materies, materia: the trunk of a tree. The use of wood in buildings, ships and furniture is as old as the pyramids – wooden artefacts at least 5000 years old have been found in them. During the sixteenth century the demand in Europe for stout oaks for shipbuilding was so great that the population of suitable trees was depleted; by the seventeenth century ships' timbers had to be imported into England from the New World. By 1800 much of Europe had been deforested by the exponential growth in the consumption of wood, a problem that programmes of reforestation have only partly overcome. Today the world production of wood is roughly the same as that of iron and steel: roughly 109 tonnes per year. This production finds many uses, in everything from musical instruments to pit props. Table 10.1 lists some of these, with the species of wood best suited for each. Much of the total production is used structurally: for beams, joists, flooring and supports which bear load. Then the properties which interest the designer are the moduli, the crushing strength and the toughness. These properties vary enormously from one wood to another: oak is more than 10 times stiffer, stronger and tougher than balsa.

Introduction and synopsis

Cork has a remarkable combination of properties. It is a light yet resilient; it is an outstanding insulator for heat and sound; it has a high coefficient of friction; and it is impervious to liquids, chemically stable and fire-resistant. Such is the demand that production now exceeds half a million tonnes a year (and 1 tonne of cork has the volume of 56 tonnes of steel).

In pre-Christian times cork was used (as we still use it today) for fishing floats and soles of shoes. When Rome was besieged by the Gauls in 400 BC, messengers crossing the Tiber clung to cork for buoyancy (Plutarch, AD 100). And ever since man has cared about wine, he has cared about cork to keep it sealed in flasks and bottles. ‘Corticum abstrictum pice demovebit amphorae’ sang Horace (27 BC) to celebrate his miraculous escape from death from a falling tree. But it was in the Benedictine Abbey at Hautvilliers where, in the seventeenth century, the technology of stopping wine bottles with clean, unsealed cork was perfected. Its elasticity and chemical stability mean that it seals the bottle without contaminating the wine, even when it must mature for many years. No better material is known, even today.

Commercial cork is the bark of an oak (Quercus suber) that grows in Portugal, Spain, Algeria and California.

Low-density, cellular solids appear widely in nature and are manufactured on a large scale by man. Aspects of their structure, and of their mechanical, thermal and other properties, have been studied in some detail by mathematicians, by physicists, by engineers and even by food technologists, each interested in one particular aspect of their geometry, or behaviour. This has led to a literature which is perhaps more scattered and diverse than that relating to any other class of engineering material: and there is not, at present, any source from which the interested reader can derive a reasonably broad and comprehensive picture. We were led to write this book in an attempt to bring together, in one text, and using a common nomenclature, a broad survey of the understanding of cellular solids.

We have been greatly helped in doing this by the advice, comments and critical readings of sections of the text, by a large number of most helpful colleagues and friends. We would particularly like to acknowledge the help of Professor C. Calladine, Dr. J. Woodhouse, Mrs. T. Shercliff, and Mr. J. Zhang of the Engineering Department, Cambridge: Dr. P. Echlin of the Botany Department, Cambridge; Ms. L. A. Demsetz, Ms. A. T. Huber, and Mr. T. C. Triantafillou of the Department of Civil Engineering, MIT; Professor L. Glicksman of the Department of Architecture, MIT; Professors B. Budiansky, J. Hutchinson, and T. McMahon of the Division of Applied Sciences, Harvard University; Professor K. E. Easterling of the University of New South Wales; Professor S. K. Maiti of the Indian Institute of Technology, Bombay.

The ten years since the first edition of this book appeared have seen a remarkable increase in interest in cellular solids. New techniques for making ceramic and metallic foams have widened the range of man-made materials and the diversity of their applications; and the continued interest in wood, cork, and cancellous bone has stimulated new experiments and models to characterise natural cellular structures. This second edition was stimulated by these developments and by the very diverse community of enthusiastic scientists and engineers who have contacted us to discuss aspects of the structure and properties of cellular solids. Each chapter of the first edition has been extensively revised and brought up to date, but this, we found, was not enough. Techniques for making foams are more fully described; recent work on the mechanical response of solid foams to multiaxial loading, to deformation under creep conditions, and to impact loading have required new sections; acoustic properties are included for the first time; and there is a new chapter on the selection of foams to meet specified design criteria and on data sources and databases for foam properties.

As with the first edition, we have been greatly helped by many generous colleagues and friends.

