Spheroidal modulus

The spheroidal modulus (So) is defined as the ratio of conduction heat flux through the vapor film to the evaporation heat flux ... 

Here, G denotes the shear modulus, and f(c/r) is a function of the ratio c/r in which c and r are the spheroidal semiaxes of the precipitate. For spheres, f(c/r= 1) = 1 = /max. For discs as well as for rods, /< 1. In principle, shear stress energies and energies arising from misfit dislocation networks also have to be added. They influence AG by additional energy terms. 

Defined in this way, the solution can be approximated by using the equations derived for the particle of defined geometry (i.e. a flat membrane or a sphere) that better resembles it. However, solution can be quite approximate. We compared the solution for oblate spheroidal particles with the approximate solution of the sphere with the redefined Thiele modulus according to Eq. 4.63 and found differences as high as 30% (Soto et al. 2002). 

Modulus VS Composition. The modulus of IPNs as a function of composition 3rields information about their relative phase continuity. The first such study on polyacrylate/polyurethane latex interpenetrating elastomeric networks (lENs) was carried out by Matsuo and co-workers (Fig. 7) (43). Note that in the midrange of composition, the modulus begins to follow the parallel Takayanagi model (44). This suggests that the stiflfer acrylic phase is becoming a continuous phase, as predicted from percolation models of three-dimensional mosaics of compressed spheroidal structures. 

There are two simplifications of the Halpin-Tsai equations, en f approaches zero, approximating spheroidal structures, the Halpin-Tsai theory converges to the inverse rule of mixtures, and provides a lower bound modulus. Equation (10.8) expresses this relation, with an appropriate change in subscripts. 

Work by Wang and Pyrz [22,23] affirms the work above. Their work begins with the Eshelby thesis described above and focuses on the development of the Mori-Tanaka approach to predict the modulus of the composites. The two morphologies that are examined are oblate spheroids (flattened spheres with the limit being a plate) and prolate spheroids (stretching or elongating a sphere with the limit being a fiber). 

One should be cautious in reviewing this work. The definition of the aspect ratio for the oblate spheroids (the thickness divided by the long dimension) is the inverse of what is described in the bulk of this chapter. There is a difference in assigning the Young s modulus in the first article (178 GPa) and the second article (167 GPa) for montmoriiionite. 178 GPa is generally accepted as the Young s modulus for montmoriiionite in the bulk of the work featured in this chapter. The bulk and the shear moduli are calculated from the model and utilized to calculate the Young s modulus. 

Figure 4. Normalized Young modulus versus porosity at various axial ratios of spheroidal pores at temperature of T = 303K, when as a parameter is used temperature of sintering.

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