Voigt bounds

This approximation is called the Voigt model, and the value of the elastic modulus is often known as the Voigt bound. The expression is identical to that for a continuous aligned fibre composite under a longitudinal load, and gives the elastic modulus when the load is applied parallel to the sheets. Similarly, if the stress is applied perpendicular to the layers, and an iso-stress condition applies (the Reuss model), the elastic modulus is ... 

Many hard materials of practical importance are polyphase materials, and it is important to formulate their effective elastic modulus. Two bounds exist for calculating the composite modulus, based on elastic deformation of the individual phases. The upper, Voigt bound, assumes compatible displacements or strains. 

These equations are analogous to the Reuss and Voigt bounds for bulk moduli and represent such wide extremes that most data can be accommodated within them. 

Experience with the effective elastic properties of polycrystalline materials has shown that the Voigt bound and the Reuss bound are for practical purposes sufficiently close [Green 1998, Hill 1952], Therefore, following Hill s recommendation [Hill 1952], it is common practice to use the arithmetic mean of the Voigt and Reuss values. 

The arithmetic mean (upper bound) corresponds to the Voigt bound of the shear and bulk modulus, Gy and Ky, respectively. 

Note that for composites we have written the Voigt boimds only for the shear modulus G and the bulk modulus K, although in the literature it is sometimes tacitly assumed that they hold also for the tensile modulus E [Eduljee McCullough 1993]. In fact, strictly speaking, the upper bound (Voigt bound) of the tensile modulus Ey corresponds to the... 

This is possible because the elastieitj standard relations must hold for all isotropic continua, whether heterogeneous on the mieroscale or not [Torquato 2002]. It will be shown below, that in the case of the dense alumina-zirconia composite ceramics (Poisson ratios 0.23 and 0.31 for alumina and zirconia, respectively) the deviation of the Voigt bound Ey... 

Then determine the value of the coefficient a via the condition that M = 0 at least for (p = (which is necessary in order not to violate the Voigt bound). In general one obtains [Pabst Gregorova 2004a, 2004c]... 

Of course, the Spriggs relation (including its special case Equation (110)), suffers from the serious principal drawback that E is not zero for = 1, i.e. the Spriggs relation necessarily violates the HS upper bound and even the Voigt bound. For this reason Hasselman [Hasselman 1962], based on previous work by Hashin [Hashin 1962], suggested a relation which can be written as... 

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.

For such materials it is possible to consider the matrix (dense host material, e.g. alumina, zirconia or AZ composites) as a homogenized medium and to invoke the Voigt bounds and the Hashin-Shtrikman upper bounds in their simple form for two-phase materials. Of course, in the case of porous materials (which can be considered as materials with an extremely, substantially infinitely, high contrast in the phase properties from the viewpoint of micromechanics) both the Reuss bound and the lower HS boimd of the effective elastic moduli degenerate to zero (in mathematical terms almost everywhere , i.e. everwhere except at the singular point = 0 where they are equal to the matrix values M (o) = ), cf. Fig. 

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...

Young s modulus data is plotted below with the following models also in evidence Voigt (Rule of Mixtures), Ress, and Hashin-Shtrikman. The Voigt bound assumes proportional stiffness... 

The upper and lower limits given for G refer to the upper (Voigt) bound Gv and the lower (Reuss) bound Gr. G of TaCoo is discussed in the text. 

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