Introduction and synopsis

Foams have unique thermal, electrical and acoustic properties. Among these are: exceptionally low thermal conductivity, making them a prime choice for thermal insulation; very low dielectric loss, allowing transmission of microwaves without attenuation or scattering; and the ability to absorb sound, suiting them as materials for noise abatement.

In this chapter we survey the thermal, electrical and acoustic properties of foams. Where possible, the underlying physical understanding of the behaviour is emphasized, since it is this which allows a degree of predictive modelling of foam properties. Case studies are used to illustrate some of the results.

Thermal properties

More foam is used for thermal insulation than for any other purpose. Closed-cell foams have the lowest thermal conductivity of any conventional non-vacuum insulation. Several factors combine to limit heat flow in foams: the low volume fraction of the solid phase; the small cell size which virtually suppresses convection and reduces radiation through repeated absorption and reflection at the cell walls; and the poor conductivity of the enclosed gas. This low thermal conductivity is exploited, at one extreme of sophistication, in the insulation for liquid oxygen rocket tanks and, at the other, in disposable cups for hot drinks.

Introduction and synopsis

Manufacturers of foams produce data-sheets, listing their properties. We have assembled a database of available foams and their suppliers, and illustrate it here. The foams and their suppliers are cataloged in the Appendix 13 A, as Tables 13.A1 and 13.A2. The first lists foams by chemistry and trade name, attaching a manufacturer or supplier code to each. The second relates this code to a company, an address, and, where possible, a telephone and fax number. The contents of the tables are based on information obtained from suppliers in 1995. Products, of course, evolve and develop, and new manufacturers and materials appear, so a completely up-to-date compilation is not possible. But this catalogue gives a starting point.

Data-plots are used to illustrate the range of foams properties. Two case studies illustrate methods of selecting foams for specific applications.

The compilation of materials and suppliers

There are three main difficulties in locating a given foam for a given engineering application. First, there are almost no standards, either national or international. Second, foams and cellular solids are marketed under weird trade names (‘Neopolen’, ‘Cellobond’) which give little guidance in identifying either their chemistry or their supplier. And third, while some large-volume foams are widely marketed by well-known multi-nationals, many are the products of small, specialized producers, not otherwise known in the market place.

Introduction and synopsis

Almost any solid can be foamed. Techniques now exist for making three-dimensional cellular solids out of polymers, metals, ceramics and even glasses. Manmade foams, manufactured on a large scale, are used for absorbing the energy of impacts (in packaging and crash protection) and in lightweight structures (in the cores of sandwich panels, for instance); their efficient use requires a detailed understanding of their mechanical behaviour. Even when the primary use is not mechanical – when the foam is used for thermal insulation, or flotation, or as a filter, for example – the strength and fracture behaviour are still important. Nature, too, uses cellular materials on a large scale. Often the primary function is mechanical, as with wood (to support the tree) or cancellous bone (to give the animal a light, stiff, frame). In other cases it may not be: the shape of the cells in leaf and stalk, or in cork and sponge, may be dictated by the need to optimize fluid transport or thermal insulation or surface area, but even here the mechanical properties are important because the cells still support the structure. And there is the consuming subject of food. Bread – The Staff of Life – and many other starch-based foods are foams. Foaming with yeast or CO2 gives the tough, leathery starch a crisp crunchiness which is attractive to bite on and chew (mechanical operations); but it also makes transporting the product more difficult because its crushing strength is reduced.

Introduction and synopsis

Packaging surrounds most things we buy or do. Food is packaged, parcels through the post are packaged, and within a car or aeroplane, we ourselves are carefully packaged. It is hard to say how much is spent on it, or the worth of the goods damaged due to inadequate packaging, but the sums involved are certainly considerable, and the potential return on any improvement is large.

The essence of protective packaging is the ability to convert kinetic energy into energy of some other sort – usually, heat – via plasticity, viscosity, visco-elasticity or friction; and this must be done whilst keeping the peak force (and thus the deceleration or acceleration) on the packaged object below the threshold which will cause damage or injury. And there is more to it than that. The direction of impact may not be predictable; then the package must offer omni-directional protection, that is, it must absorb impact from any side. Since the package must be carried with the object it protects, light weight is important. And – since much packaging is discarded – it must (almost always) be cheap.

Foams are especially good at this. The energy-absorbing capacity of a foam is compared with that of the solid of which it is made in Fig. 8.1. For the same energy-absorption, the foam always generates a lower peak force. Energy is absorbed as the cell walls bend plastically, or buckle, or fracture (depending on the material of which the foam is made), but the stress is limited by the long, flat plateau of the stress-strain curve (Figs. 4.2 and 5.1).