Cellular solids

L. J. Gibson and M. F. Ashby, Cellular Solids, 2nd edition, Butterworth-Heinemann, 1997. 

Thermal insulation is available over a wide range of temperatures, from near absolute zero (-273 C) ( 59.4°F) to perhaps 3,(1()0°C (5,432°F). Applications include residential and commercial buildings, high- or low-temperature industrial processes, ground and air vehicles, and shipping containers. The materials and systems in use can be broadly characterized as air-filled fibrous or porous, cellular solids, closed-cell polymer foams containing a gas other than air, evacuated powder-filled panels, or reflective foil systems. 

Gibson, LJ. and Ashby, M.F. 1997. Cellular Solids Structure and Properties , 2nd Ed. Cambridge University Press, Cambridge, UK. 

The behaviour and magnitude of the storage and loss moduli and yield stress as a function of applied stress or oscillatory frequency and concentration can be modelled mathematically and leads to conclusions about the structure of the material.3 For supramolecular gels, for example, their structure is not simple and may be described as cellular solids, fractal/colloidal systems or soft glassy materials. In order to be considered as gels (which are solid-like) the elastic modulus (O ) should be invariant with frequency up to a particular yield point, and should exceed G" by at least an order of magnitude (Figure 14.2). 

This newly fermented wine is cloudy from suspended materials such as yeast, protein, colloids, and fine grape cellular solids. In storage, a natural clarification, or gravity settling out, of these materials takes place. 

There are different ways to classify cellular solid foods. Here are some ... 

Table 10.1. Typical physical properties of cellular solid foods.

Soft Cellular solids with thick walls and very small air bubbles can exhibit a compressibility pattern similar to that of a dense solid see the discussion of marshmallows bellow. Marshmallows are such an example. However stmetural eoUapse unaccompanied by significant lateral expansion is a characteristic of the majority of solid food foams, regardless of whether their cells are open or closed and whether their cell wall material is brittle or complying ( soft ). 

The Compressive Stress-Strain Relationships of Typical Cellular Solids... 

Kgure 10.2. Schematic view of the three regions of the force-displacement curve of a typical cellular solid I, small deformation of the intact structure II, buckling and fracture of cell walls and III, compaction of what is increasingly collapsed cell wall material. 

Although marshmallows are cellular solids, they exhibit a deformability pattern that is atypical for cellular solids of the kinds classified by Ashby (1983). The marshmallow deformability is characterized by the absence of a prominent shoulder in the stress-strain relationships as shown schematically in Figure 10.6 (top). 

Figure 10.15. The expression of anisotropy in puffed extruded cellular solids. Notice the qualitative as well as the quantitative differenee in the force-displacement curves when determined in the longitudinal and transversal direetions.

Moisture, no doubt, is an effective plasticizer. However, its exact effect on the mechanical properties of cellular solid foods, with the possible exception of the class of soluble low molecular materials (see below), cannot be predicted on the basis of their Tg even if there were an acceptable way to determine it. This is primarily because whenever the cell wall solid is mainly made of a high molecular weight polymer, such as starch and/or protein, moisture can affect its various mechanical properties differently and in a manner that must be determined experimentally. 

Ashby s theoretical studies of the mechanical behavior of cellular solids, and those of his followers as well, serve as most useful guidelines. But one should always keep in mind that cellular foods do not have cells of rmiform size and geometry. Closed and open cells can coexist at different ratios and the former can sometimes burst open upon compression, as has been demonstrated in breads. In the future, nondestructive imaging methods to determine 3-D structures will probably provide information that will clarify the relationship between the cellular architecture, the cells properties and texture. 

In the past, many or perhaps most publications on cellular solids in the nonfood literature were the result of interest in their performance at relatively small deformations (strains). In contrast, during their mastication, foods are subjected to very large compressive strains and are then tom apart. Moreover, in engineering and biomechanics applications, the solid foam is expected to be rather inert. Or, if it does interact with the environment, this would be a slow process that takes place on a time scale of months or years. In contrast, cellular foods interact with moisture very rapidly and the resulting changes can be quite unique, depending on the amount of water soluble components in their cell walls. 

Ashby, M.F. (1983). The mechanical properties of cellular solids. Metall. Trans. A, 14, 1755-1769. 

Luyten, H., Plijter, J.J., and van Vliet, T. (2004), Crispy/erunehy erusts of cellular solid foods a literature review with discussion. J. Texture Stud., 35, 445-492. 

Peleg, M. (1993a). Calculation of the compressive stress-strain relationships of layered arrays of cellular solids using equation solving computer software. J. Cell. Plast., 29,285-293. 

Peleg, M. (1997a). Mechanical properties of dry cellular solid foods. FoodSci Technol. Int., 3, 227-240. 

Assuming that structural data are available, and that a property has been correctly measured, the next problem is to establish a relationship. Fundamental models are preferred by engineers because tlrey are based on basic principles of physics and the physical chemistry of the described phenomenon. Once it is realized that foods are essentially composite hierarchical structures, we can borrow models and theories developed for nonfood systems and apply them. A good example is the adaptation of mechanical principles for the description of cellular solids, (Gibson and Ashby 1988) to the properties of solid food foams (Attenburrow et al. 1989 Warburton et al. 1990). Examples are provided in Chapter 10. 

Gibson, L.J., and Ashby M.F. (1988). Cellular Solids Structure and Properties, Pergamon Press, Oxford. 

A clearer understanding of the relationship between foam structure and mechanical properties of solid foams has been developed by Gibson and Ashby (1988). They related the mechanical properties (e.g., strength, modulus, yield stress, fracture toughness) of idealised cellular solids to their relative density. This work considered the cell walls of solid foams as a three-dimensional network of beams (Figure 20.18) and treated their deformation in terms of classical solid mechanics, with strength and modulus related to beam thickness and length by the equations ... 

The theory indicates that the mechanical properties of the foam are dependent on the properties of the cell wall materials and their size and shape. By relating the density of the foam to its bulk mechanical properties, the slope of the fitted line (n) can give us information about the type of failure mechanism (Figure 20.19). This also indicates that the size and shape of the bubbles in a foam will have a predictable effect on the strength and fracture of the foam. Bread and extruded cereal foams have been considered as cellular solids using the Gibson and Ashby analysis, and have been shown to follow the Gibson and Ashby prediction (Keetels et al. 1996 Hayter etal. 1986). 

Cellular Solids—Structure Properties, by LJ. Gibson and M.F. Ashby, Pergamon Press, Elmsford, N.Y., 1988, 357 pp. 

This book consists of 12 well-presented chapters and explains the structure and properties of cellular solids and of the ways in which they can be exploited in engineering design. By unifying the modeling of many different types of cellular solids similarities in behavior of these diverse materials are explained by the authors. Case studies are used to... 

Cellular solids are a class of materials with low densities and novel physical, mechanical, thermal, electrical, and acoustic properties. Low-density cellular metals can feature a wide variety of topologies to include open-cell foam, closed-cell foam, hollow-sphere foam, periodic/optimized truss structures, and honeycomb. Metallic foams consist of air dispersed in a solid matrix, similar to polymer foams such as polystyrene or food foams such as whipped cream. Closed-cell foams feature solid faces such that each cell is independently sealed from its neighboring cells, whereas open-cell foams (also known as porous metals, metal sponges and truss-type materials) do not contain cell walls they only have cell edges. Hollow-sphere foams consist of an assembly of individual hollow spheres. 

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