Hashin-Shtrikman bounds

Q— Random Hollow-Sphere 0 FCC Hollow-Sphere —Octet Truss — — Hashin-Shtrikman Bound... 

Z1O2 composite (fully dense) were found to be above the upper Hashin-Shtrikman bound. Suggest possible explanations. Repeat the discussion for the case in which experimental values were below the lower bound. 

A chart of density and Young s modulus as a function of volume percent SiC is provided in Figure 2. The density follows the typical additive rule of noixtures. Over this range of SiC volume percent, the Young s modulus also appears to follow a linear relationship. When plotted in relation to the Hashin-Shtrikman bounds, however, a slight parabolic shape is observed, as shown in Figure 3. These boimds were calculated based on the equations provided by Hashin et al. [16] for Young s modulus. 

Hashin-Shtrikman Bounds for the Elastic Moduli of Two-Phase Materials... 

In contrast to the case of polycrystalline monophase materials (see above) the Voigt-Reuss bounds of two-phase materials are often too far apart to be useful for prediction purposes. Hashin and Shtrikman [Hashin Shtrikman 1963] derived the best possible bounds on the effective elastic moduli of macroscopically isotropic two-phase composites given just volume-fraction information. Actually the Hashin-Shtrikman bounds are two-point bounds, but accidentally the key integral involving this two-point information has a form that reduces to one-point information, i.e. volume-fraction information, in the case of macroscopically isotropic composites [Torquato 2002]. When and G, > G2 (the usual, so-called... 

Figure 3. Effective tensile modulus of dense alumina-zirconia composite ceramics Voigt bound (crosses slightly above the upper solid line calculated from the Voigt values of the effective shear and bulk moduli), approximate Voigt bound (according to the mixture rule, upper solid line), Reuss boimd (results of both calculations identical, crosses and lower solid curve), upper and lower Hashin-Shtrikman bounds (dashed curves) and values measured by the resonant frequency method for dense (porosity < 3 %) alumina-zirconia composite ceramics prepared by slip casting.

Figure 4. Voigt bound (solid straight line) and Hashin-Shtrikman upper bound (solid curve) for the relative tensile modulus of porous ceramics (e.g. alumina, zirconia or alumina-zirconia composites) the Reuss bound and the lower Hashin-Shtrikman bound (dashed lines along the axes) degenerate to zero...

All micromechanical relations presented in this chapter are based on the volume fraetions of the constituent phases (i.e. one-point correlation fimctions and a special part of the two-point correlation functions) as the only input information. If complete two-point and three-point information would be available (e.g. from stereological analysis or 3D tomography), the Hashin-Shtrikman bounds (i.e. the two-point bounds) could be improved in principle, whieh leads to the three-point bounds due to [Beran Molyneux 1966] and [Milton Phan-Thien 1982], cf. also [Jeulin 2001, 2002, Markov 2000, Milton 2002, Sobczyk Kirkner 2001, Torquato 2000, 2002]. Of course, in the case of dense alumina-zirconia composites it is quite... 

These bounds were constructed in regard to the SiC Si material and are merely a guide. The particle volume percents were calculated using a rule of mixtures with regard to the densities of the SiC Si samples. The Ti and A1 containing samples were estimated based on this. Expectedly, the base material follows the upper Hashin-Shtrikman bound at these volume fractions. The other materials fall within range of these bounds for the most part. They are not expected to necessarily follow these particular bounds as the other additions were not taken into account. Evidence of porosity appears in the SiC Si-... 

Under certain conditions on overall isotropy, there are narrower but algebraically more complicated bounds usually referred to as the Hashin-Shtrikman bounds (see, e.g.. Ref. 1). However, for not too elastically anisotropic single crystals, one is usually satisfied with bulk and shear moduli given by the Voigt-Reuss-Hill approximation, which has the forms... 

There are many, more or less fundamental, relations aiming at a description of conduction in inhomogeneous materials. They come under various names, and some of them are in fact identical, although that may not be immediately apparent. For instance, the Maxwell-Eucken relation cited by Williams et al. (9) and used by them to correct for the presence of TijOs in TiB2 is identical to the upper Hashin-Shtrikman bound. Generalizations of such relations to three or more phases, to ellipsoidal inclusions, etc. are reviewed in Ref. 3. 

Bounds for Elastic Moduli Voigt, Reuss, and Hashin-Shtrikman Bounds... 

FIGURE 6.26 Voigt, Reuss, Voigt-Reuss-Hill average, and Hashin-Shtrikman bounds for com-pressional modulus as function of porosity. Input parameters are quartz = 11 GPa, )Tma = 44GPa, and water 4fl=2.2GPa. [An Excel worksheet you find in the folder Elastic-mechanical, bound models on the website (see also Section 11.2) http /A>ooksite.elsevier.com/ 9780081004043/.]... 

More narrow boundaries than Voigt and Reuss can be calculated as Hashin-Shtrikman bounds (Hashin and Shtrikman, 1962a,b, 1963). Mavko et al. (1998) present the equations in a comfortable form for a two-component medium ... 

Figure 6.26 shows Voigt, Reuss, and Hashin-Shtrikman bounds for com-pressional modulus as a function of porosity. The lower Hashin-Shtrikman bound is equal the Reuss bound in case of a porous medium, where mie constituent is a fluid (shear modulus zero). Gommensen et al. (2007) therefore implemented a modified upper Hashin-Shtrikman bound this model crosses the Reuss bound at critical porosity. 

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