Gibson-Ashby model

The Gibson-Ashby model describes the relationship between the relative properties of the polymer foams and the relative density as follows ... 

Note that this exponential model results in a zero relative tensile modulus only in the limiting case of 100 % porosity. Without doubt, porosities close to 100 % can in principle be achieved, eg. in some aerogels [Gibson Ashby 1997]. However, the usual case encoimtered in practice will be a complete structural breakdown (loss of integrity) at significantly lower porosity levels. In order to allow for the possibility oi E =0 at porosities lower than 100 % (i.e. < 1), it is necessary to include a critical volume fraction (f)/ in the modulus-porosity relation, which is able to take the possible occurrence of a... 

Gibson and Ashby (a. 13) propose separate models for elastic collapse by cell edge buckling and plastic collapse by stretching of cell faces. The latter model gave a scaling relationship between the (initial) collapse stress a pi and the relative densities ... 

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. 

Figure 20.18. Generalised model for deformation of a foam material. (From Gibson and Ashby 1988.)...

In studying flows through porous media, we extend the LBM so as to study arbitrary and random approaches. Even at a first glance, many porous media, particularly the natural ones, show that the distribution of pores is random (Christakos Hristopulos, 1997), so randomness is a really important factor to describe natural and man-made porous materials they usually possess formidably complicated architecture. In modeling, reasonable idealizations have to be assumed (Dullien, 1992 Gibson and Ashby, 1988 Telega and Bielski, 2003). 

Most published literature analyzed the elastic modulus of silica aerogels by drawing inspiration from the cellular solids models. For example, Ashby and Gibson (1997) describe the open cellular foam model compressive modulus to follow power law dependence on the relative density as shown in Eq. (5.1) where C and /i are geometric constants that depend on the topological features and microstructure undergoing cell wall bending as the dominant deformation. 

Ashby and Gibson (1997) illustrated the model as a cubic array of interconnected beams as shown in Fig. 5.1 with an exponent of 2. Similarly for closed-cell foam, Ashby and Gibson approximated the relative modulus with an additional term accounted for internal gas pressure and membrane stresses on the faces as shown in Table 5.1. 

Another model was presented by Gibson and Ashby [1], based on a cubic cell model for a closed-cell foam, which takes into account the enclosed gas. As shown in Figure 1, the thickness of the edges and the faces of PP foam cell are approximately equal, which means there is no accumulation of material in the corners. Therefore, the main deformation mechanisms are the stretching of the cell walls and the compression of the enclosed gas. As a result, the elastic properties of the closed-cell foam are described by ... 

The above analytical model will give a reasonable approximation when designing a syntactic foam material. There are examples of more complicated numerical models available throughout literature if more detailed analysis is desired. We refer the reader to Gibson and Ashby (15) for analytical relationships for elastic properties for both open- and closed-celled foam that are based on the properties of the unfoamed material and relative density of the foam. 

Foam mechanical properties are often explained using Gibson and Ashby s approach [1]. Their simplified microstructural model is built from cubic cells. For closed-cell foams Gibson and Ashby predict three contributions for the Young modulus of a foam ... 

The postcollapse stress has also been modeled. Gibson and Ashby s slight modification of Rusch analysis [64] gives the following equation ... 

None of these equations considers the possible effect of microstructmal parameters such as cell size, cell shape, cell wall thickness, etc. For this reason, there has been an interest in improving the oversimphfied model of Gibson and Ashby both from an experimental and a theoretical point of view. 

For metal foams, there are also a number of models. According to Gibson and Ashby [57] and Hodge et al. [45], the relationship between the yield strength (cr) and the relative density of a foam material follows the scaling law... 

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