Reuss approximation

If, however, one assumes uniform stress throughout the same nontextured polycrystal a similar averaging procedure can be performed over the elastic-compliance tensor using the corresponding nine elastic compliance constants Sn, S12, S33, S44, S55, Sss, S12, S23, and S31. This is known as the Reuss approximation (Reuss, 1929), after Endre Reuss (1900-1968), and it yields the theoretical minimum of the elastic modulus. 

Likewise, using the relations between the elastic-compliance coefficients from Table 10.4 in Eq. 10.25, gives the Reuss approximation of the Young s modulus of a cubic crystal ... 

Using the relations between the elastic-stiffness coefficients from Table 10.3 in Eqs. 10.20 and 10.26, one may also derive the Voigt and Reuss approximations for the rigidity modulus of a cubic monocrystal. These are given by Eqs. 10.35 and 10.36, respectively ... 

It should be noted that for a polycrystal composed of cubic crystalhtes, the Voigt and Reuss approximations for the bulk modulus are equal to each other, as they should be since the bulk modulus represents a volume change but not shape change. Therefore, in a cube the deformation along the principal strain directions are the same. Hence, Eqs. 10.39 and 10.40 are equal and these equations also hold for an isotropic body. The... 

Use the relations in Tables 10.3 and 10.4 to derive the Voigt and Reuss approximations for the bulk modulus of an elastically isotropic polycrystalline aggregate composed of tetragonal monocrystals. 

Show, for an isotropic cubic polycrystal for which the Cauchy relations (cn = 3ci2 C44 = C12) hold, that the Voigt and Reuss approximations of the shear modulus reduce to ... 

Hill showed that the Voigt and Reuss approximations were, respectively, greater than and less than true polycrystalline moduli, which were then better represented, in what later became known as the VRH approximation, by the arithmetic means of these extremes. Thus, as sixmmarized by ANDERSON [And63] ... 

Similarly, the Reuss and Voigt approximations for the bulk moduh of polycrystalline aggregates composed of cubic and orthorhombic crystaUites are given by Eqs. 10.39 through 10.42 ... 

For textured samples the relation between the peak shifts and sin F becomes nonlinear and analytical expressions can be found only by approximating the texture pole distribution by d functions on some prominent sample directions. This could be a rough approximation, especially if the grain elastic interactions are not of the Reuss type, and numerical calculations of the diffraction stress factors are preferable. 

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 ... 

The densities of vibrational states for the two potentials (Fig. 15) are actually very similar, exeept for a somewhat wider gap for RIMl between the two peaks at higher frequencies, corresponding to inner modes of CO3. As for the Kieffer s model, the cut off frequencies of acoustic modes were derived from elastic constants by the Voigt-Reuss-Hill approximation[38], amounting to 51, 66 and 90 cm"i. An optic continuum ranging from 113 to 287 cm was used, and four Einstein oscillators at 708,867, 1042 and 1470 cm with appropriate weights represented the internal optical modes. The... 

Nonlinear models of rheological behavior can be approximated by step functions, whereby the existence of a finite yield stress G plays a dominant role. Three typical nonlinear models include the Saint-Venant model of ideal plastic behavior, the Prandtl-Reuss model of an elastoplastic material, and the Bingham model of viscoelastic behavior. The first model can be mechanically approximated by a sliding block, the second by a Maxwell element and a sliding block in series, and the third by a dash pot damping element and a sliding block in parallel (Figure 2.14). 

The anisotropy in N0( n)-rare gasgsystems was firstly investigated by Reuss and coworkers analyzing, with the sudden approximation, the orientational dependence of the glory structure of the integral cross section (ICS) in experiments with state selection and without state selection of NO. An estimate of the anisotropy for NO-He was also reported from low resolution total differential cross section (DCS) measurements. 

The aggregate model predicts only that the elastic constants should lie between the Reuss and Voigt average values. In polyethylene terephthalate, it is clear that the experimental compliances lie approximately midway between the two bounds. For cold-drawn fibres, it has been shown that this median condition applies almost exactly [87]. 

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.

A model system which can be used is the diacetylene polymer for which the stress-strain curve was given in Fig. 5.34. By controlling the polymerization conditions it is possible to prepare single crystal fibres which contain both monomer and polymer molecules. The monomer has a modulus of only 9 GN m along the fibre axis compared with 61 GN m" for the polymer and the partly polymerized fibres which contain both monomer and polymer molecules are found to have values of modulus between these two extremes as shown in Fig. 5.36. The variation of the modulus with the proportion of polymer (approximately equal to the conversion) can be predicted by two simple models. The first one due to Reuss assumes that the elements in the structure (i.e. the monomer and polymer molecules) are lined up in series and experience the same stress. 

The elastic moduli of a polycrystalline material can be approximately estimated from the elastic constants of the single crystal. First, the maximum and minimum values of the moduli are obtained using Voigt s and Reuss s approximations, respectively. The method in which the average value of the maximum and minimum values is adopted is called Hill s approximation, and the estimated values of Young s modulus are known to be in good agreement with the measured ones (Anderson, 1963). 

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... 

We next exemplify how the elastic constants are related to the conventional Debye temperature 0D from the low-temperature heat capacity [i.e., 0d( 3) in Table 4]. The detailed numerical integration in Eq. (5) yields 0d as shown in the first row of Table 2, using Cy from Table 1 and mass densities p calculated from the atomic volumes Qa [=V/(rL)] in Table 5. As an illustration we shall also apply Eqs. (6)-(8) together with Eqs. (19) and (20), which allow a simple calculation of 0D. Anderson (8) suggests that such an approach, with K and G estimated by the Voigt-Reuss-Hill approximation, gives 0d to within about 2% when Ah < 0.2. Here Ah = (Gv - Gr)/ (Gv + Gr) is a measure of the elastic anisotropy. Table 2 summarizes the results for 0p. 

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